LASSI: A lattice model for simulating phase transitions of multivalent proteins

Jeong-Mo Choi; Furqan Dar; Rohit V. Pappu

doi:10.1371/journal.pcbi.1007028

LASSI: A lattice model for simulating phase transitions of multivalent proteins

Choi, Jeong-Mo;Dar, Furqan;Pappu, Rohit V. 2019-10-21 00:00:00 a1111111111 Many biomolecular condensates form via spontaneous phase transitions that are driven by multivalent proteins. These molecules are biological instantiations of associative polymers that conform to a so-called stickers-and-spacers architecture. The stickers are protein-pro- OPENACCESS tein or protein-RNA interaction motifs and / or domains that can form reversible, non-cova- lent crosslinks with one another. Spacers are interspersed between stickers and their Citation: Choi J-M, Dar F, Pappu RV (2019) LASSI: A lattice model for simulating phase transitions of preferential interactions with solvent molecules determine the cooperativity of phase transi- multivalent proteins. PLoS Comput Biol 15(10): tions. Here, we report the development of an open source computational engine known as e1007028. https://doi.org/10.1371/journal. LASSI (LAttice simulation engine for Sticker and Spacer Interactions) that enables the cal- pcbi.1007028 culation of full phase diagrams for multicomponent systems comprising of coarse-grained Editor: Ozlem Keskin, Koc ¸ University, TURKEY representations of multivalent proteins. LASSI is designed to enable computationally effi- Received: April 12, 2019 cient phenomenological modeling of spontaneous phase transitions of multicomponent Accepted: October 1, 2019 mixtures comprising of multivalent proteins and RNA molecules. We demonstrate the appli- cation of LASSI using simulations of linear and branched multivalent proteins. We show that Published: October 21, 2019 dense phases are best described as droplet-spanning networks that are characterized by Copyright:© 2019 Choi et al. This is an open reversible physical crosslinks among multivalent proteins. We connect recent observations access article distributed under the terms of the Creative Commons Attribution License, which regarding correlations between apparent stoichiometry and dwell times of condensates to permits unrestricted use, distribution, and being proxies for the internal structural organization, specifically the convolution of internal reproduction in any medium, provided the original density and extent of networking, within condensates. Finally, we demonstrate that the con- author and source are credited. cept of saturation concentration thresholds does not apply to multicomponent systems Data Availability Statement: All relevant data are where obligate heterotypic interactions drive phase transitions. This emerges from the ellip- within the manuscript and its Supporting soidal structures of phase diagrams for multicomponent systems and it has direct implica- Information files. tions for the regulation of biomolecular condensates in vivo. Funding: This work was supported by grants from the US National Science Foundation (MCB- 1614766), http://nsf.gov, the Human Frontier Science Program (RGP0034/2017),http://www. hfsp.org, the US National Institutes of Health Author summary (5R01NS056114), http://nih.gov, and the St. Jude Children’s Research Hospital through the research Spatial and temporal organization of molecular matter is a defining hallmark of cellular collaborative on membraneless organelles, http:// ultrastructure and recent attention has focused on membraneless organelles, which are www.stjude.org, to RVP. The funders had no role also referred to as biomolecular condensates. Of interest are condensates that form via in study design, data collection and analysis, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 1 / 39 Simulations of phase transitions of multivalent proteins decision to publish, or preparation of the manuscript. phase transitions that combine phase separation and networking of multivalent protein and nucleic acid molecules. Building on recently recognized analogies between associative Competing interests: Rohit Pappu is a member of polymers and multivalent proteins, we have developed and deployed LASSI, an open the Scientific Advisory Board of Dewpoint Therapeutics Inc. There are no competing interests source computational engine that enables the calculation of architecture-specific phase from this affiliation. diagrams for multivalent proteins. LASSI relies on a priori identification of stickers and spacers within a multivalent protein and mapping the stickers onto a 3-dimensional lat- tice. A Monte Carlo engine that incorporates a suite of novel and established move sets enables simulations that track density inhomogeneities and changes to the extent of net- working among stickers as a function of protein concentration and interaction strengths. Calculation of distribution functions and other order parameters allow us to compute full phase diagrams for multivalent proteins modeled using a stickers-and-spacers representa- tion on simple cubic lattices. These calculations allow us to rationalize experimental observations and open the door to the design of protein architectures with bespoke phase behavior. LASSI can be deployed to study the phase behavior of multicomponent systems, which allows us to make direct contact with the physical principles underlying cellular biomolecular condensates. Introduction Biomolecular condensates organize cellular matter into non-stoichiometric assemblies of pro- teins and nucleic acids [1]. Prominent condensates include nuclear bodies [2] such as nucleoli, nuclear speckles [3, 4], and germline granules [1, 5, 6]. Condensates also form in the cyto- plasm. These include stress granules [7], membrane-anchored signaling clusters [8, 9], and bodies in post-synaptic zones [10]. All of these condensates share key features: (i) they range in size from a few hundred nanometers to tens of microns [1, 2, 11]; (ii) they are multicomponent entities comprising of hundreds of distinct types of proteins and nucleic acids; (iii) and of the hundreds of different types of molecules that make up condensates, a small number are essen- tial for the formation of condensates [1, 12]. The simplest feature that distinguishes proteins that are drivers of biomolecular condensates is the valence of interaction domains / motifs that can participate in non-covalent crosslinks [1, 12–14]. Biomolecular condensates can form and dissolve in an all-or-none manner [2, 11, 15]. The reversible formation and dissolution of condensates can be controlled by the concentrations of multivalent proteins that drive the formation of condensates; in simple two-components sys- tems comprising of macromolecules and solvent, condensates form when macromolecular concentrations cross macromolecule-specific threshold values known as saturation concentra- tions [15]. The transitions that characterize condensate formation bear the hallmarks of a sharp transition in macromolecular density, leading to the formation of a dense phase that is in equilibrium with a dilute phase. This type of transition, known as phase separation, sets up two or more coexisting phases to equalize the dense and dilute phase chemical potentials of the macromolecules across phase boundaries [15]. Phase separation is reversible and this reversibility can be achieved by (i) changes to concentrations of the driver macromolecules [9, 16], (ii) changes to solution conditions that alter the effective interaction strengths among driver molecules [17–20], (iii) altering saturation concentrations through ligand binding–a phenomenon known as polyphasic linkage [21, 22], or (iv) via biological regulation such as post-translational modifications of proteins [8, 12, 23]. Recent studies have focused on uncovering the defining features of proteins [13, 15, 17–19, 24–40] and RNA molecules [41–43] that drive phase transitions. Protein and RNA molecules PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 2 / 39 Simulations of phase transitions of multivalent proteins that drive phase transitions are biological instantiations of associative polymers [44] character- ized by a stickers-and-spacers architecture [45]. Stickers contribute to a hierarchy of specific pairwise and higher-order interactions that are either isotropic or anisotropic whereas spacers control the concentration-dependent inhomogeneities in the densities of stickers around one another. Stickers can be hot spots or sectors [46] on the surfaces of folded proteins [15, 29] or short linear motifs within intrinsically disordered regions (IDRs) [15, 24, 47]. Spacers are typi- cally IDRs that contribute through their sequence-specific effective solvation volumes to the interplay between density transitions (phase separation) and networking transitions that are better known as percolation [28, 29]. Spacers can also be folded domains that are akin to uni- formly reactive colloidal particles, although this has not yet been explored. Proteins can be mapped onto the stickers-and-spacers architecture as linear multivalent proteins, branched multivalent proteins, or some combination of the two [13, 15]. Simple two-component systems comprise of the solvent (which includes all components of the aqueous milieu) and a multivalent protein / RNA molecule. For fixed solution conditions, one can generate phase diagrams [25] as a function of protein concentration, the valence of stickers, the affinities of stickers, the sequence-specific effective solvation volumes of spacers, and the lengths / stiffness of spacers. The phase diagram can be investigated by keeping the valence of stickers, the lengths of spacers, and effective solvation volumes of spacers fixed while varying the concentration of stickers and the affinities between stickers [29]. Changes to protein concentration will enable density fluctuations and above the saturation concentration, designated as c , the density inhomogeneities lead to separation of the system into coexisting sat phases. The concentration of multivalent proteins in the dilute and dense phases will be denoted as c and c , respectively. For a given bulk concentration c that lies between sat dense bulk c and c , the fraction of molecules within each of the coexisting phases is governed by the sat dense lever rule [48]. Stickers also form reversible physical crosslinks and these crosslinks generate networks of inter-connected proteins. The number of proteins within the largest network of the system grows continuously as the protein concentration increases. Above a concentration threshold known as the percolation threshold and designated as c , the single largest network spans perc the entire system and this phenomenon is called percolation [49–51]. If the percolated net- works have the rheological properties of viscoelastic fluids, the fluids are referred to as network fluids [15, 52]. Phase separation and percolation can be coupled to one another. The coupling will depend on the values of c , c , and c relative to c . If c is smaller than all of c , c , and sat dense perc bulk bulk sat dense c , the system is in a single dilute phase with no large molecular networks (Fig 1A). If c > perc bulk c and c < c , then a system-spanning percolated network forms without phase separa- perc perc sat tion (Fig 1B). However, the system undergoes phase separation and a dense phase forms as a percolated droplet if c > (c , c ) and c < c < c (Fig 1C). Recent studies, using bulk sat perc sat perc dense three-dimensional lattice models designed to mimic the poly-SH3 and poly-PRM systems of Li et al. [16], show that sequence-specific effective solvation volumes of linkers / spacers between folded domains directly determine whether phase separation and percolation are coupled or if percolation occurs without phase separation for linear multivalent proteins [28, 29]. The cou- pling between phase separation and percolation is controlled by the extent to which spacers / linkers preferentially interact with the surrounding solvent. Theory [17, 24, 25, 27, 34, 53–59] and computations [28, 29, 43, 60–68] have important roles to play in modeling and describing the phase behavior of multivalent protein and RNA molecules. Theories provide analytical routes to explain experimental observations and to make testable predictions. On the other hand, simulations work around many of the simplify- ing assumptions that are needed to make theories analytically tractable. In doing so, they PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 3 / 39 Simulations of phase transitions of multivalent proteins Fig 1. Characteristic phases in the stickers and spacers formalism. (a) Dispersed solution phase where the polymers are uniformly mixed in solution. (b) Percolated fluid wherein the polymer chains form a percolated, system-spanning network through physical crosslinks among stickers results. (c) Droplet wherein network formation also causes the polymers to form condensed phases. (d) Two-dimensional representation of the LASSI architecture. The beads with arms denote stickers where arms denote that the monomers are capable of orientational interactions, and the curved lines connecting the monomers represent phantom tethers, which are allowed to freely overlap (implicit spacer model). Different colors denote different sticker and spacer species respectively. Note that the physical bonds are allowed to overlap (dashed circle). For the rest of this work, physical bonds will not be labeled and will only be depicted as overlapping orientational arms. https://doi.org/10.1371/journal.pcbi.1007028.g001 provide numerical routes to enable comparative assessments across different systems; they help in making testable predictions about phenomenology through what if calculations tar- geted toward specific systems; and they pave the way for designing systems with bespoke phase behavior. Phase transitions are collective phenomena that involve highly cooperative transitions of large numbers of multivalent polymers. The collective interactions that drive phase transitions are captured in terms of a small number of order parameters that are similar across disparate systems and represent a generic coarse-graining of the underlying system that defines parame- ters such as the correlation length and the sizes of cooperative units. Accordingly, practical considerations of computational tractability and rigorous considerations of identifying the rel- evant collective coordinates mandate the use of coarse-grained models for simulations of PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 4 / 39 Simulations of phase transitions of multivalent proteins Fig 2. Considerations that go into designing a coarse-grained model. As discussed in the text, the choice of a coarse-grained model has at least three ingredients. These include the type of conformational space (lattice or off-lattice), the nature of the interactions among entities that are represented in the coarse-grained description (isotropic, anisotropic or fluctuating fields), and the parameterization approach. LASSI, as described here, is based on a lattice model that uses anisotropic interactions and a phenomenological model. https://doi.org/10.1371/journal.pcbi.1007028.g002 phase transitions driven by multivalent protein and RNA molecules. We focus here on multi- valent proteins, although the methods we describe are readily adaptable to RNA molecules as well. Coarse-graining, an essential aspect of making simulations of large numbers of multivalent proteins a tractable proposition, comes in different flavors [69]. For simplicity, we divide con- siderations that go into the development of a suitable coarse-grained model into three catego- ries (Fig 2). These are (1) the type of model, (2) the types of interactions among the entities in the simulation, and (3) parameterization of the interaction potentials for the model of interest. Two distinct choices for the type of model are the choice between simulations being performed using lattice models versus off-lattice models. In either space, one or all of the molecules can be represented explicitly using architectures that represent coarse-grained mappings of the pro- tein of interest. Next, the interactions among the units that make up each protein can be mod- eled as being isotropic or anisotropic. This is true of simulations where proteins of interest are modeled explicitly. In contrast, numerical instantiations of field theoretic models model can also be brought to bear where only a single chain is modeled explicitly [60, 70]. The remaining protein and solvent molecules are modeled as fields whose fluctuations are concentration dependent [71]. The effects of all other molecules influence the phase behavior of the explicitly modeled single chain through interactions of the chain with the field. Finally, the choice of interaction potentials is the bedrock of every simulation. The functional forms and parameters for potentials can be derived using phenomenological considerations intended to enable calcu- lations of the “what if” variety–an approach that is common practice in statistical and polymer physics. One can also obtain system-specific parameters using information gleaned from atomistic simulations of smaller-scale facsimiles of the system of interest. These system-spe- cific parameters are derivable using force matching methods pioneered by Voth and coworkers [72–77] or by prescribing a functional form for the potential that describes interactions in the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 5 / 39 Simulations of phase transitions of multivalent proteins coarse-grained space and employs machine learning methods to derive the relevant parameters [74, 78]. Finally, one can adopt approaches similar to the parameterization of molecular mechanics forcefields and develop a single transferable model that should be applicable to a large number of disparate systems. Different coarse-grained simulations represent different combinations of model, interac- tion type, and parameterization. Two illustrative examples for deriving coarse-grained models for simulations of phase behavior of multivalent proteins come from the works of Ruff et al. [78] and Dignon et al. [64, 65]. Ruff et al. show how one can generate off-lattice models, of bespoke resolutions and learned parameters for isotropic potentials derived using machine learning that leverages information gleaned from atomistic simulations of individual proteins and protein oligomers. Dignon et al. also use an off-lattice model based on isotropic potentials whose parameters are designed to be transferable across disparate intrinsically disordered proteins. It is worth emphasizing that at this juncture, there is no valid reason to stipulate that one combination of approaches for deriving a coarse-grained model is superior to another combi- nation. As noted by Das et al. [67, 68], all models have distinct strengths and limitations. How- ever, for specific applications, some methods afford quantifiable computational advantages over others. In our case, we are interested in uncovering conceptual nuances of phase diagrams for multicomponent systems that comprise of multivalent proteins characterized by aniso- tropic interactions among domains / motifs. As noted above, these systems can be mapped onto a stickers-and-spacers architecture. The questions we are interested in answering pertain to the order parameters that describe phase behavior, the impact of chain connectivity and spacer effective solvation volumes on phase behavior, and the determinants of the shapes of phase diagrams of multicomponent systems where phase transitions are driven by heterotypic as well as homotypic interactions. In this context, it is noteworthy that lattice models have been adapted to model phase transitions for systems comprising of different numbers of multi- valent protein and RNA molecules [28–30, 43, 79–81]. In the present work, we provide a formal description of the design and implementation of system-specific lattice models for simulating phase transitions of multivalent proteins. The simulation engine, known as LASSI for LAttice simulation engine for Sticker and Spacer Inter- actions, formalizes the approaches that have been developed and deployed in recent studies [28–30, 79, 80]. Accordingly, LASSI combines a lattice model with anisotropic interactions among stickers and the model, at least in the current formalism, is derived based on phenome- nological considerations (Fig 2). Ongoing work shows that a machine learning methodology known as CAMELOT [78] can be adapted for using LASSI as a tool to model sequence-specific phase behavior. We describe the design of LASSI, focusing first on the overall structure of the model, the Monte Carlo sampling, and their justification for generic multivalent proteins. We further describe the calculation of order parameters for quantifying phase separation and per- colation. Then, using two specific examples of linear and branched multivalent protein sys- tems, we illustrate the deployment of LASSI to two biologically relevant systems. In both systems, we make a priori assumptions regarding the identities of stickers and spacers, which is a requirement for the deployment of LASSI. Although we focus here on systems with a few components, it should be emphasized that the design of LASSI is able to handle a wide range of multicomponent systems. Materials and methods Considerations that go into the development of a suitable lattice model include (a) the choice of the mapping between a specific multivalent protein of interest and a lattice representation, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 6 / 39 Simulations of phase transitions of multivalent proteins (b) the parameterization of the strengths and ranges of interactions for all unique pairs of beads and vacancies, (c) the design of move sets and acceptance criteria for Monte Carlo simu- lations that enable the sampling of local and collective motions of large numbers of lattice- instantiated multivalent proteins, (d) the efficient titration of key parameters such as protein concentrations and interaction strengths, and (e) the extraction of phase boundaries in terms of known and hidden collective parameters, which become the relevant order parameters for phase transitions of interest. Generating lattice representations of multivalent proteins For a given linear or branched multivalent protein, we first choose a suitable mapping between the protein degrees of freedom and a lattice representation. The conformational space is a sim- ple cubic lattice with periodic boundary conditions used to mimic a macroscopic system. Phase transitions represent the collective effects of large numbers of molecules, and simula- 3 4 tions have to include at least 10 –10 protein molecules to observe facsimiles of these collective transitions in finite sized systems [82]. Further, we need to be able to test for the effects of finite size artefacts and this requires a titration of the effects of varying the numbers of molecules. Accordingly, the lattice has to be large enough to accommodate at least 10 molecules of each type for the most dilute concentrations. Often, we might need to increase the number of mole- cules to be of O(10 ). Accordingly, a one-to-one mapping between the protein degrees of free- dom and a lattice representation would lead to a computationally intractable model. Instead, we adopt system-specific coarse-graining approaches, whereby the coarse-graining is guided by a priori rigorous or phenomenological knowledge of the identities of stickers versus spacers. For disordered proteins, the stickers within disordered regions often correspond to single amino acid residues or short linear motifs. For multivalent folded proteins, the stickers are either an entire protein domain or sectors on domain surfaces [28, 29]. Residues correspond- ing to spacers may either be modeled explicitly, where one or more spacer residues are mod- eled by a single bead on the lattice site, or be modeled as phantom tethers, where the intrinsic lengths of tethers are calibrated in terms of the numbers of lattice sites [28, 29]. In both cases, the tethers can stretch, bend, and rotate and these degrees of freedom contribute to density inhomogeneities that are the result of altered patterns of inter-sticker interactions. LASSI and bond fluctuation models The structure of LASSI is inspired by the bond fluctuation model (BFM) for lattice polymers [83]. This is a general lattice model for simulations designed to extract equilibrium conforma- tional distributions and dynamical attributes of polymers in dilute solutions as well as dense melts. There are two versions of the BFM viz., the Carmesin-Kremer BFM or CK-BFM [84] and the Shaffer BFM or S-BFM [83]. Both models are based on the use of simple cubic lattices, which discretizes the conformational space for polymers. In the CK-BFM [84], each repeating unit or monomer within a polymer is modeled as a 3-dimensional cube where the 8 corners of the cube occupy lattice sites and bond vectors con- nect pairs of monomers. Overlap of monomers is associated with an energetic penalty, and each bond vector can have up to 108 distinct directions. The choice of bond vector set encodes the geometry of the polymer and places constraints on the bond lengths and bond angles. All other interactions are governed by the inter-monomer potentials, and evolution of the system through conformational space is driven by changes to the overall potential energy. In contrast, the S-BFM places each monomer on a single lattice site. Covalently bonded monomers are connected by bonds that are constrained to be of three types, leading to chains that have bonds pffiffiffi pffiffiffi of length 1, 2 or 3 in units of lattice size. Monte Carlo moves with suitable acceptance PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 7 / 39 Simulations of phase transitions of multivalent proteins criteria can be designed for both types of BFMs. The simulations are used to generate equilib- rium conformational distributions of lattice polymers in either dilute or dense phases. The move sets control the overall polymer dynamics and the acceptance of different types of moves and the calculation of correlation functions allows one to compute dynamical quantities for lattice polymers [83]. If we were to use either of the established BFMs without modification, then each amino acid residue would be modeled as a monomer, and such an approach would be useful when the identities of stickers and spacers remain ambiguous. This approach under- lies a different simulation engine known as PIMMS [43]. LASSI is a generalization of the S-BFM that also adapts features of the CK-BFM. Given a choice of the mapping for coarse-graining, each multivalent protein is described as a chain of non-overlapping monomers viz., beads that occupy sites on a 3-dimensional cubic lattice. Note that the choice of a single site per bead is similar to that of the S-BFM, although the bead, which is a sticker or spacer monomer, need not be the monomeric unit, i.e., an amino acid res- idue in the case of proteins. Each sticker monomer is linked to its adjacent sticker on the chain via either a phantom tether or a set of spacer beads that occupy individual lattice sites [28, 29]. pffiffiffi A spacer / tether length of unity implies that adjacent monomers are within 3 lattice units of one another (Fig 1D). The choice of the spacer length will be sequence-specific or more pre- cisely, specific to the architecture of the protein of interest. Inter-monomer (sticker-sticker, sticker-spacer, and spacer-spacer) interactions are mod- eled as contact-based pairwise interactions. A sticker monomer can bind to another sticker monomer that occupies an adjacent lattice site with an interaction energy that depends on the types of both monomers. Monomers are considered to be adjacent to one another if they are pffiffiffi within a lattice distance of 3. By this criterion, each lattice site occupied by a sticker mono- mer will have 26 adjacent lattice sites. This is reminiscent of the interaction geometry of a CK-BFM for each monomer. In the current implementation of LASSI, the interactions are mutually exclusive, implying that a sticker cannot interact simultaneously with more than one other sticker, even though there are 26 adjacent sites that the interaction partner can occupy. If the sticker in question is already engaged in another inter-monomer interaction with stickers or spacers, then the unoccupied sites of the sticker will be unavailable for interaction. The combination of the geometry of the interaction sites per monomer and the single occupancy constraint leads to anisotropic interactions between sticker interactions. This feature is unique to LASSI and it is not incorporated in other variants of BFMs; this allows us to deploy LASSI for modeling heteropolymeric systems. In the context of LASSI, we note that stickers are dis- tinguished by their ability to participate in anisotropic or isotropic interactions. In contrast, explicitly modeled spacer sites only participate in isotropic interactions with other spacer or sticker sites. Furthermore, the interaction strengths involving spacers are typically weaker than those involving stickers. However, it is worth emphasizing that these distinctions only matter inasmuch as LASSI allows us to capture a numerical instantiation of the stickers-and-spacers model. For simplicity, one might simply think of LASSI as a model that has sites that are differ- entiated by whether or not they can involve themselves in anisotropic interactions, by their intrinsic site valence (a variable that we do not titrate in this work), and by the comparative magnitudes of site-site interaction strengths. Setup of simulations A system with n multivalent proteins is in reality an n+1 component system since the solvent is the implicit component. In LASSI, sites that are not occupied by protein units automatically represent solvent sites. Although the interaction potentials do not explicitly include terms between solvent and protein sites, the effective interaction strengths between pairs of protein PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 8 / 39 Simulations of phase transitions of multivalent proteins units represent an averaging over protein-protein, protein-solvent, and solvent-solvent inter- actions. The solvent sites, i.e., the sites that are not occupied by protein units, represent contri- butions from the solvent to the overall translational and mixing entropies. Simulations are initiated by randomizing the positions of protein units, subject to the constraints of chain connectivity. The parameters that are set at the start of each LASSI simulation include the total number of molecules n of type i and the size of the lattice L, from which we can calculate the total num- ber n of all protein components n ¼ n and the concentration or number density of each protein c = n /L . The setup also includes stipulations for the architectures of each protein i i such as specification of the number of monomers per chain, the overall topology of each pro- tein (linear vs. branched), the lengths of spacers, and the types of spacers (implicit / phantom vs. explicit) [28–30, 79, 80]. The number of monomers per molecule will equal the sum of the number of stickers and spacers if spacer residues are modeled explicitly. Alternatively, if spac- ers are modeled as phantom tethers, then the number of explicitly modeled monomers will equal the number of stickers. Specification of the energetics of the system includes specifica- tion of the simulation temperature in normalized units, homotypic and heterotypic interaction strengths between pairs of stickers, the energetic cost for the overlap of stickers, and the inter- action strengths between sticker and spacer sites if the spacers are modeled explicitly. Design of monte carlo move sets Our goal is to compute architecture-specific phase diagrams for systems comprising of one or more types of linear or branched multivalent proteins. This requires a simulation strategy that enables the sampling of the full spectrum of coexisting densities and networked states for mul- tivalent proteins. Accordingly, the conformations of randomly initialized systems of proteins on a simple cubic lattice are sampled via a series of Markov Chain Monte Carlo (MCMC) moves that are designed to ensure efficient sampling of changes in protein density and net- working while maintaining microscopic reversibility. We have developed and deployed a col- lection of moves and these are described below. Monte carlo sampling with biases In LASSI, we have independent contributions from two main energetic sources. Monomer units are not allowed to overlap, and this can be described by a position-dependent energy E where E = 0 or1. On the other hand, inter-monomer pairwise interactions also con- pos pos tribute to the total energy, and E denotes the sum over all of the effective pairwise inter- rot monomer interaction energies. The subscript “rot”(rotational) indicates the fact that for a pair of nearest neighbor stickers their interaction energies are actually governed by their mutual orientations. Accordingly, the total system energy in a specific configuration i is written as: E ¼ E þ E ; ð1Þ i i;pos i;rot The equilibrium probability associated with configuration i is given by the Boltzmann distribu- tion as: p / expð bEÞ ¼ expð bE Þexpð bE Þ; ð2Þ i i i;pos i;rot In Eq (2),β is the inverse of the simulation temperature in units of the Boltzmann constant (effectively, k = 1 energy unit / temperature unit). The frequency with which a transition from configuration i to j is proposed will be governed by the elements T of the targeting ij matrix T. The proposed transition is accepted / rejected based on the elements A of the ij PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 9 / 39 Simulations of phase transitions of multivalent proteins acceptance ratio matrix A. A MCMC move that transitions the system from configuration i to j defines a flow in configuration space and this flow has to satisfy microscopic reversibility: T A p ¼ T A p ; ð3Þ ij ij i ji ji j If the targeting matrix is symmetric, then T = T and the acceptance ratios such as those pre- ij ji scribed by Metropolis et al. [85] will ensure the preservation of microscopic reversibility. How- ever, if biases are incorporated into the targeting matrix, which is often essential to enhance the sampling of configurations that contribute to density inhomogeneities and the making / breaking of bonds in dense networks, then the elements of the acceptance ratio matrix have to be designed to ensure the preservation of microscopic reversibility. We deploy a general strat- egy of using biased moves to enhance the sampling of different mutual orientations among pairs of stickers. The incorporation of these orientational biases is accounted for by modifying the acceptance criterion of Metropolis et al. [85] whereby each element of A is written as: ! ! A T p ij ji j A T p ji ij i ; ð4Þ ( ) T p ji j such that :A ¼ min 1; ij T p ij i For a symmetric targeting matrix, we recover the standard acceptance ratio of Metropolis et al. [85] viz., � � � � A p p ij j j ¼ or A ¼ min 1; ; ð5Þ ij A p p ji i i Since the moves within LASSI generally involve orientational biases, the elements T are ij rewritten in terms of a Rosenbluth weighting factor W [86, 87] whereby: expð bE Þ j;rot T ¼ ; ð6Þ ij Substituting (6) into (4) leads to: � � A ¼ min 1; exp½ bðE E Þ� ; ð7Þ ij j;pos i;pos The specific form for the weighting factors W will depend on the type of move because the extent of asymmetry in the targeting matrix will depend on the nature of the bias incorporated into the biasing move that proposes a transition from i to j. The specific forms for weighting factors are discussed in the context of the move types that are introduced next. Rotational moves A monomer is in an associated or a dissociated state and in the associated state it has a speci- fied binding partner. This defines the rotational state of a monomer. To change the rotational state, we randomly pick a monomer from the system, and exhaustively sample all 26 adjacent sites to construct a list of potential binding partners. The rotational state of the monomer is changed, at random, by drawing a random integer k from a uniform distribution between [0, b], where b is the number possible binding partners available to the monomer. If k = 0, the th monomer is set to be in a dissociated state. Otherwise, the k candidate bond (reversible phys- ical crosslink) is formed and the state of the monomer is set to be in an associated state. If the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 10 / 39 Simulations of phase transitions of multivalent proteins monomer cannot be involved in a rotational interaction, as would be the case for an explicitly modeled spacer, the rotational move is rejected outright. The accessible volume for rotational interactions is within a cube of unit volume centered on the randomly chosen monomer (see S1 Fig for a 2-dimensional representation), and hence, each sticker monomer will have at most b = 3 –1 possible sites as neighbors. Since this number is not large, we sample all 26 max possible interaction sites. Local moves This move serves as the basic unit of local displacement of monomers–be they stickers or spac- ers. A randomly chosen monomer is moved from position r to r = r +Δr. Acceptance of the i j i move is predicated on the move not leading to an overlap with a site occupied by another monomer and the satisfaction of linker constraints. The choice forΔr is made by uniformly pffiffiffi sampling each component from the interval [–2,2] such thatjDrj � 2 3 (shown in S2 Fig) moves if the selected monomer is in the interior of a molecule [88], and they become analo- gous to end rotation moves if end monomers are selected [89]. Local moves have a rotational bias in LASSI and the Rosenbluth factor is calculated as fol- lows. Starting with Eq (7), we shall designate the chosen monomer by index k. In configuration i, assume that monomer k has a binding partner of index l. Typically in coarse-grained systems there are a finite number of unique monomer types, and thus it is more efficient to simply define interaction energies between different monomer types than between all monomer pairs. The energy associated with the bond between monomers k and l is written asε , where t t(k)t(l) (x) indicates the type of monomer x. The local move causes a change in binding partner, whereby the monomer k now binds to monomer m. The local move leads to a bond swap that causes a change in rotational energy, which is written as: E E ¼ ε ε ; ð8Þ i;rot j;rot tðkÞtðlÞ tðkÞtðmÞ Use of Eq (8) in Eq (6) leads to: T expð bε ÞW ij tðkÞtðlÞ i;k ¼ ; ð9Þ T expð bε ÞW ji tðkÞtðmÞ j;k In Eq (9), each Rosenbluth weight has an additional index in the subscript to indicate that the change in configuration is achieved by a change in the binding partner for the monomer k. To accelerate the creation of density inhomogeneities in supersaturated systems and facilitate the making and breaking of networks, we decompose W as: i;k ðaÞ ðdÞ W ¼ W þ W ; ð10Þ i;k i;k i;k The two terms in Eq (10) respectively represent the contributions to the Rosenbluth weights for monomers in associated (a) and dissociated states (d). First, we calculate the weight factors for the interacting monomers as a partition function over all nearest neighbor contributions, such that: ðaÞ W ¼ expð bε Þ; ð11Þ i;k tðkÞtðlÞ In Eq (11), the summation runs over all potential binding partners l (nearest neighbors) for the monomer k. To illustrate how the Rosenbluth factor is calculated, we assume that the sys- tem has only one type of interaction with the pairwise energy designated asε. If the number of nearest neighbors for monomer k in configuration i is designated as N , then the Rosenbluth k;i PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 11 / 39 Simulations of phase transitions of multivalent proteins weight factor in Eq (11) becomes: ðaÞ W ¼ N expð bεÞ; ð12Þ i;k k;i ðdÞ Setting W ¼ 1 to incorporate a bias towards associated states, and using the simplifica- i;k tion that leads to Eq (12), we rewrite Eq (9) as a definition of the acceptance criterion for the move from configuration i to j via local move involving monomer k as: ( ) N þ 1 j;k A ¼ min 1; exp½ bðE E Þ� ; ð13Þ ij;k j;pos i;pos N þ 1 i;k This choice for the acceptance criterion ensures that detailed balance is preserved while enhancing the sampling of configurations characterized by the breaking of old bonds and the forming of new ones. Reptation–or slithering snake–moves In dense configurations, it becomes difficult to realize large-scale translational or rotational motions of polymers. The slithering snake move is a Monte Carlo instantiation of reptation as first conceived by de Gennes [90]. In this move, a chain is chosen at random, and the mono- mer at one end of the target chain is moved to a new position. The remaining monomers within the target chain are then successively moved such that monomer m along the chain moves into the previous position of monomer m–1 (S3 Fig). This move relies on an inherent symmetry of chain molecules, because bond lengths between monomers are the same; if one swaps monomers across chains, the identity of the chain remains invariant. However, this move cannot be used if the molecule has heterogeneous bond lengths or if it is a branched polymer. The reptation move is rotationally biased, and this is true for every monomer in a chain. The bias is independent for each monomer and accordingly, the Rosenbluth factor for a single reptation move can be calculated from the Rosenbluth factors for each monomer-specific local move. In configuration i we obtain: Y Y W ¼ W ¼ ðN þ 1Þ; ð14Þ i i;m i;m m m In Eq (14), the product runs over all monomers m within the chain of interest. The accep- tance criterion for a reptation move takes the form: 8 9 ðN þ 1Þ > > j;m < = A ¼ min 1; exp½ bðE E Þ�; ð15Þ ij j;pos i;pos > > ðN þ 1Þ : ; i;m The inclusion of the bias for every interacting monomer, rather than just the end monomers, is to emulate how a real transiently bonded polymer would slither along its contours. Note that for strict detailed balance, the Rosenbluth factors for the two end monomers should be calcu- lated and added to the acceptance criterion, but the current implementation of LASSI uses the first trial position that satisfies the position constraints. Double pivot moves These moves swap a part of a chain with the corresponding part of another chain of the same type. A monomer is picked at random; it is denoted as i , where i is the monomer index within PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 12 / 39 Simulations of phase transitions of multivalent proteins a chain, and m is the chain index. A search is then performed around the monomer within a prescribed distance for a monomer within the same type of chain. The requirement for the search is that the monomer of interest be one index ahead along its own chain, (i+1) . Next a check is performed to ensure that the distance between i and (i+1) is within the bond con- m n straint connecting i to (i+1) , and that the distance between (i+1) and i is within the bond m m m n constraint for i and (i+1) . Each monomer (i+1) that satisfies each of these constraints is m m n stored and one of these is randomly picked for the double pivot move (S4 Fig). The move is always accepted if there is a candidate because only connectivity changes unless bonds are modeled using elastic springs. The purpose of this move is to engender large configurational changes in dense polymer melts, which approximate the dense phases formed upon phase separation. In dense regions the rate of acceptance of local moves decrease precipitously. At high enough densities, poly- mers become entangled and local moves reduce to slithering-snake moves and polymers are restricted to motions along tubes around one another [91, 92]. Therefore, rather than physi- cally moving polymers to create a change in configurations, we incorporate move sets that break and make bonds while ensuring that monomers do not overlap, and that bond con- straints are always satisfied. If two chains are close enough to each other that the bonds between two monomers can be swapped, then such the double pivot move results in a large configurational change for both chains, and for the system. Chain and cluster translation moves The chain translation move is designed to move single chains while forming new bonds at the proposed location. This move attempts to translate the center of mass of a chain i from r pffiffi 3L to r = r +Δr wherejDrj � and L is the size of the simulation cell. Multiple trial displace- j i ments are proposed until a trial position that does not result in steric clashes for the entire mol- ecule is generated. The move is then attempted. As with the slithering snake move, each monomer in the molecule that is translated will have an orientational bias. Accordingly, the Rosenbluth factors are calculated as in Eqs (14) and (15). The translational move results in large displacements for single chains and correctly biases the system for efficient sampling of configurations with alternate interaction patterns. Translational moves can also be applied to clusters of molecules. A connected cluster refers to a collection of unique chains connected via rotational interactions. A proposed move only results in a translation, and the move is readily accepted if there is no steric clash. Since no new physical bonds are created at the proposed location, the cluster remains invariant and the move is accepted. Naively this move might seem unnecessary as this move simply moves clus- ters around. However, once a physical bond has formed between two molecules, it is highly unlikely for any of the non-cluster translation moves to move the centers-of-masses of clusters closer together. Considerations for setting move set frequencies The structure of each move set serves as a guide for selecting an optimal set of frequencies. This leads to a set of heuristics that are as follows: (i) in the cluster move we pick a random chain from the system, perform a networking analysis on that chain, and then propose a dis- placement of the cluster. As the cluster size grows it is more likely that a randomly picked chain will be part of the largest cluster which itself will result in a steric clash after the proposed move. Therefore, the frequency of the cluster move should be low, if not the lowest, in the entire set. (ii) In the translational move, we pick a random chain from the system for transla- tion; as the size of the largest cluster increases it becomes less likely for a proposed translation PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 13 / 39 Simulations of phase transitions of multivalent proteins move to be accepted. However, unlike the cluster move the translation move is rotationally biased and thus results in new interactions being formed. Hence, translational moves enable single-molecule to cluster-surface interactions. Therefore, this move should be proposed more often than the cluster move, although not as often as rotational or local moves. (iii) The rota- tion move is computationally inexpensive and it enables the switching of physical bonds and should thus be proposed fairly frequently. (iv) Similarly, local moves and slithering snake moves are also rotationally biased, and they help with the local rearrangements of physical bonds. Local moves are the primary route to enable local conformational changes, and to enable local physical bond rearrangements. Therefore, local moves should be proposed most frequently. The slithering snake move is particularly effective because it allows for large local physical bond rearrangements in dense configurations. Thus, this move should also be pro- posed frequently, less so than local moves but more so than translation moves. Note that in a system where some molecules are non-linear or have heterogeneous linker lengths, the fre- quency would need to be higher since the move is rejected if an incorrect molecule is picked at random. (v) The double pivot move allows for large-scale changes to conformations within dense configurations and accordingly, this move should be proposed more frequently than both cluster and translation moves. One can track the acceptance ratios of each move over a very rough initial sweep across the relevant system parameters. Moves that are always rejected do not enable any changes in configuration and only add computational costs. Therefore, the frequency for that particular move should be lowered. This is especially the case for the cluster move in high-density systems. Identifying phase boundaries using measures for density inhomogeneities In order to detect the onset of phase separation, we can calculate excess chemical potentials using the Widom particle insertion method [93] and equalize these chemical potentials across distinct phases. This process requires a priori knowledge of the densities of both phases. An efficient variant of this approach, based on fast Fourier transforms, was recently developed and deployed by Qin and Zhou [61]. They demonstrated their method for calculations of liq- uid-liquid coexistence curves for a patchy colloid model ofγII-cyrstallin. Given that LASSI simulations are lattice-based, we instead rely on properties of pair distribution functions that help us diagnose the onset of phase separation and compute phase boundaries. Pair distribu- tion functions are helpful because phase separation is the result of the system partitioning into phases of different densities. The pair distribution, which is a reduced-dimension partition function, serves as a rigorous thermodynamic and structural measure of the average local den- sity and inhomogeneities of density. To first order, the density fluctuations are quantified by averaging over all orientations. Accordingly, the pair distribution function can be converted to a radial distribution function that allows us to probe local densities and local structural organi- zation of molecules around one another. However, normalization of the pair distribution func- tion requires some caution. The system contains polymer molecules and using a prior distribution that assumes an ideal gas of the chain monomers to normalize the pair distribu- tion function is problematic because it does not accurately capture the effects of non-idealities due to chain connectivity. We leverage the efficient sampling of polymer fluids in LASSI and obtain suitable prior distributions by simulating the system of interest in the absence of sticker-sticker interactions. (2) The pair distribution function P (r) quantifies the equilibrium distribution of distances ð2Þ between chain monomers, where r is the inter-monomer distance. If P ðrÞ denotes the prior pair distribution function calculated from simulations where the inter-sticker interactions are PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 14 / 39 Simulations of phase transitions of multivalent proteins Fig 3. Distribution functions used for calculation of density inhomogeneity. The data shown are obtained from 5 independent simulations for the A -B system with n n −5 -1 � (2) total protein concentration c = 6.89×10 voxels and reduced temperature T = 0.383. Error bars indicate standard deviations. (a) Pair distribution functions P (r) (2) and P (r), where the former is from the interacting system and the latter from the non-interacting system with chain connectivity (prior pair distribution function). (2) Note that P (r) shows two peaks, the first of which indicates dense phase formation. (b) Radial distribution function g ~ðrÞ. This captures the droplet formation by a sharp and broad peak in the beginning. The inset shows r where g ~ðrÞ intersects the line corresponding to g ~ðrÞ ¼ 1 line, delineating between the dense and solution phases. The global density inhomogeneity measure, r , is obtained by integration of absolute deviation of g ~ðrÞ from 1. https://doi.org/10.1371/journal.pcbi.1007028.g003 ignored (see Fig 3A), then the normalized radial distribution function is written as: ð2Þ P ðrÞ g ~ðrÞ ¼ ð16Þ ð2Þ P ðrÞ The function g ~ðrÞ is a direct measure of the local density of the protein of interest. Since pffiffi 3L LASSI uses periodic boundary conditions, the maximal inter-monomer distance is . Given this normalized g ~ðrÞ, we note that if the system has short-range ordering as in a canonical liq- uid, the radial distribution function will oscillate around unity but eventually approach one as r!1. Conversely, if the system undergoes a density transition, g ~ðrÞ will have two distinct spatial regimes (Fig 3B): for small r, g ~ðrÞ will be characterized by a tall and broad peak such that g ~ðrÞ > 1 until g ~ðrÞ intersects the g ~ðrÞ ¼ 1 line; this region corresponds to the dense phase and we shall denote the value of r at this intersection to be r = r . For r > r , g ~ðrÞ will be b b between 0 and 1, and for lattices that are large enough to avoid finite size artefacts, g ~ðrÞ will converge to a value lower than one and this corresponds to the density in the dilute phase region. Furthermore, g ~ðrÞ can be used to estimate the densities within the dense and dilute phases. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 15 / 39 Simulations of phase transitions of multivalent proteins To quantify the global density inhomogeneity we introduce a simple measure, r �, which is calculated as follows: pffiffi 3L � � 2 � � 1 r � ~ r ¼ jgðrÞ 1jV dr ð17Þ L L In Eq (17), V(x), the volume element for the normalized radial distance x = r/L defined as: > 2 4px ; if 0 < x � > pffiffiffi 1 2 VðxÞ ¼ ð18Þ 2pxð3 4xÞ; if < x � 2 2 pffiffiffi pffiffiffi 2 3 2xð3p 12f ðxÞþ f ðxÞÞ; if < x � 1 2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðxÞ ¼ arctanð 4x 1Þ; and ð19Þ 2xð4x 3Þ f ðxÞ ¼ 8xarctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4x 2ð4x þ 1Þ If r �� 0, the global density inhomogeneity in the system is small and this will be characteristic of a single homogeneous phase dominating the simulation volume. As r �increases beyond zero, the system accommodates density inhomogeneities. We construct coexistence curves using a cutoff value of r �= 0.025, which is universal to all systems, to delineate between a homogeneous system and one that has undergone phase separation. Quantitative assessments of finite size effects The pair distribution function is central to our calculation of density inhomogeneities and constructing coexistence curves for a system of multivalent proteins simulated using LASSI. At 3 4 the start of this section we emphasized the importance of including 10 –10 distinct molecules within the simulation cell in order to avoid finite size artefacts. Prior to presenting detailed results that mimic specific systems, we present an analysis of finite size effects that we will con- front if the requisite numbers of molecules are not included in the simulations. The data we present are for simulations of mimics of the protein FUS, specifically the A + B system intro- n n duced in the results section. The phenomenological mapping of this protein architecture onto a cubic lattice is discussed at the start of the results section. Here, we present an analysis that makes a crucial technical point about finite size effects. First, we start with simulations for ideal polymers. The data shown in Fig 4 plots the pair ð2Þ distribution function P ðrÞ extracted for simulations of ideal models of FUS-like proteins. Results are shown for simulations that use 20 chain molecules of A + B as an example of a n n small system. These results are compared to those from simulations with 100, 200, 1000, 1500, 2000, 3000, and 4000 A + B molecules, respectively. The pair distribution functions have a n n ð2Þ self-similar character and this is revealed by plotting P ðr�Þ for all of the simulations, where r is the reduced distance that accounts for the fact that for a similar concentration, the simula- tion cells are made larger (higher values of L, which is box size) as the numbers of molecules increase. This analysis shows that even for a truly ideal system, the smallest simulation com- prising of only 20 molecules will generate noisy estimates of the pair distribution function. This clearly demonstrates the problems inherent to small systems where finite size effects are PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 16 / 39 Simulations of phase transitions of multivalent proteins Fig 4. Assessment of finite size effects in simulations of ideal, non-interacting chains. (a) Pair distribution functions computed in terms of the spatial separation between chain units. The distributions are maximal at r = L/2, where L is the size of the simulation cell for a given system. Note that L increases as the number of polymers in the system increases. With the exception of the smaller systems, the ideal chains show self-similar behavior for different system sizes. (b) The data plotted in 2r pffiffi panel (a) are re-plotted in terms of the scaled variable r� ¼ where L is the size of the simulation cell for boxes with i molecules. 3L https://doi.org/10.1371/journal.pcbi.1007028.g004 accentuated. Interestingly, for the ideal chain system, all simulations with 100 or more mole- cules yield similar pair distribution functions as assessed in Fig 4B. Next we assessed the impact of finite size effects with all of the terms in the potential being included in the simulation. There are three columns, one each for different values of the � � reduced temperature T , in Fig 5. As discussed in the results section, these values of T place the system of interest in the two-phase regime, with the quench depth into the two-phase regime increasing as T increases. The first row (a to c) of Fig 5 shows the same data as Fig 4A while the second row (d to f) in Fig 5 show the normalized data like Fig 4B. Each panel shows eight unnormalized pair distribution functions, one each for the systems with 20, 100, 200, 1000, 1500, 2000, 3000, and 4000 molecules, respectively. The onset of phase separation should be manifest by the presence of a trough located (2) � � between two peaks in the profiles for P (r ). This is evident for T = 0.267 (Fig 5, panel (c)) for all systems providing the numbers of molecules are greater than or equal to 10 . This quali- tative trend is preserved even for T = 0.217, although sampling difficulties in large systems (2) � � become obvious in the noisy estimates for P (r ). At the reduced temperature of T = 0.167 we confront two problems: The small systems where the numbers of molecules are less than 10 cannot support the distinction between a proper dense phase that coexists with a dilute phase. This behavior is similar to that observed for lower quench depths i.e., higher simulation temperatures as shown in panels (b) and (c) of Fig 5. However, as the system size grows, an additional problem arises and this has to do with large clusters becoming frozen, and thus PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 17 / 39 Simulations of phase transitions of multivalent proteins Fig 5. Assessment of finite size effects in simulations with real interacting chains. Panels (a), (b), and (c), respectively are the real chain equivalents of panel (a) in Fig 4 computed for three different simulation temperatures that represent three different quench depths of the system into its two-phase regime. Panels (d), (e), and (f), respectively are the real chain equivalents of panel (b) in Fig 9 computed for three different simulation temperatures that represent three different quench depths of the system into its two-phase regime. https://doi.org/10.1371/journal.pcbi.1007028.g005 inhibiting the achievement of equilibrium. This is evident from the pair distribution functions shown in panels (a) and (d) of Fig 5 for systems where the numbers of molecules exceed 10 . To overcome this broken ergodicity and obtain reliable converged pair distribution functions, we need additional biasing potentials and temperature sweep approaches used recently [43] to break up frozen clusters and enable their coalescence. In the results that we report here, we use the system size titration to identify the reduced temperatures below which broken ergodicity becomes evident. We do not include data from these simulations in our constructions of phase diagrams. Importantly, our analysis confirms the presence of finite size effects for small sys- tems and sets a lower bound on the numbers of molecules that are needed to observe facsimiles of phase separation as diagnosed by the calculated pair distribution functions. The conclusions drawn from analysis of the pair distribution functions are reinforced in our analysis of the radial distribution function gðr�Þ shown in S5 Fig. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 18 / 39 Simulations of phase transitions of multivalent proteins Estimating the percolation transition line that delineates percolated and non-percolated networks Associative polymers form networks characterized by physical crosslinks among stickers. Accordingly, we use the concept of a cluster, viz., a collection of unique chains connected via rotational interactions, to define the extent of percolation. In polymer melt simulations, the extent of percolation, known as the gel fraction in the polymer literature, is defined as the frac- tion of polymers participating in a percolating network that spans the simulation box in at least one direction [94]. More generally, we can use the fraction of polymers that make up the single largest cluster to quantify the onset of percolation and the changes to the extent of net- working beyond the percolation threshold [95]. A molecular network cannot percolate the whole simulation cell when dilute and dense phases coexist. Accordingly, we choose the sec- ond definition for the order parameter that describes the percolation transition, and we denote this as ϕ [29]. Semenov and Rubinstein demonstrated that a percolation transition is purely a connectivity transition [45]. This implies that the identification of the percolation threshold is not achiev- able using a bona fide order parameter but instead relies on a suitable topological description. Here, we employ the midpoint of the ϕ vs. concentration curve to assess the onset of percola- tion and the percolation line or curve is obtained as the locus of points in the phase diagram for which ϕ = 0.5. In a system where finite size effects are minimized, the percolation transi- tion is sharp having either a hyperbolic or sigmoidal shape as a function of concentration. Accordingly, the location of the percolation line will be relatively robust to the choice one makes for the percolation threshold. Results We demonstrate the use of LASSI by applying it to study two archetypal systems that are known to undergo phase separation [24, 31, 33]. The systems are instantiations of linear and branched multivalent protein systems, respectively. The simulation results obtained for linear multivalent proteins illustrate how phase diagrams are generated when protein concentration (at a fixed stoichiometry) and temperature are the independent variables. In the second exam- ple that includes a branched multivalent protein and a linear peptide, the temperature is fixed, and the concentrations of the individual components are varied. The simulation parameters for both systems are summarized in Table 1. For each system, we conducted 5 independent 8 6 simulations, each of which consists of 5×10 MC steps after 5×10 equilibration steps. The data were taken over the last half of the trajectories at a frequency of 5×10 steps. Table 1. Simulation parameters for system description. FUS-like system N130 + rpL5 system (see Fig 6) (see Fig 10) Bead notations A/B: stickers A/B: stickers N: neutral spacers N: neutral hub spacers Number of stickers s s = 5, s = 5 s = 10, s = 5 i A B A B Linker length l l = 1, l = 1, l = 4 l = 1, l = 3, l = 3 ij AN BN NN NA AA BB (in lattice units) Position-dependent energy E (r , r ) 1, if r = r 1, if r = r pos 1 2 1 2 1 2 0, otherwise 0, otherwise Pairwise interaction energyε (in temperature units) ε = -3,ε = 0,ε = 0 ε = -3,ε = 0,ε = 0 ij AB ii iN AB ii iN https://doi.org/10.1371/journal.pcbi.1007028.t001 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 19 / 39 Simulations of phase transitions of multivalent proteins A FUS-like system as an example of a linear multivalent protein Wang et al. [24] recently uncovered the molecular grammar that contributes to the driving forces for phase separation of the protein FUS and a family of related proteins. They showed −1 that, to first order, c /(s s ) , where c is the measured saturation concentration of the sat Y R sat FUS family proteins and s and s are the number of tyrosine (Tyr) and arginine (Arg) resi- Y R dues, respectively. In FUS and other proteins of similar architectures, the Tyr residues are located primarily within the N-terminal disordered prion-like domain (PLD), whereas the Arg residues are located primarily within the partially disordered C-terminal RNA binding domain (RBD). Using mutagenesis experiments, Wang et al. established that Tyr and Arg residues are the stickers in the FUS family proteins. Accordingly, the zeroth-order stickers and spacers repre- sentation used to model FUS in LASSI comprises of two parts: An N-terminal mimic of the PLD encompassing Tyr residues as stickers and a C-terminal mimic of the RBD that encom- passes Arg residues as stickers. Wang et al. also measured c for a 1:1 mixture of independent sat PLDs and RBDs interacting in trans. The c for this system is approximately twice that of the sat c for full-length FUS. Given the block copolymeric architecture of FUS, we denote the PLD sat and RBD as A and B , respectively for A and B-blocks of valence n. The model system of n n PLDs and RBDs interacting in trans is denoted as A +B (Fig 6A), whereas the system mim- n n icking full-length FUS where the stickers can interact in cis and in trans is denoted as A -B n n (Fig 6B). Within A and B blocks, spacers provide a uniform spacing of six lattice sites between n n stickers along the chain. We model spacers using a hybrid approach whereby a neutral spacer Fig 6. Architecture of the linear multivalent systems. (a, b) Cartoons to depict the A +B and A -B systems, respectively. Different colors of beads denote different n n n n species of stickers. Note that A -B can be simply considered as A +B where the two different sections of the proteins are joined together. (c) Linker lengths involved n n n n in the architecture (see also Table 1). Each sticker has a neighboring spacer bead that is 1 lattice site apart whereas the neighboring spacer beads are 4 lattice sites apart. This means that consecutive stickers are 6 lattice sites apart and also that the linkers connecting the two have a positive effective solvation volume. https://doi.org/10.1371/journal.pcbi.1007028.g006 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 20 / 39 Simulations of phase transitions of multivalent proteins monomer is attached to each sticker with spacing of a single lattice site (Fig 6C). This choice was made to provide a distinction between A -B and A +B . Accordingly, the relative con- n n n n centration of neutral beads will be higher in A -B when compared to A +B . This allows us to n n n n study linker-mediated differences between the driving forces for phase separation for A -B n n vs. A +B n n Fig 7 shows phase diagrams for the A +B and A -B systems calculated using data from n n n n LASSI-based simulations. In panels (a) and (b), the ordinate quantifies the reduced tempera- k T � � B ture T calculated as T ¼ whereε is the effective energy of pairwise interactions between stickers from the A and B blocks. Panel (a) shows results for the A +B system. The bulk n n n n pffiffiffiffiffiffiffiffiffiffiffi concentration in the A +B system is quantified along the abscissa as c ¼ c c where n n bulk A B n n c and c are the bulk concentrations of A and B , respectively. Panel (b) shows the phase n n A B n n diagram for the A -B system where the abscissa represents the bulk concentration of this n n system. Experiments show that the driving forces for phase separation are roughly twice as large for the full-length FUS compared to the system comprising of a 1:1 mixture of PLDs and RBDs [24]. This feature is recapitulated in LASSI simulations. For example, the width of the two- phase regime is larger for the A -B system compared to the A +B system for all values of T n n n n as shown in panel (e) of Fig 7. The critical temperature is higher for the A -B vs. A +B sys- n n n n � � tem (T � 0:56 vs. T � 0:36, respectively). The valence of stickers is the main determinant of c c the concentration at the critical point whereas the interactions mediated by spacers determine the density inhomogeneities and the critical temperature. The impact of longer chains and increased valence of stickers per chain is also evident from the percolation threshold, which is crossed at a bulk protein concentration that is two-fold lower for the A -B system when com- n n pared to the A +B system, across all the simulation temperatures. Differences between the n n two systems are also evident in the degree of cooperativity of phase separation and the percola- tion transition as shown in panels (d) and (e) of Fig 7. For each system, the intersection of the percolation threshold line with the two-phase regime shows that the dense phase predominantly forms a percolated droplet–panels (a) and (b) in Fig 7. Therefore, while phase separation without percolation is realizable, this is not the dominant scenario for associative polymers, where phase separation and percolation are typi- cally conjoined to give rise to percolated droplets. The density of proteins in these percolated droplets is governed by the interaction strengths, modulated by T and the effective solvation volumes of spacers [28, 29]. Unlike homopolymers, which comprise entirely of stickers or spacers depending on the solvent quality, associative polymers encompass a mixture of stickers and spacers. Stickers provide the driving forces for networking via reversible crosslinks and spacers determine whether or not these driving forces lead to phase separation via condensa- tion. Indeed, the importance of sticker-driven percolation is evidenced in the persistence of percolated networks for both systems at high values of T . The observation that dense phases form percolated droplets has several implications: (1) on timescales that are concordant with or smaller than the average lifetime of physical crosslinks between stickers, the condensates will have elastic properties; this will be replaced by viscous behavior on timescales that are longer than the average lifetime of physical crosslinks [96]; (2) accordingly, condensates will have an intrinsic tendency for viscoelasticity [97] and long-lived crosslinks will cause hardening behavior as has been observed in many systems [1, 6, 7, 11, 22, 24, 39–41]; (3) the extent of crosslinking above the percolation threshold will change continu- ously with concentration [16, 29], and this will govern the overall structure, internal dynamics, and porosity of condensates; (4) reactions within condensates are likely to be constrained by the network topology formed as a result of inter-sticker interactions [98]; these constraints can PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 21 / 39 Simulations of phase transitions of multivalent proteins Fig 7. Phase behavior of the linear multivalent systems. (a, b) Phase diagrams for the A +B and A -B systems, respectively. The purple line is a 2-dimensional linear n n n n interpolation for r = 0.025, and the area encapsulated by the purple line are where the systems have large density inhomogeneities and are thus considered to be phase separated. The green line is a 2-dimensional linear interpolation for ϕ = 0.5 and thus is the proxy for the percolation line. (c, d) r and ϕ curves as a function of c c � � concentrations at T = 0.383 (solid lines in (a) and (b)). (e) Width of the two-phase regime, w(T ), as a function of the reduced temperature. Not only does the A -B n n � � system have a higher critical temperature (T ~ 0.6 vs. T ~ 0.4), but also has a wider two-phase regime than the A +B system. n n https://doi.org/10.1371/journal.pcbi.1007028.g007 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 22 / 39 Simulations of phase transitions of multivalent proteins create a variety of interesting kinetic signatures for reactions [99], including temporal memo- ries as has been demonstrated recently for a system that undergoes thermoresponsive phase behavior [66]. Clearly, any description of biochemical reactions within condensates has to account for the structural and dynamical attributes of percolated droplets that are best described as network fluids. Move set frequencies and diagnostics of converged simulations We used results from simulations of linear multivalent protein system to assess the design of LASSI. The frequencies of the different move sets for simulations of the linear multivalent pro- tein system are summarized in Table 2. Considerations that go into the design of move sets include the achievement of converged equilibrium distributions, with maximal computational efficiency, for each bulk concentration. Details of these considerations were described in the methods section. Diagnostics from short simulations are often useful to optimize the move set frequencies especially if multiple short trials are performed using very different starting configurations. Fig 8 shows the concentration dependence of acceptance ratios for each of the move set types, diagnosed for simulations of the A +B and A -B systems. The acceptance ratios show n n n n similar qualitative trends for both systems, even though there are clear quantitative differences. The move with the highest acceptance ratio in the dense regime is the double pivot move, sig- nifying that the systems are transitioning into a pure polymer melt. The second most accepted move is the local move; extrapolating from the higher concentrations it is expected that the acceptance of local moves should also become small and that the double pivot move will be the most dominant way to alter chain configurations, since even the move of an individual mono- mer will require that multiple monomers from multiple chains are moved simultaneously. Both systems have similar qualitative trends for the translation move where we see a transition from being accepted at low concentrations to not being accepted at higher concentrations. Since the proteins in the A -B system are twice long as the A +B system, the absolute accep- n n n n tance ratio of the translation move is always lower in the A -B system. n n Analysis of acceptance ratios of different move sets within droplets will be helpful for esti- mating correlation lengths and amplitudes of conformational and concentration fluctuations within droplets. Cluster moves have high acceptance ratios in the dilute regime whereas the acceptance ratio nearly vanishes as the concentration increases. This is intuitive since the like- lihood of steric clashes increases with a decrease in available volume and this is coupled to the simultaneous increase in the fraction of molecules in the largest cluster. We note here that the cluster moves have the most dramatic change in acceptance ratios from values near 1 to values near 0. However, the apparent inefficiency of cluster moves in dense configurations cannot be used as a rationale to quench such moves. In fact, as shown in panel (a) of Fig 9, phase separa- tion, diagnosed in terms of r �, is suppressed if cluster moves are not part of the move set. This Table 2. Move frequencies according to their types. They are normalized to the sum of all frequencies used in each simulation. FUS-like system N130 + rpL5 system Cluster translation move 1 1 Chain translation move 10 10 Rotation move 100 100 Local move 250 250 Reptation move 0 50 Double pivot move 50 10 https://doi.org/10.1371/journal.pcbi.1007028.t002 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 23 / 39 Simulations of phase transitions of multivalent proteins Fig 8. Analysis of acceptance ratios for different move sets. Curves with different colors indicate acceptance ratios of different types of moves. The dashed lines show the saturation concentrations. The data are obtained from simulations with T = 0.383. (a) Acceptance ratio data for the A +B system. (b) Acceptance ratio data for the n n A -B system. n n https://doi.org/10.1371/journal.pcbi.1007028.g008 highlights the importance of cluster moves for generating bona fide phase separation as these facilitate coalescence that leads to condensation. Mixtures of N130 and the rpL5 peptide as an example of a branched multivalent protein system LASSI can also be deployed to study the phase behavior of branched multivalent proteins that undergo phase separation via obligate heterotypic interactions. Examples of branched multiva- lent proteins are molecules that form symmetric, stable oligomers such as nucleophosmin 1 (NPM1) and synthetic systems such as the corelets designed by Bracha et al. [100]. NPM1 is a key molecule within the granular component (GC) of the nucleolus [101]. Three coexisting layers define the nucleolus and the GC is the outermost layer. Within the GC, NPM1 appears to form facsimiles of percolated droplets in complex ribosomal proteins with Arg-rich motifs [17, 30]. A minimalist system that mimics the phase behavior of the GC comprises of a trun- cated version of NPM1, referred to as N130, and an Arg-rich peptide, designated as rpL5 [31– 33]. Both NPM1 and N130 form symmetric pentamers in the presence of Arg-rich peptides [102]. The pentamer formed by the association of folded domains serves as a scaffold for dis- playing disordered C-terminal tails that are defined by two distinct acidic tracts. The system also features an N-terminal disordered region with a well-defined acidic motif. The FUS system is an example of a protein that undergoes phase separation via obligate homotypic interactions. This implies that the interactions necessary and sufficient for driving phase separation are encoded within the sequence of FUS and the strengths of these PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 24 / 39 Simulations of phase transitions of multivalent proteins � � Fig 9. Importance of cluster translation moves. (a) r and (b) ϕ curves for the A +B (purple) and A -B systems (green) at T = 0.383. The solid lines are identical c n n n n with the curves in panels (c) and (d) of Fig 7. The dotted lines show the simulation results under the same system conditions but the frequency for cluster translation moves is set to zero. Not only do the systems phase separate and percolate at higher saturation concentrations, but also we can see that both percolation and separation are suppressed highly. Furthermore, errors are generally higher, due to the systems being highly dependent on the initial conditions of the system. https://doi.org/10.1371/journal.pcbi.1007028.g009 interactions can be modulated by changes to solution conditions. The N130 + rpL5 system is an example of a system that undergoes phase separation mainly via obligate heterotypic inter- actions that involve interactions between residues in the acidic tracts of N130 and the Arg motifs of rpL5. This could be viewed as an example of phase separation via complex coacerva- tion, providing the heterotypic interactions are purely electrostatic in nature [31–33]. How- ever, in general, there is likely to be combination of long- and short-range interactions that contribute to the spectrum of heterotypic interactions, and hence we refer to this class of mole- cules as drivers of phase separation via obligate heterotypic interactions. In addition to demonstrating the applicability of the LASSI engine for simulations of branched systems, we use the analysis as an opportunity to highlight key conceptual features of multicomponent systems that undergo phase separation via obligate heterotypic interactions. There are three main features that are borne out in the analysis: (1) For fixed temperature, the order parameters are the concentrations of the proteins that drive phase separation via obligate heterotypic interactions. In such systems, the coexistence curves, i.e., the binodals, will have a closed loop form. These will be ellipses for two components and n-ellipsoids for systems that involve up to n obligate heterotypic interactions to drive phase separation. (2) The systems will support reentrant phase behavior as has been reported recently for protein-RNA mixtures that undergo phase separation via obligate heterotypic interactions [103]. (3) The apparent satura- tion concentration of a component molecule in a system that undergoes phase separation via obligate heterotypic interactions will show non-trivial dependencies on its bulk concentration. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 25 / 39 Simulations of phase transitions of multivalent proteins Fig 10. Architecture of an archetypal branched multivalent system. (a) Schematic to depict the overall architecture. The pentamer with 10 orange stickers represents the N130 molecule where the gray central oligomerization domain is modeled as a neutral spacer monomer, and the rpL5 peptide is modeled as a linear molecule with 2 blue stickers. (b) Relevant length scales for the architecture (see also Table 1). For the rpL5 molecule a linker length of 3 was chosen between the two stickers, and for the N130 molecule the first sticker (modeling the A1 tract) is 1 lattice site away from the hub spacer whereas the second sticker (modeling the A2 tract) is 3 lattice sites away from the first sticker. https://doi.org/10.1371/journal.pcbi.1007028.g010 These dependencies are governed directly by the slopes of the tie lines that pass through the point corresponding to the bulk concentration and intersect the binodal at coexisting concentrations that equalize the chemical potentials of all species in the dense and dilute phases. Here, we use the example of the N130 + rpL5 system to showcase the three central fea- tures of phase diagrams for systems that undergo phase separation via obligate heterotypic interactions. In the LASSI representation, N130 pentamers with disordered tails are modeled using a five-armed structure. This approach follows the strategy of Feric et al. [30], which was based on the fact that pentamers do not dissociate under conditions where NPM1 / N130 undergo phase separation. Each arm comprises of two sticker sites to mimic the presence of the A1 and A2 acidic tracts within the disordered tails of NPM1 / N130. Therefore, each N130 pentamer displays a total of ten sticker sites. The spacers between each A1 tract and the N130 core as well as between each pair of A1 and A2 tracts on a disordered tail are phantom tethers, which means that their effective solvation volumes [29] are set to zero. Each rpL5 peptide has two sticker sites corresponding to the two Arg-rich motifs along the sequence. Schematic represen- tations of the coarse-grained architecture used for N130 and rpL5 are shown in Fig 10, and the move set frequencies are summarized in Table 2. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 26 / 39 Simulations of phase transitions of multivalent proteins Percolation lines have parabolic shapes The percolation line, constructed as a function of the concentrations of two multivalent mole- cules, has an overall parabolic shape (panel (a) of Fig 11). This feature may be explained as fol- lows: Let f and f denote the fractions of N130 and rpL5 molecules that are bound in a 1 2 network; s and s will denote the valence of stickers on N130 and rpL5, respectively; for the 1 2 current implementation of the N130 + rpL5 system, s = 10 and s = 2. Based on the Flory- 1 2 Fig 11. Phase behaviors of the branched multivalent systems for T = 0.25. (a) Full phase diagram, where the purple line denotes the proxy for the binodal and the green line is the proxy for the percolation line (see also the caption for Fig 7). The phase-separated region has an elliptical shape and we have a closed loop, which demonstrates re-entrant phase behavior, whereas the percolation line has a conical shape extending into much higher densities. The solid black lines denote contours of � � constant total concentration where L1 is the lowest concentration and L3 is the highest concentration. Note that both axes are represented in the log scale. (b, c) r and ϕ curves as a function of relative stoichiometric ratio of N130 and rpL5 along the constant-concentration contours. (d) Plot ofΛ vs. the apparent stoichiometry along lines L1, L2, and L3. https://doi.org/10.1371/journal.pcbi.1007028.g011 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 27 / 39 Simulations of phase transitions of multivalent proteins Stockmayer theory [49, 51], the percolation threshold is crossed when f f (s −1)(s −1) > 1. If 1 2 1 2 we keep (s −1)(s −1) constant, the threshold concentration for percolation will be governed by 1 2 the product f f . Accordingly, if there is a large excess of N130 (component 1) compared to 1 2 rpL5 (component 2), then f ! 0 and f ! 1, and the system does not undergo a percolation 1 2 transition. In this scenario, every rpL5 molecule is crosslinked to two sticker sites from one or two N130 molecules. However, since the relative stoichiometry favors N130 molecules, there is a vast excess of un-crosslinked N130 molecules and the network cannot grow. Percolation is also inhibited when the converse situation arises, i.e., when there is a large excess of compo- nent 2 with respect to component 1. Accordingly, the percolation line takes on a parabolic form in the plane defined by the concentrations of N130 and rpL5. Binodals for systems defined by obligate heterotypic interactions will form closed loop ellipses Given that the phase behavior of the N130 + rpL5 system is driven by heterotypic interactions involving the A1 / A2 tracts from the N130 tails and the Arg-motifs from rpL5, we constructed binodals by keeping the simulation temperature fixed and varied the concentrations of N130 and rpL5 molecules. Phase diagrams defined by N130 concentration along the abscissa and rpL5 concentration along the ordinate are shown in panel (a) of Fig 11. The general shape of the binodal is comparable with that of the experimentally determined phase diagram [33], even though direct comparison is not straightforward because the scarcity of experimental data points does not yield a full binodal. The phase boundary, defined by the density transition, is an ellipse that forms a closed loop in the plane defined by the concentrations c and c of N130 and rpL5, respectively. In associa- 1 2 tive polymers, the phase behavior is governed by the affinity between stickers, the valence of stickers, and the effective solvation volumes of spacers [28, 29]. For fixed c that intersects the two-phase regime an increase in c will lead to an entry into the two-phase regime caused by a density transition as c approaches c . However, as c increases well beyond c , the joint system 2 1 2 1 exits the two-phase regime. This is because phase separation is driven by obligate heterotypic interactions and while there is a growing excess of rpL5 molecules there are not enough N130 molecules to drive the density transition via inter-sticker interactions. Similar reasoning applies to describe the reentrant behavior that will result by keeping c fixed at a value that intersects the two-phase regime and increasing c . Taken together, the parabolic percolation lines and elliptic forms for two-phase regimes define conic sections that highlight reentrant phase behavior whereby fixing the concentration of component 1 and increasing the concentration of the second species can lead to phase sepa- ration and percolation at a low threshold concentration of component 2 and exit into the one- phase, non-percolated regime beyond a second higher threshold concentration for component 2. This type of reentrant phase behavior, will be a general feature of multicomponent systems that undergo phase separation via obligate heterotypic interactions; indeed, reentrant phase behavior has been reported for a model protein + RNA system [103]. Apparent stoichiometric ratios can be different from effective stoichiometric ratios Stoichiometry of molecules that drive phase separation is another key parameter that deter- mines the functions of biomolecular condensates formed by multicomponent systems [104]. The apparent stoichiometric ratio is calculated as the ratio of the concentrations of stickers of app c s1 types s and s for N130 and rpL5, respectively such that n ¼ . However, the effective stoi- 1 2 s2 app eff chiometric ratio n can be different from n if excluded volume effects modulate the effective 12 12 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 28 / 39 Simulations of phase transitions of multivalent proteins concentration of stickers. We fit an ellipse to the two-phase boundary and determined the major axis of this ellipse. The effective stoichiometric ratio should be unity along the major axis. As shown in panel (a) of Fig 11, the major axis deviates from the line along which app app eff n ¼ 1. Therefore, n 6¼ n and angle between the major axis and the line along which 12 12 12 app n ¼ 1 quantifies the impact of excluded volume on changes to effective concentrations of stickers that in turn modifies the stoichiometric ratios. The synergy between stoichiometry and phase behavior can be analyzed by quantifying the order parameter r �and the topological parameter ϕ as a function of apparent stoichiometry for fixed bulk concentration. Along each gray line in panel (a) of Fig 11 the total concentration defined as c = (c c ) is fixed, although the stoichiometries will vary. The mean values of c along L1, bulk 1 2 bulk −2 –1 −3 –1 −4 –1 L2, and L3 are 2.09×10 (voxel ), 2.46×10 (voxel ), and 3.33×10 (voxel ), respectively and app the value of n ranges from 0.36 to 22.62 along each of L1, L2, and L3. Panels (b) and (c) in Fig 11 app show the variation of r �and ϕ as n increases along L1, L2, and L3, respectively. Along L1, the c 12 value of r �is essentially zero irrespective of stoichiometry because L1 lies is outside the two-phase regime. However, a system spanning percolated network forms for stoichiometries in the range 1.2 � ν � 13 along L1. This is because the concentrations of both components are well above the percolation threshold along L1 thus ensuring that stickers readily find one another even without a density transition. In direct contrast, along L3, we observe phase separation, characterized by val- ues of r �> 0.025 for a range of stoichiometries, but none of these support the formation of a perco- lated droplet (ϕ < 0.5 for all stoichiometries). Along L2, we observe phase separation for stoichiometries in the range 1.15� ν � 16 and percolation for stoichiometries in the range 2.14 � ν � 11.3 such that phase separation enables the formation of a percolated droplet. In panel (d) of Fig 11, we introduce a new structural parameterΛ, which we define as a convolution of r �and ϕ such that L ¼ r �� . Here, the convolution is calculated as a logical AND gate, which becomes a simple product. The parameterΛ quantifies the convolution of the density and network transition and provides an estimate of the extent to which the phase separation and percolation are coupled as the apparent stoichiometry is varied for a fixed bulk concentration. The profile ofΛ is reminiscent of profiles measured by Case et al. [104] for the dwell time of signaling molecules as a function of stoichiometric ratios that govern the forma- tion of condensates at membranes. This suggests that dwell times, which are experimentally accessible parameters, might actually be proxies for the structural features of the condensates as measured by the convolution between phase separation and percolation and the extent of network formation within the condensate. The key finding is that the combination of the bulk concentration and stoichiometric ratio (as opposed to stoichiometry alone) will determine the quench depth into the two-phase regime. This in turn determines whether a system-spanning network forms without phase sep- aration or if phase separation enables the formation of a droplet-spanning network. The struc- ture of condensates and the overall phase behavior cannot be fully described in terms of c bulk or ν alone. Instead, this requires the consideration of both parameters jointly and relative to the quench depth, which refers to the location in the two-phase regime and with respect to the percolation line. This is important because the extent of crosslinking and the time scales asso- ciated with crosslinks will determine the material properties of the condensate. This in turn should contribute to parameters such as the dwell times of clients within condensates [104]. Saturation concentrations need not be fixed parameters in multicomponent systems The concept of a saturation concentration is one of the defining hallmarks of phase separation [2, 27]. For fixed solution conditions, phase separation in a closed two-component system (or PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 29 / 39 Simulations of phase transitions of multivalent proteins pseudo one-component system) comprising of a protein and solvent is realized when the bulk concentration of the protein denoted as c exceeds a saturation concentration denoted as c . sat The presence of a saturation concentration can be quantified by measuring the concentration of protein in the coexisting dilute phase as c increases. This value will not go above c . Strik- sat ingly, the presence of a saturation concentration has been confirmed in living cells for model disordered proteins by Brangwynne and coworkers using optogenetic tools in mammalian cells [100, 105] and by Khan et al. [106] using yeast as a model system. If the protein whose phase behavior is being interrogated is part of a system where obligate heterotypic interactions drive phase separation, then whether or not the concept of a saturation concentration continues to be valid will depend on the nature of the binodals. We illustrate this by coopting the elliptic phase boundary from Fig 11 for a three-component system that comprises of N130, (component A), rpL5 (component B), plus a solvent that is implicit in the LASSI simulations. This 3-component system may be thought of as a pseudo two-component system. For fixed temperature, the top row of panels (a)-(c) in Fig 12 show three types of ellip- tical, closed loop binodals. These are constructed in a plane where [B] increases along the posi- tive direction of the abscissa and [A] increases along the positive direction of the ordinate. The bottom row in each panel shows the result to be expected were we to measure the concentra- tion of A in the dilute phase, designated as [A] , as the bulk concentration [A] is varied. In Sol each of these measurements, the concentration of B is fixed at a specific value. Slopes of tie lines within the elliptic binodals determine how [A] varies Sol with [A] in the two-phase regime For concentrations of A and B that place the pseudo two-component system in the two-phase regime–red points along each of the curves in the bottom rows of panels (a)–(c)–we find that [A] can change as [A] increases. If the tie lines are horizontal or nearly horizontal, then Sol [A] will vary linearly with [A]. Non-linear variations of [A] with [A] will result for tie lines Sol Sol with positive or negative slopes. This is shown in panel (b) for tie lines with positive slopes. If the tie lines are essentially vertical, then the standard expectation regarding the invariance of [A] with [A] within the two-phase regime is recovered. However, even in this scenario, the Sol plateau value of [A] will shift upward or downward as the value of [B] increases–the upward Sol shift is shown in panel (c) of Fig 12. Here, B acts as a bona fide ligand for A, which is the mac- romolecule. Preferential binding of B to the dilute phase leads to an increase in [A] as Sol depicted in panel (c) of Fig 12. Ligand-mediated shifts in saturation concentrations arise due to polyphasic linkage, a phenomenon first introduced by Wyman and Gill [107]. The main conclusion is that the concept of a saturation concentration, as defined for a pseudo one-component system, does not transfer over to multicomponent systems where phase transi- tions are driven by obligate heterotypic interactions. Instead, the slopes of tie lines or the geome- tries of tie planes in higher dimensional ellipsoids will have a direct bearing on inferences from measurements where the bulk concentration of a protein or RNA component is varied when condensates are observed and the concentration of the molecule of interest is quantified in the coexisting dilute phase. This insight emerges from our ability to deploy LASSI to compute full binodals for multicomponent systems. Discussion In this work, we have built on the connection between multivalent proteins and associative polymers [44, 45, 98] with their stickers-and-spacers architecture [15, 17, 24, 28, 29, 40, 43, 47] to develop and deploy LASSI, a lattice-based open source computational engine that enables the simulation of system-specific phase diagrams of single and multi-component systems. The PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 30 / 39 Simulations of phase transitions of multivalent proteins Fig 12. Slopes of tie lines within elliptic binodals are important for systems that undergo phase separation via obligate heterotypic interactions. The ellipse is � � drawn to fit the locus of points based on the value r that meets our criteria for a density transition (see main text). Data for constructing the ellipse were taken from simulations of the N130 + rpL5 system–see Fig 11(A). This ellipse is used to assess the impact of slopes of tie lines for a two-component system comprising of macromolecule A that undergoes phase separation via obligate heterotypic interactions with macromolecule B. (a) Ellipse with nearly horizontal tie lines. The vertical lines shown in green, grey, and blue correspond to fixed values for [B] along the abscissa. As [A] increases, the system traverses across the two-phase regime, delineated by the ellipse, starting outside the ellipse, crossing the ellipse, and exiting the ellipse at high concentrations of A. (b) For each fixed value of [A], the plot shows how [A] varies with [A]. The red points on each curve were extracted from within the two-phase regime, whereas the black points lie outside the two-phase regime. Sol Clearly, [A] does not stay fixed as [A] increases. (c) Equivalent plot to that shown in panel (a) for the tie lines that we obtain for the N130 + rpL5 system. (d) Sol Equivalent plot to that shown in panel (b). Note the non-linear variation of [A] as [A] increases. (e) Ellipse annotated with vertical tie lines. In this case, phase Sol separation of [A] does not depend on obligate heterotypic interactions with B, but B can bind to A and has a choice of binding preferentially to A in either the dense or dilute phase. Here, A becomes the macromolecule and B the ligand. (f) Preferential binding of the ligand to the macromolecule in its dilute phase will shift the saturation concentration, assessed in terms of [A] , upward and this shift will depend on [B]. Accordingly, the plateau value of [A] in the two-phase regime shifts to Sol Sol higher values for higher values of [B]. https://doi.org/10.1371/journal.pcbi.1007028.g012 foundations of LASSI derive from the formalism of the bond fluctuation model [83, 84, 108]. We demonstrate how canonical ensemble Monte Carlo simulations with appropriately designed move sets and analysis of order parameters derived from the distribution functions allow us to calculate coexistence curves and percolation lines as a function of protein concen- tration and interaction strengths. The choice of a lattice-based approach for coarse-graining and modeling phase behavior of multivalent proteins is guided by the advantages of lattice models [109] for polymeric systems. To titrate across the full range of volume fractions, one needs to balance considerations of finite size effects–which requires large numbers of molecules–with large simulation volumes– which makes it difficult to observe density fluctuations that can grow into density inhomoge- neities. On lattices the conformational space is discretized and the calculation of interaction PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 31 / 39 Simulations of phase transitions of multivalent proteins potentials can be made to be very efficient through the use of look up tables. Importantly, we have generalized lattice-based simulations to incorporate anisotropic interactions. LASSI allows us to query the impacts of overall and intrinsic valence of stickers, interaction strengths between stickers, the spatial ranges of these interactions, the effective solvation vol- umes and lengths of spacers, and protein concentrations. These titrations generate multidi- mensional phase diagrams. The approaches underlying LASSI have been applied to model a variety of multicomponent systems, including mimics of RNA molecules [24, 28–30, 43, 79]. What is required is the development of approaches that enable systematic coarse-graining and adaptation of machine learning based methods to parameterize interaction potentials [78]. Engineering LASSI to be interoperable to cell-based modeling suites [110] will also allow for larger scale deployment of the overall framework. The calculation of pair and higher order dis- tribution functions should afford multiscale descriptions of the structural organization of molecular components within condensates. The acceptance ratios associated with different move sets and the length scales spanned by distinct move sets open the door to analyzing the dynamics of phase separation, percolation, and the interplay between the two. Another major direction for future application of LASSI is to uncover the determinants of compositional spec- ificity of condensates [1, 12]. Supporting information S1 Fig. Two-dimensional representation of rotation move. For a given randomly selected monomer (middle orange bead), 33−1 nearest lattice sites (yellow box) are checked for possi- ble interaction candidates, where eligible candidates have a non-zero interaction energy with the selected monomer. In this figure, orange stickers interact with blue stickers and thus this sticker has 3 possible candidates. The end orientational state of the monomer is then picked using the metropolis criterion, which also includes the non-interacting state. (TIF) S2 Fig. Two-dimensional representation of local move. For a given randomly selected monomer, a new location is proposed by sampling ±2 lattice sites in each coordinate (brown box) and picking a lattice site that is empty. If an empty lattice site is found within a pre-deter- mined number of trials, the numbers of interacting candidates are calculated at the old and proposed location (yellow boxes). Then the move is accepted or rejected using the modified Metropolis criterion that considers orientational bias (see text). (TIF) S3 Fig. Two-dimensional representation of reptation move. For a given randomly selected chain that has the same linker lengths between each monomer, an end is randomly picked. Then, a version of the local move is performed where the selected end is moved to a new ran- dom location that is an empty lattice site within 2 lattice sites in each coordinate (brown box). If an empty site is found within a predetermined number of trials, the number of orientational candidates is calculated for the whole chain in the old and the new configuration (yellow boxes). The modified metropolis criterion is then used to determine if the move is accepted or rejected. Note that since the whole chain is orientationally biased, monomers may have a dif- ferent orientational state after the move is accepted, as shown in the figure. (TIF) S4 Fig. Two-dimensional representation of double pivot move. For a randomly selected monomer, a 2×2×2 cube around the monomer is searched for appropriate bridging candidates (brown box), where an appropriate bridging candidate is the next monomer from a different chain, is within a linker length of the selected monomer as shown by the dashed line PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 32 / 39 Simulations of phase transitions of multivalent proteins connecting i and (i+1) . Furthermore, the distance between (i+1) and i must also be within m n m n a linker length as depicted by the upper dashed line. A list of all possible candidates is calcu- lated and then a randomly chosen candidate is used to break and remake covalent bonds. This results in a large conformational change for both polymers. If the selected polymer is not lin- ear, the move is rejected outright. (TIF) S5 Fig. Assessments of finite size effects analyzed in terms of g ~ðr Þ . The pair distributions from Figs 9(B) and 11 are used to compute the relevant radial distribution functions. This analysis is relevant because the radial distribution functions are used to extract the value of the order parameter that detects the onset of phase separation. For systems where the number of molecules A + B molecules is greater than 200, the radial distribution functions g ~ðr Þ start n n to deviate from one another only at the lowest temperatures where broken ergodicity becomes an issue. Therefore, for the A + B system studied in this calibration, it appears that the num- n n bers of A + B molecules have to be greater than 200 in order to obtain reliable information n n about the phase behavior. (a) g ~ðr Þ extracted for different numbers of A and B molecules n n � � � for T = 0.167. (b) g ~ðr Þ extracted for different numbers of A and B molecules for T = n n � � 0.217. (c) g ~ðr Þ extracted for different numbers of A and B molecules for T = 0.267. n n (TIF) Acknowledgments We thank Tyler Harmon for his original contributions, helpful discussions, and critical insights. Ammon Posey got us thinking about rigorous analysis of closed loop phase bound- aries. We are grateful to Joshua Riback for encouraging us to refine our original discussions of phase diagrams for the N130 + rpL5 system and present the analysis reported in Fig 12. Fur- qan Dar acknowledges helpful conversations with Suman Das and Hue-Sun Chan. We have benefited immensely from conversations about phase separation, the stickers-and-spacers framework, and the design of LASSI with Clifford Brangwynne, Alex Holehouse, Anthony Hyman, Amy Gladfelter, Richard Kriwacki, Tanja Mittag, Michael Rosen, Kiersten Ruff, Andrea Soranno, and Paul Taylor. And finally, we thank Minerva Pappu for insights regarding the mathematical self-similarities of ellipsoids in higher dimensional spaces. LASSI is designed to be an open source engine that will be made available via http://pappulab.wustl.edu that will provide a hyperlink to a suitable code hosting, reviewing, and distribution site. Author Contributions Conceptualization: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Funding acquisition: Rohit V. Pappu. Investigation: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Methodology: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Software: Jeong-Mo Choi, Furqan Dar. Supervision: Rohit V. Pappu. Visualization: Furqan Dar. Writing – original draft: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Writing – review & editing: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 33 / 39 Simulations of phase transitions of multivalent proteins References 1. Banani SF, Lee HO, Hyman AA, Rosen MK. Biomolecular condensates: organizers of cellular bio- chemistry. Nature Reviews Molecular Cell Biology. 2017; 18(5):285–98. https://doi.org/10.1038/nrm. 2017.7 PMID: 28225081 2. Shin Y, Brangwynne CP. Liquid phase condensation in cell physiology and disease. Science. 2017; 357(6357):eaaf4382. https://doi.org/10.1126/science.aaf4382 PMID: 28935776 3. Lamond AI, Spector DL. Nuclear speckles: a model for nuclear organelles. Nature Reviews in Molecu- lar and Cell Biology. 2003; 4(8):605–12. Epub 2003/08/19. https://doi.org/10.1038/nrm1172 PMID: 4. Mintz PJ, Spector DL. Compartmentalization of RNA processing factors within nuclear speckles. Jour- nal of structural biology. 2000; 129(2–3):241–51. Epub 2000/05/12. https://doi.org/10.1006/jsbi.2000. 4213 PMID: 10806074. 5. Brangwynne CP, Eckmann CR, Courson DS, Rybarska A, Hoege C, Gharakhani J, et al. Germline P granules are liquid droplets that localize by controlled dissolution/condensation. Science. 2009; 324 (5935):1729–32. https://doi.org/10.1126/science.1172046 PMID: 19460965 6. Saha S, Weber CA, Nousch M, Adame-Arana O, Hoege C, Hein MY, et al. Polar positioning of phase- separated liquid compartments in cells regulated by an mRNA competition mechanism. Cell. 2016; 166(6):1572–84.e16. https://doi.org/10.1016/j.cell.2016.08.006 PMID: 27594427 7. Patel A, Lee HO, Jawerth L, Maharana S, Jahnel M, Hein MY, et al. A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation. Cell. 2015; 162(5):1066–77. https://doi.org/10. 1016/j.cell.2015.07.047 PMID: 26317470 8. Su X, Ditlev JA, Hui E, Xing W, Banjade S, Okrut J, et al. Phase separation of signaling molecules pro- motes T cell receptor signal transduction. Science. 2016; 352(6285):595–9. https://doi.org/10.1126/ science.aad9964 PMID: 27056844 9. Banjade S, Rosen MK. Phase transitions of multivalent proteins can promote clustering of membrane receptors. eLife. 2014; 3:e04123. https://doi.org/10.7554/eLife.04123 PMID: 25321392 10. Zeng M, Chen X, Guan D, Xu J, Wu H, Tong P, et al. Reconstituted Postsynaptic Density as a Molecu- lar Platform for Understanding Synapse Formation and Plasticity. Cell. 2018; 174(5):1172–87.e16. https://doi.org/10.1016/j.cell.2018.06.047 PMID: 30078712 11. Hyman AA, Weber CA, Ju ¨ licher F. Liquid-liquid phase separation in biology. Annu Rev Cell Dev Biol. 2014; 30:39–58. https://doi.org/10.1146/annurev-cellbio-100913-013325 PMID: 25288112 12. Banani SF, Rice AM, Peeples WB, Lin Y, Jain S, Parker R, et al. Compositional Control of Phase-Sep- arated Cellular Bodies. Cell. 2016; 166(3):651–63. https://doi.org/10.1016/j.cell.2016.06.010 PMID: 13. Wu H, Fuxreiter M. The Structure and Dynamics of Higher-Order Assemblies: Amyloids, Signalo- somes, and Granules. Cell. 2016; 165(5):1055–66. https://doi.org/10.1016/j.cell.2016.05.004 PMID: 14. Falkenberg CV, Carson JH, Blinov ML. Multivalent Molecules as Modulators of RNA Granule Size and Composition. Biophysical Journal. 2017; 113(2):235–45. https://doi.org/10.1016/j.bpj.2017.01.031 PMID: 28242011 15. Posey AE, Holehouse AS, Pappu RV. Phase Separation of Intrinsically Disordered Proteins. In: Rhoades E, editor. Methods in Enzymology. 611: Academic Press; 2018. p. 1–30. https://doi.org/10. 1016/bs.mie.2018.09.035 PMID: 30471685 16. Li P, Banjade S, Cheng H-C, Kim S, Chen B, Guo L, et al. Phase transitions in the assembly of multiva- lent signalling proteins. Nature. 2012; 483(7389):336–40. https://doi.org/10.1038/nature10879 PMID: 17. Wei MT, Elbaum-Garfinkle S, Holehouse AS, Chen CC, Feric M, Arnold CB, et al. Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles. Nature Chemis- try. 2017; 9(11):1118–25. https://doi.org/10.1038/nchem.2803 PMID: 29064502. 18. Zhang H, Elbaum-Garfinkle S, Langdon EM, Taylor N, Occhipinti P, Bridges AA, et al. RNA controls PolyQ protein phase transitions. Molecular Cell. 2015; 60(2):220–30. https://doi.org/10.1016/j.molcel. 2015.09.017 PMID: 26474065 19. Elbaum-Garfinkle S, Kim Y, Szczepaniak K, Chen CC-H, Eckmann CR, Myong S, et al. The disor- dered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynam- ics. Proceedings of the National Academy of Sciences USA. 2015; 112(23):7189–94. https://doi.org/ 10.1073/pnas.1504822112 PMID: 26015579 20. Molliex A, Temirov J, Lee J, Coughlin M, Kanagaraj AP, Kim HJ, et al. Phase Separation by Low Com- plexity Domains Promotes Stress Granule Assembly and Drives Pathological Fibrillization. Cell. 2015; 163(1):123–33. https://doi.org/10.1016/j.cell.2015.09.015 PMID: 26406374 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 34 / 39 Simulations of phase transitions of multivalent proteins 21. Posey AE, Ruff KM, Harmon TS, Crick SL, Li A, Diamond MI, et al. Profilin reduces aggregation and phase separation of huntingtin N-terminal fragments by preferentially binding to soluble monomers and oligomers. Journal of Biological Chemistry. 2018; 293(10):3734–46. https://doi.org/10.1074/jbc. RA117.000357 PMID: 29358329 22. Mateju D, Franzmann TM, Patel A, Kopach A, Boczek EE, Maharana S, et al. An aberrant phase tran- sition of stress granules triggered by misfolded protein and prevented by chaperone function. The EMBO Journal. 2017:e201695957. https://doi.org/10.15252/embj.201695957 PMID: 28377462 23. Rai AK, Chen J-X, Selbach M, Pelkmans L. Kinase-controlled phase transition of membraneless organelles in mitosis. Nature. 2018; 559(7713):211–6. https://doi.org/10.1038/s41586-018-0279-8 PMID: 29973724 24. Wang J, Choi J-M, Holehouse AS, Lee HO, Zhang X, Jahnel M, et al. A Molecular Grammar Governing the Driving Forces for Phase Separation of Prion-like RNA Binding Proteins. Cell. 2018; 174(3):688– 99.e16. https://doi.org/10.1016/j.cell.2018.06.006 PMID: 29961577 25. Lin Y-H, Brady J, P, Forman-Kay J, D, Chan HS. Charge pattern matching as a ‘fuzzy’ mode of molec- ular recognition for the functional phase separations of intrinsically disordered proteins. New Journal of Physics. 2017; 19(11):115003. 26. Pak CW, Kosno M, Holehouse AS, Padrick SB, Mittal A, Ali R, et al. Sequence determinants of intra- cellular phase separation by complex coacervation of a disordered protein. Mol Cell. 2016; 63(1):72– 85. https://doi.org/10.1016/j.molcel.2016.05.042 PMID: 27392146 27. Brangwynne CP, Tompa P, Pappu RV. Polymer physics of intracellular phase transitions. Nature Physics. 2015; 11(11):899–904. https://doi.org/10.1038/nphys3532 28. Harmon TS, Holehouse AS, Pappu RV. Differential solvation of intrinsically disordered linkers drives the formation of spatially organized droplets in ternary systems of linear multivalent proteins. New Journal of Physics. 2018; 20(4):045002. https://doi.org/10.1088/1367-2630/aab8d9 29. Harmon TS, Holehouse AS, Rosen MK, Pappu RV. Intrinsically disordered linkers determine the inter- play between phase separation and gelation in multivalent proteins. eLife. 2017; 6:e30294. https://doi. org/10.7554/eLife.30294 PMID: 29091028 30. Feric M, Vaidya N, Harmon TS, Mitrea DM, Zhu L, Richardson TM, et al. Coexisting liquid phases underlie nucleolar subcompartments. Cell. 2016; 165(7):1686–97. https://doi.org/10.1016/j.cell.2016. 04.047 PMID: 27212236 31. Mitrea DM, Cika JA, Stanley CB, Nourse A, Onuchic PL, Banerjee PR, et al. Self-interaction of NPM1 modulates multiple mechanisms of liquid–liquid phase separation. Nature Communications. 2018; 9 (1):842. https://doi.org/10.1038/s41467-018-03255-3 PMID: 29483575 32. Ferrolino MC, Mitrea DM, Michael JR, Kriwacki RW. Compositional adaptability in NPM1-SURF6 scaf- folding networks enabled by dynamic switching of phase separation mechanisms. Nature Communi- cations. 2018; 9(1):5064. https://doi.org/10.1038/s41467-018-07530-1 PMID: 30498217 33. Mitrea DM, Cika JA, Guy CS, Ban D, Banerjee PR, Stanley CB, et al. Nucleophosmin integrates within the nucleolus via multi-modal interactions with proteins displaying R-rich linear motifs and rRNA. eLife. 2016; 5:e13571. https://doi.org/10.7554/eLife.13571 PMID: 26836305 34. Brady JP, Farber PJ, Sekhar A, Lin Y-H, Huang R, Bah A, et al. Structural and hydrodynamic proper- ties of an intrinsically disordered region of a germ cell-specific protein on phase separation. Proceed- ings of the National Academy of Sciences. 2017; 114(39):E8194–E203. https://doi.org/10.1073/pnas. 1706197114 PMID: 28894006 35. Vernon RM, Chong PA, Tsang B, Kim TH, Bah A, Farber P, et al. Pi-Pi contacts are an overlooked pro- tein feature relevant to phase separation. eLife. 2018; 7:e31486. https://doi.org/10.7554/eLife.31486 PMID: 29424691 36. Nott Timothy J, Petsalaki E, Farber P, Jervis D, Fussner E, Plochowietz A, et al. Phase Transition of a Disordered Nuage Protein Generates Environmentally Responsive Membraneless Organelles. Molec- ular Cell. 2015; 57(5):936–47. https://doi.org/10.1016/j.molcel.2015.01.013 PMID: 25747659 37. Lin Y, Currie SL, Rosen MK. Intrinsically disordered sequences enable modulation of protein phase separation through distributed tyrosine motifs. Journal of Biological Chemistry. 2017. https://doi.org/ 10.1074/jbc.M117.800466 PMID: 28924037 38. Boeynaems S, Bogaert E, Kovacs D, Konijnenberg A, Timmerman E, Volkov A, et al. Phase Separa- tion of C9orf72 Dipeptide Repeats Perturbs Stress Granule Dynamics. Mol Cell. 2017; 65(6):1044–55. e5. https://doi.org/10.1016/j.molcel.2017.02.013 PMID: 28306503 39. Woodruff JB, Ferreira Gomes B, Widlund PO, Mahamid J, Honigmann A, Hyman AA. The Centrosome Is a Selective Condensate that Nucleates Microtubules by Concentrating Tubulin. Cell. 2017; 169 (6):1066–77.e10. https://doi.org/10.1016/j.cell.2017.05.028 PMID: 28575670 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 35 / 39 Simulations of phase transitions of multivalent proteins 40. Franzmann TM, Jahnel M, Pozniakovsky A, Mahamid J, Holehouse AS, Nu ¨ ske E, et al. Phase separa- tion of a yeast prion protein promotes cellular fitness. Science. 2018; 359(6371):eaao5654. https://doi. org/10.1126/science.aao5654 PMID: 29301985 41. Maharana S, Wang J, Papadopoulos DK, Richter D, Pozniakovsky A, Poser I, et al. RNA buffers the phase separation behavior of prion-like RNA binding proteins. Science. 2018; 360(6391):918–21. https://doi.org/10.1126/science.aar7366 PMID: 29650702 42. Langdon EM, Qiu Y, Ghanbari Niaki A, McLaughlin GA, Weidmann CA, Gerbich TM, et al. mRNA structure determines specificity of a polyQ-driven phase separation. Science. 2018; 360(6391):922–7. https://doi.org/10.1126/science.aar7432 PMID: 29650703 43. Boeynaems S, Holehouse AS, Weinhardt V, Kovacs D, Van Lindt J, Larabell C, et al. Spontaneous driving forces give rise to protein−RNA condensates with coexisting phases and complex material properties. Proceedings of the National Academy of Sciences USA. 2019; 116:7889–98. https://doi. org/10.1073/pnas.1821038116 PMID: 30926670 44. Rubinstein M, Dobrynin AV. Solutions of Associative Polymers. Trends in Polymer Science. 1997; 5 (6):181–6. 45. Semenov AN, Rubinstein M. Thermoreversible Gelation in Solutions of Associative Polymers. 1. Stat- ics. Macromolecules. 1998; 31(4):1373–85. https://doi.org/10.1021/ma970616h 46. Halabi N, Rivoire O, Leibler S, Ranganathan R. Protein Sectors: Evolutionary Units of Three-Dimen- sional Structure. Cell. 2009; 138(4):774–86. https://doi.org/10.1016/j.cell.2009.07.038 PMID: 47. Holehouse AS, Pappu RV. Functional Implications of Intracellular Phase Transitions. Biochemistry. 2018; 57(17):2415–23. https://doi.org/10.1021/acs.biochem.7b01136 PMID: 29323488 48. Rubinstein M, Colby RH. Polymer Physics. New York: Oxford University Press; 2003 2003. 49. Stockmayer WH. Theory of Molecular Size Distribution and Gel Formation in Branched-Chain Poly- mers. The Journal of Chemical Physics. 1943; 11(2):45–55. https://doi.org/10.1063/1.1723803 50. Flory PJ. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1. Journal of the American Chemical Society. 1941; 63(11):3083–90. https://doi.org/10.1021/ja01856a061 51. Flory PJ. Constitution of Three-dimensional Polymers and the Theory of Gelation. The Journal of Physical Chemistry. 1942; 46(1):132–40. https://doi.org/10.1021/j150415a016 52. Dias CS, Arau ´ jo NAM, Telo da Gama MM. Dynamics of network fluids. Advances in Colloid and Inter- face Science. 2017; 247:258–63. https://doi.org/10.1016/j.cis.2017.07.001 PMID: 28802478 53. Lin Y-H, Forman-Kay JD, Chan HS. Theories for Sequence-Dependent Phase Behaviors of Biomolec- ular Condensates. Biochemistry. 2018; 57(17):2499–508. https://doi.org/10.1021/acs.biochem. 8b00058 PMID: 29509422 54. Lin Y-H, Forman-Kay JD, Chan HS. Sequence-Specific Polyampholyte Phase Separation in Membra- neless Organelles. Physical Review Letters. 2016; 117(17):178101. https://doi.org/10.1103/ PhysRevLett.117.178101 PMID: 27824447 55. Lytle TK, Sing CE. Transfer matrix theory of polymer complex coacervation. Soft Matter. 2017; 13 (39):7001–12. https://doi.org/10.1039/c7sm01080j PMID: 28840212 56. Lytle TK, Salazar AJ, Sing CE. Interfacial properties of polymeric complex coacervates from simulation and theory. The Journal of Chemical Physics. 2018; 149(16):163315. https://doi.org/10.1063/1. 5029934 PMID: 30384702 57. Lytle TK, Sing CE. Tuning chain interaction entropy in complex coacervation using polymer stiffness, architecture, and salt valency. Molecular Systems Design & Engineering. 2018; 3(1):183–96. https:// doi.org/10.1039/C7ME00108H 58. Lytle TK, Chang L-W, Markiewicz N, Perry SL, Sing CE. Designing Electrostatic Interactions via Poly- electrolyte Monomer Sequence. ACS Central Science. 2019; 5(4):709–18. https://doi.org/10.1021/ acscentsci.9b00087 PMID: 31041391 59. Ong GMC, Sing CE. Mapping the phase behavior of coacervate-driven self-assembly in diblock copo- lyelectrolytes. Soft Matter. 2019; 15(25):5116–27. https://doi.org/10.1039/c9sm00741e PMID: 60. McCarty J, Delaney KT, Danielsen SPO, Fredrickson GH, Shea J-E. Complete Phase Diagram for Liq- uid–Liquid Phase Separation of Intrinsically Disordered Proteins. The Journal of Physical Chemistry Letters. 2019; 10(8):1644–52. https://doi.org/10.1021/acs.jpclett.9b00099 PMID: 30873835 61. Qin S, Zhou H-X. Fast Method for Computing Chemical Potentials and Liquid–Liquid Phase Equilibria of Macromolecular Solutions. The Journal of Physical Chemistry B. 2016; 120(33):8164–74. https:// doi.org/10.1021/acs.jpcb.6b01607 PMID: 27327881 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 36 / 39 Simulations of phase transitions of multivalent proteins 62. Nguemaha V, Zhou H-X. Liquid-Liquid Phase Separation of Patchy Particles Illuminates Diverse Effects of Regulatory Components on Protein Droplet Formation. Scientific Reports. 2018; 8(1):6728. https://doi.org/10.1038/s41598-018-25132-1 PMID: 29712961 63. Monahan Z, Ryan VH, Janke AM, Burke KA, Rhoads SN, Zerze GH, et al. Phosphorylation of the FUS low-complexity domain disrupts phase separation, aggregation, and toxicity. EMBO Journal. 2017: e201696394. https://doi.org/10.15252/embj.201696394 PMID: 28790177 64. Dignon GL, Zheng W, Best RB, Kim YC, Mittal J. Relation between single-molecule properties and phase behavior of intrinsically disordered proteins. Proceedings of the National Academy of Sciences. 2018; 115(40):9929–34. https://doi.org/10.1073/pnas.1804177115 PMID: 30217894 65. Dignon GL, Zheng W, Kim YC, Best RB, Mittal J. Sequence determinants of protein phase behavior from a coarse-grained model. PLOS Computational Biology. 2018; 14(1):e1005941. https://doi.org/10. 1371/journal.pcbi.1005941 PMID: 29364893 66. Roberts S, Harmon TS, Schaal JL, Miao V, Li K, Hunt A, et al. Injectable tissue integrating networks from recombinant polypeptides with tunable order. Nature Materials. 2018; 17(12):1154–63. https:// doi.org/10.1038/s41563-018-0182-6 PMID: 30323334 67. Das S, Amin AN, Lin YH, Chan HS. Coarse-grained residue-based models of disordered protein con- densates: utility and limitations of simple charge pattern parameters. Physical Chemistry Chemical Physics. 2018; 20(45):28558–74. Epub 2018/11/07. https://doi.org/10.1039/c8cp05095c PMID: 68. Das S, Eisen A, Lin YH, Chan HS. A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation. Journal of Physical Chemistry B. 2018; 122(21):5418–31. Epub 2018/02/06. https://doi.org/10.1021/acs.jpcb.7b11723 PMID: 29397728. 69. Theodorou DN. Equilibration and Coarse-Graining Methods for Polymers. In: Ferrario M, Ciccotti G, Binder K, editors. Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2. Berlin, Heidelberg: Springer Berlin Heidelberg; 2006. p. 419–48. 70. Delaney KT, Fredrickson GH. Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions. Journal of Physical Chemistry B. 2016; 120(31):7615–34. Epub 2016/ 07/15. https://doi.org/10.1021/acs.jpcb.6b05704 PMID: 27414265. 71. Duchs D, Delaney KT, Fredrickson GH. A multi-species exchange model for fully fluctuating polymer field theory simulations. The Journal of Chemical Physics. 2014; 141(17):174103. Epub 2014/11/10. https://doi.org/10.1063/1.4900574 PMID: 25381498. 72. Izvekov S, Voth GA. A multiscale coarse-graining method for biomolecular systems. Journal of Physi- cal Chemistry B. 2005; 109(7):2469–73. Epub 2006/07/21. https://doi.org/10.1021/jp044629q PMID: 73. Izvekov S, Swanson JM, Voth GA. Coarse-graining in interaction space: a systematic approach for replacing long-range electrostatics with short-range potentials. Journal of Physical Chemistry B. 2008; 112(15):4711–24. Epub 2008/03/28. https://doi.org/10.1021/jp710339n PMID: 18366209. 74. Liu P, Shi Q, Daume H 3rd, Voth GA. A Bayesian statistics approach to multiscale coarse graining. The Journal of Chemical Physics. 2008; 129(21):214114. Epub 2008/12/10. https://doi.org/10.1063/1. 3033218 PMID: 19063551. 75. Dama JF, Sinitskiy AV, McCullagh M, Weare J, Roux B, Dinner AR, et al. The Theory of Ultra-Coarse- Graining. 1. General Principles. Journal of chemical theory and computation. 2013; 9(5):2466–80. Epub 2013/05/14. https://doi.org/10.1021/ct4000444 PMID: 26583735. 76. Lu L, Dama JF, Voth GA. Fitting coarse-grained distribution functions through an iterative force-match- ing method. The Journal of Chemical Physics. 2013; 139(12):121906. Epub 2013/10/05. https://doi. org/10.1063/1.4811667 PMID: 24089718. 77. Han Y, Jin J, Wagner JW, Voth GA. Quantum theory of multiscale coarse-graining. The Journal of Chemical Physics. 2018; 148(10):102335. Epub 2018/03/17. https://doi.org/10.1063/1.5010270 PMID: 29544317. 78. Ruff K, M., Harmon TS, Pappu RV. CAMELOT: A machine learning approach for coarse-grained simu- lations of aggregation of block-copolymeric protein sequences. The Journal of Chemical Physics. 2015; 143(24):243123. https://doi.org/10.1063/1.4935066 PMID: 26723608 79. Fei J, Jadaliha M, Harmon TS, Li IT, Hua B, Hao Q, et al. Quantitative analysis of multilayer organiza- tion of proteins and RNA in nuclear speckles at super resolution. Journal of Cell Science. 2017; 130 (24):4180–92. https://doi.org/10.1242/jcs.206854 PMID: 29133588 80. Harmon TS, Holehouse AS, Pappu RV. To Mix, or To Demix, That Is the Question. Biophysical Jour- nal. 2017; 112(4):565–7. https://doi.org/10.1016/j.bpj.2016.12.031 PMID: 28256216 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 37 / 39 Simulations of phase transitions of multivalent proteins 81. Freeman Rosenzweig ES, Xu B, Kuhn Cuellar L, Martinez-Sanchez A, Schaffer M, Strauss M, et al. The Eukaryotic CO2-Concentrating Organelle Is Liquid-like and Exhibits Dynamic Reorganization. Cell. 2017; 171(1):148–62.e19. https://doi.org/10.1016/j.cell.2017.08.008 PMID: 28938114 82. Bolhuis P, Frenkel D. Numerical study of the phase diagram of a mixture of spherical and rodlike col- loids. The Journal of Chemical Physics. 1994; 101(11):9869–75. https://doi.org/10.1063/1.467953 83. Shaffer JS. Effects of chain topology on polymer dynamics: Bulk melts. The Journal of Chemical Phys- ics. 1994; 101(5):4205–13. https://doi.org/10.1063/1.467470 84. Carmesin I, Kremer K. The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules. 1988; 21(9):2819–23. https://doi.org/10.1021/ ma00187a030 85. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics. 1953; 21(6):1087–92. https://doi.org/10. 1063/1.1699114 86. Siepmann JI, Frenkel D. Configurational bias Monte Carlo: a new sampling scheme for flexible chains. Molecular Physics. 1992; 75(1):59–70. https://doi.org/10.1080/00268979200100061 87. Rosenbluth MN, Rosenbluth AW. Monte Carlo Calculation of the Average Extension of Molecular Chains. The Journal of Chemical Physics. 1955; 23(2):356–9. https://doi.org/10.1063/1.1741967 88. Mańka A, Nowicki W, Nowicka G. Monte Carlo simulations of a polymer chain conformation. The effec- tiveness of local moves algorithms and estimation of entropy. Journal of Molecular Modeling. 2013; 19 (9):3659–70. https://doi.org/10.1007/s00894-013-1875-z PMID: 23765038 89. Qin Y, Liu H-L, Hu Y. Dynamic Monte Carlo Simulation of Polymers: Cooperative Move Algorithm. Molecular Simulation. 2003; 29(10–11):649–54. https://doi.org/10.1080/0892702031000103185 90. Gennes PGd. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. The Journal of Chem- ical Physics. 1971; 55(2):572–9. https://doi.org/10.1063/1.1675789 91. Sheng Y-J, Wang M-C. Statics and dynamics of a single polymer chain confined in a tube. The Journal of Chemical Physics. 2001; 114(10):4724–9. https://doi.org/10.1063/1.1345879 92. Binder K, Paul W. Monte Carlo simulations of polymer dynamics: Recent advances. Journal of Poly- mer Science Part B: Polymer Physics. 1997; 35(1):1–31. https://doi.org/10.1002/(sici)1099-0488 (19970115)35:1<1::aid-polb1>3.0.co;2-# 93. Widom B. Some Topics in the Theory of Fluids. The Journal of Chemical Physics. 1963; 39(11):2808– 12. https://doi.org/10.1063/1.1734110 94. Rottereau M, Gimel JC, Nicolai T, Durand D. Monte Carlo simulation of particle aggregation and gela- tion: I. Growth, structure and size distribution of the clusters. The European Physical Journal E. 2004; 15(2):133–40. https://doi.org/10.1140/epje/i2004-10044-x PMID: 15517458 95. Mikes J, Dusek K. Simulation of polymer network formation by the Monte Carlo method. Macromole- cules. 1982; 15(1):93–9. https://doi.org/10.1021/ma00229a018 96. Koyama T, Tanaka H. Volume-shrinking kinetics of transient gels as a consequence of dynamic inter- play between phase separation and mechanical relaxation. Physical Review E. 2018; 98(6):062617. https://doi.org/10.1103/PhysRevE.98.062617 97. Shayegan M, Tahvildari R, Kisley L, Metera K, Michnick SW, Leslie SR. Probing inhomogeneous diffu- sion in the microenvironments of phase-separated polymers under confinement. bioRxiv. 2018:402230. https://doi.org/10.1101/402230 98. Rubinstein M, Semenov AN. Thermoreversible Gelation in Solutions of Associating Polymers. 2. Lin- ear Dynamics. Macromolecules. 1998; 31(4):1386–97. https://doi.org/10.1021/ma970617+. 99. Kopelman R. Fractal Reaction Kinetics. Science. 1988; 241(4873):1620–6. https://doi.org/10.1126/ science.241.4873.1620 PMID: 17820893 100. Bracha D, Walls MT, Wei M-T, Zhu L, Kurian M, Avalos JL, et al. Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo-Oligomerizable Seeds. Cell. 2018; 175(6):1467–80.e13. https://doi.org/10.1016/j.cell.2018.10.048 PMID: 30500534 101. Sirri V, Urcuqui-Inchima S, Roussel P, Hernandez-Verdun D. Nucleolus: the fascinating nuclear body. Histochemistry and Cell Biology. 2008; 129(1):13–31. https://doi.org/10.1007/s00418-007-0359-6 PMID: 18046571 102. Mitrea DM, Grace CR, Buljan M, Yun M-K, Pytel NJ, Satumba J, et al. Structural polymorphism in the N-terminal oligomerization domain of NPM1. Proceedings of the National Academy of Sciences. 2014; 111(12):4466–71. https://doi.org/10.1073/pnas.1321007111 PMID: 24616519 103. Banerjee PR, Milin AN, Moosa MM, Onuchic PL, Deniz AA. Reentrant Phase Transition Drives Dynamic Substructure Formation in Ribonucleoprotein Droplets. Angewandte Chemie International Edition. 2017; 56(38):11354–9. https://doi.org/10.1002/anie.201703191 PMID: 28556382 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 38 / 39 Simulations of phase transitions of multivalent proteins 104. Case LB, Zhang X, Ditlev JA, Rosen MK. Stoichiometry controls activity of phase-separated clusters of actin signaling proteins. Science. 2019; 363(6431):1093–7. https://doi.org/10.1126/science. aau6313 PMID: 30846599 105. Shin Y, Berry J, Pannucci N, Haataja MP, Toettcher JE, Brangwynne CP. Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets. Cell. 2017; 168(1):159–71.e14. https://doi.org/10.1016/j.cell.2016.11.054 PMID: 28041848 106. Khan T, Kandola TS, Wu J, Venkatesan S, Ketter E, Lange JJ, et al. Quantifying Nucleation In Vivo Reveals the Physical Basis of Prion-like Phase Behavior. Molecular Cell. 2018; 71(1):155–68.e7. https://doi.org/10.1016/j.molcel.2018.06.016 PMID: 29979963 107. Wyman J, Gill SJ. Ligand-linked phase changes in a biological system: applications to sickle cell hemoglobin. Proceedings of the National Academy of Sciences of the United States of America. 1980; 77(9):5239–42. Epub 1980/09/01. https://doi.org/10.1073/pnas.77.9.5239 PMID: 6933555; PubMed Central PMCID: PMCPmc350033. 108. Subramanian G, Shanbhag S. On the relationship between two popular lattice models for polymer melts. The Journal of Chemical Physics. 2008; 129(14):144904. https://doi.org/10.1063/1.2992047 PMID: 19045165 109. Kremer K, Binder K. Monte Carlo simulation of lattice models for macromolecules. Computer Physics Reports. 1988; 7(6):259–310. https://doi.org/10.1016/0167-7977(88)90015-9. 110. Resasco DC, Gao F, Morgan F, Novak IL, Schaff JC, Slepchenko BM. Virtual Cell: computational tools for modeling in cell biology. Wiley Interdisciplinary Reviews: Systems Biology and Medicine. 2012; 4(2):129–40. https://doi.org/10.1002/wsbm.165 PMID: 22139996 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 39 / 39 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png PLoS Computational Biology Public Library of Science (PLoS) Journal http://www.deepdyve.com/lp/public-library-of-science-plos-journal/lassi-a-lattice-model-for-simulating-phase-transitions-of-multivalent-9GK2ZHACkL

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References (224)

M. Rubinstein, A. Semenov (1998)
Thermoreversible Gelation in Solutions of Associating Polymers. 2. Linear Dynamics
Macromolecules, 31
(WuH, FuxreiterM. The Structure and Dynamics of Higher-Order Assemblies: Amyloids, Signalosomes, and Granules. Cell. 2016;165(5):1055–66. 10.1016/j.cell.2016.05.004 27203110)
WuH, FuxreiterM. The Structure and Dynamics of Higher-Order Assemblies: Amyloids, Signalosomes, and Granules. Cell. 2016;165(5):1055–66. 10.1016/j.cell.2016.05.004 27203110
WuH, FuxreiterM. The Structure and Dynamics of Higher-Order Assemblies: Amyloids, Signalosomes, and Granules. Cell. 2016;165(5):1055–66. 10.1016/j.cell.2016.05.004 27203110, WuH, FuxreiterM. The Structure and Dynamics of Higher-Order Assemblies: Amyloids, Signalosomes, and Granules. Cell. 2016;165(5):1055–66. 10.1016/j.cell.2016.05.004 27203110
G. Dignon, Wenwei Zheng, R. Best, Young Kim, J. Mittal (2018)
Relation between single-molecule properties and phase behavior of intrinsically disordered proteins
Proceedings of the National Academy of Sciences, 115
Diana Mitrea, Jaclyn Cika, Christopher Stanley, A. Nourse, Paulo Onuchic, Priya Banerjee, Aaron Phillips, Cheon‐Gil Park, A. Deniz, R. Kriwacki (2018)
Self-interaction of NPM1 modulates multiple mechanisms of liquid–liquid phase separation
Nature Communications, 9
(OngGMC, SingCE. Mapping the phase behavior of coacervate-driven self-assembly in diblock copolyelectrolytes. Soft Matter. 2019;15(25):5116–27. 10.1039/c9sm00741e 31188388)
OngGMC, SingCE. Mapping the phase behavior of coacervate-driven self-assembly in diblock copolyelectrolytes. Soft Matter. 2019;15(25):5116–27. 10.1039/c9sm00741e 31188388
OngGMC, SingCE. Mapping the phase behavior of coacervate-driven self-assembly in diblock copolyelectrolytes. Soft Matter. 2019;15(25):5116–27. 10.1039/c9sm00741e 31188388, OngGMC, SingCE. Mapping the phase behavior of coacervate-driven self-assembly in diblock copolyelectrolytes. Soft Matter. 2019;15(25):5116–27. 10.1039/c9sm00741e 31188388
Yu-Jane Sheng, Mengwei Wang (2001)
Statics and dynamics of a single polymer chain confined in a tube
Journal of Chemical Physics, 114
Ammon Posey, Kiersten Ruff, T. Harmon, Scott Crick, Aimin Li, M. Diamond, R. Pappu (2018)
Profilin reduces aggregation and phase separation of huntingtin N-terminal fragments by preferentially binding to soluble monomers and oligomers
The Journal of Biological Chemistry, 293
Yuan Lin, S. Currie, M. Rosen (2017)
Intrinsically disordered sequences enable modulation of protein phase separation through distributed tyrosine motifs
The Journal of Biological Chemistry, 292
(DasS, EisenA, LinYH, ChanHS. A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation. Journal of Physical Chemistry B. 2018;122(21):5418–31. Epub 2018/02/06. 10.1021/acs.jpcb.7b11723 .29397728)
DasS, EisenA, LinYH, ChanHS. A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation. Journal of Physical Chemistry B. 2018;122(21):5418–31. Epub 2018/02/06. 10.1021/acs.jpcb.7b11723 .29397728
DasS, EisenA, LinYH, ChanHS. A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation. Journal of Physical Chemistry B. 2018;122(21):5418–31. Epub 2018/02/06. 10.1021/acs.jpcb.7b11723 .29397728, DasS, EisenA, LinYH, ChanHS. A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation. Journal of Physical Chemistry B. 2018;122(21):5418–31. Epub 2018/02/06. 10.1021/acs.jpcb.7b11723 .29397728
Xiaolei Su, J. Ditlev, E. Hui, Wenmin Xing, Sudeep Banjade, Julia Okrut, D. King, J. Taunton, M. Rosen, R. Vale (2016)
Phase separation of signaling molecules promotes T cell receptor signal transduction
Science, 352
(BoeynaemsS, BogaertE, KovacsD, KonijnenbergA, TimmermanE, VolkovA, et al Phase Separation of C9orf72 Dipeptide Repeats Perturbs Stress Granule Dynamics. Mol Cell. 2017;65(6):1044–55.e5. 10.1016/j.molcel.2017.02.013 28306503)
BoeynaemsS, BogaertE, KovacsD, KonijnenbergA, TimmermanE, VolkovA, et al Phase Separation of C9orf72 Dipeptide Repeats Perturbs Stress Granule Dynamics. Mol Cell. 2017;65(6):1044–55.e5. 10.1016/j.molcel.2017.02.013 28306503
BoeynaemsS, BogaertE, KovacsD, KonijnenbergA, TimmermanE, VolkovA, et al Phase Separation of C9orf72 Dipeptide Repeats Perturbs Stress Granule Dynamics. Mol Cell. 2017;65(6):1044–55.e5. 10.1016/j.molcel.2017.02.013 28306503, BoeynaemsS, BogaertE, KovacsD, KonijnenbergA, TimmermanE, VolkovA, et al Phase Separation of C9orf72 Dipeptide Repeats Perturbs Stress Granule Dynamics. Mol Cell. 2017;65(6):1044–55.e5. 10.1016/j.molcel.2017.02.013 28306503
N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller (1953)
Equation of state calculations by fast computing machines
Journal of Chemical Physics, 21
(KremerK, BinderK. Monte Carlo simulation of lattice models for macromolecules. Computer Physics Reports. 1988;7(6):259–310. 10.1016/0167-7977(88)90015-9.)
KremerK, BinderK. Monte Carlo simulation of lattice models for macromolecules. Computer Physics Reports. 1988;7(6):259–310. 10.1016/0167-7977(88)90015-9.
KremerK, BinderK. Monte Carlo simulation of lattice models for macromolecules. Computer Physics Reports. 1988;7(6):259–310. 10.1016/0167-7977(88)90015-9., KremerK, BinderK. Monte Carlo simulation of lattice models for macromolecules. Computer Physics Reports. 1988;7(6):259–310. 10.1016/0167-7977(88)90015-9.
(MonahanZ, RyanVH, JankeAM, BurkeKA, RhoadsSN, ZerzeGH, et al Phosphorylation of the FUS low-complexity domain disrupts phase separation, aggregation, and toxicity. EMBO Journal. 2017:e201696394 10.15252/embj.201696394 28790177)
MonahanZ, RyanVH, JankeAM, BurkeKA, RhoadsSN, ZerzeGH, et al Phosphorylation of the FUS low-complexity domain disrupts phase separation, aggregation, and toxicity. EMBO Journal. 2017:e201696394 10.15252/embj.201696394 28790177
MonahanZ, RyanVH, JankeAM, BurkeKA, RhoadsSN, ZerzeGH, et al Phosphorylation of the FUS low-complexity domain disrupts phase separation, aggregation, and toxicity. EMBO Journal. 2017:e201696394 10.15252/embj.201696394 28790177, MonahanZ, RyanVH, JankeAM, BurkeKA, RhoadsSN, ZerzeGH, et al Phosphorylation of the FUS low-complexity domain disrupts phase separation, aggregation, and toxicity. EMBO Journal. 2017:e201696394 10.15252/embj.201696394 28790177
(MikesJ, DusekK. Simulation of polymer network formation by the Monte Carlo method. Macromolecules. 1982;15(1):93–9. 10.1021/ma00229a018)
MikesJ, DusekK. Simulation of polymer network formation by the Monte Carlo method. Macromolecules. 1982;15(1):93–9. 10.1021/ma00229a018
MikesJ, DusekK. Simulation of polymer network formation by the Monte Carlo method. Macromolecules. 1982;15(1):93–9. 10.1021/ma00229a018, MikesJ, DusekK. Simulation of polymer network formation by the Monte Carlo method. Macromolecules. 1982;15(1):93–9. 10.1021/ma00229a018
B. Widom (1963)
Some Topics in the Theory of Fluids
Journal of Chemical Physics, 39
G. Voth (2013)
The Theory of Ultra Coarse-graining
Bulletin of the American Physical Society, 2013
S. Izvekov, J. Swanson, G. Voth (2008)
Coarse-graining in interaction space: a systematic approach for replacing long-range electrostatics with short-range potentials.
The journal of physical chemistry. B, 112 15
Jingyi Fei, Mahdieh Jadaliha, T. Harmon, I. Li, Boyang Hua, Qinyu Hao, A. Holehouse, M. Reyer, Qinyu Sun, S. Freier, R. Pappu, K. Prasanth, T. Ha (2017)
Quantitative analysis of multilayer organization of proteins and RNA in nuclear speckles at super resolution
Journal of Cell Science, 130
J. Dama, Anton Sinitskiy, Martin McCullagh, J. Weare, B. Roux, A. Dinner, G. Voth (2013)
The Theory of Ultra-Coarse-Graining. 1. General Principles.
Journal of chemical theory and computation, 9 5
(RubinsteinM, SemenovAN. Thermoreversible Gelation in Solutions of Associating Polymers. 2. Linear Dynamics. Macromolecules. 1998;31(4):1386–97. 10.1021/ma970617+.)
RubinsteinM, SemenovAN. Thermoreversible Gelation in Solutions of Associating Polymers. 2. Linear Dynamics. Macromolecules. 1998;31(4):1386–97. 10.1021/ma970617+.
RubinsteinM, SemenovAN. Thermoreversible Gelation in Solutions of Associating Polymers. 2. Linear Dynamics. Macromolecules. 1998;31(4):1386–97. 10.1021/ma970617+., RubinsteinM, SemenovAN. Thermoreversible Gelation in Solutions of Associating Polymers. 2. Linear Dynamics. Macromolecules. 1998;31(4):1386–97. 10.1021/ma970617+.
De Gennes (1971)
Reptation of a Polymer Chain in the Presence of Fixed Obstacles
Journal of Chemical Physics, 55
J. McCarty, K. Delaney, S. Danielsen, G. Fredrickson, J. Shea (2019)
Complete Phase Diagram for Liquid-Liquid Phase Separation of Intrinsically Disordered Proteins.
The journal of physical chemistry letters, 10 8
(LiP, BanjadeS, ChengH-C, KimS, ChenB, GuoL, et al Phase transitions in the assembly of multivalent signalling proteins. Nature. 2012;483(7389):336–40. 10.1038/nature10879 22398450)
LiP, BanjadeS, ChengH-C, KimS, ChenB, GuoL, et al Phase transitions in the assembly of multivalent signalling proteins. Nature. 2012;483(7389):336–40. 10.1038/nature10879 22398450
LiP, BanjadeS, ChengH-C, KimS, ChenB, GuoL, et al Phase transitions in the assembly of multivalent signalling proteins. Nature. 2012;483(7389):336–40. 10.1038/nature10879 22398450, LiP, BanjadeS, ChengH-C, KimS, ChenB, GuoL, et al Phase transitions in the assembly of multivalent signalling proteins. Nature. 2012;483(7389):336–40. 10.1038/nature10879 22398450
Ammon Posey, A. Holehouse, R. Pappu (2018)
Phase Separation of Intrinsically Disordered Proteins.
Methods in enzymology, 611
M. Rosenbluth, A. Rosenbluth (1955)
MONTE CARLO CALCULATION OF THE AVERAGE EXTENSION OF MOLECULAR CHAINS
Journal of Chemical Physics, 23
Yuan Qin, Hong-Lai Liu, Ying Hu (2003)
Dynamic Monte Carlo Simulation of Polymers: Cooperative Move Algorithm
Molecular Simulation, 29
(2017)
Phase Separa- tion of C9orf72 Dipeptide Repeats Perturbs Stress Granule Dynamics
(PoseyAE, HolehouseAS, PappuRV. Phase Separation of Intrinsically Disordered Proteins In: RhoadesE, editor. Methods in Enzymology. 611: Academic Press; 2018 p. 1–30. 10.1016/bs.mie.2018.09.035 30471685)
PoseyAE, HolehouseAS, PappuRV. Phase Separation of Intrinsically Disordered Proteins In: RhoadesE, editor. Methods in Enzymology. 611: Academic Press; 2018 p. 1–30. 10.1016/bs.mie.2018.09.035 30471685
PoseyAE, HolehouseAS, PappuRV. Phase Separation of Intrinsically Disordered Proteins In: RhoadesE, editor. Methods in Enzymology. 611: Academic Press; 2018 p. 1–30. 10.1016/bs.mie.2018.09.035 30471685, PoseyAE, HolehouseAS, PappuRV. Phase Separation of Intrinsically Disordered Proteins In: RhoadesE, editor. Methods in Enzymology. 611: Academic Press; 2018 p. 1–30. 10.1016/bs.mie.2018.09.035 30471685
(QinS, ZhouH-X. Fast Method for Computing Chemical Potentials and Liquid–Liquid Phase Equilibria of Macromolecular Solutions. The Journal of Physical Chemistry B. 2016;120(33):8164–74. 10.1021/acs.jpcb.6b01607 27327881)
QinS, ZhouH-X. Fast Method for Computing Chemical Potentials and Liquid–Liquid Phase Equilibria of Macromolecular Solutions. The Journal of Physical Chemistry B. 2016;120(33):8164–74. 10.1021/acs.jpcb.6b01607 27327881
QinS, ZhouH-X. Fast Method for Computing Chemical Potentials and Liquid–Liquid Phase Equilibria of Macromolecular Solutions. The Journal of Physical Chemistry B. 2016;120(33):8164–74. 10.1021/acs.jpcb.6b01607 27327881, QinS, ZhouH-X. Fast Method for Computing Chemical Potentials and Liquid–Liquid Phase Equilibria of Macromolecular Solutions. The Journal of Physical Chemistry B. 2016;120(33):8164–74. 10.1021/acs.jpcb.6b01607 27327881
(QinY, LiuH-L, HuY. Dynamic Monte Carlo Simulation of Polymers: Cooperative Move Algorithm. Molecular Simulation. 2003;29(10–11):649–54. 10.1080/0892702031000103185)
QinY, LiuH-L, HuY. Dynamic Monte Carlo Simulation of Polymers: Cooperative Move Algorithm. Molecular Simulation. 2003;29(10–11):649–54. 10.1080/0892702031000103185
QinY, LiuH-L, HuY. Dynamic Monte Carlo Simulation of Polymers: Cooperative Move Algorithm. Molecular Simulation. 2003;29(10–11):649–54. 10.1080/0892702031000103185, QinY, LiuH-L, HuY. Dynamic Monte Carlo Simulation of Polymers: Cooperative Move Algorithm. Molecular Simulation. 2003;29(10–11):649–54. 10.1080/0892702031000103185
Yi‐Hsuan Lin, Jacob Brady, J. Forman-Kay, H. Chan (2017)
Charge pattern matching as a ‘fuzzy’ mode of molecular recognition for the functional phase separations of intrinsically disordered proteins
New Journal of Physics, 19
Suman Das, Adam Eisen, Yi‐Hsuan Lin, H. Chan (2018)
A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation.
The journal of physical chemistry. B, 122 21
(HymanAA, WeberCA, JülicherF. Liquid-liquid phase separation in biology. Annu Rev Cell Dev Biol. 2014;30:39–58. 10.1146/annurev-cellbio-100913-013325 25288112)
HymanAA, WeberCA, JülicherF. Liquid-liquid phase separation in biology. Annu Rev Cell Dev Biol. 2014;30:39–58. 10.1146/annurev-cellbio-100913-013325 25288112
HymanAA, WeberCA, JülicherF. Liquid-liquid phase separation in biology. Annu Rev Cell Dev Biol. 2014;30:39–58. 10.1146/annurev-cellbio-100913-013325 25288112, HymanAA, WeberCA, JülicherF. Liquid-liquid phase separation in biology. Annu Rev Cell Dev Biol. 2014;30:39–58. 10.1146/annurev-cellbio-100913-013325 25288112
D. Theodorou (2006)
Equilibration and coarse-graining methods for polymers
Lecture Notes in Physics, 704
(BoeynaemsS, HolehouseAS, WeinhardtV, KovacsD, Van LindtJ, LarabellC, et al Spontaneous driving forces give rise to protein−RNA condensates with coexisting phases and complex material properties. Proceedings of the National Academy of Sciences USA. 2019;116:7889–98. 10.1073/pnas.1821038116 30926670)
BoeynaemsS, HolehouseAS, WeinhardtV, KovacsD, Van LindtJ, LarabellC, et al Spontaneous driving forces give rise to protein−RNA condensates with coexisting phases and complex material properties. Proceedings of the National Academy of Sciences USA. 2019;116:7889–98. 10.1073/pnas.1821038116 30926670
BoeynaemsS, HolehouseAS, WeinhardtV, KovacsD, Van LindtJ, LarabellC, et al Spontaneous driving forces give rise to protein−RNA condensates with coexisting phases and complex material properties. Proceedings of the National Academy of Sciences USA. 2019;116:7889–98. 10.1073/pnas.1821038116 30926670, BoeynaemsS, HolehouseAS, WeinhardtV, KovacsD, Van LindtJ, LarabellC, et al Spontaneous driving forces give rise to protein−RNA condensates with coexisting phases and complex material properties. Proceedings of the National Academy of Sciences USA. 2019;116:7889–98. 10.1073/pnas.1821038116 30926670
J. Siepmann, D. Frenkel (1992)
Configurational bias Monte Carlo: a new sampling scheme for flexible chains
Molecular Physics, 75
(DiasCS, AraújoNAM, Telo da GamaMM. Dynamics of network fluids. Advances in Colloid and Interface Science. 2017;247:258–63. 10.1016/j.cis.2017.07.001 28802478)
DiasCS, AraújoNAM, Telo da GamaMM. Dynamics of network fluids. Advances in Colloid and Interface Science. 2017;247:258–63. 10.1016/j.cis.2017.07.001 28802478
DiasCS, AraújoNAM, Telo da GamaMM. Dynamics of network fluids. Advances in Colloid and Interface Science. 2017;247:258–63. 10.1016/j.cis.2017.07.001 28802478, DiasCS, AraújoNAM, Telo da GamaMM. Dynamics of network fluids. Advances in Colloid and Interface Science. 2017;247:258–63. 10.1016/j.cis.2017.07.001 28802478
V. Nguemaha, Huan‐Xiang Zhou (2018)
Liquid-Liquid Phase Separation of Patchy Particles Illuminates Diverse Effects of Regulatory Components on Protein Droplet Formation
Scientific Reports, 8
L. Case, Xu Zhang, J. Ditlev, M. Rosen (2019)
Stoichiometry controls activity of phase-separated clusters of actin signaling proteins
Science, 363
W. Richtering (2003)
Polymer Physics
Applied Rheology, 13
(ZhangH, Elbaum-GarfinkleS, LangdonEM, TaylorN, OcchipintiP, BridgesAA, et al RNA controls PolyQ protein phase transitions. Molecular Cell. 2015;60(2):220–30. 10.1016/j.molcel.2015.09.017 26474065)
ZhangH, Elbaum-GarfinkleS, LangdonEM, TaylorN, OcchipintiP, BridgesAA, et al RNA controls PolyQ protein phase transitions. Molecular Cell. 2015;60(2):220–30. 10.1016/j.molcel.2015.09.017 26474065
ZhangH, Elbaum-GarfinkleS, LangdonEM, TaylorN, OcchipintiP, BridgesAA, et al RNA controls PolyQ protein phase transitions. Molecular Cell. 2015;60(2):220–30. 10.1016/j.molcel.2015.09.017 26474065, ZhangH, Elbaum-GarfinkleS, LangdonEM, TaylorN, OcchipintiP, BridgesAA, et al RNA controls PolyQ protein phase transitions. Molecular Cell. 2015;60(2):220–30. 10.1016/j.molcel.2015.09.017 26474065
(BolhuisP, FrenkelD. Numerical study of the phase diagram of a mixture of spherical and rodlike colloids. The Journal of Chemical Physics. 1994;101(11):9869–75. 10.1063/1.467953)
BolhuisP, FrenkelD. Numerical study of the phase diagram of a mixture of spherical and rodlike colloids. The Journal of Chemical Physics. 1994;101(11):9869–75. 10.1063/1.467953
BolhuisP, FrenkelD. Numerical study of the phase diagram of a mixture of spherical and rodlike colloids. The Journal of Chemical Physics. 1994;101(11):9869–75. 10.1063/1.467953, BolhuisP, FrenkelD. Numerical study of the phase diagram of a mixture of spherical and rodlike colloids. The Journal of Chemical Physics. 1994;101(11):9869–75. 10.1063/1.467953
(FranzmannTM, JahnelM, PozniakovskyA, MahamidJ, HolehouseAS, NüskeE, et al Phase separation of a yeast prion protein promotes cellular fitness. Science. 2018;359(6371):eaao5654 10.1126/science.aao5654 29301985)
FranzmannTM, JahnelM, PozniakovskyA, MahamidJ, HolehouseAS, NüskeE, et al Phase separation of a yeast prion protein promotes cellular fitness. Science. 2018;359(6371):eaao5654 10.1126/science.aao5654 29301985
FranzmannTM, JahnelM, PozniakovskyA, MahamidJ, HolehouseAS, NüskeE, et al Phase separation of a yeast prion protein promotes cellular fitness. Science. 2018;359(6371):eaao5654 10.1126/science.aao5654 29301985, FranzmannTM, JahnelM, PozniakovskyA, MahamidJ, HolehouseAS, NüskeE, et al Phase separation of a yeast prion protein promotes cellular fitness. Science. 2018;359(6371):eaao5654 10.1126/science.aao5654 29301985
(FloryPJ. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1. Journal of the American Chemical Society. 1941;63(11):3083–90. 10.1021/ja01856a061)
FloryPJ. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1. Journal of the American Chemical Society. 1941;63(11):3083–90. 10.1021/ja01856a061
FloryPJ. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1. Journal of the American Chemical Society. 1941;63(11):3083–90. 10.1021/ja01856a061, FloryPJ. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1. Journal of the American Chemical Society. 1941;63(11):3083–90. 10.1021/ja01856a061
K. Kremer, K. Binder (1988)
Monte Carlo simulation of lattice models for macromolecules
Computer Physics Reports, 7
T. Lytle, C. Sing (2017)
Transfer matrix theory of polymer complex coacervation.
Soft matter, 13 39
K. Binder, W. Paul (1997)
Monte Carlo simulations of polymer dynamics: Recent advances
Journal of Polymer Science Part B, 35
(HarmonTS, HolehouseAS, PappuRV. Differential solvation of intrinsically disordered linkers drives the formation of spatially organized droplets in ternary systems of linear multivalent proteins. New Journal of Physics. 2018;20(4):045002 10.1088/1367-2630/aab8d9)
HarmonTS, HolehouseAS, PappuRV. Differential solvation of intrinsically disordered linkers drives the formation of spatially organized droplets in ternary systems of linear multivalent proteins. New Journal of Physics. 2018;20(4):045002 10.1088/1367-2630/aab8d9
HarmonTS, HolehouseAS, PappuRV. Differential solvation of intrinsically disordered linkers drives the formation of spatially organized droplets in ternary systems of linear multivalent proteins. New Journal of Physics. 2018;20(4):045002 10.1088/1367-2630/aab8d9, HarmonTS, HolehouseAS, PappuRV. Differential solvation of intrinsically disordered linkers drives the formation of spatially organized droplets in ternary systems of linear multivalent proteins. New Journal of Physics. 2018;20(4):045002 10.1088/1367-2630/aab8d9
A. Rai, Jia-Xuan Chen, M. Selbach, L. Pelkmans (2018)
Kinase-controlled phase transition of membraneless organelles in mitosis
Nature, 559
(LangdonEM, QiuY, Ghanbari NiakiA, McLaughlinGA, WeidmannCA, GerbichTM, et al mRNA structure determines specificity of a polyQ-driven phase separation. Science. 2018;360(6391):922–7. 10.1126/science.aar7432 29650703)
LangdonEM, QiuY, Ghanbari NiakiA, McLaughlinGA, WeidmannCA, GerbichTM, et al mRNA structure determines specificity of a polyQ-driven phase separation. Science. 2018;360(6391):922–7. 10.1126/science.aar7432 29650703
LangdonEM, QiuY, Ghanbari NiakiA, McLaughlinGA, WeidmannCA, GerbichTM, et al mRNA structure determines specificity of a polyQ-driven phase separation. Science. 2018;360(6391):922–7. 10.1126/science.aar7432 29650703, LangdonEM, QiuY, Ghanbari NiakiA, McLaughlinGA, WeidmannCA, GerbichTM, et al mRNA structure determines specificity of a polyQ-driven phase separation. Science. 2018;360(6391):922–7. 10.1126/science.aar7432 29650703
(McCartyJ, DelaneyKT, DanielsenSPO, FredricksonGH, SheaJ-E. Complete Phase Diagram for Liquid–Liquid Phase Separation of Intrinsically Disordered Proteins. The Journal of Physical Chemistry Letters. 2019;10(8):1644–52. 10.1021/acs.jpclett.9b00099 30873835)
McCartyJ, DelaneyKT, DanielsenSPO, FredricksonGH, SheaJ-E. Complete Phase Diagram for Liquid–Liquid Phase Separation of Intrinsically Disordered Proteins. The Journal of Physical Chemistry Letters. 2019;10(8):1644–52. 10.1021/acs.jpclett.9b00099 30873835
McCartyJ, DelaneyKT, DanielsenSPO, FredricksonGH, SheaJ-E. Complete Phase Diagram for Liquid–Liquid Phase Separation of Intrinsically Disordered Proteins. The Journal of Physical Chemistry Letters. 2019;10(8):1644–52. 10.1021/acs.jpclett.9b00099 30873835, McCartyJ, DelaneyKT, DanielsenSPO, FredricksonGH, SheaJ-E. Complete Phase Diagram for Liquid–Liquid Phase Separation of Intrinsically Disordered Proteins. The Journal of Physical Chemistry Letters. 2019;10(8):1644–52. 10.1021/acs.jpclett.9b00099 30873835
A. Semenov, M. Rubinstein (1998)
Thermoreversible Gelation in Solutions of Associative Polymers. 1. Statics
Macromolecules, 31
(LinY-H, BradyJ, P, Forman-KayJ, D, ChanHS. Charge pattern matching as a ‘fuzzy’ mode of molecular recognition for the functional phase separations of intrinsically disordered proteins. New Journal of Physics. 2017;19(11):115003.)
LinY-H, BradyJ, P, Forman-KayJ, D, ChanHS. Charge pattern matching as a ‘fuzzy’ mode of molecular recognition for the functional phase separations of intrinsically disordered proteins. New Journal of Physics. 2017;19(11):115003.
LinY-H, BradyJ, P, Forman-KayJ, D, ChanHS. Charge pattern matching as a ‘fuzzy’ mode of molecular recognition for the functional phase separations of intrinsically disordered proteins. New Journal of Physics. 2017;19(11):115003., LinY-H, BradyJ, P, Forman-KayJ, D, ChanHS. Charge pattern matching as a ‘fuzzy’ mode of molecular recognition for the functional phase separations of intrinsically disordered proteins. New Journal of Physics. 2017;19(11):115003.
A. Hyman, C. Weber, F. Jülicher (2014)
Liquid-liquid phase separation in biology.
Annual review of cell and developmental biology, 30
(BrachaD, WallsMT, WeiM-T, ZhuL, KurianM, AvalosJL, et al Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo-Oligomerizable Seeds. Cell. 2018;175(6):1467–80.e13. 10.1016/j.cell.2018.10.048 30500534)
BrachaD, WallsMT, WeiM-T, ZhuL, KurianM, AvalosJL, et al Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo-Oligomerizable Seeds. Cell. 2018;175(6):1467–80.e13. 10.1016/j.cell.2018.10.048 30500534
BrachaD, WallsMT, WeiM-T, ZhuL, KurianM, AvalosJL, et al Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo-Oligomerizable Seeds. Cell. 2018;175(6):1467–80.e13. 10.1016/j.cell.2018.10.048 30500534, BrachaD, WallsMT, WeiM-T, ZhuL, KurianM, AvalosJL, et al Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo-Oligomerizable Seeds. Cell. 2018;175(6):1467–80.e13. 10.1016/j.cell.2018.10.048 30500534
T. Lytle, Li Chang, Natalia Markiewicz, S. Perry, C. Sing (2019)
Designing Electrostatic Interactions via Polyelectrolyte Monomer Sequence
ACS Central Science, 5
(RubinsteinM, ColbyRH. Polymer Physics New York: Oxford University Press; 2003 2003.)
RubinsteinM, ColbyRH. Polymer Physics New York: Oxford University Press; 2003 2003.
RubinsteinM, ColbyRH. Polymer Physics New York: Oxford University Press; 2003 2003., RubinsteinM, ColbyRH. Polymer Physics New York: Oxford University Press; 2003 2003.
Yining Han, Jaehyeok Jin, J. Wagner, G. Voth (2018)
Quantum theory of multiscale coarse-graining.
The Journal of chemical physics, 148 10
C. Dias, N. Araújo, M. Gama (2017)
Dynamics of network fluids.
Advances in colloid and interface science, 247
(IzvekovS, VothGA. A multiscale coarse-graining method for biomolecular systems. Journal of Physical Chemistry B. 2005;109(7):2469–73. Epub 2006/07/21. 10.1021/jp044629q .16851243)
IzvekovS, VothGA. A multiscale coarse-graining method for biomolecular systems. Journal of Physical Chemistry B. 2005;109(7):2469–73. Epub 2006/07/21. 10.1021/jp044629q .16851243
IzvekovS, VothGA. A multiscale coarse-graining method for biomolecular systems. Journal of Physical Chemistry B. 2005;109(7):2469–73. Epub 2006/07/21. 10.1021/jp044629q .16851243, IzvekovS, VothGA. A multiscale coarse-graining method for biomolecular systems. Journal of Physical Chemistry B. 2005;109(7):2469–73. Epub 2006/07/21. 10.1021/jp044629q .16851243
T. Harmon, A. Holehouse, R. Pappu (2018)
Differential solvation of intrinsically disordered linkers drives the formation of spatially organized droplets in ternary systems of linear multivalent proteins
New Journal of Physics, 20
T. Lytle, C. Sing (2018)
Tuning chain interaction entropy in complex coacervation using polymer stiffness, architecture, and salt valency
, 3
(LytleTK, SingCE. Transfer matrix theory of polymer complex coacervation. Soft Matter. 2017;13(39):7001–12. 10.1039/c7sm01080j 28840212)
LytleTK, SingCE. Transfer matrix theory of polymer complex coacervation. Soft Matter. 2017;13(39):7001–12. 10.1039/c7sm01080j 28840212
LytleTK, SingCE. Transfer matrix theory of polymer complex coacervation. Soft Matter. 2017;13(39):7001–12. 10.1039/c7sm01080j 28840212, LytleTK, SingCE. Transfer matrix theory of polymer complex coacervation. Soft Matter. 2017;13(39):7001–12. 10.1039/c7sm01080j 28840212
R. Kopelman (1988)
Fractal Reaction Kinetics
Science, 241
(IzvekovS, SwansonJM, VothGA. Coarse-graining in interaction space: a systematic approach for replacing long-range electrostatics with short-range potentials. Journal of Physical Chemistry B. 2008;112(15):4711–24. Epub 2008/03/28. 10.1021/jp710339n .18366209)
IzvekovS, SwansonJM, VothGA. Coarse-graining in interaction space: a systematic approach for replacing long-range electrostatics with short-range potentials. Journal of Physical Chemistry B. 2008;112(15):4711–24. Epub 2008/03/28. 10.1021/jp710339n .18366209
IzvekovS, SwansonJM, VothGA. Coarse-graining in interaction space: a systematic approach for replacing long-range electrostatics with short-range potentials. Journal of Physical Chemistry B. 2008;112(15):4711–24. Epub 2008/03/28. 10.1021/jp710339n .18366209, IzvekovS, SwansonJM, VothGA. Coarse-graining in interaction space: a systematic approach for replacing long-range electrostatics with short-range potentials. Journal of Physical Chemistry B. 2008;112(15):4711–24. Epub 2008/03/28. 10.1021/jp710339n .18366209
Marina Feric, Nilesh Vaidya, T. Harmon, Diana Mitrea, Lian Zhu, Tiffany Richardson, R. Kriwacki, R. Pappu, C. Brangwynne (2016)
Coexisting Liquid Phases Underlie Nucleolar Subcompartments
Cell, 165
Chi Pak, Martyna Kosno, A. Holehouse, S. Padrick, A. Mittal, Rustam Ali, A. Yunus, David Liu, R. Pappu, M. Rosen (2016)
Sequence Determinants of Intracellular Phase Separation by Complex Coacervation of a Disordered Protein.
Molecular cell, 63 1
(MintzPJ, SpectorDL. Compartmentalization of RNA processing factors within nuclear speckles. Journal of structural biology. 2000;129(2–3):241–51. Epub 2000/05/12. 10.1006/jsbi.2000.4213 .10806074)
MintzPJ, SpectorDL. Compartmentalization of RNA processing factors within nuclear speckles. Journal of structural biology. 2000;129(2–3):241–51. Epub 2000/05/12. 10.1006/jsbi.2000.4213 .10806074
MintzPJ, SpectorDL. Compartmentalization of RNA processing factors within nuclear speckles. Journal of structural biology. 2000;129(2–3):241–51. Epub 2000/05/12. 10.1006/jsbi.2000.4213 .10806074, MintzPJ, SpectorDL. Compartmentalization of RNA processing factors within nuclear speckles. Journal of structural biology. 2000;129(2–3):241–51. Epub 2000/05/12. 10.1006/jsbi.2000.4213 .10806074
(BrangwynneCP, EckmannCR, CoursonDS, RybarskaA, HoegeC, GharakhaniJ, et al Germline P granules are liquid droplets that localize by controlled dissolution/condensation. Science. 2009;324(5935):1729–32. 10.1126/science.1172046 19460965)
BrangwynneCP, EckmannCR, CoursonDS, RybarskaA, HoegeC, GharakhaniJ, et al Germline P granules are liquid droplets that localize by controlled dissolution/condensation. Science. 2009;324(5935):1729–32. 10.1126/science.1172046 19460965
BrangwynneCP, EckmannCR, CoursonDS, RybarskaA, HoegeC, GharakhaniJ, et al Germline P granules are liquid droplets that localize by controlled dissolution/condensation. Science. 2009;324(5935):1729–32. 10.1126/science.1172046 19460965, BrangwynneCP, EckmannCR, CoursonDS, RybarskaA, HoegeC, GharakhaniJ, et al Germline P granules are liquid droplets that localize by controlled dissolution/condensation. Science. 2009;324(5935):1729–32. 10.1126/science.1172046 19460965
J. Shaffer (1994)
Effects of chain topology on polymer dynamics: Bulk melts
Journal of Chemical Physics, 101
(HarmonTS, HolehouseAS, PappuRV. To Mix, or To Demix, That Is the Question. Biophysical Journal. 2017;112(4):565–7. 10.1016/j.bpj.2016.12.031 28256216)
HarmonTS, HolehouseAS, PappuRV. To Mix, or To Demix, That Is the Question. Biophysical Journal. 2017;112(4):565–7. 10.1016/j.bpj.2016.12.031 28256216
HarmonTS, HolehouseAS, PappuRV. To Mix, or To Demix, That Is the Question. Biophysical Journal. 2017;112(4):565–7. 10.1016/j.bpj.2016.12.031 28256216, HarmonTS, HolehouseAS, PappuRV. To Mix, or To Demix, That Is the Question. Biophysical Journal. 2017;112(4):565–7. 10.1016/j.bpj.2016.12.031 28256216
(RosenbluthMN, RosenbluthAW. Monte Carlo Calculation of the Average Extension of Molecular Chains. The Journal of Chemical Physics. 1955;23(2):356–9. 10.1063/1.1741967)
RosenbluthMN, RosenbluthAW. Monte Carlo Calculation of the Average Extension of Molecular Chains. The Journal of Chemical Physics. 1955;23(2):356–9. 10.1063/1.1741967
RosenbluthMN, RosenbluthAW. Monte Carlo Calculation of the Average Extension of Molecular Chains. The Journal of Chemical Physics. 1955;23(2):356–9. 10.1063/1.1741967, RosenbluthMN, RosenbluthAW. Monte Carlo Calculation of the Average Extension of Molecular Chains. The Journal of Chemical Physics. 1955;23(2):356–9. 10.1063/1.1741967
(RuffK, M., HarmonTS, PappuRV. CAMELOT: A machine learning approach for coarse-grained simulations of aggregation of block-copolymeric protein sequences. The Journal of Chemical Physics. 2015;143(24):243123 10.1063/1.4935066 26723608)
RuffK, M., HarmonTS, PappuRV. CAMELOT: A machine learning approach for coarse-grained simulations of aggregation of block-copolymeric protein sequences. The Journal of Chemical Physics. 2015;143(24):243123 10.1063/1.4935066 26723608
RuffK, M., HarmonTS, PappuRV. CAMELOT: A machine learning approach for coarse-grained simulations of aggregation of block-copolymeric protein sequences. The Journal of Chemical Physics. 2015;143(24):243123 10.1063/1.4935066 26723608, RuffK, M., HarmonTS, PappuRV. CAMELOT: A machine learning approach for coarse-grained simulations of aggregation of block-copolymeric protein sequences. The Journal of Chemical Physics. 2015;143(24):243123 10.1063/1.4935066 26723608
(WymanJ, GillSJ. Ligand-linked phase changes in a biological system: applications to sickle cell hemoglobin. Proceedings of the National Academy of Sciences of the United States of America. 1980;77(9):5239–42. Epub 1980/09/01. 10.1073/pnas.77.9.5239 ; PubMed Central PMCID: PMCPmc350033.6933555)
WymanJ, GillSJ. Ligand-linked phase changes in a biological system: applications to sickle cell hemoglobin. Proceedings of the National Academy of Sciences of the United States of America. 1980;77(9):5239–42. Epub 1980/09/01. 10.1073/pnas.77.9.5239 ; PubMed Central PMCID: PMCPmc350033.6933555
WymanJ, GillSJ. Ligand-linked phase changes in a biological system: applications to sickle cell hemoglobin. Proceedings of the National Academy of Sciences of the United States of America. 1980;77(9):5239–42. Epub 1980/09/01. 10.1073/pnas.77.9.5239 ; PubMed Central PMCID: PMCPmc350033.6933555, WymanJ, GillSJ. Ligand-linked phase changes in a biological system: applications to sickle cell hemoglobin. Proceedings of the National Academy of Sciences of the United States of America. 1980;77(9):5239–42. Epub 1980/09/01. 10.1073/pnas.77.9.5239 ; PubMed Central PMCID: PMCPmc350033.6933555
(LamondAI, SpectorDL. Nuclear speckles: a model for nuclear organelles. Nature Reviews in Molecular and Cell Biology. 2003;4(8):605–12. Epub 2003/08/19. 10.1038/nrm1172 .12923522)
LamondAI, SpectorDL. Nuclear speckles: a model for nuclear organelles. Nature Reviews in Molecular and Cell Biology. 2003;4(8):605–12. Epub 2003/08/19. 10.1038/nrm1172 .12923522
LamondAI, SpectorDL. Nuclear speckles: a model for nuclear organelles. Nature Reviews in Molecular and Cell Biology. 2003;4(8):605–12. Epub 2003/08/19. 10.1038/nrm1172 .12923522, LamondAI, SpectorDL. Nuclear speckles: a model for nuclear organelles. Nature Reviews in Molecular and Cell Biology. 2003;4(8):605–12. Epub 2003/08/19. 10.1038/nrm1172 .12923522
Priya Banerjee, Anthony Milin, M. Moosa, Paulo Onuchic, A. Deniz (2017)
Reentrant Phase Transition Drives Dynamic Substructure Formation in Ribonucleoprotein Droplets.
Angewandte Chemie, 56 38
C. Brangwynne, P. Tompa, R. Pappu (2015)
Polymer physics of intracellular phase transitions
Nature Physics, 11
(WoodruffJB, Ferreira GomesB, WidlundPO, MahamidJ, HonigmannA, HymanAA. The Centrosome Is a Selective Condensate that Nucleates Microtubules by Concentrating Tubulin. Cell. 2017;169(6):1066–77.e10. 10.1016/j.cell.2017.05.028 28575670)
WoodruffJB, Ferreira GomesB, WidlundPO, MahamidJ, HonigmannA, HymanAA. The Centrosome Is a Selective Condensate that Nucleates Microtubules by Concentrating Tubulin. Cell. 2017;169(6):1066–77.e10. 10.1016/j.cell.2017.05.028 28575670
WoodruffJB, Ferreira GomesB, WidlundPO, MahamidJ, HonigmannA, HymanAA. The Centrosome Is a Selective Condensate that Nucleates Microtubules by Concentrating Tubulin. Cell. 2017;169(6):1066–77.e10. 10.1016/j.cell.2017.05.028 28575670, WoodruffJB, Ferreira GomesB, WidlundPO, MahamidJ, HonigmannA, HymanAA. The Centrosome Is a Selective Condensate that Nucleates Microtubules by Concentrating Tubulin. Cell. 2017;169(6):1066–77.e10. 10.1016/j.cell.2017.05.028 28575670
Sudeep Banjade, M. Rosen (2014)
Phase transitions of multivalent proteins can promote clustering of membrane receptors
eLife, 3
(SemenovAN, RubinsteinM. Thermoreversible Gelation in Solutions of Associative Polymers. 1. Statics. Macromolecules. 1998;31(4):1373–85. 10.1021/ma970616h)
SemenovAN, RubinsteinM. Thermoreversible Gelation in Solutions of Associative Polymers. 1. Statics. Macromolecules. 1998;31(4):1373–85. 10.1021/ma970616h
SemenovAN, RubinsteinM. Thermoreversible Gelation in Solutions of Associative Polymers. 1. Statics. Macromolecules. 1998;31(4):1373–85. 10.1021/ma970616h, SemenovAN, RubinsteinM. Thermoreversible Gelation in Solutions of Associative Polymers. 1. Statics. Macromolecules. 1998;31(4):1373–85. 10.1021/ma970616h
Diana Mitrea, C. Grace, M. Buljan, M. Yun, Nicholas Pytel, John Satumba, A. Nourse, Cheon‐Gil Park, M. Babu, Stephen White, Stephen White, R. Kriwacki, R. Kriwacki (2014)
Structural polymorphism in the N-terminal oligomerization domain of NPM1
Proceedings of the National Academy of Sciences, 111
T. Harmon, A. Holehouse, M. Rosen, R. Pappu (2017)
Intrinsically disordered linkers determine the interplay between phase separation and gelation in multivalent proteins
eLife, 6
(SubramanianG, ShanbhagS. On the relationship between two popular lattice models for polymer melts. The Journal of Chemical Physics. 2008;129(14):144904 10.1063/1.2992047 19045165)
SubramanianG, ShanbhagS. On the relationship between two popular lattice models for polymer melts. The Journal of Chemical Physics. 2008;129(14):144904 10.1063/1.2992047 19045165
SubramanianG, ShanbhagS. On the relationship between two popular lattice models for polymer melts. The Journal of Chemical Physics. 2008;129(14):144904 10.1063/1.2992047 19045165, SubramanianG, ShanbhagS. On the relationship between two popular lattice models for polymer melts. The Journal of Chemical Physics. 2008;129(14):144904 10.1063/1.2992047 19045165
(SiepmannJI, FrenkelD. Configurational bias Monte Carlo: a new sampling scheme for flexible chains. Molecular Physics. 1992;75(1):59–70. 10.1080/00268979200100061)
SiepmannJI, FrenkelD. Configurational bias Monte Carlo: a new sampling scheme for flexible chains. Molecular Physics. 1992;75(1):59–70. 10.1080/00268979200100061
SiepmannJI, FrenkelD. Configurational bias Monte Carlo: a new sampling scheme for flexible chains. Molecular Physics. 1992;75(1):59–70. 10.1080/00268979200100061, SiepmannJI, FrenkelD. Configurational bias Monte Carlo: a new sampling scheme for flexible chains. Molecular Physics. 1992;75(1):59–70. 10.1080/00268979200100061
(RubinsteinM, DobryninAV. Solutions of Associative Polymers. Trends in Polymer Science. 1997;5(6):181–6.)
RubinsteinM, DobryninAV. Solutions of Associative Polymers. Trends in Polymer Science. 1997;5(6):181–6.
RubinsteinM, DobryninAV. Solutions of Associative Polymers. Trends in Polymer Science. 1997;5(6):181–6., RubinsteinM, DobryninAV. Solutions of Associative Polymers. Trends in Polymer Science. 1997;5(6):181–6.
M. Rubinstein, A. Dobrynin (1997)
Solutions of associative polymers
Trends in polymer science, 5
(WeiMT, Elbaum-GarfinkleS, HolehouseAS, ChenCC, FericM, ArnoldCB, et al Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles. Nature Chemistry. 2017;9(11):1118–25. 10.1038/nchem.2803 .29064502)
WeiMT, Elbaum-GarfinkleS, HolehouseAS, ChenCC, FericM, ArnoldCB, et al Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles. Nature Chemistry. 2017;9(11):1118–25. 10.1038/nchem.2803 .29064502
WeiMT, Elbaum-GarfinkleS, HolehouseAS, ChenCC, FericM, ArnoldCB, et al Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles. Nature Chemistry. 2017;9(11):1118–25. 10.1038/nchem.2803 .29064502, WeiMT, Elbaum-GarfinkleS, HolehouseAS, ChenCC, FericM, ArnoldCB, et al Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles. Nature Chemistry. 2017;9(11):1118–25. 10.1038/nchem.2803 .29064502
C. Falkenberg, J. Carson, M. Blinov (2017)
Multivalent Molecules as Modulators of RNA Granule Size and Composition.
Biophysical journal, 113 2
T. Koyama, Hajime Tanaka (2018)
Volume-shrinking kinetics of transient gels as a consequence of dynamic interplay between phase separation and mechanical relaxation
Physical Review E
Dan Bracha, Mackenzie Walls, M. Wei, Lian Zhu, Martin Kurian, J. Avalos, Jared Toettcher, C. Brangwynne (2019)
Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo-Oligomerizable Seeds
Cell, 176
Hao Wu, M. Fuxreiter (2016)
The Structure and Dynamics of Higher-Order Assemblies: Amyloids, Signalosomes, and Granules
Cell, 165
(DuchsD, DelaneyKT, FredricksonGH. A multi-species exchange model for fully fluctuating polymer field theory simulations. The Journal of Chemical Physics. 2014;141(17):174103 Epub 2014/11/10. 10.1063/1.4900574 .25381498)
DuchsD, DelaneyKT, FredricksonGH. A multi-species exchange model for fully fluctuating polymer field theory simulations. The Journal of Chemical Physics. 2014;141(17):174103 Epub 2014/11/10. 10.1063/1.4900574 .25381498
DuchsD, DelaneyKT, FredricksonGH. A multi-species exchange model for fully fluctuating polymer field theory simulations. The Journal of Chemical Physics. 2014;141(17):174103 Epub 2014/11/10. 10.1063/1.4900574 .25381498, DuchsD, DelaneyKT, FredricksonGH. A multi-species exchange model for fully fluctuating polymer field theory simulations. The Journal of Chemical Physics. 2014;141(17):174103 Epub 2014/11/10. 10.1063/1.4900574 .25381498
Huaiying Zhang, Shana Elbaum-Garfinkle, Erin Langdon, Nicole Taylor, Patricia Occhipinti, A. Bridges, C. Brangwynne, A. Gladfelter (2015)
RNA Controls PolyQ Protein Phase Transitions.
Molecular cell, 60 2
Shambaditya Saha, C. Weber, Marco Nousch, O. Adame-Arana, C. Hoege, Marco Hein, E. Osborne-Nishimura, J. Mahamid, Marcus Jahnel, L. Jawerth, Andrej Pozniakovski, C. Eckmann, F. Jülicher, A. Hyman (2016)
Polar Positioning of Phase-Separated Liquid Compartments in Cells Regulated by an mRNA Competition Mechanism
Cell, 166
P. Mintz, D. Spector (2000)
Compartmentalization of RNA processing factors within nuclear speckles.
Journal of structural biology, 129 2-3
P. Flory (1941)
Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1
Journal of the American Chemical Society, 63
Mylene Ferrolino, Diana Mitrea, J. Michael, R. Kriwacki, R. Kriwacki (2018)
Compositional adaptability in NPM1-SURF6 scaffolding networks enabled by dynamic switching of phase separation mechanisms
Nature Communications, 9
N. Vladimirova, A. Malagoli, R. Mauri (1998)
Diffusion-driven phase separation of deeply quenched mixtures
Physical Review E, 58
(StockmayerWH. Theory of Molecular Size Distribution and Gel Formation in Branched‐Chain Polymers. The Journal of Chemical Physics. 1943;11(2):45–55. 10.1063/1.1723803)
StockmayerWH. Theory of Molecular Size Distribution and Gel Formation in Branched‐Chain Polymers. The Journal of Chemical Physics. 1943;11(2):45–55. 10.1063/1.1723803
StockmayerWH. Theory of Molecular Size Distribution and Gel Formation in Branched‐Chain Polymers. The Journal of Chemical Physics. 1943;11(2):45–55. 10.1063/1.1723803, StockmayerWH. Theory of Molecular Size Distribution and Gel Formation in Branched‐Chain Polymers. The Journal of Chemical Physics. 1943;11(2):45–55. 10.1063/1.1723803
(PoseyAE, RuffKM, HarmonTS, CrickSL, LiA, DiamondMI, et al Profilin reduces aggregation and phase separation of huntingtin N-terminal fragments by preferentially binding to soluble monomers and oligomers. Journal of Biological Chemistry. 2018;293(10):3734–46. 10.1074/jbc.RA117.000357 29358329)
PoseyAE, RuffKM, HarmonTS, CrickSL, LiA, DiamondMI, et al Profilin reduces aggregation and phase separation of huntingtin N-terminal fragments by preferentially binding to soluble monomers and oligomers. Journal of Biological Chemistry. 2018;293(10):3734–46. 10.1074/jbc.RA117.000357 29358329
PoseyAE, RuffKM, HarmonTS, CrickSL, LiA, DiamondMI, et al Profilin reduces aggregation and phase separation of huntingtin N-terminal fragments by preferentially binding to soluble monomers and oligomers. Journal of Biological Chemistry. 2018;293(10):3734–46. 10.1074/jbc.RA117.000357 29358329, PoseyAE, RuffKM, HarmonTS, CrickSL, LiA, DiamondMI, et al Profilin reduces aggregation and phase separation of huntingtin N-terminal fragments by preferentially binding to soluble monomers and oligomers. Journal of Biological Chemistry. 2018;293(10):3734–46. 10.1074/jbc.RA117.000357 29358329
Yongdae Shin, C. Brangwynne (2017)
Liquid phase condensation in cell physiology and disease
Science, 357
(MatejuD, FranzmannTM, PatelA, KopachA, BoczekEE, MaharanaS, et al An aberrant phase transition of stress granules triggered by misfolded protein and prevented by chaperone function. The EMBO Journal. 2017:e201695957 10.15252/embj.201695957 28377462)
MatejuD, FranzmannTM, PatelA, KopachA, BoczekEE, MaharanaS, et al An aberrant phase transition of stress granules triggered by misfolded protein and prevented by chaperone function. The EMBO Journal. 2017:e201695957 10.15252/embj.201695957 28377462
MatejuD, FranzmannTM, PatelA, KopachA, BoczekEE, MaharanaS, et al An aberrant phase transition of stress granules triggered by misfolded protein and prevented by chaperone function. The EMBO Journal. 2017:e201695957 10.15252/embj.201695957 28377462, MatejuD, FranzmannTM, PatelA, KopachA, BoczekEE, MaharanaS, et al An aberrant phase transition of stress granules triggered by misfolded protein and prevented by chaperone function. The EMBO Journal. 2017:e201695957 10.15252/embj.201695957 28377462
V. Sirri, S. Urcuqui-Inchima, P. Roussel, D. Hernandez-Verdun (2007)
Nucleolus: the fascinating nuclear body
Histochemistry and Cell Biology, 129
M. Shayegan, R. Tahvildari, K. Metera, Lydia Kisley, S. Michnick, Sabrina Leslie (2018)
Probing inhomogeneous diffusion in the microenvironments of phase-separated polymers under confinement
bioRxiv
S. Izvekov, G. Voth (2005)
A multiscale coarse-graining method for biomolecular systems.
The journal of physical chemistry. B, 109 7
(DelaneyKT, FredricksonGH. Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions. Journal of Physical Chemistry B. 2016;120(31):7615–34. Epub 2016/07/15. 10.1021/acs.jpcb.6b05704 .27414265)
DelaneyKT, FredricksonGH. Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions. Journal of Physical Chemistry B. 2016;120(31):7615–34. Epub 2016/07/15. 10.1021/acs.jpcb.6b05704 .27414265
DelaneyKT, FredricksonGH. Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions. Journal of Physical Chemistry B. 2016;120(31):7615–34. Epub 2016/07/15. 10.1021/acs.jpcb.6b05704 .27414265, DelaneyKT, FredricksonGH. Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions. Journal of Physical Chemistry B. 2016;120(31):7615–34. Epub 2016/07/15. 10.1021/acs.jpcb.6b05704 .27414265
P. Flory (1942)
Constitution of Three-dimensional Polymers and the Theory of Gelation.
The Journal of Physical Chemistry, 46
J. Wyman, S. Gill (1980)
Ligand-linked phase changes in a biological system: applications to sickle cell hemoglobin.
Proceedings of the National Academy of Sciences of the United States of America, 77 9
A. Holehouse, R. Pappu (2018)
Functional Implications of Intracellular Phase Transitions.
Biochemistry, 57 17
(LytleTK, SalazarAJ, SingCE. Interfacial properties of polymeric complex coacervates from simulation and theory. The Journal of Chemical Physics. 2018;149(16):163315 10.1063/1.5029934 30384702)
LytleTK, SalazarAJ, SingCE. Interfacial properties of polymeric complex coacervates from simulation and theory. The Journal of Chemical Physics. 2018;149(16):163315 10.1063/1.5029934 30384702
LytleTK, SalazarAJ, SingCE. Interfacial properties of polymeric complex coacervates from simulation and theory. The Journal of Chemical Physics. 2018;149(16):163315 10.1063/1.5029934 30384702, LytleTK, SalazarAJ, SingCE. Interfacial properties of polymeric complex coacervates from simulation and theory. The Journal of Chemical Physics. 2018;149(16):163315 10.1063/1.5029934 30384702
(FalkenbergCV, CarsonJH, BlinovML. Multivalent Molecules as Modulators of RNA Granule Size and Composition. Biophysical Journal. 2017;113(2):235–45. 10.1016/j.bpj.2017.01.031 28242011)
FalkenbergCV, CarsonJH, BlinovML. Multivalent Molecules as Modulators of RNA Granule Size and Composition. Biophysical Journal. 2017;113(2):235–45. 10.1016/j.bpj.2017.01.031 28242011
FalkenbergCV, CarsonJH, BlinovML. Multivalent Molecules as Modulators of RNA Granule Size and Composition. Biophysical Journal. 2017;113(2):235–45. 10.1016/j.bpj.2017.01.031 28242011, FalkenbergCV, CarsonJH, BlinovML. Multivalent Molecules as Modulators of RNA Granule Size and Composition. Biophysical Journal. 2017;113(2):235–45. 10.1016/j.bpj.2017.01.031 28242011
T. Khan, Tejbir Kandola, Jianzheng Wu, S. Venkatesan, Ellen Ketter, Jeffrey Lange, A. Gama, A. Box, J. Unruh, Malcolm Cook, Randal Halfmann (2019)
Quantifying Nucleation In Vivo Reveals the Physical Basis of Prion-like Phase Behavior.
Molecular cell, 73 4
K. Delaney, G. Fredrickson (2016)
Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions.
The journal of physical chemistry. B, 120 31
(DignonGL, ZhengW, KimYC, BestRB, MittalJ. Sequence determinants of protein phase behavior from a coarse-grained model. PLOS Computational Biology. 2018;14(1):e1005941 10.1371/journal.pcbi.1005941 29364893)
DignonGL, ZhengW, KimYC, BestRB, MittalJ. Sequence determinants of protein phase behavior from a coarse-grained model. PLOS Computational Biology. 2018;14(1):e1005941 10.1371/journal.pcbi.1005941 29364893
DignonGL, ZhengW, KimYC, BestRB, MittalJ. Sequence determinants of protein phase behavior from a coarse-grained model. PLOS Computational Biology. 2018;14(1):e1005941 10.1371/journal.pcbi.1005941 29364893, DignonGL, ZhengW, KimYC, BestRB, MittalJ. Sequence determinants of protein phase behavior from a coarse-grained model. PLOS Computational Biology. 2018;14(1):e1005941 10.1371/journal.pcbi.1005941 29364893
Diana Mitrea, Jaclyn Cika, C. Guy, David Ban, Priya Banerjee, Christopher Stanley, A. Nourse, A. Deniz, R. Kriwacki (2016)
Nucleophosmin integrates within the nucleolus via multi-modal interactions with proteins displaying R-rich linear motifs and rRNA
eLife, 5
Suman Das, Alan Amin, Yi‐Hsuan Lin, H. Chan (2018)
Coarse-grained residue-based models of disordered protein condensates: utility and limitations of simple charge pattern parameters.
Physical chemistry chemical physics : PCCP, 20 45
P. Bolhuis, D. Frenkel (1994)
Numerical study of the phase diagram of a mixture of spherical and rodlike colloids
Journal of Chemical Physics, 101
(Freeman RosenzweigES, XuB, Kuhn CuellarL, Martinez-SanchezA, SchafferM, StraussM, et al The Eukaryotic CO2-Concentrating Organelle Is Liquid-like and Exhibits Dynamic Reorganization. Cell. 2017;171(1):148–62.e19. 10.1016/j.cell.2017.08.008 28938114)
Freeman RosenzweigES, XuB, Kuhn CuellarL, Martinez-SanchezA, SchafferM, StraussM, et al The Eukaryotic CO2-Concentrating Organelle Is Liquid-like and Exhibits Dynamic Reorganization. Cell. 2017;171(1):148–62.e19. 10.1016/j.cell.2017.08.008 28938114
Freeman RosenzweigES, XuB, Kuhn CuellarL, Martinez-SanchezA, SchafferM, StraussM, et al The Eukaryotic CO2-Concentrating Organelle Is Liquid-like and Exhibits Dynamic Reorganization. Cell. 2017;171(1):148–62.e19. 10.1016/j.cell.2017.08.008 28938114, Freeman RosenzweigES, XuB, Kuhn CuellarL, Martinez-SanchezA, SchafferM, StraussM, et al The Eukaryotic CO2-Concentrating Organelle Is Liquid-like and Exhibits Dynamic Reorganization. Cell. 2017;171(1):148–62.e19. 10.1016/j.cell.2017.08.008 28938114
(DamaJF, SinitskiyAV, McCullaghM, WeareJ, RouxB, DinnerAR, et al The Theory of Ultra-Coarse-Graining. 1. General Principles. Journal of chemical theory and computation. 2013;9(5):2466–80. Epub 2013/05/14. 10.1021/ct4000444 .26583735)
DamaJF, SinitskiyAV, McCullaghM, WeareJ, RouxB, DinnerAR, et al The Theory of Ultra-Coarse-Graining. 1. General Principles. Journal of chemical theory and computation. 2013;9(5):2466–80. Epub 2013/05/14. 10.1021/ct4000444 .26583735
DamaJF, SinitskiyAV, McCullaghM, WeareJ, RouxB, DinnerAR, et al The Theory of Ultra-Coarse-Graining. 1. General Principles. Journal of chemical theory and computation. 2013;9(5):2466–80. Epub 2013/05/14. 10.1021/ct4000444 .26583735, DamaJF, SinitskiyAV, McCullaghM, WeareJ, RouxB, DinnerAR, et al The Theory of Ultra-Coarse-Graining. 1. General Principles. Journal of chemical theory and computation. 2013;9(5):2466–80. Epub 2013/05/14. 10.1021/ct4000444 .26583735
(TheodorouDN. Equilibration and Coarse-Graining Methods for Polymers In: FerrarioM, CiccottiG, BinderK, editors. Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2 Berlin, Heidelberg: Springer Berlin Heidelberg; 2006 p. 419–48.)
TheodorouDN. Equilibration and Coarse-Graining Methods for Polymers In: FerrarioM, CiccottiG, BinderK, editors. Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2 Berlin, Heidelberg: Springer Berlin Heidelberg; 2006 p. 419–48.
TheodorouDN. Equilibration and Coarse-Graining Methods for Polymers In: FerrarioM, CiccottiG, BinderK, editors. Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2 Berlin, Heidelberg: Springer Berlin Heidelberg; 2006 p. 419–48., TheodorouDN. Equilibration and Coarse-Graining Methods for Polymers In: FerrarioM, CiccottiG, BinderK, editors. Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2 Berlin, Heidelberg: Springer Berlin Heidelberg; 2006 p. 419–48.
(LuL, DamaJF, VothGA. Fitting coarse-grained distribution functions through an iterative force-matching method. The Journal of Chemical Physics. 2013;139(12):121906 Epub 2013/10/05. 10.1063/1.4811667 .24089718)
LuL, DamaJF, VothGA. Fitting coarse-grained distribution functions through an iterative force-matching method. The Journal of Chemical Physics. 2013;139(12):121906 Epub 2013/10/05. 10.1063/1.4811667 .24089718
LuL, DamaJF, VothGA. Fitting coarse-grained distribution functions through an iterative force-matching method. The Journal of Chemical Physics. 2013;139(12):121906 Epub 2013/10/05. 10.1063/1.4811667 .24089718, LuL, DamaJF, VothGA. Fitting coarse-grained distribution functions through an iterative force-matching method. The Journal of Chemical Physics. 2013;139(12):121906 Epub 2013/10/05. 10.1063/1.4811667 .24089718
Pu Liu, Qiang Shi, Hal Daumé, G. Voth (2008)
A Bayesian statistics approach to multiscale coarse graining.
The Journal of chemical physics, 129 21
(SirriV, Urcuqui-InchimaS, RousselP, Hernandez-VerdunD. Nucleolus: the fascinating nuclear body. Histochemistry and Cell Biology. 2008;129(1):13–31. 10.1007/s00418-007-0359-6 18046571)
SirriV, Urcuqui-InchimaS, RousselP, Hernandez-VerdunD. Nucleolus: the fascinating nuclear body. Histochemistry and Cell Biology. 2008;129(1):13–31. 10.1007/s00418-007-0359-6 18046571
SirriV, Urcuqui-InchimaS, RousselP, Hernandez-VerdunD. Nucleolus: the fascinating nuclear body. Histochemistry and Cell Biology. 2008;129(1):13–31. 10.1007/s00418-007-0359-6 18046571, SirriV, Urcuqui-InchimaS, RousselP, Hernandez-VerdunD. Nucleolus: the fascinating nuclear body. Histochemistry and Cell Biology. 2008;129(1):13–31. 10.1007/s00418-007-0359-6 18046571
Elizabeth Rosenzweig, Bin Xu, Luis Cuellar, A. Martínez-Sánchez, M. Schaffer, M. Strauss, Heather Cartwright, P. Ronceray, J. Plitzko, F. Förster, N. Wingreen, B. Engel, L. Mackinder, M. Jonikas (2017)
The Eukaryotic CO2-Concentrating Organelle Is Liquid-like and Exhibits Dynamic Reorganization
Cell, 171
Shana Elbaum-Garfinkle, Younghoon Kim, Krzysztof Szczepaniak, Carlos Chen, C. Eckmann, Sua Myong, C. Brangwynne (2015)
The disordered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynamics
Proceedings of the National Academy of Sciences, 112
Avinash Patel, Hyun Lee, L. Jawerth, Shovamayee Maharana, Marcus Jahnel, Marco Hein, S. Stoynov, J. Mahamid, Shambaditya Saha, Titus Franzmann, Andrej Pozniakovski, I. Poser, N. Maghelli, Loic Royer, Martin Weigert, E. Myers, S. Grill, D. Drechsel, A. Hyman, S. Alberti (2015)
A Liquid-to-Solid Phase Transition of the ALS Protein FUS Accelerated by Disease Mutation
Cell, 162
(BradyJP, FarberPJ, SekharA, LinY-H, HuangR, BahA, et al Structural and hydrodynamic properties of an intrinsically disordered region of a germ cell-specific protein on phase separation. Proceedings of the National Academy of Sciences. 2017;114(39):E8194–E203. 10.1073/pnas.1706197114 28894006)
BradyJP, FarberPJ, SekharA, LinY-H, HuangR, BahA, et al Structural and hydrodynamic properties of an intrinsically disordered region of a germ cell-specific protein on phase separation. Proceedings of the National Academy of Sciences. 2017;114(39):E8194–E203. 10.1073/pnas.1706197114 28894006
BradyJP, FarberPJ, SekharA, LinY-H, HuangR, BahA, et al Structural and hydrodynamic properties of an intrinsically disordered region of a germ cell-specific protein on phase separation. Proceedings of the National Academy of Sciences. 2017;114(39):E8194–E203. 10.1073/pnas.1706197114 28894006, BradyJP, FarberPJ, SekharA, LinY-H, HuangR, BahA, et al Structural and hydrodynamic properties of an intrinsically disordered region of a germ cell-specific protein on phase separation. Proceedings of the National Academy of Sciences. 2017;114(39):E8194–E203. 10.1073/pnas.1706197114 28894006
Erin Langdon, Y. Qiu, Amirhossein Niaki, Grace McLaughlin, Chase Weidmann, Therese Gerbich, Jean Smith, J. Crutchley, C. Termini, K. Weeks, Sua Myong, A. Gladfelter (2017)
mRNA structure determines specificity of a polyQ-driven phase separation
Science, 360
(LinY-H, Forman-KayJD, ChanHS. Sequence-Specific Polyampholyte Phase Separation in Membraneless Organelles. Physical Review Letters. 2016;117(17):178101 10.1103/PhysRevLett.117.178101 27824447)
LinY-H, Forman-KayJD, ChanHS. Sequence-Specific Polyampholyte Phase Separation in Membraneless Organelles. Physical Review Letters. 2016;117(17):178101 10.1103/PhysRevLett.117.178101 27824447
LinY-H, Forman-KayJD, ChanHS. Sequence-Specific Polyampholyte Phase Separation in Membraneless Organelles. Physical Review Letters. 2016;117(17):178101 10.1103/PhysRevLett.117.178101 27824447, LinY-H, Forman-KayJD, ChanHS. Sequence-Specific Polyampholyte Phase Separation in Membraneless Organelles. Physical Review Letters. 2016;117(17):178101 10.1103/PhysRevLett.117.178101 27824447
(RottereauM, GimelJC, NicolaiT, DurandD. Monte Carlo simulation of particle aggregation and gelation: I. Growth, structure and size distribution of the clusters. The European Physical Journal E. 2004;15(2):133–40. 10.1140/epje/i2004-10044-x 15517458)
RottereauM, GimelJC, NicolaiT, DurandD. Monte Carlo simulation of particle aggregation and gelation: I. Growth, structure and size distribution of the clusters. The European Physical Journal E. 2004;15(2):133–40. 10.1140/epje/i2004-10044-x 15517458
RottereauM, GimelJC, NicolaiT, DurandD. Monte Carlo simulation of particle aggregation and gelation: I. Growth, structure and size distribution of the clusters. The European Physical Journal E. 2004;15(2):133–40. 10.1140/epje/i2004-10044-x 15517458, RottereauM, GimelJC, NicolaiT, DurandD. Monte Carlo simulation of particle aggregation and gelation: I. Growth, structure and size distribution of the clusters. The European Physical Journal E. 2004;15(2):133–40. 10.1140/epje/i2004-10044-x 15517458
(FloryPJ. Constitution of Three-dimensional Polymers and the Theory of Gelation. The Journal of Physical Chemistry. 1942;46(1):132–40. 10.1021/j150415a016)
FloryPJ. Constitution of Three-dimensional Polymers and the Theory of Gelation. The Journal of Physical Chemistry. 1942;46(1):132–40. 10.1021/j150415a016
FloryPJ. Constitution of Three-dimensional Polymers and the Theory of Gelation. The Journal of Physical Chemistry. 1942;46(1):132–40. 10.1021/j150415a016, FloryPJ. Constitution of Three-dimensional Polymers and the Theory of Gelation. The Journal of Physical Chemistry. 1942;46(1):132–40. 10.1021/j150415a016
W. Stockmayer (1943)
Theory of Molecular Size Distribution and Gel Formation in Branched‐Chain Polymers
Journal of Chemical Physics, 11
(VernonRM, ChongPA, TsangB, KimTH, BahA, FarberP, et al Pi-Pi contacts are an overlooked protein feature relevant to phase separation. eLife. 2018;7:e31486 10.7554/eLife.31486 29424691)
VernonRM, ChongPA, TsangB, KimTH, BahA, FarberP, et al Pi-Pi contacts are an overlooked protein feature relevant to phase separation. eLife. 2018;7:e31486 10.7554/eLife.31486 29424691
VernonRM, ChongPA, TsangB, KimTH, BahA, FarberP, et al Pi-Pi contacts are an overlooked protein feature relevant to phase separation. eLife. 2018;7:e31486 10.7554/eLife.31486 29424691, VernonRM, ChongPA, TsangB, KimTH, BahA, FarberP, et al Pi-Pi contacts are an overlooked protein feature relevant to phase separation. eLife. 2018;7:e31486 10.7554/eLife.31486 29424691
(MitreaDM, CikaJA, GuyCS, BanD, BanerjeePR, StanleyCB, et al Nucleophosmin integrates within the nucleolus via multi-modal interactions with proteins displaying R-rich linear motifs and rRNA. eLife. 2016;5:e13571 10.7554/eLife.13571 26836305)
MitreaDM, CikaJA, GuyCS, BanD, BanerjeePR, StanleyCB, et al Nucleophosmin integrates within the nucleolus via multi-modal interactions with proteins displaying R-rich linear motifs and rRNA. eLife. 2016;5:e13571 10.7554/eLife.13571 26836305
MitreaDM, CikaJA, GuyCS, BanD, BanerjeePR, StanleyCB, et al Nucleophosmin integrates within the nucleolus via multi-modal interactions with proteins displaying R-rich linear motifs and rRNA. eLife. 2016;5:e13571 10.7554/eLife.13571 26836305, MitreaDM, CikaJA, GuyCS, BanD, BanerjeePR, StanleyCB, et al Nucleophosmin integrates within the nucleolus via multi-modal interactions with proteins displaying R-rich linear motifs and rRNA. eLife. 2016;5:e13571 10.7554/eLife.13571 26836305
T. Lytle, A. Salazar, C. Sing (2018)
Interfacial properties of polymeric complex coacervates from simulation and theory.
The Journal of chemical physics, 149 16
Titus Franzmann, Marcus Jahnel, Andrei Pozniakovsky, J. Mahamid, A. Holehouse, Elisabeth Nüske, Doris Richter, W. Baumeister, S. Grill, R. Pappu, A. Hyman, S. Alberti (2018)
Phase separation of a yeast prion protein promotes cellular fitness
Science, 359
A. Molliex, Jamshid Temirov, Jihun Lee, Maura Coughlin, A. Kanagaraj, Hong Kim, Tanja Mittag, J. Paul Taylor (2015)
Phase Separation by Low Complexity Domains Promotes Stress Granule Assembly and Drives Pathological Fibrillization
Cell, 163
Pilong Li, Sudeep Banjade, Hui-Chun Cheng, Soyeon Kim, Baoyu Chen, Liang Guo, M. Llaguno, Javoris Hollingsworth, David King, Salman Banani, Paul Russo, Q. Jiang, B. Nixon, Michael Rosen (2012)
Phase Transitions in the Assembly of Multi-Valent Signaling Proteins
Nature, 483
I. Carmesin, K. Kremer (1988)
The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions
Macromolecules, 21
(MolliexA, TemirovJ, LeeJ, CoughlinM, KanagarajAP, KimHJ, et al Phase Separation by Low Complexity Domains Promotes Stress Granule Assembly and Drives Pathological Fibrillization. Cell. 2015;163(1):123–33. 10.1016/j.cell.2015.09.015 26406374)
MolliexA, TemirovJ, LeeJ, CoughlinM, KanagarajAP, KimHJ, et al Phase Separation by Low Complexity Domains Promotes Stress Granule Assembly and Drives Pathological Fibrillization. Cell. 2015;163(1):123–33. 10.1016/j.cell.2015.09.015 26406374
MolliexA, TemirovJ, LeeJ, CoughlinM, KanagarajAP, KimHJ, et al Phase Separation by Low Complexity Domains Promotes Stress Granule Assembly and Drives Pathological Fibrillization. Cell. 2015;163(1):123–33. 10.1016/j.cell.2015.09.015 26406374, MolliexA, TemirovJ, LeeJ, CoughlinM, KanagarajAP, KimHJ, et al Phase Separation by Low Complexity Domains Promotes Stress Granule Assembly and Drives Pathological Fibrillization. Cell. 2015;163(1):123–33. 10.1016/j.cell.2015.09.015 26406374
(ShafferJS. Effects of chain topology on polymer dynamics: Bulk melts. The Journal of Chemical Physics. 1994;101(5):4205–13. 10.1063/1.467470)
ShafferJS. Effects of chain topology on polymer dynamics: Bulk melts. The Journal of Chemical Physics. 1994;101(5):4205–13. 10.1063/1.467470
ShafferJS. Effects of chain topology on polymer dynamics: Bulk melts. The Journal of Chemical Physics. 1994;101(5):4205–13. 10.1063/1.467470, ShafferJS. Effects of chain topology on polymer dynamics: Bulk melts. The Journal of Chemical Physics. 1994;101(5):4205–13. 10.1063/1.467470
(NguemahaV, ZhouH-X. Liquid-Liquid Phase Separation of Patchy Particles Illuminates Diverse Effects of Regulatory Components on Protein Droplet Formation. Scientific Reports. 2018;8(1):6728 10.1038/s41598-018-25132-1 29712961)
NguemahaV, ZhouH-X. Liquid-Liquid Phase Separation of Patchy Particles Illuminates Diverse Effects of Regulatory Components on Protein Droplet Formation. Scientific Reports. 2018;8(1):6728 10.1038/s41598-018-25132-1 29712961
NguemahaV, ZhouH-X. Liquid-Liquid Phase Separation of Patchy Particles Illuminates Diverse Effects of Regulatory Components on Protein Droplet Formation. Scientific Reports. 2018;8(1):6728 10.1038/s41598-018-25132-1 29712961, NguemahaV, ZhouH-X. Liquid-Liquid Phase Separation of Patchy Particles Illuminates Diverse Effects of Regulatory Components on Protein Droplet Formation. Scientific Reports. 2018;8(1):6728 10.1038/s41598-018-25132-1 29712961
Najeeb Halabi, O. Rivoire, S. Leibler, R. Ranganathan (2009)
Protein Sectors: Evolutionary Units of Three-Dimensional Structure
Cell, 138
C. Brangwynne, C. Eckmann, D. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher, A. Hyman (2009)
Germline P Granules Are Liquid Droplets That Localize by Controlled Dissolution/Condensation
Science, 324
(HolehouseAS, PappuRV. Functional Implications of Intracellular Phase Transitions. Biochemistry. 2018;57(17):2415–23. 10.1021/acs.biochem.7b01136 29323488)
HolehouseAS, PappuRV. Functional Implications of Intracellular Phase Transitions. Biochemistry. 2018;57(17):2415–23. 10.1021/acs.biochem.7b01136 29323488
HolehouseAS, PappuRV. Functional Implications of Intracellular Phase Transitions. Biochemistry. 2018;57(17):2415–23. 10.1021/acs.biochem.7b01136 29323488, HolehouseAS, PappuRV. Functional Implications of Intracellular Phase Transitions. Biochemistry. 2018;57(17):2415–23. 10.1021/acs.biochem.7b01136 29323488
(KoyamaT, TanakaH. Volume-shrinking kinetics of transient gels as a consequence of dynamic interplay between phase separation and mechanical relaxation. Physical Review E. 2018;98(6):062617 10.1103/PhysRevE.98.062617)
KoyamaT, TanakaH. Volume-shrinking kinetics of transient gels as a consequence of dynamic interplay between phase separation and mechanical relaxation. Physical Review E. 2018;98(6):062617 10.1103/PhysRevE.98.062617
KoyamaT, TanakaH. Volume-shrinking kinetics of transient gels as a consequence of dynamic interplay between phase separation and mechanical relaxation. Physical Review E. 2018;98(6):062617 10.1103/PhysRevE.98.062617, KoyamaT, TanakaH. Volume-shrinking kinetics of transient gels as a consequence of dynamic interplay between phase separation and mechanical relaxation. Physical Review E. 2018;98(6):062617 10.1103/PhysRevE.98.062617
A. Mańka, W. Nowicki, G. Nowicka (2013)
Monte Carlo simulations of a polymer chain conformation. The effectiveness of local moves algorithms and estimation of entropy
Journal of Molecular Modeling, 19
(PatelA, LeeHO, JawerthL, MaharanaS, JahnelM, HeinMY, et al A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation. Cell. 2015;162(5):1066–77. 10.1016/j.cell.2015.07.047 26317470)
PatelA, LeeHO, JawerthL, MaharanaS, JahnelM, HeinMY, et al A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation. Cell. 2015;162(5):1066–77. 10.1016/j.cell.2015.07.047 26317470
PatelA, LeeHO, JawerthL, MaharanaS, JahnelM, HeinMY, et al A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation. Cell. 2015;162(5):1066–77. 10.1016/j.cell.2015.07.047 26317470, PatelA, LeeHO, JawerthL, MaharanaS, JahnelM, HeinMY, et al A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation. Cell. 2015;162(5):1066–77. 10.1016/j.cell.2015.07.047 26317470
(CaseLB, ZhangX, DitlevJA, RosenMK. Stoichiometry controls activity of phase-separated clusters of actin signaling proteins. Science. 2019;363(6431):1093–7. 10.1126/science.aau6313 30846599)
CaseLB, ZhangX, DitlevJA, RosenMK. Stoichiometry controls activity of phase-separated clusters of actin signaling proteins. Science. 2019;363(6431):1093–7. 10.1126/science.aau6313 30846599
CaseLB, ZhangX, DitlevJA, RosenMK. Stoichiometry controls activity of phase-separated clusters of actin signaling proteins. Science. 2019;363(6431):1093–7. 10.1126/science.aau6313 30846599, CaseLB, ZhangX, DitlevJA, RosenMK. Stoichiometry controls activity of phase-separated clusters of actin signaling proteins. Science. 2019;363(6431):1093–7. 10.1126/science.aau6313 30846599
(HanY, JinJ, WagnerJW, VothGA. Quantum theory of multiscale coarse-graining. The Journal of Chemical Physics. 2018;148(10):102335 Epub 2018/03/17. 10.1063/1.5010270 .29544317)
HanY, JinJ, WagnerJW, VothGA. Quantum theory of multiscale coarse-graining. The Journal of Chemical Physics. 2018;148(10):102335 Epub 2018/03/17. 10.1063/1.5010270 .29544317
HanY, JinJ, WagnerJW, VothGA. Quantum theory of multiscale coarse-graining. The Journal of Chemical Physics. 2018;148(10):102335 Epub 2018/03/17. 10.1063/1.5010270 .29544317, HanY, JinJ, WagnerJW, VothGA. Quantum theory of multiscale coarse-graining. The Journal of Chemical Physics. 2018;148(10):102335 Epub 2018/03/17. 10.1063/1.5010270 .29544317
(BananiSF, LeeHO, HymanAA, RosenMK. Biomolecular condensates: organizers of cellular biochemistry. Nature Reviews Molecular Cell Biology. 2017;18(5):285–98. 10.1038/nrm.2017.7 28225081)
BananiSF, LeeHO, HymanAA, RosenMK. Biomolecular condensates: organizers of cellular biochemistry. Nature Reviews Molecular Cell Biology. 2017;18(5):285–98. 10.1038/nrm.2017.7 28225081
BananiSF, LeeHO, HymanAA, RosenMK. Biomolecular condensates: organizers of cellular biochemistry. Nature Reviews Molecular Cell Biology. 2017;18(5):285–98. 10.1038/nrm.2017.7 28225081, BananiSF, LeeHO, HymanAA, RosenMK. Biomolecular condensates: organizers of cellular biochemistry. Nature Reviews Molecular Cell Biology. 2017;18(5):285–98. 10.1038/nrm.2017.7 28225081
(KopelmanR. Fractal Reaction Kinetics. Science. 1988;241(4873):1620–6. 10.1126/science.241.4873.1620 17820893)
KopelmanR. Fractal Reaction Kinetics. Science. 1988;241(4873):1620–6. 10.1126/science.241.4873.1620 17820893
KopelmanR. Fractal Reaction Kinetics. Science. 1988;241(4873):1620–6. 10.1126/science.241.4873.1620 17820893, KopelmanR. Fractal Reaction Kinetics. Science. 1988;241(4873):1620–6. 10.1126/science.241.4873.1620 17820893
G. Ong, C. Sing (2019)
Mapping the phase behavior of coacervate-driven self-assembly in diblock copolyelectrolytes.
Soft matter, 15 25
(MitreaDM, CikaJA, StanleyCB, NourseA, OnuchicPL, BanerjeePR, et al Self-interaction of NPM1 modulates multiple mechanisms of liquid–liquid phase separation. Nature Communications. 2018;9(1):842 10.1038/s41467-018-03255-3 29483575)
MitreaDM, CikaJA, StanleyCB, NourseA, OnuchicPL, BanerjeePR, et al Self-interaction of NPM1 modulates multiple mechanisms of liquid–liquid phase separation. Nature Communications. 2018;9(1):842 10.1038/s41467-018-03255-3 29483575
MitreaDM, CikaJA, StanleyCB, NourseA, OnuchicPL, BanerjeePR, et al Self-interaction of NPM1 modulates multiple mechanisms of liquid–liquid phase separation. Nature Communications. 2018;9(1):842 10.1038/s41467-018-03255-3 29483575, MitreaDM, CikaJA, StanleyCB, NourseA, OnuchicPL, BanerjeePR, et al Self-interaction of NPM1 modulates multiple mechanisms of liquid–liquid phase separation. Nature Communications. 2018;9(1):842 10.1038/s41467-018-03255-3 29483575
(PakCW, KosnoM, HolehouseAS, PadrickSB, MittalA, AliR, et al Sequence determinants of intracellular phase separation by complex coacervation of a disordered protein. Mol Cell. 2016;63(1):72–85. 10.1016/j.molcel.2016.05.042 27392146)
PakCW, KosnoM, HolehouseAS, PadrickSB, MittalA, AliR, et al Sequence determinants of intracellular phase separation by complex coacervation of a disordered protein. Mol Cell. 2016;63(1):72–85. 10.1016/j.molcel.2016.05.042 27392146
PakCW, KosnoM, HolehouseAS, PadrickSB, MittalA, AliR, et al Sequence determinants of intracellular phase separation by complex coacervation of a disordered protein. Mol Cell. 2016;63(1):72–85. 10.1016/j.molcel.2016.05.042 27392146, PakCW, KosnoM, HolehouseAS, PadrickSB, MittalA, AliR, et al Sequence determinants of intracellular phase separation by complex coacervation of a disordered protein. Mol Cell. 2016;63(1):72–85. 10.1016/j.molcel.2016.05.042 27392146
(WidomB. Some Topics in the Theory of Fluids. The Journal of Chemical Physics. 1963;39(11):2808–12. 10.1063/1.1734110)
WidomB. Some Topics in the Theory of Fluids. The Journal of Chemical Physics. 1963;39(11):2808–12. 10.1063/1.1734110
WidomB. Some Topics in the Theory of Fluids. The Journal of Chemical Physics. 1963;39(11):2808–12. 10.1063/1.1734110, WidomB. Some Topics in the Theory of Fluids. The Journal of Chemical Physics. 1963;39(11):2808–12. 10.1063/1.1734110
(LinY, CurrieSL, RosenMK. Intrinsically disordered sequences enable modulation of protein phase separation through distributed tyrosine motifs. Journal of Biological Chemistry. 2017 10.1074/jbc.M117.800466 28924037)
LinY, CurrieSL, RosenMK. Intrinsically disordered sequences enable modulation of protein phase separation through distributed tyrosine motifs. Journal of Biological Chemistry. 2017 10.1074/jbc.M117.800466 28924037
LinY, CurrieSL, RosenMK. Intrinsically disordered sequences enable modulation of protein phase separation through distributed tyrosine motifs. Journal of Biological Chemistry. 2017 10.1074/jbc.M117.800466 28924037, LinY, CurrieSL, RosenMK. Intrinsically disordered sequences enable modulation of protein phase separation through distributed tyrosine motifs. Journal of Biological Chemistry. 2017 10.1074/jbc.M117.800466 28924037
(BanerjeePR, MilinAN, MoosaMM, OnuchicPL, DenizAA. Reentrant Phase Transition Drives Dynamic Substructure Formation in Ribonucleoprotein Droplets. Angewandte Chemie International Edition. 2017;56(38):11354–9. 10.1002/anie.201703191 28556382)
BanerjeePR, MilinAN, MoosaMM, OnuchicPL, DenizAA. Reentrant Phase Transition Drives Dynamic Substructure Formation in Ribonucleoprotein Droplets. Angewandte Chemie International Edition. 2017;56(38):11354–9. 10.1002/anie.201703191 28556382
BanerjeePR, MilinAN, MoosaMM, OnuchicPL, DenizAA. Reentrant Phase Transition Drives Dynamic Substructure Formation in Ribonucleoprotein Droplets. Angewandte Chemie International Edition. 2017;56(38):11354–9. 10.1002/anie.201703191 28556382, BanerjeePR, MilinAN, MoosaMM, OnuchicPL, DenizAA. Reentrant Phase Transition Drives Dynamic Substructure Formation in Ribonucleoprotein Droplets. Angewandte Chemie International Edition. 2017;56(38):11354–9. 10.1002/anie.201703191 28556382
Kiersten Ruff, T. Harmon, R. Pappu (2015)
CAMELOT: A machine learning approach for coarse-grained simulations of aggregation of block-copolymeric protein sequences.
The Journal of chemical physics, 143 24
D. Resasco, Fei Gao, F. Morgan, I. Novak, J. Schaff, B. Slepchenko (2012)
Virtual Cell: computational tools for modeling in cell biology
Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 4
J. Mikeš, K. Dušek (1982)
Simulation of polymer network formation by the Monte Carlo method
Macromolecules, 15
(HarmonTS, HolehouseAS, RosenMK, PappuRV. Intrinsically disordered linkers determine the interplay between phase separation and gelation in multivalent proteins. eLife. 2017;6:e30294 10.7554/eLife.30294 29091028)
HarmonTS, HolehouseAS, RosenMK, PappuRV. Intrinsically disordered linkers determine the interplay between phase separation and gelation in multivalent proteins. eLife. 2017;6:e30294 10.7554/eLife.30294 29091028
HarmonTS, HolehouseAS, RosenMK, PappuRV. Intrinsically disordered linkers determine the interplay between phase separation and gelation in multivalent proteins. eLife. 2017;6:e30294 10.7554/eLife.30294 29091028, HarmonTS, HolehouseAS, RosenMK, PappuRV. Intrinsically disordered linkers determine the interplay between phase separation and gelation in multivalent proteins. eLife. 2017;6:e30294 10.7554/eLife.30294 29091028
Salman Banani, Allyson Rice, William Peeples, Yuan Lin, Saumya Jain, R. Parker, M. Rosen (2016)
Compositional Control of Phase-Separated Cellular Bodies
Cell, 166
Shovamayee Maharana, Jie Wang, D. Papadopoulos, Doris Richter, A. Pozniakovsky, I. Poser, M. Bickle, Sandra Rizk, Jordina Guillén-Boixet, Titus Franzmann, Marcus Jahnel, Lara Marrone, Young‐Tae Chang, J. Sterneckert, P. Tomančák, A. Hyman, S. Alberti (2018)
RNA buffers the phase separation behavior of prion-like RNA binding proteins
Science, 360
(BanjadeS, RosenMK. Phase transitions of multivalent proteins can promote clustering of membrane receptors. eLife. 2014;3:e04123 10.7554/eLife.04123 25321392)
BanjadeS, RosenMK. Phase transitions of multivalent proteins can promote clustering of membrane receptors. eLife. 2014;3:e04123 10.7554/eLife.04123 25321392
BanjadeS, RosenMK. Phase transitions of multivalent proteins can promote clustering of membrane receptors. eLife. 2014;3:e04123 10.7554/eLife.04123 25321392, BanjadeS, RosenMK. Phase transitions of multivalent proteins can promote clustering of membrane receptors. eLife. 2014;3:e04123 10.7554/eLife.04123 25321392
(LytleTK, SingCE. Tuning chain interaction entropy in complex coacervation using polymer stiffness, architecture, and salt valency. Molecular Systems Design & Engineering. 2018;3(1):183–96. 10.1039/C7ME00108H)
LytleTK, SingCE. Tuning chain interaction entropy in complex coacervation using polymer stiffness, architecture, and salt valency. Molecular Systems Design & Engineering. 2018;3(1):183–96. 10.1039/C7ME00108H
LytleTK, SingCE. Tuning chain interaction entropy in complex coacervation using polymer stiffness, architecture, and salt valency. Molecular Systems Design & Engineering. 2018;3(1):183–96. 10.1039/C7ME00108H, LytleTK, SingCE. Tuning chain interaction entropy in complex coacervation using polymer stiffness, architecture, and salt valency. Molecular Systems Design & Engineering. 2018;3(1):183–96. 10.1039/C7ME00108H
(KhanT, KandolaTS, WuJ, VenkatesanS, KetterE, LangeJJ, et al Quantifying Nucleation In Vivo Reveals the Physical Basis of Prion-like Phase Behavior. Molecular Cell. 2018;71(1):155–68.e7. 10.1016/j.molcel.2018.06.016 29979963)
KhanT, KandolaTS, WuJ, VenkatesanS, KetterE, LangeJJ, et al Quantifying Nucleation In Vivo Reveals the Physical Basis of Prion-like Phase Behavior. Molecular Cell. 2018;71(1):155–68.e7. 10.1016/j.molcel.2018.06.016 29979963
KhanT, KandolaTS, WuJ, VenkatesanS, KetterE, LangeJJ, et al Quantifying Nucleation In Vivo Reveals the Physical Basis of Prion-like Phase Behavior. Molecular Cell. 2018;71(1):155–68.e7. 10.1016/j.molcel.2018.06.016 29979963, KhanT, KandolaTS, WuJ, VenkatesanS, KetterE, LangeJJ, et al Quantifying Nucleation In Vivo Reveals the Physical Basis of Prion-like Phase Behavior. Molecular Cell. 2018;71(1):155–68.e7. 10.1016/j.molcel.2018.06.016 29979963
(ShengY-J, WangM-C. Statics and dynamics of a single polymer chain confined in a tube. The Journal of Chemical Physics. 2001;114(10):4724–9. 10.1063/1.1345879)
ShengY-J, WangM-C. Statics and dynamics of a single polymer chain confined in a tube. The Journal of Chemical Physics. 2001;114(10):4724–9. 10.1063/1.1345879
ShengY-J, WangM-C. Statics and dynamics of a single polymer chain confined in a tube. The Journal of Chemical Physics. 2001;114(10):4724–9. 10.1063/1.1345879, ShengY-J, WangM-C. Statics and dynamics of a single polymer chain confined in a tube. The Journal of Chemical Physics. 2001;114(10):4724–9. 10.1063/1.1345879
(WangJ, ChoiJ-M, HolehouseAS, LeeHO, ZhangX, JahnelM, et al A Molecular Grammar Governing the Driving Forces for Phase Separation of Prion-like RNA Binding Proteins. Cell. 2018;174(3):688–99.e16. 10.1016/j.cell.2018.06.006 29961577)
WangJ, ChoiJ-M, HolehouseAS, LeeHO, ZhangX, JahnelM, et al A Molecular Grammar Governing the Driving Forces for Phase Separation of Prion-like RNA Binding Proteins. Cell. 2018;174(3):688–99.e16. 10.1016/j.cell.2018.06.006 29961577
WangJ, ChoiJ-M, HolehouseAS, LeeHO, ZhangX, JahnelM, et al A Molecular Grammar Governing the Driving Forces for Phase Separation of Prion-like RNA Binding Proteins. Cell. 2018;174(3):688–99.e16. 10.1016/j.cell.2018.06.006 29961577, WangJ, ChoiJ-M, HolehouseAS, LeeHO, ZhangX, JahnelM, et al A Molecular Grammar Governing the Driving Forces for Phase Separation of Prion-like RNA Binding Proteins. Cell. 2018;174(3):688–99.e16. 10.1016/j.cell.2018.06.006 29961577
(BrangwynneCP, TompaP, PappuRV. Polymer physics of intracellular phase transitions. Nature Physics. 2015;11(11):899–904. 10.1038/nphys3532)
BrangwynneCP, TompaP, PappuRV. Polymer physics of intracellular phase transitions. Nature Physics. 2015;11(11):899–904. 10.1038/nphys3532
BrangwynneCP, TompaP, PappuRV. Polymer physics of intracellular phase transitions. Nature Physics. 2015;11(11):899–904. 10.1038/nphys3532, BrangwynneCP, TompaP, PappuRV. Polymer physics of intracellular phase transitions. Nature Physics. 2015;11(11):899–904. 10.1038/nphys3532
Daniel Matějů, Titus Franzmann, Avinash Patel, Andrii Kopach, Edgar Boczek, Shovamayee Maharana, Hyun Lee, S. Carra, A. Hyman, S. Alberti (2017)
An aberrant phase transition of stress granules triggered by misfolded protein and prevented by chaperone function
The EMBO Journal, 36
(LytleTK, ChangL-W, MarkiewiczN, PerrySL, SingCE. Designing Electrostatic Interactions via Polyelectrolyte Monomer Sequence. ACS Central Science. 2019;5(4):709–18. 10.1021/acscentsci.9b00087 31041391)
LytleTK, ChangL-W, MarkiewiczN, PerrySL, SingCE. Designing Electrostatic Interactions via Polyelectrolyte Monomer Sequence. ACS Central Science. 2019;5(4):709–18. 10.1021/acscentsci.9b00087 31041391
LytleTK, ChangL-W, MarkiewiczN, PerrySL, SingCE. Designing Electrostatic Interactions via Polyelectrolyte Monomer Sequence. ACS Central Science. 2019;5(4):709–18. 10.1021/acscentsci.9b00087 31041391, LytleTK, ChangL-W, MarkiewiczN, PerrySL, SingCE. Designing Electrostatic Interactions via Polyelectrolyte Monomer Sequence. ACS Central Science. 2019;5(4):709–18. 10.1021/acscentsci.9b00087 31041391
Yongdae Shin, Joel Berry, N. Pannucci, M. Haataja, Jared Toettcher, C. Brangwynne (2017)
Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets
Cell, 168
(FerrolinoMC, MitreaDM, MichaelJR, KriwackiRW. Compositional adaptability in NPM1-SURF6 scaffolding networks enabled by dynamic switching of phase separation mechanisms. Nature Communications. 2018;9(1):5064 10.1038/s41467-018-07530-1 30498217)
FerrolinoMC, MitreaDM, MichaelJR, KriwackiRW. Compositional adaptability in NPM1-SURF6 scaffolding networks enabled by dynamic switching of phase separation mechanisms. Nature Communications. 2018;9(1):5064 10.1038/s41467-018-07530-1 30498217
FerrolinoMC, MitreaDM, MichaelJR, KriwackiRW. Compositional adaptability in NPM1-SURF6 scaffolding networks enabled by dynamic switching of phase separation mechanisms. Nature Communications. 2018;9(1):5064 10.1038/s41467-018-07530-1 30498217, FerrolinoMC, MitreaDM, MichaelJR, KriwackiRW. Compositional adaptability in NPM1-SURF6 scaffolding networks enabled by dynamic switching of phase separation mechanisms. Nature Communications. 2018;9(1):5064 10.1038/s41467-018-07530-1 30498217
(BinderK, PaulW. Monte Carlo simulations of polymer dynamics: Recent advances. Journal of Polymer Science Part B: Polymer Physics. 1997;35(1):1–31. 10.1002/(sici)1099-0488(19970115)35:1<1::aid-polb1>3.0.co;2-#)
BinderK, PaulW. Monte Carlo simulations of polymer dynamics: Recent advances. Journal of Polymer Science Part B: Polymer Physics. 1997;35(1):1–31. 10.1002/(sici)1099-0488(19970115)35:1<1::aid-polb1>3.0.co;2-#
BinderK, PaulW. Monte Carlo simulations of polymer dynamics: Recent advances. Journal of Polymer Science Part B: Polymer Physics. 1997;35(1):1–31. 10.1002/(sici)1099-0488(19970115)35:1<1::aid-polb1>3.0.co;2-#, BinderK, PaulW. Monte Carlo simulations of polymer dynamics: Recent advances. Journal of Polymer Science Part B: Polymer Physics. 1997;35(1):1–31. 10.1002/(sici)1099-0488(19970115)35:1<1::aid-polb1>3.0.co;2-#
(DasS, AminAN, LinYH, ChanHS. Coarse-grained residue-based models of disordered protein condensates: utility and limitations of simple charge pattern parameters. Physical Chemistry Chemical Physics. 2018;20(45):28558–74. Epub 2018/11/07. 10.1039/c8cp05095c .30397688)
DasS, AminAN, LinYH, ChanHS. Coarse-grained residue-based models of disordered protein condensates: utility and limitations of simple charge pattern parameters. Physical Chemistry Chemical Physics. 2018;20(45):28558–74. Epub 2018/11/07. 10.1039/c8cp05095c .30397688
DasS, AminAN, LinYH, ChanHS. Coarse-grained residue-based models of disordered protein condensates: utility and limitations of simple charge pattern parameters. Physical Chemistry Chemical Physics. 2018;20(45):28558–74. Epub 2018/11/07. 10.1039/c8cp05095c .30397688, DasS, AminAN, LinYH, ChanHS. Coarse-grained residue-based models of disordered protein condensates: utility and limitations of simple charge pattern parameters. Physical Chemistry Chemical Physics. 2018;20(45):28558–74. Epub 2018/11/07. 10.1039/c8cp05095c .30397688
(DignonGL, ZhengW, BestRB, KimYC, MittalJ. Relation between single-molecule properties and phase behavior of intrinsically disordered proteins. Proceedings of the National Academy of Sciences. 2018;115(40):9929–34. 10.1073/pnas.1804177115 30217894)
DignonGL, ZhengW, BestRB, KimYC, MittalJ. Relation between single-molecule properties and phase behavior of intrinsically disordered proteins. Proceedings of the National Academy of Sciences. 2018;115(40):9929–34. 10.1073/pnas.1804177115 30217894
DignonGL, ZhengW, BestRB, KimYC, MittalJ. Relation between single-molecule properties and phase behavior of intrinsically disordered proteins. Proceedings of the National Academy of Sciences. 2018;115(40):9929–34. 10.1073/pnas.1804177115 30217894, DignonGL, ZhengW, BestRB, KimYC, MittalJ. Relation between single-molecule properties and phase behavior of intrinsically disordered proteins. Proceedings of the National Academy of Sciences. 2018;115(40):9929–34. 10.1073/pnas.1804177115 30217894
Yi‐Hsuan Lin, J. Forman-Kay, H. Chan (2018)
Theories for Sequence-Dependent Phase Behaviors of Biomolecular Condensates.
Biochemistry, 57 17
(FericM, VaidyaN, HarmonTS, MitreaDM, ZhuL, RichardsonTM, et al Coexisting liquid phases underlie nucleolar subcompartments. Cell. 2016;165(7):1686–97. 10.1016/j.cell.2016.04.047 27212236)
FericM, VaidyaN, HarmonTS, MitreaDM, ZhuL, RichardsonTM, et al Coexisting liquid phases underlie nucleolar subcompartments. Cell. 2016;165(7):1686–97. 10.1016/j.cell.2016.04.047 27212236
FericM, VaidyaN, HarmonTS, MitreaDM, ZhuL, RichardsonTM, et al Coexisting liquid phases underlie nucleolar subcompartments. Cell. 2016;165(7):1686–97. 10.1016/j.cell.2016.04.047 27212236, FericM, VaidyaN, HarmonTS, MitreaDM, ZhuL, RichardsonTM, et al Coexisting liquid phases underlie nucleolar subcompartments. Cell. 2016;165(7):1686–97. 10.1016/j.cell.2016.04.047 27212236
F. Antunes, E. Marques, M. Miguel, B. Lindman (2009)
Advances in Colloid and Interface Science
(ShayeganM, TahvildariR, KisleyL, MeteraK, MichnickSW, LeslieSR. Probing inhomogeneous diffusion in the microenvironments of phase-separated polymers under confinement. bioRxiv. 2018:402230 10.1101/402230)
ShayeganM, TahvildariR, KisleyL, MeteraK, MichnickSW, LeslieSR. Probing inhomogeneous diffusion in the microenvironments of phase-separated polymers under confinement. bioRxiv. 2018:402230 10.1101/402230
ShayeganM, TahvildariR, KisleyL, MeteraK, MichnickSW, LeslieSR. Probing inhomogeneous diffusion in the microenvironments of phase-separated polymers under confinement. bioRxiv. 2018:402230 10.1101/402230, ShayeganM, TahvildariR, KisleyL, MeteraK, MichnickSW, LeslieSR. Probing inhomogeneous diffusion in the microenvironments of phase-separated polymers under confinement. bioRxiv. 2018:402230 10.1101/402230
(LiuP, ShiQ, DaumeH 3rd, VothGA. A Bayesian statistics approach to multiscale coarse graining. The Journal of Chemical Physics. 2008;129(21):214114 Epub 2008/12/10. 10.1063/1.3033218 .19063551)
LiuP, ShiQ, DaumeH 3rd, VothGA. A Bayesian statistics approach to multiscale coarse graining. The Journal of Chemical Physics. 2008;129(21):214114 Epub 2008/12/10. 10.1063/1.3033218 .19063551
LiuP, ShiQ, DaumeH 3rd, VothGA. A Bayesian statistics approach to multiscale coarse graining. The Journal of Chemical Physics. 2008;129(21):214114 Epub 2008/12/10. 10.1063/1.3033218 .19063551, LiuP, ShiQ, DaumeH 3rd, VothGA. A Bayesian statistics approach to multiscale coarse graining. The Journal of Chemical Physics. 2008;129(21):214114 Epub 2008/12/10. 10.1063/1.3033218 .19063551
Salman Banani, Hyun Lee, A. Hyman, M. Rosen (2017)
Biomolecular condensates: organizers of cellular biochemistry
Nature Reviews Molecular Cell Biology, 18
Yi‐Hsuan Lin, J. Forman-Kay, H. Chan (2016)
Sequence-Specific Polyampholyte Phase Separation in Membraneless Organelles.
Physical review letters, 117 17
(ShinY, BrangwynneCP. Liquid phase condensation in cell physiology and disease. Science. 2017;357(6357):eaaf4382 10.1126/science.aaf4382 28935776)
ShinY, BrangwynneCP. Liquid phase condensation in cell physiology and disease. Science. 2017;357(6357):eaaf4382 10.1126/science.aaf4382 28935776
ShinY, BrangwynneCP. Liquid phase condensation in cell physiology and disease. Science. 2017;357(6357):eaaf4382 10.1126/science.aaf4382 28935776, ShinY, BrangwynneCP. Liquid phase condensation in cell physiology and disease. Science. 2017;357(6357):eaaf4382 10.1126/science.aaf4382 28935776
(GennesPGd. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. The Journal of Chemical Physics. 1971;55(2):572–9. 10.1063/1.1675789)
GennesPGd. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. The Journal of Chemical Physics. 1971;55(2):572–9. 10.1063/1.1675789
GennesPGd. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. The Journal of Chemical Physics. 1971;55(2):572–9. 10.1063/1.1675789, GennesPGd. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. The Journal of Chemical Physics. 1971;55(2):572–9. 10.1063/1.1675789
Zachary Monahan, Veronica Ryan, Abigail Janke, K. Burke, S. Rhoads, Gül Zerze, Robert O’Meally, G. Dignon, Alexander Conicella, Wenwei Zheng, R. Best, R. Cole, J. Mittal, F. Shewmaker, N. Fawzi (2017)
Phosphorylation of the FUS low‐complexity domain disrupts phase separation, aggregation, and toxicity
The EMBO Journal, 36
(HalabiN, RivoireO, LeiblerS, RanganathanR. Protein Sectors: Evolutionary Units of Three-Dimensional Structure. Cell. 2009;138(4):774–86. 10.1016/j.cell.2009.07.038 19703402)
HalabiN, RivoireO, LeiblerS, RanganathanR. Protein Sectors: Evolutionary Units of Three-Dimensional Structure. Cell. 2009;138(4):774–86. 10.1016/j.cell.2009.07.038 19703402
HalabiN, RivoireO, LeiblerS, RanganathanR. Protein Sectors: Evolutionary Units of Three-Dimensional Structure. Cell. 2009;138(4):774–86. 10.1016/j.cell.2009.07.038 19703402, HalabiN, RivoireO, LeiblerS, RanganathanR. Protein Sectors: Evolutionary Units of Three-Dimensional Structure. Cell. 2009;138(4):774–86. 10.1016/j.cell.2009.07.038 19703402
T. Nott, E. Petsalaki, P. Farber, D. Jervis, E. Fussner, A. Plochowietz, Timothy Craggs, D. Bazett-Jones, T. Pawson, J. Forman-Kay, A. Baldwin (2015)
Phase Transition of a Disordered Nuage Protein Generates Environmentally Responsive Membraneless Organelles
Molecular Cell, 57
(LinY-H, Forman-KayJD, ChanHS. Theories for Sequence-Dependent Phase Behaviors of Biomolecular Condensates. Biochemistry. 2018;57(17):2499–508. 10.1021/acs.biochem.8b00058 29509422)
LinY-H, Forman-KayJD, ChanHS. Theories for Sequence-Dependent Phase Behaviors of Biomolecular Condensates. Biochemistry. 2018;57(17):2499–508. 10.1021/acs.biochem.8b00058 29509422
LinY-H, Forman-KayJD, ChanHS. Theories for Sequence-Dependent Phase Behaviors of Biomolecular Condensates. Biochemistry. 2018;57(17):2499–508. 10.1021/acs.biochem.8b00058 29509422, LinY-H, Forman-KayJD, ChanHS. Theories for Sequence-Dependent Phase Behaviors of Biomolecular Condensates. Biochemistry. 2018;57(17):2499–508. 10.1021/acs.biochem.8b00058 29509422
Lanyuan Lu, J. Dama, G. Voth (2013)
Fitting coarse-grained distribution functions through an iterative force-matching method.
The Journal of chemical physics, 139 12
G. Dignon, Wenwei Zheng, Young Kim, R. Best, J. Mittal (2017)
Sequence determinants of protein phase behavior from a coarse-grained model
PLoS Computational Biology, 14
M. Wei, Shana Elbaum-Garfinkle, A. Holehouse, Carlos Chen, Marina Feric, C. Arnold, Rodney Priestley, R. Pappu, C. Brangwynne (2017)
Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles.
Nature chemistry, 9 11
(FeiJ, JadalihaM, HarmonTS, LiIT, HuaB, HaoQ, et al Quantitative analysis of multilayer organization of proteins and RNA in nuclear speckles at super resolution. Journal of Cell Science. 2017;130(24):4180–92. 10.1242/jcs.206854 29133588)
FeiJ, JadalihaM, HarmonTS, LiIT, HuaB, HaoQ, et al Quantitative analysis of multilayer organization of proteins and RNA in nuclear speckles at super resolution. Journal of Cell Science. 2017;130(24):4180–92. 10.1242/jcs.206854 29133588
FeiJ, JadalihaM, HarmonTS, LiIT, HuaB, HaoQ, et al Quantitative analysis of multilayer organization of proteins and RNA in nuclear speckles at super resolution. Journal of Cell Science. 2017;130(24):4180–92. 10.1242/jcs.206854 29133588, FeiJ, JadalihaM, HarmonTS, LiIT, HuaB, HaoQ, et al Quantitative analysis of multilayer organization of proteins and RNA in nuclear speckles at super resolution. Journal of Cell Science. 2017;130(24):4180–92. 10.1242/jcs.206854 29133588
(MitreaDM, GraceCR, BuljanM, YunM-K, PytelNJ, SatumbaJ, et al Structural polymorphism in the N-terminal oligomerization domain of NPM1. Proceedings of the National Academy of Sciences. 2014;111(12):4466–71. 10.1073/pnas.1321007111 24616519)
MitreaDM, GraceCR, BuljanM, YunM-K, PytelNJ, SatumbaJ, et al Structural polymorphism in the N-terminal oligomerization domain of NPM1. Proceedings of the National Academy of Sciences. 2014;111(12):4466–71. 10.1073/pnas.1321007111 24616519
MitreaDM, GraceCR, BuljanM, YunM-K, PytelNJ, SatumbaJ, et al Structural polymorphism in the N-terminal oligomerization domain of NPM1. Proceedings of the National Academy of Sciences. 2014;111(12):4466–71. 10.1073/pnas.1321007111 24616519, MitreaDM, GraceCR, BuljanM, YunM-K, PytelNJ, SatumbaJ, et al Structural polymorphism in the N-terminal oligomerization domain of NPM1. Proceedings of the National Academy of Sciences. 2014;111(12):4466–71. 10.1073/pnas.1321007111 24616519
(ResascoDC, GaoF, MorganF, NovakIL, SchaffJC, SlepchenkoBM. Virtual Cell: computational tools for modeling in cell biology. Wiley Interdisciplinary Reviews: Systems Biology and Medicine. 2012;4(2):129–40. 10.1002/wsbm.165 22139996)
ResascoDC, GaoF, MorganF, NovakIL, SchaffJC, SlepchenkoBM. Virtual Cell: computational tools for modeling in cell biology. Wiley Interdisciplinary Reviews: Systems Biology and Medicine. 2012;4(2):129–40. 10.1002/wsbm.165 22139996
ResascoDC, GaoF, MorganF, NovakIL, SchaffJC, SlepchenkoBM. Virtual Cell: computational tools for modeling in cell biology. Wiley Interdisciplinary Reviews: Systems Biology and Medicine. 2012;4(2):129–40. 10.1002/wsbm.165 22139996, ResascoDC, GaoF, MorganF, NovakIL, SchaffJC, SlepchenkoBM. Virtual Cell: computational tools for modeling in cell biology. Wiley Interdisciplinary Reviews: Systems Biology and Medicine. 2012;4(2):129–40. 10.1002/wsbm.165 22139996
Sanbo Qin, Huan‐Xiang Zhou (2016)
Fast Method for Computing Chemical Potentials and Liquid-Liquid Phase Equilibria of Macromolecular Solutions.
The journal of physical chemistry. B, 120 33
(ZengM, ChenX, GuanD, XuJ, WuH, TongP, et al Reconstituted Postsynaptic Density as a Molecular Platform for Understanding Synapse Formation and Plasticity. Cell. 2018;174(5):1172–87.e16. 10.1016/j.cell.2018.06.047 30078712)
ZengM, ChenX, GuanD, XuJ, WuH, TongP, et al Reconstituted Postsynaptic Density as a Molecular Platform for Understanding Synapse Formation and Plasticity. Cell. 2018;174(5):1172–87.e16. 10.1016/j.cell.2018.06.047 30078712
ZengM, ChenX, GuanD, XuJ, WuH, TongP, et al Reconstituted Postsynaptic Density as a Molecular Platform for Understanding Synapse Formation and Plasticity. Cell. 2018;174(5):1172–87.e16. 10.1016/j.cell.2018.06.047 30078712, ZengM, ChenX, GuanD, XuJ, WuH, TongP, et al Reconstituted Postsynaptic Density as a Molecular Platform for Understanding Synapse Formation and Plasticity. Cell. 2018;174(5):1172–87.e16. 10.1016/j.cell.2018.06.047 30078712
T. Harmon, A. Holehouse, R. Pappu (2017)
To Mix, or To Demix, That Is the Question.
Biophysical journal, 112 4
(SahaS, WeberCA, NouschM, Adame-AranaO, HoegeC, HeinMY, et al Polar positioning of phase-separated liquid compartments in cells regulated by an mRNA competition mechanism. Cell. 2016;166(6):1572–84.e16. 10.1016/j.cell.2016.08.006 27594427)
SahaS, WeberCA, NouschM, Adame-AranaO, HoegeC, HeinMY, et al Polar positioning of phase-separated liquid compartments in cells regulated by an mRNA competition mechanism. Cell. 2016;166(6):1572–84.e16. 10.1016/j.cell.2016.08.006 27594427
SahaS, WeberCA, NouschM, Adame-AranaO, HoegeC, HeinMY, et al Polar positioning of phase-separated liquid compartments in cells regulated by an mRNA competition mechanism. Cell. 2016;166(6):1572–84.e16. 10.1016/j.cell.2016.08.006 27594427, SahaS, WeberCA, NouschM, Adame-AranaO, HoegeC, HeinMY, et al Polar positioning of phase-separated liquid compartments in cells regulated by an mRNA competition mechanism. Cell. 2016;166(6):1572–84.e16. 10.1016/j.cell.2016.08.006 27594427
(MańkaA, NowickiW, NowickaG. Monte Carlo simulations of a polymer chain conformation. The effectiveness of local moves algorithms and estimation of entropy. Journal of Molecular Modeling. 2013;19(9):3659–70. 10.1007/s00894-013-1875-z 23765038)
MańkaA, NowickiW, NowickaG. Monte Carlo simulations of a polymer chain conformation. The effectiveness of local moves algorithms and estimation of entropy. Journal of Molecular Modeling. 2013;19(9):3659–70. 10.1007/s00894-013-1875-z 23765038
MańkaA, NowickiW, NowickaG. Monte Carlo simulations of a polymer chain conformation. The effectiveness of local moves algorithms and estimation of entropy. Journal of Molecular Modeling. 2013;19(9):3659–70. 10.1007/s00894-013-1875-z 23765038, MańkaA, NowickiW, NowickaG. Monte Carlo simulations of a polymer chain conformation. The effectiveness of local moves algorithms and estimation of entropy. Journal of Molecular Modeling. 2013;19(9):3659–70. 10.1007/s00894-013-1875-z 23765038
Jeffrey Woodruff, Beatriz Gomes, Per Widlund, J. Mahamid, A. Honigmann, A. Hyman (2017)
The Centrosome Is a Selective Condensate that Nucleates Microtubules by Concentrating Tubulin
Cell, 169
(SuX, DitlevJA, HuiE, XingW, BanjadeS, OkrutJ, et al Phase separation of signaling molecules promotes T cell receptor signal transduction. Science. 2016;352(6285):595–9. 10.1126/science.aad9964 27056844)
SuX, DitlevJA, HuiE, XingW, BanjadeS, OkrutJ, et al Phase separation of signaling molecules promotes T cell receptor signal transduction. Science. 2016;352(6285):595–9. 10.1126/science.aad9964 27056844
SuX, DitlevJA, HuiE, XingW, BanjadeS, OkrutJ, et al Phase separation of signaling molecules promotes T cell receptor signal transduction. Science. 2016;352(6285):595–9. 10.1126/science.aad9964 27056844, SuX, DitlevJA, HuiE, XingW, BanjadeS, OkrutJ, et al Phase separation of signaling molecules promotes T cell receptor signal transduction. Science. 2016;352(6285):595–9. 10.1126/science.aad9964 27056844
(CarmesinI, KremerK. The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules. 1988;21(9):2819–23. 10.1021/ma00187a030)
CarmesinI, KremerK. The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules. 1988;21(9):2819–23. 10.1021/ma00187a030
CarmesinI, KremerK. The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules. 1988;21(9):2819–23. 10.1021/ma00187a030, CarmesinI, KremerK. The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules. 1988;21(9):2819–23. 10.1021/ma00187a030
(RaiAK, ChenJ-X, SelbachM, PelkmansL. Kinase-controlled phase transition of membraneless organelles in mitosis. Nature. 2018;559(7713):211–6. 10.1038/s41586-018-0279-8 29973724)
RaiAK, ChenJ-X, SelbachM, PelkmansL. Kinase-controlled phase transition of membraneless organelles in mitosis. Nature. 2018;559(7713):211–6. 10.1038/s41586-018-0279-8 29973724
RaiAK, ChenJ-X, SelbachM, PelkmansL. Kinase-controlled phase transition of membraneless organelles in mitosis. Nature. 2018;559(7713):211–6. 10.1038/s41586-018-0279-8 29973724, RaiAK, ChenJ-X, SelbachM, PelkmansL. Kinase-controlled phase transition of membraneless organelles in mitosis. Nature. 2018;559(7713):211–6. 10.1038/s41586-018-0279-8 29973724
R. Vernon, P. Chong, B. Tsang, Tae Kim, Alaji Bah, P. Farber, Hong Lin, J. Forman-Kay (2018)
Pi-Pi contacts are an overlooked protein feature relevant to phase separation
eLife, 7
A. Lamond, D. Spector (2003)
Nuclear speckles: a model for nuclear organelles
Nature Reviews Molecular Cell Biology, 4
G. Subramanian, S. Shanbhag (2008)
On the relationship between two popular lattice models for polymer melts.
The Journal of chemical physics, 129 14
(RobertsS, HarmonTS, SchaalJL, MiaoV, LiK, HuntA, et al Injectable tissue integrating networks from recombinant polypeptides with tunable order. Nature Materials. 2018;17(12):1154–63. 10.1038/s41563-018-0182-6 30323334)
RobertsS, HarmonTS, SchaalJL, MiaoV, LiK, HuntA, et al Injectable tissue integrating networks from recombinant polypeptides with tunable order. Nature Materials. 2018;17(12):1154–63. 10.1038/s41563-018-0182-6 30323334
RobertsS, HarmonTS, SchaalJL, MiaoV, LiK, HuntA, et al Injectable tissue integrating networks from recombinant polypeptides with tunable order. Nature Materials. 2018;17(12):1154–63. 10.1038/s41563-018-0182-6 30323334, RobertsS, HarmonTS, SchaalJL, MiaoV, LiK, HuntA, et al Injectable tissue integrating networks from recombinant polypeptides with tunable order. Nature Materials. 2018;17(12):1154–63. 10.1038/s41563-018-0182-6 30323334
(ShinY, BerryJ, PannucciN, HaatajaMP, ToettcherJE, BrangwynneCP. Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets. Cell. 2017;168(1):159–71.e14. 10.1016/j.cell.2016.11.054 28041848)
ShinY, BerryJ, PannucciN, HaatajaMP, ToettcherJE, BrangwynneCP. Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets. Cell. 2017;168(1):159–71.e14. 10.1016/j.cell.2016.11.054 28041848
ShinY, BerryJ, PannucciN, HaatajaMP, ToettcherJE, BrangwynneCP. Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets. Cell. 2017;168(1):159–71.e14. 10.1016/j.cell.2016.11.054 28041848, ShinY, BerryJ, PannucciN, HaatajaMP, ToettcherJE, BrangwynneCP. Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets. Cell. 2017;168(1):159–71.e14. 10.1016/j.cell.2016.11.054 28041848
(MaharanaS, WangJ, PapadopoulosDK, RichterD, PozniakovskyA, PoserI, et al RNA buffers the phase separation behavior of prion-like RNA binding proteins. Science. 2018;360(6391):918–21. 10.1126/science.aar7366 29650702)
MaharanaS, WangJ, PapadopoulosDK, RichterD, PozniakovskyA, PoserI, et al RNA buffers the phase separation behavior of prion-like RNA binding proteins. Science. 2018;360(6391):918–21. 10.1126/science.aar7366 29650702
MaharanaS, WangJ, PapadopoulosDK, RichterD, PozniakovskyA, PoserI, et al RNA buffers the phase separation behavior of prion-like RNA binding proteins. Science. 2018;360(6391):918–21. 10.1126/science.aar7366 29650702, MaharanaS, WangJ, PapadopoulosDK, RichterD, PozniakovskyA, PoserI, et al RNA buffers the phase separation behavior of prion-like RNA binding proteins. Science. 2018;360(6391):918–21. 10.1126/science.aar7366 29650702
M. Rottereau, J. Gimel, T. Nicolai, D. Durand (2004)
Monte Carlo simulation of particle aggregation and gelation: I. Growth, structure and size distribution of the clusters
The European Physical Journal E, 15
(MetropolisN, RosenbluthAW, RosenbluthMN, TellerAH, TellerE. Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics. 1953;21(6):1087–92. 10.1063/1.1699114)
MetropolisN, RosenbluthAW, RosenbluthMN, TellerAH, TellerE. Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics. 1953;21(6):1087–92. 10.1063/1.1699114
MetropolisN, RosenbluthAW, RosenbluthMN, TellerAH, TellerE. Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics. 1953;21(6):1087–92. 10.1063/1.1699114, MetropolisN, RosenbluthAW, RosenbluthMN, TellerAH, TellerE. Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics. 1953;21(6):1087–92. 10.1063/1.1699114
(BananiSF, RiceAM, PeeplesWB, LinY, JainS, ParkerR, et al Compositional Control of Phase-Separated Cellular Bodies. Cell. 2016;166(3):651–63. 10.1016/j.cell.2016.06.010 27374333)
BananiSF, RiceAM, PeeplesWB, LinY, JainS, ParkerR, et al Compositional Control of Phase-Separated Cellular Bodies. Cell. 2016;166(3):651–63. 10.1016/j.cell.2016.06.010 27374333
BananiSF, RiceAM, PeeplesWB, LinY, JainS, ParkerR, et al Compositional Control of Phase-Separated Cellular Bodies. Cell. 2016;166(3):651–63. 10.1016/j.cell.2016.06.010 27374333, BananiSF, RiceAM, PeeplesWB, LinY, JainS, ParkerR, et al Compositional Control of Phase-Separated Cellular Bodies. Cell. 2016;166(3):651–63. 10.1016/j.cell.2016.06.010 27374333
Jie Wang, Jeong-Mo Choi, A. Holehouse, Hyun Lee, Xiaojie Zhang, Marcus Jahnel, Shovamayee Maharana, Régis Lemaitre, Andrei Pozniakovsky, D. Drechsel, I. Poser, R. Pappu, S. Alberti, A. Hyman (2018)
A Molecular Grammar Governing the Driving Forces for Phase Separation of Prion-like RNA Binding Proteins
Cell, 174
Jacob Brady, P. Farber, A. Sekhar, Yi‐Hsuan Lin, Rui Huang, Alaji Bah, T. Nott, H. Chan, A. Baldwin, J. Forman-Kay, L. Kay (2017)
Structural and hydrodynamic properties of an intrinsically disordered region of a germ cell-specific protein on phase separation
Proceedings of the National Academy of Sciences, 114
D. Düchs, K. Delaney, G. Fredrickson (2014)
A multi-species exchange model for fully fluctuating polymer field theory simulations.
The Journal of chemical physics, 141 17
(Nott TimothyJ, PetsalakiE, FarberP, JervisD, FussnerE, PlochowietzA, et al Phase Transition of a Disordered Nuage Protein Generates Environmentally Responsive Membraneless Organelles. Molecular Cell. 2015;57(5):936–47. 10.1016/j.molcel.2015.01.013 25747659)
Nott TimothyJ, PetsalakiE, FarberP, JervisD, FussnerE, PlochowietzA, et al Phase Transition of a Disordered Nuage Protein Generates Environmentally Responsive Membraneless Organelles. Molecular Cell. 2015;57(5):936–47. 10.1016/j.molcel.2015.01.013 25747659
Nott TimothyJ, PetsalakiE, FarberP, JervisD, FussnerE, PlochowietzA, et al Phase Transition of a Disordered Nuage Protein Generates Environmentally Responsive Membraneless Organelles. Molecular Cell. 2015;57(5):936–47. 10.1016/j.molcel.2015.01.013 25747659, Nott TimothyJ, PetsalakiE, FarberP, JervisD, FussnerE, PlochowietzA, et al Phase Transition of a Disordered Nuage Protein Generates Environmentally Responsive Membraneless Organelles. Molecular Cell. 2015;57(5):936–47. 10.1016/j.molcel.2015.01.013 25747659
Menglong Zeng, Xudong Chen, Dongshi Guan, Jia Xu, Haowei Wu, P. Tong, Mingjie Zhang (2018)
Reconstituted Postsynaptic Density as a Molecular Platform for Understanding Synapse Formation and Plasticity
Cell, 174
(Elbaum-GarfinkleS, KimY, SzczepaniakK, ChenCC-H, EckmannCR, MyongS, et al The disordered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynamics. Proceedings of the National Academy of Sciences USA. 2015;112(23):7189–94. 10.1073/pnas.1504822112 26015579)
Elbaum-GarfinkleS, KimY, SzczepaniakK, ChenCC-H, EckmannCR, MyongS, et al The disordered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynamics. Proceedings of the National Academy of Sciences USA. 2015;112(23):7189–94. 10.1073/pnas.1504822112 26015579
Elbaum-GarfinkleS, KimY, SzczepaniakK, ChenCC-H, EckmannCR, MyongS, et al The disordered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynamics. Proceedings of the National Academy of Sciences USA. 2015;112(23):7189–94. 10.1073/pnas.1504822112 26015579, Elbaum-GarfinkleS, KimY, SzczepaniakK, ChenCC-H, EckmannCR, MyongS, et al The disordered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynamics. Proceedings of the National Academy of Sciences USA. 2015;112(23):7189–94. 10.1073/pnas.1504822112 26015579
S. Roberts, T. Harmon, J. Schaal, Vincent Miao, K. Li, A. Hunt, Y. Wen, T. Oas, J. Collier, R. Pappu, A. Chilkoti (2018)
Injectable tissue integrating networks from recombinant polypeptides with tunable order
Nature Materials, 17
Steven Boeynaems, A. Holehouse, V. Weinhardt, Dénes Kovács, Joris Lindt, C. Larabell, L. Bosch, Rhiju Das, P. Tompa, R. Pappu, A. Gitler (2018)
Spontaneous driving forces give rise to protein−RNA condensates with coexisting phases and complex material properties
Proceedings of the National Academy of Sciences of the United States of America, 116
T. Khan, Tejbir Kandola, Jianzheng Wu, Ellen Ketter, S. Venkatesan, Jeffrey Lange, A. Gama, A. Box, Jay Unruh, Malcolm Cook, Randal Halfmann (2017)
Quinary structure kinetically controls protein function and dysfunction
bioRxiv

Publisher: Public Library of Science (PLoS) Journal
Copyright: Copyright: © 2019 Choi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the manuscript and its Supporting Information files. Funding: This work was supported by grants from the US National Science Foundation (MCB-1614766), http://nsf.gov, the Human Frontier Science Program (RGP0034/2017),http://www.hfsp.org, the US National Institutes of Health (5R01NS056114), http://nih.gov, and the St. Jude Children’s Research Hospital through the research collaborative on membraneless organelles, http://www.stjude.org, to RVP. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: Rohit Pappu is a member of the Scientific Advisory Board of Dewpoint Therapeutics Inc. There are no competing interests from this affiliation.
ISSN: 1553-734X
eISSN: 1553-7358
DOI: 10.1371/journal.pcbi.1007028
Publisher site: See Article on Publisher Site

Abstract

a1111111111 Many biomolecular condensates form via spontaneous phase transitions that are driven by multivalent proteins. These molecules are biological instantiations of associative polymers that conform to a so-called stickers-and-spacers architecture. The stickers are protein-pro- OPENACCESS tein or protein-RNA interaction motifs and / or domains that can form reversible, non-cova- lent crosslinks with one another. Spacers are interspersed between stickers and their Citation: Choi J-M, Dar F, Pappu RV (2019) LASSI: A lattice model for simulating phase transitions of preferential interactions with solvent molecules determine the cooperativity of phase transi- multivalent proteins. PLoS Comput Biol 15(10): tions. Here, we report the development of an open source computational engine known as e1007028. https://doi.org/10.1371/journal. LASSI (LAttice simulation engine for Sticker and Spacer Interactions) that enables the cal- pcbi.1007028 culation of full phase diagrams for multicomponent systems comprising of coarse-grained Editor: Ozlem Keskin, Koc ¸ University, TURKEY representations of multivalent proteins. LASSI is designed to enable computationally effi- Received: April 12, 2019 cient phenomenological modeling of spontaneous phase transitions of multicomponent Accepted: October 1, 2019 mixtures comprising of multivalent proteins and RNA molecules. We demonstrate the appli- cation of LASSI using simulations of linear and branched multivalent proteins. We show that Published: October 21, 2019 dense phases are best described as droplet-spanning networks that are characterized by Copyright:© 2019 Choi et al. This is an open reversible physical crosslinks among multivalent proteins. We connect recent observations access article distributed under the terms of the Creative Commons Attribution License, which regarding correlations between apparent stoichiometry and dwell times of condensates to permits unrestricted use, distribution, and being proxies for the internal structural organization, specifically the convolution of internal reproduction in any medium, provided the original density and extent of networking, within condensates. Finally, we demonstrate that the con- author and source are credited. cept of saturation concentration thresholds does not apply to multicomponent systems Data Availability Statement: All relevant data are where obligate heterotypic interactions drive phase transitions. This emerges from the ellip- within the manuscript and its Supporting soidal structures of phase diagrams for multicomponent systems and it has direct implica- Information files. tions for the regulation of biomolecular condensates in vivo. Funding: This work was supported by grants from the US National Science Foundation (MCB- 1614766), http://nsf.gov, the Human Frontier Science Program (RGP0034/2017),http://www. hfsp.org, the US National Institutes of Health Author summary (5R01NS056114), http://nih.gov, and the St. Jude Children’s Research Hospital through the research Spatial and temporal organization of molecular matter is a defining hallmark of cellular collaborative on membraneless organelles, http:// ultrastructure and recent attention has focused on membraneless organelles, which are www.stjude.org, to RVP. The funders had no role also referred to as biomolecular condensates. Of interest are condensates that form via in study design, data collection and analysis, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 1 / 39 Simulations of phase transitions of multivalent proteins decision to publish, or preparation of the manuscript. phase transitions that combine phase separation and networking of multivalent protein and nucleic acid molecules. Building on recently recognized analogies between associative Competing interests: Rohit Pappu is a member of polymers and multivalent proteins, we have developed and deployed LASSI, an open the Scientific Advisory Board of Dewpoint Therapeutics Inc. There are no competing interests source computational engine that enables the calculation of architecture-specific phase from this affiliation. diagrams for multivalent proteins. LASSI relies on a priori identification of stickers and spacers within a multivalent protein and mapping the stickers onto a 3-dimensional lat- tice. A Monte Carlo engine that incorporates a suite of novel and established move sets enables simulations that track density inhomogeneities and changes to the extent of net- working among stickers as a function of protein concentration and interaction strengths. Calculation of distribution functions and other order parameters allow us to compute full phase diagrams for multivalent proteins modeled using a stickers-and-spacers representa- tion on simple cubic lattices. These calculations allow us to rationalize experimental observations and open the door to the design of protein architectures with bespoke phase behavior. LASSI can be deployed to study the phase behavior of multicomponent systems, which allows us to make direct contact with the physical principles underlying cellular biomolecular condensates. Introduction Biomolecular condensates organize cellular matter into non-stoichiometric assemblies of pro- teins and nucleic acids [1]. Prominent condensates include nuclear bodies [2] such as nucleoli, nuclear speckles [3, 4], and germline granules [1, 5, 6]. Condensates also form in the cyto- plasm. These include stress granules [7], membrane-anchored signaling clusters [8, 9], and bodies in post-synaptic zones [10]. All of these condensates share key features: (i) they range in size from a few hundred nanometers to tens of microns [1, 2, 11]; (ii) they are multicomponent entities comprising of hundreds of distinct types of proteins and nucleic acids; (iii) and of the hundreds of different types of molecules that make up condensates, a small number are essen- tial for the formation of condensates [1, 12]. The simplest feature that distinguishes proteins that are drivers of biomolecular condensates is the valence of interaction domains / motifs that can participate in non-covalent crosslinks [1, 12–14]. Biomolecular condensates can form and dissolve in an all-or-none manner [2, 11, 15]. The reversible formation and dissolution of condensates can be controlled by the concentrations of multivalent proteins that drive the formation of condensates; in simple two-components sys- tems comprising of macromolecules and solvent, condensates form when macromolecular concentrations cross macromolecule-specific threshold values known as saturation concentra- tions [15]. The transitions that characterize condensate formation bear the hallmarks of a sharp transition in macromolecular density, leading to the formation of a dense phase that is in equilibrium with a dilute phase. This type of transition, known as phase separation, sets up two or more coexisting phases to equalize the dense and dilute phase chemical potentials of the macromolecules across phase boundaries [15]. Phase separation is reversible and this reversibility can be achieved by (i) changes to concentrations of the driver macromolecules [9, 16], (ii) changes to solution conditions that alter the effective interaction strengths among driver molecules [17–20], (iii) altering saturation concentrations through ligand binding–a phenomenon known as polyphasic linkage [21, 22], or (iv) via biological regulation such as post-translational modifications of proteins [8, 12, 23]. Recent studies have focused on uncovering the defining features of proteins [13, 15, 17–19, 24–40] and RNA molecules [41–43] that drive phase transitions. Protein and RNA molecules PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 2 / 39 Simulations of phase transitions of multivalent proteins that drive phase transitions are biological instantiations of associative polymers [44] character- ized by a stickers-and-spacers architecture [45]. Stickers contribute to a hierarchy of specific pairwise and higher-order interactions that are either isotropic or anisotropic whereas spacers control the concentration-dependent inhomogeneities in the densities of stickers around one another. Stickers can be hot spots or sectors [46] on the surfaces of folded proteins [15, 29] or short linear motifs within intrinsically disordered regions (IDRs) [15, 24, 47]. Spacers are typi- cally IDRs that contribute through their sequence-specific effective solvation volumes to the interplay between density transitions (phase separation) and networking transitions that are better known as percolation [28, 29]. Spacers can also be folded domains that are akin to uni- formly reactive colloidal particles, although this has not yet been explored. Proteins can be mapped onto the stickers-and-spacers architecture as linear multivalent proteins, branched multivalent proteins, or some combination of the two [13, 15]. Simple two-component systems comprise of the solvent (which includes all components of the aqueous milieu) and a multivalent protein / RNA molecule. For fixed solution conditions, one can generate phase diagrams [25] as a function of protein concentration, the valence of stickers, the affinities of stickers, the sequence-specific effective solvation volumes of spacers, and the lengths / stiffness of spacers. The phase diagram can be investigated by keeping the valence of stickers, the lengths of spacers, and effective solvation volumes of spacers fixed while varying the concentration of stickers and the affinities between stickers [29]. Changes to protein concentration will enable density fluctuations and above the saturation concentration, designated as c , the density inhomogeneities lead to separation of the system into coexisting sat phases. The concentration of multivalent proteins in the dilute and dense phases will be denoted as c and c , respectively. For a given bulk concentration c that lies between sat dense bulk c and c , the fraction of molecules within each of the coexisting phases is governed by the sat dense lever rule [48]. Stickers also form reversible physical crosslinks and these crosslinks generate networks of inter-connected proteins. The number of proteins within the largest network of the system grows continuously as the protein concentration increases. Above a concentration threshold known as the percolation threshold and designated as c , the single largest network spans perc the entire system and this phenomenon is called percolation [49–51]. If the percolated net- works have the rheological properties of viscoelastic fluids, the fluids are referred to as network fluids [15, 52]. Phase separation and percolation can be coupled to one another. The coupling will depend on the values of c , c , and c relative to c . If c is smaller than all of c , c , and sat dense perc bulk bulk sat dense c , the system is in a single dilute phase with no large molecular networks (Fig 1A). If c > perc bulk c and c < c , then a system-spanning percolated network forms without phase separa- perc perc sat tion (Fig 1B). However, the system undergoes phase separation and a dense phase forms as a percolated droplet if c > (c , c ) and c < c < c (Fig 1C). Recent studies, using bulk sat perc sat perc dense three-dimensional lattice models designed to mimic the poly-SH3 and poly-PRM systems of Li et al. [16], show that sequence-specific effective solvation volumes of linkers / spacers between folded domains directly determine whether phase separation and percolation are coupled or if percolation occurs without phase separation for linear multivalent proteins [28, 29]. The cou- pling between phase separation and percolation is controlled by the extent to which spacers / linkers preferentially interact with the surrounding solvent. Theory [17, 24, 25, 27, 34, 53–59] and computations [28, 29, 43, 60–68] have important roles to play in modeling and describing the phase behavior of multivalent protein and RNA molecules. Theories provide analytical routes to explain experimental observations and to make testable predictions. On the other hand, simulations work around many of the simplify- ing assumptions that are needed to make theories analytically tractable. In doing so, they PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 3 / 39 Simulations of phase transitions of multivalent proteins Fig 1. Characteristic phases in the stickers and spacers formalism. (a) Dispersed solution phase where the polymers are uniformly mixed in solution. (b) Percolated fluid wherein the polymer chains form a percolated, system-spanning network through physical crosslinks among stickers results. (c) Droplet wherein network formation also causes the polymers to form condensed phases. (d) Two-dimensional representation of the LASSI architecture. The beads with arms denote stickers where arms denote that the monomers are capable of orientational interactions, and the curved lines connecting the monomers represent phantom tethers, which are allowed to freely overlap (implicit spacer model). Different colors denote different sticker and spacer species respectively. Note that the physical bonds are allowed to overlap (dashed circle). For the rest of this work, physical bonds will not be labeled and will only be depicted as overlapping orientational arms. https://doi.org/10.1371/journal.pcbi.1007028.g001 provide numerical routes to enable comparative assessments across different systems; they help in making testable predictions about phenomenology through what if calculations tar- geted toward specific systems; and they pave the way for designing systems with bespoke phase behavior. Phase transitions are collective phenomena that involve highly cooperative transitions of large numbers of multivalent polymers. The collective interactions that drive phase transitions are captured in terms of a small number of order parameters that are similar across disparate systems and represent a generic coarse-graining of the underlying system that defines parame- ters such as the correlation length and the sizes of cooperative units. Accordingly, practical considerations of computational tractability and rigorous considerations of identifying the rel- evant collective coordinates mandate the use of coarse-grained models for simulations of PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 4 / 39 Simulations of phase transitions of multivalent proteins Fig 2. Considerations that go into designing a coarse-grained model. As discussed in the text, the choice of a coarse-grained model has at least three ingredients. These include the type of conformational space (lattice or off-lattice), the nature of the interactions among entities that are represented in the coarse-grained description (isotropic, anisotropic or fluctuating fields), and the parameterization approach. LASSI, as described here, is based on a lattice model that uses anisotropic interactions and a phenomenological model. https://doi.org/10.1371/journal.pcbi.1007028.g002 phase transitions driven by multivalent protein and RNA molecules. We focus here on multi- valent proteins, although the methods we describe are readily adaptable to RNA molecules as well. Coarse-graining, an essential aspect of making simulations of large numbers of multivalent proteins a tractable proposition, comes in different flavors [69]. For simplicity, we divide con- siderations that go into the development of a suitable coarse-grained model into three catego- ries (Fig 2). These are (1) the type of model, (2) the types of interactions among the entities in the simulation, and (3) parameterization of the interaction potentials for the model of interest. Two distinct choices for the type of model are the choice between simulations being performed using lattice models versus off-lattice models. In either space, one or all of the molecules can be represented explicitly using architectures that represent coarse-grained mappings of the pro- tein of interest. Next, the interactions among the units that make up each protein can be mod- eled as being isotropic or anisotropic. This is true of simulations where proteins of interest are modeled explicitly. In contrast, numerical instantiations of field theoretic models model can also be brought to bear where only a single chain is modeled explicitly [60, 70]. The remaining protein and solvent molecules are modeled as fields whose fluctuations are concentration dependent [71]. The effects of all other molecules influence the phase behavior of the explicitly modeled single chain through interactions of the chain with the field. Finally, the choice of interaction potentials is the bedrock of every simulation. The functional forms and parameters for potentials can be derived using phenomenological considerations intended to enable calcu- lations of the “what if” variety–an approach that is common practice in statistical and polymer physics. One can also obtain system-specific parameters using information gleaned from atomistic simulations of smaller-scale facsimiles of the system of interest. These system-spe- cific parameters are derivable using force matching methods pioneered by Voth and coworkers [72–77] or by prescribing a functional form for the potential that describes interactions in the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 5 / 39 Simulations of phase transitions of multivalent proteins coarse-grained space and employs machine learning methods to derive the relevant parameters [74, 78]. Finally, one can adopt approaches similar to the parameterization of molecular mechanics forcefields and develop a single transferable model that should be applicable to a large number of disparate systems. Different coarse-grained simulations represent different combinations of model, interac- tion type, and parameterization. Two illustrative examples for deriving coarse-grained models for simulations of phase behavior of multivalent proteins come from the works of Ruff et al. [78] and Dignon et al. [64, 65]. Ruff et al. show how one can generate off-lattice models, of bespoke resolutions and learned parameters for isotropic potentials derived using machine learning that leverages information gleaned from atomistic simulations of individual proteins and protein oligomers. Dignon et al. also use an off-lattice model based on isotropic potentials whose parameters are designed to be transferable across disparate intrinsically disordered proteins. It is worth emphasizing that at this juncture, there is no valid reason to stipulate that one combination of approaches for deriving a coarse-grained model is superior to another combi- nation. As noted by Das et al. [67, 68], all models have distinct strengths and limitations. How- ever, for specific applications, some methods afford quantifiable computational advantages over others. In our case, we are interested in uncovering conceptual nuances of phase diagrams for multicomponent systems that comprise of multivalent proteins characterized by aniso- tropic interactions among domains / motifs. As noted above, these systems can be mapped onto a stickers-and-spacers architecture. The questions we are interested in answering pertain to the order parameters that describe phase behavior, the impact of chain connectivity and spacer effective solvation volumes on phase behavior, and the determinants of the shapes of phase diagrams of multicomponent systems where phase transitions are driven by heterotypic as well as homotypic interactions. In this context, it is noteworthy that lattice models have been adapted to model phase transitions for systems comprising of different numbers of multi- valent protein and RNA molecules [28–30, 43, 79–81]. In the present work, we provide a formal description of the design and implementation of system-specific lattice models for simulating phase transitions of multivalent proteins. The simulation engine, known as LASSI for LAttice simulation engine for Sticker and Spacer Inter- actions, formalizes the approaches that have been developed and deployed in recent studies [28–30, 79, 80]. Accordingly, LASSI combines a lattice model with anisotropic interactions among stickers and the model, at least in the current formalism, is derived based on phenome- nological considerations (Fig 2). Ongoing work shows that a machine learning methodology known as CAMELOT [78] can be adapted for using LASSI as a tool to model sequence-specific phase behavior. We describe the design of LASSI, focusing first on the overall structure of the model, the Monte Carlo sampling, and their justification for generic multivalent proteins. We further describe the calculation of order parameters for quantifying phase separation and per- colation. Then, using two specific examples of linear and branched multivalent protein sys- tems, we illustrate the deployment of LASSI to two biologically relevant systems. In both systems, we make a priori assumptions regarding the identities of stickers and spacers, which is a requirement for the deployment of LASSI. Although we focus here on systems with a few components, it should be emphasized that the design of LASSI is able to handle a wide range of multicomponent systems. Materials and methods Considerations that go into the development of a suitable lattice model include (a) the choice of the mapping between a specific multivalent protein of interest and a lattice representation, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 6 / 39 Simulations of phase transitions of multivalent proteins (b) the parameterization of the strengths and ranges of interactions for all unique pairs of beads and vacancies, (c) the design of move sets and acceptance criteria for Monte Carlo simu- lations that enable the sampling of local and collective motions of large numbers of lattice- instantiated multivalent proteins, (d) the efficient titration of key parameters such as protein concentrations and interaction strengths, and (e) the extraction of phase boundaries in terms of known and hidden collective parameters, which become the relevant order parameters for phase transitions of interest. Generating lattice representations of multivalent proteins For a given linear or branched multivalent protein, we first choose a suitable mapping between the protein degrees of freedom and a lattice representation. The conformational space is a sim- ple cubic lattice with periodic boundary conditions used to mimic a macroscopic system. Phase transitions represent the collective effects of large numbers of molecules, and simula- 3 4 tions have to include at least 10 –10 protein molecules to observe facsimiles of these collective transitions in finite sized systems [82]. Further, we need to be able to test for the effects of finite size artefacts and this requires a titration of the effects of varying the numbers of molecules. Accordingly, the lattice has to be large enough to accommodate at least 10 molecules of each type for the most dilute concentrations. Often, we might need to increase the number of mole- cules to be of O(10 ). Accordingly, a one-to-one mapping between the protein degrees of free- dom and a lattice representation would lead to a computationally intractable model. Instead, we adopt system-specific coarse-graining approaches, whereby the coarse-graining is guided by a priori rigorous or phenomenological knowledge of the identities of stickers versus spacers. For disordered proteins, the stickers within disordered regions often correspond to single amino acid residues or short linear motifs. For multivalent folded proteins, the stickers are either an entire protein domain or sectors on domain surfaces [28, 29]. Residues correspond- ing to spacers may either be modeled explicitly, where one or more spacer residues are mod- eled by a single bead on the lattice site, or be modeled as phantom tethers, where the intrinsic lengths of tethers are calibrated in terms of the numbers of lattice sites [28, 29]. In both cases, the tethers can stretch, bend, and rotate and these degrees of freedom contribute to density inhomogeneities that are the result of altered patterns of inter-sticker interactions. LASSI and bond fluctuation models The structure of LASSI is inspired by the bond fluctuation model (BFM) for lattice polymers [83]. This is a general lattice model for simulations designed to extract equilibrium conforma- tional distributions and dynamical attributes of polymers in dilute solutions as well as dense melts. There are two versions of the BFM viz., the Carmesin-Kremer BFM or CK-BFM [84] and the Shaffer BFM or S-BFM [83]. Both models are based on the use of simple cubic lattices, which discretizes the conformational space for polymers. In the CK-BFM [84], each repeating unit or monomer within a polymer is modeled as a 3-dimensional cube where the 8 corners of the cube occupy lattice sites and bond vectors con- nect pairs of monomers. Overlap of monomers is associated with an energetic penalty, and each bond vector can have up to 108 distinct directions. The choice of bond vector set encodes the geometry of the polymer and places constraints on the bond lengths and bond angles. All other interactions are governed by the inter-monomer potentials, and evolution of the system through conformational space is driven by changes to the overall potential energy. In contrast, the S-BFM places each monomer on a single lattice site. Covalently bonded monomers are connected by bonds that are constrained to be of three types, leading to chains that have bonds pffiffiffi pffiffiffi of length 1, 2 or 3 in units of lattice size. Monte Carlo moves with suitable acceptance PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 7 / 39 Simulations of phase transitions of multivalent proteins criteria can be designed for both types of BFMs. The simulations are used to generate equilib- rium conformational distributions of lattice polymers in either dilute or dense phases. The move sets control the overall polymer dynamics and the acceptance of different types of moves and the calculation of correlation functions allows one to compute dynamical quantities for lattice polymers [83]. If we were to use either of the established BFMs without modification, then each amino acid residue would be modeled as a monomer, and such an approach would be useful when the identities of stickers and spacers remain ambiguous. This approach under- lies a different simulation engine known as PIMMS [43]. LASSI is a generalization of the S-BFM that also adapts features of the CK-BFM. Given a choice of the mapping for coarse-graining, each multivalent protein is described as a chain of non-overlapping monomers viz., beads that occupy sites on a 3-dimensional cubic lattice. Note that the choice of a single site per bead is similar to that of the S-BFM, although the bead, which is a sticker or spacer monomer, need not be the monomeric unit, i.e., an amino acid res- idue in the case of proteins. Each sticker monomer is linked to its adjacent sticker on the chain via either a phantom tether or a set of spacer beads that occupy individual lattice sites [28, 29]. pffiffiffi A spacer / tether length of unity implies that adjacent monomers are within 3 lattice units of one another (Fig 1D). The choice of the spacer length will be sequence-specific or more pre- cisely, specific to the architecture of the protein of interest. Inter-monomer (sticker-sticker, sticker-spacer, and spacer-spacer) interactions are mod- eled as contact-based pairwise interactions. A sticker monomer can bind to another sticker monomer that occupies an adjacent lattice site with an interaction energy that depends on the types of both monomers. Monomers are considered to be adjacent to one another if they are pffiffiffi within a lattice distance of 3. By this criterion, each lattice site occupied by a sticker mono- mer will have 26 adjacent lattice sites. This is reminiscent of the interaction geometry of a CK-BFM for each monomer. In the current implementation of LASSI, the interactions are mutually exclusive, implying that a sticker cannot interact simultaneously with more than one other sticker, even though there are 26 adjacent sites that the interaction partner can occupy. If the sticker in question is already engaged in another inter-monomer interaction with stickers or spacers, then the unoccupied sites of the sticker will be unavailable for interaction. The combination of the geometry of the interaction sites per monomer and the single occupancy constraint leads to anisotropic interactions between sticker interactions. This feature is unique to LASSI and it is not incorporated in other variants of BFMs; this allows us to deploy LASSI for modeling heteropolymeric systems. In the context of LASSI, we note that stickers are dis- tinguished by their ability to participate in anisotropic or isotropic interactions. In contrast, explicitly modeled spacer sites only participate in isotropic interactions with other spacer or sticker sites. Furthermore, the interaction strengths involving spacers are typically weaker than those involving stickers. However, it is worth emphasizing that these distinctions only matter inasmuch as LASSI allows us to capture a numerical instantiation of the stickers-and-spacers model. For simplicity, one might simply think of LASSI as a model that has sites that are differ- entiated by whether or not they can involve themselves in anisotropic interactions, by their intrinsic site valence (a variable that we do not titrate in this work), and by the comparative magnitudes of site-site interaction strengths. Setup of simulations A system with n multivalent proteins is in reality an n+1 component system since the solvent is the implicit component. In LASSI, sites that are not occupied by protein units automatically represent solvent sites. Although the interaction potentials do not explicitly include terms between solvent and protein sites, the effective interaction strengths between pairs of protein PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 8 / 39 Simulations of phase transitions of multivalent proteins units represent an averaging over protein-protein, protein-solvent, and solvent-solvent inter- actions. The solvent sites, i.e., the sites that are not occupied by protein units, represent contri- butions from the solvent to the overall translational and mixing entropies. Simulations are initiated by randomizing the positions of protein units, subject to the constraints of chain connectivity. The parameters that are set at the start of each LASSI simulation include the total number of molecules n of type i and the size of the lattice L, from which we can calculate the total num- ber n of all protein components n ¼ n and the concentration or number density of each protein c = n /L . The setup also includes stipulations for the architectures of each protein i i such as specification of the number of monomers per chain, the overall topology of each pro- tein (linear vs. branched), the lengths of spacers, and the types of spacers (implicit / phantom vs. explicit) [28–30, 79, 80]. The number of monomers per molecule will equal the sum of the number of stickers and spacers if spacer residues are modeled explicitly. Alternatively, if spac- ers are modeled as phantom tethers, then the number of explicitly modeled monomers will equal the number of stickers. Specification of the energetics of the system includes specifica- tion of the simulation temperature in normalized units, homotypic and heterotypic interaction strengths between pairs of stickers, the energetic cost for the overlap of stickers, and the inter- action strengths between sticker and spacer sites if the spacers are modeled explicitly. Design of monte carlo move sets Our goal is to compute architecture-specific phase diagrams for systems comprising of one or more types of linear or branched multivalent proteins. This requires a simulation strategy that enables the sampling of the full spectrum of coexisting densities and networked states for mul- tivalent proteins. Accordingly, the conformations of randomly initialized systems of proteins on a simple cubic lattice are sampled via a series of Markov Chain Monte Carlo (MCMC) moves that are designed to ensure efficient sampling of changes in protein density and net- working while maintaining microscopic reversibility. We have developed and deployed a col- lection of moves and these are described below. Monte carlo sampling with biases In LASSI, we have independent contributions from two main energetic sources. Monomer units are not allowed to overlap, and this can be described by a position-dependent energy E where E = 0 or1. On the other hand, inter-monomer pairwise interactions also con- pos pos tribute to the total energy, and E denotes the sum over all of the effective pairwise inter- rot monomer interaction energies. The subscript “rot”(rotational) indicates the fact that for a pair of nearest neighbor stickers their interaction energies are actually governed by their mutual orientations. Accordingly, the total system energy in a specific configuration i is written as: E ¼ E þ E ; ð1Þ i i;pos i;rot The equilibrium probability associated with configuration i is given by the Boltzmann distribu- tion as: p / expð bEÞ ¼ expð bE Þexpð bE Þ; ð2Þ i i i;pos i;rot In Eq (2),β is the inverse of the simulation temperature in units of the Boltzmann constant (effectively, k = 1 energy unit / temperature unit). The frequency with which a transition from configuration i to j is proposed will be governed by the elements T of the targeting ij matrix T. The proposed transition is accepted / rejected based on the elements A of the ij PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 9 / 39 Simulations of phase transitions of multivalent proteins acceptance ratio matrix A. A MCMC move that transitions the system from configuration i to j defines a flow in configuration space and this flow has to satisfy microscopic reversibility: T A p ¼ T A p ; ð3Þ ij ij i ji ji j If the targeting matrix is symmetric, then T = T and the acceptance ratios such as those pre- ij ji scribed by Metropolis et al. [85] will ensure the preservation of microscopic reversibility. How- ever, if biases are incorporated into the targeting matrix, which is often essential to enhance the sampling of configurations that contribute to density inhomogeneities and the making / breaking of bonds in dense networks, then the elements of the acceptance ratio matrix have to be designed to ensure the preservation of microscopic reversibility. We deploy a general strat- egy of using biased moves to enhance the sampling of different mutual orientations among pairs of stickers. The incorporation of these orientational biases is accounted for by modifying the acceptance criterion of Metropolis et al. [85] whereby each element of A is written as: ! ! A T p ij ji j A T p ji ij i ; ð4Þ ( ) T p ji j such that :A ¼ min 1; ij T p ij i For a symmetric targeting matrix, we recover the standard acceptance ratio of Metropolis et al. [85] viz., � � � � A p p ij j j ¼ or A ¼ min 1; ; ð5Þ ij A p p ji i i Since the moves within LASSI generally involve orientational biases, the elements T are ij rewritten in terms of a Rosenbluth weighting factor W [86, 87] whereby: expð bE Þ j;rot T ¼ ; ð6Þ ij Substituting (6) into (4) leads to: � � A ¼ min 1; exp½ bðE E Þ� ; ð7Þ ij j;pos i;pos The specific form for the weighting factors W will depend on the type of move because the extent of asymmetry in the targeting matrix will depend on the nature of the bias incorporated into the biasing move that proposes a transition from i to j. The specific forms for weighting factors are discussed in the context of the move types that are introduced next. Rotational moves A monomer is in an associated or a dissociated state and in the associated state it has a speci- fied binding partner. This defines the rotational state of a monomer. To change the rotational state, we randomly pick a monomer from the system, and exhaustively sample all 26 adjacent sites to construct a list of potential binding partners. The rotational state of the monomer is changed, at random, by drawing a random integer k from a uniform distribution between [0, b], where b is the number possible binding partners available to the monomer. If k = 0, the th monomer is set to be in a dissociated state. Otherwise, the k candidate bond (reversible phys- ical crosslink) is formed and the state of the monomer is set to be in an associated state. If the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 10 / 39 Simulations of phase transitions of multivalent proteins monomer cannot be involved in a rotational interaction, as would be the case for an explicitly modeled spacer, the rotational move is rejected outright. The accessible volume for rotational interactions is within a cube of unit volume centered on the randomly chosen monomer (see S1 Fig for a 2-dimensional representation), and hence, each sticker monomer will have at most b = 3 –1 possible sites as neighbors. Since this number is not large, we sample all 26 max possible interaction sites. Local moves This move serves as the basic unit of local displacement of monomers–be they stickers or spac- ers. A randomly chosen monomer is moved from position r to r = r +Δr. Acceptance of the i j i move is predicated on the move not leading to an overlap with a site occupied by another monomer and the satisfaction of linker constraints. The choice forΔr is made by uniformly pffiffiffi sampling each component from the interval [–2,2] such thatjDrj � 2 3 (shown in S2 Fig) moves if the selected monomer is in the interior of a molecule [88], and they become analo- gous to end rotation moves if end monomers are selected [89]. Local moves have a rotational bias in LASSI and the Rosenbluth factor is calculated as fol- lows. Starting with Eq (7), we shall designate the chosen monomer by index k. In configuration i, assume that monomer k has a binding partner of index l. Typically in coarse-grained systems there are a finite number of unique monomer types, and thus it is more efficient to simply define interaction energies between different monomer types than between all monomer pairs. The energy associated with the bond between monomers k and l is written asε , where t t(k)t(l) (x) indicates the type of monomer x. The local move causes a change in binding partner, whereby the monomer k now binds to monomer m. The local move leads to a bond swap that causes a change in rotational energy, which is written as: E E ¼ ε ε ; ð8Þ i;rot j;rot tðkÞtðlÞ tðkÞtðmÞ Use of Eq (8) in Eq (6) leads to: T expð bε ÞW ij tðkÞtðlÞ i;k ¼ ; ð9Þ T expð bε ÞW ji tðkÞtðmÞ j;k In Eq (9), each Rosenbluth weight has an additional index in the subscript to indicate that the change in configuration is achieved by a change in the binding partner for the monomer k. To accelerate the creation of density inhomogeneities in supersaturated systems and facilitate the making and breaking of networks, we decompose W as: i;k ðaÞ ðdÞ W ¼ W þ W ; ð10Þ i;k i;k i;k The two terms in Eq (10) respectively represent the contributions to the Rosenbluth weights for monomers in associated (a) and dissociated states (d). First, we calculate the weight factors for the interacting monomers as a partition function over all nearest neighbor contributions, such that: ðaÞ W ¼ expð bε Þ; ð11Þ i;k tðkÞtðlÞ In Eq (11), the summation runs over all potential binding partners l (nearest neighbors) for the monomer k. To illustrate how the Rosenbluth factor is calculated, we assume that the sys- tem has only one type of interaction with the pairwise energy designated asε. If the number of nearest neighbors for monomer k in configuration i is designated as N , then the Rosenbluth k;i PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 11 / 39 Simulations of phase transitions of multivalent proteins weight factor in Eq (11) becomes: ðaÞ W ¼ N expð bεÞ; ð12Þ i;k k;i ðdÞ Setting W ¼ 1 to incorporate a bias towards associated states, and using the simplifica- i;k tion that leads to Eq (12), we rewrite Eq (9) as a definition of the acceptance criterion for the move from configuration i to j via local move involving monomer k as: ( ) N þ 1 j;k A ¼ min 1; exp½ bðE E Þ� ; ð13Þ ij;k j;pos i;pos N þ 1 i;k This choice for the acceptance criterion ensures that detailed balance is preserved while enhancing the sampling of configurations characterized by the breaking of old bonds and the forming of new ones. Reptation–or slithering snake–moves In dense configurations, it becomes difficult to realize large-scale translational or rotational motions of polymers. The slithering snake move is a Monte Carlo instantiation of reptation as first conceived by de Gennes [90]. In this move, a chain is chosen at random, and the mono- mer at one end of the target chain is moved to a new position. The remaining monomers within the target chain are then successively moved such that monomer m along the chain moves into the previous position of monomer m–1 (S3 Fig). This move relies on an inherent symmetry of chain molecules, because bond lengths between monomers are the same; if one swaps monomers across chains, the identity of the chain remains invariant. However, this move cannot be used if the molecule has heterogeneous bond lengths or if it is a branched polymer. The reptation move is rotationally biased, and this is true for every monomer in a chain. The bias is independent for each monomer and accordingly, the Rosenbluth factor for a single reptation move can be calculated from the Rosenbluth factors for each monomer-specific local move. In configuration i we obtain: Y Y W ¼ W ¼ ðN þ 1Þ; ð14Þ i i;m i;m m m In Eq (14), the product runs over all monomers m within the chain of interest. The accep- tance criterion for a reptation move takes the form: 8 9 ðN þ 1Þ > > j;m < = A ¼ min 1; exp½ bðE E Þ�; ð15Þ ij j;pos i;pos > > ðN þ 1Þ : ; i;m The inclusion of the bias for every interacting monomer, rather than just the end monomers, is to emulate how a real transiently bonded polymer would slither along its contours. Note that for strict detailed balance, the Rosenbluth factors for the two end monomers should be calcu- lated and added to the acceptance criterion, but the current implementation of LASSI uses the first trial position that satisfies the position constraints. Double pivot moves These moves swap a part of a chain with the corresponding part of another chain of the same type. A monomer is picked at random; it is denoted as i , where i is the monomer index within PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 12 / 39 Simulations of phase transitions of multivalent proteins a chain, and m is the chain index. A search is then performed around the monomer within a prescribed distance for a monomer within the same type of chain. The requirement for the search is that the monomer of interest be one index ahead along its own chain, (i+1) . Next a check is performed to ensure that the distance between i and (i+1) is within the bond con- m n straint connecting i to (i+1) , and that the distance between (i+1) and i is within the bond m m m n constraint for i and (i+1) . Each monomer (i+1) that satisfies each of these constraints is m m n stored and one of these is randomly picked for the double pivot move (S4 Fig). The move is always accepted if there is a candidate because only connectivity changes unless bonds are modeled using elastic springs. The purpose of this move is to engender large configurational changes in dense polymer melts, which approximate the dense phases formed upon phase separation. In dense regions the rate of acceptance of local moves decrease precipitously. At high enough densities, poly- mers become entangled and local moves reduce to slithering-snake moves and polymers are restricted to motions along tubes around one another [91, 92]. Therefore, rather than physi- cally moving polymers to create a change in configurations, we incorporate move sets that break and make bonds while ensuring that monomers do not overlap, and that bond con- straints are always satisfied. If two chains are close enough to each other that the bonds between two monomers can be swapped, then such the double pivot move results in a large configurational change for both chains, and for the system. Chain and cluster translation moves The chain translation move is designed to move single chains while forming new bonds at the proposed location. This move attempts to translate the center of mass of a chain i from r pffiffi 3L to r = r +Δr wherejDrj � and L is the size of the simulation cell. Multiple trial displace- j i ments are proposed until a trial position that does not result in steric clashes for the entire mol- ecule is generated. The move is then attempted. As with the slithering snake move, each monomer in the molecule that is translated will have an orientational bias. Accordingly, the Rosenbluth factors are calculated as in Eqs (14) and (15). The translational move results in large displacements for single chains and correctly biases the system for efficient sampling of configurations with alternate interaction patterns. Translational moves can also be applied to clusters of molecules. A connected cluster refers to a collection of unique chains connected via rotational interactions. A proposed move only results in a translation, and the move is readily accepted if there is no steric clash. Since no new physical bonds are created at the proposed location, the cluster remains invariant and the move is accepted. Naively this move might seem unnecessary as this move simply moves clus- ters around. However, once a physical bond has formed between two molecules, it is highly unlikely for any of the non-cluster translation moves to move the centers-of-masses of clusters closer together. Considerations for setting move set frequencies The structure of each move set serves as a guide for selecting an optimal set of frequencies. This leads to a set of heuristics that are as follows: (i) in the cluster move we pick a random chain from the system, perform a networking analysis on that chain, and then propose a dis- placement of the cluster. As the cluster size grows it is more likely that a randomly picked chain will be part of the largest cluster which itself will result in a steric clash after the proposed move. Therefore, the frequency of the cluster move should be low, if not the lowest, in the entire set. (ii) In the translational move, we pick a random chain from the system for transla- tion; as the size of the largest cluster increases it becomes less likely for a proposed translation PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 13 / 39 Simulations of phase transitions of multivalent proteins move to be accepted. However, unlike the cluster move the translation move is rotationally biased and thus results in new interactions being formed. Hence, translational moves enable single-molecule to cluster-surface interactions. Therefore, this move should be proposed more often than the cluster move, although not as often as rotational or local moves. (iii) The rota- tion move is computationally inexpensive and it enables the switching of physical bonds and should thus be proposed fairly frequently. (iv) Similarly, local moves and slithering snake moves are also rotationally biased, and they help with the local rearrangements of physical bonds. Local moves are the primary route to enable local conformational changes, and to enable local physical bond rearrangements. Therefore, local moves should be proposed most frequently. The slithering snake move is particularly effective because it allows for large local physical bond rearrangements in dense configurations. Thus, this move should also be pro- posed frequently, less so than local moves but more so than translation moves. Note that in a system where some molecules are non-linear or have heterogeneous linker lengths, the fre- quency would need to be higher since the move is rejected if an incorrect molecule is picked at random. (v) The double pivot move allows for large-scale changes to conformations within dense configurations and accordingly, this move should be proposed more frequently than both cluster and translation moves. One can track the acceptance ratios of each move over a very rough initial sweep across the relevant system parameters. Moves that are always rejected do not enable any changes in configuration and only add computational costs. Therefore, the frequency for that particular move should be lowered. This is especially the case for the cluster move in high-density systems. Identifying phase boundaries using measures for density inhomogeneities In order to detect the onset of phase separation, we can calculate excess chemical potentials using the Widom particle insertion method [93] and equalize these chemical potentials across distinct phases. This process requires a priori knowledge of the densities of both phases. An efficient variant of this approach, based on fast Fourier transforms, was recently developed and deployed by Qin and Zhou [61]. They demonstrated their method for calculations of liq- uid-liquid coexistence curves for a patchy colloid model ofγII-cyrstallin. Given that LASSI simulations are lattice-based, we instead rely on properties of pair distribution functions that help us diagnose the onset of phase separation and compute phase boundaries. Pair distribu- tion functions are helpful because phase separation is the result of the system partitioning into phases of different densities. The pair distribution, which is a reduced-dimension partition function, serves as a rigorous thermodynamic and structural measure of the average local den- sity and inhomogeneities of density. To first order, the density fluctuations are quantified by averaging over all orientations. Accordingly, the pair distribution function can be converted to a radial distribution function that allows us to probe local densities and local structural organi- zation of molecules around one another. However, normalization of the pair distribution func- tion requires some caution. The system contains polymer molecules and using a prior distribution that assumes an ideal gas of the chain monomers to normalize the pair distribu- tion function is problematic because it does not accurately capture the effects of non-idealities due to chain connectivity. We leverage the efficient sampling of polymer fluids in LASSI and obtain suitable prior distributions by simulating the system of interest in the absence of sticker-sticker interactions. (2) The pair distribution function P (r) quantifies the equilibrium distribution of distances ð2Þ between chain monomers, where r is the inter-monomer distance. If P ðrÞ denotes the prior pair distribution function calculated from simulations where the inter-sticker interactions are PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 14 / 39 Simulations of phase transitions of multivalent proteins Fig 3. Distribution functions used for calculation of density inhomogeneity. The data shown are obtained from 5 independent simulations for the A -B system with n n −5 -1 � (2) total protein concentration c = 6.89×10 voxels and reduced temperature T = 0.383. Error bars indicate standard deviations. (a) Pair distribution functions P (r) (2) and P (r), where the former is from the interacting system and the latter from the non-interacting system with chain connectivity (prior pair distribution function). (2) Note that P (r) shows two peaks, the first of which indicates dense phase formation. (b) Radial distribution function g ~ðrÞ. This captures the droplet formation by a sharp and broad peak in the beginning. The inset shows r where g ~ðrÞ intersects the line corresponding to g ~ðrÞ ¼ 1 line, delineating between the dense and solution phases. The global density inhomogeneity measure, r , is obtained by integration of absolute deviation of g ~ðrÞ from 1. https://doi.org/10.1371/journal.pcbi.1007028.g003 ignored (see Fig 3A), then the normalized radial distribution function is written as: ð2Þ P ðrÞ g ~ðrÞ ¼ ð16Þ ð2Þ P ðrÞ The function g ~ðrÞ is a direct measure of the local density of the protein of interest. Since pffiffi 3L LASSI uses periodic boundary conditions, the maximal inter-monomer distance is . Given this normalized g ~ðrÞ, we note that if the system has short-range ordering as in a canonical liq- uid, the radial distribution function will oscillate around unity but eventually approach one as r!1. Conversely, if the system undergoes a density transition, g ~ðrÞ will have two distinct spatial regimes (Fig 3B): for small r, g ~ðrÞ will be characterized by a tall and broad peak such that g ~ðrÞ > 1 until g ~ðrÞ intersects the g ~ðrÞ ¼ 1 line; this region corresponds to the dense phase and we shall denote the value of r at this intersection to be r = r . For r > r , g ~ðrÞ will be b b between 0 and 1, and for lattices that are large enough to avoid finite size artefacts, g ~ðrÞ will converge to a value lower than one and this corresponds to the density in the dilute phase region. Furthermore, g ~ðrÞ can be used to estimate the densities within the dense and dilute phases. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 15 / 39 Simulations of phase transitions of multivalent proteins To quantify the global density inhomogeneity we introduce a simple measure, r �, which is calculated as follows: pffiffi 3L � � 2 � � 1 r � ~ r ¼ jgðrÞ 1jV dr ð17Þ L L In Eq (17), V(x), the volume element for the normalized radial distance x = r/L defined as: > 2 4px ; if 0 < x � > pffiffiffi 1 2 VðxÞ ¼ ð18Þ 2pxð3 4xÞ; if < x � 2 2 pffiffiffi pffiffiffi 2 3 2xð3p 12f ðxÞþ f ðxÞÞ; if < x � 1 2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðxÞ ¼ arctanð 4x 1Þ; and ð19Þ 2xð4x 3Þ f ðxÞ ¼ 8xarctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4x 2ð4x þ 1Þ If r �� 0, the global density inhomogeneity in the system is small and this will be characteristic of a single homogeneous phase dominating the simulation volume. As r �increases beyond zero, the system accommodates density inhomogeneities. We construct coexistence curves using a cutoff value of r �= 0.025, which is universal to all systems, to delineate between a homogeneous system and one that has undergone phase separation. Quantitative assessments of finite size effects The pair distribution function is central to our calculation of density inhomogeneities and constructing coexistence curves for a system of multivalent proteins simulated using LASSI. At 3 4 the start of this section we emphasized the importance of including 10 –10 distinct molecules within the simulation cell in order to avoid finite size artefacts. Prior to presenting detailed results that mimic specific systems, we present an analysis of finite size effects that we will con- front if the requisite numbers of molecules are not included in the simulations. The data we present are for simulations of mimics of the protein FUS, specifically the A + B system intro- n n duced in the results section. The phenomenological mapping of this protein architecture onto a cubic lattice is discussed at the start of the results section. Here, we present an analysis that makes a crucial technical point about finite size effects. First, we start with simulations for ideal polymers. The data shown in Fig 4 plots the pair ð2Þ distribution function P ðrÞ extracted for simulations of ideal models of FUS-like proteins. Results are shown for simulations that use 20 chain molecules of A + B as an example of a n n small system. These results are compared to those from simulations with 100, 200, 1000, 1500, 2000, 3000, and 4000 A + B molecules, respectively. The pair distribution functions have a n n ð2Þ self-similar character and this is revealed by plotting P ðr�Þ for all of the simulations, where r is the reduced distance that accounts for the fact that for a similar concentration, the simula- tion cells are made larger (higher values of L, which is box size) as the numbers of molecules increase. This analysis shows that even for a truly ideal system, the smallest simulation com- prising of only 20 molecules will generate noisy estimates of the pair distribution function. This clearly demonstrates the problems inherent to small systems where finite size effects are PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 16 / 39 Simulations of phase transitions of multivalent proteins Fig 4. Assessment of finite size effects in simulations of ideal, non-interacting chains. (a) Pair distribution functions computed in terms of the spatial separation between chain units. The distributions are maximal at r = L/2, where L is the size of the simulation cell for a given system. Note that L increases as the number of polymers in the system increases. With the exception of the smaller systems, the ideal chains show self-similar behavior for different system sizes. (b) The data plotted in 2r pffiffi panel (a) are re-plotted in terms of the scaled variable r� ¼ where L is the size of the simulation cell for boxes with i molecules. 3L https://doi.org/10.1371/journal.pcbi.1007028.g004 accentuated. Interestingly, for the ideal chain system, all simulations with 100 or more mole- cules yield similar pair distribution functions as assessed in Fig 4B. Next we assessed the impact of finite size effects with all of the terms in the potential being included in the simulation. There are three columns, one each for different values of the � � reduced temperature T , in Fig 5. As discussed in the results section, these values of T place the system of interest in the two-phase regime, with the quench depth into the two-phase regime increasing as T increases. The first row (a to c) of Fig 5 shows the same data as Fig 4A while the second row (d to f) in Fig 5 show the normalized data like Fig 4B. Each panel shows eight unnormalized pair distribution functions, one each for the systems with 20, 100, 200, 1000, 1500, 2000, 3000, and 4000 molecules, respectively. The onset of phase separation should be manifest by the presence of a trough located (2) � � between two peaks in the profiles for P (r ). This is evident for T = 0.267 (Fig 5, panel (c)) for all systems providing the numbers of molecules are greater than or equal to 10 . This quali- tative trend is preserved even for T = 0.217, although sampling difficulties in large systems (2) � � become obvious in the noisy estimates for P (r ). At the reduced temperature of T = 0.167 we confront two problems: The small systems where the numbers of molecules are less than 10 cannot support the distinction between a proper dense phase that coexists with a dilute phase. This behavior is similar to that observed for lower quench depths i.e., higher simulation temperatures as shown in panels (b) and (c) of Fig 5. However, as the system size grows, an additional problem arises and this has to do with large clusters becoming frozen, and thus PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 17 / 39 Simulations of phase transitions of multivalent proteins Fig 5. Assessment of finite size effects in simulations with real interacting chains. Panels (a), (b), and (c), respectively are the real chain equivalents of panel (a) in Fig 4 computed for three different simulation temperatures that represent three different quench depths of the system into its two-phase regime. Panels (d), (e), and (f), respectively are the real chain equivalents of panel (b) in Fig 9 computed for three different simulation temperatures that represent three different quench depths of the system into its two-phase regime. https://doi.org/10.1371/journal.pcbi.1007028.g005 inhibiting the achievement of equilibrium. This is evident from the pair distribution functions shown in panels (a) and (d) of Fig 5 for systems where the numbers of molecules exceed 10 . To overcome this broken ergodicity and obtain reliable converged pair distribution functions, we need additional biasing potentials and temperature sweep approaches used recently [43] to break up frozen clusters and enable their coalescence. In the results that we report here, we use the system size titration to identify the reduced temperatures below which broken ergodicity becomes evident. We do not include data from these simulations in our constructions of phase diagrams. Importantly, our analysis confirms the presence of finite size effects for small sys- tems and sets a lower bound on the numbers of molecules that are needed to observe facsimiles of phase separation as diagnosed by the calculated pair distribution functions. The conclusions drawn from analysis of the pair distribution functions are reinforced in our analysis of the radial distribution function gðr�Þ shown in S5 Fig. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 18 / 39 Simulations of phase transitions of multivalent proteins Estimating the percolation transition line that delineates percolated and non-percolated networks Associative polymers form networks characterized by physical crosslinks among stickers. Accordingly, we use the concept of a cluster, viz., a collection of unique chains connected via rotational interactions, to define the extent of percolation. In polymer melt simulations, the extent of percolation, known as the gel fraction in the polymer literature, is defined as the frac- tion of polymers participating in a percolating network that spans the simulation box in at least one direction [94]. More generally, we can use the fraction of polymers that make up the single largest cluster to quantify the onset of percolation and the changes to the extent of net- working beyond the percolation threshold [95]. A molecular network cannot percolate the whole simulation cell when dilute and dense phases coexist. Accordingly, we choose the sec- ond definition for the order parameter that describes the percolation transition, and we denote this as ϕ [29]. Semenov and Rubinstein demonstrated that a percolation transition is purely a connectivity transition [45]. This implies that the identification of the percolation threshold is not achiev- able using a bona fide order parameter but instead relies on a suitable topological description. Here, we employ the midpoint of the ϕ vs. concentration curve to assess the onset of percola- tion and the percolation line or curve is obtained as the locus of points in the phase diagram for which ϕ = 0.5. In a system where finite size effects are minimized, the percolation transi- tion is sharp having either a hyperbolic or sigmoidal shape as a function of concentration. Accordingly, the location of the percolation line will be relatively robust to the choice one makes for the percolation threshold. Results We demonstrate the use of LASSI by applying it to study two archetypal systems that are known to undergo phase separation [24, 31, 33]. The systems are instantiations of linear and branched multivalent protein systems, respectively. The simulation results obtained for linear multivalent proteins illustrate how phase diagrams are generated when protein concentration (at a fixed stoichiometry) and temperature are the independent variables. In the second exam- ple that includes a branched multivalent protein and a linear peptide, the temperature is fixed, and the concentrations of the individual components are varied. The simulation parameters for both systems are summarized in Table 1. For each system, we conducted 5 independent 8 6 simulations, each of which consists of 5×10 MC steps after 5×10 equilibration steps. The data were taken over the last half of the trajectories at a frequency of 5×10 steps. Table 1. Simulation parameters for system description. FUS-like system N130 + rpL5 system (see Fig 6) (see Fig 10) Bead notations A/B: stickers A/B: stickers N: neutral spacers N: neutral hub spacers Number of stickers s s = 5, s = 5 s = 10, s = 5 i A B A B Linker length l l = 1, l = 1, l = 4 l = 1, l = 3, l = 3 ij AN BN NN NA AA BB (in lattice units) Position-dependent energy E (r , r ) 1, if r = r 1, if r = r pos 1 2 1 2 1 2 0, otherwise 0, otherwise Pairwise interaction energyε (in temperature units) ε = -3,ε = 0,ε = 0 ε = -3,ε = 0,ε = 0 ij AB ii iN AB ii iN https://doi.org/10.1371/journal.pcbi.1007028.t001 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 19 / 39 Simulations of phase transitions of multivalent proteins A FUS-like system as an example of a linear multivalent protein Wang et al. [24] recently uncovered the molecular grammar that contributes to the driving forces for phase separation of the protein FUS and a family of related proteins. They showed −1 that, to first order, c /(s s ) , where c is the measured saturation concentration of the sat Y R sat FUS family proteins and s and s are the number of tyrosine (Tyr) and arginine (Arg) resi- Y R dues, respectively. In FUS and other proteins of similar architectures, the Tyr residues are located primarily within the N-terminal disordered prion-like domain (PLD), whereas the Arg residues are located primarily within the partially disordered C-terminal RNA binding domain (RBD). Using mutagenesis experiments, Wang et al. established that Tyr and Arg residues are the stickers in the FUS family proteins. Accordingly, the zeroth-order stickers and spacers repre- sentation used to model FUS in LASSI comprises of two parts: An N-terminal mimic of the PLD encompassing Tyr residues as stickers and a C-terminal mimic of the RBD that encom- passes Arg residues as stickers. Wang et al. also measured c for a 1:1 mixture of independent sat PLDs and RBDs interacting in trans. The c for this system is approximately twice that of the sat c for full-length FUS. Given the block copolymeric architecture of FUS, we denote the PLD sat and RBD as A and B , respectively for A and B-blocks of valence n. The model system of n n PLDs and RBDs interacting in trans is denoted as A +B (Fig 6A), whereas the system mim- n n icking full-length FUS where the stickers can interact in cis and in trans is denoted as A -B n n (Fig 6B). Within A and B blocks, spacers provide a uniform spacing of six lattice sites between n n stickers along the chain. We model spacers using a hybrid approach whereby a neutral spacer Fig 6. Architecture of the linear multivalent systems. (a, b) Cartoons to depict the A +B and A -B systems, respectively. Different colors of beads denote different n n n n species of stickers. Note that A -B can be simply considered as A +B where the two different sections of the proteins are joined together. (c) Linker lengths involved n n n n in the architecture (see also Table 1). Each sticker has a neighboring spacer bead that is 1 lattice site apart whereas the neighboring spacer beads are 4 lattice sites apart. This means that consecutive stickers are 6 lattice sites apart and also that the linkers connecting the two have a positive effective solvation volume. https://doi.org/10.1371/journal.pcbi.1007028.g006 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 20 / 39 Simulations of phase transitions of multivalent proteins monomer is attached to each sticker with spacing of a single lattice site (Fig 6C). This choice was made to provide a distinction between A -B and A +B . Accordingly, the relative con- n n n n centration of neutral beads will be higher in A -B when compared to A +B . This allows us to n n n n study linker-mediated differences between the driving forces for phase separation for A -B n n vs. A +B n n Fig 7 shows phase diagrams for the A +B and A -B systems calculated using data from n n n n LASSI-based simulations. In panels (a) and (b), the ordinate quantifies the reduced tempera- k T � � B ture T calculated as T ¼ whereε is the effective energy of pairwise interactions between stickers from the A and B blocks. Panel (a) shows results for the A +B system. The bulk n n n n pffiffiffiffiffiffiffiffiffiffiffi concentration in the A +B system is quantified along the abscissa as c ¼ c c where n n bulk A B n n c and c are the bulk concentrations of A and B , respectively. Panel (b) shows the phase n n A B n n diagram for the A -B system where the abscissa represents the bulk concentration of this n n system. Experiments show that the driving forces for phase separation are roughly twice as large for the full-length FUS compared to the system comprising of a 1:1 mixture of PLDs and RBDs [24]. This feature is recapitulated in LASSI simulations. For example, the width of the two- phase regime is larger for the A -B system compared to the A +B system for all values of T n n n n as shown in panel (e) of Fig 7. The critical temperature is higher for the A -B vs. A +B sys- n n n n � � tem (T � 0:56 vs. T � 0:36, respectively). The valence of stickers is the main determinant of c c the concentration at the critical point whereas the interactions mediated by spacers determine the density inhomogeneities and the critical temperature. The impact of longer chains and increased valence of stickers per chain is also evident from the percolation threshold, which is crossed at a bulk protein concentration that is two-fold lower for the A -B system when com- n n pared to the A +B system, across all the simulation temperatures. Differences between the n n two systems are also evident in the degree of cooperativity of phase separation and the percola- tion transition as shown in panels (d) and (e) of Fig 7. For each system, the intersection of the percolation threshold line with the two-phase regime shows that the dense phase predominantly forms a percolated droplet–panels (a) and (b) in Fig 7. Therefore, while phase separation without percolation is realizable, this is not the dominant scenario for associative polymers, where phase separation and percolation are typi- cally conjoined to give rise to percolated droplets. The density of proteins in these percolated droplets is governed by the interaction strengths, modulated by T and the effective solvation volumes of spacers [28, 29]. Unlike homopolymers, which comprise entirely of stickers or spacers depending on the solvent quality, associative polymers encompass a mixture of stickers and spacers. Stickers provide the driving forces for networking via reversible crosslinks and spacers determine whether or not these driving forces lead to phase separation via condensa- tion. Indeed, the importance of sticker-driven percolation is evidenced in the persistence of percolated networks for both systems at high values of T . The observation that dense phases form percolated droplets has several implications: (1) on timescales that are concordant with or smaller than the average lifetime of physical crosslinks between stickers, the condensates will have elastic properties; this will be replaced by viscous behavior on timescales that are longer than the average lifetime of physical crosslinks [96]; (2) accordingly, condensates will have an intrinsic tendency for viscoelasticity [97] and long-lived crosslinks will cause hardening behavior as has been observed in many systems [1, 6, 7, 11, 22, 24, 39–41]; (3) the extent of crosslinking above the percolation threshold will change continu- ously with concentration [16, 29], and this will govern the overall structure, internal dynamics, and porosity of condensates; (4) reactions within condensates are likely to be constrained by the network topology formed as a result of inter-sticker interactions [98]; these constraints can PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 21 / 39 Simulations of phase transitions of multivalent proteins Fig 7. Phase behavior of the linear multivalent systems. (a, b) Phase diagrams for the A +B and A -B systems, respectively. The purple line is a 2-dimensional linear n n n n interpolation for r = 0.025, and the area encapsulated by the purple line are where the systems have large density inhomogeneities and are thus considered to be phase separated. The green line is a 2-dimensional linear interpolation for ϕ = 0.5 and thus is the proxy for the percolation line. (c, d) r and ϕ curves as a function of c c � � concentrations at T = 0.383 (solid lines in (a) and (b)). (e) Width of the two-phase regime, w(T ), as a function of the reduced temperature. Not only does the A -B n n � � system have a higher critical temperature (T ~ 0.6 vs. T ~ 0.4), but also has a wider two-phase regime than the A +B system. n n https://doi.org/10.1371/journal.pcbi.1007028.g007 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 22 / 39 Simulations of phase transitions of multivalent proteins create a variety of interesting kinetic signatures for reactions [99], including temporal memo- ries as has been demonstrated recently for a system that undergoes thermoresponsive phase behavior [66]. Clearly, any description of biochemical reactions within condensates has to account for the structural and dynamical attributes of percolated droplets that are best described as network fluids. Move set frequencies and diagnostics of converged simulations We used results from simulations of linear multivalent protein system to assess the design of LASSI. The frequencies of the different move sets for simulations of the linear multivalent pro- tein system are summarized in Table 2. Considerations that go into the design of move sets include the achievement of converged equilibrium distributions, with maximal computational efficiency, for each bulk concentration. Details of these considerations were described in the methods section. Diagnostics from short simulations are often useful to optimize the move set frequencies especially if multiple short trials are performed using very different starting configurations. Fig 8 shows the concentration dependence of acceptance ratios for each of the move set types, diagnosed for simulations of the A +B and A -B systems. The acceptance ratios show n n n n similar qualitative trends for both systems, even though there are clear quantitative differences. The move with the highest acceptance ratio in the dense regime is the double pivot move, sig- nifying that the systems are transitioning into a pure polymer melt. The second most accepted move is the local move; extrapolating from the higher concentrations it is expected that the acceptance of local moves should also become small and that the double pivot move will be the most dominant way to alter chain configurations, since even the move of an individual mono- mer will require that multiple monomers from multiple chains are moved simultaneously. Both systems have similar qualitative trends for the translation move where we see a transition from being accepted at low concentrations to not being accepted at higher concentrations. Since the proteins in the A -B system are twice long as the A +B system, the absolute accep- n n n n tance ratio of the translation move is always lower in the A -B system. n n Analysis of acceptance ratios of different move sets within droplets will be helpful for esti- mating correlation lengths and amplitudes of conformational and concentration fluctuations within droplets. Cluster moves have high acceptance ratios in the dilute regime whereas the acceptance ratio nearly vanishes as the concentration increases. This is intuitive since the like- lihood of steric clashes increases with a decrease in available volume and this is coupled to the simultaneous increase in the fraction of molecules in the largest cluster. We note here that the cluster moves have the most dramatic change in acceptance ratios from values near 1 to values near 0. However, the apparent inefficiency of cluster moves in dense configurations cannot be used as a rationale to quench such moves. In fact, as shown in panel (a) of Fig 9, phase separa- tion, diagnosed in terms of r �, is suppressed if cluster moves are not part of the move set. This Table 2. Move frequencies according to their types. They are normalized to the sum of all frequencies used in each simulation. FUS-like system N130 + rpL5 system Cluster translation move 1 1 Chain translation move 10 10 Rotation move 100 100 Local move 250 250 Reptation move 0 50 Double pivot move 50 10 https://doi.org/10.1371/journal.pcbi.1007028.t002 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 23 / 39 Simulations of phase transitions of multivalent proteins Fig 8. Analysis of acceptance ratios for different move sets. Curves with different colors indicate acceptance ratios of different types of moves. The dashed lines show the saturation concentrations. The data are obtained from simulations with T = 0.383. (a) Acceptance ratio data for the A +B system. (b) Acceptance ratio data for the n n A -B system. n n https://doi.org/10.1371/journal.pcbi.1007028.g008 highlights the importance of cluster moves for generating bona fide phase separation as these facilitate coalescence that leads to condensation. Mixtures of N130 and the rpL5 peptide as an example of a branched multivalent protein system LASSI can also be deployed to study the phase behavior of branched multivalent proteins that undergo phase separation via obligate heterotypic interactions. Examples of branched multiva- lent proteins are molecules that form symmetric, stable oligomers such as nucleophosmin 1 (NPM1) and synthetic systems such as the corelets designed by Bracha et al. [100]. NPM1 is a key molecule within the granular component (GC) of the nucleolus [101]. Three coexisting layers define the nucleolus and the GC is the outermost layer. Within the GC, NPM1 appears to form facsimiles of percolated droplets in complex ribosomal proteins with Arg-rich motifs [17, 30]. A minimalist system that mimics the phase behavior of the GC comprises of a trun- cated version of NPM1, referred to as N130, and an Arg-rich peptide, designated as rpL5 [31– 33]. Both NPM1 and N130 form symmetric pentamers in the presence of Arg-rich peptides [102]. The pentamer formed by the association of folded domains serves as a scaffold for dis- playing disordered C-terminal tails that are defined by two distinct acidic tracts. The system also features an N-terminal disordered region with a well-defined acidic motif. The FUS system is an example of a protein that undergoes phase separation via obligate homotypic interactions. This implies that the interactions necessary and sufficient for driving phase separation are encoded within the sequence of FUS and the strengths of these PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 24 / 39 Simulations of phase transitions of multivalent proteins � � Fig 9. Importance of cluster translation moves. (a) r and (b) ϕ curves for the A +B (purple) and A -B systems (green) at T = 0.383. The solid lines are identical c n n n n with the curves in panels (c) and (d) of Fig 7. The dotted lines show the simulation results under the same system conditions but the frequency for cluster translation moves is set to zero. Not only do the systems phase separate and percolate at higher saturation concentrations, but also we can see that both percolation and separation are suppressed highly. Furthermore, errors are generally higher, due to the systems being highly dependent on the initial conditions of the system. https://doi.org/10.1371/journal.pcbi.1007028.g009 interactions can be modulated by changes to solution conditions. The N130 + rpL5 system is an example of a system that undergoes phase separation mainly via obligate heterotypic inter- actions that involve interactions between residues in the acidic tracts of N130 and the Arg motifs of rpL5. This could be viewed as an example of phase separation via complex coacerva- tion, providing the heterotypic interactions are purely electrostatic in nature [31–33]. How- ever, in general, there is likely to be combination of long- and short-range interactions that contribute to the spectrum of heterotypic interactions, and hence we refer to this class of mole- cules as drivers of phase separation via obligate heterotypic interactions. In addition to demonstrating the applicability of the LASSI engine for simulations of branched systems, we use the analysis as an opportunity to highlight key conceptual features of multicomponent systems that undergo phase separation via obligate heterotypic interactions. There are three main features that are borne out in the analysis: (1) For fixed temperature, the order parameters are the concentrations of the proteins that drive phase separation via obligate heterotypic interactions. In such systems, the coexistence curves, i.e., the binodals, will have a closed loop form. These will be ellipses for two components and n-ellipsoids for systems that involve up to n obligate heterotypic interactions to drive phase separation. (2) The systems will support reentrant phase behavior as has been reported recently for protein-RNA mixtures that undergo phase separation via obligate heterotypic interactions [103]. (3) The apparent satura- tion concentration of a component molecule in a system that undergoes phase separation via obligate heterotypic interactions will show non-trivial dependencies on its bulk concentration. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 25 / 39 Simulations of phase transitions of multivalent proteins Fig 10. Architecture of an archetypal branched multivalent system. (a) Schematic to depict the overall architecture. The pentamer with 10 orange stickers represents the N130 molecule where the gray central oligomerization domain is modeled as a neutral spacer monomer, and the rpL5 peptide is modeled as a linear molecule with 2 blue stickers. (b) Relevant length scales for the architecture (see also Table 1). For the rpL5 molecule a linker length of 3 was chosen between the two stickers, and for the N130 molecule the first sticker (modeling the A1 tract) is 1 lattice site away from the hub spacer whereas the second sticker (modeling the A2 tract) is 3 lattice sites away from the first sticker. https://doi.org/10.1371/journal.pcbi.1007028.g010 These dependencies are governed directly by the slopes of the tie lines that pass through the point corresponding to the bulk concentration and intersect the binodal at coexisting concentrations that equalize the chemical potentials of all species in the dense and dilute phases. Here, we use the example of the N130 + rpL5 system to showcase the three central fea- tures of phase diagrams for systems that undergo phase separation via obligate heterotypic interactions. In the LASSI representation, N130 pentamers with disordered tails are modeled using a five-armed structure. This approach follows the strategy of Feric et al. [30], which was based on the fact that pentamers do not dissociate under conditions where NPM1 / N130 undergo phase separation. Each arm comprises of two sticker sites to mimic the presence of the A1 and A2 acidic tracts within the disordered tails of NPM1 / N130. Therefore, each N130 pentamer displays a total of ten sticker sites. The spacers between each A1 tract and the N130 core as well as between each pair of A1 and A2 tracts on a disordered tail are phantom tethers, which means that their effective solvation volumes [29] are set to zero. Each rpL5 peptide has two sticker sites corresponding to the two Arg-rich motifs along the sequence. Schematic represen- tations of the coarse-grained architecture used for N130 and rpL5 are shown in Fig 10, and the move set frequencies are summarized in Table 2. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 26 / 39 Simulations of phase transitions of multivalent proteins Percolation lines have parabolic shapes The percolation line, constructed as a function of the concentrations of two multivalent mole- cules, has an overall parabolic shape (panel (a) of Fig 11). This feature may be explained as fol- lows: Let f and f denote the fractions of N130 and rpL5 molecules that are bound in a 1 2 network; s and s will denote the valence of stickers on N130 and rpL5, respectively; for the 1 2 current implementation of the N130 + rpL5 system, s = 10 and s = 2. Based on the Flory- 1 2 Fig 11. Phase behaviors of the branched multivalent systems for T = 0.25. (a) Full phase diagram, where the purple line denotes the proxy for the binodal and the green line is the proxy for the percolation line (see also the caption for Fig 7). The phase-separated region has an elliptical shape and we have a closed loop, which demonstrates re-entrant phase behavior, whereas the percolation line has a conical shape extending into much higher densities. The solid black lines denote contours of � � constant total concentration where L1 is the lowest concentration and L3 is the highest concentration. Note that both axes are represented in the log scale. (b, c) r and ϕ curves as a function of relative stoichiometric ratio of N130 and rpL5 along the constant-concentration contours. (d) Plot ofΛ vs. the apparent stoichiometry along lines L1, L2, and L3. https://doi.org/10.1371/journal.pcbi.1007028.g011 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 27 / 39 Simulations of phase transitions of multivalent proteins Stockmayer theory [49, 51], the percolation threshold is crossed when f f (s −1)(s −1) > 1. If 1 2 1 2 we keep (s −1)(s −1) constant, the threshold concentration for percolation will be governed by 1 2 the product f f . Accordingly, if there is a large excess of N130 (component 1) compared to 1 2 rpL5 (component 2), then f ! 0 and f ! 1, and the system does not undergo a percolation 1 2 transition. In this scenario, every rpL5 molecule is crosslinked to two sticker sites from one or two N130 molecules. However, since the relative stoichiometry favors N130 molecules, there is a vast excess of un-crosslinked N130 molecules and the network cannot grow. Percolation is also inhibited when the converse situation arises, i.e., when there is a large excess of compo- nent 2 with respect to component 1. Accordingly, the percolation line takes on a parabolic form in the plane defined by the concentrations of N130 and rpL5. Binodals for systems defined by obligate heterotypic interactions will form closed loop ellipses Given that the phase behavior of the N130 + rpL5 system is driven by heterotypic interactions involving the A1 / A2 tracts from the N130 tails and the Arg-motifs from rpL5, we constructed binodals by keeping the simulation temperature fixed and varied the concentrations of N130 and rpL5 molecules. Phase diagrams defined by N130 concentration along the abscissa and rpL5 concentration along the ordinate are shown in panel (a) of Fig 11. The general shape of the binodal is comparable with that of the experimentally determined phase diagram [33], even though direct comparison is not straightforward because the scarcity of experimental data points does not yield a full binodal. The phase boundary, defined by the density transition, is an ellipse that forms a closed loop in the plane defined by the concentrations c and c of N130 and rpL5, respectively. In associa- 1 2 tive polymers, the phase behavior is governed by the affinity between stickers, the valence of stickers, and the effective solvation volumes of spacers [28, 29]. For fixed c that intersects the two-phase regime an increase in c will lead to an entry into the two-phase regime caused by a density transition as c approaches c . However, as c increases well beyond c , the joint system 2 1 2 1 exits the two-phase regime. This is because phase separation is driven by obligate heterotypic interactions and while there is a growing excess of rpL5 molecules there are not enough N130 molecules to drive the density transition via inter-sticker interactions. Similar reasoning applies to describe the reentrant behavior that will result by keeping c fixed at a value that intersects the two-phase regime and increasing c . Taken together, the parabolic percolation lines and elliptic forms for two-phase regimes define conic sections that highlight reentrant phase behavior whereby fixing the concentration of component 1 and increasing the concentration of the second species can lead to phase sepa- ration and percolation at a low threshold concentration of component 2 and exit into the one- phase, non-percolated regime beyond a second higher threshold concentration for component 2. This type of reentrant phase behavior, will be a general feature of multicomponent systems that undergo phase separation via obligate heterotypic interactions; indeed, reentrant phase behavior has been reported for a model protein + RNA system [103]. Apparent stoichiometric ratios can be different from effective stoichiometric ratios Stoichiometry of molecules that drive phase separation is another key parameter that deter- mines the functions of biomolecular condensates formed by multicomponent systems [104]. The apparent stoichiometric ratio is calculated as the ratio of the concentrations of stickers of app c s1 types s and s for N130 and rpL5, respectively such that n ¼ . However, the effective stoi- 1 2 s2 app eff chiometric ratio n can be different from n if excluded volume effects modulate the effective 12 12 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 28 / 39 Simulations of phase transitions of multivalent proteins concentration of stickers. We fit an ellipse to the two-phase boundary and determined the major axis of this ellipse. The effective stoichiometric ratio should be unity along the major axis. As shown in panel (a) of Fig 11, the major axis deviates from the line along which app app eff n ¼ 1. Therefore, n 6¼ n and angle between the major axis and the line along which 12 12 12 app n ¼ 1 quantifies the impact of excluded volume on changes to effective concentrations of stickers that in turn modifies the stoichiometric ratios. The synergy between stoichiometry and phase behavior can be analyzed by quantifying the order parameter r �and the topological parameter ϕ as a function of apparent stoichiometry for fixed bulk concentration. Along each gray line in panel (a) of Fig 11 the total concentration defined as c = (c c ) is fixed, although the stoichiometries will vary. The mean values of c along L1, bulk 1 2 bulk −2 –1 −3 –1 −4 –1 L2, and L3 are 2.09×10 (voxel ), 2.46×10 (voxel ), and 3.33×10 (voxel ), respectively and app the value of n ranges from 0.36 to 22.62 along each of L1, L2, and L3. Panels (b) and (c) in Fig 11 app show the variation of r �and ϕ as n increases along L1, L2, and L3, respectively. Along L1, the c 12 value of r �is essentially zero irrespective of stoichiometry because L1 lies is outside the two-phase regime. However, a system spanning percolated network forms for stoichiometries in the range 1.2 � ν � 13 along L1. This is because the concentrations of both components are well above the percolation threshold along L1 thus ensuring that stickers readily find one another even without a density transition. In direct contrast, along L3, we observe phase separation, characterized by val- ues of r �> 0.025 for a range of stoichiometries, but none of these support the formation of a perco- lated droplet (ϕ < 0.5 for all stoichiometries). Along L2, we observe phase separation for stoichiometries in the range 1.15� ν � 16 and percolation for stoichiometries in the range 2.14 � ν � 11.3 such that phase separation enables the formation of a percolated droplet. In panel (d) of Fig 11, we introduce a new structural parameterΛ, which we define as a convolution of r �and ϕ such that L ¼ r �� . Here, the convolution is calculated as a logical AND gate, which becomes a simple product. The parameterΛ quantifies the convolution of the density and network transition and provides an estimate of the extent to which the phase separation and percolation are coupled as the apparent stoichiometry is varied for a fixed bulk concentration. The profile ofΛ is reminiscent of profiles measured by Case et al. [104] for the dwell time of signaling molecules as a function of stoichiometric ratios that govern the forma- tion of condensates at membranes. This suggests that dwell times, which are experimentally accessible parameters, might actually be proxies for the structural features of the condensates as measured by the convolution between phase separation and percolation and the extent of network formation within the condensate. The key finding is that the combination of the bulk concentration and stoichiometric ratio (as opposed to stoichiometry alone) will determine the quench depth into the two-phase regime. This in turn determines whether a system-spanning network forms without phase sep- aration or if phase separation enables the formation of a droplet-spanning network. The struc- ture of condensates and the overall phase behavior cannot be fully described in terms of c bulk or ν alone. Instead, this requires the consideration of both parameters jointly and relative to the quench depth, which refers to the location in the two-phase regime and with respect to the percolation line. This is important because the extent of crosslinking and the time scales asso- ciated with crosslinks will determine the material properties of the condensate. This in turn should contribute to parameters such as the dwell times of clients within condensates [104]. Saturation concentrations need not be fixed parameters in multicomponent systems The concept of a saturation concentration is one of the defining hallmarks of phase separation [2, 27]. For fixed solution conditions, phase separation in a closed two-component system (or PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 29 / 39 Simulations of phase transitions of multivalent proteins pseudo one-component system) comprising of a protein and solvent is realized when the bulk concentration of the protein denoted as c exceeds a saturation concentration denoted as c . sat The presence of a saturation concentration can be quantified by measuring the concentration of protein in the coexisting dilute phase as c increases. This value will not go above c . Strik- sat ingly, the presence of a saturation concentration has been confirmed in living cells for model disordered proteins by Brangwynne and coworkers using optogenetic tools in mammalian cells [100, 105] and by Khan et al. [106] using yeast as a model system. If the protein whose phase behavior is being interrogated is part of a system where obligate heterotypic interactions drive phase separation, then whether or not the concept of a saturation concentration continues to be valid will depend on the nature of the binodals. We illustrate this by coopting the elliptic phase boundary from Fig 11 for a three-component system that comprises of N130, (component A), rpL5 (component B), plus a solvent that is implicit in the LASSI simulations. This 3-component system may be thought of as a pseudo two-component system. For fixed temperature, the top row of panels (a)-(c) in Fig 12 show three types of ellip- tical, closed loop binodals. These are constructed in a plane where [B] increases along the posi- tive direction of the abscissa and [A] increases along the positive direction of the ordinate. The bottom row in each panel shows the result to be expected were we to measure the concentra- tion of A in the dilute phase, designated as [A] , as the bulk concentration [A] is varied. In Sol each of these measurements, the concentration of B is fixed at a specific value. Slopes of tie lines within the elliptic binodals determine how [A] varies Sol with [A] in the two-phase regime For concentrations of A and B that place the pseudo two-component system in the two-phase regime–red points along each of the curves in the bottom rows of panels (a)–(c)–we find that [A] can change as [A] increases. If the tie lines are horizontal or nearly horizontal, then Sol [A] will vary linearly with [A]. Non-linear variations of [A] with [A] will result for tie lines Sol Sol with positive or negative slopes. This is shown in panel (b) for tie lines with positive slopes. If the tie lines are essentially vertical, then the standard expectation regarding the invariance of [A] with [A] within the two-phase regime is recovered. However, even in this scenario, the Sol plateau value of [A] will shift upward or downward as the value of [B] increases–the upward Sol shift is shown in panel (c) of Fig 12. Here, B acts as a bona fide ligand for A, which is the mac- romolecule. Preferential binding of B to the dilute phase leads to an increase in [A] as Sol depicted in panel (c) of Fig 12. Ligand-mediated shifts in saturation concentrations arise due to polyphasic linkage, a phenomenon first introduced by Wyman and Gill [107]. The main conclusion is that the concept of a saturation concentration, as defined for a pseudo one-component system, does not transfer over to multicomponent systems where phase transi- tions are driven by obligate heterotypic interactions. Instead, the slopes of tie lines or the geome- tries of tie planes in higher dimensional ellipsoids will have a direct bearing on inferences from measurements where the bulk concentration of a protein or RNA component is varied when condensates are observed and the concentration of the molecule of interest is quantified in the coexisting dilute phase. This insight emerges from our ability to deploy LASSI to compute full binodals for multicomponent systems. Discussion In this work, we have built on the connection between multivalent proteins and associative polymers [44, 45, 98] with their stickers-and-spacers architecture [15, 17, 24, 28, 29, 40, 43, 47] to develop and deploy LASSI, a lattice-based open source computational engine that enables the simulation of system-specific phase diagrams of single and multi-component systems. The PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 30 / 39 Simulations of phase transitions of multivalent proteins Fig 12. Slopes of tie lines within elliptic binodals are important for systems that undergo phase separation via obligate heterotypic interactions. The ellipse is � � drawn to fit the locus of points based on the value r that meets our criteria for a density transition (see main text). Data for constructing the ellipse were taken from simulations of the N130 + rpL5 system–see Fig 11(A). This ellipse is used to assess the impact of slopes of tie lines for a two-component system comprising of macromolecule A that undergoes phase separation via obligate heterotypic interactions with macromolecule B. (a) Ellipse with nearly horizontal tie lines. The vertical lines shown in green, grey, and blue correspond to fixed values for [B] along the abscissa. As [A] increases, the system traverses across the two-phase regime, delineated by the ellipse, starting outside the ellipse, crossing the ellipse, and exiting the ellipse at high concentrations of A. (b) For each fixed value of [A], the plot shows how [A] varies with [A]. The red points on each curve were extracted from within the two-phase regime, whereas the black points lie outside the two-phase regime. Sol Clearly, [A] does not stay fixed as [A] increases. (c) Equivalent plot to that shown in panel (a) for the tie lines that we obtain for the N130 + rpL5 system. (d) Sol Equivalent plot to that shown in panel (b). Note the non-linear variation of [A] as [A] increases. (e) Ellipse annotated with vertical tie lines. In this case, phase Sol separation of [A] does not depend on obligate heterotypic interactions with B, but B can bind to A and has a choice of binding preferentially to A in either the dense or dilute phase. Here, A becomes the macromolecule and B the ligand. (f) Preferential binding of the ligand to the macromolecule in its dilute phase will shift the saturation concentration, assessed in terms of [A] , upward and this shift will depend on [B]. Accordingly, the plateau value of [A] in the two-phase regime shifts to Sol Sol higher values for higher values of [B]. https://doi.org/10.1371/journal.pcbi.1007028.g012 foundations of LASSI derive from the formalism of the bond fluctuation model [83, 84, 108]. We demonstrate how canonical ensemble Monte Carlo simulations with appropriately designed move sets and analysis of order parameters derived from the distribution functions allow us to calculate coexistence curves and percolation lines as a function of protein concen- tration and interaction strengths. The choice of a lattice-based approach for coarse-graining and modeling phase behavior of multivalent proteins is guided by the advantages of lattice models [109] for polymeric systems. To titrate across the full range of volume fractions, one needs to balance considerations of finite size effects–which requires large numbers of molecules–with large simulation volumes– which makes it difficult to observe density fluctuations that can grow into density inhomoge- neities. On lattices the conformational space is discretized and the calculation of interaction PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 31 / 39 Simulations of phase transitions of multivalent proteins potentials can be made to be very efficient through the use of look up tables. Importantly, we have generalized lattice-based simulations to incorporate anisotropic interactions. LASSI allows us to query the impacts of overall and intrinsic valence of stickers, interaction strengths between stickers, the spatial ranges of these interactions, the effective solvation vol- umes and lengths of spacers, and protein concentrations. These titrations generate multidi- mensional phase diagrams. The approaches underlying LASSI have been applied to model a variety of multicomponent systems, including mimics of RNA molecules [24, 28–30, 43, 79]. What is required is the development of approaches that enable systematic coarse-graining and adaptation of machine learning based methods to parameterize interaction potentials [78]. Engineering LASSI to be interoperable to cell-based modeling suites [110] will also allow for larger scale deployment of the overall framework. The calculation of pair and higher order dis- tribution functions should afford multiscale descriptions of the structural organization of molecular components within condensates. The acceptance ratios associated with different move sets and the length scales spanned by distinct move sets open the door to analyzing the dynamics of phase separation, percolation, and the interplay between the two. Another major direction for future application of LASSI is to uncover the determinants of compositional spec- ificity of condensates [1, 12]. Supporting information S1 Fig. Two-dimensional representation of rotation move. For a given randomly selected monomer (middle orange bead), 33−1 nearest lattice sites (yellow box) are checked for possi- ble interaction candidates, where eligible candidates have a non-zero interaction energy with the selected monomer. In this figure, orange stickers interact with blue stickers and thus this sticker has 3 possible candidates. The end orientational state of the monomer is then picked using the metropolis criterion, which also includes the non-interacting state. (TIF) S2 Fig. Two-dimensional representation of local move. For a given randomly selected monomer, a new location is proposed by sampling ±2 lattice sites in each coordinate (brown box) and picking a lattice site that is empty. If an empty lattice site is found within a pre-deter- mined number of trials, the numbers of interacting candidates are calculated at the old and proposed location (yellow boxes). Then the move is accepted or rejected using the modified Metropolis criterion that considers orientational bias (see text). (TIF) S3 Fig. Two-dimensional representation of reptation move. For a given randomly selected chain that has the same linker lengths between each monomer, an end is randomly picked. Then, a version of the local move is performed where the selected end is moved to a new ran- dom location that is an empty lattice site within 2 lattice sites in each coordinate (brown box). If an empty site is found within a predetermined number of trials, the number of orientational candidates is calculated for the whole chain in the old and the new configuration (yellow boxes). The modified metropolis criterion is then used to determine if the move is accepted or rejected. Note that since the whole chain is orientationally biased, monomers may have a dif- ferent orientational state after the move is accepted, as shown in the figure. (TIF) S4 Fig. Two-dimensional representation of double pivot move. For a randomly selected monomer, a 2×2×2 cube around the monomer is searched for appropriate bridging candidates (brown box), where an appropriate bridging candidate is the next monomer from a different chain, is within a linker length of the selected monomer as shown by the dashed line PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 32 / 39 Simulations of phase transitions of multivalent proteins connecting i and (i+1) . Furthermore, the distance between (i+1) and i must also be within m n m n a linker length as depicted by the upper dashed line. A list of all possible candidates is calcu- lated and then a randomly chosen candidate is used to break and remake covalent bonds. This results in a large conformational change for both polymers. If the selected polymer is not lin- ear, the move is rejected outright. (TIF) S5 Fig. Assessments of finite size effects analyzed in terms of g ~ðr Þ . The pair distributions from Figs 9(B) and 11 are used to compute the relevant radial distribution functions. This analysis is relevant because the radial distribution functions are used to extract the value of the order parameter that detects the onset of phase separation. For systems where the number of molecules A + B molecules is greater than 200, the radial distribution functions g ~ðr Þ start n n to deviate from one another only at the lowest temperatures where broken ergodicity becomes an issue. Therefore, for the A + B system studied in this calibration, it appears that the num- n n bers of A + B molecules have to be greater than 200 in order to obtain reliable information n n about the phase behavior. (a) g ~ðr Þ extracted for different numbers of A and B molecules n n � � � for T = 0.167. (b) g ~ðr Þ extracted for different numbers of A and B molecules for T = n n � � 0.217. (c) g ~ðr Þ extracted for different numbers of A and B molecules for T = 0.267. n n (TIF) Acknowledgments We thank Tyler Harmon for his original contributions, helpful discussions, and critical insights. Ammon Posey got us thinking about rigorous analysis of closed loop phase bound- aries. We are grateful to Joshua Riback for encouraging us to refine our original discussions of phase diagrams for the N130 + rpL5 system and present the analysis reported in Fig 12. Fur- qan Dar acknowledges helpful conversations with Suman Das and Hue-Sun Chan. We have benefited immensely from conversations about phase separation, the stickers-and-spacers framework, and the design of LASSI with Clifford Brangwynne, Alex Holehouse, Anthony Hyman, Amy Gladfelter, Richard Kriwacki, Tanja Mittag, Michael Rosen, Kiersten Ruff, Andrea Soranno, and Paul Taylor. And finally, we thank Minerva Pappu for insights regarding the mathematical self-similarities of ellipsoids in higher dimensional spaces. LASSI is designed to be an open source engine that will be made available via http://pappulab.wustl.edu that will provide a hyperlink to a suitable code hosting, reviewing, and distribution site. Author Contributions Conceptualization: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Funding acquisition: Rohit V. Pappu. Investigation: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Methodology: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Software: Jeong-Mo Choi, Furqan Dar. Supervision: Rohit V. Pappu. Visualization: Furqan Dar. Writing – original draft: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. Writing – review & editing: Jeong-Mo Choi, Furqan Dar, Rohit V. Pappu. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 33 / 39 Simulations of phase transitions of multivalent proteins References 1. Banani SF, Lee HO, Hyman AA, Rosen MK. Biomolecular condensates: organizers of cellular bio- chemistry. Nature Reviews Molecular Cell Biology. 2017; 18(5):285–98. https://doi.org/10.1038/nrm. 2017.7 PMID: 28225081 2. Shin Y, Brangwynne CP. Liquid phase condensation in cell physiology and disease. Science. 2017; 357(6357):eaaf4382. https://doi.org/10.1126/science.aaf4382 PMID: 28935776 3. Lamond AI, Spector DL. Nuclear speckles: a model for nuclear organelles. Nature Reviews in Molecu- lar and Cell Biology. 2003; 4(8):605–12. Epub 2003/08/19. https://doi.org/10.1038/nrm1172 PMID: 4. Mintz PJ, Spector DL. Compartmentalization of RNA processing factors within nuclear speckles. Jour- nal of structural biology. 2000; 129(2–3):241–51. Epub 2000/05/12. https://doi.org/10.1006/jsbi.2000. 4213 PMID: 10806074. 5. Brangwynne CP, Eckmann CR, Courson DS, Rybarska A, Hoege C, Gharakhani J, et al. Germline P granules are liquid droplets that localize by controlled dissolution/condensation. Science. 2009; 324 (5935):1729–32. https://doi.org/10.1126/science.1172046 PMID: 19460965 6. Saha S, Weber CA, Nousch M, Adame-Arana O, Hoege C, Hein MY, et al. Polar positioning of phase- separated liquid compartments in cells regulated by an mRNA competition mechanism. Cell. 2016; 166(6):1572–84.e16. https://doi.org/10.1016/j.cell.2016.08.006 PMID: 27594427 7. Patel A, Lee HO, Jawerth L, Maharana S, Jahnel M, Hein MY, et al. A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation. Cell. 2015; 162(5):1066–77. https://doi.org/10. 1016/j.cell.2015.07.047 PMID: 26317470 8. Su X, Ditlev JA, Hui E, Xing W, Banjade S, Okrut J, et al. Phase separation of signaling molecules pro- motes T cell receptor signal transduction. Science. 2016; 352(6285):595–9. https://doi.org/10.1126/ science.aad9964 PMID: 27056844 9. Banjade S, Rosen MK. Phase transitions of multivalent proteins can promote clustering of membrane receptors. eLife. 2014; 3:e04123. https://doi.org/10.7554/eLife.04123 PMID: 25321392 10. Zeng M, Chen X, Guan D, Xu J, Wu H, Tong P, et al. Reconstituted Postsynaptic Density as a Molecu- lar Platform for Understanding Synapse Formation and Plasticity. Cell. 2018; 174(5):1172–87.e16. https://doi.org/10.1016/j.cell.2018.06.047 PMID: 30078712 11. Hyman AA, Weber CA, Ju ¨ licher F. Liquid-liquid phase separation in biology. Annu Rev Cell Dev Biol. 2014; 30:39–58. https://doi.org/10.1146/annurev-cellbio-100913-013325 PMID: 25288112 12. Banani SF, Rice AM, Peeples WB, Lin Y, Jain S, Parker R, et al. Compositional Control of Phase-Sep- arated Cellular Bodies. Cell. 2016; 166(3):651–63. https://doi.org/10.1016/j.cell.2016.06.010 PMID: 13. Wu H, Fuxreiter M. The Structure and Dynamics of Higher-Order Assemblies: Amyloids, Signalo- somes, and Granules. Cell. 2016; 165(5):1055–66. https://doi.org/10.1016/j.cell.2016.05.004 PMID: 14. Falkenberg CV, Carson JH, Blinov ML. Multivalent Molecules as Modulators of RNA Granule Size and Composition. Biophysical Journal. 2017; 113(2):235–45. https://doi.org/10.1016/j.bpj.2017.01.031 PMID: 28242011 15. Posey AE, Holehouse AS, Pappu RV. Phase Separation of Intrinsically Disordered Proteins. In: Rhoades E, editor. Methods in Enzymology. 611: Academic Press; 2018. p. 1–30. https://doi.org/10. 1016/bs.mie.2018.09.035 PMID: 30471685 16. Li P, Banjade S, Cheng H-C, Kim S, Chen B, Guo L, et al. Phase transitions in the assembly of multiva- lent signalling proteins. Nature. 2012; 483(7389):336–40. https://doi.org/10.1038/nature10879 PMID: 17. Wei MT, Elbaum-Garfinkle S, Holehouse AS, Chen CC, Feric M, Arnold CB, et al. Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles. Nature Chemis- try. 2017; 9(11):1118–25. https://doi.org/10.1038/nchem.2803 PMID: 29064502. 18. Zhang H, Elbaum-Garfinkle S, Langdon EM, Taylor N, Occhipinti P, Bridges AA, et al. RNA controls PolyQ protein phase transitions. Molecular Cell. 2015; 60(2):220–30. https://doi.org/10.1016/j.molcel. 2015.09.017 PMID: 26474065 19. Elbaum-Garfinkle S, Kim Y, Szczepaniak K, Chen CC-H, Eckmann CR, Myong S, et al. The disor- dered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynam- ics. Proceedings of the National Academy of Sciences USA. 2015; 112(23):7189–94. https://doi.org/ 10.1073/pnas.1504822112 PMID: 26015579 20. Molliex A, Temirov J, Lee J, Coughlin M, Kanagaraj AP, Kim HJ, et al. Phase Separation by Low Com- plexity Domains Promotes Stress Granule Assembly and Drives Pathological Fibrillization. Cell. 2015; 163(1):123–33. https://doi.org/10.1016/j.cell.2015.09.015 PMID: 26406374 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 34 / 39 Simulations of phase transitions of multivalent proteins 21. Posey AE, Ruff KM, Harmon TS, Crick SL, Li A, Diamond MI, et al. Profilin reduces aggregation and phase separation of huntingtin N-terminal fragments by preferentially binding to soluble monomers and oligomers. Journal of Biological Chemistry. 2018; 293(10):3734–46. https://doi.org/10.1074/jbc. RA117.000357 PMID: 29358329 22. Mateju D, Franzmann TM, Patel A, Kopach A, Boczek EE, Maharana S, et al. An aberrant phase tran- sition of stress granules triggered by misfolded protein and prevented by chaperone function. The EMBO Journal. 2017:e201695957. https://doi.org/10.15252/embj.201695957 PMID: 28377462 23. Rai AK, Chen J-X, Selbach M, Pelkmans L. Kinase-controlled phase transition of membraneless organelles in mitosis. Nature. 2018; 559(7713):211–6. https://doi.org/10.1038/s41586-018-0279-8 PMID: 29973724 24. Wang J, Choi J-M, Holehouse AS, Lee HO, Zhang X, Jahnel M, et al. A Molecular Grammar Governing the Driving Forces for Phase Separation of Prion-like RNA Binding Proteins. Cell. 2018; 174(3):688– 99.e16. https://doi.org/10.1016/j.cell.2018.06.006 PMID: 29961577 25. Lin Y-H, Brady J, P, Forman-Kay J, D, Chan HS. Charge pattern matching as a ‘fuzzy’ mode of molec- ular recognition for the functional phase separations of intrinsically disordered proteins. New Journal of Physics. 2017; 19(11):115003. 26. Pak CW, Kosno M, Holehouse AS, Padrick SB, Mittal A, Ali R, et al. Sequence determinants of intra- cellular phase separation by complex coacervation of a disordered protein. Mol Cell. 2016; 63(1):72– 85. https://doi.org/10.1016/j.molcel.2016.05.042 PMID: 27392146 27. Brangwynne CP, Tompa P, Pappu RV. Polymer physics of intracellular phase transitions. Nature Physics. 2015; 11(11):899–904. https://doi.org/10.1038/nphys3532 28. Harmon TS, Holehouse AS, Pappu RV. Differential solvation of intrinsically disordered linkers drives the formation of spatially organized droplets in ternary systems of linear multivalent proteins. New Journal of Physics. 2018; 20(4):045002. https://doi.org/10.1088/1367-2630/aab8d9 29. Harmon TS, Holehouse AS, Rosen MK, Pappu RV. Intrinsically disordered linkers determine the inter- play between phase separation and gelation in multivalent proteins. eLife. 2017; 6:e30294. https://doi. org/10.7554/eLife.30294 PMID: 29091028 30. Feric M, Vaidya N, Harmon TS, Mitrea DM, Zhu L, Richardson TM, et al. Coexisting liquid phases underlie nucleolar subcompartments. Cell. 2016; 165(7):1686–97. https://doi.org/10.1016/j.cell.2016. 04.047 PMID: 27212236 31. Mitrea DM, Cika JA, Stanley CB, Nourse A, Onuchic PL, Banerjee PR, et al. Self-interaction of NPM1 modulates multiple mechanisms of liquid–liquid phase separation. Nature Communications. 2018; 9 (1):842. https://doi.org/10.1038/s41467-018-03255-3 PMID: 29483575 32. Ferrolino MC, Mitrea DM, Michael JR, Kriwacki RW. Compositional adaptability in NPM1-SURF6 scaf- folding networks enabled by dynamic switching of phase separation mechanisms. Nature Communi- cations. 2018; 9(1):5064. https://doi.org/10.1038/s41467-018-07530-1 PMID: 30498217 33. Mitrea DM, Cika JA, Guy CS, Ban D, Banerjee PR, Stanley CB, et al. Nucleophosmin integrates within the nucleolus via multi-modal interactions with proteins displaying R-rich linear motifs and rRNA. eLife. 2016; 5:e13571. https://doi.org/10.7554/eLife.13571 PMID: 26836305 34. Brady JP, Farber PJ, Sekhar A, Lin Y-H, Huang R, Bah A, et al. Structural and hydrodynamic proper- ties of an intrinsically disordered region of a germ cell-specific protein on phase separation. Proceed- ings of the National Academy of Sciences. 2017; 114(39):E8194–E203. https://doi.org/10.1073/pnas. 1706197114 PMID: 28894006 35. Vernon RM, Chong PA, Tsang B, Kim TH, Bah A, Farber P, et al. Pi-Pi contacts are an overlooked pro- tein feature relevant to phase separation. eLife. 2018; 7:e31486. https://doi.org/10.7554/eLife.31486 PMID: 29424691 36. Nott Timothy J, Petsalaki E, Farber P, Jervis D, Fussner E, Plochowietz A, et al. Phase Transition of a Disordered Nuage Protein Generates Environmentally Responsive Membraneless Organelles. Molec- ular Cell. 2015; 57(5):936–47. https://doi.org/10.1016/j.molcel.2015.01.013 PMID: 25747659 37. Lin Y, Currie SL, Rosen MK. Intrinsically disordered sequences enable modulation of protein phase separation through distributed tyrosine motifs. Journal of Biological Chemistry. 2017. https://doi.org/ 10.1074/jbc.M117.800466 PMID: 28924037 38. Boeynaems S, Bogaert E, Kovacs D, Konijnenberg A, Timmerman E, Volkov A, et al. Phase Separa- tion of C9orf72 Dipeptide Repeats Perturbs Stress Granule Dynamics. Mol Cell. 2017; 65(6):1044–55. e5. https://doi.org/10.1016/j.molcel.2017.02.013 PMID: 28306503 39. Woodruff JB, Ferreira Gomes B, Widlund PO, Mahamid J, Honigmann A, Hyman AA. The Centrosome Is a Selective Condensate that Nucleates Microtubules by Concentrating Tubulin. Cell. 2017; 169 (6):1066–77.e10. https://doi.org/10.1016/j.cell.2017.05.028 PMID: 28575670 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 35 / 39 Simulations of phase transitions of multivalent proteins 40. Franzmann TM, Jahnel M, Pozniakovsky A, Mahamid J, Holehouse AS, Nu ¨ ske E, et al. Phase separa- tion of a yeast prion protein promotes cellular fitness. Science. 2018; 359(6371):eaao5654. https://doi. org/10.1126/science.aao5654 PMID: 29301985 41. Maharana S, Wang J, Papadopoulos DK, Richter D, Pozniakovsky A, Poser I, et al. RNA buffers the phase separation behavior of prion-like RNA binding proteins. Science. 2018; 360(6391):918–21. https://doi.org/10.1126/science.aar7366 PMID: 29650702 42. Langdon EM, Qiu Y, Ghanbari Niaki A, McLaughlin GA, Weidmann CA, Gerbich TM, et al. mRNA structure determines specificity of a polyQ-driven phase separation. Science. 2018; 360(6391):922–7. https://doi.org/10.1126/science.aar7432 PMID: 29650703 43. Boeynaems S, Holehouse AS, Weinhardt V, Kovacs D, Van Lindt J, Larabell C, et al. Spontaneous driving forces give rise to protein−RNA condensates with coexisting phases and complex material properties. Proceedings of the National Academy of Sciences USA. 2019; 116:7889–98. https://doi. org/10.1073/pnas.1821038116 PMID: 30926670 44. Rubinstein M, Dobrynin AV. Solutions of Associative Polymers. Trends in Polymer Science. 1997; 5 (6):181–6. 45. Semenov AN, Rubinstein M. Thermoreversible Gelation in Solutions of Associative Polymers. 1. Stat- ics. Macromolecules. 1998; 31(4):1373–85. https://doi.org/10.1021/ma970616h 46. Halabi N, Rivoire O, Leibler S, Ranganathan R. Protein Sectors: Evolutionary Units of Three-Dimen- sional Structure. Cell. 2009; 138(4):774–86. https://doi.org/10.1016/j.cell.2009.07.038 PMID: 47. Holehouse AS, Pappu RV. Functional Implications of Intracellular Phase Transitions. Biochemistry. 2018; 57(17):2415–23. https://doi.org/10.1021/acs.biochem.7b01136 PMID: 29323488 48. Rubinstein M, Colby RH. Polymer Physics. New York: Oxford University Press; 2003 2003. 49. Stockmayer WH. Theory of Molecular Size Distribution and Gel Formation in Branched-Chain Poly- mers. The Journal of Chemical Physics. 1943; 11(2):45–55. https://doi.org/10.1063/1.1723803 50. Flory PJ. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1. Journal of the American Chemical Society. 1941; 63(11):3083–90. https://doi.org/10.1021/ja01856a061 51. Flory PJ. Constitution of Three-dimensional Polymers and the Theory of Gelation. The Journal of Physical Chemistry. 1942; 46(1):132–40. https://doi.org/10.1021/j150415a016 52. Dias CS, Arau ´ jo NAM, Telo da Gama MM. Dynamics of network fluids. Advances in Colloid and Inter- face Science. 2017; 247:258–63. https://doi.org/10.1016/j.cis.2017.07.001 PMID: 28802478 53. Lin Y-H, Forman-Kay JD, Chan HS. Theories for Sequence-Dependent Phase Behaviors of Biomolec- ular Condensates. Biochemistry. 2018; 57(17):2499–508. https://doi.org/10.1021/acs.biochem. 8b00058 PMID: 29509422 54. Lin Y-H, Forman-Kay JD, Chan HS. Sequence-Specific Polyampholyte Phase Separation in Membra- neless Organelles. Physical Review Letters. 2016; 117(17):178101. https://doi.org/10.1103/ PhysRevLett.117.178101 PMID: 27824447 55. Lytle TK, Sing CE. Transfer matrix theory of polymer complex coacervation. Soft Matter. 2017; 13 (39):7001–12. https://doi.org/10.1039/c7sm01080j PMID: 28840212 56. Lytle TK, Salazar AJ, Sing CE. Interfacial properties of polymeric complex coacervates from simulation and theory. The Journal of Chemical Physics. 2018; 149(16):163315. https://doi.org/10.1063/1. 5029934 PMID: 30384702 57. Lytle TK, Sing CE. Tuning chain interaction entropy in complex coacervation using polymer stiffness, architecture, and salt valency. Molecular Systems Design & Engineering. 2018; 3(1):183–96. https:// doi.org/10.1039/C7ME00108H 58. Lytle TK, Chang L-W, Markiewicz N, Perry SL, Sing CE. Designing Electrostatic Interactions via Poly- electrolyte Monomer Sequence. ACS Central Science. 2019; 5(4):709–18. https://doi.org/10.1021/ acscentsci.9b00087 PMID: 31041391 59. Ong GMC, Sing CE. Mapping the phase behavior of coacervate-driven self-assembly in diblock copo- lyelectrolytes. Soft Matter. 2019; 15(25):5116–27. https://doi.org/10.1039/c9sm00741e PMID: 60. McCarty J, Delaney KT, Danielsen SPO, Fredrickson GH, Shea J-E. Complete Phase Diagram for Liq- uid–Liquid Phase Separation of Intrinsically Disordered Proteins. The Journal of Physical Chemistry Letters. 2019; 10(8):1644–52. https://doi.org/10.1021/acs.jpclett.9b00099 PMID: 30873835 61. Qin S, Zhou H-X. Fast Method for Computing Chemical Potentials and Liquid–Liquid Phase Equilibria of Macromolecular Solutions. The Journal of Physical Chemistry B. 2016; 120(33):8164–74. https:// doi.org/10.1021/acs.jpcb.6b01607 PMID: 27327881 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 36 / 39 Simulations of phase transitions of multivalent proteins 62. Nguemaha V, Zhou H-X. Liquid-Liquid Phase Separation of Patchy Particles Illuminates Diverse Effects of Regulatory Components on Protein Droplet Formation. Scientific Reports. 2018; 8(1):6728. https://doi.org/10.1038/s41598-018-25132-1 PMID: 29712961 63. Monahan Z, Ryan VH, Janke AM, Burke KA, Rhoads SN, Zerze GH, et al. Phosphorylation of the FUS low-complexity domain disrupts phase separation, aggregation, and toxicity. EMBO Journal. 2017: e201696394. https://doi.org/10.15252/embj.201696394 PMID: 28790177 64. Dignon GL, Zheng W, Best RB, Kim YC, Mittal J. Relation between single-molecule properties and phase behavior of intrinsically disordered proteins. Proceedings of the National Academy of Sciences. 2018; 115(40):9929–34. https://doi.org/10.1073/pnas.1804177115 PMID: 30217894 65. Dignon GL, Zheng W, Kim YC, Best RB, Mittal J. Sequence determinants of protein phase behavior from a coarse-grained model. PLOS Computational Biology. 2018; 14(1):e1005941. https://doi.org/10. 1371/journal.pcbi.1005941 PMID: 29364893 66. Roberts S, Harmon TS, Schaal JL, Miao V, Li K, Hunt A, et al. Injectable tissue integrating networks from recombinant polypeptides with tunable order. Nature Materials. 2018; 17(12):1154–63. https:// doi.org/10.1038/s41563-018-0182-6 PMID: 30323334 67. Das S, Amin AN, Lin YH, Chan HS. Coarse-grained residue-based models of disordered protein con- densates: utility and limitations of simple charge pattern parameters. Physical Chemistry Chemical Physics. 2018; 20(45):28558–74. Epub 2018/11/07. https://doi.org/10.1039/c8cp05095c PMID: 68. Das S, Eisen A, Lin YH, Chan HS. A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation. Journal of Physical Chemistry B. 2018; 122(21):5418–31. Epub 2018/02/06. https://doi.org/10.1021/acs.jpcb.7b11723 PMID: 29397728. 69. Theodorou DN. Equilibration and Coarse-Graining Methods for Polymers. In: Ferrario M, Ciccotti G, Binder K, editors. Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2. Berlin, Heidelberg: Springer Berlin Heidelberg; 2006. p. 419–48. 70. Delaney KT, Fredrickson GH. Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions. Journal of Physical Chemistry B. 2016; 120(31):7615–34. Epub 2016/ 07/15. https://doi.org/10.1021/acs.jpcb.6b05704 PMID: 27414265. 71. Duchs D, Delaney KT, Fredrickson GH. A multi-species exchange model for fully fluctuating polymer field theory simulations. The Journal of Chemical Physics. 2014; 141(17):174103. Epub 2014/11/10. https://doi.org/10.1063/1.4900574 PMID: 25381498. 72. Izvekov S, Voth GA. A multiscale coarse-graining method for biomolecular systems. Journal of Physi- cal Chemistry B. 2005; 109(7):2469–73. Epub 2006/07/21. https://doi.org/10.1021/jp044629q PMID: 73. Izvekov S, Swanson JM, Voth GA. Coarse-graining in interaction space: a systematic approach for replacing long-range electrostatics with short-range potentials. Journal of Physical Chemistry B. 2008; 112(15):4711–24. Epub 2008/03/28. https://doi.org/10.1021/jp710339n PMID: 18366209. 74. Liu P, Shi Q, Daume H 3rd, Voth GA. A Bayesian statistics approach to multiscale coarse graining. The Journal of Chemical Physics. 2008; 129(21):214114. Epub 2008/12/10. https://doi.org/10.1063/1. 3033218 PMID: 19063551. 75. Dama JF, Sinitskiy AV, McCullagh M, Weare J, Roux B, Dinner AR, et al. The Theory of Ultra-Coarse- Graining. 1. General Principles. Journal of chemical theory and computation. 2013; 9(5):2466–80. Epub 2013/05/14. https://doi.org/10.1021/ct4000444 PMID: 26583735. 76. Lu L, Dama JF, Voth GA. Fitting coarse-grained distribution functions through an iterative force-match- ing method. The Journal of Chemical Physics. 2013; 139(12):121906. Epub 2013/10/05. https://doi. org/10.1063/1.4811667 PMID: 24089718. 77. Han Y, Jin J, Wagner JW, Voth GA. Quantum theory of multiscale coarse-graining. The Journal of Chemical Physics. 2018; 148(10):102335. Epub 2018/03/17. https://doi.org/10.1063/1.5010270 PMID: 29544317. 78. Ruff K, M., Harmon TS, Pappu RV. CAMELOT: A machine learning approach for coarse-grained simu- lations of aggregation of block-copolymeric protein sequences. The Journal of Chemical Physics. 2015; 143(24):243123. https://doi.org/10.1063/1.4935066 PMID: 26723608 79. Fei J, Jadaliha M, Harmon TS, Li IT, Hua B, Hao Q, et al. Quantitative analysis of multilayer organiza- tion of proteins and RNA in nuclear speckles at super resolution. Journal of Cell Science. 2017; 130 (24):4180–92. https://doi.org/10.1242/jcs.206854 PMID: 29133588 80. Harmon TS, Holehouse AS, Pappu RV. To Mix, or To Demix, That Is the Question. Biophysical Jour- nal. 2017; 112(4):565–7. https://doi.org/10.1016/j.bpj.2016.12.031 PMID: 28256216 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 37 / 39 Simulations of phase transitions of multivalent proteins 81. Freeman Rosenzweig ES, Xu B, Kuhn Cuellar L, Martinez-Sanchez A, Schaffer M, Strauss M, et al. The Eukaryotic CO2-Concentrating Organelle Is Liquid-like and Exhibits Dynamic Reorganization. Cell. 2017; 171(1):148–62.e19. https://doi.org/10.1016/j.cell.2017.08.008 PMID: 28938114 82. Bolhuis P, Frenkel D. Numerical study of the phase diagram of a mixture of spherical and rodlike col- loids. The Journal of Chemical Physics. 1994; 101(11):9869–75. https://doi.org/10.1063/1.467953 83. Shaffer JS. Effects of chain topology on polymer dynamics: Bulk melts. The Journal of Chemical Phys- ics. 1994; 101(5):4205–13. https://doi.org/10.1063/1.467470 84. Carmesin I, Kremer K. The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules. 1988; 21(9):2819–23. https://doi.org/10.1021/ ma00187a030 85. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics. 1953; 21(6):1087–92. https://doi.org/10. 1063/1.1699114 86. Siepmann JI, Frenkel D. Configurational bias Monte Carlo: a new sampling scheme for flexible chains. Molecular Physics. 1992; 75(1):59–70. https://doi.org/10.1080/00268979200100061 87. Rosenbluth MN, Rosenbluth AW. Monte Carlo Calculation of the Average Extension of Molecular Chains. The Journal of Chemical Physics. 1955; 23(2):356–9. https://doi.org/10.1063/1.1741967 88. Mańka A, Nowicki W, Nowicka G. Monte Carlo simulations of a polymer chain conformation. The effec- tiveness of local moves algorithms and estimation of entropy. Journal of Molecular Modeling. 2013; 19 (9):3659–70. https://doi.org/10.1007/s00894-013-1875-z PMID: 23765038 89. Qin Y, Liu H-L, Hu Y. Dynamic Monte Carlo Simulation of Polymers: Cooperative Move Algorithm. Molecular Simulation. 2003; 29(10–11):649–54. https://doi.org/10.1080/0892702031000103185 90. Gennes PGd. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. The Journal of Chem- ical Physics. 1971; 55(2):572–9. https://doi.org/10.1063/1.1675789 91. Sheng Y-J, Wang M-C. Statics and dynamics of a single polymer chain confined in a tube. The Journal of Chemical Physics. 2001; 114(10):4724–9. https://doi.org/10.1063/1.1345879 92. Binder K, Paul W. Monte Carlo simulations of polymer dynamics: Recent advances. Journal of Poly- mer Science Part B: Polymer Physics. 1997; 35(1):1–31. https://doi.org/10.1002/(sici)1099-0488 (19970115)35:1<1::aid-polb1>3.0.co;2-# 93. Widom B. Some Topics in the Theory of Fluids. The Journal of Chemical Physics. 1963; 39(11):2808– 12. https://doi.org/10.1063/1.1734110 94. Rottereau M, Gimel JC, Nicolai T, Durand D. Monte Carlo simulation of particle aggregation and gela- tion: I. Growth, structure and size distribution of the clusters. The European Physical Journal E. 2004; 15(2):133–40. https://doi.org/10.1140/epje/i2004-10044-x PMID: 15517458 95. Mikes J, Dusek K. Simulation of polymer network formation by the Monte Carlo method. Macromole- cules. 1982; 15(1):93–9. https://doi.org/10.1021/ma00229a018 96. Koyama T, Tanaka H. Volume-shrinking kinetics of transient gels as a consequence of dynamic inter- play between phase separation and mechanical relaxation. Physical Review E. 2018; 98(6):062617. https://doi.org/10.1103/PhysRevE.98.062617 97. Shayegan M, Tahvildari R, Kisley L, Metera K, Michnick SW, Leslie SR. Probing inhomogeneous diffu- sion in the microenvironments of phase-separated polymers under confinement. bioRxiv. 2018:402230. https://doi.org/10.1101/402230 98. Rubinstein M, Semenov AN. Thermoreversible Gelation in Solutions of Associating Polymers. 2. Lin- ear Dynamics. Macromolecules. 1998; 31(4):1386–97. https://doi.org/10.1021/ma970617+. 99. Kopelman R. Fractal Reaction Kinetics. Science. 1988; 241(4873):1620–6. https://doi.org/10.1126/ science.241.4873.1620 PMID: 17820893 100. Bracha D, Walls MT, Wei M-T, Zhu L, Kurian M, Avalos JL, et al. Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo-Oligomerizable Seeds. Cell. 2018; 175(6):1467–80.e13. https://doi.org/10.1016/j.cell.2018.10.048 PMID: 30500534 101. Sirri V, Urcuqui-Inchima S, Roussel P, Hernandez-Verdun D. Nucleolus: the fascinating nuclear body. Histochemistry and Cell Biology. 2008; 129(1):13–31. https://doi.org/10.1007/s00418-007-0359-6 PMID: 18046571 102. Mitrea DM, Grace CR, Buljan M, Yun M-K, Pytel NJ, Satumba J, et al. Structural polymorphism in the N-terminal oligomerization domain of NPM1. Proceedings of the National Academy of Sciences. 2014; 111(12):4466–71. https://doi.org/10.1073/pnas.1321007111 PMID: 24616519 103. Banerjee PR, Milin AN, Moosa MM, Onuchic PL, Deniz AA. Reentrant Phase Transition Drives Dynamic Substructure Formation in Ribonucleoprotein Droplets. Angewandte Chemie International Edition. 2017; 56(38):11354–9. https://doi.org/10.1002/anie.201703191 PMID: 28556382 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 38 / 39 Simulations of phase transitions of multivalent proteins 104. Case LB, Zhang X, Ditlev JA, Rosen MK. Stoichiometry controls activity of phase-separated clusters of actin signaling proteins. Science. 2019; 363(6431):1093–7. https://doi.org/10.1126/science. aau6313 PMID: 30846599 105. Shin Y, Berry J, Pannucci N, Haataja MP, Toettcher JE, Brangwynne CP. Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets. Cell. 2017; 168(1):159–71.e14. https://doi.org/10.1016/j.cell.2016.11.054 PMID: 28041848 106. Khan T, Kandola TS, Wu J, Venkatesan S, Ketter E, Lange JJ, et al. Quantifying Nucleation In Vivo Reveals the Physical Basis of Prion-like Phase Behavior. Molecular Cell. 2018; 71(1):155–68.e7. https://doi.org/10.1016/j.molcel.2018.06.016 PMID: 29979963 107. Wyman J, Gill SJ. Ligand-linked phase changes in a biological system: applications to sickle cell hemoglobin. Proceedings of the National Academy of Sciences of the United States of America. 1980; 77(9):5239–42. Epub 1980/09/01. https://doi.org/10.1073/pnas.77.9.5239 PMID: 6933555; PubMed Central PMCID: PMCPmc350033. 108. Subramanian G, Shanbhag S. On the relationship between two popular lattice models for polymer melts. The Journal of Chemical Physics. 2008; 129(14):144904. https://doi.org/10.1063/1.2992047 PMID: 19045165 109. Kremer K, Binder K. Monte Carlo simulation of lattice models for macromolecules. Computer Physics Reports. 1988; 7(6):259–310. https://doi.org/10.1016/0167-7977(88)90015-9. 110. Resasco DC, Gao F, Morgan F, Novak IL, Schaff JC, Slepchenko BM. Virtual Cell: computational tools for modeling in cell biology. Wiley Interdisciplinary Reviews: Systems Biology and Medicine. 2012; 4(2):129–40. https://doi.org/10.1002/wsbm.165 PMID: 22139996 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007028 October 21, 2019 39 / 39

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LASSI: A lattice model for simulating phase transitions of multivalent proteins

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LASSI: A lattice model for simulating phase transitions of multivalent proteins

LASSI: A lattice model for simulating phase transitions of multivalent proteins

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