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Long-range correlation in protein dynamics: Confirmation by structural data and normal mode analysis

Long-range correlation in protein dynamics: Confirmation by structural data and normal mode analysis a1111111111 Proteins in cellular environments are highly susceptible. Local perturbations to any residue a1111111111 can be sensed by other spatially distal residues in the protein molecule, showing long-range correlations in the native dynamics of proteins. The long-range correlations of proteins contribute to many biological processes such as allostery, catalysis, and transportation. Revealing the structural origin of such long-range correlations is of great significance in OPENACCESS understanding the design principle of biologically functional proteins. In this work, based on Citation: Tang Q-Y, Kaneko K (2020) Long-range a large set of globular proteins determined by X-ray crystallography, by conducting normal correlation in protein dynamics: Confirmation by structural data and normal mode analysis. PLoS mode analysis with the elastic network models, we demonstrate that such long-range corre- Comput Biol 16(2): e1007670. https://doi.org/ lations are encoded in the native topology of the proteins. To understand how native topol- 10.1371/journal.pcbi.1007670 ogy defines the structure and the dynamics of the proteins, we conduct scaling analysis on Editor: Bert L. de Groot, Max Planck Institute for the size dependence of the slowest vibration mode, average path length, and modularity. Biophysical Chemistry, GERMANY Our results quantitatively describe how native proteins balance between order and disorder, Received: October 12, 2019 showing both dense packing and fractal topology. It is suggested that the balance between Accepted: January 21, 2020 stability and flexibility acts as an evolutionary constraint for proteins at different sizes. Over- all, our result not only gives a new perspective bridging the protein structure and its dynam- Published: February 13, 2020 ics but also reveals a universal principle in the evolution of proteins at all different sizes. Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The Author summary editorial history of this article is available here: https://doi.org/10.1371/journal.pcbi.1007670 The long-range correlated fluctuations are closely related to many biological processes of the proteins, such as catalysis, ligand binding, biomolecular recognition, and transporta- Copyright:© 2020 Tang, Kaneko. This is an open access article distributed under the terms of the tion. In this paper, we elucidate the structural origin of the long-range correlation and Creative Commons Attribution License, which describe how native contact topology defines the slow-mode dynamics of the native pro- permits unrestricted use, distribution, and teins. Our result suggests an evolutionary constraint for proteins at different sizes, which reproduction in any medium, provided the original may shed light on solving many biophysical problems such as structure prediction, multi- author and source are credited. scale molecular simulations, and the design of molecular machines. Moreover, in statisti- Data Availability Statement: All the protein cal physics, as the long-range correlations are notable signs of the critical point, unveiling structures used in this research are available from the origin of such criticality can extend our understanding of the organizing principle of a the Protein Data Bank (PDB). Related PDB-ID, large variety of complex systems. code, and the data that related to this study are provided as Supporting File. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 1 / 17 Long-range correlation in protein dynamics Funding: This research was partially supported by Introduction a Grant-in-Aid for Scientific Research (S) Proteins, including the globular, fibrous, membrane and intrinsically disordered proteins, are (15H05746) from the Japanese Society for the responsible for diverse functions in almost every process of cellular life. Globular proteins, as Promotion of Science (JSPS) and Grant-in-Aid for Scientific Research on Innovative Areas the majority type of the proteins in nature, can fold from disordered peptide chains into spe- (17H06386) from the Ministry of Education, cific three-dimensional (3D) structures on minimal-frustrated energy landscape [1–4]. Such Culture, Sports, Science and Technology (MEXT) kind of 3D structures, which are encoded by the amino acid sequences, are known as native of Japan. The funders had no role in study design, states. It is worth noting that the native state of a protein is not static, but exhibits dynamical data collection and analysis, decision to publish, or fluctuations around the energy minimum. Experiments and molecular simulations have preparation of the manuscript. shown that thermal fluctuations trigger the motions of proteins such as domain movements Competing interests: The authors have declared and allosteric transitions, which enable the biological functions of proteins such as catalysis that no competing interests exist. [5], ligand binding [6, 7], biomolecular recognition [8], and transportation [9]. Uncovering the relations between the structure and the function of proteins is a fundamental question in molecular biophysics. To answer it, the fluctuations at the native states may provide a key. One of the most fascinating properties of proteins is the long-range correlated fluctuations around the native states [10–12]. Thanks to the long-range correlations, local perturbations to any residue can be sensed by every other residue of the entire protein, even when the two sites are spatially distant. Such a property plays an important role in the functionality of the pro- teins. For example, for allosteric proteins, long-range correlations warrant the binding at one site can be transmitted to other functional sites [13, 14], and enable the high susceptibility for proteins in cellular environments. Based on the correlation analysis of structural ensembles determined by solution nuclear magnetic resonance (NMR), it was already demonstrated that the native proteins exhibit long-range correlations and high susceptibility in the native dynamics [15]. Such a phenomenon is also in line with other theoretical and experimental results, for example, the long-range conformational forces related to the hydrophobicity scales of the proteins [16–20], the fractal dimension in the oscillation spectrum [21] and configura- tion space [22], the slow relaxation of protein molecules in the solution [23, 24], the volume fluctuation of allosteric proteins [25], and the overlap between the low-frequency collective oscillation modes and large-scale conformational changes in allosteric transitions [26–30]. Accumulating evidence indicates that native proteins are not only stable enough to warrant structural robustness, but also susceptible enough to sense the signals in the milieu, and ready to perform large-scale conformational changes. However, the origin of such kind of dynamics is still unclear. In the present paper, we concentrate on the structure and the equilibrium fluctuation dynamics of a large set of globular proteins determined by X-ray crystallography, ranging from a single hairpin structure to large protein assemblies. Firstly, to elucidate the connection between the long-range correlations and protein structures, we conduct correlation analysis based on the elastic network models (ENMs) [26–30]. We find that the long-range correlations and the scaling laws can be robustly reproduced by the ENMs with different model parameters. Such a result indicates that the long-range correlations are encoded in the native topology of the proteins. Secondly, we conduct normal mode analysis [31–33] for protein molecules, ideal polymer chains, and lattice systems. A similar scaling relation holds for polymers, lattices, and proteins, but the scaling coefficients are different. Such a result shows how native proteins bal- ance between order and disorder, which resemble the physical systems near the critical point of a phase transition. Thirdly, we introduce the average path length and modularity to describe the topological characteristics of the proteins. Scaling relations are also observed between these topological descriptors and the size of the proteins. According to the result of the scaling analy- sis, we conclude that native proteins show both dense packing and fractal topology. Lastly, we focus on the size dependence of proteins’ shape. With a given chain length, the shape of a PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 2 / 17 Long-range correlation in protein dynamics protein is not random, but a most-probable shape factor always exists. Such a constraint sug- gests that native proteins balance between stability and functionality. Overall, our result not only gives a new perspective bridging the protein structure and its dynamics but also reveals a universal principle in the evolution of proteins at all different sizes. Results The critical dynamics of proteins are robustly encoded in the native structures In previous studies, based on the structural ensembles determined by solution nuclear mag- netic resonance (NMR), it was observed that the native proteins in the solution exhibit long- range correlations and high susceptibility in the dynamics [15]. The native fluctuation of pro- teins behaves as though they are near the critical point of a phase transition [34–36]. The ques- tion arises whether the critical dynamics of native proteins are encoded in the native structure or driven by other factors in the milieu. To answer this question, we employ the minimal model of proteins, the elastic network model (ENM) to conduct our analysis. In an ENM, a protein molecule is described as a set of nodes (represented by their C atoms) connected with edges of elastic springs. As shown in Fig 1A, the 3D structure of a pro- tein can be simplified as a network based on the topology of residue contacts. Note that the elastic networks are constructed only based on the spatial distances between residues. If an ENM can successfully reproduce long-range correlations in the fluctuations of the native pro- teins, then it can be concluded that the critical dynamics of proteins is encoded by the local contacts in the native structures. The correlated motions of residues can be represented by a covariance matrix, in which matrix element C ¼ hD~r � D~r i. For simplification, we conduct our analysis based on the ij i j Gaussian network model (GNM) [37, 38]. In GNM, the covariance matrix C is proportional to 3k T pseudoinverse of the Kirchhoff matrixΓ, i.e., C ¼ �½G � [26, 37]. Normalizing the ij k ij pffiffiffiffiffiffiffiffiffiffi covariance matrix, a pairwise cross correlation � ¼ C = C C an be obtained. Similar to ij ij ii jj previous works [15, 39, 40], a distance-dependent correlation function ϕ(r) can be defined by � dðr r Þ ij ij i<j averaging the correlations for residue pairs at mutual distance r, and �ðrÞ ¼ , dðr r Þ ij i<j where r denote the spatial distance between residue i and j, andδ(x) is the Dirac-delta func- ij tion selecting residue pairs at mutual distance r. Here, the correlation length ξ as the distance where ϕ(r) first decays to zero. To examine whether the correlation scales with the protein size, we sample over the protein data across different sizes. By averaging the distance-dependent correlation function ϕ(r) for a subset of proteins, we can define the averaged correlation functionhϕ(r)i to a group of pro- teins. Here, we divide the dataset into subsets according to the radius of gyration R of the pro- teins (e.g., subset {R * 12Å} contains proteins at size 11.5Å� R < 12.5Å), the distance- g g dependent correlation functions ϕ(r) for proteins at different sizes are calculated. As shown in Fig 1B, the correlation function first decreases from its maximum at short distances, crosses zero at r = ξ, continues to decline, reaches a negative minimum. As a notable sign of criticality, for proteins of different sizes, the correlation length ξ is proportional to their radius of gyration R . Therefore, the correlation functions can be scaled by the size (R ) of the proteins, and all g g the correlation functions collapse (Fig 1C). This result indicates that correlations in the native fluctuation of proteins are scale-free: No matter how large the protein molecule is, correlation length can extend to the size of the entire system. Such long-range correlation contributes to the functionality of a large variety of proteins, for example, for allosteric proteins, the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 3 / 17 Long-range correlation in protein dynamics Fig 1. The critical dynamics of proteins are robustly encoded in the native structure. (A) An illustration of the elastic network model (r = 9Å) of the protein CI2 (PDB code: 2CI2). The beads denote the residues, and the bonds denote the elastic springs in the model. (B) The correlation functions ϕ(r) for proteins at different sizes predicted by GNM with cutoff distance r = 9Å. (C) Correlation functions scaled by the radius of gyration of the proteins R . (D) For proteins of similar sizes (19.5Å� R < 20.5Å), with g g different cutoff distances r , the correlation functions ϕ(r) predicted by GNM. (E) With different cutoff distances, for proteins of different sizes, the correlation length ξ is always proportional to the size of the protein R . (F) The susceptibilityχ vs. chain length N αγ/ν shows the power-law relation:χ * N , and the scaling coefficientαγ/ν� 1 can be kept with different r (inset). https://doi.org/10.1371/journal.pcbi.1007670.g001 long-range correlation warrants the binding at one site can be transmitted to other functional sites [13, 14], even when the two sites are spatially distant. To validate the previous analysis, let us consider the parameter sensitivity in the prediction of the cross correlations in protein dynamics. The only free parameter in GNM is the cutoff distance r . With different r , the correlation would have different magnitude at short dis- C C tances; however, as shown in Fig 1D, the correlation lengths ξ keep as a constant for different cutoff distances r . As shown in Fig 1E, for cutoff distances ranging from 6 Å to 15 Å, the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 4 / 17 Long-range correlation in protein dynamics correlation length ξ is always proportional to the radius of gyration R , showing that the critical dynamics of native proteins is generally a stable property and insensitive to the selection of cut- off distances. With only short-range interactions between residues taken into account, GNM can successfully capture the long-range correlations in the native dynamics of the proteins. To have a further investigation of the criticality, it is necessary to validate the scaling rela- tions in the dynamics of proteins. Here, for illustration, we take the power-law relation between the susceptibilityχ and chain length N as an example. For protein systems, a finite- size version of susceptibilityχ is introduced to quantify the response of systems under pertur- bation [15]. It is defined as the total correlation in a unit volume within the correlation length: w ¼ � � yðx r Þ, where s denotes the shape factor of protein, and θ(x) denotes the i<j ij ij Heaviside function. Previously, based on NMR-determined protein ensembles [15], it was αγ/ν observed thatχ * N , with the scaling coefficientαγ/ν� 1 (Definitions ofα,γ and ν are listed in S1 Appendix). Here, as shown in Fig 1F, by employing the GNM, similar scaling rela- tions can also be observed. Such a result demonstrates that, no matter how large the molecule is, proteins can always have high sensitivity executing its function because the magnitude of the susceptibility grows with the chain length of the proteins. Besides, the scaling coefficients are insensitive to changes in cutoff distances (inset), demonstrating that the scale-free correla- tion of native proteins is a robust property. Our correlation analysis and scaling analysis methods can also be extended to other ver- sions of elastic network models. For example, with harmonic C potential model (HCA) [41, 42], similar scaling coefficients can also be observed (see S1 Appendix). However, some mod- els cannot correctly reproduce the scaling relations betweenχ and N, for instance, the parame- ter-free GNM (pfGNM) [43]. In fact, pfGNM fails to predict all the scaling relations in the proteins (see S1 Appendix). Previous researches already found that pfGNM can only be applied for proteins in crystalline conditions, and it will have a poor agreement to the collective motions given by molecular dynamics [42]. Such a result indicates that the scaling coefficient may help us to probe whether the protein is solvated or in a crystalline condition. The size dependence of slowest modes reveals criticality of native proteins Normal mode analysis is a practical tool to elucidate the global dynamics [31–33] and the evo- lutionary constraints [44, 45] of the proteins. Physically, the slow modes, or say, the low-fre- quency modes of a system are related to the motions with low excitation energy, long wavelengths (long-range correlation), long time scale (at the order from microseconds to sec- onds) and the large amplitude motions. Usually, the motions that correspond to the slow modes (especially the slowest nonzero mode) can have significant overlap with large displace- ment during the functional motions [46]. These functional motions usually engage relative movements of large subunits in the proteins or cooperative conformational changes of the whole proteins. Previously, the unique spectral properties of the residue contact networks have been noticed [47, 48], but the detailed differences have never been examined. To demonstrate the particularity in the spectrum of proteins, we compare the proteins with ideal polymer chains (detailed information listed in S1 Appendix) and lattice systems. Our analysis focuses on the size dependence of the slow modes. As shown in Fig 2A, for all these systems, the slowest few modes versus the system size N follow power-law distributions. Among these slow modes, we specifically focus on the eigenvalueλ which corresponds to the −z slowest nonzero mode. A similar power-lawλ * N holds for ideal polymers, lattices, and proteins. However, the scaling coefficients z are different in these systems. As shown in Fig 2A, for ideal polymer chains, the scaling coefficient z� 1.674. For face-centered cubic (fcc) lattice, by conducting normal mode analysis where atoms are connected by springs with their nearest PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 5 / 17 Long-range correlation in protein dynamics Fig 2. The slow modes of proteins are robustly defined by native structure. (A) The 1st, 2nd and the 3rd non-zero eigenvaluesλ ,λ , andλ vs. the chain length N of the proteins follows a power-law distribution. (Cutoff distance r = 1 2 3 C 9Å, and the scaling coefficients ofλ (N),λ (N), andλ (N) are 1.074, 0.900, and 0.868, respectively). As comparison, 1 2 3 similar scaling relations in lattices and ideal polymer chains are also illustrated, and the scaling coefficients are 0.728 (lattices) and 1.674 (polymer). (B) The eigenvalue of the slowest nonzero modeλ versus chain length N shows the −z scaling relation:λ * N , and the inset shows scaling coefficient z vs. the cutoff distance r . (C) For proteins at similar 1 C sizes (chain length 180� N < 220), the histogram for the eigenvalue distribution g(λ). https://doi.org/10.1371/journal.pcbi.1007670.g002 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 6 / 17 Long-range correlation in protein dynamics neighbors and 2nd nearest neighbors), we have z� 0.727. Theoretically, for lattice systems, the maximum wavelength l corresponds to the slowest elastic mode, and l is proportional to w w 1/3 the characteristic length of the system. Since the maximum wavelength l * N , one can esti- 2 2 2=3 mate that l � o � l � N , which is close to 0.727. In contrast to ideal polymers and lat- 1 1 w tices, z� 1 holds for protein molecules. The scaling relations in the slowest modes of proteins are robust to the variation in model parameters. As shown in Fig 2B, the selection of cutoff distances r would not affect the scaling coefficient z. But the robustness of the scaling coefficient cannot be attributed to that of the eigenvalue distribution. As shown in Fig 2C, selecting different r would influence the mode distribution g(λ) of native proteins. The mode distribution g(λ), especially the low-frequency part, can be enhanced by selecting a short cutoff distance r . Such a result is also consistent with previous theoretical analysis on protein elastic network and ranges of cooperativity [43], which states that with a shorter interaction range, the predicted dynamics would be more cooperative and show better overlap with the displacement in large-scale conformational changes. It is worth noting that the scaling coefficients in the size dependence of the slowest mode demonstrate that the structure of proteins stands between lattices and ideal polymer chains. For proteins, the exponent z� 1, above what is obtained from lattices (z� 0.727), and below what is obtained from polymer chains (z� 1.674). Thus, compared with ideal polymer chains, the proteins have higher structural stability, whereas compared with lattices, the proteins have higher flexibility and exhibit slower vibrations. Native proteins stand between lattices and polymers, acting as the “critical point” that separates the ordered and disordered phase. Not only are native proteins stable enough to ensure structural robustness and functional specific- ity, but also susceptible enough to sense the signals in the environment, and ready to perform large-scale conformational changes. Interestingly, staying at the critical point seems to be a common organizing principle of a large variety of biological systems [49–55]: If the system is too disordered, the system cannot stably exist; if it is too ordered, it cannot adapt or respond to perturbations from the environments. Our result of scaling analysis provides additional evi- dence to support the criticality hypothesis. Protein structure: Dense packing with fractal topology In previous sections, we demonstrated that the critical dynamics of the proteins are encoded in their native structures, and we showed that the equilibrium dynamics of protein molecules if different from lattices and polymers. How does the topology of the residue contact network encode such kind of dynamics? To answer the question, in this subsection, we will try to bridge the vibration spectrum with the architecture of the protein by mainly focusing on the issue of the network topology. In the network analysis, the average path lengthhli is one of the most important topological descriptors quantifying the total connectivity among the nodes. Here, we first focus on the scaling relations between average path lengthhli and the system size N. As shown in Fig 3A, for proteins at different sizes, there is a power-law relation between the average path lengthhli and the chain length N:hli*N , andα� 0.338, which is close to 1/3. In the calculation, the cutoff distance r is set to be 8Å. Even different cutoff distance r will lead to differenthli, but C C the scaling exponent is invariant (see S1 Appendix). The scaling relation in proteins is very similar to what in the lattice structures. Theoretically, for 3D lattices, the exponent would beα = 1/3. Such a scaling relation is confirmed in Fig 3A. While for ideal polymer chains, with an extended structure, there would be longer average path lengths, and fitting givesα� 0.675. Such a result demonstrates that the residue contact networks show similar dense packing PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 7 / 17 Long-range correlation in protein dynamics Fig 3. The protein dynamics can be quantified by topological descriptors of the residue contact network. (A) For the contact network of proteins (r = 8Å), fcc lattices and ideal polymers, the average path lengthhli vs. system size N. (B) Similarly for proteins, fcc lattice and ideal polymers, modulaity Q vs. system size N. The inset shows the log-log plot of 1 − Q vs. N. (C) For proteins at similar sizes (180� N < 220), the scattering plot (yellow dots, each dot represents a protein molecule), the binned average (red dots) and the basic trend (red curve) of the average path lengthhli vs. Q, and (D) Smallest non-zero eigenvalueλ vs. Q. https://doi.org/10.1371/journal.pcbi.1007670.g003 property as regular lattices. Both lattice and protein networks have much shorter path lengthhli than ideal polymers. Although protein and lattice share similar dense packing properties, the residue contact networks of proteins still exhibit unique properties. To demonstrate the difference between the residue contact network and the lattice networks, another measure—modularity Q is intro- duced into the study [56, 57]. Intuitively, a network that can be more easily divided into mod- ules would have a higher Q value. Modularity Q also scales as the system size increases. For a d−dimensional cubic lattice network with N nodes, theoretically, it was proved that the modu- −η larity Q versus N follows the relation: Q = 1 − K� N , where the scaling coefficient Z ¼ , dþ1 and K is a constant that depend on average degree z and dimension d [58]. For ideal polymer chains, the fitting givesη� 0.465, indicating an effective fractal dimension d � 1.15, which eff is much lower than 3. For a 3D cubic lattice, theoretically,η = 1/4. For fcc lattices, as shown in Fig 3B, fitting givesη� 0.231 < 1/4, indicating d � 3.33 > 3, that is because, in the fcc lat- eff tices, every atom has more neighbors than cubic lattice. For proteins our dataset, when taking r = 8Å, similar power law can also be observed, but the scaling coefficientη = 0.279 > 1/4. Such an exponent indicate that the proteins has an effective dimension d ¼ 1 � 2:58, eff which is lower than 3. Such a scaling coefficient displays that the residue contact networks have a fractal topology, and the fractal dimension is below 3. It is worth noting that, in this work, the fractal dimension of proteins is obtained by the scaling analysis for proteins at different sizes. The effective dimension obtained here is consistent with the fractal dimension (d� 2.7) of proteins determined by structural analysis methods (see S1 Appendix). The scaling PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 8 / 17 Long-range correlation in protein dynamics analysis of average path length reveals that the proteins have similar dense packing properties as ordered lattices, but the scaling analysis of modularity suggests that proteins exhibit fractal structures, which is similar to disordered polymer structures. In short, topological analysis demonstrates again that native of proteins balance between order and disorder. In the discussions above, by averaging the topological descriptors of proteins at similar sizes, we analyze the size dependence of topological properties. In fact, for proteins at similar sizes, topological descriptors can also play an important role in capturing the main features in the dynamics of the proteins. To illustrate that, here, we select the protein molecules with chain length 180� N < 220 from our dataset. Although these proteins have similar chain length, the structure may differ a lot. Our discussion centers around modularity Q. When the modularity Q of a protein increases, as shown in Fig 3C, the average path lengthhli also increases. This is because, in a highly modularized network, there will be few connections between different communities, on the average, it will take more steps from one node to another. As shown in Fig 3D, as the modularity Q increases, the smallest non-zero eigenvalue λ decreases, in line with the common knowledge that that modularized structures in the pro- teins contribute to slow-mode motions. Such a result is consistent with the theory of spectral graph theory. Indeed, the spectrum of the graph Laplacian is closely related to the community structures of the network [59]. Our analysis quantitatively demonstrates that modularized structures contribute to the large-scale motions and slow relaxations of the proteins. Stability-functionality constraint: The size dependence of proteins’ shape The intrinsic dynamics of proteins is encoded in their structures. Since scaling relation between the dynamics and the size of the protein is already discussed in the previous sections. We focus on the relationship between the structure and the size of the protein in this section. The shape factor s can be introduced to describe the general architecture of a protein mole- cule [15]. According to the definition, the shape factor can be understood as the residue pack- ing density within the inertia ellipsoid. When residues are tightly packed with a globular shape, the shape factor s would be large. When disordered loops or flexible linkers are connect- ing multiple domains, the shape of the molecule deviates from an ellipsoid, then s would be small. Here, for illustration, three proteins with a similar chain length 180� N < 220 but with different shape factor s are shown in Fig 4A. On the left, the receptor-binding domain of the short tail fiber (STF) is illustrated. Such a molecule has hardly any regular secondary structures likeα−helices orβ-strands [60]. The structure of such a molecule in its monomer state has a small shape factor and high modularity. To perform its functions, a knitted trimeric assembly has to be formed [60]. In the middle, there is the human molecular chaperone heat-shock pro- tein 90 (Hsp90) [61] with medium shape factor and modularity. On the right, a de novo designed helical repeat protein DHR10 is illustrated. By repeating a simple helix–loop–helix– loop structural motif, DHR10 protein is highly ordered and becomes very stable, which can stay folded even at 95˚C [62]. Generally, the proteins with larger shape factors show higher sta- bility, and the proteins with smaller shape factors show higher flexibility. Although the definition of shape factor does not introduce any detailed information on sec- ondary structures or residue contacts, the shape factor is closely related to the topological descriptors of the residue contact network. Here, statistics for the proteins with similar chain length (180� N < 220) is conducted. The scattering plot of shape factor s versus modularity Q is shown in Fig 4B. A trend line (in red) displays that as modularity Q increases, the shape fac- tor s decreases. The result is easy to understand intuitively, a protein molecule in a shape that deviates from an ellipsoid is likely to have multiple domains or have flexible linkers connecting multiple ordered regions. Interestingly, although the proteins could have very different shapes, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 9 / 17 Long-range correlation in protein dynamics Fig 4. The shape factor correlates with the chain lengths of the proteins. (A) Three proteins with similar chain lengths: (Left) The receptor-binding domain of T4 STF (PDB: 1OCY, s = 0.84, Q = 0.74); (Middle) Human Hsp90 protein (PDB: 3T0H, s = 1.77, Q = 0.65); and (Right) The DHR10 protein (PDB: 5CWG, s = 2.37, Q = 0.63). (B) For proteins at similar sizes (chain length 180� N < 220), the scattering plot (yellow dots), binned average (red dots) and the trend line (red line) of shape factor s vs. modularity Q are plotted. Besides, there are histograms of the shape factor s (right vertical) and modularity Q (top horizontal). (C) For all the proteins in our dataset, the 2D histogram (in the background) of s vs. N and the plot (in navy blue) of the most-probable shape factor s vs. chain length N. https://doi.org/10.1371/journal.pcbi.1007670.g004 for protein molecules with a specific chain length, the value of shape factor does not vary a lot. Here, in Fig 4B, histograms of the shape factor s (right vertical) and modularity Q (top hori- zontal) are plotted. The histograms show that there exists a most-probable shape factor s and � � corresponding modularity Q . Most natural proteins have shape factors close to s , exhibit a balancing behavior between stability and flexibility [21]. In fact, for proteins with different chain lengths, the most-probable shape factor s always exists, which can be recognized as a constraint in the shape of the protein. As shown in Fig 4C, it was observed that larger proteins prefer smaller shape factors. A similar relation is also observed based on NMR-determined ensembles [15]. These observations provide additional pieces of evidence to support the criticality of native proteins. The native proteins have to bal- ance between stability and flexibility. With short chain lengths, the proteins tend to have a larger shape factor to ensure a stable folded state. Accordingly, small proteins usually have higher residue packing density. However, as the chain length of the proteins increases, to exe- cute functional motions, flexibility becomes the main demand of the proteins. One good example is the designed protein DHR10 as illustrated in Fig 4A. DHR10 has high structural stability, but it is hard for such a protein to execute any biological functions. In such a situa- tion, smaller shape factors, which usually correspond with disordered loops or multi-domain structures, are demanded by the functionality. Our results suggest that the balance between stability and flexibility acts as an evolutionary constraint for proteins at different sizes. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 10 / 17 Long-range correlation in protein dynamics Discussion The long-range correlated fluctuations contribute to many biological processes of the proteins, such as allostery, catalysis, and transportation. To understand the origin of such long-range correlations, based on the elastic network model, we conduct normal mode analysis for a large dataset of globular proteins determined by X-ray crystallography. First, we predict the correlated motions for proteins at different sizes. It is observed that the correlation length of a protein can extend to the size of the whole protein, no matter how large the protein molecule is. Moreover, with different model parameters, the scale-free correlations and the scaling laws can be reproduced by the elastic networks model, which is the minimal structure-based model of native proteins. Such a result indicates that the critical dynamics characterized by the power-law relations are robustly encoded in the native topology of the proteins. Second, for proteins at different sizes, we conduct normal mode analysis and perform scal- ing analysis for the slow vibration modes of the proteins. To demonstrate the particularity in the spectrum of proteins, we compare the proteins with ideal polymer chains and lattice sys- tems. Native proteins stand between ordered lattices and disordered polymers, acting as the “critical point” that separates the ordered and disordered phase. Our result of scaling analysis provides additional evidence to support the criticality hypothesis. Third, to understand how the native topology determines the architecture and the dynam- ics of the proteins, we conduct scaling analysis for the topological descriptors and the size of the proteins. Our results demonstrate that, although proteins have similar average path length with lattice structures, the residue contact networks are more modularized. Last, we focus on the size dependence of proteins’ shape. For proteins with different chain lengths, the most-probable shape factors always exist. Larger proteins prefer smaller shape fac- tors. Such a constraint results from the balance between stability and functionality of proteins. In summary, our work quantitatively demonstrates how the native contact topology defines the long-range correlations and the slow dynamics of the native proteins. Our work not only provides quantitative scaling relations supporting the “structure-dynamics-function” para- digm but also reveals evolutionary constraints for proteins at different sizes. These results may shed light on a large variety of biophysical problems such as structure prediction, multi-scale molecular simulations, and the design of molecular machines. Materials and methods Dataset Our dataset contains 13081 proteins selected from the Protein Data Bank (PDB) [63]. The structures of these proteins are all determined by X-ray diffraction with high resolution (� 2.0Å). For every protein structure in the dataset, it contains no DNA, RNA or hybrid struc- tures; and the chain length 30� N� 1200. In our protein dataset, every two proteins share less than 30% sequence similarity. The PDB codes of all the proteins in our dataset are listed in the Supplementary Information (S1 and S2 Files). The elastic network models The elastic network models are widely applied to predict the functional dynamics of a variety of proteins and bio-machineries [26, 27, 29, 30]. With the assumption that all residue fluctua- tions are Gaussian variables distributed around their equilibrium coordinates, the Gaussian Network Model (GNM) can successfully reproduce the residue fluctuations as determined by experiments [37, 38]. For a protein consisting of of N residues, based on the native structure, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 11 / 17 Long-range correlation in protein dynamics the potential energy of the network is given by: V ¼ D~r � G � D~r ; ð1Þ GNM i ij j i;j¼1 in whichκ is a uniform force constant; D~r and D~r is the displacement of residue i and j, i j respectively; andΓ is the element of Kirchhoff matrix, or in a graph theory perspective, it is ij the graph Laplacian of the residue-residue contact network. The elements of matrixΓ is defined according to the contact topology of the native structure: for residue pair i − j, if r � ij r , thenΓ = −1; if r > r , thenΓ = 0; and for the diagonal elements,Γ = −∑ Γ = −k , C ij ij C ij ii j6¼i ij i where k denote the degree of node i. In GNM with homogenous contact strength, the only control parameter is the cutoff distance r . With a large r , residue pairs at long distances can C C interact with each other; while for smaller r , only short-range interactions are contributed to the elastic energy of the system. One may also introduce distance-dependent force constants [41–43] to refine the predictions of elastic network models. In these models, the force con- stantsκ becomes a function of the mutual distance between residue i and j. Further details ij and other variations of the elastic network models are listed in the S1 Appendix. Normal mode analysis and the spectrum of the graph laplacian Based on GNM, by diagonalizing the Kirchhoff matrixΓ, we can obtain all the eigenvalues and the corresponding eigenvectors describing the motions of every normal mode [32]. To com- pare the mode distribution for proteins of different chain lengths, the Kirchhoff (Laplacian) matrices correspond to the topology of native proteins are normalized. By normalizing all the diagonal elements as 1, we can obtain the symmetric normalized graph Laplacian [48]: 1=2 1=2 L ¼ D � G� D ; ð2Þ in which D is a matrix of all the diagonal elements of matrix D = diag[Γ ,Γ ,���Γ ], 1,1 2,2 N,N describing the local packing status of each residue. Diagonalizing matrix L, then we have L = UΛU , in which the eigenvaluesΛ = diag[λ ,λ ,λ ,���λ ] (λ � λ � λ ,� ���� λ ) 0 1 2 N−1 0 1 2 N−1 and eigenvectors U = [u , u , u ,��� u ] . The eigenvalueλ describes the frequency ω of the 0 1 2 N−1 i i i-th eigenmode (l � o ), and the eigenvector u describes the motion profile of the corre- i i sponding eigenmode. Note that the zero mode corresponds to the eigenvalueλ = 0, and eigenvector u describes the collective translational or rotational motions of the system. The code of normal mode analysis is listed in the Supplementary Code (S2 Appendix and S3 File). Shape factor To have a general description of the structure of a protein molecule, a dimensionless shape fac- tor s is defined [15]. By calculating the the moments of inertia of a protein molecule, one can Na estimate the residue packing density within the inertia ellipsoid as s ¼ , in which a = 3.8Å L L L 1 2 3 is the residue size, and L , L and L are lengths of the principal axes of the protein (L > L > 1 2 3 1 2 L ). The shape factors of the proteins in our dataset are listed in the Supplementary Data (S4 File). Average path length The average (or characteristic) path lengthhli usually works as a measure of the information transfer efficiency on a network. It is defined as the average number of steps along the shortest paths for all possible pairs of network nodes. When l denotes the shortest distance between i,j PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 12 / 17 Long-range correlation in protein dynamics node i and j, then, the average path length hli ¼ l : i;j ð3Þ NðN 1Þ i6¼j Modularity Modularity is a topological descriptor which is designed to quantify if a network can be easily divided into modules. For a network with N node and M edges, when the topology is described by the adjacency matrix A where A = 1 if and only if node i and j are connected. Modularity is ij defined as the fraction of the edges that fall within the given module minus the expected frac- tion when edges were distributed at random [56, 57]. According to the definition, one can k k i j introduce the modularity matrix B with elements B ¼ A to describe the expected num- ij ij 2M ber of edges between node pairs, in which k and k denote the degrees of node i and j, respec- i j tively. Based on matrix B, the modularity can be calculated as: ð4Þ Q ¼ Trð~x � B�~xÞ; 4M in which~x is the column vector describing the partition of a network. Vector x has elements x = ±1 indicating the modules to which the node belongs. The value of the Q lies in the range −1� Q� 1. For any given partition s of a network, one can calculate the Q corresponding to such a partition. The appropriate partition of a network would maximize the modularity Q [64]. In this work, we introduced the Louvain method [65] to partition the network and maxi- mize the value modularity Q. The code of topological analysis is listed in the Supplementary Code (S2 Appendix and S3 File). Supporting information S1 Appendix. Supplementary information. Detailed descriptions of the structural datasets involved in this research. Additional information concerning the scaling relations, generation of polymer structures, and other variations of elastic network models are also included in the Supplementary Information. (PDF) S2 Appendix. Supplementary code. The code (written in Python language) for PDB file pro- cessing, correlation analysis, normal mode analysis, and topological analysis are listed in Sup- plementary Code. (PDF) S1 File. The PDB codes and the chain length of the proteins in Dataset A (13081 proteins determined by X-ray crystallography) are listed in the file. (TXT) S2 File. The PDB codes and the chain length of the proteins in Dataset B (5078 proteins determined by solution nuclear magnetic resonance) are listed in the file. (TXT) S3 File. A Jupyter Notebook version of the supplementary code. (ZIP) S4 File. The data (chain length N, radius of gyration R , average path lengthhli, smallest non-zero eigenvalueλ , shape factor s and susceptibilityχ) for all the proteins in our PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 13 / 17 Long-range correlation in protein dynamics dataset are listed in the file. (TXT) Author Contributions Conceptualization: Qian-Yuan Tang. Data curation: Qian-Yuan Tang. Methodology: Qian-Yuan Tang. Supervision: Kunihiko Kaneko. Validation: Kunihiko Kaneko. Writing – original draft: Qian-Yuan Tang. Writing – review & editing: Kunihiko Kaneko. References 1. Go N. Theoretical studies of protein folding. Annu Rev Biophys Bioeng. 1983; 12(1): 183–210. https:// doi.org/10.1146/annurev.bb.12.060183.001151 PMID: 6347038 2. Onuchic JN, Luthey-Schulten Z, Wolynes PG. Theory of protein folding: the energy landscape perspec- tive. Annu Rev Phys Chem. 1997; 48(1): 545–600. https://doi.org/10.1146/annurev.physchem.48.1. 545 PMID: 9348663 3. Rao F, Caflisch A. The protein folding network. J Mol Biol. 2004; 342(1): 299–306. https://doi.org/10. 1016/j.jmb.2004.06.063 PMID: 15313625 4. Banavar JR, Maritan A. Physics of proteins. Annu Rev Biophys Biomol Struct. 2007; 36: 261–280. https://doi.org/10.1146/annurev.biophys.36.040306.132808 PMID: 17477839 5. Welch GR, Somogyi B, Damjanovich S. The role of protein fluctuations in enzyme action: a review. Prog Biophys Mol Biol. 1982; 39: 109–146. https://doi.org/10.1016/0079-6107(83)90015-9 PMID: 6. Whitten ST, Hilser VJ. Local conformational fluctuations can modulate the coupling between proton binding and global structural transitions in proteins. Proc Natl Acad Sci USA. 2005; 102(12): 4282– 4287. https://doi.org/10.1073/pnas.0407499102 PMID: 15767576 7. Bowman GR, Geissler PL. Equilibrium fluctuations of a single folded protein reveal a multitude of poten- tial cryptic allosteric sites. Proc Natl Acad Sci USA. 2012; 109(29): 11681–11686. https://doi.org/10. 1073/pnas.1209309109 PMID: 22753506 8. Boehr DD, Nussinov R, Wright PE. The role of dynamic conformational ensembles in biomolecular rec- ognition. Nat Chem Biol. 2009; 5(11): 789. https://doi.org/10.1038/nchembio.232 PMID: 19841628 9. Shrivastava IH, Jiang J, Amara SG, Bahar I. Time-resolved mechanism of extracellular gate opening and substrate binding in a glutamate transporter. J Biol Chem. 2008; 283(42): 28680–28690. https:// doi.org/10.1074/jbc.M800889200 PMID: 18678877 10. Berendsen HJ, Hayward S. Collective protein dynamics in relation to function. Curr Opin Struct Biol. 2000; 10(2): 165–169. https://doi.org/10.1016/s0959-440x(00)00061-0 PMID: 10753809 11. Zhou Y, Cook M, Karplus M. Protein motions at zero-total angular momentum: the importance of long- range correlations. Biophys J. 2000; 79(6): 2902–2908. https://doi.org/10.1016/S0006-3495(00)76527-1 PMID: 11106598 12. Fenwick RB, Esteban-Martin S, Richter B, Lee D, Walter KF, Milovanovic D, et al. Weak long-range cor- related motions in a surface patch of ubiquitin involved in molecular recognition. J Amer Chem Soc. 2011; 133(27): 10336–10339. https://doi.org/10.1021/ja200461n 13. Motlagh HN, Wrabl JO, Li J, Hilser VJ. The ensemble nature of allostery. Nature. 2014; 508(7496): 331–339. https://doi.org/10.1038/nature13001 PMID: 24740064 14. Sumbul F, Acuner-Ozbabacan SE, Haliloglu T. Allosteric dynamic control of binding. Biophys J. 2015; 109(6): 1190–1201. https://doi.org/10.1016/j.bpj.2015.08.011 PMID: 26338442 15. Tang QY, Zhang YY, Wang J, Wang W, Chialvo DR. Critical Fluctuations in the Native State of Proteins. Phys Rev Lett. 2017; 118(8): 088102. https://doi.org/10.1103/PhysRevLett.118.088102 PMID: PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 14 / 17 Long-range correlation in protein dynamics 16. Moret MA, Zebende GF. Amino acid hydrophobicity and accessible surface area. Phys Rev E. 2007; 75(1): 011920. https://doi.org/10.1103/PhysRevE.75.011920 17. Moret MA. Self-organized critical model for protein folding. Physica A. 2011; 390(17): 3055–3059. https://doi.org/10.1016/j.physa.2011.04.008 18. Phillips JC. Fractals and self-organized criticality in proteins. Physica A. 2014; 415: 440–448. 19. Phillips JC. Scaling and self-organized criticality in proteins I. Proc Natl Acad Sci USA. 2009; 106(9): 3107–3112. https://doi.org/10.1073/pnas.0811262106 PMID: 19218446 20. Phillips JC. Scaling and self-organized criticality in proteins II. Proc Natl Acad Sci USA. 2009; 106(9): 3113–3118. https://doi.org/10.1073/pnas.0811308105 PMID: 19124778 21. Reuveni S, Granek R, Klafter J. Proteins: coexistence of stability and flexibility. Phys Rev Lett. 2008; 100(20): 208101. https://doi.org/10.1103/PhysRevLett.100.208101 PMID: 18518581 22. Neusius T, Daidone I, Sokolov IM, Smith JC. Subdiffusion in peptides originates from the fractal-like structure of configuration space. Phys Rev Lett. 2008; 100(18): 188103. https://doi.org/10.1103/ PhysRevLett.100.188103 PMID: 18518418 23. Lu HP, Xun L, Xie XS. Single-molecule enzymatic dynamics. Science. 1998; 282(5395): 1877–1882. https://doi.org/10.1126/science.282.5395.1877 PMID: 9836635 24. Hu X, Hong L, Smith MD, Neusius T, Cheng X, Smith JC. The dynamics of single protein molecules is non-equilibrium and self-similar over thirteen decades in time. Nat Phys. 2016; 12: 171–174. https:// doi.org/10.1038/nphys3553 25. Law AB, Sapienza PJ, Zhang J, Zuo X, Petit CM. Native State Volume Fluctuations in Proteins as a Mechanism for Dynamic Allostery. J Amer Chem Soc. 2017; 139(10): 3599–3602. https://doi.org/10. 1021/jacs.6b12058 26. Bahar I, Atilgan AR, Demirel MC, Erman B. Vibrational dynamics of folded proteins: significance of slow and fast motions in relation to function and stability. Phys Rev Lett. 1998; 80(12): 2733. https://doi.org/ 10.1103/PhysRevLett.80.2733 27. Bahar I, Lezon TR, Yang LW, Eyal E. Global dynamics of proteins: bridging between structure and func- tion. Annu Rev Biophys. 2010; 39, 23–42. https://doi.org/10.1146/annurev.biophys.093008.131258 PMID: 20192781 28. Meireles L, Gur M, Bakan A, Bahar I. Pre-existing soft modes of motion uniquely defined by native con- tact topology facilitate ligand binding to proteins. Protein Sci. 2011; 20(10), 1645–1658. https://doi.org/ 10.1002/pro.711 PMID: 21826755 29. Yang L, Song G, Jernigan RL. How well can we understand large-scale protein motions using normal modes of elastic network models?. Biophys J. 2007; 93(3): 920–929. https://doi.org/10.1529/biophysj. 106.095927 PMID: 17483178 30. Flechsig H, Togashi Y. Designed elastic networks: Models of complex protein machinery. Intl J Mol Sci. 2018; 19(10): 3152. https://doi.org/10.3390/ijms19103152 31. Ichiye T, Karplus M. Collective motions in proteins: a covariance analysis of atomic fluctuations in molecular dynamics and normal mode simulations. Proteins. 1991; 11(3): 205–217. https://doi.org/10. 1002/prot.340110305 PMID: 1749773 32. Case DA. Normal mode analysis of protein dynamics. Curr Opin Struct Biol. 1994; 4(2): 285–290. https://doi.org/10.1016/S0959-440X(94)90321-2 33. Wako H, Endo S. Normal mode analysis as a method to derive protein dynamics information from the Protein Data Bank. Biophys Rev. 2017; 9(6): 877–893. https://doi.org/10.1007/s12551-017-0330-2 PMID: 29103094 34. Stanley HE. Phase transitions and critical phenomena. Oxford: Clarendon Press; 1971. 35. Goldenfeld N. Lectures on phase transitions and the renormalization group. Boca Raton: CRC Press; 36. Bak P. How nature works: the science of self-organized criticality. New York: Copernicus Press; 37. Haliloglu T, Bahar I, Erman B. Gaussian dynamics of folded proteins. Phys Rev Lett. 1997; 79(16): 3090. https://doi.org/10.1103/PhysRevLett.79.3090 38. Haliloglu T, Erman B. Analysis of correlations between energy and residue fluctuations in native pro- teins and determination of specific sites for binding. Phys Rev Lett. 2009; 102(8): 088103. https://doi. org/10.1103/PhysRevLett.102.088103 PMID: 19257794 39. Cavagna A, Cimarelli A, Giardina I, Parisi G, Santagati R, Stefanini F, et al. Scale-free correlations in starling flocks. Proc Natl Acad Sci USA. 2010; 107(26): 11865–11870. https://doi.org/10.1073/pnas. 1005766107 PMID: 20547832 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 15 / 17 Long-range correlation in protein dynamics 40. Attanasi A, Cavagna A, Del Castello L, Giardina I, Melillo S, Parisi L, et al. Finite-size scaling as a way to probe near-criticality in natural swarms. Phys Rev Lett. 2014; 113(23): 238102. https://doi.org/10. 1103/PhysRevLett.113.238102 PMID: 25526161 41. Hinsen K. Structural flexibility in proteins: impact of the crystal environment. Bioinformatics. 2007; 24(4): 521–528. https://doi.org/10.1093/bioinformatics/btm625 PMID: 18089618 42. Fuglebakk E, Reuter N, Hinsen K. Evaluation of protein elastic network models based on an analysis of collective motions. J Chem Theor Comp. 2013; 9(12): 5618–5628. https://doi.org/10.1021/ct400399x 43. Yang L, Song G, Jernigan RL. Protein elastic network models and the ranges of cooperativity. Proc Natl Acad Sci USA. 2009; 106(30): 12347–12352. https://doi.org/10.1073/pnas.0902159106 PMID: 44. Rivoire O. Parsimonious evolutionary scenario for the origin of allostery and coevolution patterns in pro- teins. Phys Rev E. 2019; 100: 032411. https://doi.org/10.1103/PhysRevE.100.032411 PMID: 45. Eckmann JP, Rougemont J, Tlusty T. Colloquium: Proteins: The physics of amorphous evolving matter. Rev Mod Phys. 2019; 91: 031001. https://doi.org/10.1103/RevModPhys.91.031001 46. Lehnert U, Echols N, Milburn D, Engelman D, Gerstein M, Normal modes for predicting protein motions: a comprehensive database assessment and associated Web tool. Protein Sci. 2005; 14(3): 633–643. https://doi.org/10.1110/ps.04882105 PMID: 15722444 47. Atilgan AR, Turgut D, Atilgan C. Screened nonbonded interactions in native proteins manipulate optimal paths for robust residue communication. Biophys J. 2007; 92(9): 3052–3062. https://doi.org/10.1529/ biophysj.106.099440 PMID: 17293401 48. Atilgan C, Okan OB, Atilgan AR. Network-based models as tools hinting at nonevident protein function- ality. Annu Rev Biophys. 2012; 41: 205–225. https://doi.org/10.1146/annurev-biophys-050511-102305 PMID: 22404685 49. Mora T, Bialek W. Are biological systems poised at criticality?. J Stat Phys. 2011; 144(2): 268–302. https://doi.org/10.1007/s10955-011-0229-4 50. Honerkamp-Smith AR, Veatch SL, Keller SL. An introduction to critical points for biophysicists: observa- tions of compositional heterogeneity in lipid membranes. Biochim Biophys Acta. 2009; 1788(1): 53–63. https://doi.org/10.1016/j.bbamem.2008.09.010 PMID: 18930706 51. Chialvo DR. Emergent complex neural dynamics. Nat Phys. 2010; 6(10): 744–750. https://doi.org/10. 1038/nphys1803 52. Furusawa C, Kaneko K. Zipf’s law in gene expression. Phys Rev Lett. 2003; 90(8): 088102. https://doi. org/10.1103/PhysRevLett.90.088102 PMID: 12633463 53. Furusawa C, Kaneko K. Adaptation to optimal cell growth through self-organized criticality. Phys Rev Lett. 2012; 108(20): 208103. https://doi.org/10.1103/PhysRevLett.108.208103 PMID: 23003193 54. Chate ´ H, Muñoz M. Viewpoint: Insect Swarms Go Critical. Physics. 2014; 7: 120. https://doi.org/10. 1103/Physics.7.120 55. Muñoz MA. Colloquium: Criticality and dynamical scaling in living systems. Rev Mod Phys. 2018; 90(3): 031001. https://doi.org/10.1103/RevModPhys.90.031001 56. Newman ME, Girvan M. Finding and evaluating community structure in networks. Phys Rev E. 2004; 69(2): 026113. https://doi.org/10.1103/PhysRevE.69.026113 57. Newman ME. Modularity and community structure in networks. Proc Natl Acad Sci USA. 2006; 103(23): 8577–8582. https://doi.org/10.1073/pnas.0601602103 PMID: 16723398 58. Guimera R, Sales-Pardo M, Amaral LAN. Modularity from fluctuations in random graphs and complex networks. Phys Rev E. 2004; 70(2): 025101. https://doi.org/10.1103/PhysRevE.70.025101 59. Newman ME. Detecting community structure in networks. Euro Phys J B. 2004; 38(2): 321–330. https://doi.org/10.1140/epjb/e2004-00124-y 60. Thomassen E, Gielen G, Schu ¨ tz M, Schoehn G, Abrahams JP, Miller S, et al. The structure of the receptor-binding domain of the bacteriophage T4 short tail fibre reveals a knitted trimeric metal-binding fold. J Mol Biol. 2003; 331(2): 361–373. https://doi.org/10.1016/s0022-2836(03)00755-1 PMID: 61. Li J, Sun L, Xu C, Yu F, Zhou H, Zhao Y, et al. Structure insights into mechanisms of ATP hydrolysis and the activation of human heat-shock protein 90. Acta Biochim Biophys Sin. 2012; 44(4): 300–306. https://doi.org/10.1093/abbs/gms001 PMID: 22318716 62. Brunette TJ, Parmeggiani F, Huang PS, Bhabha G, Ekiert DC, Tsutakawa SE, et al. Exploring the repeat protein universe through computational protein design. Nature, 2015; 528(7583): 580–584. https://doi.org/10.1038/nature16162 PMID: 26675729 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 16 / 17 Long-range correlation in protein dynamics 63. Berman HM, Westbrook J, Feng Z, Gilliland G, Bhat TN, Weissig H, et al. The protein data bank. Nucleic Acids Res. 2000; 28(1): 235–242. https://doi.org/10.1093/nar/28.1.235 PMID: 10592235 64. Newman ME. Spectral methods for community detection and graph partitioning. Phys Rev E. 2013; 88(4): 042822. https://doi.org/10.1103/PhysRevE.88.042822 65. Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E. Fast unfolding of communities in large networks. J Stat Mech. 2008; 2008(10): P10008. https://doi.org/10.1088/1742-5468/2008/10/P10008 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 17 / 17 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png PLoS Computational Biology Public Library of Science (PLoS) Journal

Long-range correlation in protein dynamics: Confirmation by structural data and normal mode analysis

Tang, Qian-Yuan; Kaneko, Kunihiko
PLoS Computational Biology , Volume 16 (2) – Feb 13, 2020
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Copyright: © 2020 Tang, Kaneko. 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 the protein structures used in this research are available from the Protein Data Bank (PDB). Related PDB-ID, code, and the data that related to this study are provided as Supporting File. Funding: This research was partially supported by a Grant-in-Aid for Scientific Research (S) (15H05746) from the Japanese Society for the Promotion of Science (JSPS) and Grant-in-Aid for Scientific Research on Innovative Areas (17H06386) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.
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Abstract

a1111111111 Proteins in cellular environments are highly susceptible. Local perturbations to any residue a1111111111 can be sensed by other spatially distal residues in the protein molecule, showing long-range correlations in the native dynamics of proteins. The long-range correlations of proteins contribute to many biological processes such as allostery, catalysis, and transportation. Revealing the structural origin of such long-range correlations is of great significance in OPENACCESS understanding the design principle of biologically functional proteins. In this work, based on Citation: Tang Q-Y, Kaneko K (2020) Long-range a large set of globular proteins determined by X-ray crystallography, by conducting normal correlation in protein dynamics: Confirmation by structural data and normal mode analysis. PLoS mode analysis with the elastic network models, we demonstrate that such long-range corre- Comput Biol 16(2): e1007670. https://doi.org/ lations are encoded in the native topology of the proteins. To understand how native topol- 10.1371/journal.pcbi.1007670 ogy defines the structure and the dynamics of the proteins, we conduct scaling analysis on Editor: Bert L. de Groot, Max Planck Institute for the size dependence of the slowest vibration mode, average path length, and modularity. Biophysical Chemistry, GERMANY Our results quantitatively describe how native proteins balance between order and disorder, Received: October 12, 2019 showing both dense packing and fractal topology. It is suggested that the balance between Accepted: January 21, 2020 stability and flexibility acts as an evolutionary constraint for proteins at different sizes. Over- all, our result not only gives a new perspective bridging the protein structure and its dynam- Published: February 13, 2020 ics but also reveals a universal principle in the evolution of proteins at all different sizes. Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The Author summary editorial history of this article is available here: https://doi.org/10.1371/journal.pcbi.1007670 The long-range correlated fluctuations are closely related to many biological processes of the proteins, such as catalysis, ligand binding, biomolecular recognition, and transporta- Copyright:© 2020 Tang, Kaneko. This is an open access article distributed under the terms of the tion. In this paper, we elucidate the structural origin of the long-range correlation and Creative Commons Attribution License, which describe how native contact topology defines the slow-mode dynamics of the native pro- permits unrestricted use, distribution, and teins. Our result suggests an evolutionary constraint for proteins at different sizes, which reproduction in any medium, provided the original may shed light on solving many biophysical problems such as structure prediction, multi- author and source are credited. scale molecular simulations, and the design of molecular machines. Moreover, in statisti- Data Availability Statement: All the protein cal physics, as the long-range correlations are notable signs of the critical point, unveiling structures used in this research are available from the origin of such criticality can extend our understanding of the organizing principle of a the Protein Data Bank (PDB). Related PDB-ID, large variety of complex systems. code, and the data that related to this study are provided as Supporting File. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 1 / 17 Long-range correlation in protein dynamics Funding: This research was partially supported by Introduction a Grant-in-Aid for Scientific Research (S) Proteins, including the globular, fibrous, membrane and intrinsically disordered proteins, are (15H05746) from the Japanese Society for the responsible for diverse functions in almost every process of cellular life. Globular proteins, as Promotion of Science (JSPS) and Grant-in-Aid for Scientific Research on Innovative Areas the majority type of the proteins in nature, can fold from disordered peptide chains into spe- (17H06386) from the Ministry of Education, cific three-dimensional (3D) structures on minimal-frustrated energy landscape [1–4]. Such Culture, Sports, Science and Technology (MEXT) kind of 3D structures, which are encoded by the amino acid sequences, are known as native of Japan. The funders had no role in study design, states. It is worth noting that the native state of a protein is not static, but exhibits dynamical data collection and analysis, decision to publish, or fluctuations around the energy minimum. Experiments and molecular simulations have preparation of the manuscript. shown that thermal fluctuations trigger the motions of proteins such as domain movements Competing interests: The authors have declared and allosteric transitions, which enable the biological functions of proteins such as catalysis that no competing interests exist. [5], ligand binding [6, 7], biomolecular recognition [8], and transportation [9]. Uncovering the relations between the structure and the function of proteins is a fundamental question in molecular biophysics. To answer it, the fluctuations at the native states may provide a key. One of the most fascinating properties of proteins is the long-range correlated fluctuations around the native states [10–12]. Thanks to the long-range correlations, local perturbations to any residue can be sensed by every other residue of the entire protein, even when the two sites are spatially distant. Such a property plays an important role in the functionality of the pro- teins. For example, for allosteric proteins, long-range correlations warrant the binding at one site can be transmitted to other functional sites [13, 14], and enable the high susceptibility for proteins in cellular environments. Based on the correlation analysis of structural ensembles determined by solution nuclear magnetic resonance (NMR), it was already demonstrated that the native proteins exhibit long-range correlations and high susceptibility in the native dynamics [15]. Such a phenomenon is also in line with other theoretical and experimental results, for example, the long-range conformational forces related to the hydrophobicity scales of the proteins [16–20], the fractal dimension in the oscillation spectrum [21] and configura- tion space [22], the slow relaxation of protein molecules in the solution [23, 24], the volume fluctuation of allosteric proteins [25], and the overlap between the low-frequency collective oscillation modes and large-scale conformational changes in allosteric transitions [26–30]. Accumulating evidence indicates that native proteins are not only stable enough to warrant structural robustness, but also susceptible enough to sense the signals in the milieu, and ready to perform large-scale conformational changes. However, the origin of such kind of dynamics is still unclear. In the present paper, we concentrate on the structure and the equilibrium fluctuation dynamics of a large set of globular proteins determined by X-ray crystallography, ranging from a single hairpin structure to large protein assemblies. Firstly, to elucidate the connection between the long-range correlations and protein structures, we conduct correlation analysis based on the elastic network models (ENMs) [26–30]. We find that the long-range correlations and the scaling laws can be robustly reproduced by the ENMs with different model parameters. Such a result indicates that the long-range correlations are encoded in the native topology of the proteins. Secondly, we conduct normal mode analysis [31–33] for protein molecules, ideal polymer chains, and lattice systems. A similar scaling relation holds for polymers, lattices, and proteins, but the scaling coefficients are different. Such a result shows how native proteins bal- ance between order and disorder, which resemble the physical systems near the critical point of a phase transition. Thirdly, we introduce the average path length and modularity to describe the topological characteristics of the proteins. Scaling relations are also observed between these topological descriptors and the size of the proteins. According to the result of the scaling analy- sis, we conclude that native proteins show both dense packing and fractal topology. Lastly, we focus on the size dependence of proteins’ shape. With a given chain length, the shape of a PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 2 / 17 Long-range correlation in protein dynamics protein is not random, but a most-probable shape factor always exists. Such a constraint sug- gests that native proteins balance between stability and functionality. Overall, our result not only gives a new perspective bridging the protein structure and its dynamics but also reveals a universal principle in the evolution of proteins at all different sizes. Results The critical dynamics of proteins are robustly encoded in the native structures In previous studies, based on the structural ensembles determined by solution nuclear mag- netic resonance (NMR), it was observed that the native proteins in the solution exhibit long- range correlations and high susceptibility in the dynamics [15]. The native fluctuation of pro- teins behaves as though they are near the critical point of a phase transition [34–36]. The ques- tion arises whether the critical dynamics of native proteins are encoded in the native structure or driven by other factors in the milieu. To answer this question, we employ the minimal model of proteins, the elastic network model (ENM) to conduct our analysis. In an ENM, a protein molecule is described as a set of nodes (represented by their C atoms) connected with edges of elastic springs. As shown in Fig 1A, the 3D structure of a pro- tein can be simplified as a network based on the topology of residue contacts. Note that the elastic networks are constructed only based on the spatial distances between residues. If an ENM can successfully reproduce long-range correlations in the fluctuations of the native pro- teins, then it can be concluded that the critical dynamics of proteins is encoded by the local contacts in the native structures. The correlated motions of residues can be represented by a covariance matrix, in which matrix element C ¼ hD~r � D~r i. For simplification, we conduct our analysis based on the ij i j Gaussian network model (GNM) [37, 38]. In GNM, the covariance matrix C is proportional to 3k T pseudoinverse of the Kirchhoff matrixΓ, i.e., C ¼ �½G � [26, 37]. Normalizing the ij k ij pffiffiffiffiffiffiffiffiffiffi covariance matrix, a pairwise cross correlation � ¼ C = C C an be obtained. Similar to ij ij ii jj previous works [15, 39, 40], a distance-dependent correlation function ϕ(r) can be defined by � dðr r Þ ij ij i<j averaging the correlations for residue pairs at mutual distance r, and �ðrÞ ¼ , dðr r Þ ij i<j where r denote the spatial distance between residue i and j, andδ(x) is the Dirac-delta func- ij tion selecting residue pairs at mutual distance r. Here, the correlation length ξ as the distance where ϕ(r) first decays to zero. To examine whether the correlation scales with the protein size, we sample over the protein data across different sizes. By averaging the distance-dependent correlation function ϕ(r) for a subset of proteins, we can define the averaged correlation functionhϕ(r)i to a group of pro- teins. Here, we divide the dataset into subsets according to the radius of gyration R of the pro- teins (e.g., subset {R * 12Å} contains proteins at size 11.5Å� R < 12.5Å), the distance- g g dependent correlation functions ϕ(r) for proteins at different sizes are calculated. As shown in Fig 1B, the correlation function first decreases from its maximum at short distances, crosses zero at r = ξ, continues to decline, reaches a negative minimum. As a notable sign of criticality, for proteins of different sizes, the correlation length ξ is proportional to their radius of gyration R . Therefore, the correlation functions can be scaled by the size (R ) of the proteins, and all g g the correlation functions collapse (Fig 1C). This result indicates that correlations in the native fluctuation of proteins are scale-free: No matter how large the protein molecule is, correlation length can extend to the size of the entire system. Such long-range correlation contributes to the functionality of a large variety of proteins, for example, for allosteric proteins, the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 3 / 17 Long-range correlation in protein dynamics Fig 1. The critical dynamics of proteins are robustly encoded in the native structure. (A) An illustration of the elastic network model (r = 9Å) of the protein CI2 (PDB code: 2CI2). The beads denote the residues, and the bonds denote the elastic springs in the model. (B) The correlation functions ϕ(r) for proteins at different sizes predicted by GNM with cutoff distance r = 9Å. (C) Correlation functions scaled by the radius of gyration of the proteins R . (D) For proteins of similar sizes (19.5Å� R < 20.5Å), with g g different cutoff distances r , the correlation functions ϕ(r) predicted by GNM. (E) With different cutoff distances, for proteins of different sizes, the correlation length ξ is always proportional to the size of the protein R . (F) The susceptibilityχ vs. chain length N αγ/ν shows the power-law relation:χ * N , and the scaling coefficientαγ/ν� 1 can be kept with different r (inset). https://doi.org/10.1371/journal.pcbi.1007670.g001 long-range correlation warrants the binding at one site can be transmitted to other functional sites [13, 14], even when the two sites are spatially distant. To validate the previous analysis, let us consider the parameter sensitivity in the prediction of the cross correlations in protein dynamics. The only free parameter in GNM is the cutoff distance r . With different r , the correlation would have different magnitude at short dis- C C tances; however, as shown in Fig 1D, the correlation lengths ξ keep as a constant for different cutoff distances r . As shown in Fig 1E, for cutoff distances ranging from 6 Å to 15 Å, the PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 4 / 17 Long-range correlation in protein dynamics correlation length ξ is always proportional to the radius of gyration R , showing that the critical dynamics of native proteins is generally a stable property and insensitive to the selection of cut- off distances. With only short-range interactions between residues taken into account, GNM can successfully capture the long-range correlations in the native dynamics of the proteins. To have a further investigation of the criticality, it is necessary to validate the scaling rela- tions in the dynamics of proteins. Here, for illustration, we take the power-law relation between the susceptibilityχ and chain length N as an example. For protein systems, a finite- size version of susceptibilityχ is introduced to quantify the response of systems under pertur- bation [15]. It is defined as the total correlation in a unit volume within the correlation length: w ¼ � � yðx r Þ, where s denotes the shape factor of protein, and θ(x) denotes the i<j ij ij Heaviside function. Previously, based on NMR-determined protein ensembles [15], it was αγ/ν observed thatχ * N , with the scaling coefficientαγ/ν� 1 (Definitions ofα,γ and ν are listed in S1 Appendix). Here, as shown in Fig 1F, by employing the GNM, similar scaling rela- tions can also be observed. Such a result demonstrates that, no matter how large the molecule is, proteins can always have high sensitivity executing its function because the magnitude of the susceptibility grows with the chain length of the proteins. Besides, the scaling coefficients are insensitive to changes in cutoff distances (inset), demonstrating that the scale-free correla- tion of native proteins is a robust property. Our correlation analysis and scaling analysis methods can also be extended to other ver- sions of elastic network models. For example, with harmonic C potential model (HCA) [41, 42], similar scaling coefficients can also be observed (see S1 Appendix). However, some mod- els cannot correctly reproduce the scaling relations betweenχ and N, for instance, the parame- ter-free GNM (pfGNM) [43]. In fact, pfGNM fails to predict all the scaling relations in the proteins (see S1 Appendix). Previous researches already found that pfGNM can only be applied for proteins in crystalline conditions, and it will have a poor agreement to the collective motions given by molecular dynamics [42]. Such a result indicates that the scaling coefficient may help us to probe whether the protein is solvated or in a crystalline condition. The size dependence of slowest modes reveals criticality of native proteins Normal mode analysis is a practical tool to elucidate the global dynamics [31–33] and the evo- lutionary constraints [44, 45] of the proteins. Physically, the slow modes, or say, the low-fre- quency modes of a system are related to the motions with low excitation energy, long wavelengths (long-range correlation), long time scale (at the order from microseconds to sec- onds) and the large amplitude motions. Usually, the motions that correspond to the slow modes (especially the slowest nonzero mode) can have significant overlap with large displace- ment during the functional motions [46]. These functional motions usually engage relative movements of large subunits in the proteins or cooperative conformational changes of the whole proteins. Previously, the unique spectral properties of the residue contact networks have been noticed [47, 48], but the detailed differences have never been examined. To demonstrate the particularity in the spectrum of proteins, we compare the proteins with ideal polymer chains (detailed information listed in S1 Appendix) and lattice systems. Our analysis focuses on the size dependence of the slow modes. As shown in Fig 2A, for all these systems, the slowest few modes versus the system size N follow power-law distributions. Among these slow modes, we specifically focus on the eigenvalueλ which corresponds to the −z slowest nonzero mode. A similar power-lawλ * N holds for ideal polymers, lattices, and proteins. However, the scaling coefficients z are different in these systems. As shown in Fig 2A, for ideal polymer chains, the scaling coefficient z� 1.674. For face-centered cubic (fcc) lattice, by conducting normal mode analysis where atoms are connected by springs with their nearest PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 5 / 17 Long-range correlation in protein dynamics Fig 2. The slow modes of proteins are robustly defined by native structure. (A) The 1st, 2nd and the 3rd non-zero eigenvaluesλ ,λ , andλ vs. the chain length N of the proteins follows a power-law distribution. (Cutoff distance r = 1 2 3 C 9Å, and the scaling coefficients ofλ (N),λ (N), andλ (N) are 1.074, 0.900, and 0.868, respectively). As comparison, 1 2 3 similar scaling relations in lattices and ideal polymer chains are also illustrated, and the scaling coefficients are 0.728 (lattices) and 1.674 (polymer). (B) The eigenvalue of the slowest nonzero modeλ versus chain length N shows the −z scaling relation:λ * N , and the inset shows scaling coefficient z vs. the cutoff distance r . (C) For proteins at similar 1 C sizes (chain length 180� N < 220), the histogram for the eigenvalue distribution g(λ). https://doi.org/10.1371/journal.pcbi.1007670.g002 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 6 / 17 Long-range correlation in protein dynamics neighbors and 2nd nearest neighbors), we have z� 0.727. Theoretically, for lattice systems, the maximum wavelength l corresponds to the slowest elastic mode, and l is proportional to w w 1/3 the characteristic length of the system. Since the maximum wavelength l * N , one can esti- 2 2 2=3 mate that l � o � l � N , which is close to 0.727. In contrast to ideal polymers and lat- 1 1 w tices, z� 1 holds for protein molecules. The scaling relations in the slowest modes of proteins are robust to the variation in model parameters. As shown in Fig 2B, the selection of cutoff distances r would not affect the scaling coefficient z. But the robustness of the scaling coefficient cannot be attributed to that of the eigenvalue distribution. As shown in Fig 2C, selecting different r would influence the mode distribution g(λ) of native proteins. The mode distribution g(λ), especially the low-frequency part, can be enhanced by selecting a short cutoff distance r . Such a result is also consistent with previous theoretical analysis on protein elastic network and ranges of cooperativity [43], which states that with a shorter interaction range, the predicted dynamics would be more cooperative and show better overlap with the displacement in large-scale conformational changes. It is worth noting that the scaling coefficients in the size dependence of the slowest mode demonstrate that the structure of proteins stands between lattices and ideal polymer chains. For proteins, the exponent z� 1, above what is obtained from lattices (z� 0.727), and below what is obtained from polymer chains (z� 1.674). Thus, compared with ideal polymer chains, the proteins have higher structural stability, whereas compared with lattices, the proteins have higher flexibility and exhibit slower vibrations. Native proteins stand between lattices and polymers, acting as the “critical point” that separates the ordered and disordered phase. Not only are native proteins stable enough to ensure structural robustness and functional specific- ity, but also susceptible enough to sense the signals in the environment, and ready to perform large-scale conformational changes. Interestingly, staying at the critical point seems to be a common organizing principle of a large variety of biological systems [49–55]: If the system is too disordered, the system cannot stably exist; if it is too ordered, it cannot adapt or respond to perturbations from the environments. Our result of scaling analysis provides additional evi- dence to support the criticality hypothesis. Protein structure: Dense packing with fractal topology In previous sections, we demonstrated that the critical dynamics of the proteins are encoded in their native structures, and we showed that the equilibrium dynamics of protein molecules if different from lattices and polymers. How does the topology of the residue contact network encode such kind of dynamics? To answer the question, in this subsection, we will try to bridge the vibration spectrum with the architecture of the protein by mainly focusing on the issue of the network topology. In the network analysis, the average path lengthhli is one of the most important topological descriptors quantifying the total connectivity among the nodes. Here, we first focus on the scaling relations between average path lengthhli and the system size N. As shown in Fig 3A, for proteins at different sizes, there is a power-law relation between the average path lengthhli and the chain length N:hli*N , andα� 0.338, which is close to 1/3. In the calculation, the cutoff distance r is set to be 8Å. Even different cutoff distance r will lead to differenthli, but C C the scaling exponent is invariant (see S1 Appendix). The scaling relation in proteins is very similar to what in the lattice structures. Theoretically, for 3D lattices, the exponent would beα = 1/3. Such a scaling relation is confirmed in Fig 3A. While for ideal polymer chains, with an extended structure, there would be longer average path lengths, and fitting givesα� 0.675. Such a result demonstrates that the residue contact networks show similar dense packing PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 7 / 17 Long-range correlation in protein dynamics Fig 3. The protein dynamics can be quantified by topological descriptors of the residue contact network. (A) For the contact network of proteins (r = 8Å), fcc lattices and ideal polymers, the average path lengthhli vs. system size N. (B) Similarly for proteins, fcc lattice and ideal polymers, modulaity Q vs. system size N. The inset shows the log-log plot of 1 − Q vs. N. (C) For proteins at similar sizes (180� N < 220), the scattering plot (yellow dots, each dot represents a protein molecule), the binned average (red dots) and the basic trend (red curve) of the average path lengthhli vs. Q, and (D) Smallest non-zero eigenvalueλ vs. Q. https://doi.org/10.1371/journal.pcbi.1007670.g003 property as regular lattices. Both lattice and protein networks have much shorter path lengthhli than ideal polymers. Although protein and lattice share similar dense packing properties, the residue contact networks of proteins still exhibit unique properties. To demonstrate the difference between the residue contact network and the lattice networks, another measure—modularity Q is intro- duced into the study [56, 57]. Intuitively, a network that can be more easily divided into mod- ules would have a higher Q value. Modularity Q also scales as the system size increases. For a d−dimensional cubic lattice network with N nodes, theoretically, it was proved that the modu- −η larity Q versus N follows the relation: Q = 1 − K� N , where the scaling coefficient Z ¼ , dþ1 and K is a constant that depend on average degree z and dimension d [58]. For ideal polymer chains, the fitting givesη� 0.465, indicating an effective fractal dimension d � 1.15, which eff is much lower than 3. For a 3D cubic lattice, theoretically,η = 1/4. For fcc lattices, as shown in Fig 3B, fitting givesη� 0.231 < 1/4, indicating d � 3.33 > 3, that is because, in the fcc lat- eff tices, every atom has more neighbors than cubic lattice. For proteins our dataset, when taking r = 8Å, similar power law can also be observed, but the scaling coefficientη = 0.279 > 1/4. Such an exponent indicate that the proteins has an effective dimension d ¼ 1 � 2:58, eff which is lower than 3. Such a scaling coefficient displays that the residue contact networks have a fractal topology, and the fractal dimension is below 3. It is worth noting that, in this work, the fractal dimension of proteins is obtained by the scaling analysis for proteins at different sizes. The effective dimension obtained here is consistent with the fractal dimension (d� 2.7) of proteins determined by structural analysis methods (see S1 Appendix). The scaling PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 8 / 17 Long-range correlation in protein dynamics analysis of average path length reveals that the proteins have similar dense packing properties as ordered lattices, but the scaling analysis of modularity suggests that proteins exhibit fractal structures, which is similar to disordered polymer structures. In short, topological analysis demonstrates again that native of proteins balance between order and disorder. In the discussions above, by averaging the topological descriptors of proteins at similar sizes, we analyze the size dependence of topological properties. In fact, for proteins at similar sizes, topological descriptors can also play an important role in capturing the main features in the dynamics of the proteins. To illustrate that, here, we select the protein molecules with chain length 180� N < 220 from our dataset. Although these proteins have similar chain length, the structure may differ a lot. Our discussion centers around modularity Q. When the modularity Q of a protein increases, as shown in Fig 3C, the average path lengthhli also increases. This is because, in a highly modularized network, there will be few connections between different communities, on the average, it will take more steps from one node to another. As shown in Fig 3D, as the modularity Q increases, the smallest non-zero eigenvalue λ decreases, in line with the common knowledge that that modularized structures in the pro- teins contribute to slow-mode motions. Such a result is consistent with the theory of spectral graph theory. Indeed, the spectrum of the graph Laplacian is closely related to the community structures of the network [59]. Our analysis quantitatively demonstrates that modularized structures contribute to the large-scale motions and slow relaxations of the proteins. Stability-functionality constraint: The size dependence of proteins’ shape The intrinsic dynamics of proteins is encoded in their structures. Since scaling relation between the dynamics and the size of the protein is already discussed in the previous sections. We focus on the relationship between the structure and the size of the protein in this section. The shape factor s can be introduced to describe the general architecture of a protein mole- cule [15]. According to the definition, the shape factor can be understood as the residue pack- ing density within the inertia ellipsoid. When residues are tightly packed with a globular shape, the shape factor s would be large. When disordered loops or flexible linkers are connect- ing multiple domains, the shape of the molecule deviates from an ellipsoid, then s would be small. Here, for illustration, three proteins with a similar chain length 180� N < 220 but with different shape factor s are shown in Fig 4A. On the left, the receptor-binding domain of the short tail fiber (STF) is illustrated. Such a molecule has hardly any regular secondary structures likeα−helices orβ-strands [60]. The structure of such a molecule in its monomer state has a small shape factor and high modularity. To perform its functions, a knitted trimeric assembly has to be formed [60]. In the middle, there is the human molecular chaperone heat-shock pro- tein 90 (Hsp90) [61] with medium shape factor and modularity. On the right, a de novo designed helical repeat protein DHR10 is illustrated. By repeating a simple helix–loop–helix– loop structural motif, DHR10 protein is highly ordered and becomes very stable, which can stay folded even at 95˚C [62]. Generally, the proteins with larger shape factors show higher sta- bility, and the proteins with smaller shape factors show higher flexibility. Although the definition of shape factor does not introduce any detailed information on sec- ondary structures or residue contacts, the shape factor is closely related to the topological descriptors of the residue contact network. Here, statistics for the proteins with similar chain length (180� N < 220) is conducted. The scattering plot of shape factor s versus modularity Q is shown in Fig 4B. A trend line (in red) displays that as modularity Q increases, the shape fac- tor s decreases. The result is easy to understand intuitively, a protein molecule in a shape that deviates from an ellipsoid is likely to have multiple domains or have flexible linkers connecting multiple ordered regions. Interestingly, although the proteins could have very different shapes, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 9 / 17 Long-range correlation in protein dynamics Fig 4. The shape factor correlates with the chain lengths of the proteins. (A) Three proteins with similar chain lengths: (Left) The receptor-binding domain of T4 STF (PDB: 1OCY, s = 0.84, Q = 0.74); (Middle) Human Hsp90 protein (PDB: 3T0H, s = 1.77, Q = 0.65); and (Right) The DHR10 protein (PDB: 5CWG, s = 2.37, Q = 0.63). (B) For proteins at similar sizes (chain length 180� N < 220), the scattering plot (yellow dots), binned average (red dots) and the trend line (red line) of shape factor s vs. modularity Q are plotted. Besides, there are histograms of the shape factor s (right vertical) and modularity Q (top horizontal). (C) For all the proteins in our dataset, the 2D histogram (in the background) of s vs. N and the plot (in navy blue) of the most-probable shape factor s vs. chain length N. https://doi.org/10.1371/journal.pcbi.1007670.g004 for protein molecules with a specific chain length, the value of shape factor does not vary a lot. Here, in Fig 4B, histograms of the shape factor s (right vertical) and modularity Q (top hori- zontal) are plotted. The histograms show that there exists a most-probable shape factor s and � � corresponding modularity Q . Most natural proteins have shape factors close to s , exhibit a balancing behavior between stability and flexibility [21]. In fact, for proteins with different chain lengths, the most-probable shape factor s always exists, which can be recognized as a constraint in the shape of the protein. As shown in Fig 4C, it was observed that larger proteins prefer smaller shape factors. A similar relation is also observed based on NMR-determined ensembles [15]. These observations provide additional pieces of evidence to support the criticality of native proteins. The native proteins have to bal- ance between stability and flexibility. With short chain lengths, the proteins tend to have a larger shape factor to ensure a stable folded state. Accordingly, small proteins usually have higher residue packing density. However, as the chain length of the proteins increases, to exe- cute functional motions, flexibility becomes the main demand of the proteins. One good example is the designed protein DHR10 as illustrated in Fig 4A. DHR10 has high structural stability, but it is hard for such a protein to execute any biological functions. In such a situa- tion, smaller shape factors, which usually correspond with disordered loops or multi-domain structures, are demanded by the functionality. Our results suggest that the balance between stability and flexibility acts as an evolutionary constraint for proteins at different sizes. PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 10 / 17 Long-range correlation in protein dynamics Discussion The long-range correlated fluctuations contribute to many biological processes of the proteins, such as allostery, catalysis, and transportation. To understand the origin of such long-range correlations, based on the elastic network model, we conduct normal mode analysis for a large dataset of globular proteins determined by X-ray crystallography. First, we predict the correlated motions for proteins at different sizes. It is observed that the correlation length of a protein can extend to the size of the whole protein, no matter how large the protein molecule is. Moreover, with different model parameters, the scale-free correlations and the scaling laws can be reproduced by the elastic networks model, which is the minimal structure-based model of native proteins. Such a result indicates that the critical dynamics characterized by the power-law relations are robustly encoded in the native topology of the proteins. Second, for proteins at different sizes, we conduct normal mode analysis and perform scal- ing analysis for the slow vibration modes of the proteins. To demonstrate the particularity in the spectrum of proteins, we compare the proteins with ideal polymer chains and lattice sys- tems. Native proteins stand between ordered lattices and disordered polymers, acting as the “critical point” that separates the ordered and disordered phase. Our result of scaling analysis provides additional evidence to support the criticality hypothesis. Third, to understand how the native topology determines the architecture and the dynam- ics of the proteins, we conduct scaling analysis for the topological descriptors and the size of the proteins. Our results demonstrate that, although proteins have similar average path length with lattice structures, the residue contact networks are more modularized. Last, we focus on the size dependence of proteins’ shape. For proteins with different chain lengths, the most-probable shape factors always exist. Larger proteins prefer smaller shape fac- tors. Such a constraint results from the balance between stability and functionality of proteins. In summary, our work quantitatively demonstrates how the native contact topology defines the long-range correlations and the slow dynamics of the native proteins. Our work not only provides quantitative scaling relations supporting the “structure-dynamics-function” para- digm but also reveals evolutionary constraints for proteins at different sizes. These results may shed light on a large variety of biophysical problems such as structure prediction, multi-scale molecular simulations, and the design of molecular machines. Materials and methods Dataset Our dataset contains 13081 proteins selected from the Protein Data Bank (PDB) [63]. The structures of these proteins are all determined by X-ray diffraction with high resolution (� 2.0Å). For every protein structure in the dataset, it contains no DNA, RNA or hybrid struc- tures; and the chain length 30� N� 1200. In our protein dataset, every two proteins share less than 30% sequence similarity. The PDB codes of all the proteins in our dataset are listed in the Supplementary Information (S1 and S2 Files). The elastic network models The elastic network models are widely applied to predict the functional dynamics of a variety of proteins and bio-machineries [26, 27, 29, 30]. With the assumption that all residue fluctua- tions are Gaussian variables distributed around their equilibrium coordinates, the Gaussian Network Model (GNM) can successfully reproduce the residue fluctuations as determined by experiments [37, 38]. For a protein consisting of of N residues, based on the native structure, PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 11 / 17 Long-range correlation in protein dynamics the potential energy of the network is given by: V ¼ D~r � G � D~r ; ð1Þ GNM i ij j i;j¼1 in whichκ is a uniform force constant; D~r and D~r is the displacement of residue i and j, i j respectively; andΓ is the element of Kirchhoff matrix, or in a graph theory perspective, it is ij the graph Laplacian of the residue-residue contact network. The elements of matrixΓ is defined according to the contact topology of the native structure: for residue pair i − j, if r � ij r , thenΓ = −1; if r > r , thenΓ = 0; and for the diagonal elements,Γ = −∑ Γ = −k , C ij ij C ij ii j6¼i ij i where k denote the degree of node i. In GNM with homogenous contact strength, the only control parameter is the cutoff distance r . With a large r , residue pairs at long distances can C C interact with each other; while for smaller r , only short-range interactions are contributed to the elastic energy of the system. One may also introduce distance-dependent force constants [41–43] to refine the predictions of elastic network models. In these models, the force con- stantsκ becomes a function of the mutual distance between residue i and j. Further details ij and other variations of the elastic network models are listed in the S1 Appendix. Normal mode analysis and the spectrum of the graph laplacian Based on GNM, by diagonalizing the Kirchhoff matrixΓ, we can obtain all the eigenvalues and the corresponding eigenvectors describing the motions of every normal mode [32]. To com- pare the mode distribution for proteins of different chain lengths, the Kirchhoff (Laplacian) matrices correspond to the topology of native proteins are normalized. By normalizing all the diagonal elements as 1, we can obtain the symmetric normalized graph Laplacian [48]: 1=2 1=2 L ¼ D � G� D ; ð2Þ in which D is a matrix of all the diagonal elements of matrix D = diag[Γ ,Γ ,���Γ ], 1,1 2,2 N,N describing the local packing status of each residue. Diagonalizing matrix L, then we have L = UΛU , in which the eigenvaluesΛ = diag[λ ,λ ,λ ,���λ ] (λ � λ � λ ,� ���� λ ) 0 1 2 N−1 0 1 2 N−1 and eigenvectors U = [u , u , u ,��� u ] . The eigenvalueλ describes the frequency ω of the 0 1 2 N−1 i i i-th eigenmode (l � o ), and the eigenvector u describes the motion profile of the corre- i i sponding eigenmode. Note that the zero mode corresponds to the eigenvalueλ = 0, and eigenvector u describes the collective translational or rotational motions of the system. The code of normal mode analysis is listed in the Supplementary Code (S2 Appendix and S3 File). Shape factor To have a general description of the structure of a protein molecule, a dimensionless shape fac- tor s is defined [15]. By calculating the the moments of inertia of a protein molecule, one can Na estimate the residue packing density within the inertia ellipsoid as s ¼ , in which a = 3.8Å L L L 1 2 3 is the residue size, and L , L and L are lengths of the principal axes of the protein (L > L > 1 2 3 1 2 L ). The shape factors of the proteins in our dataset are listed in the Supplementary Data (S4 File). Average path length The average (or characteristic) path lengthhli usually works as a measure of the information transfer efficiency on a network. It is defined as the average number of steps along the shortest paths for all possible pairs of network nodes. When l denotes the shortest distance between i,j PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 12 / 17 Long-range correlation in protein dynamics node i and j, then, the average path length hli ¼ l : i;j ð3Þ NðN 1Þ i6¼j Modularity Modularity is a topological descriptor which is designed to quantify if a network can be easily divided into modules. For a network with N node and M edges, when the topology is described by the adjacency matrix A where A = 1 if and only if node i and j are connected. Modularity is ij defined as the fraction of the edges that fall within the given module minus the expected frac- tion when edges were distributed at random [56, 57]. According to the definition, one can k k i j introduce the modularity matrix B with elements B ¼ A to describe the expected num- ij ij 2M ber of edges between node pairs, in which k and k denote the degrees of node i and j, respec- i j tively. Based on matrix B, the modularity can be calculated as: ð4Þ Q ¼ Trð~x � B�~xÞ; 4M in which~x is the column vector describing the partition of a network. Vector x has elements x = ±1 indicating the modules to which the node belongs. The value of the Q lies in the range −1� Q� 1. For any given partition s of a network, one can calculate the Q corresponding to such a partition. The appropriate partition of a network would maximize the modularity Q [64]. In this work, we introduced the Louvain method [65] to partition the network and maxi- mize the value modularity Q. The code of topological analysis is listed in the Supplementary Code (S2 Appendix and S3 File). Supporting information S1 Appendix. Supplementary information. Detailed descriptions of the structural datasets involved in this research. Additional information concerning the scaling relations, generation of polymer structures, and other variations of elastic network models are also included in the Supplementary Information. (PDF) S2 Appendix. Supplementary code. The code (written in Python language) for PDB file pro- cessing, correlation analysis, normal mode analysis, and topological analysis are listed in Sup- plementary Code. (PDF) S1 File. The PDB codes and the chain length of the proteins in Dataset A (13081 proteins determined by X-ray crystallography) are listed in the file. (TXT) S2 File. The PDB codes and the chain length of the proteins in Dataset B (5078 proteins determined by solution nuclear magnetic resonance) are listed in the file. (TXT) S3 File. A Jupyter Notebook version of the supplementary code. (ZIP) S4 File. The data (chain length N, radius of gyration R , average path lengthhli, smallest non-zero eigenvalueλ , shape factor s and susceptibilityχ) for all the proteins in our PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 13 / 17 Long-range correlation in protein dynamics dataset are listed in the file. (TXT) Author Contributions Conceptualization: Qian-Yuan Tang. Data curation: Qian-Yuan Tang. Methodology: Qian-Yuan Tang. Supervision: Kunihiko Kaneko. Validation: Kunihiko Kaneko. Writing – original draft: Qian-Yuan Tang. Writing – review & editing: Kunihiko Kaneko. References 1. Go N. Theoretical studies of protein folding. Annu Rev Biophys Bioeng. 1983; 12(1): 183–210. https:// doi.org/10.1146/annurev.bb.12.060183.001151 PMID: 6347038 2. Onuchic JN, Luthey-Schulten Z, Wolynes PG. Theory of protein folding: the energy landscape perspec- tive. Annu Rev Phys Chem. 1997; 48(1): 545–600. https://doi.org/10.1146/annurev.physchem.48.1. 545 PMID: 9348663 3. Rao F, Caflisch A. The protein folding network. J Mol Biol. 2004; 342(1): 299–306. https://doi.org/10. 1016/j.jmb.2004.06.063 PMID: 15313625 4. Banavar JR, Maritan A. Physics of proteins. Annu Rev Biophys Biomol Struct. 2007; 36: 261–280. https://doi.org/10.1146/annurev.biophys.36.040306.132808 PMID: 17477839 5. Welch GR, Somogyi B, Damjanovich S. The role of protein fluctuations in enzyme action: a review. Prog Biophys Mol Biol. 1982; 39: 109–146. https://doi.org/10.1016/0079-6107(83)90015-9 PMID: 6. Whitten ST, Hilser VJ. Local conformational fluctuations can modulate the coupling between proton binding and global structural transitions in proteins. Proc Natl Acad Sci USA. 2005; 102(12): 4282– 4287. https://doi.org/10.1073/pnas.0407499102 PMID: 15767576 7. Bowman GR, Geissler PL. Equilibrium fluctuations of a single folded protein reveal a multitude of poten- tial cryptic allosteric sites. Proc Natl Acad Sci USA. 2012; 109(29): 11681–11686. https://doi.org/10. 1073/pnas.1209309109 PMID: 22753506 8. Boehr DD, Nussinov R, Wright PE. The role of dynamic conformational ensembles in biomolecular rec- ognition. Nat Chem Biol. 2009; 5(11): 789. https://doi.org/10.1038/nchembio.232 PMID: 19841628 9. Shrivastava IH, Jiang J, Amara SG, Bahar I. Time-resolved mechanism of extracellular gate opening and substrate binding in a glutamate transporter. J Biol Chem. 2008; 283(42): 28680–28690. https:// doi.org/10.1074/jbc.M800889200 PMID: 18678877 10. Berendsen HJ, Hayward S. Collective protein dynamics in relation to function. Curr Opin Struct Biol. 2000; 10(2): 165–169. https://doi.org/10.1016/s0959-440x(00)00061-0 PMID: 10753809 11. Zhou Y, Cook M, Karplus M. Protein motions at zero-total angular momentum: the importance of long- range correlations. Biophys J. 2000; 79(6): 2902–2908. https://doi.org/10.1016/S0006-3495(00)76527-1 PMID: 11106598 12. Fenwick RB, Esteban-Martin S, Richter B, Lee D, Walter KF, Milovanovic D, et al. Weak long-range cor- related motions in a surface patch of ubiquitin involved in molecular recognition. J Amer Chem Soc. 2011; 133(27): 10336–10339. https://doi.org/10.1021/ja200461n 13. Motlagh HN, Wrabl JO, Li J, Hilser VJ. The ensemble nature of allostery. Nature. 2014; 508(7496): 331–339. https://doi.org/10.1038/nature13001 PMID: 24740064 14. Sumbul F, Acuner-Ozbabacan SE, Haliloglu T. Allosteric dynamic control of binding. Biophys J. 2015; 109(6): 1190–1201. https://doi.org/10.1016/j.bpj.2015.08.011 PMID: 26338442 15. Tang QY, Zhang YY, Wang J, Wang W, Chialvo DR. Critical Fluctuations in the Native State of Proteins. Phys Rev Lett. 2017; 118(8): 088102. https://doi.org/10.1103/PhysRevLett.118.088102 PMID: PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 14 / 17 Long-range correlation in protein dynamics 16. Moret MA, Zebende GF. Amino acid hydrophobicity and accessible surface area. Phys Rev E. 2007; 75(1): 011920. https://doi.org/10.1103/PhysRevE.75.011920 17. Moret MA. Self-organized critical model for protein folding. Physica A. 2011; 390(17): 3055–3059. https://doi.org/10.1016/j.physa.2011.04.008 18. Phillips JC. Fractals and self-organized criticality in proteins. Physica A. 2014; 415: 440–448. 19. Phillips JC. Scaling and self-organized criticality in proteins I. Proc Natl Acad Sci USA. 2009; 106(9): 3107–3112. https://doi.org/10.1073/pnas.0811262106 PMID: 19218446 20. Phillips JC. Scaling and self-organized criticality in proteins II. Proc Natl Acad Sci USA. 2009; 106(9): 3113–3118. https://doi.org/10.1073/pnas.0811308105 PMID: 19124778 21. Reuveni S, Granek R, Klafter J. Proteins: coexistence of stability and flexibility. Phys Rev Lett. 2008; 100(20): 208101. https://doi.org/10.1103/PhysRevLett.100.208101 PMID: 18518581 22. Neusius T, Daidone I, Sokolov IM, Smith JC. Subdiffusion in peptides originates from the fractal-like structure of configuration space. Phys Rev Lett. 2008; 100(18): 188103. https://doi.org/10.1103/ PhysRevLett.100.188103 PMID: 18518418 23. Lu HP, Xun L, Xie XS. Single-molecule enzymatic dynamics. Science. 1998; 282(5395): 1877–1882. https://doi.org/10.1126/science.282.5395.1877 PMID: 9836635 24. Hu X, Hong L, Smith MD, Neusius T, Cheng X, Smith JC. The dynamics of single protein molecules is non-equilibrium and self-similar over thirteen decades in time. Nat Phys. 2016; 12: 171–174. https:// doi.org/10.1038/nphys3553 25. Law AB, Sapienza PJ, Zhang J, Zuo X, Petit CM. Native State Volume Fluctuations in Proteins as a Mechanism for Dynamic Allostery. J Amer Chem Soc. 2017; 139(10): 3599–3602. https://doi.org/10. 1021/jacs.6b12058 26. Bahar I, Atilgan AR, Demirel MC, Erman B. Vibrational dynamics of folded proteins: significance of slow and fast motions in relation to function and stability. Phys Rev Lett. 1998; 80(12): 2733. https://doi.org/ 10.1103/PhysRevLett.80.2733 27. Bahar I, Lezon TR, Yang LW, Eyal E. Global dynamics of proteins: bridging between structure and func- tion. Annu Rev Biophys. 2010; 39, 23–42. https://doi.org/10.1146/annurev.biophys.093008.131258 PMID: 20192781 28. Meireles L, Gur M, Bakan A, Bahar I. Pre-existing soft modes of motion uniquely defined by native con- tact topology facilitate ligand binding to proteins. Protein Sci. 2011; 20(10), 1645–1658. https://doi.org/ 10.1002/pro.711 PMID: 21826755 29. Yang L, Song G, Jernigan RL. How well can we understand large-scale protein motions using normal modes of elastic network models?. Biophys J. 2007; 93(3): 920–929. https://doi.org/10.1529/biophysj. 106.095927 PMID: 17483178 30. Flechsig H, Togashi Y. Designed elastic networks: Models of complex protein machinery. Intl J Mol Sci. 2018; 19(10): 3152. https://doi.org/10.3390/ijms19103152 31. Ichiye T, Karplus M. Collective motions in proteins: a covariance analysis of atomic fluctuations in molecular dynamics and normal mode simulations. Proteins. 1991; 11(3): 205–217. https://doi.org/10. 1002/prot.340110305 PMID: 1749773 32. Case DA. Normal mode analysis of protein dynamics. Curr Opin Struct Biol. 1994; 4(2): 285–290. https://doi.org/10.1016/S0959-440X(94)90321-2 33. Wako H, Endo S. Normal mode analysis as a method to derive protein dynamics information from the Protein Data Bank. Biophys Rev. 2017; 9(6): 877–893. https://doi.org/10.1007/s12551-017-0330-2 PMID: 29103094 34. Stanley HE. Phase transitions and critical phenomena. Oxford: Clarendon Press; 1971. 35. Goldenfeld N. Lectures on phase transitions and the renormalization group. Boca Raton: CRC Press; 36. Bak P. How nature works: the science of self-organized criticality. New York: Copernicus Press; 37. Haliloglu T, Bahar I, Erman B. Gaussian dynamics of folded proteins. Phys Rev Lett. 1997; 79(16): 3090. https://doi.org/10.1103/PhysRevLett.79.3090 38. Haliloglu T, Erman B. Analysis of correlations between energy and residue fluctuations in native pro- teins and determination of specific sites for binding. Phys Rev Lett. 2009; 102(8): 088103. https://doi. org/10.1103/PhysRevLett.102.088103 PMID: 19257794 39. Cavagna A, Cimarelli A, Giardina I, Parisi G, Santagati R, Stefanini F, et al. Scale-free correlations in starling flocks. Proc Natl Acad Sci USA. 2010; 107(26): 11865–11870. https://doi.org/10.1073/pnas. 1005766107 PMID: 20547832 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 15 / 17 Long-range correlation in protein dynamics 40. Attanasi A, Cavagna A, Del Castello L, Giardina I, Melillo S, Parisi L, et al. Finite-size scaling as a way to probe near-criticality in natural swarms. Phys Rev Lett. 2014; 113(23): 238102. https://doi.org/10. 1103/PhysRevLett.113.238102 PMID: 25526161 41. Hinsen K. Structural flexibility in proteins: impact of the crystal environment. Bioinformatics. 2007; 24(4): 521–528. https://doi.org/10.1093/bioinformatics/btm625 PMID: 18089618 42. Fuglebakk E, Reuter N, Hinsen K. Evaluation of protein elastic network models based on an analysis of collective motions. J Chem Theor Comp. 2013; 9(12): 5618–5628. https://doi.org/10.1021/ct400399x 43. Yang L, Song G, Jernigan RL. Protein elastic network models and the ranges of cooperativity. Proc Natl Acad Sci USA. 2009; 106(30): 12347–12352. https://doi.org/10.1073/pnas.0902159106 PMID: 44. Rivoire O. Parsimonious evolutionary scenario for the origin of allostery and coevolution patterns in pro- teins. Phys Rev E. 2019; 100: 032411. https://doi.org/10.1103/PhysRevE.100.032411 PMID: 45. Eckmann JP, Rougemont J, Tlusty T. Colloquium: Proteins: The physics of amorphous evolving matter. Rev Mod Phys. 2019; 91: 031001. https://doi.org/10.1103/RevModPhys.91.031001 46. Lehnert U, Echols N, Milburn D, Engelman D, Gerstein M, Normal modes for predicting protein motions: a comprehensive database assessment and associated Web tool. Protein Sci. 2005; 14(3): 633–643. https://doi.org/10.1110/ps.04882105 PMID: 15722444 47. Atilgan AR, Turgut D, Atilgan C. Screened nonbonded interactions in native proteins manipulate optimal paths for robust residue communication. Biophys J. 2007; 92(9): 3052–3062. https://doi.org/10.1529/ biophysj.106.099440 PMID: 17293401 48. Atilgan C, Okan OB, Atilgan AR. Network-based models as tools hinting at nonevident protein function- ality. Annu Rev Biophys. 2012; 41: 205–225. https://doi.org/10.1146/annurev-biophys-050511-102305 PMID: 22404685 49. Mora T, Bialek W. Are biological systems poised at criticality?. J Stat Phys. 2011; 144(2): 268–302. https://doi.org/10.1007/s10955-011-0229-4 50. Honerkamp-Smith AR, Veatch SL, Keller SL. An introduction to critical points for biophysicists: observa- tions of compositional heterogeneity in lipid membranes. Biochim Biophys Acta. 2009; 1788(1): 53–63. https://doi.org/10.1016/j.bbamem.2008.09.010 PMID: 18930706 51. Chialvo DR. Emergent complex neural dynamics. Nat Phys. 2010; 6(10): 744–750. https://doi.org/10. 1038/nphys1803 52. Furusawa C, Kaneko K. Zipf’s law in gene expression. Phys Rev Lett. 2003; 90(8): 088102. https://doi. org/10.1103/PhysRevLett.90.088102 PMID: 12633463 53. Furusawa C, Kaneko K. Adaptation to optimal cell growth through self-organized criticality. Phys Rev Lett. 2012; 108(20): 208103. https://doi.org/10.1103/PhysRevLett.108.208103 PMID: 23003193 54. Chate ´ H, Muñoz M. Viewpoint: Insect Swarms Go Critical. Physics. 2014; 7: 120. https://doi.org/10. 1103/Physics.7.120 55. Muñoz MA. Colloquium: Criticality and dynamical scaling in living systems. Rev Mod Phys. 2018; 90(3): 031001. https://doi.org/10.1103/RevModPhys.90.031001 56. Newman ME, Girvan M. Finding and evaluating community structure in networks. Phys Rev E. 2004; 69(2): 026113. https://doi.org/10.1103/PhysRevE.69.026113 57. Newman ME. Modularity and community structure in networks. Proc Natl Acad Sci USA. 2006; 103(23): 8577–8582. https://doi.org/10.1073/pnas.0601602103 PMID: 16723398 58. Guimera R, Sales-Pardo M, Amaral LAN. Modularity from fluctuations in random graphs and complex networks. Phys Rev E. 2004; 70(2): 025101. https://doi.org/10.1103/PhysRevE.70.025101 59. Newman ME. Detecting community structure in networks. Euro Phys J B. 2004; 38(2): 321–330. https://doi.org/10.1140/epjb/e2004-00124-y 60. Thomassen E, Gielen G, Schu ¨ tz M, Schoehn G, Abrahams JP, Miller S, et al. The structure of the receptor-binding domain of the bacteriophage T4 short tail fibre reveals a knitted trimeric metal-binding fold. J Mol Biol. 2003; 331(2): 361–373. https://doi.org/10.1016/s0022-2836(03)00755-1 PMID: 61. Li J, Sun L, Xu C, Yu F, Zhou H, Zhao Y, et al. Structure insights into mechanisms of ATP hydrolysis and the activation of human heat-shock protein 90. Acta Biochim Biophys Sin. 2012; 44(4): 300–306. https://doi.org/10.1093/abbs/gms001 PMID: 22318716 62. Brunette TJ, Parmeggiani F, Huang PS, Bhabha G, Ekiert DC, Tsutakawa SE, et al. Exploring the repeat protein universe through computational protein design. Nature, 2015; 528(7583): 580–584. https://doi.org/10.1038/nature16162 PMID: 26675729 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 16 / 17 Long-range correlation in protein dynamics 63. Berman HM, Westbrook J, Feng Z, Gilliland G, Bhat TN, Weissig H, et al. The protein data bank. Nucleic Acids Res. 2000; 28(1): 235–242. https://doi.org/10.1093/nar/28.1.235 PMID: 10592235 64. Newman ME. Spectral methods for community detection and graph partitioning. Phys Rev E. 2013; 88(4): 042822. https://doi.org/10.1103/PhysRevE.88.042822 65. Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E. Fast unfolding of communities in large networks. J Stat Mech. 2008; 2008(10): P10008. https://doi.org/10.1088/1742-5468/2008/10/P10008 PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007670 February 13, 2020 17 / 17

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