Bulletin of Mathematical Biology

Sainudiin, Raazesh; Thornton, Kevin; Harlow, Jennifer; Booth, James; Stillman, Michael; Yoshida, Ruriko; Griffiths, Robert; McVean, Gil; Donnelly, Peter

doi: 10.1007/s11538-010-9605-5pmid: 21181503

Evaluating the likelihood function of parameters in highly-structured population genetic models from extant deoxyribonucleic acid (DNA) sequences is computationally prohibitive. In such cases, one may approximately infer the parameters from summary statistics of the data such as the site-frequency-spectrum (SFS) or its linear combinations. Such methods are known as approximate likelihood or Bayesian computations. Using a controlled lumped Markov chain and computational commutative algebraic methods, we compute the exact likelihood of the SFS and many classical linear combinations of it at a non-recombining locus that is neutrally evolving under the infinitely-many-sites mutation model. Using a partially ordered graph of coalescent experiments around the SFS, we provide a decision-theoretic framework for approximate sufficiency. We also extend a family of classical hypothesis tests of standard neutrality at a non-recombining locus based on the SFS to a more powerful version that conditions on the topological information provided by the SFS.

Bóna, Miklós; Sitharam, Meera; Vince, Andrew

doi: 10.1007/s11538-010-9606-4pmid: 21174231

This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the capsid is modeled by a polyhedron whose facets represent the monomers. The assembly process is modeled by a rooted tree, the leaves representing the facets of the polyhedron, the root representing the assembled polyhedron, and the internal vertices representing intermediate stages of assembly (subsets of facets). Besides its virological motivation, the enumeration of orbits of trees under the action of a finite group is of independent mathematical interest. If G is a finite group acting on a finite set X, then there is a natural induced action of G on the set $\mathcal{T}_{X}$ of trees whose leaves are bijectively labeled by the elements of X. If G acts simply on X, then |X|:=|X n |=n⋅|G|, where n is the number of G-orbits in X. The basic combinatorial results in this paper are (1) a formula for the number of orbits of each size in the action of G on $\mathcal{T}_{X_{n}}$ , for every n, and (2) a simple algorithm to find the stabilizer of a tree $\tau\in\mathcal{T} _{X}$ in G that runs in linear time and does not need memory in addition to its input tree. These results help to clarify the effect of symmetry on the probability and number of assembly pathways for icosahedral viral capsids, and more generally for any finite, symmetric macromolecular assembly.

Siebert, Heike

doi: 10.1007/s11538-010-9609-1pmid: 21170598

A discrete model of a biological regulatory network can be represented by a discrete function that contains all available information on interactions between network components and the rules governing the evolution of the network in a finite state space. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem. In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in some region of state space. We state rules to explicitly construct attractors of the system from subnetwork attractors. Using the results, we formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the graph representing the structure of the system. In addition, we discuss approaches to finding symbolic steady states. We focus both on dynamics derived via synchronous as well as asynchronous update rules. Lastly, we illustrate the results by analyzing a model of T helper cell differentiation.

Blöchl, Florian; Wittmann, Dominik; Theis, Fabian

doi: 10.1007/s11538-010-9604-6pmid: 21181504

Signaling networks are abundant in higher organisms. They play pivotal roles, e.g., during embryonic development or within the immune system. In this contribution, we study the combined effect of the various kinetic parameters on the dynamics of signal transduction. To this end, we consider hierarchical complex systems as prototypes of signaling networks. For given topology, the output of these networks is determined by an interplay of the single parameters. For different kinetics, we describe this by algebraic expressions, the so-called effective parameters.