On equilibrium properties of evolutionary multi-player games with random payoff matrices

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman-Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.     

Interaction of fibronectin with heparin in model extracellular matrices: role of arginine residues and sulfate groups

The interaction of heparin with the NH2-terminal domain of human plasma fibronectin was studied by using matrix-driven translocation, an assay for the adhesion of extracellular macromolecules with cell or particle surfaces within artificial collagen matrices. Partial desulfation of heparin rendered it ineffective in competitively inhibiting the interaction of the fibronectin NH2-terminal domain with heparin-coated particles, suggesting a role for sulfate groups of heparin in the interaction. Analysis of the fibronectin domain in terms of its primary structure, its proposed organization into type I modules, and its hydrophilic and flexible segments led to the identification of several arginine-containing sites of potential interaction with the sulfate groups of heparin. Modification of increasing numbers of arginine side chains with 1,2-cyclohexanedione under mild conditions eventually led to decreases in translocation-promoting activity, and of heparin binding capacity as measured in a gel-shift assay, but the major portions of these functions were retained even when the four most accessible arginines (attributed to sites in and adjacent to the large loops of the type I modules) were modified. With the modification of additional arginines (attributed to sites in the small loops), both functions were lost. The peptide Gly-Arg-Gly, corresponding to a repeated determinant at the tips of two small loops, inhibited translocation, but arginine alone did not. Cleavage of the large loops by CNBr also led to loss of translocation-promoting activity. The correspondence between the molecular determinants of matrix-driven translocation and those previously found for mesenchymal morphogenesis indicates the utility of this system in the analysis of adhesive interactions of biological importance.     

A tale of two matrices: multivariate approaches in evolutionary biology

Two symmetric matrices underlie our understanding of microevolutionary change. The first is the matrix of nonlinear selection gradients (gamma) which describes the individual fitness surface. The second is the genetic variance-covariance matrix (G) that influences the multivariate response to selection. A common approach to the empirical analysis of these matrices is the element-by-element testing of significance, and subsequent biological interpretation of pattern based on these univariate and bivariate parameters. Here, I show why this approach is likely to misrepresent the genetic basis of quantitative traits, and the selection acting on them in many cases. Diagonalization of square matrices is a fundamental aspect of many of the multivariate statistical techniques used by biologists. Applying this, and other related approaches, to the analysis of the structure of gamma and G matrices, gives greater insight into the form and strength of nonlinear selection, and the availability of genetic variance for multiple traits.     

Dynamic properties of water bound to matrices of natural DNA and model complexes

From experimental data on the hydration energetics of nucleic acids obtained by differential scanning calorimetry under isothermal conditions, dielectric relaxation time tau d and "free volume" Vf occupied by water molecules in hydration shells of natural DNA and model polyribonucleotides were calculated. In addition, systems consisting of dinucleotides ApA, TpT, UpU, TpU, UpT and water clusters of various sizes (from 20 to 400 water molecules) were studied by Monte Carlo computer simulation. It was shown that, as water content in systems increases, the dynamic characteristics of bound water obtained with both methods approached the values for bulk water.     

Two major groups of colicin factors: Their evolutionary significance

Summary Eleven colicin factors have been placed in two groups defined by a number of physiological criteria such as the effect of the host recA (recombination-proficiency) allele on colicin titres and the maximum number of copies of the colicin factor per chromosome. The fundamental difference between the two groups may lie in the molecular weight of the plasmid DNA: one group is about 5x10⁶, the other about 70x10⁶. The colicins specified by members of each group are also related. Colicin factors within the same group may therefore resemble each other because they are descended from the same ancestral plasmid which was either EK-like of low molecular weight or BIV-like of high molecular weight.Aided by grants from the Science Research Council and the Central Research Fund of the University of London.     

Network properties, species abundance and evolution in a model of evolutionary ecology

We study the evolution of the network properties of a populated network embedded in a genotype space characterized by either a low or a high number of potential links, with particular emphasis on the connectivity and clustering. Evolution produces two distinct types of network. When a specific genotype is only able to influence a few other genotypes, the ecosystem consists of separate non-interacting clusters (i.e. isolated compartments) in genotype space. When different types may influence a large number of other sites, the network becomes one large interconnected cluster. The distribution of interaction strengths--but not the number of connections--changes significantly with time. We find that the species abundance is only realistic for a high level of species connectivity. This suggests that real ecosystems form one interconnected whole in which selection leads to stronger interactions between the different types. Analogies with niche and neutral theory and assembly models are also considered.     

Genomic analysis of the TRIM family reveals two groups of genes with distinct evolutionary properties

Background  

The TRIM family is composed of multi-domain proteins that display the Tripartite Motif (RING, B-box and Coiled-coil) that can be associated with a C-terminal domain. TRIM genes are involved in ubiquitylation and are implicated in a variety of human pathologies, from Mendelian inherited disorders to cancer, and are also involved in cellular response to viral infection.     

Mutation matrices and physical-chemical properties: Correlations and implications

To investigate how the properties of individual amino acids result in proteins with particular structures and functions, we have examined the correlations between previously derived structure-dependent mutation rates and changes in various physical-chemical properties of the amino acids such as volume, charge, α-helical and β-sheet propensity, and hydrophobicity. In most cases we found the ΔG of transfer from octanol to water to be the best model for evolutionary constraints, in contrast to the much weaker correlation with the ΔG of transfer from cyclohexane to water, a property found to be highly correlated to changes in stability in site-directed mutagenesis studies. This suggests that natural evolution may follow different rules than those suggested by results obtained in the laboratory. A high degree of conservation of a surface residue's relative hydrophobicity was also observed, a fact that cannot be explained by constraints on protein stability but that may reflect the consequences of the reverse-hydrophobic effect. Local propensity, especially α-helical propensity, is rather poorly conserved during evolution, indicating that non-local interactions dominate protein structure formation. We found that changes in volume were important in specific cases, most significantly in transitions among the hydrophobic residues in buried locations. To demonstrate how these techniques could be used to understand particular protein families, we derived and analyzed mutation matrices for the hypervariable and framework regions of antibody light chain V regions. We found a surprisingly high conservation of hydrophobicity in the hypervariable region, possibly indicating an important role for hydrophobicity in antigen recognition. Proteins 27:336–344, 1997. © 1997 Wiley-Liss, Inc.     

Memory in retroviral quasispecies: experimental evidence and theoretical model for human immunodeficiency virus

Viral quasispecies may possess a molecular memory of their past evolutionary history, imprinted on minority components of the mutant spectrum. Here we report experimental evidence and a theoretical model for memory in retroviral quasispecies in vivo. Apart from replicative memory associated with quasispecies dynamics, retroviruses may harbour a "cellular" or "anatomical" memory derived from their integrative cycle and the presence of viral reservoirs in body compartments. Three independent sets of data exemplify the two kinds of memory in human immunodeficiency virus type 1 (HIV-1). The data provide evidence of re-emergence of sequences that were hidden in cellular or anatomical compartments for extended periods of infection, and recovery of a quasispecies from pre-existing genomes. We develop a three-component model that incorporates the essential features of the quasispecies dynamics of retroviruses exposed to selective pressures. Significantly, a numerical study based on this model is in agreement with the experimental data, further supporting the existence of both replicative and reservoir memory in retroviral quasispecies.     

Covariance reducing models: An alternative to spectral modelling of covariance matrices

We introduce covariance reducing models for studying the samplecovariance matrices of a random vector observed in differentpopulations. The models are based on reducing the sample covariancematrices to an informational core that is sufficient to characterizethe variance heterogeneity among the populations. They possessuseful equivariance properties and provide a clear alternativeto spectral models for covariance matrices.     

The topology of the possible: formal spaces underlying patterns of evolutionary change

The current implementation of the Neo-Darwinian model of evolution typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space equipped with a notion of distance, for example, a Euclidean vector space. Recent computational work on a biophysical genotype-phenotype model based on the folding of RNA sequences into secondary structures suggests a rather different picture. If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and is formalized by an unfamiliar structure, known as a pre-topology. Patterns of phenotypic evolution-such as punctuation, irreversibility, modularity--result naturally from the properties of this space. The classical framework, however, addresses these patterns by exclusively invoking natural selection on suitably imposed fitness landscapes. We propose to extend the explanatory level for phenotypic evolution from fitness considerations alone to include the topological structure of phenotype space as induced by the genotype-phenotype map. We introduce the mathematical concepts and tools necessary to formalize the notion of accessibility pre-topology relative to which we can speak of continuity in the genotype-phenotype map and in evolutionary trajectories. We connect the factorization of a pre-topology into a product space with the notion of phenotypic character and derive a condition for factorization. Based on anecdotal evidence from the RNA model, we conjecture that this condition is not globally fulfilled, but rather confined to regions where the genotype-phenotype map is continuous. Equivalently, local regions of genotype space on which the map is discontinuous are associated with the loss of character autonomy. This is consistent with the importance of these regions for phenotypic innovation. The intention of the present paper is to offer a perspective, a framework to implement this perspective, and a few results illustrating how this framework can be put to work. The RNA case is used as an example throughout the text.     

Effect of anisotropic bending rigidity and finite twisting rigidity on statistical properties of DNA model filaments

The persistence length and effective long-range bending rigidity are derived for a discrete model of an anisotropically bending filament and shown to be independent of the torsional rigidity. The twisting persistence length is found to be independent of the anisotropic bending rigidity. Other statistical properties are briefly discussed, including the dependence of tangent vector projections on contour length. The dependence of a tensor contraction on contour length is derived for an isotropically bending filament with no equilibrium twist.     

Tangled nature: a model of evolutionary ecology

We discuss a simple model of co-evolution. In order to emphasize the effect of interaction between individuals, the entire population is subjected to the same physical environment. Species are emergent structures and extinction, origination and diversity are entirely a consequence of co-evolutionary interaction between individuals. For comparison, we consider both asexual and sexually reproducing populations. In either case, the system evolves through periods of hectic reorganization separated by periods of coherent stable coexistence.     

Amyloid peptides derived from CsgA and FapC modify the viscoelastic properties of biofilm model matrices

The bacterial biofilm is a complex environment of cells, which secrete a matrix made of various components, mainly polysaccharides and proteins. An understanding of the precise role of these components in the stability and dynamics of biofilm architecture would be a great advantage for the improvement of anti-biofilm strategies. Here, artificial biofilm matrices made of polysaccharides and auto-assembled peptides were designed, and the influence of bacterial amyloid proteins on the mechanical properties of the biofilm matrix was studied. The model polysaccharides methylcellulose and alginate and peptides derived from the amyloid proteins curli and FapC found in biofilms of Enterobacteriaceae and Pseudomonas, respectively, were used. Rheological measurements showed that the amyloid peptides do not prevent the gelation of the polysaccharides but influence deformation of the matrices under shear stress and modify the gel elastic response. Hence the secretion of amyloids could be for the biofilm a way of adapting to environmental changes.