Extreme Selection Unifies Evolutionary Game Dynamics in Finite and Infinite Populations

We show that when selection is extreme—the fittest strategy always reproduces or is imitated—the unequivalence between the possible evolutionary game scenarios in finite and infinite populations resolves, in the sense that the three generic outcomes—dominance, coexistence, and mutual exclusion—emerge in well-mixed populations of any size. We consider the simplest setting of a 2-player-2-strategy symmetric game and the two most common microscopic definitions of strategy spreading—the frequency-dependent Moran process and the imitation process by pairwise comparison—both in the case allowing any intensity of selection. We show that of the seven different invasion and fixation scenarios that are generically possible in finite populations—fixation being more or less likely to occur and rapid compared to the neutral game—the three that are possible in large populations are the same three that occur for sufficiently strong selection: (1) invasion and fast fixation of one strategy; (2) mutual invasion and slow fixation of one strategy; (3) no invasion and no fixation. Moreover (and interestingly), in the limit of extreme selection 2 becomes mutual invasion and no fixation, a case not possible for finite intensity of selection that better corresponds to the deterministic case of coexistence. In the extreme selection limit, we also derive the large population deterministic limit of the two considered stochastic processes.     

The Evolutionary Dynamics of Protein-Protein Interaction Networks Inferred from the Reconstruction of Ancient Networks

Cellular functions are based on the complex interplay of proteins, therefore the structure and dynamics of these protein-protein interaction (PPI) networks are the key to the functional understanding of cells. In the last years, large-scale PPI networks of several model organisms were investigated. A number of theoretical models have been developed to explain both the network formation and the current structure. Favored are models based on duplication and divergence of genes, as they most closely represent the biological foundation of network evolution. However, studies are often based on simulated instead of empirical data or they cover only single organisms. Methodological improvements now allow the analysis of PPI networks of multiple organisms simultaneously as well as the direct modeling of ancestral networks. This provides the opportunity to challenge existing assumptions on network evolution. We utilized present-day PPI networks from integrated datasets of seven model organisms and developed a theoretical and bioinformatic framework for studying the evolutionary dynamics of PPI networks. A novel filtering approach using percolation analysis was developed to remove low confidence interactions based on topological constraints. We then reconstructed the ancient PPI networks of different ancestors, for which the ancestral proteomes, as well as the ancestral interactions, were inferred. Ancestral proteins were reconstructed using orthologous groups on different evolutionary levels. A stochastic approach, using the duplication-divergence model, was developed for estimating the probabilities of ancient interactions from today''s PPI networks. The growth rates for nodes, edges, sizes and modularities of the networks indicate multiplicative growth and are consistent with the results from independent static analysis. Our results support the duplication-divergence model of evolution and indicate fractality and multiplicative growth as general properties of the PPI network structure and dynamics.     

Evolutionary Game Dynamics in Populations with Heterogenous Structures

Evolutionary graph theory is a well established framework for modelling the evolution of social behaviours in structured populations. An emerging consensus in this field is that graphs that exhibit heterogeneity in the number of connections between individuals are more conducive to the spread of cooperative behaviours. In this article we show that such a conclusion largely depends on the individual-level interactions that take place. In particular, averaging payoffs garnered through game interactions rather than accumulating the payoffs can altogether remove the cooperative advantage of heterogeneous graphs while such a difference does not affect the outcome on homogeneous structures. In addition, the rate at which game interactions occur can alter the evolutionary outcome. Less interactions allow heterogeneous graphs to support more cooperation than homogeneous graphs, while higher rates of interactions make homogeneous and heterogeneous graphs virtually indistinguishable in their ability to support cooperation. Most importantly, we show that common measures of evolutionary advantage used in homogeneous populations, such as a comparison of the fixation probability of a rare mutant to that of the resident type, are no longer valid in heterogeneous populations. Heterogeneity causes a bias in where mutations occur in the population which affects the mutant''s fixation probability. We derive the appropriate measures for heterogeneous populations that account for this bias.     

Stochastic Fluctuations Through Intrinsic Noise in Evolutionary Game Dynamics

A one-step (birth–death) process is used to investigate stochastic noise in an elementary two-phenotype evolutionary game model based on a payoff matrix. In this model, we assume that the population size is finite but not fixed and that all individuals have, in addition to the frequency-dependent fitness given by the evolutionary game, the same background fitness that decreases linearly in the total population size. Although this assumption guarantees population extinction is a globally attracting absorbing barrier of the Markov process, sample trajectories do not illustrate this result even for relatively small carrying capacities. Instead, the observed persistent transient behavior can be analyzed using the steady-state statistics (i.e., mean and variance) of a stochastic model for intrinsic noise that assumes the population does not go extinct. It is shown that there is good agreement between the theory of these statistics and the simulation results. Furthermore, the ESS of the evolutionary game can be used to predict the mean steady state.     

Continuous Probabilistic Analysis to Evolutionary Game Dynamics in Finite Populations

Evolutionary game dynamics of two strategies in finite population is studied by continuous probabilistic approach. Besides frequency dependent selection, mutation was also included in this study. The equilibrium probability density functions of abundance, expected time to extinction or fixation were derived and their numerical solutions are calculated as illustrations. Meanwhile, individual-based computer simulations are also done. A comparison reveals the consistency between theoretical analysis and simulations.     

蛋白质相互作用网络进化分析研究进展

近年来,随着高通量实验技术的发展和广泛应用,越来越多可利用的蛋白质相互作用网络数据开始出现．这些数据为进化研究提供了新的视角．从蛋白质、蛋白质相互作用、模体、模块直到整个网络五个层次,综述了近年来蛋白质相互作用网络进化研究领域的主要进展,侧重于探讨蛋白质相互作用、模体、模块直到整个网络对蛋白质进化的约束作用,以及蛋白质相互作用网络不同于随机网络特性的起源和进化等问题．总结了前人工作给学术界的启示,探讨了该领域未来可能的发展方向．     

Evolutionary Capacitance and Control of Protein Stability in Protein-Protein Interaction Networks

In addition to their biological function, protein complexes reduce the exposure of the constituent proteins to the risk of undesired oligomerization by reducing the concentration of the free monomeric state. We interpret this reduced risk as a stabilization of the functional state of the protein. We estimate that protein-protein interactions can account for of additional stabilization; a substantial contribution to intrinsic stability. We hypothesize that proteins in the interaction network act as evolutionary capacitors which allows their binding partners to explore regions of the sequence space which correspond to less stable proteins. In the interaction network of baker''s yeast, we find that statistically proteins that receive higher energetic benefits from the interaction network are more likely to misfold. A simplified fitness landscape wherein the fitness of an organism is inversely proportional to the total concentration of unfolded proteins provides an evolutionary justification for the proposed trends. We conclude by outlining clear biophysical experiments to test our predictions.     

Simple Topological Features Reflect Dynamics and Modularity in Protein Interaction Networks

The availability of large-scale protein-protein interaction networks for numerous organisms provides an opportunity to comprehensively analyze whether simple properties of proteins are predictive of the roles they play in the functional organization of the cell. We begin by re-examining an influential but controversial characterization of the dynamic modularity of the S. cerevisiae interactome that incorporated gene expression data into network analysis. We analyse the protein-protein interaction networks of five organisms, S. cerevisiae, H. sapiens, D. melanogaster, A. thaliana, and E. coli, and confirm significant and consistent functional and structural differences between hub proteins that are co-expressed with their interacting partners and those that are not, and support the view that the former tend to be intramodular whereas the latter tend to be intermodular. However, we also demonstrate that in each of these organisms, simple topological measures are significantly correlated with the average co-expression of a hub with its partners, independent of any classification, and therefore also reflect protein intra- and inter- modularity. Further, cross-interactomic analysis demonstrates that these simple topological characteristics of hub proteins tend to be conserved across organisms. Overall, we give evidence that purely topological features of static interaction networks reflect aspects of the dynamics and modularity of interactomes as well as previous measures incorporating expression data, and are a powerful means for understanding the dynamic roles of hubs in interactomes.     

Groups on Groups: Some Dynamics and Possible Resolution of the Units of Selection Debates in Evolutionary Biology

David Hull's analysis of conceptual change in science, as presentedin his book, Science as a Process (1988), provides a useful framework for understanding one of the scientific controversies in which he actively and constructively intervened, the units of selectiondebates in evolutionary biology. What follows is a brief overview ofthose debates and some reflections on them.     

Local Replicator Dynamics: A Simple Link Between Deterministic and Stochastic Models of Evolutionary Game Theory

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner’s dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.     

Cyber War Game in Temporal Networks

In a cyber war game where a network is fully distributed and characterized by resource constraints and high dynamics, attackers or defenders often face a situation that may require optimal strategies to win the game with minimum effort. Given the system goal states of attackers and defenders, we study what strategies attackers or defenders can take to reach their respective system goal state (i.e., winning system state) with minimum resource consumption. However, due to the dynamics of a network caused by a node’s mobility, failure or its resource depletion over time or action(s), this optimization problem becomes NP-complete. We propose two heuristic strategies in a greedy manner based on a node’s two characteristics: resource level and influence based on k-hop reachability. We analyze complexity and optimality of each algorithm compared to optimal solutions for a small-scale static network. Further, we conduct a comprehensive experimental study for a large-scale temporal network to investigate best strategies, given a different environmental setting of network temporality and density. We demonstrate the performance of each strategy under various scenarios of attacker/defender strategies in terms of win probability, resource consumption, and system vulnerability.     

Extrapolating Weak Selection in Evolutionary Games

In evolutionary games, reproductive success is determined by payoffs. Weak selection means that even large differences in game outcomes translate into small fitness differences. Many results have been derived using weak selection approximations, in which perturbation analysis facilitates the derivation of analytical results. Here, we ask whether results derived under weak selection are also qualitatively valid for intermediate and strong selection. By “qualitatively valid” we mean that the ranking of strategies induced by an evolutionary process does not change when the intensity of selection increases. For two-strategy games, we show that the ranking obtained under weak selection cannot be carried over to higher selection intensity if the number of players exceeds two. For games with three (or more) strategies, previous examples for multiplayer games have shown that the ranking of strategies can change with the intensity of selection. In particular, rank changes imply that the most abundant strategy at one intensity of selection can become the least abundant for another. We show that this applies already to pairwise interactions for a broad class of evolutionary processes. Even when both weak and strong selection limits lead to consistent predictions, rank changes can occur for intermediate intensities of selection. To analyze how common such games are, we show numerically that for randomly drawn two-player games with three or more strategies, rank changes frequently occur and their likelihood increases rapidly with the number of strategies . In particular, rank changes are almost certain for , which jeopardizes the predictive power of results derived for weak selection.     

Using Evolutionary Conserved Modules in Gene Networks as a Strategy to Leverage High Throughput Gene Expression Queries

Background

Large-scale gene expression studies have not yielded the expected insight into genetic networks that control complex processes. These anticipated discoveries have been limited not by technology, but by a lack of effective strategies to investigate the data in a manageable and meaningful way. Previous work suggests that using a pre-determined seed-network of gene relationships to query large-scale expression datasets is an effective way to generate candidate genes for further study and network expansion or enrichment. Based on the evolutionary conservation of gene relationships, we test the hypothesis that a seed network derived from studies of retinal cell determination in the fly, Drosophila melanogaster, will be an effective way to identify novel candidate genes for their role in mouse retinal development.

Methodology/Principal Findings

Our results demonstrate that a number of gene relationships regulating retinal cell differentiation in the fly are identifiable as pairwise correlations between genes from developing mouse retina. In addition, we demonstrate that our extracted seed-network of correlated mouse genes is an effective tool for querying datasets and provides a context to generate hypotheses. Our query identified 46 genes correlated with our extracted seed-network members. Approximately 54% of these candidates had been previously linked to the developing brain and 33% had been previously linked to the developing retina. Five of six candidate genes investigated further were validated by experiments examining spatial and temporal protein expression in the developing retina.

Conclusions/Significance

We present an effective strategy for pursuing a systems biology approach that utilizes an evolutionary comparative framework between two model organisms, fly and mouse. Future implementation of this strategy will be useful to determine the extent of network conservation, not just gene conservation, between species and will facilitate the use of prior biological knowledge to develop rational systems-based hypotheses.     

Remarks on the Evolutionary Effect of Natural Selection

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.     

Convergent Evolutionary Paths in Biological and Technological Networks

Technology seems to follow a different type of evolutionary dynamic when compared with biological systems. As pointed out by Francois Jacob, evolution takes place by means of extensive tinkering and does not foresee the future. Engineers will typically have a well-defined purpose and are not—in principle—constrained by the available technological constraints. However, the truth is that technological change shares much more than we might suspect with the patterns and processes displayed by evolution. Using case studies from both protein maps and large-scale software networks, we show that several key traits, such as scale-free structure and modularity, are shared by both man-made and biological evolving systems. Surprisingly, we find convergent evolution in several key features of software systems, indicating that strong constraints are at work. Such constraints force engineers to extensively reuse already constructed parts, thus de facto tinkering with their designs in a way similar to the duplication–diversification mechanism driving genome growth. The evolution of these systems reveals that well-defined patterns are obtained “for free.” Some of them can be properly interpreted as technological spandrels.     

The Impact of Macroscopic Epistasis on Long-Term Evolutionary Dynamics

Genetic interactions can strongly influence the fitness effects of individual mutations, yet the impact of these epistatic interactions on evolutionary dynamics remains poorly understood. Here we investigate the evolutionary role of epistasis over 50,000 generations in a well-studied laboratory evolution experiment in Escherichia coli. The extensive duration of this experiment provides a unique window into the effects of epistasis during long-term adaptation to a constant environment. Guided by analytical results in the weak-mutation limit, we develop a computational framework to assess the compatibility of a given epistatic model with the observed patterns of fitness gain and mutation accumulation through time. We find that a decelerating fitness trajectory alone provides little power to distinguish between competing models, including those that lack any direct epistatic interactions between mutations. However, when combined with the mutation trajectory, these observables place strong constraints on the set of possible models of epistasis, ruling out many existing explanations of the data. Instead, we find that the data are consistent with a “two-epoch” model of adaptation, in which an initial burst of diminishing-returns epistasis is followed by a steady accumulation of mutations under a constant distribution of fitness effects. Our results highlight the need for additional DNA sequencing of these populations, as well as for more sophisticated models of epistasis that are compatible with all of the experimental data.