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.     

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.     

Fixation of Strategies for an Evolutionary Game in Finite Populations

A stochastic evolutionary dynamics of two strategies given by 2x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with site dependent hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.     

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.     

Strategy Selection in Evolutionary Game Dynamics on Group Interaction Networks

Evolutionary game theory provides an appropriate tool for investigating the competition and diffusion of behavioral traits in biological or social populations. A core challenge in evolutionary game theory is the strategy selection problem: Given two strategies, which one is favored by the population? Recent studies suggest that the answer depends not only on the payoff functions of strategies but also on the interaction structure of the population. Group interactions are one of the fundamental interactive modes within populations. This work aims to investigate the strategy selection problem in evolutionary game dynamics on group interaction networks. In detail, the strategy selection conditions are obtained for some typical networks with group interactions. Furthermore, the obtained conditions are applied to investigate selection between cooperation and defection in populations. The conditions for evolution of cooperation are derived for both the public goods game and volunteer’s dilemma game. Numerical experiments validate the above analytical results.     

Catalysis of Protein Folding by Chaperones Accelerates Evolutionary Dynamics in Adapting Cell Populations

Although molecular chaperones are essential components of protein homeostatic machinery, their mechanism of action and impact on adaptation and evolutionary dynamics remain controversial. Here we developed a physics-based ab initio multi-scale model of a living cell for population dynamics simulations to elucidate the effect of chaperones on adaptive evolution. The 6-loci genomes of model cells encode model proteins, whose folding and interactions in cellular milieu can be evaluated exactly from their genome sequences. A genotype-phenotype relationship that is based on a simple yet non-trivially postulated protein-protein interaction (PPI) network determines the cell division rate. Model proteins can exist in native and molten globule states and participate in functional and all possible promiscuous non-functional PPIs. We find that an active chaperone mechanism, whereby chaperones directly catalyze protein folding, has a significant impact on the cellular fitness and the rate of evolutionary dynamics, while passive chaperones, which just maintain misfolded proteins in soluble complexes have a negligible effect on the fitness. We find that by partially releasing the constraint on protein stability, active chaperones promote a deeper exploration of sequence space to strengthen functional PPIs, and diminish the non-functional PPIs. A key experimentally testable prediction emerging from our analysis is that down-regulation of chaperones that catalyze protein folding significantly slows down the adaptation dynamics.     

Feedback between Population and Evolutionary Dynamics Determines the Fate of Social Microbial Populations

The evolutionary spread of cheater strategies can destabilize populations engaging in social cooperative behaviors, thus demonstrating that evolutionary changes can have profound implications for population dynamics. At the same time, the relative fitness of cooperative traits often depends upon population density, thus leading to the potential for bi-directional coupling between population density and the evolution of a cooperative trait. Despite the potential importance of these eco-evolutionary feedback loops in social species, they have not yet been demonstrated experimentally and their ecological implications are poorly understood. Here, we demonstrate the presence of a strong feedback loop between population dynamics and the evolutionary dynamics of a social microbial gene, SUC2, in laboratory yeast populations whose cooperative growth is mediated by the SUC2 gene. We directly visualize eco-evolutionary trajectories of hundreds of populations over 50–100 generations, allowing us to characterize the phase space describing the interplay of evolution and ecology in this system. Small populations collapse despite continual evolution towards increased cooperative allele frequencies; large populations with a sufficient number of cooperators “spiral” to a stable state of coexistence between cooperator and cheater strategies. The presence of cheaters does not significantly affect the equilibrium population density, but it does reduce the resilience of the population as well as its ability to adapt to a rapidly deteriorating environment. Our results demonstrate the potential ecological importance of coupling between evolutionary dynamics and the population dynamics of cooperatively growing organisms, particularly in microbes. Our study suggests that this interaction may need to be considered in order to explain intraspecific variability in cooperative behaviors, and also that this feedback between evolution and ecology can critically affect the demographic fate of those species that rely on cooperation for their survival.     

Invasion Waves in Populations with Excitable Dynamics

Whilst the most obvious mechanism for a biological invasion is the occupation of a new territory as a result of direct ingress by individuals of the invading population, a more subtle “invasion” may occur without significant motion of invading individuals if the population dynamics in a predator prey scenario has an “excitable” character. Here, “excitable” means that a local equilibrium state, either of coexistence of predator and prey, or of prey only, may, when disturbed by a small perturbation, switch to a new, essentially invaded state. In an invasion of this type little spatial movement of individuals occurs, but a wave of rapid change of population level nevertheless travels through the invaded territory. In this article we summarise and review recent modelling research which shows that the macroscopic features of these invasion waves depend strongly on the detailed spatial dynamics of the predator–prey relationship; the models assume simple (linear) diffusion and pursuit-evasion, represented by (non-linear) cross-diffusion, as examples. In the context of plankton population dynamics, such waves may be produced by sudden injections of nutrient and consequent rapid increase in plankton populations, brought about, for example, by the upwelling caused by a passing atmospheric low pressure system.     

Evolutionary Dynamics of Insertion Sequences in Relation to the Evolutionary Histories of the Chromosome and Symbiotic Plasmid Genes of Rhizobium etli Populations

Insertion sequences (IS) are mobile genetic elements that are distributed in many prokaryotes. In particular, in the genomes of the symbiotic nitrogen-fixing bacteria collectively known as rhizobia, IS are fairly abundant in plasmids or chromosomal islands that carry the genes needed for symbiosis. Here, we report an analysis of the distribution and genetic conservation of the IS found in the genome of Rhizobium etli CFN42 in a collection of 87 Rhizobium strains belonging to populations with different geographical origins. We used PCR to generate presence/absence profiles of the 39 IS found in R. etli CFN42 and evaluated whether the IS were located in consistent genomic contexts. We found that the IS from the symbiotic plasmid were frequently present in the analyzed strains, whereas the chromosomal IS were observed less frequently. We then examined the evolutionary dynamics of these strains based on a population genetic analysis of two chromosomal housekeeping genes (glyA and dnaB) and three symbiotic sequences (nodC and the two IS elements). Our results indicate that the IS contained within the symbiotic plasmid have a higher degree of genomic context conservation, lower nucleotide diversity and genetic differentiation, and fewer recombination events than the chromosomal housekeeping genes. These results suggest that the R. etli populations diverged recently in Mexico, that the symbiotic plasmid also had a recent origin, and that the IS elements have undergone a process of cyclic infection and expansion.Insertion sequences (IS) are the smallest transposable elements found in prokaryotes (usually less than 3 kb in size). They encode a transposase and may also encode small hypothetical proteins (4, 9). IS are distinguished by their ability to move within prokaryotic replicons, including both the chromosome and plasmids, and to copy themselves into various genomic sites. In this manner, IS elements can inactivate or alter the expression of adjacent genes (4). When IS occur in two or more identical copies within a genome, they can participate in various types of genetic rearrangements (e.g., duplications, inversions, and deletions), suggesting that IS may play an important role in the evolution of their hosts by promoting genomic plasticity (34). Due to these evolutionary dynamics, the diversity and distribution of IS elements differ greatly between taxa and even within strains of the same species (27).Various theories have been proposed to explain the evolution of IS elements in laboratory model strains and environmental bacterial populations (8, 18, 25, 29). Two main hypotheses seek to explain how these elements are maintained over the long term in their host genomes. The first proposes that they occasionally generate beneficial mutations and therefore may represent a selective advantage to their hosts (34). The second suggests that IS elements are genomic parasites that are maintained by their high rate of transposition and might be disseminated among different bacterial lineages by horizontal gene transfer (HGT). Data supporting the second hypothesis have shown that some IS elements may transpose at high rates upon entering a new host (42). After the initial infection, however, purifying selection may continuously remove these elements from the genome. Thus, IS may undergo an infection-expansion-extinction cycle that allows them to remain in different bacterial populations within the gene pool (42). These two hypotheses are not contradictory, and the evolutionary dynamics and distribution of IS may differ greatly depending on several factors, including (most notably) the rate of transposition and HGT, as well as selective pressures, population size, and the host''s habitat (6, 18, 21, 25, 27, 29).In the nitrogen-fixing symbiotic bacteria of the genera Rhizobium, Sinorhizobium, Mesorhizobium, Bradyrhizobium (of the alphaproteobacteria), Cupriavidus, and Burkholderia (of the betaproteobacteria), IS are particularly abundant in symbiotic plasmids (pSym) and symbiotic chromosomal islands (SI) (2, 12, 14, 19, 20, 43). SI and pSym include most of the genes needed to establish symbiosis in the roots of leguminous plants through nodule formation and nitrogen fixation (11). It is generally believed that these elements entered the rhizobial genomes through HGT (39, 40). Comparative genomic analyses have shown that both pSym and SI are highly variable, with the exception of a common set of genes encoding factors critical to nitrogen fixation (nif) and nodulation (nod) (5, 14). SI and pSym have been found to have lower GC contents and different codon usages than the corresponding chromosomal and nonsymbiotic plasmid sequences, suggesting that they were recently acquired by HGT.Some of these symbiotic elements, such as in pSym of Rhizobium etli CFN42 and the SI of Mesorhizobium loti, are conjugative and mobile (30, 32). Genomic analysis of R. etli CFN42 revealed the presence of 39 IS belonging to different families (14); they were found in the chromosome (11 IS); the 371-kb symbiotic plasmid (13 IS); the smaller 192-kb conjugative plasmid, p42a (13 IS); and two other plasmids, p42b and p42c (2 IS). Interestingly, this particular strain shows no evidence of IS disrupting open reading frames (ORFs) or having transpositional activity. However, another 42 incomplete IS may be found in the chromosome, pSym, and the conjugative plasmid; these incomplete sequences are truncated or contain stop codons in their coding sequences.Here, we focused on the dynamics and distribution of IS in different populations of the nitrogen-fixing symbiont R. etli. Since the maintenance of IS in bacterial species might depend on their transpositional activities and horizontal transfer rates, the identification of IS in the same genomic contexts across different strains of the same species could provide new insights into their persistence and divergence over short evolutionary periods. To examine the evolutionary dynamics of IS in natural populations of R. etli, we characterized the distributions, genomic contexts, and sequence variations of IS in isolates of R. etli from three populations with different origins, as well as in some other Rhizobium species. More specifically, we used PCR to generate presence/absence profiles of the 39 IS found in R. etli CFN42 in a collection of 87 strains representing different geographical sites and a gradient of domestication of the bacterial host, the common bean (Phaseolus vulgaris). We also evaluated whether the IS were conserved in the same genomic context relative to their position in R. etli CFN42 and determined the nucleotide sequences of two IS found in most of the isolates. Several population genetic tests applied to these IS, another pSym gene (nodC), and two chromosomal housekeeping genes (dnaB and glyA) suggest that these two IS elements have been inherited vertically and represent recent components of the R. etli gene pool. Finally, the present study strongly suggests that symbiotic plasmids have a recent origin within the R. etli populations.     

Evolutionary Dynamics of HTLV-I

Using mathematical models to describe the in vivo dynamics of HTLV-I infection, an explanation is offered for the slow rate of evolution of HTLV-I relative to HIV-1. In agreement with experimental findings, it is assumed that cell activation is required for successful replication in T helper cells and that HTLV-I induces a significant degree of bystander activation. It is found that the rate of evolution of HTLV-I is limited by the restricted availability of activated uninfected T cells, both at high and low proviral loads. This limits the within-host sequence diversity of HTLV-I and may therefore account for the slow rate of evolution of the virus in the population. Specific differences in the in vivo dynamics of HTLV-I and HIV-1 are identified which may account for the discrepancy in the rate of evolution of these two retroviruses. Received: 7 September 1999 / Accepted: 6 December 1999     

Evolutionary Relationship of DNA Sequences in Finite Populations

With the aim of analyzing and interpreting data on DNA polymorphism obtained by DNA sequencing or restriction enzyme technique, a mathematical theory on the expected evolutionary relationship among DNA sequences (nucleons) sampled is developed under the assumption that the evolutionary change of nucleons is determined solely by mutation and random genetic drift. The statistical property of the number of nucleotide differences between randomly chosen nucleons and that of heterozygosity or nucleon diversity is investigated using this theory. These studies indicate that the estimates of the average number of nucleotide differences and nucleon diversity have a large variance, and a large part of this variance is due to stochastic factors. Therefore, increasing sample size does not help reduce the variance significantly. The distribution of sample allele (nucleomorph) frequencies is also studied, and it is shown that a small number of samples are sufficient in order to know the distribution pattern.     

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.     

Evolutionary Implications of Changing Nutritional Patterns in Human Populations

Malnutrition is a widespread problem in the tropical regions of the world, the same areas which are believed to be man's ancestral home. Much of the adaptive complex characterizing contemporary Homo sapiens was assembled during the period of at least partial reliance on dietary intake of animal protein. Adjustments to low protein intake are most difficult during the period of growth and development. Selection against individuals unable to make suitable adjustments exerts pressure on the human population to retain adaptability while maintaining appropriate body proportions and sexual dimorphisms for body size .     

Evolutionary Dynamics of Ralstonia solanacearum

We investigated the genetic diversity, extent of recombination, natural selection, and population divergence of Ralstonia solanacearum samples obtained from sources worldwide. This plant pathogen causes bacterial wilt in many crops and constitutes a serious threat to agricultural production due to its very wide host range and aggressiveness. Five housekeeping genes, dispersed around the chromosome, and three virulence-related genes, located on the megaplasmid, were sequenced from 58 strains belonging to the four major phylogenetic clusters (phylotypes). Whereas genetic variation is high and consistent for all housekeeping loci studied, virulence-related gene sequences are more diverse. Phylogenetic and statistical analyses suggest that this organism is a highly diverse bacterial species containing four major, deeply separated evolutionary lineages (phylotypes I to IV) and a weaker subdivision of phylotype II into two subgroups. Analysis of molecular variations showed that the geographic isolation and spatial distance have been the significant determinants of genetic variation between phylotypes. R. solanacearum displays high clonality for housekeeping genes in all phylotypes (except phylotype III) and significant levels of recombination for the virulence-related egl and hrpB genes, which are limited mainly to phylotype strains III and IV. Finally, genes essential for species survival are under purifying selection, and those directly involved in pathogenesis might be under diversifying selection.     

Evolutionary Dynamics of Spore Killers

Spore killing in ascomycetes is a special form of segregation distortion. When a strain with the Killer genotype is crossed to a Sensitive type, spore killing is expressed by asci with only half the number of ascospores as usual, all surviving ascospores being of the Killer type. Using population genetic modeling, this paper explores conditions for invasion of Spore killers and for polymorphism of Killers, Sensitives and Resistants (which neither kill, nor get killed), as found in natural populations. The models show that a population with only Killers and Sensitives can never be stable. The invasion of Killers and stable polymorphism only occur if Killers have some additional advantage during the process of spore killing. This may be due to the effects of local sib competition or some kind of ``heterozygous' advantage in the stage of ascospore formation or in the short diploid stage of the life cycle. This form of segregation distortion appears to be essentially different from other, well-investigated forms, and more field data are needed for a better understanding of spore killing.     

Evolutionary Aspects of Enzyme Dynamics

The role of evolutionary pressure on the chemical step catalyzed by enzymes is somewhat enigmatic, in part because chemistry is not rate-limiting for many optimized systems. Herein, we present studies that examine various aspects of the evolutionary relationship between protein dynamics and the chemical step in two paradigmatic enzyme families, dihydrofolate reductases and alcohol dehydrogenases. Molecular details of both convergent and divergent evolution are beginning to emerge. The findings suggest that protein dynamics across an entire enzyme can play a role in adaptation to differing physiological conditions. The growing tool kit of kinetics, kinetic isotope effects, molecular biology, biophysics, and bioinformatics provides means to link evolutionary changes in structure-dynamics function to the vibrational and conformational states of each protein.     

Selection for High Adult Body Weight in Drosophila Populations with Different Structures

Selection for high adult body weight in Drosophila melanogaster was practiced for 18 generations in three selection lines. These lines were genetically similar and of equal size but different in population structure. One line represented a large mass-selected, random-mating population, while the other two lines simulated large populations that had been subdivided into partial isolates or demes. Mass selection and random mating occurred within each deme. These two subdivided lines were different only in the rate of effective migration among the demes (5% and 10%). Selection intensities of approximately 20% were applied to these populations. A fourth line served as a random mating control. Heritability of adult body weight in the base population was estimated to be 0.58± 0.22. The results indicate that significantly greater responses were achieved in the subdivided lines than in the large mass-selected line, in spite of the fact that larger selection differentials were applied to the latter. No significant differences in response were observed between the two subdivided lines. Wright (1930, 1931) postulated that selection would be most efficient in subdivided populations with limited interdeme migration. The present findings appear to support this theory.