The ancestral selection graph for a Λ-asymmetric Moran model

Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ-reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ-asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.

A geometric approach to the evolution of altruism

Fisher's geometric model provides a powerful tool for making predictions about key properties of Darwinian adaptation. Here, I apply the geometric model to predict differences between the evolution of altruistic versus nonsocial phenotypes. I recover Kimura's prediction that probability of fixation is greater for mutations of intermediate size, but I find that the effect size that maximises probability of fixation is relatively small in the context of altruism and relatively large in the context of nonsocial phenotypes, and that the overall probability of fixation is lower for altruism and is higher for nonsocial phenotypes. Accordingly, the first selective substitution is expected to be smaller, and to take longer, in the context of the evolution of altruism. These results strengthen the justification for employing streamlined social evolutionary methodologies that assume adaptations are underpinned by many genes of small effect.

Eco-evolutionary dynamics in finite network-structured populations with migration

We consider the effect of network structure on the evolution of a population. Models of this kind typically consider a population of fixed size and distribution. Here we consider eco-evolutionary dynamics where population size and distribution can change through birth, death and migration, all of which are separate processes. This allows complex interaction and migration behaviours that are dependent on competition. For migration, we assume that the response of individuals to competition is governed by tolerance to their group members, such that less tolerant individuals are more likely to move away due to competition. We look at the success of a mutant in the rare mutation limit for the complete, cycle and star networks. Unlike models with fixed population size and distribution, the distribution of the individuals per site is explicitly modelled by considering the dynamics of the population. This in turn determines the mutant appearance distribution for each network. Where a mutant appears impacts its success as it determines the competition it faces. For low and high migration rates the complete and cycle networks have similar mutant appearance distributions resulting in similar success levels for an invading mutant. A higher migration rate in the star network is detrimental for mutant success because migration results in a crowded central site where a mutant is more likely to appear.

Evolutionary game dynamics with non-uniform interaction rates in finite population

In this study, we extend evolutionary game dynamics with non-uniform interaction rates to the situation with finite population. Our main goal is to show how the fixation probability is influenced by the non-uniform interaction rates under weak selection. Based on the diffusion approximation of the Moran process and assumption of weak selection, the stochastic dynamic properties of a two-phenotype game with non-uniform interaction rates in a finite population are investigated. By the analysis of some cases, we show that the non-uniform interaction rates may result in the potential evolutionary complexity of game dynamics in finite population.

Fixation and effective size in a haploid-diploid population with asexual reproduction

The majority of population genetic theory assumes fully haploid or diploid organisms with obligate sexuality, despite complex life cycles with alternating generations being commonly observed. To reveal how natural selection and genetic drift shape the evolution of haploid-diploid populations, we analyze a stochastic genetic model for populations that consist of a mixture of haploid and diploid individuals, allowing for asexual reproduction and niche separation between haploid and diploid stages. Applying a diffusion approximation, we derive the fixation probability and describe its dependence on the reproductive values of haploid and diploid stages, which depend strongly on the extent of asexual reproduction in each phase and on the ecological differences between them.

Inclusive fitness and Hamilton's rule in a stochastic environment

The evolution of cooperation in Prisoner's Dilemmas with additive random cost and benefit for cooperation cannot be accounted for by Hamilton's rule based on mean effects transferred from recipients to donors weighted by coefficients of relatedness, which defines inclusive fitness in a constant environment. Extensions that involve higher moments of stochastic effects are possible, however, and these are connected to a concept of random inclusive fitness that is frequency-dependent. This is shown in the setting of pairwise interactions in a haploid population with the same coefficient of relatedness between interacting players. In an infinite population, fixation of cooperation is stochastically stable if a mean geometric inclusive fitness of defection when rare is negative, while fixation of defection is stochastically unstable if a mean geometric inclusive fitness of cooperation when rare is positive, and these conditions are generally not equivalent. In a finite population, the probability for cooperation to ultimately fix when represented once exceeds the probability under neutrality or the corresponding probability for defection if the mean inclusive fitness of cooperation when its frequency is 1/3 or 1/2, respectively, exceeds 1. All these results rely on the simplifying assumption of a linear fitness function. It is argued that meaningful applications of random inclusive fitness in complex settings (multi-player game, diploidy, population structure) would generally require conditions of weak selection and additive gene action.

The effect of the opting-out strategy on conditions for selection to favor the evolution of cooperation in a finite population

We consider a Prisoner's Dilemma (PD) that is repeated with some probability 1-ρ only between cooperators as a result of an opting-out strategy adopted by all individuals. The population is made of N pairs of individuals and is updated at every time step by a birth-death event according to a Moran model. Assuming an intensity of selection of order 1/N and taking 2N² birth-death events as unit of time, a diffusion approximation exhibiting two time scales, a fast one for pair frequencies and a slow one for cooperation (C) and defection (D) frequencies, is ascertained in the limit of a large population size. This diffusion approximation is applied to an additive PD game, cooperation by an individual incurring a cost c to the individual but providing a benefit b to the opponent. This is used to obtain the probability of ultimate fixation of C introduced as a single mutant in an all D population under selection, which can be compared to the probability under neutrality, 1/(2N), as well as the corresponding probability for a single D introduced in an all C population under selection. This gives conditions for cooperation to be favored by selection. We show that these conditions are satisfied when the benefit-to-cost ratio, b/c, exceeds some increasing function of ρ that is approximately given by (1+ρ)/(1-ρ). This condition is more stringent, however, than the condition for tit-for-tat (TFT) to be favored against always-defect (AllD) in the absence of opting-out.

The effects of phenotypic plasticity on the fixation probability of mutant cancer stem cells

The cancer stem cell hypothesis claims that tumor growth and progression are driven by a (typically) small niche of the total cancer cell population called cancer stem cells (CSCs). These CSCs can go through symmetric or asymmetric divisions to differentiate into specialised, progenitor cells or reproduce new CSCs. While it was once held that this differentiation pathway was unidirectional, recent research has demonstrated that differentiated cells are more plastic than initially considered. In particular, differentiated cells can de-differentiate and recover their stem-like capacity. Two recent papers have considered how this rate of plasticity affects the evolutionary dynamic of an invasive, malignant population of stem cells and differentiated cells into existing tissue (Mahdipour-Shirayeh et al., 2017; Wodarz, 2018). These papers arrive at seemingly opposing conclusions, one claiming that increased plasticity results in increased invasive potential, and the other that increased plasticity decreases invasive potential. Here, we show that what is most important, when determining the effect on invasive potential, is how one distributes this increased plasticity between the compartments of resident and mutant-type cells. We also demonstrate how these results vary, producing non-monotone fixation probability curves, as inter-compartmental plasticity changes when differentiated cell compartments are allowed to continue proliferating, highlighting a fundamental difference between the two models. We conclude by demonstrating the stability of these qualitative results over various parameter ranges. Keywords: cancer stem cells, plasticity, de-differentiation, fixation probability.

Diffusion approximation for an age-class-structured population under viability and fertility selection with application to fixation probability of an advantageous mutant

In this paper, we ascertain the validity of a diffusion approximation for the frequencies of different types under recurrent mutation and frequency-dependent viability and fertility selection in a haploid population with a fixed age-class structure in the limit of a large population size. The approximation is used to study, and explain in terms of selection coefficients, reproductive values and population-structure coefficients, the differences in the effects of viability versus fertility selection on the fixation probability of an advantageous mutant.

Evolutionary game dynamics of the Wright-Fisher process with different selection intensities

Evolutionary game dynamics in finite populations can be described by a frequency-dependent, stochastic Wright-Fisher process. The fitness of individuals in a population is not only linked to environmental conditions but also tightly coupled to the types and frequencies of competitors, leading to different types of individuals with different selection intensities. We studied a 2 × 2 symmetric game in a finite population and established a dynamic model of the Wright-Fisher process by introducing different selection intensities for different strategies. Thus, we provided another effective way to study the evolutionary dynamics of a finite population and obtained the analytical expressions of fixation probabilities under weak selection. The fixation probability of a strategy is not only related to a game matrix but also to different selection intensities. The conditions required for natural selection to favor one strategy and for that strategy to be an evolutionary stable strategy (ESS_N) are specified in our model. We compared our results with those of a Moran dynamic process with different selection intensities to explore these two processes better. In the two processes, the conditions conducive to the strategy's taking fixation are the same. By simulation analysis, the dynamic relationships between the fixation probabilities and selection intensities were intuitively observed in the prisoner's dilemma, coordination, and coexistence games. The fixation probability of the cooperative strategy in the prisoner's dilemma decreases with the increase of its own selection intensity. In the coexistence and coordination games, the fixation probability of the cooperative strategy increases with its own selection intensity. For the three types of games, the fixation probability of the cooperative strategy decreases with the increase of the selection intensity of the defection strategy.

Methods for approximating stochastic evolutionary dynamics on graphs

Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes.

From Fixation Probabilities to d-player Games: An Inverse Problem in Evolutionary Dynamics

The probability that the frequency of a particular trait will eventually become unity, the so-called fixation probability, is a central issue in the study of population evolution. Its computation, once we are given a stochastic finite population model without mutations and a (possibly frequency dependent) fitness function, is straightforward and it can be done in several ways. Nevertheless, despite the fact that the fixation probability is an important macroscopic property of the population, its precise knowledge does not give any clear information about the interaction patterns among individuals in the population. Here we address the inverse problem: from a given fixation pattern and population size, we want to infer what is the game being played by the population. This is done by first exploiting the framework developed in Chalub and Souza (J Math Biol 75:1735-1774, 2017), which yields a fitness function that realises this fixation pattern in the Wright-Fisher model. This fitness function always exists, but it is not necessarily unique. Subsequently, we show that any such fitness function can be approximated, with arbitrary precision, using d-player game theory, provided d is large enough. The pay-off matrix that emerges naturally from the approximating game will provide useful information about the individual interaction structure that is not itself apparent in the fixation pattern. We present extensive numerical support for our conclusions.

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Disentangling eco-evolutionary effects on trait fixation

In population genetics, fixation of traits in a demographically changing population under frequency-independent selection has been extensively analysed. In evolutionary game theory, models of fixation have typically focused on fixed population sizes and frequency-dependent selection. A combination of demographic fluctuations with frequency-dependent interactions such as Lotka-Volterra dynamics has received comparatively little attention. We consider a stochastic, competitive Lotka-Volterra model with higher order interactions between two traits. The emerging individual-based model allows for stochastic fluctuations in the frequencies of the two traits and the total population size. We calculate the fixation probability of a trait under differing competition coefficients. This fixation probability resembles, qualitatively, the deterministic evolutionary dynamics. Furthermore, we partially disentangle the selection effects into their ecological and evolutionary components. We find that changing the evolutionary selection strength also changes the population dynamics and vice versa. Thus, a clean separation of the ecological and evolutionary effects is not possible. Instead, our results imply a nested interaction of the evolutionary and ecological effects. The entangled eco-evolutionary processes thus cannot be ignored when determining fixation properties in a co-evolutionary system.

Fixation probabilities in populations under demographic fluctuations

We study the fixation probability of a mutant type when introduced into a resident population. We implement a stochastic competitive Lotka-Volterra model with two types and intra- and interspecific competition. The model further allows for stochastically varying population sizes. The competition coefficients are interpreted in terms of inverse payoffs emerging from an evolutionary game. Since our study focuses on the impact of the competition values, we assume the same net growth rate for both types. In this general framework, we derive a formula for the fixation probability [Formula: see text] of the mutant type under weak selection. We find that the most important parameter deciding over the invasion success of the mutant is its death rate due to competition with the resident. Furthermore, we compare our approximation to results obtained by implementing population size changes deterministically in order to explore the parameter regime of validity of our method. Finally, we put our formula in the context of classical evolutionary game theory and observe similarities and differences to the results obtained in that constant population size setting.

Effect of cellular de-differentiation on the dynamics and evolution of tissue and tumor cells in mathematical models with feedback regulation

Tissues are maintained by adult stem cells that self-renew and also differentiate into functioning tissue cells. Homeostasis is achieved by a set of complex mechanisms that involve regulatory feedback loops. Similarly, tumors are believed to be maintained by a minority population of cancer stem cells, while the bulk of the tumor is made up of more differentiated cells, and there is indication that some of the feedback loops that operate in tissues continue to be functional in tumors. Mathematical models of such tissue hierarchies, including feedback loops, have been analyzed in a variety of different contexts. Apart from stem cells giving rise to differentiated cells, it has also been observed that more differentiated cells can de-differentiate into stem cells, both in healthy tissue and tumors, aspects of which have also been investigated mathematically. This paper analyses the effect of de-differentiation on the basic and evolutionary dynamics of cells in the context of tissue hierarchy models that include negative feedback regulation of the cell populations. The models predict that in the presence of de-differentiation, the fixation probability of a neutral mutant is lower than in its absence. Therefore, if de-differentiation occurs, a mutant with identical parameters compared to the wild-type cell population behaves like a disadvantageous mutant. Similarly, the process of de-differentiation is found to lower the fixation probability of an advantageous mutant. These results indicate that the presence of de-differentiation can lower the rates of tumor initiation and progression in the context of the models considered here.

Modeling Cell Dynamics in Colon and Intestinal Crypts: The Significance of Central Stem Cells in Tumorigenesis

Colon and intestinal crypts have been widely chosen to study cell dynamics because of their fairly simple structures. In the colon and intestinal crypts, stem cells (SCs) are located at very bottom of the crypt, fully differentiated cells (FDs) are located in the top of the crypt, and transit-amplifying cells (TAs) are in the middle of the crypt between FDs and SCs. Recently, it has been discovered that there are two types of stem cells in the intestinal crypts: central stem cells (CeSCs) and border stem cells. To investigate dynamics of mutants in colon and intestinal crypts, we develop a four-compartmental stochastic model, which includes two SC compartments, and TAs and FDs compartments. We calculate the probability of the progeny of marked or mutant cells located at each of these compartments taking over the entire crypt or being washed out from the crypt. We found that the progeny of CeSCs will take over the entire crypt with a probability close to one. Interestingly, the progeny of advantageous mutant TAs and FDs will be washed out faster than disadvantageous mutants. Saliently, the model predicts that the time that the progeny of wild-type central stem cells will take over the mouse intestinal crypt is around 60 days, which is in perfect agreement with an experimental observation.

Evolution of highly fecund haploid populations

We consider a model of viability selection in a highly fecund haploid population with sweepstakes reproduction. We use simulations to estimate the time until the allelic type with highest fitness has reached high frequency in a finite population. We compare the time between two reproduction modes of high and low fecundity. We also consider the probability that the allelic type with highest fitness is lost from the population before reaching high frequency. Our simulation results indicate that highly fecund populations can evolve faster (in some cases much faster) than populations of low fecundity. However, high fecundity and sweepstakes reproduction also confer much higher risk of losing the allelic type with highest fitness from the population by chance. The impact of selection on driving alleles to high frequency varies depending on the trait value conferring highest fitness; in some cases the effect of selection can hardly be detected.

Fixation probability of a nonmutator in a large population of asexual mutators

In an adapted population of mutators in which most mutations are deleterious, a nonmutator that lowers the mutation rate is under indirect selection and can sweep to fixation. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large population of asexual mutators. We show that when beneficial mutations are absent, the fixation probability is a nonmonotonic function of the mutation rate of the mutator: it first increases sublinearly and then decreases exponentially. We also find that beneficial mutations can enhance the fixation probability of a nonmutator. Our analysis is relevant to an understanding of recent experiments in which a reduction in the mutation rates has been observed.

Role of epistasis on the fixation probability of a non-mutator in an adapted asexual population

The mutation rate of a well adapted population is prone to reduction so as to have a lower mutational load. We aim to understand the role of epistatic interactions between the fitness affecting mutations in this process. Using a multitype branching process, the fixation probability of a single non-mutator emerging in a large asexual mutator population is analytically calculated here. The mutator population undergoes deleterious mutations at constant, but at a much higher rate than that of the non-mutator. We find that antagonistic epistasis lowers the chances of mutation rate reduction, while synergistic epistasis enhances it. Below a critical value of epistasis, the fixation probability behaves non-monotonically with variation in the mutation rate of the background population. Moreover, the variation of this critical value of the epistasis parameter with the strength of the mutator is discussed in the appendix. For synergistic epistasis, when selection is varied, the fixation probability reduces overall, with damped oscillations.

Exact numerical calculation of fixation probability and time on graphs

The Moran process on graphs is a popular model to study the dynamics of evolution in a spatially structured population. Exact analytical solutions for the fixation probability and time of a new mutant have been found for only a few classes of graphs so far. Simulations are time-expensive and many realizations are necessary, as the variance of the fixation times is high. We present an algorithm that numerically computes these quantities for arbitrary small graphs by an approach based on the transition matrix. The advantage over simulations is that the calculation has to be executed only once. Building the transition matrix is automated by our algorithm. This enables a fast and interactive study of different graph structures and their effect on fixation probability and time. We provide a fast implementation in C with this note (Hindersin et al., 2016). Our code is very flexible, as it can handle two different update mechanisms (Birth-death or death-Birth), as well as arbitrary directed or undirected graphs.

Social evolution and genetic interactions in the short and long term

The evolution of social traits remains one of the most fascinating and feisty topics in evolutionary biology even after half a century of theoretical research. W.D. Hamilton shaped much of the field initially with his 1964 papers that laid out the foundation for understanding the effect of genetic relatedness on the evolution of social behavior. Early theoretical investigations revealed two critical assumptions required for Hamilton's rule to hold in dynamical models: weak selection and additive genetic interactions. However, only recently have analytical approaches from population genetics and evolutionary game theory developed sufficiently so that social evolution can be studied under the joint action of selection, mutation, and genetic drift. We review how these approaches suggest two timescales for evolution under weak mutation: (i) a short-term timescale where evolution occurs between a finite set of alleles, and (ii) a long-term timescale where a continuum of alleles are possible and populations evolve continuously from one monomorphic trait to another. We show how Hamilton's rule emerges from the short-term analysis under additivity and how non-additive genetic interactions can be accounted for more generally. This short-term approach reproduces, synthesizes, and generalizes many previous results including the one-third law from evolutionary game theory and risk dominance from economic game theory. Using the long-term approach, we illustrate how trait evolution can be described with a diffusion equation that is a stochastic analogue of the canonical equation of adaptive dynamics. Peaks in the stationary distribution of the diffusion capture classic notions of convergence stability from evolutionary game theory and generally depend on the additive genetic interactions inherent in Hamilton's rule. Surprisingly, the peaks of the long-term stationary distribution can predict the effects of simple kinds of non-additive interactions. Additionally, the peaks capture both weak and strong effects of social payoffs in a manner difficult to replicate with the short-term approach. Together, the results from the short and long-term approaches suggest both how Hamilton's insight may be robust in unexpected ways and how current analytical approaches can expand our understanding of social evolution far beyond Hamilton's original work.

Games among relatives revisited

We present a simple model for the evolution of social behavior in family-structured, finite sized populations. Interactions are represented as evolutionary games describing frequency-dependent selection. Individuals interact more frequently with siblings than with members of the general population, as quantified by an assortment parameter r, which can be interpreted as "relatedness". Other models, mostly of spatially structured populations, have shown that assortment can promote the evolution of cooperation by facilitating interaction between cooperators, but this effect depends on the details of the evolutionary process. For our model, we find that sibling assortment promotes cooperation in stringent social dilemmas such as the Prisoner's Dilemma, but not necessarily in other situations. These results are obtained through straightforward calculations of changes in gene frequency. We also analyze our model using inclusive fitness. We find that the quantity of inclusive fitness does not exist for general games. For special games, where inclusive fitness exists, it provides less information than the straightforward analysis.

Fast and asymptotic computation of the fixation probability for Moran processes on graphs

Evolutionary dynamics has been classically studied for homogeneous populations, but now there is a growing interest in the non-homogeneous case. One of the most important models has been proposed in Lieberman et al. (2005), adapting to a weighted directed graph the process described in Moran (1958). The Markov chain associated with the graph can be modified by erasing all non-trivial loops in its state space, obtaining the so-called Embedded Markov chain (EMC). The fixation probability remains unchanged, but the expected time to absorption (fixation or extinction) is reduced. In this paper, we shall use this idea to compute asymptotically the average fixation probability for complete bipartite graphs K(n,m). To this end, we firstly review some recent results on evolutionary dynamics on graphs trying to clarify some points. We also revisit the 'Star Theorem' proved in Lieberman et al. (2005) for the star graphs K(1,m). Theoretically, EMC techniques allow fast computation of the fixation probability, but in practice this is not always true. Thus, in the last part of the paper, we compare this algorithm with the standard Monte Carlo method for some kind of complex networks.

Fluctuation driven fixation of cooperative behavior

Cooperative behaviors are defined as the production of common goods benefitting all members of the community at the producer's cost. They could seem to be in contradiction with natural selection, as non-cooperators have an increased fitness compared to cooperators. Understanding the emergence of cooperation has necessitated the development of concepts and models (inclusive fitness, multilevel selection, etc.) attributing deterministic advantages to this behavior. In contrast to these models, we show here that cooperative behaviors can emerge by taking into account only the stochastic nature of evolutionary dynamics: when cooperative behaviors increase the population size, they also increase the genetic drift against non-cooperators. Using the Wright-Fisher models of population genetics, we compute exactly this increased genetic drift and its consequences on the fixation probability of both types of individuals. This computation leads to a simple criterion: cooperative behavior dominates when the relative increase in population size caused by cooperators is higher than the selection pressure against them. This is a purely stochastic effect with no deterministic interpretation.

Stochastic evolution of staying together

Staying together means that replicating units do not separate after reproduction, but remain attached to each other or in close proximity. Staying together is a driving force for evolution of complexity, including the evolution of multi-cellularity and eusociality. We analyze the fixation probability of a mutant that has the ability to stay together. We assume that the size of the complex affects the reproductive rate of its units and the probability of staying together. We examine the combined effect of natural selection and random drift on the emergence of staying together in a finite sized population. The number of states in the underlying stochastic process is an exponential function of population size. We develop a framework for any intensity of selection and give closed form solutions for special cases. We derive general results for the limit of weak selection.

Reproductive value in graph-structured populations

Evolutionary graph theory has grown to be an area of intense study. Despite the amount of interest in the field, it seems to have grown separate from other subfields of population genetics and evolution. In the current work I introduce the concept of Fisher's (1930) reproductive value into the study of evolution on graphs. Reproductive value is a measure of the expected genetic contribution of an individual to a distant future generation. In a heterogeneous graph-structured population, differences in the number of connections among individuals translate into differences in the expected number of offspring, even if all individuals have the same fecundity. These differences are accounted for by reproductive value. The introduction of reproductive value permits the calculation of the fixation probability of a mutant in a neutral evolutionary process in any graph-structured population for either the moran birth-death or death-birth process.

Fixation probability of mobile genetic elements such as plasmids

Mobile genetic elements such as plasmids are increasingly becoming thought of as evolutionarily important. Being horizontally transmissible is generally assumed to be beneficial for a gene. Using several simple modelling approaches we show that in fact being horizontally transferable is just as important for fixation as being beneficial to the host, in line with other results. We find fixation probability is approximately 2(s+β), where s is the increased (vertical) fitness provided by the gene, and β the rate of horizontal transfer when rare. This result comes about because when the gene is rare, almost all individuals in the population are possible recipients of horizontal transfer. The ability to horizontally transfer could thus cause a deleterious gene to become fixed in a population even without hitchhiking. Our findings provide further evidence for the importance and ubiquity of mobile genetic elements, particularly in microorganisms.

The two-mutant problem: clonal interference in evolutionary graph theory

In large asexual populations, clonal interference, whereby different beneficial mutations compete to fix in the population simultaneously, may be the norm. Results extrapolated from the spread of individual mutations in homogeneous backgrounds are found to be misleading in such situations: clonal interference severely inhibits the spread of beneficial mutations. In contrast with results gained in systems with just one mutation striving for fixation at any one time, the spatial structure of the population is found to be an important factor in determining the fixation probability when there are two beneficial mutations.