Impact of demography on extinction/fixation events

In this article we consider diffusion processes modeling the dynamics of multiple allelic proportions (with fixed and varying population size). We are interested in the way alleles extinctions and fixations occur. We first prove that for the Wright–Fisher diffusion process with selection, alleles get extinct successively (and not simultaneously), until the fixation of one last allele. Then we introduce a very general model with selection, competition and Mendelian reproduction, derived from the rescaling of a discrete individual-based dynamics. This multi-dimensional diffusion process describes the dynamics of the population size as well as the proportion of each type in the population. We prove first that alleles extinctions occur successively and second that depending on population size dynamics near extinction, fixation can occur either before extinction almost surely, or not. The proofs of these different results rely on stochastic time changes, integrability of one-dimensional diffusion processes paths and multi-dimensional Girsanov’s tranform.

     

Absorption and fixation times for neutral and quasi-neutral populations with density dependence

We study a generalisation of Moran’s population-genetic model that incorporates density dependence. Rather than assuming fixed population size, we allow the number of individuals to vary stochastically with the same events that change allele number, according to a logistic growth process with density dependent mortality. We analyse the expected time to absorption and fixation in the ‘quasi-neutral’ case: both types have the same carrying capacity, achieved through a trade-off of birth and death rates. Such types would be competitively neutral in a classical, fixed-population Wright–Fisher model. Nonetheless, we find that absorption times are skewed compared to the Wright–Fisher model. The absorption time is longer than the Wright–Fisher prediction when the initial proportion of the type with higher birth rate is large, and shorter when it is small. By contrast, demographic stochasticity has no effect on the fixation or absorption times of truly neutral alleles in a large population. Our calculations provide the first analytic results on hitting times in a two-allele model, when the population size varies stochastically.     

Mathematische Modelle in der Populationsgenetik

In mathematical population genetics, the influence of selection and mutation on the evolution of a population is modelled. Because all populations and particularly the samples used for their analysis are finite, the stochastic nature of these models plays an important role. Relevant genetic models include the Wright–Fisher model and the coalescence model for the genealogy of samples, as well as the infinite alleles model and the infinite sites model for the mutation processes superimposed upon these genealogies.     

Allele fixation in a dynamic metapopulation: Founder effects vs refuge effects

The fixation of mutant alleles has been studied with models assuming various spatial population structures. In these models, the structure of the metapopulation that we call the “landscape” (number, size and connectivity of subpopulations) is often static. However, natural populations are subject to repetitive population size variations, fragmentation and secondary contacts at different spatiotemporal scales due to geological, climatic and ecological processes. In this paper, we examine how such dynamic landscapes can alter mutant fixation probability and time to fixation. We consider three stochastic landscape dynamics: (i) the population is subject to repetitive bottlenecks, (ii) to the repeated alternation of fragmentation and fusion of demes with a constant population carrying capacity, (iii) idem with a variable carrying capacity. We show by deriving a variance, a coalescent and a harmonic mean population effective size, and with simulations that these landscape dynamics generate repetitive founder effects which counteract selection, thereby decreasing the fixation probability of an advantageous mutant but accelerate fixation when it occurs. For models (ii) and (iii), we also highlight an antagonistic “refuge effect” which can strongly delay mutant fixation. The predominance of either founder effects or refuge effects determines the time to fixation and mainly depends on the characteristic time scales of the landscape dynamics.     

The free energy method and the Wright–Fisher model with 2 alleles

We systematically investigate the Wright–Fisher model of population genetics with the free energy functional formalism of statistical mechanics and in the light of recent mathematical work on the connection between Fokker–Planck equations and free energy functionals. In statistical physics, entropy increases, or equivalently, free energy decreases, and the asymptotic state is given by a Gibbs-type distribution. This also works for the Wright–Fisher model when rewritten in divergence to identify the correct free energy functional. We not only recover the known results about the stationary distribution, that is, the asymptotic equilibrium state of the model, in the presence of positive mutation rates and possibly also selection, but can also provide detailed formulae for the rate of convergence towards that stationary distribution. In the present paper, the method is illustrated for the simplest case only, that of two alleles.     

Anecdotal,Historical and Critical Commentaries on Genetics: Wright and Fisher on Inbreeding and Random Drift

Sewall Wright and R. A. Fisher often differed, including on the meaning of inbreeding and random gene frequency drift. Fisher regarded them as quite distinct processes, whereas Wright thought that because his inbreeding coefficient measured both they should be regarded as the same. Since the effective population numbers for inbreeding and random drift are different, this would argue for the Fisher view.SEWALL Wright and R. A. Fisher were central figures in mathematical population genetics; along with J. B. S. Haldane they effectively invented the field and dominated it for many years. On most issues the three were in agreement. In particular, all favored a neo-Darwinian gradualist approach and believed in the importance of a mathematical theory for understanding the evolutionary process. Yet on a few questions Fisher and Wright differed profoundly and argued vehemently. Fisher was contentious and was often involved in controversy, frequently attacking his opponents mercilessly. Wright, in contrast, was very gentle to most people. But there were a few exceptions and Fisher was one. Haldane mostly stayed out of the arguments between them.One question on which the two disagreed was the importance of random gene frequency drift and its role in Wright''s shifting-balance theory of evolution. Wright thought that a structured population with many partially isolated subpopulations, within which there was random drift and among which there was an appropriate amount of migration, offered the greatest chance for evolutionary novelty and could greatly increase the speed of evolution. Fisher thought that a large panmictic population offered the best chance for advantageous genes and gene combinations to spread through the population, unimpeded by random processes. They also disagreed on dominance, Fisher believing that it evolved by selection of dominance modifiers and Wright that it was a consequence of the nature of gene action. These differences were widely argued by population geneticists in the middle third of the twentieth century, and the interested community divided into two camps. Although the issues are not settled, Wright''s shifting-balance theory has less support than it formerly had. As for dominance, there is general quantitative disagreement with Fisher''s explanation of modifiers, but other mechanisms (e.g., selection for more active alleles) have to some extent replaced it. Wright''s theory remains popular and has been generalized and extended (Kacser and Burns 1973).     

An introduction to the mathematical structure of the Wright–Fisher model of population genetics

In this paper, we develop the mathematical structure of the Wright–Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker–Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.     

Weak seed banks influence the signature and detectability of selective sweeps

Seed banking (or dormancy) is a widespread bet-hedging strategy, generating a form of population overlap, which decreases the magnitude of genetic drift. The methodological complexity of integrating this trait implies it is ignored when developing tools to detect selective sweeps. But, as dormancy lengthens the ancestral recombination graph (ARG), increasing times to fixation, it can change the genomic signatures of selection. To detect genes under positive selection in seed banking species it is important to (1) determine whether the efficacy of selection is affected, and (2) predict the patterns of nucleotide diversity at and around positively selected alleles. We present the first tree sequence-based simulation program integrating a weak seed bank to examine the dynamics and genomic footprints of beneficial alleles in a finite population. We find that seed banking does not affect the probability of fixation and confirm expectations of increased times to fixation. We also confirm earlier findings that, for strong selection, the times to fixation are not scaled by the inbreeding effective population size in the presence of seed banks, but are shorter than would be expected. As seed banking increases the effective recombination rate, footprints of sweeps appear narrower around the selected sites and due to the scaling of the ARG are detectable for longer periods of time. The developed simulation tool can be used to predict the footprints of selection and draw statistical inference of past evolutionary events in plants, invertebrates, or fungi with seed banks.     

Dynamics of Fst for the island model

F(st) is a measure of genetic differentiation in a subdivided population. Sewall Wright observed that F(st)=1/1+2Nm in a haploid diallelic infinite island model, where N is the effective population size of each deme and m is the migration rate. In demonstrating this result, Wright relied on the infinite size of the population. Natural populations are not infinite and therefore they change over time due to genetic drift. In a finite population, F(st) becomes a random variable that evolves over time. In this work we ask, given an initial population state, what are the dynamics of the mean and variance of F(st) under the finite island model? In application both of these quantities are critical in the evaluation of F(st) data. We show that after a time of order N generations the mean of F(st) is slightly biased below 1/1+2Nm. Further we show that the variance of F(st) is of order 1/d where d is the number of demes in the population. We introduce several new mathematical techniques to analyze coalescent genealogies in a dynamic setting.     

The impact of host-cell dynamics on the fixation probability for lytic viruses

Lytic viruses are obligate parasites whose population dynamics are necessarily coupled to the dynamics of their host-cell population. The adaptation rate of these viruses has attracted considerable scientific interest, as they are a key model organism in experimental evolution. Nevertheless, to date mathematical models of experimental evolution have largely ignored the host-cell population. In this paper we incorporate two important features of host-cell dynamics—the possibility of clearance or death of an infected cell before lysis, and the possibility of changing host-cell density—into previous models for the fixation probability of lytic viruses. We compute the fixation probabilities of rare alleles that confer reproductive benefit through either an increase in attachment rate or burst size, or a reduction in lysis time. We find that host-cell clearance significantly reduces the fixation probabilities of all types of beneficial mutations, having the largest impact on mutations which reduce the lysis time, but has only modest effects on the pattern of fixation probabilities previously observed. We further predict that exponential growth of the host-cell population preferentially selects for mutations that affect burst size or lysis time, and exacerbates the sensitive dependence of fixation probabilities on the time between population bottlenecks. Even when burst size and lysis time are constrained to vary together, our results suggest that lytic viruses should readily adapt to optimize these traits to the timing between population bottlenecks.     

Generalized population models and the nature of genetic drift

The Wright–Fisher model of allele dynamics forms the basis for most theoretical and applied research in population genetics. Our understanding of genetic drift, and its role in suppressing the deterministic forces of Darwinian selection has relied on the specific form of sampling inherent to the Wright–Fisher model and its diffusion limit. Here we introduce and analyze a broad class of forward-time population models that share the same mean and variance as the Wright–Fisher model, but may otherwise differ. The proposed class unifies and further generalizes a number of population-genetic processes of recent interest, including the Λ and Cannings processes. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We characterize in detail the behavior of standard population-genetic quantities across this family of generalized models. Some quantities, such as heterozygosity, remain unchanged; but others, such as neutral absorption times and fixation probabilities under selection, deviate by orders of magnitude from the Wright–Fisher model. We show that generalized population models can produce startling phenomena that differ qualitatively from classical behavior — such as assured fixation of a new mutant despite the presence of genetic drift. We derive the forward-time continuum limits of the generalized processes, analogous to Kimura’s diffusion limit of the Wright–Fisher process, and we discuss their relationships to the Kingman and non-Kingman coalescents. Finally, we demonstrate that some non-diffusive, generalized models are more likely, in certain respects, than the Wright–Fisher model itself, given empirical data from Drosophila populations.     

Conservation corridors affect the fixation of novel alleles

Corridors are a popular tool for conservation of small populations. However, two purported benefits of corridors, increasing gene flow and providing a means for the recolonization of extinct patches of habitat (population rescue), may have unappreciated impacts on the likelihood that a new allele will become incorporated (fixed) within a population. Using a simulation model, I demonstrate that connecting a stable, isolated population with a population that requires periodic rescue (due to extinction via natural or anthropogenic disturbance) can affect fixation of alleles in the stable population, largely by changing the effective population size Ne of the two-patch complex. When disturbance is rare, connecting the two patches with corridors can increase fixation of beneficial alleles and increase loss of harmful alleles. However, the opposite occurs when rates of disturbance are high: corridors can promote fixation of harmful alleles and reduce fixation of beneficial alleles. Because the impact of corridors hinges upon disturbance frequency (i.e. rate of population rescue), population growth rate, movement rates, and habitat quality, different species are likely to have different responses to corridor-mediated fixation, even if the species reside within the same ecological community. By changing fixation, corridors could thus either promote adaptation or extinction.     

Understanding evolutionary and ecological dynamics using a continuum limit

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual‐based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright–Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi‐stationary distribution in a finite population.     

Approximate Joint Diagonalization and Geometric Mean of Symmetric Positive Definite Matrices

We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed with the Fisher information metric. The resulting mean has several important invariance properties and has proven very useful in diverse engineering applications such as biomedical and image data processing. While for two SPD matrices the mean has an algebraic closed form solution, for a set of more than two SPD matrices it can only be estimated by iterative algorithms. However, none of the existing iterative algorithms feature at the same time fast convergence, low computational complexity per iteration and guarantee of convergence. For this reason, recently other definitions of geometric mean based on symmetric divergence measures, such as the Bhattacharyya divergence, have been considered. The resulting means, although possibly useful in practice, do not satisfy all desirable invariance properties. In this paper we consider geometric means of covariance matrices estimated on high-dimensional time-series, assuming that the data is generated according to an instantaneous mixing model, which is very common in signal processing. We show that in these circumstances we can approximate the Fisher information geometric mean by employing an efficient AJD algorithm. Our approximation is in general much closer to the Fisher information geometric mean as compared to its competitors and verifies many invariance properties. Furthermore, convergence is guaranteed, the computational complexity is low and the convergence rate is quadratic. The accuracy of this new geometric mean approximation is demonstrated by means of simulations.     

Fixation probability and time in subdivided populations

New alleles arising in a population by mutation ultimately are either fixed or lost. Either is possible, for both beneficial and deleterious alleles, because of stochastic changes in allele frequency due to genetic drift. Spatially structured populations differ from unstructured populations in the probability of fixation and the time that this fixation takes. Previous results have generally made many assumptions: that all demes contribute to the next generation in exact proportion to their current sizes, that new mutations are beneficial, and that new alleles have additive effects. In this article these assumptions are relaxed, allowing for an arbitrary distribution among demes of reproductive success, both beneficial and deleterious effects, and arbitrary dominance. The effects of population structure can be expressed with two summary statistics: the effective population size and a variant of Wright's F(ST). In general, the probability of fixation is strongly affected by population structure, as is the expected time to fixation or loss. Population structure changes the effective size of the species, often strongly downward; smaller effective size increases the probability of fixing deleterious alleles and decreases the probability of fixing beneficial alleles. On the other hand, population structure causes an increase in the homozygosity of alleles, which increases the probability of fixing beneficial alleles but somewhat decreases the probability of fixing deleterious alleles. The probability of fixing new beneficial alleles can be simply described by 2hs(1 - F(ST))N(e)/N(tot), where hs is the change in fitness of heterozygotes relative to the ancestral homozygote, F(ST) is a weighted version of Wright's measure of population subdivision, and N(e) and N(tot) are the effective and census sizes, respectively. These results are verified by simulation for a broad range of population structures, including the island model, the stepping-stone model, and a model with extinction and recolonization.     

The network structure affects the fixation probability when it couples to the birth-death dynamics in finite population

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.     

Population growth enhances the mean fixation time of neutral mutations and the persistence of neutral variation

A fundamental result of population genetics states that a new mutation, at an unlinked neutral locus in a randomly mating diploid population, has a mean time of fixation of ～4N(e) generations, where N(e) is the effective population size. This result is based on an assumption of fixed population size, which does not universally hold in natural populations. Here, we analyze such neutral fixations in populations of changing size within the framework of the diffusion approximation. General expressions are derived for the mean and variance of the fixation time in changing populations. Some explicit results are given for two cases: (i) the effective population size undergoes a sudden change, representing a sudden population expansion or a sudden bottleneck; (ii) the effective population changes linearly for a limited period of time and then remains constant. Additionally, a lower bound for the mean time of fixation is obtained for an effective population size that increases with time, and this is applied to exponentially growing populations. The results obtained in this work show, among other things, that for populations that increase in size, the mean time of fixation can be enhanced, sometimes substantially so, over 4N(e,0) generations, where N(e,0) is the effective population size at the time the mutation arises. Such an enhancement is associated with (i) an increased probability of neutral polymorphism in a population and (ii) an enhanced persistence of high-frequency neutral variation, which is the variation most likely to be observed.     

EFFECTIVE POPULATION SIZE IN A CONTINUOUSLY DISTRIBUTED POPULATION

An individual-based simulation model was created to examine genetic variability, time until fixation and spatial genetic structure in a continuously distributed population. Previous mathematical models for continuously distributed populations have the difficulty that the assumption of independent reproduction and independent dispersal of offspring cause clumped spatial distribution and thus violate an assumption of random spatial distribution. In this study, this problem is avoided by considering the dispersal behavior of offspring. The simulation results showed that the inbreeding effective population size estimated by the rate of decrease of heterozygosity during the first 15 generations corresponds to the neighborhood size calculated by the standard deviation of the dispersal distance (σ_T). This inbreeding effective population size does not greatly change with the area of simulation when the densities and σ_T are the same. However, the inbreeding effective population size estimated by heterozygosity using the first 500 generations is larger than the neighborhood size calculated by the dispersal distance and increases with the area of simulation with the same densities. The variance effective population size, estimated by time until fixation of alleles, increases with dispersal distance (σ_T) and with the area of simulation given the same densities. The inbreeding effective population size and variance effective population size were smaller than the actual population size unless σ_T is sufficiently large (2 σ_T > approximate L/2, where L is a side of the simulation square).