The fixation probability of two competing beneficial mutations

Suppose that a beneficial mutation is undergoing a selective sweep when another beneficial mutation arises at a linked locus. We study the fixation probability of the double mutant, i.e., one (produced by recombination) that carries both mutations. Previous analysis works well for the case where the earlier beneficial mutation confers a greater selective advantage than the later mutation, but not so well in the opposite case. We present an approach to approximating the fixation probability in the case where the later mutation confers a greater selective advantage.     

The fixation probability of a beneficial allele in a population dividing by binary fission

We derive formulae for the fixation probability, P, of a rare benefical allele segregating in a population of fixed size which reproduces by binary fission, in terms of the selection coefficient for the beneficial allele, s. We find that an earlier result P 4s does not depend on the assumption of binary fission, but depends on an assumption about the ordering of events in the life cycle. We find that P 2s for mutations occurring during chromosome replication and P 2.8s for mutations occurring at random times between replication events.     

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.     

On the fixation process of a beneficial mutation in a variable environment

A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak.     

Estimating the time since the fixation of a beneficial allele

The fixation of a beneficial allele in a population leaves a well-characterized signature in patterns of nucleotide variation at linked sites. This signature can be used to estimate the time since fixation from patterns of polymorphism in extant individuals. I introduce a method to assess the support in polymorphism data for a recent episode of directional positive selection and to estimate the time since fixation. I summarize the polymorphism data by three statistics that carry information about levels of diversity, the allele frequency spectrum, and the extent of allelic associations. Simulations are then used to obtain a sample from the posterior distribution of the time since fixation, conditional on the observed summaries. I test the performance of the approach on simulated data and apply it to the gene tb1 in maize. The data support the recent fixation of a favored allele, consistent with what is known about the importance of tb1 in the domestication process of maize.     

Fixation dynamics of beneficial alleles in prokaryotic polyploid chromosomes and plasmids

Theoretical population genetics has been mostly developed for sexually reproducing diploid and for monoploid (haploid) organisms, focusing on eukaryotes. The evolution of bacteria and archaea is often studied by models for the allele dynamics in monoploid populations. However, many prokaryotic organisms harbor multicopy replicons—chromosomes and plasmids—and theory for the allele dynamics in populations of polyploid prokaryotes remains lacking. Here, we present a population genetics model for replicons with multiple copies in the cell. Using this model, we characterize the fixation process of a dominant beneficial mutation at 2 levels: the phenotype and the genotype. Our results show that depending on the mode of replication and segregation, the fixation of the mutant phenotype may precede genotypic fixation by many generations; we term this time interval the heterozygosity window. We furthermore derive concise analytical expressions for the occurrence and length of the heterozygosity window, showing that it emerges if the copy number is high and selection strong. Within the heterozygosity window, the population is phenotypically adapted, while both alleles persist in the population. Replicon ploidy thus allows for the maintenance of genetic variation following phenotypic adaptation and consequently for reversibility in adaptation to fluctuating environmental conditions.     

The genetic effective size of a metapopulation

The structure of a population over time, space and categories of social and sexual role governs its ability to retain genetic variation in the face of drift. A metapopulation is an extreme form of spatial structure in which loosely coupled local populations 'turnover', that is, suffer extinction followed by recolonization from elsewhere within the metapopulation. These local populations turn over with a characteristic half-life. Based on a simulation model that incorporates both realistic features of population ecology and population genetics, the ability of such a metapopulation to retain genetic variation, which may be defined as proportional to its so-called effective population size, denoted N_e(^meta), can be one to two orders of magnitude lower than the maximum total number of individuals in the system. N_e(meta) depends on the persistence time associated with longevity of local populations (the turnover half-life), the average number of local populations extant in the metapopulation and the gene flow between local populations. Habitat fragmentation, which can create a metapopulation from a formerly continuously distributed species, may have unappreciated large genetic consequences for species impacted by human development.     

The limiting behaviour of a mainland-island metapopulation

Stochastic patch occupancy models (SPOMs) are a class of discrete time Markov chains used to model the presence/absence of a population in a collection of habitat patches. This class of model is popular with ecologists due to its ability to incorporate important factors of the habitat patch network such as connectivity and distance between patches as well as heterogeneity in patch characteristics. We present an asymptotic examination of a simple type of SPOM called the mainland-island model. In this model a single patch called the mainland is connected to a large number of smaller patches called islands and each island is only connected to the mainland. We discuss the limiting behaviour of the SPOM as the number of islands increases and the size of the islands decrease relative to the mainland. We demonstrate that a variety of limiting behaviours is possible depending on the scaling of the island size and on the heterogeneity of habitat quality.     

Gene genealogies in a metapopulation

A simple genealogical process is found for samples from a metapopulation, which is a population that is subdivided into a large number of demes, each of which is subject to extinction and recolonization and receives migrants from other demes. As in the migration-only models studied previously, the genealogy of any sample includes two phases: a brief sample-size adjustment followed by a coalescent process that dominates the history. This result will hold for metapopulations that are composed of a large number of demes. It is robust to the details of population structure, as long as the number of possible source demes of migrants and colonists for each deme is large. Analytic predictions about levels of genetic variation are possible, and results for average numbers of pairwise differences within and between demes are given. Further analysis of the expected number of segregating sites in a sample from a single deme illustrates some previously known differences between migration and extinction/recolonization. The ancestral process is also amenable to computer simulation. Simulation results show that migration and extinction/recolonization have very different effects on the site-frequency distribution in a sample from a single deme. Migration can cause a U-shaped site-frequency distribution, which is qualitatively similar to the pattern reported recently for positive selection. Extinction and recolonization, in contrast, can produce a mode in the site-frequency distribution at intermediate frequencies, even in a sample from a single deme.     

Synchronism in a metapopulation model

We consider a spatially explicit metapopulation model with interaction among the two nearest neighbors to relate, with a simple mathematical expression, chaos in the local, uncoupled, populations, the degree of interaction among patches, size of the metapopulation, and the stability of the synchronized attractor. Since synchronism is strongly correlated with extinction, our results can provide useful information on factors leading to population extinction.     

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.     

Evolution of migration in a metapopulation

In this paper a general deterministic discrete-time metapopulation model with a finite number of habitat patches is analysed within the framework of adaptive dynamics. We study a general model and prove analytically that (i) if the resident populations state is a fixed point, then the resident strategy with no migration is an evolutionarily stable strategy, (ii) a mutant population with no migration can invade any resident population in a fixed point state, (iii) in the uniform migration case the strategy not to migrate is attractive under small mutational steps so that selection favours low migration. Some of these results have been previously observed in simulations, but here they are proved analytically in a general case. If the resident population is in a two-cyclic orbit, then the situation is different. In the uniform migration case the invasion behaviour depends both on the type of the residents attractor and the survival probability during migration. If the survival probability during migration is low, then the system evolves towards low migration. If the survival probability is high enough, then evolutionary branching can happen and the system evolves to a situation with several coexisting types. In the case of out-of-phase attractor, evolutionary branching can happen with significantly lower survival probabilities than in the in-phase attractor case. Most results in the two-cyclic case are obtained by numerical simulations. Also, when migration is not uniform we observe in numerical simulations in the two-cyclic orbit case selection for low migration or evolutionary branching depending on the survival probability during migration.     

The effective size of a metapopulation living in a heterogeneous patch network

I analyze stochastic patch occupancy models (SPOMs), which record habitat patches as empty or occupied. A problem with SPOMs has been that if the spatial structure of a heterogeneous habitat patch network is taken into account, the computational effort needed to analyze a SPOM grows as a power of 2n, where n is the number of habitat patches. I propose a computationally feasible approximation method, which approximates the behavior of a heterogeneous SPOM by an "ideal" metapopulation inhabiting a network of identical and equally connected habitat patches. The transformation to the ideal metapopulation is based on weighting the individual patch occupancies by the dynamic values of the habitat patches, which may be calculated from the deterministic mean-field approximation of the original SPOM. Conceptually, the method resembles the calculation of the effective size of a population in the context of population genetics. I demonstrate how the method may be applied to SPOMs with flexible structural assumptions and with spatially correlated and temporally varying parameter values. I apply the method to a real habitat patch network inhabited by the Glanville fritillary butterfly, illustrating that the metapopulation dynamics of this species are essentially driven by temporal variability in the environmental conditions.     

Cultural niche construction in a metapopulation

Cultural niche construction is the process by which certain evolving cultural traits form a cultural niche that affects the evolution of other genetic and cultural traits [Laland, K., et al., 2001. Cultural niche construction and human evolution. J. Evol. Biol. 14, 22-33; Ihara, Y., Feldman, M., 2004. Cultural niche construction and the evolution of small family size. Theor. Popul. Biol. 65, 105-111]. In this study we focus on cultural niche construction in a metapopulation (a population of populations), where the frequency of one cultural trait (e.g. the level of education) determines the transmission rate of a second trait (e.g. the adoption of fertility reduction preferences) within and between populations. We formulate the Metapopulation Cultural Niche Construction (MPCNC) model by defining the cultural niche induced by the first trait as the construction of a social interaction network on which the second trait may percolate. Analysis of the model reveals dynamics that are markedly different from those observed in a single population, allowing, for example, different (or even opposing) dynamics in each population. In particular, this model can account for the puzzling phenomenon reported in previous studies [Bongaarts, J., Watkins, S., 1996. Social interactions and contemporary fertility transitions. Popul. Dev. Rev. 22 (4), 639-682] that the onset of the demographic transition in different countries occurred at ever lower levels of development.     

Fixation of new alleles and the extinction of small populations: drift load, beneficial alleles, and sexual selection

With a small effective population size, random genetic drift is more important than selection in determining the fate of new alleles. Small populations therefore accumulate deleterious mutations. Left unchecked, the effect of these fixed alleles is to reduce the reproductive capacity of a species, eventually to the point of extinction. New beneficial mutations, if fixed by selection, can restore some of this lost fitness. This paper derives the overall change in fitness due to fixation of new deleterious and beneficial alleles, as a function of the distribution of effects of new mutations and the effective population size. There is a critical effective size below which a population will on average decline in fitness, but above which beneficial mutations allow the population to persist. With reasonable estimates of the relevant parameters, this critical effective size is likely to be a few hundred. Furthermore, sexual selection can act to reduce the fixation probability of deleterious new mutations and increase the probability of fixing new beneficial mutations. Sexual selection can therefore reduce the risk of extinction of small populations.     

Neutral additive genetic variance in a metapopulation

For neutral, additive quantitative characters, the amount of additive genetic variance within and among populations is predictable from Wright's FST, the effective population size and the mutational variance. The structure of quantitative genetic variance in a subdivided metapopulation can be predicted from results from coalescent theory, thereby allowing single-locus results to predict quantitative genetic processes. The expected total amount of additive genetic variance in a metapopulation of diploid individual is given by 2Ne sigma m2 (1 + FST), where FST is Wright's among-population fixation index, Ne is the eigenvalue effective size of the metapopulation, and sigma m2 is the mutational variance. The expected additive genetic variance within populations is given by 2Ne sigma e2(1-FST), and the variance among demes is given by 4FSTNe sigma m2. These results are general with respect to the types of population structure involved. Furthermore, the dimensionless measure of the quantitative genetic variance among populations, QST, is shown to be generally equal to FST for the neutral additive model. Thus, for all population structures, a value of QST greater than FST for neutral loci is evidence for spatially divergent evolution by natural selection.     

Dispersal-related life-history trade-offs in a butterfly metapopulation

1. Recent studies on butterflies have documented apparent evolutionary changes in dispersal rate in response to climate change and habitat change. These studies often assume a trade-off between dispersal rate (or flight capacity) and reproduction, which is the rule in wing-dimorphic species but might not occur equally in wing-monomorphic species such as butterflies. 2. To investigate the relationship between dispersal rate and fecundity in the Glanville fritillary butterfly Melitaea cinxia we recorded lifetime individual movements, matings, ovipositions, and maximal life span in a large (32 x 26 m) population cage in the field. Experimental material was obtained from 20 newly established and 20 old local populations within a large metapopulation in the Aland Islands in Finland. 3. Females of the Glanville fritillary from newly established populations are known to be more dispersive in the field, and in the cage they showed significantly greater mobility, mated earlier, and laid more egg clutches than females from old populations. The dispersive females from new populations exhibited no reduced lifetime fecundity in the cage, but they had a shorter maximal life span than old-population females. 4. These results challenge the dispersal-fecundity trade-off for nonmigratory butterflies but instead suggest a physiological trade-off between high metabolic performance and reduced maximal life span. High metabolic performance may explain high rates of dispersal and oviposition in early life. 5. In fragmented landscapes, an ecological trade-off exists between being more dispersive and hence spending more time in the landscape matrix vs. having more time for reproduction in the habitat. We estimate with a dispersal model parameterized for the Glanville fritillary that the lifetime egg production is 4% smaller on average in the more dispersive butterflies in a representative landscape, with much variation depending on landscape structure in the neighbourhood of the natal patch, from--26 to 45% in the landscape analysed in this paper.