Formal properties of the probability of fixation: identities, inequalities and approximations
David M. McCandlish, Charles L. Epstein, Joshua B. Plotkin
(Submitted on 5 Dec 2013)
The formula for the probability of fixation of a new mutation is widely used in theoretical population genetics and molecular evolution. Here we derive a series of identities, inequalities and approximations for the exact probability of fixation of a new mutation under the Moran process (equivalent results hold for the approximate probability of fixation for the Wright-Fisher process after an appropriate change of variables). We show that the behavior of the logarithm of the probability of fixation is particularly simple when the selection coefficient is measured as a difference of Malthusian fitnesses, and we exploit this simplicity to derive several inequalities and approximations. We also present a comprehensive comparison of both existing and new approximations for the probability of fixation, highlighting in particular approximations that result in a reversible Markov chain when used to model the dynamics of evolution under weak mutation.
Ploidy and the Predictability of Evolution in Fisher’s Geometric Model
Sandeep Venkataram, Diamantis Sellis, Dmitri A Petrov
Predicting adaptive evolutionary trajectories is a primary goal of evolutionary biology. One can differentiate between forward and backward predictability, where forward predictability measures the likelihood of the same adaptive trajectory occurring in independent evolutions and backward predictability measures the likelihood of a particular adaptive path given the knowledge of starting and final states. Recent studies have attempted to measure both forward and backward predictability using experimental evolution in asexual haploid microorganisms. Similar experiments in diploid organisms have not been conducted. Here we simulate adaptive walks using Fisher’s Geometric Model in haploids and diploids and find that adaptive walks in diploids are less forward- and more backward-predictable than adaptive walks in haploids. We argue that the difference is due to the ability of diploids in our simulations to generate transiently stable polymorphisms and to allow adaptive mutations of larger phenotypic effect. As stable polymorphisms can be generated in both haploid and diploid natural populations through a number of mechanisms, we argue that inferences based on experiments in which adaptive walks proceed through succession of monomorphic states might miss many of the key features of adaptation.
The effect of linkage on establishment and survival of locally beneficial mutations
Simon Aeschbacher, Reinhard Buerger
(Submitted on 25 Nov 2013)
When organisms adapt to spatially heterogeneous environments, selection may drive divergence at multiple genes. If populations under divergent selection also exchange migrants, we expect genetic differentiation to be high at selected loci, relative to the baseline caused by migration and genetic drift. Indeed, empirical studies have found peaks of putatively adaptive differentiation. These are highly variable in length, some of them extending over several hundreds of thousands of base pairs. How can such `islands of differentiation’ be explained? Physical linkage produces elevated levels of differentiation at loci close to genes under selection. However, whether this is enough to account for the observed patterns of divergence is not well understood. Here, we investigate the fate of a locally beneficial mutation that arises in linkage to an existing migration-selection polymorphism and derive two important quantities: the probability that the mutation becomes established, and the expected time to its extinction. We find that intermediate levels of recombinations are sometimes favourable, and that physical linkage can lead to strongly elevated invasion probabilities and extinction times. We provide a rule of thumb for when this is the case. Moreover, we quantify the long-term effect of polygenic local adaptation on linked neutral variation.
Computational inference beyond Kingman’s coalescent
Jere Koskela, Paul Jenkins, Dario Spano
(Submitted on 22 Nov 2013)
Full likelihood inference under Kingman’s coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) method have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general Λ- and Ξ-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites Λ- and Ξ-coalescents, and use them to obtain “approximately optimal” IS and PAC algorithms for Λ-coalescents, yielding substantial gains in efficiency over existing methods.
Population genetic consequences of the Allee effect and the role of offspring-number variation
Meike J. Wittmann, Wilfried Gabriel, Dirk Metzler
(Submitted on 21 Nov 2013)
A strong demographic Allee effect in which the expected population growth rate is negative below a certain critical population size can cause high extinction probabilities in small introduced populations. However, many species are repeatedly introduced to the same location and eventually one population may overcome the Allee effect by chance. With the help of stochastic models, we investigate how much genetic diversity such successful populations harbour on average and how this depends on offspring-number variation, an important source of stochastic variability in population size. We find that with increasing variability, the Allee effect increasingly promotes genetic diversity in successful populations. Successful Allee-effect populations with highly variable population dynamics escape rapidly from the region of small population sizes and do not linger around the critical population size. Therefore, they are exposed to relatively little genetic drift. We show that here—unlike in classical population genetics models—the role of offspring-number variation cannot be accounted for by an effective-population-size correction. Thus, our results highlight the importance of detailed biological knowledge, in this case on the probability distribution of family sizes, when predicting the evolutionary potential of newly founded populations or when using genetic data to reconstruct their demographic history.
Validity of covariance models for the analysis of geographical variation
Gilles Guillot, René Schilling, Emilio Porcu, Moreno Bevilacqua
(Submitted on 17 Nov 2013)
Due to the availability of large molecular data-sets, covariance models are increasingly used to describe the structure of genetic variation as an alternative to more heavily parametrised biological models. We focus here on a class of parametric covariance models that received sustained attention lately and show that the conditions under which they are valid mathematical models have been overlooked so far. We provide rigorous results for the construction of valid covariance models in this family. We also outline how to construct alternative covariance models for the analysis of geographical variation that are both mathematically well behaved and easily implementable.
Genetic diversity in introduced populations with Allee effect
Meike J. Wittmann, Wilfried Gabriel, Dirk Metzler
(Submitted on 18 Nov 2013)
A phenomenon that strongly influences the demography of small introduced populations and thereby potentially their genetic diversity is the Allee effect, a reduction in population growth rates at small population sizes. We take a stochastic modeling approach to investigate levels of genetic diversity in populations that successfully overcame a strong demographic Allee effect, a scenario in which populations smaller than a certain critical size are expected to decline. Our results indicate that compared to successful populations without Allee effect, successful Allee-effect populations tend to 1) derive from larger founder population sizes and thus have a higher initial amount of genetic variation, 2) spend fewer generations at small population sizes where genetic drift is particularly strong, and 3) spend more time around the critical population size and thus experience more drift there. Altogether, the Allee effect can either increase or decrease genetic diversity, depending on the average founder population size. In the case of multiple introduction events, there is an additional increase in diversity because Allee-effect populations tend to derive from a larger number of introduction events than other populations. Finally, we show that given genetic data from sufficiently many populations, we can statistically infer the critical population size.
Stochastic dynamics of adaptive trait and neutral marker driven by eco-evolutionary feedbacks
Sylvain Billiard (GEPV), Regis Ferriere (CNRS UMR 7625,), Sylvie Méléard (CMAP), Viet Chi Tran (LPP)
(Submitted on 23 Oct 2013)
How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. Population dynamics are driven by births and deaths, mutations at birth, and competition between individuals. Trait values influence ecological processes (demographic events, competition), and competition generates selection on trait variation, thus closing the eco-evolutionary feedback loop. The demographic effects of the trait are also expected to influence the generation and maintenance of neutral variation. We consider a large population limit with rare mutation, under the assumption that the neutral marker mutates faster than the trait under selection. We prove the convergence of the stochastic individual-based process to a new measure-valued diffusive process with jumps that we call Substitution Fleming-Viot Process (SFVP). When restricted to the trait space this process is the Trait Substitution Sequence first introduced by Metz et al. (1996). During the invasion of a favorable mutation, a genetical bottleneck occurs and the marker associated with this favorable mutant is hitchhiked. By rigorously analysing the hitchhiking effect and how the neutral diversity is restored afterwards, we obtain the condition for a time-scale separation; under this condition, we show that the marker distribution is approximated by a Fleming-Viot distribution between two trait substitutions. We discuss the implications of the SFVP for our understanding of the dynamics of neutral variation under eco-evolutionary feedbacks and illustrate the main phenomena with simulations. Our results highlight the joint importance of mutations, ecological parameters, and trait values in the restoration of neutral diversity after a selective sweep.
Cryptic Genetic Variation Can Make Irreducible Complexity a Common Mode of Adaptation
Meredith V. Trotter, Daniel B. Weissman, Grant I. Peterson, Kayla M. Peck, Joanna Masel
(Submitted on 22 Oct 2013)
The existence of complex (multiple-step) genetic adaptations that are “irreducible” (i.e., all partial combinations are less fit than the original genotype) is one of the longest standing problems in evolutionary biology. In standard genetics parlance, these adaptations require the crossing of a wide adaptive valley of deleterious intermediate stages. Here we demonstrate, using a simple model, that evolution can cross wide valleys to produce “irreducibly complex” adaptations by making use of previously cryptic mutations. When revealed by an evolutionary capacitor, previously cryptic mutants have higher initial frequencies than do new mutations, bringing them closer to a valley-crossing saddle in allele frequency space. Moreover, simple combinatorics imply an enormous number of candidate combinations exist within available cryptic genetic variation. We model the dynamics of crossing of a wide adaptive valley after a capacitance event using both numerical simulations and analytical approximations. Although individual valley crossing events become less likely as valleys widen, by taking the combinatorics of genotype space into account, we see that revealing cryptic variation can cause the frequent evolution of complex adaptations. This finding also effectively dismantles “irreducible complexity” as an argument against evolution by providing a general mechanism for crossing wide adaptive valleys.
Sex-specific recombination rates and allele frequencies affect the invasion of sexually antagonistic variation on autosomes
Minyoung Wyman, Mark Wyman
(Submitted on 19 Oct 2013)
The introduction and persistence of novel sexually antagonistic alleles can depend upon factors that differ between males and females. Understanding the conditions for invasion in a two-locus model can elucidate these processes. For instance, selection can act differently upon the sexes, or sex-linkage can facilitate the invasion of genetic variation with opposing fitness effects between the sexes. Two factors that deserve further attention are recombination rates and allele frequencies — both of which can vary substantially between the sexes. We find that sex-specific recombination rates in a two-locus diploid model can affect the invasion outcome of sexually antagonistic alleles and that the sex-averaged recombination rate is not necessarily sufficient to predict invasion. We confirm that the range of permissible recombination rates is smaller in the sex benefitting from invasion and larger in the sex harmed by invasion. However, within the invasion space, male recombination rate can be greater than, equal to, or less than female recombination rate in order for a male-benefit, female-detriment allele to invade (and similarly for a female-benefit, male-detriment allele). We further show that a novel, sexually antagonistic allele that is also associated with a lowered recombination rate can invade more easily when present in the double heterozygote genotype. Finally, we find that sexual dimorphism in resident allele frequencies can impact the invasion of new sexually antagonistic alleles at a second locus. Our results suggest that accounting for sex-specific recombination rates and allele frequencies can determine the difference between invasion and non-invasion of novel sexually antagonistic alleles in a two-locus model.