The evolution of moment generating functions for the Wright Fisher model of population genetics
Tat Dat Tran, Julian Hofrichter, Juergen Jost
(Submitted on 21 Jan 2014)

We derive and apply a partial differential equation for the moment generating function of the Wright-Fisher model of population genetics.

Coalescence 2.0: a multiple branching of recent theoretical developments and their applications
Aurelien Tellier, Christophe Lemaire
(Submitted on 21 Jan 2014)

Population genetics theory has laid the foundations for genomics analyses including the recent burst in genome scans for selection and statistical inference of past demographic events in many prokaryote, animal and plant species. Identifying SNPs under natural selection and underpinning species adaptation relies on disentangling the respective contribution of random processes (mutation, drift, migration) from that of selection on nucleotide variability. Most theory and statistical tests have been developed using the Kingman coalescent theory based on the Wright-Fisher population model. However, these theoretical models rely on biological and life-history assumptions which may be violated in many prokaryote, fungal, animal or plant species. Recent theoretical developments of the so called multiple merger coalescent models are reviewed here ({\Lambda}-coalescent, beta-coalescent, Bolthausen-Snitzman, {\Xi}-coalescent). We explicit how these new models take into account various pervasive ecological and biological characteristics, life history traits or life cycles which were not accounted in previous theories such as 1) the skew in offspring production typical of marine species, 2) fast adapting microparasites (virus, bacteria and fungi) exhibiting large variation in population sizes during epidemics, 3) the peculiar life cycles of fungi and bacteria alternating sexual and asexual cycles, and 4) the high rates of extinction-recolonization in spatially structured populations. We finally discuss the relevance of multiple merger models for the detection of SNPs under selection in these species, for population genomics of very large sample size and advocate to potentially examine the conclusion of previous population genetics studies.

The existence and abundance of ghost ancestors in biparental populations
Simon Gravel, Mike Steel
(Submitted on 15 Jan 2014)

In a randomly-mating biparental population of size N there are, with high probability, individuals who are genealogical ancestors of every extant individual within approximately log2(N) generations into the past. We use this result of Chang to prove a curious corollary under standard models of recombination: there exist, with high probability, individuals within a constant multiple of log2(N) generations into the past who are simultaneously (i) genealogical ancestors of {\em each} of the individuals at the present, and (ii) genetic ancestors to {\em none} of the individuals at the present. Such ancestral individuals – ancestors of everyone today that left no genetic trace — represent `ghost’ ancestors in a strong sense. In this short note, we use simple analytical argument and simulations to estimate how many such individuals exist in Wright-Fisher populations.

Entropy Rates of the Multidimensional Moran Processes and Generalizations
Marc Harper
(Submitted on 13 Jan 2014)

The interrelationships of the fundamental biological processes natural selection, mutation, and stochastic drift are quantified by the entropy rate of Moran processes with mutation, measuring the long-run variation of a Markov process. The entropy rate is shown to behave intuitively with respect to evolutionary parameters such as monotonicity with respect to mutation probability (for the neutral landscape), relative fitness, and strength of selection. Strict upper bounds, depending only on the number of replicating types, for the entropy rate are given and the neutral fitness landscape attains the maximum in the large population limit. Various additional limits are computed including small mutation, weak and strong selection, and large population holding the other parameters constant, revealing the individual contributions and dependences of each evolutionary parameter on the long-run outcomes of the processes.

Evolution of female choice and age-dependent male traits with paternal germ-line mutation
Joel James Adamson
(Submitted on 11 Dec 2013)

Several studies question the adaptive value of female preferences for older males. Theory and evidence show that older males carry more deleterious mutations in their sperm than younger males carry. These mutations are not visible to females choosing mates. Germ-line mutations could oppose preferences for “good genes.” Choosy females run the risk that offspring of older males will be no more attractive or healthy than offspring of younger males. Germ-line mutations could pose a particular problem when females can only judge male trait size, rather than assessing age directly. I ask whether or not females will prefer extreme traits, despite reduced offspring survival due to age-dependent mutation. I use a quantitative genetic model to examine the evolution of female preferences, an age-dependent male trait, and overall health (“condition”). My dynamical equation includes mutation bias that depends on the generation time of the population. I focus on the case where females form preferences for older males because male trait size depends on male age. My findings agree with good genes theory. Females at equilibrium always select above-average males. The trait size preferred by females directly correlates with the direct costs of the preference. Direct costs can accentuate the equilibrium preference at a higher rate than mutational parameters. Females can always offset direct costs by mating with older, more ornamented males. Age-dependent mutation in condition maintains genetic variation in condition and thereby maintains the selective value of female preferences. Rather than eliminating female preferences, germ-line mutations provide an essential ingredient in sexual selection.

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.