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.

Non-monotonic effects of migration in populations with balancing selection
Pierangelo Lombardo, Andrea Gambassi, Luca Dall’Asta
(Submitted on 18 Oct 2013)

Balancing selection is recognized as a prominent evolutionary force responsible for the maintenance of genetic diversity in natural populations. We quantify its influence on the evolution of a subdivided population, investigating how the mean-fixation time (MFT) depends on the migration rate among subpopulations. We identify a threshold in the strength of the balancing selection above which the MFT changes its qualitative behavior compared to that of neutral populations, developing an unexpected non-monotonic dependence on the migration rate. This feature carries over into an analogous behavior of the heterozygosity, which is an index of the biodiversity of the population.

General triallelic frequency spectrum under demographic models with variable population size
Paul A. Jenkins, Jonas W. Mueller, Yun S. Song
(Submitted on 13 Oct 2013)

It is becoming routine to obtain datasets on DNA sequence variation across several thousands of chromosomes, providing unprecedented opportunity to infer the underlying biological and demographic forces. Such data make it vital to study summary statistics which offer enough compression to be tractable, while preserving a great deal of information. One well-studied summary is the site frequency spectrum—the empirical distribution, across segregating sites, of the sample frequency of the derived allele. However, most previous theoretical work has assumed that each site has experienced at most one mutation event in its genealogical history, which becomes less tenable for very large sample sizes. In this work we obtain, in closed-form, the predicted frequency spectrum of a site that has experienced at most two mutation events, under very general assumptions about the distribution of branch lengths in the underlying coalescent tree. Among other applications, we obtain the frequency spectrum of a triallelic site in a model of historically varying population size. We demonstrate the utility of our formulas in two settings: First, we show that triallelic sites are more sensitive to the parameters of a population that has experienced historical growth, suggesting that they will have use if they can be incorporated into demographic inference. Second, we investigate a recently proposed alternative mechanism of mutation in which the two derived alleles of a triallelic site are created simultaneously within a single individual, and we develop a test to determine whether it is responsible for the excess of triallelic sites in the human genome.

Non-identifiability of identity coefficients at biallelic loci
Miklós Csűrös
(Submitted on 13 Oct 2013)

Shared genealogies introduce allele dependencies in diploid genotypes, as alleles within an individual or between different individuals will likely match when they originate from a recent common ancestor. At a locus shared by a pair of diploid individuals, there are nine combinatorially distinct modes of identity-by-descent (IBD), capturing all possible combinations of coancestry and inbreeding. A distribution over the IBD modes is described by the nine associated probabilities, known as (Jacquard’s) identity coefficients. The genetic relatedness between two individuals can be succinctly characterized by the identity coefficients corresponding to the joint genealogy. The identity coefficients (together with allele frequencies) determine the distribution of joint genotypes at a locus. At a locus with two possible alleles, identity coefficients are not identifiable because different coefficients can generate the same genotype distribution.
We analyze precisely how different IBD modes combine into identical genotype distributions at diallelic loci. In particular, we describe IBD mode mixtures that result in identical genotype distributions at all allele frequencies, implying the non-identifiability of the identity coefficients from independent loci. Our analysis yields an exhaustive characterization of relatedness statistics that are always identifiable. Importantly, we show that identifiable relatedness statistics include the kinship coefficient (probability that a random pair of alleles are identical by descent between individuals) and inbreeding-related measures, which can thus be estimated from genotype distributions at independent loci.

Chaos and Unpredictability in Evolution
Iaroslav Ispolatov, Michael Doebeli
(Submitted on 24 Sep 2013)

The possibility of complicated dynamic behaviour driven by non-linear feedbacks in dynamical systems has revolutionized science in the latter part of the last century. Yet despite examples of complicated frequency dynamics, the possibility of long-term evolutionary chaos is rarely considered. The concept of “survival of the fittest” is central to much evolutionary thinking and embodies a perspective of evolution as a directional optimization process exhibiting simple, predictable dynamics. This perspective is adequate for simple scenarios, when frequency-independent selection acts on scalar phenotypes. However, in most organisms many phenotypic properties combine in complicated ways to determine ecological interactions, and hence frequency-dependent selection. Therefore, it is natural to consider models for the evolutionary dynamics generated by frequency-dependent selection acting simultaneously on many different phenotypes. Here we show that complicated, chaotic dynamics of long-term evolutionary trajectories in phenotype space is very common in a large class of such models when the dimension of phenotype space is large, and when there are epistatic interactions between the phenotypic components. Our results suggest that the perspective of evolution as a process with simple, predictable dynamics covers only a small fragment of long-term evolution. Our analysis may also be the first systematic study of the occurrence of chaos in multidimensional and generally dissipative systems as a function of the dimensionality of phase space.