On the role of epistasis in adaptation
David M. McCandlish, Jakub Otwinowski, Joshua B. Plotkin
Subjects: Populations and Evolution (q-bio.PE)
Although the role of epistasis in evolution has received considerable attention from experimentalists and theorists alike, it is unknown which aspects of adaptation are in fact sensitive to epistasis. Here, we address this question by comparing the evolutionary dynamics on all finite epistatic landscapes versus all finite non-epistatic landscapes, under weak mutation. We first analyze the fitness trajectory — that is, the time course of the expected fitness of a population. We show that for any epistatic fitness landscape and choice of starting genotype, there always exists a non-epistatic fitness landscape and starting genotype that produces the exact same fitness trajectory. Thus, surprisingly, the presence or absence of epistasis is irrelevant to the first-order dynamics of adaptation. On the other hand, we show that the time evolution of the variance in fitness across replicate populations can be sensitive to epistasis: some epistatic fitness landscapes produce variance trajectories that cannot be produced by any non-epistatic landscape. Likewise, the mean substitution trajectory — that is, the expected number of mutations that fix over time — is also sensitive to epistasis. These results on identifiability have direct implications for efforts to infer epistasis from the types of data often measured in experimental populations.
The role of standing variation in geographic convergent adaptation
Peter L. Ralph, Graham Coop
The extent to which populations experiencing shared selective pressures adapt through a shared genetic response is relevant to many questions in evolutionary biology. In a number of well studied traits and species, it appears that convergent evolution within species is common. In this paper, we explore how standing, deleterious genetic variation contributes to convergent genetic responses in a geographically spread population, extending our previous work on the topic. Geographically limited dispersal slows the spread of each selected allele, hence allowing other alleles — newly arisen mutants or present as standing variation — to spread before any one comes to dominate the population. When such alleles meet, their progress is substantially slowed — if the alleles are selectively equivalent, they mix slowly, dividing the species range into a random tessellation, which can be well understood by analogy to a Poisson process model of crystallization. In this framework, we derive the geographic scale over which a typical allele is expected to dominate, the time it takes the species to adapt as a whole, and the proportion of adaptive alleles that arise from standing variation. Finally, we explore how negative pleiotropic effects of alleles before an environment change can bias the subset of alleles that get to contribute to a species adaptive response. We apply the results to the many geographically localized G6PD deficiency alleles thought to confer resistance to malaria, whose large mutational target size and deleterious effects make them likely candidates to have been present as deleterious standing variation. We find the numbers and geographic spread of these alleles matches our predictions reasonably well, which suggest that these arose both from standing variation and new mutations since the advent of malaria. Our results suggest that much of adaptation may be geographically local even when selection pressures are wide-spread. We close by discussing the implications of these results for arguments of species coherence and the nature of divergence between species.
On the unfounded enthusiasm for soft selective sweeps
Jeffrey D. Jensen
Underlying any understanding of the mode, tempo, and relative importance of the adaptive process in the evolution of natural populations is the notion of whether adaptation is mutation-limited. Two very different population genetic models have recently been proposed in which the rate of adaptation is not strongly limited by the rate at which newly arising beneficial mutations enter the population. This review discusses the theoretical underpinnings and requirements of these models, as well as the experimental insights on the parameters of relevance. Importantly, empirical and experimental evidence to date challenges the recent enthusiasm for invoking these models to explain observed patterns of variation in humans and Drosophila.
Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates
(Submitted on 8 Sep 2014)
The evolution of the allelic proportion x of a biallelic locus subject to the forces of mutation and drift is investigated in a diffusion model, assuming small scaled mutation rates. The overall scaled mutation rate is parametrized with θ=(μ1+μ0)N and the ratio of mutation rates with α=μ1/(μ1+μ0)=1−β. The equilibrium density of this process is beta with parameters αθ and βθ. Away from equilibrium, the transition density can be expanded into a series of modified Jacobi polynomials. If the scaled mutation rates are small, i.e., θ≪1, it may be assumed that polymorphism derives from mutations at the boundaries. A model, where the interior dynamics conform to the pure drift diffusion model and the mutations are entering from the boundaries is derived. In equilibrium, the density of the proportion of polymorphic alleles, \ie\ x within the polymorphic region [1/N,1−1/N], is αβθ(1x+11−x)=αβθx(1−x), while the mutation bias α influences the proportion of monomorphic alleles at 0 and 1. Analogous to the expansion with modified Jacobi polynomials, a series expansion of the transition density is derived, which is connected to Kimura’s well known solution of the pure drift model using Gegenbauer polynomials. Two temporal and two spatial regions are separated. The eigenvectors representing the spatial component within the polymorphic region depend neither on the on the scaled mutation rate θ nor on the mutation bias α. Therefore parameter changes, e.g., growing or shrinking populations or changes in the mutation bias, can be modeled relatively easily, without the change of the eigenfunctions necessary for the series expansion with Jacobi polynomials.
The general recombination equation in continuous time and its solution
Ellen Baake, Michael Baake, Majid Salamat
(Submitted on 4 Sep 2014)
The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and solve the latter recursively for generic sets of parameters. We also briefly discuss the singular cases, and how to extend the solution to this situation.
Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution
Ute Lenz, Sandra Kluth, Ellen Baake, Anton Wakolbinger
(Submitted on 2 Sep 2014)
In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let x denote today’s frequency of the beneficial type, and given x, let h(x) be the probability that, among all individuals of today’s population, the individual whose progeny will eventually take over in the population is of the beneficial type. Fearnhead [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38-54] and Taylor [Taylor, J. E., 2007. The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab. 12, 808-847] obtained a series representation for h(x). We develop a construction that contains elements of both the ancestral selection graph and the lookdown construction and includes pruning of certain lines upon mutation. Besides interest in its own right, this construction allows a transparent derivation of the series coefficients of h(x) and gives them a probabilistic meaning.
Continuous and Discontinuous Phase Transitions in Quantitative Genetics: the role of stabilizing selective pressure
Annalisa Fierro, Sergio Cocozza, Antonella Monticelli, Giovanni Scala, Gennaro Miele
(Submitted on 2 Sep 2014)
By using the tools of statistical mechanics, we have analyzed the evolution of a population of N diploid hermaphrodites in random mating regime. The population evolves under the effect of drift, selective pressure in form of viability on an additive polygenic trait, and mutation. The analysis allows to determine a phase diagram in the plane of mutation rate and strength of selection. The involved pattern of phase transitions is characterized by a line of critical points for weak selective pressure (smaller than a threshold), whereas discontinuous phase transitions characterized by metastable hysteresis are observed for strong selective pressure. A finite size scaling analysis suggests the analogy between our system and the mean field Ising model for selective pressure approaching the threshold from weaker values. In this framework, the mutation rate, which allows the system to explore the accessible microscopic states, is the parameter controlling the transition from large heterozygosity (disordered phase) to small heterozygosity (ordered one).