Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates

Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates

Claus Vogl
(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

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

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

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).

Fixation in large populations: a continuous view of a discrete problem

Fixation in large populations: a continuous view of a discrete problem

Fabio A. C. C. Chalub, Max O. Souza
(Submitted on 27 Aug 2014)

We study fixation in large, but finite populations with two types, and dynamics governed by birth-death processes. By considering a restricted class of such processes, which includes most classical evolutionary processes, we derive a continuous approximation for the probability of fixation that is valid beyond the weak-selection (WS) limit. Indeed, in the derivation three regimes naturally appear: selection-driven, balanced, and quasi-neutral — the latter two require WS, while the former can appear with or without WS. From the continuous approximations, we then obtain asymptotic approximations for evolutions with at most one equilibrium, in the selection-driven regime, that does not preclude a weak-selection regime. As an application, we show that the fixation pattern for the Hawk and Dove game satisfies what we term the one-half law: if the Evolutionary Stable Strategy (ESS) is outside a small interval around $\sfrac{1}{2}$, the fixation is of dominance type. We also show that outside of the weak-selection regime the dynamics of large populations can have very little resemblance to the infinite population case. In addition, we also show results for the case of two equilibria. Finally, we present a continuous restatement of the definition of an ESSN strategy, that is valid for large populations. We then present two applications of this restatement: we obtain a definition valid in the quasi-neutral regime that recovers the one-third law under linear fitness and, as a generalisation, we introduce the concept of critical-frequency.

Emergent speciation by multiple Dobzhansky-Muller incompatibilities

Emergent speciation by multiple Dobzhansky-Muller incompatibilities

Tiago , Kevin E. Bassler, Ricardo B. R. Azevedo

The Dobzhansky-Muller model posits that incompatibilities between alleles at different loci cause speciation. However, it is known that if the alleles involved in a Dobzhansky-Muller incompatibility (DMI) between two loci are neutral, the resulting reproductive isolation cannot be maintained in the presence of either mutation or gene flow. Here we propose that speciation can emerge through the collective effects of multiple neutral DMIs that cannot, individually, cause speciation-a mechanism we call emergent speciation. We investigate emergent speciation using a haploid neutral network model with recombination. We find that certain combinations of multiple neutral DMIs can lead to speciation. Complex DMIs and high recombination rate between the DMI loci facilitate emergent speciation. These conditions are likely to occur in nature. We conclude that the interaction between DMIs may be a root cause of the origin of species.

Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model

Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model

Yuri S. Semenov, Artem S. Novozhilov
(Submitted on 19 Aug 2014)

We reformulate the eigenvalue problem for the selection–mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples and theoretical findings are compared with numerical calculations.