Two linked loci under mutation-selection balance and Muller’s ratchet

Highlights

• Mutation-selection balance at two linked loci can be fully investigated.

• Dynamics of loss of wild-type alleles depend on the type of epistasis.

• After the Muller's ratchet clicks, a deleterious allele is fixed by selection.

Abstract

We report complete analysis of a deterministic model of deleterious mutations and negative selection against them at two haploid loci without recombination. As long as mutation is a weaker force than selection, mutant alleles remain rare at the only stable equilibrium, and otherwise, a variety of dynamics are possible. If the mutation-free genotype is absent, generally the only stable equilibrium is the one that corresponds to fixation of the mutant allele at the locus where it is less deleterious. This suggests that fixation of a deleterious allele that follows a click of the Muller's ratchet is governed by natural selection, instead of random drift.

Introduction

Negative selection against deleterious mutations occurs in all populations. As long as random drift can be ignored, such selection leads, with rare exceptions (Yang and Kondrashov, 2003), to deterministic mutation-selection equilibrium. Analysis of this equilibrium at a single locus represents one of the foundational results of population genetics (Haldane, 1937, Muller, 1950, Moran, 1976, Kingman, 1977, Bürger, 1998).

Of course, selection can simultaneously act at multiple loci. Mutation and selection at two loci were treated in a number of papers (see, for example, Eshel and Feldman, 1970, Karlin and McGregor, 1971). Initially, these analyses were mostly concerned with situations where at least some mutations can be advantageous. Strictly deleterious mutations were studied by Feldman (1971) (see also Otto and Feldman, 1997), but only under the assumption that mutation rates are sufficiently low, so that wild-type alleles always persist in the population which converges to the polymorphic mutation-selection equilibrium.

The balance between deleterious mutations and negative selection against them at many loci in an infinite population was first investigated by Kimura et al. (1966). Hermisson et al. (2002) described a number phenomena that are possible in this situation, including the loss of wild-type alleles at some loci due to mutational pressure. However, these analyses assumed that all the mutable loci are identical in terms of both the mutation rates and the effect of mutant alleles on fitness.

Here, we address two questions. First, we provide a complete, straightforward description of the dynamic phenomena that may lead to loss of wild-type allele(s) in a simple model of two totally linked loci in an infinite haploid population. This model is simple enough to be analyzed completely, but still rich enough to admit a number of qualitatively different regimes. The mode of epistasis between the two loci, synergistic vs. diminishing returns, is the key factor affecting dynamics of the population. Second, we consider the dynamics of this model under an assumption that the mutation-free genotype is initially absent, perhaps being lost due to random drift (a click of the Muller’s ratchet). This analysis demonstrates that fixation of a mutant allele at one of the loci that occurs soon after every click of the Muller’s ratchet (Charlesworth and Charlesworth, 1997), is an inherently deterministic effect and generally does not depend on random drift.

Model

Consider two non-recombining haploid diallelic loci, A and B, where allele A mutates into a deleterious allele a with the rate μ and allele B mutates into a deleterious allele b with the rate $\nu$. Frequencies of genotypes of AB, Ab, aB, and ab are 1-x-y-z, x, y, and z, respectively, and their fitnesses are 1, 1-t, 1-s, and (1-s)*(1-t)*(1-k), respectively, where s and t are coefficients of selection against mutant alleles at loci A and B, and k characterizes epistasis. Then, assuming discrete generations, genotype frequencies in successive generations are connected by the following system of difference equations:

${\mathrm{X}}_{\mathrm{n}+1}=\frac{\left(x+\left(1-x-y-z\right)\nu \left(1-\mu \right)-x\mu \right)\left(1-t\right)}{\stackrel{-}{W}}$

$${\mathrm{y}}_{\mathrm{n}+1}=\frac{\left(y+\left(1-x-y-z\right)\mu \left(1-\nu \right)-y\nu \right)\left(1-s\right)}{\stackrel{-}{W}}$$

${\mathrm{z}}_{\mathrm{n}+1}=\frac{\left(z+y\nu +x\mu +\left(1-x-y-z\right)\mu \nu \right)\left(1-s\right)\left(1-t\right)\left(1-k\right)}{\stackrel{-}{W}}$,where

$\stackrel{-}{W}=1-\left(x+\left(1-x-y-z\right)\nu \left(1-\mu \right)-x\mu \right)-\left(y+\left(1-x-y-z\right)\mu \left(1-\nu \right)-y\nu \right)-\left(z+y\nu +x\mu +\left(1-x-y-z\right)\mu \nu \right)+\left(x+\left(1-x-y-z\right)\nu \left(1-\mu \right)-x\mu \right)\left(1-t\right)+\left(y+\left(1-x-y-z\right)\mu \left(1-\nu \right)-y\nu \right)\left(1-s\right)+\left(z+y\nu +x\mu +\left(1-x-y-z\right)\mu \nu \right)\left(1-s\right)\left(1-t\right)\left(1-k\right)$is the mean population fitness. Equilibria of this system and their stability were studied using Wolfram Mathematica. All computer code used in this research is available at https://github.com/khudyakovaks/Two-linked-loci-under-mutation-selection-balance-and-Muller-s-ratchet.

Because our model lacks recombination, it is formally equivalent to a one-locus model with four alleles. However, explicit consideration of its two diallelic loci makes its dynamics easier to grasp.