Time to fixation in the presence of recombination
Introduction
Sexual reproduction is ubiquitous in nature — most eukaryotes reproduce sexually and genetic mixing is common in some bacterias (Otto and Lenormand, 2002, Rice, 2002). However, asexual reproduction is not entirely absent. Microbes such as viruses and bacteria reproduce asexually most of the time, ancient asexuals (Judson and Normark, 1996) which have remained exclusively asexual for millions of years persist and human mitochondrial DNA has not recombined for a few million years (Loewe, 2006). It is then natural to ask: under what conditions is one or the other mode of reproduction preferred?
A detailed study of theoretical models has been helpful in identifying some relevant parameters and conditions. A parameter which plays a crucial role in the evolution of sex and recombination is epistatic interaction amongst gene loci (Kouyos et al., 2007). Experiments have shown that the individual locus do not always contribute independently to the fitness of the whole sequence (Weinreich, 2000, Phillips, 2008) and the deviation of the fitness from the independent loci model is a measure of the epistatic interactions. The nature of epistasis is important in determining whether a mode of reproduction may be viable. For instance, in the absence of back mutations and recombination, a finite asexual population evolving on a nonepistatic fitness landscape accumulates deleterious mutations irreversibly (Muller’s ratchet) (Muller, 1964, Felsenstein, 1974). But the degeneration can be effectively halted if synergistic epistasis is present (Kondrashov, 1994, Jain, 2008). On complex multipeaked fitness landscapes that incorporate sign epistasis (Weinreich et al., 2005), the effect of sex has been seen to depend on the detailed topography of the fitness landscape (Kondrashov and Kondrashov, 2001, de Visser et al., 2009).
The epistatic interactions play an important role in infinitely large populations as well. In a two locus model with the four possible sequences denoted by , , and and with respective fitnesses , recombination reduces the frequency of the favorable mutant when epistasis parameter is positive but increases the frequency for negative (Eshel and Feldman, 1970). In this article, we ask: in an infinitely large recombining population, if all the population is initially located at the sequence , how much time does it take to get fixed to the double mutant with fitness ? The fixation time is expected to decrease with recombination for negative epistasis and increase for positive epistasis (Otto et al., 1994, Feldman et al., 1996). These qualitative trends are understandable from the results of Eshel and Feldman (1970): for , as recombination acts in favor of the double mutant, it will get fixed faster than in the asexual case, while the reverse holds for the case.
The main purpose of this article is to find analytical expressions for the fixation time . To this end, we develop a new method to handle the inherently nonlinear equations obeyed by the genotype frequencies in the presence of recombination (see Section 2). The basic idea of our approach is that, at any instant, only one of the genotypes dominate, so that the equations can be expanded perturbatively in powers of the ratio of the non-dominant genotype frequency to the dominant one.
The rest of the article is organised as follows. We first define the model under consideration in Section 2. The dynamics of the population frequencies for various choices of epistasis are discussed in Section 3. The fixation time, defined as the time at which the double mutant frequency reaches a given finite fraction, is calculated in Section 4. The effect of initial conditions on fixation time is considered in Section 5. The last section discusses our results which are summarised in Table 1.
Section snippets
Model
We consider a two locus model with sequences denoted by , , and and respective fitnesses . The population at these sequences evolves according to mutation, selection and recombination dynamics. In such models, several schemes have been used to implement these basic processes, such as recombination followed by mutation and then selection (Feldman et al., 1996), selection, mutation and then recombination (Kouyos et al., 2006) and selection, mutation and
Time evolution of populations
As we shall see, the dynamics of population ’s can be divided in following three dynamical phases: (i) (phase I) (ii) (phase II) and (iii) (phase III). Thus we can expand Eqs. (6), (7), (8) in powers of in phase I, in phase II and similarly, in phase III. The timescale at which a phase ends is obtained by matching the solutions of the relevant populations in the two phases. In the crossover region, however, the above assumptions
Fixation time
As seen in the last section, the unnormalised populations ’s vary exponentially (or faster) with time so that the normalised population will reach unity asymptotically. Therefore, we define the fixation time as the time when the population fraction where . In terms of ’s, this condition gives where in the above equation is the population fraction in the Phase III at . Another reason why is that for , the above equation cannot be
Initial condition with nonzero linkage disequilibrium
So far, we have discussed the population dynamics starting with an initial condition in which only one genotype has a nonzero population. In this section, we consider the situation when a small finite frequency at the intermediate loci is also present at ; i.e. . As the analytical method presented in the last sections assumes that all but one frequency is rare at a given time, it seems difficult to obtain analytical results. Therefore, we present numerical
Conclusions
In this article, we have studied the dynamics of a 2 locus model in which the population evolves deterministically under mutation, selection and recombination. As the recombination process makes the equations nonlinear, in general it is difficult to study such problems analytically. Here we have developed an analytical method to find the fixation time to the best locus for various fitness schemes.
The fixation time is one of the measures for judging whether recombination is beneficial for a
Acknowledgment
The author is grateful to J. Krug for introducing her to this problem and useful discussions. She also thanks S.-C. Park for comments on the manuscript and KITP, Santa Barbara for hospitality where a part of this work was done.
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