Fig 1.
Different studies on distribution of fitness effects of beneficial mutations demonstrate an exponential form.
Y-axis: Frequency of beneficial alleles (arbitrary units), DFE(s,t)π(s) in Eq 1. X-axis: Mutation gain in fitness due to a beneficial mutation (selection coefficient). Symbols represent results obtained for different sites of the genome in experiments on Escherichia coli [1], Pseudomonas fluorescens [2], poliovirus synonymous mutations, poliovirus non-synonymous mutations [3], poliovirus low MOI [4], E. coli acetamide (ACT), propionamide (PR), and isobutyramide (IB) [5].
Fig 2.
Deleterious alleles with higher values of fitness cost, s, are the first to be depleted during the process of adaptation (single run).
(A-E) Evolution of a sample of 102 sequences. Violet dots: better-fit alleles, yellow dots: less-fit alleles. X-axis: the cost in fitness, s, multiplied by 100. The values of s are randomly distributed with the half-Gaussian distribution, s>0, with the average sav = 0.05. Genomic sites are ordered by the value of s. Y-axis: genome number in the sample. The initial population is randomized with the average frequency of deleterious alleles fin = 0.2. Time points in generations are shown. (F) Evolution of the genome distribution in fitness. X-axis: the effective number of deleterious alleles, defined as k = −W/sav. where W is genome fitness. Different colors show discrete time intervals from 0 to 5. The Vertical grey line shows the most-fit class of genomes at t = 0. The emergence of clonal structure in (A-E) coincides with the transition from the selection of pre-existing sequences to the traveling wave regime. The other parameters: population size N = 104, number of sites L = 100, genomic mutation rate μL = 0.05. In the text, we study the ensemble-average allelic frequency averaged over many runs.
Fig 3.
The frequency of deleterious alleles decays exponentially with their fitness effect, with the slope increasing in time.
(A) Analytic prediction for the frequency of deleterious alleles from Eq 5 agrees with Monte-Carlo simulation. X-axis: Mutation cost of deleterious allele at a genomic site, s. Y-axis: Frequency of deleterious alleles at such a site, f(s). The mutant frequency f is averaged over 20 random simulation runs, the straight lines are linear regression. Different colors show different times, symbols are simulation, and lines are analytic prediction (Eq 5). The numbers on the curves are the values of the slope. Parameters as in Fig 2. (B) The slope of the distribution of deleterious alleles β, analytic (blue lines) and simulation (purple lines), as a function of time, t. Parameters fin and sav, if different from those in Fig 2, are shown on the legend. The log-slope for the simulated curves of mutant frequency in (A) is obtained by an exponential fit. We observe that the deviation of the simulated slope from the analytic prediction Eq 5 at long times coincides with the establishment of the traveling regime, which occurs later for smaller sav (Fig 2F). At long times, the traveling wave prediction, Eq 8, applies (dashed blue lines). Grey diagonal shows β(t) = t. Parameters are as in the legend of Fig 2.