Interference Effects of Deleterious and Beneficial Mutations in Large Asexual Populations

Abstract

Linked beneficial and deleterious mutations are known to decrease the fixation probability of a favorable mutation in large asexual populations. While the hindering effect of strongly deleterious mutations on adaptive evolution has been well studied, how weakly deleterious mutations, either in isolation or with superior beneficial mutations, influence the rate of adaptation has not been fully explored. When the selection against the deleterious mutations is weak, the beneficial mutant can fix in many genetic backgrounds, besides the one it arose on. Here, taking this factor into account, I obtain an accurate analytical expression for the fixation probability of a beneficial mutant in an asexual population at mutation-selection balance. I then exploit this result along with clonal interference theory to investigate the joint effect of linked beneficial and deleterious mutations on the rate of adaptation, and identify parameter regions where it is reduced due to interference by either beneficial or deleterious or both types of mutations. I also study the evolution of mutation rates in adapting asexual populations, and find that linked beneficial mutations have a stronger influence than the deleterious mutations on mutator fixation.

Keywords: clonal interference; direct selection; indirect selection; mutation rates.

Figure 1

Fixation probability $p_i$ of a beneficial mutation when it arises in a genetic background with i deleterious mutations. The points are obtained by numerically iterating the exact relation (1) (squares) and the quadratic-approximation (5) (circles). The analytical expression (8) (solid line), and the upper and lower bounds (dashed lines) given by (D.1) are also shown. The parameters are $s_d={10}^{-3},u_d=8s_d,U_d=4u_d,s_b=0.04$.

Figure 2

Variation of total fixation probability P with the deleterious mutation rate $U_d$ of the mutator allele. The data shown in circles is obtained by numerically iterating (5) while the dashed and solid lines, respectively, show the analytical expression (9) and the upper bound (D.2). The parameters are $u_d=8\times {10}^{-3},s_b=0.04$ and $s_d={10}^{-3}$ (main), ${10}^{-4}$ (inset).

Figure 3

Main: Variation of total fixation probability P with deleterious effect $s_d$ for $U_d=u_d=0.9s_b$ (circles) and $1.1s_b$ (triangles), and $s_b=0.005$. Inset: Total fixation probability as a function of beneficial effect $s_b$ for $U_d=u_d=0.01$. In both figures, the numerical solution of (5) is shown by points, and the analytical result (9) by lines.

Figure 4

Substitution rate $E\left[k_b\right]$ as a function of mean beneficial effect ${\overline{s}}_b$ for $u_d=5\times {10}^{-3}$ and $u_b={10}^{-12}$. The solid and dashed lines show the substitution rate (13) for $s_d\to 0$, while the points show the results obtained using (9) and (12) for $s_d={10}^{-3}$ (circles) and $5\times {10}^{-4}$ (squares). For $N={10}^{11}$, clonal interference occurs for ${\overline{s}}_b/u_d>0.73$, while it remains absent for all ${\overline{s}}_b$ for $N={10}^8$.

Figure 5

Substitution rate $E\left[k_b\right]$ as a function of deleterious mutation rate $u_d$ for $N={10}^9$ (solid) and $6\times {10}^{12}$ (dashed). The dot-dashed curve and dotted curve, respectively, show the substitution rate when there is no interference and when only clonal interference occurs. The other parameters are ${\overline{s}}_b={10}^{-2},p_b={10}^{-12}$ for $s_d={10}^{-3}$ (circles) and $5\times {10}^{-4}$ (squares). For $N=6\times {10}^{12}$, (15) and (16) predict that clonal interference occurs for $0.63<u_d/{\overline{s}}_b<1.49$.

Figure 6

Regions in the space of scaled population size $N{\overline{s}}_b$ and scaled deleterious mutation rate $u_d/{\overline{s}}_b,$ where linked mutations interfere when deleterious effects are weak. The dashed and dotted lines, respectively, show (15) and (16) with beneficial mutation rate $u_b={10}^{-12}{u}_d,$ and the solid horizontal line separates the regimes where interference by deleterious mutations can be neglected and where it is effective. The boundaries separating these evolutionary regimes for $s_d/{\overline{s}}_b\to 0,{10}^{-2},{10}^{-1}$ are depicted, respectively, by circles, squares and triangles. The points where $u_d/s_d<1$ are not shown as the analytical expression (9) does not hold for such parameters.

Figure 7

Adaptation rate R as a function of population size N for $u_d=5\times {10}^{-3},u_b={10}^{-12}$ and $s_d=5\times {10}^{-4}$ (points) for various ${\overline{s}}_b$. The lines show the result obtained using (18) for $s_d\to 0$. The population size above which clonal interference operates is found to be $3.2\hspace{0.17em}\times \hspace{0.17em}{10}^{10},6.9\hspace{0.17em}\times \hspace{0.17em}{10}^{10},1.1\hspace{0.17em}\times \hspace{0.17em}{10}^{11}$ for ${\overline{s}}_b/u_d=4,\hspace{0.17em}1,\hspace{0.17em}0.7$, respectively.

Figure 8

Main: Critical population size $N_c$ above which mutators are favored when clonal interference is taken into account (filled symbols) and on neglecting it (open symbols). The data for strongly deleterious mutations (circles, $s_d=0.1$) and weakly deleterious mutations (triangles, $s_d\to 0$) are obtained by numerically solving (23) and (24), respectively. The parameters are $u_d={10}^{-4},u_b={10}^{-6},{\overline{s}}_b=0.01$. Inset: Scaled mutation rate at which the rate of adaptation (18) for $s_d\to 0$ is maximum for ${\overline{s}}_b=0.01$ for the fraction of beneficial mutations $p_b={10}^{-10}$ (diamonds) and ${10}^{-12}$ (square).