Accumulation of beneficial mutations in one dimension

When beneficial mutations are relatively common, competition between multiple unfixed mutations can reduce the rate of fixation in well-mixed asexual populations. We introduce a one-dimensional model with a steady accumulation of beneficial mutations. We find a transition between periodic selection and multiple-mutation regimes. In the multiple-mutation regime, the increase of fitness along the lattice bears a striking similarity to surface growth phenomena, with power-law growth and saturation of the interface width. We also find significant differences compared to the well-mixed model. In our lattice model, the transition between regimes happens at a much lower mutation rate due to slower fixation times in one dimension. Also, the rate of fixation is reduced with increasing mutation rate due to the more intense competition, and it saturates with large population size.

Figure 1

All mutations shown are assumed to be beneficial compared to the wild type. (a) Clonal interference: mutation A has to compete with mutations B, C, and D, reducing its chances of fixation compared to the case when there is only one mutation. If mutation A fixates, mutations B, C, and D are “wasted,”slowing down the rate of fixation. (b) Multiple-mutation effect: mutation AB arises in a population with mutation A, increasing mutation A's chances of fixation.

Figure 2

Logarithm of the fitness varies in spatial position. This interface moves with velocity $v$ and has a standard deviation $\sigma$. $N=1000$, $U={10}^{-3}$, and $s=0.01$. Shown are snapshots separated by $10000$ generations.

Figure 3

Fixation times were determined by planting single mutations with $s=0.01$ (circles), $s=0.025$ (squares), and $s=0.05$ ($\times$'s), averaged over 1000 fixations. Lines indicate $t_{\text{fix}}=2N/s$.

Figure 4

(a) Fixation rate $R$ vs mutation rate $U$ with $N=512$ and $s=0.01$ averaged over ${10}^3$ fixations for the 1D model (circles) and well-mixed model (squares). The dotted line indicates the transition between the single-fixation regime and the multiple-mutation regime for the spatial model, and the dot-dashed line indicates the transition for the well-mixed Wright-Fisher model. The solid line is the noninterfering fixation rate (4), and the dashed line is the neutral fixation rate, $R_n=U$. (b) The difference between the 1D model (circles) and the well-mixed model (squares) is clearer with $R/U-1$, which approaches zero as $R$ approaches the neutral fixation rate $R_n=U$. As guides, the dashed and dotted lines indicate power-law relations with exponents $-0.25$ and $-0.3$, respectively.

Figure 5

Fixation rate, $R$, vs system size for the 1D model (circles) and the well-mixed model (squares) with $U={10}^{-5}$. $v$ quickly saturates in 1D but diverges for the well-mixed model. Solid line is the noninterfering fixation rate (4).

Figure 6

Time evolution of the standard deviation of the fitnesses $\sigma$ with $N=512$, $s=0.01$ and $U={10}^{-3}$ (circles), $U={10}^{-4}$ (squares), and $U={10}^{-5}$ (crosses). The dashed lines are $\sim t^{1/2}$. The width approaches a stationary value as a power law with time $\sigma \sim t^{\beta }$, where $\beta$ is the growth exponent. $\beta$ is approximately $1/2$ for high $U$, and increases slightly as $U$ gets smaller.

Figure 7

Top: Stationary standard deviation ${\lim }_{t\to \infty }\sigma \left(t\right)$ vs mutation rate $U$ with $N=512$ and $s=0.01$ averaged over ${10}^5$ mutations. The dashed line is $\sim U^{1/3}$. Bottom: width increases with system size as a power law with a characteristic exponent, with $U={10}^{-5}$. The dashed line is $\sim N^{0.45}$.