Suppression of Beneficial Mutations in Dynamic Microbial Populations

Quantitative predictions for the spread of mutations in bacterial populations are essential to interpret evolution experiments and to improve the stability of synthetic gene circuits. We derive analytical expressions for the suppression factor for beneficial mutations in populations that undergo periodic dilutions, covering arbitrary population sizes, dilution factors, and growth advantages in a single stochastic model. We find that the suppression factor grows with the dilution factor and depends nontrivially on the growth advantage, resulting in the preferential elimination of mutations with certain growth advantages. We confirm our results by extensive numerical simulations.

Figure 1

Typical trajectories of the model Eq. (1) for (a) a dynamic population undergoing repeated pruning and (b) a constant population. Parameters are $\alpha =1$, $N_s=10$, $T=\log 10$, $\mu ={10}^{-3}$, resulting in $f=10$, $N_c=39$. (c) Fixation times in constant (crosses) and dynamic (squares) populations from 10 000 stochastic simulations using an accelerated algorithm [18]. (d) Fixation time ratio. Solid lines in (c) and (d) indicate exact numerical values from Markov models for a constant population and a population pruned when reaching a fixed size $f{N}_s$.

Figure 2

(a) Fixation probabilities from numerical simulations (symbols) compared to diffusion approximation, Eqs. (5) and (4) (lines). (b) Numerical ${\tau }_d/{\tau }_c$ (symbols) and diffusion approximation (lines). Colored dashed lines show initial slope at $s=0$ according to Eq. (6); gray dashed line is the asymptotic ratio $\mathrm{\Delta }$, Eq. (7). For all data in (a) and (b) $f=20$, ${\xi }^2=1$, $m_s=1$. (c),(d) Numerical $p$ and ${\tau }_d/{\tau }_c$ (symbols) compared to branching process approximation, Eqs. (9) and (11) (lines). The $y$ intercept in (d) is also $\mathrm{\Delta }$.

Figure 3

Fixation probabilities and ratios in the multistage model. (a) ${\tau }_d/{\tau }_c$ for small $s$ for different $k$ in numerical simulations (symbols). Lines indicate the slope predicted by Eq. (6) with ${\xi }^2=2(\log 2{)}^2/k$. (b) ${\tau }_d/{\tau }_c$ from numerical simulations for larger $s$. (c) $p_c$ and $p_d$ as functions of the dilution factor $f$. (d) ${\tau }_d/{\tau }_c$ for the data shown in (c) compared to the analytical approximation, Eq. (7). Parameters are $N_s=50$, $f=20$ (a), (b) and $N_s=20$, $s=0.2$ (c), (d).