Molecular Evolution under Fitness Fluctuations

Molecular evolution is a stochastic process governed by fitness, mutations, and reproductive fluctuations in a population. Here, we study evolution where fitness itself is stochastic, with random switches in the direction of selection at individual genomic loci. As the correlation time of these fluctuations becomes larger than the diffusion time of mutations within the population, fitness changes from an annealed to a quenched random variable. We show that the rate of evolution has its maximum in the crossover regime, where both time scales are comparable. Adaptive evolution emerges in the quenched fitness regime (evidence for such fitness fluctuations has recently been found in genomic data). The joint statistical theory of reproductive and fitness fluctuations establishes a conceptual connection between evolutionary genetics and statistical physics of disordered systems.

Figure 1

Evolution under fluctuating selection, mutations, and genetic drift. One trajectory $x(t)$ of the stochastic process (1) with scaled parameters $2N{\mu }_0=0.02$ and $2N{\sigma }_0=2$, the direction of selection is indicated by shading (the fitter allele appears unshaded). (a) Selection with micro-evolutionary (annealed) fluctuations ($1/{\gamma }_0\ll {\tau }_0$): Evolution can be described as a homogeneous diffusion process. Time is shown in units of $2N$. (b) Selection with intermediate fluctuations ($1/{\gamma }_0\sim {\tau }_0$): The diffusion picture breaks down due to large frequency changes within one correlation interval of selection. (c) Selection with macro-evolutionary (quenched) fluctuations ($1/{\gamma }_0\gg {\tau }_0$): The process reaches a nonequilibrium, time-irreversible stationary state with adaptations as surplus of advantageous substitutions (▵) over deleterious ones (▴). Time is now shown in coarse-grained units $1/{\mu }_0$. (d) Time-independent selection (${\gamma }_0=0$): The process reaches an evolutionary equilibrium with an equal number of advantageous and deleterious substitutions, hence without adaptations.

Figure 2

(a) Population variance $\overline{\Delta }$, (b) substitution rate $\overline{u}$, and (c) degree of adaptation $\overline{H}$ at stationarity under fluctuating selection, shown as a function of the fluctuation rate $\gamma$ for $\sigma =40$. Numerical results (data points) together with analytical results of diffusion theory [14] (dashed lines) and of quenched fluctuation theory [16] (full lines) valid in the micro-evolutionary regime ($\gamma \gg 1/\tau$) and the macro-evolutionary regime ($\gamma \ll 1/\tau$), respectively. (d) Crossover scaling: $2\overline{u}/\sigma \mu$ is plotted against $\gamma \tau$ for $\mu =0.0028$, 0.0084, 0.025 (data points, from top to bottom) and $\sigma =20$, 40 (squares, points), showing a data collapse in the crossover region $\gamma \sim 1/\tau$. Diffusion theory (dashed lines) and quenched theory (solid lines) are seen to be valid for $\gamma \gtrsim 1/\tau$ and $\gamma \lesssim 1/\tau$, respectively.