Allele fixation probability in a Moran model with fluctuating fitness landscapes

Evolution on changing fitness landscapes (seascapes) is an important problem in evolutionary biology. We consider the Moran model of finite population evolution with selection in a randomly changing, dynamic environment. In the model, each individual has one of the two alleles, wild type or mutant. We calculate the fixation probability by making a proper ansatz for the logarithm of fixation probabilities. This method has been used previously to solve the analogous problem for the Wright-Fisher model. The fixation probability is related to the solution of a third-order algebraic equation (for the logarithm of fixation probability). We consider the strong interference of landscape fluctuations, sampling, and selection when the fixation process cannot be described by the mean fitness. Such an effect appears if the mutant allele has a higher fitness in one landscape and a lower fitness in another, compared with the wild type, and the product of effective population size and fitness is large. We provide a generalization of the Kimura formula for the fixation probability that applies to these cases. When the mutant allele has a fitness (dis-)advantage in both landscapes, the fixation probability is described by the mean fitness.

Figure 1

The fixation probability $p$ for a mutant with initial conditions defined in the first landscape versus the selection parameter $s$. The parameter values are $N=100,s_1=s/\left(2N\right),s_2=s/N,d_1=d_2=s^2/N^2$. The solid line corresponds to the numerical simulation, and the dotted line to the analytical solution. The analytical and numerical results coincide with the accuracy better than $1\%$ with the result obtained as the average fixation probability over two landscapes or with the fixation probability for average fitness.

Figure 2

The fixation probability versus the initial number $i$ of mutants with evolution starting in the first environment. The parameter values are $N=100,s_1=0.02,s_2=0.04,p_1=1-0.001,p_2=p_1$. The solid line corresponds to the numerical result and the dotted line to the analytical result. The analytical solution coincides with the numerical result with a 0.2% accuracy.

Figure 3

The fixation probability $p$ for the initial conditions in the first landscape case versus the selection parameter $s$. The parameter values are $N=100,s_1=-s/N,s_2=s/N,d_1=d_2=s^2/N^2$. The solid line corresponds to the results of the numerical simulation. The upper dashed line corresponds to the averaged fixation probability over the two landscapes [slow transition approximation, Eq. (5) is averaged via the landscape probabilities], and the dots to the analytical solution. The lower dashed line is the fixation probability with averaged fitness for the fast transition rate approximation [given by Eq. (5) with a selection coefficient by Eq. (20)]. The analytical result has a relative accuracy better than 10% at $s=20$, while the fast transition approximation gives an error $\sim 200\%$.

Figure 4

The fixation probability $p$ versus the parameter $s$ for a mutant with evolution starting in the first landscape. The parameter values are $N=1000,s_1=-s/N,s_2=s/N,d_1=d_2=s^2/N^2$. The solid line corresponds to the numerical result, and the dots correspond to the analytical solution. The dashed line corresponds to the averaged fixation probability over the two landscapes.