Dynamics of Fixation of Advantageous Mutations

We investigate the process of fixation of advantageous mutations in an asexual population. We assume that the effect of each beneficial mutation is exponentially distributed with mean value $\omega_{med}=1/\beta$. The model also considers that the effect of each new deleterious mutation reduces the fitness of the organism independent on the previous number of mutations. We use the branching process formulation and also extensive simulations to study the model. The agreement between the analytical predictions and the simulational data is quite satisfactory. Surprisingly, we observe that the dependence of the probability of fixation $P_{fix}$ on the parameter $\omega_{med}$ is precisely described by a power-law relation, $P_{fix} \sim \omega_{med}^{\gamma}$. The exponent $\gamma$ is an increase function of the rate of deleterious mutations $U$, whereas the probability $P_{fix}$ is a decreasing function of $U$. The mean value $\omega_{fix}$ of the beneficial mutations which reach ultimate fixation depends on $U$ and $\omega_{med}$. The ratio $\omega_{fix}/\omega_{med}$ increases as we consider higher values of mutation value $U$ in the region of intermediate to large values of $\omega_{med}$, whereas for low $\omega_{med}$ we observe the opposite behavior.