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 ωmed=1/β. 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 Pfix on the parameter ωmed is precisely described by a power-law relation, Pfix ωmedγ. The exponent γ is an increasing function of the rate of deleterious mutations U, whereas the probability Pfix is a decreasing function of U. The mean value ωfix of the beneficial mutations which reach ultimate fixation depends on U and ωmed. The ratio ωfix/ωmed increases as we consider higher values of mutation value U in the region of intermediate to large values of ωmed, whereas for low ωmed we observe the opposite behavior