Fixation probability and the crossing time in the Wright–Fisher multiple alleles model

The fixation probability and crossing time in the Wright–Fisher multiple alleles model, which describes a finite haploid population, were calculated by switching on an asymmetric sharply-peaked landscape with a positive asymmetric parameter, r, such that the reversal allele of the optimal allele has higher fitness than the optimal allele. The fixation probability, which was evaluated as the ratio of the first arrival time at the reversal allele to the origination time, was double the selective advantage of the reversal allele compared with the optimal allele in the strong selection region, where the fitness parameter, k, is much larger than the critical fitness parameter, kc. The crossing time in a finite population for r>0 and k<kc was the same as the crossing time in an infinite population for the same r. The critical fitness parameter designated the fitness parameter, where the crossing time in a finite population begins to deviate from the crossing time in an infinite population. This suggested the deterministic criterion such that additional offspring with the reversal allele in the first generation should be greater than one individual in an asymmetric sharply-peaked landscape. It was also found that the crossing time in a finite population for r>0 and k≫kc scaled as a power law in the fitness parameter with a similar scaling exponent as the crossing time in an infinite population for r=0, and that the critical fitness parameter decreased with increasing sequence length with a fixed population size.