Allelic Entropy and Times until Loss or Fixation of Neutral Polymorphisms

Abstract

In the diffusion approximation of the neutral Wright–Fisher model, the expected time until fixation or loss of a neutral allele is proportional to the initial entropy of the distribution of the allele in the population. No explanation is known for this coincidence. In this paper, we show that the rate of entropy dissipation is proportional to the number of segregating alleles. Since the final fixed state has zero entropy, the expected lifetime of segregating alleles is proportional to the initial entropy in the system. We show that classical formulae on the average time to loss of segregating alleles and the expected time to fixation of the last segregating allele stem from these properties of the diffusion process. We also extend our results to the case of population size changing in time. The dissipation of heterozygosity and entropy shows that superlinear population growth leads to infinite expected fixation times, i.e., neutral alleles in fast-growing populations could segregate forever without ever becoming fixed or disappearing by genetic drift.

Keywords

• diffusion approximation
• fixation time
• fixation probability
• genetic drift
• superlinear population growth
• allelic entropy