2006/04/08 by Steven N. Evans, Evans, Steven N., Yelena Shvets +3
Biochemistry, Genetics and Molecular Biology · Medicine · #Evolution and Genetic Dynamics #FOS: Biological sciences #FOS: Mathematics #Genetic diversity and population structure #Mathematical and Theoretical Epidemiology and Ecology Models #Populations and Evolution (q-bio.PE) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.q-bio/0604010
openalex publication_date 2006/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A forward diffusion equation describing the evolution of the allele frequency spectrum is presented. The influx of mutations is accounted for by imposing a suitable boundary condition. For a Wright-Fisher diffusion with or without selection and varying population size, the boundary condition is limx \downarrow 0 x f(x,t)=θρ(t), where f(⋅,t) is the frequency spectrum of derived alleles at independent loci at time t and ρ(t) is the relative population size at time t. When population size and selection intensity are independent of time, the forward equation is equivalent to the backwards diffusion usually used to derive the frequency spectrum, but the forward equation allows computation of the time dependence of the spectrum both before an equilibrium is attained and when population size and selection intensity vary with time. From the diffusion equation, we derive a set of ordinary differential equations for the moments of f(⋅,t) and express the expected spectrum of a finite sample in terms of those moments. We illustrate the use of the forward equation by considering neutral and selected alleles in a highly simplified model of human history. For example, we show that approximately 30% of the expected heterozygosity of neutral loci is attributable to mutations that arose since the onset of population growth in roughly the last 150,000 years.