dorsal/arxiv
View SchemaNon-equilibrium theory of the allele frequency spectrum
| Authors | Steven N. Evans, Yelena Shvets, Montgomery Slatkin |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0604010 |
| URL | https://arxiv.org/abs/q-bio/0604010 |
Abstract
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 $\lim_{x \downarrow 0} x f(x,t)=\theta \rho(t)$, where $f(\cdot,t)$ is the frequency spectrum of derived alleles at independent loci at time $t$ and $\rho(t)$ 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 $f(\cdot,t)$ 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.
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"abstract": "A forward diffusion equation describing the evolution of the allele frequency\nspectrum is presented. The influx of mutations is accounted for by imposing a\nsuitable boundary condition. For a Wright-Fisher diffusion with or without\nselection and varying population size, the boundary condition is $\\lim_{x\n\\downarrow 0} x f(x,t)=\\theta \\rho(t)$, where $f(\\cdot,t)$ is the frequency\nspectrum of derived alleles at independent loci at time $t$ and $\\rho(t)$ is\nthe relative population size at time $t$. When population size and selection\nintensity are independent of time, the forward equation is equivalent to the\nbackwards diffusion usually used to derive the frequency spectrum, but the\nforward equation allows computation of the time dependence of the spectrum both\nbefore an equilibrium is attained and when population size and selection\nintensity vary with time. From the diffusion equation, we derive a set of\nordinary differential equations for the moments of $f(\\cdot,t)$ and express the\nexpected spectrum of a finite sample in terms of those moments. We illustrate\nthe use of the forward equation by considering neutral and selected alleles in\na highly simplified model of human history. For example, we show that\napproximately 30% of the expected heterozygosity of neutral loci is\nattributable to mutations that arose since the onset of population growth in\nroughly the last $150,000$ years.",
"arxiv_id": "q-bio/0604010",
"authors": [
"Steven N. Evans",
"Yelena Shvets",
"Montgomery Slatkin"
],
"categories": [
"q-bio.PE",
"math.PR"
],
"title": "Non-equilibrium theory of the allele frequency spectrum",
"url": "https://arxiv.org/abs/q-bio/0604010"
},
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