dorsal/arxiv
View SchemaExact Solution of the Multi-Allelic Diffusion Model
| Authors | G. Baxter, R. A. Blythe, A. J. McKane |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0508045 |
| URL | https://arxiv.org/abs/q-bio/0508045 |
| DOI | 10.1016/j.mbs.2007.01.001 |
| Journal | Mathematical Biosciences (2007) v209 pp124-70 |
Abstract
We give an exact solution to the Kolmogorov equation describing genetic drift for an arbitrary number of alleles at a given locus. This is achieved by finding a change of variable which makes the equation separable, and therefore reduces the problem with an arbitrary number of alleles to the solution of a set of equations that are essentially no more complicated than that found in the two-allele case. The same change of variable also renders the Kolmogorov equation with the effect of mutations added separable, as long as the mutation matrix has equal entries in each row. Thus this case can also be solved exactly for an arbitrary number of alleles. The general solution, which is in the form of a probability distribution, is in agreement with the previously known results--which were for the cases of two and three alleles only. Results are also given for a wide range of other quantities of interest, such as the probabilities of extinction of various numbers of alleles, mean times to these extinctions, and the means and variances of the allele frequencies. To aid dissemination, these results are presented in two stages: first of all they are given without derivations and too much mathematical detail, and then subsequently derivations and a more technical discussion are provided.
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"abstract": "We give an exact solution to the Kolmogorov equation describing genetic drift\nfor an arbitrary number of alleles at a given locus. This is achieved by\nfinding a change of variable which makes the equation separable, and therefore\nreduces the problem with an arbitrary number of alleles to the solution of a\nset of equations that are essentially no more complicated than that found in\nthe two-allele case. The same change of variable also renders the Kolmogorov\nequation with the effect of mutations added separable, as long as the mutation\nmatrix has equal entries in each row. Thus this case can also be solved exactly\nfor an arbitrary number of alleles. The general solution, which is in the form\nof a probability distribution, is in agreement with the previously known\nresults--which were for the cases of two and three alleles only. Results are\nalso given for a wide range of other quantities of interest, such as the\nprobabilities of extinction of various numbers of alleles, mean times to these\nextinctions, and the means and variances of the allele frequencies. To aid\ndissemination, these results are presented in two stages: first of all they are\ngiven without derivations and too much mathematical detail, and then\nsubsequently derivations and a more technical discussion are provided.",
"arxiv_id": "q-bio/0508045",
"authors": [
"G. Baxter",
"R. A. Blythe",
"A. J. McKane"
],
"categories": [
"q-bio.PE",
"cond-mat.stat-mech"
],
"doi": "10.1016/j.mbs.2007.01.001",
"journal_ref": "Mathematical Biosciences (2007) v209 pp124-70",
"title": "Exact Solution of the Multi-Allelic Diffusion Model",
"url": "https://arxiv.org/abs/q-bio/0508045"
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