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Authored Appendix to Cumulant Dynamics of a Population under Multiplicative Selection, Mutation, and Drift by M. Rattray and J. L. Shapiro

Authored Appendix to Cumulant Dynamics of a Population under Multiplicative Selection, Mutation, and Drift by M. Rattray and J. L. Shapiro
Authored Appendix to Cumulant Dynamics of a Population under Multiplicative Selection, Mutation, and Drift by M. Rattray and J. L. Shapiro
We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.
28-32
Prügel-Bennett, A.
b107a151-1751-4d8b-b8db-2c395ac4e14e
Prügel-Bennett, A.
b107a151-1751-4d8b-b8db-2c395ac4e14e

Prügel-Bennett, A. (2001) Authored Appendix to Cumulant Dynamics of a Population under Multiplicative Selection, Mutation, and Drift by M. Rattray and J. L. Shapiro. Theoretical Population Biology, 60 (1), 28-32.

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Abstract

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.

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Published date: 2001
Additional Information: This is an authored appendix
Organisations: Southampton Wireless Group

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Local EPrints ID: 256410
URI: http://eprints.soton.ac.uk/id/eprint/256410
PURE UUID: 21182af4-8a41-4e18-ac5c-b1f59dff2b9b

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Date deposited: 12 Mar 2004
Last modified: 24 Mar 2020 17:47

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Author: A. Prügel-Bennett

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