Moments, cumulants, and polygenic dynamics.

*(English)*Zbl 0760.92014The paper describes the evolution of polygenic traits subject to selection and mutation by a deterministic continuous-time model with a continuum of possible alleles. It assumes random mating, no sex differences, additive gene action, linkage equilibrium and Hardy-Weinberg proportions. Using a Taylor series expansion of phenotypic fitness, the binomial theorem and generating functions explicit expressions for the marginal fitness values of alleles are derived. Differential equations for the change of cumulants of the allelic frequency distribution at a particular locus and for the cumulants of the distributions of genotypic and phenotypic values are given. The results are illustrated by two concrete examples considering linear increasing fitness and quadratically deviating fitness.

The conceptual idea behind the present paper is the use of cumulants instead of moments to describe the dynamics of polygenic systems. The most important advantage of cumulants compared with moments is that, under the assumption of global linkage equilibrium, the dynamics of genotypic and phenotypic values follow immediately from the dynamics of allele frequencies simply by summing over all loci. For simple fitness functions, the structure of the cumulant equations is simpler than that of the corresponding moment equations even for the allele-frequency dynamics. However, the dependence of the rate of change of the \(n\)-th cumulant from cumulants up to order \(n+K\) (where \(K\) is the highest term of the Taylor series) and the dependence of the exact phenotypic dynamics from genetic details cannot be removed through the use of cumulants. Nevertheless the present derivation shows that cumulants are a much more convenient tool for investigating polygenic traits.

The conceptual idea behind the present paper is the use of cumulants instead of moments to describe the dynamics of polygenic systems. The most important advantage of cumulants compared with moments is that, under the assumption of global linkage equilibrium, the dynamics of genotypic and phenotypic values follow immediately from the dynamics of allele frequencies simply by summing over all loci. For simple fitness functions, the structure of the cumulant equations is simpler than that of the corresponding moment equations even for the allele-frequency dynamics. However, the dependence of the rate of change of the \(n\)-th cumulant from cumulants up to order \(n+K\) (where \(K\) is the highest term of the Taylor series) and the dependence of the exact phenotypic dynamics from genetic details cannot be removed through the use of cumulants. Nevertheless the present derivation shows that cumulants are a much more convenient tool for investigating polygenic traits.

Reviewer: H.-P.Altenburg (Mannheim)

##### MSC:

92D15 | Problems related to evolution |

92D10 | Genetics and epigenetics |

45K05 | Integro-partial differential equations |

##### Keywords:

quantitative trait; evolution of polygenic traits; selection; mutation; deterministic continuous-time model; random mating; additive gene action; Hardy-Weinberg proportions; Taylor series expansion of phenotypic fitness; binomial theorem; generating functions; explicit expressions; marginal fitness values of alleles; change of cumulants; allelic frequency distribution; linear increasing fitness; quadratically deviating fitness; global linkage equilibrium
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