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Mutation and recombination with tight linkage. (English) Zbl 1010.92031
Summary: An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form.

MSC:
92D10 Genetics and epigenetics
92D15 Problems related to evolution
15A69 Multilinear algebra, tensor calculus
15A99 Basic linear algebra
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