Lyapunov Equation and Mathematical Population Genetics
Marek Kimmel
Rice University
USA
We derive new results giving mathematical properties of
functions of allele frequencies under the time-continuous
Fisher-Wright-Moran model with mutations of the general Markov-chain form.
The matrix R(t) (possibly infinite) of the joint distributions of the
types of a pair of alleles sampled from the population at time t,
satisfies a matrix differential equation of the form
dR(t)/dt=[Q*R(t)+R(t)Q]-[1/(2N)]R(t)+[1/(2N)]Pi (t), where Q is
the intensity matrix of the Markov chain, Pi(t) is its diagonalized
probability distribution, and N is the effective population size. This
is the Lyapunov differential equation, known in the control theory.
Investigation of behavior of its solutions leads to consideration of
tensor products of transition (Markov) semigroups. Semigroup theory
methods allow to prove asymptotic results for the model, also in the cases
when the population size does not stay constant. If population is composed
of a number of disjoint subpopulations, the asymptotics depend on the
growth rate of the population. Special cases of the model include stepwise
mutation models with and without allele size constraints, and with
directional bias of mutations. Allele state changes caused by
recombinatorial misalignment and more complex sequence conversion patterns
also can be incorporated in this model. The methodology developed can also
be applied to model coevolution of disease and marker loci, of further use
for linkage disequilibrium mapping of disease genes.
6.12.2002
