The coalescent, Fisher-Wright-Moran model, and Lyapunov Equation
Marek Kimmel
Dept. of Statistics, Rice University, Houston, USA
with Adam Bobrowski, Ranajit Chakraborty and Ovide Arino
Abstract
We provide new results for the time-continuous 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^T R(t)+R(t)Q]-(1/(2N))R(t)+(1/(2N))P(t), where Q is the
intensity matrix of the Markov chain, P(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 case when the
population size did not stay constant. Special cases of the
model described above include stepwise mutation models with and without
allele size constraints, and directional bias of mutations. Allele state
changes caused by recombinatorial misalignment and more complex sequence
conversion patterns also can be incorporated in this model. As an
application, we investigate the influence of size constraints on
evolutionary dynamics of microsatellite loci. We conclude that observed
differences of variability between neutral and selective microsatellite loci
are less likely due to allele-size constraints than to different mutation
rates. Consequences for population dynamics of certain genetic diseases
(such as fragile X or myotonic dystrophy) are discussed.
