TY - JOUR

T1 - A semigroup representation and asymptotic behavior of certain statistics of the fisher-wright-moran coalescent

AU - Bobrowski, Adam

AU - Kimmel, Marek

AU - Arino, Ovide

AU - Chakraborty, Ranajit

N1 - Funding Information:
We thank an anonymous referee who informed us of the papers by O'Brien (1982, 1985) and contributed a number of helpful remarks. This work was supported by US Public Health Service research grants GM 41399 and GM 45861 (to RC) and GM 58545 (to RC and MK) and by the Keck's Center for Computational Biology at Rice University (MK). AB is on leave from the Department of Mathematics at the Lublin Technical University, Poland. MK did part of his research in April and May 1998, when he was a long-term visitor at the Gothenburg Stochastic Centre, supported by the Swedish Foundation for Strategic Research.

PY - 2001

Y1 - 2001

N2 - 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)]Π(t) where Q is the intensity matrix of the Markov chain, II(t) is its diagonalized probability distribution, and N is the effective population size. This is the Lyapunov differential equation, known in control theory. Investigation of behavior of its solutions leads to consideration of tensor products of transition (Markov) semigroups. Semigroup theory methods allow proofs of 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.

AB - 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)]Π(t) where Q is the intensity matrix of the Markov chain, II(t) is its diagonalized probability distribution, and N is the effective population size. This is the Lyapunov differential equation, known in control theory. Investigation of behavior of its solutions leads to consideration of tensor products of transition (Markov) semigroups. Semigroup theory methods allow proofs of 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.

UR - http://www.scopus.com/inward/record.url?scp=2042540848&partnerID=8YFLogxK

U2 - 10.1016/S0169-7161(01)19010-3

DO - 10.1016/S0169-7161(01)19010-3

M3 - Review article

AN - SCOPUS:2042540848

VL - 19

SP - 215

EP - 247

JO - Handbook of Statistics

JF - Handbook of Statistics

SN - 0169-7161

ER -