Microsatellite evolution: Markov transition functions for a suite of models

Abstract

This paper takes from the collection of models considered by Whittaker et. al. (2003) derived from direct observation of microsatellite mutation in parent-child pairs and provides analytical expressions for the probability distributions for the change in number of repeats over any given number of generations. The mathematical framework for this analysis is the theory of Markov processes. We find these expressions using two approaches, approximating by circulant matrices and solving a partial differential equation satisfied by the generating function. The impact of the differing choice of models is examined using likelihood estimates for time to most recent common ancestor. The analysis presented here may play a role in elucidating the connections between these two approaches and shows promise in reconciling differences between estimates for mutation rates based on Whittaker's approach and methods based on phylogenetic analyses.

On the size distribution of private microsatellite alleles

“…Indeed, more complex mutation models, allowing for directional bias, multistep mutations, length-dependent mutation rates, or a combination of these factors could potentially be considered (e.g. Calabrese & Durrett, 2003; Whittaker et al , 2003; Watkins, 2007). In general, however, we did not need a more complex model to explain the core observation that private alleles frequently lie on the edges of the size distribution.…”

Detecting past changes of effective population size

“…) in which a proportion c of the mutational events gives rise to a variation of two motifs so that m 1 = 1− c and m 2 = c , and the Geometrical Stepwise Mutation model (Whittaker et al. ; Watkins ) for which m r = (1 − c ) c r −1 .…”

“…Two other models are considered, needing an additional parameter c < 1 to fix them: a special case of the Two Phase model (Di Rienzo et al 1994) in which a proportion c of the mutational events gives rise to a variation of two motifs so that m 1 = 1Àc and m 2 = c, and the Geometrical Stepwise Mutation model (Whittaker et al 2003;Watkins 2006) for which m r = (1 À c)c rÀ1 .…”

A Markov chain description of the stepwise mutation model: Local and global behaviour of the allele process

“…The SMM assumes that, in one generation, the repeat number can only increase or decrease by at most one, usually with equal probability. More refined models have been proposed that include mutations of greater length, mutation rates that depend upon repeat number, or the additional introduction of point mutations (Di Rienzo et al, 1994;Garza et al, 1995;Feldman et al, 1997;Zhivotovsky et al, 1997;Kruglyak et al, 1998;Durrett and Kruglyak, 1999;Falush and Iwasa, 1999;Calabrese et al, 2001); for an overview, see Watkins (2007) or Calabrese and Sainudiin (2005). As yet, however, it has remained controversial to what extent these models approximate the reality (Chambers and MacAvoy, 2000;Whittaker et al, 2003;Sainudiin et al, 2004;Cornuet et al, 2006).…”

“…Markov processes have been applied before to the characterisation of microsatellite mutation models by Watkins (2007). In contrast to Kingman (1976), who used the analytic tool of characteristic functions, we will apply the stochastic method of recurrence of Markov chains.…”