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Register a new userAn exactly solved model for mutation, recombination and selection
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Authors
Michael Baake
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It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from. Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Mobius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.
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We investigate a continuous time, probability measure-valued dynamical system that describes the process of mutation-selection balance in a context where the population is infinite, there may be infinitely many loci, and there are weak assumptions on selective costs. Our model arises when we incorporate very general recombination mechanisms into a previous model of mutation and selection from Steinsaltz, Evans and Wachter (2005) and take the relative strength of mutation and selection to be sufficiently small. The resulting dynamical system is a flow of measures on the space of loci. Each such measure is the intensity measure of a Poisson random measure on the space of loci: the points of a realization of the random measure record the set of loci at which the genotype of a uniformly chosen individual differs from a reference wild type due to an accumulation of ancestral mutations. Our motivation for working in such a general setting is to provide a basis for understanding mutation-driven changes in age-specific demographic schedules that arise from the complex interaction of many genes, and hence to develop a framework for understanding the evolution of aging. We establish the existence and uniqueness of the dynamical system, provide conditions for the existence and stability of equilibrium states, and prove that our continuous-time dynamical system is the limit of a sequence of discrete-time infinite population mutation-selection-recombination models in the standard asymptotic regime where selection and mutation are weak relative to recombination and both scale at the same infinitesimal rate in the limit.
We study a continuous-time dynamical system that models the evolving distribution of genotypes in an infinite population where genomes may have infinitely many or even a continuum of loci, mutations accumulate along lineages without back-mutation, added mutations reduce fitness, and recombination occurs on a faster time scale than mutation and selection. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete time models, were presented in earlier work by Evans, Steinsaltz, and Wachter for quite general selective costs. Here we study a special case where the selective cost of a genotype with a given accumulation of ancestral mutations from a wild type ancestor is a sum of costs attributable to each individual mutation plus successive interaction contributions from each $k$-tuple of mutations for $k$ up to some finite ``degree. Using ideas from complex chemical reaction networks and a novel Lyapunov function, we establish that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions. That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.
Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.
Pedigrees are directed acyclic graphs that represent ancestral relationships between individuals in a population. Based on a schematic recombination process, we describe two simple Markov models for sequences evolving on pedigrees - Model R (recombinations without mutations) and Model RM (recombinations with mutations). For these models, we ask an identifiability question: is it possible to construct a pedigree from the joint probability distribution of extant sequences? We present partial identifiability results for general pedigrees: we show that when the crossover probabilities are sufficiently small, certain spanning subgraph sequences can be counted from the joint distribution of extant sequences. We demonstrate how pedigrees that earlier seemed difficult to distinguish are distinguished by counting their spanning subgraph sequences.
Some recent developments in the study of exactly solved lattice models in statistical mechanics are briefly reviewed. These include work with Debbie Bennett-Wood and Aleks Owczarek on polymers at surfaces (cond-mat/9805148) and with Katherine Seaton on the calculation of correlation lengths and the E_8 mass spectrum of the dilute A_3 lattice model (hep-th/9712121).
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