# An Exactly Solved Model for Mutation, Recombination and Selection

@article{Baake2003AnES, title={An Exactly Solved Model for Mutation, Recombination and Selection}, author={Michael Baake and Ellen Baake}, journal={Canadian Journal of Mathematics}, year={2003}, volume={55}, pages={3 - 41} }

Abstract 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…

## 37 Citations

Mutation-Selection Balance with Recombination: Convergence to Equilibrium for Polynomial Selection Costs

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- 2009

It is established that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions and that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.

Single-crossover recombination in discrete time

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This work considers a particular case of recombination in discrete time, allowing only for single crossovers, and transforms the equations to a solvable system in a two-step procedure: linearisation followed by diagonalisation.

The Moran model of population genetics : case studies with recombination and selection

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- 2012

The Moran model is a widespread model for a finite population in the field of population genetics.
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This paper shows how to do a direct general solution at the level of the corresponding ODE itself to the nonlinear recombination equation from population genetics, and how to extend the approach to the discrete-time case as well.

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The process of recombination in population genetics, in its
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Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated and the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.

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The Moran model is extended to include general recombination and mutation and shows that, in the case without resampling, the expectations of products of marginal processes defined via partitions of sites form a closed hierarchy, which is exhaustively described by a finite system of differential equations.

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- 2014

The ancestry of single individuals from the present population is traced back by a random tree, whose branching events correspond to the splitting of the sequence due to recombination, and the probabilities of the topologies of the ancestral trees are calculated.

A MUTATION-SELECTION MODEL FOR GENERAL GENOTYPES WITH RECOMBINATION

- Biology
- 2006

It is shown that the new model arises from the haploid model when recombination is added, in the limit as generations per unit time go to infinity, and selection strength and mutation per generation go to 0.

The recombination equation for interval partitions

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The general deterministic recombination equation in continuous time is analysed for various lattices, with special emphasis on the lattice of interval (or ordered) partitions and the corresponding solution for interval partitions is derived and analysed in detail.

## References

SHOWING 1-10 OF 51 REFERENCES

Mathematical structures in population genetics

- Mathematics
- 1992

In the theory of population genetics, fundamental results on its dynamical processes and equilibrium laws have emerged during the last few decades. This monograph systematically reviews these…

STATIONARY DISTRIBUTIONS UNDER MUTATION- SELECTION BALANCE: STRUCTURE AND PROPERTIES

- Mathematics
- 1996

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume…

Evolution processes with continuity of types

- MathematicsAdvances in Applied Probability
- 1972

The objective of this work is to study the long range evolutionary traits in a population with an infinite number of types; we are especially interested in the asymptotic rate of evolution, variance…

Central equilibria in multilocus systems. I. Generalized nonepistatic selection regimes.

- BiologyGenetics
- 1979

Exact analytic conditions for existence and stability of a multilocus Hardy-Weinberg (H-W) polymorphic equilibrium configuration are ascertained and it is established that the central H-W polymorphism is stable only if the component loci are "over-dominant" and sufficient recombination is in force.

Multilocus dynamics under haploid selection

- MathematicsJournal of mathematical biology
- 1997

If haploid selection is additive then the fundamental theorem is established even with an estimate for the change in the mean fitness, and exponential convergence to an equilibrium is proved.

The decay of linkage disequilibrium under random union of gametes: how to calculate Bennett's principal components.

- MathematicsTheoretical population biology
- 2000

It is shown that the transformation from the allelic moments to Bennett's variables and the inverse transformation always have the form that Bennett claimed, and general recursions for calculating the coefficients in the forward transformation and the coefficient in the inverse Transformation are presented.

Markov population processes

- MathematicsJournal of Applied Probability
- 1969

Summary The processes of the title have frequently been used to represent situations involving numbers of individuals in different categories or colonies. In such processes the state at any time is…

General two-locus selection models: some objectives, results and interpretations.

- BiologyTheoretical population biology
- 1975