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}
}
  • M. Baake, E. Baake
  • Published 28 October 2002
  • Mathematics
  • Canadian Journal of Mathematics
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… 
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References

SHOWING 1-10 OF 51 REFERENCES
Mathematical structures in population genetics
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
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
The evolution of a population under recombination: how to linearise the dynamics
Evolution processes with continuity of types
  • I. Eshel
  • Mathematics
    Advances 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.
TLDR
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
TLDR
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.
  • K. Dawson
  • Mathematics
    Theoretical population biology
  • 2000
TLDR
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
  • J. Kingman
  • Mathematics
    Journal 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.
  • S. Karlin
  • Biology
    Theoretical population biology
  • 1975
Increasing propagation of chaos for mean field models
...
...