Single-crossover recombination and ancestral recombination trees

@article{Baake2014SinglecrossoverRA,
  title={Single-crossover recombination and ancestral recombination trees},
  author={Ellen Baake and Ute Wangenheim},
  journal={Journal of Mathematical Biology},
  year={2014},
  volume={68},
  pages={1371-1402}
}
We consider the Wright–Fisher model for a population of $$N$$ individuals, each identified with a sequence of a finite number of sites, and single-crossover recombination between them. We trace back the ancestry of single individuals from the present population. In the $$N \rightarrow \infty $$ limit without rescaling of parameters or time, this ancestral process is described by a random tree, whose branching events correspond to the splitting of the sequence due to recombination. With the help… 
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References

SHOWING 1-10 OF 94 REFERENCES
Deterministic and stochastic aspects of single-crossover recombination
TLDR
A closed solution of the deterministic continuous-time system, for the important special case of single crossovers, is presented and an analogous deterministic discrete-time dynamics is provided, in terms of its generalised eigenvalues and a simple recursion for the corresponding coefficients.
Single-Crossover Dynamics: Finite versus Infinite Populations
TLDR
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.
Single-crossover recombination in discrete time
TLDR
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.
Mutation and recombination with tight linkage
  • E. Baake
  • Mathematics
    Journal of mathematical biology
  • 2001
Abstract. An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every
Recombination, gene conversion, and identity-by-descent at three loci.
Approximating the coalescent with recombination
TLDR
This work introduces a simplification of the coalescent process in which coalescence between lineages with no overlapping ancestral material is banned and the resulting process has a simple Markovian structure when generating genealogies sequentially along a sequence, yet has very similar properties to the full model.
Properties of a neutral allele model with intragenic recombination.
  • R. Hudson
  • Mathematics
    Theoretical population biology
  • 1983
The evolution of a population under recombination: how to linearise the dynamics
Ancestral Inference from Samples of DNA Sequences with Recombination
TLDR
Ancestral inference procedures are discussed for estimating recombination and mutation rates; estimating the times to the most recent common ancestors along the sequences; estimating ages of mutations; and estimating the number of recombination events in the ancestry of the sample.
...
...