Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes

Alberti F, Baake E (2021)
Documenta Mathematica 26: 743-793.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
The deterministic selection-recombination equation describes the evolution of the genetic type composition of a population under selection and recombination in a law of large numbers regime. So far, an explicit solution has seemed out of reach; only in the special case of three sites with selection acting on one of them has an approximate solution been found, but without an obvious path to generalisation. We use both an analytical and a probabilistic, genealogical approach for the case of an arbitrary number of neutral sites linked to one selected site. This leads to a recursive integral representation of the solution. Starting from a variant of the ancestral selection-recombination graph, we develop an efficient genealogical structure, which may, equivalently, be represented as a weighted partitioning process, a family of Yule processes with initiation and resetting, and a family of initiation processes. We prove them to be dual to the solution of the differential equation forward in time and thus obtain a stochastic representation of the deterministic solution, along with the Markov semigroup in closed form.
Stichworte
Moran model with selection and recombination; selection-recombination; differential equation; ancestral selection-recombination graph; interactive particle system; duality; population genetics
Erscheinungsjahr
2021
Zeitschriftentitel
Documenta Mathematica
Band
26
Seite(n)
743-793
ISSN
1431-0643
Page URI
https://pub.uni-bielefeld.de/record/2957017

Zitieren

Alberti F, Baake E. Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes. Documenta Mathematica . 2021;26:743-793.
Alberti, F., & Baake, E. (2021). Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes. Documenta Mathematica , 26, 743-793. https://doi.org/10.25537/dm.2021v26.743-793
Alberti, F., and Baake, E. (2021). Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes. Documenta Mathematica 26, 743-793.
Alberti, F., & Baake, E., 2021. Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes. Documenta Mathematica , 26, p 743-793.
F. Alberti and E. Baake, “Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes”, Documenta Mathematica , vol. 26, 2021, pp. 743-793.
Alberti, F., Baake, E.: Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes. Documenta Mathematica . 26, 743-793 (2021).
Alberti, Frederic, and Baake, Ellen. “Solving the Selection-Recombination Equation: Ancestral Lines and Dual Processes”. Documenta Mathematica 26 (2021): 743-793.

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