Deterministic and stochastic aspects of single-crossover recombination

TL;DR

This paper reviews mathematical models of genetic recombination, providing new solutions for deterministic and stochastic systems, especially focusing on single-crossover scenarios, and analyzing their linearity and eigenstructure.

Contribution

It presents a closed-form solution for the deterministic continuous-time model of single-crossover recombination and explores its extension to stochastic and discrete-time models.

Findings

Closed solution for continuous-time deterministic model

Analysis of linearity in stochastic setting

Solution of discrete-time dynamics via eigenvalues

Abstract

This contribution is concerned with mathematical models for the dynamics of the genetic composition of populations evolving under recombination. Recombination is the genetic mechanism by which two parent individuals create the mixed type of their offspring during sexual reproduction. The corresponding models are large, nonlinear dynamical systems (for the deterministic treatment that applies in the infinite-population limit), or interacting particle systems (for the stochastic treatment required for finite populations). We review recent progress on these difficult problems. In particular, we present a closed solution of the deterministic continuous-time system, for the important special case of single crossovers; we extract an underlying linearity; we analyse how this carries over to the corresponding stochastic setting; and we provide a solution of the analogous deterministic discrete-time dynamics, in terms of its generalised eigenvalues and a simple recursion for the corresponding coefficients.