Single--crossover recombination in discrete time

TL;DR

This paper analyzes the discrete-time single-crossover recombination process, providing a new explicit solution method that overcomes the challenges posed by nonlinear dynamics, contrasting with the continuous-time case.

Contribution

It introduces a novel two-step solution approach for discrete-time single-crossover recombination dynamics, combining linearisation and diagonalisation techniques.

Findings

Explicit solutions for all times in single-crossover dynamics

Insights into the difficulties of discrete-time recombination models

Comparison with continuous-time models showing different solvability

Abstract

Modelling the process of recombination leads to a large coupled nonlinear dynamical system. Here, we consider a particular case of recombination in {\em discrete} time, allowing only for {\em single crossovers}. While the analogous dynamics in {\em continuous} time admits a closed solution, this no longer works for discrete time. A more general model (i.e. without the restriction to single crossovers) has been studied before and was solved algorithmically by means of Haldane linearisation. Using the special formalism introduced by Baake and Baake (2003), we obtain further insight into the single-crossover dynamics and the particular difficulties that arise in discrete time. We then transform the equations to a solvable system in a two-step procedure: linearisation followed by diagonalisation. Still, the coefficients of the second step must be determined in a recursive manner, but once this is done for a given system, they allow for an explicit solution valid for all times.