Repeat distributions from unequal crossovers

TL;DR

This paper analyzes a class of recombination models with single crossovers that produce equilibrium distributions of repeated genetic sequences, employing advanced mathematical tools to understand their nonlinear, infinite-dimensional dynamics.

Contribution

It introduces a mathematical framework for analyzing nonlinear recombination models with geometric repeat distributions using measure theory and functional analysis.

Findings

Models reach equilibrium distributions of geometric type.

Analysis reveals nonlinear, quadratic dynamics akin to Markov chains.

Mathematical tools provide insights into complex genetic recombination processes.

Abstract

It is a well-known fact that genetic sequences may contain sections with repeated units, called repeats, that differ in length over a population, with a length distribution of geometric type. A simple class of recombination models with single crossovers is analysed that result in equilibrium distributions of this type. Due to the nonlinear and infinite-dimensional nature of these models, their analysis requires some nontrivial tools from measure theory and functional analysis, which makes them interesting also from a mathematical point of view. In particular, they can be viewed as quadratic, hence nonlinear, analogues of Markov chains.