Deterministic and Stochastic Aspects of Single-crossover Recombination

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.Mathematics Subject Classification (2000). Primary 92D10, 34L30; Secondary 37N25, 06A07, 60J25.

Haldane linearisation done right: Solving the nonlinear recombination equation the easy way

“…The recombination equation is a well-known dynamical system from mathematical population genetics [15,10,9,3], which describes the evolution of the genetic composition of a population that evolves under recombination. The genetic composition is described via a probability distribution (or measure) on a space of sequences of finite length, and recombination is the genetic mechanism in which two parent individuals are involved in creating the mixed sequence of their offspring during sexual reproduction.…”

Erratum and addendum to: The general recombination equation in continuous time and its solution

“…The resulting ODE system appears difficult to handle, due to the large number of possible states and the nonlinearity of the right-hand side. In previous papers [6,5,4,26,3,7,8], we have concentrated on a special case, namely, the situation in which at most one crossover happens at any given time. That is, we restricted attention to ordered partitions into two parts, corresponding to the sites before and after a single-crossover point.…”

The recombination equation for interval partitions

“…One quantity that requires a different approach is the decay rate of a partition, which needs a 'bottom up' recursion. The decay rates ψ = ψ S are defined recursively via those of the subsystems for non-empty U ⊆ S. One has [3,5] (2)…”

“…Moreover, the recombination ODE admits a reduction to a finite-dimensional nonlinear system of ODEs via a suitable ansatz in form of a finite convex combination of probability measures that derive from the initial condition by the application of a finite set of (also nonlinear) operators, which are known as recombinators. The explicit solution of this ODE system was recently achieved in [5], building on previous work [7,6,3,15], in a recursive way. The solution formula derived there was explicit enough to establish the exponentially fast convergence of the solution to a unique equilibrium that only depends on the initial condition, as expected from the previously known cases [7] as well as the general theory [9]; see also the indroduction of [5] and references therein for more.…”