[Paper Review] Moment closure in a Moran model with recombination

Ellen Baake, Thiemo Hustedt|arXiv (Cornell University)|May 4, 2011

Advanced Thermodynamics and Statistical Mechanics14 references3 citations

TL;DR

This paper extends the Moran model with recombination to include general recombination and mutation, demonstrating that the dynamics of moment hierarchies close exactly after a finite number of steps in the absence of genetic drift. Surprisingly, this exact moment closure is lost when resampling (genetic drift) is introduced, disrupting the finite hierarchy despite the system remaining linear in structure.

ABSTRACT

We extend the Moran model with single-crossover recombination to include general recombination and mutation. We show that, in the case without resampling, the expectations of products of marginal processes defined via partitions of sites form a closed hierarchy, which is exhaustively described by a finite system of differential equations. One thus has the exceptional situation of moment closure in a nonlinear system. Surprisingly, this property is lost when resampling (i.e., genetic drift) is included.

Motivation & Objective

• To investigate whether moment closure occurs in a stochastic Moran model with general recombination and mutation.
• To determine whether the hierarchy of moment dynamics closes after a finite number of steps, avoiding infinite hierarchies typical in nonlinear systems.
• To examine the impact of genetic drift (resampling) on moment closure, particularly in relation to linkage disequilibrium and correlation decay.
• To explore the structural conditions under which exact moment closure is preserved or lost in stochastic population genetic models.

Proposed method

• The authors model the population using a continuous-time Markov process on a finite type space defined by site-wise alleles and recombination events.
• They define marginal processes via partitions of sites and track the expectations of products of these processes.
• Using the generator of the Markov process, they derive a system of coupled ordinary differential equations (ODEs) for the moment dynamics.
• The analysis shows that in the absence of resampling, the moment hierarchy closes after finitely many steps due to the partition-based structure of the processes.
• When resampling is included, the derivation reveals that higher-order moments emerge, breaking the closure.
• The method explicitly compares the dynamics of correlation functions (e.g., linkage disequilibrium) under recombination and resampling, showing exponential decay.

Experimental results

Research questions

• RQ1 Does the inclusion of general recombination in the Moran model lead to a finite, closed system of ODEs for moment dynamics?
• RQ2 Can exact moment closure be achieved in the absence of genetic drift, despite nonlinear interactions from recombination?
• RQ3 How does the introduction of resampling (genetic drift) affect the closure of the moment hierarchy?
• RQ4 What is the role of partition-based marginal processes in enabling exact moment closure?
• RQ5 Why is moment closure lost when resampling is added, even though both recombination and resampling are linear in structure?

Key findings

• In the absence of resampling, the moment hierarchy for products of marginal processes closes after a finite number of steps, yielding a finite system of ODEs.
• This exact moment closure is preserved when mutation is included, though the processes can no longer be described by site partitions.
• The inclusion of resampling destroys moment closure, as higher-order moments continue to emerge upon differentiation.
• The linkage disequilibrium term $\mathbb{E}[1,2]_tN - \mathbb{E}[1]_t\mathbb{E}[2]_t$ decays exponentially under both recombination and resampling.
• For three-site systems, the derivative of $\mathbb{E}[1]_t[2]_t[3]_t$ contains terms like $\mathbb{E}[1]^2_t\mathbb{E}[1,2,3]_t$, which generate higher-order moments, proving the hierarchy does not close.
• The system exhibits a stochastic analogue of Haldane linearisation, but only in the absence of genetic drift.