[Paper Review] The Renormalization Group and the Dynamics of Genetic Systems

Christopher R. Stephens|arXiv (Cornell University)|Oct 11, 2002

Origins and Evolution of Life9 references20 citations

TL;DR

This paper applies the Renormalization Group (RG) framework to genetic systems, showing that recombination induces a natural coarse-graining of genetic configurations into hierarchical building blocks (BBs), leading to a Feynman diagrammatic representation of dynamics. The key contribution is a perturbative, diagrammatic formulation that enables exact resummation of recombination effects, yielding a closed-form solution for population dynamics under weak selection and strong recombination, where correlations decay and 1-schemata become effective degrees of freedom.

ABSTRACT

In this brief article I show how the notion of coarse graining and the Renormalization Group enter naturally in the dynamics of genetic systems, in particular in the presence of recombination. I show how the latter induces a dynamics wherein coarse grained and fine grained degrees of freedom are naturally linked as a function of time leading to a hierarchical dynamics that has a Feynman-diagrammatic representation. I show how this coarse grained formulation can be exploited to obtain new results.

Motivation & Objective

• To establish a connection between the Renormalization Group (RG) and the dynamics of genetic systems, particularly in the presence of recombination.
• To demonstrate that recombination naturally induces a coarse-graining of genetic configurations into hierarchical building blocks (BBs).
• To develop a Feynman diagrammatic formulation for the dynamics of genetic populations under recombination.
• To derive exact results for population dynamics by resumming the full perturbative series in the recombination rate.
• To show that under strong recombination and weak selection, the effective degrees of freedom reduce to 1-schemata, simplifying the dynamics.

Proposed method

• Coarse-graining is defined via schemata, with building blocks (BBs) representing subsets of loci, and recombination acting as a natural coarse-graining operation.
• The dynamics is formulated in terms of a time-evolving probability distribution over genotypes, with recombination expressed as a binary interaction between parent strings via a recombination mask.
• A perturbative expansion is constructed in the number of recombination events, leading to a diagrammatic representation where internal lines represent propagators and vertices represent recombination interactions.
• Feynman rules are derived: propagators depend on fitness and recombination probability, and vertices are weighted by fitness ratios and delta functions enforcing bit conservation.
• The full diagrammatic series is exactly resummed in the continuous-time limit for a flat fitness landscape, yielding a closed-form expression for the probability of any string at time t.
• The solution is expressed in terms of initial probabilities over BBs, showing that the asymptotic dynamics is non-perturbative in the recombination rate.

Experimental results

Research questions

• RQ1 How does recombination induce a natural coarse-graining of genetic configurations in population dynamics?
• RQ2 Can the dynamics of recombination be represented using a Feynman diagrammatic formalism with well-defined rules?
• RQ3 What is the exact solution for the time evolution of genotype probabilities under strong recombination and weak selection?
• RQ4 Why is the asymptotic behavior of the system non-perturbative in the recombination rate, and how can this be captured exactly?
• RQ5 Under what conditions do 1-schemata become the effective degrees of freedom in the genetic dynamics?

Key findings

• The full diagrammatic series for recombination dynamics can be exactly resummed in the continuous-time limit under a flat fitness landscape, yielding a closed-form solution.
• The exact solution is given by equation (16), which expresses the probability of a string at time t as a sum over contributions from all possible numbers of initial building blocks, weighted by exponential decay factors.
• For a given string, the initial conditions are encoded in probabilities over 1-schemata (e.g., P(11,0) and P(1*,0)P(*1,0) for the string 11).
• The effective degrees of freedom in the asymptotic regime are 1-schemata when recombination is strong and selection is weak, indicating that correlations between loci decay over time.
• The dynamics is non-perturbative in the recombination rate, meaning that the asymptotic behavior cannot be captured by a finite-order perturbation series.
• The use of building blocks (BBs) as fundamental degrees of freedom leads to a simpler, exact formulation compared to the string basis, which involves complex combinations of dynamical factors.