Stable cycling in quasi-linkage equilibrium: Fluctuating dynamics under gene conversion and selection

Area of Science:

• Population genetics
• Molecular evolution
• Mathematical biology

Background:

• Genetic systems with multiple loci can display complex evolutionary dynamics, deviating from simple fitness increase.
• Stable cycling of allele frequencies, where populations fluctuate between near extinction and fixation, is a known phenomenon in multi-locus genetics.
• Recognition-dependent double-strand breaks and repair mechanisms during recombination hotspots provide a biologically relevant context for studying complex genetic dynamics.

Purpose of the Study:

• To investigate the dynamics of a genetic system inspired by recombination hotspot mechanisms.
• To analyze the emergence of sustained cycling and heteroclinic cycles within the quasi-linkage equilibrium (QLE) manifold.
• To analytically determine the stability of these cycles in both discrete and continuous time approximations.

Main Methods:

• Development of a genetic model incorporating slow-fast dynamics leading to convergence on the QLE manifold.
• Derivation of a closed-form approximation for the QLE manifold to simplify model analysis.
• Analytical calculation of the stability of the heteroclinic cycle for the simplified model.

Main Results:

• The genetic model converges to a QLE manifold, where sustained cycling dynamics are possible.
• A heteroclinic cycle, characterized by alternating near extinction and near fixation of allele frequencies, was identified.
• The discrete-time model demonstrated a stable cycle, while the continuous-time approximation showed an unstable cycle.

Conclusions:

• Complex evolutionary dynamics, including sustained cycling, are possible within the framework of quasi-linkage equilibrium.
• The study highlights the importance of considering slow-fast dynamics in multi-locus genetic systems.
• The findings provide analytical insights into the stability of genetic cycles under different temporal approximations.