Equilibrium state of molecular breeding

TL;DR

This paper analyzes the equilibrium distribution of binding affinities in a molecular breeding model, deriving approximate solutions across different selection regimes and validating them with numerical results.

Contribution

It provides a mathematical framework for understanding the equilibrium state of molecular breeding, including approximate solutions for large sequence lengths.

Findings

Derived an equation for the equilibrium distribution of binding affinity.

Obtained approximate solutions in strong, intermediate, and weak selection regimes.

Validated solutions through comparison with numerical results.

Abstract

We investigate the equilibrium state of the model of Peng, \textit{et al.} for molecular breeding. In the model, a population of DNA sequences is successively culled by removing the sequences with the lowest binding affinity to a particular target sequence. The remaining sequences are then amplified to restore the original population size, undergoing some degree of point-substitution of nucleotides in the process. Working in the infinite population size limit, we derive an equation for the equilibrium distribution of binding affinity, here modeled by the number of matches to the target sequence. The equation is then solved approximately in the limit of large sequence length, in the three regimes of strong, intermediate and weak selection. The approximate solutions are verified via comparison to exact numerical results.