Using General Algebra to Model the Directed Evolution of an Asexual Population

Humans have long employed directed evolution (DE) to engineer desired biological traits. In this paper, I introduce an algebraic framework that provides a quantitative representation of the general phenotypic traits of asexual populations, enabling the systematic modeling of DE processes. Within this framework, key evolutionary quantities such as the time required for a DE to reach a desired trait or the probability of arriving at a target algebra can be computed or qualitatively analyzed using principles from the evolutionary dynamics of an asexual population. As illustrative examples of trait-representing algebras, I evolve an integer, a two-dimensional vector, a four-dimensional vector with modulo-4 elements, a 2 by 2 matrix, and a cyclic algebra. The generations needed to reach the objective algebra in the DE simulations were consistent with those predicted by the theoretical analysis. Furthermore, I propose a method for mathematically designing evolutionary pathways that minimize the generations needed to reach the desired algebra, offering a key criterion for improving the efficiency of DE. Finally, I discuss how this algebraic approach can be applied in practical experimental setups and outline directions for future research in algebraic modeling of evolution.