Solution of the Crow-Kimura and Eigen models for alphabets of arbitrary size by Schwinger spin coherent states

TL;DR

This paper extends the analytical solutions of the Crow-Kimura and Eigen models of molecular evolution to arbitrary alphabet sizes using a functional integral approach, revealing phase transition behaviors that depend on alphabet size.

Contribution

It provides a general solution for these models with any alphabet size, including derivations of maximum principles and phase transition analysis for complex fitness functions.

Findings

Solutions for arbitrary alphabet sizes using a maximum principle.

Phase transition order changes from second to first as alphabet size increases.

Analytic results for various fitness functions including sharp peak and polynomial forms.

Abstract

To represent the evolution of nucleic acid and protein sequence, we express the parallel and Eigen models for molecular evolution in terms of a functional integral representation with an $h$-letter alphabet, lifting the two-state, purine/pyrimidine assumption often made in quasi-species theory. For arbitrary $h$ and a general mutation scheme, we obtain the solution of this model in terms of a maximum principle. Euler's theorem for homogeneous functions is used to derive this `thermodynamic' formulation of evolution. The general result for the parallel model reduces to known results for the purine/pyrimidine $h=2$ alphabet and the nucleic acid $h=4$ alphabet for the Kimura 3 ST mutation scheme. Examples are presented for the $h=4$ and $h=20$ cases. We derive the maximum principle for the Eigen model for general $h$. The general result for the Eigen model reduces to a known result for $h=2$. Examples are presented for the nucleic acid $h=4$ and the amino acid $h=20$ alphabet. An error catastrophe phase transition occurs in these models, and the order of the phase transition changes from second to first order for smooth fitness functions when the alphabet size is increased beyond two letters to the generic case. As examples, we analyze the general analytic solution for sharp peak, linear, quadratic, and quartic fitness functions.