On the behavior of the leading eigenvalue of Eigen's evolutionary matrices

TL;DR

This paper analyzes the properties of the leading eigenvalue of Eigen's matrices in evolutionary models, providing exact expressions, bounds, and implications for mutation rates and error thresholds in biological systems.

Contribution

It offers new exact formulas and bounds for the leading eigenvalue of Eigen's matrices, enhancing understanding of mutation effects in evolutionary dynamics.

Findings

Exact expressions for eigenvalue and derivatives at specific points

Minimum eigenvalue occurs in [0,0.5] interval

Bounds on eigenvalues for single peaked landscapes

Abstract

We study general properties of the leading eigenvalue $\overline{w}(q)$ of Eigen's evolutionary matrices depending on the probability $q$ of faithful reproduction. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for $\overline{w}(q),\overline{w}'(q),\overline{w}''(q)$ for $q=0,0.5,1$ and prove that the absolute minimum of $\overline{w}(q)$, which always exists, belongs to the interval $[0,0.5]$. For the specific case of a single peaked landscape we also find lower and upper bounds on $\overline{w}(q)$, which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact. Therefore, the discussions of the error threshold applied to biological systems must take this fact into account.