Rigorous mathematical analysis of the quasispecies model: From Manfred Eigen to the recent developments

TL;DR

This paper reviews the rigorous mathematical analysis of the classical quasispecies model, highlighting recent progress, open questions, and potential directions for future research in understanding population fitness and distribution.

Contribution

It provides a comprehensive overview of recent mathematical results on the quasispecies model and discusses unresolved biological and mathematical questions.

Findings

Mathematical results on the Eigen and Crow--Kimura models

Identification of open problems in quasispecies theory

Emphasis on the need for further analysis of fitness landscapes

Abstract

We review the major progress in the rigorous analysis of the classical quasispecies model that usually comes in two related but different forms: the Eigen model and the Crow--Kimura model. The model itself was formulated almost 50 years ago, and in its stationary form represents an easy to formulate eigenvalue problem. Notwithstanding the simplicity of the problem statement, we still lack full understanding of the behavior of the mean population fitness and the quasispecies distribution for an arbitrary fitness landscape. Our main goal in this review is two-fold: First, to highlight a number of impressive mathematical results, including some of the recent ones, which pertain to the mathematical development of the quasispecies theory. Second, to emphasize that, despite these 50 years of vigorous research, there are still very natural both biological and mathematical questions that remain to be addressed within the quasispecies framework. Our hope is that at least some of the approaches we review in this text can be of help for anyone embarking on further analysis of the quasispecies model.