Variational ansatz for quasispecies in the Eigen model

TL;DR

This paper introduces a variational approach to analyze the error threshold and universal scaling behaviors of quasispecies in the Eigen model, validated by numerical methods and applicable to short genome lengths.

Contribution

A novel variational ansatz is proposed for the Eigen model, accurately predicting quasispecies properties and uncovering universal scaling near the error threshold.

Findings

Correctly predicts survival population of wild-type sequences

Reveals universal scaling behaviors near the error threshold

Shows excellent agreement with numerical methods

Abstract

We investigate the error threshold for the emergence of quasispecies in the Eigen model. By mapping to to an effective Hamiltonian ruled by the "imaginary-time" Schr\"odinger equation, a variational ansatz is proposed and applied to calculate various quantities associated with the quasispecies. The variational ansatz gives correct predictions for the survival population of the wild-type sequence and also reveals an unexpected universal scaling behaviors near the error threshold. We check the validity of the variational ansatz by numerical methods and find excellent agreement. Though the emergence of the scaling behaviors is not yet fully understood, it is remarkable that the universal scaling function reigns even for relatively short genome length such as L=16. Further investigations may reveal the mechanism of the universal scaling and extract the essential ingredients for the emergence of the quasispecies in molecular evolution.