Field theory for a reaction-diffusion model of quasispecies dynamics

TL;DR

This paper develops a field theory approach to model the error threshold in quasispecies dynamics of RNA viruses, revealing critical behavior and exponents that could be experimentally observed, and highlighting the role of spatial structure.

Contribution

It introduces a reaction-diffusion field theory for the quasispecies model, connecting it to universality classes and providing a new perspective on the error catastrophe.

Findings

Identifies the universality class of the reaction-diffusion model.

Derives critical exponents characterizing the error threshold.

Suggests spatial effects can alter mean field predictions.

Abstract

RNA viruses are known to replicate with extremely high mutation rates. These rates are actually close to the so-called error threshold. This threshold is in fact a critical point beyond which genetic information is lost through a second-order phase transition, which has been dubbed the ``error catastrophe.'' Here we explore this phenomenon using a field theory approximation to the spatially extended Swetina-Schuster quasispecies model [J. Swetina and P. Schuster, Biophys. Chem. {\bf 16}, 329 (1982)], a single-sharp-peak landscape. In analogy with standard absorbing-state phase transitions, we develop a reaction-diffusion model whose discrete rules mimic the Swetina-Schuster model. The field theory representation of the reaction-diffusion system is constructed. The proposed field theory belongs to the same universality class than a conserved reaction-diffusion model previously proposed [F. van Wijland {\em et al.}, Physica A {\bf 251}, 179 (1998)]. From the field theory, we obtain the full set of exponents that characterize the critical behavior at the error threshold. Our results present the error catastrophe from a new point of view and suggest that spatial degrees of freedom can modify several mean field predictions previously considered, leading to the definition of characteristic exponents that could be experimentally measurable.