Analytical study of an exclusive genetic switch

TL;DR

This paper analyzes a minimal model of an exclusive genetic switch, providing exact solutions, mean field theory, and perturbative methods to understand its bistability and symmetry-breaking behavior.

Contribution

It introduces an exact perturbative approach and mean field theory for the exclusive genetic switch, revealing conditions for bistability and symmetry breaking.

Findings

Bistability occurs generically in the model.

Mean field theory is exact at low binding/unbinding rates.

Exact solutions are derived for extreme binding/unbinding rate limits.

Abstract

The nonequilibrium stationary state of an exclusive genetic switch is considered. The model comprises two competing species and a single binding site which, when bound to by a protein of one species, causes the other species to be repressed. The model may be thought of as a minimal model of the power struggle between two competing parties. Exact solutions are given for the limits of vanishing binding/unbinding rates and infinite binding/unbinding rates. A mean field theory is introduced which is exact in the limit of vanishing binding/unbinding rates. The mean field theory and numerical simulations reveal that generically bistability occurs and the system is in a symmetry broken state. An exact perturbative solution which in principle allows the nonequilibrium stationary state to be computed is also developed and computed to first and second order.