Bifurcation and criticality

Abstract

Equilibrium and nonequilibrium systems exhibit power-law singularities close to their critical and bifurcation points respectively. A recent study has shown that biochemical nonequilibrium models with positive feedback belong to the universality class of the mean-field Ising model. Through a mapping between the two systems, effective thermodynamic quantities like temperature, magnetic field and order parameter can be expressed in terms of biochemical parameters. In this paper, we demonstrate the equivalence using a simple deterministic approach. As an illustration we consider a model of population dynamics exhibiting the Allee effect for which we determine the exact phase diagram. We further consider a two-variable model of positive feedback, the genetic toggle, and discuss the conditions under which the model belongs to the mean-field Ising universality class. In the biochemical models, the supercritical pitchfork bifurcation point serves as the critical point. The dynamical behaviour predicted by the two models is in qualitative agreement with experimental observations and opens up the possibility of exploring critical point phenomena in laboratory populations and synthetic biological circuits.