Calculation of switching times in the genetic toggle switch and other bistable systems

Abstract

Genetic circuits with feedback such as the toggle switch often exhibit bistability, namely, two stable states with rare spontaneous transitions between them. These systems can be characterized by the average time between such transitions (referred to as the switching time). However, commonly used deterministic models, based on rate equations, do not account for these fluctuation-induced transitions. Stochastic methods, such as the direct integration of the master equation, do account for the transitions. However, they cannot be used to evaluate the switching time. In order to obtain the switching time, one needs to use Monte Carlo simulations. These methods require the accumulation of statistical data, which limits their accuracy. They may become infeasible when the switching time is long. Here we present an accurate and efficient method for the calculation of the switching time. The method consists of coupled recursion equations for the transition times between microscopic states of the system. Using a suitable definition of the two macroscopic bistable states (in terms of the microscopic states) and the probabilities obtained from the master equation, the method provides the switching time between the two states of the system. The method is demonstrated for the genetic toggle switch. It can be used to evaluate the switching times in a broad range of bistable and multistable systems. We also show that it is suitable for the evaluation of the oscillation periods in oscillatory systems such as the repressilator.