Exact calculation of stationary solution and parameter sensitivity analysis of stochastic continuous time Boolean models

Solutions to stochastic Boolean models are usually estimated by Monte Carlo simulations, but as the state space of these models can be enormous, there is an inherent uncertainty about the accuracy of Monte Carlo estimates and whether simulations have reached all asymptotic solutions. Moreover, these models have timescale parameters (transition rates) that the probability values of stationary solutions depend on in complex ways that have not been analyzed yet in the literature. These two fundamental uncertainties call for an exact calculation method for this class of models. We show that the stationary probability values of the attractors of stochastic (asynchronous) continuous time Boolean models can be exactly calculated. The calculation does not require Monte Carlo simulations, instead it uses an exact matrix calculation method previously applied in the context of chemical kinetics. Using this approach, we also analyze the under-explored question of the effect of transition rates on the stationary solutions and show the latter can be sensitive to parameter changes. The analysis distinguishes processes that are robust or, alternatively, sensitive to parameter values, providing both methodological and biological insights.