Dynamical bimodality in equilibrium monostable systems

Stochastic time-dependent enzyme kinetics: Closed-form solution and transient bimodality

We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.

Master equation for a bistable chemical system with perturbed particle velocity distribution function

We present a modified master equation for a homogeneous gaseous reactive system which includes nonequilibrium corrections due to the reaction-induced perturbation of the particle velocity distribution function. For the Schlögl model, the modified stochastic approach predicts nonequilibrium-induced transitions between different dynamical regimes, including the transformation of a monostable system into a bistable one, and vice versa. These predictions are confirmed by the comparison with microscopic simulations using the direct simulation Monte Carlo method. Compared to microscopic simulations of the particle dynamics, the modified master equation approach proves to be much more efficient.

Transient multimodality in the presence of potential fluctuations

When a stochastic system evolves from its unstable state the phenomenon of transient multimodality may appear, namely for a period of time, probability distribution develops a larger number of maxima than it possesses in the initial and in the stationary states. We investigate the influence of potential fluctuations generated by a Gaussian white noise on this effect. Some general conditions which suffice for the occurrence of noise induced transient multimodality are presented. The opposite phenomenon, namely the noise suppression of transient multimodality, is shown as well.