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.