Approximation of the Chemical Master Equation using conditional moment closure and time-scale separation

To describe the stochastic behavior of biomolecular systems, the Chemical Master Equation (CME) is widely used. The CME gives a complete description of the evolution of a system's probability distribution. However, in general, the CME's dimension is very large or even infinite, so analytical solutions may be difficult to write and analyze. To handle this problem, based on the fact that biomolecular systems are time-scale separable, we approximate the CME with another CME that describes the dynamics of the slow species only. In particular, we assume that the number of each molecular species is bounded, although it may be very large. We thus write Ordinary Differential Equations (ODEs) of the slow-species counts' marginal probability distribution and of the fast-species counts' firstN conditional moments. Here, N is an arbitrary (possibly small) number, which can be chosen to compromise between approximation accuracy and the computational burden associated with simulating or analyzing a high dimensional system. Then we apply conditional moment closure and timescale separation to approximate the first N conditional moments of the fast-species counts as functions of the slow-species counts. By subsituting these functions on the right-hand side of the ODEs that describes the marginal probability distribution of the slow-species counts, we can approximate the original CME with a lower dimensional CME. We illustrate the application of this method on an enzymatic and a protein binding reaction.