The Enhanced Finite State Projection algorithm, using conditional moment closure and time-scale separation

Abstract

The Chemical Master Equation (CME) is commonly used to describe the stochastic behavior of biomolecular systems. However, in general, the CME's dimension is very large or infinite, so analytical solutions may be difficult to achieve. To handle this problem, the Finite State Projection (FSP) algorithm can be used. However, when multiple time scales exist, which is common in biomolecular systems, the FSP algorithm also suffers from the computational issue. To deal with this problem, we propose the Enhanced Finite State Projection (EFSP) algorithm, which combines the original FSP algorithm and the model reduction technique that we developed, to approximate an infinite dimensional CME with a finite dimensional CME that contains the slow species only. We quantify the approximation error between the slow-species counts' marginal probability distribution of the original CME and those of the approximated CME, and prove that this error becomes smaller as 3 (the EFSP error) or E (time-scale separation between the fast and slow species) decreases. Unlike other time-scale separation methods, which rely on the fast-species counts' stationary conditional probability distributions, our model reduction technique relies on only the first few conditional moments of the fast-species counts. This is possible because we apply conditional moment closure to close the fast-species counts' dynamics, which provides a significant computation advantage. The benefit of our algorithm is illustrated through a protein binding reaction and a toggle switch.