Method of conditional moments (MCM) for the Chemical Master Equation

Abstract

The time-evolution of continuous-time discrete-state biochemical processes is governed by the Chemical Master Equation (CME), which describes the probability of the molecular counts of each chemical species. As the corresponding number of discrete states is, for most processes, large, a direct numerical simulation of the CME is in general infeasible. In this paper we introduce the method of conditional moments (MCM), a novel approximation method for the solution of the CME. The MCM employs a discrete stochastic description for low-copy number species and a moment-based description for medium/high-copy number species. The moments of the medium/high-copy number species are conditioned on the state of the low abundance species, which allows us to capture complex correlation structures arising, e.g., for multi-attractor and oscillatory systems. We prove that the MCM provides a generalization of previous approximations of the CME based on hybrid modeling and moment-based methods. Furthermore, it improves upon these existing methods, as we illustrate using a model for the dynamics of stochastic single-gene expression. This application example shows that due to the more general structure, the MCM allows for the approximation of multi-modal distributions.

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

“…This observation was originally stated in Ref. 11 where the authors considered a specific example with m = 1 and n = 2 and deduced that Table II in Ref. 11).…”

“…Recently, it has been argued that in such cases, the stochastic dynamics of the network can be described more accurately with fewer equations through the use of conditional moments. 11,12 Here, I show that the result that less a) Electronic mail: jruess@ist.ac.at equations are needed is only true if the specific structure of the reaction network is not taken into account in the derivation of the classical unconditional moment equations. I provide a formula to determine how many classical moment equations are minimally needed to compute all moments up to any desired order L, and show that, for all L, this number is smaller than the number of conditional moment equationsin line with the mathematical intuition that more accurate descriptions of the whole underlying probability distribution cannot be obtained with less equations.…”

“…This seems to be the reason why a different formulation of models of gene expression is more prevalent in the field. 11,21 In this formulation, the promoter is represented by m + 1 binary variables X 1 , . .…”

“…, m have so far only been treated in the same way as any other species and the specific properties in (3) were neglected. 11 This does not lead to incorrect results, but the derived moment equations are unnecessarily complicated. In the following, I will explain how smaller systems of moment equations can be obtained by using the structure in (3) and what the implications of these equations are for general systems with partially finite state space.…”

Reconstructing dynamic molecular states from single-cell time series

“…Moreover, the proposed method can be extended to a hybrid framework (e.g. [45][46][47]) where a diffusion approximation can be performed for some states and reactions. This is especially interesting for multi-scale cellular processes, for instance in signal transduction coupled to gene expression where different abundance scales of molecules are involved.…”