Dimensionality Reduction of the Chemical Master Equation

Abstract

The dynamics of biochemical systems show significant variability when the reactant populations are small. Standard approaches via deterministic modeling exclude such variability. A well established stochastic model, the Chemical master equation (CME), describes the dynamics of biochemical systems by representing the time evolution of the probability distribution of species' discrete states in a well-mixed reaction volume. However, the dimension of the CME (i.e.~the number of transition states in the system) rapidly grows as the molecular population and number of reactions in the network increases. Also, the dynamics of biochemical systems typically vary over a wide range of time scales: a phenomenon referred to as stiffness. Large dimensions and stiffness pose challenges to numerical analysis of system behavior. By eliminating the fast modes, which correspond to fast time scales that are often not experimentally observed, a model reduction can be achieved. In our work, we apply such a model reduction to the CME. The slow and fast modes of the system correspond to small and large eigenvalues of the transition matrix of the CME. By a transformation, we exclude the fast modes to arrive at a truncated model. We propose a method based on eigenbasis transformations that provide efficient approximations that are accurate beyond a short initial time interval. We also present efficient algorithms for generation of the CME from a network and for computation of eigenbases. Finally, we describe how this reduction approach can be implemented to provide efficient time-step identification in a well-established scheme for an approximation of the CME (the so-called finite state projection).