Constrained approximation of effective generators for multiscale stochastic reaction networks and application to conditioned path sampling

Abstract

Efficient analysis and simulation of multiscale stochastic systems of chemical kinetics is an ongoing area for research, and is the source of many theoretical and computational challenges. In this paper, we present a significant improvement to the constrained approach, which is a method for computing effective dynamics of slowly changing quantities in these systems, but which does not rely on the quasi-steady-state assumption (QSSA). The QSSA can cause errors in the estimation of effective dynamics for systems where the difference in timescales between the "fast" and "slow" variables is not so pronounced.This new application of the constrained approach allows us to compute the effective generator of the slow variables, without the need for expensive stochastic simulations. This is achieved by finding the null space of the generator of the constrained system. For complex systems where this is not possible, or where the constrained subsystem is itself multiscale, the constrained approach can then be applied iteratively. This results in breaking the problem down into finding the solutions to many small eigenvalue problems, which can be efficiently solved using standard methods.Since this methodology does not rely on the quasi steady-state assumption, the effective dynamics that are approximated are highly accurate, and in the case of systems with only monomolecular reactions, are exact. We will demonstrate this with some numerics, and also use the effective generators to sample paths of the slow variables which are conditioned on their endpoints, a task which would be computationally intractable for the generator of the full system.

Product-Form Stationary Distributions for Deficiency Zero Networks with Non-mass Action Kinetics

“…2005). However, more recent improvements in stochastic averaging can lead to fast subsystems with non-mass action kinetics, and this observation was the motivation for the present work (Cotter 2016; Cotter and Erban 2016; Cotter et al. 2011).…”

“…First, we consider a motivating example arising from model reduction, through constrained averaging (Cotter 2016; Cotter and Erban 2016; Cotter et al. 2011), of the following system:where the intensity functions are placed next to the reaction arrows.…”

“…Readers interested in seeing how this is derived should refer to Cotter (2016). This network is weakly reversible and has a deficiency of zero.…”

“…Another option involves finding the distribution by finding the null space of the adjoint of the generator (see the discussion in and around (5)). However, as the state space of (18) will typically be huge, the latter method often involves truncating the state space and approximating the actual distribution with that of the stationary distribution of the truncated system (Cotter 2016). Both approaches will lead to approximation errors and varying amounts of computational cost.…”

“…As opposed to the specific models we considered in the previous examples, here we present a more general framework in which the specific form of the propensity function for the reaction is arbitrary. This situation is common when undertaking certain types of averaging arguments (Cotter 2016). …”