Multilevel Hybrid Split-Step Implicit Tau-Leap

TL;DR

This paper introduces a multilevel hybrid split-step implicit tau-leap method for efficient stochastic simulation of biochemical systems with multiple timescales, improving stability and speed over existing methods.

Contribution

It proposes a novel multilevel Monte Carlo approach combining explicit and implicit tau-leap methods to handle stability issues in multiscale stochastic biochemical simulations.

Findings

The method improves simulation efficiency for systems with fast and slow reactions.

Numerical examples demonstrate enhanced stability and computational speed.

The approach outperforms traditional explicit tau-leap in complex systems.

Abstract

In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics is dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (Explicit-TL) method, can be very slow. Implicit approximations have been developed to improve numerical stability and provide efficient simulation algorithms for those systems. Here, we propose an efficient Multilevel Monte Carlo (MLMC) method in the spirit of the work by Anderson and Higham (2012). This method uses split-step implicit tau-leap (SSI-TL) at levels where the SSI-TL method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.