Chemical master equation parameter exploration using DMRG

TL;DR

This paper uses tensor network methods, specifically DMRG, to efficiently compute steady-state distributions of complex chemical reaction networks, revealing stochastic effects and bistability beyond mean-field approximations.

Contribution

It introduces a tensor network approach, utilizing DMRG, to analyze steady-states of CRNs, enabling parameter exploration and capturing stochastic fluctuations.

Findings

Successfully computed steady-states of a seven-species CRN using DMRG.

Identified bistability in the gene toggle switch model as a function of parameters.

Demonstrated the method's ability to analyze stochastic effects beyond mean-field theory.

Abstract

Well-mixed chemical reaction networks (CRNs) contain many distinct chemical species with copy numbers that fluctuate in correlated ways. While those correlations are typically monitored via Monte Carlo sampling of stochastic trajectories, there is interest in systematically approximating the joint distribution over the exponentially large number of possible microstates using tensor networks or tensor trains. We exploit the tensor network strategy to determine when the steady state of a seven-species gene toggle switch CRN model supports bistability as a function of two decomposition rates, both parameters of the kinetic model. We highlight how the tensor network solution captures the effects of stochastic fluctuations, going beyond mean field and indeed deviating meaningfully from a mean-field analysis. The work furthermore develops and demonstrates several technical advances that will allow steady-states of broad classes of CRNs to be computed in a manner conducive to parameter exploration. We show that the steady-state distributions can be computed via the ordinary density matrix renormalization group (DMRG) algorithm despite having a non-Hermitian rate operator with a small spectral gap, we illustrate how that steady-state distribution can be efficiently projected to an order parameter that identifies bimodality, and we employ excited-state DMRG to calculate a relaxation timescale for the bistability.