Computation of stationary distributions in stochastic models of cellular processes with molecular memory

Abstract

Modeling stochastic dynamics of intracellular processes has long rested on Markovian (i.e., memoryless) hypothesis. However, many of these processes are non-Markovian (i.e., memorial) due to, e.g., small reaction steps involved in synthesis or degradation of a macroscopic molecule. When interrogating aspects of a cellular network by experimental measurements (e.g., by singlemolecule and single-cell measurement technologies) of network components, a key need is to develop efficient approaches to simulate and compute joint distributions of these components. To cope with this computational challenge, we develop two efficient algorithms: stationary generalized Gillespie algorithm and stationary generalized finite state projection, both being established based on a stationary generalized chemical master equation. We show how these algorithms can be combined in a streamlined procedure for evaluation of non-Markovian effects in a general cellular network. Stationary distributions are evaluated in two models of constitutive and bursty gene expressions as well as a model of genetic toggle switch, each considering molecular memory. Our approach significantly expands the capability of stochastic simulation to investigate gene regulatory network dynamics, which has the potential to advance both understanding of molecular systems biology and design of synthetic circuits. Cellular systems are driven by interactions between subsystems via time-stamped discrete events, involving numerous components and reaction steps and spanning several time scales. Such biochemical reactions are subject to inherent noise due to the small numbers of molecules. Also, they could involve several small steps, creating a memory between individual events. Delineating these molecular stochasticity and memory of biomolecular networks are continuing challenges for molecular systems biology. We present a novel approach to compute the probability distribution in stochastic models of cellular processes with molecular memory based on stationary generalized chemical master equation. We map a stochastic system with memory onto a Markovian model with effective reaction propensity functions.