Efficient calculation of steady state probability distribution for stochastic biochemical reaction network

Abstract

The Steady State (SS) probability distribution is an important quantity needed to characterize the steady state behavior of many stochastic biochemical networks. In this paper, we propose an efficient and accurate approach to calculating an approximate SS probability distribution from solution of the Chemical Master Equation (CME) under the assumption of the existence of a unique deterministic SS of the system. To find the approximate solution to the CME, a truncated state-space representation is used to reduce the state-space of the system and translate it to a finite dimension. The subsequent ill-posed eigenvalue problem of a linear system for the finite state-space can be converted to a well-posed system of linear equations and solved. The proposed strategy yields efficient and accurate estimation of noise in stochastic biochemical systems. To demonstrate the approach, we applied the method to characterize the noise behavior of a set of biochemical networks of ligand-receptor interactions for Bone Morphogenetic Protein (BMP) signaling. We found that recruitment of type II receptors during the receptor oligomerization by itself doesn't not tend to lower noise in receptor signaling, but regulation by a secreted co-factor may provide a substantial improvement in signaling relative to noise. The steady state probability approximation method shortened the time necessary to calculate the probability distributions compared to earlier approaches, such as Gillespie's Stochastic Simulation Algorithm (SSA) while maintaining high accuracy.

Enzyme sequestration by the substrate: An analysis in the deterministic and stochastic domains

“…Our formulation extends a Proposition made by Karim et al [ 44 ]. The theoretical development of this tool, along with its associated theorems, can be found in S1 Text , S1.8.…”

Understanding the discrete genetic toggle switch phenomena using a discrete ‘nullcline’ construct inspired by the Markov chain tree theorem

“…Although k in Fig. 5 can be found numerically directly by calculating the null space of matrix A for each k, we use a formulation of ours proved in [19], extending a result by Karim et al [13]. In our formulation A q = b, where A is the principal submatrix of A after the removal of the row and of the column corresponding to the equilibrium microstate j without loss of generality.…”

Selected articles from the IEEE International Workshop on Genomic Signal Processing and Statistics (GENSIPS'2011)

“…The proposed technique is validated using gene expression and ER α Chip-seq data from the MCF-7 cell line. Fine-scale modeling of genetic regulatory networks using stochastic master equation models entails a huge computational complexity and Karim et al [ 5 ] presents a computationally inexpensive way to generate the steady state distributions for stochastic master equation models. In Wang et al [ 6 ], a quantitative mathematical model to predict macrophage activation patterns following myocardial infarction is reported.…”