Chemical Master Equation Empirical Moment Closure

Ammar A

doi:10.4172/2329-6577.1000155

Abstract

The numerical solution of the Chemical Master Equation (CME) governing gene regulatory networks and cell signaling processes remains a challenging task due to its complexity, exponentially growing with the number of species involved. When considering separated representations of the probability distribution function within the Proper Generalized Decomposition-PGD-frame-work the complexity of the CME grows only linearly with the number of state space dimensions. In order to speed up calculations moment-based descriptions are usually preferred, however these descriptions involve the necessity of using closure relations whose impact on the calculated solution is most of time unpredictable. In this work we propose an empirical closure, fitted from the solution of the chemical master equation, the last solved within the PGD framework.

Highlights

Simulating the behavior of gene regulatory networks is a formidable task for several reasons
Under some weak hypotheses the system can be considered as Markovian, and can be modeled by the so-called Chemical Master Equation (CME) [3], which is no more than a set of ordinary differential equations stating the conservation of the probability distribution function - pdf -P in time:
For a given system, we solve the CME by using the PGD in order to circumvent the curse of dimensionality and evaluate the third order moment according to Equation (11) and we choose the alpha parameters in Hegland et al [16] to provide the best fitting

Summary

Introduction

Simulating the behavior of gene regulatory networks is a formidable task for several reasons. Despite the fact of being able to solve the CME, its solution requires a significant amount of computation, and the simulation of a variety of scenarios remains a challenging issue because its computational complexity For alleviating such a computational complexity an appealing route consists of calculating the moments of the probability distribution function instead of the pdf itself. To avoid the exponentially growing complexity of the problem with the number of state space dimensions, the method approximates the variable of interest, say u, as a finite sum of separable functions: n i =1 The reason for this particular choice motivated the method itself that is conceived as a greedy algorithm that computes one sum at a time and one product at a time, within a fixed-point, alternating directions algorithm. The PGD avoids the exponential complexity with respect to the problem dimension

A PGD approach to gene regulatory networks simulation

Numerical Results

Conclusions