Analysis of tensor methods for stochastic models of gene regulatory networks

TL;DR

This paper provides a detailed error analysis of the tensor-structured parametric analysis (TPA) method for simulating stochastic gene regulatory networks, focusing on its accuracy and validity across various approximation steps.

Contribution

It offers a comprehensive error analysis of the TPA method, including modeling, discretization, and tensor rounding errors, with computational illustrations.

Findings

Error bounds for TPA approximations

Validation on death-birth process example

Application to high-dimensional reaction chain

Abstract

The tensor-structured parametric analysis (TPA) has been recently developed for simulating and analysing stochastic behaviours of gene regulatory networks [Liao et. al., 2015]. The method employs the Fokker-Planck approximation of the chemical master equation, and uses the Quantized Tensor Train (QTT) format, as a low-parametric tensor-structured representation of classical matrices and vectors, to approximate the high-dimensional stationary probability distribution. This paper presents a detailed error analysis of all approximation steps of the TPA regarding validity and accuracy, including modelling error, artificial boundary error, discretization error, tensor rounding error, and algebraic error. The error analysis is illustrated using computational examples, including the death-birth process and a 50-dimensional isomerization reaction chain.