Steady‐state analysis of genetic regulatory networks modelled by probabilistic Boolean networks

Ilya Shmulevich,Edward R. Dougherty,Wei Zhang,Ilya Gluhovsky,Ronaldo F. Hashimoto

doi:10.1002/cfg.342

Ilya Shmulevich, Edward R. Dougherty + Show 3 more

Open Access

https://doi.org/10.1002/cfg.342

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Abstract

Probabilistic Boolean networks (PBNs) have recently been introduced as a promising class of models of genetic regulatory networks. The dynamic behaviour of PBNs can be analysed in the context of Markov chains. A key goal is the determination of the steady-state (long-run) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the long-term influence of a gene on another gene or determine the long-term joint probabilistic behaviour of a few selected genes. Because matrix-based methods quickly become prohibitive for large sizes of networks, we propose the use of Monte Carlo methods. However, the rate of convergence to the stationary distribution becomes a central issue. We discuss several approaches for determining the number of iterations necessary to achieve convergence of the Markov chain corresponding to a PBN. Using a recently introduced method based on the theory of two-state Markov chains, we illustrate the approach on a sub-network designed from human glioma gene expression data and determine the joint steadystate probabilities for several groups of genes.

Highlights

Modelling of genetic regulatory networks is becoming increasingly widespread for gaining insight into the underlying processes of living systems
It can be envisioned that some gene expression events have been initiated at different stages at the lowand mid-grade brain tumours and may eventually have led to the state when insulin-like growth factor binding protein 2 (IGFBP2) is activated
The goal is that the use of gene expression profiles, along with the construction of the gene networks using models such as probabilistic Boolean networks (PBNs), will make it possible to predict whether the convergence to events such as IGFBP2 activation state will occur

Summary

Introduction

Modelling of genetic regulatory networks is becoming increasingly widespread for gaining insight into the underlying processes of living systems. The goal is that the use of gene expression profiles, along with the construction of the gene networks using models such as PBNs, will make it possible to predict whether (and when) the convergence to events such as IGFBP2 activation state will occur. Monte Carlo methods represent a viable alternative to numerical matrix-based methods for obtaining steady-state distributions Speaking, this consists of running the Markov chain for a sufficiently long time, until convergence to the stationary distribution is reached, and observing the proportion of time the process spends in the parts of the state space that represent the information of interest, such as the joint stationary distribution of several specific genes. We select subsamples x1(G), x1(2G), . . . and x2(G), x2(2G), . . . and use the Kolmogorov–Smirnov statistic with the lexicographical ordering to define the indicator: K

M max η

Experimental results

Concluding remarks