Matrix Formalism to Describe Functional States of Transcriptional Regulatory Systems

Erwin P Gianchandani,Nathan D Price,Andrew R Joyce,Bernhard O Palsson,Jason A Papin,Reka Albert

doi:10.1371/journal.pcbi.0020101

Erwin P Gianchandani, Nathan D Price + Show 4 more

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https://doi.org/10.1371/journal.pcbi.0020101

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Abstract

Complex regulatory networks control the transcription state of a genome. These transcriptional regulatory networks (TRNs) have been mathematically described using a Boolean formalism, in which the state of a gene is represented as either transcribed or not transcribed in response to regulatory signals. The Boolean formalism results in a series of regulatory rules for the individual genes of a TRN that in turn can be used to link environmental cues to the transcription state of a genome, thereby forming a complete transcriptional regulatory system (TRS). Herein, we develop a formalism that represents such a set of regulatory rules in a matrix form. Matrix formalism allows for the systemic characterization of the properties of a TRS and facilitates the computation of the transcriptional state of the genome under any given set of environmental conditions. Additionally, it provides a means to incorporate mechanistic detail of a TRS as it becomes available. In this study, the regulatory network matrix, R, for a prototypic TRS is characterized and the fundamental subspaces of this matrix are described. We illustrate how the matrix representation of a TRS coupled with its environment (R*) allows for a sampling of all possible expression states of a given network, and furthermore, how the fundamental subspaces of the matrix provide a way to study key TRS features and may assist in experimental design.

Highlights

With the delineation of multiple genome sequences, there is an increased interest in understanding how the genes within a given genome are regulated through complex transcriptional regulatory networks (TRNs)
This paper presents a novel approach for describing a complete transcriptional regulatory system (TRS), including inputs and outputs to the set of internal reactions defined by the TRN, in a functional matrix form that connects environmental cues to transcriptional responses
The matrix formalism for representing TRSs was evaluated using a small-scale reconstruction of the TRN of the lac operon in E. coli

Summary

Introduction

With the delineation of multiple genome sequences, there is an increased interest in understanding how the genes within a given genome are regulated through complex transcriptional regulatory networks (TRNs). Several approaches have been used to characterize features of TRNs, including Bayesian networks [2], Boolean networks [3,4,5,6], and stochastic equations [7] (see [8] for a review of many such methods) While most of these methods have been applied to relatively small systems due to a lack of relevant data, there are notable exceptions (for examples, see [9,10,11,12]). An integrated analysis of metabolic and regulatory networks in Escherichia coli was performed [9] through dual perturbation experiments [13] This systematic approach to reconstructing and interrogating the integrated network of E. coli led to the novel characterization of multiple regulatory rules and an expansion of a genome-scale TRN, based on a model-driven analysis of multiple high-throughput datasets

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