Symbolic reachability analysis of genetic regulatory networks using discrete abstractions

doi:10.1016/j.automatica.2007.08.004

Automatica

Volume 44, Issue 4, April 2008, Pages 982-989

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Abstract

We use hybrid-systems techniques for the analysis of reachability properties of a class of piecewise-affine (PA) differential equations that are particularly suitable for the modeling of genetic regulatory networks. More specifically, we introduce a hyperrectangular partition of the state space that forms the basis for a discrete abstraction preserving the sign of the derivatives of the state variables. The resulting discrete transition system provides a qualitative description of the network dynamics that is well-adapted to available experimental data and that can be efficiently computed in a symbolic manner from inequality constraints on the parameters.

Introduction

A class of piecewise-affine (PA) differential equations introduced by Glass and Kauffman (1973) in the seventies has been shown particularly suitable for modeling so-called genetic regulatory networks, networks of genes, proteins, small molecules, and their mutual interactions that are involved in the control of intracellular processes. The dynamics of these networks is hybrid in nature, in the sense that the continuous evolution of the concentration of proteins and other molecules is punctuated by discrete changes in the activity of genes coding for the proteins. This switch-like character of gene regulation is well-captured by the PA models, which have the additional advantage that the qualitative dynamics of the systems is relatively simple to analyze, even in higher dimensions, without the use of numerical values for the kinetic parameters. Given that such information is usually absent in molecular biology, the PA models have been found to be a valuable tool for the practical analysis of complex genetic regulatory networks, which would be difficult to handle with more conventional nonlinear models.

The dynamical properties of the class of PA models considered here have been the subject of active research for more than three decades (e.g., Glass and Kauffman, 1973, Belta et al., 2005, Edwards, 2000, Ghosh and Tomlin, 2004, Gouzé and Sari, 2002, Mestl et al., 1995, see Batt, Ropers, de Jong, Page, & Geiselmann, 2007 for further references). In our previous work (de Jong et al., 2004), we have made a contribution to the analysis of these PA models by showing how to use differential inclusions to deal with discontinuities in the righthand side of the equations. Moreover, we have proposed algorithms and tools to compute a discrete representation of the state space dynamics in the form of a state transition graph.

In this note, we carry the analysis of the PA models further on a number of points, borrowing concepts and techniques from the field of hybrid systems. First, we partition the state space into hyperrectangular regions in which the time derivatives of the solutions have a unique sign pattern. In a second step, this partition motivates the definition of a discrete abstraction (Alur, Henzinger, Lafferriere, & Pappas, 2000) that leads to a discrete transition system providing a finer-grained description of the qualitative dynamics of the system than was hitherto possible and which is better adapted to currently available experimental data. Third, we give rules for the symbolic computation of the discrete state transition system from inequality constraints on the parameters. The implementation of these rules has been shown to scale up to large and complex PA models of genetic regulatory networks.

A long version of this note, containing the proofs of all lemmas and propositions, as well as examples of the application of the method to an actual biological network, is available as supplemental material on the INRIA web site (Batt et al., 2007). This work extends a short and preliminary version of the paper presented at the HSCC conference (Batt et al., 2005).

Section snippets

PA systems

The dynamics of genetic regulatory networks can be described by PA differential equation models using step functions to account for regulatory interactions (Glass and Kauffman, 1973, Mestl et al., 1995). Fig. 1 gives an example of the PA model of a simple two-gene network. Below we define the models and review some mathematical properties.

We denote by $x = (x_{1}, \dots, x_{n})^{'} \in Ω$ a vector of cellular protein concentrations, where $Ω = Ω_{1} \times \dots \times Ω_{n} \subset R_{⩾ 0}^{n}$ is a bounded n-dimensional hyperrectangular state space region.

Flow domain partition

The mismatch between the mathematical analysis and the experimental data calls for a finer partitioning of the state space, which can then provide the basis for a more adequate abstraction criterion. Along these lines, the regular and singular mode domains distinguished above are repartitioned by means of the $(n - 1)$ -dimensional hyperplanes corresponding to the focal concentrations.

Definition 2 Flow domain partition

$D^{M}$ is the hyperrectangular partition of a mode domain $M \in M$ induced by ${ψ_{i} (M)}$ if M is regular, and by ${ψ_{i} (M^{'}) ∣ M^{'} \in R (M)$

Qualitative PA transition systems

As a preparatory step, we define a continuous transition system having the same reachability properties as the original PA system $Σ$ . Consider $x \in D$ and $x^{'} \in D^{'}$ , where $D, D^{'} \in D$ are flow domains. If there exists a solution $ξ (t)$ of $Σ$ passing through x at time $τ \in R_{⩾ 0}$ and reaching $x^{'}$ at time $τ^{'} \in R_{> 0} \cup {\infty}$ , without leaving $D \cup D^{'}$ in the time interval $[τ, τ^{'}]$ , then the absolute continuity of $ξ (t)$ implies that D and $D^{'}$ are either equal or contiguous. We consequently distinguish three types of continuous

Symbolic computation of qualitative PA transition system

The computation of the qualitative PA transition system $Σ - QTS$ is greatly simplified by the fact that the domains D and the focal sets $Ψ (M)$ are hyperrectangular, which allows them to be expressed as product sets, i.e., $D = D_{1} \times \dots \times D_{n}$ and $Ψ (M) = Ψ_{1} (M) \times \dots \times Ψ_{n} (M)$ . As a consequence, the computation can be carried out for each dimension separately. In this section, we will describe rules to determine the set of states $D$ , the satisfaction relation $⊨_{\sim_{Ω}}$ , and the transition relation $\to_{\sim_{Ω}}$ .

Discussion

We have presented a method for the qualitative analysis of a class of PA models that has been well-studied in mathematical biology. By defining a qualitative abstraction preserving the sign pattern of the derivatives of the state variables, the PA model is transformed into a discrete transition system whose properties can be analyzed by means of classical model-checking tools. The discrete transition system provides a conservative approximation of the qualitative dynamics of the system and can

Acknowledgements

The authors acknowledge financial support from the ARC initiative at INRIA (GDyn), the ACI IMPBio initiative of the French Ministry for Research (BacAttract), and the NEST programme of the European Commission (Hygeia, NEST 4995). We would like to thank Jean-Luc Gouzé, Tewfik Sari and Giancarlo Ferrari-Trecate for helpful comments on previous versions of this paper.

Grégory Batt obtained a BSc in Molecular and Cellular Biology from Ecole Normale Supérieure in Lyon and an MSc in Computer Science from the Joseph Fourier University (UJF) in Grenoble (France). After his PhD in Computer Science at UJF and INRIA Grenoble-Rhône-Alpes, obtained in 2005, he has continued his research on the interface of hybrid systems theory and systems biology with post-doctoral fellowships at Boston University and VERIMAG (CNRS, UJF). He is currently research scientist at

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Hidde de Jong obtained MSc degrees in Computer Science, Philosophy of Science, and Management Science from the University of Twente (the Netherlands) and completed a PhD thesis in Computer Science at the same university. He joined INRIA in 1998 and is currently a senior research scientist in the bioinformatics and biological modeling group at INRIA Grenoble-Rhône-Alpes (Grenoble).

Michel Page obtained a PhD thesis in Computer Science from the Institut National Polytechnique in Grenoble (INPG) in 1991. He is currently an associate professor in Computer Science at the Pierre Mendès-France University (Grenoble) and also affiliated with INRIA Grenoble-Rhône-Alpes, where he works on the development of modeling and simulation tools for bioinformatics.

Hans Geiselmann obtained a PhD in Molecular Biology from the University of Oregon (USA) in 1989. He has been an assistant professor at the University of Geneva and, since 1998, a full professor in Microbiology at UJF, where he explores interdisciplinary approaches to study the molecular mechanisms underlying the control of bacterial gene expression.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Michael Henson under the direction of Editor Frank Allgöwer.

¹: Present Address: INRIA Paris-Rocquencourt, 78153 Le Chesnay, France.

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Brief paperSymbolic reachability analysis of genetic regulatory networks using discrete abstractions☆

Abstract

Introduction

Section snippets

PA systems

Flow domain partition

Qualitative PA transition systems

Symbolic computation of qualitative PA transition system

Discussion

Acknowledgements

Bulletin of Mathematical Biology

Physica D

Journal of Theoretical Biology

Journal of Theoretical Biology

Discrete abstractions of hybrid systems

Proceedings of the IEEE

Validation of qualitative models of genetic regulatory networks by model checking

Bioinformatics

Brief paper
Symbolic reachability analysis of genetic regulatory networks using discrete abstractions☆