Analyzing Stationary States of Gene Regulatory Network Using Petri Nets - Biological Petri Nets - page 143

Analyzing Stationary States of Gene
Regulatory Network Using Petri Nets
Anna Gambin, Sławomir Lasota and Michał Rutkowski ∗
Institute of Informatics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
ABSTRACT: We introduce and formally define the notion of a stationary state for Petri nets. We also propose a fully automatic
method for finding such states. The procedure makes use of the Presburger arithmetic to describe all the stationary states.
Finally we apply this novel approach to find stationary states of a gene regulatory network describing the flower morphogenesis
of A. thaliana . This shows that the proposed method can be successfully applied in the study of biological systems.
KEYWORDS: Petri net, stationary state, gene regulatory network, Presburger arithmetic, flower morphogenesis, Arabidopsis
thaliana
INTRODUCTION
Recent research in the field of molecular biology yields large amounts of data. Analysis of that data
brought much better understanding of the processes that govern the life cycle of the cells. At the base of
all these processes lie chemical reactions. A major drawback of experiments conducted in laboratories
is that they are time and money consuming. For this reason computer analysis is very promising.
Petri nets is a mathematical formalism used in modeling concurrent and distributed systems. The
theory has been studied for over 40 years and is well developed. For a biologist Petri nets might be
of interest because their graphical representation is very similar to that of chemical systems [see Peleg
et al. , 2005; Hardy and Robillard, 2004]. Moreover, a stochastic extension of Petri nets defines the
same stochastic process as the one defined by the Chemical Master Equation , which is commonly used
to describe the dynamics of chemical systems. This implies that apart from numerical analysis and
simulations, one might derive conclusions about a chemical system, using the theory of Petri nets. Some
successful application of stochastic Petri nets is reported in [Goss and Peccoud, 1998].
One of central goals of contemporary molecular biology is to understand the cellular processes
described by time series of thousands of gene expression measurements. The interactions between
genes can be represented in a graphical form as a gene regulatory network. Some previous work on
applications of Petri nets to regulatory networks can be found in [Matsuno et al. , 2000]. The crucial
task in modeling gene regulation is the analysis of stationary states of the network. Multistationarity
(i.e. the property of systems whose structure induces two or more distinct steady states) can account for
∗ Corresponding author. E-mail: [e-mail] .