Petri Nets for Systems and Synthetic Biology
Monika Heiner
1
, David Gilbert
2
, and Robin Donaldson
2
1
Department of Computer Science, Brandenburg University of Technology
Postbox 10 13 44, 03013 Cottbus, Germany
monika.heiner@tu-cottbus.de
2
Bioinformatics Research Centre, University of Glasgow
Glasgow G12 8QQ, Scotland, UK
drg@brc.dcs.gla.ac.uk, radonald@brc.dcs.gla.ac.uk
Abstract. We give a description of a Petri net-based framework for
modelling and analysing biochemical pathways, which unifies the qualita-
tive, stochastic and continuous paradigms. Each perspective adds its con-
tribution to the understanding of the system, thus the three approaches
do not compete, but complement each other. We illustrate our approach
by applying it to an extended model of the three stage cascade, which
forms the core of the ERK signal transduction pathway. Consequently
our focus is on transient behaviour analysis. We demonstrate how quali-
tative descriptions are abstractions over stochastic or continuous descrip-
tions, and show that the stochastic and continuous models approximate
each other. Although our framework is based on Petri nets, it can be
applied more widely to other formalisms which are used to model and
analyse biochemical networks.
1 Motivation
Biochemical reaction systems have by their very nature three distinctive charac-
teristics. (1) They are inherently bipartite, i.e. they consist of two types of game
players, the species and their interactions. (2) They are inherently concurrent,
i.e. several interactions can usually happen independently and in parallel. (3)
They are inherently stochastic, i.e. the timing behaviour of the interactions is
governed by stochastic laws. So it seems to be a natural choice to model and
analyse them with a formal method, which shares exactly these distinctive char-
acteristics: stochastic Petri nets.
However, due to the computational efforts required to analyse stochastic mod-
els, two abstractions are more popular: qualitative models, abstracting away from
any time dependencies, and continuous models, commonly used to approximate
stochastic behaviour by a deterministic one. We describe an overall framework
to unify these three paradigms, providing a family of related models with high
analytical power.
The advantages of using Petri nets as a kind of umbrella formalism are seen
in the following:
M. Bernardo, P. Degano, and G. Zavattaro (Eds.): SFM 2008, LNCS 5016, pp. 215–264, 2008.
c© Springer-Verlag Berlin Heidelberg 2008

216 M. Heiner, D. Gilbert, and R. Donaldson
– intuitive and executable modelling style,
– true concurrency (partial order) semantics, which may be lessened to inter-
leaving semantics to simplify analyses,
– mathematically founded analysis techniques based on formal semantics,
– coverage of structural and behavioural properties as well as their relations,
– integration of qualitative and quantitative analysis techniques,
– reliable tool support.
This chapter can be considered as a tutorial in the step-wise modelling and anal-
ysis of larger biochemical networks as well as in the structured design of systems
of ordinary differential equations (ODEs). The qualitative model is introduced
as a supplementary intermediate step, at least from the viewpoint of the bio-
chemist accustomed to quantitative modelling only, and serves mainly for model
validation since this cannot be performed on the continuous level, and is gen-
erally much harder to do on the stochastic level. Having successfully validated
the qualitative model, the quantitative models are derived from the qualitative
one by assigning stochastic or deterministic rate functions to all reactions in the
network. Thus the quantitative models preserve the structure of the qualitative
one, and the stochastic Petri net describes a system of stochastic reaction rate
equations (RREs), and the continuous Petri net is nothing else than a structured
description of ODEs.
systems biology: modelling as formal knowledge representation
synthetic biology: modelling for system construction
biosystem
natural
biosystem
synthetic
observed
behaviour
predicted
behaviour
model
(blueprint)
desired
behaviour
design construction
verification
verification
observed
behaviour
predicted
behaviour
wetlab
model-based
experiment design
experiments
formalizing
understanding
wetlab
experiments
model
(knowledge)
Fig. 1. The role of formal models in systems biology and synthetic biology