# PetriNets

*In the following we give a brief description of our Petri net based framework. For more background information see References*.

### Framework

Our framework comprises a family of related Petri net models sharing the underlying plain network structure, but differing in kinetic details (quantitative information) and/or representation density (structural information).

Models can be converted into each other (import/export relation) which obviously may involve loss or gain of kinetic or structural information.

### Qualitative Petri Nets (QPN)

are bipartite directed multi-graphs with two types of nodes: places and transitions; arcs can be annotated with integer weights. A possible interpretation in systems biology is:

*places*- biochemical entities (passive system elements);*transitions*- biochemical reactions (active system elements);*arcs*- go from substrates to reactions, and from reactions to their products;*arc weights*- stoichiometries.

Places host *tokens* representing movable quantities. One token on a specific place could stand for, e.g., one molecule, or a certain amount of substance of the biochemical entity modelled by this place.

Tokens move through the net by the firing (occurrence) of enabled transitions. The *firing rule* follows the well-known interpretation of stoichiometric equations.

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### Stochastic Petri Nets (SPN)

associate firing rates to transitions of the underlying qualitative Petri net and can be seen as a high-level description of a Continuous-Time Markov Chain (CTMC). The sojourn time in a state follows a negative exponential distribution, which depends on the firing rates, which in turn may be state-dependent.

### Continuous Petri Nets (CPN)

replace the discrete, stochastic behaviour by a continuous, deterministic one: a firing transition continuously removes substance from its pre-places and continuously adds substance to its post-places. The strength of flow is determined by the typically state-dependent firing rate - the deterministic counterpart to the stochastic rate. A CPN can be read as a structured specification of a system of Ordinary Differential Equations (ODEs).

**One and the same quantitative (kinetic) model can be read either stochastically or continuously**, with no changes required, besides some scaling of kinetic constants for higher order reactions.

### Coloured Petri Nets

are a form of high-level nets and a well-established modelling formalism, which allow the description of similar network structures in a concise way.

Tokens can be distinguished via their colours. This allows for the discrimination of species (molecules, metabolites, proteins, secondary substances, genes, etc.). In addition, colours can be used to distinguish between sub-populations of a species in different locations (cytosol, nucleus and so on).

- Each place gets a
*colour set*, specifying the kind of tokens which can reside on the place. - Each transition gets a
*guard*, specifying which coloured tokens are required for firing. - Each arc gets an
*arc inscription*specifying the colour of tokens flowing through it.

The colouring principle can be equally applied to qualitative, stochastic and continuous Petri nets. We denote the coloured counterparts by QPN^{c}, SPN^{c}, and CPN^{c}.

### Folding and Unfolding

Coloured Petri nets with finite colour sets can be automatically unfolded into uncoloured Petri nets, which then allows the application of all of the existing powerful standard Petri net analysis techniques. Vice versa, uncoloured Petri nets can be folded into coloured Petri nets, if partitions of the place and transition sets are given. These partitions of the uncoloured net define the colour sets of the coloured net.

The conversion between uncoloured and coloured Petri nets changes the style of representation, but does not change the actual net structure of the underlying reaction network.

*latest update: December 12, 2011, at 11:06 AM*