In the graphical notation each place is represented ascircleand each transition asrectangle and from each and every place pi has exact Fi,j arcs lead to transistion tj. Thus, from transitiontjhas exactBi,jarrows lead to placepi.
Bis also calledbackward matrixand consequentlyFis calledforward matrix. From the view of the places: Arrows starting from a place, means forward directed, so they are writ-ten in the forward matrix. On the other hand looking back from a place, means looking along the arc to a place, the places are written in the backward matrix.
Definition 3.2 Let (P, T,F,B) be a Petri net. ThusF : P ×T ∪T ×P → Nis the arc function, defined as∀x, y ∈P ∪T:
F(x, y) :=
Fi,j, ifx=piandy=tj Bi,j, ifx=tj andy=pi
Forx∈P ∪T we call•x:={y∈P ∪T|F(y, x)≥1}pre-set ofx, x•:={y∈P ∪T|F(x, y)≥1}post-set ofx.
Analogue forX ⊆P ∪T :•X :=S
x∈X•x,X•:=S
x∈Xx•.
Typically, Petri nets are also defined as (P, T, F) nets. This illustration is in principle equivalent to definition 3.1. If a Petri net does not contain multiple arcs, thenF can be considered as a subset ofP ×T ∪T ×P. Ingeneralized Petri netsare explicitly multiple arcs allowed.
Definition 3.3 LetN = (P, T,F,B)respectivelyN = (P, T, F)be a Petri net. The state space of those Petri nets isNP. A mappings:P →Nis calledstateormarking. Ifs(pi) = k we can say that placepihasktoken in states. We can also write states as column vector.
Graphically token are marked as dots in a place.
The dynamic behavior of a Petri net is expressed by changing markings. A marking changes when a transition fires, and a transition may fire when it is enabled.
Definition 3.4 LetN = (P, T,F,B),s ∈NP a state ofN ant ∈T a transition ofN. We callt s-enabled (transition may fire in states) if
s≥F(t)
(i.e.∀p∈P :s(p)≥Fp,t=F(p, t)) Continuing,tfires from statesto states0, if
? s≥F(t)
? s0 =s−F(t) +B(t)
State changes are carried out by firing enabled transitions. In an ordinary Petri net a transition is enabled when all its input places have at least one token. When an enabled transitiontis fired, tokens are consumed by each input place oftand tokens are placed at each output place oft. Finally, it gives a new state.
3.4.2 Petri Net Models for Biological Networks
During the 90s, when the field of bioinformatics was getting stronger, modeling and sim-ulation of metabolic pathways became a fundamental research topic. The first Petri net application for metabolic simulation was published by R[RML93] and H
[Hof94]. They showed that formalism of Petri nets is useful to model simple metabolic networks. The first papers already illustrated that in addition to the possibility of mod-eling parallel processes. One important advantage of this formalism is that the Petri net simulation can start with a discrete model. Moreover, these discrete models can be ex-tended to a quantitative model step-by-step and at any point in time [HT98].
Consequently, Petri nets are useful for the representation of biochemical reactions and metabolic pathways. Figure 3.12 shows a model of an abstract enzyme-controlled reac-tion, which is the basic element of metabolic pathways. First of all the paper of R
[RML93] addressed the modeling of metabolic pathways as Petri nets. The disadvantage of this application is that kinetic effects cannot be included in the model. Based on this paper the work of H[Hof94] showed that a Petri net can also model gene-controlled metabolic networks and cell communication processes. The idea of Hand T
- [HT98] was to include the kinetic effects. Therefore, they introduce the functional Petri net (FPN), which allows the kinetic simulation of metabolic networks by placing specific functions to the arcs. Kinetic effects of biological networks can be simulated can be simulated by using FPNs. Thus, any qualitative Petri net model of a biological network can be extended by using the FPN. Of course quantitative experimental data (for instance quantitative proteomic data or kinetic data) can be included. In general, extensions of more complexity lead to high-level Petri-Nets. These high-level nets include following type of nets.
? Hierarchical Petri nets: Allow a place or a transition being a place-holder for a pre-defined net.
? Coloured Petri nets: Allow different kinds of tokens in one net. The tokens can have a complex internal structure. Accordingly, transitions can have more complex firing rules.
? Stochastic Petri nets: Places and transitions may be assigned with a probability dis-tribution.
? Timed Petri nets: Assign a delay time to each transition.
(a) Start configuration of the enzymatic reaction shows an abstract glucose molecule, an abstract enzyme hexokinase and energy (ATP).
(b) Final condition of the Petri net shows the re-sult of the biochemical reaction.
Figure 3.12: The enzyme-catalyze process of glucose into glucose-6-phosphate.)
? Continuous Petri nets: They do not have tokens, but introduce the concept of con-tinuous flow. Using this concept, it is no longer a natural number that is assigned to a place, but a real number. Transitions can be equipped with a complex mathemat-ical function implying actual assignment of places. Hence, continuous Petri nets are expedient for modeling of metabolic pathways with regard to the concentration flow.
? Hybrid Petri nets: A mixture of ordinary (discrete) and continuous Petri nets. Places and transitions can be either discrete or continuous so that they deal with natural numbers (tokens) or a continuous flow. This type of high-level Petri net extends the continuous net with the useful ability to model logical coherences.
Other publications published by M[MDNM00] showed that higher Petri nets, in particular hybrid Petri nets, present the formalism which is useful to model and simulate complex biological networks. The motivation to define the hybrid functional Petri net (HFPN) was to allow the quantitative modeling of biological networks, which is still one of the major tasks of Systems Biology today. Having more and more quantitative molecular data the realistic simulation of biological networks required more extensions. The usage of real numbers is one important aspect. The HFPN is an extension of theHybrid Petri net(HPN) [AD98] and hybrid dynamic net (HDN) [Dra98]. Additionally, HFPN has the feature of the functional Petri nets and allows assigning a function of values to any arc.
Based on these ideas several simulation tools were created and implemented, such as the Cell Illustrator.
3.4.3 Cell Illustrator
The Cell Illustrator1is a promising and widely used tool for biological network modeling and simulation. A first version of the Cell Illustrator (CI) is available since mid 2003 and has been evolved fromGenome Object Viewer[NDMM05]. Since this version CI has been developed to a standard software for modeling and simulation of pathways.
Figure 3.13: Visualization of the apoptosis pathway with the Cell Illustrator.
The CI employs the HFPN (described in section 3.4.2) as a basic architecture. CSML (Cell System Markup Language) as description language was developed for the CI. The goal of CSML 3.0 is to create a usable XML format for visualization, modeling and simulation of biological pathways that are based on HFPNs. Moreover, CSML is able to store biological
1http://www.cellillustrator.com/
images, result sets of simulation, as well as biological annotations and therefore it is more powerful than CellML or SBML. Both formats, SBML and CellML are widely used and many models are already available, so it is possible to import via conversion tools these formats into the CI [NDMM05].
However, the CI enables scientist to draw, edit, model and simulate complex biological systems and networks. Regarding to the HFPNs it is possible to create pathways of dis-crete, continuous and generic entities and transitions, as well as typical, inhibitory and
“test” arcs. A “test arc” can be used like typical arcs. In contrast a “test arc” does not consume any tokens of the place at the source of the arc by firing. An inhibitory arc can be used to represent the function of “repress” in gene regulation. A graphical representa-tion of all elements in a HFPN is shown in figure 3.14. For each and every element in the pathway the CI provides miscellaneous properties. Starting from labeling, initial concen-tration, shape and color, up to biological functions, database identifier and images every setting is possible. Figure 3.13 shows an example network of CI tool. Biological networks modeled with the Cell Illustrator can be visualized and animated using Cell Animator™.
Additionally, the CI since version 3.0 includes a library of SVG graphical elements, en-hancements in the Petri net simulation engine, improved charts, external references, and gene network view. In summary, Cell Illustrator provides biologists, biochemists, and sci-entists with comprehensive visual representations of biochemical processes which form large and complex networks.
Figure 3.14: Graphical representation of HFPN components.