Petri net modelling of biological networks

1. Petri net modelling of biological networks Presenter : Chi-Yun Cheng

2. Outline • Introduction • Petri Net Basics • Petri Net Extensions

3. Introduction • In recent years, the biological data is huger and more complex. It is a hard work to analyze these interaction networks. • Petri Net (PN) is proposed a graphical and mathematical formalism, it can be suitable for the modelling and the analysis of concurrent, asynchronous, distributed system. • With various extensions during these years, a large amount of work has been done on theoretical developments and PNs have been successfully applied to a wide range of applications.

4. Outline • Introduction • Petri Net Basics • Petri Net Extensions

5. Petri Net Basic • PN is a Directed-bipartite graph with two different types of nodes: Places and transitions. • Places represent resources of the system. • Transitions correspond to events that can change the state of the resources. • Weighted arcs connect places and transitions to depict the relations between resources and events. • A place is connected to a transition. • A transition is connected to a place.

6. Petri Net Basic • At any time of the evolution of PN , the places hold 0 or a positive number of tokens. • The definition of a PN includes the specification of an initial marking, which allocates a number of tokens to each place. • A transition is enabled if its input contain at least the required number of tokens(the weight of the arc). • The firing of an enabled transition will consume the tokens of its input places and produce the tokens of its output places.

7. An PN Example • The three figure illustrate the firing process from the initial marking enabling the firing of transition t1, which in turn enables the firing t2.

8. An PN Example • The algebraic description of the PN :

9. An PN Example • Given the initial marking, the dynamical behaviour can be described by the marking graph.(Above shows the reachability of M0 )

10. Proporties of PN • For now, some typical properties are described with their possible interpretations in the context of biological networks: • Boundedness : means that whatever the initial marking and the evolution of the net, the number of tokens in each place in bounded. • P-invariants : P-invariants are sets of places for which the weighted sum of tokens is constant independently of the sequence firing.

11. Proporties of PN • T-invariants : T-invariants are firing sequences, which reproduce a marking. In biological terms, it may represents cyclic behaviors. • Reachability : It describes the evolution of a initial marking(M). This property may be relevant for biological networks, it ensures the existence of a path from initial state to a desired state. • Liveness : It insures that it is always possible to fire any transition ultimately. In other words, it guarantees that an event can eventually occur.

12. Outline • Introduction • Petri Net Basics • Petri Net Extensions

13. Extensions and Purposes • Colored Petri Net (CPN) • It assign data values to the tokens(defining color sets), and expressions are attached to the arcs, these define the constraints on the token values. • With CPNs, the complex systems can be reduced. • Hybrid Petri Net (HPN) • It allows the coexistence of both continuous and discrete process, they include discrete places and continuous places associated with real variables. • Discrete transitions fire after a determined delay, while enabled continuous transitions fire continuously at a given rate.

14. Extensions and Purposes • PNs used in Biochemical Networks • In 1997, Reddy et al. have shown that standard PNs allow the representation of the essential components in biochemical pathways. • Metabolic network can be analysed by drawing extensive relationships between traditional biochemical models and PNs. • Using CPNs to simulate enzymatic reaction chains. • ... more purposes will be detected…

15. Extensions and Purposes • PN modelling of different basic reactions: • Synthesis A + B → AB • Decomposition AB → A + B • Catalyzed reaction A + B + E → AB + E • Inhibited reaction A + B → AB • Reversible reaction A + 2B → C C → A + 2B I=0

16. Extensions and Purposes • Other purposes • Deriving a standard PN model from a Boolean regulatory network. • Defining integrated models of regulated biochemical pathways. • There was also proposed a method to determine relevant transition delays to derive a timed PN model used in signalling networks.

17. Thank you