Arcs are allowed only between places and transitions and vice versa, but never between places or between transitions. The arc labels multiplicities, weights correspond to the stoichiometric numbers in the reaction equations. Fluxes are modelled as token flows through the firing of transitions. A transition is enabled to fire, if the token amount of all pre-places satisfies the token amount required by the labels of the arcs connecting these pre-places with the considered transition.
When the reaction takes place the transition fires , the multiplicity of a given arc indicates the token amount, which has to flow through the arc from a pre-place to the transition or from the transition to a post-place, respectively. The distribution of tokens over the places represents the marking of the net and characterizes a certain state of the system, i. The initial marking is the marking of the start state for any animation or marking-dependent analysis.
For more detailed explanations see Heiner et al.
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Model validation aims at checking the constructed model for system inconsistencies and deriving statements on structural and dynamic system properties, which reflect correctly the behaviour of the system in reality. In the following we introduce those model properties in terms of Petri net terminology, which are expected to be meaningful for metabolic network analysis.
Afterwards, they are applied to our case study. The three orthogonal dynamic properties of a Petri net are liveness, reversibility and boundedness. An important expected property of a MPN is the infinite net behaviour in the sense that no part of the network will ever stop working, if a sufficient amount of input substrates enters the net.
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An unintended interruption of the substance fluxes is likely to indicate a modelling error. In order to validate the net we check it for dead transitions. A transition is dead at a given system state, if no state can be reached anymore, where the transition is enabled. A transition is live at a given state, if no state can be reached, where the transition is dead. A net is live, if all its transitions are live in the initial marking. This includes that there is no dead state reachable, where no transition at all is enabled, i. A net is reversible, if the initial system state can be reached again from each reachable state.
This excludes obviously irreversible system behaviour like unexpected burning out or accumulation of certain compounds. Let us consider boundedness, which is essential for any steady state network analysis. A net is bounded, if there exists a positive integer number, k , which represents an upper bound for the number of tokens on each place in all states of the net.
If the net is bounded in every initial marking, it is said to be structurally bounded. If a net is bounded, the number of reachable states is finite. Let us turn to analysis techniques to decide dynamic properties. Basically, three different ways of net analyses can be distinguished, first, the structural analysis, second, the invariant analysis and third, the reachability analysis. Structural analysis aims at discovering certain net structures that allow conclusions on dynamic properties.
It starts with the calculation of elementary properties, as e. The first three properties give information on stoichiometric relations of biochemical networks. A net is ordinary, if all arc weights, i. A net is homogenous, if, for any place, all arcs starting at this place have the same weights. These two properties hold typically in signal transduction pathways. A net is conservative, if for each transition the sum of input arc weights is equal to the sum of output arc weights.
Application of Petri net based analysis techniques to signal transduction pathways
Therefore, all reactions work in a token-preserving way. If a net is conservative, it is bounded. A net is pure, if no transition exists, for which a pre-place is also a post-place. MPNs are pure as long as the reaction enzymes are not modelled explicitly. A net is static conflict-free, if there are no transitions sharing a pre-place.
A MPN is usually not free of static conflicts, because compounds may be used by several reactions. Static conflicts may result in dynamic conflicts, establishing alternative system behaviour, if the tokens in shared places are not sufficient for the concurrent firing of all their enabled post-transitions. A network representing a biochemical metabolism should be connected, but must not be strongly connected. Other interesting structural properties are structural deadlocks and traps.
A set of places, which cannot get a new token as soon as it has lost all its tokens, is called a structural deadlock. In this case, every transition, which puts tokens into the place set, has also a pre-place in this set. Therefore, an insufficient marking of this place set makes all its post-transitions to dead ones. In MPNs, structural deadlocks indicate sets of compounds requiring a sufficient initial amount of tokens to make the net live.
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In biological context it means that the modelled metabolic network cannot run successfully without some metabolites being available at certain concentrations. The opposite situation that not all tokens can be removed from a set of places, is called a trap. This occurs, if every transition, which subtracts tokens from this place set, has also a post-place in this set, and thus returns always at least one token to this set. In biochemical systems, traps can represent sets of compounds for metabolite storage.
The invariant analysis was introduced by Lautenbach and comprises the structural computation of dynamic net properties. Usually, an invariant is a property, which is valid in all states while the system is working. A transition-invariant T-invariant is a multi- set of transitions, the firing of which reproduces a given system state. In biological context, a T-invariant represents a set of chemical reactions reproducing a given distribution of chemical compounds, e. A T-invariant corresponds to a possible pathway through the network, which is defined by the net representation of the T-invariant, i.
If there is a T-invariant, which covers all transitions, then all reactions may contribute to some pathway. Please note, we do not use the graph-theoretical term path for the interpretation of T-invariants, but the term pathway.
A path in graph-theoretical sense is defined as a sequence of consecutive arcs through a given graph. In contrast, a pathway can exhibit branches describing alternatives and concurrencies in the corresponding subnet representation. A pathway can degenerate to a path. A place-invariant P-invariant is a set of places, where the weighted sum of tokens remains constant while the net is working. P-invariants are structural deadlocks as well as traps and represent conservation relationships. Hence, if there is a P-invariant, which covers all places, then the net is bounded.
To compute net invariants we need the incidence matrix of the net, which corresponds to the stoichiometric matrix of the reaction system.
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The matrix elements contain the change of the token amount on a place, when the corresponding transition fires Fig. Please note, a non-pure net structure is not uniquely defined by its incidence matrix. Therefore, the results do not depend on the initial marking. Owing to the meaning of a token flow, only those integer solutions are of interest, where all numbers of a solution vector are positive integers. These generally infinite sets of positive integer solutions can be obtained by calculating the uniquely defined basis of the solution space.
Such a basis consists of the so-called minimal invariants, which are pairwise linearly independent from each other, and the greatest common divisor of all elements of a solution vector is one. Minimal T-invariants correspond basically to the elementary modes.
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Each T-invariant, i. In the following we consider only minimal invariants, which are therefore called invariants for short. The reachability analysis is based on the construction of the reachability graph or just the set of reachable states of the net. The result depends generally on the initial marking. The nodes of a reachability graph represent all possible states of the net. Thus, it can be finite only for bounded nets. The arcs connecting nodes of a reachability graph are labelled by single firing transitions and give altogether a graph representation of all possible single-step firing sequences.
Consequently, concurrent behaviour is described by enumerating all interleaving firing sequences. Realizable T-invariants, where the current marking actually allows the firing of the transitions of the T-invariant, correspond to cycles in the reachability graph, possibly to a cycle of length one a firing transition without marking change. Realizable T-invariants with inherent concurrency correspond to several cycles in the reachability graph, one for each interleaving sequence. Based on the reachability graph, many detailed dynamic properties can be decided, e.
Summarily, there exist various algorithms for reachability analysis, using different principles and data structures. Models of biochemical systems are often unbounded in the first place. To decide unboundedness, an always finite reduced version of the reachability graph exists, the so-called coverability graph. It is generally possible to transform an unbounded system into a bounded one, still reflecting the steady state behaviour, by adding artificial nodes and arcs, which remove the output substances and convert them into input substances in correspondence with the total equation of the network.
But often, even if the system is bounded, the large and complex biochemical networks exhibit too many states to be represented explicitly, or even symbolically. To date there exists no method in systems biology to compute all reachable states in really complex networks. The complexity of any algorithm to construct explicitly the reachability or coverability graph is known to be not primitive recursive.
Even more sophisticated representation techniques, like symbolic state space representations, are of exponential complexity in the worst case. Hence, for smaller networks, especially for signal transduction pathways, all system states might be computable. In our example, like in other metabolic networks, the high stoichiometric numbers e. About this book Driven by the request for increased productivity, flexibility, and competitiveness, modern civilization increasingly has created high-performance discrete event dynamic systems DEDSs.
Show all. Petri Nets Pages Wang, Jiacun. Show next xx. Read this book on SpringerLink. Recommended for you. PAGE 1. A net structure is called safe if there are no contact situations in its marking class. A net structure is called secure if there are no transjunctions in its marking class. A transjunction is a conjunction of conditions across a transition: an input condition is marked and also an output condition. Tokens denoting electromagnetic signals might interfere. Some nets allow two or more independent processes kept apart by a mechanism of mutual exclusion.
This mechanism ensures that only one of the processes can enter a critical phase on each repetition. The working of the mechanism is described below: the mutual exclusion is embodied by a dead transition which can never occur. Dead transitions are called facts because they represent factual statements about the marking class.
The schedule on the left can be misused by a fast competitor through usurping the system all for himself. To ensure a fair competition, the cycles must be linked by a buffer. A buffer of capacity 1 would enforce alternating use; in practice, larger buffers allow more freedom. The example above Figure 6 indicates the general method to define all valid logical statements about the marking class by a set of facts. This highly abstract and laborious procedure, which Petri performed in cooperation with Konrad Zuse, is as follows:.
The background of this table is an Integer Minkowski Grid, implemented between a pulsating light source and a mirror. The yellow patches are cut out and their opposite sides are glued together to form a torus on which certain tokens particles oscillate periodically in specific modes Figure 7. LT-compatible means that the Lorentz transformation can be applied literally to the patterns, while this is not immediately possible for the degenerate patterns.
It turned out, to the surprise of the inventors, that those periodical patterns of movement corresponded one-to-one to a complete set of loss-free computing primitives by changing the orientation of arrows. Moreover, loss-free means that they have exact reverses each is even its own reverse! The detailed procedure for constructing the Quine Transfer which performs the conditional interchange of two bits are shown in Figure 9.
There exist systematic mappings to generate information flow graphs from basic nets Figure 9.
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