Pain Signaling - A Case Study of the Modular Petri Net Modeling Concept with Prospect to a Protein-Oriented Modeling Platform

Abstract

The construction of monolithic pathway models, as well as their coupling, curation and the integration of new data is arduous and inconvenient. The modular Petri net modeling concept we present here shows one way to manage these di culties. In our concept, proteins are represented as functional units by Petri net submodels with a de ned structure and connection interface, called modules. Each module integrates all publicly available information about its intramolecular changes and interactions with other molecules. Hence, a module corresponds to an interactive review written in a formalized language. This allows to intuitively understand the functionality of a protein. Modules of interacting proteins communicate through matching subnets, which renders the automatic generation of molecular networks possible. Here, we demonstrate the applicability and advantages of our concept on pain signaling. The molecular mechanisms involved in pain signaling are complex and poorly understood. To enhance our understanding of the mechanisms and to get an impression of the functional interactions among the involved pathways, we systematically build a model from modules of pain-relevant proteins. We also o er a prospect of a platform to organize approved curated modules in order to generate molecular networks. Hopefully, our concept helps bridging the gap between experimental bioscientists and theoretically oriented systems biologists.

Keywords

• Petri net
• modular approach
• pain signaling
• molecular networks

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The modeling of large molecular networks in systems biology is a challenging process, as well as their steady curation and improvement. Our modular Petri net modeling concept described here, o ers a way to handle these challenges by combing Petri net modeling with a modular approach. Modular approaches have already conquered the eld of systems biology [ 1 ]. In the biological context, just monolithic pathways have been regarded as single entities up to now. In our concept, proteins are represented as functional units by a Petri net with a de ned structure and connection interface, called module. Each module comprises and integrates scattered information about individual proteins, its intramolecular changes and interactions with other molecules. Therefore, a module is equivalent to an interactive review article written in a formalized language with the help of Petri nets. The graphical notation of the underlying mathematical model allows to intuitively understand the modeled protein. Modules of interacting proteins communicate through identical matching subnets, the connection interface. The coupling of monolithic pathway models is far from trivial in contrast to the combination of protein modules described here. The de ned connection interface of each module allows to easily generate a comprehensive model of a molecular network from a set of modules. However, it has been proven that Petri nets are ideally suited to describe biological processes by their very nature. The Petri net formalism provides a mathematical language to describe parallel and concurrent processes of bipartite systems [ 2 ]. Therefore, we choose Petri nets to describe the molecular network of pain signaling (case study). Pain signaling comprises complex and diverse molecular mechanisms of parallel, convergent and concurrent processes. Up to now, a large variety of molecular pain mediators is known Figure 1. Nevertheless, especially the intracellular plethora of pain signaling cascades triggered by membrane components is underinvestigated and therefore partly unknown [ 3 ]. The challenge to represent pain signaling in terms of a comprehensive Petri net model has led to the development of our modular Petri net modeling concept and a modular network describing pain signaling in the peripheral terminals of dorsal root ganglion (DRG) neurons Figure 1. Our concept is not at all limited to pain signaling, the application to other molecular networks is straightforward. However, by studying pain signaling it turned out that our concept is well suited to handle large molecular networks. Since last year, we improved and extended our modular Petri net modeling concept presented here (compare [ 4 ]). In addition, we developed rst ideas to manage the modules after curation by bioscientists in a database and thus, provide a platform to the scienti c community. The platform also facilitates the automatic generation of a model of a molecular network from a collection of approved curated modules. 2

Petri Net Formalism

Petri nets o er a mathematical modeling language to describe concurrent and parallel processes, as well as communication and synchronization in bipartite systems. The graphical notation and construction of Petri nets allows to intuitively model such processes while being formally and mathematically consistent. Therefore, Petri nets are ideally suited to describe biological processes by their very nature [ 2 ]. The standard Petri net [ 2 ] consists of four elements: places, transitions, arcs and tokens. In biological systems places correspond to species (chemical compounds) and transitions describe the action occurring among the species ((bio-)chemical reactions). Arcs specify the relations between places and transitions. Tokens refer to the amount (discrete number, concentration) of a species. Further, transitions are allowed to re (enabled) if all pre-places are su ciently marked. By ring of transition it deletes tokens from its pre-places and produces tokens on its post-places.

De nition 1 (Petri net). 1 A Petri net is a quadruple N = (P, T, f, m0), where: { P, T are nite, non empty, disjoint sets. P is the set of places. T is the set of transitions. { f: ((P T) [ (T P) ! N0 de nes the set of directed arcs, weighted by non-negative integer values { m0: P ! N0 gives the initial marking.

One bene t of Petri nets is the formal analysis of the network structure. The topological properties of a Petri net are also meaningful in a biological context and give valuable hints to validate the network structure [ 2 ]. Important criteria to validate a biological Petri net are liveness, boundedness, reversibility, as well as T- and P-Invariants [ 2 ].

De nition 2 (Boundedness). 1 { A place p is k-bounded if there exists a positive integer number k, which represents an upper bound for the number of tokens on this place in all reachable markings of the Petri net: 9k 2 N0 : 8m 2 [m0i : m (p) k. { A Petri net is k-bounded if all its places are k-bounded.

{ A Petri net is structurally bounded if it is bounded in any initial marking. De nition 3 (Liveness). 1 { A transition t is dead in the marking m if it is not enabled in any marking m0 reachable from: @m0 2 [mi : m0(t). { A transition t is live if it is not dead in any marking reachable from m0. { A marking m is dead if there is no transitions which is enabled in m. { A Petri net is deadlock-free if there are no reachable dead markings. { A Petri net is live if each transitions is live.

De nition 4 (Reversibility). 1 A Petri net is reversible if the initial marking can be reached again from each reachable marking: 8m 2 [m0i : m0 2 [mi. De nition 5 (P-invariants, T-invariants). 1 { The incidence matrix of N is a matrix C : P T ! Z, indexed by P and T, such that C(p; t) = f (t; p) f (p t). { A place vector (transition vector) is a vector x : P ! Z, indexed by P (y : T ! Z, indexed by T) { A place vector (transition vector) is called P-invariant (T-invariant) if it is a nontrivial nonnegative integer solution of the linear equation system x C = 0 (C y = 0). { The set of nodes corresponding to an invariant's nonzero entries are called the support of this invariant x, written as supp(x). { An invariant x is called minimal if @ invariant z: supp(z) supp(x), i.e. its support does not contain the support of any other invariant z, and the greatest common divisor of all nonzero entries of x is 1. { A net is covered by P-Invariants (T-invariants) if every place (transition) belongs to a P-invariant (T-invariant).

Thereby, we can determine if the model of the molecular network contains deadstates, is live or if it can reset its initial state (reversible). To ensure the mass conversation the coverage by P-invariants and the boundedness of the Petri net must be considered. P-invariants describe sets of related species or states of a species. Boundedness ascertains that no species in nitely accumulates in the network. T-invariants contain a set of actions/reactions to reset its initial state [ 2 ].

Several specialized Petri net classes like qualitative, stochastic, continuous, hybrid Petri nets and their colored counterparts are available to describe di erent scenarios and to consider di erent simulative approaches. All network classes are convertible into each other without changing the network structure. This allows the application of the same powerful analysis techniques to the underlying qualitative structure for all Petri net network classes [ 2 ].

In particular, we use stochastic Petri nets to describe the inherently stochastic nature of biological processes. In addition to the standard Petri net, ring rates are assigned to each transition, which are determined by random variables depending on the probability distribution. Therefore, we use stochastic simulation to investigate the dynamic behavior by the time-dependent token ow [ 2 ]. De nition 6 (Stochastic Petri Net). 1 A biochemically interpreted stochastic Petri net is a quintuple SPNBio = (P, T, f, v, m0), where: { P, T are nite, non empty, disjoint sets. P is the set of places. T is the set of transitions. { f: ((P T) [ (T P) ! N0 de nes the set of directed arcs, weighted by non-negative integer values { v: T ! H is a function, which assigns a stochastic hazard function ht to each transition t, whereby H := St2T n htj ht : Nj0 tj ! N+o is the set of all stochastic hazard functions, and v(t) = ht for all transitions t 2 T.

{ m0: P ! N0 gives the initial marking. 3

Modular Modeling Concept

The enormous amount of regulative events in pain signaling Figure 1 results into the development of a modular modeling concept, which considers every protein as functional independent unit. The basic concept presented last year (see reference [ 4 ]) has now been improved and extended to a de ned modeling concept. The suggested modular modeling concept uses Petri nets to allow the assembling of molecular networks from functional Petri net submodels of the involved proteins with a de ned structure and connection interface, called modules. A module re ects all the intramolecular changes of a protein and its interactions with other molecules as reported in the literature. Therefore, a module comprises wide-spread information about each protein. Figure 3 and 4 show exemplary the modules of two proteins and their regulation. Non-proteins (ions, second messenger, energy equivalents etc.) are contained in the modules as interactants and indirectly connect the proteins. The structure and the dynamic behavior of each module has to ful ll certain criteria to be valid to meet the requirements of our modular Petri net modeling concept Table 1. After positive validation, the modules can be easily linked to a modular network by the de ned connection interface of each module. Additionally, we are now able to predict the properties of the complex modular network from the topological properties of the combined modules Table 1. In this section, we explain all steps needed for the construction of a single module from literature and the assembling of the modular network from the constructed modules. 3.1

Network Structure and Properties of a Module

We construct modules based on the information about the structure of a protein, interactions with other components and intramolecular changes during its regulation given in the literature. Each place of a module corresponds to a speci c state of a functional protein domain (phosphorylation site, catalytic and 1 Mathematical de nitions are taken from [ 2 ]. inhibitory domain etc.) or a speci c state of a non-protein (free or bound, substrate or product etc.). In this context, a transition describes a shift between two di erent states of a protein domain or non-protein by a molecular action (binding/dissociation, (de-)phosphorylation, conformational changes, substrate processing etc.). Each module is constructed in such a way, that it obeys criteria important for biological networks, which are given in detail in [ 2 ] and summarized in Table 1. All states of a protein domain and all states of a non-protein constitute a P-invariant (see also Section 2). Therefore, the whole module must be covered by P-invariants. In this context, P-invariants ensure mass conservation. Sequential state shifts that restore an initial state of a protein domain or a non-protein form T-invariants (see also Section 2). Consequently, all Tinvariants are covered by P-invariants. Places connected with a transition by a double arc (see Figure 3 and 4) indicate molecular states responsible for other state shifts without changing itself. To prohibit external sinks and sources of protein domains, the modules are not bounded by transitions. However, a module might well be bounded by places. Boundary places have their origin in modules of other proteins or represent non-proteins, which are consumed or produced. The principle of double entry-bookkeeping is a part of the modular modeling concept, since every module has to contain all interactions with other molecules. Thus, modules of two interacting proteins share identical matching subnets describing the interaction mechanism.

Properties Module Modular network

A Properties that must be ful lled for each module

Ordinary Just natural elementary regulation steps are considered. Therefore, the arc weights are uniformly set to "1".

Homogeneous Due to ordinary: state-shifts produce (consume) the same amount of tokens on each place.

Connected Among all di erent states of protein domains

Covered with aapAnaptdsrheotntteoooifnnr-r.epeprlraoettseeedinntssttaehtxeeissitnotfaetrareplelraaotstetedinosntderuomcintaudirinreeocortf ftnAurelaltlnwplslorfeerodkrpa,bebbrylteeiaceltaslouamstrehoeedtdhumielreyeoscd.aturlear

P-invariants of a non-protein must form a P-invariant. In

consequence, the module must be covered with

P-invariants.

Boundedness The coverage with P-invariants causes boundedness of each module and avoids the in nite accumulation of tokens in a module.

B Properties that must not be ful lled for each module

Pure Every module contains states of a protein domain responsible for other state shifts without changing itself. Therefore, each module contains double arcs.

Boundary The modules are not bounded by transitions

Transitions to avoid external sinks and sources of protein domains and non-proteins.

Conservative The formation of protein complexes results in

a non-token-preservingly ring of transitions.

Static con ict A module contains at least one state of a profree tein domain or a non-protein attending on multiple state shifts.

Strongly cov- A module contains two sequential actions re

ered with producing the initial state of the involved pro

T-Invariants tein domain or non-protein.

C Properties that are variable among all modules

Dead Transi- Depending on the initial marking. tion

Dead states

All properties are direct transferable to the modular network, because they are ful lled by all modules.

Depending on the initial marking.

Depending on the speci c regulation of a pro

tein, the respective module contains at least The modular network has no one set of sequential state shifts acting inde- dead state if a least one pendent of all other actions. In this case, the module has no dead state.

module has no dead state.

Diyctnafrmeeic con- tunDmelaeieomnpd,deiuccnoleedcr.niotnoCangtionionicnsnsttehsatq.ithbueeietsnshtopiltefyhtc,seitrihnceatrcmhetgeioourdnleausstlpieionenhctatoihsvfeenaomsapdomrdyoe--- cdTmoyhonnedauimmcletoi.cdcouconlantrainincestt wafroedreyk,niiafsmonniocet

Boundary Depending on the speci c regulation of a pro

places tein, the respective module contains places corresponding to protein domains of other intertavhceitrsisnibgclaypserc,ohtatehninegsedmoroindnutolhnee-phrreaosstpeebincotsuivntehdaamtryoadrpuellaei.crreIesn-. tArlalnosffetrheedsetoprtohpeemrtoiedsuclaarn be The following properties cannot be ful lled if network, if at least one a module has at least one boundary place: nmeotdwuolrekihnatshaetmleoadsutloanre boundary place that does not gain a pretransition after module coupling. { Strongly connected { Non-blocking multiplicity { Covered with T-Invariants { Siphon-Trap Property { Liveness 3.2

Validation of a Module

The topological properties can be used for the validation of each module and locating inconsistencies in the module structure [ 2 ]. Each topological property has a signi cant biological interpretation. A set of those properties must be fullled, another set must not be ful lled and a third set is variable depending on the unique function and structure of each module Table 1 (see [ 2 ] for further explanations). After the construction of a module, its structure is checked whether it obeys the given criteria given in Table 1. Each module is also subjected to stochastic simulation studies. The kinetic function of each transition can be de ned by known kinetic parameters (binding, dissociations, and a nity constants) or more complex kinetic equations (michaelis menten, hill kinetic). If kinetic information are not available, the kinetic parameters can be determined by trial and error or by more sophisticated parameter estimation methods. The observed dynamic behavior, i.e. the time-dependent token- ow (see also references [ 4,5,6 ]), must in principle re ect the modeled e ector function. A module is valid if its structure con rms the given topological properties and the dynamic behavior re ects the experimental obtained time curves. 3.3

Generation of a Modular Network

Next, the modules are connected by identical places of non-proteins and identical matching subnets among the modules. All shared elements have to be indicated as such in each module by declaring the included transitions and places as logical nodes (connection interface). Afterwards, the modules can be combined to one comprehensive simulative model. No further interventions are required. The coupling procedure does not a ect the structure and properties of each module. Even the kinetic of the modules are kept and inherited to the resulting modular network. Hence, simulation with the modular network can be performed right after its generation. 3.4

Properties of the Modular Network

A new important achievement of the modular modeling concept is the determination of properties of the complex network from submodels, which might also be interesting for other Petri net applications. Due to the de ned structure, the resulting uniform topological properties and de ned connection interface of the modules, we are able to predict the properties of the modular network. Obviously, the respective non-variable properties among the modules can be transferred one-on-one to the modular network Table 1. From the comparison of the ful llment of each variable property among the modules it can be deduced, whether the respective property holds for the modular network (see also Table 1 for more details). Therefore, all properties of the modular network are derivable from the respective set of modules that assemble the modular network. Computational analyses with the place/transition analyzer Charlie [ 7 ] con rm the predicted properties (not shown here, see [ 6 ] for more details). 4

A Model Relevant for Pain Signaling

Pain signaling comprises complex and diverse processes. Therefore, a plethora of proteins and other components participate in the molecular regulation of painful sensations (see also Figure 1). Several members of the G-protein-coupled receptor family (GPCRs) are involved in pain signaling like opioid, cannabinoid, muscarinic, prostaglandine and -2-adrenergic receptors. The GPCRs act through their G-proteins on adenylyl cyclases (Type VIII, V, I), phospholipase C and ion-channels. Numerous protein kinases regulate pain signaling among them are di erent isoforms of PKA (RII ), PKC ( , , ), CaMK (II, IV). Opponents of the protein kinases are protein phosphatase 2A and calcineurin. Calmodulin is an important calcium-binding protein interacting with other pain-related protein like the voltage dependent calcium channels (CaV1.2, CaV1.3, CaV3.3) and the capsaicin receptor (TRPV1). Second messenger like DAG, Ca2+ and cAMP are also important components in the context of pain signaling and indirectly link proteins [ 3 ].

Each of those pain-relevant proteins is represented by its respective module. In total, 38 modules of pain-relevant proteins have been derived from clinical pain literature (all references can be found in [ 6 ]). New modules have been added and old modules have been updated by formulating the mechanisms in more detail since last year (e.g. compare the modules in Figure 3 and 4 with the respective module shown in [ 4 ]). Exemplary, we show the mechanisms of the Gi-protein activation by the -opioid receptor Figure 2. Both, the -opioid receptor and the Gi-protein are represented by functional connectable modules (see Figure 3 and 4). The biological meaning of each transition refering to the steps shown in Figure 2 are given in Table 2. The resulting modular network Figure 5 generated from the set of modules of pain-relevant proteins consists of 713 places and 775 transitions spread over 325 pages with a nesting depth of 4. The top level of the modular network shown in Figure 5 contains all non-proteins (logic places in grey) and modules of painrelevant proteins represented by single macro places (boxed circles). Figure 3 and 4 depict two of these modules exemplarily. All components are arranged by their localization in the nociceptor. Due to the complexity of the regulation events, the modules are hierarchically designed to conserve the neat-arrangement o ered by Petri nets. The modules communicate through identical matching subnets among them on lower levels and non-proteins. Therefore, the interaction among the displayed components are not visible on the top level of the modular network in Figure 5. Here, we added arcs (no Petri net element) to the top-level shown in Figure 5 to indicate the interactions among the components involved in pain signaling. Figure 5 illustrates the high degree of interactivity among the components which was not obvious from the literature. The authors of reference [ 3 ] discuss whether the pathways involved in pain signaling are parallel or convergent. The interaction scheme in Figure 5 clearly indicates that the involved pathways are highly convergent and in uence each other. Several feedback loops are contained (not shown here) to regulate the cAMP- and Ca2+- level and the membrane voltage, which are important for the sensitization of the nociceptor and the initiation of action potentials resulting into painful sensations. Therefore, the regulation of pain signaling is very complex and the resulting dynamic behavior is not trivial at all. The modules of the pain-relevant proteins are still in the curation process by the pain community. Since kinetic data is still rare in the pain signaling context, we have not been yet able to parameterize the modules and therefore to perform reliable simulation studies. The topological properties of the achieved modular network con rm the predicted properties of a common modular network given in Table 2. 5

Work in Progress: Establishing a Protein-Module

Platform

We developed rst basic ideas of a new protein-orientated modeling platform to open our modular Petri net modeling concept and the modules to the scienti c community. This platform and the modular Petri net modeling concept provide the framework to organize the connectable modules in a database and to generate computational models of molecular networks from a central collection of approved curated modules.

Additionally to the Petri net of each module, a dataset will be provided in our database to characterize each module and the represented protein. The dataset contains information about the author and curator, the names of all places and transitions and their meaning, references to relevant literature used for the construction of the module, a list of open issues and information about the protein (accession number, gene symbol, synonyms, taxonomic classi cation, involved pathways etc.) extracted from the UniProt database [ 8 ].

The strict obedience of a naming convention for each node is the most important prerequisite to correctly link the modules by identical matching subnets and logical places of non-proteins. The identity of each module is determined by the unique accession number for proteins provided by UniProt [ 8 ]. The accession number will be used as pre x for all nodes but is not shown in the graph. More readable gene symbols for each protein, which are also provided from UniProt [ 8 ], are used as nicknames combined with common abbreviations of domains and their states to de ne a unique name for each place. The unique name of a transition is created from the gene symbols of the involved proteins and a counter. As a consequence, the module coupling is insensitive against author and version of a module, but sensitive against the organism. Also, the name of each node can be automatically generated. Checks will be integrated to prove the correctness of the names and thereby the connectivity of the module to its interactants. To control the generation of a network from modules an interaction matrix is provided Figure 6. An interaction matrix, derived from the places of all modules, indicates which modules can be coupled by a common set of nodes. Thus, it controls the module coupling. First of all, the user adjusts the stringency at the organism level. Based on the interaction matrix, the generation of a modular network from modules might now occur in two ways: (a) pathway-oriented suggestion of a set interacting proteins, (b) iterative search of interactants from a chosen start protein. The pathway-oriented generation of modular networks will be achieved by tags that are added to the dataset of each module referring to involved pathways (e.g. pain signaling) and localization (e.g. DRG neuron, nociceptor). In a next step the connectivity of the modules has to be proven by the interaction matrix. Figure 7 illustrates the application of the iterative search algorithm to generate a submodel of the model shown in Figure 5. Both possibilities lead to a list of interacting proteins showing all suitable modules. The user chooses for every protein the preferred module, if di erent versions of a module for one protein exist. Based on the module selection a comprehensive modular network is generated. The user can now execute simulations with the model and apply advanced structural analysis methods to investigate the model of the molecular network. The model relevant for pain signaling integrates the knowledge of approximately 320 scienti c articles within 38 valid modules of important molecular pain actors in the nociceptor. The application of the modular modeling concept to the complex network of pain signaling proves its ability to cope with the speci c demands of large and complex molecular networks. Our experience with biologists con rm the need of a molecule-oriented modeling concept. So far, the explained modular modeling concept is appreciated by our cooperation partners. It supports their work and improves the modeling of molecular networks. The concept is suited to test di erent hypothesis by exchanging di erent versions of a module in the modular network. Thus, the approach shed new light on molecular mechanisms. In a next straightforward step, the modules and therewith the complete modular network can be parameterized by experimental data, which are at the moment unfortunately still rare. After parameterization the model will be investigated by in silico experiments. The analysis of e ects of systematic perturbation on the dynamic behavior of the model will help to pinpoint promising targets for the pain therapy. The extension of the model to colored Petri nets [ 9 ] enables the consideration of multiple copies of the pain-related proteins and of DRG neuron populations. The application of hybrid Petri nets [ 10 ] allows the combination of the current model with continuous models describing the generation of action potentials in neurons. Sophisticated structural analysis methods will be applied to the model to screen for non-obvious properties that are de ned by the Petri net structure. By this, we expect new information about multiple steady states, bifurcations and feedback loops that mainly determine the dynamic behavior of a network. At least, the validated and parameterized model and the mentioned investigations should contribute to the development of a mechanism-based pain therapy.

The modular modeling concept as such o ers a lot of promising advantages and opportunities. All constructed modules can be easily reused in any other biological systems. Hence, the extension of the modular modeling concept to other biological systems is worthwhile. Every module by itself pools the currently spread knowledge about a protein and its interactants. The process of translating information about a protein into a module reveals missing interrelations. The modular modeling concept avoids inconsistencies in the entire complex modular network by constructing and validating rst small independent submodels. The coupling procedure of the modules to an entire modular network by natural matching nodes is e ortless. For di erent reasons, monolithic pathway models organized for example in the BioModels database [ 11 ] cannot be easily updated and combined with each other to give a more comprehensive model. Advantageously, the properties of the modular network can be deduced from the de ned properties of the modules. The modeler and the curator just need to concentrate on one protein and its interactants. Also, the user has not inevitably to deal with the whole pathway and the theoretical concept itself.

The Petri net formalism itself o ers quite few advantages against ODE models. As mentioned before, qualitative, continuous, stochastic and hybrid Petri nets as well as their colored counterparts are convertible in to each other without changing the qualitative structure. ODE systems do not o er the possibility to consider a model from such a range of corresponding sites without reconstructing the set of equations. Due to the graphical visualization of molecular networks by Petri nets, a bioscientist can intuitively understand the modeled mechanisms. This does not count for the mathematical representation of ODE systems. In case of ODE systems, the user has to deal with three di erent representations of a molecular network which do not obviously correspond to each other: (a) structure of the biological network, (b) the mathematical equations and (c) the implementation of those. Besides, the transformation of ODE systems into Petri nets is not unique. Various Petri nets can be constructed based on an ODE system [ 12 ]. The compact mathematical structure of an ODE might hide important biological information. Structural analysis techniques are sensitive to the respective structure of a Petri net. Therefore, the application of those techniques to Petri nets obtained from the variable transformation of ODEs leads also to variable results, which have to be treated with care [ 12 ].

The modular principle of the modeling concept o ers some more possibilities to apply and extend the concept. The generated modular core network can be extended by gene expression, degradation and translocation modules. Even homoand hetero-multimeric protein complexes can be modeled in detail with an extension modular modeling concept. Further, we plan to couple the network reconstruction for Petri nets [ 13 ] with the modular modeling concept. Here, nodes of the reconstructed network can be matched with corresponding modules. In addition, the minimal causal Petri nets reconstructed from experimental time series are extended with modules of the involved proteins.

The establishment of a platform for protein-modules and the opportunity to generate models of molecular networks form approved curated modules supports the switch from monolithic modeling to modular modeling of biological systems. Such a platform simpli es the exchange of data and knowledge among bioscientists by concentrating biological information about proteins and their interactants in the structure of the modules. Thereby, easing the access to systems biology for wetlab bioscientist. 7

Acknowledgement

This work is supported by the "Modeling Pain Switches" (MOPS) program of Federal Ministry of Education and Research (Funding Number: 0315449F). We thank Monika Heiner and co-workers for the outstanding cooperation on Petri nets and the software supply of Snoopy [ 14 ] and Charlie [ 7 ]. Further we thank the members of the MOPS consortia for supporting collaborations. 8

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