A Deductive Compositional Approach to Petri Nets for Systems Biology Draft

A Deductive Compositional Approach to Petri Nets for Systems Biology Draft Ozan Kahramanoğulları ozank@doc.ic.ac.uk Centre for Integrative Systems Biology, Imperial College London Abstract. We introduce the language CP, a compositional language for place transition petri nets for the purpose of modelling signalling pathways in complex biological systems. We give the operational semantics of the language CP by means of a proof theoretical deductive system which extends multiplicative exponential linear logic with a self-dual non-commutative logical operator. This allows to express parallel and sequential composition of processes at the same syntactic level as in process algebra, and perform logical reasoning on these processes. We demonstrate the use of the language on a model of a signaling pathway for Fc receptor-mediated phagocytosis. 1 Introduction The view of biological systems as complex reactive systems is becoming more and more wide-spread. As new techniques emerge, their applications shed light to behavior and interrelations of biological components of complex biological systems. By means of abstractions from biological data, these symbolic computational methods allow to formally represent interactions and reactions of the components of a biological system. This makes it possible to build computational models for hypothesis generation and analysis. Symbolic computational methods, with different representation schemes, expressive capabilities, and tools, have been proposed by various authors. Stochastic π-calculus [12, 17, 18, 15] and PEPA are examples [9] to process algebra based approaches. Other approaches include the PRISM language [8], petri nets [16], and K-calculus [4]. Pathway logic [25] is another approach based on petri nets. The advantage of symbolic computational methods, in comparison to, e.g., methods based on differential equations (see, e.g., [5]), is their compositionality: In the context of modelling, from an analytic bottom-up point of view, compositionality is the idea of considering parts of a complex biological system in isolation, and this way building models and composing parts gradually to build more and more complex models. From a dual top-down point of view, compositionality allows to add more and more detail to components so that models can be considered at lower and lower levels of abstraction, while moving from greater parts of building blocks towards smaller components. An example for

this would be moving from protein complexes to proteins, then to amino-acids, and then to chemical compounds, etc. The combination of these two perspectives of modelling (top-down and bottom-up) then results in the so called middle-out approach to modelling. Petri nets [14], which was originally conceived as a language for studying concurrent information systems, provide a formal platform for representing the events in the cellular signalling pathways (see, e.g., [16]). Petri nets have a graphical representation which resembles conventional representations of biochemical networks. Thus, they can communicate the biological data in a way that is natural for biologists while remaining in formal grounds. Petri nets have been used to model signalling pathways and simple genetic networks (see, e.g., [13]). Petri nets are well suited for analyzing causality and independence of components which are involved in a signalling cascade. However, petri nets lack a broadly accepted formal compositional semantics which would allow to model and analyze biological systems compositionally. In this paper, we introduce a compositional language, called CP, for place transition petri nets with infinite place capacities (see Section 2). The language CP is equipped with sequential and parallel composition and a proof theoretical operational semantics. This way, concurrent signals in a biological pathway can be represented as in process algebra at the same syntactic level and logical deductive reasoning can be performed on these processes 1. Because biological signals have durations the synchronization is not performed by means of a hand-shake operation as in process algebra, but by means of common successors and predecessors of concurrent processes due to the causality relation between processes. The nondeterministic choice is embedded into the operational semantics of the underlying logic. As an example for an application of the language CP, through out the paper we consider the following signalling cascade which occurs during phagocytosis where cells engulf particles of greater size: During Fc receptor-mediated phagocytosis, the Rho GTP binding proteins Cdc42 and Rac are activated by means of a signalling cascade that results in two concurrent pathways of actin polymerization [23]. The Cdc42 signal results in a branching structure of actin, whereas Rac has a linear structure. These two processes act in concert and this way extend the cytoskeleton around the engulfed particle. These two processes are initiated by a common signalling pathway, however Cdc42 signal is more dominant at the early stages of actin polymerization, whereas Rac signal becomes more dominant at the later stages [23]. Thus, although their initiation is synchronized, because they have durations, this synchronization does not occur in a handshake-manner, but instead by means of their common successors and predecessors. The language CP is obtained by encoding the petri nets in a logic which extends multiplicative exponential linear logic with a self-dual non-commutative logical operator. The self-dual non-commutative operator, called seq, resembles the prefixing of the process algebra. This way, while parallel composition of processes is mapped to the par operator of linear logic, the sequential composition 1 In a deductive setting provided by the underlying proof theoretical operational semantics, computations are performed by means of bottom-up proof constructions.

of processes is naturally mapped to the operator seq. Thus, in language CP, parallel and sequential composition of the processes are expressed at the same logical level. This allows to use the underlying deductive system not only as an elegant interface to the language CP, but to do logical reasoning in an interesting and useful way without sacrificing from mathematical purity. Space restrictions do not permit us to give the proofs of the results here, we refer to [11]. 2 P/T ω Petri Nets We resort to the representation of petri nets as multiset rewriting systems (see, e.g., [3]). A multiset rewriting system M over a set F is a set of multiset rewrite rules of the form a : a.pre a.post where a is the name of the rule, and a.pre and a.post are multisets over F, called preset and postset of a 2. Given a multiset M, a rule a is enabled at M if a.pre is a submultiset of M. If a is enabled at M, the application of a on M produces the multsiet M = M a.pre a.post. We write in this case M a M. A place/transition petri net with infinite place capacities (P/T ω net) is a pair N = (F, M), where F is a set of places and M is a multiset rewriting system over F. The elements of M are called transitions. A P/T ω system is a pair (N,M) where M is a multiset over F called marking. The application of an enabled transition to a marking is referred to as the firing of that transition. Example 1. In Figure 1, we see a P/T ω petri net model for enzyme kinetics, given by the chemical reaction E + S ES E + P. In this reaction an enzyme and substrate can react to form an enzyme-substrate complex (es) which can evolve to a state where an enzymatic product is produced and the enzyme is recovered (ep) or the system can return to the state where the enzyme and the substrate coexist (se). In Figure 1, the state of the system is given by the marking M. The transition es is enabled, and the transitions se and ep are not enabled. 3 Syntax of the Language CP In this section, we introduce the syntax of the language CP. Definition 1 (process structure). A process structure is a structure generated by P ::= a [ P, P ] P; P where denotes the empty process and a denotes atoms representing transition names. [, ] and ; denote parallel and sequential composition, respectively. 2 Multisets are denoted by the curly brackets { and }. The empty multiset is denoted by.,, and denote the multiset operations corresponding to the usual set operations,, and, respectively.

e s es se es ep p N 1 = { es : { e, s } { es }, se : { es } { e, s }, ep : { es } { e, p } } M = { e, e, s, s } Fig. 1. Enzyme kinetics as a P/T ω net Example 2. Consider the process structure P = es; [se,ep]. In P, following the firing of es, the transitions se and ep are fired concurrently. Definition 2 (transition structure). Given a transition a : { c 1,...,c p } { e 1,...,e q }, the transition structure a, denoted by Q (possibly indexed), is a structure of the following form: ( c 1,..., c p );a; [e 1,...,e q ] Definition 3 (net structure). Given a P/T ω system (N,M) where M = { r 1,...,r n }, let Q1,...,Q s be the transition structures for all the transitions in N. Let P be a process structure. The net structure (denoted by R) for (N,M) with P is defined as follows: [?Q 1,...,?Q s, P; [r 1,...,r n ] ] Example 3. The net structure for the enzyme kinetics example, given in Section 2 and depicted in Figure 1, where P =, is as follows: [? (ē, s);es;es,? ēs;se; [e,s],? ēs;ep; [e,p],e,e,s,s] 4 Operational Semantics We define the operational semantics of the language CP, introduced in the previous section, by means of the proof theoretical deductive system NEL (noncommutative exponential linear logic). 4.1 System NEL System NEL [7, 20] is a conservative extension of multiplicative exponential linear logic with the rules mix and mix0 (see, e.g., [1]) and a non-commutative selfdual logical operator, resembling prefixing in the process algebra. System NEL cannot be designed in a standard sequent calculus, because a notion of deep rewriting is necessary in order to derive all the provable formulae of system NEL [26]. System NEL is designed within the proof theoretic methodology of

S{ } S([R, U], T) S [R, U]; [T, V ] ai s q S[a, ā] S[(R, T), U] S[ R; T, U; V ] S{![R, T ]} p S[!R,?T ] S{ } S[?R, R] w b S{?R} S{?R} Fig. 2. System NEL deep inference (see, e.g., [6, 2, 21]) which allows such a deep rewriting. Although it is unknown whether multiplicative exponential linear logic is decidable or not, in [22], Straßburger showed that system NEL is Turing-complete. Before introducing system NEL, let us see the logical expressions of system NEL, that we call structures. Structures are entities which share properties of formulae and sequents of the sequent calculus. Definition 4. There are infinitely many atoms, denoted with a,a,b,ab,abc,... NEL structures are generated by R ::= a [ R, R ] ( R, R ) R; R?R!R R Every atom is a structure. denotes the negation of a structure. [, ] and (, ) are called par and copar and they denote the operators and of linear logic, respectively. Par and copar are De Morgan duals of each other., is called seq, it is self-dual, i.e., R,T = R, T. Par and copar are associative and commutative, whereas seq is associative but not commutative. is the unit for par and copar, and it is left-unit and right-unit for seq. The exponentials! and? of linear logic are De Morgan duals of each other. On the NEL structures, we also impose the equalities??r =?R,!!R = R,? =, and! =. All NEL structures can be equivalently considered in negation normal form, negation can always be pushed inwards to atoms. Example 4. In order to see the relationship between system NEL and multiplicative exponential linear logic consider the structure! [(?a,b),ā,! b] which corresponds to!((?a b) ā! b). However, the logical operator seq does not have an equivalent in multiplicative exponential linear logic. Definition 5. The system in Figure 2 is called non-commutative exponential linear logic, or system NEL. Definition 6. An instance of an inference rule of the form S{T } ρ S{R} is the inference rule ρ where the structure schemes R and T and the context scheme S{ }

are replaced with structures and a structure context, respectively, that respect their scheme. A derivation is either a structure or a finite chain of instances of inference rules. A derivation with premise T and conclusion R will be written T as. A proof is a derivation with the premise. NEL R For a detailed discussion on proof theory of NEL and the precise relation between NEL and MELL, the reader is referred to [7, 20, 6, 10]. 4.2 Proof Theoretical Operational Semantics In this subsection, we will define the operational semantics of the language CP as analytic bottom-up proof constructions. In a bottom-up 3 proof construction, derivations and proofs are constructed by starting from the conclusion, and by going up by applications of the inference rules. Let us first define the notion of securing which will be used in the rest of the paper. Definition 7 (securing). A securing, denoted with S, is a structure of the form a 1 ;...;a n where n 1. For a net N, we say that M results from applying S = a 1 ;...;a n to marking M if there are transitions a 1,...,a n N and markings M 1,...,M n such that M a1 a M 1 a 1... n Mn and M n = M. We then write Φ(S,M) = M. Example 5. Consider the net structure in Example 3. es;ep;es is a securing. Definition 8 (firing). The following rule is called firing: S[? ( c 1,..., c p );a; E, P;a; [E,R] ] firing S[? ( c1,..., c p );a; E, P; [c 1,...,c p,r] ] In an instance of the rule firing, ( c 1,..., c p );a; E is called the active transition structure. For any pair (N,M) and securing S, we can compute the marking resulting from applying the securing to the marking by applying the rule firing bottom-up. Example 6. The net structure given in Example 3 is the conclusion of the derivation below. By means of bottom-up proof construction, we compute the marking resulting from applying the securing es;ep;es to M = { e,e,s,s } as follows: [? (ē, s);es;es,? ēs;se; [e,s],? ēs;ep; [e,p], es;ep;es; [e,es,p] ] firing [? (ē, s);es;es,? ēs;se; [e,s],? ēs;ep; [e,p], es;ep; [e,e,s,p] ] firing [? (ē, s);es;es,? ēs;se; [e,s],? ēs;ep; [e,p], es; [e,s,es] ] firing [? (ē, s);es;es,? ēs;se; [e,s],? ēs;ep; [e,p],e,e,s,s] 3 The use of the word bottom-up here is purely mechanical, and should not be confused with the conceptual use of this word in the introduction.

Lemma 1. The rule firing is derivable for system NEL. Proof. Take the following derivation: S[? ( c 1,..., c p );a; E, P;a; [E,R] ] q S[? ( c1,..., c p );a; E, P; [ a; E,R] ] i S[? ( c1,..., c p );a; E, P; [ [c 1,...,c p,( c 1,..., c p )];a; E,R] ] q S[? ( c1,..., c p );a; E, P; [c 1,...,c p, ( c 1,..., c p );a; E,R] ] q S[? ( c1,..., c p );a; E, ( c 1,..., c p );a; E, P; [c 1,...,c p,r] ] b S[? ( c 1,..., c p );a; E, P; [c 1,...,c p,r] ] Definition 9 (sequential). The following rule is called sequential composition: S C;P 1 ;P 2 ; [E 1,E 2 ] sequential S[ C;P1 ; [r 1,...,r m,e 1 ], ( r 1,..., r m );P 2 ;E 2 ] Definition 10 (parallel). The following rule is called parallel composition: S (C 1,C 2 ); [P 1,P 2 ]; [E 1,E 2 ] parallel S[ C 1 ;P 1 ;E 1, C 2 ;P 2 ;E 2 ] Lemma 2. The rules sequential and parallel are derivable for system NEL. Proof. Take the following derivations, respectively: S C;P 1 ;P 2 ; [E 1,E 2 ] q S C;P1 ; [ P 2 ;E 2,E 1 ] i S C;P1 ; [ [E,Ē];P 2;E 2,E 1 ] q S C;P1 ; [E, Ē;P 2;E 2,E 1 ] q S[ C;P1 ; [E,E 1 ], Ē;P 2;E 2 ] S (C 1,C 2 ); [P 1,P 2 ]; [E 1,E 2 ] s S [C1,C 2 ]; [P 1,P 2 ]; [E 1,E 2 ] q S [C1,C 2 ]; [ P 1 ;E 1, P 2 ;E 2 ] q S[ C 1 ;P 1 ;E 1, C 2 ;P 2 ;E 2 ] Example 7. Let us consider the net structure of Example 3. We obtain the process structure [es,es]; [ep,ep] by using parallel and sequential composition. (ē,ē, s, s); [es,es]; [ep,ep] ; [e,e,p,p] sequential [ (ē,ē, s, s); [es,es]; [es,es], (ēs,ēs); [ep,ep]; [e,e,p,p] ] parallel [ (ē, s);es;es, (ē, s);es;es, (ēs,ēs); [ep,ep]; [e,e,p,p] ] b ;b ;w [? (ē, s);es;es, (ēs,ēs); [ep,ep]; [e,e,p,p] ] parallel [? (ē, s);es;es, ēs;ep; [e,p], ēs;ep; [e,p] ] b ;b ;w [? (ē, s);es;es,? ēs;ep; [e,p] ] The definitions of sequential and parallel composition allow to lift to notion of firing from transitions to processes. By composing the transitions with each other, we can treat processes as atomic transitions. This way, we can treat transitions and processes, thus nets, compositionally.

e enz syk p syk N 2 = { enz : { e, e, s, s } { e, e, p, p }, s syk : { p, p } { syk } } M = { e, e, s, s } Fig. 3. A model of the activation of the protein Syk as a result of the phosphorylation of two tyrosine residues on the immunoreceptor tyrosine-based activation motif. Example 8. During Fc receptor-mediated phagocytosis [24], after the binding of the Fc with the Fc receptor on the cell surface, two tyrosine residues on the ITAM (immunoreceptor tyrosine-based activation motif) located on the cytoplasmic tail of the Fc receptor gets phosphorylated in an enzymatic reaction catalyzed by the protein Src. Then the protein Syk binds to these two phosphorylated domains and transmits the signal further. Let us represent the binding of the Syk to the phosphorylated tyrosine residues with the following transition: syk : { p,p } { syk }. Let us denote with enz the process [es,es]; [ep,ep] obtained as the premise of the derivation in Example 7, i.e., enz = [es,es]; [ep,ep]. This way, we treat the net given in Example 7 as a single transition enz. The transition enz models the phosphorylation of the two residues by an enzymatic reaction. We then define a net with transitions and an initial marking as depicted in Figure 3. We get the following derivation: enz;syk; [syk,e,e] firing [ (ē,ē, s, s);enz;syk; [e,e,syk], e,e,s,s] sequential [ (ē,ē, s, s);enz; [e,e,p,p], (p,p);syk;syk,e,e,s,s] In the derivation above, the instance of the rule sequential composes the two shaded transition structures and the instance of the rule firing applies the resulting transition structure to the given marking. The following two theorems bring the above ideas to formal grounds. Theorem 1. Let R be a net structure of a P/T ω system (N,M). For a marking M = { r 1,...,r n } and transitions a1,...,a k N, if M results from applying a 1 ;...;a k to M then there is a derivation a 1,...,a k ; [r 1,...,r n ] NEL. R Theorem 2. Let R be a net structure of a P/T ω system (N,M). If there is a P; [r 1,...,r n ] S process P and a derivation then for any derivation {q }, R NEL R we have that { r 1,...,r n } results from applying S to M.

4.3 Securing Equivalence A process structure provides a canonical representation of a class of securings that can be replaced with each other in any context. For a given process structure, these securings are determined by the derivations of the form {q }. S P Definition 11. Given a process structure P, securings S 1 and S 2 are P-equivalent if there are the derivations S 1 {q } P and S 2 {q }. P Proposition 1. If securings S 1 and S 2 are P-equivalent, then for any marking M, Φ(S 1,M) = Φ(S 2,M) = Φ(P,M). Theorem 3. For a process structure P, securings S 1 and S 2 are P-equivalent if {ai,q } {ai,q } and only if there are proofs and. [P, S1 ] [P, S2 ] Example 9. Let P = es; [se,ep], then the securings es;se;ep and es;ep;se are P-equivalent, because we have the two proofs below. ai [ep, ep] ai [ep, [se, se]; ep ] q [se, ep, se; ep ] ai [es,ēs]; [se,ep, se; ep ] q [ es; [se,ep], ēs; se; ep ] ai [se, se] ai [se, [ep, ep]; se ] q [se, ep, ep; se ] ai [es,ēs]; [se,ep, ep; se ] q [ es; [se,ep], ēs; ep; se ] 4.4 Petri Net Reachability By resorting to Theorem 2, it is possible to formulate petri net reachability queries for a given P/T ω net, i.e., for two markings M and M, is there a process P such that M results from applying P to M?. We can search for securings by exploring a search space that is constructed by consequent applications of the rule firing. However, when we are searching for processes which are constructed by means of parallel and sequential composition, the situation is quite different: Because any two processes can be composed in parallel, it is difficult to come up with a straight-forward strategy that delivers a process. In the following, we will present a procedure for obtaining a maximally parallel process from a securing. Definition 12. Let be a derivation, built by the instances of the rule firing only, and the conclusion is a net structure R. We label the atoms in every transition structure in with the name of that transition. Whenever there is an instance of the rule firing, the labels of the atoms of the active transition structure

cdc cdc ba N pha 3 = { cdc : { cdc } { ba }, ph rac : { rac } { la }, rac rac la pha : { ba, la } { ph } } M = { cdc, rac } Fig. 4. A model of branching (ba) and linear (la) actin polymerization resulting in phagocytosis (pha). are extended with a natural number that does not occur with the same transition name elsewhere in the derivation. Let Label denote the set of all the labels occurring in. The function µ on R {firing} R firing R is defined as follows. If is a net structure R, then µ( ) =. Otherwise in the bottom-most instance of the rule firing in, let l 1,...,l s be the label of the atoms in the marking and k be the label of the active transition structure. µ( ) = {(l 1,k),...,(l s,k) } µ R {firing} R. Given a derivation, the securing order of, denoted by Sec, is the transitive closure of µ( ). Example 10. Let be the derivation below where the conclusion is the net structure for the petri net depicted in Figure 4. This net models phagocytosis (ph) as an outcome of branching (ba) and linear (la) actin polymerization, represented by the transition pha. The polymerization of actin along these two pathways causes the extension of cytoskeleton around the engulfed particle. The branching and linear actin polymerization result from activation of the Rho GTP binding proteins Cdc42 (cdc) and Rac (rac), respectively [24]. In Figure 4, the activation of these proteins are represented by the transitions cdc and rac, respectively. [? cdc;cdc;ba,? rac;rac;la,? ( ba, la);pha;ph, cdc;rac;pha; [ph] ] firing [? cdc;cdc;ba,? rac;rac;la,? ( ba, la);pha;ph, cdc;rac; [ba,la] ] firing [? cdc;cdc;ba,? rac; rac;la,? ( ba, la);pha;ph, cdc; [ba,rac] ] firing [? cdc; cdc;ba,? rac;rac;la,? ( ba, la);pha;ph,cdc,rac]

Following Definition 12, we get µ( ) = Sec = {(cdc,pha),(rac,pha)}. By using this information, we can compose the causally independent transitions cdc and rac in parallel, and obtain the process in the premise of the following derivation: ( cdc, rac); [cdc,rac];pha ;ph sequential [ ( cdc, rac); [cdc,rac]; [ba,la], ( ba, la);pha;ph ] parallel [ cdc;cdc;ba, rac;rac;la, ( ba, la);pha;ph ] Definition 13. A securing S is induced by Sec if the following holds: There is a strict total order Lin such that Sec Lin and the transition names that appear in S are exactly those in Lin obeying the order given by Lin, that is, if (a,b) Lin then a appears on the left of b in S. Example 11. Let Sec = {(cdc,pha),(rac,pha)}. Then the securings cdc;rac;pha and cdc;rac;pha are both induced by Sec. Theorem 4. Let R and R be net structures of P/T ω systems (N,M) and (N,M ), respectively, such that there is a derivation R R induced by Sec if and only if Φ(S,M) = M. {firing}. S is a securing Definition 14. Given a securing order Sec, two securings S 1 and S 2 are Secequivalent if S 1 and S 2 are both induced by Sec. Corollary 1. Let Sec be a securing order. For any marking M, if securings S 1 and S 2 are Sec-equivalent then Φ(S 1,M) = Φ(S 2,M). Example 12. Let Sec = {(cdc,pha),(rac,pha)}. Then the securings cdc;rac;pha and cdc; rac; pha are Sec-equivalent. 5 Process Structures Revisited Securing orders are strict partial orders that provide canonical representations of sets of securings. Analogously, a process structure P gives a canonical representation of securings, determined by all the derivations {q }. In other words, a S P process structure provides a syntactical representation of a partial order of transitions. In contrast to the securing orders, the syntactic representation of process structures sets a boundary which is meaningful from the point of view of concurrent messages (or signalling pathways). A partial order which is represented by a process structure is an N-free partial order: Definition 15. A partial order A A is N-free (series-parallel) if and only if, for all a,b,c,d A, {(a,b),(c,d),(c,b) } implies (a,d). N-free closure of a partial order is the smallest N-free partial order containing.

Example 13. Consider the partial orders denoted by the graphs below, which are given in the notation of event structures [27], where nodes denote events and arrows denote dependencies with respect to causality between events: The one on the left is an N-free partial order, whereas the one on the right is not. From the point of view of concurrent messages, a representation of signalling as N-free partial orders is meaningful: When meet and join of two processes are considered as points in time, these N-free partial orders provide a representation of synchronization of processes: Because the representation of resources provides a model of dependencies, processes with a common meet and join can be executed concurrently. However, such an observation is impossible in a partial order that is not N-free. A securing order is not necessarily an N-free order. Thus, although a securing order provides a canonical representation of a class of securings, N- free closures of securing orders needs to be considered when modeling concurrent computations. Because the process structures allow the representation of only N-free partial orders, they are well suited for modeling concurrent computations and also signalling pathways. Example 14. Consider the net depicted in Figure 5: During Fc receptor-mediated phagocytosis, following the binding of the Fc receptor (fcr) with the Fc (fc) which results in FcR-Fc complex (f cc) and initiates a signalling cascade (fcc), the Rho GTP protein Rac becomes activated. However, the activation of Rac (rac) is also causally dependent on the prior activation of Syk (syk) by the pathway given in Example 8, because Syk activates a guanosine-nucleotide exchange factor (GEF) which activates Cdc42 (vav). Furthermore, another Rho GTP binding protein Cdc42 (cdc) becomes activated by an unknown GEF (x) as a result of Fc receptor and Fc binding [24]. In Figure 5, sk denotes the process enz;syk given in Example 8 which activates Syk. This results in a situation which is depicted in the following graph where nodes denote transitions and arrows denote causal dependencies in the event structures notation [27] of a possible securing order. fcc sk x vav In this graph, we observe that the transitions fcc and sk are partially ordered because they are independent from each other due to the resources that they require to be executed. Similarly the pairs x, vav and sk, x are partially ordered. Because such partially ordered transitions can be performed in any order, this graph provides a canonical representation of the following securings: fcc;sk;x;vav fcc;sk;vav;x fcc;x;sk;vav sk;fcc;x;vav sk;fcc;vav;x

fcr fcc x x cdc fc s e fcc sk syk vav rac N 4 = { fcc : { fcr, fc } { x, fcc }, sk : { e, e, s, s } { syk, e, e }, x : { x } { cdc }, vav : { fcc, syk } { rac } } Fig. 5. A model of signalling during phagocytosis which is initiated with the Fc receptor Fc binding. However, if one considers the concurrent executions, we observe that if the transitions fcc and sk are performed concurrently, then sk and x cannot be performed concurrently because x requires fcc to be executed. In this system, the possible concurrent executions are the ones that are given by the process structures below. fcc; [sk,x];vav [fcc,sk]; [x,vav] It is important to observe that these process structures denote N-free closures of the partial order that is given by the securing order depicted above. Their graphical representations are depicted as the following Hasse-diagrams, respectively: fcc fcc sk sk x vav x vav These two process structures indicate two equally possible flows of the signal in this model. Example 15. We compose N 1, N 2, N 3, and N 4, given in Figure 1, Figure 3, Figure 4, and Figure 5, respectively, and obtain the following process as a model for Fc receptor mediated phagocytosis. Following Example 14, we make a choice with respect to N 4 since there are two options, and we choose the process [fcc, sk]; [x, vav]. By composing these process structures sequentially, we obtain [ [es, es]; [ep, ep]; syk, fcc]; [x, vav]; [cdc, rac]; pha. By means of deductive analysis, we also observe that the process structure [x, vav]; [cdc, rac] can be replaced with [ x; cdc, vav; rac ] because they have the same preset and postset and latter process structure is more parallelized than the former. 6 Discussion We introduced a compositional language, called CP, for place transition petri nets for modelling biological signalling pathways. The language CP is equipped

with parallel and sequential composition and a proof theoretical deductive operational semantics that allows to perform logical reasoning on the processes which are being analyzed. The operational semantics is obtained by encoding the multiset rewriting representation of the place transition petri nets in an extension of multiplicative exponential linear logic with a self-dual non-commutative operator called seq. While parallel composition of processes is naturally mapped to the par operator of linear logic, the seq operator serves as a data structure to represent sequential composition at the same logical level with the parallel composition. This allows the models to be built by starting from any level of abstraction and extending the model compositionally, also by adding more data at lower levels of abstraction at will. In language CP, the process structures which are obtained by composing processes are N-free partial orders. These N-free partial orders provide an explicit representation of possible signalling pathways, and a platform for analysis, where the transitions can be composed in many different ways including those resulting in non-n-free partial orders as in Example 14. Because of the causal dependencies captured by the petri nets due to the resources which enable transitions, the common predecessors and successors of processes provide a synchronization mechanism. Such a view of language CP is also in agreement with the event structure [27] view of the processes, which is a topic of on going work. When the synchronization mechanism of language CP is compared with the synchronization in process algebra, e.g., π-calculus [12], in language CP we are not restricted to binary interactions, because more than two processes can share predecessors and successors. However, binary reactions as hand-shake synchronization can be simulated by extending the definition of transition structures with a synchronization token as in the following example. Example 16. Consider the two transitions p : { a } { c } and q : { b } { d }. We synchronize these two transitions over the name x as in the following derivation: parallel (ā, b); [p,q]; [c,d] ai (ā, b); [p,q]; [x, x]; [c,d] q (ā, b); [ p;x, q; x ]; [c,d] [ ā;p;x;c, b;q; x;d ] The petri net formalism is well developed with a variety of languages and tools, including those for quantitative analysis. Integrating and adapting these ideas language CP is a topic of future work. Another direction of future investigation is considering more involved data structures without departing from mathematical rigor by exploiting the logical operators and exponentials of system NEL, e.g., for representing compartments as in [19], or to represent the DNA by means of sequential composition.

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