Formal Methods and Systems Biology: The Calculus of Looping Sequences

Transcription

1 Formal Methods and Systems Biology: The Calculus of Looping Sequences Paolo Milazzo Dipartimento di Informatica, Università di Pisa, Italy Verona January 22, 2008 Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

2 Outline of the talk 1 Introduction Cells are complex interactive systems The EGF pathway and the lac operon 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 Bisimulations in CLS A labeled semantics for CLS Bisimulations in CLS Bisimulations applied to the CLS model of the lac operon 4 CLS variants Stochastic CLS LCLS 5 Future Work and References Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

3 Cells: complex systems of interactive components Two classifications of cell: procaryotic eucaryotic Main actors: membranes proteins DNA/RNA ions, macromolecules,... Interaction networks: metabolic pathways signaling pathways gene regulatory networks Computer Science can provide biologists with formalisms for the description of interactive systems and tools for their analysis. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

4 Examples of interaction networks: the EGF pathway phosphorylations EGF EGFR nucleus SHC CELL MEMBRANE Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

5 Examples of interaction networks: the lac operon DNA i p o z y a mrna proteins R lac Repressor beta-gal. permease transacet. i p o z y a a) RNA Polimerase R NO TRANSCRIPTION i p o z y a b) RNA Polimerase R LACTOSE TRANSCRIPTION Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

6 Outline of the talk 1 Introduction Cells are complex interactive systems The EGF pathway and the lac operon 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 Bisimulations in CLS A labeled semantics for CLS Bisimulations in CLS Bisimulations applied to the CLS model of the lac operon 4 CLS variants Stochastic CLS LCLS 5 Future Work and References Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

7 The Calculus of Looping Sequences (CLS) We assume an alphabet E. Terms T and Sequences S of CLS are given by the following grammar: T ::= S S ::= ɛ ( ) L S T T T a S S where a is a generic element of E, and ɛ is the empty sequence. The operators are: S S : Sequencing ( ) L S : Looping (S is closed and it can rotate) T 1 T 2 : Containment (T 1 contains T 2 ) T T : Parallel composition (juxtaposition) Actually, looping and containment form a single binary operator ( S ) L T. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

8 Examples of Terms (i) (ii) (iii) ( a b c ) L ɛ ( a b c ) L ( d e ) L ɛ ( a b c ) L (f g ( d e ) L ɛ) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

9 Structural Congruence The Structural Congruence relations S and T are the least congruence relations on sequences and on terms, respectively, satisfying the following rules: S 1 (S 2 S 3 ) S (S 1 S 2 ) S 3 S ɛ S ɛ S S S T 1 T 2 T T 2 T 1 T 1 (T 2 T 3 ) T (T 1 T 2 ) T 3 T ɛ T T ( ɛ ) L ( ) L ( ) L ɛ T ɛ S1 S 2 T T S2 S 1 T We write for T. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

10 CLS Patterns Let us consider variables of three kinds: term variables (X, Y, Z,...) sequence variables ( x, ỹ, z,...) element variables (x, y, z,...) Patterns P and Sequence Patterns SP of CLS extend CLS terms and sequences with variables: P ::= SP SP ::= ɛ a ( ) L SP P P P X SP SP x x where a is a generic element of E, ɛ is the empty sequence, and x, x and X are generic element, sequence and term variables The structural congruence relation extends trivially to patterns Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

11 Rewrite Rules Pσ denotes the term obtained by replacing any variable in T with the corresponding term, sequence or element. Σ is the set of all possible instantiations σ A Rewrite Rule is a pair (P, P ), denoted P P, where: P, P are patterns variables in P are a subset of those in P A rule P P can be applied to all terms Pσ. Example: a x a b x b can be applied to a c a (producing b c b) cannot be applied to a c c a Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

12 Formal Semantics Given a set of rewrite rules R, evolution of terms is described by the transition system given by the least relation satisfying P P R Pσ ɛ Pσ P σ T T T T T T T T ( ) L ( ) L S T S T and closed under structural congruence. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

13 CLS modeling examples: the EGF pathway (1) phosphorylations EGF EGFR nucleus SHC CELL MEMBRANE Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

14 CLS modeling examples: the EGF pathway (2) First steps of the EGF signaling pathway up to the binding of the signal-receptor dimer to the SHC protein The EGFR,EGF and SHC proteins are modeled as the alphabet symbols EGFR, EGF and SHC, respectively The cell is modeled as a looping sequence (representing its external membrane): EGF EGF ( EGFR EGFR EGFR EGFR ) L (SHC SHC) Rewrite rules modeling the first steps of the pathway: EGF ( EGFR x ) L ( ) L X CMPLX x X (R1) ( ) L ( ) L CMPLX x CMPLX ỹ X DIM x ỹ X (R2) ( DIM x ) L X ( DIMp x ) L X (R3) ( DIMp x ) L (SHC X ) ( DIMpSHC x ) L X (R4) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

15 CLS modeling examples: the EGFR pathway (2) A possible evolution of the system: EGF EGF ( EGFR EGFR EGFR EGFR ) L (SHC SHC) (R1) EGF ( EGFR CMPLX EGFR EGFR ) L (SHC SHC) (R1) ( EGFR CMPLX EGFR CMPLX ) L (SHC SHC) (R2) ( EGFR DIM EGFR ) L (SHC SHC) (R3) ( EGFR DIMp EGFR ) L (SHC SHC) (R4) ( EGFR DIMpSHC EGFR ) L SHC Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

16 CLS modeling examples: the lac operon (1) DNA i p o z y a mrna proteins R lac Repressor beta-gal. permease transacet. i p o z y a a) RNA Polimerase R NO TRANSCRIPTION i p o z y a b) RNA Polimerase R LACTOSE TRANSCRIPTION Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

17 CLS modeling examples: the lac operon (2) Ecoli ::= ( m ) L (laci lacp laco lacz lacy laca polym) Rules for DNA transcription/translation: laci x laci x repr polym x lacp ỹ x PP ỹ x PP laco ỹ x lacp PO ỹ x PO lacz ỹ x laco PZ ỹ x PZ lacy ỹ x lacz PY ỹ betagal x PY laca x lacy PA perm x PA x laca transac polym (R1) (R2) (R3) (R4) (R5) (R6) (R7) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

18 CLS modeling examples: the lac operon (3) Ecoli ::= ( m ) L (laci lacp laco lacz lacy laca polym) Rules to describe the binding of the lac Repressor to gene o, and what happens when lactose is present in the environment of the bacterium: repr x laco ỹ x RO ỹ LACT ( m x ) L ( ) L X m x (X LACT ) x RO ỹ LACT x laco ỹ RLACT ) L ( ) L ( x (perm X ) perm x X LACT ( perm x ) L ( ) L X perm x (LACT X ) betagal LACT betagal GLU GAL (R8) (R9) (R10) (R11) (R12) (R13) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

19 CLS modeling examples: the lac operon (4) Ecoli ::= ( m ) L (laci lacp laco lacz lacy laca polym) Example: Ecoli LACT LACT ( m ) L (laci lacp laco lacz lacy laca polym repr) LACT LACT ( m ) L (laci lacp RO lacz lacy laca polym) LACT LACT ( m ) L (laci lacp laco lacz lacy laca polym RLACT ) LACT ( perm m ) L (laci A betagal transac polym RLACT ) LACT ( perm m ) L (laci A betagal transac polym RLACT GLU GAL) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

20 Outline of the talk 1 Introduction Cells are complex interactive systems The EGF pathway and the lac operon 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 Bisimulations in CLS A labeled semantics for CLS Bisimulations in CLS Bisimulations applied to the CLS model of the lac operon 4 CLS variants Stochastic CLS LCLS 5 Future Work and References Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

21 Bisimulations Bisimilarity is widely accepted as the finest extensional behavioral equivalence one may impose on systems. Two systems are bisimilar if they can perform step by step the same interactions with the environment. Properties of a system can be verified by assessing the bisimilarity with a system known to enjoy them. Bisimilarities need semantics based on labeled transition relations capturing the potential interactions with the environment. In process calculi, transitions are usually labeled with actions. In CLS labels are contexts in which rules can be applied. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

22 Labeled semantics Contexts C are given by the following grammar: C ::= C T T C ( S ) L C where T T and S S. Context is called the empty context. Given a set of rewrite rules R R, the labeled semantics of CLS is the labeled transition system given by the following inference rules: (rule appl) P P R C[T ] Pσ T ɛ σ Σ C C T C P σ (cont) T T ( S ) L T ( S ) L T (par) T C T C C P T T C T T where C P are contexts that do not include ( S ) L C and the dual version of the (par) rule is omitted. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

23 Bisimulations in CLS (1) A binary relation R on terms is a strong bisimulation if, given T 1, T 2 such that T 1 RT 2, the two following conditions hold: T 1 C T 1 = T 2 s.t. T 2 C T 2 and T 1 RT 2 T 2 C T 2 = T 1 s.t. T 1 C T 1 and T 2 RT 1. The strong bisimilarity is the largest of such relations. A binary relation R on terms is a weak bisimulation if, given T 1, T 2 such that T 1 RT 2, the two following conditions hold: C T 1 T 1 = T 2 s.t. T C 2 = T 2 and T 1 RT 2 C T 2 T 2 = T 1 s.t. T C 1 = T 1 and T 2 RT 1. The weak bisimilarity is the largest of such relations. Theorem: Strong and weak bisimilarities are congruences. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

24 Bisimulations in CLS (2) Consider the following set of rewrite rules: R = { a b c, d b e, e e, c e, f a } We have that a d, because and f d, because On the other hand, f e and f e. f a b c e e... d b e e... a b c e e... e e e... Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

25 Bisimulations in CLS (3) Let us consider systems (T, R)... A binary relation R is a strong bisimulation on systems if, given (T 1, R 1 ) and (T 2, R 2 ) such that (T 1, R 1 )R(T 2, R 2 ): R 1 : T 1 C T 1 = T 2 s.t. R 2 : T 2 C T 2 and (T 1, R 1)R(T 2, R 2) R 2 : T 2 C T 2 = T 1 s.t. R 1 : T 1 C T 1 and (R 2, T 2 )R(R 1, T 1 ). The strong bisimilarity on systems is the largest of such relations. A binary relation R is a weak bisimulation on systems if, given (T 1, R 1 ) and (T 2, R 2 ) such that (T 1, R 1 )R(T 2, R 2 ): R 1 : T 1 C T 1 = T 2 s.t. R 2 : T 2 C = T 2 and (T 1, R 1)R(T 2, R 2) R 2 : T 2 C T 2 = T 1 s.t. R 1 : T 1 C = T 1 and (T 2, R 2)R(T 1, R 1) The weak bisimilarity on systems is the largest of such relations. Strong and weak bisimilarities on systems are NOT congruences. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

26 Bisimulations in CLS (4) Consider the following sets of rewrite rules R 1 = {a b c} R 2 = {a d c, b e c} We have that a, R 1 e, R 2 because R 1 : a b c R 2 : e b c and b, R 1 d, R 2, because R 1 : b a c R 2 : d a c but a b, R 1 e d, R 2, because R 1 : a b c R 2 : c d Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

27 Applying bisimulations to the lac operon By using the weak bisimilarity on systems we can prove that from the state in which the repressor is bound to the DNA we can reach a state in which the enzymes are synthesized only if lactose appears in the environment. We replace rule x RO ỹ LACT x laco ỹ RLACT (R10) with ( w ) L ( x RO ỹ LACT X ) START ( w ) L ( x laco ỹ RLACT X ) (R10bis) The obtained model is bisimilar to (T 1, R) where R is T 1 LACT T 2 (R1 ) T 2 START T 3 (R3 ) T 2 LACT T 2 (R2 ) T 3 LACT T 3 (R4 ) that is a system satisfying the property. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

28 Some variants of CLS Full CLS The looping operator can be applied to any term Terms such as ( a ( b ) L ) L c d are allowed CLS+ More realistic representation of the fluid nature of membranes: the looping operator can be applied to parallel compositions of sequences Can be encoded into CLS Stochastic CLS The application of a rule consumes a stochastic quantity of time LCLS (CLS with Links) Description of protein protein interactions at the domain level Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

29 Outline of the talk 1 Introduction Cells are complex interactive systems The EGF pathway and the lac operon 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 Bisimulations in CLS A labeled semantics for CLS Bisimulations in CLS Bisimulations applied to the CLS model of the lac operon 4 CLS variants Stochastic CLS LCLS 5 Future Work and References Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

30 Background: the kinetics of chemical reactions Usual notation for chemical reactions: l 1 S l ρ S ρ k k 1 l 1P l γp γ where: S i, P i are molecules (reactants) l i, l i are stoichiometric coefficients k, k 1 are the kinetic constants The kinetics is described by the law of mass action: d[p i ] dt = l i k[s 1 ] l1 [S ρ ] lρ }{{} reaction rate l i k 1 [P 1 ] l 1 [Pγ ] l γ }{{} reaction rate Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

31 Background: Gillespie s simulation algorithm represents a chemical solution as a multiset of molecules computes the reaction rate a µ by multiplying the kinetic constant by the number of possible combinations of reactants Example: chemical solution with X 1 molecules S 1 and X 2 molecules S 2 reaction R 1 : S 1 + S 2 2S 1 rate a 1 = ( X 1 1 )( X2 1 ) k1 = X 1 X 2 k 1 reaction R 2 : 2S 1 S 1 + S 2 rate a 2 = ( X 1 2 ) k2 = X 1(X 1 1) 2 k 2 Given a set of reactions {R 1,... R M } and a current time t The time t + τ at which the next reaction will occur is randomly chosen with τ exponentially distributed with parameter M ν=1 a ν; The reaction R µ that has to occur at time t + τ is randomly chosen a with probability µ P M. ν=1 aν At each step t is incremented by τ and the chemical solution is updated. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

32 Stochastic CLS (1) Stochastic CLS incorporates Gillespie s stochastic framework into the semantics of CLS Two main problems: What is a reactant in Stochastic CLS? A subterm of a term T is a term T ɛ such that T C[T ] for some context C A reactant is an occurence of a subterm What happens with variables? ( ) L ( ) L We consider a rule a (b X ) c X as a reaction between a molecule a on a membrane and any molecule b contained in the membrane. The semantics has to count how many times b occurs in the instantiation of X Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

33 Stochastic CLS (2) Let us assume the syntax of Full CLS... Given a finite set of stochastic rewrite rules R, the semantics of Stochastic R,T,r,b CLS is the least transition relation closed wrt and satisfying by the following inference rules: R : P L k PR R σ Σ P L σ R,P Lσ,k comb(p L,σ),1 P R σ T 1 R,T,r,b T 2 T 1 T 3 R,T,r,b binom(t,t 1,T 3) T 2 T 3 R,T,r,b T 1 T 2 T 1 (T 1 ) L R,(T 1) T L T 3,r b,1 3 (T 2 ) L T 3 R,T,r,b T 2 (T 3 ) L R,(T 3) T L T 1,r b,1 1 (T 3 ) L T 2 The transition system obtained can be easily transformed into a Continuous Time Markov Chain Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

34 A Stochastic CLS model of the lac operon (1) DNA i p o z y a mrna proteins R lac Repressor beta-gal. permease transacet. i p o z y a a) RNA Polimerase R NO TRANSCRIPTION i p o z y a b) RNA Polimerase R LACTOSE TRANSCRIPTION Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

35 A Stochastic CLS model of the lac operon (2) Transcription of DNA, binding of lac Repressor to gene o, and interaction between lactose and lac Repressor: laci x 0.02 laci x Irna Irna 0.1 Irna repr polym x lacp ỹ 0.1 x PP ỹ x PP ỹ 0.01 polym x lacp ỹ x PP laco ỹ 20.0 polym Rna x lacp laco ỹ Rna 0.1 Rna betagal perm transac repr x laco ỹ 1.0 x RO ỹ x RO ỹ 0.01 repr x laco ỹ repr LACT RLACT RLACT 0.1 repr LACT (S1) (S2) (S3) (S4) (S5) (S6) (S7) (S8) (S9) (S10) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

36 A Stochastic CLS model of the lac operon (3) The behaviour of the three enzymes for lactose degradation: ( x ) L (perm X ) 0.1 ( perm x ) L X LACT ( perm x ) L X ( perm x ) L (LACT X ) (S11) (S12) betagal LACT betagal GLU GAL (S13) Degradation of all the proteins and mrna involved in the process: perm ɛ (S14) betagal transac Irna ɛ (S16) repr 0.01 ɛ (S18) Rna RLACT LACT (S20) ɛ (S15) ɛ (S17) 0.01 ɛ (S19) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

37 Simulation results (1) betagal perm perm on membrane Number of elements Time (sec) Production of enzymes in the absence of lactose ( m ) L (laci A 30 polym 100 repr) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

38 Simulation results (2) betagal perm perm on membrane Number of elements Time (sec) Production of enzymes in the presence of lactose 100 LACT ( m ) L (laci A 30 polym 100 repr) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

39 Simulation results (3) LACT (env.) LACT (inside) GLU Number of elements Time (sec) Degradation of lactose into glucose 100 LACT ( m ) L (laci A 30 polym 100 repr) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

40 Outline of the talk 1 Introduction Cells are complex interactive systems The EGF pathway and the lac operon 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 Bisimulations in CLS A labeled semantics for CLS Bisimulations in CLS Bisimulations applied to the CLS model of the lac operon 4 CLS variants Stochastic CLS LCLS 5 Future Work and References Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

41 Modeling proteins at the domain level To model a protein at the domain level in CLS it would be natural to use a sequence with one symbol for each domain The binding between two elements of two different sequences, cannot be expressed in CLS LCLS extends CLS with labels on basic symbols two symbols with the same label represent domains that are bound to each other example: a b 1 c d e 1 f Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

42 Syntax of LCLS Terms T and Sequences S of LCLS are given by the following grammar: T ::= S S ::= ɛ ( ) L S T T T a a n S S where a is a generic element of E, and n is a natural number. Patterns P and sequence patterns SP of LCLS are given by the following grammar: P ::= SP SP ::= ɛ a ( ) L SP P P P X a n SP SP x x x n where a is an element of E, n is a natural number and X, x and x are elements of TV, SV and X, respectively. Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

43 Well formedness of LCLS terms and patterns (1) A 1 ( B1 1 B2 2) L C1 C22 C3 A 1 ( B ) L C 1 A 1 B 1 C 1 Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

44 Well formedness of LCLS terms and patterns (2) An LCLS term (or pattern) is well formed if and only if a label occurs no more than twice, and in the content of a looping sequence a label occurs either zero or two times Type system for well formedness: 1. (, ) = ɛ 2. (, ) = a 3. (, {n} ) = a n 4. (, ) = x 5. (, {n} ) = x n 6. (, ) = x 7. (, ) = X ( ) ( ) N1, N 1 = SP1 N2, N 2 = SP2 N 1 N 2 = N 1 N 2 = N 1 N 2 ( = N1 N 2 (N 1 N 2 ), (N 1 N 2 ) \ (N 1 N 2 )) = SP 1 SP 2 ( ) ( ) N1, N 1 = P1 N2, N 2 = P2 N 1 N 2 = N 1 N 2 = N 1 N 2 ( = N1 N 2 (N 1 N 2 ), (N 1 N 2 ) \ (N 1 N 2 )) = P 1 P ( ) ( ) N1, N 1 = SP N2, N 2 = P N1 N 2 = N 1 N 2 = N 1 N 2 = N 2 N 1 ( N1 N 2, N 1 \ ) ( ) L N 2 = SP P Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

45 Application of rewrite rules We would like to ensure that the application of a rewrite rule to a well formed term preserves well formedness not trivial: well formedness can be easily violated e.g. the rewrite rule a a 1 applied to ( b ) L a produces ( b ) L a 1 A compartment safe rewrite rule is such that it does not add/remove occurrences of variables it does not moves variables from one compartment (content of a looping sequence) to another one The application of a compartment safe rewrite rule preserves well formedness To apply a compartment unsafe rewrite rule we require that its patterns are CLOSED its variables are instantiated with CLOSED terms Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

46 The semantics of LCLS Given a set of compartment safe rewrite rules R CS and a set of compartemnt unsafe rewrite rules R CU, the semantics of LCLS is given by the following rules (appcs) (appcu) P 1 P 2 R CS P 1 σ ɛ σ Σ α A P 1 ασ P 2 ασ P 1 P 2 R CU P 1 σ ɛ σ Σ wf α A P 1 ασ P 2 ασ (par) (cont) T 1 T 1 L(T 1 ) L(T 2 ) = {n 1,..., n M } n 1,..., n M fresh T 1 T 2 T 1 {n 1,..., n M/n 1,..., n M } T 2 T T L(S) L(T ) = {n 1,..., n M } n 1,..., n M fresh ( S ) L T ( S ) L T { n 1,..., n M/n 1,..., n M } where α is link renaming, L(T ) the set of links occurring twice in the top level compartment of T Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

47 Main theoretical result Theorem (Subject Reduction) Given a set of well formed rewrite rules R and a well formed term T T T = T well formed Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

48 An LCLS model of the EGF pathway (1) phosphorylations EGF EGFR nucleus SHC CELL MEMBRANE Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

49 An LCLS model of the EGF pathway (2) We model the EGFR protein as the sequence R E1 R E2 R I 1 R I 2 R E1 and R E2 are two extra cellular domains R I 1 and R I 2 are two intra cellular domains The rewrite rules of the model are EGF ( R E1 x ) L X EGF 1 ( R 1 E1 x ) L X (R1) ( R 1 E1 R E2 x R 2 E1 R E2 ỹ ) L X ( R 1 E1 R 3 E2 x R 2 E1 R 3 E2 ỹ ) L X (R2) ( R 1 E2 R I 1 x ) L X ( R 1 E2 PR I 1 x ) L X (R3) ( R 1 E2 PR I 1 R I 2 x R 1 E2 PR I 1 R I 2 ỹ ) L (SHC X ) ( R 1 E2 PR I 1 R 2 I 2 x R 1 E2 PR I 1 R I 2 ỹ ) L (SHC 2 X ) (R4) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

50 An LCLS model of the EGF pathway (3) Let us write EGFR for R E1 R E2 R I 1 R I 2 A possible evolution of the system is EGF EGF `EGFR EGFR EGFR L (SHC SHC) (R1) (R1) (R2) (R3) (R3) (R4) EGF 1 EGF `R E1 R 1 E2 R I 1 R I 2 EGFR EGFR L (SHC SHC) EGF 1 EGF 2 `R E1 R 1 E2 R I 1 R I 2 EGFR RE1 R 2 E2 R I 1 R I 2 L (SHC SHC) EGF 1 EGF 2 `R E1 R 1 E2 R 3 I 1 R I 2 EGFR RE1 R 2 E2 R 3 I 1 R I 2 L (SHC SHC) EGF 1 EGF 2 `R E1 R 1 E2 PR 3 I 1 R I 2 EGFR RE1 R 2 E2 R 3 I 1 R I 2 L (SHC SHC) EGF 1 EGF 2 `R E1 R 1 E2 PR 3 I 1 R I 2 EGFR RE1 R 2 E2 PR 3 I 1 R I 2 L (SHC SHC) EGF 1 EGF 2 `R E1 R 1 E2 PR 3 I 1 RI 4 2 EGFR RE1 R 2 E2 PR 3 I 1 R I 2 L (SHC 4 SHC) Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

51 Outline of the talk 1 Introduction Cells are complex interactive systems The EGF pathway and the lac operon 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 Bisimulations in CLS A labeled semantics for CLS Bisimulations in CLS Bisimulations applied to the CLS model of the lac operon 4 CLS variants Stochastic CLS LCLS 5 Future Work and References Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

52 Current and future work We developed a stochastic simulator based on Stochastic CLS currently, we are developing an intermediate language for stochastic simulation of biological systems (ssmsr) high level formalisms (Stochastic CLS, π calculus, etc...) can be translated into ssmsr we plan to develop analysis and verification techniques for ssmsr In order to model cell division and differentiation, tissues, etc... we are developing a spatial extension of CLS in which terms are placed and can move in a 2D/3D space We are translating Kohn s Molecular Interaction Maps into CLS Moreover: we plan to study other behavioural equivalences (traces, testing,... ) we plan to use CLS to study (in collaboration with biologists) retinal cell development and differentiation Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53

53 References P. Milazzo. Qualitative and Quantitative Formal Modeling of Biological Systems, PhD Thesis, Università di Pisa. R. Barbuti, A. Maggiolo-Schettini, P. Milazzo, P. Tiberi and A. Troina. Stochastic CLS for the Modeling and Simulation of Biological Systems. Submitted. R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and A. Troina. Bisimulations in Calculi Modelling Membranes. Formal Aspects of Computing, in press. R. Barbuti, G. Caravagna, A. Maggiolo-Schettini and P. Milazzo. An Intermediate Language for the Simulation of Biological Systems. Int. Workshop From Biology To Concurrency and back (FBTC 07), ENTCS 194 (3), pages 19 34, R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and A. Troina. The Calculus of Looping Sequences for Modeling Biological Membranes. Invited paper at the 8th Workshop on Membrane Computing (WMC8), LNCS 4860, pages 54 76, Springer, R. Barbuti, A. Maggiolo-Schettini and P. Milazzo. Extending the Calculus of Looping Sequences to Model Protein Interaction at the Domain Level. Int. Symposium on Bioinformatics Research and Applications (ISBRA 07), LNBI 4463, pages , Springer, R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and A. Troina. A Calculus of Looping Sequences for Modelling Microbiological Systems. Fundamenta Informaticae, volume 72, pages 21 35, Paolo Milazzo (Università di Pisa) Formal Methods and Systems Biology Verona January 22, / 53