The Calculus of Looping Sequences

Transcription

1 The Calculus of Looping Sequences Roberto Barbuti, Giulio Caravagna, Andrea MaggioloSchettini, Paolo Milazzo, Giovanni Pardini Dipartimento di Informatica, Università di Pisa, Italy Bertinoro June 7, 2008 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

2 Outline of the talk 1 Introduction Cells are complex interactive systems 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 CLS variants Stochastic CLS LCLS Spatial CLS 4 Future Work and References Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

3 Cells: complex systems of interactive components 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. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

4 Outline of the talk 1 Introduction Cells are complex interactive systems 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 CLS variants Stochastic CLS LCLS Spatial CLS 4 Future Work and References Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

5 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. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

6 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

7 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. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

8 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

9 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

10 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. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

11 CLS modeling examples: the EGF pathway (1) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

12 CLS modeling examples: the EGF pathway (2) phosphorylations EGF EGFR nucleus SHC CELL MEMBRANE Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

13 CLS modeling examples: the EGF pathway (3) 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

14 CLS modeling examples: the EGFR pathway (4) 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

15 CLS modeling examples: the lac operon (1) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

16 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

17 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 x RO ỹ repr x laco ỹ repr LACT RLACT RLACT repr LACT ) 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) (R14) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

18 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 Spatial CLS Description of physical size and position of entities Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

19 Outline of the talk 1 Introduction Cells are complex interactive systems 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 CLS variants Stochastic CLS LCLS Spatial CLS 4 Future Work and References Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

20 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

21 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 M. ν=1 aν At each step t is incremented by τ and the chemical solution is updated. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

22 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

23 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

24 A Stochastic CLS model of the lac operon (1) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

25 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

26 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

27 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

28 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

29 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

30 A Stochastic CLS model of the Quorum Sensing (1) It is recognised that many bacteria have the ability of modulating their gene expressions according to their population density. This process is called quorum sensing. A diffusible small molecules (called autoinducers) One or more transcriptional activator proteins (R-proteins) located within the cell The autoinducer can cross freely the cellular membrane The R-protein by itself is not active without the autoinducer. The autoinducer molecule can bind to the R-protein to form an autoinducer/r-protein complex. The autoinducer/r-protein complex binds to the DNA enhancing the transcription of specific genes. These genes regulate both the production of specific behavioural traits and the production of the autoinducer and of the R-protein. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

31 A Stochastic CLS model of the Quorum Sensing (2) At low cell density, the autoinducer is synthesized at basal levels and diffuse in the environment where it is diluted. With high cell density the concentration of the autoinducer increases. Beyond a treshold the autoinducer is produced autocatalytically. The autocatalytic production results in a dramatic increase of product concentration. dna ++ LasB LasR LasI LasR 3-oxo-C12-HSL Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

32 A Stochastic CLS model of the Quorum Sensing (3) The behaviour of a single bacterium: 20 laso lasr lasi laso lasr lasi LasR (R1) laso lasr lasi 5 laso lasr lasi LasI (R2) LasI 8 LasI 3oxo (R3) 3oxo LasR R 3R 400 3oxo LasR (R4-R5) 3R laso lasr lasi RO lasr lasi (R6) 3RO lasr lasi 10 3R laso lasr lasi (R7) 3RO lasr lasi RO lasr lasi LasR (R8) 300 3RO lasr lasi 3RO lasr lasi LasI (R9) ( ) L 30 m (3oxo X ) 3oxo ( m ) L X (R10) 3oxo ( m ) L 1 ( ) L X m (3oxo X ) (R11) LasI 1 ɛ LasR 1 ɛ 3oxo 1 ɛ (R12-S14) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

33 Simulation results (1) 120 autoinducer 100 Number of elements Time (sec) Production of the autoinducer by a single bacterium Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

34 Simulation results (2) 120 autoinducer 100 Number of elements Time (sec) Production of the autoinducer by a population of five bacteria Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

35 Simulation results (3) 600 autoinducer 500 Number of elements Time (sec) Production of the autoinducer by a population of twenty bacteria Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

36 Outline of the talk 1 Introduction Cells are complex interactive systems 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 CLS variants Stochastic CLS LCLS Spatial CLS 4 Future Work and References Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

37 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

38 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. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

39 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

40 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

41 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

42 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

43 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 Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

44 An LCLS model of the EGF pathway (1) phosphorylations EGF EGFR nucleus SHC CELL MEMBRANE Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

45 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) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

46 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 ( RE1 R 1 E2 R I 1 R I 2 EGFR EGFR )L (SHC SHC) EGF 1 EGF 2 ( RE1 R 1 E2 R I 1 R I 2 EGFR RE1 R 2 ) L E2 R I 1 R I 2 (SHC SHC) EGF 1 EGF 2 ( RE1 R 1 E2 R 3 I 1 R I 2 EGFR RE1 R 2 E2 R 3 ) L I 1 R I 2 (SHC SHC) EGF 1 EGF 2 ( RE1 R 1 E2 PR 3 I 1 R I 2 EGFR RE1 R 2 E2 R 3 ) L I 1 R I 2 (SHC SHC) EGF 1 EGF 2 ( RE1 R 1 E2 PR 3 I 1 R I 2 EGFR RE1 R 2 E2 PR 3 ) L I 1 R I 2 (SHC SHC) EGF 1 EGF 2 ( RE1 R 1 E2 PR 3 I 1 RI 4 2 EGFR RE1 R 2 E2 PR 3 ) L I 1 R I 2 (SHC 4 SHC) Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

47 Outline of the talk 1 Introduction Cells are complex interactive systems 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 CLS variants Stochastic CLS LCLS Spatial CLS 4 Future Work and References Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

48 Spatial CLS Spatial CLS extends the CLS with space and time: elements can be associated with spheres in a 2D/3D space, and each sphere represents: the space occupied by the element the space available inside membrane elements can move autonomously during the passage of time the applicability of reactions can be constrained to the position of the elements involved Spatial CLS can be especially useful to describe biological processes where the behaviour depends on the exact position of the elements, in order to obtain faithful representation of their evolution. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

49 Syntax of Spatial CLS Terms T, branes B and sequences S of Spatial CLS are defined as: T ::= λ B ::= ( S ) d S ::= ɛ ( S ) a d B B S S ( B ) L d T T T where a is a generic element of the alphabet E, ɛ is the empty sequence, and λ is the empty term. Parameter d describes the spatial information of elements. Elements can be: positional, when d = [p, m], r non-positional, when d =, r where p is the center of the sphere, r the radius, and m denotes a movement function. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

50 Representing the movement A movement function gives new position p of the element, reached after a time interval δt from the current time t: where p = m fun (p, r, x, l, t, δt) p: current position of the element (at time t) r: radius of the element x: where the element appears: on the surface or inside a membrane l: radius of the containing membrane or Example Linear motion: m fun (p, r, x, l, t, δt) = q + vt Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

51 Example of term Term T 1 = ( a ) [(1,2),m 1 ],0.5 ( b c d ) L [(4,3),m 2 ],2 ( a ) [( 1,0),m 3 ],0.5 can be represented graphically as: Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

52 Semantics Rewrite rules are of the form [ f c ] P L k PR where k is the reaction rate parameter (like in Stochastic CLS) f c are the application constraints, which express the applicability of the rule according to the position of the elements A system evolves by performing a sequence of steps, where each step is composed of two phases: 1 at most one rewrite rule is applied, among those enable in the state 2 the elements are moved according to their movement functions Non-positional elements are assumed homogeneously distributed, thus the rate of reactions involving them are in accordance to the Law of Mass Action. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

53 Resolving space conflicts During the evolution of the system, conflicts between the space occupied by different elements may arise. In such cases, the elements are rearranged, as if they push each other. Example Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

54 Cell proliferation Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

55 A model of cell proliferation The initial state of the system is described by the following term: T = ( b ) L,50 ( m ) L [(0,0),m 1 ],10 ( n ) L (cr g1 g 2 g 3 cr g 4 g 5 ) The subterms appearing in T represent: ( ) b L,50 : the space available ( ) L m : the membrane of the cell [(0,0),m 1 ],10 ( ) L: n the nucleus cr...: the chromosomes Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

56 A model of cell proliferation The initial state of the system is described by the following term: T = ( b ) L,50 ( m ) L [(0,0),m 1 ],10 ( n ) L (cr g1 g 2 g 3 cr g 4 g 5 ) The subterms appearing in T represent: ( ) b L,50 : the space available ( ) L m : the membrane of the cell [(0,0),m 1 ],10 ( ) L: n the nucleus cr...: the chromosomes Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

57 A model of cell proliferation The initial state of the system is described by the following term: T = ( b ) L,50 ( m ) L [(0,0),m 1 ],10 ( n ) L (cr g1 g 2 g 3 cr g 4 g 5 ) The subterms appearing in T represent: ( ) b L,50 : the space available ( ) L m : the membrane of the cell [(0,0),m 1 ],10 ( ) L: n the nucleus cr...: the chromosomes Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

58 A model of cell proliferation The initial state of the system is described by the following term: T = ( b ) L,50 ( m ) L [(0,0),m 1 ],10 ( n ) L (cr g1 g 2 g 3 cr g 4 g 5 ) The subterms appearing in T represent: ( ) b L,50 : the space available ( ) L m : the membrane of the cell [(0,0),m 1 ],10 ( ) L: n the nucleus cr...: the chromosomes Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

59 A model of cell proliferation The initial state of the system is described by the following term: T = ( b ) L,50 ( m ) L [(0,0),m 1 ],10 ( n ) L (cr g1 g 2 g 3 cr g 4 g 5 ) The subterms appearing in T represent: ( ) b L,50 : the space available ( ) L m : the membrane of the cell [(0,0),m 1 ],10 ( ) L: n the nucleus cr...: the chromosomes Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

60 Rewrite rules The evolution of the system is modeled by the following rewrite rules: R 1 : [ r = 7 ] R 2 : [ r = 10 ] R 3 : [ r = 14 ] ( ) L m [p,f ],r X 0.33 ( m ) L [p,f ],10 X ( ) L m [p,f ],r X 0.25 ( m ) L [p,f ],14 X ( ) ( L (n m u ) ) L 0.5 X ( m ) ( L (ndup u ) ) L X R 4 : ( n dup ) L (cr x X ) ( n dup ) L (2cr x X ) R 5 : ( ) L 0.17 n dup (2cr x 2cr ỹ) ( ) L ( ) L n (cr x cr ỹ) n (cr x cr ỹ) R 6 : ( m ) ( L (n [(x,y),f ],r ) L ( ) ) L 1 X n Y ( ) L m [(x 5,y),f ],7 ( n ) L ( ) L X m [(x+5,y),f ],7 ( n ) L Y The rates are expressed with respect to a time unit of 1 hour. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

61 Outline of the talk 1 Introduction Cells are complex interactive systems 2 The Calculus of Looping Sequences (CLS) Definition of CLS The EGF pathway and the lac operon in CLS 3 CLS variants Stochastic CLS LCLS Spatial CLS 4 Future Work and References Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

62 Current and future work We developed a stochastic simulator based on Stochastic CLS We have defined 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 We are translating Kohn s Molecular Interaction Maps (MIM) into CLS We plan to define the evolution of CLS terms by means of operators based on MIM. Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59

63 References R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and G. Pardini. Spatial Calculus of Looping Sequences. Submitted. R. Barbuti, A. Maggiolo-Schettini, P. Milazzo, P. Tiberi and A. Troina. Stochastic CLS for the Modeling and Simulation of Biological Systems. Transactions on Computational Systems Biology, in press. R. Barbuti, G. Caravagna, A. Maggiolo-Schettini and P. Milazzo. An Intermediate Language for the Stochastic Simulation of Biological Systems. Theoretical Computer Science, in press. R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and A. Troina. Bisimulations in Calculi Modelling Membranes. Formal Aspects of Computing, in press. R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and A. Troina. The Calculus of Looping Sequences for Modeling Biological Membranes. WMC8, LNCS 4860, 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. ISBRA 07, LNBI 4463, , Springer, R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and A. Troina. A Calculus of Looping Sequences for Modelling Microbiological Systems. Fundamenta Informaticae, volume 72, 21 35, Roberto Barbuti (Università di Pisa) The Calculus of Looping Sequences Bertinoro June 7, / 59