1. François Fages CPCV, March 2004 Constraint-based Model Checking of Hybrid Systems: A First Experiment in Systems Biology François Fages, INRIA Rocquencourt http://contraintes.inria.fr/ Joint work with and Nathalie Chabrier-Rivier Sylvain Soliman In collaboration with ARC CPBIO http://contraintes.inria.fr/cpbio Alexander Bockmayr, Vincent Danos, Vincent Schächter et al.

2. François Fages CPCV, March 2004 Current revolution in Biology Elucidation of high-level biological processes in terms of their biochemical basis at the molecular level. Mass production of genomic and post-genomic data: ARN expression, protein synthesis, protein-protein interactions,… Need for a strong parallel effort on the formal representation of biological processes: Systems Biology. Need for formal tools for modeling and reasoning about their global behavior.

3. François Fages CPCV, March 2004 Formalisms for modeling biochemical systems Diagrammatic notation Boolean networks [Thomas 73] Milner’s pi–calculus [Regev-Silverman-Shapiro 99-01, Nagasali et al. 00] Concurrent transition systems [Chabrier-Chiaverini-Danos-Fages-Schachter 03] Biochemical abstract machine BIOCHAM [Chabrier-Fages-Soliman 03] Pathway logic [Eker-Knapp-Laderoute-Lincoln-Meseguer-Sonmez 02] Bio-ambients [Regev-Panina-Silverman-Cardelli-Shapiro 03] Differential equations Hybrid Petri nets [Hofestadt-Thelen 98, Matsuno et al. 00] Hybrid automata [Alur et al. 01, Ghosh-Tomlin 01] Hybrid concurrent constraint languages [Bockmayr-Courtois 01]

4. François Fages CPCV, March 2004 Our goal Beyond simulation, provide formal tools for querying, validating and completing biological models. Our proposal: Use of temporal logic CTL as a query language for models of biological processes; Use of concurrent transition systems for their modeling; Use of symbolic and constraint-based model checkers for automatically evaluating CTL queries in qualitative and quantitative models. Use of inductive logic programming for learning models In course, learn and teach bits of biology with logic programs.

5. François Fages CPCV, March 2004 Plan of the talk 1. Introduction 2. The Biochemical Abstract Machine BIOCHAM Simple algebra of cell compounds Modeling reactions with concurrent transition systems 3. Temporal logic CTL as a query language Example of the MAPK signaling pathway Symbolic model-checking with NuSMV in BIOCHAM 4.Kinetics models Constraint-based model checking with DMC 5. Conclusion and perspectives

6. François Fages CPCV, March 2004 2. A Simple Algebra of Cell Molecules Small molecules: covalent bonds (outer electrons shared) 50-200 kcal/mol 70% water 1% ions 6% amino acids (20), nucleotides (5), fats, sugars, ATP, ADP, … Macromolecules: hydrogen bonds, ionic, hydrophobic, Waals 1-5 kcal/mol Stability and bindings determined by the number of weak bonds: 3D shape 20% proteins (50-10 4 amino acids) RNA (10 2 -10 4 nucleotides AGCU) DNA (10 2 -10 6 nucleotides AGCT)

7. François Fages CPCV, March 2004 Formal proteins Cyclin dependent kinase 1 Cdk1 (free, inactive) Complex Cdk1-Cyclin B Cdk1–CycB (low activity) Phosphorylated form Cdk1~{thr161}-CycB at site threonine 161 (high activity) (BIOCHAM syntax)

8. François Fages CPCV, March 2004 Algebra of Cell Molecules E ::= Name|E-E|E~{E,…,E}|(E) S ::= _|E+S Names : molecules, proteins, #gene binding sites, abstract @processes… - : binding operator for protein complexes, gene binding sites, … Associative and commutative. ~{…} : modification operator for phosphorylated sites, … Set (Associative, Commutative, Idempotent). + : solution operator, “soup aspect”, Assoc. Comm. Idempotent, Neutral _ No membranes, no transport formalized. Bitonal calculi [Cardelli 03].

9. François Fages CPCV, March 2004 Concurrent Transition Syst. of Biochemical Reactions Enzymatic reactions: R ::= S=>S | S=[E]=>S | S=[R]=>S | S S | S S (where A B stands for A=>B B=>A and A=[C]=>B for A+C=>B+C, etc.) define a concurrent transition system over integers denoting the multiplicity of the molecules (multiset rewriting). One can associate a finite abstract CTS over boolean state variables denoting the presence/absence of molecules which correctly over-approximates the set of all possible behaviors a reaction A+B=>C+D is translated with 4 rules for possible consumption: A+B  A+B+C+D A+B   A+B +C+D A+B   A+  B+C+D A+B  A+  B+C+D

10. François Fages CPCV, March 2004 Six Rule Schemas Complexation: A + B => A-B Decomplexation A-B => A + B Cdk1+CycB => Cdk1–CycB Phosphorylation: A =[C]=> A~{p} Dephosphorylation A~{p} =[C]=> A Cdk1–CycB =[Myt1]=> Cdk1~{thr161}-CycB Cdk1~{thr14,tyr15}-CycB =[Cdc25~{Nterm}]=> Cdk1-CycB Synthesis: _ =[C]=> A. _ =[#Ge2-E2f13-Dp12]=> CycA Degradation: A =[C]=> _. CycE =[@UbiPro]=> _ (not for CycE-Cdk2 which is stable)

11. François Fages CPCV, March 2004 3. Temporal Logic CTL as a Query Language Computation Tree Logic Choice Time E exists A always X next time EX(  )AX(  ) F finally EF(  )  AG(  ) AF(  ) liveness G globally EG(  )  AF(   ) AG(  ) safety U until E (    U   )A (    U   )

12. François Fages CPCV, March 2004 Biological Queries About reachability: Given an initial state init, can the cell produce some protein P? init  EF(P) Which are the states from which a set of products P1,..., Pn can be produced simultaneously? EF(P1^…^Pn) About pathways: Can the cell reach a state s while passing by another state s 2 ? init  EF(s 2 ^EFs) Is state s 2 a necessary checkpoint for reaching state s?  EF(  s 2 U s) Can the cell reach a state s without violating some constraints c? init  EF(c U s)

13. François Fages CPCV, March 2004 Biological Queries About stability: Is a certain (partially described) state s a stable state? s  AG(s) s  AG(s) (s denotes both the state and the formula describing it). Is s a steady state (with possibility of escaping) ? s  EG(s) Can the cell reach a stable state? init  EF(AG(s)) not a LTL formula. Must the cell reach a stable state? init  AF(AG(s)) What are the stable states? Not expressible in CTL [Chan 00]. Can the system exhibit a cyclic behavior w.r.t. the presence of P ? init  EG((P  EF  P) ^ (  P  EF P))

14. François Fages CPCV, March 2004 MAPK Signaling Pathway RAF + RAFK RAF-RAFK. RAF~{p1} + RAFPH RAF~{p1}-RAFPH. MEK~$P + RAF~{p1} MEK~$P-RAF~{p1} where p2 not in $P. MEKPH + MEK~{p1}~$P MEK~{p1}~$P-MEKPH. MAPK~$P + MEK~{p1,p2} MAPK~$P-MEK~{p1,p2} where p2 not in $P. MAPKPH + MAPK~{p1}~$P MAPK~{p1}~$P-MAPKPH. RAF-RAFK => RAFK + RAF~{p1}. RAF~{p1}-RAFPH => RAF + RAFPH. MEK~{p1}-RAF~{p1} => MEK~{p1,p2} + RAF~{p1}. MEK-RAF~{p1} => MEK~{p1} + RAF~{p1}. MEK~{p1}-MEKPH => MEK + MEKPH. MEK~{p1,p2}-MEKPH => MEK~{p1} + MEKPH. MAPK-MEK~{p1,p2} => MAPK~{p1} + MEK~{p1,p2}. MAPK~{p1}-MEK~{p1,p2} => MAPK~{p1,p2} + MEK~{p1,p2}. MAPK~{p1}-MAPKPH => MAPK + MAPKPH. MAPK~{p1,p2}-MAPKPH => MAPK~{p1} + MAPKPH.

15. François Fages CPCV, March 2004 MAPK Signaling Pathway MEK~{p1} is a checkpoint for producing MAPK~{p1,p2} biocham: !E(!MEK~{p1} U MAPK~{p1,p2}) True The PH complexes are not compulsory for the cascade biocham: !E(!MEK~{p1}-MEKPH U MAPK~{p1,p2}) false Step 1 rule 15 Step 2 rule 1 RAF-RAFK present Step 3 rule 21 RAF~{p1} present Step 4 rule 5 MEK-RAF~{p1} present Step 5 rule 24 MEK~{p1} present Step 6 rule 7 MEK~{p1}-RAF~{p1} present Step 7 rule 23 MEK~{p1,p2} present Step 8 rule 13 MAPK-MEK~{p1,p2} present Step 9 rule 27 MAPK~{p1} present Step 10 rule 15 MAPK~{p1}-MEK~{p1,p2} present Step 11 rule 28 MAPK~{p1,p2} present

16. François Fages CPCV, March 2004 Mammalian Cell Cycle Control Map [Kohn 99]

17. François Fages CPCV, March 2004 Mammalian Cell Cycle Control Benchmark 700 rules, 165 proteins and genes, 500 variables, 2 500 states. BIOCHAM NuSMV model-checker time in seconds: Initial state G2Query:Time: compiling29 Reachability G1EF CycE2 Reachability G1EF CycD1.9 Reachability G1EF PCNA-CycD1.7 Checkpoint for mitosis complex  EF (  Cdc25~{Nterm} U Cdk1~{Thr161}-CycB) 2.2 Cycle EG ( (CycA  EF  CycA)  (  CycA  EF CycA)) 31.8

18. François Fages CPCV, March 2004 4. Kinetics Models Enzymatic reactions with rates k 1 k 2 k 3 E+S  k1 C  k2 E+P E+S  k3 C can be compiled by the law of mass action into a system of Michaelis-Menten Ordinary Differential Equations (non-linear) dE/dt = -k 1 ES+(k 2 +k 3 )C dS/dt = -k 1 ES+k 3 C dC/dt = k 1 ES-(k 2 +k 3 )C dP/dt = k 2 C

19. François Fages CPCV, March 2004 MAPK kinetics model

20. François Fages CPCV, March 2004 Gene Interaction Networks Gene interaction example [Bockmayr-Courtois 01] Hybrid Concurrent Constraint Programming HCC [Saraswat et al.] 2 genes x and y. Hybrid linear approximation dx/dt = 0.01 – 0.02*x if y < 0.8 dx/dt = – 0.02*x if y ≥ 0.8 dy/dt = 0.01*x

21. François Fages CPCV, March 2004 Concurrent Transition System Time discretization using Euler’s method: y < 0.8  x’ = x + dt*(0.01-0.02*x), y’ = y + dt*0.01*x y ≥ 0.8  x’ = x + dt*(0.01-0.02*x), y’ = y + dt*0.01*x Initial condition: x=0, y=0. CLP(R) program (dt=1) Init :- X=0, Y=0, p(X,Y). p(X,Y):-X>=0, Y>=0, Y<0.8, X1=X-0.02*X+0.01, Y1=Y+0.01*X, p(X1,Y1). p(X,Y):-X>=0, Y>=0, Y>=0.8, X1=X-0.02*X, Y1=Y+0.01*X, p(X1,Y1).

22. François Fages CPCV, March 2004 Proving CTL properties by computing fixpoints of CLP programs Theorem [Delzanno Podelski 99] EF(f)=lfp(T P  {p(x):-f} ), EG(f)=gfp(T P  f ). Safety property AG(  f) iff  EF(f) iff init  lfp(T P  {f} ) Liveness property AG(f1  AF(f2)) iff init  lfp(T P  f1  gfp(T P  {f2} ) ) Implementation in Sicstus-Prolog CLP(R,B) [Delzanno 00]

23. François Fages CPCV, March 2004 Deductive Model Checker DMC: Gene Interaction r(init, p(s_s,A,B), {A=0,B=0}). r(p(s_s,A,B), p(s_s,C,D), {A>=0,B>=0.8,C=A-0.02*A,D=B+0.01*A}). r(p(s_s,A,B), p(s_s,C,D), {A>=0,B>=0,B<0.8, C=A-0.02*A+0.01,D=B+0.01*A}). | ?- prop(P,S). P = unsafe, S = p:s*(x>=0.6) | ?- ti. Property satisfied. Execution time 0.0 | ?- ls. s(0, p(s_s,A,_), {A>=0.6}, 1, (0,0)).

24. François Fages CPCV, March 2004 Gene interaction (continued) | ?- prop(P,S). P = unsafe, S = p:s*(x>=0.2) ? | ?- ti. Property NOT satisfied. Execution time 1.5 | ?- ls. s(0, p(s_s,A,_), {A>=0.2}, 1, (0,0)). s(1, p(s_s,A,B), {B =-0.0,A>=0.19387755102040816}, 2, (2,1)). … s(26, p(s_s,A,B), {B>=0.0,A>=0.0, B+0.1982676351105516*A<0.7741338175552753}, 27, (2,26)). s(27, init, {}, 28, (1,27)).

25. François Fages CPCV, March 2004 Conclusion and Perspectives The biochemical abstract machine BIOCHAM provides: a first-order-rule-based language for modeling biochemical systems a powerful query language based on temporal logic CTL Implementation in Prolog + model-checker NuSMV + Constraint-based model checker DMC for Ordinary Differential Equations (Euler method) models of metabolic and signaling pathways, cell-cycle control,… Combination of boolean models with ODE models Proof of concept, issue of scaling-up: efficient constraints, abstractions STREP APrIL 2: learning of reaction weights and rules. http://www.rewerse.net EU 6th PCRD NoE REWERSE semantic web for bioinformatics http://www.rewerse.net