François Fages
CPCV, March 2004
Constraint-based Model Checking ofHybrid Systems:A First Experiment in Systems BiologyFrançois 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.
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 ofbiological processes: Systems Biology.
•Need for formal tools for modeling and reasoning about their globalbehavior.
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-Schachter03]
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]
François Fages
CPCV, March 2004
Our goal
Beyond simulation, provide formal tools for querying, validating andcompleting biological models.
Our proposal:
•Use of temporal logic CTL as a query language for models of biologicalprocesses;
•Use of concurrent transition systems for their modeling;
•Use of symbolic and constraint-based model checkers for automaticallyevaluating 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.
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
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 (50-104 amino acids)
• RNA (102-104 nucleotides AGCU)
• DNA (102-106 nucleotides AGCT)
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)
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].
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<=[E]=>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 themultiplicity of the molecules (multiset rewriting).
One can associate a finite abstract CTS over boolean state variablesdenoting 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+BA+B+C+D A+BA+B +C+D
A+BA+B+C+D A+BA+B+C+D
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)
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 )
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 beproduced simultaneously? EF(P1^…^Pn)
About pathways:
•Can the cell reach a state s while passing by another state s2?init EF(s2^EFs)
•Is state s2 a necessary checkpoint for reaching state s? EF(s2U s)
•Can the cell reach a state s without violating some constraints c?init EF(c U s)
François Fages
CPCV, March 2004
Biological Queries
About stability:
•Is a certain (partially described) state s a stable state? sAG(s)sAG(s) (s denotes both the state and the formula describing it).
•Is s a steady state (with possibility of escaping) ? sEG(s)
•Can the cell reach a stable state? initEF(AG(s))not a LTL formula.
•Must the cell reach a stable state? initAF(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))
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.
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
François Fages
CPCV, March 2004
Mammalian Cell Cycle Control Map [Kohn 99]
François Fages
CPCV, March 2004
Mammalian Cell Cycle Control Benchmark
700 rules, 165 proteins and genes, 500 variables, 2500 states.
BIOCHAM NuSMV model-checker time in seconds:
Initial state G2
Query:
Time:
compiling
29
Reachability G1
EF CycE
2
Reachability G1
EF CycD
1.9
Reachability G1
EF PCNA-CycD
1.7
Checkpoint
for mitosis complex
EF ( Cdc25~{Nterm}
U Cdk1~{Thr161}-CycB)
2.2
Cycle
EG ( (CycA EF CycA)
( CycA EF CycA))
31.8
François Fages
CPCV, March 2004
4. Kinetics Models
Enzymatic reactions with rates k1 k2 k3
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 = -k1ES+(k2+k3)C
dS/dt = -k1ES+k3C
dC/dt = k1ES-(k2+k3)C
dP/dt = k2C
François Fages
CPCV, March 2004
MAPK kinetics model
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
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).
François Fages
CPCV, March 2004
Proving CTL properties by computing fixpoints ofCLP programs
Theorem [Delzanno Podelski 99]
EF(f)=lfp(TP{p(x):-f}),
EG(f)=gfp(TPf ).
Safety property AG(f) iff EF(f) iff initlfp(TP{f})
Liveness property AG(f1AF(f2)) iff initlfp(TPf1gfp(T P{f2} ) )
Implementation in Sicstus-Prolog CLP(R,B) [Delzanno 00]
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)).
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<0.8,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)).
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-basedmodel 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