Constraint-based Model Checking of Hybrid Systems: A First Experiment in Systems Biology Fran

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Constraint-based Model Checking of Hybrid
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.

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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.

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Formalisms for modeling biochemical systems

• Diagrammatic notation
• Boolean networks Thomas 73
• Milners picalculus 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-Me
  seguer-Sonmez 02
• Bio-ambients Regev-Panina-Silverman-Cardelli-Shap
  iro 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

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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.

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Plan of the talk

• Introduction
• The Biochemical Abstract Machine BIOCHAM
• Simple algebra of cell compounds
• Modeling reactions with concurrent transition
  systems
• Temporal logic CTL as a query language
• Example of the MAPK signaling pathway
• Symbolic model-checking with NuSMV in BIOCHAM
• Kinetics models
• Constraint-based model checking with DMC
• Conclusion and perspectives

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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)

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Formal proteins

• Cyclin dependent kinase 1 Cdk1
• (free, inactive)
• Complex Cdk1-Cyclin B Cdk1CycB
• (low activity)
• Phosphorylated form Cdk1thr161-CycB
• at site threonine 161
• (high activity)
• 
  (BIOCHAM syntax)

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Algebra of Cell Molecules

• E NameE-EEE,,E(E) S _ES
• Names molecules, proteins, gene binding sites,
  abstract _at_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.

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Concurrent Transition Syst. of Biochemical
Reactions

• Enzymatic reactions
• R SgtS SEgtS SRgtS SltgtS
  SltEgtS
• (where AltgtB stands for AgtB BgtA and ACgtB
  for ACgtBC, 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 ABgtCD is translated with 4 rules
  for possible consumption
• AB?ABCD AB??AB CD
• AB??A?BCD AB?A?BCD

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Six Rule Schemas

• Complexation A B gt A-B
  Decomplexation A-B gt A B
• Cdk1CycB gt Cdk1CycB
• Phosphorylation A Cgt Ap
  Dephosphorylation Ap Cgt A
• Cdk1CycB Myt1gt Cdk1thr161-CycB
• Cdk1thr14,tyr15-CycB Cdc25Ntermgt
  Cdk1-CycB
• Synthesis _ Cgt A.
• _ Ge2-E2f13-Dp12gt CycA
• Degradation A Cgt _.
• CycE _at_UbiProgt _ (not for CycE-Cdk2 which
  is stable)

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3. Temporal Logic CTL as a Query Language

• Computation Tree Logic

Choice Time E exists  A always
X next time EX(f) AX(f)
F finally EF(f) ? AG(?f) AF(f) liveness
G globally EG(f) ? AF(? f) AG(f) safety
U until E (f1 U f2) A (f1 U f2)
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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(P1Pn)
• About pathways
• Can the cell reach a state s while passing by
  another state s2? init ? EF(s2EFs)
• 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)

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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))

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MAPK Signaling Pathway

• RAF RAFK ltgt RAF-RAFK.
• RAFp1 RAFPH ltgt RAFp1-RAFPH.
• MEKP RAFp1 ltgt MEKP-RAFp1
• where p2 not in P.
• MEKPH MEKp1P ltgt MEKp1P-MEKPH.
• MAPKP MEKp1,p2 ltgt MAPKP-MEKp1,p2
• where p2 not in P.
• MAPKPH MAPKp1P ltgt MAPKp1P-MAPKPH.
• RAF-RAFK gt RAFK RAFp1.
• RAFp1-RAFPH gt RAF RAFPH.
• MEKp1-RAFp1 gt MEKp1,p2 RAFp1.
• MEK-RAFp1 gt MEKp1 RAFp1.
• MEKp1-MEKPH gt MEK MEKPH.
• MEKp1,p2-MEKPH gt MEKp1 MEKPH.
• MAPK-MEKp1,p2 gt MAPKp1 MEKp1,p2.
• MAPKp1-MEKp1,p2 gt MAPKp1,p2
  MEKp1,p2.
• MAPKp1-MAPKPH gt MAPK MAPKPH.
• MAPKp1,p2-MAPKPH gt MAPKp1 MAPKPH.

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MAPK Signaling Pathway

• MEKp1 is a checkpoint for producing
  MAPKp1,p2
• biocham !E(!MEKp1 U MAPKp1,p2)
• True
• The PH complexes are not compulsory for the
  cascade
• biocham !E(!MEKp1-MEKPH U MAPKp1,p2)
• false
• Step 1 rule 15
• Step 2 rule 1 RAF-RAFK present
• Step 3 rule 21 RAFp1 present
• Step 4 rule 5 MEK-RAFp1 present
• Step 5 rule 24 MEKp1 present
• Step 6 rule 7 MEKp1-RAFp1 present
• Step 7 rule 23 MEKp1,p2 present
• Step 8 rule 13 MAPK-MEKp1,p2 present
• Step 9 rule 27 MAPKp1 present
• Step 10 rule 15 MAPKp1-MEKp1,p2 present
• Step 11 rule 28 MAPKp1,p2 present

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Mammalian Cell Cycle Control Map Kohn 99
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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 (? Cdc25Nterm U Cdk1Thr161-CycB) 2.2
Cycle EG ( (CycA ? EF ? CycA) ? (? CycA ? EF CycA)) 31.8
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4. Kinetics Models

• Enzymatic reactions with rates k1 k2 k3
• ES ?k1 C ?k2 EP
• ES ?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(k2k3)C
• dS/dt -k1ESk3C
• dC/dt k1ES-(k2k3)C
• dP/dt k2C

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MAPK kinetics model
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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 0.02x if y lt 0.8
• dx/dt 0.02x if y 0.8
• dy/dt 0.01x

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Concurrent Transition System

• Time discretization using Eulers method
• y lt 0.8 ? x x dt(0.01-0.02x) , y y
  dt0.01x
• y 0.8 ? x x dt(0.01-0.02x) , y y
  dt0.01x
• Initial condition x0, y0.
• CLP(R) program (dt1)
• Init - X0, Y0, p(X,Y).
• p(X,Y)-Xgt0, Ygt0, Ylt0.8,
• X1X-0.02X0.01, Y1Y0.01X,
  p(X1,Y1).
• p(X,Y)-Xgt0, Ygt0, Ygt0.8,
• X1X-0.02X, Y1Y0.01X,
  p(X1,Y1).

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Proving CTL properties by computing fixpoints of
CLP programs
Theorem Delzanno Podelski 99
EF(f)lfp(TP?p(x)-f), EG(f)gfp(TP?f
). Safety property AG(?f) iff ?EF(f) iff
init?lfp(TP?f) Liveness property
AG(f1?AF(f2)) iff init?lfp(TP?f1?gfp(T P?f2 )
) Implementation in Sicstus-Prolog CLP(R,B)
Delzanno 00
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Deductive Model Checker DMC Gene Interaction

• r(init, p(s_s,A,B), A0,B0).
• r(p(s_s,A,B), p(s_s,C,D), Agt0,Bgt0.8,CA-0.02A,
  DB0.01A).
• r(p(s_s,A,B), p(s_s,C,D), Agt0,Bgt0,Blt0.8,
• 
  CA-0.02A0.01,DB0.01A).
• ?- prop(P,S).
• P unsafe, S ps(xgt0.6)
• ?- ti.
• Property satisfied. Execution time 0.0
• ?- ls.
• s(0, p(s_s,A,_), Agt0.6, 1, (0,0)).

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Gene interaction (continued)

• ?- prop(P,S).
• P unsafe, S ps(xgt0.2) ?
• ?- ti.
• Property NOT satisfied. Execution time 1.5
• ?- ls.
• s(0, p(s_s,A,_), Agt0.2, 1, (0,0)).
• s(1, p(s_s,A,B), Blt0.8,Bgt-0.0,Agt0.1938775510204
  0816, 2, (2,1)).
• s(26, p(s_s,A,B), Bgt0.0,Agt0.0,
• B0.1982676351105516Alt0.7741338175552753,
  27, (2,26)).
• s(27, init, , 28, (1,27)).

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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