Semantical and Algorithmic Aspects of the Living Franois Fages Constraint Programming projectteam, I

1
Semantical and Algorithmic Aspects of the
LivingFrançois FagesConstraint Programming
project-team, INRIA Paris-Rocquencourt

• To tackle the complexity of biological systems,
  investigate
• Programming Theory Concepts
• Formal Methods of Circuit and Program
  Verification
• Automated Reasoning Tools
• Prototype Implementation in the Biochemical
  Abstract Machine BIOCHAM
• modeling environment available at
  http//contraintes.inria.fr/BIOCHAM

2
Systems Biology ?

• Systems Biology aims at systems-level
  understanding which
• requires a set of principles and methodologies
  that links the
• behaviors of molecules to systems characteristics
  and functions.
• H. Kitano, ICSB 2000
• Analyze (post-)genomic data produced with
  high-throughput technologies (stored in databases
  like GO, KEGG, BioCyc, etc.)
• Integrate heterogeneous data about a specific
  problem
• Understand and Predict behaviors or interactions
  in large networks of genes and proteins.
• Systems Biology Markup Language (SBML) exchange
  format for reaction models

3
Issue of Abstraction

• Models are built in Systems Biology with two
  contradictory perspectives

4
Issue of Abstraction

• Models are built in Systems Biology with two
  contradictory perspectives
• 1) Models for representing knowledge the more
  concrete the better

5
Issue of Abstraction

• Models are built in Systems Biology with two
  contradictory perspectives
• 1) Models for representing knowledge the more
  concrete the better
• 2) Models for making predictions the more
  abstract the better !

6
Issue of Abstraction

• Models are built in Systems Biology with two
  contradictory perspectives
• 1) Models for representing knowledge the more
  concrete the better
• 2) Models for making predictions the more
  abstract the better !
• These perspectives can be reconciled by
  organizing formalisms and models into hierarchies
  of abstractions.
• To understand a system is not to know everything
  about it but to know
• abstraction levels that are sufficient for
  answering questions about it

7
Semantics of Living Processes ?

• Formally, the behavior of a system depends on
  our choice of observables.
• ?
  ?

Mitosis movie Lodish et al. 03
8
Boolean Semantics

• Formally, the behavior of a system depends on
  our choice of observables.
• Presence/absence of molecules
• Boolean transitions

0
1
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Continuous Differential Semantics

• Formally, the behavior of a system depends on
  our choice of observables.
• Concentrations of molecules
• Rates of reactions

x
ý
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Stochastic Semantics

• Formally, the behavior of a system depends on
  our choice of observables.
• Numbers of molecules
• Probabilities of reaction

n
?
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Temporal Logic Semantics

• Formally, the behavior of a system depends on
  our choice of observables.
• Presence/absence of molecules
• Temporal logic formulas

F x F (x F (? x y)) FG (x v y)
F
x
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Constraint Temporal Logic Semantics

• Formally, the behavior of a system depends on
  our choice of observables.
• Concentrations of molecules
• Constraint LTL temporal formulas

F (x gt0.2) F (x gt0.2 F (xlt0.1 ygt0.2)) FG
(xgt0.2 v ygt0.2)
F
xgt1
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Language-based Approaches to Cell Systems Biology

• Qualitative models from diagrammatic notation to
• Boolean networks Kaufman 69, Thomas 73
• Petri Nets Reddy 93, Chaouiya 05
• Process algebra pcalculus Regev-Silverman-Shapir
  o 99-01, Nagasali et al. 00
• Bio-ambients Regev-Panina-Silverman-Cardelli-Shap
  iro 03
• Pathway logic Eker-Knapp-Laderoute-Lincoln-Mesegu
  er-Sonmez 02
• Reaction rules Chabrier-Fages 03
  Chabrier-Chiaverini-Danos-Fages-Schachter 04
• Quantitative models from ODEs and stochastic
  simulations to
• Hybrid Petri nets Hofestadt-Thelen 98, Matsuno
  et al. 00
• Hybrid automata Alur et al. 01, Ghosh-Tomlin 01
  HCC Bockmayr-Courtois 01
• Stochastic pcalculus Priami et al. 03
  Cardelli et al. 06
• Reaction rules with continuous time dynamics
  Fages-Soliman-Chabrier 04

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Overview of the Talk

• Rule-based Modeling of Biochemical Systems
• Syntax of molecules, compartments and reactions
• Semantics at three abstraction levels boolean,
  differential, stochastic
• Cell cycle control models
• Temporal Logic Language for Formalizing
  Biological Properties
• CTL for the boolean semantics
• Constraint LTL for the differential semantics
• PCTL for the stochastic semantics
• Automated Reasoning Tools
• Inferring kinetic parameter values from
  Constraint-LTL specification
• Inferring reaction rules from CTL specification
• L. Calzone, N. Chabrier, F. Fages, S. Soliman.
  TCSB VI, LNBI 422068-94. 2006.

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Molecules of the living

• Small molecules covalent bonds 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
• 20 proteins (50-104 amino acids)
• RNA (102-104 nucleotides AGCU)
• DNA (102-106 nucleotides AGCT)

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Structure Levels of Proteins

• 1) Primary structure word of n amino acids
  residues (20n possibilities)
• linked with C-N bonds
• Example INRIA
• Isoleucine-asparagiNe-aRginine-Isoleucine-Alanine

17
Structure Levels of Proteins

• 1) Primary structure word of n amino acids
  residues (20n possibilities)
• linked with C-N bonds
• Example INRIA
• Isoleucine-asparagiNe-aRginine-Isoleucine-Alanine
• 2) Secondary word of m a-helix, b-strands,
  random coils, (3m-10m)
• stabilized by hydrogen
  bonds H---O

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Structure Levels of Proteins

• 1) Primary structure word of n amino acids
  residues (20n possibilities)
• linked with C-N bonds
• Example INRIA
• Isoleucine-asparagiNe-aRginine-Isoleucine-Alanine
• 2) Secondary word of m a-helix, b-strands,
  random coils, (3m-10m)
• stabilized by hydrogen
  bonds H---O
• 3) Tertiary 3D structure spatial folding
• 
  stabilized by
• 
  hydrophobic
• 
  interactions

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Syntax of 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)
• mitosis promotion factor
  

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BIOCHAM Syntax of Objects

• E compound E-E Ep1,,pn
• Compound molecule, gene binding site, abstract
  _at_process
• - binding operator for protein complexes, gene
  binding sites,
• Associative and commutative.
• modification operator for phosphorylated
  sites,
• Set of modified sites
  (Associative, Commutative, Idempotent).
• O E Elocation
• Location symbolic compartment (nucleus,
  cytoplasm, membrane, )
• S _ OS
• solution operator (Associative, Commutative,
  Neutral _)

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Elementary Reaction Rules

• Complexation A B gt A-B
  Decomplexation A-B gt A B
• cdk1cycB gt cdk1cycB

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Elementary Reaction Rules

• Complexation A B gt A-B
  Decomplexation A-B gt A B
• cdk1cycB gt cdk1cycB
• Phosphorylation A Cgt Ap
  Dephosphorylation Ap Cgt A
• Cdk1-CycB Myt1gt Cdk1thr161-CycB
• Cdk1thr14,tyr15-CycB Cdc25Ntermgt
  Cdk1-CycB

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Elementary Reaction Rules

• Complexation A B gt A-B
  Decomplexation A-B gt A B
• cdk1cycB gt cdk1cycB
• Phosphorylation A Cgt Ap
  Dephosphorylation Ap Cgt A
• Cdk1-CycB Myt1gt Cdk1thr161-CycB
• Cdk1thr14,tyr15-CycB Cdc25Ntermgt
  Cdk1-CycB
• Synthesis _ Cgt A.
  Degradation A Cgt _.
• _ E2-E2f13-Dp12gt CycA cycE _at_UbiProgt _
• 
  (not for cycE-cdk2 which is stable)

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Elementary Reaction Rules

• Complexation A B gt A-B
  Decomplexation A-B gt A B
• cdk1cycB gt cdk1cycB
• Phosphorylation A Cgt Ap
  Dephosphorylation Ap Cgt A
• Cdk1-CycB Myt1gt Cdk1thr161-CycB
• Cdk1thr14,tyr15-CycB Cdc25Ntermgt
  Cdk1-CycB
• Synthesis _ Cgt A.
  Degradation A Cgt _.
• _ E2-E2f13-Dp12gt CycA cycE _at_UbiProgt _
• 
  (not for cycE-cdk2 which is stable)
• Transport AL1 gt AL2
• Cdk1p-CycBcytoplasm gt Cdk1p-CycBnucleus
  

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From Syntax to Semantics

• R SgtS S Ogt S S ltgt S S ltOgt S
• where A Cgt B stands
  for AC gt BC
• A ltgt B
  stands for AgtB and BgtA, etc.
• kinetic for R (import/export
  SBML format)
• In SBML no semantics (exchange format)

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From Syntax to Semantics

• R SgtS S Ogt S S ltgt S S ltOgt S
• where A Cgt B stands
  for AC gt BC
• A ltgt B
  stands for AgtB and BgtA, etc.
• kinetic for R (import/export
  SBML format)
• In SBML no semantics (exchange format)
• In BIOCHAM three abstraction levels
• Boolean Semantics presence-absence of molecules
• Concurrent Transition System (asynchronous,
  non-deterministic)

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From Syntax to Semantics

• R SgtS S Ogt S S ltgt S S ltOgt S
• where A Cgt B stands
  for AC gt BC
• A ltgt B
  stands for AgtB and BgtA, etc.
• kinetic for R (import/export
  SBML format)
• In SBML no semantics (exchange format)
• In BIOCHAM three abstraction levels
• Boolean Semantics presence-absence of molecules
• Concurrent Transition System (asynchronous,
  non-deterministic)
• Differential Semantics concentration
• Ordinary Differential Equations or Hybrid system
  (deterministic)

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From Syntax to Semantics

• R SgtS S Ogt S S ltgt S S ltOgt S
• where A Cgt B stands
  for AC gt BC
• A ltgt B
  stands for AgtB and BgtA, etc.
• kinetic for R (import/export
  SBML format)
• In SBML no semantics (exchange format)
• In BIOCHAM three abstraction levels
• Boolean Semantics presence-absence of molecules
• Concurrent Transition System (asynchronous,
  non-deterministic)
• Differential Semantics concentration
• Ordinary Differential Equations or Hybrid system
  (deterministic)
• Stochastic Semantics number of molecules
• Continuous time Markov chain

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1. Differential Semantics

• Associates to each molecule its concentration
  Ai Ai / volume ML-1
• volume of diffusion

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1. Differential Semantics

• Associates to each molecule its concentration
  Ai Ai / volume ML-1
• volume of compartment
• Compiles a set of rules ei for SigtSI i1,,n
  (by default ei is MA(1))
• into the system of ODEs (or hybrid automaton)
  over variables A1,,Ak
• dA/dt Sni1 ri(A)ei - Snj1 lj(A)ej
• where ri(A) (resp. li(A)) is the stoichiometric
  coefficient of A in Si (resp. Si)
• multiplied by the volume ratio of the location of
  A.

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1. Differential Semantics

• Associates to each molecule its concentration
  Ai Ai / volume ML-1
• volume of compartment
• Compiles a set of rules ei for SigtSI i1,,n
  (by default ei is MA(1))
• into the system of ODEs (or hybrid automaton)
  over variables A1,,Ak
• dA/dt Sni1 ri(A)ei - Snj1 lj(A)ej
• where ri(A) (resp. li(A)) is the stoichiometric
  coefficient of A in Si (resp. Si)
• multiplied by the volume ratio of the location of
  A.
• volume_ratio (15,n),(1,c).
• mRNAcycAn ltgt mRNAcycAc.
• means 15Vn Vc and is equivalent to
  15mRNAcycAn ltgt mRNAcycAc.

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Numerical Integration

• Adaptive step size 4th order Runge-Kutta can be
  weak for stiff systems
• Rosenbrock implicit method using the Jacobian
  matrix ?xi/?xj
• computes a (clever) discretization of time
• and a time series of concentrations and their
  derivatives
• (t0, X0, dX0/dt), (t1, X1, dX1/dt), , (tn, Xn,
  dXn/dt),
• at discrete time points

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2. Stochastic Semantics

• Associate to each molecule its number Ai in its
  location of volume Vi

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2. Stochastic Semantics

• Associate to each molecule its number Ai in its
  location of volume Vi
• Compile the rule set into a continuous time
  Markov chain
• over vector states X(A1,, Ak)
• and where the transition rate ti for the
  reaction ei for SigtSI
• (giving probability after normalization) is
  obtained from ei by replacing concentrations by
  molecule numbers

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2. Stochastic Semantics

• Associate to each molecule its number Ai in its
  location of volume Vi
• Compile the rule set into a continuous time
  Markov chain
• over vector states X(A1,, Ak)
• and where the transition rate ti for the
  reaction ei for SigtSI
• (giving probability after normalization) is
  obtained from ei by replacing concentrations by
  molecule numbers
• Stochastic simulation Gillespie 76, Gibson 00
• computes realizations as time series (t0, X0),
  (t1, X1), , (tn, Xn),

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3. Boolean Semantics

• Associate to each molecule a Boolean denoting its
  presence/absence in its location

37
3. Boolean Semantics

• Associate to each molecule a Boolean denoting its
  presence/absence in its location
• Compile the rule set into an asynchronous
  transition system

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3. Boolean Semantics

• Associate to each molecule a Boolean denoting its
  presence/absence in its location
• Compile the rule set into an asynchronous
  transition system where a reaction like ABgtCD
  is translated into 4 transition rules taking into
  account the possible complete consumption of
  reactants
• AB?ABCD
• AB??AB CD
• AB?A?BCD
• AB??A?BCD

39
3. Boolean Semantics

• Associate to each molecule a Boolean denoting its
  presence/absence in its location
• Compile the rule set into an asynchronous
  transition system where a reaction like ABgtCD
  is translated into 4 transition rules taking into
  account the possible complete consumption of
  reactants
• AB?ABCD
• AB??AB CD
• AB?A?BCD
• AB??A?BCD
• Necessary for over-approximating the possible
  behaviors under the stochastic/discrete semantics
  (abstraction N ? zero, non-zero)

40
Hierarchy of Semantics
abstraction
Theory of abstract Interpretation
Cousot Cousot POPL77 Fages Soliman
TCSc07
Boolean model
Discrete model
Differential model
Stochastic model
Models for answering queries The more abstract
the better Optimal abstraction w.r.t. query
Syntactical model
concretization
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Query what are the stationary states ?
Boolean circuit analysis
abstraction
abstraction
Discrete circuit analysis
Boolean model
abstraction
Jacobian circuit analysis
Discrete model
abstraction
Differential model
Positive circuits are a necessary condition for
multistationarity Thomas 73 de Jong 02 Soulé
03 Remy Ruet Thieffry 05
Stochastic model
Syntactical model
concretization
42
Type Inference / Type Checking
abstraction
Fages Soliman CMSB06
Boolean model
Discrete model
Differential model
Influence graph of proteins Protein functions
(kinase, phosphatase,) Compartments topology
Stochastic model
Syntactical model
concretization
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Type Inference / Type Checking
abstraction
Fages Soliman CMSB06
Boolean model
Influence graph of proteins (activation/inhibition
)
Discrete model
Differential model
Influence graph of proteins Protein functions
(kinase, phosphatase,) Compartments topology
Stochastic model
Syntactical model
concretization
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Cell Cycle G1 ? DNA Synthesis ? G2 ? Mitosis

• G1 CdK4-CycD S Cdk2-CycA
  G2,M Cdk1-CycA
• Cdk6-CycD
  Cdk1-CycB (MPF)
• Cdk2-CycE
  

45
Example Cell Cycle Control Model Tyson 91

• MA(k1) for _ gt Cyclin.
• MA(k2) for Cyclin gt _.
• MA(K7) for Cyclinp1 gt _.
• MA(k8) for Cdc2 gt Cdc2p1.
• MA(k9) for Cdc2p1 gtCdc2.
• MA(k3) for CyclinCdc2p1 gt
  Cdc2p1-Cyclinp1.
• MA(k4p) for Cdc2p1-Cyclinp1 gt
  Cdc2-Cyclinp1.
• k4Cdc2-Cyclinp12Cdc2p1-Cyclinp1
  for
• Cdc2p1-Cyclinp1 Cdc2-Cyclinp1 gt
  Cdc2-Cyclinp1.
• MA(k5) for Cdc2-Cyclinp1 gt Cdc2p1-Cyclinp
  1.
• MA(k6) for Cdc2-Cyclinp1 gt Cdc2Cyclinp1.

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Interaction Graph
47
Stochastic Simulation
48
Differential Simulation
49
Boolean Simulation
50
(No Transcript)
51
Mammalian Cell Cycle Control Map Kohn 99
52
Transcription of Kohns Map

• _ E2F13-DP12-gE2 gt cycA.
• ...
• cycB APCp1 gt_.
• cdk1p1,p2,p3 cycA gt cdk1p1,p2,p3-cycA.
• cdk1p1,p2,p3 cycB gt cdk1p1,p2,p3-cycB.
• ...
• cdk1p1,p3-cycA Wee1 gt cdk1p1,p2,p3-cycA
  .
• cdk1p1,p3-cycB Wee1 gt cdk1p1,p2,p3-cycB
  .
• cdk1p2,p3-cycA Myt1 gt cdk1p1,p2,p3-cycA
  .
• cdk1p2,p3-cycB Myt1 gt cdk1p1,p2,p3-cycB
  .
• ...
• cdk1p1,p2,p3 cdc25Cp1,p2 gt
  cdk1p1,p3.
• cdk1p1,p2,p3-cycA cdc25Cp1,p2 gt
  cdk1p1,p3-cycA.
• cdk1p1,p2,p3-cycB cdc25Cp1,p2 gt
  cdk1p1,p3-cycB.

165 proteins and genes, 500 variables, 800 rules
Chiaverini Danos 02
53
Overview of the Talk

• Rule-based Modeling of Biochemical Systems
• Syntax of molecules, compartments and reactions
• Semantics at three abstraction levels boolean,
  differential, stochastic
• Cell cycle control models
• Temporal Logic Language for Formalizing
  Biological Properties
• CTL for the boolean semantics
• Constraint LTL for the differential semantics
• PCTL for the stochastic semantics
• Automated Reasoning Tools
• Inferring kinetic parameter values from
  Constraint-LTL specification
• Inferring reaction rules from CTL specification

54
A Logical Paradigm for Systems Biology

• Biological model Transition System
• Biological property Temporal Logic Formula
• Biological validation Model-checking
• Formalize properties of the biological system in
• Computation Tree Logic CTL for the boolean
  semantics
• Linear Time Logic with numerical constraints for
  the concentration semantics
• Probabilistic CTL with numerical constraints for
  the stochastic semantics
• Evaluate the formulas by model checking
  techniques
• Lincoln et al. PSB02 Chabrier Fages CMSB03
  Bernot et al. TCS04

55
A Logical Paradigm for Systems Biology

• Biological model Transition System
• Biological property Temporal Logic Formula
• Biological validation Model-checking
• In the Biochemical Abstract Machine environment,
• Model BIOCHAM
  
• - Boolean - simulation
  
• - Differential
• - Stochastic
• (SBML)

56
A Logical Paradigm for Systems Biology

• Biological model Transition System
• Biological property Temporal Logic Formula
• Biological validation Model-checking
• In the Biochemical Abstract Machine environment,
• Model BIOCHAM
  Biological Properties
• - Boolean - simulation
  - CTL
• - Differential - query evaluation
  - LTL with constraints
• - Stochastic
  - PCTL with constraints
• (SBML)

57
A Logical Paradigm for Systems Biology

• Biological model Transition System
• Biological property Temporal Logic Formula
• Biological validation Model-checking
• In the Biochemical Abstract Machine environment,
• Model BIOCHAM
  Biological Properties
• - Boolean - simulation
  - CTL
• - Differential - query evaluation
  - LTL with constraints
• - Stochastic
  - PCTL with constraints
• (SBML)

58
A Logical Paradigm for Systems Biology

• Biological model Transition System
• Biological property Temporal Logic Formula
• Biological validation Model-checking
• In the Biochemical Abstract Machine environment,
• Model BIOCHAM
  Biological Properties
• - Boolean - simulation
  - CTL
• - Differential - query evaluation
  - LTL with constraints
• - Stochastic - rule learning
  - PCTL with constraints
• (SBML) - parameter search

59
2.1 Computation Tree Logic CTL

• Extension of propositional (or first-order) logic
  with operators for time and choices Clarke et
  al. 99

60
Biological Properties formalized in CTL (1/3)

• About reachability
• Can the cell produce some protein P?
  reachable(P)EF(P)

61
Biological Properties formalized in CTL (1/3)

• About reachability
• Can the cell produce some protein P?
  reachable(P)EF(P)
• Can the cell produce P, Q and not R?
  reachable(PQ?R)

62
Biological Properties formalized in CTL (1/3)

• About reachability
• Can the cell produce some protein P?
  reachable(P)EF(P)
• Can the cell produce P, Q and not R?
  reachable(PQ?R)
• Can the cell always produce P? AG(reachable(P))

63
Biological Properties formalized in CTL (1/3)

• About reachability
• Can the cell produce some protein P?
  reachable(P)EF(P)
• Can the cell produce P, Q and not R?
  reachable(PQ?R)
• Can the cell always produce P? AG(reachable(P))
• About pathways
• Can the cell reach a (partially described) set
  of states s while passing by another set of
  states s2? EF(s2EFs)

64
Biological Properties formalized in CTL (1/3)

• About reachability
• Can the cell produce some protein P?
  reachable(P)EF(P)
• Can the cell produce P, Q and not R?
  reachable(PQ?R)
• Can the cell always produce P? AG(reachable(P))
• About pathways
• Can the cell reach a (partially described) set
  of states s while passing by another set of
  states s2? EF(s2EFs)
• Is it possible to produce P without Q? E(?Q U P)

65
Biological Properties formalized in CTL (1/3)

• About reachability
• Can the cell produce some protein P?
  reachable(P)EF(P)
• Can the cell produce P, Q and not R?
  reachable(PQ?R)
• Can the cell always produce P? AG(reachable(P))
• About pathways
• Can the cell reach a (partially described) set
  of states s while passing by another set of
  states s2? EF(s2EFs)
• Is it possible to produce P without Q? E(?Q U P)
• Is (set of) state s2 a necessary checkpoint for
  reaching (set of) state s?
• checkpoint(s2,s) ?E(?s2U s)

66
Biological Properties formalized in CTL (1/3)

• About reachability
• Can the cell produce some protein P?
  reachable(P)EF(P)
• Can the cell produce P, Q and not R?
  reachable(PQ?R)
• Can the cell always produce P? AG(reachable(P))
• About pathways
• Can the cell reach a (partially described) set
  of states s while passing by another set of
  states s2? EF(s2EFs)
• Is it possible to produce P without Q? E(?Q U P)
• Is (set of) state s2 a necessary checkpoint for
  reaching (set of) state s?
• checkpoint(s2,s) ?E(?s2U s)
• Is s2 always a checkpoint for s? AG(?s -gt
  checkpoint(s2,s))

67
Biological Properties formalized in CTL (2/3)

• About stationarity
• Is a (set of) state s a stable state? stable(s)
  AG(s)

68
Biological Properties formalized in CTL (2/3)

• About stationarity
• Is a (set of) state s a stable state? stable(s)
  AG(s)
• Is s a steady state (with possibility of
  escaping) ? steady(s)EG(s)

69
Biological Properties formalized in CTL (2/3)

• About stationarity
• Is a (set of) state s a stable state? stable(s)
  AG(s)
• Is s a steady state (with possibility of
  escaping) ? steady(s)EG(s)
• Can the cell reach a stable state s?
  EF(stable(s)) not in LTL

70
Biological Properties formalized in CTL (2/3)

• About stationarity
• Is a (set of) state s a stable state? stable(s)
  AG(s)
• Is s a steady state (with possibility of
  escaping) ? steady(s)EG(s)
• Can the cell reach a stable state s?
  EF(stable(s)) not in LTL
• Must the cell reach a stable state s?
  AG(stable(s))

71
Biological Properties formalized in CTL (2/3)

• About stationarity
• Is a (set of) state s a stable state? stable(s)
  AG(s)
• Is s a steady state (with possibility of
  escaping) ? steady(s)EG(s)
• Can the cell reach a stable state s?
  EF(stable(s)) not in LTL
• Must the cell reach a stable state s?
  AG(stable(s))
• What are the stable states? Not expressible in
  CTL. Needs to combine CTL with search (e.g.
  constraint programming Thieffry et al. 05 )

72
Biological Properties formalized in CTL (3/3)

• About oscillations
• Can the system exhibit a cyclic behavior w.r.t.
  the presence of P ? oscil(P) EG(F ?P F P)
• CTL formula that can be approximated in CTL
  by
• oscil(P) EG((P ? EF ?P) (?P ? EF P))
• (necessary but not sufficient condition for
  oscillation)

73
Biological Properties formalized in CTL (3/3)

• About oscillations
• Can the system exhibit a cyclic behavior w.r.t.
  the presence of P ? oscil(P) EG((P ? EF ?P)
  (?P ? EF P))
• (necessary but not sufficient condition)
• Can the system loops between states s and s2 ?
• loop(P,Q) EG((s ? EF s2) (s2 ? EF s))

74
Symbolic Model-Checking

• Still for finite Kripke structures, use boolean
  constraints to represent
• sets of states as a boolean constraint c(V)
• the transition relation as a boolean constraint
  r(V,V)
• Binary Decision Diagrams BDD Bryant 85 provide
  canonical forms to Boolean formulas (decide
  Boolean equivalence)
• (x?y)?(y?z)?(z?x)
• and
• (x?z)?(z?y)?(y?x)
• are equivalent, they
• have the same BDD(x,y,z)

75
Mammalian Cell Cycle Control Map Kohn 99
76
Cell Cycle Model-Checking (with BDD NuSMV)

• biocham check_reachable(cdk46p1,p2-cycDp1).
  
• Ei(EF(cdk46p1,p2-cycDp1)) is true
• biocham check_checkpoint(cdc25Cp1,p2,
  cdk1p1,p3-cycB).
• Ai(!(E(!(cdc25Cp1,p2) U cdk1p1,p3-cycB)))
  is true
• biocham nusmv(Ai(AG(!(cdk1p1,p2,p3-cycB) -gt
  checkpoint(Wee1, cdk1p1,p2,p3-cycB))))).
• Ai(AG(!(cdk1p1,p2,p3-cycB)-gt!(E(!(Wee1) U
  cdk1p1,p2,p3-cycB)))) is false
• biocham why.
• -- Loop starts here
• cycB-cdk1p1,p2,p3 is present
• cdk7 is present
• cycH is present
• cdk1 is present
• Myt1 is present
• cdc25Cp1 is present
• rule_114 cycB-cdk1p1,p2,p3cdc25Cp1gtcycB-
  cdk1p2,p3.
• cycB-cdk1p2,p3 is present
• cycB-cdk1p1,p2,p3 is absent
• rule_74 cycB-cdk1p2,p3Myt1gtcycB-cdk1p1,p2
  ,p3.
• cycB-cdk1p2,p3 is absent

77
Mammalian Cell Cycle Control Benchmark

• 500 variables, 2500 states. 800 rules.
• BIOCHAM NuSMV model-checker time in sec.
  Chabrier et al. TCS 04

78
2.2 LTL with Constraints for the Differential
Semantics

• Constraints over concentrations and derivatives
  as FOL formulae over the reals
• M gt 0.2
• MP gt Q
• d(M)/dt lt 0

79
LTL with Constraints for the Differential
Semantics

• Constraints over concentrations and derivatives
  as FOL formulae over the reals
• M gt 0.2
• MP gt Q
• d(M)/dt lt 0
• Linear Time Logic LTL operators for time X, F, U,
  G
• F(Mgt0.2)
• FG(Mgt0.2)
• F (Mgt2 F (d(M)/dtlt0 F (Mlt2
  d(M)/dtgt0 F(d(M)/dtlt0))))
• oscil(M,n) defined as at least n alternances of
  sign of the derivative
• Period(A,75) ? t ?v F(T t A v
  d(A)/dt gt 0 X(d(A)/dt lt 0)
• F(T t 75 A v
  d(A)/dt gt 0 X(d(A)/dt lt 0)))

80
How to Evaluate a Constraint LTL Formula ?

• Consider the ODEs of the concentration semantics
  dX/dt f(X)

81
How to Evaluate a Constraint LTL Formula ?

• Consider the ODEs of the concentration semantics
  dX/dt f(X)
• Numerical integration methods produce a
  discretization of time (adaptive step size
  Runge-Kutta or Rosenbrock method for stiff syst.)
• The trace is a linear Kripke structure
• (t0,X0,dX0/dt), (t1,X1,dX1/dt), ,
  (tn,Xn,dXn/dt),
• over concentrations and their derivatives at
  discrete time points
• Evaluate the formula on that Kripke structure
  with a model checking alg.

82
Simulation-Based Constraint LTL Model Checking

• Hypothesis 1 the initial state is completely
  known
• Hypothesis 2 the formula can be checked over a
  finite period of time 0,T
• Run the numerical integration from 0 to T
  producing
  values at a finite sequence of time points
• Iteratively label the time points with the
  sub-formulae of f that are true
• Add f to the time points where a FOL formula
  f is true,
• Add F f (X f) to the (immediate) previous
  time points labeled by f,
• Add f1 U f2 to the predecessor time points
  of f2 while they satisfy f1,
• (Add G f to the states satisfying f until
  T)
• Model checker and numerical integration methods
  implemented in Prolog

83
Constraint-LTL Instanciation Algo. Fages Rizk
CMSB07
84
2.3 PCTL Model Checker for the Stochastic
Semantics

• Compute the probability of realisation of a TL
  formula (with constraints) by Monte Carlo method
• Perform several stochastic simulations
• Evaluate the probability of realization of the TL
  formula
• Costly
• PRISM Kwiatkowska et al. 04 PCTL model
  checker based on BDDs or using Monte Carlo method.

85
Overview of the Talk

• Rule-based Modeling of Biochemical Systems
• Syntax of molecules, compartments and reactions
• Semantics at three abstraction levels boolean,
  differential, stochastic
• Cell cycle control models
• Temporal Logic Language for Formalizing
  Biological Properties
• CTL for the boolean semantics
• Constraint LTL for the differential semantics
• PCTL for the stochastic semantics
• Automated Reasoning Tools
• Inferring kinetic parameter values from
  Constraint-LTL specification
• Inferring reaction rules from CTL specification

86
Example Cell Cycle Control Model Tyson 91

• MA(k1) for _ gt Cyclin.
• MA(k2) for Cyclin gt _.
• MA(K7) for Cyclinp1 gt _.
• MA(k8) for Cdc2 gt Cdc2p1.
• MA(k9) for Cdc2p1 gtCdc2.
• MA(k3) for CyclinCdc2p1 gt
  Cdc2p1-Cyclinp1.
• MA(k4p) for Cdc2p1-Cyclinp1 gt
  Cdc2-Cyclinp1.
• k4Cdc2-Cyclinp12Cdc2p1-Cyclinp1
  for
• Cdc2p1-Cyclinp1 Cdc2-Cyclinp1 gt
  Cdc2-Cyclinp1.
• MA(k5) for Cdc2-Cyclinp1 gt Cdc2p1-Cyclinp
  1.
• MA(k6) for Cdc2-Cyclinp1 gt Cdc2Cyclinp1.

87
3.1 Inferring Parameters from Temporal Properties

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• 
  oscil(Cdc2-Cyclinp1,3),150).

88
3.1 Inferring Parameters from Temporal Properties

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• 
  oscil(Cdc2-Cyclinp1,3),150).
• First values found
• parameter(k3,10).
• parameter(k4,70).

89
3.1 Inferring Parameters from Temporal Properties

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• oscil(Cdc2-Cyclinp1,3)
  F(Cdc2-Cyclinp1gt0.15), 150).
• First values found
• parameter(k3,10).
• parameter(k4,120).

90
3.1 Inferring Parameters from LTL Specification

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• 
  period(Cdc2-Cyclinp1,35), 150).
• First values found
• parameter(k3,10).
• parameter(k4,280).

91
Linking the Cell and Circadian Cycles through Wee1
Cell cycle
Leloup and Goldbeter (1999)
Wee1P
Wee1
.. ..
preMPF
MPF
APC
APC
Cdc25P
.. ..
.. ..
Cdc25
.. ..
L
L. Calzone, S. Soliman 2006
92
PCN
Wee1m
BN
Wee1
MPF
Cdc25
93
Condition on Wee1/Cdc25 for the Entrainment in
Period
Entrainment in period constraint expressed in LTL
with the period formula
94
3.2. Inferring Rules from Temporal Properties

• Given
• a BIOCHAM model (background knowledge)
• a set of properties formalized in temporal logic
• learn
• revisions of the reaction model, i.e. rules to
  delete and rules to add such that the revised
  model satisfies the properties

95
Model Revision from Temporal Properties

• Background knowledge T BIOCHAM model
• reaction rule language complexation,
  phosphorylation,
• Examples f biological properties formalized in
  temporal logic language
• Reachability
• Checkpoints
• Stable states
• Oscillations
• Bias R Reaction rule patterns or parameter
  ranges
• Kind of rules to add or delete
• Find a revision T of T such that T f

96
Model Revision Algorithm

• General idea of constraint programming replace a
  generate-and-test algorithm by a
  constrain-and-generate algorithm.
• Anticipate whether one has to add or remove a
  rule.
• Positive ECTL formula if false, remains false
  after removing a rule
• EF(f) where f is a boolean formula (pure state
  description)
• Oscil(f)
• Negative ACTL formula if false, remains false
  after adding a rule
• AG(f) where f is a boolean formula,
• Checkpoint(a,b) E(aUb)
• Remove a rule on the path given by the model
  checker (why command)
• Unclassified CTL formulae

97
Example Cell Cycle Control Model Tyson 91

• MA(k1) for _ gt Cyclin.
• MA(k2) for Cyclin gt _.
• MA(K7) for Cyclinp1 gt _.
• MA(k8) for Cdc2 gt Cdc2p1.
• MA(k9) for Cdc2p1 gtCdc2.
• MA(k3) for CyclinCdc2p1 gt
  Cdc2p1-Cyclinp1.
• MA(k4p) for Cdc2p1-Cyclinp1 gt
  Cdc2-Cyclinp1.
• k4Cdc2-Cyclinp12Cdc2p1-Cyclinp1
  for
• Cdc2p1-Cyclinp1 Cdc2-Cyclinp1 gt
  Cdc2-Cyclinp1.
• MA(k5) for Cdc2-Cyclinp1 gt Cdc2p1-Cyclinp
  1.
• MA(k6) for Cdc2-Cyclinp1 gt Cdc2Cyclinp1.

98
Automatic Generation of True CTL Properties

• Ei(reachable(Cyclin)))
• Ei(reachable(!(Cyclin))))
• Ai(oscil(Cyclin)))
• Ei(reachable(Cdc2p1)))
• Ei(reachable(!(Cdc2p1))))
• Ai(oscil(Cdc2p1)))
• Ai(AG(!(Cdc2p1)-gtcheckpoint(Cdc2,Cdc2p1))))
• Ei(reachable(Cdc2-Cyclinp1,p2)))
• Ei(reachable(!(Cdc2-Cyclinp1,p2))))
• Ai(oscil(Cdc2-Cyclinp1,p2)))
• Ei(reachable(Cdc2-Cyclinp1)))
• Ei(reachable(!(Cdc2-Cyclinp1))))
• Ai(oscil(Cdc2-Cyclinp1)))
• Ai(AG(!(Cdc2-Cyclinp1)-gtcheckpoint(Cdc2-Cyclin
  p1,p2,Cdc2-Cyclinp1)))
• Ei(reachable(Cdc2)))
• Ei(reachable(!(Cdc2))))
• Ai(oscil(Cdc2)))
• Ei(reachable(Cyclinp1)))
• Ei(reachable(!(Cyclinp1))))

99
Rule Deletion

• biocham delete_rules(Cdc2 gt Cdc2p1).
• biocham check_all.
• First formula not satisfied
• Ei(EF(Cdc2-Cyclinp1))

100
Model Revision from Temporal Properties

• biocham revise_model.
• Rules to delete
• Rules to add
• Cdc2 gt Cdc2p1.

101
Model Revision from Temporal Properties

• biocham revise_model.
• Rules to delete
• Rules to add
• Cdc2 gt Cdc2p1.
• biocham learn_one_addition.
• (1) Cdc2 gt Cdc2p1.
• (2) Cdc2 Cdc2gt Cdc2p1.
• (3) Cdc2 Cyclingt Cdc2p1.

102
Conclusion

• Temporal logic with constraints is powerful
  enough to express both qualitative and
  quantitative biological properties of systems

103
Conclusion

• Temporal logic with constraints is powerful
  enough to express both qualitative and
  quantitative biological properties of systems
• Three levels of abstraction in BIOCHAM
• Boolean semantics CTL formulas (rule
  learning)
• Differential semantics LTL with constraints over
  reals (parameter search)
• Stochastic semantics Probabilistic CTL with
  integer constraints

104
Conclusion

• Temporal logic with constraints is powerful
  enough to express both qualitative and
  quantitative biological properties of systems
• Three levels of abstraction in BIOCHAM
• Boolean semantics CTL formulas (rule
  learning)
• Differential semantics LTL with constraints over
  reals (parameter search)
• Stochastic semantics Probabilistic CTL with
  integer constraints
• Parameter search from temporal properties proved
  useful and complementary to bifurcation theory
  tools (Xppaut)

105
Conclusion

• Temporal logic with constraints is powerful
  enough to express both qualitative and
  quantitative biological properties of systems
• Three levels of abstraction in BIOCHAM
• Boolean semantics CTL formulas (rule
  learning)
• Differential semantics LTL with constraints over
  reals (parameter search)
• Stochastic semantics Probabilistic CTL with
  integer constraints
• Parameter search from temporal properties proved
  useful and complementary to bifurcation theory
  tools (Xppaut)
• Rule inference from temporal properties still in
  infancy, to be optimized and improved by types
  (e.g. protein functions, computed by abstract
  interpretation)

106
Collaborations

• STREP APrIL2 Luc de Raedt, Univ. Freiburg,
  Stephen Muggleton, IC ,
• Learning in a probabilistic logic setting
  (finished)
• ARC MOCA
• modularity, compositionality and abstraction
• NoE REWERSE semantic web, François Bry, Münich,
  Rolf Backofen,
• Connecting Biocham to gene and protein ontologies
  (types)
• STREP TEMPO Cancer chronotherapies, INSERM
  Villejuif, F. Lévi Bang J. Clairambault,
  Contraintes S. Soliman
• Coupled models of cell cycle, circadian cycle,
  cytotoxic drugs.
• INRA Tours E. Reiter, D. Heitzler, Sysiphe F.
  Clément
• Model of FSH signalling.