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François Fages Rocquencourt, Sep Semantical and Algorithmic Aspects of the Living François Fages Constraint 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

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François Fages Rocquencourt, Sep 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

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François Fages Rocquencourt, Sep Issue of Abstraction Models are built in Systems Biology with two contradictory perspectives :

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François Fages Rocquencourt, Sep Issue of Abstraction Models are built in Systems Biology with two contradictory perspectives : 1) Models for representing knowledge : the more concrete the better

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François Fages Rocquencourt, Sep 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 !

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François Fages Rocquencourt, Sep 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

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François Fages Rocquencourt, Sep Semantics of Living Processes ? Formally, “the” behavior of a system depends on our choice of observables. ? ? Mitosis movie [Lodish et al. 03]

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François Fages Rocquencourt, Sep Boolean Semantics Formally, “the” behavior of a system depends on our choice of observables. Presence/absence of molecules Boolean transitions 01

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François Fages Rocquencourt, Sep Continuous Differential Semantics Formally, “the” behavior of a system depends on our choice of observables. Concentrations of molecules Rates of reactions xý

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François Fages Rocquencourt, Sep 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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François Fages Rocquencourt, Sep Temporal Logic Semantics Formally, “the” behavior of a system depends on our choice of observables. Presence/absence of molecules Temporal logic formulas Fx F x F (x ^ F ( x ^ y)) FG (x v y) …

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François Fages Rocquencourt, Sep Constraint Temporal Logic Semantics Formally, “the” behavior of a system depends on our choice of observables. Concentrations of molecules Constraint LTL temporal formulas Fx>1x>1 F (x >0.2) F (x >0.2 ^ F (x 0.2)) FG (x>0.2 v y>0.2) …

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François Fages Rocquencourt, Sep 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 π–calculus [Regev-Silverman-Shapiro 99-01, Nagasali et al. 00] Bio-ambients [Regev-Panina-Silverman-Cardelli-Shapiro 03] Pathway logic [Eker-Knapp-Laderoute-Lincoln-Meseguer-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 π–calculus [Priami et al. 03] [Cardelli et al. 06] Reaction rules with continuous time dynamics [Fages-Soliman-Chabrier 04]

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François Fages Rocquencourt, Sep Overview of the Talk 1.Rule-based Modeling of Biochemical Systems 1.Syntax of molecules, compartments and reactions 2.Semantics at three abstraction levels: boolean, differential, stochastic 3.Cell cycle control models 2.Temporal Logic Language for Formalizing Biological Properties 1.CTL for the boolean semantics 2.Constraint LTL for the differential semantics 3.PCTL for the stochastic semantics 3.Automated Reasoning Tools 1.Inferring kinetic parameter values from Constraint-LTL specification 2.Inferring reaction rules from CTL specification L. Calzone, N. Chabrier, F. Fages, S. Soliman. TCSB VI, LNBI 4220:

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François Fages Rocquencourt, Sep Molecules of the living Small molecules: covalent bonds 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 ( amino acids) RNA ( nucleotides AGCU) DNA ( nucleotides AGCT)

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François Fages Rocquencourt, Sep Structure Levels of Proteins 1) Primary structure: word of n amino acids residues (20 n possibilities) linked with C-N bonds Example: INRIA Isoleucine-asparagiNe-aRginine-Isoleucine-Alanine

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François Fages Rocquencourt, Sep Structure Levels of Proteins 1) Primary structure: word of n amino acids residues (20 n possibilities) linked with C-N bonds Example: INRIA Isoleucine-asparagiNe-aRginine-Isoleucine-Alanine 2) Secondary: word of m helix, strands, random coils,… (3 m -10 m ) stabilized by hydrogen bonds H---O

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François Fages Rocquencourt, Sep Structure Levels of Proteins 1) Primary structure: word of n amino acids residues (20 n possibilities) linked with C-N bonds Example: INRIA Isoleucine-asparagiNe-aRginine-Isoleucine-Alanine 2) Secondary: word of m helix, strands, random coils,… (3 m -10 m ) stabilized by hydrogen bonds H---O 3) Tertiary 3D structure: spatial folding stabilized by hydrophobic interactions

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François Fages Rocquencourt, Sep Syntax of 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) mitosis promotion factor

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François Fages Rocquencourt, Sep BIOCHAM Syntax of Objects E == compound | E-E | E~{p1,…,pn} Compound : molecule, #gene binding site, - : 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 | E::location Location : symbolic compartment (nucleus, cytoplasm, membrane, …) S == _ | O+S + : solution operator (Associative, Commutative, Neutral _)

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François Fages Rocquencourt, Sep Elementary Reaction Rules Complexation: A + B => A-B Decomplexation A-B => A + B cdk1+cycB => cdk1–cycB

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François Fages Rocquencourt, Sep Elementary Reaction Rules 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

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François Fages Rocquencourt, Sep Elementary Reaction Rules 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. Degradation: A =[C]=> _. _ =[#E2-E2f13-Dp12]=> CycA cycE _ (not for cycE-cdk2 which is stable)

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François Fages Rocquencourt, Sep Elementary Reaction Rules 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. Degradation: A =[C]=> _. _ =[#E2-E2f13-Dp12]=> CycA cycE _ (not for cycE-cdk2 which is stable) Transport: A::L1 => A::L2 Cdk1~{p}-CycB::cytoplasm => Cdk1~{p}-CycB::nucleus

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François Fages Rocquencourt, Sep From Syntax to Semantics R ::= S=>S | S =[O]=> S | S S | S S where A =[C]=> B stands for A+C => B+C A B stands for A=>B and B=>A, etc. | kinetic for R (import/export SBML format) In SBML : no semantics (exchange format)

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François Fages Rocquencourt, Sep From Syntax to Semantics R ::= S=>S | S =[O]=> S | S S | S S where A =[C]=> B stands for A+C => B+C A B stands for A=>B and B=>A, etc. | kinetic for R (import/export SBML format) In SBML : no semantics (exchange format) In BIOCHAM : three abstraction levels 1.Boolean Semantics: presence-absence of molecules 1.Concurrent Transition System (asynchronous, non-deterministic)

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François Fages Rocquencourt, Sep From Syntax to Semantics R ::= S=>S | S =[O]=> S | S S | S S where A =[C]=> B stands for A+C => B+C A B stands for A=>B and B=>A, etc. | kinetic for R (import/export SBML format) In SBML : no semantics (exchange format) In BIOCHAM : three abstraction levels 1.Boolean Semantics: presence-absence of molecules 1.Concurrent Transition System (asynchronous, non-deterministic) 2.Differential Semantics: concentration 1.Ordinary Differential Equations or Hybrid system (deterministic)

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François Fages Rocquencourt, Sep From Syntax to Semantics R ::= S=>S | S =[O]=> S | S S | S S where A =[C]=> B stands for A+C => B+C A B stands for A=>B and B=>A, etc. | kinetic for R (import/export SBML format) In SBML : no semantics (exchange format) In BIOCHAM : three abstraction levels 1.Boolean Semantics: presence-absence of molecules 1.Concurrent Transition System (asynchronous, non-deterministic) 2.Differential Semantics: concentration 1.Ordinary Differential Equations or Hybrid system (deterministic) 3.Stochastic Semantics: number of molecules 1.Continuous time Markov chain

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François Fages Rocquencourt, Sep Differential Semantics Associates to each molecule its concentration [A i ]= | A i | / volume ML -1 volume of diffusion …

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François Fages Rocquencourt, Sep Differential Semantics Associates to each molecule its concentration [A i ]= | A i | / volume ML -1 volume of compartment Compiles a set of rules { e i for S i =>S’ I } i=1,…,n (by default e i is MA(1)) into the system of ODEs (or hybrid automaton) over variables {A 1,…,A k } dA/dt = Σ n i=1 r i (A)*ei - Σ n j=1 l j (A)*e j where r i (A) (resp. l i (A)) is the stoichiometric coefficient of A in S i (resp. S’ i ) multiplied by the volume ratio of the location of A.

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François Fages Rocquencourt, Sep Differential Semantics Associates to each molecule its concentration [A i ]= | A i | / volume ML -1 volume of compartment Compiles a set of rules { e i for S i =>S’ I } i=1,…,n (by default e i is MA(1)) into the system of ODEs (or hybrid automaton) over variables {A 1,…,A k } dA/dt = Σ n i=1 r i (A)*ei - Σ n j=1 l j (A)*e j where r i (A) (resp. l i (A)) is the stoichiometric coefficient of A in S i (resp. S’ i ) multiplied by the volume ratio of the location of A. volume_ratio (15,n),(1,c). mRNAcycA::n mRNAcycA::c. means 15*Vn = Vc and is equivalent to 15*mRNAcycA::n mRNAcycA::c.

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François Fages Rocquencourt, Sep Numerical Integration Adaptive step size 4th order Runge-Kutta can be weak for stiff systems Rosenbrock implicit method using the Jacobian matrix ∂x’ i /∂x j computes a (clever) discretization of time and a time series of concentrations and their derivatives (t 0, X 0, dX 0 /dt), (t 1, X 1, dX 1 /dt), …, (t n, X n, dX n /dt), … at discrete time points

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François Fages Rocquencourt, Sep Stochastic Semantics Associate to each molecule its number |A i | in its location of volume V i

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François Fages Rocquencourt, Sep Stochastic Semantics Associate to each molecule its number |A i | in its location of volume V i Compile the rule set into a continuous time Markov chain over vector states X=(|A 1 |,…, |A k |) and where the transition rate τ i for the reaction e i for S i =>S’ I (giving probability after normalization) is obtained from e i by replacing concentrations by molecule numbers

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François Fages Rocquencourt, Sep Stochastic Semantics Associate to each molecule its number |A i | in its location of volume V i Compile the rule set into a continuous time Markov chain over vector states X=(|A 1 |,…, |A k |) and where the transition rate τ i for the reaction e i for S i =>S’ I (giving probability after normalization) is obtained from e i by replacing concentrations by molecule numbers Stochastic simulation [Gillespie 76, Gibson 00] computes realizations as time series (t 0, X 0 ), (t 1, X 1 ), …, (t n, X n ), …

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François Fages Rocquencourt, Sep Boolean Semantics Associate to each molecule a Boolean denoting its presence/absence in its location

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François Fages Rocquencourt, Sep 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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François Fages Rocquencourt, Sep 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 A+B=>C+D is translated into 4 transition rules taking into account the possible complete consumption of reactants: A+B A+B+C+D A+B A+B +C+D A+B A+ B+C+D A+B A+ B+C+D

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François Fages Rocquencourt, Sep 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 A+B=>C+D is translated into 4 transition rules taking into account the possible complete consumption of reactants: A+B A+B+C+D A+B A+B +C+D A+B A+ B+C+D A+B A+ B+C+D Necessary for over-approximating the possible behaviors under the stochastic/discrete semantics (abstraction N {zero, non-zero})

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François Fages Rocquencourt, Sep Hierarchy of Semantics Stochastic model Differential model Discrete model abstraction concretization Boolean model Theory of abstract Interpretation [Cousot Cousot POPL’77] [Fages Soliman TCSc’07] Syntactical model Models for answering queries: The more abstract the better Optimal abstraction w.r.t. query

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François Fages Rocquencourt, Sep Query: what are the stationary states ? Stochastic model Differential model Discrete model abstraction concretization Boolean model Syntactical model Jacobian circuit analysis Discrete circuit analysis Boolean circuit analysis abstraction Positive circuits are a necessary condition for multistationarity [Thomas 73] [de Jong 02] [Soul é 03] [Remy Ruet Thieffry 05]

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François Fages Rocquencourt, Sep Type Inference / Type Checking Stochastic model Differential model Discrete model abstraction concretization Boolean model Syntactical model [Fages Soliman CMSB ’ 06] Influence graph of proteins Protein functions (kinase, phosphatase, … ) Compartments topology

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François Fages Rocquencourt, Sep Type Inference / Type Checking Stochastic model Differential model Discrete model abstraction concretization Boolean model Syntactical model [Fages Soliman CMSB ’ 06] Influence graph of proteins Protein functions (kinase, phosphatase, … ) Compartments topology Influence graph of proteins (activation/inhibition)

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François Fages Rocquencourt, Sep Cell Cycle: G1 DNA Synthesis G2 Mitosis G1: CdK4-CycD S: Cdk2-CycA G2,M: Cdk1-CycA Cdk6-CycD Cdk1-CycB (MPF) Cdk2-CycE

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François Fages Rocquencourt, Sep Example: Cell Cycle Control Model [Tyson 91] MA(k1) for _ => Cyclin. MA(k2) for Cyclin => _. MA(K7) for Cyclin~{p1} => _. MA(k8) for Cdc2 => Cdc2~{p1}. MA(k9) for Cdc2~{p1} =>Cdc2. MA(k3) for Cyclin+Cdc2~{p1} => Cdc2~{p1}-Cyclin~{p1}. MA(k4p) for Cdc2~{p1}-Cyclin~{p1} => Cdc2-Cyclin~{p1}. k4*[Cdc2-Cyclin~{p1}]^2*[Cdc2~{p1}-Cyclin~{p1}] for Cdc2~{p1}-Cyclin~{p1} =[Cdc2-Cyclin~{p1}] => Cdc2-Cyclin~{p1}. MA(k5) for Cdc2-Cyclin~{p1} => Cdc2~{p1}-Cyclin~{p1}. MA(k6) for Cdc2-Cyclin~{p1} => Cdc2+Cyclin~{p1}.

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François Fages Rocquencourt, Sep Interaction Graph

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François Fages Rocquencourt, Sep Stochastic Simulation

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François Fages Rocquencourt, Sep Differential Simulation

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François Fages Rocquencourt, Sep Boolean Simulation

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François Fages Rocquencourt, Sep. 2007

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François Fages Rocquencourt, Sep Mammalian Cell Cycle Control Map [Kohn 99]

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François Fages Rocquencourt, Sep Transcription of Kohn’s Map _ =[ E2F13-DP12-gE2 ]=> cycA.... cycB =[ APC~{p1} ]=>_. cdk1~{p1,p2,p3} + cycA => cdk1~{p1,p2,p3}-cycA. cdk1~{p1,p2,p3} + cycB => cdk1~{p1,p2,p3}-cycB.... cdk1~{p1,p3}-cycA =[ Wee1 ]=> cdk1~{p1,p2,p3}-cycA. cdk1~{p1,p3}-cycB =[ Wee1 ]=> cdk1~{p1,p2,p3}-cycB. cdk1~{p2,p3}-cycA =[ Myt1 ]=> cdk1~{p1,p2,p3}-cycA. cdk1~{p2,p3}-cycB =[ Myt1 ]=> cdk1~{p1,p2,p3}-cycB.... cdk1~{p1,p2,p3} =[ cdc25C~{p1,p2} ]=> cdk1~{p1,p3}. cdk1~{p1,p2,p3}-cycA =[ cdc25C~{p1,p2} ]=> cdk1~{p1,p3}-cycA. cdk1~{p1,p2,p3}-cycB =[ cdc25C~{p1,p2} ]=> cdk1~{p1,p3}-cycB. 165 proteins and genes, 500 variables, 800 rules [Chiaverini Danos 02]

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François Fages Rocquencourt, Sep Overview of the Talk 1.Rule-based Modeling of Biochemical Systems 1.Syntax of molecules, compartments and reactions 2.Semantics at three abstraction levels: boolean, differential, stochastic 3.Cell cycle control models 2.Temporal Logic Language for Formalizing Biological Properties 1.CTL for the boolean semantics 2.Constraint LTL for the differential semantics 3.PCTL for the stochastic semantics 3.Automated Reasoning Tools 1.Inferring kinetic parameter values from Constraint-LTL specification 2.Inferring reaction rules from CTL specification

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François Fages Rocquencourt, Sep 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. PSB’02] [Chabrier Fages CMSB’03] [Bernot et al. TCS’04] …

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François Fages Rocquencourt, Sep 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)

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François Fages Rocquencourt, Sep 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)

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François Fages Rocquencourt, Sep 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)

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François Fages Rocquencourt, Sep 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

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François Fages Rocquencourt, Sep Computation Tree Logic CTL Extension of propositional (or first-order) logic with operators for time and choices [Clarke et al. 99] Choice Time E exists A always X next time EX( f ) ¬ AX( ¬ f ) AX( f ) F finally EF( f ) ¬ AG( ¬ f ) AF( f ) G globally EG( f ) ¬ AF( ¬ f ) AG( f ) U until E ( f 1 U f 2 )A ( f 1 U f 2 )

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François Fages Rocquencourt, Sep Biological Properties formalized in CTL (1/3) About reachability: Can the cell produce some protein P? reachable(P)==EF(P)

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François Fages Rocquencourt, Sep 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(P^Q^ R)

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François Fages Rocquencourt, Sep 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(P^Q^ R) Can the cell always produce P? AG(reachable(P))

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François Fages Rocquencourt, Sep 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(P^Q^ 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 s 2 ? EF(s 2 ^EFs)

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François Fages Rocquencourt, Sep 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(P^Q^ 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 s 2 ? EF(s 2 ^EFs) Is it possible to produce P without Q? E( Q U P)

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François Fages Rocquencourt, Sep 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(P^Q^ 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 s 2 ? EF(s 2 ^EFs) Is it possible to produce P without Q? E( Q U P) Is (set of) state s 2 a necessary checkpoint for reaching (set of) state s? checkpoint(s 2,s)== E( s 2 U s)

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François Fages Rocquencourt, Sep 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(P^Q^ 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 s 2 ? EF(s 2 ^EFs) Is it possible to produce P without Q? E( Q U P) Is (set of) state s 2 a necessary checkpoint for reaching (set of) state s? checkpoint(s 2,s)== E( s 2 U s) Is s 2 always a checkpoint for s? AG( s -> checkpoint(s 2,s))

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François Fages Rocquencourt, Sep Biological Properties formalized in CTL (2/3) About stationarity: Is a (set of) state s a stable state? stable(s)== AG(s)

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François Fages Rocquencourt, Sep 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)

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François Fages Rocquencourt, Sep 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

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François Fages Rocquencourt, Sep 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))

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François Fages Rocquencourt, Sep 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] )

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François Fages Rocquencourt, Sep 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)

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François Fages Rocquencourt, Sep 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))

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François Fages Rocquencourt, Sep Symbolic Model-Checking Still for finite Kripke structures, use boolean constraints to represent 1.sets of states as a boolean constraint c(V) 2.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)

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François Fages Rocquencourt, Sep Mammalian Cell Cycle Control Map [Kohn 99]

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François Fages Rocquencourt, Sep Cell Cycle Model-Checking (with BDD NuSMV) biocham: check_reachable(cdk46~{p1,p2}-cycD~{p1}). Ei(EF(cdk46~{p1,p2}-cycD~{p1})) is true biocham: check_checkpoint(cdc25C~{p1,p2}, cdk1~{p1,p3}-cycB). Ai(!(E(!(cdc25C~{p1,p2}) U cdk1~{p1,p3}-cycB))) is true biocham: nusmv(Ai(AG(!(cdk1~{p1,p2,p3}-cycB) -> checkpoint(Wee1, cdk1~{p1,p2,p3}-cycB))))). Ai(AG(!(cdk1~{p1,p2,p3}-cycB)->!(E(!(Wee1) U cdk1~{p1,p2,p3}-cycB)))) is false biocham: why. -- Loop starts here cycB-cdk1~{p1,p2,p3} is present cdk7 is present cycH is present cdk1 is present Myt1 is present cdc25C~{p1} is present rule_114 cycB-cdk1~{p1,p2,p3}=[cdc25C~{p1}]=>cycB-cdk1~{p2,p3}. cycB-cdk1~{p2,p3} is present cycB-cdk1~{p1,p2,p3} is absent rule_74 cycB-cdk1~{p2,p3}=[Myt1]=>cycB-cdk1~{p1,p2,p3}. cycB-cdk1~{p2,p3} is absent cycB-cdk1~{p1,p2,p3} is present

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François Fages Rocquencourt, Sep Mammalian Cell Cycle Control Benchmark 500 variables, states. 800 rules. BIOCHAM NuSMV model-checker time in sec. [Chabrier et al. TCS 04] Initial state G2Query:Time: compiling29 s Reachability G1EF CycE2 s Reachability G1EF CycD1.9 s Reachability G1EF PCNA-CycD1.7 s Checkpoint for mitosis complex EF ( Cdc25~{Nterm} U Cdk1~{Thr161}-CycB) 2.2 s Oscillation EG ( (CycA => EF CycA) ^ ( CycA => EF CycA)) 31.8 s

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François Fages Rocquencourt, Sep LTL with Constraints for the Differential Semantics Constraints over concentrations and derivatives as FOL formulae over the reals: [M] > 0.2 [M]+[P] > [Q] d([M])/dt < 0

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François Fages Rocquencourt, Sep LTL with Constraints for the Differential Semantics Constraints over concentrations and derivatives as FOL formulae over the reals: [M] > 0.2 [M]+[P] > [Q] d([M])/dt < 0 Linear Time Logic LTL operators for time X, F, U, G F([M]>0.2) FG([M]>0.2) F ([M]>2 & F (d([M])/dt 0 & F(d([M])/dt<0)))) 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 > 0 & X(d([A])/dt < 0) & F(T = t + 75 & [A] = v & d([A])/dt > 0 & X(d([A])/dt < 0)))…

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François Fages Rocquencourt, Sep How to Evaluate a Constraint LTL Formula ? Consider the ODE’s of the concentration semantics dX/dt = f(X)

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François Fages Rocquencourt, Sep How to Evaluate a Constraint LTL Formula ? Consider the ODE’s 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: (t 0,X 0,dX 0 /dt), (t 1,X 1,dX 1 /dt), …, (t n,X n,dX n /dt), … over concentrations and their derivatives at discrete time points Evaluate the formula on that Kripke structure with a model checking alg.

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François Fages Rocquencourt, Sep 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] 1.Run the numerical integration from 0 to T producing values at a finite sequence of time points 2.Iteratively label the time points with the sub-formulae of that are true: Add to the time points where a FOL formula is true, Add F (X ) to the (immediate) previous time points labeled by Add U to the predecessor time points of while they satisfy (Add G to the states satisfying until T) Model checker and numerical integration methods implemented in Prolog

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François Fages Rocquencourt, Sep Constraint-LTL Instanciation Algo. [Fages Rizk CMSB’07]

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François Fages Rocquencourt, Sep 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.

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François Fages Rocquencourt, Sep Overview of the Talk 1.Rule-based Modeling of Biochemical Systems 1.Syntax of molecules, compartments and reactions 2.Semantics at three abstraction levels: boolean, differential, stochastic 3.Cell cycle control models 2.Temporal Logic Language for Formalizing Biological Properties 1.CTL for the boolean semantics 2.Constraint LTL for the differential semantics 3.PCTL for the stochastic semantics 3.Automated Reasoning Tools 1.Inferring kinetic parameter values from Constraint-LTL specification 2.Inferring reaction rules from CTL specification

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François Fages Rocquencourt, Sep Example: Cell Cycle Control Model [Tyson 91] MA(k1) for _ => Cyclin. MA(k2) for Cyclin => _. MA(K7) for Cyclin~{p1} => _. MA(k8) for Cdc2 => Cdc2~{p1}. MA(k9) for Cdc2~{p1} =>Cdc2. MA(k3) for Cyclin+Cdc2~{p1} => Cdc2~{p1}-Cyclin~{p1}. MA(k4p) for Cdc2~{p1}-Cyclin~{p1} => Cdc2-Cyclin~{p1}. k4*[Cdc2-Cyclin~{p1}]^2*[Cdc2~{p1}-Cyclin~{p1}] for Cdc2~{p1}-Cyclin~{p1} =[Cdc2-Cyclin~{p1}] => Cdc2-Cyclin~{p1}. MA(k5) for Cdc2-Cyclin~{p1} => Cdc2~{p1}-Cyclin~{p1}. MA(k6) for Cdc2-Cyclin~{p1} => Cdc2+Cyclin~{p1}.

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François Fages Rocquencourt, Sep Inferring Parameters from Temporal Properties biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, oscil(Cdc2-Cyclin~{p1},3),150).

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François Fages Rocquencourt, Sep Inferring Parameters from Temporal Properties biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, oscil(Cdc2-Cyclin~{p1},3),150). First values found : parameter(k3,10). parameter(k4,70).

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François Fages Rocquencourt, Sep Inferring Parameters from Temporal Properties biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, oscil(Cdc2-Cyclin~{p1},3) & F([Cdc2-Cyclin~{p1}]>0.15), 150). First values found : parameter(k3,10). parameter(k4,120).

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François Fages Rocquencourt, Sep Inferring Parameters from LTL Specification biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, period(Cdc2-Cyclin~{p1},35), 150). First values found: parameter(k3,10). parameter(k4,280).

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François Fages Rocquencourt, Sep Leloup and Goldbeter (1999) MPF preMPF Wee1 Wee1P Cdc25 Cdc25P APC.. Cell cycle Linking the Cell and Circadian Cycles through Wee1 BMAL1/CLOCK PER/CRY Circadian cycle Wee1 mRNA L [L. Calzone, S. Soliman 2006]

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François Fages Rocquencourt, Sep PCN Wee1m Wee1 MPF BN Cdc25

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François Fages Rocquencourt, Sep entrainment Condition on Wee1/Cdc25 for the Entrainment in Period Entrainment in period constraint expressed in LTL with the period formula

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François Fages Rocquencourt, Sep 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

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François Fages Rocquencourt, Sep Model Revision from Temporal Properties Background knowledge T: BIOCHAM model reaction rule language: complexation, phosphorylation, … Examples φ: 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’ |= φ

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François Fages Rocquencourt, Sep 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(φ) where φ is a boolean formula (pure state description) Oscil(φ) Negative ACTL formula: if false, remains false after adding a rule AG(φ) where φ 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

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François Fages Rocquencourt, Sep Example: Cell Cycle Control Model [Tyson 91] MA(k1) for _ => Cyclin. MA(k2) for Cyclin => _. MA(K7) for Cyclin~{p1} => _. MA(k8) for Cdc2 => Cdc2~{p1}. MA(k9) for Cdc2~{p1} =>Cdc2. MA(k3) for Cyclin+Cdc2~{p1} => Cdc2~{p1}-Cyclin~{p1}. MA(k4p) for Cdc2~{p1}-Cyclin~{p1} => Cdc2-Cyclin~{p1}. k4*[Cdc2-Cyclin~{p1}]^2*[Cdc2~{p1}-Cyclin~{p1}] for Cdc2~{p1}-Cyclin~{p1} =[Cdc2-Cyclin~{p1}] => Cdc2-Cyclin~{p1}. MA(k5) for Cdc2-Cyclin~{p1} => Cdc2~{p1}-Cyclin~{p1}. MA(k6) for Cdc2-Cyclin~{p1} => Cdc2+Cyclin~{p1}.

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François Fages Rocquencourt, Sep Automatic Generation of True CTL Properties Ei(reachable(Cyclin))) Ei(reachable(!(Cyclin)))) Ai(oscil(Cyclin))) Ei(reachable(Cdc2~{p1}))) Ei(reachable(!(Cdc2~{p1})))) Ai(oscil(Cdc2~{p1}))) Ai(AG(!(Cdc2~{p1})->checkpoint(Cdc2,Cdc2~{p1})))) Ei(reachable(Cdc2-Cyclin~{p1,p2}))) Ei(reachable(!(Cdc2-Cyclin~{p1,p2})))) Ai(oscil(Cdc2-Cyclin~{p1,p2}))) Ei(reachable(Cdc2-Cyclin~{p1}))) Ei(reachable(!(Cdc2-Cyclin~{p1})))) Ai(oscil(Cdc2-Cyclin~{p1}))) Ai(AG(!(Cdc2-Cyclin~{p1})->checkpoint(Cdc2-Cyclin~{p1,p2},Cdc2-Cyclin~{p1}))) Ei(reachable(Cdc2))) Ei(reachable(!(Cdc2)))) Ai(oscil(Cdc2))) Ei(reachable(Cyclin~{p1}))) Ei(reachable(!(Cyclin~{p1})))) Ai(oscil(Cyclin~{p1}))) Ai(AG(!(Cyclin~{p1})->checkpoint(Cdc2-Cyclin~{p1},Cyclin~{p1}))))

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François Fages Rocquencourt, Sep Rule Deletion biocham: delete_rules(Cdc2 => Cdc2~{p1}). biocham: check_all. First formula not satisfied Ei(EF(Cdc2-Cyclin~{p1}))

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François Fages Rocquencourt, Sep Model Revision from Temporal Properties biocham: revise_model. Rules to delete: Rules to add: Cdc2 => Cdc2~{p1}.

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François Fages Rocquencourt, Sep Model Revision from Temporal Properties biocham: revise_model. Rules to delete: Rules to add: Cdc2 => Cdc2~{p1}. biocham: learn_one_addition. (1) Cdc2 => Cdc2~{p1}. (2) Cdc2 =[Cdc2]> Cdc2~{p1}. (3) Cdc2 =[Cyclin]> Cdc2~{p1}.

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François Fages Rocquencourt, Sep Conclusion Temporal logic with constraints is powerful enough to express both qualitative and quantitative biological properties of systems

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François Fages Rocquencourt, Sep 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

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François Fages Rocquencourt, Sep 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)

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François Fages Rocquencourt, Sep 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)

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François Fages Rocquencourt, Sep 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.

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