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François Fages Les Houches, avril 2007 Formal Verification of Dynamical Models and Application to Cell Cycle Control François Fages, Sylvain Soliman Constraint Programming Group, INRIA Rocquencourt mailto:Francois.Fages@inria.fr http://contraintes.inria.fr/ Main idea: to master the complexity of biological systems investigate Programming Language 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

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François Fages Les Houches, avril 2007 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 big networks of genes or proteins. Systems Biology Markup Language (SBML) : exchange format for reaction models

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François Fages Les Houches, avril 2007 Issue of Abstraction Models are built in Systems Biology with two contradictory perspectives :

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 Overview of the Lecture 1.Rule-based Language for Modeling Biochemical Systems 1.Syntax of molecules, compartments and reactions 2.Semantics at three abstraction levels: boolean, differential, stochastic 3.Simple examples of signal transduction, transcription, cell-cell interaction 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 3.Type inference by abstract interpretation L. Calzone, N. Chabrier, F. Fages, S. Soliman. TCSB VI, LNBI 4220:68-94. 2006.

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François Fages Les Houches, avril 2007 Formal Proteins Cyclin dependent kinase 1 Cdk1 (free, inactive) Complex Cdk1-Cyclin B Cdk1–CycB (low activity preMPF) Phosphorylated form Cdk1~{thr161}-CycB at site threonine 161 (high activity MPF) [Alberts et al 2002]

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François Fages Les Houches, avril 2007 Syntax of BIOCHAM Objects E == name | E-E | E~{p1,…,pn}

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François Fages Les Houches, avril 2007 Syntax of BIOCHAM Objects E == name | E-E | E~{p1,…,pn} name : molecule, #gene binding site, abstract @process… - : binding operator for protein complexes, gene bindings, … Associative and commutative. ~{…} : modification operator for phosphorylated sites, acetylated, etc… Set of modified sites (associative, commutative, idempotent).

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François Fages Les Houches, avril 2007 Syntax of BIOCHAM Objects E == name | E-E | E~{p1,…,pn} name : molecule, #gene binding site, abstract @process… - : binding operator for protein complexes, gene bindings, … Associative and commutative. ~{…} : modification operator for phosphorylated sites, acetylated, etc… Set of modified sites (associative, commutative, idempotent). O == E | E::location Location : symbolic compartment (nucleus, cytoplasm, cell …)

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François Fages Les Houches, avril 2007 Syntax of BIOCHAM Objects E == name | E-E | E~{p1,…,pn} name : molecule, #gene binding site, abstract @process… - : binding operator for protein complexes, gene bindings, … Associative and commutative. ~{…} : modification operator for phosphorylated sites, acetylated, etc… Set of modified sites (associative, commutative, idempotent). O == E | E::location Location : symbolic compartment (nucleus, cytoplasm, cell …) S == _ | O+S + : solution multiset operator (associative, commutative, neutral element _)

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François Fages Les Houches, avril 2007 Basic Rule Schemas Complexation: A + B => A-B Decomplexation A-B => A + B cdk1+cycB => cdk1–cycB

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François Fages Les Houches, avril 2007 Basic 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

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François Fages Les Houches, avril 2007 Basic 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. Degradation: A =[C]=> _. _ =[#Ge2-E2f13-Dp12]=> CycA cycE =[@UbiPro]=> _ (not for cycE-cdk2 which is stable)

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François Fages Les Houches, avril 2007 Basic 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. Degradation: A =[C]=> _. _ =[#Ge2-E2f13-Dp12]=> CycA cycE =[@UbiPro]=> _ (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 Les Houches, avril 2007 From Syntax to Semantics R ::= S=>S

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François Fages Les Houches, avril 2007 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)

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 1. Differential Semantics Associates to each molecule its concentration [A i ]= | A i | / volume ML -1 volume of diffusion …

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François Fages Les Houches, avril 2007 1. 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 Les Houches, avril 2007 1. 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 Les Houches, avril 2007 1. Differential Semantics 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 (t 0, X 0, dX 0 /dt), (t 1, X 1, dX 1 /dt), …, (t n, X n, dX n /dt), … of concentrations and their derivatives at discrete time points

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François Fages Les Houches, avril 2007 2. Stochastic Semantics Associates to each molecule its number |A i | in its location

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François Fages Les Houches, avril 2007 2. Stochastic Semantics Associates to each molecule its number |A i | in its location Compiles the rule set into a continuous time Markov chain over vector states (|A 1 |,…, |A k |) where the transition rate τ i for the reaction e i for S i =>S’ I (giving probability after normalization) is

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François Fages Les Houches, avril 2007 2. Stochastic Semantics Associates to each molecule its number |A i | in its location Compiles the rule set into a continuous time Markov chain over vector states (|A 1 |,…, |A k |) where the transition rate τ i for the reaction e i for S i =>S’ I (giving probability after normalization) is [Gillespie 76, Gibson 00] where Vi is the volume where the reaction occurs and K is Avogadro number τ i = e i for reactions of the form A =>..., τ i = ei /Vi×K for reactions of the form A+B=>..., τ i = 2 × ei /Vi×K for reactions of the form A+A=>...,

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François Fages Les Houches, avril 2007 2. Stochastic Semantics Associates to each molecule its number |A i | in its location Compiles the rule set into a continuous time Markov chain over vector states (|A 1 |,…, |A k |) where the transition rate τ i for the reaction e i for S i =>S’ I (giving probability after normalization) is [Gillespie 76, Gibson 00] where Vi is the volume where the reaction occurs and K is Avogadro number τ i = e i for reactions of the form A =>..., τ i = ei /Vi×K for reactions of the form A+B=>..., τ i = 2 × ei /Vi×K for reactions of the form A+A=>..., Computes realizations as time series (t 0, X 0 ), (t 1, X 1 ), …, (t n, X n ), …

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François Fages Les Houches, avril 2007 2. Stochastic Semantics Associates to each molecule its number |A i | in its location Compiles the rule set into a continuous time Markov chain over vector states (|A 1 |,…, |A k |) where the transition rate τ i for the reaction e i for S i =>S’ I (giving probability after normalization) is [Gillespie 76, Gibson 00] where Vi is the volume where the reaction occurs and K is Avogadro number τ i = e i for reactions of the form A =>..., τ i = ei /Vi×K for reactions of the form A+B=>..., τ i = 2 × ei /Vi×K for reactions of the form A+A=>..., The differential semantics is an abstraction of the stochastic one [Gillespie 76]

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François Fages Les Houches, avril 2007 3. Boolean Semantics Associates to each molecule a Boolean denoting its presence/absence in its location

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François Fages Les Houches, avril 2007 3. Boolean Semantics Associates to each molecule a Boolean denoting its presence/absence in its location Compiles the rule set into an asynchronous transition system

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François Fages Les Houches, avril 2007 3. Boolean Semantics Associates to each molecule a Boolean denoting its presence/absence in its location Compiles 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 Les Houches, avril 2007 3. Boolean Semantics Associates to each molecule a Boolean denoting its presence/absence in its location Compiles 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 to over-approximate the possible behaviors under : 1.the stochastic semantics : trivial abstraction N {zero, non-zero} 2.the differential semantics : harder to relate mathematically

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François Fages Les Houches, avril 2007 Hierarchy of Semantics Stochastic model Differential model Discrete model abstraction concretization Boolean model Theory of abstract Interpretation [Cousot Cousot 77]

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François Fages Les Houches, avril 2007 Cell Signaling Receptors: RTK tyrosine kinase, G-protein coupled, Notch…

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François Fages Les Houches, avril 2007 Cell Signaling Receptors: RTK tyrosine kinase, G-protein coupled, Notch… Signals: hormones insulin, adrenaline, steroids, EGF, … membrane proteins Delta, … nutriments, light, pressure …

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François Fages Les Houches, avril 2007 Cell Signaling Receptors: RTK tyrosine kinase, G-protein coupled, Notch… Signals: hormones insulin, adrenaline, steroids, EGF, … membrane proteins Delta, … nutriments, light, pressure … L + R L-R

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François Fages Les Houches, avril 2007 Cell Signaling Receptors: RTK tyrosine kinase, G-protein coupled, Notch… Signals: hormones insulin, adrenaline, steroids, EGF, … membrane proteins Delta, … nutriments, light, pressure … L + R L-R L-R + L-R L-R-L-R

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François Fages Les Houches, avril 2007 Cell Signaling Receptors: RTK tyrosine kinase, G-protein coupled, Notch… Signals: hormones insulin, adrenaline, steroids, EGF, … membrane proteins Delta, … nutriments, light, pressure … L + R L-R L-R + L-R L-R-L-R RAS-GDP =[L-R-L-R]=> RAS-GTP

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François Fages Les Houches, avril 2007 MAPK Signaling Pathways Input: RAF activated by the receptor RAF-p14-3-3 + RAS-GTP => RAF + p14-3-3 + RAS-GDP Output: MAPK~{T183,Y185} moves to the nucleus phosphorylates a transcription factor which stimulates gene transcription

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François Fages Les Houches, avril 2007 MAPK Signalling Network [Levchencko et al 2000] (MA(1),MA(0.4)) for RAF + RAFK RAF-RAFK. MA(0.1) for RAF-RAFK => RAFK + RAF~{p1}.

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François Fages Les Houches, avril 2007 MAPK Signalling Network [Levchencko et al 2000] (MA(1),MA(0.4)) for RAF + RAFK RAF-RAFK. MA(0.1) for RAF-RAFK => RAFK + RAF~{p1}. RAF~{p1} + RAFPH RAF~{p1}-RAFPH. RAF~{p1}-RAFPH => RAF + RAFPH. MEK~$P + RAF~{p1} MEK~$P-RAF~{p1} where p2 not in $P. MEK~{p1}-RAF~{p1} => MEK~{p1,p2} + RAF~{p1}. $P pattern variable for sites or molecules

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François Fages Les Houches, avril 2007 MAPK Signalling Network [Levchencko et al 2000] (MA(1),MA(0.4)) for RAF + RAFK RAF-RAFK. MA(0.1) for RAF-RAFK => RAFK + RAF~{p1}. RAF~{p1} + RAFPH RAF~{p1}-RAFPH. RAF~{p1}-RAFPH => RAF + RAFPH. MEK~$P + RAF~{p1} MEK~$P-RAF~{p1} where p2 not in $P. MEK~{p1}-RAF~{p1} => MEK~{p1,p2} + RAF~{p1}. MEK-RAF~{p1} => MEK~{p1} + RAF~{p1}. MEKPH + MEK~{p1}~$P MEK~{p1}~$P-MEKPH. MEK~{p1}-MEKPH => MEK + MEKPH. MEK~{p1,p2}-MEKPH => MEK~{p1} + MEKPH. MAPK~$P+MEK~{p1,p2} MAPK~$P-MEK~{p1,p2} where p2 not in $P. MAPKPH + MAPK~{p1}~$P MAPK~{p1}~$P-MAPKPH. MAPK~{p1}-MAPKPH => MAPK + MAPKPH. MAPK~{p1,p2}-MAPKPH => MAPK~{p1} + MAPKPH. MAPK-MEK~{p1,p2} => MAPK~{p1} + MEK~{p1,p2}. MAPK~{p1}-MEK~{p1,p2} => MAPK~{p1,p2}+MEK~{p1,p2}. $P pattern variable for sites or molecules

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François Fages Les Houches, avril 2007 Bipartite Protein-Reaction Graph of MAPK GraphViz http://www.research.att.co/sw/tools/graphviz

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François Fages Les Houches, avril 2007 Numerical Simulation of MAPK Signaling

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François Fages Les Houches, avril 2007 Boolean Simulation (random) of MAPK Signaling

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François Fages Les Houches, avril 2007 Five MAP Kinase Pathways in Budding Yeast (Saccharomyces Cerevisiae)

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François Fages Les Houches, avril 2007 Transcription: DNA pre-mRNA mRNA Protein Activation: transcription factors bind to the regulatory region of the gene #E2 + E2F13-DP12 #E2-E2F13-DP12

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François Fages Les Houches, avril 2007 Transcription: DNA pre-mRNA mRNA Protein Activation: transcription factors bind to the regulatory region of the gene #E2 + E2F13-DP12 #E2-E2F13-DP12 Transcription: RNA polymerase copies the DNA from start to stop positions into a single stranded pre-mature messenger RNA _ _ =[#E2-E2F13-DP12]=> pRNAcycA

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François Fages Les Houches, avril 2007 Transcription: DNA pre-mRNA mRNA Protein Activation: transcription factors bind to the regulatory region of the gene #E2 + E2F13-DP12 #E2-E2F13-DP12 Transcription: RNA polymerase copies the DNA from start to stop positions into a single stranded pre-mature messenger RNA _ _ =[#E2-E2F13-DP12]=> pRNAcycA (Alternative) splicing: non coding regions of pRNA are removed giving mature messenger mRNA pRNAcycA => mRNAcycA

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François Fages Les Houches, avril 2007 Transcription: DNA pre-mRNA mRNA Protein Activation: transcription factors bind to the regulatory region of the gene #E2 + E2F13-DP12 #E2-E2F13-DP12 Transcription: RNA polymerase copies the DNA from start to stop positions into a single stranded pre-mature messenger RNA _ _ =[#E2-E2F13-DP12]=> pRNAcycA (Alternative) splicing: non coding regions of pRNA are removed giving mature messenger mRNA pRNAcycA => mRNAcycA Transport: mRNA moves from the nucleus to the cytoplasm mRNAcycA => mRNAcycA::c

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François Fages Les Houches, avril 2007 Transcription: DNA pre-mRNA mRNA Protein Activation: transcription factors bind to the regulatory region of the gene #E2 + E2F13-DP12 #E2-E2F13-DP12 Transcription: RNA polymerase copies the DNA from start to stop positions into a single stranded pre-mature messenger RNA _ _ =[#E2-E2F13-DP12]=> pRNAcycA (Alternative) splicing: non coding regions of pRNA are removed giving mature messenger mRNA pRNAcycA => mRNAcycA Transport: mRNA moves from the nucleus to the cytoplasm mRNAcycA => mRNAcycA::c Translation: mRNA binds to a ribosome and synthesizes its protein mRNAcycA::c + Ribosome::c mRNAcycA-Ribosome::c _ =[mRNAcycA-Ribosome::c]=> cycA::c

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François Fages Les Houches, avril 2007 Numerical Simulation with Default Mass Action Law

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François Fages Les Houches, avril 2007 Boolean simulation (random) of transcription

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François Fages Les Houches, avril 2007 Cell Differentiation by Delta-Notch Signaling Xenopus embryonic skin [Ghosh, Tomlin 2001] At the steady state, a cell has either the Delta phenotype or the Notch

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François Fages Les Houches, avril 2007 Lateral Inhibition through Delta-Notch Signaling Delta production is triggered by low Notch concentration in the same cell if [N::c1] D::c1.

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François Fages Les Houches, avril 2007 Lateral Inhibition through Delta-Notch Signaling Delta production is triggered by low Notch concentration in the same cell if [N::c1] D::c1. Notch production is triggered by high Delta levels in neigboring cells if [D::c21]+[D::c23]+[D::c12]+[D::c32]>0.2 then 1,[N::c22] for _ N::c22.

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François Fages Les Houches, avril 2007 Lateral Inhibition through Delta-Notch Signaling Delta production is triggered by low Notch concentration in the same cell if [N::c1] D::c1. Notch production is triggered by high Delta levels in neigboring cells if [D::c21]+[D::c23]+[D::c12]+[D::c32]>0.2 then 1,[N::c22] for _ N::c22.

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François Fages Les Houches, avril 2007 Overview of the Lecture 1.Rule-based Language for Modeling Biochemical Systems 1.Syntax of molecules, compartments and reactions 2.Semantics at three abstraction levels: boolean, differential, stochastic 3.Examples of signal transduction, transcription, cell-cell interaction 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 3.Type inference by abstract interpretation

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François Fages Les Houches, avril 2007 A Logical Paradigm for Systems Biology Biological model = Transition System Biological property = Temporal Logic Formula Biological validation = Model-checking Express properties 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

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François Fages Les Houches, avril 2007 2.1 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 Temporal Logic Querying of MAPK Signaling Pathway MEK~{p1} is a checkpoint for the cascade, i.e. producing MAPK~{p1,p2} biocham: checkpoint(MEK~{p1}, MAPK~{p1,p2}). !E(!MEK~{p1} U MAPK~{p1,p2}) is True The phosphatase PH complexes are not checkpoints biocham: checkpoint(MEK~{p1}-MEKPH, MAPK~{p1,p2}). !E(!MEK~{p1}-MEKPH U MAPK~{p1,p2}) is false Biocham : why. 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

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François Fages Les Houches, avril 2007 Generation of CTL properties & Model Reduction biocham: add_genCTL. … 70 properties of reachability, oscillation and checkpoints are generated biocham: reduce_model. 1: deleting RAF-RAFK=>RAF+RAFK 2: deleting RAFPH-RAF~{p1}=>RAFPH+RAF~{p1} 3: deleting MEK-RAF~{p1}=>MEK+RAF~{p1} 4: deleting MEKPH-MEK~{p1}=>MEKPH+MEK~{p1} 5: deleting MAPK-MEK~{p1,p2}=>MAPK+MEK~{p1,p2} 6: deleting MAPKPH-MAPK~{p1}=>MAPKPH+MAPK~{p1} 7: deleting MEK~{p1}-RAF~{p1}=>MEK~{p1}+RAF~{p1} 8: deleting MEKPH-MEK~{p1,p2}=>MEKPH+MEK~{p1,p2} 9: deleting MAPK~{p1}-MEK~{p1,p2}=>MAPK~{p1}+MEK~{p1,p2 10: deleting MAPKPH-MAPK~{p1,p2}=>MAPKPH+MAPK~{p1,p2}

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François Fages Les Houches, avril 2007 Basic Model-Checking Algorithm Model Checking is an algorithm for computing, in a given finite Kripke structure K the set of states satisfying a CTL formula: {s S : s |= }. Represent K as a (finite) graph and iteratively label the nodes with the subformulas of which are true in that node. Add to the states satisfying Add EF (EX ) to the (immediate) predecessors of states labeled by Add E( U ) to the predecessor states of while they satisfy Add EG to the states for which there exists a path leading to a non trivial strongly connected component of the subgraph of states satisfying Thm. CTL model checking is P-complete, model checking alg in O(|K|*| |).

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François Fages Les Houches, avril 2007 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, TAUT is co-NP) (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 Les Houches, avril 2007 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 Les Houches, avril 2007

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François Fages Les Houches, avril 2007 Mammalian Cell Cycle Control Map [Kohn 99]

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 Mammalian Cell Cycle Control Benchmark 500 variables, 2 500 states. 800 rules. BIOCHAM NuSMV model-checker time in sec. [Chabrier et al. TCS 04] Initial state G2Query:Time: compiling29 Reachability G1EF CycE2 Reachability G1EF CycD1.9 Reachability G1EF PCNA-CycD1.7 Checkpoint for mitosis complex EF ( Cdc25~{Nterm} U Cdk1~{Thr161}-CycB) 2.2 Oscillation EG ( (CycA EF CycA) ( CycA EF CycA)) 31.8

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François Fages Les Houches, avril 2007 2.2 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Monte Carlo method.

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François Fages Les Houches, avril 2007 Overview of the Lecture 1.Rule-based Language for Modeling Biochemical Systems 1.Syntax of molecules, compartments and reactions 2.Semantics at three abstraction levels: boolean, differential, stochastic 3.Examples of signal transduction, transcription, cell-cell interaction 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 3.Type inference by abstract interpretation

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 Interaction Graph

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François Fages Les Houches, avril 2007 Concentration Simulation

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François Fages Les Houches, avril 2007 Boolean Simulation

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 RR INRIA 2006

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François Fages Les Houches, avril 2007 PCN Wee1m Wee1 MPF BN Cdc25

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 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

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 Cell Cycle Kinetic Model [Tyson 91]

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François Fages Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 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 Les Houches, avril 2007 Use of Types in Model Revision Use of types to limit the search to biologically possible rules and to speed- up the search process The types of protein functions (kinase, phosphatase etc.) restrict the rules where the protein can appear The types of influences (activation, inhibition) similalry restrict the possible rules. Type inference by abstract interpretation [Fages Soliman CMSB’06] Inference of the influence graph from the reaction model, either from the differential semantics (Jacobian) or from the syntax (pattern matching)

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François Fages Les Houches, avril 2007 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 (inefficient) Parameter search from temporal properties proved useful and complementary to bifurcation theory and tools (Xppaut) Rule inference from temporal properties still in its infancy, to be optimized and improved by types computed by abstract interpretation

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François Fages Les Houches, avril 2007 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 (use of types) STREP TEMPO : Cancer chronotherapies, INSERM Villejuif, Francis Lévi Coupled models of cell cycle, circadian cycle, cytotoxic drugs. INRA Tours : FSH signalling, Eric Reiter, Frédérique Clément, Domitille

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