François Fages Les Houches, avril 2007 Formal Verification of Dynamical Models and Application to Cell Cycle Control François Fages, Sylvain Soliman Constraint

François Fages Les Houches, avril 2007

Formal Verification of Dynamical Models and Application to Cell Cycle Control

François Fages, Sylvain SolimanConstraint Programming Group, INRIA Rocquencourt

mailto:[email protected]://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 BIOCHAMmodeling environment available at http://contraintes.inria.fr/BIOCHAM


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


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 !





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


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]


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 Properties1. CTL for the boolean semantics

2. Constraint LTL for the differential semantics

3. PCTL for the stochastic semantics

3. Automated Reasoning Tools1. Inferring kinetic parameter values from Constraint-LTL specification

2. Inferring reaction rules from CTL specification

3. Type inference by abstract interpretationL. Calzone, N. Chabrier, F. Fages, S. Soliman. TCSB VI, LNBI 4220:68-94. 2006.


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]


Syntax of BIOCHAM Objects

E == name | E-E | E~{p1,…,pn}



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).



E == name | E-E | E~{p1,…,pn}






O == E | E::location

Location: symbolic compartment (nucleus, cytoplasm, cell …)



E == name | E-E | E~{p1,…,pn}






O == E | E::location

Location: symbolic compartment (nucleus, cytoplasm, cell …)

S == _ | O+S

+ : solution multiset operator (associative, commutative, neutral element _)


Basic Rule Schemas

Complexation: A + B => A-B Decomplexation A-B => A + B cdk1+cycB => cdk1–cycB


Basic Rule Schemas


Phosphorylation: A =[C]=> A~{p} Dephosphorylation A~{p} =[C]=> A

Cdk1-CycB =[Myt1]=> Cdk1~{thr161}-CycB

Cdk1~{thr14,tyr15}-CycB =[Cdc25~{Nterm}]=> Cdk1-CycB


Basic Rule Schemas





Synthesis: _ =[C]=> A. Degradation: A =[C]=> _. _ =[#Ge2-E2f13-Dp12]=> CycA cycE =[@UbiPro]=> _

(not for cycE-cdk2 which is stable)


Basic Rule Schemas





Synthesis: _ =[C]=> A. Degradation: A =[C]=> _. _ =[#Ge2-E2f13-Dp12]=> CycA cycE =[@UbiPro]=> _

(not for cycE-cdk2 which is stable)

Transport: A::L1 => A::L2Cdk1~{p}-CycB::cytoplasm => Cdk1~{p}-CycB::nucleus


From Syntax to Semantics

R ::= S=>S



R ::= S=>S | S =[O]=> S | S <=> S | S <=[O]=> 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)








1. Boolean Semantics: presence-absence of molecules 1. Concurrent Transition System (asynchronous, non-

deterministic)

2. Differential Semantics: concentration• Ordinary Differential Equations or Hybrid system

(deterministic)








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)

• Stochastic Semantics: number of molecules • Continuous time Markov chain


1. Differential Semantics

Associates to each molecule its concentration [Ai]= | Ai| / volume ML-1

volume of diffusion …




volume of compartment

Compiles a set of rules { ei for Si=>S’I }i=1,…,n (by default ei is MA(1))

into the system of ODEs (or hybrid automaton) over variables {A1,…,Ak}

dA/dt = Σni=1 ri(A)*ei - Σn

j=1 lj(A)*ej

where ri(A) (resp. li(A)) is the stoichiometric coefficient of A in Si (resp. S’i)

multiplied by the volume ratio of the location of A.




volume of compartment

Compiles a set of rules { ei for Si=>S’I }i=1,…,n (by default ei is MA(1))

into the system of ODEs (or hybrid automaton) over variables {A1,…,Ak}

dA/dt = Σni=1 ri(A)*ei - Σn

j=1 lj(A)*ej

where ri(A) (resp. li(A)) is the stoichiometric coefficient of A in Si (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.



Numerical integration

• Adaptive step size 4th order Runge-Kutta (can be weak for stiff systems)

• Rosenbrock implicit method using the Jacobian matrix ∂x’ i/∂xj

computes a (clever) discretization of time and a time series

(t0, X0, dX0/dt), (t1, X1, dX1/dt), …, (tn, Xn, dXn/dt), …

of concentrations and their derivatives at discrete time points


2. Stochastic Semantics

Associates to each molecule its number |Ai| in its location




Compiles the rule set into a continuous time Markov chain

over vector states (|A1|,…, |Ak|) where the transition rate τi for the reaction ei for Si=>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

τi = ei for reactions of the form A =>...,

τi = ei /Vi×K for reactions of the form A+B=>...,

τi = 2 × ei /Vi×K for reactions of the form A+A=>...,










Computes realizations as time series (t0, X0), (t1, X1), …, (tn, Xn), …










The differential semantics is an abstraction of the stochastic one [Gillespie 76]


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




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+BA+B+C+D

A+BA+B +C+D

A+BA+B+C+D

A+BA+B+C+D




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+BA+B+C+D

A+BA+B +C+D

A+BA+B+C+D

A+BA+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


Hierarchy of Semantics

Stochastic model

Differential model

Discrete model

abstraction

concretization

Boolean model

Theory of

abstract Interpretation [Cousot Cousot 77]


Cell Signaling

Receptors: RTK tyrosine kinase, G-protein coupled, Notch…


Cell Signaling


Signals: hormones insulin, adrenaline, steroids, EGF, … membrane proteins Delta, … nutriments, light, pressure …


Cell Signaling



L + R <=> L-R


Cell Signaling



L + R <=> L-R

L-R + L-R <=> L-R-L-R


Cell Signaling



L + R <=> L-R

L-R + L-R <=> L-R-L-R

RAS-GDP =[L-R-L-R]=> RAS-GTP


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


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





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


Bipartite Protein-Reaction Graph of MAPK

GraphVizhttp://www.research.att.co/sw/tools/graphviz


Numerical Simulation of MAPK Signaling


Boolean Simulation (random) of MAPK Signaling


Five MAP Kinase Pathways in Budding Yeast

(Saccharomyces Cerevisiae)


Transcription: DNA pre-mRNA mRNA Protein

• Activation: transcription factors bind to the regulatory region of the gene

#E2 + E2F13-DP12 <=> #E2-E2F13-DP12




#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




#E2 + E2F13-DP12 <=> #E2-E2F13-DP12



• (Alternative) splicing: non coding regions of pRNA are removed giving mature messenger mRNA pRNAcycA => mRNAcycA




#E2 + E2F13-DP12 <=> #E2-E2F13-DP12




• Transport: mRNA moves from the nucleus to the cytoplasm mRNAcycA => mRNAcycA::c




#E2 + E2F13-DP12 <=> #E2-E2F13-DP12




• 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


Numerical Simulation with Default Mass Action Law


Boolean simulation (random) of transcription


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


Lateral Inhibition through Delta-Notch Signaling

Delta production is triggered by low Notch concentration in the same cell

if [N::c1]<0.5 then 1,[D::c1] for _<=>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.





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.



1. Rule-based Language for Modeling Biochemical Systems 1. Syntax of molecules, compartments and reactions2. Semantics at three abstraction levels: boolean, differential,

stochastic

3. Examples of signal transduction, transcription, cell-cell interaction

2. Temporal Logic Language for Formalizing Biological Properties1. CTL for the boolean semantics2. Constraint LTL for the differential semantics3. PCTL for the stochastic semantics

3. Automated Reasoning Tools1. Inferring kinetic parameter values from Constraint-LTL

specification2. Inferring reaction rules from CTL specification

3. Type inference by abstract interpretation


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


2.1 Computation Tree Logic CTL

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

Time

Non-determinism E, A

F,G,U EF

EU

AG

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

untilE (f1 U f2) A (f1 U f2)


Biological Properties formalized in CTL (1/3)

About reachability:

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



About reachability:


• Can the cell produce P, Q and not R? reachable(P^Q^R)



About reachability:



• Can the cell always produce P? AG(reachable(P))



About reachability:




About pathways:

• Can the cell reach a (partially described) set of states s while passing by another set of states s2? EF(s2^EFs)



About reachability:




About pathways:


• Is it possible to produce P without Q? E(Q U P)



About reachability:




About pathways:


• 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)



About reachability:




About pathways:


• 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 -> checkpoint(s2,s))



About stationarity:

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



About stationarity:


• Is s a steady state (with possibility of escaping) ? steady(s)==EG(s)



About stationarity:



• Can the cell reach a stable state s? EF(stable(s)) not in LTL



About stationarity:




• Must the cell reach a stable state s? AG(stable(s))



About stationarity:




• 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] )



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)



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))


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 checkpointsbiocham: checkpoint(MEK~{p1}-MEKPH , MAPK~{p1,p2}).!E(!MEK~{p1}-MEKPH U MAPK~{p1,p2}) is falseBiocham : why.Step 1 rule 15 Step 2 rule 1 RAF-RAFK presentStep 3 rule 21 RAF~{p1} presentStep 4 rule 5 MEK-RAF~{p1} presentStep 5 rule 24 MEK~{p1} presentStep 6 rule 7 MEK~{p1}-RAF~{p1} presentStep 7 rule 23 MEK~{p1,p2} presentStep 8 rule 13 MAPK-MEK~{p1,p2} presentStep 9 rule 27 MAPK~{p1} presentStep 10 rule 15 MAPK~{p1}-MEK~{p1,p2} presentStep 11 rule 28 MAPK~{p1,p2} present


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}


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:

{sS : 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|*||).


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)


Cell Cycle: G1 DNA Synthesis G2 Mitosis

G1: CdK4-CycD S: Cdk2-CycA G2,M: Cdk1-CycA

Cdk6-CycD Cdk1-CycB (MPF)

Cdk2-CycE



Mammalian Cell Cycle Control Map [Kohn 99]


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]


Cell Cycle Model-Checking (with BDD NuSMV)

biocham: check_reachable(cdk46~{p1,p2}-cycD~{p1}). Ei(EF(cdk46~{p1,p2}-cycD~{p1})) is truebiocham: check_checkpoint(cdc25C~{p1,p2}, cdk1~{p1,p3}-cycB). Ai(!(E(!(cdc25C~{p1,p2}) U cdk1~{p1,p3}-cycB))) is truebiocham: 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 falsebiocham: 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 presentrule_114 cycB-cdk1~{p1,p2,p3}=[cdc25C~{p1}]=>cycB-cdk1~{p2,p3}. cycB-cdk1~{p2,p3} is present cycB-cdk1~{p1,p2,p3} is absentrule_74 cycB-cdk1~{p2,p3}=[Myt1]=>cycB-cdk1~{p1,p2,p3}. cycB-cdk1~{p2,p3} is absent cycB-cdk1~{p1,p2,p3} is present


Mammalian Cell Cycle Control Benchmark

500 variables, 2500 states. 800 rules.

BIOCHAM NuSMV model-checker time in sec. [Chabrier et al. TCS 04]

Initial state G2 Query: Time:

compiling 29

Reachability G1 EF CycE 2

Reachability G1 EF CycD 1.9

Reachability G1 EF PCNA-CycD 1.7

Checkpoint

for mitosis complex

EF ( Cdc25~{Nterm}

U Cdk1~{Thr161}-CycB)

2.2

Oscillation EG ( (CycA EF CycA) ( CycA EF CycA))

31.8


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


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 ([M]<2 & 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)))…


How to Evaluate a Constraint LTL Formula ?

• Consider the ODE’s of the concentration semantics dX/dt = f(X)


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:

(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.


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


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.



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 tra nsduction, transcription, cell-cell interaction

2. Temporal Logic Language for Formalizing Biological Properties1. CTL for the boolean semantics

2. Constraint LTL for the differential semantics

3. PCTL for the stochastic semantics

3. Automated Reasoning Tools1. Inferring kinetic parameter values from Constraint-LTL

specification

2. Inferring reaction rules from CTL specification

3. Type inference by abstract interpretation


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}.


Interaction Graph


Concentration Simulation


Boolean Simulation


Inferring Parameters from Temporal Properties

biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20,

oscil(Cdc2-Cyclin~{p1},3),150).




oscil(Cdc2-Cyclin~{p1},3),150).

First values found :

parameter(k3,10).

parameter(k4,70).




oscil(Cdc2-Cyclin~{p1},3) & F([Cdc2-Cyclin~{p1}]>0.15), 150).

First values found :

parameter(k3,10).

parameter(k4,120).


Inferring Parameters from LTL Specification


period(Cdc2-Cyclin~{p1},35), 150).

First values found:

parameter(k3,10).

parameter(k4,280).


Leloup and Goldbeter (1999)

MPF preMPF

Wee1

Wee1P

Cdc25

Cdc25PAPC

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


PCN

Wee1m

Wee1MPF

BN

Cdc25


entrainmententrainment

Condition on Wee1/Cdc25 for the Entrainment in Period

Entrainment in period constraint expressed in LTL with the period formula


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


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


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


Cell Cycle Kinetic Model [Tyson 91]


Automatic Generation of True CTL PropertiesEi(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}))))


Rule Deletion

biocham: delete_rules(Cdc2 => Cdc2~{p1}).

biocham: check_all.

First formula not satisfied

Ei(EF(Cdc2-Cyclin~{p1}))


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}.


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)


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


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

Documents

François Fages Les Houches, avril 2007 Formal Verification of Dynamical Models and Application to Cell Cycle Control François Fages, Sylvain Soliman Constraint