To tackle the complexity of biological systems, investigate Programming Theory Concepts

François Fages Rocquencourt, Sep. 2007

Semantical and Algorithmic Aspects of the Living

François FagesConstraint Programming project-team,

INRIA Paris-Rocquencourt

To tackle the complexity of biological systems, investigate• Programming Theory Concepts• Formal Methods of Circuit and Program Verification• Automated Reasoning Tools

Prototype Implementation in the Biochemical Abstract Machine 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 large networks of genes and 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


Semantics of Living Processes ?

Formally, “the” behavior of a system depends on our choice of observables.

? ?

Mitosis movie [Lodish et al. 03]


Boolean Semantics


Presence/absence of molecules

Boolean transitions

0 1


Continuous Differential Semantics


Concentrations of molecules

Rates of reactions

x ý


Stochastic Semantics


Numbers of molecules

Probabilities of reaction

n


Temporal Logic Semantics


Presence/absence of molecules

Temporal logic formulas

F xF x

F (x ^ F ( x ^ y))

FG (x v y)

…


Constraint Temporal Logic Semantics


Concentrations of molecules

Constraint LTL temporal formulas

F x>1F (x >0.2)

F (x >0.2 ^ F (x<0.1 ^ y>0.2))

FG (x>0.2 v y>0.2)

…


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 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 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

L. Calzone, N. Chabrier, F. Fages, S. Soliman. TCSB VI, LNBI 4220:68-94. 2006.


Molecules of the living

Small molecules: covalent bonds 50-200 kcal/mol

• 70% water

• 1% ions

• 6% amino acids (20), nucleotides (5),

• fats, sugars, ATP, ADP, …

Macromolecules: hydrogen bonds, ionic, hydrophobic, Waals 1-5 kcal/mol

• 20% proteins (50-104 amino acids)

• RNA (102-104 nucleotides AGCU)

• DNA (102-106 nucleotides AGCT)


Structure Levels of Proteins

1) Primary structure: word of n amino acids residues (20n possibilities)

linked with C-N bonds

Example: INRIA

Isoleucine-asparagiNe-aRginine-Isoleucine-Alanine





Example: INRIA


2) Secondary: word of m helix, strands, random coils,… (3m-10m)

stabilized by hydrogen bonds H---O





Example: INRIA


2) Secondary: word of m helix, strands, random coils,… (3m-10m)

stabilized by hydrogen bonds H---O

3) Tertiary 3D structure: spatial folding

stabilized by

hydrophobic

interactions


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


BIOCHAM Syntax of Objects

E == compound | E-E | E~{p1,…,pn}

Compound: molecule, #gene binding site, abstract @process…

- : binding operator for protein complexes, gene binding sites, …

Associative and commutative.

~{…}: modification operator for phosphorylated sites, …

Set of modified sites (Associative, Commutative, Idempotent).

O == E | E::location

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

S == _ | O+S

+ : solution operator (Associative, Commutative, Neutral _)


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 =[@UbiPro]=> _

(not for cycE-cdk2 which is stable)







Synthesis: _ =[C]=> A. Degradation: A =[C]=> _. _ =[#E2-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 | 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 of concentrations and their derivatives

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

at discrete time points


2. Stochastic Semantics

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




Compile the rule set into a continuous time Markov chain

over vector states X=(|A1|,…, |Ak|)

and where the transition rate τi for the reaction ei for Si=>S’I

(giving probability after normalization) is obtained from ei by replacing concentrations by molecule numbers




Compile the rule set into a continuous time Markov chain

over vector states X=(|A1|,…, |Ak|)

and where the transition rate τi for the reaction ei for Si=>S’I

(giving probability after normalization) is obtained from ei by replacing concentrations by molecule numbers

Stochastic simulation [Gillespie 76, Gibson 00]

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


3. Boolean Semantics

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




Compile the rule set into an asynchronous transition system




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+BA+B+C+D

A+BA+B +C+D

A+BA+B+C+D

A+BA+B+C+D




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+BA+B+C+D

A+BA+B +C+D

A+BA+B+C+D

A+BA+B+C+D

Necessary for over-approximating the possible behaviors under the stochastic/discrete semantics (abstraction N {zero, non-zero})


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


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 analysisabstraction

abstraction

abstraction

Positive circuits are a

necessary condition

for multistationarity

[Thomas 73] [de Jong 02]

[Soulé 03]

[Remy Ruet Thieffry 05]


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


Type Inference / Type Checking

Stochastic model

Differential model

Discrete model

abstraction

concretization

Boolean model

Syntactical

model

[Fages Soliman CMSB’06]


Protein functions

(kinase, phosphatase,…)

Compartments topology


(activation/inhibition)


Cell Cycle: G1 DNA Synthesis G2 Mitosis

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

Cdk6-CycD Cdk1-CycB (MPF)

Cdk2-CycE


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


Stochastic Simulation


Differential Simulation


Boolean Simulation



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]






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



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

specification



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] …






In the Biochemical Abstract Machine environment,

Model: BIOCHAM

- Boolean - simulation

- Differential

- Stochastic

(SBML)







Model: BIOCHAM Biological Properties:

- Boolean - simulation - CTL

- Differential - query evaluation - LTL with constraints

- Stochastic - PCTL with constraints

(SBML)










- Stochastic - PCTL with constraints

(SBML)










- Stochastic - rule learning - PCTL with constraints

(SBML) - parameter search


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))


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)


Mammalian Cell Cycle Control Map [Kohn 99]


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 s

Reachability G1 EF CycE 2 s

Reachability G1 EF CycD 1.9 s

Reachability G1 EF PCNA-CycD 1.7 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


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


Constraint-LTL Instanciation Algo. [Fages Rizk CMSB’07]


2.3 PCTL Model Checker for the Stochastic Semantics

Compute the probability of realisation of a TL formula (with constraints) by Monte Carlo method

Perform several stochastic simulations

Evaluate the probability of realization of the TL formula

Costly…

PRISM [Kwiatkowska et al. 04] : PCTL model checker based on BDDs or using Monte Carlo method.






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



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

specification
















3.1 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).


3.1 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 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















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}))



biocham: revise_model.

Rules to delete:

Rules to add:

Cdc2 => Cdc2~{p1}.



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}.


Conclusion

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


Conclusion


• 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


Conclusion






• Parameter search from temporal properties proved useful and complementary to bifurcation theory tools (Xppaut)


Conclusion






• 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)


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.

Date post:	07-Jan-2016
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