Joint work with and

François Fages Evry, April 2004

The Biochemical Abstract Machine BIOCHAMLogic programming steps towards formal biology

François Fages, INRIA Rocquencourt http://contraintes.inria.fr/

Joint work with andNathalie Chabrier-Rivier Sylvain Soliman

ARC CPBIO “Process Calculi and Biology of Molecular Networks”Alexander Bockmayr, LORIA Nancy, Vincent Danos, CNRS Paris PPS,Vincent Schächter, Genoscope Evry, et al.http://contraintes.inria.fr/cpbio/


Current Revolution in Systems Biology

• Elucidation of high-level biological processes

in terms of their biochemical basis at the molecular level.

• Mass production of genomic and post-genomic data:

ARN expression, protein synthesis, protein-protein interactions,…

• Need for a strong parallel effort on the formal representation of biological processes: Systems Biology.

• Need for formal tools for modeling and reasoning about their global behavior.


Formalisms for Modeling Biochemical Systems

• Diagrammatic notation • Boolean networks [Thomas 73]• Milner’s pi–calculus [Regev-Silverman-Shapiro 99-01, Nagasali et al. 00]• Concurrent transition systems [Chabrier-Chiaverini-Danos-Fages-Schachter

03] Biochemical abstract machine BIOCHAM [Chabrier-Fages-Soliman 03] Pathway logic [Eker-Knapp-Laderoute-Lincoln-Meseguer-Sonmez 02]• Bio-ambients [Regev-Panina-Silverman-Cardelli-Shapiro 03]

• Differential equations • Hybrid Petri nets [Hofestadt-Thelen 98, Matsuno et al. 00]• Hybrid automata [Alur et al. 01, Ghosh-Tomlin 01]• Hybrid concurrent constraint languages [Bockmayr-Courtois 01]


Our Goal

Beyond simulation, provide formal tools for querying, validating and completing biological models.

Our proposal:

• Use of temporal logic CTL as a query language for models of biological processes;

• Use of concurrent transition systems for their modeling;

• Use of symbolic and constraint-based model checkers for automatically evaluating CTL queries in qualitative and quantitative models.

• Use of inductive logic programming for learning models

In course, learn and teach bits of biology with logic programs.


References

A wonderful textbook:Molecular Cell Biology. 5th Edition, 1100 pages+CD, Freeman Publ.Lodish, Berk, Zipursky, Matsudaira, Baltimore, Darnell. Nov. 2003.

Genes and signals. Ptashne, Gann. CSHL Press. 2002.

Modeling dynamic phenomena in molecular and cellular biology. Segel. Cambridge Univ. Press. 1987.

Modeling and querying bio-molecular interaction networks. Chabrier, Chiaverini, Danos, Fages, Schächter. To appear in TCS. 2004.

The biochemical abstract machine BIOCHAM. Chabrier, Fages, Soliman. http://contraintes.inria.fr/BIOCHAM


Plan of the Talk

1. Introduction

2. A simple algebra of cell molecules

3. Concurrent transition systems of biochemical reactions• Example of the mammalian cell cycle control

4. Temporal logic CTL as a query language• Computational results with BIOCHAM

5. Learning reaction rules• An experiment with inductive logic programming

6. Kinetics models• Simulation with differential equations

• Hybrid systems

7. Conclusion


2. A Simple Algebra of Cell Molecules

Small molecules: covalent bonds (outer electrons shared) 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

Stability and bindings determined by the number of weak bonds: 3D shape

• 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

ICLP

Isoleucine Cysteine Leucine Proline

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


Formal 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)

(BIOCHAM syntax)


Abstraction of gene expression DNA RNA protein

DNA: word over 4 nucleotides Adenine, Guanine, Cytosine, Thymine

double helix of pairs A--T and C---G

Replication: DNA synthesis

Genes: parts of DNA

Transcription: RNA copying from a gene

#ERCC1-(PRB-JUN-CFOS)


BIOCHAM Algebra of Cell Molecules

E ::= Name|E-E|E~{E,…,E}|(E) S ::= _|E+S

Names: molecules, proteins, #gene binding sites, abstract @processes…

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

Associative and commutative.

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

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

+ : solution operator, “soup aspect”, Assoc. Comm. Idempotent, Neutral _

No membranes, no transport formalized. Bitonal calculi [Cardelli 03].


Plan of the talk

1. Introduction


3. Concurrent transition systems of biochemical reactions• Example of the mammalian cell cycle control

4. Temporal logic CTL as a query language• Computational results with BIOCHAM

5. Learning reaction rules• An experiment with inductive logic programming

6. Kinetics models• Simulation with differential equations

• Hybrid systems

7. Conclusion


3. Concurrent Transition Syst. of Biochemical Reactions

Enzymatic reactions:

R ::= S=>S | S=[E]=>S | S=[R]=>S | S<=>S | S<=[E]=>S

(where A<=>B stands for A=>B B=>A and A=[C]=>B for A+C=>B+C, etc.)

define a concurrent transition system over integers denoting the multiplicity of the molecules (multiset rewriting).

One can associate a finite abstract CTS over boolean state variables denoting the presence/absence of molecules

which correctly over-approximates the set of all possible behaviors

a reaction A+B=>C+D is translated with 4 rules for possible consumption:

A+BA+B+C+D A+BA+B +C+D

A+BA+B+C+D A+BA+B+C+D


Six 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.

_ =[#Ge2-E2f13-Dp12]=> CycA

Degradation: A =[C]=> _.

CycE =[@UbiPro]=> _ (not for CycE-Cdk2 which is stable)


MAPK Signaling Pathway

RAF + RAFK <=> RAF-RAFK.RAF~{p1} + RAFPH <=> RAF~{p1}-RAFPH.MEK~$P + RAF~{p1} <=> MEK~$P-RAF~{p1} where p2 not in $P.MEKPH + MEK~{p1}~$P <=> MEK~{p1}~$P-MEKPH.MAPK~$P + MEK~{p1,p2} <=> MAPK~$P-MEK~{p1,p2} where p2 not in $P.MAPKPH + MAPK~{p1}~$P <=> MAPK~{p1}~$P-MAPKPH.

RAF-RAFK => RAFK + RAF~{p1}.RAF~{p1}-RAFPH => RAF + RAFPH.MEK~{p1}-RAF~{p1} => MEK~{p1,p2} + RAF~{p1}.MEK-RAF~{p1} => MEK~{p1} + RAF~{p1}.MEK~{p1}-MEKPH => MEK + MEKPH.MEK~{p1,p2}-MEKPH => MEK~{p1} + MEKPH.MAPK-MEK~{p1,p2} => MAPK~{p1} + MEK~{p1,p2}.MAPK~{p1}-MEK~{p1,p2} => MAPK~{p1,p2} + MEK~{p1,p2}.MAPK~{p1}-MAPKPH => MAPK + MAPKPH.MAPK~{p1,p2}-MAPKPH => MAPK~{p1} + MAPKPH.






Cell Cycle: G1 DNA Synthesis G2 Mitosis

G1: CdK4-CycD

Cdk6-CycD

Cdk2-CycE

S: Cdk2-CycA

G2

M: Cdk1-CycA

Cdk1-CycB


Mammalian Cell Cycle Control Map [Kohn 99]


Kohn’s map detail for Cdk2

Complexation with CycA and CycE

Phosphorylation sites PY15 and P

Biocham Rules:

cdk2~$P + cycA-$C => cdk2~$P-cycA-$C

where $C in {_,cks1} .

cdk2~$P + cycE~$Q-$C => cdk2~$P-cycE~$Q-$C

where $C in {_,cks1} .

p57 + cdk2~$P-cycA-$C => p57-cdk2~$P-cycA-$C

where $C in {_, cks1}.

cycE-$C =[cdk2~{p2}-cycE-$S]=> cycE~{T380}-$C

where $S in {_, cks1} and $C in {_, cdk2~?, cdk2~?-cks1}

147-2733 rules, 165 proteins and genes, 500 variables, 2500 states.


Plan of the talk

1. Introduction2. A simple algebra of cell molecules3. Concurrent transition systems of biochemical reactions

• Example of the mammalian cell cycle control4. Temporal logic CTL as a query language

• Expressivity and computational results5. Learning reaction rules

• An experiment with inductive logic programming6. Kinetics models

• Simulation with differential equations• Hybrid systems

7. Conclusion


4. Temporal Logic CTL as a Query Language

Computation Tree Logic

Time

Non-determinism E, A

F,G,U EF

EU

AG

Choice

Time

E

exists

A

always

X

next time

EX() AX()

F

finally

EF()

AG()

AF()

liveness

G

globally

EG()

AF( )

AG()

safety

U

untilE (U ) A (U )


Kripke Structures

A Kripke structure K is a triple (S; R; L) where S is a set of states, and RSxS is a total relation.

s |= if is true in s,

s |= E if there is a path from s such that |= ,

s |= A if for every path from s, |= ,

|= if s |= where s is the starting state of ,

|= X if 1 |= ,

|= F if there exists k >0 such that k |= ,

|= G if for every k >0, k |= ,

|= U iff there exists k>0 such that k |= for all j < k j |= Following [Emerson 90] we identify a formula to the set of states which

satisfy it ~ {sS : s |= }.


Symbolic Model Checking

Model Checking is an algorithm for computing, in a given finite Kripke structure the set of states satisfying a CTL formula: {sS : s |= }.

Basic algorithm: represent K as a 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

Symbolic model checking: use OBDDs to represent states and transitions as boolean formulas (S is finite).


Biological Queries (1/3)

About reachability:

• Given an initial state init, can the cell produce some protein P? init EF(P)

• Which are the states from which a set of products P1,. . . , Pn can be produced simultaneously? EF(P1^…^Pn)

About pathways:

• Can the cell reach a state s while passing by another state s2? init EF(s2^EFs)

• Is state s2 a necessary checkpoint for reaching state s? EF(s2U s)

• Is it possible to produce P without using nor creating Q? EF(Q U s)• Can the cell reach a state s without violating some constraints c? init EF(c U s)



About stability:

• Is a certain (partially described) state s a stable state? sAG(s) sAG(s) (s denotes both the state and the formula describing it).

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

• Can the cell reach a stable state? initEF(AG(s))not a LTL formula.

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

• What are the stable states? Not expressible in CTL [Chan 00].

• Can the system exhibit a cyclic behavior w.r.t. the presence of P ? init EG((P EF P) ^ (P EF P))



About the correctness of the model:

• Can one see the inaccuracies of the model and correct them?

Exhibit a counterexample pathway or a witness. Suggest refinements of the model or biological experiments to validate/invalidate the property of the model.

About durations:

• How long does it take for a molecule to become activated?

• In a given time, how many Cyclins A can be accumulated?

• What is the duration of a given cell cycle’s phase?

CTL operators abstract from durations. Time intervals can be modeled in FO by adding numerical arguments for start times and durations.







MEK~{p1} is a checkpoint for producing MAPK~{p1,p2}biocham: !E(!MEK~{p1} U MAPK~{p1,p2})True

The PH complexes are not compulsory for the cascadebiocham: !E(!MEK~{p1}-MEKPH U MAPK~{p1,p2})falseStep 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


Mammalian Cell Cycle Control Benchmark

700 rules, 165 proteins and genes, 500 variables, 2500 states.

BIOCHAM NuSMV model-checker time in seconds:

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

Cycle EG ( (CycA EF CycA) ( CycA EF CycA))

31.8


Plan of the talk

1. Introduction


3. Concurrent transition systems of biochemical reactions


• Computational results with BIOCHAM5. Learning reaction rules


• Simulation with differential equations• Hybrid systems

7. Conclusion


5. Learning Reaction Weights and Rules

Idea 1: learning reaction weights from temporal propertiesreaction weights restricts the non-determinism (Markov models)

Idea 2: learn reaction rules from temporal properties of the system.Learning of cell cycle reaction rules from reachability properties and

counterexamples with Progol [Muggleton 00].reaction([m_CP,m_Y],[m_pM]).reaction([m_CP],[m_C2]).% reaction([m_pM],[m_M]). reaction([m_M],[m_C2,m_YP]).reaction([m_C2],[m_CP]).reaction([m_YP],[]).reaction([],[m_Y]).pathway(S1,S2) :- same(S1,S2).pathway(S1,S2) :- reaction(L1,L2), transition(S1,L1,S3,L2),

pathway(S3,S2).


Inductive Logic Programming

reaction([m_pM],[m_M]) learned…6th PCRD APRIL 2 “Applications of Probabilistic Inductive Logic Progr.” Luc de

Raedt, Univ. Freiburg, Stephen Muggleton, Imperial College London.

pathway([m_CP,m_Y],[m_M]).

pathway([m_CP,m_Y],[m_M,m_pM]).

pathway([m_CP,m_Y],[m_M,m_Y]).

pathway([m_CP,m_Y],[m_M,m_Y,m_pM]).

pathway([m_CP,m_Y],[m_M,m_CP]).

pathway([m_CP,m_Y],[m_M,m_CP,m_Y]).

pathway([m_CP,m_Y],[m_M,m_CP,m_pM]).

pathway([m_CP,m_Y],[m_M,m_CP,m_Y,m_pM]).

pathway([m_pM],[m_C2,m_YP]).

pathway([m_pM],[m_M,m_C2,m_YP]).

pathway([m_pM],[m_pM,m_C2,m_YP]).

pathway([m_pM],[m_M,m_pM,m_C2,m_YP]).

:-pathway([],[m_C2]).

:-pathway([],[m_CP]).

:-pathway([],[m_C2,m_CP]).

:-pathway([],[m_M]).

:-pathway([],[m_YP]).

:-pathway([],[m_YP, m_Y]).

:-pathway([],[m_Y,m_pM]).

:-pathway([],[m_CP,m_pM]).

:-pathway([],[m_Y,m_M]).

:-pathway([m_CP, m_C2],[m_YP]).

:-pathway([m_CP],[m_YP]).

:-pathway([m_C2],[m_YP]).

:-pathway([m_Y],[]).


Plan of the talk

1. Introduction


3. Concurrent transition systems of biochemical reactions


• Computational results with BIOCHAM5. Learning reaction rules


• Simulation with differential equations• Hybrid system

7. Conclusion


6. Kinetics Models

Enzymatic reactions with rates k1 k2 k3

E+S k1 C k2 E+P

E+S k3 C

can be compiled by the law of mass action into a system of

Michaelis-Menten Ordinary Differential Equations (non-linear)

dE/dt = -k1ES+(k2+k3)C

dS/dt = -k1ES+k3C

dC/dt = k1ES-(k2+k3)C

dP/dt = k2C


MAPK kinetics model


Gene Interaction Networks

Gene interaction example [Bockmayr-Courtois 01]

Hybrid Concurrent Constraint Programming HCC [Saraswat et al.]

2 genes x and y.

Hybrid linear approximation

dx/dt = 0.01 – 0.02*x if y < 0.8

dx/dt = – 0.02*x if y ≥ 0.8

dy/dt = 0.01*x


Concurrent Transition System

Time discretization using Euler’s method:

y < 0.8 x’ = x + dt*(0.01-0.02*x) , y’ = y + dt*0.01*x

y ≥ 0.8 x’ = x + dt*(0.01-0.02*x) , y’ = y + dt*0.01*x

Initial condition: x=0, y=0.

Translation into a CLP(R) program (dt=1)Init :- X=0, Y=0, p(X,Y).

p(X,Y):-X>=0, Y>=0, Y<0.8,

X1=X-0.02*X+0.01, Y1=Y+0.01*X, p(X1,Y1).

p(X,Y):-X>=0, Y>=0, Y>=0.8,

X1=X-0.02*X, Y1=Y+0.01*X, p(X1,Y1).


Proving CTL properties by computing fixpoints of Constraint Logic Programs

Theorem [Delzanno Podelski 99]

EF(f)=lfp(TP{p(x):-f}),

EG(f)=gfp(TPf ).

Safety property AG(f) iff EF(f) iff initlfp(TP{f})

Liveness property AG(f1AF(f2)) iff initlfp(TPf1gfp(T P{f2} ) )

Implementation in Sicstus-Prolog CLP(R,B) [Delzanno 00]


Deductive Model Checker DMC: Gene Interaction

r(init, p(s_s,A,B), {A=0,B=0}).r(p(s_s,A,B), p(s_s,C,D), {A>=0,B>=0.8,C=A-

0.02*A,D=B+0.01*A}).r(p(s_s,A,B), p(s_s,C,D), {A>=0,B>=0,B<0.8, C=A-0.02*A+0.01,D=B+0.01*A}).| ?- prop(P,S).P = unsafe, S = p:s*(x>=0.6) | ?- ti.Property satisfied. Execution time 0.0 | ?- ls.s(0, p(s_s,A,_), {A>=0.6}, 1, (0,0)).


Demonstration DMC (continued)

| ?- prop(P,S).P = unsafe, S = p:s*(x>=0.2) ?| ?- ti. Property NOT satisfied. Execution time 1.5| ?- ls.s(0, p(s_s,A,_), {A>=0.2}, 1, (0,0)).s(1, p(s_s,A,B), {B<0.8,B>=-0.0,A>=0.19387755102040816}, 2,

(2,1)).…s(26, p(s_s,A,B), {B>=0.0,A>=0.0, B+0.1982676351105516*A<0.7741338175552753}, 27,

(2,26)). s(27, init, {}, 28, (1,27)).


Conclusion

The biochemical abstract machine BIOCHAM provides:

• A first-order-rule-based language for modeling biochemical systems

• A powerful query language based on temporal logic CTL

• Models of complex biochemical processes,

intracellular and extracellular signaling,

cell-cycle control,… a repository of models http://contraintes.inria.fr/CMBSlib

• Implementation in Prolog + model-checkers NuSMV and DMC

Learning techniques investigated in APrIL 2

• PILP-based learning of reaction weights from temporal properties

• PILP-based learning of reaction rules from temporal properties


Perspectives

Collaboration with biologists on BIOCHAM models of the cell-cycle control

• Colon cancer therapies, Domenjoud, UHP Nancy

• Chronotherapies, Clairambault, INSERM

Hybrid concurrent constraint logic programming [Bockmayr Courtois 01, Saraswat 04]

Multi-scale molecular-electro-physiological models [Sorine et al. 03]

http://www-rocq.inria.fr/sosso/icema2

http://www.sci.sdsu.edu/movies

Joint work with and

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