overview of the lectures

François Fages MPRI Bio-info 2005

Formal Biology of the Cell

Modeling, Computing and Reasoning with Constraints

François Fages, Constraint Programming Group,

INRIA Rocquencourt mailto:[email protected]://contraintes.inria.fr/


Overview of the Lectures

1. Introduction. Formal molecules and reactions in BIOCHAM.

2. Formal biological properties in temporal logic. Symbolic model-checking.

3. Continuous dynamics. Kinetics models.

4. Computational models of the cell cycle control [L. Calzone].

5. Mixed models of the cell cycle and the circadian cycle [L. Calzone].

6. Machine learning reaction rules from temporal properties.

7. Constraint-based model checking. Learning kinetic parameter values.

8. Constraint Logic Programming approach to protein structure prediction.


A Logical Paradigm for Systems Biology

Biological property = Temporal Logic Formula

Biological validation = Model-checking

Initial state : experimental conditions, wild-life/mutated organisms,…

reachable(P)==EF(P)

checkpoint(s2,s)== E(s2U s)

stable(s)== AG(s)

steady(s)==EG(s)

oscil(P)== EG((P EF P) ^ (P EF P))

Reach threshold concentration : F([M]>0.2) On derivative : F(d([M])>0.2)

Reach and stays above threshold : FG([M]>0.2)

oscil(P,n)==F(d([M])/dt>0 & F(d([M])/dt<0 & … )) n times


Kripke Semantics of CTL

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

s |= if propositional formula 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 |= }.


CTL Equivalence of Boolean Models

For a class C of CTL formulae,

given two Kripke structures K=(S,R), K’=(S,R’) and an initial state s

K ~C K’ iff {C : K,s|= } = {C : K’,s|= }

Which model transformations preserve a class of CTL properties?

Model refinement or simplification preserving a CTL specification

Which model transformations can make a CTL property true?

Learning of rules to add or to delete to satisfy a CTL specification

Which CTL properties are satisfied by a Biocham model? genCTL(pattern)


CTL Equivalence for a Simple Enzymatic Reaction

Two Biocham models: M1={A+B<=>D, D=>A+C} or M2={B =[A]=> C}

D having no other occurrence in M1.

Let and ψ be two propositional formulae.

Proposition If M2 |= EF() then M1 |= EF().Moreover, if A and B do not appear negatively (i.e. under an odd number of negations) in and D does not appear at all in , then M1 |= EF() implies M2 |= EF().

Proposition If A and B do not appear negatively in ψ and D does not appear in ψ,then M2 |= ￢ E( ￢ U ψ) implies M1 |= ￢ E( ￢ U ψ). If A and B do not appear negatively in and D does not appear in ,then M1 |= ￢ E( ￢ U ψ) implies M2 |= ￢ E( ￢ U ψ).


Positive and Negative CTL Formulae

Let K = (S,R,L) and K’ = (S,R’,L) be two Kripke structures such that RR’

.

Def. An ECTL (positive) formula is a CTL formula with no occurrence of A (nor negative occurrence of E).

Def. An ACTL (negative) formula is a CTL formula with no occurrence of E (nor negative occurrence of A).

Proposition For any ECTL formula , if K’ |≠ then K |≠ .

Since RR’ all paths in K are also paths in K’, hence for a positive formula , if K |= then K’ |= , which shows the proposition.

Proposition For any ACTL formula , if K |≠ then K’ |≠ .

By duality ￢ is a positive formula, hence if K |= ￢ then K’ |= ￢


Example of Qu’s Model of Cell Cycle

_=>Cyclin.

Cyclin=>_.

Cyclin+Cdc2~{p1}=>Cdc2~{p1}-Cyclin~{p1}.

Cdc2~{p1}-Cyclin~{p1}=>Cdc2-Cyclin~{p1}.

Cdc2~{p1}-Cyclin~{p1}=[Cdc2-Cyclin~{p1}]=>Cdc2-Cyclin~{p1}.

Cdc2-Cyclin~{p1}=>Cdc2~{p1}-Cyclin~{p1}.

Cdc2-Cyclin~{p1}=>Cdc2+Cyclin~{p1}.

Cyclin~{p1}=>_.

Cdc2=>Cdc2~{p1}.

Cdc2~{p1}=>Cdc2.

present(Cdc2,1). make_absent_not_present.


Aut. Generation of a CTL Specification in Qu’s Model

Enumerate all CTL formulae (of some pattern) that are true in a model.? genCTL(elementary_properties).Ai(oscil(C25)), Ei(reachable(C25~{p1})), Ei(reachable(!(C25~{p1}))), Ai(oscil(C25~{p1})),Ei(reachable(Wee1)), Ei(reachable(!(Wee1))), Ai(oscil(Wee1)),Ei(reachable(Wee1~{p1})),Ei(reachable(!(Wee1~{p1}))), Ai(oscil(Wee1~{p1})),Ai(AG(!(Wee1~{p1})->checkpoint(Wee1,Wee1~{p1}))),Ei(reachable(CKI)), Ei(reachable(!(CKI))), Ai(oscil(CKI)),Ei(reachable(CKI-CycB-CDK~{p1})),Ei(reachable(!(CKI-CycB-CDK~{p1}))),Ai(oscil(CKI-CycB-CDK~{p1})),Ei(reachable((CKI-CycB-CDK~{p1})~{p2})), Ei(reachable(!((CKI-CycB-

CDK~{p1})~{p2}))),Ai(oscil((CKI-CycB-CDK~{p1})~{p2})), Ai(AG(!((CKI-CycB-CDK~{p1})~{p2}) ->checkpoint(CKI-CycB-CDK~{p1},CKI-CycB-

CDK~{p1})~{p2}))}).


Generation of Rule Additions for a CTL Specification

Enumerate all rules (of some pattern) that satisfy a CTL specification

? delete_rule(Cyclin+Cdc2~{p1}=>Cdc2~{p1}-Cyclin~{p1}).

? learn_one_addition(elementary_interaction_rules).

(1) Cyclin+Cdc2~{p1}=[Cdc2]=>Cdc2~{p1}-Cyclin~{p1}

(2) Cyclin+Cdc2~{p1}=[Cyclin]=>Cdc2~{p1}-Cyclin~{p1}

(3) Cyclin+Cdc2~{p1}=>Cdc2~{p1}-Cyclin~{p1}

(4) Cyclin+Cdc2~{p1}=[Cdc2~{p1}]=>Cdc2~{p1}-Cyclin~{p1}

Similarly enumerate all rule deletions that satisfy or preserve a CTL spec.

? learn_one_deletion(reaction_pattern, spec_CTL)

? reduce_model(spec_CTL)


Learning Model Revision from Temporal Properties

• Theory T: BIOCHAM model • molecule declarations

• interaction rules: complexation, phosphorylation, …

• Examples φ: CTL specification of biological properties• Reachability

• Checkpoints

• Stable states

• Oscillations

• Bias R: Rule pattern• Kind of rules to add or delete

Find a revision T’ of T such that T’ |= φ


Theory 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)

• 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• Loop(a)= AG((a EFa)^(a EFa))


Theory Revision Algorithm Rules

Initial state: <(0, 0, 0), (E,U,A), R>

E transition: <(E,U,A), (E{e},U,A), R> <(E{e},U,A), (E,U,A),R> if R |= e

E’ transition: <(E,U,A), (E {e},U,A), R> <(E {e},U,A), (E,U,A),R {r}>

if R |≠ e and f {e} E U A, K {r} |= f



Initial state: <(0, 0, 0), (E,U,A), R>E transition: <(E,U,A), (E{e},U,A), R> <(E{e},U,A), (E,U,A),R> if R |= eE’ transition: <(E,U,A), (E {e},U,A), R> <(E {e},U,A), (E,U,A),R {r}> if R |≠ e and f {e} E U A, K {r} |= fU transition: <(E,U,A), (0,U {u},A), R > <(E,U {u},A), (0,U,A),R> if R |= uU’ transition: <(E,U,A), (0,U {u},A), R > <(E,U {u},A), (0,U,A),R {r}> if R|≠u and f {u} E U A, R {r} |= fU” transition: <(E,U,A), (0,U {u},A), R Re > <(E,U {u},A),(0,U,A), R> if K, si|≠u and f {u} E U A, R |= f



Initial state: <(0, 0, 0), (E,U,A), R>

E transition: <(E,U,A), (E{e},U,A), R> <(E{e},U,A), (E,U,A),R> if R |= e

E’ transition: <(E,U,A), (E {e},U,A), R> <(E {e},U,A), (E,U,A),R {r}>

if R |≠ e and f {e} E U A, K {r} |= f

U transition: <(E,U,A), (0,U {u},A), R > <(E,U {u},A), (0,U,A),R> if R |= u

U’ transition: <(E,U,A), (0,U {u},A), R > <(E,U {u},A), (0,U,A),R {r}>

if R|≠u and f {u} E U A, R {r} |= f

U” transition: <(E,U,A), (0,U {u},A), R Re > <(E,U {u},A),(0,U,A), R>

if K, si|≠u and f {u} E U A, R |= f

A transition: <(E,U,A), (0, 0,A {a}), R > <(E,U,A {a}), (Ep,Up,A),R> if R |= a

A’ transition: <(EEp,UUp,A),(0,0,A{a}), RRe><(E,U,A{a}),(Ep,Up,A),R> if R|≠ a, f {u} [ E U A, R |= f and Ep Up is the set of formulae no longer satisfied after the deletion of the rules in Re.


Termination and Correctness

Proposition The model revision algorithm terminates. If the terminal configuration

is of the form < (E,U,A), (0,0,0), R > then the model R satisfies the

initial CTL specification.

Proof The termination of the algorithm is proved by considering the lexicographic

ordering over the couple < a, n > where a is the number of unsatisfied

ACTL formulae, and n is the number of unsatisfied ECTL and UCTL formulae.

Each transition strictly decreases either a, or lets a unchanged and strictly

decreases n.

The correction of the algorithm comes from the fact that each transition

maintains only true formulae in the satisfied set, and preserves the complete

CTL specification in the union of the satisfied set and the untreated set.


Incompleteness

Two reasons:

1) The satisfaction of ECTL and UCTL formula is searched by adding only one rule to the model (transition E’ and U’)

2) The Kripke structure associated to a Biocham set of rules adds loops on terminal states. Hence adding or removing a rule may have an opposite deletion or addition of the loops.

Just a heuristic algorithm…


Example in Qu’s Model of Cell Cycle

biocham:add_spec(Ai(AG((CycB-CDK~{p1})) ->checkpoint(C25~{p1,p2},CycB-CDK~{p1})))).biocham: revise_model.SuccessTime: 441.00 s40 properties treatedModifications found:Deletion(s):k5*[CycB-CDK~{p1,p2}] for CycB-CDK~{p1,p2}=>CycB-CDK~{p1}.k15*[CKI-CycB-CDK~{p1}] for CKI-CycB-CDK~{p1}=>CKI+CycB-

CDK~{p1}.Addition(s):


Example of Model Refinement

? Add_spec({ Ei(reachable(CycE)), Ei(reachable(CycE-CDKp)),

Ei(reachable(CycE-CDKp~{p1})), Ei(reachable(CKI-CycE-CDKp)), Ei(reachable(!(CKI-CycE-CDKp))), Ai(oscil(CycE-CDKp)), Ai(oscil(CycE-CDK~{p1})), Ai(loop(CycE-CDKp,(CycE-CDKp)~{p1}))}).

? Revise_model.

Deletion(s):

Addition(s):

CKI+CycE-CDKp=>CKI-CycE-CDKp.

CDKp+CycE=>CycE-CDKp.

CycE-CDKp=>(CycE-CDK)~{p1}.

(CycE-CDKp)~{p1}=>CycE-CDKp.

CKI+CycE-CDKp=>CKI-CycE-CDKp.


Rule Inference in Cell Cycle Control

[Tyson et al. 91] model over 6 variables,

initial state present(cdc2).

_ => cyclin.

cdc2˜{p} + cyclin => cdc2˜{p}-cyclin˜{p}.

cdc2˜{p}-cyclin˜{p} =>cdc2-cyclin˜{p}. ERASED

cdc2-cyclin˜{p} => cdc2 + cyclin˜{p}.

cyclin˜{p} => _.

cdc2 <=> cdc2˜{p}.


Rule Inference in Cell Cycle Control (cont.)

CTL specification of biological properties:

Activation of the kinase-cyclin (MPF) complex

reachable(cdc2-cyclin˜{p}).

Oscillation of the cycle’s phase:

loop(cyclin & cyclin˜{p} & !(cdc2-cyclin˜{p})).


Rule Inference in Cell Cycle Control (cont.)

? learn([$Q=>$P where $P in complexes and $Q in complexes]).

_=>cdc2-cyclin˜{p}

cyclin=>cdc2-cyclin˜{p}

cdc2˜{p}-cyclin˜{p}=>cdc2-cyclin˜{p}

? learn([$qp=>$q where $q in complexes and $qp modif $q]).


Adding temporal specification checkpoint(cdc2˜{p},cdc2-cyclin˜{p}).

? learn([$Q=>$P where $P in complexes and $Q in complexes]).



Model Refinement by Theory Revision

Hypothetical model of three proteins MA, MB, MC

reachable(MA), reachable(MB), reachable(MC).

reachable(¬MA), reachable(¬MB), reachable(¬MC).

Oscillations are observed experimentally.

loop(MA), loop(MB), loop(MC).

Interactions between protein are unknown.

Simplest boolean model:

_<=>MA.

_<=>MB.

_<=>MC.


Model Refinement by Theory Revision (cont.)

MC is needed for the disappearance of MB: checkpoint(MC,!MB).

? checkpoint(MC,!MB).

false

? Why.

… MB=>_ …

? delete_rules(MB=>_).

? learn(elementary_interaction_pattern).

MB+MC=>MB˜{p}+MC.

MB+MC=>MC.

MB+MC=>MB-MC.

? Add_rule(MB+MC=>MB˜{p}+MC).



MA is needed for the disappearance of MC: checkpoint(MA,!MC).

? checkpoint(MA,!MC).

False

? Why

… MC => _ …

? delete_rule(MC => _ ).

? Learn(elementary_interaction_pattern(MC)).

MC+MA=>MA-MC

MC+MA=>MC~{p}+MA

MC+MA=>MA

? Add_rule(MC+MA=>MC~{p}+MA ).



MB is needed for the disappearance of MA: checkpoint(MB,!MA).? checkpoint(MB,!MA).False? Why… MA => _ …? delete_rule(MA => _ ).? Learn(elementary_interaction_pattern(MA)).MC+MA=>MA-MC MC+MA=>MC~{p}+MA MC+MA=>MA ? Add_rule(MC+MA=>MC~{p}+MA )._=>MA. MA=[MB]=>_._=>MB. MB=[MC]=>MB˜{p}._=>MC. MC=[MA]=>MC˜{p}.

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