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François Fages MPRI Bio-info 2005 Formal Biology of the Cell Modeling, Computing and Reasoning with Constraints François Fages, Constraint Programming Group, INRIA Rocquencourt

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François Fages MPRI Bio-info 2005 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.

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François Fages MPRI Bio-info 2005 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(s 2,s)== E( s 2 U 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

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François Fages MPRI Bio-info 2005 Kripke Semantics of CTL A Kripke structure K is a triple (S,R) where S is a set of states, and R SxS 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 ~ {s S : s |= }.

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François Fages MPRI Bio-info 2005 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)

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François Fages MPRI Bio-info 2005 CTL Equivalence for a Simple Enzymatic Reaction Two Biocham models: M 1 ={A+B D, D=>A+C} or M 2 ={B =[A]=> C} D having no other occurrence in M 1. Let and ψ be two propositional formulae. Proposition If M 2 |= EF( ) then M 1 |= 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 M 1 |= EF( ) implies M 2 |= EF( ). Proposition If A and B do not appear negatively in ψ and D does not appear in ψ,then M 2 |= ￢ E( ￢ U ψ) implies M 1 |= ￢ E( ￢ U ψ). If A and B do not appear negatively in and D does not appear in ,then M 1 |= ￢ E( ￢ U ψ) implies M 2 |= ￢ E( ￢ U ψ).

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François Fages MPRI Bio-info 2005 Positive and Negative CTL Formulae Let K = (S,R,L) and K’ = (S,R’,L) be two Kripke structures such that R R’. 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 R R’ 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’ |= ￢

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François Fages MPRI Bio-info 2005 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.

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François Fages MPRI Bio-info 2005 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}))}).

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François Fages MPRI Bio-info 2005 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)

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François Fages MPRI Bio-info 2005 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

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François Fages MPRI Bio-info 2005 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 EF a)^( a EFa))

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François Fages MPRI Bio-info 2005 Theory Revision Algorithm Rules Initial state: E transition: if R |= e E’ transition: if R |≠ e and f {e} E U A, K {r} |= f

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François Fages MPRI Bio-info 2005 Theory Revision Algorithm Rules Initial state: E transition: if R |= e E’ transition: if R |≠ e and f {e} E U A, K {r} |= f U transition: if R |= u U’ transition: if R|≠u and f {u} E U A, R {r} |= f U” transition: if K, si|≠u and f {u} E U A, R |= f

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François Fages MPRI Bio-info 2005 Theory Revision Algorithm Rules Initial state: E transition: if R |= e E’ transition: if R |≠ e and f {e} E U A, K {r} |= f U transition: if R |= u U’ transition: if R|≠u and f {u} E U A, R {r} |= f U” transition: if K, si|≠u and f {u} E U A, R |= f A transition: if R |= a A’ transition: 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.

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François Fages MPRI Bio-info 2005 Termination and Correctness Proposition The model revision algorithm terminates. If the terminal configuration is of the form 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 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.

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François Fages MPRI Bio-info 2005 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…

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François Fages MPRI Bio-info 2005 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. Success Time: s 40 properties treated Modifications 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):

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François Fages MPRI Bio-info 2005 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.

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François Fages MPRI Bio-info 2005 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}.

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François Fages MPRI Bio-info 2005 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})).

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François Fages MPRI Bio-info 2005 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 $q]). cdc2˜{p}-cyclin˜{p}=>cdc2-cyclin˜{p} Adding temporal specification checkpoint(cdc2˜{p},cdc2-cyclin˜{p}). ? learn([$Q=>$P where $P in complexes and $Q in complexes]). cdc2˜{p}-cyclin˜{p}=>cdc2-cyclin˜{p}

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François Fages MPRI Bio-info 2005 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.

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François Fages MPRI Bio-info 2005 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).

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François Fages MPRI Bio-info 2005 Model Refinement by Theory Revision (cont.) 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 ).

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François Fages MPRI Bio-info 2005 Model Refinement by Theory Revision (cont.) 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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