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François Fages MPRI Bio-info 2007 Formal Biology of the Cell Inferring Reaction Rules from Temporal Properties François Fages, Constraint Programming Group, INRIA Rocquencourt mailto:Francois.Fages@inria.fr http://contraintes.inria.fr/

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François Fages MPRI Bio-info 2007 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. Kinetic models. 4.Learning kinetic parameter values. Constraint-based model checking. 5.Abstract Interpretation for systems biology I: hierarchy of semantics 6.Abstract Interpretation for systems biology II: types 7.Locations, transport and intercellular signalling 8.Inferring reaction rules from temporal properties 9.… 10.Protein structure prediction in constraint logic programming

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François Fages MPRI Bio-info 2007 A Logical Paradigm for Systems Biology Biological model = Transition System Biological property = Temporal Logic Formula Biological validation = Model-checking Initial state : experimental conditions, wild-life/mutated organisms,… Boolean semantics (propositionnal-CTL) reachable(P) = EF(P) checkpoint(s 2,s) = E( s 2 U s) stable(s) = AG(s) steady(s) = EG(s) oscil(P) = EG(F P ^ F P) Differential semantics (constraint-LTL) Reach threshold concentration : F([M]>0.2) on derivative : F(d([M]/dt)>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 2007 Learning Model Revision from Temporal Properties Theory T: BIOCHAM model molecule declarations reaction 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 2007 Kripke Semantics of CTL* Kripke structure K=(S,R) where S is a set of states and R SxS is total. 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 |= , |= U iff there exists k ≥ 0 such that k |= for all j < k j |= |= W iff j j |= or k ≥ 0 k |= and j < k j |= F = (true U |= F if there exists k ≥ 0 such that k |= , G = ( W false |= G if for every k ≥ 0, k |=

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François Fages MPRI Bio-info 2007 Duality in CTL* E = A X = X U = W F = G CTL*(X) : fragment of CTL* without U, W, F, G CTL*(U) : fragment of CTL* without X CTL : fragment of CTL* with E, A immediately before X, F, G, U, W can be identified to the set of states where it is true ~ {s S : s |= } LTL : fragment of CTL* without E, A LTL(U) : fragment of LTL without X LTL(F) : fragment of LTL without X, U, W

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François Fages MPRI Bio-info 2007 Complexity of Model-checking and Satisfiability Model-checking Satisfiability given an explicit Kripke structure K given a formula , does there exist and a formula , does K,s |= ? a structure K,s such that K,s |= ? LTL, LTL(U) : Pspace complete Pspace complete LTL(F) : NP-complete NP-complete CTL : Ptime DetExpTime complete CTL* : Pspace complete DetExpExpTime complete

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François Fages MPRI Bio-info 2007 Simple Model of Cell Cycle Control [Tyson et al. 91] model over 6 variables, initial state present(cdc2). _=>Cyclin. Cyclin=>_. Cyclin+Cdc2~{p1}=>Cdc2-Cyclin~{p1,p2}. Cdc2-Cyclin~{p1,p2}=>Cdc2-Cyclin~{p1}. Cdc2-Cyclin~{p1,p2}=[Cdc2-Cyclin~{p1}]=>Cdc2-Cyclin~{p1}. Cdc2-Cyclin~{p1}=>Cdc2-Cyclin~{p1,p2}. Cdc2-Cyclin~{p1}=>Cyclin~{p1}+Cdc2. Cyclin~{p1}=>_. Cdc2=>Cdc2~{p1}. Cdc2~{p1}=>Cdc2.

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François Fages MPRI Bio-info 2007 (Aut. Generated) CTL Specification of the Model biocham: add_genCTL. reachable(Cyclin). reachable(!(Cyclin)). oscil(Cyclin). reachable(Cdc2~{p1}). reachable(!(Cdc2~{p1})). checkpoint(Cdc2, Cdc2~{p1}). oscil(Cdc2). … reachable(Cyclin~{p1}). reachable(!(Cyclin~{p1})) oscil(Cyclin~{p1}). checkpoint(Cdc2-Cyclin~{p1}, Cyclin~{p1}).

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François Fages MPRI Bio-info 2007 Model Compression biocham: reduce_model. 1: deleting Cyclin=>_ 2: deleting Cdc2-Cyclin~{p1,p2}=[Cdc2-Cyclin~{p1}]=>Cdc2- Cyclin~{p1} 3: deleting Cdc2-Cyclin~{p1}=>Cdc2-Cyclin~{p1,p2} 4: deleting Cdc2~{p1}=>Cdc2 After reduction, 6 rules remain corresponding to the bias ? => ? Deletion(s): Cyclin=>_. Cdc2-Cyclin~{p1,p2}=[Cdc2-Cyclin~{p1}]=>Cdc2-Cyclin~{p1}. Cdc2-Cyclin~{p1}=>Cdc2-Cyclin~{p1,p2}. Cdc2~{p1}=>Cdc2.

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François Fages MPRI Bio-info 2007 Theory Revision biocham: delete_rules(Cdc2=>Cdc2~{p1}). Cdc2=>Cdc2~{p1} biocham: revise_model. 1: adding Cdc2-Cdc2~{p1}=>Cdc2+Cdc2~{p1} 2: adding Cdc2=>_ 2: backtracking on previous add -> deleting Cdc2=>_ 2: adding Cdc2=[Cyclin]=>_ 2: backtracking on previous add -> deleting Cdc2=[Cyclin]=>_ 2: adding Cdc2=[Cdc2-Cdc2~{p1}]=>_ 3: adding Cdc2=>Cdc2~{p1} 4: deleting Cdc2=[Cdc2-Cdc2~{p1}]=>_ 5: deleting Cdc2-Cdc2~{p1}=>Cdc2+Cdc2~{p1} Modifications found: Deletion(s): Addition(s): Cdc2=>Cdc2~{p1}.

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François Fages MPRI Bio-info 2007 Search for all Solutions biocham: learn_one_addition(elementary_interaction_rules). Time: 5.00 s Rules tested: 112 Good rules to be added: 2 Cdc2=>Cdc2~{p1} Cdc2=[Cyclin]=>Cdc2~{p1}

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François Fages MPRI Bio-info 2007 CTL Equivalence of Boolean Models For a class C of CTL formulae, and an initial state s, two Kripke structures K=(S,R), K’=(S,R’) are equivalent K ~ C K’ iff { C : K,s|= } = { C : K’,s|= }

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François Fages MPRI Bio-info 2007 CTL Equivalence of Boolean Models For a class C of CTL formulae, and an initial state s, two Kripke structures K=(S,R), K’=(S,R’) are equivalent 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

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François Fages MPRI Bio-info 2007 CTL Equivalence for a Simple Enzymatic Reaction Two Biocham models: M 1 = {A+B D, D=>A+C} M M 2 = {B =[A]=> C} M D having no occurrence in M nor in the initial state s, is atomic.

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François Fages MPRI Bio-info 2007 CTL Equivalence for a Simple Enzymatic Reaction Two Biocham models: M 1 = {A+B D, D=>A+C} M M 2 = {B =[A]=> C} M D having no occurrence in M nor in the initial state s, is atomic. Proposition If M 2,s |= EF( ) then M 1,s |= EF( ). Proof In M 2 the transitions A+B A+C (resp. A+B A+B+C) can be replaced in M 1 by A+B D A+C (resp. A+B B+D B+A+C).

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François Fages MPRI Bio-info 2007 CTL Equivalence for a Simple Enzymatic Reaction Two Biocham models: M 1 = {A+B D, D=>A+C} M M 2 = {B =[A]=> C} M D having no occurrence in M nor in the initial state s, is atomic. Proposition If M 2,s |= EF( ) then M 1,s |= EF( ). Proposition If M 1,s |= EF( ) then M 2,s |= EF( ) whenever A and B do not appear negatively (i.e. under an odd number of negations) in and D does not appear at all in . Proof Let be a path in M 1 such that k |= . If k does not contain D then one can easily mimick with ’ in M 2 such that ’ k’ = k for some k’≤k. Otherwise, the last transition on D is either D D+A+C and can be replaced by D A+C, or A+B D and can be erased. In both cases the path is mimicked in M 2.

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François Fages MPRI Bio-info 2007 CTL Equivalence for a Simple Enzymatic Reaction Two Biocham models: M 1 = {A+B D, D=>A+C} M M 2 = {B =[A]=> C} M D having no occurrence in M nor in the initial state s, ψ atomic. Proposition If M 2,s |= ￢ E( ￢ U ψ) then M 1,s |= ￢ E( ￢ U ψ) whenever A and B do not appear negatively in ψ and D does not appear positively in ψ

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François Fages MPRI Bio-info 2007 CTL Equivalence for a Simple Enzymatic Reaction Two Biocham models: M 1 = {A+B D, D=>A+C} M M 2 = {B =[A]=> C} M D having no occurrence in M nor in the initial state s, ψ atomic. Proposition If M 2,s |= ￢ E( ￢ U ψ) then M 1,s |= ￢ E( ￢ U ψ) whenever A and B do not appear negatively in ψ and D does not appear positively in ψ Proposition If M 1,s |= ￢ E( ￢ U ψ) implies M 2,s |= ￢ E( ￢ U ψ) A and B do not appear negatively in and D does not appear positively in

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François Fages MPRI Bio-info 2007 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). Ex. : reachability EF( ), steady EG( ) Def. An ACTL (negative) formula is a CTL formula with no occurrence of E (nor negative occurrence of A). Ex. : checkpoint E( 2 U ), stable AG( )

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François Fages MPRI Bio-info 2007 Monotonicity of Positive ECTL Formulae Let K = (S,R) and K’ = (S,R’) be two Kripke structures such that R R’. Proposition For any ECTL formula , if K’,s |≠ then K,s |≠ . Proof We show that K,s |= implies K’,s |= by induction on the proof of If is propositionnal, s |= hence K’,s |= ; If = 1& 2 (resp. 1| 2) then by induction K’,s|= 1 and (resp. or) K’,s|= 2. If =EX then K, |= X 1 for some path in K, hence in K’, so K, 1 |= 1 and by induction K’, 1 |= 1 hence K’, |= X 1 If =E( U 2) then K, |= 1 U 2 for some path in K, hence in K’, so there exists k K, k |= 2 and for all j

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François Fages MPRI Bio-info 2007 Anti-monotonicity of Negative ECTL Formulae Let K = (S,R) and K’ = (S,R’) be two Kripke structures such that R R’. Proposition For any ACTL formula , if K,s |≠ then K’,s |≠ . Proof If K,s |≠ then K,s |= where is an ECTL formula. By the previous proposition, K’,s |= hence K’,s |≠ .

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François Fages MPRI Bio-info 2007 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 Reachability, stability Need to add rules Negative ACTL formula: if false, remains false after adding a rule Checkpoints Need to remove a rule on the path given by the model checker Unclassified CTL formulae oscil(a)= AG((a EF a)^( a EFa))

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François Fages MPRI Bio-info 2007 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 2007 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 2007 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 2007 Termination Proposition The model revision algorithm terminates. 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 a, or lets a unchanged and strictly decreases n.

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François Fages MPRI Bio-info 2007 Correctness Proposition If the terminal configuration is of the form then the model R satisfies the initial CTL specification. Proof 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 2007 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 those loops.

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François Fages MPRI Bio-info 2007 Optimisations Restrict the search space for rules to add by: Considering type information on molecular species Kinase(A) B=[A]=>B~{p}. for any B Phosphatase(A) B~{p}=[A]=>B. for any B Kinase(A,B) Phosphatase(A,B) Considering the influence graph between molecular species Activates(A,B) _=[A]=>B. A+B’=>B. B~{p}=[A]=>B. B’=[A]=>B. Inhibits(A,B) B=[A]=>_. A+B=>A-B. B=[A]=>B~{p}. B=[A]=>B’. Considering the topology of locations Neighbor(L,L’) A:L+…=>B:L’+…

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François Fages MPRI Bio-info 2006 Formal Biology of the Cell Modeling, Computing and Reasoning with Constraints François Fages, Constraints Group, INRIA.

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