Expressiveness I ssues in Calculi for Artificial Biochemistry A � r C 1 +…+C n A ::= τ @r;C 1 |…|C n + b@s; 0 A+B � s D 1 +…+D m B ::= b@s;D 1 |…|D m What is the computational What is the computational power of this calculus? power of this calculus? Based on joint work with Gianluigi Zavattaro University of Bologna Luca Cardelli
Plan of the talk � Basic Chemistry and Basic Biochemistry � Biochemistry = Chemistry + complexation � Chemical Ground Form (CGF) � A process algebra for basic chemistry � Biochemical Ground Form (BGF) � A process algebra for basic biochemistry � Considered TERMINATION problems: � Existential termination in CGF (DECIDABLE) � Existential termination in BGF (UNDECIDIBLE) � Universal termination in CGF � Nondeterministic -all computations terminate- (DECIDABLE) � Probabilistic -terminate with probability 1- (UNDECIDABLE) Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Basic Chemistry � Molecules belong to Species � Behavior described by reactions: � Monomolecular: A � C 1 +…+C n � Bimolecular: A+B � D 1 +…+D m C 1 D 1 B A … … A C n D m Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Basic Biochemistry � Molecules form and modify complexes � by means of association and dissociation M M M M Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Plan of the talk � Basic Chemistry and Basic Biochemistry � Biochemistry = Chemistry + complexation � Chemical Ground Form (CGF) � A process algebra for basic chemistry � Biochemical Ground Form (BGF) � A process algebra for basic biochemistry � Considered TERMINATION problems: � Existential termination in CGF (DECIDABLE) � Existential termination in BGF (UNDECIDIBLE) � Universal termination in CGF � Nondeterministic -all computations terminate- (DECIDABLE) � Probabilistic -terminate with probability 1- (UNDECIDABLE) Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Chemical Ground Forms � Stochastic variant of Milner’s CCS, with an equivalent graphical notation (Stochastic Collective Automata) A send a receive b internal action … … … B 1 B n D 1 D s C 1 C m Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Chemical Ground Forms � Stochastic variant of Milner’s CCS, with an equivalent graphical notation (Stochastic Collective Automata) A !a ?b τ … … … B 1 B n D 1 D s C 1 C m Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Why stochastic… � Actions take (a variable amount of) time � Each action has an associated rate r � Internal delay: τ @r � Pr(internal delay < t) = 1-e -rt � Synchronization between complementary actions: ?a@r, !a@r � Pr(synchronization time < t) = 1-e -rt Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 1 !a@r τ @s ?a@r !b@r ?b@r τ @s � Starting population: A|A’ Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 1 !a@r τ @s ?a@r !b@r ?b@r τ @s a* τ b* τ � Starting population: A|A’ Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 2 !a@r τ @s ?a@r !b@r ?b@r � Starting population: A|A’ Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 2 !a@r τ @s ?a@r !b@r ?b@r … a n τ b n � Starting population: A|A’ Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
CGF = Basic Chemistry [TCS08] Continuous-State Continuous Semantics Chemistry = CGF BC Discrete-State Discrete = Semantics Chemistry Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
A nice example !c A+B � B+B C B+C � C+C ?a ?c C+A � A+A A B ?b !a !b Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
with a nice behaviour… Discrete-State Continuous-State Semantics Semantics Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Plan of the talk � Basic Chemistry and Basic Biochemistry � Biochemistry = Chemistry + complexation � Chemical Ground Form (CGF) � A process algebra for basic chemistry � Biochemical Ground Form (BGF) � A process algebra for basic biochemistry � Considered TERMINATION problems: � Existential termination in CGF (DECIDABLE) � Existential termination in BGF (UNDECIDIBLE) � Universal termination in CGF � Nondeterministic -all computations terminate- (DECIDABLE) � Probabilistic -terminate with probability 1- (UNDECIDABLE) Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Polymerization � Monomers associate and dissociate M M M M Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Association and Dissociation � How to model the actin-like monomer behavior? M f M l M r M M M M b Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Association and Dissociation � How to model the actin-like monomer behavior? M f %?a &!a %!a &?a !a ?a M l M r M %?a &!a &?a !a ?a !a ?a M M M b %!a Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Association histories � Each association has a unique key � Keys are stored in the molecule’s history %?a &!a %!a &?a (!a,k) (?a,k) M M %?a &!a &?a M %!a Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Association histories � Each association has a unique key � Keys are stored in the molecule’s history %?a &!a %!a &?a (!a,s) (?a,k) M M M %?a &!a &?a %!a (?a,s)(!a,k) Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Association histories � Each association has a unique key � Keys are stored in the molecule’s history Not possible! %?a &!a s ≠ k %!a &?a (!a,s) (?a,k) M M M %?a &!a &?a %!a (?a,s)(!a,k) Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Association histories � Each association has a unique key � Keys are stored in the molecule’s history Possible! %?a &!a k=k %!a &?a (!a,s) (?a,k) M M M %?a &!a &?a %!a (?a,s)(!a,k) Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Association histories � Each association has a unique key � Keys are stored in the molecule’s history %?a &!a %!a &?a (!a,s) (?a,s) M M %?a &!a &?a M %!a Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Plan of the talk � Basic Chemistry and Basic Biochemistry � Biochemistry = Chemistry + complexation � Chemical Ground Form (CGF) � A process algebra for basic chemistry � Biochemical Ground Form (BGF) � A process algebra for basic biochemistry � Considered TERMINATION problems: � Existential termination in CGF (DECI DABLE) � Existential termination in BGF (UNDECIDIBLE) � Universal termination in CGF � Nondeterministic -all computations terminate- (DECIDABLE) � Probabilistic -terminate with probability 1- (UNDECIDABLE) Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Existential termination for CGF � Given a CGF system, decide whether there exists a computation leading to a deadlock Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 1: does it terminate? YES !a@r τ @s ?a@r !b@r ?b@r τ @s a* τ b* τ � Starting population: A|A’ Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 2: does it terminate? !a@r YES τ @s ?a@r !b@r ?b@r … a n τ b n � Starting population: A|A’ Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 3: does it terminate? !c A+B � B+B 100 B+C � C+C ?a ?c C+A � A+A 900 500 ?b !a !b Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
with a nice behaviour… Discrete-State Continuous-State Semantics Semantics Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
with a nice behaviour… 1600 1400 1200 1000 800 600 400 200 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 But in a longer simulation… Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Example 3: does it terminate? !c YES A+B � B+B 1500 B+C � C+C ?a ?c C+A � A+A 0 0 ?b !a !b Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
Decidability of termination � We reduce existential termination for CGF to termination for Petri Nets � Petri Nets is an interesting infinite state system in which many properties (reachability, coverability, termination, divergence,…) are decidable Expressiveness Issues in Calculi for Artificial Biochemistry SFM-08:Bio - 7.6.08
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