Expressiveness I ssues in Calculi for Artificial Biochemistry A r C - - PowerPoint PPT Presentation

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Expressiveness I ssues in Calculi for Artificial Biochemistry A r C - - PowerPoint PPT Presentation

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Expressiveness I ssues in Calculi for Artificial Biochemistry A r C 1 ++C n A ::= @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

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Based on joint work with Luca Cardelli Gianluigi Zavattaro University of Bologna

Expressiveness I ssues in Calculi for Artificial Biochemistry

A r C1+…+Cn A+B s D1+…+Dm A ::= τ@r;C1|…|Cn + b@s;0 B ::= b@s;D1|…|Dm What is the computational power of this calculus? What is the computational power of this calculus?

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Basic Chemistry

Molecules belong to Species Behavior described by reactions:

Monomolecular:

A C1+…+Cn

Bimolecular:

A+B D1+…+Dm A C1 Cn

…

B D1 Dm

…

A

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Basic Biochemistry

Molecules form and modify complexes

by means of association and dissociation

M M M M

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Chemical Ground Forms

A send a receive b

Stochastic variant of Milner’s CCS, with

an equivalent graphical notation (Stochastic Collective Automata) …

B1 Bn

…

C1 Cm

internal action …

D1 Ds

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Chemical Ground Forms

A

Stochastic variant of Milner’s CCS, with

an equivalent graphical notation (Stochastic Collective Automata) …

B1 Bn

…

C1 Cm

…

D1 Ds !a ?b

τ

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Example 1

Starting population: A|A’

τ@s τ@s !a@r ?b@r ?a@r !b@r

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Example 1

Starting population: A|A’

τ@s τ@s a*τb*τ !a@r ?b@r ?a@r !b@r

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Example 2

Starting population: A|A’

τ@s !a@r ?b@r ?a@r !b@r

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Example 2

Starting population: A|A’

τ@s anτbn … !a@r ?b@r ?a@r !b@r

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

CGF = Basic Chemistry [TCS08]

CGF Discrete-State Semantics Continuous-State Semantics = = Discrete Chemistry BC Continuous Chemistry

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

C B A

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A

A nice example

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

with a nice behaviour…

Discrete-State Semantics Continuous-State Semantics

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Polymerization

Monomers associate and dissociate

M M M M

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

How to model the actin-like monomer

behavior? Mf Ml Mr Mb M M M

Association and Dissociation

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Association and Dissociation

How to model the actin-like monomer

behavior? Mf Ml Mr Mb M !a ?a M !a ?a M !a ?a &!a &!a &?a &?a %?a %?a %!a %!a

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Association histories

&!a &!a &?a &?a %?a %?a %!a %!a M M M

(!a,k) (?a,k)

Each association has a unique key

Keys are stored in the molecule’s history

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Association histories

&!a &!a &?a &?a %?a %?a %!a %!a M M M

(!a,s) (?a,k) (?a,s)(!a,k)

Each association has a unique key

Keys are stored in the molecule’s history

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Association histories

&!a &!a &?a &?a %?a %?a %!a %!a M M M

(!a,s) (?a,k)

Not possible! s≠k

(?a,s)(!a,k)

Each association has a unique key

Keys are stored in the molecule’s history

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Association histories

&!a &!a &?a &?a %?a %?a %!a %!a M M M

(!a,s) (?a,k)

Possible! k=k

(?a,s)(!a,k)

Each association has a unique key

Keys are stored in the molecule’s history

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Association histories

&!a &!a &?a &?a %?a %?a %!a %!a M M

(!a,s) (?a,s)

M

Each association has a unique key

Keys are stored in the molecule’s history

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SLIDE 25

SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Existential termination for CGF

Given a CGF system, decide whether

there exists a computation leading to a deadlock

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Example 1: does it terminate?

Starting population: A|A’

τ@s τ@s a*τb*τ !a@r ?b@r ?a@r !b@r YES

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Example 2: does it terminate?

Starting population: A|A’

τ@s anτbn … !a@r ?b@r ?a@r YES !b@r

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A

Example 3: does it terminate?

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

with a nice behaviour…

Discrete-State Semantics Continuous-State Semantics

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

with a nice behaviour…

200 400 600 800 1000 1200 1400 1600 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

But in a longer simulation…

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

1500

!a ?a !c ?c !b ?b

Example 3: does it terminate?

A+B B+B B+C C+C C+A A+A YES

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Petri nets

A Petri net is a triple

A finite set of Places A finite set of

Transitions: pairs of multisets of places (preset,postset)

An initial marking

(multiset of places)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Petri nets

A transition is enabled

when it is possible to

consume tokens in the preset

When a transition fires

tokens are placed in the

postset

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Petri nets

A transition is enabled

when it is possible to

consume tokens in the preset

When a transition fires

tokens are placed in the

postset

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

A Petri net semantics for CGF

One place for each Species One transition for each reaction

τ@s τ@s !a@r ?b@r ?a@r !b@r

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

A Petri net semantics for CGF

One place for each Species One transition for each reaction

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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 (UNDECI DI BLE) Universal termination in CGF

Nondeterministic -all computations terminate- (DECIDABLE) Probabilistic -terminate with probability 1- (UNDECIDABLE)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Turing completeness of BGF

In BGF we model

Random Access Machines: [Min67]

Registers: r1 … rn hold natural numbers Program: sequence of numbered

instructions

i: I nc(rj): add 1 to the content of rj and go to

the next instruction

i: DecJump(rj,s): if the content of rj is not 0

then decrease by 1 and go to the next instruction; otherwise jump to instruction s

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Registers as Linearly growing polymer

Initially empty register rj: a seed Zj Increment on rj: produce a new monomer

and associate it to the polymer

Decrement on rj: remove last monomer

Zj Rj Rj

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

RAM encoding

i: Inc(rj) k: DecJump(rj,s) Ii !incj Ii+1 ?ackj Ik !decj Ik+1 ?ackj Is !zeroj register rj:

Zj Rj

!lj ?lj ?lj Zj ?zeroj &?lj %?lj ?incj Rj !ackj &!lj &?lj %?lj ?decj %!lj !ack

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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- (DECI DABLE) Probabilistic -terminate with probability 1- (UNDECIDABLE)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Petri Nets strike back…

In Petri nets, termination of all

computations is decidable

the translation from CGF to Petri nets

allows us to prove that (nondeterministic) universal termination in CGF is decidable

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A Example 3: does it (nondeterministically) universally terminate?

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A Example 3: does it (nondeterministically) universally terminate?

900-899-899-900.. 500-501-500-500.. 100-100-101-100..

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A NO Example 3: does it (nondeterministically) universally terminate?

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SLIDE 48

SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

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- (UNDECI DABLE)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Probabilistic universal termination

Given a CGF system, decide whether

the probability for the system to terminate is 1

This corresponds to checking whether

there exists an infinite computation with associated probability > 0

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A Example 3: does it (probabilistically) universally terminate?

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A Example 3: does it (probabilistically) universally terminate?

900-899-899-900.. 500-501-500-500.. 100-100-101-100..

What is the probability

  • f this computation?

What is the probability

  • f this computation?
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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A Example 3: does it (probabilistically) universally terminate?

900-899-899-900.. 500-501-500-500.. 100-100-101-100..

Probability = 0

(as for all infinite computations)

Probability = 0

(as for all infinite computations)

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

100 500 900

!a ?a !c ?c !b ?b A+B B+B B+C C+C C+A A+A Example 3: does it (probabilistically) universally terminate? YES

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Is probabilistic universal termination decidable?

It is undecidable [Concur08] The overall proof includes the proof of

the following interesting result:

even if RAMs cannot be deterministically

modeled in CGF (remember Petri nets modeling of CGF), they can be probabilistically approximated up to any arbitrarily small error ε

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Approximate RAM modeling

i: Inc(rj) k: DecJump(rj,s) Ii Ii+1 Ik !decj Ik+1 Is rj with content nj: Rj ?decj Rj … Rj Rj

τ τ

nj instances

Problem: wrong jump!

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Approximate RAM modeling

i: Inc(rj) k: DecJump(rj,s) Ii Ii+1 Ik Is rj with content nj: Rj ?decj Rj … Rj Rj

τ τ

!inh

τ τ

!inh nj instances Inh … Inh Inh ?inh h instances !decj Ik+1

p < 1/ h2 But in an unbounded computation, with infinitely many DecJump’s, the prob. of a wrong jump is 1

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Approximate RAM modeling

i: Inc(rj) k: DecJump(rj,s) Ii Ii+1 Ik !decj Ik+1 Is rj with content nj: Rj ?decj Rj … Rj Rj

τ τ

!inh

τ τ

!inh Inh nj instances Inh … Inh Inh ?inh h instances

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Approximate RAM modeling

i: Inc(rj) k: DecJump(rj,s) Ii Ii+1 Ik !decj Ik+1 Is rj with content nj: Rj ?decj Rj … Rj Rj

τ τ

!inh

τ τ

!inh Inh nj instances Inh … Inh Inh ?inh h instances

I ncrementing the occurrences

  • f Inh the prob. of a wrong jump is

<

∞ k=h 1 k2

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SFM-08:Bio - 7.6.08 Expressiveness Issues in Calculi for Artificial Biochemistry

Related work

  • Magnasco. Chemical Kinetics is Turing Universal.

Phys Rev Lett. 1997

Exploit different reaction rates to model “finite logical circuits

with unbounded memory” using unbounded chemical species

Liekens and Fernando. Turing Complete Catalytic

Particle Computers. In Proc. ECAL’07. 2007

Approximate bounded computations of RAMs

Soloveichik et al. Computation with Finite

Stochastic Chemical Reaction Networks. In Nat.

  • Computing. 2008

Approximate also unbounded computations of RAMs

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References

  • Cardelli. On process rate semantics. To appear in Theoretical

Computer Science. 2008

Definition of CGF and proof of equivalence with chemical kinetics

  • Cardelli. Artificial Biochemistry. In Proc. Algorithmic

Bioprocesses ’08. To appear in LNCS. 2008

Informal introduction of association/dissociation mechanisms

Cardelli and Zavattaro. On the computational power of

  • biochemistry. In Proc. AB’08. To appear in LNCS. 2008

Definition of BGF and proof of Turing completeness

Zavattaro and Cardelli. Termination problems in chemical

  • kinetics. In Proc. Concur’08. To appear in LNCS. 2008

Decidability and nondecidability of nondeterministic and probabilistic

versions of properties in CGF