Simulating Biological Systems in the Stochastic Pi-calculus

Machine ➢ Formalise how the simulator works (program speci cation). ➢ Prove properties about the simulator. ➢ Give greater con dence in the simulation results. ➢ Improve on existing simulators. Andrew Phillips Microsoft Research 2004 23 Machine Data Structures ➢ Machine syntax νn1 νn2 ...νnN (Σ1 ::Σ2 :: ... ::ΣM :: []) : V,U ::= νn V Restriction p A List A,B ::= [] Empty p Σ::A Summation Andrew Phillips Microsoft Research 2004 24 Machine Encoding ➢ Encoding (P ): (P ) , P ◦ [] ➢ Construction (P ◦ V ): n 6∈ fn(P ) ⇒ P ◦ (νn V ) , νn (P ◦ V ) 0 ◦A , A (P | Q) ◦A , P ◦Q ◦A n 6∈ fn(P ◦A) ⇒ (νm P ) ◦A , νn (P{n/m} ◦A) !π.P ◦A , (π.(P | !π.P ) + 0) ◦A (π.P + Σ) ◦A , (π.P + Σ)::A Andrew Phillips Microsoft Research 2004 25 Machine Execution ➢ Reduction (V −→ V ′): V −→ V ′ ⇒ νn V −→ νn V ′ ∣∣∣∣ A Â (x(m).P + Σ)::A′ ∧A′ Â (x〈n〉.Q + Σ′) ::A′′ ⇒ A −→ P{n/m} ◦Q ◦A ′′ ➢ Selection: A Â A A Â Σ′ ::A′ ⇒ Σ::A Â Σ′ ::Σ::A′ Σ::A Â (π′.P ′ + Σ′) ::A ⇒ (π.P + Σ) :: A Â (π′.P ′ + π.P + Σ′) ::A Andrew Phillips Microsoft Research 2004 26 Stochastic Machine ➢ Machine can be easily extended with rates: V r −→ V ′ ⇒ νnr V r −→ νnr V ′ ∣∣∣∣∣ x = Next(A) ∧A Â (x(m).P + Σ)::A′ ∧A′ Â (xr〈n〉.Q + Σ′) ::A′′ ⇒ A r −→ P{n/m} ◦Q ◦A′′ ➢ Choose next reaction Next(A) using a stochastic algorithm (Gillespie) Andrew Phillips Microsoft Research 2004 27 Channel Activity ➢ Activity = number of possible interactions on a given channel: Actx(A) = (Inx(A) ∗Outx(A))−Mixx(A) ➢ Inx(A) = the number of unguarded inputs on channel x in A. ➢ Outx(A) = the number of unguarded outputs on channel x in A. ➢ Mixx(A) = the sum of Inx(Σi)×Outx(Σi) for each summation Σi in A. Andrew Phillips Microsoft Research 2004 28 Gillespie: Choosing the Next Reaction Next(A) 1. For all x ∈ fn(A) calculate axr = Actxr(A) ∗ r 2. Store non-zero values of axr in a list (xμ, aμ), where μ ∈ 1...M . 3. Calculate a0 = ∑M ν=0 aν 4. Randomly generate n1 and n2 ∈ [0, 1] and calculate τ and μ such that: τ = (1/a0) ln(1/n1) μ−1 ∑ ν=1 aν < n2a0 ≤ μ ∑ ν=1 aν 5. Next(A) = xμ and Delay(A) = τ . Andrew Phillips Microsoft Research 2004 29 Correctness of the Machine ➢ Safety: no runtime errors (no crashes) Lemma 1. ∀V.V ∈ PiM ∧ V −→ V ′ ⇒ V ′ ∈ PiM ➢ Soundness: machine only performs valid executions steps (behaves well) Theorem 1. ∀V.V ∈ PiM ∧ V −→ V ′ ⇒ [V ] −→ [V ′] ➢ Completeness: machine accurately executes all behaviours of the calculus Theorem 2. ∀P.P ∈ Pi ∧ P −→ P ′ ⇒ (P ) −→≡ (P ′). ➢ Termination: machine does not loop forever unnecessarily Theorem 3. ∀P.P ∈ Pi ∧ P 6−→⇒ (P ) 6−→ Andrew Phillips Microsoft Research 2004 30 Stochastic Correctness ➢ Theorems easily extend to reductions with rates ( r −→) ➢ Need to take into account the number of possible interactions on a channel: P1 , xr〈n〉.P + xr〈n〉.P | x(m).Q P2 , xr〈n〉.P | x(m).Q ➢ Reduction in P1 is twice as fast as the reduction in P2 ➢ Ensure that the reactions in the machine have the same rates as in the calculus Proposition 1. ∀V ∈ PiM.Appxr(V ) = Appxr([V ]) Proposition 2. ∀P ∈ Pi.Appxr(P ) = Appxr((P )) Andrew Phillips Microsoft Research 2004 31 Implementation ➢ Abstract Machine maps almost directly to program code ➢ Implemented a polymorphic type system and type checker ➢ Correctness of the machine gives greater con dence in the simulation results Andrew Phillips Microsoft Research 2004 32 Conclusion ➢ Presented a graphical representation for pi-calculus: ❑ Precise, compositional, executable descriptions. ❑ Used to model regulatory systems at the molecular level. ➢ Presented an abstract machine for the stochastic pi-calculus: ❑ Proof of correctness (safety, soundness, completeness, termination). ❑ Maps readily to program code. ➢ Built a simulator based on the machine. Andrew Phillips Microsoft Research 2004 33 Safety Proof Lemma 2. ∀V.V ∈ PiM ∧ V −→ V ′ ⇒ V ′ ∈ PiM Proof. By Lemma 3, Lemma 4 and by induction on reduction in PiM. 2 Lemma 3. ∀A ∈ PiM.A Â B ⇒ B ∈ PiM Proof. By induction on selection in PiM. 2 Lemma 4. ∀V.∀P.V ∈ PiM ∧ P ∈ Pi ⇒ P ◦ V ∈ PiM Proof. By induction on construction in PiM. 2 Andrew Phillips Microsoft Research 2004 34 Soundness Proof Lemma 5. ∀V.V ∈ PiM ⇒ [V ] ∈ Pi Proof. By induction on decoding in PiM. 2 Theorem 4. ∀V.V ∈ PiM ∧ V −→ V ′ ⇒ [V ] −→ [V ′] Proof. By Lemma 6, Lemma 7 and by induction on reduction in PiM. 2 Lemma 6. ∀A.A ∈ PiM ∧A Â B ⇒ [A] ≡ [B] Proof. By induction on selection in PiM. 2 Lemma 7. ∀V.∀P.V ∈ PiM ∧ P ∈ Pi ⇒ [P ◦ V ] ≡ P | [V ] Proof. By induction on construction in PiM. 2 [νn V ] , νn [V ] (1) [[]] , 0 (2) [Σ::A] , Σ | [A] (3) Andrew Phillips Microsoft Research 2004 35 Completeness Proof Lemma 8. ∀V.V ∈ PiM∧U ≡ V ∧V −→ V ′ ⇒ ∃U ′.U −→ U ′∧U ′ ≡ V ′ Proof. By induction on structural congruence in PiM. 2 Theorem 5. ∀P.P ∈ Pi ∧ P −→ P ′ ⇒ (P ) −→≡ (P ′). Proof. By Lemma 9 and by induction on reduction in Pi, where the rule for parallel composition is expanded over the remaining rules. 2 Lemma 9. P ≡ Q ⇒ (P ) ≡ (Q) Proof. By induction on structural congruence in Pi. 2 V ≡α U ⇒ V ≡ U n 6∈ fn(V ) ⇒ νn V ≡ V νx νy V ≡ νy νx V Andrew Phillips Microsoft Research 2004 36 Σ::Σ′ ::A ≡ Σ′ ::Σ::A A ≡ A′ ⇒ Σ::A ≡ Σ::A′ (π.P + π′.P ′ + Σ)::A ≡ (π′.P ′ + π.P + Σ)::A Σ::A ≡ Σ′ ::A ⇒ (π.P + Σ)::A ≡ (π.P + Σ′) ::A Andrew Phillips Microsoft Research 2004 37 Termination Proof Theorem 6. ∀P.P ∈ Pi ∧ P 6−→⇒ (P ) 6−→ Proof. By Theorem 4 and by basic relationships between encoding and decoding. 2 Andrew Phillips Microsoft Research 2004 38 Link Encoding ➢ Encoding uses restriction, replication, parallel composition and communication. ➢ A linked node → a replicated input on a fresh channel x, in parallel with an output on x ➢ A link to the node → an output on x. ➢ E.g.: Σ Σ π π ’ ’ = Σ Σ π π ’ ’ new x ! x() x<> x<> Andrew Phillips Microsoft Research 2004 39 Graphical Semantics x P x(m) P’ ’ x P x(m) P’ ’ x P x(m) P’ x P x(m) P’ Σ Σ ! Σ ! ! Σ ! ➢ Requires some imagination: for substituting names and for cloning graphs. Andrew Phillips Microsoft Research 2004 40