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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Probabilistic Analysis of Discrete Orgnisations Chemical System to in PRISM PRISM model Discrete organisations and PRISM analysis C Good N Kamaleson C Mu M Puljiz

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

  • rganisations

and PRISM analysis Discussions

Probabilistic Analysis of Discrete Orgnisations in PRISM

C Good N Kamaleson C Mu M Puljiz D Parker J Rowe

School of Computer Science, University of Birmingham

29 April, 2015

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

  • rganisations

and PRISM analysis Discussions

1 Translate Chemical System to PRISM model 2 Discrete organisations and PRISM analysis 3 Discussions

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

  • rganisations

and PRISM analysis Discussions

Outline

1 Translate Chemical System to PRISM model 2 Discrete organisations and PRISM analysis 3 Discussions

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Remind: reaction systems

Reaction systems M, R

  • the set of all possible species M
  • the set of all possible reactions among all possible species

R = PM(M) × PM(M)

  • let R ⊆ M denote the set of reactants and P ⊆ M

denote the set of products

  • the dynamics describes how the reactions are applied to a

collection of species

Example

  • M = {a, b, c}
  • R = {a+2b → ∅, a+c → 2b+c, b+c → a+c, 2c → ∅}

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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RS and Transition System

RS = M, R and F = (Q, Σ, q0, δ)

  • q = {s → N|s ∈ M}, Q = {q};
  • Σ = R;
  • δ(q, σ) = q′, if q, q′ ∈ Q, σ = PM(R) → PM(P),

R ⊆ Dom(q), P ⊆ Dom(q′), and ∀(s ∈ P ∪ R).(q′(s) − q(s) = ♯(s ∈ P) − ♯(s ∈ R));

  • The transition sequences of F should be equivalent to all

possible trajectory of movements of RS.

Example

  • M = {a, b, c}
  • R = {a+2b → ∅, a+c → 2b+c, b+c → a+c, 2c → ∅}

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Translate txt to PRISM code

//in the modelling language @species a=1 b=2 c=1 @parameters rA=1 rB=1 @reactions @r=r1 a+b+b -> 0 rA*a*b @r=r2 a+c -> b+b+c rA*a*c @r=r3 b+c -> a+c rB*b*c @r=r4 c+c -> 0 rA*c //translation txt to PRISM model ctmc const int MAX_AMOUNT = 5; formula total = a + b + c; init total <= MAX_AMOUNT endinit // Model parameters const double rA = 1; // rA const double rB = 1; // rB module RN a : [0..MAX_AMOUNT]; b : [0..MAX_AMOUNT]; c : [0..MAX_AMOUNT]; // r1: a+2b -> 0 [r1] (rA*a*b > 0) & (a > 0) & (b > 1) & (total<= MAX_AMOUNT)

  • > rA*a*b : (a’=a-1) & (b’=b-2);

// r2: a+c -> 2b+c [r2] (rA*a*c > 0) & (a > 0) & (c > 0) & (total+1<= MAX_AMOUNT)

  • > rA*a*c : (a’=a-1) & (b’=b+2) & (c’=c);

// r3: b+c -> a+c [r3] (rB*b*c > 0) & (b > 0) & (c > 0) & (total<= MAX_AMOUNT)

  • > rB*b*c : (a’=a+1) & (b’=b-1) & (c’=c);

// r4: 2c -> 0 [r4] (rA*c > 0) & (c > 1) & (total<= MAX_AMOUNT)

  • > rA*c : (c’=c-2);

endmodule 5 / 18

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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SBML-to-PRISM Translation

SBML-to-PRISM Translation

  • Systems Biology Markup Language (SBML) is an

XML-based format for representing models of biochemical reaction networks.

  • PRISM includes a (prototype) tool to translate

specifications in SBML to model descriptions in the PRISM language.

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Translate SBML to PRISM code

// File generated by automatic SBML-to-PRISM conversion // Original SBML file: examples/bioModels/BIOMD0000000004_SBML-L2V1.xml ctmc const int MAX_AMOUNT = 3; // Compartment size const double cell = 1.0; formula total = C + M + X + MI + XI; init total <= MAX_AMOUNT endinit // Model parameters const double V1 =0; // V1 const double V3 =0; // V3 const double VM1 = 3; // VM1 const double VM3 = 1; // VM3 const double Kc = 0.5; // Kc // Parameters for reaction reactions const double vi = 0.025; //for r1 const double kd = 0.01; //for r2 const double vd = 0.25; //for r3 const double Kd = 0.02; //for r3 const double K1 = 0.005; //for r4 const double V2 = 1.5; //for r5 const double K2 = 0.005; //for r5 const double K3 = 0.005; //for r6 const double K4 = 0.005; //for r7 const double V4 = 0.5; //for r7 7 / 18

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Translate SBML to PRISM code

module RN C : [0..MAX_AMOUNT]; M : [0..MAX_AMOUNT]; X : [0..MAX_AMOUNT]; MI : [0..MAX_AMOUNT]; XI : [0..MAX_AMOUNT]; // reaction1 (creation of cyclin):

  • > C

[reaction1] ((cell*vi) > 0) & (total+1<= MAX_AMOUNT) -> (cell*vi) : (C’=C+1); // reaction2 (default degradation of cyclin): C -> [reaction2] ((C*cell*kd) > 0) & (C > 0) & (total<= MAX_AMOUNT) -> (C*cell*kd) : (C’=C-1); // reaction3 (cdc2 kinase triggered degration of cyclin): [reaction3] ((C*cell*vd*X*(func(pow,(C+Kd),-1))) > 0) & (C > 0) & (total<= MAX_AMOUNT)

  • > (C*cell*vd*X*(func(pow,(C+Kd),-1))) : (C’=C-1);

// reaction4 (activation of cdc2 kinase): MI -> M [reaction4] ((cell*MI*V1*(func(pow,(K1+MI),-1))) > 0) & (MI > 0) & (total<= MAX_AMOUNT)

  • > (cell*MI*V1*(func(pow,(K1+MI),-1))) : (M’=M+1) & (MI’=MI-1);

// reaction5 (deactivation of cdc2 kinase): M -> MI [reaction5] ((cell*M*V2*(func(pow,(K2+M),-1))) > 0) & (M > 0) & (total<= MAX_AMOUNT)

  • > (cell*M*V2*(func(pow,(K2+M),-1))) : (M’=M-1) & (MI’=MI+1);

[reaction6] ((cell*V3*XI*(func(pow,(K3+XI),-1))) > 0) & (XI > 0) & (total<= MAX_AMOUNT)

  • > (cell*V3*XI*(func(pow,(K3+XI),-1))) : (X’=X+1) & (XI’=XI-1);

[reaction7] ((cell*V4*_X*(func(pow,(K4+_X),-1))) > 0) & (_X > 0) & (total<= MAX_AMOUNT)

  • > (cell*V4*_X*(func(pow,(K4+_X),-1))) : (_X’=_X-1) & (XI’=XI+1);

endmodule 8 / 18

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

  • rganisations

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Outline

1 Translate Chemical System to PRISM model 2 Discrete organisations and PRISM analysis 3 Discussions

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Discrete organisations

[Kreyssig et al’14]

Definition: organisation

A subset of M is a chemical organisation if it is closed and self-maintaining.

Definition: discrete organisation and generator

A subset of speices D of M is called discrete organisation if there is a state s such that D is the domain of the accessible states from s, and there is a sequence of transitions (σ1, . . . , σk) such that s′ = (σk ◦ · · · ◦ σ1)(s) satisfies: ∀M ∈ D.s′(M) ≥ s(M) and each reaction rules are firable within D. State s is called generator of the discrete

  • rganisation.

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Discrete organisations

[Kreyssig et al’14]

Lemma

Every (continuous) organisations is a discrete organisation.

Definition: purely discrete organisation (pdorg)

Discrete organisations which are not found in the continous theory.

Definition: connected purely discrete organisation

A purely discrete organisations is connected if there is a generator s of D s.t. (D, RAcc(s)) is connected as a continuous chemical organisation.

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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SCCs and BSCCs

Strongly connected components (SCC)

A strongly connected component of a directed graph G is a maximal set of vertices T ⊆ V such that for every pair of vertices s and s′, there is a directed path from s to s′ and a directed path from s′ to s.

Bottom strongly connected components (BSCC)

A bottom strongly connected component (BSCC) is an SCC T from which no state outside T is reachable from T.

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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SCCs and BSCCs in PRISM

Example: 18 SCCs, 17 BSCCs

(0,0,0) 1 1 (0,0,1) 1 2 (0,0,2) 2 3 (0,0,3) 3 4 (0,0,4) 4 5 (0,0,5) 5 6 (0,1,0) 1 7 (0,1,1) 22 (1,0,1) 1 12 (0,2,1) 1 8 (0,1,2) 2 23 (1,0,2) 2 13 (0,2,2) 2 21 (1,0,0) 2 9 (0,1,3) 3 24 (1,0,3) 3 3 14 (0,2,3) 3 10 (0,1,4) 4 25 (1,0,4) 4 4 11 (0,2,0) 1 27 (1,1,1) 2 16 (0,3,1) 1 37 (2,0,1) 1 2 28 (1,1,2) 4 17 (0,3,2) 2 26 (1,1,0) 2 38 (2,0,2) 2 3 29 (1,1,3) 6 3 39 (2,0,3) 3 15 (0,3,0) 1 31 (1,2,1) 3 2 19 (0,4,1) 1 41 (2,1,1) 2 2 32 (1,2,2) 6 2 30 (1,2,0) 2 42 (2,1,2) 4 18 (0,4,0) 1 34 (1,3,1) 4 3 44 (2,2,1) 3 20 (0,5,0) 1 1 1 2 4 36 (2,0,0) 2 3 2 2 47 (3,0,1) 1 40 (2,1,0) 2 48 (3,0,2) 2 33 (1,3,0) 3 4 50 (3,1,1) 2 35 (1,4,0) 4 1 1 3 46 (3,0,0) 2 43 (2,2,0) 4 53 (4,0,1) 1 45 (2,3,0) 6 1 49 (3,1,0) 1 1 51 (3,2,0) 6 52 (4,0,0) 1 54 (4,1,0) 1 55 (5,0,0) 1

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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SCCs and generators

Connections: SCCs and generators

  • states in SCCs (denoted by S1): generators of the
  • rganisations
  • states in SCCs with positive number of reactants (denoted

by S2) and make each reaction rules be firable: generators

  • f pdorgs
  • states in S1 \S2: generators of the rest of the organisations

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Mass probability to reach SCCs?

Observation

  • Every run will eventually reach a BSCC and stay there.
  • Non-BSCC will be left at some point with probability 1.

SCC reduction?

  • Probability to reach organisations: is it useful to look at

the whole probability mass provided by SCCs?

  • Approaches:
  • Markov chain: compute probabilities (Daws’04,

HanKatoen’08)

  • Bottom-up computing

(AbrahamJasenWimmerKatoenBecker’10)

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

  • rganisations

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Outline

1 Translate Chemical System to PRISM model 2 Discrete organisations and PRISM analysis 3 Discussions

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Discssions

Questions

  • Translate the reaction systems into PRISM
  • SCCs and BSCCs in PRISM: any further connections with

discrete orgnisations and generators?

  • Any information we can derive from PRISM’s analysis of

the SCCs?

  • Mass probability calculations between good SCCs?
  • Suggestions for further work?

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Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to PRISM model Discrete

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Discssions

Quantitative properties of interests? e.g.,

  • Probability that states labelled y is reached: P =?[F y],

e.g., states labelled y in which B molecules form a strict majority

  • Probability of reaching pdorg (A, B, C):

P =?[F (A > 0) ∧ (B > 0) ∧ (C > 0)]

  • Probability of reaching a generator?: P =?[F s], states

labelled s can be any states in SCCs

  • Probability that a present of A molecules is reached

without passing through the state labelled x: P =?[¬x U (A > 0)], e.g., states labelled x can be generators containing 2B molecules and 1A molecule

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