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 Discussions D Parker J Rowe School of Computer Science, University of Birmingham 29 April, 2015 1 / 18
Probabilistic Analysis of Discrete Orgnisations in PRISM Translate Chemical System to 1 Translate Chemical System to PRISM model PRISM model Discrete organisations and PRISM analysis Discussions 2 Discrete organisations and PRISM analysis 3 Discussions 2 / 18
Probabilistic Analysis of Outline Discrete Orgnisations in PRISM Translate Chemical System to PRISM model 1 Translate Chemical System to PRISM model Discrete organisations and PRISM analysis Discussions 2 Discrete organisations and PRISM analysis 3 Discussions 2 / 18
Probabilistic Analysis of Remind: reaction systems Discrete Orgnisations in PRISM Reaction systems �M , R� Translate Chemical • the set of all possible species M System to PRISM model • the set of all possible reactions among all possible species Discrete R = P M ( M ) × P M ( M ) organisations and PRISM analysis • let R ⊆ M denote the set of reactants and P ⊆ M Discussions 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 +2 b → ∅ , a + c → 2 b + c , b + c → a + c , 2 c → ∅} 3 / 18
Probabilistic Analysis of RS and Transition System Discrete Orgnisations in PRISM RS = �M , R� and F = ( Q , Σ , q 0 , δ ) Translate Chemical • q = { s �→ N | s ∈ M} , Q = { q } ; System to PRISM model • Σ = R ; Discrete • δ ( q , σ ) = q ′ , if q , q ′ ∈ Q , σ = P M ( R ) → P M ( P ), organisations and PRISM analysis R ⊆ Dom ( q ), P ⊆ Dom ( q ′ ), and Discussions ∀ ( 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 +2 b → ∅ , a + c → 2 b + c , b + c → a + c , 2 c → ∅} 4 / 18
Probabilistic Analysis of Translate txt to PRISM code Discrete Orgnisations in PRISM //in the modelling language //translation txt to PRISM model Translate @species ctmc Chemical a=1 const int MAX_AMOUNT = 5; System to b=2 formula total = a + b + c; PRISM model c=1 init total <= MAX_AMOUNT endinit @parameters Discrete rA=1 // Model parameters organisations rB=1 const double rA = 1; // rA and PRISM @reactions const double rB = 1; // rB analysis @r=r1 a+b+b -> 0 module RN Discussions rA*a*b a : [0..MAX_AMOUNT]; @r=r2 b : [0..MAX_AMOUNT]; a+c -> b+b+c c : [0..MAX_AMOUNT]; rA*a*c @r=r3 // r1: a+2b -> 0 b+c -> a+c [r1] (rA*a*b > 0) & (a > 0) & (b > 1) & (total<= MAX_AMOUNT) rB*b*c -> rA*a*b : (a’=a-1) & (b’=b-2); @r=r4 // r2: a+c -> 2b+c c+c -> 0 [r2] (rA*a*c > 0) & (a > 0) & (c > 0) & (total+1<= MAX_AMOUNT) rA*c -> 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
Probabilistic Analysis of SBML-to-PRISM Translation Discrete Orgnisations in PRISM Translate Chemical System to PRISM model SBML-to-PRISM Translation Discrete organisations • Systems Biology Markup Language (SBML) is an and PRISM analysis XML-based format for representing models of biochemical Discussions reaction networks. • PRISM includes a (prototype) tool to translate specifications in SBML to model descriptions in the PRISM language. 6 / 18
Probabilistic Analysis of Translate SBML to PRISM code Discrete Orgnisations in PRISM // File generated by automatic SBML-to-PRISM conversion // Original SBML file: examples/bioModels/BIOMD0000000004_SBML-L2V1.xml ctmc Translate const int MAX_AMOUNT = 3; Chemical System to // Compartment size PRISM model const double cell = 1.0; Discrete organisations formula total = C + M + X + MI + XI; and PRISM init total <= MAX_AMOUNT endinit analysis // Model parameters Discussions 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
Probabilistic Analysis of Translate SBML to PRISM code Discrete Orgnisations in PRISM module RN C : [0..MAX_AMOUNT]; M : [0..MAX_AMOUNT]; Translate X : [0..MAX_AMOUNT]; MI : [0..MAX_AMOUNT]; Chemical XI : [0..MAX_AMOUNT]; System to PRISM model // reaction1 (creation of cyclin): -> C [reaction1] ((cell*vi) > 0) & (total+1<= MAX_AMOUNT) -> (cell*vi) : (C’=C+1); Discrete organisations // reaction2 (default degradation of cyclin): C -> and PRISM [reaction2] ((C*cell*kd) > 0) & (C > 0) & (total<= MAX_AMOUNT) -> (C*cell*kd) : (C’=C-1); analysis // reaction3 (cdc2 kinase triggered degration of cyclin): Discussions [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
Probabilistic Analysis of Outline Discrete Orgnisations in PRISM Translate Chemical System to PRISM model 1 Translate Chemical System to PRISM model Discrete organisations and PRISM analysis Discussions 2 Discrete organisations and PRISM analysis 3 Discussions 9 / 18
Probabilistic Analysis of Discrete organisations Discrete Orgnisations in PRISM [Kreyssig et al’14] Translate Chemical Definition: organisation System to PRISM model A subset of M is a chemical organisation if it is closed and Discrete organisations self-maintaining. and PRISM analysis Definition: discrete organisation and generator Discussions 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 ( σ 1 , . . . , σ k ) such that s ′ = ( σ k ◦ · · · ◦ σ 1 )( s ) satisfies: ∀ M ∈ D . s ′ ( M ) ≥ s ( M ) and each reaction rules are firable within D . State s is called generator of the discrete organisation. 10 / 18
Probabilistic Analysis of Discrete organisations Discrete Orgnisations in PRISM [Kreyssig et al’14] Translate Chemical System to Lemma PRISM model Discrete Every (continuous) organisations is a discrete organisation. organisations and PRISM analysis Definition: purely discrete organisation (pdorg) Discussions 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 , R Acc ( s ) ) is connected as a continuous chemical organisation. 11 / 18
Probabilistic Analysis of SCCs and BSCCs Discrete Orgnisations in PRISM Translate Chemical System to Strongly connected components (SCC) PRISM model Discrete A strongly connected component of a directed graph G is a organisations and PRISM maximal set of vertices T ⊆ V such that for every pair of analysis vertices s and s ′ , there is a directed path from s to s ′ and a Discussions 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 . 12 / 18
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