A Unifying Framework for Modelling and Analysing Biochemical Pathways Using Petri Nets

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1 A Unifying Framework for Modelling and Analysing Biochemical Pathways Using Petri Nets Groningen, NL, London, UK, Cottbus, DE BME Tutorial, Part 4 Paris, June 2009

2 Framework Qualitative Stochastic Continuous

3 Framework Qualitative Time-free Timed, Quantitative Stochastic Continuous Discrete State Space Continuous State Space

4 Framework Qualitative Time-free Timed, Quantitative Abstraction Abstraction Stochastic Continuous Discrete State Space Continuous State Space

5 Framework Qualitative Time-free Timed, Quantitative Abstraction Stochastic Approximation of Hazard function, type (1) Approximation by Hazard function, type (2) Abstraction Continuous Discrete State Space Continuous State Space

6 (Qualitative) Petri Net Definition : A place/transition Petri net is a quadruple PN = (P,T,f,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions)

7 (Qualitative) Petri Net Definition : A place/transition Petri net is a quadruple PN = (P,T,f,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) N 0 (weighted directed arcs)

8 (Qualitative) Petri Net Definition : A place/transition Petri net is a quadruple PN = (P,T,f,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) N 0 (weighted directed arcs) m 0 : P N 0 (initial marking)

9 (Qualitative) Petri Net Definition : A place/transition Petri net is a quadruple PN = (P,T,f,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) N 0 (weighted directed arcs) m 0 : P N 0 (initial marking) Interleaving Semantics : reachability graph / CTL, LTL Partial Order Semantics : prefix / CTL, LTL

10 Stochastic Petri Net Definition : A biochemically interpreted stochastic Petri net is a quintuple SPN Bio = (P,T,f,v,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) N 0 (weighted directed arcs) m 0 : P N 0 (initial marking)

11 Stochastic Petri Net Definition : A biochemically interpreted stochastic Petri net is a quintuple SPN Bio = (P,T,f,v,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) N 0 (weighted directed arcs) m 0 : P N 0 (initial marking) v : T H (stochastic firing rate functions) with { } - H := h t h t : N t 0 R +,t T - v(t) = h t for all transitions t T

12 Stochastic Petri Net Definition : A biochemically interpreted stochastic Petri net is a quintuple SPN Bio = (P,T,f,v,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) N 0 (weighted directed arcs) m 0 : P N 0 (initial marking) v : T H (stochastic firing rate functions) with { } - H := h t h t : N t 0 R +,t T - v(t) = h t for all transitions t T Semantics : Continuous Time Markov Chain / CSL, PLTLc

13 Stochastic Petri Net Interpretation of tokens : tokens = molecules tokens = concentration levels

14 Stochastic Petri Net Specialised stochastic firing rate function, two examples : molecules semantics h t := c t p t ( ) m(p) f (p,t) (1) concentration levels semantics h t := k t N p t ( m(p) N ) (2)

15 Stochastic Petri Net - Semantics a transition t is enabled as usual : p t : m(p) > 0. an enabled transition t has to wait transition s waiting time is an exponentially distributed random variable X t with the probability density function : f Xt (τ) = λ t (m) e ( λt(m) τ), τ 0. while waiting, the transition may loose its enabled state firing itself does not consume time, follows standard firing rule Semantics : Continuous Time Markov Chain / CSL, PLTLc

16 Continuous Petri Net Definition : A biochemically interpreted continuous Petri net is a quintuple CPN Bio = (P,T,f,v,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) R + 0 (weighted directed arcs) m 0 : P R + 0 (initial marking)

17 Continuous Petri Net Definition : A biochemically interpreted continuous Petri net is a quintuple CPN Bio = (P,T,f,v,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) R + 0 (weighted directed arcs) m 0 : P R + 0 (initial marking) v : T H (continuous firing rate functions) with - H := { h t h t : R t R +,t T } - v(t) = h t for all transitions t T

18 Continuous Petri Net Definition : A biochemically interpreted continuous Petri net is a quintuple CPN Bio = (P,T,f,v,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) f : ((P T) (T P)) R + 0 (weighted directed arcs) m 0 : P R + 0 (initial marking) v : T H (continuous firing rate functions) with - H := { h t h t : R t R +,t T } - v(t) = h t for all transitions t T Semantics : ODEs / LTLc

19 Continuous Petri Net - Semantics A continuous transition t is enabled in m, if p t : m(p) > 0. each place p subject to changes gets its own equation dm (p) dt = t p f (t,p) v (t) t p f (p,t) v (t), each equation corresponds to a line in the incidence matrix Semantics : ODEs / LTLc

20 Continuous Petri Net - Semantics A continuous transition t is enabled in m, if p t : m(p) > 0. each place p subject to changes gets its own equation dm (p) dt = t p f (t,p) v (t) t p f (p,t) v (t), each equation corresponds to a line in the incidence matrix continuous Petri net = structured description of ODEs. Semantics : ODEs / LTLc

21 Continuous Petri Net - Example Raf-1Star m1 m2 RKIP k1 k2 ERK-PP m9 m3 Raf-1Star_RKIP k8 k3 k4 k11 m8 MEK-PP_ERK m4 Raf-1Star_RKIP_ERK-PP m11 RKIP-P_RP k6 k7 k5 k9 k10 m7 MEK-PP m5 ERK m6 RKIP-P m10 RP

22 Continuous Petri Net - Example dm3 = dt Raf-1Star m1 m2 RKIP k1 k2 ERK-PP m9 m3 Raf-1Star_RKIP k8 k3 k4 k11 m8 MEK-PP_ERK m4 Raf-1Star_RKIP_ERK-PP m11 RKIP-P_RP k6 k7 k5 k9 k10 m7 MEK-PP m5 ERK m6 RKIP-P m10 RP

23 Continuous Petri Net - Example dm3 = + r1 dt + r4 Raf-1Star m1 m2 RKIP k1 k2 ERK-PP m9 m3 Raf-1Star_RKIP k8 k3 k4 k11 m8 MEK-PP_ERK m4 Raf-1Star_RKIP_ERK-PP m11 RKIP-P_RP k6 k7 k5 k9 k10 m7 MEK-PP m5 ERK m6 RKIP-P m10 RP

24 Continuous Petri Net - Example dm3 = + r1 dt + r4 - r2 - r3 ERK-PP Raf-1Star m1 k1 RKIP m2 k2 m9 m3 Raf-1Star_RKIP k8 k3 k4 k11 m8 MEK-PP_ERK m4 Raf-1Star_RKIP_ERK-PP m11 RKIP-P_RP k6 k7 k5 k9 k10 m7 MEK-PP m5 ERK m6 RKIP-P m10 RP

25 Continuous Petri Net - Example dm3 = + k1 * m1 * m2 dt + r4 - r2 - r3 ERK-PP Raf-1Star m1 k1 RKIP m2 k2 m9 m3 Raf-1Star_RKIP k8 k3 k4 k11 m8 MEK-PP_ERK m4 Raf-1Star_RKIP_ERK-PP m11 RKIP-P_RP k6 k7 k5 k9 k10 m7 MEK-PP m5 ERK m6 RKIP-P m10 RP

26 Continuous Petri Net - Example dm3 = + k1 * m1 * m2 dt + k4 * m4 - k2 * m3 - k3 * m3 * m9 ERK-PP Raf-1Star m1 k1 RKIP m2 k2 m9 m3 Raf-1Star_RKIP k8 k3 k4 k11 m8 MEK-PP_ERK m4 Raf-1Star_RKIP_ERK-PP m11 RKIP-P_RP k6 k7 k5 k9 k10 m7 MEK-PP m5 ERK m6 RKIP-P m10 RP

27 Continuous Petri Net - Example, ODEs dm 1 dt dm 2 dt dm 3 dt dm 4 dt dm 5 dt dm 6 dt = r 2 + r 5 r 1 = r 2 + r 11 r 1 = r 1 + r 4 r 2 r 3 = r 3 r 4 r 5 = r 5 + r 7 r 6 = r 5 + r 10 r 9 dm 7 dt dm 8 dt dm 9 dt dm 10 dt dm 11 dt = r 7 + r 8 r 6 = r 6 r 7 r 8 = r 4 + r 8 r 3 = r 10 + r 11 r 9 = r 9 r 10 r 11

28 Continuous Petri Net - Example, ODEs r 1 = k 1 m 1 m 2 r 7 = k 7 m 8 r 2 = k 2 m 3 r 3 = k 3 m 3 m 9 r 4 = k 4 m 4 r 5 = k 5 m 4 r 8 = k 8 m 8 r 9 = k 9 m 6 m 10 r 10 = k 10 m 11 r 11 = k 11 m 11 r 6 = k 6 m 5 m 7

29 References M Heiner, D Gilbert, R Donaldson : Petri Nets for Systems and Synthetic Biology; SFM 2008, Springer LNCS 5016, pp. 215Ð264, M Heiner; R Donaldson ; D Gilbert : Petri Nets for Systems Biology; in MS Iyengar (ed.) : Symbolic Systems Biology : Theory and Methods, Jones & Bartlett Publishers, LLC, in Press. D Gilbert, M Heiner, S Rosser, R Fulton, X Gu, M Trybilo : A Case Study in Model-driven Synthetic Biology; IFIP WCC 2008/BICC 2008, Springer Boston, IFIP, Vol. 268, pp , D Gilbert, M Heiner, S Lehrack : ical Pathways Using Petri Nets ; Proc. CMSB 2007, LNCS/LNBI 4695, pp

30 Qualitative Petri Net - Extensions (Qualitative) Petri net + special arcs : read arc inhibitor arc equal arc reset arc t0 t1 2 2 p0 p t3 t2 read/inhibitor/equal arcs : influence on enabling, tested place is not changed by the firing reset arc : no influence on enabling, tested place is reset to zero

31 Stochastic Petri Net - Extensions special arcs : read, inhibitor, equal, reset immediate transitions - highest priority deterministic transitions : deterministic waiting time/delay - relative time points scheduled transitions : start(repetition)end schedule specifies absolute points of the simulation time where the transition may fire, if enabled remark : immediate transition deterministic transition with zero waiting time

32 Extended Stochastic Petri Net Definition, part 1 : A biochemically interpreted deterministic and stochastic Petri net is a septuple DSPN Bio = (P,T,f,g,v,d,m 0 ), where P, T - finite, non empty, disjoint sets (places, transitions) The set T is the union of three disjunctive transition sets : T := T stoch T im T timed with : T stoch - set of stochastic transitions with exponentially distributed waiting time, T im - set of immediate transitions, T timed - set of transitions with deterministic waiting time.... continued on next slide

33 Extended Stochastic Petri Net Definition, part 2 : f : ((P T) (T P)) N 0 (weighted directed arcs) g : (P T) N 0 (weighted directed inhibitor arcs) v : T H (stochastic firing rate functions) with - H := { } t T h t h t : N t 0 R + - v(t) = h t for all transitions t T d : T timed R + (deterministic waiting time) m 0 : P N 0 (initial marking) Semantics : CTMC? simulative MC/PLTLc

34 Extended Stochastic Petri Nets, Examples EX1 : time-controlled inflow/outflow input A B output input - scheduled(11,1,20) output - scheduled(31,1,40) t1 t2 Marking Time A B

35 Extended Stochastic Petri Net, Examples EX2 : time-controlled inflow/outflow input A t1 t2 B switch_output_off output output_on input - scheduled(11,1,20) switch_output_on - scheduled(20,20,_simend) switch_output_on Marking Time A B

36 Extended Stochastic Petri Net, Examples EX3 : Token-controlled Inflow input A t1 t2 B Marking A B 100 t1 - BioMassAction(0.1) t2 - BioMassAction(0.005) Time

37 Extended Stochastic Petri Net, Examples EX4 : Token-controlled Inflow t1 input_on input_off input A B 10 switch_on switch_off 30 t2 output input - delay(0.5) output - scheduled(5,5,_simend) Marking Time A input_on

38 Extended Stochastic Petri Net, Examples EX5 : time-controlled switch between deterministic and stochastic behaviour stochastic_on t_stoch switch_to_stoch switch_to_det 1000 A det_on t_det B Marking A B switch_to_det - scheduled(10) switch_to_stoch - scheduled(30) Time

39 Extended Stochastic Petri Net, Examples EX6 : token-controlled switch between deterministic and stochastic behaviour switch_to_stoch 500 stochastic_on 700 switch_to_det det_on t_stoch A 1000 B t_det Marking Time A B

40 Framework Molecules/Levels CTL, LTL Qualitative Time-free Timed, Quantitative Molecules/Levels Stochastic rates CSL Abstraction Stochastic Approximation of Hazard function, type (1) Approximation by Hazard function, type (2) Abstraction Continuous Concentrations Deterministic rates LTLc Discrete State Space Continuous State Space

41 Tools model construction BioNessi (London) Snoopy (CB)

42 Tools model construction BioNessi (London) Snoopy (CB) qualitative analysis INA (HUB), Charlie (CB) BDD-CTL (Boolean), IDD-CTL (integer) (CB)

43 Tools model construction BioNessi (London) Snoopy (CB) qualitative analysis INA (HUB), Charlie (CB) BDD-CTL (Boolean), IDD-CTL (integer) (CB) stochastic analysis PRISM/CSL (Oxford), IDD-CSL (CB) MC2/PLTLc (Glasgow)

44 Tools model construction BioNessi (London) Snoopy (CB) qualitative analysis INA (HUB), Charlie (CB) BDD-CTL (Boolean), IDD-CTL (integer) (CB) stochastic analysis PRISM/CSL (Oxford), IDD-CSL (CB) MC2/PLTLc (Glasgow) continuous analysis MATLAB BIOCHAM/LTLc (INRIA/Paris), MC2/LTLc (Glasgow)

45 Tools - Snoopy s Export Features Latex MetaTool SBML Level 2 Continuous PN Extended PN APNN INA LoLA Maria PEP Prod Tina Stochastic PN Petri Net INA tim INA tmd PRISM SMART Music PN Modulo PN Time PN IDD-CSL all net classes export to EPS; MIF; Xfig all Petri net classes are read by Charlie

46 ... some more examples deg_mr 10 mr 10 transc_dr transc_dr_a transl_r deg_c Circadian Clock ; PRISM web page dr_a rel_r 10 bind_r dr r deg_r 10 deactive c places : 9 transitions : 16 deg_a a 10 rel_a bind_a 10 transl_a da_a da deg_ma transc_da_a transc_da ma

47 ... some more examples not covered by T invariants 20 k3 G P3 GP3 Receptor k4 Survival k1 KdStar k0 k6 KdStar k5 Regeneration KdStarG KdStarGP3 Proliferation k2 KdStarGStar k7 P3 k11 k10..., Balbo : Angiogenesis Process ; CMSB 2009 k16 KdStarGStarP3k KdStar GStarP3kP3 KdStarGStarP3 GStarP3 P3k P3k P3k Pg Pg Pg k17 k22 k23 k12 k14 k9 k13 k8 k50 k49 k51 k44 k43 k38 k37 KdStar Pg k32 k31 KdStarGStarP3kP3 KdStar GStarPgP3 KdStar KdStarGStarPgP3 KdStarGStarPg KdStarPg places : 39 transitions : 64 k18 KdStarGStarP3kStar P2 k15 k24 KdStarGStarP3kStarP3 P2 P2 k52 k45 KdStarGStarPgStarP3 P2 k39 KdStarGStarPgStar P2 k33 KdStarPgStar P2 k19 k20 k25 k26 k57 k56 k47 k46 k41 k40 k35 k34 KdStarGStarP3kStarP2 KdStarGStarP3kStarP3P2 Pt k55 PtP2 k63 KdStarGStarPgStarP3P2 KdStarGStarPgStarP2 KdStarPgStarP2 P3 k21 k27 PtP3 k59 k58 DAGE E k48 k42 k36 k54 k53 PtP3P2 k60 k61 k62 Akt P3 DAG k29 k28 AktP3 k30 AktStar

48 ... some more examples C:Gi/Gq Protein Aktivierung Ã¼ber B2 R D:Gq Protein Aktivierung Ã¼ber m R B B2 R m R Ach Galphaq GDP r0 6 r0 8 Galphaq GTP B:Gi Protein Aktivierung Ã¼ber CB R E:Gs Protein Aktivierung Ã¼ber beta2ar CB,AEA(EX) r0 4 CB R Gq Protein(I,GDP) beta2ar r0 10 A Gs Protein (I,GDP) Gi Protein (I,GDP) Pain-related pathways ; MOSS project, MD project group, 2009 places : 132 transitions : 152 A:Gi Protein Aktivierung Ã¼ber mu OR F:Gs Protein Aktivierung Ã¼ber PGE2 R mu OR O PGE2 R PGE2 r0 2 r0 12 Galphai GTPGalphas GTP H:IP3/DAG Synthese Galphai GDP G:Regulation AC Galphas GDP IP3 DAG camp I:cAMP Abbau L:Ca Pumpe S:IP3 Rezeptor/Channel Ca2+(EX) Ca2+(ER) Ca2+(IN) K:Regulation VACCs T:RYR Regulation PKA_kat J:Aktivierung der PKA U:SERCA PKC(A) N:Aktivierung PKC V:Synaptische AusschÃ¼ttung R:Regulation TRPV1 CaMKII(A) O:Aktivierung CaMKII Calcineurin(A) P:Aktivierung Calcineurin AEA(IN) Q:AEA Regulation

49 ... some more examples C. elegance vulval development : places : 206, transitions : Miyano et al., BMC Syst. Biol ; - Bonzanni et al., Bioinformatics 2009 ; LIN3_Anchorcell parameter AC p_s1 2 m1 p_d1 lin15 v4_lin15 v5_lin15 v6_lin15 v7_lin15 v8_lin15 LIN15_hyp7 v4_lin15_hyp7 v5_lin15_hyp7 v6_lin15_hyp7 v7_lin15_hyp7 v8_lin15_hyp7 p_s10 m29 p_d27 4_p_s10 m29 4_p_d27 5_p_s10 m29 5_p_d27 6_p_s10 m29 6_p_d27 7_p_s10 m29 7_p_d27 8_p_s10 m29 8_p_d p_s9 p_d26 4_p_s9 4_p_d26 5_p_s9 5_p_d26 6_p_s9 6_p_d26 7_p_s9 7_p_d26 8_p_s9 8_p_d26 LIN_3_from_hyp7 m28 v4_lin_3_from_hyp7 m28 v5_lin_3_from_hyp7 m28 v6_lin_3_from_hyp7 m28 v7_lin_3_from_hyp7 m28 v8_lin_3_from_hyp7 m28 VPC_3 VPC_4 VPC_5 VPC_6 VPC_7 VPC_8 p_u2 4_p_u2 5_p_u2 6_p_u2 7_p_u2 8_p_u2 LS_molecule_E v4_ls_molecule_e v5_ls_molecule_e v6_ls_molecule_e v7_ls_molecule_e v8_ls_molecule_e m270 m270 m270 m270 m270 m270

50 ... some more examples C. elegance vulval development : LIN_3_from_hyp7 LIN3_Anchorcell lin12_wt lin12_gf m28 m1 p_s3_wt p_s3_gf p24 p17 v4_ls_molecule_e p_s2 p_d5 p18 LIN12_NotchR_extracellular m270 p_d2 LET_23 m2 p3 LIN3_LET23 m3 p4 m4 p5 LIN3_LET23_dimer m5 p_d6 LIN_12 m20 LIN_12R_LScomplex m21 p19 m22 p_d9 LIN3_LET23_p_dimer p_d3 p_d4 SEM_5_active 4 p_s4 LIN12_NotchR_intracellular p_d8 p_d10 SEM_5 m6 p6 m7 p_d11 m23 LET60_active p_s5 p_d12 LET_60 m8 p7 m9 p_d13 m10 pd14 p_d15 m11 p_s6 MPK_1 p8 MPK_1_active_C 20 LST_inhibitor m27 p_d25 3_VPC_Nucleus m19 p_d28 LS_molecule p12 LS_molecule_E m270

51 ... some more examples C. elegance vulval development : v7_lin12_gf m11 v7_mpk_1_active_c v7_lin_12 m20 v7_mpk_1_active_n v7_lin12_notchr_intracellular 7_p13 m15 m12 7_p9 7_p23 m23 v7_lin_1_a 7_p_d16 v7_lin12_notchr_intracellular_n v7_lst_inhibitor m27 v7_lin31_lin1_complex 7_pd17 7_p10 m25 7_p_d23 m24 7_p20 m13 m14 v7_lin_31_a v7_lag_1 9 7_p21 v7_p_fate_2 m30 7_p_u2 7_ps7 7_pd18 7_p_d19 7_p14 m17 7_p_s87_p_d22 v7_lst 20 m16 v7_ls_mrna v7_lst_mrna m26 7_p_d24 7_p_u1 7_pd20 v7_vulval_gene 7_p11 7_pd21 v7_p_fate_1 m18 7_p15 7_p22 m19 v7_ls_molecule m5 v7_lin3_let23_p_dimer

CSL model checking of biochemical networks with Interval Decision Diagrams

CSL model checking of biochemical networks with Interval Decision Diagrams Brandenburg University of Technology Cottbus Computer Science Department http://www-dssz.informatik.tu-cottbus.de/software/mc.html