MILANO, JUNE Monika Heiner Brandenburg University of Technology Cottbus
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1 MILANO, JUNE 2013 WHAT CAN PETRI NETS DO 4 MULTISCALE SYSTEMS BIOLOGY? Monika Heiner Brandenburg University of Technology Cottbus
2 PETRI NETS - T - THE BIG PROS umbrella with unifying power readable & unambigious -> fault avoidant model construction locality - causality - concurrency compositional, hierarchical notations -> logical and macro nodes executable -> to experience the model, spec. causality -> interpretation in qualitative / stochastic / continuous / hybrid paradigms Petri net theory -> P/T-invariants, partial order interpretation of T-invariants, conclusions CTI/CPI -> behavioural properties -> STP, reduction rules,...
3 THE PETRI NET FRAMEWORK
4 KEY IDEA MODELS SHARING STRUCTURE QUANTITATIVE MODEL = QUALITATIVE MODEL + RATE FUNCTIONS (KINETICS)
5 CRUCIAL POINT STATE-DEPENDENT RATE FUNCTIONS STOCHASTIC RATES CONTINUOUS RATES LAMBDA OF EXPONENTIAL WAITING TIME STRENGTH OF CONTINUOUS FLOW CTMC ODES -> supported by, e.g., COPASI, Dizzy,..., Snoopy
6 FRAMEWORK 2010 LTS / PO CTL, LTL QUALITATIVE net reduction, SC, SB, CPI, CTI, ADT sets STP, bad siphons, etc. time-free timed, quantitative STOCHASTIC CTMC CSL, PLTLC abstraction extension approximation approximation extension abstraction CONTINUOUS ODES LTLC discrete state space HYBRID continuous state space
7 STOCHASTIC PETRI NETS (SPN)
8 STOCHASTIC PETRI NETS, BASICS transitions r i get a stochastic waiting time -> exponential distribution with parameter lambda state-dependent lambda defined by rate function v i (r i ) -> any arithmetic function including the transition s pre-places as integer variables and user-defined real-valued parameters -> modifier arcs -> popular kinetics: mass-action semantics, level semantics semantics: Continuous Time Markov Chain (CTMC) -> reachability graph + state transition rates analysis -> standard Markov analysis techniques: transient, steady state -> stochastic simulation algorithms (SSA), e.g. Gillespie s SSA
9 RATE FUNCTIONS mass-action semantics h t := c t p t ( ) m(p) f (p,t) level semantics h t := k t N p t ( m(p) N )
10 LEVEL CONCEPT
11 CONTINUOUS PETRI NETS (CPN)
12 CONTINUOUS PETRI NETS, BASICS transitions r i fire continuously rate functions v i (r i ) -> any arithmetic function including the transition s pre-places as real-valued variables and user-defined real-valued parameters real-valued tokens -> concentrations semantics: set of Ordinary Differential Equations (ODEs) -> uniquely defined, but not vice versa -> [SOLIMAN, HEINER 2010] -> typically non-linear simulation (numerical integration) -> stiff / unstiff solvers
13 CONTINUOUS PETRI NET DEFINES ODES s1 Raf-1Star s2 RKIP k1 k2 s9 ERK-PP s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 MEK-PP_ERK s4 Raf-1Star_RKIP_ERK-PP s11 RKIP-P_RP k6 k7 k5 k9 k10 s7 MEK-PP s5 ERK s6 RKIP-P s10 RP
14 CONTINUOUS PETRI NET DEFINES ODES ds3 = dt s1 Raf-1Star s2 RKIP k1 k2 s9 ERK-PP s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 MEK-PP_ERK s4 Raf-1Star_RKIP_ERK-PP s11 RKIP-P_RP k6 k7 k5 k9 k10 s7 MEK-PP s5 ERK s6 RKIP-P s10 RP
15 CONTINUOUS PETRI NET DEFINES ODES ds3 dt = + v1 + v4 s1 Raf-1Star s2 RKIP k1 k2 s9 ERK-PP s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 MEK-PP_ERK s4 s11 RKIP-P_RP Raf-1Star_RKIP_ERK-PP k6 k7 k5 k9 k10 s7 s5 s6 s10 MEK-PP ERK RKIP-P RP
16 CONTINUOUS PETRI NET DEFINES ODES ds3 dt = + v1 + v4 - v2 - v3 ERK-PP s1 Raf-1Star k1 s2 k2 RKIP s9 s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 MEK-PP_ERK s4 s11 RKIP-P_RP Raf-1Star_RKIP_ERK-PP k6 k7 k5 k9 k10 s7 s5 s6 s10 MEK-PP ERK RKIP-P RP
17 CONTINUOUS PETRI NET DEFINES ODES ds3 dt = + k1 * s1 * s2 + v4 - v2 - v3 ERK-PP s1 Raf-1Star k1 s2 k2 RKIP s9 s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 MEK-PP_ERK s4 s11 RKIP-P_RP Raf-1Star_RKIP_ERK-PP k6 k7 k5 k9 k10 s7 s5 s6 s10 MEK-PP ERK RKIP-P RP
18 CONTINUOUS PETRI NET DEFINES ODES ds3 dt = + k1 * s1 * s2 + k4 * s4 s1 Raf-1Star s2 RKIP - k2 * s3 - k3 * s3 * s9 k1 k2 ERK-PP s9 s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 MEK-PP_ERK s4 s11 RKIP-P_RP Raf-1Star_RKIP_ERK-PP k6 k7 k5 k9 k10 s7 s5 s6 s10 MEK-PP ERK RKIP-P RP
19 GENERALIZED HYBRID PETRI NETS (GHPN) GHPN = XSPN + CPN [H [HERAJY, H, HEINER H 2010] XSPN - Extended Generalized Stochastic Petri Nets -> discrete places -> discrete transitions: stochastic, immediate, deterministically delayed, scheduled -> special arcs: read, inhibitor, equal, reset CPN - Continuous Petri Nets -> continuous places -> continuous transitions r5 r1 -> special arcs: read, inhibitor Struct hybrid simulation engine -> static partitioning r4 gen r3 1e-05 temp -> dynamic partitioning r6 r2
20 BIO PETRI NETS - SOME EXAMPLES
21 EX1 - Glycolysis and Pentose Phosphate Pathway Ru5P 4 Xu5P [Reddy 1993] 1 2 GSSG 2 NADPH R5P 6 S7P GAP 7 E4P F6P 8 4 GSH 2 NADP + -> INTERPRETATION? Gluc ATP 9 ADP G6P 10 F6P ATP FBP ADP DHAP GAP 15 NAD + + Pi NAD + NADH ATP ADP ATP ADP NADH Lac 20 Pyr 19 PEP 18 2PG 17 3PG 16 1,3-BPG
22 EX1 - Glycolysis and Pentose Phosphate Pathway GSSG NADPH Ru5P Xu5P S7P E4P GAP F6P R5P ATP ADP [Reddy 1993] [Heiner 1998]... [Koch, Heiner 2010] GSH NADP+ Gluc 9 ATP NAD+ NADH 10 G6P ADP ATP ADP F6P ATP 11 ADP FBP 12 DHAP ATP GAP ADP 15 Pi NAD+ Pi NADH Lac 20 Pyr 19 PEP 18 2PG 17 3PG 16 1,3-BPG
23 EX1 - Glycolysis and Pentose Phosphate Pathway GSSG NADPH Ru5P Xu5P S7P E4P GAP F6P R5P ATP ADP [Reddy 1993] [Heiner 1998]... [Koch, Heiner 2010] GSH NADP+ Gluc 9 ATP NAD+ NADH 10 G6P ADP ATP ADP F6P ATP 11 ADP FBP 12 DHAP ATP GAP ADP 15 Pi NAD+ Pi NADH Lac 20 Pyr 19 PEP 18 2PG 17 3PG 16 1,3-BPG
24 EX2 - APOPTOSIS IN MAMMALIAN CELLS Fas-Ligand Apoptotic_Stimuli FADD Procaspase-8 Bax_Bad_Bim s8 s7 Bcl-2_Bcl-xL Apaf-1 s1 Caspase-8 BidC-Terminal Bid s5 s9 s6mitochondrion CytochromeC s10 datp/atp s2 Procaspase-3 (m20) Caspase-3 s3 s13 Caspase-9 Procaspase-9 s11 DFF DFF40-Oligomer CleavedDFF45 s12 (m22) DNA s4 DNA-Fragment [GON 2003] [HEINER, KOCH, WILL 2004]
25 EX2 - APOPTOSIS IN MAMMALIAN CELLS Fas-Ligand Apoptotic_Stimuli FADD Procaspase-8 Bax_Bad_Bim s8 s7 Bcl-2_Bcl-xL Apaf-1 s1 Caspase-8 BidC-Terminal Bid s5 s9 s6mitochondrion CytochromeC s10 datp/atp s2 Procaspase-3 (m20) Caspase-3 s3 s13 Caspase-9 Procaspase-9 s11 DFF DFF40-Oligomer CleavedDFF45 s12 (m22) DNA s4 DNA-Fragment [GON 2003] [HEINER, KOCH, WILL 2004]
26 EX3 - RKIP SIGNALLING PATHWAY one pathway Ras Mitogens Growth factors receptor Receptor P P Raf P kinase P P MEK P P ERK cytoplasmic substrates Elk SAP Gene
27 EX3 - RKIP SIGNALLING PATHWAY [Cho et al. 2003]
28 EX3 - RKIP SIGNALLING PATHWAY, PETRI NET mi -> si Raf-1Star s1 s2 RKIP k1 k2 s9 ERK-PP s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 k6 MEK-PP_ERK k7 s4 Raf-1Star_RKIP_ERK-PP k5 s11 k9 RKIP-P_RP k10 [HEINER, GILBERT 2006] s7 MEK-PP s5 ERK s6 RKIP-P s10 RP [HEINER, DONALDSON, GILBERT 2010]
29 EX3 - RKIP SIGNALLING PATHWAY, HIERARCHICAL PETRI NET Raf-1Star s1 s2 RKIP k1_k2 s9 ERK-PP s3 Raf-1Star_RKIP k8 k3_k4 k11 s8 MEK-PP_ERK k6_k7 s4 Raf-1Star_RKIP_ERK-PP k5 s11 RKIP-P_RP k9_k10 [HEINER, GILBERT 2006] s7 MEK-PP s5 ERK s6 RKIP-P s10 RP [HEINER, DONALDSON, GILBERT 2010]
30 EX3 - RKIP SIGNALLING PATHWAY, HIERARCHICAL PETRI NET Raf-1Star s1 s2 RKIP k1_k2 s9 ERK-PP s3 Raf-1Star_RKIP k8 k3_k4 k11 s8 MEK-PP_ERK k6_k7 s4 Raf-1Star_RKIP_ERK-PP k5 s11 RKIP-P_RP k9_k10 [HEINER, GILBERT 2006] s7 MEK-PP s5 ERK s6 RKIP-P s10 RP [HEINER, DONALDSON, GILBERT 2010]
31 EX4 - SIGNALLING CASCADE RasGTP Raf RafP Phosphatase1 MEK MEKP MEKPP Phosphatase2 ERK ERKP ERKPP Phosphatase3
32 EX4 - SIGNALLING CASCADE RasGTP Raf_RasGTP Raf k1/k2 k6 k3 RafP k4/k5 RafP_Phase1 Phase1 MEK_RafP MEKP_RafP k7/k8 k9 k10/k11 k12 MEK MEKP MEKPP k18 k16/k17 k15 k13/k14 MEKP_Phase2 MEKPP_Phase2 Phase2 ERK ERK_MEKPP k21 k19/k20 ERKP ERKP_MEKPP k24 k22/k23 ERKPP [GILBERT, HEINER, LEHRACK 2007] k28/k29 k25/k26 k30 k27 ERKP_Phase3 ERKPP_Phase3 Phase3 [HEINER, GILBERT, DONALDSON 2008]
33 EX4 - SIGNALLING CASCADE RasGTP Raf Raf_RasGTP k1/k2 k6 RafP_Phase1 Phase1 k3 MEK RafP k4/k5 MEK_RafP MEKP_Phase2 MEKP_RafP CPI -> BND k7/k8 k18 k9 MEKP k16/k17 Phase2 k10/k11 k15 k12 MEKPP k13/k14 MEKPP_Phase2 ERK ERK_MEKPP k19/k20 k30 k21 ERKP ERKP_Phase3 Phase3 ERKP_MEKPP k22/k23 k27 k24 ERKPP ERKPP_Phase3 [GILBERT, HEINER, LEHRACK 2007] NOT ES & STP -> NO DST k28/k29 k25/k26 [HEINER, GILBERT, DONALDSON 2008]
34 EX5 - LAC OPERON InhibitorRnaDegradation InhibitorDegradation IOp InhibitorTranscription InhibitorTranslation I 50 Idna Irna InhibitorBinding/Dissociation LactoseInhibitorBinding/Dissociation ILactose RnapBinding/Dissociation Op RnapOp 100 Rnap Transcription LactoseInhibitorDegradation Lactose Intervention Rna RnaDegradation Conversion Translation Z ZDegradation [WILKINSON 2006]
35 EX5 - LAC OPERON reduced net structure while preserving liveness & boundedness A B liveness becomes obvious example of two simple reduction rules t p p t t live p unbounded t not live p bounded
36 ABOUT THE RELATION QUALITATIVE VS CONTINUOUS
37 EX8 - HYPOXIA [YU ET AL ] : ( , 95,;<5=/'!# : ( ,!5,;<5=/' 67 #-7.545,81+ 95,;<5= 9!:$%&30(!"!"#$%&'(!"#$%$!"&!"#)1!2$%&3'(!"#)8!%&34(!"#6!)9!:%&3;( : ( ,!5,;<5=/(!"#$%&'() *+,-./ $95.,!"+!",$%$!"'!#(!"" <+,*!&$%$!+ *+,-$%&4(!##$%$!#"!"( "=>?@<?ABC>?D'!"#)*+,-$%&.(!"#)*+,-)1!2$%&3.(!"-!"#6!)*+,-$%&35(!'!")%$!"* 67 "=>?@<?ABC>?D7!)$%$!*!+/$%&5(!#'$%$!&(!,!"#)*+,-)!+/$%&0(!-!"#6!)*+,-)!+/$%&77(!- "=>?@<?ABC>?D3
38 EX8 - HYPOXIA k1 k4 H1F k2 S3 k13 PHD k12 S12 H1F:PHD S13 ARNT S4 Oxygen O2 k14 k22 H1FOH S14 H1FOH:VAL S18 k18 k19 S17 VHL k20 [H [HEINER, SRIRAM 2010] 2010] k3 Oxygen O2 k21 k15 S5 H1F:ARNT k6 k16 S15 k17 H1F:ARNT:PHD HRE k30 S16 H1FOH:ARNT k5 S6 k29 S7 H1F:ARNT:HRE S22 H1FOH:ARNT:HRE
39 EX8 - HYPOXIA (a) 100 (b) k5, k6, k29, k HIF (steady state values) (c) (b) + k (d) (c) + k4, k (e) (d) + k (f) (e) + k Oxygen
40 EX8 - HYPOXIA k2 Oxygen O2 H1FOH:VAL S18 k1 H1F S3 k12 H1F:PHD S13 k14 H1FOH S14 k18 k20 k3 S12 PHD ARNT S4 k22 S17 VHL Oxygen O2 S5 H1F:ARNT k15 S15 k17 H1F:ARNT:PHD S16 H1FOH:ARNT
41 EX8 - HYPOXIA B C k2 k12, k14 k1 A H1F S3 PHD S12 H1FOH S14 E k18, k20 D k3, k15, k17, k22
42 EX8 - HYPOXIA B C k1 A H1F S3 k2 k12, k14 PHD S12 H1FOH S14 k18, k20 E D k3, k15, k17, k22 B C B C k1 A H1F S3 k2 k12, k14 PHD S12 H1FOH S14 k18, k20 E k1 A H1F S3 k2 k12, k14 PHD S12 H1FOH S14 k18, k20 E D D k3, k15, k17, k22 k3, k15, k17, k22
43 ABOUT THE RELATION STOCHASTIC VS CONTINUOUS
44 STOCHASTIC SIMULATION
45 STOCHASTIC SIMULATION
46 STOCHASTIC SIMULATION
47 DETERMINISTIC SIMULATION
48 EX4 - RKIP SIGNALLING PATHWAY, PETRI NET Raf-1Star s1 s2 RKIP k1 k2 s9 ERK-PP s3 Raf-1Star_RKIP k8 k3 k4 k11 s8 k6 MEK-PP_ERK k7 s4 Raf-1Star_RKIP_ERK-PP k5 s11 k9 RKIP-P_RP k10 [HEINER, GILBERT 2006] s7 MEK-PP s5 ERK s6 RKIP-P s10 RP [HEINER, DONALDSON, GILBERT 2010]
49 EX4 - RKIP, REACHABILITY GRAPH (STS) simple algorithm nodes : system states arcs : the (single) firing transition single step firing rule r5 m13 r10 r8 r7 m5 r10 r9 m12 r6 r9 r8 r11 m4 r9 r6 m6 r11 r8 m1 r2 r1 r10 m7 r7 r7 m11 r2 r8 r11 m8 r6 r1 r1 r6 m10 r2 m9 r7 m2 r4 r3 m3
50 EX4 - RKIP, QUANTITATIVE ANALYSIS 13 Good state configurations Cho et al Biochemist Distribution of `bad' steady states as euclidean distances from the `good' final steady state 13 good state configurations the bad ones
51 EX4 - RKIP, QUANTITATIVE ANALYSIS
52 EX4 - RKIP, QUANTITATIVE ANALYSIS
53 EX4 - RKIP, QUANTITATIVE ANALYSIS
54 BUT, TRANSITION SPN -> CPN MAY COME WITH COUNTERINTUITIVE EFFECTS.
55 EX9 - ABSOLUT CONCENTRATION ROBUSTNESS (ACR) ACR: steady state value of variable (place) does not depend on total mass, only on kinetic constants -> [SHINAR, FEINBERG 2010] simple example mass-action kinetics r 1 : A + B -> 2B v 1 (r 1 ) = k 1 AB r 2 : B -> A v 2 (r 2 ) = k 2 B ODEs da/dt = v 2 - v 1 = k 2 B - k 1 AB db/dt = v 1 - v 2 = k 1 AB - k 2 B ACR r 1 2 A r 2 B BAD SIPHON CPI: m 0 (A) + m 0 (B) = total steady state da/dt = k 2 B - k 1 AB = 0 db/dt = k 1 AB - k 2 B = 0 A r 1 2 B -> steady_state(a) = k 2 /k 1 r 2 steady_state(b) = total - k 2 /k 1
56 SUMMARY
57 SUMMARY representation of bio networks by Petri nets -> partial order representation -> better comprehension -> formal semantics -> sound analysis techniques -> unifying view purposes -> animation -> to experience the model -> model validation against consistency criteria -> to increase confidence -> qualitative / quantitative behaviour prediction -> experiment design, new insights step-wise model development -> qualitative model -> discrete Petri nets -> discrete quantitative model -> stochastic Petri nets -> continuous quantitative model -> continuous Petri nets = ODEs, hybrid models -> locality and space -> coloured Petri nets
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