ECMTB, J ULY 2005 PN & Systems Biology P ETRI N ETS AS P ARTIAL O RDER S EMANTICS FOR B IOCHEMICAL N ETWORKS Monika Heiner Ina Koch Brandenburg University of Technology Technical University of Applied Sciences Cottbus Berlin Dep. of CS WG of Bioinformatics monika.heiner@informatik.tu-cottbus.de July 2005
F RAMEWORK PN & Systems Biology bionetworks knowledge qualitative modelling understanding Petri net theory (invariants) animation / qualitative model validation analysis models model qualitative checking behaviour prediction quantitative quantitative parameters modelling understanding RG SLI animation / quantitative model validation LP analysis /simulation models ODEs quantitative behaviour prediction monika.heiner@informatik.tu-cottbus.de July 2005
PN & Systems Biology P ETRI N ETS & P ARTIAL O RDER S EMANTICS monika.heiner@informatik.tu-cottbus.de July 2005
P ETRI N ETS , B ASICS - THE S TRUCTURE PN & Systems Biology atomic actions -> Petri net transitions -> chemical reactions ❑ 2 NAD + + 2 H 2 O -> 2 NADH + 2 H + + O 2 NADH NAD + 2 2 input output 2 H + r1 compounds compounds 2 H 2 O O 2 hyperarcs 2 NAD + 2 NADH 2 H + 2 H 2 O O 2 monika.heiner@informatik.tu-cottbus.de July 2005
P ETRI N ETS , B ASICS - THE S TRUCTURE PN & Systems Biology atomic actions -> Petri net transitions -> chemical reactions ❑ 2 NAD + + 2 H 2 O -> 2 NADH + 2 H + + O 2 NADH NAD + 2 2 input output 2 H + r1 compounds compounds 2 H 2 O pre-conditions post-conditions O 2 local conditions -> Petri net places -> chemical compounds ❑ multiplicities -> Petri net arc weights -> stoichiometric relations ❑ monika.heiner@informatik.tu-cottbus.de July 2005
P ETRI N ETS , B ASICS - THE S TRUCTURE PN & Systems Biology atomic actions -> Petri net transitions -> chemical reactions ❑ 2 NAD + + 2 H 2 O -> 2 NADH + 2 H + + O 2 NADH NAD + 2 2 input output 2 H + r1 compounds compounds 2 H 2 O O 2 local conditions -> Petri net places -> chemical compounds ❑ multiplicities -> Petri net arc weights -> stoichiometric relations ❑ condition’s state -> token(s) in its place -> available amount (e.g. mol) ❑ system state -> marking -> compounds distribution ❑ monika.heiner@informatik.tu-cottbus.de July 2005
P ETRI N ETS , B ASICS - THE B EHAVIOUR PN & Systems Biology atomic actions -> Petri net transitions -> chemical reactions ❑ 2 NAD + + 2 H 2 O -> 2 NADH + 2 H + + O 2 NADH NAD + 2 2 input output 2 H + r1 compounds compounds 2 H 2 O O 2 monika.heiner@informatik.tu-cottbus.de July 2005
P ETRI N ETS , B ASICS - THE B EHAVIOUR PN & Systems Biology atomic actions -> Petri net transitions -> chemical reactions ❑ 2 NAD + + 2 H 2 O -> 2 NADH + 2 H + + O 2 NADH NAD + 2 2 input output 2 H + r1 compounds compounds 2 H 2 O O 2 FIRING NADH NAD + 2 2 2 H + r1 2 H 2 O O 2 monika.heiner@informatik.tu-cottbus.de July 2005
P ETRI N ETS , B ASICS - THE B EHAVIOUR PN & Systems Biology atomic actions -> Petri net transitions -> chemical reactions ❑ 2 NAD + + 2 H 2 O -> 2 NADH + 2 H + + O 2 NADH NAD + 2 2 input output 2 H + r1 compounds compounds 2 H 2 O O 2 TOKEN GAME FIRING NADH NAD + 2 2 DYNAMIC BEHAVIOUR 2 H + r1 (substance/signal flow) 2 H 2 O O 2 monika.heiner@informatik.tu-cottbus.de July 2005
P ARTIAL O RDER VERSUS I NTERLEAVING S EMANTICS PN & Systems Biology order between r1 - r2 and r1 - r3 ❑ b d -> causality x < y r2 -> dependency a r1 no order between r2 , r3 ❑ c e r3 -> concurrency x || y -> independency partial order run ❑ r2 r1 r3 monika.heiner@informatik.tu-cottbus.de July 2005
P ARTIAL O RDER VERSUS I NTERLEAVING S EMANTICS PN & Systems Biology order between r1 - r2 and r1 - r3 ❑ b d -> causality x < y r2 -> dependency a r1 no order between r2 , r3 ❑ c e r3 -> concurrency x || y -> independency partial order run ❑ r2 r1 r3 -> PARTIAL ORDER SEMANTICS “true concurrency semantics” all partially ordered runs monika.heiner@informatik.tu-cottbus.de July 2005
P ARTIAL O RDER VERSUS I NTERLEAVING S EMANTICS PN & Systems Biology order between r1 - r2 and r1 - r3 ❑ b d -> causality x < y r2 -> dependency a r1 no order between r2 , r3 ❑ c e r3 -> concurrency x || y -> independency possible interleaving runs ❑ partial order run ❑ -> r1 - r2 - r3 r2 r1 -> r1 - r3 - r2 r3 totally ordered runs ❑ -> PARTIAL ORDER SEMANTICS “true concurrency semantics” all partially ordered runs monika.heiner@informatik.tu-cottbus.de July 2005
P ARTIAL O RDER VERSUS I NTERLEAVING S EMANTICS PN & Systems Biology order between r1 - r2 and r1 - r3 ❑ b d -> causality x < y [ x-y ] r2 -> dependency a r1 no order between r2 , r3 ❑ c e r3 -> concurrency x || y -> independency possible interleaving runs ❑ partial order run ❑ -> r1 - r2 - r3 r2 r1 -> r1 - r3 - r2 r3 totally ordered runs ❑ -> PARTIAL ORDER SEMANTICS “true concurrency semantics” -> INTERLEAVING SEMANTICS all partially ordered runs all totally ordered runs monika.heiner@informatik.tu-cottbus.de July 2005
B IOCHEMICAL P ETRI NETS , S UMMARY PN & Systems Biology biochemical networks ❑ -> networks of (abstract) chemical reactions biochemically interpreted Petri net ❑ -> partial order sequences of chemical reactions (= elementary actions) transforming input into output compounds / signals [ respecting the given stoichiometric relations, if any ] -> set of all pathways from the input to the output compounds / signals [ respecting the stoichiometric relations, if any ] pathway ❑ -> self-contained partial order sequence of elementary (re-) actions monika.heiner@informatik.tu-cottbus.de July 2005
PN & Systems Biology T HE R UNNING E XAMPLE monika.heiner@informatik.tu-cottbus.de July 2005
T HE RKIP P ATHWAY PN & Systems Biology [Cho et al., CMSB 2003] monika.heiner@informatik.tu-cottbus.de July 2005
T HE RKIP P ATHWAY , P ETRI N ET PN & Systems Biology Raf-1Star RKIP m1 m2 k2 k1 ERK-PP m9 m3 Raf-1Star_RKIP k4 k3 k8 k11 m11 m8 m4 RKIP-P_RP MEK-PP_ERK Raf-1Star_RKIP_ERK-PP k7 k5 k6 k10 k9 m7 m5 m6 m10 ERK RP RKIP-P MEK-PP monika.heiner@informatik.tu-cottbus.de July 2005
T HE RKIP P ATHWAY , H IERARCHICAL P ETRI N ET PN & Systems Biology Raf-1Star RKIP m1 m2 k1_k2 ERK-PP m9 Raf-1Star_RKIP m3 k3_k4 k8 k11 m8 m11 m4 RKIP-P_RP MEK-PP_ERK Raf-1Star_RKIP_ERK-PP k9_k10 k5 k6_k7 m7 m6 m5 m10 MEK-PP ERK RP RKIP-P monika.heiner@informatik.tu-cottbus.de July 2005
T HE RKIP P ATHWAY , H IERARCHICAL P ETRI N ET PN & Systems Biology Raf-1Star RKIP initial marking m1 m2 -> unique -> constructed k1_k2 ERK-PP m9 Raf-1Star_RKIP m3 k3_k4 k8 k11 m8 m11 m4 RKIP-P_RP MEK-PP_ERK Raf-1Star_RKIP_ERK-PP k9_k10 k5 k6_k7 m7 m6 m5 m10 MEK-PP ERK RP RKIP-P monika.heiner@informatik.tu-cottbus.de July 2005
PN & Systems Biology Q UALITATIVE A NALYSES monika.heiner@informatik.tu-cottbus.de July 2005
T- INVARIANTS & P ARTIAL O RDER S EMANTICS PN & Systems Biology Lautenbach, 1973 ❑ T-invariants ❑ -> multisets of transitions , ≠ , ≥ -> integer solutions x of -> Parikh vector Cx = 0 x 0 x 0 minimal T-invariants ❑ -> there is no T-invariant with a smaller support -> sets of transitions -> gcD of all entries is 1 any T-invariant is a non-negative linear combination of minimal ones ❑ -> multiplication with a positive integer ∑ kx aixi = -> addition i -> Division by gcD -> elementary modes [Schuster 1993] monika.heiner@informatik.tu-cottbus.de July 2005
T- INVARIANTS , I NTERPRETATIONS PN & Systems Biology T-invariants = (multi-) sets of transitions ❑ -> zero effect on marking -> reproducing a marking / system state two interpretations ❑ 1. relative transition firing rates of transitions occuring permanently & concurrently -> steady state behaviour 2. partially ordered transition sequence -> behaviour understanding of transitions occuring one after the other -> substance / signal flow a T-invariant defines a subnet -> partial order structure ❑ -> the T-invariant’s transitions (the support), + all their pre- and post-places + the arcs in between -> pre-sets of supports = post-sets of supports monika.heiner@informatik.tu-cottbus.de July 2005
T- INVARIANTS , T HE RKIP P ATHWAY PN & Systems Biology -> non-trivial T-invariant Raf-1Star RKIP + four trivial ones for m1 m2 reversible reactions k1 ERK-PP m9 Raf-1Star_RKIP m3 k3 k8 k11 m8 m11 m4 RKIP-P_RP MEK-PP_ERK Raf-1Star_RKIP_ERK-PP k5 k9 k6 m7 m6 m5 m10 MEK-PP ERK RP RKIP-P monika.heiner@informatik.tu-cottbus.de July 2005
T- INVARIANT ’ S R UN m1 m2 RKIP Raf-1Star k1 partial order structure ❑ m3 Raf-1Star_RKIP T-invariant’s unfolding ❑ to describe its behaviour monika.heiner@informatik.tu-cottbus.de
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