P ETRI N ETS AS P ARTIAL O RDER S EMANTICS FOR B IOCHEMICAL N ETWORKS - - PowerPoint PPT Presentation

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P ETRI N ETS AS P ARTIAL O RDER S EMANTICS FOR B IOCHEMICAL N ETWORKS - - PowerPoint PPT Presentation

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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

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

ECMTB, JULY 2005

PETRI NETS AS PARTIAL ORDER SEMANTICS

FOR BIOCHEMICAL NETWORKS

Monika Heiner Brandenburg University of Technology Cottbus

  • Dep. of CS

Ina Koch Technical University of Applied Sciences Berlin WG of Bioinformatics

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

FRAMEWORK

qualitative models bionetworks knowledge qualitative modelling quantitative modelling quantitative models quantitative parameters animation / analysis animation / analysis /simulation understanding model validation qualitative behaviour prediction understanding model validation quantitative behaviour prediction (invariants) model checking Petri net theory

ODEs

LP SLI RG

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PETRI NETS & PARTIAL ORDER SEMANTICS

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PETRI NETS, BASICS - THE STRUCTURE

❑ atomic actions

  • > Petri net transitions
  • > chemical reactions

input compounds

  • utput

compounds

2 2 2 2 r1 O2 H+ NADH H2O NAD+

2 NAD+ + 2 H2O -> 2 NADH + 2 H+ + O2

O2 H+ NADH H2O NAD+

hyperarcs

2 2 2 2

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PETRI NETS, BASICS - THE STRUCTURE

❑ atomic actions

  • > Petri net transitions
  • > chemical reactions

❑ local conditions

  • > Petri net places
  • > chemical compounds

❑ multiplicities

  • > Petri net arc weights
  • > stoichiometric relations

input compounds

  • utput

compounds

2 2 2 2 r1 O2 H+ NADH H2O NAD+

2 NAD+ + 2 H2O -> 2 NADH + 2 H+ + O2 pre-conditions post-conditions

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PETRI NETS, BASICS - THE STRUCTURE

❑ atomic actions

  • > Petri net transitions
  • > chemical reactions

❑ 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

input compounds

  • utput

compounds

2 2 2 2 r1 O2 H+ NADH H2O NAD+

2 NAD+ + 2 H2O -> 2 NADH + 2 H+ + O2

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PETRI NETS, BASICS - THE BEHAVIOUR

❑ atomic actions

  • > Petri net transitions
  • > chemical reactions

input compounds

  • utput

compounds

2 2 2 2 r1 O2 H+ NADH H2O NAD+

2 NAD+ + 2 H2O -> 2 NADH + 2 H+ + O2

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PETRI NETS, BASICS - THE BEHAVIOUR

❑ atomic actions

  • > Petri net transitions
  • > chemical reactions

input compounds

  • utput

compounds

2 2 2 2 r1 O2 H+ NADH H2O NAD+

2 NAD+ + 2 H2O -> 2 NADH + 2 H+ + O2

2 2 2 2 r1 O2 H+ NADH H2O NAD+

FIRING

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PETRI NETS, BASICS - THE BEHAVIOUR

❑ atomic actions

  • > Petri net transitions
  • > chemical reactions

input compounds

  • utput

compounds

2 2 2 2 r1 O2 H+ NADH H2O NAD+

2 NAD+ + 2 H2O -> 2 NADH + 2 H+ + O2

2 2 2 2 r1 O2 H+ NADH H2O NAD+

FIRING TOKEN GAME DYNAMIC BEHAVIOUR

(substance/signal flow)

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PARTIAL ORDER VERSUS INTERLEAVING SEMANTICS

r3 r2 r1 e d c b a

❑

  • rder between r1 - r2 and r1 - r3
  • > causality

x < y

  • > dependency

❑ no order between r2 , r3

  • > concurrency

x || y

  • > independency

❑ partial order run r2 r3 r1

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PARTIAL ORDER VERSUS INTERLEAVING SEMANTICS

r3 r2 r1 e d c b a

❑

  • rder between r1 - r2 and r1 - r3
  • > causality

x < y

  • > dependency

❑ no order between r2 , r3

  • > concurrency

x || y

  • > independency

❑ partial order run

  • > PARTIAL ORDER SEMANTICS

“true concurrency semantics” all partially ordered runs r2 r3 r1

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PARTIAL ORDER VERSUS INTERLEAVING SEMANTICS

❑ possible interleaving runs

  • > r1 - r2 - r3
  • > r1 - r3 - r2

❑ totally ordered runs

r3 r2 r1 e d c b a

❑

  • rder between r1 - r2 and r1 - r3
  • > causality

x < y

  • > dependency

❑ no order between r2 , r3

  • > concurrency

x || y

  • > independency

❑ partial order run

  • > PARTIAL ORDER SEMANTICS

“true concurrency semantics” all partially ordered runs r2 r3 r1

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PARTIAL ORDER VERSUS INTERLEAVING SEMANTICS

❑ possible interleaving runs

  • > r1 - r2 - r3
  • > r1 - r3 - r2

❑ totally ordered runs

  • > INTERLEAVING SEMANTICS

all totally ordered runs

r3 r2 r1 e d c b a

❑

  • rder between r1 - r2 and r1 - r3
  • > causality

x < y [ x-y ]

  • > dependency

❑ no order between r2 , r3

  • > concurrency

x || y

  • > independency

❑ partial order run

  • > PARTIAL ORDER SEMANTICS

“true concurrency semantics” all partially ordered runs r2 r3 r1

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

BIOCHEMICAL PETRI NETS, SUMMARY

❑ 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
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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

THE RUNNING EXAMPLE

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

THE RKIP PATHWAY [Cho et al., CMSB 2003]

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

THE RKIP PATHWAY, PETRI NET

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

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

THE RKIP PATHWAY, HIERARCHICAL PETRI NET

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 MEK-PP_ERK m8 ERK-PP m9 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9_k10 k6_k7 k3_k4 k1_k2

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

THE RKIP PATHWAY, HIERARCHICAL PETRI NET

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 MEK-PP_ERK m8 ERK-PP m9 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9_k10 k6_k7 k3_k4 k1_k2

initial marking

  • > unique
  • > constructed
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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

QUALITATIVE ANALYSES

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

T-INVARIANTS & PARTIAL ORDER SEMANTICS

❑ Lautenbach, 1973 ❑ T-invariants

  • > multisets of transitions
  • > integer solutions x of
  • > Parikh vector

❑ 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
  • > addition
  • > Division by gcD
  • >

elementary modes [Schuster 1993] Cx 0 x 0 x ≥ , ≠ , = kx aixi i

∑

=

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

T-INVARIANTS, INTERPRETATIONS

❑ T-invariants = (multi-) sets of transitions

  • > zero effect on marking
  • > reproducing a marking / system state

❑ two interpretations

  • 1. relative transition firing rates
  • f transitions occuring permanently & concurrently
  • > steady state behaviour
  • 2. partially ordered transition sequence
  • > behaviour understanding
  • f 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
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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

T-INVARIANTS, THE RKIP PATHWAY

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 MEK-PP_ERK m8 ERK-PP m9 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1

  • > non-trivial T-invariant

+ four trivial ones for reversible reactions

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monika.heiner@informatik.tu-cottbus.de

T-INVARIANT’S RUN

❑ partial order structure ❑ T-invariant’s unfolding to describe its behaviour

Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k1

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monika.heiner@informatik.tu-cottbus.de

T-INVARIANT’S RUN

❑ partial order structure ❑ T-invariant’s unfolding to describe its behaviour

Raf-1Star_RKIP_ERK-PP m4 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k3 k1 ERK_PP m9

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monika.heiner@informatik.tu-cottbus.de

T-INVARIANT’S RUN

❑ partial order structure ❑ T-invariant’s unfolding to describe its behaviour

k5 RKIP-P m6 ERK m5 Raf-1Star_RKIP_ERK-PP m4 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k3 k1 Raf-1Star m1 ERK_PP m9

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monika.heiner@informatik.tu-cottbus.de

T-INVARIANT’S RUN

❑ partial order structure ❑ T-invariant’s unfolding to describe its behaviour

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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monika.heiner@informatik.tu-cottbus.de

T-INVARIANT’S RUN

❑ partial order structure ❑ T-invariant’s unfolding to describe its behaviour ❑ labelled condition / event net

  • > events (boxes)
  • transition occurences
  • > conditions (circles)
  • involved compounds

❑

  • ccurence net
  • > acyclic
  • > no backward branching

conditions

  • > infinite

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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monika.heiner@informatik.tu-cottbus.de

MARKING AND CUTS

❑ marking

  • > interleaving semantics
  • > system state
  • > nodes in reachability graph

❑ cut

  • > partial order semantics
  • > maximal set of

concurrent conditions

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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monika.heiner@informatik.tu-cottbus.de

MARKING AND CUTS

❑ marking

  • > interleaving semantics
  • > system state
  • > nodes in reachability graph

❑ cut

  • > partial order semantics
  • > maximal set of

concurrent conditions ❑ examples

  • > initial marking
  • k11

k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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SLIDE 31

monika.heiner@informatik.tu-cottbus.de

MARKING AND CUTS

❑ marking

  • > interleaving semantics
  • > system state
  • > nodes in reachability graph

❑ cut

  • > partial order semantics
  • > maximal set of

concurrent conditions ❑ examples

  • > initial marking
  • > marking reached by

k1-k3

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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monika.heiner@informatik.tu-cottbus.de

MARKING AND CUTS

❑ marking

  • > interleaving semantics
  • > system state
  • > nodes in reachability graph

❑ cut

  • > partial order semantics
  • > maximal set of

concurrent conditions ❑ examples

  • > initial marking
  • > marking reached by

k1-k3-k5-k9

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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monika.heiner@informatik.tu-cottbus.de

MARKING AND CUTS

❑ marking

  • > interleaving semantics
  • > system state
  • > nodes in reachability graph

❑ cut

  • > partial order semantics
  • > maximal set of

concurrent conditions ❑ examples

  • > initial marking
  • > marking reached by

k1-k3-k5-(k6-k8 || k9-k11)

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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SLIDE 34

monika.heiner@informatik.tu-cottbus.de

MARKING AND CUTS

❑ marking

  • > interleaving semantics
  • > system state
  • > nodes in reachability graph

❑ cut

  • > partial order semantics
  • > maximal set of

concurrent conditions ❑ examples

  • > initial marking
  • > marking reached by

k1-k3-k5-(k6-k8 || k9-k11)

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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monika.heiner@informatik.tu-cottbus.de

PARTIAL ORDER SEMANTICS

❑ all (infinite) partial order runs ❑ any prefix

  • > finite
  • > feasible

❑ complete prefix

  • > markings = cuts

k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK

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monika.heiner@informatik.tu-cottbus.de

PARTIAL ORDER SEMANTICS

❑ all (infinite) partial order runs ❑ any prefix

  • > finite
  • > feasible

❑ complete prefix

  • > markings = cuts

Raf-1Star_RKIP m3 k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK k1

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monika.heiner@informatik.tu-cottbus.de

PARTIAL ORDER SEMANTICS

❑ all (infinite) partial order runs ❑ any prefix

  • > finite
  • > feasible

❑ complete prefix

  • > markings = cuts

❑ new cuts

  • > k1-k3-k5-k9-k11-k1

Raf-1Star_RKIP m3 k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK k1

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SLIDE 38

monika.heiner@informatik.tu-cottbus.de

PARTIAL ORDER SEMANTICS

❑ all (infinite) partial order runs ❑ any prefix

  • > finite
  • > feasible

❑ complete prefix

  • > markings = cuts

❑ new cuts

  • > k1-k3-k5-k9-k11-k1
  • > k1-k3-k5-(k6 || k9-k11-k1)

Raf-1Star_RKIP m3 k11 k8 k5 RP m10 RKIP-P m6 ERK m5 MEK-PP m7 RKIP-P_RP m11 Raf-1Star_RKIP_ERK-PP m4 m8 Raf-1Star_RKIP m3 RKIP m2 Raf-1Star m1 k9 k6 k3 k1 Raf-1Star m1 ERK_PP m9 MEK-PP m9 ERK-PP m7 RP m10 RKIP m2 MEK-PP_ERK k1

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SLIDE 39

monika.heiner@informatik.tu-cottbus.de

PARTIAL ORDER SEMANTICS

❑ infinite partial order run

  • f the essential T-invariant

❑ short-cut notation

k11 k8 k5 k9 k6 k3 k1 k1 k11 k8 k9 k6 k1 k5 k3

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SLIDE 40

monika.heiner@informatik.tu-cottbus.de

PARTIAL ORDER SEMANTICS

❑ infinite partial order run

  • f the essential T-invariant

❑ short-cut notation ❑ DISCLAIMER

  • > behaviour induced by

trivial t-invariants not considered here

k11 k8 k5 k9 k6 k3 k1 k1 k11 k8 k9 k6 k1 k5 k3

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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

PARTIAL ORDER SEMANTICS, ALGORITHMS & TOOLS

❑ 1-bounded nets

  • > complete prefix construction
  • > CTL0 - model checking
  • > very efficient if moderate amount of dynamic conflicts

❑ bounded / unbounded nets

  • > algorithms:

yes

  • > tools:

?

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SLIDE 42

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

SUMMARY

❑ representation of bionetworks by Petri nets

  • > partial order representation
  • > better comprehension
  • > formal semantics
  • > various sound analysis techniques
  • > unifying view

❑ purposes

  • > animation
  • > to experience the model
  • > model validation against consistency criteria
  • > to increase confidence
  • > qualitative / quantitative behaviour prediction
  • > new insights

❑ two-step model development

  • > qualitative models
  • > discrete Petri nets
  • > quantitative models -> continuous Petri nets = ODEs
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PN & Systems Biology monika.heiner@informatik.tu-cottbus.de July 2005

THANKS !