BY T- INVARIANTS Monika Heiner INRIA / Rocquencourt, projet - - PDF document

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BY T- INVARIANTS Monika Heiner INRIA / Rocquencourt, projet - - PDF document

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MOCA W ORKSHOP , P ARIS , J UNE 2007 PN & Systems Biology M ODULARIZATION BY T- INVARIANTS Monika Heiner INRIA / Rocquencourt, projet CONTRAINTES on sabbatical leave from BTU Cottbus, Dept. of CS monika.heiner@informatik.tu-cottbus.de

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SLIDE 1

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

MOCA WORKSHOP, PARIS, JUNE 2007

MODULARIZATION

BY T-INVARIANTS

Monika Heiner INRIA / Rocquencourt, projet CONTRAINTES

  • n sabbatical leave from BTU Cottbus, Dept. of CS

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

STRUCTURE OF THE TALK

  • What are T-invariants ?
  • > formal definition
  • > interpretations
  • abstract dependent transition sets
  • > define logical building blocks
  • applications
  • > glycolysis
  • > potato tuber
  • > apoptosis
  • summary
  • > pros & cons
  • > outlook - further case studies necessary
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SLIDE 2

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

T- INVARIANTS,

A CRASH COURSE

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

INCIDENCE MATRIX C

  • a representation of the net structure

=> stoichiometric matrix

  • matrix entry cij:

token change in place pi by firing of transition tj

  • matrix column Δtj:

vector describing the change of the whole marking by firing of tj

  • side-conditions are neglected

P T t1 tj tm p1 pi pn

cij

cij = (pi, tj) = F(tj,pi) - F(pi, tj) = Δ tj(pi)

. . . . . . . . .

C =

Δtj Δtj = Δ tj(*)

enzyme b a enzyme-catalysed reaction x x

cij = 0 j i

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SLIDE 3

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

T-INVARIANTS, BASICS

  • Lautenbach, 1973
  • > Schuster, 1993
  • T-invariant x
  • > multiset of transitions
  • > integer solution of
  • > Parikh vector
  • support of a T-invariant x -> supp(x)
  • > set of transitions
  • > set of transitions involved, i.e.
  • minimal T-invariants
  • > there is no T-invariant with a smaller support
  • > 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

Cx 0 x 0 x ≥ , ≠ , = x i

( )

≠ kx aixi i

∑

=

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

T-INVARIANTS, INTERPRETATIONS

  • T-invariants = (multi-) sets of transitions = Parikh vector
  • > zero effect on marking
  • > reproducing a marking / system state
  • two interpretations
  • 1. partially ordered transition sequence
  • > behaviour understanding
  • f transitions occuring one after the other
  • > substance / signal flow
  • 2. relative transition firing rates
  • > steady state behaviour
  • f transitions occuring permanently & concurrently
  • > steady state behaviour
  • a T-invariant defines a connected subnet
  • > 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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SLIDE 4

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

T-INVARIANTS, EX1

2 3 cons_C cons_B r2 r1 prod_A C B A

1 1 2

T-INVARIANT 1 T-INVARIANT 2

A ->2 B, 2 A -> 3 C

2 3 cons_C cons_B r2 r1 prod_A C B A

2 1 3

2 2

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

T-INVARIANTS, EX2 trivial min. T-invariants (5)

  • boundary transitions of

auxiliary compounds

  • > (g_a, r_a), (g_b, r_b),

(g_c, r_c)

  • reversible reactions
  • > (r5, r5_rev), (r8, r8_rev)

non-trivial min. T-invariants (7)

  • covering

boundary transitions of input / output compounds

  • > i/o-T-invariants
  • inner cycles

2 28 29 29 r_a g_a r_c g_c r_b g_b r_K g_A r11 r5 r8 r3 r10 r9 r4 r7 r6 r2 r1 a b c a K b c c b d a F c b c b H G E D C A B

  • > CTI
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SLIDE 5

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

T-INVARIANTS, EX2

  • i/o-T-invariant, example

12 | 0.r1 : 1 | 1.r2 : 1, | 3.r8_rev : 1, | 4.r6 : 1, | 5.r7 : 1, | 9.r9 : 2, | 12.r11 : 2, | 13.g_A : 1, | 14.r_K : 2, | 15.g_b : 4, | 18.r_c : 4, | 20.r_a : 4

  • sum equation

A + 4b -> 2K +4a + 4c

2 28 29 29 r_a g_a r_c g_c r_b g_b r_K g_A r11 r5 r8 r3 r10 r9 r4 r7 r6 r2 r1 a b c a K b c c b d a F c b c b H G E D C A B 2x 2x 2x 4x 4x 4x

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

MODULARIZATION

BY T-INVARIANTS

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SLIDE 6

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

ABSTRACT DEPENDENT TRANSITION SETS (ADT-SETS)

  • dependency relation:

two transitions depend on each other, if they always appear together in all minimal T-invariants

  • equivalence relation in the transition set T, leading to a partition of T
  • > reflexive
  • > symmetric
  • > transitive
  • the equivalence classes A represent ADT-sets
  • > maximal common transition sets (MCT-sets)
  • classification of all transitions based on the T-invariants’ support
  • variations
  • > with / without trivial T-invariants
  • > whole / partial set of T-invariants

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

ADT-SETS, FORMAL DEFINITION

Let X denote the set of all (non-trivial) minimal t-invariants x of a given PN.

  • Two transitions i and j belong to the same ADT-set,

if they participate in exactly the same minimal T-invariants, i.e., : Equally, we can define the following.

  • A transition set

is called an ADT-set, if : x X ∈ ( ) i j , T ∈ ( ) ∀ , ∀ supp x ( ) i ( ) supp x ( ) j ( ) = A T ⊆ x X ∈ ( ) ∀ A supp x ( ) ⊆ A supp x ( ) ∩ ∨ ∅ =

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SLIDE 7

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

ADT-SETS, INTERPRETATION

  • ADT-sets

minimal T-invariants

  • > disjunctive subnets
  • > overlapping subnets
  • > not necessarily connected
  • > connected
  • interpretation
  • > structural decomposition into rather small subnets
  • > smallest biologically meaningful functional units
  • > building blocks

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

ADT-SETS, EX

2 3 cons_C cons_B r2 r1 prod_A C B A

1 1 2

T-INVARIANT 1 T-INVARIANT 2

A ->2 B, 2 A -> 3 C

2 3 cons_C cons_B r2 r1 prod_A C B A

1 1 3

2 2

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SLIDE 8

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

BIO PETRI NETS, SOME EXAMPLES

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

EX1 - Glycolysis and Pentose Phosphate Pathway

Ru5P 4 5 Xu5P R5P 6 S7P GAP 7 E4P F6P 8 GAP 15 NAD+ + Pi G6P F6P 10 ATP ADP FBP 11 12 DHAP 13 14 ATP ADP 9 Gluc 1,3-BPG ATP ADP 16 ATP ADP 19 NAD+ NADH 20 3PG 17 2PG PEP 18 Pyr Lac 2 NADP+ 2 NADPH 4 GSH 2 3 1 2 GSSG NADH

[Reddy 1993]

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SLIDE 9

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

EX1 - Glycolysis and Pentose Phosphate Pathway

[Reddy 1993]

NADP+ NADPH GSSG GSH Ru5P Xu5P R5P S7P GAP GAP E4P F6P F6P Gluc G6P FBP DHAP 1,3−BPG 3PG 2PG PEP Pyr Lac NAD+ NAD+ NADH NADH Pi Pi ATP ATP ATP ATP ATP ADP ADP ADP ADP ADP 15 16 17 18 19 20 13 14 12 11 10 9 2 1 3 4 5 6 7 8 2 2 2 2

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

EX2 - Carbon Metabolism in Potato Tuber

[KOCH; JUNKER; HEINER 2005]

Suc eSuc Glc Frc UDPglc G6P F6P G1P UDP UDP UTP S6P PP PP starch AMP ATP ATP ATP ATP ATP ATP ADP 29 ADP 29 ADP 29 ADP 29 ADP 29 ADP 29 Pi 28 Pi 28 Pi 28 Pi 28 SucTrans Inv HK FK SPP StaSy(b) Glyc(b) ATPcons(b) PPase rstarch geSuc SuSy SPS PGI PGM NDPkin UGPase AdK 29 29 28 2 2 2

ADT-sets build without trivial T-invariants

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SLIDE 10

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

EX3: APOPTOSIS IN MAMMALIAN CELLS

Fas−Ligand FADD Procaspase−8 Caspase−8 Bid BidC−Terminal Bax_Bad_Bim Apoptotic_Stimuli Bcl−2_Bcl−xL CytochromeC dATP/ATP Apaf−1 (m20) (m22) Procaspase−9 Caspase−9 Procaspase−3 Caspase−3 DFF CleavedDFF45 DFF40−Oligomer DNA DNA−Fragment Mitochondrion s1 s7 s9 s8 s5 s10 s11 s2 s12 s13 s3 s4 s6

[HEINER; KOCH; WILL 2004] [GON 2003]

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

A BIT OF HISTORY

  • Andrea Sackmann

Modelling and simulation of signal transduction pathways in Saccharomyces cerevisiae using Petri net theory (in German); Diploma Thesis, Univ. Greifswald & TFH Berlin, April 2005

  • Sackmann, Heiner, Koch

Application of Petri net based analysis techniques to signal transduction pathways; Journal BMC Bioinformatics 2006, 7:482.

  • Katja Winder

A structural characterization of a Petri net’s T-invariant set (in German); Diploma Thesis, BTU Cottbus, March 2006

  • Eva Grafahrend-Belau et al.

Modularization of biochemical networks based on classification of Petri net T-invariants; work in progress

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SLIDE 11

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

SUMMARY

  • PROS
  • > algorithmically defined
  • > various interpretations
  • > static analysis technique (RG not constructed),

works also for unbounded models

  • CONS
  • > computation of all T-invariants required
  • > in the worst-case: exponentially many
  • - especially helpful for analyzing bio Petri nets --
  • related work
  • > modular computation of invariants;

see slides “Pathway Analysis of Biochemical Networks with Petri Nets”, FU Berlin, DFG-Research Center Matheon, May 2006

PN & Systems Biology monika.heiner@informatik.tu-cottbus.de June 2007

THANKS !

HHTP://WWW-DSSZ.INFORMATIK.TU-COTTBUS.DE