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

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 June 2007 S TRUCTURE OF THE T ALK PN & Systems Biology 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 monika.heiner@informatik.tu-cottbus.de June 2007

PN & Systems Biology T- INVARIANTS , A CRASH COURSE monika.heiner@informatik.tu-cottbus.de June 2007 I NCIDENCE M ATRIX C PN & Systems Biology a representation of the net structure => stoichiometric matrix � P T t1 . . . . . . tj tm p1 C = cij = (pi, tj) = F(tj,pi) - F(pi, tj) = Δ tj(pi) cij pi Δ tj = Δ tj(*) Δ tj . . . pn matrix entry cij : � token change in place pi by firing of transition tj matrix column Δ tj : � vector describing the change of the whole marking by firing of tj side-conditions are neglected � cij = 0 enzyme i x x j enzyme-catalysed a b reaction monika.heiner@informatik.tu-cottbus.de June 2007

T- INVARIANTS , B ASICS PN & Systems Biology Lautenbach, 1973 -> Schuster, 1993 � T-invariant x -> multiset of transitions � , ≠ , ≥ -> integer solution of Cx = 0 x 0 x 0 -> Parikh vector support of a T-invariant x -> supp(x) -> set of transitions � ≠ x i 0 -> 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 ∑ kx = aixi -> addition i -> Division by gcD monika.heiner@informatik.tu-cottbus.de June 2007 T- INVARIANTS , I NTERPRETATIONS PN & Systems Biology T-invariants = (multi-) sets of transitions = Parikh vector � -> zero effect on marking -> reproducing a marking / system state two interpretations � -> behaviour understanding 1. partially ordered transition sequence of transitions occuring one after the other -> substance / signal flow 2. relative transition firing rates -> steady state behaviour of 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 monika.heiner@informatik.tu-cottbus.de June 2007

T- INVARIANTS , E X 1 PN & Systems Biology A ->2 B, 2 A -> 3 C prod_A prod_A 1 2 A A 2 2 1 1 r1 r2 r1 r2 2 3 2 3 B C B C 2 3 cons_C cons_C cons_B cons_B T - INVARIANT 2 T - INVARIANT 1 monika.heiner@informatik.tu-cottbus.de June 2007 T- INVARIANTS , E X 2 PN & Systems Biology -> CTI trivial min. T-invariants (5) g_A A boundary transitions of � auxiliary compounds r1 -> (g_a, r_a), (g_b, r_b), r2 r4 a B (g_c, r_c) r3 reversible reactions � r5 -> (r5, r5_rev), (r8, r8_rev) C D F E b b non-trivial min. T-invariants (7) r6 r7 c c r8 H G covering � a g_a r_a b boundary transitions of a 28 29 c r9 r10 input / output compounds c 2 b 29 r_b r11 g_b -> i/o-T-invariants d b K c r_c g_c inner cycles r_K � monika.heiner@informatik.tu-cottbus.de June 2007

T- INVARIANTS , E X 2 PN & Systems Biology i/o-T-invariant, example � g_A 12 | 0.r1 : 1 A | 1.r2 : 1, r1 | 3.r8_rev : 1, r2 | 4.r6 : 1, r4 a B | 5.r7 : 1, r3 | 9.r9 : 2, r5 | 12.r11 : 2, D C F E | 13.g_A : 1, b b r6 r7 | 14.r_K : 2, c c r8 | 15.g_b : 4, G H 4x a | 18.r_c : 4, g_a r_a b a 29 28 | 20.r_a : 4 c 2x r10 r9 c 2 b 4x 29 r_b g_b r11 d b 2x sum equation � K 4x c g_c r_c A + 4b -> 2K +4a + 4c r_K 2x monika.heiner@informatik.tu-cottbus.de June 2007 PN & Systems Biology M ODULARIZATION BY T- INVARIANTS monika.heiner@informatik.tu-cottbus.de June 2007

A BSTRACT D EPENDENT T RANSITION S ETS (ADT- SETS ) PN & Systems Biology 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 monika.heiner@informatik.tu-cottbus.de June 2007 ADT- SETS , F ORMAL D EFINITION PN & Systems Biology 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., ∀ ( ∈ ) , ∀ ( , ∈ ) ( ) i ( ) ( ) j ( ) x X i j T : supp x = supp x Equally, we can define the following. ⊆ A transition set is called an ADT-set, if A T � ∀ ( ∈ ) ⊆ ( ) ∨ ∩ ( ) ∅ x X : A supp x A supp x = monika.heiner@informatik.tu-cottbus.de June 2007

ADT- SETS , I NTERPRETATION PN & Systems Biology 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 monika.heiner@informatik.tu-cottbus.de June 2007 ADT- SETS , E X PN & Systems Biology A ->2 B, 2 A -> 3 C prod_A prod_A 1 1 A A 2 2 1 1 r1 r1 r2 r2 2 3 2 3 B C B C 2 3 cons_C cons_C cons_B cons_B T - INVARIANT 2 T - INVARIANT 1 monika.heiner@informatik.tu-cottbus.de June 2007

PN & Systems Biology B IO P ETRI NETS , S OME E XAMPLES monika.heiner@informatik.tu-cottbus.de June 2007 E X 1 - Glycolysis and Pentose Phosphate Pathway PN & Systems Biology [Reddy 1993] 4 Ru5P Xu5P 5 S7P E4P 6 7 8 2 NADPH 2 GSSG GAP F6P R5P 2 3 1 2 NADP + 4 GSH 9 10 11 12 Gluc G6P F6P FBP GAP 13 14 ATP ADP ATP ADP NAD + DHAP + Pi 15 NAD + NADH NADH ATP ADP ATP ADP 20 19 18 17 16 Lac Pyr 2PG 3PG PEP 1,3-BPG monika.heiner@informatik.tu-cottbus.de June 2007

E X 1 - Glycolysis and Pentose Phosphate Pathway PN & Systems Biology [Reddy 1993] Xu5P 4 E4P S7P Ru5P 6 7 8 ATP GSSG NADPH GAP F6P 5 R5P 2 1 2 3 ADP 2 2 2 GSH NADP+ Pi F6P Gluc FBP GAP 12 9 10 11 13 G6P 14 NAD+ ATP ATP ADP ADP DHAP Pi 15 NADH NAD+ NADH ATP ADP ATP ADP 20 19 18 17 1,3 − BPG 16 Lac Pyr 2PG PEP 3PG monika.heiner@informatik.tu-cottbus.de June 2007 E X 2 - Carbon Metabolism in Potato Tuber PN & Systems Biology geSuc eSuc SucTrans SPP Inv Suc SuSy 28 UDP Pi SPS Glc Frc S6P UDPglc ATP ATP HK FK UDP 29 29 ADP ADP PGI PP F6P UGPase 29 28 28 29 ADP Pi Glyc(b) ATP 29 ATP G6P ATPcons(b) 29 NDPkin G1P ADP UTP [K OCH ; J UNKER ; H EINER 2005] PGM ATP StaSy(b) 28 Pi 29 ADP ATP 2 2 starch 28 29 2 AMP Pi AdK PP PPase ADP ADT-sets build without trivial T-invariants rstarch monika.heiner@informatik.tu-cottbus.de June 2007

E X 3: A POPTOSIS IN M AMMALIAN C ELLS PN & Systems Biology Fas − Ligand Apoptotic_Stimuli s7 Bax_Bad_Bim FADD Procaspase − 8 Apaf − 1 Bcl − 2_Bcl − xL s8 CytochromeC BidC − Terminal s1 Bid dATP/ATP s9 s6 s10 Mitochondrion Caspase − 8 s5 Procaspase − 3 s2 (m20) Caspase − 9 Caspase − 3 s13 s11 Procaspase − 9 s3 DFF CleavedDFF45 DFF40 − Oligomer (m22) s12 s4 DNA DNA − Fragment [H EINER ; K OCH ; WI LL 2004] [GON 2003] monika.heiner@informatik.tu-cottbus.de June 2007 A B IT OF H ISTORY PN & Systems Biology 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 monika.heiner@informatik.tu-cottbus.de June 2007