FROM PETRI NETS TUTORIAL, PART II A STRUCTURED APPROACH... TO DIFFERENTIAL EQUATIONS ISMB, JULY 2008

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1 ISMB, JULY 008 A STRUCTURED APPROACH... TUTORIAL, PART II FROM PETRI NETS TO DIFFERENTIAL EQUATIONS Monika Heiner Brandenburg University of Technology Cottbus, Dept. of CS

2 FRAMEWORK: SYSTEMS BIOLOGY MODELLING = FORMAL KNOWLEDGE REPRESENTATION wetlab experiments observed behaviour formalizing understanding natural biosystem model model validation wetlab experiments predicted behaviour model-based experiment design MODEL VALIDATION = CONFIDENCE INCREASE

3 WHAT KIND OF MODEL SHOULD BE USED?

4 NETWORK REPRESENTATIONS, EX1 -> FORMAL SEMANTICS? Rap1 camp GEF camp B-Raf MEK1, ERK1, AdCyc ATP camp α camp AMP PDE nucleus transcription factors camp PKA Receptor heterotrimeric G-protein e.g. 7-TMR γ α β γ β MKP tyrosine kinase shc camp grb PKA ERK1, SOS MEK Ras Raf-1 Akt cell membrane Ras PI-3 K cytosol Rac PAK

5 NETWORK REPRESENTATIONS, EX -> READABILITY?

6 NETWORK REPRESENTATIONS informal cartoon-like representations -> readability -> fault avoidance formal = mathematical representations -> analysability WHY NOT BOTH? & EXECUTABILITY

7 PETRI NETS - AN INFORMAL CRASH COURSE

8 PETRI NETS, BASICS NAD + + H O -> NADH + H + + O NAD + H O r1 NADH H + O hyper-arcs NAD + NADH H O H + O

9 PETRI NETS, BASICS - THE STRUCTURE atomic actions -> transitions -> chemical reactions NAD + + H O -> NADH + H + + O input compounds NAD + H O r1 NADH H + O output compounds local conditions -> places -> chemical compounds multiplicities -> arc weights -> stoichiometric relations condition s state -> token(s) -> available amount (e.g. mol) system state -> marking -> compounds distribution PN = (P, T, F, m 0 ), F: (P x T) U (T x P) -> N 0, m 0 : P -> N 0

10 PETRI NETS, BASICS - THE FIRING RULE an action can happen, if -> prerequisite -> all preconditions are fulfilled (corresponding to the arc weights) if an action happens, then -> firing behaviour -> tokens are removed from all preconditions (corresponding to the arc weights), and -> tokens are added to all postconditions (corresponding to the arc weights) action happens (firing of a transition) -> model assumptions -> atomic -> time-less

11 PETRI NETS, BASICS - THE BEHAVIOUR atomic actions -> transitions -> chemical reactions NAD + + H O -> NADH + H + + O input compounds NAD + H O r1 NADH H + O output compounds FIRING TOKEN GAME NAD + H O r1 NADH H + O DYNAMIC BEHAVIOUR (substance/signal flow) STATE SPACE

12 TYPICAL BASIC STRUCTURES I B A A --> B + C A A + B --> C C r1 r C B A --> B, A --> C A r3 r4 B C A --> C, B --> C A B r5 r6 C

13 TYPICAL BASIC STRUCTURES II A --> B A <--> B r1 A r1 B A B A r1, r B r E A --> B E A <--> B E r1 E E A r1 B A B A r1, r B r

14 TYPICAL BASIC STRUCTURES III E A <--> A E --> B enzymatic reaction, mass-action approach 1 r1 E A A E r3 B r E E A r1, r A E r3 B A MA1 B

15 TYPICAL BASIC STRUCTURES IV metabolic networks -> substance flows e1 e e3 r1 r r3 signal transduction networks -> signal flows r1 r r3

16 TYPICAL BASIC STRUCTURES IV metabolic networks -> substance flows e1 e e3 r1 r r3 INPUT OUTPUT COMPOUND COMPOUND signal transduction networks -> signal flows INPUT SIGNAL r1 r OUTPUT SIGNAL r3 -> OPEN / CLOSED SYSTEMS

17 PETRI NET ELEMENTS, INTERPRETATIONS METABOLIC NETWORKS SIGNAL TRANSDUCTION NETWORKS GENE REGULATORY NETWORKS transitions -> (reversible, stoichiometric) chemical reactions, -> enzyme-catalysed conversions of metabolites, proteins,... -> complexations / decomplexations, de- / phosphorylations,... places -> (primary, secondary) chemical compounds, -> (various states of) proteins, protein complex, genes,... tokens -> molecules, moles, -> concentration levels, gene expression levels,... (e.g., high / low = present / not present, or any finite number)

18 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

19 BIO PETRI NETS - SOME EXAMPLES

20 EX1 - Glycolysis and Pentose Phosphate Pathway 4 Ru5P Xu5P [Reddy 1993] GSSG NADPH GSH NADP R5P S7P 7 GAP E4P 8 F6P Gluc 9 ATP ADP 10 G6P F6P 11 1 FBP ATP ADP DHAP GAP NAD + + Pi NAD + NADH ATP ADP ATP ADP NADH Lac Pyr PEP PG 3PG 16 1,3-BPG

21 EX1 - Glycolysis and Pentose Phosphate Pathway Xu5P 4 Ru5P GSSG NADPH S7P E4P ATP [Reddy 1993] R5P GAP F6P ADP GSH NADP+ Gluc F6P FBP GAP Pi G6P NAD ATP ADP ATP ADP DHAP Pi 15 NAD NAD+ NADH ATP ADP ATP ADP ,3 BPG Lac Pyr PEP PG 3PG

22 EX - Carbon Metabolism in Potato Tuber gesuc esuc SucTrans SPP Inv Suc SuSy 8 UDP Pi SPS Glc S6P UDPglc Frc HK FK ATP ATP 9 ADP 9 ADP UDP PGI PP F6P UGPase 8 9 Pi 8 9 ADP Glyc(b) ATP 9 ATP G6P 9 ATPcons(b) [KOCH; JUNKER; HEINER 005] StaSy(b) starch PGM ATP 9 ADP PP PPase G1P UTP NDPkin ATP 8 Pi AMP AdK ADP 8 Pi 9 ADP rstarch

23 EX3: APOPTOSIS IN MAMMALIAN CELLS Fas Ligand Apoptotic_Stimuli s7 Bax_Bad_Bim FADD Procaspase 8 Bcl _Bcl xl Apaf 1 s8 s1 Caspase 8 Bid BidC Terminal CytochromeC datp/atp s6 s9 Mitochondrion s10 s5 s Procaspase 3 (m0) Caspase 9 Caspase 3 s13 s11 s3 Procaspase 9 DFF DFF40 Oligomer CleavedDFF45 (m) s1 s4 [GON 003] DNA DNA Fragment [HEINER; KOCH; WILL 004]

24 EX4 - SWITCH CYCLE HALOBACTERIUM SALINARUM [Marwan; Oesterhelt 1999]

25 _p10_ no_hv487 CheY P on off _p7_ EX4 - SWITCH CYCLE HALOBACTERIUM SALINARUM CheB P CheB _p37 p36_ CheB P _p0 p1_ CheB SR_I_373 SR_I_373Me SR_I_587Me SR_I_587 SR_II360_50Me _p33 p35 p34 p3 p4 p p3 p5_ SR_II480 SR_II480Me SR_II360_50 on _p8 p7_ off _p9_ no_hv580 hv580 hv487 _p30_ CheY P CheA _p7_ on _p6 p3 p6 p4 p5_ off CheR no_hv373 hv373 SR_I_510Me SR_I_510 _p9 p8_ CheA P CheY _p31_ CheR k1_cw k_cw Tstop_cw _p p1 p0 p19_ Rcw Ccw k0_cw 44 Acw Scw 44 _p14_ 44 co_cheyp co_cheyp Conf1 _p14_ 44 _p1_ 44 ka4 ka3 ka ka1 kd1 kd kd3 kd4 t1 t11 t1 t _p13 p11_ CheYPbound Conf co_cheyp _p14_ k0_ccw Sccw Rccw Cccw Accw 44 _p15 p16 p17 p18_ CheA _p30 p9_ CheA P

26 EX5 - SIGNALLING CASCADE RasGTP Raf RafP Phosphatase1 MEK MEKP MEKPP Phosphatase ERK ERKP ERKPP Phosphatase3

27 EX5 - SIGNALLING CASCADE RasGTP Raf_RasGTP k1/k k3 Raf RafP k6 k4/k5 RafP_Phase1 MEK_RafP MEKP_RafP Phase1 k7/k8 k9 k1 k10/k11 MEK MEKP MEKPP k16/k17 k13/k14 k18 k15 MEKP_Phase MEKPP_Phase ERK_MEKPP ERKP_MEKPP Phase k1 k4 k19/k0 k/k3 ERK ERKP ERKPP k30 k8/k9 k5/k6 k7 ERKP_Phase3 ERKPP_Phase3 Phase3

28 QUALITATIVE ANALYSES

29 TYPICAL PETRI NET QUESTIONS How many tokens can reside at most in a given place? -> (0, 1, k, oo) -> BOUNDEDNESS How often can a transition fire? -> (0-times, n-times, oo-times) -> LIVENESS How often can a system state be reached? -> never -> UNREACHABLE -> SAFETY PROPERTIES -> n-times -> REPRODUCIBLE -> always reachable again -> REVERSIBLE (HOME STATE) -> reversible initial state -> REVERSIBILITY Are there behaviourally invariant subnet structures? -> token conservation -> P - INVARIANTS -> token distribution reproduction -> T - INVARIANTS... and many more -> temporal logics (CTL, LTL)

30 ANALYSIS TECHNIQUES static analyses -> no state space construction -> structural properties (graph theory) -> P / T - invariants (linear algebra) dynamic analyses -> total / partial state space construction -> analysis of general behavioural system properties, i.e. boundedness, liveness, reversibility -> model checking of special behavioural system properties, e.g. reachability of a given (sub-) system state (with constraints), reproducability of a given (sub-) system state (with constraints) => expressed in temporal logics (CTL / LTL), as very flexible & powerful query language

31 BIONETWORKS, VALIDATION validation criterion 1 -> all expected structural properties hold -> all expected general behavioural properties hold validation criterion -> initial marking construction -> CPI (if closed model) -> no minimal P-invariant without biological interpretation validation criterion 3 -> CTI -> no minimal T-invariant without biological interpretation -> no known biological behaviour without corresponding T-invariant validation criterion 4 -> all expected special behavioural properties hold -> temporal-logic properties -> TRUE

32 NOW WE ARE READY FOR SOPHISTICATED QUANTITATIVE ANALYSES!

33 QUANTITATIVE ANALYSIS quantitative model = qualitative model + quantitative parameters -> known or estimated quantitative parameters typical quantitative parameters of bionetworks -> compound concentrations -> real numbers -> reaction rates / fluxes -> concentration-dependent continuous Petri nets = ODEs v1 = k1*a*e E k1 continuous nodes! da / dt = -v1 + v da E / dt = v1 - v - v3 A A E k3 B k v3 = k3*a E v = k*a E db / dt = v3 de / dt = -v1 + v + v3

34 THE RKIP PATHWAY, CONTINUOUS PETRI NET dm3 = + k1 * m1 * m dt + k4 * m4 - k * m3 - k3 * m3 * m9 ERK-PP Raf-1Star m1 k1 m k RKIP m9 m3 Raf-1Star_RKIP k8 k3 k4 k11 m8 MEK-PP_ERK m4 m11 RKIP-P_RP Raf-1Star_RKIP_ERK-PP k6 k7 k5 k9 k10 m7 MEK-PP m5 m6 m10 ERK RKIP-P RP

35 THE QUALITATIVE MODEL BECOMES THE STRUCTURED DESCRIPTION OF THE QUANTITATIVE MODEL!