MATHEMATICAL MODELLING OF BIOCHEMICAL NETWORKS

Transcription

1 FMP BERLIN, FEBRUARY 007 MATHEMATICAL MODELLING OF BIOCHEMICAL NETWORKS WITH PETRI NETS Monika Heiner Brandenburg University of Technology Cottbus Dept. of CS MODEL- BASED SYSTEM ANALYSIS Problem system system properties CONSTRUCTION technical system verification requirement specification Petrinetz model model properties

2 MODEL- BASED SYSTEM ANALYSIS Problem system system properties UNDERSTANDING biological system validation behaviour prediction known properties unknown properties Petrinetz model model properties WHAT KIND OF MODEL SHOULD BE USED?

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

4 BIO NETWORKS, SOME PROBLEMS knowledge -> PROBLEM 1 -> uncertain -> growing, changing -> distributed over independent data bases, papers, journals,... various, mostly ambiguous representations -> PROBLEM -> verbose descriptions -> diverse graphical representations -> contradictory and / or fuzzy statements network structure -> PROBLEM 3 -> tend to grow fast -> dense, apparently unstructured -> hard to read - models are full of assumptions - BIO NETWORKS, SOME PROBLEMS

5 FRAMEWORK bionetworks knowledge qualitative modelling qualitative models animation / analysis understanding model validation qualitative behaviour prediction Petri net theory (invariants) model checking quantitative modelling quantitative parameters quantitative models animation / analysis /simulation understanding model validation quantitative behaviour prediction Markov chains ODEs PETRI NETS - AN INFORMAL CRASH COURSE

6 PETRI NETS, BASICS NAD + + H O -> + H + + O NAD + r1 H + H O O hyper-arcs NAD + H O H + O PETRI NETS, BASICS - THE STRUCTURE atomic actions -> transitions -> chemical reactions NAD + + H O -> + H + + O input compounds NAD + H O r1 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

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

8 TYPICAL BASIC STRUCTURES metabolic networks -> substance flows e1 e e3 r1 r r3 signal transduction networks -> signal flows r1 r r3 PETRI NET ELEMENTS, INTERPRETATIONS METABOLIC NETWORKS SIGNAL TRANSDUCTION NETWORKS GENE REGULATORY NETWORKS transitions -> (reversible, stoichiometric) chemical reactions, -> enzyme-catalyzed conversions of metabolites, proteins,... -> complexations/decomplexations, de-/phosphorylations,... places -> (primary, seocndary) chemical compounds, -> (various states of) proteins, protein complex, genes,... tokens -> molecules, moles, -> concentration levels, gene expression levels,... (e.g., high/low = present/not present)

9 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 BIO PETRI NETS - SOME EXAMPLES

10 EX1 - Glycolysis and Pentose Phosphate Pathway Ru5P 4 Xu5P [Reddy 1993] 1 GSSG 4 GSH NH 3 N + 5 R5P 6 S7P GAP 7 E4P F6P 8 Gluc 9 G6P 10 F6P 11 1 FBP DHAP GAP NAD + + Pi NAD + Lac 0 Pyr 19 PEP 18 PG 17 3PG 16 1,3-BPG EX1 - Glycolysis and Pentose Phosphate Pathway GSSG NH Ru5P 4 Xu5P S7P E4P [Reddy 1993] R5P GAP F6P GSH N+ Gluc 9 G6P 10 F6P 11 FBP 1 GAP Pi NAD NAD+ DHAP 15 Pi NAD Lac 0 Pyr 19 PEP 18 PG 17 3PG 16 1,3 BPG

11 EX - Carbon Metabolism in Potato Tuber gesuc esuc SucTrans SPP Inv Suc SuSy 8 UDP Pi SPS Glc Frc UDPglc S6P HK FK 9 9 UDP PGI PP F6P UGPase 8 Pi Glyc(b) 9 G6P 9 cons(b) [KOCH; JUNKER; HEINER 005] StaSy(b) 9 PGM G1P UTP NDPkin 8 Pi starch 8 9 PP PPase Pi AMP AdK rstarch EX3: APOPTOSIS IN MAMMALIAN CELLS Fas Ligand Apoptotic_Stimuli s7 FADD Procaspase 8 Bax_Bad_Bim Bcl _Bcl xl Apaf 1 s8 s1 Bid BidC Terminal s6 Mitochondrion CytochromeC s9 s10 d/ Caspase 8 s5 Procaspase 3 s (m0) Caspase 3 s13 Caspase 9 s11 s3 Procaspase 9 DFF DFF40 Oligomer CleavedDFF45 s1 (m) [GON 003] DNA s4 DNA Fragment [HEINER; KOCH; WILL 004]

12 EX4 - SWITCH CYCLE HALOBACTERIUM SALINARUM [Marwan; Oesterhelt 1999] EX4 - SWITCH CYCLE HALOBACTERIUM SALINARUM CheB P CheB _p37 p36_ CheB P _p0 p1_ CheB SR_I_587 SR_I_373 SR_I_373Me SR_I_587Me SR_II360_50Me _p33 p35 p34 p3 p4 p p3 p5_ SR_II480 SR_II360_50 SR_II480Me on _p10_ on _p9 p7_ no_hv580 off _p8_ hv580 no_hv487 hv487 _p30_ CheY P _p30_ CheA off CheA _p7_ on CheR _p6 p4_ no_hv373 off _p3_ hv373 SR_I_510 _p6_ SR_I_510Me _p5 p9 p9 p8_ CheA P CheA P CheY CheR _p31_ k1_cw k_cw Tstop_cw _p p1 p0 p19_ k0_cw Rcw 44 Ccw Acw Scw _p14_ CheY P _p7_ co_cheyp co_cheyp Conf1 44 _p14_ 44 _p1_ ka4 ka3 ka ka1 kd1 kd kd3 kd4 t1 t11 t1 t _p13 p11_ CheYPbound Conf co_cheyp _p14_ k0_ccw Rccw _p15_ 44 Cccw _p16_ Accw _p17_ Sccw _p18_ k1_ccw k_ccw Tstop_ccw

13 QUALITATIVE ANALYSES 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 behavourally invariant net structures? -> token conservation -> P - INVARIANTS -> token distribution reproduction -> T - INVARIANTS... and many more -> temporal logics

14 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, e.g. boundedness, liveness, reversibility,... -> model checking of special behavioural system properties, e.g. reachability of a given (sub-) system state (with constaints), reproducability of a given (sub-) system state (with constraints) expressed in temporal logics (CTL / LTL), very flexible, powerful querry language CASE STUDIES -> CREDITS gene regulatory networks bacteriophage lambda -> C. Chaouiya, D. Thieffry / Univ. Marseille signal-transduction networks RKIP/MEK-ERK signalling pathway -> David Gilbert / Univ. Glasgow yeast pheromone pathway -> Andrea Sackmann, Ina Koch / TFH Berlin G1/S - phase in mammalian cells -> Thomas Kaunath, Ina Koch / TFH Berlin E. coli pathway -> Nina Kramer, Ina Koch / TFH Berlin lipoprotein metabolism (liver) -> Daniel Schrödter / BTU Cottbus apoptosis in mammalian cells -> Jürgen Will / BTU Cottbus blood coagulation, hemostasis -> Gerry Neumann / BTU Cottbus switch cycle halobacterium salinarum -> Wolfgang Marwan / MPI Magdeburg metabolic networks glycolysis in humans carbon metabolism in potato tuber -> Thomas Runge / BTU Cottbus -> Björn Junker / IPK Gatersleben

15 SUMMARY representation of bionetworks by Petri nets -> partial order representation -> better comprehension -> formal semantics -> sound analysis techniques -> unifying view purposes -> animation -> to experience the model -> model validation against consistency criteria -> to increase confidence -> qualitative / quantitative behaviour prediction -> experiment design, new insights step-wise model development -> qualitative model -> discrete Petri nets -> discrete quantitative model -> stochastic Petri nets -> continuous quantitative model -> continuous Petri nets = ODEs OUTLOOK THANKS! HHTP://WWW-DSSZ.INFORMATIK.TU-COTTBUS.DE