[PPT] - A S TRUCTURED A PPROACH . . . T UTORIAL , P ART II F ROM P ETRI N ETS PowerPoint Presentation

SLIDE 1

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

ISMB, JULY 2008

A STRUCTURED APPROACH . . . TUTORIAL, PART II FROM PETRI NETS

TO DIFFERENTIAL EQUATIONS

Monika Heiner Brandenburg University of Technology Cottbus, Dept. of CS

SLIDE 2

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

FRAMEWORK: SYSTEMS BIOLOGY

biosystem natural model

bserved

behaviour predicted behaviour wetlab model-based experiment design experiments formalizing understanding wetlab experiments model validation

MODELLING = FORMAL KNOWLEDGE REPRESENTATION MODEL VALIDATION = CONFIDENCE INCREASE

SLIDE 3

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

WHAT KIND OF MODEL

SHOULD BE USED?

SLIDE 4

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

NETWORK REPRESENTATIONS, EX1

Raf-1 M EK ER K1,2 M EK1,2 ERK1,2 B-R af Rap1

cAM P G EF

Akt Receptor

e.g. 7-TMR

α β γ

tyrosine kinase

β γ SO S shc grb2 Ras PAK R ac PI-3 K R as

cA M P

PKA

cA M P

PD E

cA M P A M P

α AdC yc

cA M P A TP

PKA

cA M P

M KP transcription factors

nucleus

cell m em brane cytosol

heterotrim eric G

protein

Raf-1 Raf-1 M EK M EK ER K1,2 ER K1,2 M EK1,2 M EK1,2 ERK1,2 ERK1,2 B-R af B-R af Rap1 Rap1

cAM P G EF cAM P G EF

Akt Akt Receptor

e.g. 7-TMR

α β γ α β γ α β γ β β γ

tyrosine kinase

β β γ SO S SO S shc shc grb2 grb2 Ras Ras PAK PAK R ac R ac PI-3 K PI-3 K R as R as

cA M P cA M P

PKA

cA M P

PKA PKA

cA M P cA M P

PD E

cA M P A M P

PD E PD E

cA M P A M P cA M P cA M P A M P

α AdC yc

cA M P A TP

α AdC yc AdC yc

cA M P A TP cA M P cA M P A TP

PKA

cA M P

PKA PKA

cA M P cA M P

M KP M KP transcription factors transcription factors

nucleus

cell m em brane cytosol

heterotrim eric G

protein
>

FORMAL SEMANTICS ?

SLIDE 5

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

NETWORK REPRESENTATIONS, EX2

>

R E A D A B I L I T Y

?

SLIDE 6

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

NETWORK REPRESENTATIONS

informal cartoon-like representations
> readability
> fault avoidance
formal = mathematical representations
> analysability

WHY NOT BOTH ? & EXECUTABILITY

SLIDE 7

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

PETRI NETS -

AN INFORMAL CRASH COURSE

SLIDE 8

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

PETRI NETS, BASICS

2

2 2 2 r1 O2 H+ NADH H2O NAD+

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

O2 H+ NADH H2O NAD+

hyper-arcs

2 2 2 2

SLIDE 9

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

PETRI NETS, BASICS - THE STRUCTURE

atomic actions
> transitions
> chemical reactions
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, m0),

F: (P x T) U (T x P) -> N0, m0: P -> N0 input compounds

utput

compounds

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

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

SLIDE 10

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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

SLIDE 11

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

PETRI NETS, BASICS - THE BEHAVIOUR

atomic actions
> 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)

STATE SPACE

SLIDE 12

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

TYPICAL BASIC STRUCTURES I

A B C B A C A B C A B C r1 r2 r3 r4 r5 r6

A --> B, A --> C A --> B + C A + B --> C A --> C, B --> C

SLIDE 13

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

TYPICAL BASIC STRUCTURES II

A B A B B A B A E A B E E B A r1 r1 r2 r1 r1 r2 r1, r2 r1, r2

A --> B A <--> B A --> B E A <--> B E

SLIDE 14

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

TYPICAL BASIC STRUCTURES III A <--> A|E --> B E

E B A|E A A A|E B E A B E r3 r2 r1 r3 r1, r2 MA1

enzymatic reaction, mass-action approach 1

SLIDE 15

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

TYPICAL BASIC STRUCTURES IV

metabolic networks
> substance flows
signal transduction

networks

> signal flows

r3 r2 r1 e3 e2 e1 r3 r2 r1

SLIDE 16

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

TYPICAL BASIC STRUCTURES IV

metabolic networks
> substance flows
signal transduction

networks

> signal flows
> OPEN / CLOSED SYSTEMS

r3 r2 r1 e3 e2 e1

INPUT OUTPUT COMPOUND COMPOUND

r3 r2 r1

INPUT SIGNAL OUTPUT SIGNAL

SLIDE 17

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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)

SLIDE 18

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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

SLIDE 19

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

BIO PETRI NETS - SOME EXAMPLES

SLIDE 20

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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]

SLIDE 21

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

EX1 - Glycolysis and Pentose Phosphate Pathway

[Reddy 1993]

Pi Pi 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 ATP ATP ATP ATP ATP ADP ADP ADP ADP ADP NAD NAD+ NAD NADH 15 16 17 18 19 20 13 14 12 11 10 9 2 1 3 4 5 6 7 8 2 2 2 2

SLIDE 22

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

EX2 - Carbon Metabolism in Potato Tuber

Suc eSuc Glc Frc UDPglc G6P F6P G1P UDP UDP UTP ATP ATP ATP ATP ATP ATP ADP 29 ADP 29 ADP 29 ADP 29 ADP 29 ADP 29 S6P Pi 28 Pi 28 Pi 28 Pi 28 PP PP starch AMP 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

[KOCH; JUNKER; HEINER 2005]

SLIDE 23

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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]

SLIDE 24

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

EX4 - SWITCH CYCLE HALOBACTERIUM SALINARUM

[Marwan; Oesterhelt 1999]

SLIDE 25

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

EX4 - SWITCH CYCLE HALOBACTERIUM SALINARUM

CheB−P _p0_ CheB _p1_ SR_II360_520 _p2_ SR_II360_520Me _p3_ SR_II480 _p4_ SR_II480Me _p5_ CheR _p6_ CheY−P _p7_ CheY−P _p7_ CheY _p8_ hv487 _p9_ no_hv487 _p10_ Conf2 _p11_ Conf1 _p12_ 44 CheYPbound _p13_ co_CheYP _p14_ 44 co_CheYP _p14_ 44 co_CheYP _p14_ 44 Rccw _p15_ Cccw _p16_ Accw _p17_ Sccw _p18_ Scw _p19_ Acw _p20_ Ccw _p21_ Rcw _p22_ hv373 _p23_ no_hv373 _p24_ SR_I_510Me _p25_ SR_I_510 _p26_ no_hv580 _p27_ hv580 _p28_ CheA−P _p29_ CheA−P _p29_ CheA _p30_ CheA _p30_ CheR _p31_ SR_I_587Me _p32_ SR_I_587 _p33_ SR_I_373Me _p34_ SR_I_373 _p35_ CheB _p36_ CheB−P _p37_

ff
n

t22 t12 kd4 kd3 kd2 ka3 ka2 ka4 t21 t11 kd1 ka1 k0_ccw Tstop_cw k2_cw k1_cw k0_cw

ff
n
n
ff

44 44 44 44

SLIDE 26

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

EX5 - SIGNALLING CASCADE

Raf RafP MEKP MEKPP MEK ERKP ERKPP ERK Phosphatase3 Phosphatase1 Phosphatase2 RasGTP

SLIDE 27

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

EX5 - SIGNALLING CASCADE

Raf RasGTP Raf_RasGTP RafP RafP_Phase1 MEK_RafP MEKP_RafP MEKP_Phase2 MEKPP_Phase2 ERK ERK_MEKPP ERKP_MEKPP ERKP MEKPP ERKPP_Phase3 ERKP_Phase3 MEKP ERKPP Phase2 Phase3 MEK Phase1 k3 k6 k21 k18 k9 k12 k15 k24 k27 k30 k7/k8 k1/k2 k4/k5 k10/k11 k16/k17 k22/k23 k19/k20 k13/k14 k28/k29 k25/k26

SLIDE 28

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

QUALITATIVE ANALYSES

SLIDE 29

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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)

SLIDE 30

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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

SLIDE 31

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

BIONETWORKS, VALIDATION

validation criterion 1
> all expected structural properties hold
> all expected general behavioural properties hold
validation criterion 2
> 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

SLIDE 32

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

NOW WE ARE READY

FOR SOPHISTICATED QUANTITATIVE ANALYSES !

SLIDE 33

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

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 dA / dt = -v1 + v2

continuous nodes !

E B A|E A k3 k2 k1

v2 = k2*A|E v3 = k3*A|E dA|E / dt = v1 - v2 - v3 dB / dt = v3 dE / dt = -v1 + v2 + v3

SLIDE 34

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

THE RKIP PATHWAY, CONTINUOUS 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

dm3 = + k1 * m1 * m2 dt + k4 * m4

k2 * m3
k3 * m3 * m9

SLIDE 35

PN & Systems Biology monika.heiner@tu-cottbus.de July 2008

THE QUALITATIVE MODEL

BECOMES THE STRUCTURED DESCRIPTION OF THE QUANTITATIVE MODEL !