from petri ets to differential quations filepn & systems biology...

PN & Systems Biology

May 2005

MSBF, MAY 2005

TO IONS

H NALYSIS

y Cottbus

[email protected]

FROM PETRI NETS DIFFERENTIAL EQUAT

AN INTEGRATIVE APPROAC

FOR BIOCHEMICAL NETWORK A

Monika Heiner

Brandenburg University of Technolog

Dept. of CS


May 2005

MODEL- BASED SYSTEM ANALYSIS

systemproperties

modelproperties

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Petrinetzmodel

Problemsystem


May 2005


systemproperties

modelproperties

ementication

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Petrinetzmodel

Problemsystem

technicalsystem

requirspecif

verificationCONSTRUCTION


May 2005


systemproperties

modelproperties

wn

nown perties

perties

[email protected]

Petrinetzmodel

Problemsystem

biologicalsystem

kno

unkpro

provalidation

behaviour prediction

UNDERSTANDING


May 2005

DEL ?

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WHAT KIND OF MO

SHOULD BE USED


May 2005

NETWORK REPRESENTATIONS, EX1

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May 2005


NTICS ?

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-> FORMAL SEMA


May 2005


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May 2005


ITY ?

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-> READABIL


May 2005

BIONETWORKS, SOME PROBLEMS

-> PROBLEM 1

-> PROBLEM 2

ls, . . .

-> PROBLEM 3

NS << - -

[email protected]

❑ various, mostly ambiguous representations

-> verbose descriptions

-> diverse graphical representations

-> contradictory and / or fuzzy statements

❑ knowledge

-> uncertain

-> growing, changing

-> distributed over various data bases, papers, journa

❑ network structures

-> tend to grow fast

-> dense, apparently unstructured

-> hard to read

- - >> models are full of ASSUMPTIO


May 2005

FRAMEWORK

ding

dation

ve r prediction ODEs

[email protected]

bionetworks knowledge

quantitative modelling

quantitative models

animation /analysis /simulation

understan

model vali

quantitatibehaviou


May 2005

FRAMEWORK

ding

lidation

prediction

ding

dation

ve r prediction

(invariants)

modelchecking

Petri net theory

ODEs

[email protected]

quantitativeparameters


qualitative modelling

qualitative models


quantitative models

animation /analysis


understan

model va

qualitativebehaviour

understan

model vali

quantitatibehaviou


May 2005

FRAMEWORK

ding

lidation

prediction

ding

dation

ve r prediction

(invariants)

modelchecking

Petri net theory

ODEsLPSLIRG

[email protected]


qualitative modelling

qualitative models


quantitative models

quantitativeparameters

animation /analysis


understan

model va

qualitativebehaviour

understan

model vali

quantitatibehaviou


May 2005

OURSE

[email protected]

PETRI NETS -AN INFORMAL CRASH C


May 2005

PETRI NETS, BASICS - THE STRUCTURE

hemical reactions

output compounds

+ + O2

[email protected]

❑ atomic actions -> Petri net transitions -> c

input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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


May 2005


hemical reactions

output compounds

+ + O2

H

[email protected]


input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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

O2

H+

NAD

H2O

NAD+

hyperarcs

2

22

2


May 2005


hemical reactions

hemical compounds

output compounds

+ + O2

post-conditions

[email protected]


❑ local conditions -> Petri net places -> c

input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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

pre-conditions


May 2005


hemical reactions

hemical compounds

toichiometric relations

output compounds

+ + O2

[email protected]



❑ multiplicities -> Petri net arc weights -> s

input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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


May 2005


hemical reactions

hemical compounds


vailable amount (e.g. mol)

ompounds distribution

output compounds

+ + O2

[email protected]




❑ condition’s state -> token(s) in its place -> a

❑ system state -> marking -> c

input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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


May 2005


hemical reactions

hemical compounds


vailable amount (e.g. mol)

ompounds distribution

P -> N0

output compounds

+ + O2

[email protected]




❑ condition’s state -> token(s) in its place -> a

❑ system state -> marking -> c

❑ PN = (P, T, F, m0), F: (P x T) U (T x P) -> N0, m0:

input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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


May 2005

PETRI NETS, BASICS - THE BEHAVIOUR

hemical reactions

output compounds

+ + O2

[email protected]


input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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


May 2005


hemical reactions

output compounds

+ + O2

[email protected]


input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

FIRING


May 2005


hemical reactions

output compounds

+ + O2

TOKEN GAME

DYNAMIC BEHAVIOUR

(substance flow)

[email protected]


input compounds

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

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

2

2

2

2

r1

O2

H+

NADH

H2O

NAD+

FIRING


May 2005

TYPICAL BASIC STRUCTURES

r3r2

e3e2

r3

r2

r1

[email protected]

❑ metabolic networks

-> substance flows

❑ signal transduction networks

-> signal flows

r1

e1


May 2005

THE RUNNING EXAMPLE - THE RKIP PATHWAY

[Cho et al.,CMSB 2003]

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May 2005

THE RKIP PATHWAY, PETRI NET

k11

k10k9

RPm10

RKIP-P_RPm11

P

[email protected]

k8

k5k7k6

k4k3

k2k1

RKIP-P

m6

ERK

m5

MEK-PP

m7

Raf-1Star_RKIP_ERK-PP

m4MEK-PP_ERKm8

ERK-PP

m9 Raf-1Star_RKIm3

RKIP

m2

Raf-1Star

m1


May 2005

THE RKIP PATHWAY, HIERARCHICAL PETRI NET

k11

RPm10

RKIP-P_RPm11

P

k9_k10

[email protected]

k8

k5

RKIP-P

m6

ERKm5

MEK-PP

m7


m4MEK-PP_ERKm8

ERK-PP

m9 Raf-1Star_RKIm3

RKIP

m2

Raf-1Star

m1

k6_k7

k3_k4

k1_k2


May 2005

THE RKIP PATHWAY, HIERARCHICAL PETRI NET

k11

RPm10

RKIP-P_RPm11

P

k9_k10

[email protected]

k8

k5

RKIP-P

m6

ERKm5

MEK-PP

m7


m4MEK-PP_ERKm8

ERK-PP

m9 Raf-1Star_RKIm3

RKIP

m2

Raf-1Star

m1

k6_k7

k3_k4

k1_k2

initial marking


May 2005

BIOCHEMICAL PETRI NETS, SUMMARY

ementary actions) y ]

y (re-) actions

[email protected]

❑ biochemical networks

-> networks of (abstract) chemical reactions

❑ biochemically interpreted Petri net

-> partial order sequences of chemical reactions (= eltransforming input into output compounds / signals[ respecting the given stoichiometric relations, if an

-> set of all pathwaysfrom the input to the output compounds / signals [ respecting the stoichiometric relations, if any ]

❑ pathway

-> self-contained partial order sequence of elementar

❑ typical basic assumption

-> steady state behaviour


May 2005
[email protected]
QUALITATIVE

ANALYSES


May 2005

ANALYSIS TECHNIQUES, OVERVIEW

nstruction

space construction (RG)

erties,te [with constraints],state [with constraints]

[email protected]

❑ static analyses -> no state space co

-> structural properties (graph theory)

-> P / T - invariants (linear algebra),

❑ dynamic analyses -> total/partial state

-> analysis of general behavioural system properties,

e.g. boundedness, liveness, reversibility, . . .

-> model checking of special behavioural system prope.g. reachability of a given (sub-) system sta

reproducability of a given (sub-) system

expressed in temporal logics (CTL / LTL),very flexible, powerful querry language


May 2005

INCIDENCE MATRIX C

=> stoichiometric matrix

g by firing of tj

j,pi) - F(pi, tj) = ∆ tj(pi)

enzyme

MB2-catalysed ction

x

cij = 0

j

i

[email protected]

❑ a representation of the net structure

❑ matrix entry cij:token change in place pi by firing of transition tj

❑ matrix column ∆tj:vector describing the change of the whole markin

❑ side-conditions are neglected

PT t1 tj tm

p1

pi

pn

cij cij = (pi, tj) = F(t

. . . . . .

...

C =

∆tj ∆tj = ∆ tj(*)

MB1 enzymerea

x


May 2005

T-INVARIANTS, BASICS

> multisets of transitions

> Parikh vector

> sets of transitions

n of minimal ones

kx aixii∑=

[email protected]

❑ Lautenbach, 1973

❑ T-invariants -

-> integer solutions x of -

❑ 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 combinatio

-> multiplication with a positive integer

-> addition

-> Division by gcD

❑ Covered by T-Invariants (CTI)

-> each transition belongs to a T-invariant

-> BND & LIVE => CTI (necessary condition)

Cx 0 x 0 x 0≥,≠,=


May 2005

T-INVARIANTS, INTERPRETATION

RG

ving sequences

invariant, ence

> partial order structure

[email protected]

❑ T-invariants = (multi-) sets of transitions

-> zero effect on marking

-> reproducing a marking / system state

-> steady state substance flows / reaction rates

-> elementary modes [Schuster 1993]

❑ realizable T-invariants correspond to cycles in the

-> RG: concurrent transitions -> all transitions’ interlea

-> if there are concurrent transitions in a realizable T-then there is a RG cycle for each interleaving sequ

-> analogously for conflicts

❑ a T-invariant defines a 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


May 2005

T-INVARIANTS, THE RKIP PATHWAY

k11

k5

RPm10

RKIP-P

m6

RKIP-P_RPm11

RKIP_ERK-PP

Raf-1Star_RKIP

RKIP

m2

k9

k3

k1

[email protected]

k8

ERKm5

MEK-PP

m7

Raf-1Star_

m4MEK-PP_ERKm8

ERK-PP

m9 m3

Raf-1Star

m1

k6

-> non-trivial T-invariant

+ four trivial ones for reversible reactions


May 2005

RUN OF THE NON-TRIVIAL T-INVARIANT

k11

k5

RPm10

RKIP-Pm6

RKIP-P_RPm11

Raf-1Star_RKIP_ERK-PPm4

Raf-1Star_RKIPm3

RKIPm2af-1Star

k9

k3

k1

Raf-1Starm1

RK-PP RPm10RKIPm2

RK

[email protected]

❑ partial order structure

❑ T-invariant’s unfoldingto describe its behaviour

❑ labelled condition / event net

-> events - transition occurences

-> conditions - input / output compounds

❑ partial order semantics

-> a net’s all partial order runs

k8

ERKm5

MEK-PPm7

m8

Rm1

k6

ERK_PPm9

MEK-PP m9 Em7

MEK-PP_E


May 2005

P-INVARIANTS, BASICS

> multisets of places

> sets of places

n of minimal ones

ky aiyii∑=

[email protected]

❑ Lautenbach, 1973

❑ P-invariants -

-> integer solutions y of

❑ minimal P-invariants

-> there is no P-invariant with a smaller support -

-> gcD of all entries is 1

❑ any P-invariant is a non-negative linear combinatio

-> multiplication with a positive integer

-> addition

-> Division by gcD

❑ Covered by P-Invariants (CPI)

-> each transition belongs to a P-invariant

-> CPI => BND (sufficient condition)

yC 0 y 0 y 0≥,≠,=


May 2005

P-INVARIANTS, INTERPRETATION

weighted sum of tokens on

rom a P-invariant’s placetokens to a P-invariant’s place

m reachable from m0

[email protected]

❑ the firing of any transition has no influence on the the P-invariant’s places

-> for all t: the effect of the arcs, removing tokens fis equal to the effect of the arcs, adding

❑ set of places with

-> a constant weighted sum of tokens for all markingsym = ym0

-> token / compound preservation

❑ a place belonging to a P-invariant is bounded

-> CPI - sufficient condition for BND

❑ a P-invariant defines a subnet

-> the P-invariant’s places (the support), + all their pre- and post-transitions+ the arcs in between

-> pre-sets of supports = post-sets of supports


May 2005

P-INVARIANTS, THE RKIP PATHWAY

k11

k5

RPm10

RKIP-P

m6

RKIP-P_RPm11


m4

Raf-1Star_RKIPm3

RKIP

m2

tar

k9_k10

k3_k4

k1_k2

k11

k5

RPm10

RKIP-P

m6

K

RKIP-P_RPm11


m4K

Raf-1Star_RKIPm3

RKIP

m2

Star

k9_k10

k3_k4

k1_k2

[email protected]

k11k8

k5

RPm10

RKIP-P

m6

ERK

m5

MEK-PP

m7

RKIP-P_RPm11


m4MEK-PP_ERKm8

ERK-PP

m9 Raf-1Star_RKIPm3

RKIP

m2

Raf-1Star

m1

k9_k10k6_k7

k3_k4

k1_k2

k8

ERK

m5

MEK-PP

m7

MEK-PP_ERKm8

ERK-PP

m9

Raf-1S

m1

k6_k7

k8

ERm5

MEK-PP

m7

MEK-PP_ERm8

ERK-PP

m9

Raf-1

m1

k6_k7

P-INV1: MEK

P-INV2: RAF-1STAR

P-INV3: RP

P-INV4: ERK

P-INV5: RKIP


May 2005

CONSTRUCTION OF THE INITIAL MARKING

[email protected]

❑ each P-invariant gets at least one token

-> P-invariants are structural deadlocks and traps

❑ all (non-trivial) T-invariants get realizable

-> to make the net live

❑ minimal marking

-> minimization of the state space

❑ assumption: top-to-bottom reading of the figure

-> but, all reachable markings are equivalent (= produce same state space)

-> UNIQUE INITIAL MARKING


May 2005

STATIC ANALYSIS, SUMMARY

tion subnet (cycle)

lic behaviour

pF0 MG SM FC EFC ESN N N N N YV L&SY Y

[email protected]

❑ structural properties

❑ CPI

-> structural bounded (SB)

-> each P-invariant represents a substance conserva

❑ CTI

-> Live & BND -> CTI

-> 4 trivial T-invariants for reversible reactions

-> 1 non-trivial T-invariant describing the essential cyc

❑ DTP & ES -> Live

INAORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 Y Y Y Y N N Y Y N N N DTP CPI CTI B SB REV DSt BSt DTr DCF L L Y Y Y Y Y Y N ? N N Y


May 2005

DYNAMIC ANALYSIS - REACHABILITY GRAPH

-> single step firing rule

-> interleaving semantics

-> model checking

[email protected]

❑ simple construction algorithm

-> nodes - system states

-> arcs - the (single) firing transition

❑ unbounded Petri net -> infinite RGbounded Petri net -> finite RG

❑ concurrency

-> enumeration of all interleaving sequences

❑ branching arcs in the RG

-> conflict OR concurrency

❑ RG tend to be very large

-> automatic evaluation necessary

❑ worst case: over-exponential growth

-> alternative analyses techniques ?


May 2005

MODEL CHECKING, EXAMPLES

?

nor using MEK-PP ?

nce of RKIP ?

) );

[email protected]

❑ property 1

Is a given (sub-) marking (system state) reachable

EF ( ERK * RP );

❑ property 2

Liveness of transition k8 ?

AG EF ( MEK-PP_ERK );

❑ property 3

Is it possible to produce ERK-PP neither creating

E ( ! MEK-PP U ERK-PP );

❑ property 4

Is there cyclic behaviour w.r.t. the presence / abse

EG ( ( RKIP -> EF ( ! RKIP ) ) * ( ! RKIP -> EF ( RKIP )


May 2005

QUALITATIVE ANALYSIS RESULTS, SUMMARY

-> static analysis

l interpretation

the essential behaviour

-> dynamic analysis

ND

IVE

EV

ofessional understanding

[email protected]

❑ structural decisions of behavioural properties

-> CPI -> BND

-> ES & DTP -> LIVE

❑ CPI & CTI

-> all minimal T-invariant / P-invariants enjoy biologica

-> non-trivial T-invariant -> partial order description of

❑ reachability graph

-> finite -> B

-> the only SCC contains all transitions -> L

-> one Strongly Connected Component (SCC) -> R

❑ model checking -> requires pr

-> all expected properties are valid


May 2005

QUALITATIVE ANALYSIS RESULTS, SUMMARY

-> static analysis

l interpretation

the essential behaviour

-> dynamic analysis

ND

IVE

EV

rofessional understanding

VE MODEL

[email protected]

❑ structural decisions of behavioural properties

-> CPI -> BND

-> ES & DTP -> LIVE

❑ CPI & CTI

-> all minimal T-invariant / P-invariants enjoy biologica

-> non-trivial T-invariant -> partial order description of

❑ reachability graph

-> finite -> B

-> the only SCC contains all transitions -> L

-> one Strongly Connected Component (SCC) -> R

❑ model checking -> requires p

-> all expected properties are valid

-> VALIDATED QUALITATI


May 2005

BIONETWORKS, VALIDATION

ion

ng T-invariant

tion (?)

[email protected]

❑ validation criterion 0

-> all expected structural properties hold

-> all expected general behavioural properties hold


-> CTI

-> no minimal T-invariant without biological interpretat

-> no known biological behaviour without correspondi


-> CPI

-> no minimal P-invariant without biological interpreta


-> all expected special behavioural properties hold

-> temporal-logic properties -> TRUE


May 2005

DY ED SES !

[email protected]

NOW WE ARE REA

FOR SOPHISTICAT

QUANTITATIVE ANALY


May 2005

QUANTITATIVE ANALYSIS

ive parameters

ndent

uous nodes !

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❑ quantitative model = qualitative model + quantitat

-> BUT: quantitative parameters often unknown

❑ typical quantitative parameters of bionetworks

-> compound concentrations -> real numbers

-> reaction rates / fluxes -> concentration-depe

❑ continuous Petri nets

p1Cont

p2Cont

p3Contt1Contm1

m2

m3

v1 = k1*m1*m2

d [p1Cont] / dt = d [p2Cont] / dt = - v1d [p3Cont] / dt = v1 - v2

contin

ODEs

v2 = k2*m3

t2Cont


May 2005

EXAMPLE - MICHAELIS-MENTEN REACTION

50 100 150 200

t

03333

06667

1

50 100 150 200

t

03333

06667

1

[email protected]

0

0,

0,

0,m1

0

0,

0,

0,m2

-> Visual Object Nets-> GON / cell illustrator (?)

s p tCont

m1

v = 0.005 * m1 / (0.1375 + m1)

m2

Vmax = 0.005 (maximal reaction rate)Km = 0.1375 (Michaelis constant)

d[s]/dt = d[p]/dt = Vmax*[s]/(Km+[s])dm1/dt = dm2/dt = Vmax*m1/(Km+m1)


May 2005

ODEL

RIPTION MODEL !

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THE QUALITATIVE MBECOMES

THE STRUCTURAL DESC

OF THE QUANTITATIVE


May 2005

CHALLENGES

l order semantics

der development)

Es

[email protected]

❑ extensions

-> read arcs -> interleaving / partia

-> inhibitor arcs !? -> Turing power !

❑ efficient computation of minimal invariants

-> exponential complexity

-> compositional / step-wise refinement approach (un

❑ analysis of unbounded nets

-> besides T-invariant analysis ?

❑ model checking

-> relevant properties ?

❑ comparision: continuous / hybrid Petri nets <-> OD

-> Petri net simulation versus classical ODEs solver

-> is there a winner (for certain structures) ?


May 2005

SUMMARY

sound analysis techniques

> to experience the model

> to increase confidence

> new insights

DEs

es

[email protected]

❑ representation of bionetworks by Petri nets

-> partial order representation

-> formal semantics -> various

-> unifying view

❑ purposes

-> animation -

-> model validation against consistency criteria -

-> qualitative / quantitative behaviour prediction -

❑ two-step model development

-> qualitative model -> discrete Petri nets

-> quantitative model -> continuous Petri nets = O

❑ many challenging questions for analysis techniqu

-> qualitative as well as quantitative ones


May 2005
[email protected]
THANKS !

from petri ets to differential quations filepn & systems biology...

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