Gilbert & Heiner 1

From Petri Nets to Differential EquationsAn Integrative Approach

for Biochemical Network Analysis

David Gilbertdrg@brc.dcs.gla.ac.uk

Bioinformatics Research Centre, University of Glasgow

and

Monika Heinermonika.heiner@informatik.tu-cottbus.de

Department of Computer Science, Brandenburg University of Technology

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Long-term vision• Current trial-and-error drug prescription

procedure– "adjust" an individual to a given drug by

testing reaction over several weeks

• Vision: model-based drug prescriptionas part of medical patient treatment bya physician:– what and how much is necessary to

substitute a givenunderfunction/malfunction

• Drug discovery process:– in-silico quantitative modelling before

major development of drug

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Biological function?

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… by interaction in networks

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System

Model

SystemProperties

ModelProperties

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System

Model

SystemProperties

ModelProperties

Construction technicalsystem

requirementspecification

verification

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System

Model

SystemProperties

ModelProperties

Understanding biologicalsystem

knownproperties

validation

behaviourprediction

unknownproperties

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Raf-1

MEKERK1,2MEK1,2

ERK1,2

B-Raf

Rap1cAMP GEF

Akt

Receptor e.g. 7-TMR

αβ

γtyrosine kinaseβ

γ

SOSshcgrb2

Ras

PAK

RacPI-3 K

Ras

cAMP

PKAcAMP

PDE

cAMP AMP

αAdCyc

cAMPATP

PKAcAMP

MKP

transcription factors

nucleus

cell membrane

cytosol

heterotrimeric G-protein

Biochemical networks

What happens?Why does it happen ?

How is specificity achieved?

We can describe the general topology and single biochemical steps.However, we do not understand the network function as a whole.

FORMAL SEMANTICS?

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READABILITY?

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What is a biochemical network model?

1. Structure graphQUALITATIVE

2. Kinetics (if you can) reaction ratesd[Raf1*]/dt = k1*m1*m2 + k2*m3 + k5*m4 QUANTITATIVEk1 = 0.53; k2 = 0.0072; k5 = 0.0315

3. Initial conditions marking , concentrations[Raf1*]t=0= 2 µMolar

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Bionetworks: some problems

knowledge PROBLEM 1⇒ uncertain⇒ growing, changing⇒ time-consuming wet-lab experiments⇒ some data estimated⇒ results distributed over independent data bases, papers, journals, . . .

various, mostly ambiguous representations PROBLEM 2⇒ 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 ⇐

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Framework

quantitativemodelling

quantitativemodels

animation /analysis / simulation

understanding

model validation

quantitativebehaviour prediction}

Bionetworksknowledge

ODEs

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Framework

quantitativemodelling

quantitativemodels

animation /analysis / simulation

understanding

model validation

quantitativebehaviour prediction} ODEs

qualitativemodelling

qualitativemodels

animation /analysis

understanding

model validation

qualitativebehaviour prediction} Petri net theory

(invariants)

Model checking

Bionetworksknowledge

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Framework

quantitativemodelling

quantitativemodels

animation /analysis / simulation

understanding

model validation

quantitativebehaviour prediction} ODEs

qualitativemodelling

qualitativemodels

animation /analysis

understanding

model validation

qualitativebehaviour prediction} Petri net theory

(invariants)

Model checking

Reachability graphLinear unequationsLinear programming

quantitativeparameters

Bionetworksknowledge

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• Proliferation• Cell cycle• Differentiation• Transformation• Survival• Cytoskeleton• Adhesion• Motility• etc.

Mitogens Growth factors

Receptorreceptor

Ras kinase

RafP P

P

P

MEKP

ERKP P

cytoplasmic substrates

ElkSAP Gene

Ras-Raf-MEK-ERK signalling pathway

…one pathway……mediates many functions,and hence diseases…

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Network building block - enzymatic reaction

k1 k2

k3

E/P

P

Pp

E

!

E +Pk2

" # #

k1# $ # E | P

k3# $ # E +Pp

P Pp

E

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Signalling cascade of enzymatic reactions

P3 P3pOutput

P1

Input SignalX

P1p

P2pP2

Product become enzyme at next stage

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Continuous system & modelEffect of addition of EGF vs NGF

ProliferationDifferentiation

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Case study: small model networkRKIP inhibited ERK pathway

Cho et al, CMSB03Protein

(concentration)

Reaction+ rate

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k1 k2

k3 k4

k5k6 k7

k8

k9 k10

k11

m1Raf-1*

m3 Raf-1*/RKIP

m2RKIP

m4Raf-1*/RKIP/ERK-PP

m9

ERK-PP

m5

ERK

m8 MEK-PP/ERK

m7

MEK-PPm6

RKIP-P

m10

RP

m11 RKIP-P/RP

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k1/k2

k3/k4

k5k6/k7

k8

k9/k10

k11

m1Raf-1*

m3 Raf-1*/RKIP

m2RKIP

m4Raf-1*/RKIP/ERK-PP

m9

ERK-PP

m5

ERK

m8 MEK-PP/ERK

m7

MEK-PPm6

RKIP-P

m10

RP

m11 RKIP-P/RP

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Step 1 - Qualitative analysis

• Structural properties Tool supported

• General behavioural properties:– Boundedness– Liveness– Reversability

• Place & transition invariants:– CTI– CPI

• Model checking of temporal-logic properties:– special behavioural properties

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Systematic constructionof the minimal marking

• Each P-invariant gets at least one token

• In signal transduction:– exactly 1 token, corresponding to species conservation– token in least active state

• All (non-trivial) T-invariants become realizable– to make the net live

• Minimal marking– minimization of the state space

⇒ UNIQUE INITIAL MARKING ⇐

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P-invariants

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T-invariants

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Partial order run of thenon-trivial T-invariant

• Partial order structure

• Illustrates causal & concurrentbehaviour

• Labelled condition/event net– Events - transition occurrences– Conditions: input/output

compounds

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CTL queries

1. There are reachable states where ERK isphosphorylated and RKIP is not phosphorylated.EF [ (ERK-PP ∨ RAF1*/RKIP/ERK-PP) ∧ RKIP ]

2. The phosphorylation of ERK does not depend onthe phosphorylation of RKIP.EG [ ERK → E (¬RKIP-P ∨ RKIP-P/RP) U ERK-PP]

3. A cyclic behaviour w.r.t. RKIP is possible forever.AG [ (RKIP → EF (¬RKIP)) ∧ (¬RKIP → EF (RKIP)) ]

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Qualitative analysis - summary• Validation criterion 0

– all expected structural properties hold– all expected general behavioural properties hold

• Validation criterion 1– CTI– no minimal T-invariant without biological interpretation– no known biological behaviour without corresponding

T-invariant

• Validation criterion 2– CPI– no minimal P-invariant without biological interpretation

• Validation criterion 3– all expected special behavioural properties expressed by

temporal logic formulae hold

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Step 2: quantitative analysis

• Derive differential equations usinggraph structure & rate constants

• Add (13 alternative) initial conditionsderived via qualitative analysis

• Compute steady-states to show“equivalence” of alternatives

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From (time-less) discrete to(timed) continuous Petri nets

• Keep the structure

• Add rate function to each transition– Pre-places appear as parameters → state dependent– Function defines the continuous flow

• Each place gets 1 token (real number)– Represents current continuous concentration

• Continuous state space

⇒ Continuous Pn defines a system of ODEs ⇐

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k1 k2

k3 k4

k5k6 k7

k8

k9 k10

k11

m1Raf-1*

m3 Raf-1*/RKIP

m2RKIP

m4Raf-1*/RKIP/ERK-PP

m9

ERK-PP

m5

ERK

m8 MEK-PP/ERK

m7

MEK-PPm6

RKIP-P

m10

RP

m11 RKIP-P/RP

dm3/dt =

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k1 k2

k3 k4

k5k6 k7

k8

k9 k10

k11

m1Raf-1*

m3 Raf-1*/RKIP

m2RKIP

m4Raf-1*/RKIP/ERK-PP

m9

ERK-PP

m5

ERK

m8 MEK-PP/ERK

m7

MEK-PPm6

RKIP-P

m10

RP

m11 RKIP-P/RP

dm3/dt = + r1 + r4 - r2 - r3

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k1 k2

k3 k4

k5k6 k7

k8

k9 k10

k11

m1Raf-1*

m3 Raf-1*/RKIP

m2RKIP

m4Raf-1*/RKIP/ERK-PP

m9

ERK-PP

m5

ERK

m8 MEK-PP/ERK

m7

MEK-PPm6

RKIP-P

m10

RP

m11 RKIP-P/RP

dm3/dt = + k1*m1*m2 + r4 - r2 - r3

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k1 k2

k3 k4

k5k6 k7

k8

k9 k10

k11

m1Raf-1*

m3 Raf-1*/RKIP

m2RKIP

m4Raf-1*/RKIP/ERK-PP

m9

ERK-PP

m5

ERK

m8 MEK-PP/ERK

m7

MEK-PPm6

RKIP-P

m10

RP

m11 RKIP-P/RP

dm3/dt = + k1*m1*m2 + k4*m4 - k2*m3 - k3*m3*m9

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Validation via ODE simulation

Distribution of `bad' steady states as euclideandistances from the `good' final steady state

13 ‘Good’ stateconfigurations

Cho et al Biochemist

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Quantitative analysis - summary

• Reasonable initial states of the quantitative modelcan be constructed by help of the qualitative model

• All “live” discrete states result in the same steadystate

• No other (theoretically possible) initial states are“close” to this steady state

• Hold for this self-contained case study– Other case studies confirm these findings

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Overall summary

• We use a 2-step approach;– qualitative model to validate the structure;– qualitative & quantitative model share the

same structure

• Continuous pn are structured notations forODEs

• Many open questions…

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…Outlook

• Deeper insights into the relation between thediscrete and the continuous world

• The whole truth about biomolecular systems⇒ inherently stochastic behaviour!!!

• Continuous models are just an approximation ofreality

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Acknowledgements

• Rainer Breitling (Groningem)

• Alex Tovchigrechko (Cottbus)

• Simon Rogers (Glasgow)

Support from the UK DTI BioscienceBeacons programme

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David Gilbertdrg@brc.dcs.gla.ac.uk

Bioinformatics Research Centre, University of Glasgow

and

Monika Heinermonika.heiner@informatik.tu-cottbus.de

Department of Computer Science, Brandenburg University of Technology