From cell biology to Petri nets

Rainer Breitling, Groningen, NLDavid Gilbert, London, UK

Monika Heiner, Cottbus, DE

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Biology = Concentrations

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The simplest chemical reaction

A B

irreversible, one-molecule reactionexamples: all sorts of decay processes, e.g. radioactive, fluorescence, activated receptor returning to inactive stateany metabolic pathway can be described by a combination of processes of this type (including reversible reactions and, in some respects, multi-molecule reactions)

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The simplest chemical reaction

A Bvarious levels of description:

homogeneous system, large numbers of molecules = ordinary differential equations, kineticssmall numbers of molecules = probabilistic equations, stochasticsspatial heterogeneity = partial differential equations, diffusionsmall number of heterogeneously distributed molecules = single-molecule tracking (e.g. cytoskeleton modelling)

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Kinetics Description

Imagine a box containing N molecules.How many will decay during time t? k*N

Imagine two boxes containing N/2 molecules each.How many decay? k*N

Imagine two boxes containing N molecules each.How many decay? 2k*N

In general:

)(*)( tndt

tdn λ=−

Main idea: Molecules don’t talk

teNtn λ−= 0)(⇔differential equation (ordinary, linear, first-order)

exact solution (in more complex cases replaced by a numerical approximation)

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Kinetics Description

If you know the concentration at one time, you can calculate it for any other time! (and this really works)

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Probabilistic Description

Probability of decay of a single molecule in some small time interval:

Probability of survival in Δt:

Probability of survival for some time t:

Transition to large number of molecules:

tp Δ= λ1

tpp Δ−=−= λ11 12

tx

xe

xtp λλ −

∞→=−= )1(lim

teNtn λ−= 0)(

)()(0 tneN

dttdn t λλ λ −=−= −

or

Main idea: Molecules are isolated entities without memory

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Probabilistic Description – 2

Probability of survival of a single molecule for some time t:

Probability that exactly x molecules survive for some time t:

Most likely number to survive to time t:

tx

xe

xtp λλ −

∞→=−= )1(lim

⎟⎟⎠

⎞⎜⎜⎝

⎛−= −−−

xN

eep xNtxtx

00)1()( λλ

tx eNpx λ−= 0)|max(

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Probabilistic Description – 3

Decay rate: Probability of decay:Probability distribution of n

surviving molecules at time t:Description:Time: t -> wait dt -> t+dt

Molecules:n -> no decay -> nn+1 -> one decay -> n

λntn =Λ ),(dttnp ),(Λ=

),( tnP

]),(1)[,(),1(),1(

),(

dttntnPdttntnP

dttnP

Λ−++Λ+

=+

Final Result (after some calculating): The same as in the previous probabilistic description

Markov Model (pure death!)

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Petri Net representation

?

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Some (Bio)Chemical Conventions

Concentration of Molecule A = [A], usually in units mol/litre (molar)

Rate constant = k, with indices indicating constants for various reactions (k1, k2...)

Therefore:A B

][][][1 Ak

dtBd

dtAd

−=−=

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Reversible, Single-Molecule Reaction

A B, or A B || B A, or Differential equations:

][][][

][][][

21

21

BkAkdtBd

BkAkdtAd

−=

+−=forward reverse

Main principle: Partial reactions are independent!

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Reversible, single-molecule reaction – 2

Differential Equation:

Equilibrium (=steady-state):

equiequi

equi

equiequi

equiequi

Kkk

BA

BkAkdtBd

dtAd

==

=+−

==

1

2

21

][][

0][][

0][][

][][][

][][][

21

21

BkAkdtBd

BkAkdtAd

−=

+−=

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Irreversible, two-molecule reaction

A+B CDifferential equations:

]][[][

][][][

BAkdtAd

dtCd

dtBd

dtAd

−=

−==

Underlying idea: Reaction probability = Combined probability that both [A] and [B] are in a “reactive mood”:

]][[][][)()()( *2

*1 BAkBkAkBpApABp ===

The last piece of the puzzle

Non-linear!

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A simple metabolic pathway

A B C+DDifferential equations:

-k3[C][D]+k2[B][D]=

-k3[C][D]+k2[B][C]=

+k3[C][D]-k2[B]+k1[A][B]=

-k1[A][A]=

reverseforwarddecayd/dt

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Metabolic Networks as Bigraphs

A B C+D

-k3[C][D]+k2[B][D]

-k3[C][D]+k2[B][C]

+k3[C][D]-k2[B]+k1[A][B]

-k1[A][A]

reverseforwarddecayd/dt

A BC

Dk1 k2 k3

-110D

-110C

1-11B

00-1A

k3k2k1

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Petri nets

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Petri nets

http://www-dssz.informatik.tu-cottbus.de/web_animation/pn_demos_flat-nets.html

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Qualitative Petri-Net Modelling & Analysis

Graphicalrepresentation -Snoopy

• Qualitative analysis  Charlie– Unbounded, live & reversible

– Covered by T invariants

– P invariants 

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Biological description bigraph differential equations

KEGG

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Biological description bigraph ODEs

EC 1.1.1.2

substance A substance B

A Bk1

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Biological description bigraph ODEs

EC 1.1.1.2

substance A substance B

A Bk

EA EBk*k1 k2

E

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A special case: enzyme reactions

In a quasi steady state, we can assume that [ES] is constant. Then:

If we now define a new constant Km (Michaelis constant), we get:

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A special case: enzyme reactions

Substituting [E] (free enzyme) by the total enzyme concentration we get:

Hence, the reaction rate is:

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A special case: enzyme reactions

Underlying assumptions of the Michaelis-Menten approximation:

• Free diffusion, random collisions of infinite number of molecules

• Irreversible reactions

• Quasi steady state

In cell signaling pathways, all three assumptions will be frequently violated:

• Reactions of rather rare molecules happen at membranes and on scaffold structures

• Reactions happen close to equilibrium and both reactions have non-zero fluxes

• Enzymes are themselves substrates for other enzymes, concentrations change rapidly, d[ES]/dt ≈ d[P]/dt

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Cell signaling pathways

Fig. courtesy of W. Kolch

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Metabolic pathways vs. Signaling Pathways

E1

(initial substrate)S

S’

E2

E3

S’’

S’’’(final product)

Metabolic

S1

Input SignalX

P2S2

S3 P3Output

Signaling cascade

P1

Product become enzyme at next stageClassical enzyme-product pathway

(can you give the mass-action equations?)

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Metabolic pathways vs. Signalling Pathways

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Cell signaling pathways

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Cell signaling pathways

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Cell signaling pathways

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Cell signaling pathways

Common components:Receptors binding to ligands

R(inactive) + L RL(active)

Proteins forming complexesP1 + P2 P1P2-complex

Proteins acting as enzymes on other proteins (e.g., phosphorylation by kinases)

P1 + K P1* + K

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MA1: Mass action for enzymatic reaction

A: substrate B: productE: enzymeE|A substrate-enzyme complex

E+Ak2

← ⎯ ⎯

k1⎯ → ⎯ E | A k3⎯ → ⎯ E + B

A B

E

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MA2 model

EBEBEAEAk

k

k

k

k +⎯→⎯+ ⎯⎯→⎯

⎯⎯⎯←

⎯→⎯

⎯⎯←

1'

2'

31

2

|| '

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MA3 model

EBEBEAEAk

k

k

k

k

k

++ ⎯⎯→⎯

⎯⎯⎯←

⎯⎯→⎯

⎯⎯⎯←

⎯→⎯

⎯⎯←

1'

2'

3'

4'

1

2

||

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Cell signaling pathways – feedback loops

Breitling / 37

Cell signaling pathways – feedback loops

Fig. courtesy of W. Kolch

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Feedback loops in Petri Nets

RpR

S1

RR

P1

P2

RRpRRp

RpR

S1

RR

P1

P2

RRp

RpR

S1

RR

P1

P2

RRp

RpR

S1

RR

P1

P2

(a)

(c)

(b)

(d)

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Feedback loops in Petri Nets

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Feedback loops in Petri Nets

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…and added inhibitor

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Many PN modelling challengings remain…

Lack of parametersQualitative vs. Continuous PN

Small molecule numbersDeterministics vs. Stochastic models

Spatial heterogeneity???

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Cell signaling pathways

Fig. courtesy of W. Kolch

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5

X

Stochastic vs. Continuous

5

X

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Stochastic model checkingTwo Reaction Model

First a simple model of two reactions:

A BC D

Assess property:P=?[ A = $X { A = D } ]

“What is the probability that, when A and D first equal each other, they both have $X number of molecules?”

0.01

0.1

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Two Reaction ModelCo

ncen

tration

TimeD

A

Property:P=?[ A = $X { A = D } ]

*

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Two Reaction Model

Set reactants to 10 molecules (model bound to 10 molecules)

Simulate with Gillespie 1,000 times and model check each output

Number of simulations which are true over total number of simulations is the probability.

Also checked the continuous model and the answer is the solid line.

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Two Reaction Model

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Spatial heterogeneity

concentrations are different in different places, n = f(t,x,y,z)diffusion superimposed on chemical reactions:

partial differential equation

diffusion)()(

±−=∂

∂xyz

xyz tnttn

λ

Breitling / 50

Spatial heterogeneity

one-dimensional case (diffusion only, and conservation of mass)

xn(t,x)K

xxxtnK

xt

xtn

∂∂

−=

∂Δ+∂

−=

−=Δ∂

∂

inflow

),(outflow

outflowinflow),(

Δx

inflow outflow

Breitling / 51

Spatial heterogeneity – 2

)()()(:reaction chemicaln with Combinatio

),,,(),,,(:dimensions for three Shorthand

),(),(:equationdiffusion get oequation t aldifferenti toTransition

),(),(),(

2

2

2

2

tnKtnttn

zyxtnKt

zyxtn

xxtnK

txtn

xxtnK

xxxtnKx

txtn

∇+−=∂

∂

∇=∂

∂

∂∂

=∂

∂

∂∂

−∂

Δ+∂=Δ

∂∂

λ

Breitling / 52

Acknowledgements

David Gilbert, Brunel University, London

Monika Heiner, Cottbus University, Germany

Robin Donaldson, Glasgow University

The Groningen Bioinformatics Centre (Netherlands) is expanding its young and successful team.

Several PhD and Postdoc positions are available for creative bioinformaticians with an interest in Systems Biology, Metabolomics, Proteomics, Quantitative Genetics, Network Reconstruction, Dynamic Modelling…

For more information and to apply……visit www.rug.nl/gbic…e-mail r.breitling@rug.nl…talk to Rainer Breitling at Petri Nets 2009

Recent GBiC papers: Breitling R et al. New surveyor tools for charting microbial metabolic maps Nature Reviews Microbiology (2008). Fu J et al. MetaNetwork: a computational protocol for the genetic study of metabolic networks Nature Protocols (2007). Swertz MA et al. Beyond standardization: dynamic software infrastructures for systems biology Nature Reviews Genetics(2007). Keurentjes JJB et al. The genetics of plant metabolism Nature Genetics (2006). Hoeller D et al. Regulation of ubiquitin-binding proteins by monoubiquitylation Nature Cell Biology(2006). Bystrykh L et al. Uncovering regulatory pathways that effect hematopoietic stem cell function using ’genetical genomics’ Nature Genetics (2005).