Modelling Cell Signalling Pathways in PEPA

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Modelling Cell Signalling Pathways in PEPA

Muffy CalderDepartment of Computing Science

University of Glasgow

Jane Hillston and Stephen GilmoreLaboratory for Foundations of Computer Science

University of Edinburgh

February 2005

2

Cell Signalling or Signal Transduction*

• fundamental cell processes (growth, division, differentiation, apoptosis) determined by signalling

• most signalling via membrane receptors

signalling molecule

receptor

gene effects

* movement of signal from outside cell to inside

3

Abbreviations and notes•7-TMR: seven trans-membrane receptor•small G-proteins: Rap1, Ras, Rac; active when GTP bound•cAMP-GEF: cAMPactivated GTP-Exchange-Factor•AdCyc: Adenylate cyclase•PDE: Phhosphodiesterase•PKA: cAMP activated protein kinase•adaptor proteins: shc, grb2•SOS: Son-of-Sevenless, a GEF for Ras•PI-3 K: Phosphatidylinositol-3 kinase•Akt: a kinase activated by PI-3 K via PI-3 and another kinase, PDK•PAK: a kinased activated by binding to Rac•MKP: MAPK phosphatase, dephosphorylates MAPKs

activation inhibition phosphorylationactivation inhibition phosphorylation

cell membraneReceptor

e.g. 7-TMRcell membrane

Receptore.g. 7-TMR

Receptore.g. 7-TMR

heterotrimericG-protein

cytosol



cytosol

tyrosinekinase

tyrosinekinase

SOSshc

grb2SOSSOSshcshc

grb2grb2

RasRas

Raf-1Raf-1Raf-1

MEKERK1,2

MEKMEKERK1,2ERK1,2

PI-3 K

Ras

PI-3 K

PI-3 K

RasRasAktAktAkt

PAK

Rac

PAKPAK

RacRac

PKAcAMP

PKAcAMP

PKAPKAcAMPcAMP

AdCyc

cAMPATP

AdCycAdCyc

cAMPATP cAMPcAMPATP

PKAcAMP

PKAPKAcAMPcAMP

cAMPGEF

cAMP

cAMPGEFcAMPGEF

cAMPcAMPRap1Rap1

MEK1,2

ERK1,2

B-Raf

MEK1,2

ERK1,2

MEK1,2MEK1,2

ERK1,2ERK1,2

B-RafB-RafB-Raf

PDE

cAMP AMP

PDE

cAMP AMP

PDEPDE

cAMP AMPcAMPcAMP AMP

nucleus

transcriptionfactors

nucleusnucleus




MKPMKPMKP

A little more complex.. pathways/networks

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RKIP Inhibited ERK Pathway

proteins/complexes

forward /backward

reactions (associations/disassociations)

products

(disassociations)

m1, m2 .. concentrations of

proteins

k1,k2 ..: rate (performance)

coefficients

m1

Raf-1*

m2

k1

m3 Raf-1*/RKIP

m12

MEK

k12/k13

m7

MEK-PP

k6/k7

m5

ERK

m8MEK-PP/ERK-P

k8

m9

ERK-PP

k3

m4

k5

m6

RKIP-P

m10

RP

k9/k10

m11

RKIP-P/RP

k11

m2

k1

m3

k3

Raf-1*/RKIP/ERK-PP

m2

RKIP

k1/k2

m3

K3/k4

k15

m13

k14

From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.

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Pathways have computational

content!

Producers and

consumers.

Feedback.

m1

Raf-1*

m2

k1

m3 Raf-1*/RKIP

m12

MEK

k12/k13

m7

MEK-PP

k6/k7

m5

ERK

m8MEK-PP/ERK-P

k8

m9

ERK-PP

k3

m4

k5

m6

RKIP-P

m10

RP

k9/k10

m11

RKIP-P/RP

k11

m2

k1

m3

k3

Raf-1*/RKIP/ERK-PP

m2

RKIP

k1/k2

m3

k3/k4

k15

m13

k14

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Why not useprocess algebras for

modelling?

High level formalisms that make interactions

andevent rates

explicit.

m1

Raf-1*

m2

k1

m3 Raf-1*/RKIP

m12

MEK

k12/k13

m7

MEK-PP

k6/k7

m5

ERK

m8MEK-PP/ERK-P

k8

m9

ERK-PP

k3

m4

k5

m6

RKIP-P

m10

RP

k9/k10

m11

RKIP-P/RP

k11

m2

k1

m3

k3

Raf-1*/RKIP/ERK-PP

m2

RKIP

k1/k2

m3

k3/k4

k15

m13

k14

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Process algebra(for dummies)

High level descriptions of interaction, communication and synchronisation

Event Prefix .PChoice P1 + P2Synchronisation P1 |l| P2 ¬( l) independent concurrent (interleaved)

actions l synchronised actionConstant A = P assign names to components

Relations (bisimulation ) Laws P1 + P2 P2 + P1

a

b c

aaaaa

c bbbc

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PEPA

Markovian process algebra invented by Jane Hillston with workbench by Stephen Gilmore.

PEPA descriptions denote continuous Markov chains.

Prefix (,).PChoice P1 + P2 competition between components (race)Cooperation/ P1 |l| P1 ¬(a l) independent concurrent (interleaved) actionsSynchronisation a l shared action, at rate of slowestConstant A = P assign names to components

Performance of Action

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

P(t

)

tetP 1)(

is a rate, from which a probability is derived - exponential distribution.

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Modelling the ERK Pathway in PEPA

• Each reaction is modelled by an event, which has a performance coefficient.

• Each protein is modelled by a process which synchronises others involved in a reaction.

(reagent-centric view)

• Each sub-pathway is modelled by a process which synchronises with other sub-pathways.

(pathway-centric view)

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Signalling Dynamics

m1

P1

m2

P2

k1/k2

m5

P5

K6/k7

m6

P6

m4

P5/P6

Reaction Producer(s) Consumer(s)

k1react {P2,P1} {P1/P2}

k2react {P1/P2} {P2,P1}

k3product {P1/P2} {P5}

…

k1react will be a 3-way synchronisation,

k2react will be a 3-way synchronisation,

k3product will be a 2-way synchronisation.

k4

m3

k3

P1/P2

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Reagent View

m1

P1

m2

P2

k1/k2

m5

P5

k6/k7

m6

P6

m4

P5/P6

Model whether or not a reagent can participate in a reaction (observable/unobservable): each reagent gives rise to a pair of definitions.

P1H = (k1react,k1). P1L

P1L = (k2react,k1). P2H


P2L = (k2react,k2). P2H + (k4react). P2H

P1/P2H = (k2react,k2). P1/P2L + (k3react, k3). P1/P2L

P1/P2L = (k1react,k1). P1/P2H

P5H = (k6react,k6). P5L + (k4react,k4). P5L

P5L = (k3react,k3). P5H +(k7react,k7). P5H


P6L = (k7react,k7). P6H

P5/P6H = (k7react,k7). P5/P6L

P5/P6L = (k6react,k6) . P5/P6H

k4

m3

k3

P1/P2

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Reagent view:

Raf-1*H = (k1react,k1). Raf-1*L + (k12react,k12). Raf-1*L

Raf-1*L = (k5product,k5). Raf-1*H +(k2react,k2). Raf-1*H + (k13react,k13). Raf-1*H + (k14product,k14). Raf-1*H

…

(26 equations)

m1

Raf-1*

m2

k1

m3 Raf-1*/RKIP

m12

MEK

k12/k13

m7

MEK-PP

k6/k7

m5

ERK

m8MEK-PP/ERK-P

k8

m9

ERK-PP

k3

m4

k5

m6

RKIP-P

m10

RP

k9/k10

m11

RKIP-P/RP

k11

m2

k1

m3

k3

Raf-1*/RKIP/ERK-PP

m2

RKIP

k1/k2

m3

k3/k4

k15

m13

k14

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Reagent Viewmodel configuration

Raf-1*H |k1react,k12react,k13react,k5product,k14product|

RKIPH | k1react,k2react,k11product |

Raf-1*H/RKIPL |k3react,k4react|

Raf-1*/RKIP/ERK-PPL |k3react,k4react,k5product|

ERK-PL |k5product,k6react,k7react|

RKIP-PL |k9react,k10react|

RKIP-PL|k9react,k10react|

RKIP-P/RPL|k9react,k10react,k11product|

RPH||

MEKL|k12react,k13react,k15product|

MEK/Raf-1*L|k14product|

MEK-PPH |k8product,k6react,k7react|

MEK-PP/ERKL|k8product|

MEK-PPH|k8product|

ERK-PPH

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Pathway View

m1

P1

m2

P2

k1/k2

m5

P5

K6/k7

m6

P6

m4

P5/P6

Model chains of behaviour flow.

Two pathways, corresponding to initial concentrations:

Path10 = (k1react,k1). Path11

Path11 = (k2react).Path10 + (k3product,k3).Path12

Path12 = (k4product,k4).Path10 + (k6react,k6).Path13

Path13 = (k7react,k7).Path12

Path20 = (k6react,k6). Path21

Path21 = (k7react,k6).Path20

Pathway view: model configuration

Path10 | k6react,k7react | Path20

(much simpler!)

k4

m3

k3

P1/P2

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Pathway view:

Pathway10 = (k9react,k9). Pathway11

Pathway11 = (k11product,k11). Pathway10 + (k10react,k10). Pathway10

…

(5 pathways)

m1

Raf-1*

m2

k1

m3 Raf-1*/RKIP

m12

MEK

k12/k13

m7

MEK-PP

k6/k7

m5

ERK

m8MEK-PP/ERK-P

k8

m9

ERK-PP

k3

m4

k5

m6

RKIP-P

m10

RP

k9/k10

m11

RKIP-P/RP

k11

m2

k1

m3

k3

Raf-1*/RKIP/ERK-PP

m2

RKIP

k1/k2

m3

k3/k4

k15

m13

k14

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What is the difference between the two views/models?

• reagent-centric view is a fine grained view

• pathway-centric view is a coarse grained view

– reagent-centric is easier to derive from data– pathway-centric allows one to build up networks from already

known components

The two models are equivalent!

The equivalence proof, based on bisimulation between steady state solutions, unites two views of the same biochemical pathway.

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1 0.041350790041564812 0.0208061151023106323 0.073467759296928994 0.0069353717007701525 0.065161040166416726 0.037375466220971197 0.0113367157494711948 0.0360482059335932869 0.00463984157716770810 0.00569139435096023711 0.0413845661862080312 0.002582808982032050513 0.00480778362079702414 0.0481712379850729615 0.01864067106983505516 0.01674353961951514217 0.0216287435105674518 0.002891255249280381619 0.00497023810042315820 0.0207678071832230221 0.184005485148599922 0.00884605267233758523 0.0141321835645967824 0.003048222164904722425 0.002084470415146022326 0.2047732923318231227 0.0964257689104687428 0.0012831731450123965

Reagent view

1 0.041350790041563532 0.0208061151023106043 0.073467759296924194 0.0069353717007698345 0.065161040166412626 0.037375466220967837 0.0113367157494708898 0.036048205933591569 0.00569139435095978710 0.00463984157716754311 0.0413845661862075212 0.0481712379850750513 0.002582808982031824614 0.0186406710698350415 0.00480778362079673716 0.0167435396195150717 0.02076780718322434518 0.02162874351056822219 0.1840054851486054920 0.00289125524928003821 0.00884605267233746422 0.00497023810042342423 0.01413218356459749924 0.2047732923318296425 0.0964257689104713926 0.003048222164904605327 0.002084470415145398328 0.0012831731450119671

Pathway view

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State space of reagent and pathway model

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What do you do with these two models?

-investigate properties of the underlying Markov model.

Generate steady-state probability distribution (using linear algebra) and then perform;

-Transient analysis

e.g. analysis to determine whether a state will be reached.

OR

-Steady state analysis (more appropriate here)

e.g. analysis of the steady state solution.

Note: there isn’t one steady state, but a very large “cycle”!

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Quantitative Analysis

Effect of increasing the rate of k1 on k8product throughput (rate x probability)i.e. effect of binding of RKIP to Raf-1* on ERK-PP.

Increasing the rate of binding of RKIP to Raf-1* dampens down the k8product reactions, i.e. it dampens down the ERK pathway.

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Quantitative Analysis – by logicSteady state analysis

Formula S=? [ERK_PP_H_STATE = 0]

PRISM result (after translation):

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Quantitative Analysis – by logicNow reduce backward rates (.53)

Formula S=? [ERK_PP_H_STATE = 0]

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Reagent view and ODEs

m1

P1

m2

P2

k1/k2

m5

P5

K6/k7

m6

P6

m4

P5/P6

Activity matrix

k1 k2 k3 k4 k5 k6 k7

P1 -1 +1 0 0 0 0 0

P2 -1 +1 0 +1 0 0 0

P1/P2 +1 -1 0 0 0 0 0

P5 0 0 +1 -1 0 -1 +1

P6 0 0 0 0 0 -1 +1

P5/P6 0 0 0 0 0 +1 -1

Column: corresponds to a single reaction.

Row: correspond to a reagent; entries indicate whether the concentration is +/- for that reaction.

mass action dynamics:

dm1 = - k1*m1*m2 + k2*m3 (nonlinear)dt

Reagent views tells us producer or consumer.

k4

m3

k3

P1/P2

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Big Picture

Benefits

Interactions

Relative change

Abstraction

Behaviour patterns

Quantitative analysis

stochasticprocess algebra

pathwayview

reagentview

mass actiondifferential equations

Continuous timeMarkov chains

abstractionpathway composition

throughput analysis

denote

derive

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Bigger Picture

Benefits

Interactions

Relative change

Abstraction

Behaviour patternsQuantitative analysis

stochasticprocess algebra

pathwayview Matlab

reagentview

mass actiondifferential equations

multilevel reagent view

PRISM

Continuous timeMarkov chains

experimentaldata

simulate

logic

denote

derive

abstractionpathway composition

throughput analysis

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Further Challenges

• Derivation of the reagent-centric model from experimental data

• Quantification of abstraction over networks – zoom in or out

• Other dynamics (inhibition)

• Functional rates

• Very large scale pathways

• Model spatial dynamics (vesicles).

Modelling Cell Signalling Pathways in PEPA

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