Modelling Biochemical Pathways in PEPA

1

Modelling Biochemical Pathways in PEPA

Muffy CalderDepartment of Computing Science

University of Glasgow

Joint work with Jane Hillston and Stephen Gilmore

October 2004

2

Are you in the right room?

Yes, this is computing science!

Question

Can we apply computing science theory and tools to biochemical pathways?

If so,

What analysis do these new models offer?How do these models relate to traditional ones?What are the implications for life scientists?What are the implications for computing science?

3

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

4

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

αβγ

heterotrimericG-protein

cytosol

αβγ

heterotrimericG-protein

αβγααβγββγγ

heterotrimericG-protein

cytosol

tyrosinekinaseβ

γtyrosinekinaseββ

γγSOSshc

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

αAdCyc

cAMPATP

ααAdCycAdCyc

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

transcriptionfactors

transcriptionfactors

transcriptionfactors

MKPMKPMKP

A little more complex.. pathways/networks

5

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RKIP Inhibited ERK Pathwaym1

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

k15

m13

k14

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

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RKIP Inhibited ERK Pathwaym1

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

k15

m13

k14

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

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RKIP protein expression is reduced in breast cancers

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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

k15

m13

k14

10

RKIP Inhibited ERK Pathway

This network seems to be very similar

to producer/consumer networks.

Why not to try usingprocess algebras for

modelling?

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

k15

m13

k14

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Why process algebras for pathways?

• Process algebras are high level formalisms that make interactions and constraints explicit. Structure becomes apparent.

• Reasoning about livelocks and deadlocks.

• Reasoning with (temporal) logics.

• Equivalence relations between high level descriptions.

• Stochastic process algebras allow performance analysis.

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Process algebra(for dummies)

High level descriptions of interaction, communication and synchronisation

Event α (simple), α!34 (data offer), α?x (data receipt)

Prefix α.S

Choice S + S

Synchronisation P |l| P α l independent concurrent (interleaved) actions

α l synchronised action

Constant A = S assign names to components

Laws P1 + P2 P2 + P1

Relations (bisimulation) a

b c

aaaaa

c bbbc

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PEPA Process algebra with performance, invented by

Jane Hillston

Prefix (α,r).S

Choice S + S competition between components (race)

Cooperation/ P |l| P a l independent concurrent (interleaved) actions

Synchronisation a l shared action, at rate of slowest

Constant A = S assign names to components

P ::= S | P |l| P

S ::= (α,r).S | S+S | A

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Performance of Action

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.55 1.1 1.65 2.2 2.75 3.3 3.85 4.4 4.95 5.5 6.05 6.6 7.15 7.7 8.25 8.8 9.35 9.9

10.4511

11.55 12.112.6513.213.75 14.314.8515.415.95

t

P(t)

Rates is a rate, from which a probability is derived

tetP −−=1)(

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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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Modelling Signalling Dynamics

• There is an important difference between computing science networks and biochemical networks

• We have to distinguish between the individual and the population.

• Previous approaches have modelled at molecular level (individual)– Simulation– State space explosion– Relation to population (what can be inferred?)

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Signalling Dynamics

m1

P1

m2

P2

k1/k2

m5

P5

k6/k7

m6

P6

m4

P5/P6

Reagent view: model whether or not a reagent can participate in a reaction (observable/unobservable).

k4

m3

k3

P1/P2

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Signalling Dynamics

m1

P1

m2

P2

k1/k2

m5

P5

k6/k7

m6

P6

m4

P5/P6

Reagent view: 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

P2H = (k1react,k1). P2L

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

P6H = (k6react,k6). P6L

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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Signalling Dynamics

m1

P1

m2

P2

k1/k2

m5

P5

K6/k7

m6

P6

m4

P5/P6

Reagent view: model configuration

P1H |k1react,k2react|

P2H | k1react,k2react,k4react |

P1/P2L |k1react,k2react,k3react|

P5L |k3react,k6react,k4react|

P6H |k6react,k7react|

P5/P6L

Assuming initial concentrations of m1,m2,m6.

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

k15

m13

k14

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Signalling DynamicsReagent view: model 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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Signalling Dynamics

m1

P1

m2

P2

k1/k2

m5

P5

K6/k7

m6

P6

m4

P5/P6

Pathway view: model chains of behaviour flow

k4

m3

k3

P1/P2

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Signalling Dynamics

m1

P1

m2

P2

k1/k2

m5

P5

K6/k7

m6

P6

m4

P5/P6

Pathway view: 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

k15

m13

k14

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Pathway view: model configuration

Pathway10 |k12react,k13react,k14product| Pathway40

|k3react,k4react,k5product,k6react,k7react,k8product| Pathway30

|k1react,k2react,k3react,k4react,k5product| Pathway20

|k9react,k10react,k11product| Pathway10

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What is the difference?

• 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

Formal proof shows that those two models are equivalent!

This equivalence proof, based on bisimulation, unites two views of the same biochemical pathway.

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state reagent-view s1 Raf-1*H, RKIPH,Raf-1*/RKIPL,Raf-1*/RKIPERK-PPL, ERKL,RKIP-PL, RKIP-P/RPL, RPH, MEKL,MEK/Raf-1*L,MEK-PPH,MEK-PP/ERKL/ERK-PPH

pathway view Pathway50,Pathway40,Pathway20,Pathway10

s2 …

.

.

.

s28

(28 states)

State space of reagent and pathway model

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State space of reagent and pathway model

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Quantitative Analysis

Generate steady-state probability distribution (using linear algebra).

1. Use state finder (in reagent model) to aggregate probabilities.

Example increase k1 from 1 to 100 and the probability of being in a state with ERK-PPH

drops from .257 to .005

2. Perform throughput analysis (in pathway model)

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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

32

Quantitative Analysis

Effect of increasing the rate of k1 on k14product throughput (rate x probability)i.e. effect of binding of RKIP to Raf-1* on MEK-PP

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Quantitative Analysis - Conclusion

Increasing the rate of binding of RKIP to Raf-1* dampens down the k14product and k8product reactions,

In other words,

it dampens down the ERK pathway.

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Signalling Dynamics

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.

k4

m3

k3

P1/P2

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Signalling Dynamics

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

Differential equations

Each row is labelled by a protein concentration. One equation per row.

For row r,

dr = column c A[r,c]) * row x f(A[x,c])

dt

where f(A[x,c]) = if (A[x,c]== -) then x else 1

a rate is a product of the rate constant and current concentration of substrates consumed.

k4

m3

k3

P1/P2

36

Signalling Dynamics

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

Differential equations (mass action)

dm1 = - k1 + k2 (two terms)

dt

k4

m3

k3

P1/P2

37

Signalling Dynamics

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

Differential equations (mass action)

dm1 = - k1*m1*m2 + k2

dt

k4

m3

k3

P1/P2

38

Signalling Dynamics

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

Differential equations (mass action)

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

dt

k4

m3

k3

P1/P2

39

Signalling Dynamics

m1

P1

m2

P2

k1/k2

m5

P5

K6/k7

m6

P6

m4

P5/P6

Differential equations (mass action)

For RKIP inhibited ERK pathway, change in Raf-1* is:

k4

m3

k3

P1/P2dm1 = - k1*m1*m2 + k2*m3 + k5*m4 – k12*m1*m12 dt +k13*m13 + k14*m13

(catalysis, inhibition, etc. )

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Discussion & Conclusions• Regent-centric view

– probabilities of states (H/L)– differential equations– fit with data

• Pathway-centric view– simpler model – building blocks, modularity approach

– no further information is gained from having multiple levels.

• Life science – (some) see potential of an interaction approach

• Computing science– individual/population view– continuous, traditional mathematics

41

Further Challenges

• Derivation of the reagent-centric model from experimental data.

• Derivation of pathway-centric models from reagent-centric models and vice-versa.

• Quantification of abstraction over networks – “chop” off bits of network

• Model spatial dynamics (vesicles).

42

The End

Thank you.