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1

Modelling BiochemicalPathways in PEPA

Muffy CalderDepartment of Computing Science

University of Glasgow

Joint work with Jane Hillston and Stephen GilmoreOctober 2004

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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?

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Cell Signalling or Signal Transduction*• fundamental cell processes (growth, division, differentiation, apoptosis) determined bysignalling

• most signalling via membrane receptors

signalling molecule

receptor

gene effects

* movement of signal from outside cell to inside4

A little more complex.. pathways/networks

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5 6

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-Pm10

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-Pm10

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-Pm10

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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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-Pm10

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 andconstraints 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 andsynchronisation

Event α (simple), α!34 (data offer), α?x (data receipt)Prefix α.SChoice S + SSynchronisation P |l| P α ε l independent concurrent (interleaved) actions α ε l synchronised actionConstant A = S assign names to components

Laws P1 + P2 ≅ P2 + P1Relations ≅ (bisimulation)

ab c

aaaaa

c bbbc≅ ≅

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PEPAProcess algebra with performance, invented by Jane

Hillston

Prefix (α,r).SChoice S + S competition between components (race)Cooperation/ P |l| P a ε l independent concurrent (interleaved) actionsSynchronisation a ε l shared action, at rate of slowestConstant 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

0

0.55

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

10.45

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11.55

12.1

12.65

13.2

13.75

14.3

14.85

15.4

15.95

t

P(t

)

Ratesλ 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 performancecoefficient.

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

(reagent-centric view)

• Each sub-pathway is modelled by a process which synchronises withother 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

5

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Modelling Signalling Dynamics

• There is an important difference between computing science networks andbiochemical 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

P1m2

P2

k1/k2

m5

P5

k6/k7

m6

P6

m4

P5/P6

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

k4

m3

k3

P1/P2

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Signalling Dynamics

m1

P1m2

P2

k1/k2

m5

P5

k6/k7

m6

P6

m4

P5/P6

Reagent view: model whether or not a reagent can participate in areaction (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

P1m2

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

6

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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-Pm10

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

P1m2

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

P1m2

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-Pm10

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 ofthe same biochemical pathway. 28

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.

Exampleincrease 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

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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 thek14product and k8product reactions,

In other words,

it dampens down the ERK pathway.

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Signalling Dynamics

m1

P1m2

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 theconcentration is +/- for that reaction.

k4

m3

k3

P1/P2

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Signalling Dynamics

m1

P1m2

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 = Σ column c A[r,c]) * Π 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 ofsubstrates consumed.

k4

m3

k3

P1/P2

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Signalling Dynamics

m1

P1m2

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

10

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Signalling Dynamics

m1

P1m2

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

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Signalling Dynamics

m1

P1m2

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

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Signalling Dynamics

m1

P1m2

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*m12dt +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

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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).

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The End

Thank you.