Modelling Biochemical Pathways in PEPA

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Modelling Biochemical Pathways in PEPA
Muffy CalderDepartment of Computing Science
University of GlasgowJoint 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 by
  signalling
• most signalling via membrane receptors
• signalling molecule
• receptor
• gene effects

movement of signal from outside cell to inside
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A little more complex.. pathways/networks
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(No Transcript)
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RKIP Inhibited ERK Pathway
Raf-1
MEK
RKIP
m1
m2
m2
m2
m12
k1
k1
k1/k2
K12/k13
ERK-PP
k11
k15
m3
m9
m3
m3
Raf-1/RKIP
m13
k3
k3
k3
k8
m11
RKIP-P/RP
MEK-PP/ERK-P
m4
m8
Raf-1/RKIP/ERK-PP
k14
k5
k9/k10
k6/k7
m5
m6
m7
m10
MEK-PP
ERK
RKIP-P
RP
From paper by Cho, Shim, Kim, Wolkenhauer,
McFerran, Kolch, 2003.
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RKIP Inhibited ERK Pathway
Raf-1
MEK
RKIP
m1
m2
m2
m2
m12
k1
k1
k1/k2
k12/k13
ERK-PP
k11
k15
m3
m9
m3
m3
Raf-1/RKIP
m13
k3
k3
k3
k8
m11
RKIP-P/RP
MEK-PP/ERK-P
m4
m8
Raf-1/RKIP/ERK-PP
k14
k5
k9/k10
k6/k7
m5
m6
m7
m10
MEK-PP
ERK
RKIP-P
RP
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
Raf-1
MEK
RKIP

proteins/complexes forward
/backward
reactions (associations/disassoc
iations) products
(disassociations) m1, m2 .. concentrations of
proteins k1,k2 .. rate
(performance) coefficients
m1
m2
m2
m2
m12
k1
k1
k1/k2
k12/k13
ERK-PP
k11
k15
m3
m9
m3
m3
Raf-1/RKIP
m13
k3
k3
k3
k8
m11
RKIP-P/RP
MEK-PP/ERK-P
m4
m8
Raf-1/RKIP/ERK-PP
k14
k5
k9/k10
k6/k7
m5
m6
m7
m10
MEK-PP
ERK
RKIP-P
RP
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RKIP Inhibited ERK Pathway
Raf-1
MEK
RKIP
m1
m2
m2
m2
m12
This network seems to be very similar to
producer/consumer networks. Why not to try
using process algebras for modelling?
k1
k1
k1/k2
k12/k13
ERK-PP
k11
k15
m3
m9
m3
m3
Raf-1/RKIP
m13
k3
k3
k3
k8
m11
RKIP-P/RP
MEK-PP/ERK-P
m4
m8
Raf-1/RKIP/ERK-PP
k14
k5
k9/k10
k6/k7
m5
m6
m7
m10
MEK-PP
ERK
RKIP-P
RP
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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 a (simple), a!34 (data offer),
  a?x (data receipt)
• Prefix a.S
• Choice S S
• Synchronisation P l P a e l independent
  concurrent (interleaved) actions
• a e l
  synchronised action
• Constant A S assign names to components
• Laws P1 P2 _at_ P2 P1
• Relations _at_ (bisimulation)
  

a
a
a
a
a
a
_at_
b
_at_
c
b
c
b
b
c
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PEPA

• Process algebra with performance, invented by
  Jane Hillston
• Prefix (a,r).S
• Choice S S competition between components
  (race)
• Cooperation/ P l P a e l independent
  concurrent (interleaved) actions
• Synchronisation a e l shared
  action, at rate of slowest
• Constant A S assign names to components
• P S P l P
• S (a,r).S
  SS A

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Rates

• l is a rate, from which a probability is derived

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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
P1
P2
m2
m1
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.
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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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
Reagent view model whether or not a reagent can
participate in a reaction (observable/unobservable
).
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
k6/k7
m6
m5
P6
P5
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Signalling Dynamics
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
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
k6/k7
m6
m5
P6
P5
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Signalling Dynamics
Reagent view model configuration P1H
k1react,k2react P2H k1react,k2react,k4r
eact P1/P2L k1react,k2react,k3react
P5L k3react,k6react,k4r
eact P6H
k6react,k7react
P5/P6L Assuming initial concentrations
of m1,m2,m6.
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Raf-1
MEK
RKIP
m1
m2
m2
m2
m12
k1
k1
k1/k2
k12/k13
ERK-PP
k11
k15
m3
m9
m3
m3
Raf-1/RKIP
m13
k3
k3
k3
k8
m11
RKIP-P/RP
MEK-PP/ERK-P
m4
m8
Raf-1/RKIP/ERK-PP
k14
k5
k9/k10
k6/k7
m5
m6
m7
m10
MEK-PP
ERK
RKIP-P
RP
Reagent view Raf-1H (k1react,k1). Raf-1L
(k12react,k12). Raf-1L Raf-1L
(k5product,k5). Raf-1H (k2react,k2). Raf-1H
(k13react,k13). Raf-1H (k14product,k14).
Raf-1H (26 equations)
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Signalling Dynamics
Reagent view model configuration Raf-1H
k1react,k12react,k13react,k5product,k14product
RKIPH k1react,k2react,k11product
Raf-1H/RKIPL k3react,k4react
Raf-1/RKIP/ERK-PPL k3react,k4react,k5product
ERK-PL k5product,k6react,k7rea
ct RKIP-PL
k9react,k10react
RKIP-PLk9react,k10react
RKIP-P/RPLk9react,k10react,k11product
RPH
MEKLk12react,k13react,k15prod
uct
MEK/Raf-1Lk14product
MEK-PPH k8product,k6react,k7re
act
MEK-PP/ERKLk8product

MEK-PPHk8product
ERK-PPH
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Signalling Dynamics
Pathway view model chains of behaviour flow
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Signalling Dynamics
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!)
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Raf-1
MEK
RKIP
m2
m2
m1
m2
m12
k1
k1
k1/k2
k12/k13
ERK-PP
k11
k15
m3
m3
m3
m9
Raf-1/RKIP
m13
k3
k3
k3
k8
m11
RKIP-P/RP
MEK-PP/ERK-P
m4
m8
Raf-1/RKIP/ERK-PP
k14
k5
k9/k10
k6/k7
m5
m6
m7
m10
MEK-PP
ERK
RKIP-P
RP
Pathway view Pathway10 (k9react,k9).
Pathway11 Pathway11 (k11product,k11). Pathway10
(k10react,k10). Pathway10 (5 pathways)
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Pathway view model configuration Pathway10
k12react,k13react,k14product Pathway40
k3react,k4react,k5product,k6react,k7
react,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-1H,
RKIPH,Raf-1/RKIPL,Raf-1/RKIPERK-PPL,
ERKL,RKIP-PL, RKIP-P/RPL, RPH,
MEKL,MEK/Raf-1L,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

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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 the k14product and k8product
  reactions,
• In other words,
• it dampens down the ERK pathway.

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Signalling Dynamics
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.
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Signalling Dynamics
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 S column c Ar,c) P
row x f(Ax,c) dt where f(Ax,c) if
(Ax,c -) then x else 1 a rate is a product
of the rate constant and current concentration of
substrates consumed.
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Signalling Dynamics
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
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Signalling Dynamics
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
- k1m1m2 k2 dt
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Signalling Dynamics
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
- k1m1m2 k2m3 (nonlinear) dt
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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Signalling Dynamics
Differential equations (mass action) For
RKIP inhibited ERK pathway, change in Raf-1 is
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
dm1 - k1m1m2 k2m3 k5m4 k12m1m12
dt k13m13 k14m13
(catalysis, inhibition, etc. )
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5
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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.