Modelling Cell Signalling Pathways in PEPA

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Modelling Cell Signalling Pathways in PEPA
Muffy CalderDepartment of Computing Science
University of GlasgowJane Hillston and Stephen
GilmoreLaboratory for Foundations of Computer
Science University of EdinburghFebruary 2005
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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

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/k4
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
Pathways have computational content! Producers
and consumers. Feedback.
k1
k1
k1/k2
k12/k13
ERK-PP
k11
k15
m3
m9
m3
m3
Raf-1/RKIP
m13
k3
k3
k3/k4
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
Why not use process algebras for modelling? High
level formalisms that make interactions and eve
nt rates explicit.
k1
k1
k1/k2
k12/k13
ERK-PP
k11
k15
m3
m9
m3
m3
Raf-1/RKIP
m13
k3
k3
k3/k4
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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Process algebra(for dummies)

• High level descriptions of interaction,
  communication and synchronisation
• Event a
• Prefix a.P
• Choice P1 P2
• Synchronisation P1 l P2 (a e l) independent
  concurrent (interleaved) actions
• 
  a e l synchronised action
• Constant A P assign names to components
• Relations _at_ (bisimulation )
• Laws P1 P2 _at_ P2 P1

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

• Markovian process algebra invented by Jane
  Hillston with workbench by Stephen Gilmore.
• PEPA descriptions denote continuous Markov
  chains.
• Prefix (a,l).P
• Choice P1 P2 competition between components
  (race)
• Cooperation/ P1 l P1 (a e l) independent
  concurrent (interleaved) actions
• Synchronisation a e l shared
  action, at rate of slowest
• Constant A P assign names to
  components

l 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
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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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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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.
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/k4
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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Reagent View
model configuration Raf-1H k1react,k12react,k13
react,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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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/k4
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 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
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

• Reagent view

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 isnt 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 logic

• Steady state analysis
• Formula S? ERK_PP_H_STATE 0
• PRISM result (after translation)

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Quantitative Analysis by logic

• Now reduce backward rates (.53)
• Formula S? ERK_PP_H_STATE 0

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Reagent view and ODEs
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 -
k1m1m2 k2m3 (nonlinear) dt Reagent
views tells us producer or consumer.
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
K6/k7
m6
m5
P6
P5

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Big Picture
abstraction pathway composition
stochastic process algebra
throughput analysis
pathway view
reagent view
mass action differential equations
derive
denote

• Benefits
• Interactions
• Relative change
• Abstraction
• Behaviour patterns
• Quantitative analysis

Continuous time Markov chains
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Bigger Picture
abstraction pathway composition
experimental data
stochastic process algebra
throughput analysis
pathway view
reagent view
mass action differential equations
derive
Matlab
multilevel reagent view
denote
simulate

• Benefits
• Interactions
• Relative change
• Abstraction
• Behaviour patterns
• Quantitative analysis

PRISM
logic
Continuous time Markov chains
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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).