1. Modelling Cell Signalling Pathways in PEPA
Muffy Calder Department of Computing Science University of Glasgow Jane Hillston and Stephen Gilmore Laboratory 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. A little more complex.. pathways/networks

5. RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
proteins/complexes
forward /backward
reactions (associations/disassociations)
products
(disassociations)
m1, m2 .. concentrations of
proteins
k1,k2 ..: rate (performance)
coefficients
m12 m1 m2 m2 m2
k1/k2 k1 k1
k12/k13
ERK-PP
k11
k15 m9 m3 m3 m3
Raf-1*/RKIP
m13
K3/k4 k3 k3 k8
m11
RKIP-P/RP
MEK-PP/ERK-P m4 m8
Raf-1*/RKIP/ERK-PP
k14 k5
k9/k10
k6/k7 m7 m5 m6
m10
MEK-PP
ERK
RKIP-P RP
From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.

6. RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12 m1 m2 m2 m2
Pathways have
computational
content!
Producers
and
consumers.
Feedback.
k1/k2 k1 k1
k12/k13
ERK-PP
k11
k15 m9 m3 m3 m3
Raf-1*/RKIP
m13
k3/k4 k3 k3 k8
m11
RKIP-P/RP
MEK-PP/ERK-P m4 m8
Raf-1*/RKIP/ERK-PP
k14 k5
k9/k10
k6/k7 m7 m5 m6
m10
MEK-PP
ERK
RKIP-P RP

7. RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12 m1 m2 m2 m2
Why not use
process algebras for
modelling?
High level formalisms
that make
interactions
and
event rates
explicit.
k1/k2 k1 k1
k12/k13
ERK-PP
k11
k15 m9 m3 m3 m3
Raf-1*/RKIP
m13
k3/k4 k3 k3 k8
m11
RKIP-P/RP
MEK-PP/ERK-P m4 m8
Raf-1*/RKIP/ERK-PP
k14 k5
k9/k10
k6/k7 m7 m5 m6
m10
MEK-PP
ERK
RKIP-P RP

8. 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 @ (bisimulation )
Laws P1 + P2 + P1 a a a a a a @ @ b c c b b b c

9. 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.

10. 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)

11. Signalling Dynamics P2 P1 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 m5 m6 P6 P5

12. 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 m5 m6 P6 P5

13. Reagent View Model configuration P2 P1 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 m5 m6 P6 P5

14. MEK Raf-1* RKIP m12 m1 m2 m2 m2 k1 k1 ERK-PP m9 m3 m3 m3 Raf-1*/RKIP
RKIP-P/RP
MEK-PP/ERK-P m4 m8
Raf-1*/RKIP/ERK-PP
k14 k5
k9/k10
k6/k7 m7 m5 m6
m10
MEK-PP
ERK
RKIP-P RP
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)

15. Reagent 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

16. Pathway View Model chains of behaviour flow. P2 P1
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 m5 m6 P6 P5

17. MEK Raf-1* RKIP m12 m1 m2 m2 m2 k1 k1 ERK-PP m9 m3 m3 m3 Raf-1*/RKIP
RKIP-P/RP
MEK-PP/ERK-P m4 m8
Raf-1*/RKIP/ERK-PP
k14 k5
k9/k10
k6/k7 m7 m5 m6
m10
MEK-PP
ERK
RKIP-P RP
Pathway view:
Pathway10 = (k9react,k9). Pathway11
Pathway11 = (k11product,k11). Pathway10 + (k10react,k10). Pathway10 …
(5 pathways)

18. 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

19. 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.

20. Reagent view Pathway view
Reagent view
Pathway view

21. State space of reagent and pathway model

22. 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”!

23. 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.

24. Quantitative Analysis – by logic
Steady state analysis
Formula S=? [ERK_PP_H_STATE = 0]
PRISM result (after translation):

25. Quantitative Analysis – by logic
Now reduce backward rates (.53)
Formula S=? [ERK_PP_H_STATE = 0]

26. Reagent view and ODEs Activity matrix k1 k2 k3 k4 k5 k6 k7 P2 P1
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*m (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 m5 m6 P6 P5

27. 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

28. 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
logic
PRISM
Continuous time
Markov chains

29. 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).