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
1
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
2
A little more complex.. pathways/networks
3
4
RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12
m1
m2
proteins/complexes
k1/k2
k1
k12/k13
forward /backward
ERK-PP
k15
reactions
(associations/disassociations)
k11
m3 Raf-1*/RKIP
m9
m13
K3/k4
k3
k8
MEK-PP/ERK-P
products
m11
(disassociations)
m4
Raf-1*/RKIP/ERK-PP
m8
k14
k5
m1, m2 .. concentrations of
k9/k10
proteins
k6/k7
m7
m5
m6
m10
MEK-PP
ERK
RKIP-P
RP
k1,k2 ..: rate (performance)
coefficients
5
From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.
RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12
m1
m2
Pathways have
computational
content!
k1/k2
k1
k12/k13
ERK-PP
k15
k11
m3 Raf-1*/RKIP
m9
m13
k3/k4
k3
k8
MEK-PP/ERK-P
m11
RKIP-P/RP
m4
Raf-1*/RKIP/ERK-PP
m8
k14
k5
Producers
and
consumers.
Feedback.
k9/k10
k6/k7
m7
m5
m6
m10
MEK-PP
ERK
RKIP-P
RP
6
RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
k12/k13
ERK-PP
k15
Why not use
process algebras for
modelling?
k11
m3 Raf-1*/RKIP
m9
m13
k3/k4
k3
k8
MEK-PP/ERK-P
m11
RKIP-P/RP
m4
Raf-1*/RKIP/ERK-PP
m8
k14
k5
k9/k10
k6/k7
m7
m5
m6
m10
MEK-PP
ERK
RKIP-P
RP
High level formalisms
that make
interactions
and
event rates
explicit.
7
Process algebra
(for dummies)
High level descriptions of interaction, communication and
synchronisation
Event
Prefix
Choice
Synchronisation
a
a.P
Constant
P1 + P2
P1 |l| P2 ¬(a e l) independent concurrent (interleaved) actions
a e l synchronised action
A=P
assign names to components
Relations
Laws
@ (bisimulation )
P1 + P2 @ P2 + P1
a
b
@
c
a
c
a
b
@
a
c
a
b
a
b
8
PEPA
Markovian process algebra invented by Jane Hillston with
workbench by Stephen Gilmore.
PEPA descriptions denote continuous Markov chains.
(a,l).P
P1 + P2 competition between components (race)
P1 |l| P1 ¬(a e l) independent concurrent (interleaved) actions
a e l shared action, at rate of slowest
A=P
assign names to components
Prefix
Choice
Cooperation/
Synchronisation
Constant
Performance of Action
1
l is a rate, from which a probability is derived exponential distribution.
0.9
0.8
0.7
0.5
P(t )  1  e lt
0.4
0.3
0.2
0.1
t
11
11
.5
5
12
.1
12
.6
5
13
.2
13
.7
5
14
.3
14
.8
5
15
.4
15
.9
5
9.
9
10
.4
5
8.
8
9.
35
7.
7
8.
25
6.
6
7.
15
5.
5
6.
05
4.
4
4.
95
3.
3
3.
85
2.
2
2.
75
1.
1
1.
65
0
0
0.
55
P(t)
0.6
9
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)
10
Signalling Dynamics
P2
P1
m2
m1
k1/k2
k4
P1/P2
m3
Reaction
Producer(s)
Consumer(s)
k1react
{P2,P1}
{P1/P2}
k2react
{P1/P2}
{P2,P1}
k3product
{P1/P2}
{P5}
…
P5/P6
m4
k3
k1react will be a 3-way synchronisation,
K6/k7
k2react will be a 3-way synchronisation,
m5
m6
P5
P6
k3product will be a 2-way synchronisation.
11
Reagent View
P2
P1
m2
m1
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
k1/k2
k4
P2H = (k1react,k1). P2L
P1/P2
P2L = (k2react,k2). P2H + (k4react). P2H
m3
P1/P2H = (k2react,k2). P1/P2L + (k3react, k3). P1/P2L
P5/P6
P1/P2L = (k1react,k1). P1/P2H
m4
k3
P5H = (k6react,k6). P5L + (k4react,k4). P5L
k6/k7
P5L = (k3react,k3). P5H +(k7react,k7). P5H
m5
m6
P6H = (k6react,k6). P6L
P5
P6
P6L = (k7react,k7). P6H
P5/P6H = (k7react,k7). P5/P6L
P5/P6L = (k6react,k6) . P5/P6H
12
Reagent View
Model configuration
P2
P1
m2
m1
P1H |k1react,k2react|
P2H | k1react,k2react,k4react |
P1/P2L |k1react,k2react,k3react|
k1/k2
P5L |k3react,k6react,k4react|
k4
P6H |k6react,k7react|
P1/P2
m3
P5/P6L
P5/P6
Assuming initial concentrations of m1,m2,m6.
m4
k3
K6/k7
m5
m6
P5
P6
13
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
k12/k13
ERK-PP
k15
k11
m3 Raf-1*/RKIP
m9
m13
k3/k4
k3
k8
MEK-PP/ERK-P
m11
RKIP-P/RP
m4
Raf-1*/RKIP/ERK-PP
m8
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)
14
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
15
Pathway View
Model chains of behaviour flow.
P2
P1
m2
m1
Two pathways, corresponding to initial concentrations:
Path10 = (k1react,k1). Path11
k1/k2
Path11 = (k2react).Path10 + (k3product,k3).Path12
k4
Path12 = (k4product,k4).Path10 + (k6react,k6).Path13
P1/P2
Path13 = (k7react,k7).Path12
m3
Path20 = (k6react,k6). Path21
P5/P6
Path21 = (k7react,k6).Path20
m4
k3
K6/k7
Pathway view: model configuration
m5
m6
P5
P6
Path10 | k6react,k7react | Path20
(much simpler!)
16
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
k12/k13
ERK-PP
k15
k11
m3 Raf-1*/RKIP
m9
m13
k3/k4
k3
k8
MEK-PP/ERK-P
m11
RKIP-P/RP
m4
Raf-1*/RKIP/ERK-PP
m8
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)
17
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
18
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.
19
1 0.04135079004156481
2 0.020806115102310632
3 0.07346775929692899
4 0.006935371700770152
5 0.06516104016641672
6 0.03737546622097119
7 0.011336715749471194
8 0.036048205933593286
9 0.004639841577167708
10 0.005691394350960237
11 0.04138456618620803
12 0.0025828089820320505
13 0.004807783620797024
14 0.04817123798507296
15 0.018640671069835055
16 0.016743539619515142
17 0.02162874351056745
18 0.0028912552492803816
19 0.004970238100423158
20 0.02076780718322302
21 0.1840054851485999
22 0.008846052672337585
23 0.01413218356459678
24 0.0030482221649047224
25 0.0020844704151460223
26 0.20477329233182312
27 0.09642576891046874
28 0.0012831731450123965
Reagent view
1 0.04135079004156353
2 0.020806115102310604
3 0.07346775929692419
4 0.006935371700769834
5 0.06516104016641262
6 0.03737546622096783
7 0.011336715749470889
8 0.03604820593359156
9 0.005691394350959787
10 0.004639841577167543
11 0.04138456618620752
12 0.04817123798507505
13 0.0025828089820318246
14 0.01864067106983504
15 0.004807783620796737
16 0.01674353961951507
17 0.020767807183224345
18 0.021628743510568222
19 0.18400548514860549
20 0.002891255249280038
21 0.008846052672337464
22 0.004970238100423424
23 0.014132183564597499
24 0.20477329233182964
25 0.09642576891047139
26 0.0030482221649046053
27 0.0020844704151453983
28 0.0012831731450119671
Pathway view
20
State space of reagent and pathway model
21
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”!
22
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.
23
Quantitative Analysis – by logic
Steady state analysis
Formula
S=? [ERK_PP_H_STATE = 0]
PRISM result (after translation):
24
Quantitative Analysis – by logic
Now reduce backward rates (.53)
Formula
S=? [ERK_PP_H_STATE = 0]
25
Reagent view and ODEs
Activity matrix
P2
P1
m2
k1
m1
k1/k2
k4
P1/P2
m3
P5/P6
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.
m4
Row: correspond to a reagent; entries indicate whether the
concentration is +/- for that reaction.
k3
K6/k7
mass action dynamics:
m5
m6
P5
P6
dm1 = - k1*m1*m2 + k2*m3
dt
(nonlinear)
Reagent views tells us producer or consumer.
26
Big Picture
abstraction
pathway
composition
stochastic
process algebra
pathway
view
throughput
analysis
reagent
view
derive
mass action
differential
equations
denote
Continuous time
Markov chains
Benefits
Interactions
Relative change
Abstraction
Behaviour patterns
Quantitative analysis
27
Bigger Picture
abstraction
pathway
composition
experimental
data
stochastic
process algebra
reagent
view
pathway
view
denote
logic
throughput
analysis
derive
multilevel
reagent view
mass action
differential
equations
Matlab
simulate
PRISM
Continuous time
Markov chains
Benefits
Interactions
Relative change
Abstraction
Behaviour patterns
Quantitative analysis
28
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).
29