1
Modelling Cell Signalling Pathways in PEPA
Muffy CalderDepartment of Computing Science
University of Glasgow
Jane Hillston and Stephen GilmoreLaboratory 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
Abbreviations and notes•7-TMR: seven trans-membrane receptor•small G-proteins: Rap1, Ras, Rac; active when GTP bound•cAMP-GEF: cAMPactivated GTP-Exchange-Factor•AdCyc: Adenylate cyclase•PDE: Phhosphodiesterase•PKA: cAMP activated protein kinase•adaptor proteins: shc, grb2•SOS: Son-of-Sevenless, a GEF for Ras•PI-3 K: Phosphatidylinositol-3 kinase•Akt: a kinase activated by PI-3 K via PI-3 and another kinase, PDK•PAK: a kinased activated by binding to Rac•MKP: MAPK phosphatase, dephosphorylates MAPKs
activation inhibition phosphorylationactivation inhibition phosphorylation
cell membraneReceptor
e.g. 7-TMRcell membrane
Receptore.g. 7-TMR
Receptore.g. 7-TMR
heterotrimericG-protein
cytosol
heterotrimericG-protein
heterotrimericG-protein
cytosol
tyrosinekinase
tyrosinekinase
SOSshc
grb2SOSSOSshcshc
grb2grb2
RasRas
Raf-1Raf-1Raf-1
MEKERK1,2
MEKMEKERK1,2ERK1,2
PI-3 K
Ras
PI-3 K
PI-3 K
RasRasAktAktAkt
PAK
Rac
PAKPAK
RacRac
PKAcAMP
PKAcAMP
PKAPKAcAMPcAMP
AdCyc
cAMPATP
AdCycAdCyc
cAMPATP cAMPcAMPATP
PKAcAMP
PKAPKAcAMPcAMP
cAMPGEF
cAMP
cAMPGEFcAMPGEF
cAMPcAMPRap1Rap1
MEK1,2
ERK1,2
B-Raf
MEK1,2
ERK1,2
MEK1,2MEK1,2
ERK1,2ERK1,2
B-RafB-RafB-Raf
PDE
cAMP AMP
PDE
cAMP AMP
PDEPDE
cAMP AMPcAMPcAMP AMP
nucleus
transcriptionfactors
nucleusnucleus
transcriptionfactors
transcriptionfactors
transcriptionfactors
MKPMKPMKP
A little more complex.. pathways/networks
4
5
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-P
m10
RP
k9/k10
m11
RKIP-P/RP
k11
m2
k1
m3
k3
Raf-1*/RKIP/ERK-PP
m2
RKIP
k1/k2
m3
K3/k4
k15
m13
k14
From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.
6
RKIP Inhibited ERK Pathway
Pathways have computational
content!
Producers and
consumers.
Feedback.
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-P
m10
RP
k9/k10
m11
RKIP-P/RP
k11
m2
k1
m3
k3
Raf-1*/RKIP/ERK-PP
m2
RKIP
k1/k2
m3
k3/k4
k15
m13
k14
7
RKIP Inhibited ERK Pathway
Why not useprocess algebras for
modelling?
High level formalisms that make interactions
andevent rates
explicit.
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-P
m10
RP
k9/k10
m11
RKIP-P/RP
k11
m2
k1
m3
k3
Raf-1*/RKIP/ERK-PP
m2
RKIP
k1/k2
m3
k3/k4
k15
m13
k14
8
Process algebra(for dummies)
High level descriptions of interaction, communication and synchronisation
Event Prefix .PChoice P1 + P2Synchronisation P1 |l| P2 ¬( l) independent concurrent (interleaved)
actions l synchronised actionConstant A = P assign names to components
Relations (bisimulation ) Laws P1 + P2 P2 + P1
a
b c
aaaaa
c bbbc
9
PEPA
Markovian process algebra invented by Jane Hillston with workbench by Stephen Gilmore.
PEPA descriptions denote continuous Markov chains.
Prefix (,).PChoice P1 + P2 competition between components (race)Cooperation/ P1 |l| P1 ¬(a l) independent concurrent (interleaved) actionsSynchronisation a l shared action, at rate of slowestConstant A = P assign names to components
Performance of Action
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
P(t
)
tetP 1)(
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
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
12
Reagent View
m1
P1
m2
P2
k1/k2
m5
P5
k6/k7
m6
P6
m4
P5/P6
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
k4
m3
k3
P1/P2
13
Reagent View
m1
P1
m2
P2
k1/k2
m5
P5
K6/k7
m6
P6
m4
P5/P6
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
14
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-P
m10
RP
k9/k10
m11
RKIP-P/RP
k11
m2
k1
m3
k3
Raf-1*/RKIP/ERK-PP
m2
RKIP
k1/k2
m3
k3/k4
k15
m13
k14
15
Reagent Viewmodel 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
m1
P1
m2
P2
k1/k2
m5
P5
K6/k7
m6
P6
m4
P5/P6
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
17
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-P
m10
RP
k9/k10
m11
RKIP-P/RP
k11
m2
k1
m3
k3
Raf-1*/RKIP/ERK-PP
m2
RKIP
k1/k2
m3
k3/k4
k15
m13
k14
18
model configuration
Pathway10 |k12react,k13react,k14product| Pathway40
|k3react,k4react,k5product,k6react,k7react,k8product| Pathway30
|k1react,k2react,k3react,k4react,k5product| Pathway20
|k9react,k10react,k11product| Pathway10
Pathway View
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
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
Reagent view
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
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 logicSteady state analysis
Formula S=? [ERK_PP_H_STATE = 0]
PRISM result (after translation):
25
Quantitative Analysis – by logicNow reduce backward rates (.53)
Formula S=? [ERK_PP_H_STATE = 0]
26
Reagent view and ODEs
m1
P1
m2
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 the concentration is +/- for that reaction.
mass action dynamics:
dm1 = - k1*m1*m2 + k2*m3 (nonlinear)dt
Reagent views tells us producer or consumer.
k4
m3
k3
P1/P2
27
Big Picture
Benefits
Interactions
Relative change
Abstraction
Behaviour patterns
Quantitative analysis
stochasticprocess algebra
pathwayview
reagentview
mass actiondifferential equations
Continuous timeMarkov chains
abstractionpathway composition
throughput analysis
denote
derive
28
Bigger Picture
Benefits
Interactions
Relative change
Abstraction
Behaviour patternsQuantitative analysis
stochasticprocess algebra
pathwayview Matlab
reagentview
mass actiondifferential equations
multilevel reagent view
PRISM
Continuous timeMarkov chains
experimentaldata
simulate
logic
denote
derive
abstractionpathway composition
throughput analysis
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).
Top Related