Transcript PowerPoint Presentation - Modelling the ERK Signalling

Modelling Biochemical
Pathways in PEPA
Muffy Calder
Department of Computing Science
University of Glasgow
Joint work with Jane Hillston and Stephen Gilmore
October 2004
1
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?
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
4
5
RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
K12/k13
ERK-PP
k15
k11
m3 Raf-1*/RKIP
m9
m13
k3
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
6
From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.
RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
k12/k13
ERK-PP
k15
k11
m3 Raf-1*/RKIP
m9
m13
k3
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
7
From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.
RKIP protein expression is reduced in breast cancers
8
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
k3
k8
MEK-PP/ERK-P
products
m11
(disassociations)
RKIP-P/RP
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
9
RKIP Inhibited ERK Pathway
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
k12/k13
ERK-PP
k15
This network seems
to be very similar
to producer/consumer
networks.
k11
m3 Raf-1*/RKIP
m9
m13
k3
k3
k8
MEK-PP/ERK-P
m11
RKIP-P/RP
m4
Raf-1*/RKIP/ERK-PP
m8
k14
k5
Why not to try using
process algebras for
modelling?
k9/k10
k6/k7
m7
m5
m6
m10
MEK-PP
ERK
RKIP-P
RP
10
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.
11
Process algebra
(for dummies)
High level descriptions of interaction, communication and
synchronisation
Event
Prefix
Choice
Synchronisation
a (simple), a!34 (data offer),
a.S
Constant
A=S
Laws
Relations
P1 + P2 @ P2 + P1
@ (bisimulation)
S+S
P |l| P
a?x (data receipt)
a e l independent concurrent (interleaved) actions
a e l synchronised action
assign names to components
a
b
@
c
a
c
a
b
@
a
c
a
b
a
b
12
PEPA
Process algebra with performance, invented by
Jane Hillston
Prefix
Choice
Cooperation/
Synchronisation
Constant
(a,r).S
S+S
P |l| P
A=S
competition between components (race)
a e l independent concurrent (interleaved) actions
a e l shared action, at rate of slowest
assign names to components
P ::= S | P |l| P
S ::= (a,r).S | S+S | A
13
Rates
l is a rate, from which a probability is derived
Performance of Action
1
0.9
0.8
0.7
0.5
0.4
0.3
P(t ) 1  elt
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
4.
4
4.
95
5.
5
6.
05
2.
2
2.
75
3.
3
3.
85
0
0
0.
55
1.
1
1.
65
P(t)
0.6
14
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)
15
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.
16
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?)
17
Signalling Dynamics
P2
P1
m2
m1
Reagent view: model whether or not a reagent can participate in
a reaction (observable/unobservable).
k1/k2
k4
P1/P2
m3
P5/P6
m4
k3
k6/k7
m5
m6
P5
P6
18
Signalling Dynamics
P2
P1
m2
m1
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
k1/k2
P2H = (k1react,k1). P2L
k4
P2L = (k2react,k2). P2H + (k4react). P2H
P1/P2
m3
P1/P2H = (k2react,k2). P1/P2L + (k3react, k3). P1/P2L
P1/P2L = (k1react,k1). P1/P2H
P5/P6
m4
P5H = (k6react,k6). P5L + (k4react,k4). P5L
k3
P5L = (k3react,k3). P5H +(k7react,k7). P5H
k6/k7
P6H = (k6react,k6). P6L
m5
m6
P5
P6
P6L = (k7react,k7). P6H
P5/P6H = (k7react,k7). P5/P6L
P5/P6L = (k6react,k6) . P5/P6H
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Signalling Dynamics
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
20
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
k12/k13
ERK-PP
k15
k11
m3 Raf-1*/RKIP
m9
m13
k3
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)
21
Signalling Dynamics
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
22
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
m5
m6
P5
P6
23
Signalling Dynamics
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!)
24
MEK
Raf-1*
RKIP
m12
m1
m2
k1/k2
k1
k12/k13
ERK-PP
k15
k11
m3 Raf-1*/RKIP
m9
m13
k3
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)
25
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
26
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.
27
State space of reagent and pathway model
state
s1
reagent-view
Raf-1*H, RKIPH,Raf-1*/RKIPL,Raf-1*/RKIPERK-PPL, ERKL,RKIP-PL, RKIP-P/RPL,
RPH, MEKL,MEK/Raf-1*L,MEK-PPH,MEK-PP/ERKL/ERK-PPH
pathway view
Pathway50,Pathway40,Pathway20,Pathway10
s2
…
.
.
.
s28
(28 states)
28
State space of reagent and pathway model
29
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)
30
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
31
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
32
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.
33
Signalling Dynamics
Activity matrix
P2
P1
m2
k1
m1
k1/k2
k4
P1/P2
m3
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
P5/P6
m4
Column: corresponds to a single reaction.
k3
Row: correspond to a reagent; entries indicate whether the
concentration is +/- for that reaction.
K6/k7
m5
m6
P5
P6
34
Signalling Dynamics
Activity matrix
P2
P1
m2
m1
k1
k1/k2
k4
P1/P2
m3
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
P5/P6
m4
Differential equations
k3
K6/k7
Each row is labelled by a protein concentration. One equation per row.
For row r,
m5
m6
P5
P6
dr
=
S column c A[r,c]) * P row x f(A[x,c])
dt
where f(A[x,c]) = if (A[x,c]== -) then x else 1
a rate is a product of the rate constant and current
concentration of substrates consumed.
35
Signalling Dynamics
Activity matrix
P2
P1
m2
k1
m1
k1/k2
k4
P1/P2
m3
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
P5/P6
m4
Differential equations (mass action)
k3
K6/k7
dm1 = - k1
+ k2
(two terms)
dt
m5
m6
P5
P6
36
Signalling Dynamics
Activity matrix
P2
P1
m2
k1
m1
k1/k2
k4
P1/P2
m3
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
P5/P6
m4
Differential equations (mass action)
k3
K6/k7
dm1 = - k1*m1*m2 + k2
dt
m5
m6
P5
P6
37
Signalling Dynamics
Activity matrix
P2
P1
m2
k1
m1
k1/k2
k4
P1/P2
m3
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
P5/P6
m4
Differential equations (mass action)
k3
K6/k7
dm1 = - k1*m1*m2 + k2*m3
(nonlinear)
dt
m5
m6
P5
P6
38
Signalling Dynamics
P2
P1
m2
m1
Differential equations (mass action)
For RKIP inhibited ERK pathway, change in Raf-1* is:
k1/k2
k4
P1/P2
dm1 = - k1*m1*m2 + k2*m3 + k5*m4 – k12*m1*m12
dt
+k13*m13 + k14*m13
m3
P5/P6
(catalysis, inhibition, etc. )
m4
k3
K6/k7
m5
m6
P5
P6
39
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
40
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).
41
The End
Thank you.
42