modelling dynamic modularisation - Brunel University...

Bioinformatics

David GilbertBioinformatics Research Centre

www.brc.dcs.gla.ac.ukDepartment of Computing Science, University of Glasgow

Modelling dynamicbehaviour:

Modularisation

(c) David Gilbert 2008 Modularisation 2

Lecture outline

• Modelling Enzymatic Reactions • Signal Transduction Cascades • Modelling:

– Building blocks & composition/decomposition – Positive & negative feedback– Modelling the effect of drug inhibitors

• Analysing the behaviour of models


Motivation

• Quantitative models of biochemical networks are a centralcomponent of modern systems biology.

• Building & managing these complex models is a majorchallenge that can benefit from the application of formalmethods adopted from theoretical computing science.

• Hence - a general introduction to the field of formal modelling– emphasizes the intuitive biochemical basis of the modelling process,– accessible for an audience with a background in computing science

and/or model engineering.


Some (Bio)Chemical Conventions

Concentration of Molecule A = [A], usually in units mol/litre(molar)

Rate constant = k, with indices indicating constants for variousreactions (k1, k2...)

Therefore:AB

][][][

1 Akdt

Bd

dt

Ad!=!=


Reversible, Single-MoleculeReaction

A B, or A B || B A, orDifferential equations:

][][][

][][][

21

21

BkAkdt

Bd

BkAkdt

Ad

!=

+!=

forward reverse

Main principle: Partial reactions are independent!


Irreversible, two-molecule reaction

A+BCDifferential equations:

]][[][

][][][

BAkdt

Ad

dt

Cd

dt

Bd

dt

Ad

!=

!==

Underlying idea: Reaction probability = Combined probability that both [A]and [B] are in a “reactive mood”:

]][[][][)()()( *

2

*

1 BAkBkAkBpApABp ===

The last piece of the puzzle

Non-linear!


Biological description bigraph ODEs


Mass action, MA1 model

• A: substrate,• B: product• E: enzyme• E|A substrate-enzyme complex

!

E +Ak2

" # #

k1# $ # E | A

k3# $ # E + B

A B

E

A B

E

E|A


Differential equations


MA2, MA3 models

!

A+Ek2

" # #

k1# $ # A | E

k3# $ # B | E

k 5" # #

k4# $ # B +E

!

A+Ek2

" # #

k1# $ # A | E

k6

" # #

k3# $ # B | E

k 5" # #

k4# $ # B +E

MA2

MA3


Multiple substrates


Michaelis-Menten

!

V =Vmax

"[A]

(KM

+[A])

kcat

=Vmax

[ET]

d[A]

dt= #

d[B]

dt= #k

cat"[E

T]"

[A]

(KM

+[A])

V : Reaction velocityKM : Michaelis constant - concentration of substrate at

which reaction rate is half max value[ET] : total enzyme concentration


AssumptionsIt is critical to note that the Michaelis-Menten equation only

holds at the initial stage of a reaction before theconcentration of the product is appreciable, and makesthe following assumptions:

1. No product reverts to initial substrate

2. MM Equation holds at initial stage of reaction beforeconcentration of product is appreciable

3. [E] << [A]


Metabolic pathways vs Signalling Pathways(can you give the mass-action equations?)

E1

(initial substrate)S

S’

E2

E3

S’’

S’’’(final product)

Metabolic

S1

Input SignalX

P2S2

S3 P3Output

Signalling cascade

P1

Product become enzyme at next stageClassical enzyme-product pathway


A special case: enzyme reactionsUnderlying assumptions of the Michaelis-Menten approximation:

• Free diffusion, random collisions

• Irreversible reactions

• Quasi steady state

In cell signaling pathways, all three assumptions will be frequently violated:

• Reactions happen at membranes and on scaffold structures

• Reactions happen close to equilibrium and both reactions have non-zero fluxes

• Enzymes are themselves substrates for other enzymes, concentrations changerapidly, d[ES]/dt ≈ d[P]/dt


Cell signaling pathways



• Common components:– Receptors binding to ligands

• R(inactive) + L RL(active)

– Proteins forming complexes• P1 + P2 P1P2-complex

– Proteins acting as enzymes on other proteins (e.g.,phosphorylation by kinases)

• P1 + K P1* + K



Fig. courtesy of W. Kolch


MAPK Pathway• Responds to wide range of stimuli:

cytokines, growth factors, neurotransmitters,cellular stress and cell adherence,…

• Pivotal role in many key cellular processes:– growth control in all its variations,– cell differentiation and survival– cellular adaptation to chemical and physical

stress.

• Deregulated in various diseases: cancer;immunological, inflammatory anddegenerative syndromes,

• Represents an important drug target.

STIMULUS


Mass action for enzymatic reaction -phosphorylation

• R: substrate,• Rp: product (phosphorylated R)• S1: enzyme (kinase)• R|S1 substrate-enzyme complex

!

R+S1

k2" # #

k1# $ # R | S

1

k3# $ # Rp + S

1

R Rp

S1


Phosphorylation - dephosphorylation stepMass action MA1

• R: unphosphorylated form• Rp: phosphorylated form• S: kinase• P: phosphotase• R|S unphosphorylated+kinase complex• R|P unphosphorylated+phosphotase complex

R Rp

S

P

!

R+Sk2

" # #

k1# $ # R | S

k3# $ # Rp + S

R+Pkr3" # # Rp | P

kr2# $ #

kr1" # # Rp +P



dydt = [-k1*S*R + k2*RS + k3*RS % S

-k1*S*R + k2*RS + k3*RpP % R +k1*S*R - k2*RS - k3*RS % RS

-kr1*P*Rp + kr2*RpP + k3*RS % Rp -kr1*P*Rp + kr2*RpP + kr3*RpP % P +k1*Rp*P - kr2*RpP - kr3*RpP ]; % RpP

!

R+S k1" # " R | S Rp +P kr1" # " Rp | P

R | S k2" # " R+ S Rp | P kr2" # " Rp +P

R | Sk

3" # " Rp + S Rp | Pkr

3" # " R +P


Phosphorylation - dephosphorylation loopMass action MA2

R Rp

S1

S2

!

R+S1

k2" # #

k1# $ # R | S

1

k3# $ # Rp | S1

k 5" # #

k4# $ # Rp + S

1

R+S2

kr 5# $ #

kr4" # # R | S

2

kr3" # # Rp | S2

kr2# $ #

kr1" # # Rp + S

2

• R: unphosphorylated form• Rp: phosphorylated form• S1: kinase• S2: phosphotase• R|S1 unphosphorylated+kinase complex• Rp|S1 phosphorylated+kinase complex• R|S2 unphosphorylated+phosphotase complex• Rp|S2 phosphorylated+phosphotase complex



R Rp

S

P

!

R+Sk2

" # #

k1# $ # R | S

k6" # #

k3# $ # Rp | S

k 5" # #

k4# $ # Rp + S

R+Pkr5

# $ #

kr4" # # R | P

kr 6# $ # #

kr3" # # Rp | P

kr2# $ #

kr1" # # Rp +P

• R: unphosphorylated form• Rp: phosphorylated form• S: kinase• P: phosphotase• R|S unphosphorylated+kinase complex• R|P unphosphorylated+phosphotase complex


Michaelis-Menten equation forphosphorylation-dephosphorylation

• Assumptions:1. No product reverts to initial substrate2. MM Equation holds at initial stage of reaction before concentration of product is

appreciable3. [Enzyme] << [Substrate]

• Km is [Substrate] at which the reaction rate is half its maximum value• dRp/dt == reaction rate V• k3 x S == Vmax for the forward reaction• k3’ == Vmax for the reverse reaction (Phosphotase is ignored)• Km1 == (k2+k3)/k1 (k’s from mass-action 1)

!

V = k3"[S]"

[R]

(Km1 +[R])# k

3'"

[Rp ]

(Km2 +[Rp ]) RpR

S

P


Questions• Is Michaelis-Menten adequate for phosphorylation pathways?• Is Mass Action sufficient/correct for these pathways?• What is the effect of negative feedback?• Can we confirm the ‘negative feedback amplifer’ behaviour in

both MM and MA models• Can oscillators be built?• Overall, what are the rules for component-based construction?


Compositionvertical & horizontal

RpR

S1

RRpRR

P1

P2

RpR

S

Rpp

P2-stage cascade

1-stage cascadedouble phosphorylation step


Phosphorylation cascade:2-stage, Mass Action MA1

RpR

S1

RRpRR

!

R+S1

k2" # #

k1# $ # R | S

1

k3# $ # Rp + S

1

R+S2

k3©

" # # R | S2

k2 ©# $ # #

k1©" # # # Rp + S

2

RR+ Rpkk2

" # #

kk1# $ # RR | Rp

kk3# $ # RRp + Rp

RR+ SS2

kk3©

" # # # RR | SS2

kk2©# $ # #

kk1©" # # # RRp + SS

2


Phosphorylation cascade:2-stage, Michaelis-Menten

RpR

S1

RRpRR

!

dRp

dt=k3" S

1" R

Km1 + R#

k3'"Rp

Km2 + Rp

dRRp

dt=kk

3" Rp " RR

KKm1 + RR#

kk3'"RRp

KKm2 + RRp


Phosphorylation cascade:3-stage, Mass-Action MA1

RpR

S1

RRpRR

RRRpRRR

!

R+S1

k2" # #

k1# $ # R | S

1

k3# $ # Rp + S

1

R+S2

k3'

" # # Rp | S2k2 '

# $ #

k1 '" # # Rp + S

2

RR + Rpkk2

" # #

kk1# $ # RR |Rp

kk3# $ # RRp + Rp

RR+ SS2

kk3'

" # # RRp | SS2kk2 '

# $ # #

kk1 '" # # # RRp + SS

2

RRR + RRpkkk2

" # # #

kkk1# $ # # RRR |RRp

kkk3# $ # RRRp + RRp

RRR+ SSS2

kkk3'

" # # # RRRp | SSS2kkk2 '

# $ # #

kkk1 '" # # # RRRp + SSS

2


Phosphorylation cascade:3-stage, Michaelis-Menten

RpR

S1

RRpRR

RRRpRRR

!

dRp

dt=k3" S

1" R

Km1 + R#

k3'"Rp

Km2 + Rp

dRRp

dt=kk

3" Rp " RR

KKm1 + RR#

kk3'"RRp

KKm2 + RRp

dRRRp

dt=kkk

3" RRp " RRR

KKKm1 + RRR#

kkk3'"RRRp

KKKm2 + RRRp


Levchenko et al. “Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signalingand reduce its threshold properties”. Proc Natl Acad Sci USA, 97(11):5818–5823, 2000.

Example: Levchenko


Schoeberl et al 2002 model

Schoeberl et al. (2002), Computational modeling of the dynamics of the MAP kinase cascade activated bysurface and internalized EGF receptors, Nature Biotechnology 20, 370-375


Phosphorylation cascade + negative feedback:2-stage, Mass Action MA1

RpR

S1

RRpRR

!

RRp+S1ki '

" # # #

ki# $ # RRp | S1

R+S1

k2" # #

k1# $ # R | S

1

k3# $ # Rp + S

1

R+S2

k3'

" # # Rp | S2k2 '

# $ # #

k1 '" # # # Rp + S

2

RR+ Rpkk2

" # #

kk1# $ # RR | Rp

kk3# $ # RRp + Rp

RR+ SS2

kk3'

" # # # RRp | SS2kk2 '

# $ # #

kk1 '" # # # RRp + SS

2


Phosphorylation cascade + negative feedback:2-stage, Michaelis-Menten

• Using Competitive Inhibition• Ki is the dissociation constant for the SI complex

RpR

S1

RRpRR

!

dRp

dt=

k3" S

1" R

Km1 " 1+RRp

Ki

#

$ %

&

' ( + R

)k3©"Rp

Km2 + Rp

dRRp

dt=kk

3" Rp " RR

KKm1 + RR)kk

3©"RRp

KKm2 + RRp

!

V =Vmax

"[S]

[S]+Km" 1+

[I ]

[Ki]

#

$ %

&

' (


Phosphorylation cascade + negative feedback: 3-stage, Mass Action, MA1

RpR

S1

RRpRR

RRRpRRR

!

RRRp+S1ki©

" # # #

ki# $ # RRRp | S1

R+S1

k2" # #

k1# $ # R | S

1

k3# $ # Rp + S

1

R+S2

k3©

" # # Rp | S2k2 ©

# $ # #

k1©" # # # Rp + S

2

RR+ Rpkk2

" # #

kk1# $ #

RR | Rp

kk3# $ # RRp + Rp

RR+ SS2

kk3©

" # # # RRp | SS2kk2©

# $ # #

kk1©" # # # RRp + SS

2

RRR+ RRpkkk2

" # # #

kkk1# $ # #

RRR | RRp


RRR+ SSS2

kkk3©

" # # # RRRp | SSS2kkk2 ©

# $ # #

kkk1©" # # # RRRp + SSS

2


Phosphorylation cascade + negativefeedback: 3-stage, Michaelis-Menten

RpR

S1

RRpRR

RRRpRRR

• Using Competitive Inhibition• Ki is the dissociation constant for the SI complex

!

V =Vmax

"[S]

[S]+Km" 1+

[I ]

[Ki]

#

$ %

&

' (

!

dRp

dt=

k3" S

1" R

Km1 " 1+RRRp

Ki

#

$ %

&

' ( + R

)k3©"Rp

Km2 + Rp

dRRp

dt=kk

3" Rp " RR

KKm1 + RR)kk

3©"RRp

KKm2 + RRp

dRRRp

dt=kkk

3" RRp " RRR

KKKm1 + RRR)kkk

3©"RRRp

KKKm2 + RRRp


Oscillations! Phosphorylation cascade + negative feedback:3-stage, Mass Action

ConditionsS1=3Inhibitor=0.5


Kholodenko Model


Kholodenko Simulation

Time = 500 Seconds Time = 5000 Seconds

Simulation from Richard Orton


[Ras] [Ras*] (m1)

Input

[Raf] [Raf*] (m2)

[MEK] [MEK-pp](m3)

[ERK] [ERK-pp](m4)

v1

v2

v3

v4

v5

v6

v7

v8

Neg

ativ

e feedb

ack

[RKIP] [RKIPp] (m5)v10

Positive feedback

v9

Modeling and Analysis of Two Feedback Loop Dynamics in Ras/Raf-1/MEK/ERK Signaling PathwayKwang-Hyun Cho, Sung-Young Shin, Walter Kolch, Olaf Wolkenhauer. ICSB 2004

Combination of positive & negative feedbackMathematical Model


No Feedback Positive Feedback

Negative Feedback Positive & Negative Feedback

Combination of positive & negative feedback: Simulation


Combination of positive & negative feedback:Simulation vs. Experimental Data

0 20’ 40’ 1h 2h 3h 4h 6h TPA

ERK-pp (activated ERK)

total ERK

Western blots COS1 cell lysates

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [hour]

Normal.[unitless]

Comparison of experimentad data and simulation result

m5(erk-pp)

raw-erk

Simulation

Experiment


‘Real cascade & feedback’

Ras

Raf

MEK

ERK

U0126


Is the ERK pathway anegative feedback amplifier?

Sauro HM, Kholodenko BN.Quantitative analysis of signaling networks.Prog Biophys Mol Biol. 2004 Sep;86(1):5-43.


Negative Feedback Amplifier• A negative feedback amplifier stems from the field of electronics and consists of an amplifier

with a negative feedback loop from the output of the amplifier to its input.

• The negative feedback loop results in a system that is much more robust to disturbances in theamplifier.

• The negative feedback amplifier was invented in 1927 by Harold Black of Western Electric andwas originally used for reducing distortion in long distance telephone lines.

• The negative feedback amplifier is now a key electrical component used in a wide variety ofapplications

Input Amplifier

Feedback

Output


Negative Feedback Amplifier

Input

Amplifier

Negative Feedback Loop

Output

Input After Feedback

AF1

Auy

+=

y = Aee = u – Fy

y = A (u – Fy)y = Au – AFyy + AFy = Auy (1 + AF) = Au

Steady State Equation

yA

F

u

-

e Ae


Standard Amplifier

Au y+ +

y=A*u

Amplifier (A) gain

Out

put

(y)


Amplifier (A) gain

Out

put

(y)

yA

F

u + +

-

y=A*u/(1+A*F)

yA

F

u + +

-

y=A*u/(1+A*F)

The negative feedback impartssignalling robustness

A large change in amplifier gain leads to a small change in output (y)


FeedbackO

utpu

t In

crea

sing

->

<- Amplifier Decreasing

Feedback Increasing ->

Increasing Feedback


Standard Amplifier

Au y+ +

y=A*u


yA

F

u + +

-

y=A*u/(1+A*F)

yA

F

u + +

-

y=A*u/(1+A*F)

The negative feedback impartssignalling robustness

Time

Out

put

(y)

Sudden drop in Amplifier(A) gain

Δy Output

Sudden drop in Amplifier(A) gain

Out

put

(y)

Time

Δy Output


Application to Biology• The ERK cascade is a well known biological amplifier and

contains numerous negative feedback loops.

• At first sight, it has the correct structure to be a negativefeedback amplifier.

• If the ERK cascade is a negative feedback amplifier itshould be robust to disturbances within the cascade.

• From a biological point of view, these disturbances couldbe caused by drugs, such as U0126, aimed at decreasingthe activity of the ERK cascade.

• This suggests that these drugs will be relatively ineffective.

• In fact, current drugs aimed at decreasing the activity of theMAPK pathway have proved less efficient in in vivoapplications than anticipated from in vitro inhibitionassays.

Sauro & Kholodenko (2004)


Raf/MEK/ERK amplifies the signal

femtomol987.110.9NIH 3T3

ratio5.92.91

ratio90.71

femtomol21.210.63.6COS1

Concentrationper cell

ERKMEKRaf-1Cell line

NIH CO

SRec. GST-BXB

195

118907055

3833

WB: Raf-1

195118

9070

55

38

33

NIH CO

SRec. MEK1-His

WB: MEK

195

118907055

3833

NIH

CO

SRec. GST-ERK

WB: ERK


How to test if the ERK pathway is aNFA?

Ras-GTP

Raf-1

MEK1/2

ERK1/2

Neg

ativ

e Fe

edba

ck

U0126

Generate input:Stimulate with GF

Measure signal output:i.e. ERK phosphorylation

Remove negative feedback

“Disturb the Amplifier”:Use a MEK inhibitior, suchas U0126


Hypothesis: Braking the feedback should sensitisethe ERK pathway to MEK-inhibitor

Ras-GTP

Raf-1

MEK1/2

ERK1/2

Neg

ativ

e Fe

edba

ck

U0126

Ras-GTP

Raf-1

MEK1/2

ERK1/2

U0126

phos

pho-

ERK

MEK inhibitor

Feedback intact Feedback removed


How to test if the ERK pathway is aNFA?

Strategy

In vivo system that allows usto compare feedback broken to feedback intact model.

Computational Model ofERK pathway with/withoutfeedback


Computational Modeling 1:Build the model

• Non-linear ordinary differentialequations (ODE’s).

• ODE’s were solved using Math Lab andGepasi.

• Models are based on the Schoeberl et al.(2002) model

• Mass Action Kinetics instead of MichaelisMenten

• Kinetic parameters are from literature,previous models and “guesstimates”

Schoeberl et al. (2002), Computational modeling of the dynamics of the MAP kinase cascade activated bysurface and internalized EGF receptors, Nature Biotechnology 20, 370-375


Amplifier / negative feedback• Model amplifier strength by

– Adding inhibitor to 2nd stage– Modifying kk3, kkk3 [I.e. modifying rate of production of RRp, RRRp]– Add/remove cascade elements

• Then plot amp strength versus output, e.g. [U] vs [RRRp]• ?Model feedback strength by

– Leaving out feedback loop– varying ki, and plot ki vs [RRRp]

• Notes: avoid saturation; use signal in linear range; ?modeldegradation in S1 signal?


Feedback brokenFeedback intact

Computational Modeling 2:

Results

Prediction: Braking the feedback modulates drug response


Sensitivity of kinetic parameters is decreased due toNegative Feedback

Computational Modeling 2:

Results


EGFR

Sos

Ras

Raf

MEK

ERK

The experimental systemsNegative feedback

loops intact

RasV12

Raf

MEK

ERK

One feedback loopeliminated by

constitutively activeRasV12 mutant

BXB-ER

4-OHT

MEK

ERK

Both feedback loopseliminated by BXB-ER(4-OHT regulatable

Raf-1 mutant)

U0126 U0126 U0126 MEKinhibitor

4557W EGFR inhibitor


BXB-ER ER hormone bindingCR3

642

P

S

HA

ER hormone bindingER hormone bindingER hormone bindingCR3CR3

642

P

S642

P

S642

PP

S

HA

ERK feedback phosphorylation sites

Raf-1 CR3CR2CR1

289

P

S

296

P

301

P

S S

29

P

S

43

P

S

642

P

S

CR3CR2CR1

289

P

S

296

P

301

P

S S

289

P

289

PP

S

296

P

296

PP

301

P

301

PP

S S

29

P

S

43

P

S

29

P

S

29

PP

S

43

P

S

43

PP

S

642

P

S

642

P

S

642

PP

S

Regulatory Domain Kinase Domain

Breaking the ERK feedback with BXBER

Raf-1 stimulated with EGF

BXB-ER stimulated with 4-OHT(4-Hydroxy Tamoxifen, a synthetic estrogen)

0

1

2

3

4

5

6

7

8

9

0 5 10 20 40 80 120 min

ppER

K le

vels


Ablation of feedback by BXBERdecreases robustness to MEK-inhibitor

U0126Computer Simulation


Experiment

Ablation of feedback by BXBERdecreases robustness to MEK-inhibitor

U0126


0 10 20 40 80 min stimulation

pERK1/2, +EGF

pERK1/2, + BXBER/4HT

U0126 added

Signal recovery after MEK inhibition

Simulation Experiment


Implications for drug targeting

• The aim of a drug is to cause a disruption to the network in such away that it restores the network to its ‘healthy’ wild-type state.

• Targets must be susceptible to disruption for the drug to have anyeffect.

• The analysis of feedback suggests that targets inside the feedbackloop will prove difficult drug targets because any attempt to disturbthese targets will be resisted by the feedback loop.


Summary

• Modelling Enzymatic Reactions •• Signal Transduction Cascades •• Modelling:

– Building blocks & composition/decomposition – Positive & negative feedback, & drug inhibitors

• Analysing the behaviour of models

modelling dynamic modularisation - Brunel University...

Documents

Transcript of modelling dynamic modularisation - Brunel University...