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A structured approach Part III Biological applications David Gilbert Bioinformatics Research Centre University of Glasgow, Glasgow, UK drg@brc.dcs.gla.ac.uk Biological Applications 1 MAPK Pathway Responds to wide range of stimuli:

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A structured approach … Part III Biological applications

David Gilbert

Bioinformatics Research Centre University of Glasgow, Glasgow, UK

Biological Applications

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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 and degenerative syndromes,

  • Represents an important drug target.

STIMULUS

Biological Applications

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MA1: Mass action for enzymatic reaction

  • phosphorylation
  • A: substrate
  • B: product (phosphorylated A)
  • E: enzyme (kinase)
  • E|A substrate-enzyme complex

E+A

k2

←  

k1

 →  E | A k3

 →  E + B

A B E

Biological Applications

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Differential equations

Enzymatic reaction

A+E

k2

←  

k1

 →  A | E k3

 →  B + E

Biological Applications

d[A] dt = −k1 ×[A]×[E] + k2 ×[A | E] d[A | E] dt = +k1 ×[A]×[E] − k2 ×[A | E] − k3 ×[A | E] d[B] dt = + k3 ×[A | E] d[E] dt = −k1 ×[A]×[E] + k2 ×[A | E] + k3 ×[A | E]

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MA2 model

Biological Applications

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MA3 model

Biological Applications

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Multiple substrates

Biological Applications

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Metabolic pathways vs Signalling Pathways

(Petri Nets)

Biological Applications

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Mass action for enzymatic reaction - phosphorylation

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

R Rp S1

Biological Applications

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Phosphorylation - dephosphorylation step Mass action model 1

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

R Rp S P

Biological Applications

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Phosphorylation - dephosphorylation loop Mass action model 2

R Rp S1 S2

  • 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

Biological Applications

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Phosphorylation - dephosphorylation step Mass action (all singing/dancing)

R Rp S P

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

Biological Applications

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Michaelis-Menten equation for phosphorylation-dephosphorylation

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

product is appreciable

  • 3. [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)

Rp R S P

Biological Applications

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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?

Biological Applications

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15 Biological Applications

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Composition Vertical & horizontal

Rp R S1 RRp RR P1 P2 Rp R S Rpp P

2-stage cascade 1-stage cascade double phosphorylation step

Biological Applications

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Composition Vertical & horizontal

Biological Applications

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Two stage, double phosphorylation

Biological Applications

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Engineering Biochemical Network models

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Phosphorylation cascade: 2-stage, Mass Action model 1

Rp R S1 RRp RR

R+S1

k2

←  

k1

 →  R | S1 k3

 →  Rp + S1 R+P

1 k3'

←   R | P

1

k2 '

 → 

k1 '

←   Rp + P 1

RR + Rp

kk2

←  

kk1

 →  RR | Rp kk3

 →  RRp + Rp RR+ P

2 kk3'

←   RR | P

2

kk2 '

 →  

kk1 '

←    RRp + P 2 BioSysBio08

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Engineering Biochemical Network models

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Phosphorylation cascade: 2-stage, Michaelis-Menten

Rp R S1 RRp RR

dRp dt = k3 × S1 × R Km1 + R − k3'×Rp Km2 + Rp dRRp dt = kk3 × Rp × RR KKm1 + RR − kk3'×RRp KKm2 + RRp

BioSysBio08

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Engineering Biochemical Network models

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3-stage Phosphorylation cascade (Mass Action)

R+S1

k2

←  

k1

 →  R | S1 k3

 →  Rp + S1 R+P

1 k3'

←   Rp | P

1

k2 '

 → 

k1 '

←   Rp + P 1

RR + Rp

kk2

←  

kk1

 →  RR | Rp kk3

 →  RRp + Rp RR+ P

2 kk3'

←   RRp | P

2

kk2 '

 →  

kk1 '

←    RRp + P 2

RRR + RRp

kkk2

←   

kkk1

 →   RRR | RRp kkk3

 →  RRRp + RRp RRR+ P

3 kkk3'

←    RRRp | P

3

kkk2 '

 →  

kkk 1 '

←    RRRp + P 3

Rp R S1 RRp RR RRRp RRR P1 P2 P3

BioSysBio08

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Engineering Biochemical Network models

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Phosphorylation cascade: 3-stage, Michaelis-Menten

Rp R S1 RRp RR RRRp RRR

dRp dt = k3 × S1 × R Km1 + R − k3'×Rp Km2 + Rp dRRp dt = kk3 × Rp × RR KKm1 + RR − kk3'×RRp KKm2 + RRp dRRRp dt = kkk3 × RRp × RRR KKKm1 + RRR − kkk3'×RRRp KKKm2 + RRRp

BioSysBio08

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3-stage

Biological Applications

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Phosphorylation cascade + feedback

Rp R S1 RR P1 P2 RRp RRp Rp R S1 RR P1 P2 RRp Rp R S1 RR P1 P2 RRp Rp R S1 RR P1 P2

Biological Applications

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25 Biological Applications

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26 Biological Applications

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Engineering Biochemical Network models

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Phosphorylation cascade + negative feedback: 2-stage, Mass Action model 1

Rp R S1 RRp RR

RRp+S1

ki '

←  

ki

 →  RRp | S1

R+S1

k2

←  

k1

 →  R | S1 k3

 →  Rp + S1 R+P

1 k3'

←   R | P

1

k2 '

 → 

k1 '

←   Rp + P 1

RR + Rp

kk2

←  

kk1

 →  RR | Rp kk3

 →  RRp + Rp RR+ P

2 kk3'

←   RR | P

2

kk2 '

 →  

kk1 '

←    RRp + P 2 BioSysBio08

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Engineering Biochemical Network models

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

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

Rp R S1 RRp RR

dRp dt = k3 × S1 × R Km1 × 1+ RRp Ki       + R − k3'×Rp Km2 + Rp dRRp dt = kk3 × Rp × RR KKm1 + RR − kk3'×RRp KKm2 + RRp

V =Vmax × [S] [S]+ Km × 1+ [I] [Ki]      

BioSysBio08

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Engineering Biochemical Network models

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Phosphorylation cascade + negative feedback: 3-stage, Mass Action, model 1

Rp R S1 RRp RR RRRp RRR

RRRp+S1

ki '

←  

ki

 →  RRRp | S1

R+S1

k2

←  

k1

 →  R | S1 k3

 →  Rp + S1 R+P

1 k3'

←   Rp | P

1

k2 '

 → 

k1 '

←   Rp + P 1

RR + Rp

kk2

←  

kk1

 →  RR | Rp kk3

 →  RRp + Rp RR+ P

2 kk3'

←   RRp | P

2

kk2 '

 →  

kk1 '

←    RRp + P 2

RRR + RRp

kkk2

←   

kkk1

 →   RRR | RRp kkk3

 →  RRRp + RRp RRR+ P

3 kkk3'

←    RRRp | P

3

kkk2 '

 →  

kkk 1 '

←    RRRp + P 3

BioSysBio08

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Engineering Biochemical Network models

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Phosphorylation cascade + negative feedback: 3-stage, Michaelis-Menten

Rp R S1 RRp RR RRRp RRR

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

V =Vmax × [S] [S]+ Km × 1+ [I] [Ki]      

dRp dt = k3 × S1 × R Km1 × 1+ RRRp Ki       + R − k3'×Rp Km2 + Rp dRRp dt = kk3 × Rp × RR KKm1 + RR − kk3'×RRp KKm2 + RRp dRRRp dt = kkk3 × RRp × RRR KKKm1 + RRR − kkk3'×RRRp KKKm2 + RRRp

BioSysBio08

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3-stage, negative feedback

Biological Applications

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Engineering Biochemical Network models

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Phosphorylation cascade + negative feedback: 3-stage, Inhibitor on 2nd stage, Mass Action

RRRp+S1

ki '

←  

ki

 →  RRRp | S1

R+S1

k2

←  

k1

 →  R | S1 k3

 →  Rp + S1 R+P

1 k3'

←   Rp | P

1

k2 '

 → 

k1 '

←   Rp + P 1

RR + Rp

kk2

←  

kk1

 →  RR | Rp kk3

 →  RRp + Rp RR+ P

2 kk3'

←   RRp | P

2

kk2 '

 →  

kk1 '

←    RRp + P 2

RRR + RRp

kkk2

←   

kkk1

 →   RRR | RRp kkk3

 →  RRRp + RRp RRR+ P

3 kkk3'

←    RRRp | P

3

kkk2 '

 →  

kkk 1 '

←    RRRp + P 3

U + RR

ku2

←  

ku1

 →   U | RR

U + RR p

ku2

←  

ku1

 →   U | RR p

U | RR + Rp

kk2

←  

kk1

 →  U | RR | Rp kk3

 →  U | RRp + Rp U | RR+ P

2 kk3'

←   U | RR p| P

2

kk2 '

 →  

kk1 '

←    U | RRp + P 2

Rp R S1 RRp RR RRRp RRR U|RRp U|RR

BioSysBio08

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3-stage, negative feedback + inhibitor

Biological Applications

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‘Real cascade & feedback’

Ras Raf MEK ERK U0126

Biological Applications

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Is the ERK pathway a negative feedback amplifier?

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

Biological Applications

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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 the amplifier.

  • The negative feedback amplifier was invented in 1927 by Harold Black of

Western Electric and was 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 of applications

Input

Amplifier Feedback

Output

Biological Applications

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Negative Feedback Amplifier

Input Amplifier Negative Feedback Loop Output Input After Feedback

y = Ae e = u – Fy y = A (u – Fy) y = Au – AFy y + AFy = Au y (1 + AF) = Au

Steady State Equation

y A F u

  • e

Ae Biological Applications

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38 Standard Amplifier

A u y + + y=A*u Amplifier (A) gain Output (y)

Negative Feedback Amplifier

Amplifier (A) gain Output (y)

The negative feedback imparts signalling robustness

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

Biological Applications

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Feedback

Output Increasing -> <- Amplifier Decreasing Feedback Increasing -> Increasing Feedback

Biological Applications

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40 Standard Amplifier

A u y + + y=A*u

Negative Feedback Amplifier

The negative feedback imparts signalling robustness

Time Output (y) Sudden drop in Amplifier (A) gain

Δy Output

Sudden drop in Amplifier (A) gain Output (y) Time

Δy Output

Biological Applications

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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

negative feedback amplifier.

  • If the ERK cascade is a negative feedback

amplifier it should be robust to disturbances within the cascade.

  • From a biological point of view, these

disturbances could be caused by drugs, such as U0126, aimed at decreasing the activity of the ERK cascade.

  • This suggests that these drugs will be relatively

ineffective.

  • In fact, current drugs aimed at decreasing the

activity of the MAPK pathway have proved less efficient in in vivo applications than anticipated from in vitro inhibition assays.

Sauro & Kholodenko (2004)

Biological Applications

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Raf/MEK/ERK amplifies the signal

Cell line Raf-1 MEK ERK

Concentration per cell

COS1

3.6 10.6 21.2 femtomol 1 2.9 5.9 ratio

NIH 3T3

10.9 7.1 98 femtomol 1 0.7 9 ratio

NIH COS

  • Rec. GST-BXB

195 118 90 70 55 38 33

WB: Raf-1

195 118 90 70 55 38 33

NIH COS

  • Rec. MEK1-His

WB: MEK

195 118 90 70 55 38 33

NIH COS

  • Rec. GST-ERK

WB: ERK

Biological Applications

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How to test if the ERK pathway is a NFA?

Ras-GTP Raf-1 MEK1/2 ERK1/2 Negative Feedback

U0126

Generate input: Stimulate with GF Measure signal output: i.e. ERK phosphorylation Remove negative feedback “Disturb the Amplifier”: Use a MEK inhibitior, such as U0126 Biological Applications

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

Ras-GTP Raf-1 MEK1/2 ERK1/2 Negative Feedback

U0126

Ras-GTP Raf-1 MEK1/2 ERK1/2

U0126

phospho-ERK MEK inhibitor

Feedback intact Feedback removed

Biological Applications

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How to test if the ERK pathway is a NFA?

Strategy

In vivo system that allows us to compare feedback broken to feedback intact model. Computational Model of ERK pathway with/without feedback

Biological Applications

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Computational Modeling 1: Build the model

  • Non-linear ordinary differential

equations (ODE’s).

  • ODE’s were solved using Math

Lab and Gepasi.

  • Models are based on the

Schoeberl et al. (2002) model

  • Mass Action Kinetics instead of

Michaelis Menten

  • Kinetic parameters are from

literature, previous models and “guesstimates”

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

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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; ?

model degradation in S1 signal?

Biological Applications

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Feedback broken Feedback intact

Computational Modeling 2: Results

Prediction: Braking the feedback modulates drug response

Biological Applications

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Sensitivity of kinetic parameters is decreased due to Negative Feedback

Computational Modeling 2: Results

Biological Applications

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EGFR Sos Ras Raf MEK ERK

The experimental systems

Negative feedback loops intact

RasV12 Raf MEK ERK

One feedback loop eliminated by constitutively active RasV12 mutant

BXB-ER 4-OHT MEK ERK

Both feedback loops eliminated by BXB-ER (4-OHT regulatable Raf-1 mutant) U0126 U0126 U0126 MEK

inhibitor

4557W EGFR inhibitor Biological Applications

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BXB-ER

ERK feedback phosphorylation sites

Raf-1

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)

5 10 20 40 80 120 min

ppERK levels Biological Applications

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Ablation of feedback by BXBER decreases robustness to MEK-inhibitor U0126

Computer Simulation

Biological Applications

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Experiment

Ablation of feedback by BXBER decreases robustness to MEK- inhibitor U0126

Biological Applications

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0 10 20 40 80 min stimulation pERK1/2, +EGF pERK1/2, + BXBER/4HT U0126 added

Signal recovery after MEK inhibition

Simulation Experiment Biological Applications

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Kholodenko – negative feedback oscillator

Engineering Biochemical Network models

BioSysBio08 55

Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades. Kholodenko BN., Eur J Biochem 2000 Mar;267(6):1583-8

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Oscillations! Phosphorylation cascade + negative feedback: 3-stage, Inhibitor on 2nd stage, Mass Action

Conditions S1=3 Inhibitor=0.5

Biological Applications

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

Combination of positive & negative feedback

Mathematical Model

Biological Applications

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58 No Feedback Positive Feedback Negative Feedback Positive & Negative Feedback

Combination of positive & negative feedback: Simulation

Biological Applications

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

20’ 40’ 1h 2h 3h 4h 6h TPA ERK-pp (activated ERK) total ERK

Western blots COS1 cell lysates

Simulation Experiment

Biological Applications