Extracting Essential Features of Biological Networks

Natalie Arkus, Michael P. Brenner

School of Engineering and Applied Sciences

Harvard University

Model

Explanations

Predictions

Empirical

System

Biological

System

BiologicalSystem

Model

A B A

B

Courtesy of http://www.london-nano.com, Guillaume Charras

Map Kinase Pathway

Nerve growth factor signalingImportin nuclear protein import

p53 Pathway

BiologicalSystem

Model

A B A

B

•Many nonlinear coupled equations → can’t solve analytically

•Many unknown parameters → many possible solutions

Biological

System

Complicated

Model

Explanations

Predictions

? Analysis?

Current Methods

•Numerical simulation

B = f(A)

A

B

not falsifiable!

X

Complicated

Model

Simple

Model

XCurrent Methods: Another Option

Input Output

Explanations!Predictions!

Biological

System

Captures everything

Knowingly ignores biology

Too complicated to fully analyze

Can be fully analyzed

A C

B

Complicated

Model

Explanations

Predictions

Simple

Model

?math

Courtesy of BB310 Molecular Genetics Webpage from strath.ac.uk

e. Coli heat shock response system

El Samad et al., PNAS, 102, 2736 (2005)

What is the role of feedback loops in heat shock response?

Biological

System

Courtesy of BB310 Molecular Genetics Webpage from strath.ac.uk

Heat Shock Response (HSR):

Proteins unfold/misfold and malfunction

σ32 is upregulated

Heat shock gene (hsg) transcription

↑ Heat shock proteins (hsp’s)

Ex. DnaK, FtsH

Refold and degrade unfolded proteins

Feedback Loop:

DnaK (chaperone) sequesters σ32 (transcription factor)

→ decreases rate of hsg transcription

Another Feedback Loop:

Proteases (FtsH, HslVU) degrade σ32 (transcription factor)

→ decreases rate of hsg transcription

El Samad et al., PNAS, 102, 2736 (2005)

1st Feedback Loop

2nd Feedback Loop

1) 2 feedback loop model

23 ODEs, 8 AEs, 60 parameters

2) 1 feedback loop model

14 ODEs, 5 AEs, 39 parameters

3) 0 feedback loop model

13 ODEs, 5 AEs, 37 parameters

→ 11 ODEs, 20 AEs, 48 parameters

→ 5 ODEs, 14 AEs, 33 parameters

→ 5 ODEs, 13 AEs, 32 parameters

They reduced these systems a priori by assuming that all binding reactions were fast

Differential Equations = ODEs

Algebraic Equations = AEs

•What is the response time?•How do feedback loops ([σ32:DnaK], [FtsHt],…) effect the response time?

but are not equipped to answer such questions…

Can ask such questions…

1) Separation of scales

→ Reduction in the # of differential equations

2) Dominant Balance

≈ 0

3)

Let us focus on 1 feedback loop model as an example…

Differential Equations (ODEs)

Algebraic Equations (AEs)Reduction Method:

1Feedback Loop ModelTranscription & Translation Equations

Algebraic Binding Equations Mass Balance (Conservation) Equations

1) Separation of scales

→ Reduction in the # of differential equations

2) Dominant Balance

≈ 0

3)

Reduction Method

Look for a separation of time scales:

Transcription & Translation Equations

0.5

0.03

0.5

1.4

~100

Only 1 slow variable!

→ 1 ODE, 18 AEs, 29 parameters

Temperature upshift

Temperature upshift

1)

Separation of scales

→ Reduction in the # of differential equations

2) Dominant Balance

≈ 0

3)

Reduction Method

Solving Algebraic Components

Algebraic System:

One Example

→ σ32 sequestration hardly effects DnaKf levels!

X

X

1)

Separation of scales

→ Reduction in the # of differential equations

2) Dominant Balance

≈ 0

3)

Reduction Method

.

(after many dominant balances)..

✓

•How do feedback loops ([σ32:DnaK], [FtsHt],…) effect the response time?

With reduced system, are equipped to answer questions of interest…

Reduced Model for all Feedback Loops:

Effect of 2 feedback loops

Effect of 1st feedback loop

Complicated

Model

Explanations

Predictions

Simple

Modelmath

Biological

System

What Sets the Time of Heat Shock Response?

El Samad et al.'s conclusion: Response time decreases as number of feedback loops increase.

Is response time feedback- or parameter-dependent?

Temperature upshift

High [DnaKt] Limit:

Low [DnaKt] Limit:

(using linear [DnaKf] approximation)

Response of folded proteins is a feedback-loop independent property

Response time set by when [DnaKt] = 1.9*10^4

Reduced Model for all Feedback Loops:

Feedback loops → slower response time

How can the response time decrease with additional feedback loops?

Production TermDegradation Term

B > 0 → smaller production term → slower response time

C > 0 → smaller production term → slower response time

A = effect of 0F loopB = effect of 1F loopC = combined effect of 1F and 2F loops

Changes in Network Topology and Parameter Values Cause Models with More Feedback Loops to Respond Faster

For the same value of A, feedback loops slower response time

However, the topology of the σ32t equation changes in the 2 feedback loop

model

a different expression for the effective parameter A (the 0F term) in the 2 feedback loop model

Will be encompassed within C

Parameter changes across the feedback loop models

Translation of [mRNA(DnaK)]

Degradation of [σ32]

Effect of parameter changes is unclear in full model

0 feedback loop: 1 feedback loop: 2 feedback loop:

Effect of Parameter Changes Is Apparent in Reduced Model

Reduced Model for all Feedback Loops:

*

*

If is the same over the 3 feedback loop models and in a certain parameter regime

1 and 0 feedback loop models respond quicker.

Constructing Reduced Models Allows One to Extract Essential Biological

ComponentsHere, the effect of topology and parameters were decoupled

And it was shown, for example, that response time is a parameter dependent and not a feedback loop dependent property

Is this system special, were we just lucky?

Wnt signaling pathway

Lee et al, PLoS Biology, 1, 116 (2003)

(Protein network involved in embryogenesis and cancer)

System Is Not Special…

Curves a-d:

Curve d:

Conclusionssimple models with all relevant biological components

•Back and forth with experiments

Courtesy of BB310 Molecular Genetics Webpage from strath.ac.uk

31 equations

1 equation

14 equations

3 equations

Yeast Cell Cycle (Tyson et al, 2004)

62 equations 17 equations

testable, falsifiable!

Future Directions

{ Reduced Model 1, Reduced Model 2, Reduced Model 3, …}

f(dimenionless parameters) ?

Courtesy of cancerworld.org

Can we explain a biological system in a way that experiments alone can not?