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Extracting Essential Features of Biological Networks. Natalie Arkus, Michael P. Brenner. School of Engineering and Applied Sciences Harvard University. Model. Empirical System. Biological System. Explanations. Predictions. Biological System. Model. B. A. B. A.

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extracting essential features of biological networks

Extracting Essential Features of Biological Networks

Natalie Arkus, Michael P. Brenner

School of Engineering and Applied Sciences

Harvard University

slide2

Model

Empirical

System

Biological

System

Explanations

Predictions

slide3

Biological

System

Model

B

A

B

A

slide4

Nerve growth factor signaling

Importin nuclear protein import

Map Kinase Pathway

p53 Pathway

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

slide5

Biological

System

Model

B

A

B

A

slide6

Biological

System

Complicated

Model

X

Explanations

Predictions

?

Analysis?

B

A

  • Many nonlinear coupled equations → can’t solve analytically
  • Many unknown parameters → many possible solutions

B = f(A)

Current Methods

  • Numerical simulation

not falsifiable!

slide7

X

Current Methods: Another Option

Biological

System

Complicated

Model

Simple

Model

Explanations!

Predictions!

Input

Output

slide8

A

C

B

Knowingly ignores biology

Can be fully analyzed

Captures everything

Too complicated to fully analyze

slide9

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

?

Simple

Model

Biological

System

Complicated

Model

math

Explanations

Predictions

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?

slide10

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

slide11

Feedback Loop:

DnaK (chaperone) sequestersσ32 (transcription factor)

→ decreases rate of hsg transcription

slide12

Another Feedback Loop:

Proteases (FtsH, HslVU) degradeσ32 (transcription factor)

→ decreases rate of hsg transcription

slide13

1st Feedback Loop

2nd Feedback Loop

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

Differential Equations = ODEs

Algebraic Equations = AEs

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

  • 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

slide14

What is the response time?

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

Can ask such questions…

but are not equipped to answer such questions…

slide15

Differential Equations (ODEs)

Reduction Method:

Algebraic Equations (AEs)

1) Separation of scales

→ Reduction in the # of differential equations

≈ 0

2)Dominant Balance

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

3)

1feedback loop model
1Feedback Loop Model

Transcription & Translation Equations

Mass Balance (Conservation) Equations

Algebraic Binding Equations

slide17

Reduction Method

1) Separation of scales

→ Reduction in the # of differential equations

≈ 0

2)Dominant Balance

3)

slide18

Look for a separation of time scales:

Transcription & Translation Equations

0.5

Only 1 slow variable!

0.03

0.5

1.4

~100

slide19

Temperature upshift

Temperature upshift

→ 1 ODE, 18 AEs, 29 parameters

slide20

1)

Separation of scales

→ Reduction in the # of differential equations

Reduction Method

≈ 0

2)Dominant Balance

3)

one example
One Example

→ σ32 sequestration hardly effects DnaKf levels!

slide23

X

X

slide25

1)

Separation of scales

→ Reduction in the # of differential equations

Reduction Method

≈ 0

2)Dominant Balance

3)

slide26

.

.

(after many dominant balances)

.

slide28

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

  • How do feedback loops ([σ32:DnaK], [FtsHt],…) effect the response time?
slide29

Reduced Model for all Feedback Loops:

Effect of 1st feedback loop

Effect of 2 feedback loops

slide30

Simple

Model

Biological

System

Complicated

Model

math

Explanations

Predictions

what sets the time of heat shock response
What Sets the Time of Heat Shock Response?

Temperature upshift

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

Is response time feedback- or parameter-dependent?

slide32

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

High [DnaKt] Limit:

Low [DnaKt] Limit:

(using linear [DnaKf] approximation)

Response of folded proteins is a feedback-loop independent property

slide33

Reduced Model for all Feedback Loops:

Degradation Term

Production Term

A = effect of 0F loop

B = effect of 1F loop

C = combined effect of 1F and 2F loops

B > 0 → smaller production term → slower response time

C > 0 → smaller production term → slower response time

Feedback loops → slower response time

How can the response time decrease with additional feedback loops?

slide34

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

slide35

Parameter changes across the feedback loop models

Translation of [mRNA(DnaK)]

Degradation of [σ32]

Effect of parameter changes is unclear in full model

slide36

Effect of Parameter Changes Is Apparent in Reduced Model

Reduced Model for all Feedback Loops:

0 feedback loop:

1 feedback loop:

2 feedback loop:

slide37

*

*

slide38

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

 1 and 0 feedback loop models respond quicker.

slide39

Constructing Reduced Models Allows One to Extract Essential Biological Components

Here, 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?

slide40

System Is Not Special…

Wnt signaling pathway

(Protein network involved in embryogenesis and cancer)

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

slide41

Curves a-d:

Curve d:

conclusions
Conclusions

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

  • simple models with all relevant biological components
    • Back and forth with experiments

testable, falsifiable!

31 equations

 1 equation

14 equations

 3 equations

Yeast Cell Cycle (Tyson et al, 2004)

62 equations  17 equations

future directions
Future Directions

f(dimenionless parameters) ?

{ Reduced Model 1,

…}

Reduced Model 2,

Reduced Model 3,

slide44

Courtesy of cancerworld.org

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