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?