Extracting Essential Features
of Biological Networks
Natalie Arkus, Michael P. Brenner
School of Engineering and Applied
Sciences
Harvard University
Biological
Empirical
Model
System
Explanations
Predictions
Biological
System
Model
B
A
B
A
Nerve growth factor signaling
Importin nuclear protein import
Map Kinase Pathway
p53 Pathway
Courtesy of http://www.london-nano.com, Guillaume Charras
Biological
System
Model
B
A
B
A
Biological
Complicated
System
Model
X
Explanations
Predictions
?
Analysis?
B
A
•Many nonlinear coupled equations → can’t solve analytically
•Many unknown parameters → many possible solutions
Current Methods
•Numerical simulation
not falsifiable!
B = f(A)
X
Biological
Complicated
System
Model
Current Methods: Another Option
Simple
Model
Explanations!
Predictions!
Input
Output
A
C
B
Knowingly ignores biology
Can be fully analyzed
Captures everything
Too complicated to fully analyze
Biological
Complicated
?
Simple
System
Model
math
Model
Explanations
Predictions
e. Coli heat shock response system
El Samad et al., PNAS, 102, 2736 (2005)
Courtesy of BB310 Molecular Genetics Webpage
from strath.ac.uk
What is the role of feedback loops in heat shock response?
Heat Shock Response (HSR):
Proteins unfold/misfold
and malfunction
σ32 is upregulated
Heat shock gene
(hsg) transcription
Courtesy of BB310 Molecular Genetics Webpage
from strath.ac.uk
↑ 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
Differential Equations = ODEs
Algebraic Equations = AEs
1) 2 feedback loop model
23 ODEs, 8 AEs, 60 parameters
2nd Feedback Loop
They reduced these systems a priori by
assuming that all binding reactions were fast
→ 11 ODEs, 20 AEs, 48 parameters
2) 1 feedback loop model
14 ODEs, 5 AEs, 39 parameters
→ 5 ODEs, 14 AEs, 33 parameters
3) 0 feedback loop model
13 ODEs, 5 AEs, 37 parameters
→ 5 ODEs, 13 AEs, 32 parameters
•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…
Differential Equations (ODEs)
Reduction Method:
Algebraic Equations (AEs)
1) Separation of scales
→ Reduction in the # of differential equations
≈0
2) Dominant Balance
3)
Let us focus on 1
feedback loop model as
an example…
1Feedback Loop Model
Transcription & Translation Equations
Algebraic Binding Equations
Mass Balance (Conservation) Equations
Reduction Method
1) Separation of scales
→ Reduction in the # of differential equations
≈0
2) Dominant Balance
3)
Look for a separation of time scales:
Transcription & Translation Equations
0.5
Only 1 slow
variable!
0.03
0.5
1.4
~100
Temperature
upshift
→ 1 ODE, 18 AEs, 29 parameters
Temperature
upshift
Reduction Method
1)
Separation of scales
→ Reduction in the # of differential equations
≈0
2) Dominant Balance
3)
Solving Algebraic Components
Algebraic System:
One Example
→ σ32 sequestration hardly effects DnaKf levels!
X
X
Reduction Method
1)
Separation of scales
→ Reduction in the # of differential equations
≈0
2) Dominant Balance
3)
.
. (after many dominant balances)
.
✓
With reduced system, are
equipped to answer questions of
interest…
•How do feedback loops
([σ32:DnaK], [FtsHt],…) effect the
response time?
Reduced Model for all Feedback Loops:
Effect of 1st
feedback loop
Effect of 2
feedback loops
Biological
Complicated
System
Model
Simple
math
Explanations
Predictions
Model
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?
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
Reduced Model for all Feedback Loops:
Degradation Term
Production Term
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?
A = effect of 0F loop
B = effect of 1F loop
C = 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
Will be
encompassed
within C
 a different
expression for
the effective
parameter A
(the 0F term) in
the 2 feedback
loop model
Parameter changes across the
feedback loop models
Translation of Degradation of
[mRNA(DnaK)]
[σ32]
Effect of parameter changes is
unclear in full model
Effect of Parameter Changes Is Apparent in Reduced Model
Reduced Model for all Feedback Loops:
0 feedback loop:
1 feedback loop:
2 feedback loop:
*
*
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
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?
System Is Not Special…
Wnt signaling pathway
(Protein network involved in embryogenesis and
cancer)
Lee et al, PLoS Biology, 1, 116 (2003)
Curves a-d:
Curve d:
Conclusions
simple models with all
relevant biological
components
•Back and forth
with experiments
testable,
falsifiable!
31 equations
14 equations
 1 equation
 3 equations
Courtesy of BB310 Molecular Genetics
Webpage from strath.ac.uk
Yeast Cell Cycle (Tyson et al, 2004)
62 equations  17 equations
Future Directions
f(dimenionless parameters) ?
{ Reduced Model 1,
Reduced Model 2,
Reduced Model 3,
…}
Can we explain a biological system in a way that
experiments alone can not?
Courtesy of cancerworld.org