Connecting Function and Topology
(of small biological circuits)
Chao Tang
University of California, San Francisco
International Workshop and Conference on Network Science, Queens, NY, May 22, 2007
Collaborators
Wenzhe Ma
(Center for Theoretical Biology
Peking University
UCSF)
Prof. Qi Ouyang
Prof. Luhua Lai
(CTB, PKU)
Function follows
form!
Form follows
function!
(As of 5/19/2007)
“Function Follows Form” -- 29,100 hits
“Form Follows Function” -- 363,000 hits
Form follows function
Function follows form
Function and form in biology
Macroscopic
? ? ? ?
Organismic
Microscopic
Molecular
Bistability
Oscillation
[A]
t
A
Patterning
Signal transduction
Homeostasis
Adaptation
Cell polarization
Cell division
……
[A]
t
A
A
Gene cascade of segmentation
What kinds of networks can perform this function?
Why did nature pick the one in fly?
How would i design it?
Need at least two components
Enumerate all 2-node networks
4x2=8 edges
E
E
W
W
……
A
A
A
B
B
B
3 possibilities per edge
38=6561 networks
……
Model of regulation
A
n,k
B

B
Define B 
,
V
An
An  k n
dB
An
B
V n

dt
A  kn 
then
An
1 n
A  kn
n/4k
k
A1
A2
B
dB 1
An
 ( n
 B)
n
dt  A  k
A
dB 1
 ( H ( Ai , ki , ni )  B )
dt  B i
A
B
A
An example
…
A
A
A
B
B
B
…
n
dA 1
k1
 ( n
 A)
n
dt  A B + k1
n
n
dB 1
k2
Aout
 ( n
 B)
n
n
n
dt  B A + k2 Aout + k3
Q=fraction of parameter space that can perform the function
Distribution of Q values
What are these 45 networks?
Skeletons and families
Essential
Neutral
Bad
Very bad
Three and half topological features:
Positive loop on E
Positive loop on W
Mutual intercellular activation of E and W
Mutual repression if extracellular loop
Topology follows function
E
dA
An
A
V n

n
dt
A k

E
An
An  k n
W
A
W
A
…
W
W
E
E
W
W
W
E
E
W
W
W
E
E
W
…
Coarse-graining the biological network
3-node networks
E
3x6=18 edges
318=387,420,489 networks
S
S
Only two extracellular signaling
315=14,348,907
W
W
E
Distribution of Q values
?
Functional modules
Bistability
Sharp boundaries
Bistability
Modules for 3-node networks
108 possible combinations
44 combinations form the skeletons
for all robust networks (Q>0.1)
Q=0.66
Q=0.66
Q=0.48
Q=0.63
Q=0.59
Q=0.58
Q=0.50
Q=0.63
Q=0.34
Q=0.29
Q=0.26
Family size versus Q value
Skeletons with larger Q
have larger family size
Essential
Neutral
Bad
Very bad
Q values of the modules
E
E
W
W
Q = QE×QW×QB ?
E module
W module
B module
W
E
W
E
Two candidates for bionetwork
?
Derek Lessing and Roel Nusse, (1998) Development 125, 1469-1476
Marita Buescher, et al. (2004) Current Biology, 14, 1694-1702
Hsiu-Hsiang Lee and Manfred Frasch, Development 127, 5497-5508 (2000)
?
ptc mutant
wild type

W
E
W
E
W
E
patched mutant
W
continuous Hh signaling
W
E
E
W
W
W
zw3(shaggy) mutant
wild type
W

E
W
E
W
E

zw3 mutant
W
continuous Wingless signaling
E
E
E
E
W
E
Mutant tests for the two candidates






Wild type
patched mutant
zw3 mutant, or ectopic expression of Wg
W
W
E
W
E
W
E
E
E
E
W
E
W
W
W
W
E
W
E
W
E
E
Why fly picked this one?
Q=0.61
Q=0.36
The best without any direct auto positive loop
Summary
•
Robust functionality drastically limits network topology.
•
Modular structure originates from subfunctions
•
Modularity provides combinatorial variability
– Evolvability and pleiotropy
•
The one selected by nature may be optimized under biological
constraints
– Hh and Wg signaling are utilized in other functions
•
More complex functions from simpler modules
– Examples in transcription control and protein domains
– Hierarchical build up of modules
•
Simplicity of biological systems
Molecular Systems Biology 2, 70 (2007)