Connecting Function and Topology(of small biological circuits)

International Workshop and Conference on Network Science, Queens, NY, May 22, 2007

Chao Tang

University of California, San Francisco

Collaborators

Prof. Qi OuyangProf. Luhua Lai (CTB, PKU)

Wenzhe Ma (Center for Theoretical BiologyPeking UniversityUCSF)

Form follows function!

Function follows form!

“Function Follows Form” -- 29,100 hits

“Form Follows Function” -- 363,000 hits

(As of 5/19/2007)

Form follows function

Function follows form

Function and form in biology

Molecular

MicroscopicMacroscopic

Organismic

? ? ? ?

PatterningSignal transductionHomeostasisAdaptationCell polarizationCell division… …

BistabilityOscillation

[A]

t

A 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

E

W

E

W

4x2=8 edges

3 possibilities per edge

38=6561 networks

A

B

A

B

A

B

… … … …

Model of regulation

B

kAAV

dtdB

nn

n

A B

)(1 BkA

AdtdB

nn

n

,VBB Define then

n,k

k

n/4k

A

nn

n

kAA

A

nn

n

kAA

1A B

)),,((1 BnkAHdtdB

iiii

B

A1

A2B

An example

A

B

A

B

A

B

)(1

)(1

32

2

1

1

Bk+A

Ak+A

kdtdB

Ak+B

kdtdA

nnout

nout

nn

n

B

nn

n

A

Q=fraction of parameter space that can perform the function

… …

Distribution of Q values

What are these 45 networks?

Skeletons and families

EssentialNeutralBadVery bad

Three and half topological features:Positive loop on EPositive loop on WMutual intercellular activation of E and WMutual repression if extracellular loop

Topology follows function

…… E W E W EW

E

W

A

nn

n

kAA

A

A

kAAV

dtdA

nn

n

E

W

…… E W E W EW WE

WE

WE

W WW

Coarse-graining the biological network

3-node networks

E

S

W

E

S

W

3x6=18 edges318=387,420,489 networks

Only two extracellular signaling315=14,348,907

Distribution of Q values

?

Bistability

Bistability

Sharp boundaries

Functional modules

Modules for 3-node networks

108 possible combinations

44 combinations form the skeletons for all robust networks (Q>0.1)

Q=0.63Q=0.59 Q=0.58

Q=0.50

Q=0.48Q=0.34

Q=0.66Q=0.66 Q=0.63

Q=0.26Q=0.29

Family size versus Q value

Skeletons with larger Q have larger family size

EssentialNeutralBadVery bad

Q values of the modules

E E

W W

W E W E

E module

W module

B module

Q = QE×QW×QB ?

Two candidates for bionetwork

Derek Lessing and Roel Nusse, (1998) Development 125, 1469-1476Marita Buescher, et al. (2004) Current Biology, 14, 1694-1702Hsiu-Hsiang Lee and Manfred Frasch, Development 127, 5497-5508 (2000)

?

?

ptc mutant

E WW E

wild type

E WW EW W WEW

patched mutant

continuous Hh signaling

zw3(shaggy) mutant

E WW E

wild type

continuous Wingless signaling

E WW EWE

zw3 mutant

E E E

Mutant tests for the two candidates

Wild type E WW E

patched mutant E WW EW W WEW

zw3 mutant, or ectopic expression of Wg E WW EWEE E E

Why fly picked this one?

The best without any direct auto positive loop

Q=0.61 Q=0.36

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)