Class 12: Communities
Dr. Baruch Barzel
Network Science: Motifs March 28, 2011
A Closer Look at Networks
The bird’s eye view:
The detailed view:
P(k ), k , d , C , r
k , C , M ij
A Closer Look at Networks
P(k ), k , d , C , r
Intermediate view:
k , C , M ij
Motifs and Sub-graphs
Sub-graph: a connected graph
consisting of a subset of the
nodes and links of a network
Motifs and Sub-graphs
Sub-graph: a connected graph
consisting of a subset of the
nodes and links of a network
Motifs: Sub-graphs that have a
significantly higher density in the real
network than in the randomized
version of the studied network
R. Milo et al., Science 298, 824 (2002)
Motifs and Sub-graphs
Sub-graph: a connected graph
consisting of a subset of the
nodes and links of a network
Motifs: Sub-graphs that have a
significantly higher density in the real
network than in the randomized
version of the studied network
Randomized networks: Ensemble of maximally random networks
preserving the degree distribution of the original network
R. Milo et al., Science 298, 824 (2002)
Motifs in Realistic Networks
Motifs: Sub-graphs that have
a significantly higher density in
the real network than in the
randomized version of the
studied network
R. Milo et al., Science 298, 824 (2002)
Motifs in Realistic Networks
Motifs in Realistic Networks
Motifs in Biological Networks
Regulatory Network
Protein protein
interaction network
Metabolic Network
Genes are connected if one regulates the expression of the other
Motifs in Biological Networks
Regulatory Network
Y
Activation
Y
X
Inhibition
Y
X
X
Gene x
Genes are connected if one regulates the expression of the other
Motifs in Biological Networks
Regulatory Network
Feed-back
loop
Auto-regulation
Y
X
X
Tong et al. Science 298, 799 (2002)
The Significance of Motifs
Evolutionary conservation of sub-graphs
Natural selection aims to maintain function
Function is typically not carried by single
components, but rather by a network of
interacting subunits
We expect a tendency towards the
evolutionary conservation of sub-networks
that are capable of carrying biological
function
Tong et al. Science 298, 799 (2002)
Auto-regulation
Auto-regulation
X
Mortality or Degradation
could do the jobs
X
X
X
X
X
X
X
Rosenfeld et al. J. Mol. Biol. 323, 785 (2002)
The Auto-regulation Advantage
Auto-regulation vs. Protein Degradation
X
X
Rosenfeld et al. J. Mol. Biol. 323, 785 (2002)
The Auto-regulation Advantage
Protein Degradation
X
dn
 g  wn(t )
dt
n(t )  Time dependent concentrat ion
n0  Steady - state concentrat ion
g
0  g  wn0  n0 
w
g  Generation rate
w  Degradatio n rate
 0  Binding rate
dn
dn
 dt 
 wdt  ln( n0  n)  w(t  t0 )
g  wn
n0  n

n(t )  n0 1  e  wt

1  Unbinding rate

k 0
1
 deg
ln 2

w
Rosenfeld et al. J. Mol. Biol. 323, 785 (2002)
The Auto-regulation Advantage
n(t )  Time dependent concentrat ion
n0  Steady - state concentrat ion
Protein Auto-regulation
X
 dn
 dt  g (1  r )  dn   0 n(1  r )  1r
 dr
   0 n(1  r )  1r
 dt
Steady state for r:
dr
kn
0r 
dt
1  kn
g  Generation rate
w  Degradatio n rate
 0  Binding rate
1  Unbinding rate

k 0
1
 1  1  4kn0
n
dn
g

 wn  kn2  n  n0  0  n 
 0
dt 1  kn
2k
k

 wdt 
1  kndn
kn  n  n0
2
0
 n(t )  n0 1  e  2 wt
X
1
 deg
ln 2

w
X
g
Rosenfeld et al. J. Mol. Biol. 323, 785 (2002)
The Auto-regulation Advantage
n(t )  Time dependent concentrat ion
n0  Steady - state concentrat ion
Protein Auto-regulation
X
 dn
 dt  g (1  r )  dn   0 n(1  r )  1r
 dr
   0 n(1  r )  1r
 dt
Steady state for r:
dr
kn
0r 
dt
1  kn
g  Generation rate
w  Degradatio n rate
 0  Binding rate
1  Unbinding rate

k 0
1
 1  1  4kn0
n
dn
g

 wn  kn2  n  n0  0  n 
 0
dt 1  kn
2k
k

 wdt 
1  kndn
kn  n  n0
2
0
 n(t )  n0 1  e  2 wt
X
1
ln 2

w
X
g
Rosenfeld et al. J. Mol. Biol. 323, 785 (2002)
The Auto-regulation Advantage
n(t )  Time dependent concentrat ion
n0  Steady - state concentrat ion
Protein Auto-regulation
X
n(t )  n0 1  e 2 wt
1  e  2 wt 
 AR
1
3
 e  2 wt 
2
4
0
1
1  Unbinding rate

k 0
1
ln 43

2w
 AR 12 ln 43

 0.2
 deg ln 2
X
g  Generation rate
w  Degradatio n rate
 0  Binding rate
 deg 
ln 2
w
X
g
Rosenfeld et al. J. Mol. Biol. 323, 785 (2002)
The Auto-regulation Advantage
Auto-regulation vs. Protein Degradation
X
X
 AR 12 ln 43

 0.2
 deg ln 2
nAR (t )  n0 1  e 2 wt

ndeg (t )  n0 1  e wt

Auto-regulation is an efficient
scheme to achieve fast
response time to external stimuli
Rosenfeld et al. J. Mol. Biol. 323, 785 (2002)
The Auto-regulation Advantage
Auto-regulation vs. Protein Degradation
Uri Alon Nature Reviews 8, 450 (2007)
The Feed-Forward Loop
X
X
Y
Y
Z
Z
Mangan and Alon PNAS 100, 21 (2003)
The Feed-Forward Loop
X
Y
Z
Arbinose system
Flagella
Galactose utilization
Uri Alon Nature Reviews 8, 450 (2007)
Function oF the Feed Forward Loop
Filtering of spurious
spikes, and
detecting persistent
stimuli
Uri Alon Nature Reviews 8, 450 (2007)
Function oF the Feed Forward Loop
Sign Sensitive Delay
AND – Causes a rise time delay
OR – Results in turn-off delay
Uri Alon Nature Reviews 8, 450 (2007)
Function oF the Feed Forward Loop
Coherent FFL:
Sign Sensitive Delay
AND – Causes a rise time
delay
OR – Results in turn-off delay
Incoherent FFL:
Persistent Stimulus results in a
spike of expression
Uri Alon Nature Reviews 8, 450 (2007)
Topologically Induced Motifs
What determines the number of sub-graphs in biological networks?
(n,m)
Transcription
Metabolic
Protein
E. coli
S. cerevisiae
E. coli
S. cerevisiae
S. cerevisiae
(3,2)
12
19
101
72
70
(3,3)
0.30
0.31
5.0
5.8
4.1
(4,3)
169
220
4412
2041
2395
(4,6)
0.00
0.00
0.44
0.77
0.97
(5,4)
2492
2587
2.1x105
5.9x104
1.2x105
(5,10)
0.00
0.00
0.055
0.20
0.66
(6,5)
3.2x104
2.8x104
8.8x106
1.5x106
5.7x106
(6,15)
0.00
0.00
0.00
0.03
0.36
(7,6)
3.4x105
2.7x105
3.5x108
3.7x107
2.4x108
(7,21)
0.00
0.00
0.00
0.00
0.00
Vázquez et al. PNAS 101, 52 (2004)
Numerical Parameterization
Numerical description of sub-graphs
Includes n nodes
A central node connected to all others
Has m edges
(n,m)
Denoted by: (n,m)
(4,3)
(3,2)
(3,3)
(4,6)
(5,4)
(n, m)
Feed Forward Loop
(1,1)
(5,10)
(6,5)
(6,15)
X
(7,6)
(7,21)
Vázquez et al. PNAS 101, 52 (2004)
The Local and the Global Views
Vázquez et al. PNAS 101, 52 (2004)
Think Global - Act Local
P( k ) ~ k 
C (k ) ~ k 
(n, m)
You need a node
with a degree of at
least n – 1
You need m – (n – 1)
links between its
neighbors
Vázquez et al. PNAS 101, 52 (2004)
Think Global - Act Local
How frequent will the motifs (n,m) be?
n  1 Edges for the central node
m  (n  1) Edges between its n  1 selected neighbors
For a node with a degree of k:
 P( k ) ~ k 
 k  ( mn1)
n 1( m  n 1)



N
(
k
)
~
k

k

nm
 n  1

C
(
k
)
~
k



So for the whole network:
k max
k max
k 1
k 1
  n ( m  n 1)
N nm ~ N  P(k ) N nm (k ) ~ N  k   n1( mn1) ~ Nkmax
  n  (m  n  1)  0  N nm  N

  n  (m  n  1)  0  N nm  N
Vázquez et al. PNAS 101, 52 (2004)
Think Global - Act Local
How frequent will the motifs (n,m) be?
  n  (m  n  1)  0  N nm  N

  n  (m  n  1)  0  N nm  N
 1   
m  1   n  1  
   
m
m
n
n
 1   
m  1   n  1  
   
Vázquez et al. PNAS 101, 52 (2004)
Interplay Between Scales
The abundance of motifs is dictated by the scaling exponents
The scaling exponents are dictated by the frequency of motifs
Vázquez et al. PNAS 101, 52 (2004)
The Sub-graph Degree Distribution
P(T) = The probability that a node participates in exactly T triangles
 P (k ) ~ k 


C (k ) ~ k
T (k )  C (k )
k (k  1)
~ k 2
2


P (T ) ~  P (k ) T  k 2 dk
dT ~ k 3 dk , k ~ T 2
1
P (T ) ~  T (k )  T  T (k ) 

2
dT (k )
T (k )
3
2
P (T ) ~ T 
 1
  1
2 
Vázquez et al. PNAS 101, 52 (2004)
Sub-graph Giant Component
At what point will the sub-graphs become sparse enough to
break the network down
Vázquez et al. PNAS 101, 52 (2004)
Sub-graph Giant Component
At what point will the sub-graphs become sparse enough to
break the network down
Vázquez et al. PNAS 101, 52 (2004)
Network Motifs
Recurring sub-graphs that poses a
functional benefit
Shown to have functional roles in
biological networks
The large scale attributes and the
local interaction patterns are closely
related