Download Protein Interaction Networks: a Mathematical approach Dr A Annibale

Protein Interaction Networks: a
Mathematical approach
Dr A Annibale
Department of Mathematics Taster Day, 29 June 2012
Protein Interaction Networks: a Mathematical approach – p. 1
Introduction
Although the title may sound biological..
Protein Interaction Networks: a Mathematical approach – p. 2
Introduction
Although the title may sound biological..
.. I am a
Mathematician
Protein Interaction Networks: a Mathematical approach – p. 2
Introduction
Although the title may sound biological..
.. I am a
Mathematician
I do collaborate with biologists..
Protein Interaction Networks: a Mathematical approach – p. 2
Introduction
Although the title may sound biological..
.. I am a
Mathematician
I do collaborate with biologists..
and this is not so strange..
Protein Interaction Networks: a Mathematical approach – p. 2
Introduction
Although the title may sound biological..
.. I am a
Mathematician
I do collaborate with biologists..
and this is not so strange..
Many problems in biology and medical sciences are
tackled by multidisciplinary teams which include
mathematicians
Protein Interaction Networks: a Mathematical approach – p. 2
Introduction
Although the title may sound biological..
.. I am a
Mathematician
I do collaborate with biologists..
and this is not so strange..
Many problems in biology and medical sciences are
tackled by multidisciplinary teams which include
mathematicians
Maths can help greatly bio-medical sciences
Protein Interaction Networks: a Mathematical approach – p. 2
Introduction
Although the title may sound biological..
.. I am a
Mathematician
I do collaborate with biologists..
and this is not so strange..
Many problems in biology and medical sciences are
tackled by multidisciplinary teams which include
mathematicians
Maths can help greatly bio-medical sciences
understand complex data in a quantitative and
systematic way
Protein Interaction Networks: a Mathematical approach – p. 2
Introduction
Although the title may sound biological..
.. I am a
Mathematician
I do collaborate with biologists..
and this is not so strange..
Many problems in biology and medical sciences are
tackled by multidisciplinary teams which include
mathematicians
Maths can help greatly bio-medical sciences
understand complex data in a quantitative and
systematic way
modelling complex systems as the biological ones
Protein Interaction Networks: a Mathematical approach – p. 2
Overview
Networks as infrastructure of signalling processes
Graphs: definitions and topology measures
Tailored random graph ensembles
Conclusions
Protein Interaction Networks: a Mathematical approach – p. 3
Overview
Networks as infrastructure of signalling processes
Graphs: definitions and topology measures
Tailored random graph ensembles
Conclusions
Protein Interaction Networks: a Mathematical approach – p. 4
Protein interaction networks (PIN)
Signalling processes,
signalling transduction
Transport
Protein complexes
(interact for a long time)
Protein modification
(e.g. protein kinase)
Protein Interaction Networks: a Mathematical approach – p. 5
Networks as infrastructure of
signalling process
Protein Interaction Networks: a Mathematical approach – p. 6
Networks as infrastructure of
signalling process
Errors in e.g. cellular information
processing are responsible for
diseases as cancer, autoimmunity
or diabetes
Protein Interaction Networks: a Mathematical approach – p. 6
Networks as infrastructure of
signalling process
Errors in e.g. cellular information
processing are responsible for
diseases as cancer, autoimmunity
or diabetes
Understand underlying structure
of signalling processes and how
changes in the signalling networks
may affect the flow of information
Protein Interaction Networks: a Mathematical approach – p. 6
Normal or abnormal?
Tools to quantify structure of a network
Protein Interaction Networks: a Mathematical approach – p. 7
Normal or abnormal?
Tools to quantify structure of a network
Compare networks, detect abnormalities:
Is this signalling network abnormal?
Protein Interaction Networks: a Mathematical approach – p. 7
Normal or abnormal?
Tools to quantify structure of a network
Compare networks, detect abnormalities:
Is this signalling network abnormal?
Assess significance of an observed pattern: “trivial” or
“special” element of structure?
Protein Interaction Networks: a Mathematical approach – p. 7
Normal or abnormal?
Tools to quantify structure of a network
Compare networks, detect abnormalities:
Is this signalling network abnormal?
Assess significance of an observed pattern: “trivial” or
“special” element of structure?
Crucially, our answers depend on our model of
“normal”/“random”!
Protein Interaction Networks: a Mathematical approach – p. 7
Normal or abnormal?
Tools to quantify structure of a network
Compare networks, detect abnormalities:
Is this signalling network abnormal?
Assess significance of an observed pattern: “trivial” or
“special” element of structure?
Crucially, our answers depend on our model of
“normal”/“random”!
Define good “reference” or Null models
Protein Interaction Networks: a Mathematical approach – p. 7
Normal or abnormal?
Tools to quantify structure of a network
Compare networks, detect abnormalities:
Is this signalling network abnormal?
Assess significance of an observed pattern: “trivial” or
“special” element of structure?
Crucially, our answers depend on our model of
“normal”/“random”!
Define good “reference” or Null models
Generate null models as ‘proxies’ to analyse processes:
hypothesis testing and prediction
Protein Interaction Networks: a Mathematical approach – p. 7
Overview
Networks as infrastructure of signalling processes
Graphs: definitions and topology measures
Tailored random graph ensembles
Conclusions
Protein Interaction Networks: a Mathematical approach – p. 8
Definitions
size N : number of ’nodes’, i, j, k = 1...N
Protein Interaction Networks: a Mathematical approach – p. 9
Definitions
size N : number of ’nodes’, i, j, k = 1...N
(
1 if link i −−j;
’links’:
cij =
0 otherwise
Protein Interaction Networks: a Mathematical approach – p. 9
Definitions
size N : number of ’nodes’, i, j, k = 1...N
(
1 if link i −−j;
’links’:
cij =
0 otherwise

Connectivity matrix




c=




c11
c21
..
.
c12
c22
..
.
···
···
..
.
c1j
c2j
..
.
···
···
..
.
c1N
c2N
..
.
ci1
..
.
ci2
..
.
···
..
.
cij
..
.
···
..
.
ciN
..
.
cN 1 cN 2 · · ·
cN j · · ·
cN N
Protein Interaction Networks: a Mathematical approach – p. 9










Definitions
size N : number of ’nodes’, i, j, k = 1...N
(
1 if link i −−j;
’links’:
cij =
0 otherwise

Connectivity matrix




c=




c11
c21
..
.
c12
c22
..
.
···
···
..
.
c1j
c2j
..
.
···
···
..
.
c1N
c2N
..
.
ci1
..
.
ci2
..
.
···
..
.
cij
..
.
···
..
.
ciN
..
.
cN 1 cN 2 · · ·
cN j · · ·
cN N
symmetry, i.e. graph undirected cij = cji ∀ i, j
Protein Interaction Networks: a Mathematical approach – p. 9










Definitions
size N : number of ’nodes’, i, j, k = 1...N
(
1 if link i −−j;
’links’:
cij =
0 otherwise

Connectivity matrix




c=




c11
c21
..
.
c12
c22
..
.
···
···
..
.
c1j
c2j
..
.
···
···
..
.
c1N
c2N
..
.
ci1
..
.
ci2
..
.
···
..
.
cij
..
.
···
..
.
ciN
..
.
cN 1 cN 2 · · ·
cN j · · ·
cN N
symmetry, i.e. graph undirected cij = cji ∀ i, j
no self-interaction, i.e. cii = 0 ∀i
Protein Interaction Networks: a Mathematical approach – p. 9










1
3
•
•
2
•
•
4



c=

0
1
1
0
1
0
0
1
1
0
0
1
0
1
1
0





Protein Interaction Networks: a Mathematical approach – p. 10
1
3
•
•
2
•
•
4



c=

0
1
1
0
1
0
0
1
1
0
0
1
0
1
1
0





So, how different are these two?
Protein Interaction Networks: a Mathematical approach – p. 10
Local structure measures
degree ki (nr of partners): ki =
✟
•
2
•3
✟
✟
❅
❅
ki = 4
1
•
❅•
i
P
j cij
•4
5
•
❆
❆
❆•
7
•
6
Protein Interaction Networks: a Mathematical approach – p. 11
Local structure measures
degree ki (nr of partners): ki =
✟
•
2
✟
❅
❅
ki = 4
1
•
✟•3
❅•
i
P
•4
5
•
❆
❆
❆•
7
•
P
j cij
= summation symbol: add ci1 , ci2 , . . .
ci1 = ci2 = ci4 = ci5 = 1
ci3 = ci6 = ci7 = 0
Only the neighbours of i contribute
6
Protein Interaction Networks: a Mathematical approach – p. 11
Local structure measures
degree ki (nr of partners): ki =
✟
•
2
✟
❅
❅
ki = 4
1
•
✟•3
❅•
i
P
•4
5
•
❆
❆
❆•
7
•
P
j cij
= summation symbol: add ci1 , ci2 , . . .
ci1 = ci2 = ci4 = ci5 = 1
ci3 = ci6 = ci7 = 0
Only the neighbours of i contribute
6
generalized degrees:
(1) P
ki = j cij ,
(2) P
ki = js cij cjs ,
etc
Protein Interaction Networks: a Mathematical approach – p. 11
Local structure measures
degree ki (nr of partners): ki =
✟
•
2
✟
❅
❅
ki = 4
1
✟•3
❅•
i
P
•4
5
•
•
❆
❆
❆•
7
P
j cij
= summation symbol: add ci1 , ci2 , . . .
•
ci1 = ci2 = ci4 = ci5 = 1
ci3 = ci6 = ci7 = 0
Only the neighbours of i contribute
6
generalized degrees:
(2)
(1) P
ki = j cij ,
(2) P
ki = js cij cjs ,
etc
ki : Contributions only from j, s such that
i•
•j
•s
Protein Interaction Networks: a Mathematical approach – p. 11
Local structure measures
degree ki (nr of partners): ki =
✟
•
2
✟
❅
❅
ki = 4
1
✟•3
❅•
i
P
•4
5
•
•
❆
❆
❆•
7
P
j cij
= summation symbol: add ci1 , ci2 , . . .
•
ci1 = ci2 = ci4 = ci5 = 1
ci3 = ci6 = ci7 = 0
Only the neighbours of i contribute
6
generalized degrees:
(2)
(1) P
ki = j cij ,
(2) P
ki = js cij cjs ,
etc
ki : Contributions only from j, s such that
i•
•j
•s
(ℓ)
(ki : nr of paths of length ℓ away from i)
Protein Interaction Networks: a Mathematical approach – p. 11
clustering coefficient Ci
number of connected pairs among neighbours of i
Ci =
number of pairs among neighbours of i
❆❆
2•
❅
❅
1•
❆•3
❅•
•4
i
❆❆
❆
Ci = 1/6
Protein Interaction Networks: a Mathematical approach – p. 12
clustering coefficient Ci
number of connected pairs among neighbours of i
Ci =
number of pairs among neighbours of i
❆❆
2•
❅
❅
1•
❆•3
❅•
•4
i
❆❆
❆
→ Possible pairs among neighbours of i:
(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)
Ci = 1/6
Protein Interaction Networks: a Mathematical approach – p. 12
clustering coefficient Ci
number of connected pairs among neighbours of i
Ci =
number of pairs among neighbours of i
❆❆
2•
❅
❅
1•
❆•3
❅•
•4
i
❆❆
❆
→ Possible pairs among neighbours of i:
(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)
→only (3, 4) is connected
Ci = 1/6
Protein Interaction Networks: a Mathematical approach – p. 12
Friendship network
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
How many friends
do you have?
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
How many friends
do you have?
Do your friends
know each other?
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
How many friends
do you have?
Do your friends
know each other?
How many people
are within a distance
of 2 links from you?
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
How many friends
do you have?
Do your friends
know each other?
How many people
are within a distance
of 2 links from you?
Answer: Calculate
Degree ki
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
How many friends
do you have?
Do your friends
know each other?
How many people
are within a distance
of 2 links from you?
Answer: Calculate
Degree ki
Clustering Coefficient Ci
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
How many friends
do you have?
Do your friends
know each other?
How many people
are within a distance
of 2 links from you?
Answer: Calculate
Degree ki
Clustering Coefficient Ci
(2)
Generalized degree ki
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
How many friends
do you have?
Do your friends
know each other?
How many people
are within a distance
of 2 links from you?
Answer: Calculate
Degree ki
Clustering Coefficient Ci
(2)
Generalized degree ki
Impractical for large N !
Protein Interaction Networks: a Mathematical approach – p. 13
Friendship network
from a “not-egocentric” point of view
Protein Interaction Networks: a Mathematical approach – p. 14
Friendship network
from a “not-egocentric” point of view
How many friends do people
have on average?
Protein Interaction Networks: a Mathematical approach – p. 14
Friendship network
from a “not-egocentric” point of view
How many friends do people
have on average?
Picking up at random one person
what is the probability of their
having 10 friends?
Protein Interaction Networks: a Mathematical approach – p. 14
Friendship network
from a “not-egocentric” point of view
How many friends do people
have on average?
Picking up at random one person
what is the probability of their
having 10 friends?
What is the average path length
between two people in the world?
Protein Interaction Networks: a Mathematical approach – p. 14
Friendship network
from a “not-egocentric” point of view
How many friends do people
have on average?
Picking up at random one person
what is the probability of their
having 10 friends?
What is the average path length
between two people in the world?
Six degrees of separation?
Protein Interaction Networks: a Mathematical approach – p. 14
Friendship network
from a “not-egocentric” point of view
How many friends do people
have on average?
Picking up at random one person
what is the probability of their
having 10 friends?
What is the average path length
between two people in the world?
Six degrees of separation?
How likely is that hubs
interact with each other?
Protein Interaction Networks: a Mathematical approach – p. 14
Global Structure Measures
Average degree hki =
1
N
P
i ki
Protein Interaction Networks: a Mathematical approach – p. 15
Global Structure Measures
Average degree hki =
1
N
P
i ki
Degree frequency e.g. histogram of degree sequence
(k1 , k2 , ..., kN )
f (k)
N = 24
k
Protein Interaction Networks: a Mathematical approach – p. 15
Global Structure Measures
Average degree hki =
1
N
P
i ki
Degree frequency e.g. histogram of degree sequence
(k1 , k2 , ..., kN )
f (k)
N = 24
k
X
f (k) = N
k
Protein Interaction Networks: a Mathematical approach – p. 15
Degree Distribution (size independent)
f (k)
p(k) =
N
p(k) =probability of drawing a node with degree k
Protein Interaction Networks: a Mathematical approach – p. 16
Degree Distribution (size independent)
f (k)
p(k) =
N
p(k) =probability of drawing a node with degree k
f (k)
p(k)
k
k
Protein Interaction Networks: a Mathematical approach – p. 16
Degree Distribution (size independent)
f (k)
p(k) =
N
p(k) =probability of drawing a node with degree k
f (k)
p(k)
k
X
k
k
p(k) =
X f (k)
k
N
=
=1
N
N
Protein Interaction Networks: a Mathematical approach – p. 16
Facebook, PINs etc: shape of p(k)?
hki = 50:
0.05
0.04
Poissonian (random):
0.03
p(k) = e−hki hkik /k!
0.02
0.01
20
40
60
80
100
friends
or this?
0.04
hki = 50:
0.03
Power law (preferential attachment):
‘hubs’
0.02
p(k) ∼ k −γ
0.01
❄
0
20
40
60
80
100
friends
Protein Interaction Networks: a Mathematical approach – p. 17
Degree Correlations
W (k, k ′ ): prob link between nodes with degrees (k, k ′ )
❅
ki = k
•
❅
❅
❅
cij = 1?
❅
• kj = k ′
❅
❅
❅
Protein Interaction Networks: a Mathematical approach – p. 18
Degree Correlations
W (k, k ′ ): prob link between nodes with degrees (k, k ′ )
❅
ki = k
•
❅
❅
❅
cij = 1?
❅
• kj = k ′
❅
❅
❅
cij independent of k, k ′ : W (k, k ′ ) = W0 (k, k ′ )
Protein Interaction Networks: a Mathematical approach – p. 18
Degree Correlations
W (k, k ′ ): prob link between nodes with degrees (k, k ′ )
❅
ki = k
•
❅
❅
❅
cij = 1?
❅
• kj = k ′
❅
❅
❅
cij independent of k, k ′ : W (k, k ′ ) = W0 (k, k ′ )
cij = 1 more likely if k, k ′ : W (k, k ′ ) > W0 (k, k ′ )
Protein Interaction Networks: a Mathematical approach – p. 18
Degree Correlations
W (k, k ′ ): prob link between nodes with degrees (k, k ′ )
❅
ki = k
•
❅
❅
❅
cij = 1?
❅
• kj = k ′
❅
❅
❅
cij independent of k, k ′ : W (k, k ′ ) = W0 (k, k ′ )
cij = 1 more likely if k, k ′ : W (k, k ′ ) > W0 (k, k ′ )
cij = 1 less likely if k, k ′ : W (k, k ′ ) < W0 (k, k ′ )
Protein Interaction Networks: a Mathematical approach – p. 18
Degree Correlations
W (k, k ′ ): prob link between nodes with degrees (k, k ′ )
❅
ki = k
•
❅
❅
❅
cij = 1?
❅
• kj = k ′
❅
❅
❅
cij independent of k, k ′ : W (k, k ′ ) = W0 (k, k ′ )
cij = 1 more likely if k, k ′ : W (k, k ′ ) > W0 (k, k ′ )
cij = 1 less likely if k, k ′ : W (k, k ′ ) < W0 (k, k ′ )

′  = 1 for all k, k ′ uncorrelated
W
(k,
k
)
′
⇒ Define : Π(k, k ) =
W0 (k, k ′ )  6= 1
correlated
Protein Interaction Networks: a Mathematical approach – p. 18
Protein interaction networks
Π(k, k′ )
p(k)
0.30
cam jejuni PIN
N = 1325
hki = 17.50
0.20
0.10
Pi
0.00
0.30
0
10
20
30
40
50
1.5
60
1.4
1.3
human PIN
1.2
1.1
1.0
0.20
0.9
k‘
0.8
0.7
0.6
N = 9463
hki = 7.40
0.5
0.10
0.4
0.3
0.2
0.1
0.0
0.00
0
10
20
30
40
50
60
k
how ‘special’ is a network with topology {p, Π}?
‘null models’ with topology {p, Π}?
distance between networks {p, Π} and {p′ , Π′ }?
Protein Interaction Networks: a Mathematical approach – p. 19
Overview
Networks as infrastructure of signalling processes
Graphs: definitions and topology measures
Tailored random graph ensembles
Conclusions
Protein Interaction Networks: a Mathematical approach – p. 20
Ensembles of random graphs
Random graphs are graphs where each link i−−j has a
prescribed probability Prob(cij ), so c with Prob(c)
Protein Interaction Networks: a Mathematical approach – p. 21
Ensembles of random graphs
Random graphs are graphs where each link i−−j has a
prescribed probability Prob(cij ), so c with Prob(c)
An ensemble of random graphs is defined by a set G of
allowed graphs and a probability Prob(c) that tells how
likely is to draw a graph c from G
Protein Interaction Networks: a Mathematical approach – p. 21
Ensembles of random graphs
Random graphs are graphs where each link i−−j has a
prescribed probability Prob(cij ), so c with Prob(c)
An ensemble of random graphs is defined by a set G of
allowed graphs and a probability Prob(c) that tells how
likely is to draw a graph c from G
G={all possible graphs}
✬
✘✘
✘
✘
✾
c✘
✫
✩
✘
✘✘✘
Prob(c)
✪
Protein Interaction Networks: a Mathematical approach – p. 21
Ensembles of random graphs
Random graphs are graphs where each link i−−j has a
prescribed probability Prob(cij ), so c with Prob(c)
An ensemble of random graphs is defined by a set G of
allowed graphs and a probability Prob(c) that tells how
likely is to draw a graph c from G
G={all possible graphs}
✬
✘✘
✘
✘
✾
c✘
Useful?
✫
✩
✘
✘✘✘
Prob(c)
✪
Protein Interaction Networks: a Mathematical approach – p. 21
Ensembles of random graphs
Random graphs are graphs where each link i−−j has a
prescribed probability Prob(cij ), so c with Prob(c)
An ensemble of random graphs is defined by a set G of
allowed graphs and a probability Prob(c) that tells how
likely is to draw a graph c from G
G={all possible graphs}
✬
✘✘
✘
✘
✾
c✘
Useful?
✫
✩
✘
✘✘✘
Prob(c)
✪
Hard to measure/handle all microcopic variables {cij }
Protein Interaction Networks: a Mathematical approach – p. 21
Ensembles of random graphs
Random graphs are graphs where each link i−−j has a
prescribed probability Prob(cij ), so c with Prob(c)
An ensemble of random graphs is defined by a set G of
allowed graphs and a probability Prob(c) that tells how
likely is to draw a graph c from G
G={all possible graphs}
✬
✘✘
✘
✘
✾
c✘
Useful?
✫
✩
✘
✘✘✘
Prob(c)
✪
Hard to measure/handle all microcopic variables {cij }
But for large N expect microcopic details less important
(any glass of water freezes at T = 0◦ Celsius!)
Protein Interaction Networks: a Mathematical approach – p. 21
Ensembles of random graphs
Random graphs are graphs where each link i−−j has a
prescribed probability Prob(cij ), so c with Prob(c)
An ensemble of random graphs is defined by a set G of
allowed graphs and a probability Prob(c) that tells how
likely is to draw a graph c from G
G={all possible graphs}
✬
✘✘
✘
✘
✾
c✘
Useful?
✫
✩
✘
✘✘✘
Prob(c)
✪
Hard to measure/handle all microcopic variables {cij }
But for large N expect microcopic details less important
(any glass of water freezes at T = 0◦ Celsius!)
Assume probability distribution Prob(c) ⇒ average
behaviour of macroscopic process
Protein Interaction Networks: a Mathematical approach – p. 21
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✫
all possible graphs
✩
✪
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
✫
✫
✩
all possible graphs
hki = ...
✩
✪
✪
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
✬
✫
✫
✫
✩
all possible graphs
hki = ...
p(k) = . . .
✩
✩
✪
✪
✪
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
✩
all possible graphs
✬
✬
k = (k1 , . . . , kN )
✫
✫
✫
✫
hki = ...
✩
✩
p(k) = . . .
✩
✪
✪
✪
✪
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki)
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki), Prob(c|p(k))
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki), Prob(c|p(k)), Prob(c|k)
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki), Prob(c|p(k)), Prob(c|k): Easy
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki), Prob(c|p(k)), Prob(c|k): Easy
Prob(c|k, W )
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki), Prob(c|p(k)), Prob(c|k): Easy
Prob(c|k, W ), Prob(c|p, W )
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki), Prob(c|p(k)), Prob(c|k): Easy
Prob(c|k, W ), Prob(c|p, W ): Less easy...
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored Random Graphs Ensembles
Tailored random graph ensemble: demand that all
graphs c in the ensemble have prescribed properties
(so Prob(c) = 0 if c does not satisfy those properties)
✬
Demand properties
✬
hki
p(k)
k = (k1 , . . . , kN )
W (k, k ′ )
Need to specify:
✬
✬
★
✩
all possible graphs
hki = ...
✧
✫
✫
✫
✫
✩
p(k) = . . .
✩
✥
k = (k1 , . . . , kN )
W (k, k′ ) = . . .
✩
✦✪
✪
✪
✪
Prob(c|hki), Prob(c|p(k)), Prob(c|k): Easy
Prob(c|k, W ), Prob(c|p, W ): Less easy...
Beyond?
Protein Interaction Networks: a Mathematical approach – p. 22
Tailored graph ensembles as proxies
protein
network
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
e.g. p(k), W (k, k ′ ), . . .
Measure
properties
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
Measure
properties
e.g. p(k), W (k, k ′ ), . . .
define
random graph ensemble
with same properties
❄
✬
✩
✫
✪
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
e.g. p(k), W (k, k ′ ), . . .
Measure
properties
define
random graph ensemble
with same properties
Prob(c|p, W, . . .) =
❄
✬
✩
✫
✪
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
e.g. p(k), W (k, k ′ ), . . .
Measure
properties
✛
analyze processes
graph ensemble
define
random graph ensemble
with same properties
Prob(c|p, W, . . .) =
❄
✬
✩
✫
✪
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
e.g. p(k), W (k, k ′ ), . . .
Measure
properties
hypothesis testing
and prediction
✛
analyze processes
graph ensemble
define
random graph ensemble
with same properties
Prob(c|p, W, . . .) =
❄
✬
✩
✫
✪
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
e.g. p(k), W (k, k ′ ), . . .
Measure
properties
hypothesis testing
and prediction
✛
analyze processes
graph ensemble
define
random graph ensemble
with same properties
Prob(c|p, W, . . .) =
❄
✬
✩
✫
✪
How special?: count graphs in the ensemble!
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
e.g. p(k), W (k, k ′ ), . . .
Measure
properties
hypothesis testing
and prediction
✛
analyze processes
graph ensemble
define
random graph ensemble
with same properties
Prob(c|p, W, . . .) =
❄
✬
✩
✫
✪
How special?: count graphs in the ensemble!
Distance between graphs cA , cB = Distance between
Prob(c|pA , WA . . .), Prob(c|pB , WB , . . .)
Protein Interaction Networks: a Mathematical approach – p. 23
Tailored graph ensembles as proxies
protein
network
✲
e.g. p(k), W (k, k ′ ), . . .
Measure
properties
hypothesis testing
and prediction
✛
analyze processes
graph ensemble
define
random graph ensemble
with same properties
Prob(c|p, W, . . .) =
❄
✬
✩
✫
✪
How special?: count graphs in the ensemble!
Distance between graphs cA , cB = Distance between
Prob(c|pA , WA . . .), Prob(c|pB , WB , . . .)
Generating graphs with right probabilities is highly non
trivial
Protein Interaction Networks: a Mathematical approach – p. 23
Detection biases
PI may be detected via different experiments
Protein Interaction Networks: a Mathematical approach – p. 24
Detection biases
PI may be detected via different experiments
Assume we have ℓ biologial species; M experiments;
Protein Interaction Networks: a Mathematical approach – p. 24
Detection biases
PI may be detected via different experiments
Assume we have ℓ biologial species; M experiments;
Biological
c1
c2
..
.
cℓ
Protein Interaction Networks: a Mathematical approach – p. 24
Detection biases
PI may be detected via different experiments
Assume we have ℓ biologial species; M experiments;
Biological
Experimental data
✿
✘✘
✘
✘
✘
✲
c1 ❳
❳❳
❳❳
③
1
c.11 experiment
.
..
..
cM
1 experiment M
c2
..
.
cℓ
Protein Interaction Networks: a Mathematical approach – p. 24
Detection biases
PI may be detected via different experiments
Assume we have ℓ biologial species; M experiments;
Biological
Experimental data
✿
✘✘
✘
✘
✘
✲
c1 ❳
❳❳
❳❳
③
1
c.11 experiment
.
..
..
cM
1 experiment M
Different experiments yield
different data
Experiments are imperfect!
c2
..
.
cℓ
Protein Interaction Networks: a Mathematical approach – p. 24
Detection biases
PI may be detected via different experiments
Assume we have ℓ biologial species; M experiments;
Biological
Experimental data
✿
✘✘
✘
✘
✘
✲
c1 ❳
❳❳
❳❳
③
1
c.11 experiment
.
..
..
cM
1 experiment M
✿
✘✘✘
✘
c2 ❳
✘
✲
❳❳
❳❳
③
c.12
..
cM
2
..
.
Different experiments yield
different data
Experiments are imperfect!
cℓ
Protein Interaction Networks: a Mathematical approach – p. 24
Detection biases
PI may be detected via different experiments
Assume we have ℓ biologial species; M experiments;
Biological
Experimental data
✿
✘✘
✘
✘
✘
✲
c1 ❳
❳❳
❳❳
③
1
c.11 experiment
.
..
..
cM
1 experiment M
✿
✘✘✘
✘
c2 ❳
✘
✲
❳❳
❳❳
③
c.12
..
cM
2
..
.
Different experiments yield
different data
Experiments are imperfect!
1
c
✘
✿
✘✘ ..ℓ
✘
✘
✲ .
cℓ ❳
❳❳
❳❳
③ cM
ℓ
Protein Interaction Networks: a Mathematical approach – p. 24
Detection biases
PI may be detected via different experiments
Assume we have ℓ biologial species; M experiments;
Biological
Experimental data
✿
✘✘
✘
✘
✘
✲
c1 ❳
❳❳
❳❳
③
1
c.11 experiment
.
..
..
cM
1 experiment M
✿
✘✘✘
✘
c2 ❳
✘
✲
❳❳
❳❳
③
c.12
..
cM
2
..
.
Different experiments yield
different data
Experiments are imperfect!
1
c
✘
✿
✘✘ ..ℓ
✘
✘
✲ .
cℓ ❳
❳❳
❳❳
③ cM
ℓ
Infer true underlying biological PINs from
noisy/inconsistent data? Need mathematical model!
Protein Interaction Networks: a Mathematical approach – p. 24
Overview
Networks as infrastructure of signalling processes
Graphs: definitions and topology measures
Tailored random graph ensembles
Conclusions
Protein Interaction Networks: a Mathematical approach – p. 25
Conclusions
Networks as infrastructure of signalling processes
Protein Interaction Networks: a Mathematical approach – p. 26
Conclusions
Networks as infrastructure of signalling processes
Maths proved useful to:
Quantify structure of networks
Define “good” reference models for hypothesis
testing and quantitative predictions
Assess significance of observed patterns
Define distance between graphs
Generate random graphs with the right probabilities
Network inference/reconstruction from noisy and
incosistent data
Protein Interaction Networks: a Mathematical approach – p. 26
Conclusions
Networks as infrastructure of signalling processes
Maths proved useful to:
Quantify structure of networks
Define “good” reference models for hypothesis
testing and quantitative predictions
Assess significance of observed patterns
Define distance between graphs
Generate random graphs with the right probabilities
Network inference/reconstruction from noisy and
incosistent data
Maths of 21st century helps bio-medical sciences
Protein Interaction Networks: a Mathematical approach – p. 26
Conclusions
Networks as infrastructure of signalling processes
Maths proved useful to:
Quantify structure of networks
Define “good” reference models for hypothesis
testing and quantitative predictions
Assess significance of observed patterns
Define distance between graphs
Generate random graphs with the right probabilities
Network inference/reconstruction from noisy and
incosistent data
Maths of 21st century helps bio-medical sciences
Thanks!
Protein Interaction Networks: a Mathematical approach – p. 26