Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
......OMICS
..OMICS..
Today, people talk about:
Genomics
Variomics
Transcriptomics
Proteomics
Interactomics
Regulomics
Metabolomics
Many more at:
http://www.genomicglossaries.com/content/omes.asp
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
OMICS AND
HIGH-THROUGHPUT METHODS
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
WHICH EXPERIMENTAL METHODS ARE
SUITABLE FOR: Genomics?
NGS sequencing, DNA Microarrays
Variomics?
NGS sequencing, DNA Microarrays
Transcriptomics?
NGS sequencing, DNA Microarrays
Proteomics?
2D gels, Protein arrays, Mass Spectrometry, Isotope tag
Protein-protein Interactomics?
Yeast 2-hybrid, affìnity purification + Mass Spec
Protein-DNA Interactomics?
Chromatine immunoprecipitation on chip, ChIP-Seq
Metabolomics?
Mass Spec, NMR
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
WHICH ARE THE PROS AND CONS OF THE
ALTERNATIVE METHODS?
TO WHICH LEVEL DO THE DIFFERENT
METHODS AGREE?
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
INTERACTOMICS: FINDING THE
INTERACTING PROTEINS
Yeast two hybrids
CHARACTERIZATION OF PHYSICAL
INTERACTIONS
Obligation
obligate (protomers only found/function together)
non-obligate (protomers can exist/function alone)
Time of interaction
permanent (complexes, often obligate)
strong transient (require trigger, e.g. G proteins)
weak transient (dynamic equilibrium)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
ol
EXAMPLES: GPCR
obligate, permanent
non-obligate,
strong transient
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
APPROACHES BY INTERACTION TYPE
Physical Interactions
Yeast two hybrid screens
Affinity purification (mass spec)
Other measures of ‘association’
Genetic interactions (double deletion mutants)
Genomic context (STRING)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
YEAST TWO-HYBRID METHOD
Y2H assays interactions in vivo.
Uses property that transcription
factors generally have separable
transcriptional activation (AD) and
DNA binding (DBD) domains.
A functional transcription factor can
be created if a separately expressed
AD can be made to interact with a
DBD.
A protein ‘bait’ B is fused to a DBD
and screened against a library of
protein “preys”, each fused to a AD.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
YEAST TWO-HYBRID METHOD
Ito et al., Trends Biotechnol. 19, S23 (2001)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Protein Interaction Network of Yeast
Uetz et al, Nature 2000
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Protein Interaction Network of Drosophila
Giot et al, Science 2003
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
ISSUES WITH Y2H
Strengths High sensitivity (transient & permanent PPIs)
Takes place in vivo
Independent of endogenous expression
Weaknesses: False positive interactions Auto-activation
‘sticky’ prey
Detects “possible interactions” that may not take place under real physiological conditions
May identify indirect interactions (A-C-B)
Weaknesses: False negatives interactions Similar studies often reveal very different sets of interacting proteins (i.e.
False negatives)
May miss PPIs that require other factors to be present (e.g. ligands, proteins, PTMs)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
DIFFERENT Y2H EXPERIMENTS GIVE
DIFFERENT RESULTS….
Deane et al, Mol Cell Proteomics 1:349 (2002)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
DIFFERENT Y2H EXPERIMENTS GIVE
DIFFERENT RESULTS….
Deane et al, Mol Cell Proteomics 1:349 (2002)
A Venn diagram illustrates the overlap between the datasets in YEAST-DIP.
Each oval represents a high throughput Y2H study, and the overlaps
between the Y2H studies are given at the intersections. The number in
parentheses represents those interactions that have been determined by
small scale methods (see "Experimental Procedures" for more details).
Thus, the numbers within parentheses represent the INT set. Notice the
small overlap among the datasets.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Y2H FOR MEMBRANE PROTEINS
Fields, FEBS Journal 272, 5391 (2005)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
EXERCISE: Y2H
Draw the correspondent graph
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
INTERACTOMICS: FINDING THE
INTERACTING PROTEINS
Mass spectrometry
PROTEIN INTERACTIONS BY IMMUNO-PRECIPITATION
FOLLOWED BY MASS SPECTROMETRY
Start with affinity purification of a single epitope-tagged
protein
This enriched sample typically has a low enough
complexity to be fractionated on a standard
polyacrylamide gel.
Individual bands can be excised from the gel and
identified with mass spectrometry.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico
Bio
logy -
20
14
-2
01
5
PROTEIN INTERACTIONS BY IMMUNO-PRECIPITATION
FOLLOWED BY MASS SPECTROMETRY
Kumar & Snyder, Nature 415, 123 (2002)
TANDEM AFFINITY PURIFICATION
LA Huber Nature Reviews Molecular Cell Biology 4, 74-80 (2003)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
AFFINITY PURIFICATION
Strengths
• High specificity
• Well suited for detecting permanent or
strong transient interactions (complexes)
• Detects real, physiologically relevant
PPIs
Weaknesses
• Less suited for detecting weaker
transient interactions (low sensitivity)
• May miss complexes not present under
the given experimental conditions (low
sensitivity)
• May identify indirect interactions (A-C-B)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Franzot & Carugo, J Struct Funct Biol 4, 245 (2004)
Y2H
Y2H
MS
MS
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Titz et al, Exp Review Proteomics, 2004
Caveat: different experiments give different results P
ier L
uig
i Ma
rtelli-
Syste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
DIFFERENT INFORMATION HAVE TO BE
CROSSED TO LOWER THE ERROR RATE
The fraction of interactions in
which both partners have the
same protein
localization. Here, only proteins
clearly assigned to a single
category are considered
Von Mering, Nature,2002
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
INTERACTOMICS: FINDING THE
INTERACTING PROTEINS
Genomic methods
Rost et al.Cellular Molecular Life Sciences, 2003, 60:2637-2650
http://cubic.bioc.columbia.edu/papers/2003_rev_func/paper.html
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
REGULOMICS: FINDING THE
TRANSCRIPTION NETWORK
ChIP-chip: Chromatine ImmunoPrecipitation on chip
ChIPSeq: Chromatine ImmunoPrecipitation coupled to
NGS
CHIP-CHIP MEASUREMENT OF PROTEIN-
DNA INTERACTIONS
Simon et al., Cell 2001
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
CHIP-CHIP MEASUREMENT OF PROTEIN-
DNA INTERACTIONS
Lee et al., Science 2002
CHIP-SEQ MEASUREMENT OF PROTEIN-
DNA INTERACTIONS
Szalkowski, A.M, and Schmid, C.D.(2010).
Rapid innovation in ChIP-seq peak-calling algorithms is
outdistancing banchmarking efforts.
Briefings in Bioinfomatics.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
MAPPING TRANSCRIPTION FACTOR BINDING
SITES
Harbison C., Gordon B., et al. Nature 2004
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
PROMOTER ARCHITECTURES
Harbison C., Gordon B., et al. Nature 2004
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
TRANCRIPTION NETWORKS
Babu et al., Curr. Opin. Struct. Biol. 14, 283 (2004)
REGULATION OF TRANSCRIPTION FACTORS
Lee et al., Science 298, 799 (2002)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
METABOLOMICS: FINDING
THE CORRELATIONS
AMONG METABOLITESChromatography-Mass spectroscopy, NMR
Separation methodsGas chromatography, especially when interfaced with mass spectrometry
(GC-MS), is one of the most widely used and powerful methods It offers very
high chromatographic resolution, but requires chemical derivatization for many
biomolecules: only volatile chemicals can be analysed without derivatization.
(Some modern instruments allow '2D' chromatography, using a short polar
column after the main analytical column, which increases the resolution still
further.) Some large and polar metabolites cannot be analysed by GC.
High performance liquid chromatography (HPLC). Compared to GC, HPLC
has lower chromatographic resolution, but it does have the advantage that a
much wider range of analytes can potentially be measured.
Capillary electrophoresis (CE). CE has a higher theoretical separation
efficiency than HPLC, and is suitable for use with a wider range of metabolite
classes than is GC. As for all electrophoretic techniques, it is most appropriate
for charged analytes.
Wikipedia: Metabolomics
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Detection methodsMass spectrometry (MS) is used to identify and to quantify metabolites after
separation by GC, HPLC (LC-MS), or CE. GC-MS is the most 'natural' combination of
the three, and was the first to be developed. In addition, mass spectral fingerprint
libraries exist or can be developed that allow identification of a metabolite according
to its fragmentation pattern. MS is both sensitive (although, particularly for HPLC-MS,
sensitivity is more of an issue as it is affected by the charge on the metabolite, and
can be subject to ion suppression artifacts) and can be very specific. There are also a
number of studies which use MS as a stand-alone technology: the sample is infused
directly into the mass spectrometer with no prior separation, and the MS serves to
both separate and to detect metabolites.
Nuclear magnetic resonance (NMR) spectroscopy. NMR is the only detection
technique which does not rely on separation of the analytes, and the sample can thus
be recovered for further analyses. All kinds of small molecule metabolites can be
measured simultaneously - in this sense, NMR is close to being a universal detector.
The main advantages of NMR are high analytical reproducibility and simplicity of
sample preparation. Practically, however, it is relatively insensitive compared to mass
spectrometry-based techniques.
Wikipedia: Metabolomics
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
GC-MS
Weckwerth, Annu. Rev. Plant Biol. 54, 669 (2003)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
FINDING CORRELATION BETWEEN THE
METABOLITE CONTENT
Weckwerth, Annu. Rev. Plant Biol. 54, 669 (2003)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
METABOLITE NETWORK
Weckwerth, Annu. Rev. Plant Biol. 54, 669 (2003)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
IS THE CORRELATION
EVALUATION SUFFICIENT?
COLLECT THE MEASURES FOR VARIABLES
X AND Y
X Y
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
… …
xn yn
Covariance:
yyxxn
YX i
n
i
i
11
1),cov(
Linear regression
XaYb
YXa
bXaY
XY
x
XY
XY
,cov
2
If correlation is significant enough: X Y
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
COLLECT THE MEASURES FOR VARIABLES
Y AND Z
Y Z
y1 z1
y2 z2
y3 z3
y4 z4
y5 z5
… …
yn zn
Covariance:
zzyyn
XY i
n
i
i
11
1),cov(
Linear regression
YaZc
ZYa
cYaZ
YZ
Y
YZ
YZ
,cov
2
Y Z
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
If correlation is significant enough:
WHAT ABOUT THE X AND Z?
In general: 'cXaYaZ XZYZ
Y ZX
IF we suppose that Z depends on X ONLY indirectly,
via Y, aXZ=0 :
dXadXaacbXaacYaZ XZXYYZXYYZYZ ~''
So, Z and X have a regression with coefficient aXYaYZ
222
,cov,cov,cov~
YXX
XZ
ZYYXZXa
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
WHAT ABOUT THE X AND Z?Y ZX
ZYYXZYYX
ZX
ZYYXZX
ZXY
Y
,,,cov,cov
,
,cov,cov,cov
2
2
X and Z have a correlation index equal to the
product of correlation indexes, even if there is not
direct relation between them.
Correlation is not sufficient to establish the direct
relation between the variables
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
EXAMPLE
σ2X=4 , σ2
Y=3 , cov(X,Y) =2
σ2Y=3 , σ2
Z=6 , cov(Z,Y) =1.5
If X and Z are not directly dependent:
So the overall covariance matrix is:
13
5.12,cov,cov,cov
2
Y
ZYYXZX
65.11
5.132
124
COV
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
EXAMPLE
65.11
5.132
124
COV
Gaussian model:
xCOVx
COV
COVxT 1
2
1
2
3 2
1exp
2
1),|(
The inverse of the correlation matrix (called
precision matrix, K) is involved
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
EXAMPLE
65.11
5.132
124
COV
190.0095.00
095.0548.0250.0
0250.0375.01COVK
The element of the precision matrix corresponding
to the non directly related variable vanishes!
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
“DIRECT LINKS” ARE RECOVERED FROM THE
PRECISION MATRIX
Given a set of sample describe by variables
X1 X2 X3 X4 … XN
Compute the Covariance Matrix
Compute the Precision Matrix (K) as the inverse of
the covariance matrix
The partial correlation indexes between pairs of
variables Xi Xj, with i≠j is
jjii
ij
jiKK
KXX
,~
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
PARTIAL CORRELATION COEFFICIENT
Formally, the partial correlation between X and Y
given a set of n controlling variables Z = {Z1, Z2, ...,
Zn}, written ρXY|Z, is the correlation between the
residuals RX and RY resulting from the linear
regression of X with Z and of Y with Z, respectively.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
APPLICATION IN METABOLOMICS ANALYSIS
In our new approach we propose the application of a Gaussian graphical model
(GGM), an undirected probabilistic graphical model estimating the conditional
dependence between variables. GGMs are based on partial correlation
coefficients, that is pairwise Pearson correlation coefficients conditioned against
the correlation with all other metabolites. We first demonstrate the general validity
of the method and its advantages over regular correlation networks with computer-
simulated reaction systems. Then we estimate a GGM on data from a large
human population cohort, covering 1020 fasting blood serum samples with
151 quantified metabolites. The GGM is much sparser than the correlation
network, shows a modular structure with respect to metabolite classes, and is
stable to the choice of samples in the data set. On the example of human fatty acid
metabolism, we demonstrate for the first time that high partial correlation
coefficients generally correspond to known metabolic reactions. This feature
is evaluated both manually by investigating specific pairs of high-scoring
metabolites, and then systematically on a literature-curated model of fatty acid
synthesis and degradation. Our method detects many known reactions along with
possibly novel pathway interactions, representing candidates for further
experimental examination.
Krumsiek et al. BMC Systems Biology 2011, 5:21
http://www.biomedcentral.com/1752-0509/5/21
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Krumsiek et al. BMC Systems Biology 2011, 5:21
http://www.biomedcentral.com/1752-0509/5/21
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Krumsiek et al. BMC Systems Biology 2011, 5:21
http://www.biomedcentral.com/1752-0509/5/21
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Krumsiek et al. BMC Systems Biology 2011, 5:21
http://www.biomedcentral.com/1752-0509/5/21
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Krumsiek et al. BMC Systems Biology 2011, 5:21
http://www.biomedcentral.com/1752-0509/5/21
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
APPLICATION IN PROTEIN STRUCTURE
PREDICTION
Marks DS, Colwell LJ, Sheridan R, Hopf TA, Pagnani A, et al. (2011). PLoS
ONE 6(12): e28766. doi:10.1371/journal.pone.0028766
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
APPLICATION IN PROTEIN STRUCTURE
PREDICTION
Marks DS, Colwell LJ, Sheridan R, Hopf TA, Pagnani A, et al. (2011). PLoS
ONE 6(12): e28766. doi:10.1371/journal.pone.0028766
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
APPLICATION IN PROTEIN STRUCTURE
PREDICTION
Marks DS, Colwell LJ, Sheridan R, Hopf TA, Pagnani A, et al. (2011). PLoS
ONE 6(12): e28766. doi:10.1371/journal.pone.0028766
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
APPLICATION IN PROTEIN STRUCTURE
PREDICTION
See also PSICOV:
Jones D, Buchan DWA, Cozzetto D, Pontil M, Bioinformatics 28:184-190 (2012)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
How to Describe a System As a Whole?
Networks - The Language of Complex Systems
Air Transportation Network Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The World Wide Web Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Fragment of a Social Network(Melburn, 2004)
Friendship among 450 people in Canberra
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Biological NetworksA. Intra-Cellular Networks
Protein interaction networks
Metabolic Networks
Signaling Networks
Gene Regulatory Networks
Composite networksNetworks of Modules, Functional Networks Disease networks
B. Inter-Cellular NetworksNeural Networks
C. Organ and Tissue Networks
D. Ecological Networks
E. Evolution Network
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Protein Interaction Network of Yeast
Yeast two hybrid
Uetz et al., Nature 2000
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Source: ExPASy
Metabolic Networks Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Gene Regulation Networks
Abdollahi A et al., PNAS 2007
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Networks derived from networks
Goh,..,Barabasi (2007) PNAS 104:8685
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Networks derived from networks
Goh,..,Barabasi (2007) PNAS 104:8685
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
protein-gene
interactions
protein-protein
interactions
PROTEOME
GENOME
Citrate Cycle
METABOLISM
Bio-chemical
reactions
L-A Barabasi
miRNA
regulation?-
-
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
What is a Network?
Network is a mathematical structure composed of points connected by lines
Network Theory <-> Graph Theory
Network Graph
Nodes Vertices (points)
Links Edges (Lines)
A network can be build for any functional system
System vs. Parts = Networks vs. Nodes
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The 7 bridges of Königsberg
The question is whether it is possible to walk with a route that crosses each bridge exactly once.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The representation of Euler
The shape of a graph may be distorted inany way without changing the graph itself,so long as the links between nodes areunchanged. It does not matter whetherthe links are straight or curved, orwhether one node is to the left or right ofanother.
In 1736 Leonhard Euler formulated the problem in terms of abstracted the case of Königsberg:
1) by eliminating all features except the landmasses and the bridges connecting them;
2) by replacing each landmass with a dot (vertex) and each bridge with a line (edge).
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
3
3
5 3
The solution depends on the node degree
In a continuous path crossing the edges exactly once, each visited node requires an edge for entering and a different edge for exiting (except for the start and the end nodes).
A path crossing once each edge is called Eulerian path.It possible IF AND ONLY IF there are exactly two or zero nodes of odd degree. Since the graph corresponding to Königsberg has four nodes of odd degree, it cannot have an Eulerian path.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The solution depends on the node degree
If there are two nodes of odd degree, those must be the starting and ending points of an Eulerian path.
2
3
4
5
1
Start
6
End
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Hamiltonian paths
Find a path visiting each node exactly one
Conditions of existence for Hamiltonian paths are not simple
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Hamiltonian pathsP
ier L
uig
i Ma
rtelli-
Syste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Graph nomenclature
Graphs can be simple or multigraphs, depending on whetherthe interaction between two neighboring nodes is unique or can be multiple, respectively.
A node can have or not self loops
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Graph nomenclature
Networks can be undirected or directed, depending on whetherthe interaction between two neighboring nodes proceeds in bothdirections or in only one of them, respectively.
The specificity of network nodes and links can be quantitativelycharacterized by weights
2.5
2.5
7.3 3.3 12.7
8.1
5.4
Vertex-Weighted Edge-Weighted
1 2 3 4 5 6
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Networks having no cycles are termed trees. The more
cycles the network has, the more complex it is.
A network can be connected (presented by a single
component) or disconnected (presented by several disjoint
components).
connected disconnected
trees
cyclic graphs
Graph nomenclatureP
ier L
uig
i Ma
rtelli-
Syste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Paths
Stars
Cycles
Complete Graphs
Graph nomenclatureP
ier L
uig
i Ma
rtelli-
Syste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Large graphs = NetworksP
ier L
uig
i Ma
rtelli-
Syste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
• Vertex degree distribution (the degree of a vertex is the number of vertices connected with it via an edge)
Statistical features of networksP
ier L
uig
i Ma
rtelli-
Syste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
• Clustering coefficient: the average proportion of neighbours of a vertex that are themselves neighbours
Statistical features of networks
Node4 Neighbours (N)
2 Connections among the Neighbours
6 possible connections among the Neighbours(Nx(N-1)/2)
Clustering for the node = 2/6
Clustering coefficient: Average over all the nodes
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
• Clustering coefficient: the average proportion of neighbours of a vertex that are themselves neighbours
Statistical features of networks
C=0
C=0
C=0
C=1
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Given a pair of nodes, compute the shortest path between them
• Average shortest distance between two vertices
• Diameter: maximal shortest distance
Statistical features of networks
How many degrees of separation are they between two random people in the world, when friendship networks are considered?
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Wutchy, Ravasz, Barabasi. In: Complex Systems in Biomedicine, Kluwer 2003
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
How to compute the shortest path between home and work?
Edge-weighted Graph
The exaustive search can be too much time-consuming
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Initialization:Fix the distance between “Casa” and “Casa”
equal to 0Compute the distance between “Casa” and its
neighboursSet the distance between “Casa” and its NON-
neighbours equal to ∞
Fixed nodesNON –fixed nodes
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Iteration (1):Search the node with the minimum distance
among the NON-fixed nodes and Fix its distance, memorizing the incoming direction
Fixed nodesNON –fixed nodes
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
4
Iteration (2):
Update the distance of NON-fixed nodes, starting from the fixed distances
The Dijkstra’s algorithm
Fixed nodesNON –fixed nodes
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Fixed nodesNON –fixed nodes
The updated distance is different from the previous one
Iteration:
Fix the NON-fixed nodes with minimum distance
Update the distance of NON-fixed nodes, starting from the fixed distances.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Fixed nodesNON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum distance
Update the distance of NON-fixed nodes, starting from the fixed distances.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Fixed nodesNON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum distance
Update the distance of NON-fixed nodes, starting from the fixed distances.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Fixed nodesNON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum distance
Update the distance of NON-fixed nodes, starting from the fixed distances.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Fixed nodesNON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum distance
Update the distance of NON-fixed nodes, starting from the fixed distances.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The Dijkstra’s algorithm
Fixed nodesNON –fixed nodes
Conclusion:
The label of each node represents the minimal distance from the starting node
The minimal path can be reconstructed with a back-tracing procedure
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
• Average shortest distance between two vertices (L)
• Diameter: maximal shortest distance
Statistical features of networks
• Vertex degree (k) distribution
• Clustering coefficient (C)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Two reference models for networks
Regular network (lattice) Random network (Erdös+Renyi, 1959)
Each edge is randomly set with probability p
Regular connections
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Two reference models for networksComparing networks with the same total number of nodes (N) and edges (E)
Degree (k) distribution
Poisson distribution
!k
ekPk
Exp decay
Average shortest path
Average clustering
≈ N High ≈ log (N) Low
≈1.5 (s-1)/(2s-1) High
with s=E/N
≈2s/N Low
Some examples for real networks
Network size vertex degree
shortest path
Shortest path in fitted random graph
Clustering Clustering in random graph
Film actors 225,226 61 3.65 2.99 0.79 0.00027
MEDLINE coauthorship
1,520,251 18.1 4.6 4.91 0.43 1.8 x 10-4
E.Coli substrate graph
282 7.35 2.9 3.04 0.32 0.026
C.Elegans neuron network
282 14 2.65 2.25 0.28 0.05
Real networks are not regular (low shortest path)Real networks are not random (high clustering)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Adding randomness in a regular network
Random changes in edges
OR
Addition of random links
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Adding randomness in a regular network(rewiring)
Networks with high clustering (like regular ones) and low path length (like random ones) can be obtained:SMALL WORLD NETWORKS (Strogatz and Watts, 1999)
Small World Networks
A small amount of random shortcuts can decrease the path length, still maintaining a high clustering: this model “explains” the 6-degrees of separations in human friendship network
What about the degree distribution in real networks?
Both random and small world models predict an approximate Poisson distribution: most of the values are near the mean;Exponential decay when k gets higher: P(k) ≈ e-k, for large k.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
What about the degree distribution in real networks?
In 1999, modelling the WWW (pages: nodes; link: edges), Barabasi and Albert discover a slower than exponential decay:
P(k) ≈ k-a with 2 < a < 3, for large k
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Scale-free networks
Networks that are characterized by a power-law degree distribution are highly non-uniform: most of the nodes have only a few links. A few nodes with a very large number of links, which are often called hubs, hold these nodes together. Networks with a power degree distribution are called scale-free
hubs
It is the same distribution of wealth following Pareto’s 20-80 law:Few people (20%) possess most of the wealth (80%), most of the people (80%) possess the rest (20%)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Hubs
Attacks to hubs can rapidly destroy the network
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Three non biological scale-free networks
Albert and Barabasi, Science, 1999
Note the log-log scaleLINEAR PLOT
kAkPkAkP loglog)(log)(
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Wutchy, Ravasz, Barabasi. In: Complex Systems in Biomedicine, Kluwer 2003
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
AttackTolerance
Complex systems maintain their basic functions
even under errors and failures
(cell mutations; Internet router breakdowns)
node failure
Albert and Barabasi, Rev Mod Phys, 2002
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Attack Tolerance
• Robust. For <3, removing
nodes does not break
network into islands.
• Very resistant to random
attacks, but attacks targeting
key nodes are more
dangerous. P
ath
Length
Albert and Barabasi, Rev Mod Phys, 2002
Targeted
attackRandom
attack
Targeted
attackRandom
attack
Targeted
attackRandom
attack
Targeted
attackRandom
attack
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
How can a scale-free network emerge?
Network growth models: start with one vertex.
How can a scale-free network emerge?
Network growth models: new vertex attaches to existing vertices by preferential attachment: vertex tends choose vertex according to vertex degree
In economy this is called Matthew’s effect: The rich get richerThis explain the Pareto’s distribution of wealth
How can a scale-free network emerge?
Network growth models: hubs emerge(in the WWW: new pages tend to link to existing, well linked pages)
Metabolic pathways are scale-free
Hubs are pyruvate, coenzyme A….
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Protein interaction networks are scale-free
Uetz et al., Nature 2000
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Protein interaction networks are scale-free
Albert R, J Cell Sci, 2005
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Protein interaction networks are scale-free
Degree is in some measure related to phenotypic effect upon gene knock-out
Red : lethalGreen: non lethalYellow: Unknown
Uetz et al., Nature 2000
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Proteins with ≤ 6 neighbours 21% are essential (lethality upon knock-out)
Proteins with ≥ 15 neighbours 62% are essential (lethality upon knock-out)
Are central proteins essential? Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Titz et al, Exp Review Proteomics, 2004
Caveat: different experiments give different results P
ier L
uig
i Ma
rtelli-
Syste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
How can a scale-free interaction network emerge?
Gene duplication (and differentiation): duplicated genes give origin to a protein that interacts with the same proteins as the original protein (and then specializes its functions)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Wutchy, Ravasz, Barabasi. In: Complex Systems in Biomedicine, Kluwer 2003
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Trancription networks
Babu et al., Curr. Opin. Struct. Biol. 14, 283 (2004)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The incoming connectivity is the number of transcription factors regulating a target gene, which quantifies the combinatorial effect of gene regulation.
The fraction of target genes with a given incoming connectivity decreases exponentially.
Most target genes are regulated by similar numbers of factors (93% of genes are regulated by 1–4 factors in yeast).
Trancription networks
Babu et al., Curr. Opin. Struct. Biol. 14, 283 (2004)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
The outgoing connectivity is the number of target genes regulated by each transcription factor.
It is distributed according to a power law.This is indicative of a hub-containing network structure, in which a select few transcription factors participate in the regulation of a disproportionately large number of target genes. These hubs can be viewed as ‘global regulators’, as opposed to the remaining transcription factors that can be considered ‘fine tuners’.
In the transcriptional network in yeast, regulatoryhubs have a propensity to be lethal if removed.
Trancription networks
Babu et al., Curr. Opin. Struct. Biol. 14, 283 (2004)
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
These mechanisms alone cannot explain the evolution of network motifs and the scale-free topology.
Babu et al., Curr. Opin. Struct. Biol. 14, 283 (2004)
Caveat on the use of the scale-free theory
The same noisy data can be fitted in different ways
Keller, BioEssays 2006
x
dzzfxF )()( has to be used: more discriminative
)1()( Cxxf
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Caveat on the use of the scale-free theory
Finding a scale-free behaviour do NOT imply the growth with preferential attachment mechanism
A sub-net of a non-free-scale network can have a scale-free behaviour
Keller, BioEssays 2006
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Hierarchical networks
Standard free scale models have low clustering: a modular hierarchical model accounts for high clustering, low average path and scale-freeness
Wutchy, Ravasz, Barabasi. In: Complex Systems in Biomedicine, Kluwer 2003
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Wutchy, Ravasz, Barabasi. In: Complex Systems in Biomedicine, Kluwer 2003
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Wutchy, Ravasz, Barabasi. In: Complex Systems in Biomedicine, Kluwer 2003
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Hierarchical Modularity
Metabolic Networks Protein Networks
E. Ravasz et al., Science, 2002
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Hierarchical structures in directed networks
master regulators (nodes with zero in-degree),
workhorses (nodes with zero out-degree),
middle managers (nodes with nonzero in- and out-degree).
Yan & Gerstein, PNAS 2010
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Yan & Gerstein, PNAS 2010
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Bhardvaj, Yan, Gerstein PNAS 2010
Sc: Yeast
Hs: Homo
Rr: Rat
Mm: Mouse
Ec: E.coli
Mt: Mycobacteriun tubercolosis
Ph: Phosphorilation
Mo: Modification
Tr: Trancriptional regulation
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Bhardvaj, Yan, Gerstein, PNAS, 2010
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Sub-graphs more represented than expected
Motifs
209 bi-fan motifs found in the E.coli regulatory network
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Measures of centrality
Degree centrality is defined as the number of links incident upon a node
Betweenness is the ratio between the number of shortest paths passing through a given vertex over the number of shortest pairs.
Closeness is defined as the mean shortest path between a vertex v and all other vertices reachable from it.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Measures of centrality
AB
C
Which is the node with the highest degree centrality?Which is the node with the highest closeness?Which is the node with the highest betweenness?
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Del Rio et al.,
BMC Systems Biology 2009, 3:102
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Community structure
subsets of vertices within which vertex–vertex connections are dense,
but between which connections are less dense.
Girvan and Newman, PNAS, 2002
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Detecting communities
Betweenness can be computed also fo edges: ratio between the number of shortest paths passing through a given edge over the number of shortest pairs. bottleneck of the communication though the network
GIRVAN NEWMAN ALGORITHM
1. Calculate the betweenness for all edges in the network.2. Remove the edge with the highest betweenness.3. Recalculate betweennesses for all edges affected by theremoval.4. Repeat from step 2 until no edges remain.
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Girvan and Newman, PNAS, 2002
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Community clustering of protein-protein interaction networks
Dunn et al, BMC Bioinformatics, 2005
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Community clustering of protein-protein interaction networks
Dunn et al, BMC Bioinformatics, 2005
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Community clustering of protein-protein interaction networks
Dunn et al, BMC Bioinformatics, 2005
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Community clustering of protein-protein interaction networks
Dunn et al, BMC Bioinformatics, 2005
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Science 298, 2002
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Geometric structure for networks
Geometric random networks
Higham, et al , Bioinformatics, 2008
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Algorithm for embedding the
graph in a metric space
Higham, et al , Bioinformatics, 2008
IS = ?
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Higham, et al , Bioinformatics, 2008
YHC: Yeast
ER: Random
ER-DD: Random with the
same degree distribution
as YHC
GEO-3D: Geometric in
3D
GEO-3D-10%: GEO-3D
with 10% noise
SF: Scale free
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Higham, et al , Bioinformatics, 2008
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6
Barabasi and Oltvai (2004) Network Biology: understanding the cell’s functional
organization. Nature Reviews Genetics 5:101-113
Stogatz (2001) Exploring complex networks. Nature 410:268-276
Hayes (2000) Graph theory in practice. American Scientist 88:9-13/104-109
Mason and Verwoerd (2006) Graph theory and networks in Biology
Keller (2005) Revisiting scale-free networks. BioEssays 27.10: 1060-1068
Pie
r Lu
igi M
arte
lli-S
yste
ms a
nd
in S
ilico B
iolo
gy -
20
15
-2
01
6