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Systems Biology. What Is It?
 A branch of science that seeks to integrate
different levels of information to understand
how biological systems function.
 L. Hood: “Systems biology defines and analyses
the interrelationships of all of the elements in a
functioning system in order to understand how the
system works.”
 It is not (only) the number and properties of
system elements but their relations!!
More on Systems Biology
Essence of living systems is flow of mass,
energy, and information in space and time.
The flow occurs along specific networks
 Flow of mass and energy (metabolic networks)
 Flow of information involving DNA (transcriptional
regulation networks)
 Flow of information not involving DNA (signaling networks)
The Goal of Systems Biology:
To understand the flow of mass, energy,
and information in living systems.
Networks and the Core Concepts
of Systems Biology
(i)
Complexity emerges at all levels of the
hierarchy of life
(ii) System properties emerge from interactions
of components
(iii) The whole is more than the sum of the parts.
(iv) Applied mathematics provides approaches to
modeling biological systems.
How to Describe a System
As a Whole?
Networks - The Language
of Complex Systems
Air Transportation Network
The World Wide Web
Fragment of a Social Network
(Melburn, 2004)
Friendship among 450 people in Canberra
Biological Networks
A. Intra-Cellular Networks
Protein interaction networks
Metabolic Networks
Signaling Networks
Gene Regulatory Networks
Composite networks
Networks of Modules, Functional Networks
Disease networks
B. Inter-Cellular Networks
Neural Networks
C. Organ and Tissue Networks
D. Ecological Networks
E. Evolution Network
The Protein Interaction Network of Yeast
Yeast two hybrid
Uetz et al, Nature 2000
Metabolic Networks
Source: ExPASy
Gene Regulation Networks
L-A Barabasi
GENOME
miRNA
regulation?
_____________________
-
protein-gene
interactions
PROTEOME
protein-protein
interactions
METABOLISM
Bio-chemical
reactions
Citrate Cycle
Functional Networks
Yeast: 1400 proteins, 232 complexes, nine functional groups of complexes
Cell Cycle
13
111
Cell Polarity & Structure
7
25
Transcription/DNA
77
Maintenance/Chromatin
14
11
30 16 27
Structure
187
55
740
Number of protein complexes
8
61
40
Number of proteins
15 19
7
22
94
43
221
33
73
83
Number of shared proteins
37
(Data A.-M. Gavin
et al. (2002) Nature
415,141-147)
Intermediate
and Energy
Metabolism
103
65
11
Signaling
13
20
125
20
147
53
41 321
35
19
97
28
RNA
Metabolism 692
24
172
12
160
299
75
5
9
260
6
75
Membrane
Biogenesis &
Turnover
49
33
419
Protein Synthesis
and Turnover
Protein RNA / Transport
D. Bonchev, Chemistry & Biodiversity 1(2004)312-326
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
The 7 bridges of Königsberg
The question is whether it is possible to walk with a route that
crosses each bridge exactly once.
The representation of Euler
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).
The shape of a graph may be distorted in
any way without changing the graph itself,
so long as the links between nodes are
unchanged. It does not matter whether
the links are straight or curved, or
whether one node is to the left or right of
another.
The solution depends on the node degree
3
3
5
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).
3
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.
The solution depends on the node degree
End
3
2
6
5
1
4
Start
If there are two nodes of odd degree, those must be the starting
and ending points of an Eulerian path.
Hamiltonian paths
Find a path visiting each node exactly one
Conditions of existence for Hamiltonian paths are not simple
Hamiltonian paths
Graph nomenclature
 Graphs can be simple or multigraphs, depending on whether
the interaction between two neighboring nodes is unique or can be multiple,
respectively.
 A node can have or not self loops
Graph nomenclature
 Networks can be undirected or directed, depending on whether
the interaction between two neighboring nodes proceeds in both
directions or in only one of them, respectively.

1
2
3
4
5
6
 The specificity of network nodes and links can be quantitatively
characterized by weights
2.5
7.3
3.3
12.7
5.4
8.1
2.5
Vertex-Weighted
Edge-Weighted
Graph nomenclature
A network can be connected (presented by a single
component) or disconnected (presented by several disjoint
components).
connected
disconnected
Networks having no cycles are termed trees. The more
cycles thenetwork has, the more complex it is.
trees
cyclic graphs
Graph nomenclature
Paths
Stars
Cycles
Complete Graphs
Large graphs = Networks
Statistical features of networks
• Vertex degree distribution (the degree of a
vertex is the number of vertices connected
with it via an edge)
Statistical features of networks
• Clustering coefficient: the average proportion
of neighbours of a vertex that are themselves
neighbours
Node
4 Neighbours (N)
2 Connections among
the Neighbours
Clustering for the node = 2/6
Clustering coefficient: Average over all the nodes
6 possible connections
among the Neighbours
(Nx(N-1)/2)
Statistical features of networks
• Clustering coefficient: the average proportion
of neighbours of a vertex that are themselves
neighbours
C=0
C=0
C=0
C=1
Statistical features of networks
Given a pair of nodes, compute the shortest
path between them
• Average shortest distance between two
vertices
• Diameter: maximal shortest distance
How many degrees of separation are they between two
random people in the world, when friendship networks
are considered?
How to compute the shortest path
between home and work?
Edge-weighted Graph
The exaustive search can be too much time-consuming
The Dijkstra’s algorithm
Fixed nodes
NON –fixed nodes
Initialization:
Fix the distance between “Casa” and “Casa”
equal to 0
Compute the distance between “Casa” and its
neighbours
Set the distance between “Casa” and its NONneighbours equal to ∞
The Dijkstra’s algorithm
Fixed nodes
NON –fixed nodes
Iteration (1):
Search the node with the minimum distance
among the NON-fixed nodes and Fix its distance,
memorizing the incoming direction
The Dijkstra’s algorithm
Fixed nodes
NON –fixed nodes
4
Iteration (2):
Update the distance of NON-fixed nodes,
starting from the fixed distances
The Dijkstra’s algorithm
Fixed nodes
NON –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.
The Dijkstra’s algorithm
Fixed nodes
NON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum
distance
Update the distance of NON-fixed nodes,
starting from the fixed distances.
The Dijkstra’s algorithm
Fixed nodes
NON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum
distance
Update the distance of NON-fixed nodes,
starting from the fixed distances.
The Dijkstra’s algorithm
Fixed nodes
NON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum
distance
Update the distance of NON-fixed nodes,
starting from the fixed distances.
The Dijkstra’s algorithm
Fixed nodes
NON –fixed nodes
Iteration:
Fix the NON-fixed nodes with minimum
distance
Update the distance of NON-fixed nodes,
starting from the fixed distances.
The Dijkstra’s algorithm
Fixed nodes
NON –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
Statistical features of networks
• Vertex degree distribution
• Clustering coefficient
• Average shortest distance between two
vertices
• Diameter: maximal shortest distance
Two reference models for networks
Regular network (lattice)
Regular connections
Random network (Erdös+Renyi, 1959)
Each edge is randomly
set with probability p
Two reference models for networks
Comparing networks with the same number of nodes (N) and edges
Poisson distribution
Pk   e
Degree distribution

k
k!
Exp decay
Average shortest path
Average connectivity
≈N
high
≈ log (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)
Adding randomness in a regular network
Random changes
in edges
OR
Addition of
random links
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.
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
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%)
Hubs
Attacks to hubs can rapidly destroy the network
Three non biological scale-free networks
P(k )  A  k
Note the log-log scale
LINEAR PLOT

 log P(k )  log A   log k
Albert and Barabasi, Science 1999
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 richer
This 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….
Protein interaction networks are scale-free
Degree is in some
measure related to
phenotypic effect
upon gene knock-out
Red : lethal
Green: non lethal
Yellow: Unknown
Caveat: different experiments give different
results
Titz et al, Exp Review Proteomics, 2004
How can a scale-free 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)
Caveat on the use of the scale-free theory
The same noisy data can be
fitted in different ways
A sub-net of a non-free-scale network can have a scale-free
behaviour
Finding a scale-free behaviour do NOT imply the growth with
preferential attachment mechanism
Keller, BioEssays 2006
Hierarchical networks
Standard free scale models have low clustering: a modular
hierarchical model accounts for high clustering, low average
path and scale-freeness
Modules
Sub-graphs more represented than expected
209 bi-fan motifs found in the E.coli regulatory network
Summary
 Many complex networks in nature and technology
have common features.
 They differ considerably from random networks
of the same size
 By studying network structure and dynamics, and
by using comparative network analysis, one can
get answers of important biological questions.
Fundamental Biological
Questions to Answer
(i) Which interactions and groups of interactions are likely
to have equivalent functions across species?
(ii) Based on these similarities, can we predict new functional
information about proteins and interactions that are poorly
characterized?
(iii) What do these relationships tell us about the
evolution of proteins, networks and whole species?
(iv) How to reduce the noise in biological data: Which
interactions represent true binding events?
False-positive interaction is unlikely to be reproduced
across the interaction maps of multiple species.
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