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I corsi vengono integrati e conterranno grosso modo due moduli: SB: systems biology

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The Dijkstra’s algorithm

I corsi vengono integrati e conterranno grosso modo due moduli:

SB: systems biology

ML: machine learning

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!!

The Goal of Systems Biology:

To understand the flow of mass, energy,

and information in living systems.

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 ofinformation involving DNA (transcriptional
- regulationnetworks)

- Flow of information not involving DNA (signaling networks)

Networks and the Core Concepts

of Systems Biology

- Complexity emerges at all levels of the
- hierarchy of life

- 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.

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

Biological Networks

C. Organ and Tissue Networks

D. Ecological Networks

E. Evolution Network

Metabolic Networks

Source: ExPASy

L-A Barabasi

protein-gene interactions

PROTEOME

protein-protein interactions

METABOLISM

Bio-chemical reactions

Citrate Cycle

GENOME

miRNA

regulation?

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

- -

Cell Cycle

Cell Polarity & Structure

7

Number of protein complexes

13

111

8

61

25

40

Number of proteins

Transcription/DNA

Maintenance/Chromatin

Structure

77

19

15

Number of shared proteins

14

11

7

30

16

27

22

Intermediate

and Energy

Metabolism

187

55

740

43

221

94

33

73

83

37

103

65

11

Signaling

Membrane

Biogenesis &

Turnover

13

20

125

20

147

53

35

321

19

41

299

49

596

75

97

Protein Synthesis

and Turnover

28

692

33

419

RNA

Metabolism

260

24

172

75

12

160

Protein RNA / Transport

Functional NetworksYeast: 1400 proteins, 232 complexes, nine functional groups of complexes

(Data A.-M. Gavin

et al. (2002) Nature

415,141-147)

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:
- by eliminating all features except the landmasses and the bridges connecting them;
- 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

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

5

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.

End

3

2

6

5

1

4

Start

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.

Hamiltonian paths

Find a path visiting each node exactly one

Conditions of existence for Hamiltonian paths are not simple

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 byweights

2.5

12.7

7.3

3.3

5.4

8.1

2.5

Vertex-Weighted

Edge-Weighted

Graph nomenclature

trees

cyclic graphs

- 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.

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)

6 possible connections among the Neighbours

(Nx(N-1)/2)

2 Connections among the Neighbours

Statistical features of networks

Clustering for the node = 2/6

Clustering coefficient: Average over all the nodes

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

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?

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 NON-neighbours 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

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 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.

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

Average shortest distance between two vertices

Diameter: maximal shortest distance

Statistical features of networks

- Vertex degree distribution

- Clustering coefficient

Two reference models for networks

Regular network (lattice)

Random network (Erdös+Renyi, 1959)

Regular connections

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

Degree distribution

Exp decay

Average shortest path

≈ N

≈ log (N)

high

Average connectivity

low

Some examples for real networks

Real networks are not regular (low shortest path)

Real networks are not random (high clustering)

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

Note the log-log scale

LINEAR PLOT

Albert and Barabasi, Science 1999

How can a scale-free network emerge?

Network growthmodels: 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.

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

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