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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.

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I corsi vengono integrati e conterranno grosso modo due moduli:

SB: systems biology

ML: machine learning


Systems Biology. What Is It? moduli:

  • 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: moduli:

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 moduli:

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.


How to Describe a System moduli:

As a Whole?

Networks - The Language

of Complex Systems




Fragment of a Social Network moduli:

(Melburn, 2004)

Friendship among 450 people in Canberra


A. Intra-Cellular Networks moduli:

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


The Protein Interaction Network of Yeast moduli:

Yeast two hybrid

Uetz et al, Nature 2000


Metabolic Networks moduli:

Source: ExPASy



L-A Barabasi moduli:

protein-gene interactions

PROTEOME

protein-protein interactions

METABOLISM

Bio-chemical reactions

Citrate Cycle

GENOME

miRNA

regulation?

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

- -


Functional networks

Cell Cycle moduli:

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 Networks

Yeast: 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? moduli:

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 moduli:

The question is whether it is possible to walk with a route that crosses each bridge exactly once.


The representation of Euler moduli:

  • 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 moduli:

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 moduli:

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 moduli:

Find a path visiting each node exactly one

Conditions of existence for Hamiltonian paths are not simple



Graph nomenclature moduli:

  • 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 moduli:

  • 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 moduli:

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.


Graph nomenclature moduli:

Paths

Stars

Cycles

Complete Graphs



Vertex degree distribution moduli: (the degree of a vertex is the number of vertices connected with it via an edge)

Statistical features of networks


Clustering coefficient moduli:: 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 moduli:: 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? them

Edge-weighted Graph

The exaustive search can be too much time-consuming


The Dijkstra’s algorithm them

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 them

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 them

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 them

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 them

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 them

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 them

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 them

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 them

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 them

Diameter: maximal shortest distance

Statistical features of networks

  • Vertex degree distribution

  • Clustering coefficient


Two reference models for networks them

Regular network (lattice)

Random network (Erdös+Renyi, 1959)

Regular connections

Each edge is randomly set with probability p


Two reference models for networks them

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
Some examples for real networks them

Real networks are not regular (low shortest path)

Real networks are not random (high clustering)


Adding randomness in a regular network them

Random changes in edges

OR

Addition of random links


Adding randomness in a regular network them

(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 them

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? them

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? them

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 them

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 them

Attacks to hubs can rapidly destroy the network


Three non biological scale-free networks them

Note the log-log scale

LINEAR PLOT

Albert and Barabasi, Science 1999


How can a scale-free network emerge? them

Network growthmodels: start with one vertex.


How can a scale-free network emerge? them

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? them

Network growth models: hubs emerge

(in the WWW: new pages tend to link to existing, well linked pages)


Metabolic pathways are scale-free them

Hubs are pyruvate, coenzyme A….


Protein interaction networks are scale-free them

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 them

Titz et al, Exp Review Proteomics, 2004


How can a scale-free network emerge? them

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 them

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 them

Standard free scale models have low clustering: a modular hierarchical model accounts for high clustering, low average path and scale-freeness


Modules them

Sub-graphs more represented than expected

209 bi-fan motifs found in the E.coli regulatory network


Summary them

  • 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: them

(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) them 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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