341: introduction to bioinformatics dr. nataša pržulj department of computing imperial college...

341: Introduction to Bioinformatics

Dr. Nataša PržuljDepartment of ComputingImperial College [email protected]

Winter 2011

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Topics

Introduction to biology (cell, DNA, RNA, genes, proteins) Sequencing and genomics (sequencing technology, sequence

alignment algorithms) Functional genomics and microarray analysis (array technology,

statistics, clustering and classification) Introduction to biological networks Introduction to graph theory Network properties

Global: network/node centralities Local: network motifs and graphlets

Network models Network/node clustering Network comparison/alignment Software tools for network analysis Interplay between topology and biology 2

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Topics






Network properties: summary of last class

Network Comparisons: Large network comparison is computationally hard due to NP-completeness of the underlying subgraph isomorphism problem:

• Given 2 graphs G and H as input, determine whether G contains a subgraph that is isomorphic to H.

Thus, network comparisons rely on easily computable heuristics (approximate solutions), called “network properties”

Network properties can roughly & historically be divided in two categories:

1.Global network properties: give an overall view of the network, but might not be detailed enough to capture complex topological characteristics of large networks.

2.Local network properties: more detailed network descriptors which usually encompass larger number of constraints, thus reducing degrees of freedom in which the networks being compared can vary.

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1. Global Network Properties

Readings: Chapter 3 of “Analysis of biological networks” by Junker and Schreiber.

Global Network Properties:1) Degree distribution2) Average clustering coefficient3) Clustering spectrum4) Average Diameter5) Spectrum of shortest path lengths6) Centralities

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2. Local Network PropertiesReadings: Chapter 5 of “Analysis of Biological Networks” by Junker and Schreiber.

1) Network motifs

2) GraphletsTwo network comparison measures based on graphlets:

2.1) Relative Graphlet Frequency Distance between two networks 2.2) Graphlet Degree Distribution Agreement between two networks


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1) Network motifs (Uri Alon’s group, ’02-’04)

http://www.weizmann.ac.il/mcb/UriAlon/

Also, see Pajek, MAVisto, and FANMOD

N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free

or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.

2) Graphlets2.1) Reltive graphlet frequency distance between two networks

2) Graphlets2.1) Graphlet degree distribution agreement between two networks

N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” ECCB, Bioinformatics, vol. 23, pg. e177-e183, 2007.

T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.

Graphlet Degree (GD) vectors, or “node signatures”


T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.

Signature Similarity Measure between nodes u and v


Software that implements many of these networkproperties and compares networks with respect to them: GraphCrunchhttp://bio-nets.doc.ic.ac.uk/graphcrunch/

Software that implements many of these networkproperties and compares networks with respect to them: GraphCrunchhttp://bio-nets.doc.ic.ac.uk/graphcrunch2/

Another Software: Cytoscapehttp://www.cytoscape.org/

T. Milenković and N. Pržulj, “Uncovering Biological Network Function via Graphlet Degree Signatures,” Cancer Informatics, 2008:6 257-273, 2008 (Highly Visible).

Examples of signatures and signature similarities:

40%SMD1

PMA1

YBR095C



90%*

SMD1

SMB1RPO26


*Statistically significant threshold at ~85%


Later we will see how to use this and other techniquesto link network structure with biological function

N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.

Generalize Degree Distribution of a network

The degree distribution measures:• the number of nodes “touching” k edges for each value of k

N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.

/ sqrt(2) ( to make it between 0 and 1)

This is called Graphlet Degree Distribution (GDD) Agreement between networks G and H.

Software that implements many of these networkproperties and compares networks with respect to them: GraphCrunchhttp://bio-nets.doc.ic.ac.uk/graphcrunch/

Software that implements many of these networkproperties and compares networks with respect to them: GraphCrunchhttp://bio-nets.doc.ic.ac.uk/graphcrunch2/

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Topics




Network/node centralities Network motifs


What is a network (graph) model?

Does the model network fit the data?

Use network properties:LocalGlobal

Why? “Hardness” of graph theoretic problems

E.g. NP-completeness of subgraph isomorphism Cannot exactly compare/align networks

• Use heuristics (approximate solutions)

Exact comparison inappropriate in biology• Due to biological variation

Noise revise models as data sets evolve

Why model networks?

Understand laws reproduction/predictions

Network models have already been used in biological applications:Network motifs

(Shen-Orr et al., Nature Genetics 2002, Milo et al., Science 2002)

De-noising of PPI network data (Kuchaiev et al., PLoS Comp. Biology, 2009)

Guiding biological experiments (Lappe and Holm, Nature Biotechnology, 2004)

Development of computationally easy algorithms for PPI nets that are computationally intensive on graphs in general (Przulj et al., Bioinformatics, 2006)

Network models

We will cover the following network models:

I. Erdos–Renyi random graphs

II. Generalized random graphs (with the same degree distribution as the data networks)

III. Small-world networks

IV. Scale-free networks

V. Hierarchical model

VI. Geometric random graphs

VII. Stickiness index-based network model

Erdos–Renyi random graphs (ER)

Model a data network G(V,E) with |V|=n and |E|=m An ER graph that models G is constructed as follows:

It has n nodes Edges are added between pairs of nodes uniformly at

random with the same probability p Two (equivalent) methods for constructing ER graphs:

Gn,p: pick p so that the resulting model network has m edges

Gn,m: pick randomly m pairs of nodes and add edges between them with probability 1


Number of edges, |E|=m, in Gn,p is:

Average degree is:


Many properties of ER can be proven theoretically (See: Bollobas, "Random Graphs," 2002)

Example: When m=n/2,suddenly the giant component

emerges, i.e.:• One connected component of the network has

O(n) nodes• The next largest connected component has

O(log(n)) nodes


The degree distribution is binomial:

For large n, this can be approximated with Poisson distribution:

where z is the average degree However, currently available biological

networks have power-law degree distribution


Clustering coefficient, C, of ER is low (for low p)

C=p, since probability p of connecting any two nodes in an ER graph is the same, regardless of whether the nodes are neighbors

However, biological networks have high clustering coefficients


Average diameter of ER graphs is small It is equal to

Biological networks also have small average diameters

Summary

Generalized random graphs (ER-DD)

Preserve the degree distribution of data

(“ER-DD”)

Constructed as follows: An ER-DD network has n nodes

(so does the data) Edges are added between pairs of nodes using

the “stubs method”


The “stubs method” for constructing ER-DD graphs: The number of “stubs” (to be filled by edges) is

assigned to each node in the model network according to the degree distribution of the real network to be modeled

Edges are created between pairs of nodes with

“available” stubs picked at random After an edge is created, the number of stubs left

available at the corresponding “end nodes” of the edges is decreased by one

Multiple edges between the same pair of nodes are not allowed


Summary

2 global network properties are matched by ER-DD How about local network properties (graphlet frequencies)?

Low-density graphlets are over-represented in ER and ER-DD However, data have lots of dense graphlets, since they have

high clustering coefficients

Small-world networks (SW)

Watts and Strogatz, 1998 Created from regular ring lattices by random

rewiring of a small percentage of their edges E.g.

Small-world networks (SW)

SW networks have: High clustering coefficients – introduced by “ring

regularity” Large average diameters of regular lattices – fixed

by randomly re-wiring a small percentage of edges

Summary

Scale-free networks (SF)

Power-law degree distributions: P(k) = k−γ

γ > 0; 2 < γ < 3


Different models exist, e.g.:

Preferential Attachment Model (SF-BA)(Barabasi-Albert, 1999)

Gene Duplication and Mutation Model (SF-GD)(Vazquez et al., 2003)


Preferential Attachment Model (SF-BA) “Growth” model: nodes are added to an existing network New nodes preferentially attach to existing nodes with

probability proportional to the degrees of the existing nodes; e.g.:

This is repeated until the size of SF network matches

the size of the data “Rich getting richer” The starting network strongly influences the properties

of the resulting network (F. Hormozdiari, et al., PLoS Computational Biology, 3(7):e118, July 2007. )

SF-BA: particularly effective at describing Internet


Gene Duplication and Mutation Model (SF-GD)

Biologically motivated

Attempts to mimic gene duplication and mutation processes


Gene Duplication and Mutation Model (SF-GD) At each time step, a node is added to the network as follows:


Summary

Hierarchical model

Preserves network “modularity” via a fractal-like generation of the network

Hierarchical model

These graphs do not match any biological data and are highly unlikely to be found in data sets

Geometric random graphs

“Uniform” geometric random graphs (GEO)N. Przulj lab, 2004-2010

Geometric gene duplication and mutation model (GEO-GD)

N. Przulj et al., PSB 2010


“Uniform” geometric random graphs (GEO) Take any metric space and, using a uniform random

distribution, place nodes within the space If any nodes are within radius r (calculated via any

chosen distance norm for the space), they will be connected

Choose r so that the size of the GEO network matches that of the data

There are many possible metric spaces (e.g., Euclidean space)

There are many possible distance norms (e.g. the Euclidean distance, the Chessboard distance, and the Manhattan/Taxi Driver distance)


“Uniform” geometric random graphs (GEO)

Summary



Gene duplications and mutations can be used to guide the growth process in geometric graph



This variant also reproduces graphlet properties of the empirical dataset

Also, these networks have power-law degree distributions

-GD

Stickiness index-based network model(N. Przulj and D. Higham, Journal of the Royal Society Interface, vol 3, num 10, pp 711 - 716, 2006.)

Based on the stickiness index: A number based on the a protein’s normalized degree in a PPI

network Used to summarize the abundance and popularity of binding

domains of a protein

Assumption: a high degree protein has many binding domains

However, remember “date” vs. “party” hubs

A pair of proteins is more likely to interact under this model if both proteins have high stickiness indices

The probability of an edge between two nodes is the product of their stickiness indices

Stickiness index-based network model

“Sticky networks” have the expected degree distribution of the data

Also, they mimic well the clustering coefficients and the diameters of real-world networks

Summary

Software that implements many of these network models and evaluates their fit to data networks with respect to a variety of network properties (but there are others): GraphCrunch: http://bio-nets.doc.ic.ac.uk/graphcrunch/

Software that implements many of these network models and evaluates their fit to data networks with respect to a variety of network properties (but there are others): GraphCrunch: http://bio-nets.doc.ic.ac.uk/graphcrunch2/

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Topics






341: introduction to bioinformatics dr. nataša pržulj department of computing imperial college...

Documents

networks network properties

network analysis interplay

class network comparisons

large network comparison

network comparison measures

detailed network descriptors

network motifs uri alons

class slide