Lecture 4 1.Protein Function prediction using network concepts 2.Hierarchical Clustering.

Lecture 4

1.Protein Function prediction using network concepts

2.Hierarchical Clustering

Topology of Protein-protein interaction is informative but further analysis can reveal other information.

A popular assumption, which is true in many cases is that similar function proteins interact with each other.

Based on these assumption, we have developed methods to predict protein functions and protein complexes from the PPI networks mainly based on cluster analysis.

Cluster Analysis

Cluster Analysis, also called data segmentation, implies grouping or segmenting a collection of objects into subsets or "clusters", such that those within each cluster are more closely related to one another than objects assigned to different clusters.

In the context of a graph densely connected nodes are considered as clusters

Visually we can detect two clusters in this graph

K-cores of Protein-Protein Interaction Networks

Definition

Let, a graph G=(V, E) consists of a finite set of nodes V and a finite set of edges E.

A subgraph S=(V, E) where V V and E E is a k-core or a core of order k of G if and only if v V: deg(v) k within S and S is the maximal subgraph of this property.

1-core graph: The degree of all nodes are one or more

Graph G

Concept of a k-core graph

2-core graph: The degree of all nodes are two or more


3-core graph: The degree of all nodes are three or more

The 3-core is the highest k-core subgraph of the graph G

Graph G

Analyzing protein-protein interaction data obtained from different sources, G. D. Bader and C.W.V. Hogue, Nature biotechnology, Vol 20, 2002

Application of a k-core graph

Protein function prediction using k-core graphs

Hishigaki, H., Nakai, K., Ono, T., Tanigami, A., and Tagaki, T. Assessment of prediction accuracy of protein function from protein-protein interaction data. Yeast 18, 523-531 (2001)

Reported similar results..

Schwikowski, B., Uetz, P. and Fields, S. A network of protein-protein interactions in yeast. Nature Biotech. 18, 1257-1261 (2000)

Deals with a network of 2039 proteins and 2709 interactions.

65% of interactions occurred between protein pairs with at least one common function

Introduction : Function prediction

14

HypothesisUnknown function proteins that form densely connected subgraph with proteins of a particular function may belong to that functional group.

Introduction : Function prediction

UNCLASSIFIED PROTEINS

CLASS A

UNCLASSIFIED PROTEINS

CLASS A

We utilize this concept by determining k-cores of strategically constructed sub-networks.

Prediction of Protein Functions Based on K-cores of Protein-Protein Interaction Networks

“Prediction of Protein Functions Based on K-cores of Protein-Protein Interaction Networks and Amino Acid Sequences”, Md. Altaf-Ul-Amin, Kensaku Nishikata, Toshihiro Koma, Teppei Miyasato, Yoko Shinbo, Md. Arifuzzaman, Chieko Wada, Maki Maeda, Taku Oshima, Hirotada Mori, Shigehiko Kanaya The 14th International Conference on Genome Informatics December 14-17, 2003, Yokohama Japan.

Total 3007 proteins and 11531 interactions

Around 2000 are unknown function proteins

Highest K-core of this total graph is not so helpful

E.Coli PPI network

10-core graph—the highest k-core of the E.Coli PPI network

We separate 1072 interactions (out of 11531) involving protein synthesis and function unknown proteins.

P. S. U. F.

P. S. P. S.

Unknown

Function unknown Proteins of this 6-kore graph are likely to be involved in protein synthesis

Extending the k-core based function prediction method and its application to PPI data of Arabidopsis thaliana

Protein Function Prediction based on k-cores of Interaction Networks, Norihiko Kamakura, Hiroki Takahashi, Kensuke Nakamura, Shigehiko Kanaya and Md. Altaf-Ul-Amin, Proceedings of 2010 International Conference on Bioinformatics and Biomedical Technology (ICBBT 2010)

21

Materials and Methods : Dataset All PPI data of Arabidopsis thaliana

•3118 interactions involving 1302 proteins.

• Collected from databases and scientific literature by our laboratory.

Green= Unknown proteins

(289 proteins)

Pink= Known proteins

(1013 proteins)

22

function names number of proteinsCELL CYCLE AND DNA PROCESSING 69CELL FATE 5CELL RESCUE, DEFENSE AND VIRULENCE 32CELLULAR COMMUNICATION/ SIGNAL TRANSDUCTION MECHANISM 171CONTROL OF CELLULAR ORGANIZATION 3DEVELOPMENT (Systemic) 9ENERGY 51Endoplasmic reticulum biogenesis 4METABOLISM 120Mitochondria biogenesis 4PROTEIN ACTIVITY REGULATION 1PROTEIN FATE (folding, modification, destination) 112PROTEIN SYNTHESIS 20REGULATION OF / INTERACTION WITH CELLULAR ENVIRONMENT 1STORAGE PROTEIN 1SYSTEMIC REGULATION OF / INTERACTION WITH ENVIRONMENT 2TRANSCRIPTION 362TRANSPORT FACILITATION 46UNCLASSIFIED PROTEINS 289

Materials and Methods : Dataset

Functional groups in the network The PPI dataset contains proteins of 19 different functions according to the first level categories of the KNApSAcK database.

23

Materials and Methods : DatasetThe trends of interactions in the context of functional

similarity

function name No No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19METABOLISM 1 72 23 1 9 10 0 1 0 67 0 29 0 4 3 0 0 0 0 0UNCLASSIFIED PROTEINS 2 23 82 19 166 279 9 3 4 189 0 35 0 35 16 0 0 0 0 1CELL RESCUE, DEFENSE AND VIRULENCE 3 1 19 9 15 7 0 0 0 38 0 1 0 3 4 0 0 0 0 0TRANSCRIPTION 4 9 166 15 689 64 6 1 0 354 0 2 3 22 7 0 0 0 1 0PROTEIN FATE (folding, modification, destination) 5 10 279 7 64 137 0 9 2 20 0 22 2 7 5 0 0 0 0 0DEVELOPMENT (Systemic) 6 0 9 0 6 0 1 0 0 1 0 0 0 0 2 0 0 0 0 0CELL FATE 7 1 3 0 1 9 0 1 0 2 0 0 0 0 1 0 0 0 0 0PROTEIN SYNTHESIS 8 0 4 0 0 2 0 0 17 2 0 1 0 1 1 0 0 0 0 0CELLULAR COMMUNICATION/ SIGNAL TRANSDUCTION MECHANISM9 67 189 38 354 20 1 2 2 374 0 24 0 35 11 0 0 1 1 0Mitochondria biogenesis 10 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0ENERGY 11 29 35 1 2 22 0 0 1 24 0 64 0 3 8 0 0 0 0 0SYSTEMIC REGULATION OF / INTERACTION WITH ENVIRONMENT 12 0 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0CELL CYCLE AND DNA PROCESSING 13 4 35 3 22 7 0 0 1 35 0 3 0 44 2 2 0 0 0 0TRANSPORT FACILITATION 14 3 16 4 7 5 2 1 1 11 0 8 0 2 17 0 2 0 0 3CONTROL OF CELLULAR ORGANIZATION 15 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0REGULATION OF / INTERACTION WITH CELLULAR ENVIRONMENT 16 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0PROTEIN ACTIVITY REGULATION 17 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0STORAGE PROTEIN 18 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0Endoplasmic reticulum biogenesis 19 0 1 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 6

Diagonal elements show number of interactions between similar function proteins.

24

Materials And Methods : Flowchart of the method

Input: A PPI network

Make a sub-network corresponding to a functional group

Determine k-cores and assign the corresponding function to the unknown proteins included in the k-cores(for k =3 or more)

Output: Predicted functions for some unknown proteins

Remove the components consisting of only unknown proteins

25

Results : Subnetworks

we do not consider in this work the sub-networks that contain less than 100 interactions.And finally I consider subnetworks corresponding to 9 functional classes.

Subnetwork Name Number of interactions

26

Subnetwork extraction

Cellular communication-Cellular communication

Cellular communication-Unknown,

Unknown-Unknown

Total 603 interactions

We extracted the following 3 types of interactions.

Results : Subnetwork corresponding to cellular communication

As an example here we show the subnetworks and k-cores corresponding to cellular communication.

27

1-core

Results : Subnetwork corresponding to cellular communication

The red nodes : known proteins.The green nodes : unknown proteins.

28

2-core 3-core

The red color nodes represent known proteins, the green color nodes represent function unknown proteins.

Results : k-cores corresponding to cellular communication

The red nodes : known proteins.The green nodes : unknown proteins.

29

4-core 5-core

The red nodes : known proteins

The green nodes : unknown proteins.

6-core 7-core

This figure implies that determination of k-cores in strategically constructed sub-networks can reveal which unknown proteins are densely connected to proteins of a particular functional class.

Results : k-cores corresponding to cellular communication

30

k-core 2 k-core 3 k-core 4 k-core 5 k-core 6 k-core 7 k-core 8

cell_cycle 11 7

cell_rescue 4

cellular_communication 37 33 23 15 12 8

energy 5 2 2 2 2 2 2

metabo 5 1 1

protein_fate 69 35 25 25 15 10

protein_synthesis 2

transcription 33 24 14 11 8 8

transport_facilitation 2

total 129 88 64 52 36 27 2

The number of unknown genes included in different k-cores corresponding to different functional groups

Results : Function Predictions

31

Most proteins have been assigned unique functions and some have been assigned multiple functions

2-core

3-core

Prediction based on 2-cores, 3-cores and 4-cores

Results : Function Predictions

4-core

Most proteins have been assigned unique functions

PROTEIN SYNTHESIS

METABOLISM

TRANSCRIPTION

PROTEIN FATE (folding, modification, destination)

TRANSPORT FACILITATION

CELL CYCLE AND DNA PROCESSING

ENERGY

CELL RESCUEM, SEFENSE AND VIRULENCE

CELLULAR COMMUNICATIO/SIGNAL TRANDUCTION

PROTEIN SYNTHESISPROTEIN SYNTHESIS

METABOLISMMETABOLISM

TRANSCRIPTIONTRANSCRIPTION

PROTEIN FATE (folding, modification, destination)PROTEIN FATE (folding, modification, destination)

TRANSPORT FACILITATIONTRANSPORT FACILITATION

CELL CYCLE AND DNA PROCESSINGCELL CYCLE AND DNA PROCESSING

ENERGYENERGY

CELL RESCUEM, SEFENSE AND VIRULENCECELL RESCUEM, SEFENSE AND VIRULENCE

CELLULAR COMMUNICATIO/SIGNAL TRANDUCTIONCELLULAR COMMUNICATIO/SIGNAL TRANDUCTION

32

Assessment of Predictions

However to assess statistically, we constructed 1000 random graphs consisting of the same 1,302 proteins but I inserted 3,118 edges randomly and constructed subnetworks.

When k is much larger than one, the effect of false positives is greatly reduced.

As most of the function predicted proteins are still unknown their annotations do not contain clear information on their functions.

Cell Cycle Cell Rescue Cellular Communication

Energy Metabolism Protein fate

Protein Synthesis

Transcription Transport

The box plots show the distribution of k-cores with respect to their size in 1000 graphs corresponding to each sub-network and the filled triangles show the size of k-cores in real PPI sub-networks.

Assessment of Predictions

3434

Assessment of Predictions•it can be theoretically concluded that the existence of higher order k-core graphs in PPI sub-networks compared to in the random graphs of the same size are likely to be because of interaction between similar function proteins. •Therefore we assume that the function prediction based on k-cores for the value of k greater than highest possible value of k for corresponding random graphs are statistically significant predictions.• Based on this we predicted the functions of 67 proteins(list is available online at http://kanaya.naist.jp/Kcore/supplementary/Function_prediction.xls.

http://kanaya.naist.jp/Kcore/supplementary/Function_prediction.xls

http://kanaya.naist.jp/Kcore/supplementary/Function_prediction.xls

“Prediction of Protein Functions Based on Protein-Protein Interaction Networks: A Min-Cut Approach”, Md. Altaf-Ul-Amin, Toshihiro Koma, Ken Kurokawa, Shigehiko Kanaya, Proceedings of the Workshop on Biomedical Data Engineering (BMDE), Tokyo, Japan, pp. 37-43, April 3-4, 2005.

Outline

•Introduction

•The concept of Min-Cut

•Problem Formulation

•A Heuristic Method

•Evaluation of the Proposed Method

•Conclusions

Introduction

After the complete sequencing of several genomes, the challenging problem now is to determine the functions of proteins

1) Determining protein functions experimentally

2) Using various computational methods

a) sequence

b) structure

c) gene neighborhood

d) gene fusions

e) cellular localization

f) protein-protein interactions

Present work predicts protein functions based on protein-protein interaction network.

Introduction

For the purpose of prediction, we consider the interactions of

•function-unknown proteins with function-known proteins and

• function-unknown proteins with function-unknown proteins

In the context of the whole network.

Hence we call the proposed approach a Min-Cut approach.

Introduction

Majority of protein-protein interactions are between similar function protein pairs.

Therefore,

We assign function-unknown proteins to different functional groups in such a way so that the number of inter-group interactions becomes the minimum.

Outline

•Introduction





•Conclusions

U4

K2K6

K4

K3

K1K8

K5U1

U2

U3

The concept of Min-Cut

G1

G2

A typical and small network of known and unknown proteins

U4

KK

K

K

KK

KU1

U2

U3

G1

G2


Unknown proteins assigned to known groups based on

majority interactions

U4

KK

K

K

KK

KU1

U2

U3

G1

G2


Number of CUT = 4

U4

KK

K

K

KK

KU1

U2

U3

G1

G2


An alternative assignment of unknown proteins

U4

KK

K

K

KK

KU1

U2

U3

G1

G2


Number of CUT = 2

For every assignment of unknown proteins, there is a value of CUT.

Min-cut approach looks for an assignment for which the number of CUT is minimum.

Outline

•Introduction





•Conclusions

Problem Formulation

L e t 1G , 2G , … … . . , nG a r e n s e t s / g r o u p s o f f u n c t i o n -k n o w n p r o t e i n s s u c h t h a t a l l p r o t e i n s o f a g r o u p a r e o f s i m i l a r f u n c t i o n . M u l t i p l e f u n c t i o n p r o t e i n s a r e m e m b e r s o f m o r e t h a n o n e g r o u p . T h e r e f o r e , t h e s e t o f a l l f u n c t i o n - k n o w n p r o t e i n s 1

nk kG G . T h e s e t o f

f u n c t i o n - u n k n o w n p r o t e i n s i s d e n o t e d b y U . ( , )N V E i s a g r a p h / n e t w o r k w h e r e iv V i s a n o d e r e p r e s e n t i n g a p r o t e i n a n d ( , )i j i je v v E i s a n e d g e r e p r e s e n t i n g … … .

Here we explain some points with a typical example.

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

( , )N V E

V= set of all nodes

E =set of all edges

G={K1, K2, K3, K4, K5, K6, K7, K8, K9, K10}

U={U1, U2, U3, U4, U5, U6, U7, U8}

Problem Formulation

U´= {U1, U2, U3, U4, U5, U6, U7}

Problem Formulation

We generate U´ U such that each protein of U´ is connected in N with at least one protein of group G by a path of length 1 or length 2.

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

For this assignment of unknown proteins, the CUT= 6

Interactions between known protein pairs can never be part of CUT

Problem FormulationWe can assign proteins of U´ to different groups and calculate CUT

The problem we are trying to solve is to assign the proteins of set U´ to known groups G1 , G2 ,…….., G3 in such a way so that the CUT becomes the minimum.

Problem Formulation

Outline

•Introduction





•Conclusions

•The problem under hand is a variant of network partitioning problem.

•It is known that network partitioning problems are NP-hard.

•Therefore, we resort to some heuristics to find a solution as better as it is possible.

A Heuristic Method

A Heuristic Method min_cut = |E|

iteration = 0

Make a table for each protein of U containing maximum 3 IDs of respective priority groups

Assign each protein of Uto some randomly or intentionally chosen group from among its priority groups

Calculate CUT

CUT < min_cut

iteration = iteration + 1

iteration < max_value

min_cut = CUT Record the current

assignment

Print min_cut, corresponding assignment and Exit

YES

NO

NO

YES

U1

U2

U3

U4

U5

U6

U7

U1 G2 G1 x

U2

U3

U4

U5

U6

U7

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

A Heuristic Method

U1 has one path of length 1 with G2 and two paths of length two with G1

U1 G2 G1 x

U2 G2 G1 x

U3 G2 G1 x

U4 G1 G2 G3

U5

U6

U7

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

A Heuristic Method

U4 has two paths of length 1 with G1, one path of length one with G2 and one path of length two with G3.

U1 G2 G1 x

U2 G2 G1 x

U3 G2 G1 x

U4 G1 G2 G3

U5 G1 G2 G3

U6 G1 G3 G2

U7 G3 G2 x

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

A Heuristic Method

U1 G2 G1 x

U2 G2 G1 x

U3 G2 G1 x

U4 G1 G2 G3

U5 G1 G2 G3

U6 G1 G3 G2

U7 G3 G2 x

A Heuristic Method min_cut = |E|

iteration = 0



Calculate CUT

CUT < min_cut




assignment


YES

NO

NO

YES

U1 G2 G1 x

U2 G2 G1 x

U3 G2 G1 x

U4 G1 G2 G3

U5 G1 G2 G3

U6 G1 G3 G2

U7 G3 G2 x

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

A Heuristic Method

By assigning all the unknown proteins to respective height priority groups, CUT = 6

U1 G2 G1 x

U2 G2 G1 x

U3 G2 G1 x

U4 G1 G2 G3

U5 G1 G2 G3

U6 G1 G3 G2

U7 G3 G2 x

A Heuristic Method

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3


U1 G2 G1 x

U2 G2 G1 x

U3 G2 G1 x

U4 G1 G2 G3

U5 G1 G2 G3

U6 G1 G3 G2

U7 G3 G2 x

U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3


A Heuristic Method

Outline

•Introduction





•Conclusions

Evaluation of the Proposed Approach

•The proposed method is a general one and can be applied to any organism and any type of functional classification.

•Here we applied it to yeast Saccharomyces cerevisiae protein-protein interaction network

•We obtain the protein-protein interaction data from ftp://ftpmips.gsf.de/yeast/PPI/ which contains 15613 genetic and physical interactions.

YAR019c YMR001c

YAR019c YNL098c

YAR019c YOR101w

YAR019c YPR111w

YAR027w YAR030c

YAR027w YBR135w

YAR031w YBR217w

------------- -------------

------------- -------------

Total 12487 pairs

We discard self-interactions and extract a set of 12487 unique binary interactions involving 4648 proteins.


A network of 12487 interactions and 4648 proteins is reasonably big


Name of functional class # of

proteins METABOLISM 984 ENERGY 260 CELL CYCLE AND DNA PROCESSING

690

TRANSCRIPTION 842 PROTEIN SYNTHESIS 381 PROTEIN FATE (folding, modification, destination)

631

PROTEIN WITH BINDING FUNCTION OR COFACTOR REQUIREMENT (structural or catalytic)

39

PROTEIN ACTIVITY REGULATION 27 CELLULAR TRANSPORT, TRANSPORT FACILITATION AND TRANSPORT ROUTES

719

CELLULAR COMMUNICATION/SIGNAL TRANSDUCTION MECHANISM

94

CELL RESCUE, DEFENSE AND VIRULENCE

296

INTERACTION WITH THE CELLULAR ENVIRONMENT

336

TRANSPOSABLE ELEMENTS, VIRAL AND PLASMID PROTEINS

118

BIOGENESIS OF CELLULAR COMPONENTS

451

CELL TYPE DIFFERENTIATION 339

We collect from http://mips.gsf.de/genre/proj/yeast/index.jsp the classification data


Name of functional class # of proteins

METABOLISM 984 ENERGY 260 CELL CYCLE AND DNA PROCESSING

690


631


39


719


94


296


336


118


451


•The proposed approach is intended to predict the functions of function-unknown proteins.

•However, by predicting the functions of function-unknown proteins, it is not possible to determine the correctness of the predictions.

•We consider around 10% randomly selected proteins of each group of Table 1 as function-unknown proteins.


Name of functional class # of

proteins METABOLISM 984 ENERGY 260 CELL CYCLE AND DNA PROCESSING

690


631


39


719


94


296


336


118


451


•The union of 10% of all groups consists of 604 proteins. This is the unknown group U.

•The union of the rest 90% of each of the functional groups constitutes the set of known proteins G. There are total 3783 proteins in G.

•We generate U´ U such that each protein of U´ is connected in N with at least one protein of group G by a path of length 1 or length 2. There are 470 proteins in U´ .

•We predicted functions of these 470 proteins using the proposed method.


min_cut = |E| iteration = 0



Calculate CUT

CUT < min_cut




assignment


YES

NO

NO

YES

We applied this algorithm using Max_value=50000 to predict the functions 470 proteins.


•We cannot guarantee that minimum CUT corresponds to maximum successful prediction.

•However, the trends of the results of the Figure above shows that it is very likely that the lower is the value of CUT the greater is the number of successful predictions


We then examine the relation of successful predictions with the number of degrees of the proteins in the network .


U4

K2K6

K3

K4

K1K7

K5U1

U2

U3

K10

K8

K9U7

U5

U8

U6

G1

G2

G3

Degree of U4 =7

Degree of U7=3

We then examine the relation of successful predictions with the number of degrees of the proteins in the network .


Degree Number of proteins

Successful prediction

Percentage

1 128 39 30.46 2 80 39 48.75 3 60 32 53.33 4 33 24 72.72 5 23 15 65.21 6 24 14 58.33 7 17 12 70.58

>7 105 71 67.61 Total 470 246 52.34

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8

Degree

Suc

cess

Per

cent

age

•The success rate of prediction is as low as 30.46% for proteins that have only one degree in the interaction network.

•However it is 67.61% for proteins that have degrees 8 or more.

•This implies that the reliability of the prediction can be improved by providing reasonable amount of interaction information


Hierarchical clustering

Hierarchical Clustering

AtpB AtpAAtpG AtpEAtpA AtpHAtpB AtpHAtpG AtpHAtpE AtpH

Data is not always available as binary relations as in the case of protein-protein interactions where we can directly apply network clustering algorithms.

In many cases for example in case of microarray gene expression analysis the data is multivariate type.

An Introduction to Bioinformatics Algorithms by Jones & Pevzner

We can convert multivariate data into networks and can apply network clustering algorithm about which we will discuss in some later class.

If dimension of multivariate data is 3 or less we can cluster them by plotting directly.



However, when dimension is more than 3, we can apply hierarchical clustering to multivariate data.

In hierarchical clustering the data are not partitioned into a particular cluster in a single step. Instead, a series of partitions takes place.

Some data reveal good cluster structure when plotted but some data do not.

Data plotted in 2 dimensions


Hierarchical clustering is a technique that organizes elements into a tree.

A tree is a graph that has no cycle.

A tree with n nodes can have maximum n-1 edges.

A Graph A tree


Hierarchical Clustering is subdivided into 2 types

1. agglomerative methods, which proceed by series of fusions of the n objects into groups,

2. and divisive methods, which separate n objects successively into finer groupings.

Agglomerative techniques are more commonly used

Data can be viewed as a single cluster containing all objects to n clusters each containing a single object .


Distance measurementsThe Euclidean distance between points and

, in Euclidean n-space, is defined as:

Euclidean distance between g1 and g2

0622.81640

)910()08()1010( 222



In stead of Euclidean distance correlation can also be used as a distance measurement.

For biological analysis involving genes and proteins, nucleotide and or amino acid sequence similarity can also be used as distance between objects


•An agglomerative hierarchical clustering procedure produces a series of partitions of the data, Pn, Pn-1, ....... , P1. The first Pn consists of n single object 'clusters', the last P1, consists of single group containing all n cases. •At each particular stage the method joins together the two clusters which are closest together (most similar). (At the first stage, of course, this amounts to joining together the two objects that are closest together, since at the initial stage each cluster has one object.)



Differences between methods arise because of the different ways of defining distance (or similarity) between clusters.


How can we measure distances between clusters?

Single linkage clustering

Distance between two clusters A and B, D(A,B) is computed as

D(A,B) = Min { d(i,j) : Where object i is in cluster A and object j is cluster B}


Complete linkage clustering

Distance between two clusters A and B, D(A,B) is computed as D(A,B) = Max { d(i,j) : Where object i is in cluster A and

object j is cluster B}


Average linkage clustering

Distance between two clusters A and B, D(A,B) is computed as D(A,B) = TAB / ( NA * NB)

Where TAB is the sum of all pair wise distances between objects of cluster A and cluster B. NA and NB are the sizes of the clusters

A and B respectively.

Total NA * NB edges


Average group linkage clustering

Distance between two clusters A and B, D(A,B) is computed as D(A,B) = = Average { d(i,j) : Where observations i and j are in

cluster t, the cluster formed by merging clusters A and B }

Total n(n-1)/2 edges


Alizadeh et al. Nature 403: 503-511 (2000).


Classifying bacteria based on 16s rRNA sequences.

Lecture 4 1.Protein Function prediction using network concepts 2.Hierarchical Clustering.

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Transcript of Lecture 4 1.Protein Function prediction using network concepts 2.Hierarchical Clustering.