LECTURE 3Introduction to PCA and PLSK-mean clustering

Protein function prediction using network concepts

Network Centrality measures

Handling Multivariate data

Multivariate data example

Student Math Chem Phy Bio Eco Soc

A 7 8 7 8 7 7

B 8 7 7 6 8 7

C 9 7 8 7 6 7

D 7 7 7 7 9 8

E 7 6 6 6 8 8

F 7 7 7 7 8 8

G 6 6 6 7 7 7

H 9 8 8 6 6 6

I 8 8 8 7 6 6

J 7 7 6 6 8 9

Principle Component Analysis (PCA) and Partial Least Square (PLS)

• Two major common effects of using PCA or PLS Convert a group of correlated predictive variables to a group

of independent variables Construct a “strong” predictive variable from several “weaker”

predictive variables

• Major difference between PCA and PLS PCA is performed without a consideration of the target

variable. So PCA is an unsupervised analysis PLS is performed to maximiz the correlation between the

target variable and the predictive variables. So PLS is a supervised analysis

A(n x p)

X(n x p)

PCA PLS

Y(n x q)

PC(n x p)

T(n x c)

U(n x c)max cov.

1 12

1 Decomposition step

2 Regression step

A = data matrixPC = principal component matrixn = # of observationsp = # of variables

n = # of observationsp = # of predictorsq = # of responsesc = # of extracted factors

X = matrix of predictorsY = matrix of responsesT = factors of predictorsU = factors of responses

Principle Component Analysis (PCA)

In Principal Component Analysis, we look for a few linear combinations of the predictive variables which can be used to summarize the data without loosing too much information.

Intuitively, Principal components analysis is a method of extracting information from a higher dimensional data by projecting it to a lower dimension.

Example: Consider the scatter plot of a 3-dimentional data (3 variables). Data across the 3

variables are higly correlated and majority of the points cluster around the center of the space. This is also the direction of the 1st PC and all 3 variables roughly imposes equal loading to 1st PC.

PC1 = – 0.56 (X1) – 0.57 (X2) – 0.59 (X3)

Properties of Principal Components

• Var(PCi) = i

• Cov(PCi,PCj) = 0

• Var(PC1) Var(PC2) … Var(PCp)

Here is the sorted eigenvalue of the variance-co-varience matrix

Numerical ExampleStudent Math Chem Phy Bio Eco Soc

A 7 8 7 8 7 7

B 8 7 7 6 8 7

C 9 7 8 7 6 7

D 7 7 7 7 9 8

E 7 6 6 6 8 8

F 7 7 7 7 8 8

G 6 6 6 7 7 7

H 9 8 8 6 6 6

I 8 8 8 7 6 6

J 7 7 6 6 8 9

The following is the high school grade of 10 students on 6 subjects (scale 1-10)

• Math = Mathematics• Chem = Chemistry• Phy = Phisics• Bio = Biology• Eco = Economy• Soc = Sociology

ResultsPC1 PC2 PC3 PC4 PC5 PC6

Eigenvalue 3.020 0.708 0.497 0.219 0.167 0.023

Proportion 0.652 0.153 0.107 0.047 0.036 0.005

Cumulative 0.652 0.804 0.912 0.959 0.995 1

Eigenvectors

Math 0.461 0.621 -0.088 0.168 0.267 -0.542

Chem 0.302 -0.059 -0.594 0.016 -0.740 -0.074

Phy 0.428 0.110 -0.365 -0.064 0.386 0.720

Bio 0.054 -0.666 -0.410 0.248 0.445 -0.355

Eco -0.533 0.271 -0.526 -0.559 0.185 -0.140

Soc -0.475 0.286 -0.248 0.771 -0.020 0.192

Partial Least Squares (PLS)

• Unlike PCA, the PLS technique works by successively extracting factors from both predictive and target variables such that covariance between the extracted factors is maximized

• Decomposition step X = TWt + E Y = UVt + F

• Regression step Y = TB + D = XWB + D = XBPLS + D; BPLS = WB

Numerical ExampleStudent Math Chem Phy Bio Eco Soc GPA

A 7 8 7 8 7 7 2.9

B 8 7 7 6 8 7 3.1

C 9 7 8 7 6 7 3.6

D 7 7 7 7 9 8 3.3

E 7 6 6 6 8 8 3.0

F 7 7 7 7 8 8 2.9

G 6 6 6 7 7 7 3.2

H 9 8 8 6 6 6 3.4

I 8 8 8 7 6 6 2.8

J 7 7 6 6 8 9 3.5

The following is the high school grade of 10 students on 6 subjects (scale 1-10)

• Math = Mathematics• Chem = Chemistry• Phy = Phisics• Bio = Biology• Eco = Economy• Soc = Sociology

and the corresponding GPA score during undergraduate level.

Objective: Can we use information of student’s performance during high school to predict their GPA score when they enter undergraduate level?

K-mean clustering

Source: “Clustering Challenges in Biological Networks” edited by S. Butenko et. al.

Source:Teknomo, Kardi. K-Means Clustering Tutorials http:\\people.revoledu.com\kardi\ tutorial\

kMean\

1. Initial value of centroids: Suppose we use medicine A and medicine B as the first centroids. Let c1 and c2 denote the coordinate of the centroids, then c1 = (1,1) and c2 = (2,1)

Protein function prediction using network concepts

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

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

Concept of a k-core graph

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

Concept of a k-core graph

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

Concept of a k-core graph

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

•A model of Internet topology using k-shell decomposition, Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt, and Eran Shir, PNAS, vol. 104, no. 27, 2007

Notice that the concept of a K-core graph has been applied to very different areas of research.

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

33

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)

40

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)

41

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.

42

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.

43

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

44

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

45

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.

46

1-core

Results : Subnetwork corresponding to cellular communication

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

47

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.

48

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

49

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

50

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

51

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

5353

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.

“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

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

•The concept of Min-Cut

•Problem Formulation

•A Heuristic Method

•Evaluation of the Proposed Method

•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

The concept of Min-Cut

Unknown proteins assigned to known groups based on

majority interactions

U4

KK

K

K

KK

KU1

U2

U3

G1

G2

The concept of Min-Cut

Number of CUT = 4

U4

KK

K

K

KK

KU1

U2

U3

G1

G2

The concept of Min-Cut

An alternative assignment of unknown proteins

U4

KK

K

K

KK

KU1

U2

U3

G1

G2

The concept of Min-Cut

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

•The concept of Min-Cut

•Problem Formulation

•A Heuristic Method

•Evaluation of the Proposed Method

•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

•The concept of Min-Cut

•Problem Formulation

•A Heuristic Method

•Evaluation of the Proposed Method

•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

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 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

For this assignment of unknown proteins, the CUT= 7

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

For this assignment of unknown proteins, the CUT= 4

A Heuristic Method

Outline

•Introduction

•The concept of Min-Cut

•Problem Formulation

•A Heuristic Method

•Evaluation of the Proposed Method

•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.

Evaluation of the Proposed Approach

A network of 12487 interactions and 4648 proteins is reasonably big

Evaluation of the Proposed Approach

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

Evaluation of the Proposed Approach

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

•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.

Evaluation of the Proposed Approach

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

•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.

Evaluation of the Proposed Approach

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

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

Evaluation of the Proposed Approach

•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

Evaluation of the Proposed Approach

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

Evaluation of the Proposed Approach

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 .

Evaluation of the Proposed Approach

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

Evaluation of the Proposed Approach

Centrality measures of nodes

Centrality measures

Within graph theory and network analysis, there are various measures of the centrality of a vertex within a graph that determine the relative importance of a vertex within the graph.

•Degree centrality

•Betweenness centrality

•Closeness centrality

•Eigenvector centrality

•Subgraph centrality

We will discuss on the following centrality measures:

Degree centrality

Degree centrality is defined as the number of links incident upon a node i.e. the number of degree of the node

Degree centrality is often interpreted in terms of the immediate risk of the node for catching whatever is flowing through the network (such as a virus, or some information).

Degree centrality of the blue nodes are higher

Betweenness centrality

The vertex betweenness centrality BC(v) of a vertex v is defined as follows:

Here σuw is the total number of shortest paths between node u and w and σuw(v) is number of shortest paths between node u and w that pass node v

Vertices that occur on many shortest paths between other vertices have higher betweenness than those that do not.

a

db f

e

c

Betweenness centrality σuw σuw(v) σuw/σuw(v)

(a,b) 1 0 0

(a,d) 1 1 1

(a,e) 1 1 1

(a,f) 1 1 1

(b,d) 1 1 1

(b,e) 1 1 1

(b,f) 1 1 1

(d,e) 1 0 0

(d,f) 1 0 0

(e,f) 1 0 0

Betweenness centrality of node c=6

Betweenness centrality of node a=0 Calculation for node c

Hue (from red=0 to blue=max) shows the node betweenness.

Betweenness centrality

•Nodes of high betweenness centrality are important for transport.

•If they are blocked, transport becomes less efficient and on the other hand if their capacity is improved transport becomes more efficient.

•Using a similar concept edge betweenness is calculated.

http://en.wikipedia.org/wiki/Betweenness_centrality#betweenness

Closeness centrality

The farness of a vortex is the sum of the shortest-path distance from the vertex to any other vertex in the graph.The reciprocal of farness is the closeness centrality (CC).

Here, d(v,t) is the shortest distance between vertex v and vertex t

Closeness centrality can be viewed as the efficiency of a vertex in spreading information to all other vertices

vVt

tvdvCC

\

),(

1)(

Eigenvector centralityLet A is the adjacency matrix of a graph and λ is the largest eigenvalue of A and x is the corresponding eigenvector then

The ith component of the eigenvector x then gives the eigenvector centrality score of the ith node in the network.

From (1)

N

jjjii xAx

1,

1

•Therefore, for any node, the eigenvector centrality score be proportional to the sum of the scores of all nodes which are connected to it. •Consequently, a node has high value of EC either if it is connected to many other nodes or if it is connected to others that themselves have high EC

-----(1)N×N N×1 N×1

|A-λI|=0, where I is an identity matrix

Subgraph centrality

the number of closed walks of length k starting and ending on vertex i in the network is given by the local spectral moments μ k (i), which are simply defined as the ith diagonal entry of the kth power of the adjacency matrix, A:

Closed walks can be trivial or nontrivial and are directly related to the subgraphs of the network.

Subgraph Centrality in Complex Networks, Physical Review E 71, 056103(2005)

0 1 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 1 0 1 0 0 0 0 0 0 0 0

0 1 0 1 1 1 0 0 0 0 0 0 0 0

0 1 1 0 1 1 0 1 0 0 0 0 0 0

0 0 1 1 0 1 0 0 0 0 0 0 0 0

0 1 1 1 1 0 1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 1 0 0 0

0 0 0 1 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 1 0 1 0 0 1 1

0 0 0 0 0 0 0 0 1 0 1 0 1 1

0 0 0 0 0 0 1 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1 1 0 1 0 1

0 0 0 0 0 0 0 0 1 1 0 0 1 0

M =

Muv = 1 if there is an edge between

nodes u and v and 0 otherwise.

Subgraph centrality

Adjacency matrix

1 0 1 1 0 1 0 0 0 0 0 0 0 0

0 4 2 2 3 2 1 1 0 0 0 0 0 0

1 2 4 3 2 3 1 1 0 0 0 0 0 0

1 2 3 5 2 3 1 0 1 0 0 0 0 0

0 3 2 2 3 2 1 1 0 0 0 0 0 0

1 2 3 3 2 5 0 1 0 0 1 0 0 0

0 1 1 1 1 0 2 0 0 1 0 0 0 0

0 1 1 0 1 1 0 2 0 1 0 0 1 1

0 0 0 1 0 0 0 0 4 2 1 1 2 2

0 0 0 0 0 0 1 1 2 4 0 1 2 2

0 0 0 0 0 1 0 0 1 0 2 0 1 1

0 0 0 0 0 0 0 0 1 1 0 1 0 1

0 0 0 0 0 0 0 1 2 2 1 0 4 2

0 0 0 0 0 0 0 1 2 2 1 1 2 3

M2 =

(M2)uv for uv represents the

number of common neighbor of the nodes u and v.

local spectral moment

Subgraph centrality

The subgraph centrality of the node i is given by

Let λ be the main eigenvalue of the adjacency matrix A. It can be shown that

Thus, the subgraph centrality of any vertex i is bounded above by

Subgraph centrality

Table 2. Summary of results of eight real-world complex networks.

Software Open AccessExploration of biological network centralities with CentiBiNBjörn H Junker, Dirk Koschützki* and Falk SchreiberAddress: Department of Molecular Genetics, Leibniz Institute of Plant Genetics and Crop Plant Research (IPK), Corrensstr. 3, 06466 Gatersleben, GermanyEmail: Björn H Junker - junker@ipk-gatersleben.de; Dirk Koschützki* - koschuet@ipk-gatersleben.de; Falk Schreiber - schreibe@ipkgatersleben.deBMC Bioinformatics 2006, 7:219 doi:10.1186/1471-2105-7-219

Node Degree Closeness Eccentricity Betweeness Eigenvector

n1 1 0.04 0.2 0 0.215411

n2 1 0.04 0.2 0 0.215411

n3 1 0.04 0.2 0 0.215411

n4 5 0.058824 0.25 21 0.5717

n5 2 0.05 0.25 0 0.394844

n6 3 0.0625 0.333333 20 0.476217

n7 3 0.055556 0.333333 18.5 0.297338

n8 2 0.041667 0.25 3.5 0.156459

n9 2 0.033333 0.2 0.5 0.117904

n10 2 0.041667 0.25 3.5 0.156459

Centrality values of the nodes of an example graph calculated using CentiBiN