1. Comparative Genomics (Network Biology)
Today’s lecture will cover the following three topics
On finding clusters in undirected simple graphs: application to protein complex detection 
DPClus software tool
Concept of Line Graphs

2. On finding clusters in undirected simple graphs: application to protein complex detection
Outline
Introduction
Some basic concepts
The proposed algorithm
The DPClus software
Results & Discussion
Conclusions

3. Introduction
There is no universal definition of a cluster.
But clustering is an important issue.
Consequently there are diverse definitions and various methods.
The major purpose of clustering is finding cohesive groups.
Here, we are going to discuss a graph clustering algorithm.

4. Introduction
Regarding a graph, a cluster is a subgraph whose nodes are densely connected with each other compared to their connections with other nodes in the graph.
This is a flexible definition of a cluster.
Intuitively, we can recognize two clusters in this arbitrary graph.
But it is difficult to draw a big graph revealing its clusters.

5. Introduction
An E. coli protein-protein interaction network---consisting of proteins and interactions (From Mori Lab NAIST, Japan)
Some algorithm is needed to detect locally dense regions……

6. Introduction
Md. Altaf-Ul-Amin, Yoko Shinbo, Kenji Mihara, Ken Kurokawa and Shigehiko Kanaya, “Development and implementation of an algorithm for detection of protein complexes in large interaction networks”, BMC Bioinformatics 7:207, April 2006.

7. Some basic concepts
It is likely that two nodes belong to the same cluster have more common neighbors than two nodes that are not

8. Some basic concepts
It is likely that two nodes belong to the same cluster have more common neighbors than two nodes that are not

9. It is easy to realize that d = |E|/|E|max = 2*|E|/|N|*(|N|-1).
Some basic concepts
The density d of a cluster is the ratio of the number of edges present in it and the maximum possible number of edges in it.
It is easy to realize that d = |E|/|E|max = 2*|E|/|N|*(|N|-1).
d is a real number ranging from 0 to 1.

10. Some basic concepts
d=0.9
d=1.0
Density of the total graph = 0.241
The density of the complexes are relatively higher

11. Some basic concepts
Considering density alone is not enough
Both the graphs consist of 8 nodes and both are of density 0.5
But one of them seems to be a single cluster while the other is divided into two clusters
Such situations can be tackled by keeping track of the periphery

12. Some basic concepts
The cluster property of any node n with respect to any cluster k of density dk and size Nk is defined as follows:
cpnk=|Enk|/(dk* |Nk|)
Here, |Enk| is the total number of edges between the node n and each of the nodes of cluster k.
Cluster property of node f = 0.2
Cluster property of node f  0.57

13. The proposed Algorithm
The proposed algorithm is a sequential constructive algorithm:
It initializes the complex/cluster by choosing a seed node.
It then repeatedly add other nodes on the basis of priority and some conditions.
The major methods of the algorithm
Choosing a seed node.
Selecting a priority node.
Checking necessary conditions before adding a node to a complex.

14. The proposed Algorithm
Inputs to the algorithm are:
The associated matrix of the network.
A minimum threshold density for the generated clusters.
A parameter to determine how we separate a complex from its periphery.
Output of the algorithm are :
Overlapping/non-overlapping complexes whose densities are more or equal to the given density.

15. Flowchart of the proposed Algorithm-

16. The proposed Algorithm
M =
Muv = 1 if there is an edge between nodes u and v and 0 otherwise.

17. The proposed Algorithm
M2 =
(M2)uv for uv represents the number of common neighbor of the nodes u and v.

18. The proposed Algorithm
M2 =
(M2)uv for uv represents the number of common neighbor of the nodes u and v.

19. The proposed Algorithm2 3 2 2 3 2 2 2 2 2 2 2 2 3 2
The weights of edges are derived by squaring the associated matrix of the graph

20. The proposed Algorithm10 6 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
The weights of nodes (sum of the weights of the connecting edges)

21. The proposed Algorithm10
Seed 6 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges P1 2 1 P3 3 P4 P5
Neighbors

22. The proposed Algorithm10 6 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges P3 3 1 P5 P1 2 P4
cp of P3 = 1
Neighbors

23. The proposed Algorithm10 6
d=1.0 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges P1 4 2 P4 P5 6 P7 1
Neighbors

24. The proposed Algorithm10 6
d=1.0 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges P5 6 2 P1 4 P4 P7 1
cp of P5 = 1
Neighbors

25. The proposed Algorithm10 6
d=1.0 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges P1 4 2 P4 P6 1 P7
cp of P1 = 1
Neighbors

26. The proposed Algorithm10 6
d=1.0 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges P0 1 P4 4 2 P6 P7
Neighbors

27. The proposed Algorithm10 6
d=1.0 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges P4 4 2 P0 1 P6 P7
cp of P4 = 0.75
Neighbors

28. The proposed Algorithm10
d=0.9 6 2 6 3 2 2 3 10 2 2 2 2 2 6 6 2 2 2 3 6 6 10 2
Sum of edge weights
# of edges
cp-value P0 1
~0.22 P6 P7
Neighbors

29. The proposed Algorithm6 2 6 2 2
Seed 2 2 6 2 6
The remaining graph

30. The proposed Algorithm6 2 6 2 2 2 2 6 2 6

31. The proposed Algorithm6 2 6 2 2 2 2 6 2 6

32. The proposed Algorithm6 2 6 2 2 2 2 6 2 6

33. The proposed Algorithm
The remaining graph

34. The proposed Algorithm
Clustering by the proposed algorithm

35. Example A B D C E L F H G K J I
(ⅰ)

36. 1. Input and Initialized cpin=0.4, din = 0.6B D C E L F H G K J I
(ⅰ)

37. 1. Seed Selection-1: calculation of weights of edgesB D C E L F H G K J I 2 3 1

38. 1. Seed selection-2: Calculation of weights of nodesB D C E L F H G K J I
(ⅲ)
クラスター1のシード選択 2 3 1 6 10 8 4
Selected seed

39. 2. Cluster formation-1 Calculation of weights of nodesB D C E L F H G K J I
(ⅳ) 2 3 1
Cluster 1
d1=1
クラスター１の形成
Candidate merged to Cluster 1 1

40. 2. Cluster formation-2 Candidate merged to Cluster 1 4
Check thresholds  OK
d1=1/1=1 > 0.6 A 4
cpC1=1/(1*1)=1 > 0.4 (cpin ) 2 B 2 2 C 2 1 D 2 L 1 3 E 1 F G I K H
クラスター１の形成
(ⅴ) J

41. 2. Cluster formation-3 Cluster 1 d1=3/3=1 A 2 cpA1=2/(1x2)=1>0.4 6B D C E L F H G K J I
(ⅵ)
クラスター１の形成
Cluster 1 d1=3/3=1 2
cpA1=2/(1x2)=1>0.4 6 2 2 1 2 1 3 1

42. 2. Cluster formation-4 Candidate merged to Cluster 1
Check thresholds  OK
d1=1/1=1 > 0.6 A
cpB1=3/(1x3)=1 > 0.4 (cpin ) B C 1 3 D 2 L 1
Candidate merged to Cluster 1 E F 1 G I K H
クラスター1の形成
(ⅶ) J

43. 2. Cluster formation-5 Candidate merged to Cluster 1
Check thresholds  OK
d1=8/10=0.8 > 0.6 A
cpL1=2/(1*4)=0.5 > 0.4 (cpin ) B C D L 1 1 E
Candidate merged to Cluster 1 F 2 G I K H
クラスター1の形成
(ⅷ) J

44. 2. Cluster formation-6 Candidate merged to Cluster 1
Check thresholds  OK
d1=10/15=0.67 > 0.6 A
cpE1=2/(0.8*5)=0.6 > 0.4 (cpin ) B C D L E F G I K
Candidate merged to Cluster 1 H
クラスター1の探索
(ⅸ) J

45. 2. Cluster formation-7 Check thresholds  Out d1=11/12=0.52 < 0.6 A
cpE1=1/(0.52*6)=0.32 < 0.4 (cpin ) B C D L E F G I K H
クラスター1の探索
(ⅸ) J

46. 2. Cluster formation-8 Check thresholds  Out d1=11/12=0.52 < 0.6 A
cpF1=1/(0.52*6)=0.32 < 0.4 (cpin ) B C D L E F G I K H
クラスター1の探索
(ⅸ) J

47. 2. Cluster formation-8 Check thresholds  Out d1=11/12=0.52 < 0.6 A
cpF1=1/(0.52*6)=0.0 < 0.4 (cpin ) B C D L E F G I K H
クラスター1の探索
(ⅸ) J

48. 2. Cluster formation-9: Remove the edges and nodes belonging to Cluster 1K J I
(ⅹ)
クラスター1を削除

49. Results of Density Periphery ClusteringA B C
Cluster 1
d1=10/15=0.67 D L E F
Cluster 3
d3=3/3=1
Cluster 2
d2=3/3=1 G I K H ⅰ 終了
(ⅹ) J

50. Results: Complexes in the E. coli PPI Network
DIP:339N GroEL DIP:1081N PrnP
DIP:1025N CarB DIP:1026N CarA
DIP:539N MalG DIP:508N MalE
DIP:124N XerD DIP:726N XerC
DIP:367N PntB DIP:366N PntA
DIP:342N SbcC DIP:572N Gam
The network of E. coli proteins consists of 363 interactions involving a total of 336 proteins

51. Results: Complexes in the E. coli PPI Network
components of RNA polymerase (RpoA, RpoB, RpoC, Rsd, RpoZ RpoD, RpoN, FliA)

52. Results: Complexes in the E. coli PPI Network
components of ATP synthetase (AtpA, AtpB, AtpE, AtpF, AtpG, AtpH, AtpL);

53. Results: Complexes in the E. coli PPI Network
Proteins involved in cell division (FtsQ, FtsI, FtsW, FtsN, FtsK and FtsL)

54. Results: Complexes in the E. coli PPI Network
components of DNA polymerase (DnaX, HolA, HolB, HolD, and HolC);

55. Results: Complexes in the S. cerevisiae PPI Network
We extract a set of unique binary interactions involving 4648 proteins by discarding self-interactions of the PPI data obtained from ftp://ftpmips.gsf.de/yeast/PPI/.

56. Results: Details of a Group of Predicted Complexes
Information on the complexes that are of size 6 of the set generated using din=0.7, cpin=0.50 and non-overlapping mode.
We considered 15 functional classes: (1) Cell cycle and DNA processing, (2) Protein with binding function or cofactor requirement (structural or catalytic), (3) Protein fate (folding, modification, destination), (4) Biogenesis of cellular components, (5) Cellular transport, transport facilitation and transport routes, (6) Metabolism, (7) Interaction with the cellular environment, (8) Transcription, (9) Energy, (10) Cell rescue, defense and virulence, (11) Cell type differentiation, (12) Cellular communication/signal transduction mechanism, (13) Protein activity regulation, (14) Protein synthesis, and (15) Transposable elements, viral and plasmid proteins

57. Results: Hypergeometric distribution
N= Total number of proteins in the network
F= Number of proteins of a functional group in the network
C= Number of proteins in a cluster
k= Number of proteins of a functional group in a cluster
The p-value of a cluster implies the probability that the proteins of the cluster have been randomly selected
The lower the p-value the higher the statistical significance

58. P-value & Hyper geometric distribution
3 green and 4 red balls
Put them in a box
Randomly choose any 3
P1(# of red ball is 1) =
P0(# of red ball is 0) =
P3(# of red ball is 3) =
P2(# of red ball is 2) =
Notice that, P0 +P1+P2+P3=1

59. P-value & Hyper geometric distribution
P1(# of red ball is 1) =
P0(# of red ball is 0) =
P3(# of red ball is 3) =
P2(# of red ball is 2) = 1 3 2

60. P-value & Hyper geometric distribution
P1(# of red ball is 1) =
P0(# of red ball is 0) =
P3(# of red ball is 3) =
P2(# of red ball is 2) =
P(# of red ball ≤ 1)= P0 +P1
P(# of red ball ≥ 2)=1-(P0 +P1)
P(# of red ball ≥ k)=1-(P0 +P1+…+Pk-1)
N=7, F=4, C=3

61. Results: Details of a Group of Predicted Complexes
Information on the complexes that are of size 6 of the set generated using din=0.7, cpin=0.50 and non-overlapping mode.
Protein YDR425w of complex 19 is related to cellular transport and YIP1, YGL198w, YGL161c and GCS1 are related to vesicular transport. Hence, we predict the function-unknown protein YPL095c of this complex is a transport related protein most likely related to vesicular transport.

62. Conclusions
In this work, we present an algorithm to detect locally dense regions in undirected simple graphs.
The algorithm can be used to detect protein complexes in large protein-protein interaction networks or co-expressed gene clusters based on microarray data.
It can also be used for protein/gene function prediction by way of finding complexes/clusters in networks consisting of function known and function unknown proteins.
Also, DPClus can be applied to other networks where finding cohesive groups is an agenda.
The DPClus software is available at

63. 2. The DPClus Software
The DPClus software has been developed based on the proposed algorithm.
Md. Altaf-Ul-Amin, Hisashi Tsuji, Ken Kurokawa, Hiroko Asahi, Yoko Shinbo, Shigehiko Kanaya, “DPClus: A Density-periphery Based Graph Clustering Software Mainly Focused on Detection of Protein Complexes in Interaction Networks”, Journal of Computer Aided Chemistry , Vol.7, , 2006.
The DPClus software is available at

64. The DPClus Software
The main window of DPClus

65. Corresponding network List of edges
The DPClus Software
The input file format
AtpB AtpA
AtpG AtpE
AtpA AtpH
AtpB AtpH
AtpG AtpH
AtpE AtpH
Adjacency matrix
AtpA AtpB, AtpH
AtpB AtpA , AtpH
AtpH AtpB, AtpA, AtpG, AtpE
AtpG AtpH, AtpE
AtpE AtpG
Corresponding network
List of edges
Adjacency list

66. The DPClus Software Output file format
ClusterLength of cluster 1 is: 8
RpoA
RpoB
RpoC
Rsd
RpoZ
RpoD
RpoN
FliA
ClusterLength of cluster 2 is: 8
AtpH
AtpG
AtpB
AtpA
AtpF
AtpL
AtpE
AtpB(A)
ClusterLength of cluster 3 is: 5

67. The DPClus Software
Intra cluster edges are green and inter cluster edges are red
Nodes have been arranged by dragging

68. Hierarchical graph of the clusters
The DPClus Software
Click
Hierarchical graph of the clusters

69. The DPClus Software
Clustering of microarray data
Sample microarray data
To apply DPCcus, we need to convert this data to a network

70. The DPClus Software Experiment ID Gene-Gene correlation Genes
Select highly correlated gene pairs
At3g10060 At3g54150
At3g10060 At3g63140
At3g10060 At5g07020
Edges of a Network

71. The DPClus Software # of experiments 626 Threshold correlation 0.95
cp value 0.5
density value 0.9
Minimum cluster size 3

72. The DPClus Software Electron transport clusters Ribosomal protein
Photosynthesis clusters

73. Line Graphs
Given a graph G, its line graph L(G) is a graph such that
each vertex of L(G) represents an edge of G; and
two vertices of L(G) are adjacent if and only if their corresponding
edges share a common endpoint ("are adjacent") in G.
                
Graph G
Vertices in L(G) constructed
from edges in G
Added edges in L(G)
The line graph L(G)

74. The chemical graphs G1 and G2 are shown in Figure a,
Line Graphs
RASCAL: Calculation of Graph Similarity using Maximum Common Edge Subgraphs
By JOHN W. RAYMOND1, ELEANOR J. GARDINER2 AND PETER WILLETT2
THE COMPUTER JOURNAL, Vol. 45, No. 6, 2002
The above paper has introduced a new graph similarity calculation procedure for comparing labeled graphs.
The chemical graphs G1 and G2 are shown in Figure a,
and their respective line graphs are depicted in Figure b.

75. A star is transformed into a clique
Line Graphs
Detection of Functional Modules From
Protein Interaction Networks
By Jose B. Pereira-Leal,1 Anton J. Enright,2 and Christos A. Ouzounis1
PROTEINS: Structure, Function, and Bioinformatics 54:49–57 (2004)
Transforming a network of proteins to a network of interactions. a) Schematic representation illustrating a graph representation of protein interactions: nodes correspond to proteins and edges to interactions. b) Schematic representation illustrating the transformation of the protein graph connected by interactions to an interaction graph connected by proteins. Each node represents a binary interaction and edges represent shared proteins. Note that labels that are not shared correspond to terminal nodes in (a)
A star is transformed into a clique