Dense Subgraphs with Restrictions & Applications to Gene Annotations Graphs

Samir KhullerUniversity of Maryland

Joint Work withBarna Saha, Allie Hoch, Louiqa Raschid, Xiao-Ning Zhang

RECOMB 2010

Story of the collaborationLouiqa Raschid (BMGT)Life Sciences Data Mgt.

Samir Khuller (CS)Algorithms

Barna Saha (CS)Ph.D. student

Allie Hoch (CS)Undergrad

Xiao-Ning Zhang (Bio)User evaluation

TIME

TAIR Annotation Example

gene

annotations

AT1G15550GA4

GO:0016707 gibberellin 3-beta-dioxygenase activity

GO:0009686 gibberellin biosynthetic process

GO:0009739 response to gibberellin stimulus

GO:0009639 response to red or far red light

GO:0008134 transcription factor binding

GO:0010114 response to red light

PO:0019018 embryo axis

PO:0009046 flower

PO:0009005 root

PO:0009001 fruit

PO:0020001 ovary placenta

PO:0020148 shoot apical meristem

PO:0020030 cotyledon

PO:0009064 receptacle

PO:0003011 root vascular system

PO:0000014 rosette leaf

PO:0004723 sepal vascular system

PO:0009047 stem

PO:0020141 stem node

PO:0009009 embryo

PO:0004714 terminal floral bud

PO:0009025 leaf

PO:0007057 0 germination

PO:0007131 seedling growth

PO:0009067 filament

GO:0009740 gibberellic acid mediated signalling

GO:0005737 cytoplasm

GO-(gene)-PO tri-partite graph






GO:0008135 biological process

GO OntologyGO Ontology

PO:0019018 embryo axisPO:0009046 flower

PO:0009005 root

PO:0009001 fruit








PO:0009047 stem


PO:0009009 embryoPO:0004714 terminal floral bud

PO:0009025 leaf

PO:0009067 filament

Plant structurePO OntologyPO Ontology

Gene Annotation GraphGene Annotation Graph

Construct graphs for each gene using their GO, PO annotations

Combine the graphs of several genes into one single weighted graph

Gene 1

Gene 2

Gene 3

Gene 4

GO 1

GO 2

GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

Biologists would like to find patterns in gene annotation graphs – but these are huge!

Need to allow biologists some control over the kind of patterns that are computed

Would like to find biologically meaningful patterns Gene

1

Gene 2

Gene 3

Gene 4

GO 1

GO 2

GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

Node

Edge

AT1G15550GA4








PO:0009046 flower

PO:0009005 root

PO:0009001 fruit








PO:0009047 stem


PO:0009009 embryo


PO:0009025 leaf



PO:0009067 filament



GO-(gene)-PO tri-partite graph








PO:0009046 flower

PO:0009005 root

PO:0009001 fruit








PO:0009047 stem


PO:0009009 embryo


PO:0009025 leaf



PO:0009067 filament



GO-PO bipartite graph

Gene Annotation GraphGene Annotation Graph

Construct complete bipartite graph for each gene using their GO, PO annotations

Combine the bipartite graphs of several genes into one single weighted graph

GO 1

GO 2

GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

1

2 1

11

3

3

2

31

1

1

2

How can we extract knowledge? Cliques – these might give us some

biological information – but this is a stringent reqmt.

However clique finding is well known to be really hard (NP-hard, hard to approximate).

Why not look for “dense regions”? Note that the notion of density could be

defined for hyper-edges as well, but for our purposes this does not do as well.

5

3 4

2

7

6

1The density of {1,2,3,4,5,6,7} = 9/7 = 1.28

The density of {1,2,3,4} = 6/4 = 1.5

The densest subgraph is {1,2,3,4}.

How do we compute the densest subgraph?

Surprisingly, this can be solved optimally in polynomial time!

[Goldberg 84, Lawler 76, Queyranne 75]

Easily extends to weighted graphs.

1

sum of weights of edges in the induced subgraphGraph density = number of nodes in the induced subgraph

Dense Subgraphs in Gene Dense Subgraphs in Gene Annotation GraphAnnotation Graph A collection of GO-PO terms that appear together in the

underlying genes.

GO 1

GO 2

GO 3

GO 4

PO 1

PO 2

PO 3

PO 4

1

2 1

11

3

3

2

31

1

1

2

(GO3,PO1),(GO3,PO2),(GO3,PO4),(GO4,PO1),(GO4,PO2),(GO4,PO4) appear frequently in the 4 genes

Are all dense subgraphs biologically meaningful ?◦ How do we allow biologists to have some control over the

kind of dense subgraphs that are computed ?◦ Putting size constraints makes the problem intractable

immediately. Densest subgraph of size >=k. NP-hard, but can get 2 approximation [Khuller,

Saha] Densest subgraph of size <=k. NP-hard and no good approximations known


kind of dense subgraphs that are computed.◦ In fact we can impose both restrictions at the same time!

Restrictions in dense subgraph computation

Distance Restricted

Subset Restricted

GO terms and similarly PO terms that appear must be biologically related

Certain GO, PO terms must appear in the returned subgraph


kind of dense subgraphs that are computed ?


Distance Restricted

Subset Restricted

GO terms that appear in the densest subgraph must be close in the GO ontology graph and similarly for the PO terms

Distance threshold = 1 This means that some sets of nodes are not allowed to

coexist in the final solution: {GO1 ,GO2}, {GO1,GO4}, {PO1 ,PO4}, {PO1,PO2},{PO2,PO3,}.

The final solution is {GO2, GO3, GO4, PO2, PO4}, which has a density of .8.

GO1

GO2

GO3

GO4

PO1

PO2

PO1

PO3

PO4

PO2

PO3

PO4

GO2

GO1

GO3

GO4

For arbitrary ontology graph structure◦ NP Hard even to approximate it reasonably

Reduction from Independent set problem◦ Factor 2 relaxation of distance threshold is enough to get a

solution with density as high as the optimum Trees, Interval Graphs, Each edge participates in

small number of cycles◦ Polynomial time algorithm to compute the optimum


kind of dense subgraphs that are computed ?


Distance Restricted

Subset Restricted

Given a subset of GO, PO terms compute the densest subgraph containing them.

8

2 3 4 5 6

7

1 2 2 2 1 1

1 11 1

1

3

•This set must be in the graph: {5,6}

•Density of {1,2,3,4} = (3+2+2+2)/4 = 2.25– Doesn’t contain {5,6}

•Density of {5,6,7,8} = 6/4 = 1.5 (Satisfies subset requirement)

•Density of {1,2,3,4,5,6,7,8} = (2+3+2+2+1*7)/8 = 2.0 (Best answer)

Polynomial time algorithm to compute the optimum solution

A graph may contain multiple subgraphs of equal (or close to equal) density

Computing just one subgraph may not be sufficient Compute all subgraphs close to maximum density Extension of Picard and Queyranne’s result

◦ Polynomial time algorithm to find almost all dense subgraphs given the number of such subgraphs is polynomial in the number of vertices.

Can be extended to consider both distance and subset restriction

8

2 3 4 5 6

7

1 2 2 2 1 1

1 11 1

2

3

9

2

2

•Density of {1,2,3,4} = 9/8 = 2.25

•Density of {5,6,7,8,9} = 11/5 = 2.

•Density of {1,2,3,4,5,6,7,8,9} = 21/9 = 2.333

•The entire graph is the densest subgraph, but {1,2,3,4} and {5,6,7,8,9} are “almost” dense subgraphs

23

4

5

1 6

9

7

8

58

3

7

4

2

6

1

GO-OntologyPO-Ontology

Distance Threshold=2

2

3

4

5

1

6

9

7

8

2

3

4

5

1

6

7

8


Guess two nodes in each ontology that appears in the optimum solution and have maximum distance

23

4

5

1 6

9

7

8

58

3

7

4

2

6

1

GO-Ontology PO-Ontology

2

3

4

5

1

6

9

7

8

2

3

4

5

1

6

7

8


Compute all the nodes which are within distance threshold from both the guessed nodes

23

4

5

1 6

9

7

8

58

3

7

4

2

6

1


2

3

4

5

1

6

9

7

8

2

3

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5

1

6

7

8


In the gene-annotation bipartite graph consider only the chosen nodes and compute the densest subgraph

23

4

5

1 6

9

7

8

58

3

7

4

2

6

1


2

3

4

5

1

6

9

7

8

2

3

4

5

1

6

7

8



23

4

5

1 6

9

7

8

58

3

7

4

2

6

1


5

6

9

7

2

4

5



23

4

5

1 6

9

7

8

58

3

7

4

2

6

1


5

6

9

7

2

4

5


23

4

5

1 6

9

7

8

58

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1


5

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4

5

Proof of optimality:Any node not chosen can not be in the optimum solutionAll the nodes chosen are within distance threshold

Guess a small subset of nodes from the optimum Choose candidate nodes by considering distance from the

guessed nodes Compute the densest subgraph by restricting the gene

annotation graph to only the chosen nodes

Following Goldberg’s algorithm (not explained here for brevity), a new graph, or network, is created with “directed” edges. The new graph can be thought of as a network of pipes in which water can flow only one way in each pipe. If edges were “undirected” (as in the previous graph) water could flow in both directions.

The min cut is computed in order to find the densest subgraph. All nodes on the “source” side of the cut are part of the densest subgraph. (The source is where all the “water” flows from)

1 3

2 4

source sink

1

1

11

1

1

1

1

1

1

Original Graph:

1

2

3

5

2

1

2

3

5

2

sourcesink

7

7

9

6

4

7

Edges from source to original nodes: m’= sum of all edges in graph

Edge from original node i to sink: m’ + 2g – degree(i)

Since the source is not the only node on the source side of the cut, the guess is too low.

g = guess = 2

For this problem we modified Lawler’s method of finding densest subgraphs. Let’s assume that we have a graph in which we want to force {5,6} to be in the final solution.

The guess “g” is iteratively updated, as in Goldberg’s algorithm until the min cut is calculated and there is more than one possible solution, one contains just {s’ and s} and the other specifies the densest subgraph.

10 Photomorphogenesis genes

CIB5 CRY2 HFR1 COP1 PHOT1 PHOT2 HY5 SHB1 CRY1 CIB1

66 GO CV terms. 41 PO CV terms; 2230 GO-PO edges.

Generate distance restricted dense subgraph. GO distance = 2. PO distance = 3. Dense subgraph with 3 GO terms & 13 PO terms

Photomorphogenesis ExperimentPhotomorphogenesis Experiment

HFR1

COP1

PHOT1

PHOT2

HY5

13 PO CV terms 3 GO CV termsSet of 10 genes

CRY2

CIB5

SHB1

CIB1

CRY1

(partial) dense subgraph; 3 GO terms; 13 PO terms; 10 genes

0 annotation edges

8

26

12

13

13

12

13

2

13

Photomorphogenesis ExperimentPhotomorphogenesis Experiment

GO CV Terms PO CV Terms5634-nucleus:cellular-component 13-cauline leaf:plant structure 9010-seed:plant

structure

5794-Golgi apparatus;cellular-comp 37-shoot apex:plant struture 9025-leaf:plant structure

5773-vacuole:cellular-component 8034-leaf whorl:plant structure 9031-sepal:plant structure

9005-root;plant struture 9032-petal-plant structure

9006- shhot:plant structure 9047-stem:plant structure

9009-embryo;plant structure 20030-cotyledon:plant structure

20038: petiole:plant structure

5634-13 5634-37 5773-13 5773-37

HFR1 (AT1G02340) 1 0 0 0

CRY2 (AT1G04400) 1 1 1 1

CIB5 (AT1G26260) 1 1 0 0

COP1 (AT2G32950) 1 1 0 0

PHOT1 (AT3G45780) 0 0 1 1

CRY1 (AT4G08920) 1 1 0 0

SHB1 (AT4G25350) 1 0 0 0

HY5 (AT5G11260) 1 1 0 0

PHOT2 (AT5G5840) 0 0 0 0

CIB1 (AT4G34530) 0 0 0 0

Potential Discovery

Genes CRY2 and PHOT1 are both observed in the dense subgraph with the following two GO and PO combinations: 5773: vacuole: cellular_component 13: cauline leaf; plant_structure 37: shoot apex; plant_structure (5773, 13) (5773, 37) This patterns has not been reported in the literature. Two independent studies [Kang et al. Planta 08, Ohgishi PNAS 04] have suggested that there may be some functional interactions between the members of PHOT1 and CRY2 in vacuole

Validation - Generate subset restricted dense subgraph. Add 10 control genes. 2 GO terms: 5634 and 5773. 2 PO terms: 13 cauline leaf; plant_structure and 37 shoot apex. Dense subgraph with 2 GO terms, 12 PO terms User validated that the missing PO term and additional control genes and edges were acceptable changes from the distance restricted dense subgraph to the subset restricted dense subgraph.

Photomorphogenesis Photomorphogenesis Experiment with Control GenesExperiment with Control Genes

Identifying dense subgraphs with distance and subset restriction may help in identifying interesting biological patterns

Potential Applications in other domains:◦ Distance restricted dense subgraph for community detection◦ Subset restricted dense subgraph in PPI network for deriving protein

complexes Ranking almost all dense subgraphs

Documents

Dense Subgraphs with Restrictions & Applications to Gene Annotations Graphs