Download Improving Biological Significance of Gene Expression Biclusters

Improving Biological Significance of Gene Expression
Biclusters with Key Missing Genes
∗
Shufan Ji
Xing Tian
Jin Chen
Department of Computer
Science
Beihang University
Beijing, China 100191
Department of Computer
Science
Beihang University
Beijing, China 100191
[email protected]
[email protected]
Department of Energy Plant
Research Laboratory
Department of Computer
Science and Engineering
Michigan State University
East Lansing, MI
[email protected]
ABSTRACT
1.
Identifying condition-specific co-expressed gene groups is critical for gene functional and regulatory analysis. However,
given that genes with critical functions (such as transcription factors) may not co-express with their target genes, it is
insufficient to uncover gene functional associations only from
gene expression data. In this paper, we propose a novel integrative biclustering approach to build high quality biclusters
from gene expression data, and to identify critical missing
genes in biclusters based on Gene Ontology as well. Our
approach delivers a complete inter- and intra-bicluster functional relationship, thus provides biologists a clear picture
for gene functional association study. We experimented with
the Yeast cell cycle and Arabidopsis cold-response gene expression datasets. Experimental results show that a clear
inter- and intra-bicluster relationship is identified, and the
biological significance of the biclusters is considerably improved.
Gene expression clustering is a key step towards gene
functional and regulatory analysis at the genome level [36].
Genes that have similar expression patterns under certain
subsets of experimental conditions can be grouped together,
and are considered to be functionally related [14, 8, 41].
Biclustering analysis, that identifies condition-specific coexpressed patterns from gene expression data, is essential to
better understanding of the biological interactions between
genes that are not apparent when clustered globally [7, 9].
In the last decade, many biclustering algorithms have been
proposed to simultaneously cluster genes and experimental
conditions together [7, 39, 40, 42, 13, 19, 21, 6, 9]. However,
it has been shown that although these biclustering methods
can capture gene groups that function together under a subset of experimental conditions, there is a lot of potential to
improve towards uncovering gene regulation relationships,
mainly because of the following two reasons:
Categories and Subject Descriptors
H.3.3 [Information Search and Retrieval]: Clustering;
J.3 [Life and Medical Sciences]: Biology and genetics
General Terms
INTRODUCTION
1. Biclustering based on gene expression data would inevitably miss certain functionally related genes, mainly
because some functionally related genes do not coexpress. For example, transcription factors (TF) usually do not co-express with their target genes, and if a
key TF is missing from a bicluster, the functional association among the related genes is difficult to identify.
Algorithms
Keywords
bi-clustering, Gene Ontology, gene expression, missing gene,
biological network
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http://dx.doi.org/10.1145/2808719.2808747.
2. It is the inter- and intra- cluster relationship rather
than the clustered genes that contributes towards better interpretation of the overall picture of gene regulatory networks [17, 22, 32, 2]. Although hierarchical
biclustering algorithms[13, 21, 6] can reveal the interbicluster relationships, to our knowledge no one has
extensively studied both the inter- and intra-bicluster
relationships.
In summary, some functionally related genes may be missing in a bicluster. As a result, the intra-bicluster connections
are weakened with reduced functional association. The gene
regulatory networks built upon such biclusters would then
be incomplete and disconnected.
In order to generate and interpret biological hypotheses
on gene transcriptional regulations, a biclustering algorithm
that can 1) build high quality biclusters from gene expression
data, 2) reveal inter- and intra-bicluster relationships, and
1. New Concept of Biclustering. We picture the
whole skeleton of biclusters. The proposed concept
of bicluster skeleton (i.e. inter- and intra- cluster relationships) could reveal the hierarchical subset/superset
of a gene bicluster and the relationships among genes
within a bicluster. Moreover, we improve the bicluster
quality by adding new genes and new connections into
biclusters to reinforce the intra-bicluster relationships.
root
leaves
seeds
Experimental results on Yeast cell cycle [7] and Arabidopsis cold-response microarray datasets [37] show that SKB
could build a clear structure of the inter- and intra-bicluster
relationship, consequently the mined biclusters have higher
biological significance than existing methods. Comparing
with existing biclustering algorithms, we have made four
main contributions:
cluster graphs
2. Novel Way to Employ Domain Knowledge. It
is generally difficult to integrate mining data with domain knowledge in unsupervised learning models. Work
to date either combines domain knowledge with mining
data to form a complicated feature space, or takes domain knowledge as a post-processing filter [23, 30, 35,
12, 4]. However, the former approach suffers from the
complicated combination of heterogeneous features, while
the latter approach results in data loss. In this paper,
we employ the domain knowledge as an independent
data space, onto which the mining results are mapped
for further improvement.
Figure 1: Framework of SKB. In the upper part,
SKB mines biclusters from gene expression data and
reveals the inter-bicluster relationship with a hierarchical tree. In the tree, each node is a bicluster, shown with gene expression changing trend. In
the lower part, SKB reveals the intra-bicluster relationships by identifying missing genes (yellow) that
could bridge the functionally distinct genes in each
bicluster, so as to uncover the otherwise hidden gene
associations.
3) identify missing genes and missing gene associations, is
highly expected.
In this paper, we propose to identify gene expression biclusters with a novel biclustering algorithm called SKeleton
Biclustering (SKB). SKB not only builds biclusters and reveals bicluster skeleton (inter-bicluster and intra-bicluster
relationships), but also identifies relevant missing genes that
can bridge the functionally distinct biclustered genes, in order to uncover the otherwise hidden gene associations within
biclusters. As shown in Figure 1, SKB has three phases. In
phase one, a hierarchical biclustering method QHB [13] is
adopted to generate biclusters from gene expression data,
which reveals the hierarchical inter-bicluster relationship by
a dendrogram. The inter-bicluster relationships are revealed
by a hierarchical tree. In phase two, each bicluster is converted into a graph (made up of several disconnected subgraphs) by linking the genes based on their functional similarity, which is calculated with Gene Ontology (GO) annotation [1, 10]. Gene connections within the bicluster is
defined as the intra-bicluster relationship. In phase three, a
graph mining algorithm is proposed to efficiently add new
genes, if any, to each bicluster, to re-connect the disconnected subgraphs generated in phase 2, in order to enhance
the intra-bicluster relationship. Therefore, the overall functional connectivity of the bicluster is increased.
3. Improved Biological Significance. SKB improves
the biological significance of biclusters by reinforcing
the gene functional associations. Different from most
existing biclustering algorithms, our method can identify key missing genes that express differently but function associatively with biclustered genes, thus delivers
a complete skeleton of biclusters.
4. Efficient Algorithm. Both gene biclustering and
missing gene identification are computationally expensive. We introduce a novel algorithm to achieve the
goal efficiently.
2.
RELATED WORK
The biclustering algorithm on gene expression data was
first introduced by Cheng and Church 2000 [7] to simultaneously cluster both genes and experimental conditions, which
captures the coherence of a subset of gene under a subset of
experimental conditions [7]. A two-phase iterative strategy
is used to identify biclusters in the data: 1) remove rows
and columns from the gene expression matrix to compose
a submetrix with the mean squared residue score below a
threshold δ; 2) add the deleted rows and columns to the
submatrix as along as the mean squared residue score does
not exceed the threshold. δ-cluster model [40] was later proposed to accelerate the mining process with a move-based
algorithm FLOC, which randomly generates initial biclusters and iteratively optimizes biclusters by adding/deleting
genes or conditions. [39] further accelerated the biclustering
process by employing a depth-first algorithm to mine pClusters with a user specified minimum pScore. However, those
biclustering algorithms fail to generate deterministic mining
results.
A deterministic algorithm DBF was proposed [42] to further improve the quality and efficiency of biclustering. DBF
employs a frequent closed pattern mining algorithm to generate “good seeds” with better pattern similarity and then
refines the seeds by adding genes and conditions to achieve a
low MSR score as well as large bicluster volume. A minimum
row variance threshold is set to remove biclusters with trivial
changes in trends. In particular, on a binary gene expression matrix, we can view the biclustering formulation as the
search for all maximal bicliques in a bipartite graph, where
the nodes are genes or conditions and an edge connecting a
gene gi and a condition cj indicates gene gi responses in the
condition cj [38]. Subsequently, the biclustering problem
on a binary matrix can solved by the backtracking BronKerbosch algorithm for maximal clique enumeration [5].
While these algorithms can generate biclusters with similar trends, they are limited in several ways. First, these
schemes typically employ a similarity score (e.g., MSR and
pScore) to determine the quality of biclusters, which, however, cannot adequately capture the trend consistency of
biclusters. Second, these algorithms usually generate biclusters based on selected “seeds” that cover only a small part
of the whole dataset. As such, interesting patterns may be
missed resulting in loss of relevant information. Third, the
seed improvement process follows the hill-climbing paradigm,
and hence, is computationally expensive. To overcome these
limitations, a quick hierarchical biclustering algorithm called
QHB was proposed to generate consistent-trends biclusters
(a subsets of genes homogeneously expressed in a subset of
conditions) and produces a hierarchical tree to reveal the
inter-bicluster relations [13].
More recently, model-based biclustering algorithms [20,
21], which approximate the data density by a mixture of
Gaussian distributions, adopt a hierarchical structure of biclusters to define a flexible distribution of data density. In
[6], a graphical model was adopted to identify hierarchical
biclusters. The resulting tree structure of conditions, where
each node corresponds to a bicluster, provides an interpretation basis for the grouping [6]. LateBiclustering identifies biclusters with time lags, thus recovering genes that are
missed because they co-expressed with delays [11]. DeBi improves the biological significance of biclusters by discovering
maximum size homogeneous biclusters, in which each gene
is strongly associated with a subset of samples, providing
functionally more coherent gene sets compared to standard
clustering or biclustering algorithms [33].
Note that biclustering methods solely based on gene expression data would inevitably leave some functionally related genes out of the biclusters [28], and thus weaken the
overall gene relationships. Hence, incorporating biological
domain knowledge in biclustering [23, 30, 35] has been recognized as a reliable way to enhance the interpretability of
biclustered genes. Gene Ontology (GO) [1, 10] has now
been accepted as the de facto language for the attribute
description of biological entities in three key domains that
are shared by all organisms, namely molecular function, biological process and cellular component [29, 24, 27, 26, 25].
In each of these domains, the corresponding GO is structured as a directed acyclic graph to reflect the complex
hierarchy of biological terminologies. To incorporate Gene
Ontology to increase biclustering performance, we propose
a biclustering algorithm called Skeleton Biclustering (SKB)
that could build high quality biclusters with clear inter- and
intra-bicluster relationships, as well as identify missing genes
and missing gene associations.
Table 1: Original Data Matrix A
A c1
c2
c3
c4
g1 2.4
2.95 2.45 2.99
g2 1.95 1.71 1
0.29
g3 0.5
1.1
0.38 1.56
Table 2: Slope Angle Matrix A0
A0 c1 c2
c2 c3
c3 c4
g1 28.81◦
−26.57◦ 28.37◦
g2 −13.50◦ −35.37◦ −35.37◦
g3 30.96◦
−35.75◦ 49.72◦
Table 3: Binary Matrix A00 (BinningT hreshold =
26.5◦ )
A00 c1 c2 (c1 c2 )0 c2 c3 (c2 c3 )0 c3 c4 (c3 c4 )0
g1
0
1
1
0
0
1
g2
0
0
1
0
1
0
g3
0
1
1
0
0
1
3.
METHOD
In this section, we introduce the three phases of SKB respectively. The overall framework of SKB is in Figure 1.
3.1
SKB phase 1, inter-bicluster relationship
identification
We employ an efficient top-down hierarchical biclustering algorithm QHB [13] (without modification) as the first
phase of SKB to identify biclusters from gene expression
data. We generate biclusters with consistent rising / falling
trends and produce a hierarchical tree to reveal the interbicluster relationship. Here we adopt QHB algorithm in
that it could deliver a well-organized hierarchical tree with
each node representing a bicluster, from which we could easily identify the subset/superset and the nearest neighbors of
a bicluster, which will facilitate gene family studies. Now
that we will briefly describe the algorithm in two steps with
a running example.
Step 1: Trend Separation
In Step 1, different changing trends of gene expression data
are separated. Firstly, the gene expression matrix A (Table 1), where rows represent genes and columns represent
experimental conditions, is transformed into a slope angle
matrix A0 (Table 2) to reflect different changing trends when
the experimental conditions change. Then A0 is binned
into a binary matrix A00 (Table 3) where the rising(01)
and falling(10) trends are separated into two consecutive
columns; trends with trivial change(00) are filtered out.
Step 2: Matrix Partition
In Step 2, genes with similar changing trends (rising and
falling simultaneously) are grouped into biclusters through
a hierarchical partition tree in Figure 2. From matrix A00 ,
the partitioning process constructs a hierarchical tree where
all valuable upper level information is kept intact and propagated into the lower level. This helps in preventing any
information loss. The binning and partitioning ensures that
genes with consistent trends under condition transitions are
kept together in the same biclusters while genes with incon-
A''
g1
g2
g3
A''
g2
g3
c1c2 (c1c2)' c2c3 (c2c3)' c3c4 (c3c4)'
0
1
1
0
0
1
0
0
1
0
1
0
0
1
1
0
0
1
c1c2 (c1c2)' c2c3 (c2c3)' c3c4 (c3c4)'
0
0
1
0
1
0
0
1
1
0
0
1
NULL
A''
g2
g3
NULL
c2c3
1
1
c3c4
1
0
NULL
A''
g1
g3
A''
g1
g2
g3
Gene Distance Measure
Biologically, many genes are involved in multiple cellular
processes and therefore labeled with more than one GO
term. Thus, the distance between two genes is defined as
the minimum dissimilarity between any two GO terms annotating the two genes [34]. Let Tgi and Tgj be the set of
GO terms annotating gene gi and gj respectively, the gene
distance measure is defined as:
(c1c2)' c2c3 (c3c4)'
1
1
1
0
1
0
1
1
1
(c1c2)' c2c3 (c3c4)'
1
1
1
1
1
1
NULL
Figure 2: Matrix (A00 ) Partition. Genes with similar
changing trends (rising and falling simultaneously)
are grouped into biclusters through a hierarchical
partition tree, where all valuable upper level information is kept intact and propagated into the lower
level.
sistent trends are separated into different biclusters. According to QHB, the biclustering results can be further refined to reflect the similar magnitudes of change by the same
binning and partitioning procedures, although only different
changing trends are considered here for simplicity.
This scheme helps maintain the bicluster quality effectively, while the simultaneous grouping of multiple genes
and conditions makes the process rather efficient. Moreover,
the hierarchical partition tree reveals to users a clear interbicluster relationships. Based on the tree, users can navigate
up or down the tree to get a more general or detailed insight
into biclusters.
3.2
SKB phase 2, gene distance measure and
bicluster graph generation
In this phase, each gene expression bicluster (on the leaf
level of the hierarchical biclustering tree) is converted into an
undirected graph by linking functionally associated genes together. In general, two genes are considered to be functionally associated if they share at least one biological feature.
Here we adopt GO annotation for the function association
measure, defined as gene distance. Thus, genes satisfying a
certain distance threshold are linked together. Clearly, the
algorithm could be easily extended to adopt other domain
knowledge as gene distance measure.
GO Term Similarity Measure
As the GO terms are not equally informative [29], for each
gene set, specific weights are assigned to the GO terms [18]:
the weight of a GO term is defined as the occurrence ratio
of the GO term (with all its descendants) to the total terms
in the gene set. That is, for any GO term t ∈ T , where
T is the full set ofPGO terms, the term weight is defined as
w(t) = (f req(t)+ d∈Dt f req(d))/N , where f req(t) denotes
the number of occurrences of GO term t; Dt is the set of
descendants of GO term t in T ; and N is the total number
of term occurrences.
Given two GO terms ta and tb and their weights w(ta )
and w(tb ), their term similarity score sim(ta , tb ) is defined
in Equation 1 [31]. Note that tab is the nearest common
ancestor of ta and tb , and w(tab ) is its weight.
sim(ta , tb ) =
2 × ln w(tab ) × (1 − w(tab ))
ln w(ta ) + ln w(tb )
(1)
d(gi , gj ) =
min
ta ∈Tgi ,tb ∈Tgj
(1 − sim(ta , tb ))
(2)
Bicluster Graph Generation
To reveal the biological relationship among the biclustered
genes, a bicluster S is converted into an undirected bicluster
graph G(V, E), with the vertex gi ∈ V representing gene gi ,
and the edge e(gi , gj ) ∈ E indicating that the gene distance
d(gi , gj ) is smaller than the predefined distance threshold
σ. The graph G is either connected or made up of several
disconnected components, each of which is a connected subgraph of G (see Algorithm 1 line 4-9).
3.3
SKB phase 3, adding new genes to improve
biological significance
Without key functional genes, the originally associated
genes in resulting bicluster graph might be separated into
different components. As for the example in Figure 3, without gene gx (function f1 , f2 ), gene ga (function f1 , f3 ) is
separated with gene gb (function f2 , f4 ) due to non-similar
functions (long gene-to-gene distance). gx could bridge the
gap of ga and gb , in that it shares the same function f1 , f2
with ga , gb respectively. Therefore, given gene ga and gb in
a bicluster graph G(V, E), there might exist a third gene gx ,
such that d(ga , gb ) > d(ga , gx ) + d(gb , gx ). According to this
property, adding new genes to the bicluster graph might help
to connect the otherwise disconnected components, leading
to the enriched functional associations within biclusters.
The search space of new genes, which is set as the whole
genome in our study, could be defined according to different
applications. Given the whole gene set I, let S be a biclustered gene set, G(V, E) be its bicluster graph, and C =
{C1 , C2 , . . . , Cn } be the disconnected component set of G.
Two disconnected components Ci (Vi , Ei ) and Cj (Vj , Ej ) are
defined as “connectable” if and only if there exists gx ∈ I − S
such that min(d(gi , gx ) + d(gj , gx )) < σ (the gene distance
threshold), where gi ∈ Vi and gj ∈ Vj .
The strategy to identify new genes is to search for the
smallest set of new genes that connect all the connectable
components in a bicluster graph (see pseudocode in Algorithm 1 line 10-22). We decided to identify the smallest set
of new genes because of three reasons. First, we notice that
owing to incomplete biological knowledge in GO annotation,
introducing too many new gene could cause the bicluster to
be biased toward known information. Specifically, if two
genes share the same function that is not yet known or annotated, the distance between them will be artificially large.
Therefore, we decided to add the minimum number of new
genes as a constraint to weaken such bias towards known information. Second, by minimizing the number of new genes
to be included, we are maximizing the number of components each new gene connects to. This results in new genes
that are more informative. Third, the objective of adding
gx
{f1, f2}
Ci
Cj
ga
gb
{f1, f3}
{f2, f4}
cluster graph G
Figure 3: Effect of Key Gene. Adding key genes to
a bicluster graph may help connect the otherwise
disconnected components, leading to the enriched
functional associations within biclusters.
Algorithm 1 SKB phase 2 and 3
1: procedure AddNewGenes(T, I, σ)
2:
Input: S - bicluster genes; T - the full set of GO
terms; I - the full set of genes in gene expression data;
σ - gene distance threshold;
3:
Output: Vx - new gene set; G0 - the new cluster
graph;
4:
V = ∅; E = ∅;
. Initialization
5:
for each gene pair gi , gj ∈ S (i 6= j) do
.
Bicluster graph generation
6:
if getDistance(gi , gj , T ) < σ then
7:
V = V ∪ {gi , gj }; E = E ∪ {< gi , gj >};
8:
end if
9:
end for
10:
G = (V, E);
11:
C0 = getComponent(G);
12:
if |C0 | = 1 then . No need to insert missing genes
for a connected graph
13:
Vx = ∅; G0 = G;
14:
else
15:
C = ∅;
16:
for each component pair Ci , Cj ∈ C0 do
17:
if (Connectable(Ci , Cj ) = true) C = C ∪
{Ci , Cj };
18:
end for
19:
(Vx , Ex ) = getExtraGenes(C);
. Identify
missing genes
20:
G0 = (V ∪ Vx , E ∪ Ex );
21:
end if
22:
return Vx and G0 ;
23: end procedure
Algorithm 2 getExtraGenes(C)
1: Input: S - bicluster genes; T - the full set of GO terms;
I - the full set of genes in gene expression data; σ - gene
distance threshold; C - the set of connectable components; c - a constant large number;
2: Output: Vx - new gene set; Ex - edge set that connect
vertices in Vx & V ;
3: Vx = ∅; Ex = ∅; Λ = ∅; Cx = ∅;
4: for each gene g ∈ I − S do . Number of components g
is connectable to
5:
N (g) = getConnComp(g, C, σ);
6:
if (N (g) ≥ 2) Λ = Λ ∪ {g};
7: end for
8: while Λ 6= ∅ do
9:
remove g with max N (g) from Λ;
10:
Vx = Vx ∪ {g}; Cx = Cx ∪ {g};
11:
for each component Ci ∈ C that g connects to do
12:
gi = getM inDist(Ci , g); Ex = Ex ∪ {< g, gi >};
13:
Cx = Cx ∪ Ci ; remove Ci from C;
14:
end for
15:
C = C ∪ {Cx }; Cx = ∅;
16:
for each gene g ∈ Λ do
17:
N (g) = getConnComp(g, C, σ);
18:
if (N (g) < 2) remove g from Λ;
19:
end for
20: end while
21: return Vx and Ex ;
For a given bicluster S, instead of testing every subset of
I − S exhaustively, which is a NP problem, we develop a
heuristic algorithm to efficiently identify the local optimized
smallest new gene set. Let Vx be the initial new gene set.
The algorithm to identify Vx consists of two steps. First,
candidate genes in I − S − Vx are sorted in descending order according to the number of components that it could
connect in C. Candidate genes that could connect less than
two components are ignored (see Algorithm 2 line 4-7). Second, the top candidate gene is added to Vx , connecting as
many components in C as possible. Then the whole process
is repeated until all components in C are connected or no
more candidate gene exists (see Algorithm 2 line 8-20). This
heuristic algorithm has polynomial time complexity.
For example, given a cluster graph with 9 disconnected
components C0 , C1 , . . . , C8 in Figure 4, and five new gene
candidates g1 , g2 , . . . , g5 . In each iteration, the numbers of
components that each gene could connect are listed in Table 4. As for this example, g5 is firstly ignored since it cannot
connect any two components. Meanwhile, g1 is added to Vx
in that it connects the most number of the components, i.e.,
6. In the second iteration, the number of connections for
each remaining gene candidate is recalculated after g1 connects C0 , C1 , C2 , C3 , C5 , C6 together. Similarly, g3 is added
to Vx , connecting together two more components C7 , C8 with
C5 . And finally, g4 is added to connect C4 with the others.
As such, the whole process terminates with Vx = {g1 , g3 , g4 }
to connect all the components together.
4.
new genes is to understand the relationships among the biclustered genes, and too many new genes may reduce the
gene expression coherence.
EXPERIMENTAL RESULTS
We performed experiments on two datasets, the Yeast cell
cycle microarray data [7] and the Arabidopsis cold-response
microarray data [37], with two GO categories, molecular
C0
C8
g5
C2
g1
Gene Ontology
category
Biological Process
Molecular Function
Biological Process
Molecular Function
new genes
per cluster
5.1
2.5
9.9
7.6
FOS
QHB Rand
2.68
2.51
1.20
1.09
8.42
2.16
8.12
2.90
C3
C6
g2
C5
Arabi
-dopsis
Yeast
g3
g4
C4
Process
C7
Table 5: SKB performance summary
dataset
C1
transcription
transport
Table 4: Number of connections for candidate new
genes
Candidate
The value of N(g)
new gene
Iter 1 Iter 2 Iter 3 Iter 4
g1
6X
g2
4
1
g3
3
3X
g4
2
2
2X
g5
1
-
function (MF) and biological process (BP) adopted as domain knowledge [10]. Note that the proportion of functional
annotations, esp. BP, are not complete for most genomes.
For example, only 37% of the Arabidopsis genes have annotations with experimental evidence [15], leading to challenges in using GO in gene clustering. We tested the performance of SKB on both rich annotated genome (yeast)
and sparsely annotated genome (Arabidopsis). The results
showed that the performance of SKB is robust.
The Yeast microarray dataset contains 2884 genes whose
expressions are altered during cell cycle under 17 time points.
Among them, 1649 genes have at least one MF annotation, and 1973 genes have at least one BP annotation. The
Arabidopsis microarray dataset contains 2255 cold-response
genes under 6 time points with cold treatment at 4◦ C. Among
them, 1142 genes have at least one MF annotation, and 975
genes have at least one BP annotation. Only non-IEA GO
annotations are used in the experiments.
4.1
Results of SKB
In SKB phase 1 of hierarchical biclustering, we mined biclusters and inter-cluster relationships, requiring that each
bicluster has at least 50 genes and 5 conditions. For the
Yeast dataset, 114 biclusters were mined with average 56.7
genes on each bicluster. For the Arabidopsis dataset, 44
biclusters were mined with average 55.0 genes on each bicluster. By drawing the hierarchical biclustering trees, the
inter-cluster relationships are revealed.
In SKB phase 2 of bicluster graph generation, we set distance thresholds σ for each gene expression dataset under
different GO categories, such that for any edge in a bicluster
graph, the false discovery rate of its distance value is smaller
DNA/RNA binding
transporter activity
Component
Figure 4: Illustrative example for new gene identification.
Function
TF activity
nucleus
New genes
cell wall
All cold-response genes
0
5
10
15
20
25
30
Percentage
Figure 5: The functional characterization distribution of the new genes.
than 0.01. Hence, all the edges in each bicluster graph are
statistically sound. With the given distance thresholds, we
convert all of the 114 Yeast and 44 Arabidopsis biclusters
into bicluster graphs. Note that all of the un-annotated
genes are removed from the bicluster graphs before further
processing.
In SKB phase 3 of new gene identification, a set of new
genes are identified for each bicluster (see details in Table 5).
On average, for Yeast biclusters, 7 and 10 new genes were
found per cluster with MF and BP annotations. For Arabidopsis biclusters, 3 and 5 new genes were found per cluster
with MF and BP annotations.
4.2
Performance evaluation
The performance of SKB was tested on two ways: 1) a GO
based measure on the identification of functionally related
missing genes, and 2) network connectivity on the identification of hidden key genes. In addition, the biological
significance of the new genes were tested with functional
characterization.
Effectiveness of SKB on the identification of functionally related missing genes
To qualitatively show the biological significance of a gene
cluster graph, we define the Functionally Overall Similarity
measure (FOS) in Equation 3:
|V |
X
F OS(G) = 1 −
f (gi , gj )
i,j=1
1
|V |(|V
2
| − 1)
(3)
Disease
Signaling
Unknown
TFs
CPK28
LHY1
Circadian
Rhythm
CCA1
MPK3
Cell Expansion
CBF1
Cold regulators
& Targets
CBF3
CBF
CBF
CBF2
Figure 6: Case study. Red colored are clustered genes, and yellow colored are new genes. To study why
the different processes co-express under cold treatment and how these processes may work together, SKB
introduces 7 new genes, in which MPK3 acts as a central node to bridge C1 with other components, providing
clues for how the processes may be functioning together in cold conditions.
8
d(gi , gj ) X
if< gi , gj >∈ E
>
>
<
min
d(u,
v)
if< gi , gj >∈
/ E & Φ 6= ∅
f (gi , gj ) =
φ∈Φ(gi ,gj )
>
(u,v)∈φ
>
:
1
otherwise
(4)
where Φ(gi , gj ) denotes the set of paths connecting gi and
gj ; f (gi , gj ) is the gene distance value if gi and gj are directly connected, or the sum of distance along the shortest
path connecting gi and gj if such path exists. If no such
connections exist between two genes, the value of f (gi , gj )
is 1.
The score of FOS indicates the overall functional similarity of a bicluster using GO annotation. The value close to
1 means that, for most of the genes in cluster graph G, the
overall pair-wise similarity of the connected is high, or the
similarity value along the path to connect the gene pair is
high. In contrast, the value of F OS(G) close to 0 means
most of the genes are dissimilar from each other.
To evaluate the effectiveness of SKB in identifying functionally related missing genes, we compared SKB against
QHB on FOS scores of the resulting biclusters. We chose
QHB to compare because that i) QHB performs well in mining biclusters with consistent trends and trends with similar
degrees of fluctuations; ii) such comparison shows the ef-
fectiveness of the new genes since SKB generates the same
biclusters as QHB. To evaluate SKB’s strategy to identify
new genes, we also compared SKB with a random selection
process, in which we randomly choose the same number of
new genes as SKB does from the candidate gene set and
add them to each bicluster. Experimental results in Table 5
show that the FOS scores of the biclusters are considerably
increased comparing to QHB and the random selection. For
example, by introducing an average of 5.1 new genes to the
Arabidopsis biclusters with BP annotations, comparing to
QHB and the random selection, the FOS score is increased
by 2.67 and 2.51 times, respectively.
Furthermore, in order to verify the performance of SKB
independent to GO annotation, we adopt 11 known cold
related motifs [3] and test their existence in the Arabidopsis cold-response genes. Experiment shows that 80.0% new
genes share the same cold motifs with the biclustered genes
with BP annotation. In contrast, randomly picked new
genes only share 47.2% same cold motifs with the biclustered genes. The results correlate well with our biological
knowledge that the new genes should have the same cold
motifs as the biclustered genes that they connect, since all
the biclustered genes are co-expressed under cold treatment.
Graph connectivity on the identification of hidden key
genes
We further tested whether the selection of new genes is
significantly different from random with a leave-k-out approach. The SKB phase 2 on the Arabidopsis dataset generated a bicluster graph with 975 nodes. In the graph, we
identified and then deleted 255 key genes with the minimal
cut algorithm [16]. In addition, we removed 56 marginal
genes with the lowest node betweenness scores. This leavek-out process separated the whole network into 25 disconnected subnetworks. Let the candidate new genes be all the
removed genes, we tested whether the SKB phase 3 is able
to re-discover any bridge genes to connect the disconnected
components in the resulting graph. The results showed that
SKB added 13 missing genes, all of which are the bridge
genes, and thus reduced the number of disconnected subgraphs from 25 to 7. In contrast, a random addition of the
same amount (13) of new genes from the candidate gene set
results in 22.0 disconnected subgraphs (the average of 1000
repetitions, with standard deviation at 1.1). The results
indicate that the SKB results are far from random (t-test
p-value< 0.001).
4.3
Functional characterization of the missing genes in Arabidopsis
The functional characterization of the new genes for Arabidopsis shows in Figure 5 that there is an enrichment of TFs
in the new genes compared to all the genes in the microarray data. TFs are regulate transcriptions and are generally
located in nucleus. A TF usually does not co-express substantially with its target genes. Hence, such TFs are usually
missing from biclusters based on expression data. The significant and consistent rise of TFs in the new gene set from
the highly noisy background on all the three GO categories
indicates that SKB is biologically sound.
A case study on Arabidopsis dataset is shown in Figure 6.
An Arabidopsis bicluster has 26 annotated genes (red). Using BP annotation, the cluster graph is formed with one
large component belonging to the cold regulatory process,
denoted as C1 , and the other 10 small components. These
11 components are disconnected because they are conceptually dissimilar based on GO annotation. To study why
the different processes co-express under cold treatment and
how these processes may work together, SKB introduces 7
new genes (yellow) that can connect the biclustered genes, in
which M P K3 acts as a central node to bridge C1 with other
components, providing clues for how the processes may be
functioning together in cold conditions.
5.
DISCUSSION AND CONCLUSIONS
In this paper, we introduce SKB, a method to effectively
improve the biological significance of gene expression biclustering. The delineation of functional relationships and incorporation of such missing genes may help biologists to discover biological processes that are important in a given study
and provide clues for how the processes may function together. Experimental results show that SKB can reveal the
inter- and intra-bicluster relationships efficiently and accurately, and greatly improve the biological significance of the
biclusters. Note that, SKB achieves better performance on
yeast than on Arabidopsis, probably because of the different
ratio of Gene Ontology annotations of the two organisms.
It may be argued that instead of designing a new algorithm to add new genes, a connected cluster graph may be
easily obtained by simply increasing the distance threshold.
To address this question, we increased the distance thresholds until the same number of components as those connected by SKB was reached. To reach this point, the distance threshold was increased on average from 0.122 to 0.665
for the Arabidopsis dataset with BP annotations. Consequently, the resultant cluster graphs are cliques or clique-like
dense graphs, in which the edges are not statistically significant and numerous weak gene associations are present. By
contrast, the distance threshold in SKB is constantly low
and all the edges are guaranteed to be statistically significant, revealing that SKB can not only recognize functional
modules in biclusters, but also increase the intra-cluster
relationships and filter weak gene associations effectively.
Therefore, SKB is significantly better on graph representation and noise control than simply increasing the distance
threshold.
It may also be argued that the effort to introduce new
gene can be replaced by simply joining two related biclusters because the biclusters are highly overlapped and new
genes found for one bicluster may already exist in another
bicluster. To address this question, we tested how many
new genes identified by SKB were already included in the
biclusters. For Arabidopsis, only 12 out of 50 new genes
are overlapped with biclustered genes using BP annotation.
This indicates that the majority of the new genes cannot
be captured by joining related biclusters. Therefore, SKB
is more effective than joining existing biclusters because it
finds considerably more new genes.
In the framework of SKB, we chose QHB because it is
effective in identifying hierarchical inter-bicluster relationships, suitable for our skeleton bicluster expectation. With
simple extension, SKB can adopt any biclustering method
as its first step. We will extend the capability of SKB in two
ways in the future. First, heterogeneous domain knowledge,
including protein motifs, cis-elements, sequence alignment
and others, will be utilized to improve the bicluster performance in addition to the GO annotations. Second, SKB will
be further extended to enhance the performance of a gene
search engine by grouping functionally similar genes, recommending additional genes that are not in the search result,
and visualizing the search result graphically.
Competing interests
[13]
[14]
none.
Authors’ contributions
[15]
JC conceived the approach. SJ and XT implemented the
algorithm and conducted all the experiments. All wrote the
manuscript.
Acknowledgement
[16]
This research was supported by Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (award
number DE-FG02-91ER20021).
[17]
6.
REFERENCES
[1] M. Ashburner, C. Ball, J. Blake, et al. Gene ontology:
tool for the unification of biology. Nat Genet,
25(1):25–29, 2000.
[2] S. Awad, N. Panchy, S.-K. Ng, and J. Chen. Inferring
the regulatory interaction models of transcription
factors in transcriptional regulatory networks. Journal
of bioinformatics and computational biology,
10(05):1250012, 2012.
[3] C. Benedict, M. Geisler, J. Trygg, N. Huner, and
V. Hurry. Consensus by democracy. using
meta-analyses of microarray and genomic data to
model the cold acclimation signaling pathway in
arabidopsis. Plant Physiol, 141(4):1219–1232, 2006.
[4] G. M. Boratyn, S. Datta, and S. Datta. Incorporation
of biological knowledge into distance for clustering
genes. Bioinformation, 1(10):396, 2007.
[5] C. Bron and J. Kerbosch. Algorithm 457: finding all
cliques of an undirected graph. Comm ACM,
16(9):575–577, 1973.
[6] J. Caldas and S. Kaski. Hierarchical generative
biclustering for microrna expression analysis. J Comp
Bio, 18(3):251–261, 2011.
[7] Y. Cheng and G. Church. Biclustering of expression
data. In ISMB, pages 93–103, 2000.
[8] P. D’haeseleer. How does gene expression clustering
work? Nat Biotech, 23(12):1499–1502, 2005.
[9] J. Flores, I. Inza, P. Larrañaga, and B. Calvo. A new
measure for gene expression biclustering based on
non-parametric correlation. Comput Meth Prog Bio,
112(3):367–397, 2013.
[10] T. Gene Ontology Consortium. Gene ontology
annotations and resources. Nucl Acid Res,
41(D1):D530–D535, 2013.
[11] J. Goncalves and S. Madeira. Latebiclustering:
Efficient heuristic algorithm for time-lagged bicluster
identification. IEEE ACM T Comput Bi,
11(5):801–813, 2014.
[12] D. Huang and W. Pan. Incorporating biological
knowledge into distance-based clustering analysis of
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
microarray gene expression data. Bioinformatics,
22(10):1259–1268, 2006.
L. Ji, K. Mock, and K. Tan. Quick hierarchical
biclustering on microarray gene expression data. In
BIBE, pages 110–120, 2006.
J. Jun, S. Chung, and D. McLeod. Subspace clustering
of microarray data based on domain transformation.
In VLDB Workshop on Data Mining in
Bioinformatics, 2006.
P. Lamesch, T. Z. Berardini, D. Li, D. Swarbreck,
C. Wilks, R. Sasidharan, R. Muller, K. Dreher, D. L.
Alexander, M. Garcia-Hernandez, et al. The
arabidopsis information resource (tair): improved gene
annotation and new tools. Nucleic acids research,
40(D1):D1202–D1210, 2012.
U. Lauther. A min-cut placement algorithm for
general cell assemblies based on a graph
representation. In Proceedings of the 16th Design
Automation Conference, pages 1–10, 1979.
Y. Lazebnik. Can a biologist fix a radio?–or, what i
learned while studying apoptosis. Cancer Cell,
2(3):179–182, 2002.
P. Lord, D. Robert, B. Andy, and A. Carole.
Investigating semantic similarity measures across the
gene ontology: the relationship between sequence and
annotation. Bioinformatics, 19(10):1275–83, 2002.
S. Madeira, M. Teixeira, I. Sá-Correia, and
A. Oliveira. Identification of regulatory modules in
time series gene expression data using a linear time
biclustering algorithm. IEEE ACM T Comput Bi,
7(1):153–165, 2010.
F. Martella, M. Alfò, and M. Vichi. Biclustering of
gene expression data by an extension of mixtures of
factor analyzers. Intl J Biostat, 4(1):1–19, 2008.
M. Miller, L. Barkal, K. Jeng, A. Herrlich, M. Moss,
L. Griffith, and D. Lauffenburger. Proteolytic activity
matrix analysis (prama) for simultaneous
determination of multiple protease activities. Integ
Biol, 3(4):422–438, 2011.
M. E. Newman. The structure and function of
complex networks. SIAM review, 45(2):167–256, 2003.
W. Pan. Incorporating gene functions as priors in
model-based clustering of microarray gene expression
data. Bioinformatics, 22(7):795–801, 2006.
J. Peng, J. Chen, and Y. Wang. Identifying
cross-category relations in gene ontology and
constructing genome-specific term association
networks. BMC bioinformatics, 14(Suppl 2):S15, 2013.
J. Peng, H. Li, Q. Jiang, Y. Wang, and J. Chen. An
integrative approach for measuring semantic
similarities using gene ontology. BMC systems biology,
8(Suppl 5):S8, 2014.
J. Peng, S. Uygun, T. Kim, Y. Wang, S. Y. Rhee, and
J. Chen. Measuring semantic similarities by combining
gene ontology annotations and gene co-function
networks. BMC bioinformatics, 16(1):44, 2015.
J. Peng, Y. Wang, and J. Chen. Towards integrative
gene functional similarity measurement. BMC
bioinformatics, 15(Suppl 2):S5, 2014.
A. Prelić, S. Bleuler, P. Zimmermann, A. Wille,
P. Bühlmann, W. Gruissem, L. Hennig, L. Thiele, and
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
E. Zitzler. A systematic comparison and evaluation of
biclustering methods for gene expression data.
Bioinformatics, 22(9):1122–1129, 2006.
S. Rhee, V. Wood, K. Dolinski, and S. Draghici. Use
and misuse of the gene ontology annotations. Nat Rev
Genet, 9(7):509–515, 2008.
M. Robinson, J. Grigull, N. Mohammad, and
T. Hughes. Funspec: a web-based cluster interpreter
for yeast. BMC Bioinformatics, 3(35), 2002.
A. Schlicker et al. A new measure for functional
similarity of gene products based on gene ontology.
BMC Bioinformatics, 7:302, 2006.
E. Segal, M. Shapira, A. Regev, D. Pe’er, D. Botstein,
D. Koller, and N. Friedman. Module networks:
identifying regulatory modules and their
condition-specific regulators from gene expression
data. Nature genetics, 34(2):166–176, 2003.
A. Serin and M. Vingron. Debi: Discovering
differentially expressed biclusters using a frequent
itemset approach. Algorithms for Molecular Biology,
6(1):18, 2011.
J. Sevilla et al. Correlation between gene expression
and go semantic similarity. In IEEE ACM T Comput
Bi, volume 2, pages 330–333, 2005.
F. Sohler, D. Hanisch, and R. Zimmer. New methods
for joint analysis of biological networks and expression
data. Bioinformatics, 20(10):1517–1521, 2004.
M. E. Studham, A. Tjärnberg, T. E. Nordling,
S. Nelander, and E. L. Sonnhammer. Functional
association networks as priors for gene regulatory
network inference. Bioinformatics, 30(12):i130–i138,
2014.
J. Vogel, D. Zarka, H. A. Van B., S. Fowler, and
M. Thomashow. Roles of the cbf2 and zat12
transcription factors in configuring the low
temperature transcriptome of arabidopsis. 41:195–211,
2005.
O. Voggenreiter, S. Bleuler, and W. Gruissem. Exact
biclustering algorithm for the analysis of large gene
expression data sets. BMC Bioinformatics, 13(Suppl
18):A10, 2012.
H. Wang, W. Wang, J. Yang, and P. Yu. Clustering by
pattern similarity in large data sets. In SIGMOD,
pages 394–405, 2002.
J. Yang, W.Wang, H.Wang, and P. Yu. δ-clusters:
Capturing subspace correlation in a large data set. In
ICDE, pages 517–528, 2002.
Y. Yang, L. Han, Y. Yuan, J. Li, N. Hei, and
H. Liang. Gene co-expression network analysis reveals
common system-level properties of prognostic genes
across cancer types. Nat Comm, 5, 2014.
Z. Zhang, M. Teo, B. Ooi., and K. Tan. Mining
deterministic biclusters in gene expresssion data. In
BIBE, pages 283–290, 2004.