A biclustering algorithm based on a Bicluster Enumeration Tree: application to DNA microarray data
- Wassim Ayadi^{1, 2}Email author,
- Mourad Elloumi^{1} and
- Jin-Kao Hao^{2}
https://doi.org/10.1186/1756-0381-2-9
© Ayadi et al; licensee BioMed Central Ltd. 2009
Received: 20 July 2009
Accepted: 16 December 2009
Published: 16 December 2009
Abstract
Background
In a number of domains, like in DNA microarray data analysis, we need to cluster simultaneously rows (genes) and columns (conditions) of a data matrix to identify groups of rows coherent with groups of columns. This kind of clustering is called biclustering. Biclustering algorithms are extensively used in DNA microarray data analysis. More effective biclustering algorithms are highly desirable and needed.
Methods
We introduce BiMine, a new enumeration algorithm for biclustering of DNA microarray data. The proposed algorithm is based on three original features. First, BiMine relies on a new evaluation function called Average Spearman's rho (ASR). Second, BiMine uses a new tree structure, called Bicluster Enumeration Tree (BET), to represent the different biclusters discovered during the enumeration process. Third, to avoid the combinatorial explosion of the search tree, BiMine introduces a parametric rule that allows the enumeration process to cut tree branches that cannot lead to good biclusters.
Results
The performance of the proposed algorithm is assessed using both synthetic and real DNA microarray data. The experimental results show that BiMine competes well with several other biclustering methods. Moreover, we test the biological significance using a gene annotation web-tool to show that our proposed method is able to produce biologically relevant biclusters. The software is available upon request from the authors to academic users.
Background
DNA microarray technology is a revolutionary method enabling the measurement of expression levels of at least thousands of genes in a single experiment under diverse experimental conditions. This technology has found numerous applications in research and applied areas like biology, drug discovery, toxicological study and diseases diagnosis.
DNA microarray data is typically represented by a matrix where each cell represents the gene expression level of a gene under a particular experimental condition. One important analysis task of microarray data concerns the simultaneous identification of groups of genes that show similar expression patterns across specific groups of experimental conditions (samples) [1]. Such an application can be addressed by a biclustering process whose aim is to discover coherent biclusters. That is, a bicluster is a subset of genes and conditions of the original expression matrix where the selected genes present a coherent behavior under all the experimental conditions contained in the bicluster.
More generally, biclustering has also applications in other domains such as text mining [2, 3], target marketing [4, 5], markets search [6], search in databases [7, 8] and analyzing foreign exchange data [9].
where f is an objective function measuring the quality, i.e., degree of coherence, of a group of biclusters and BC(M) is the set of all the possible groups of biclusters associated with M.
Clearly, biclustering is a highly combinatorial problem with a search space of order of O(2 ^{|I|+|J|}). In the general case, biclustering is known to be NP-hard [1]. Consequently, most of the algorithms used to discover biclusters are based on heuristics to explore partially the combinatorial search space. The existing algorithms for biclustering can roughly be classified into two large families: systematic search methods and stochastic search methods (also called metaheuristic methods). Representative examples of systematic search methods include, among others, greedy algorithms [1, 10–14], divide and conquer algorithms [7, 15] and enumeration algorithms [16–18]. On the other hand, among the metaheuristic methods, we can mention neighbourhood-based algorithms like simulated annealing [19], GRASP [20], evolutionary and hybrid algorithms [21–24]. A recent review of various biclustering algorithms for biological data analysis is provided in [25].
Since the biclustering problem is a NP-hard problem and no single existing algorithm is completely satisfactory for solving the problem. It is useful to seek more effective algorithms for better solutions. In this paper, we introduce a new enumeration algorithm for biclustering of DNA microarray data, called BiMine. Our algorithm is based on three original features. First, BiMine relies on a new evaluation function called Average Spearman's rho (ASR) which is used to guide effectively the exploration of the search space. Second, BiMine uses a new tree structure, called Bicluster Enumeration Tree (BET), to represent conveniently the different biclusters discovered during the enumeration process. Third, to avoid the combinatorial explosion of the search tree, BiMine introduces a parametric rule that allows the enumeration process to cut tree branches that cannot lead to good biclusters.
To assess the performance of the proposed BiMine algorithm, we show computational results obtained on both synthetic and real datasets and compare our results with those from four state-of-the-art biclustering algorithms. Moreover, to evaluate the biological relevance of our resulting biclusters, we carry out a practical validation with respect to a specific Gene Ontology (GO) annotation with the help of a popular web tool.
Methods
A New Evaluation Function of Biclustering
Like any search algorithm, BiMine needs an evaluation function to assess the quality of a candidate bicluster. One possibility is to use the so-called Mean Squared Residue (MSR) function [1]. Indeed, since its introduction, MSR has largely been used by biclustering algorithms, see for instance [11, 13, 20–22, 26, 27]. However, MSR is known to be deficient to assess correctly the quality of certain types of biclusters [14, 28, 29]. In a recent work, Teng and Chan [14] proposed another function for bicluster evaluation called Average Correlation Value (ACV). However, the performance of ACV is known to be sensitive to errors [13].
where (resp. ) is the rank of (resp. ).
where:
ρ _{ i, j }(i ≠ j) is the Spearman's rank correlation associated with the row indices i and j in the bicluster (I', J'). ρ _{ k, l }(k ≠ l) is the Spearman's rank correlation associated with the column indices k and l in the bicluster (I', J').
With Spearman's rank correlation, a high (resp. low) value, close to 1 (resp. close to -1), indicates that the data is strongly (resp. weakly) correlated between two vectors [30]. As shown above, ASR also takes values from [-1..1]. A high (resp. low) ASR value, close to 1 (resp. close to -1), indicates that the genes/conditions of the bicluster are strongly (resp. weakly) correlated.
Furthermore, in the next subsection, we want to assess the quality of the proposed ASR evaluation function in comparison with two popular functions MSR and ACV.
Studies of the ASR Evaluation Function
We compare the ASR evaluation function with Mean Squared Residue (MSR) [1]. As mentioned previously, MSR is probably the most popular evaluation function and largely used in the literature. As a second reference function, we use Average Correlation Value (ACV) which was proposed very recently in [14].
ASR versus MSR and ACV.
Biclusters | M _{1} | M _{2} | M _{3} | M _{4} | M _{5} | M _{6} | M _{7} |
---|---|---|---|---|---|---|---|
Evaluation Functions | |||||||
MSR | 0 | 0 | 0 | 0 | 0.62 | 2.425 | 131.87 |
ACV | 1 | 1 | 1 | 1 | 1 | 1 | 0.84 |
ASR | 1 | 1 | 1 | 1 | 1 | 1 | 0.99 |
Concerning MSR, a low (resp. high) value, close to 0 (resp. higher than a fixed threshold), indicates that the genes/conditions of the bicluster are strongly (resp. weakly) correlated.
Concerning ACV, a high (resp. low) value, close to 1 (resp. close to 0), indicates that the genes/conditions of the bicluster are strongly (resp. weakly) correlated.
According to Table 1, the ASR, ACV and MSR functions are perfect to assess the quality of biclusters M _{1}, M _{2}, M _{3} and M _{4}. However, MSR is deficient on M _{6} and M _{7}, confirming the claim that MSR may have trouble on certain types of biclusters [14, 28, 29]. On the other hand, ASR and ACV are perfect to assess the quality of biclusters M _{5}and M _{6} but ASR is slightly better than ACV when applied on M _{7}.
BiMine Algorithm
We present now our biclustering algorithm called BiMine which uses ASR as its evaluation function and a new structure, called Bicluster Enumeration Tree (BET) to represent the different biclusters associated with a data matrix. We describe first the main procedure for building biclusters and then show an illustrative example to ease the understanding of the algorithm.
Let M be a data matrix, by using our algorithm, we operate in three steps: During the first step, we preprocess the data matrix M. During the second step, we construct a BET associated with M. Finally, during the last step, we identify the best biclusters.
Preprocessing
Data matrix M'.
C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | |
---|---|---|---|---|---|---|
I_{1} | 10 | 20 | 5 | 15 | 40 | 18 |
I_{2} | 20 | 40 | 10 | 30 | 24 | 20 |
I_{3} | 23 | 12 | 8 | 15 | 29 | 50 |
I_{4} | 4 | 8 | 2 | 6 | 5 | 5 |
I_{5} | 15 | 25 | 8 | 12 | 29 | 50 |
Data matrix M after preprocess.
C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | |
---|---|---|---|---|---|---|
I_{1} | 10 | 20 | 5 | 15 | 40 | - |
I_{2} | 20 | 40 | 10 | 30 | - | 20 |
I_{3} | - | 12 | 8 | 15 | 29 | 50 |
I_{4} | 4 | 8 | 2 | 6 | - | - |
I_{5} | 15 | - | 8 | 12 | 29 | 50 |
By considering only non-missing values, we minimize the loss of information in the data matrix. This way of preprocessing missing values should be contrasted with other techniques. For instance, in [31], where the whole row is removed if the row contains at least one missing value or in [32], where the whole column is removed if it contains at least 5% of missing values. Furthermore, BiMine operates directly on the raw data matrix without resorting to a discretization of data, reducing thus the risk of loss of information.
Building Bicluster Enumeration Tree
After the preprocessing step, we construct a Bicluster Enumeration Tree (BET) that represents every possible bicluster that can be made from M. Compared to other data structure, BET permits to represent the maximum number of significant biclusters and the links that exist between these biclusters. Since the number of possible biclusters (nodes of BET) increases exponentially, BiMine employs parametric rules to help the enumeration process to close (or cut) a tree node. Intuitively, a node is cut down if the quality of the bicluster represented by this node is below a fixed threshold.
To describe formally our BiMine algorithm, let us define in the following the needed notations:
n _{ i }: i th node order containing biclusters.
n _{ i }.g _{ i }: genes of n _{ i }.
n _{ i }.Cg _{ i }: conditions of n _{ i }.
bic: bicluster.
δ: threshold used in Equation 4.
Threshold: quality threshold according to ASR.
BET-tree (Figure 2 (Function 2)) creates recursively the BET (Line 13) and generates the set of the best biclusters. The i ^{th} child of a node is made up, on the one hand, of the union of the genes of the father node and the genes of the i ^{th} uncle node, starting from the right side of the father. On the other hand, it is made up of the intersection of the conditions of the father and those of the i ^{th} uncle starting from the right side of the father (Line 4-12). If the ASR value associated with the i ^{th} child is smaller than or equal to the given Threshold, then this child will be ignored (Line 6-11).
Notice that this parametric pruning rule based on a quality threshold is fully justified in this context. Indeed, if the current bicluster is not good enough, then it is useless to keep it because expanding such a bicluster leads certainly to biclusters of worse quality. From this point of view, the pruning rule shares similar principles largely applied in optimization methods like Dynamic Programming. In addition, this pruning rule is essential in reducing the tree size and remains indispensable for handling large datasets.
Finally, the union of the leaves of the constructed BET that are not included in other leaves and have at least two genes represents a good group of biclusters (Line 8-9).
Proposition 2: Time complexity of BiMine is O(2^{ n } mlog(m)), where n is the number of rows and m is the number of columns of the data matrix.
Proof: Time complexity of the first step of BiMine is O(nm). Indeed, this step is achieved via a scanning of the whole data matrix M that is of size nm.
Time complexity of the second step of BiMine is O(2^{ n } mlog(m)). Actually, in the worst case, we have 2^{ n }nodes in the BET, representing the possible clusters of genes, each of which is associated with m conditions. On the other hand, since the conditions of the node are sorted, the construction of the intersection of two subsets of conditions of size m boils down to the search of m elements in a sorted array of size m. This can be done via a dichotomic search with a time complexity O(mlog(m)). Hence, the time complexity of the second step of BiMine is O(2^{ n } mlog(m)). Thus, The time complexity of BiMine is O(2^{ n } mlog(m)).
Illustrative Example
The second level of the BET is made up of nodes that are the union of genes and the intersection of conditions in the first level.
Results
In this section, we assess the BiMine algorithm on both synthetic and real DNA microarray data. We have implemented our algorithm in Java programming language. We compare BiMine results with the results of four prominent biclustering algorithms used by the community, named as: CC [1], OPSM [10], ISA [33] and Bimax [15]. For these reference algorithms, we have used Biclustering Analysis Toolbox (BicAT) which is a recent software platform for clustering-based data analysis that integrates all these biclustering algorithms [34].
Synthetic Data
Data Sets
According to [14, 19, 35], we generated randomly two types of synthetic datasets of size (I, J) = (200, 20). Different types of biclusters are embedded like constant columns, additive, multiplicative and coherent evolution biclusters. The first (resp. second) dataset contains biclusters without (resp. with) overlapping. To obtain statistically stable results, for each type of datasets, we generated 10 problem instances by randomly inserting the biclusters at different places in the data matrix.
Comparison Criteria
with
S _{ cb }= Portion size of biclusters correctly extracted
with
S _{ ncb }= Portion size of biclusters not correctly extracted
Tot _{ size }= Total size of corrected biclusters
The ratio θ _{ Shared }(resp. θ _{ NotShared }) expresses the percent of shared (resp. not shared) biclusters volume which corresponds (resp. not corresponds) with the real biclusters. In fact, when θ _{ Shared }(resp. θ _{ NotShared }) is equal to 100% the algorithm extracts the corrected (resp. not corrected) biclusters. A perfect solution have θ _{ Shared }= 100% and θ _{ NotShared }= 0%.
Protocol for Experiments
For our biclustering algorithm, we have fixed δ = 0.2 and threshold of ASR = 0.85. The parameter settings used for the four reference algorithms are the default values as used in [12]. We run all the algorithms and we select the 4 biclusters obtained by each algorithm which best fit the 4 real biclusters. We compute the θ _{ Shared }and the θ _{ NotShared }for each algorithm to show the averaged percentage of volume of the resulting biclusters which is shared and not shared with the real biclusters. The objective of this experiment is to determine which algorithm is able to extract all implanted biclusters.
BiMine results and comparison with other algorithms in synthetic data without overlapped biclusters.
Algorithms | θ _{ Shared } | θ _{ NotShared } |
---|---|---|
CC | 18.21% | 36.57% |
OPSM | 46.39% | 74.42% |
ISA | 39.38% | 5.31% |
Bimax | 58.18% | 21.39% |
BiMine | 100% | 33.03% |
As we can see in Table 4, BiMine can extract 100% of implanted biclusters with an extra volume that represent 33,03% of implanted biclusters. In fact, to obtain a new bicluster, combining two biclusters provide an extra volume only on conditions but give exactly the correct number of genes. However, the best of the studied algorithms, i.e., Bimax, can extract only 58.18% of implanted biclusters with 21.39% of extra volume. CC uses the MSR function of the selected elements as the biclustering criterion. When the signal of the implanted biclusters is weak, the greedy nature of CC may delete some rows and columns of the implanted biclusters in the beginning of the algorithm and miss the deleted rows and columns in the output biclusters. ISA uses only up-regulated and down-regulated constant expression values in its biclustering algorithm. When coherent biclusters exist, ISA may miss some rows and columns of the implanted biclusters. OPSM seeks only up and down regulation expression values with coherent evolution. Its performance decreases when there exist scenarios constant biclusters. The discretization preprocessing used by Bimax cannot identify the elements in the coherent biclusters. Hence, the algorithm cannot find exactly the implanted biclusters.
BiMine results and comparison with other algorithms in synthetic data with overlapped biclusters.
Algorithms | θ _{ Shared } | θ _{ NotShared } |
---|---|---|
CC | 9.21% | 47.94% |
OPSM | 42.87% | 49.31% |
ISA | 23.28% | 23.97% |
Bimax | 34.07% | 3.43% |
BiMine | 85.35% | 41.78% |
As we can see in Table 5, the results with BiMine present the highest coverage of the correct biclusters. In fact, BiMine can extract 85.35% of implanted biclusters with an extra volume that represent 41.78% of implanted biclusters. However, the best of the studied algorithms, i.e., OPSM, can extract only 42.87% of implanted biclusters with 49.31% of extra volume. To find overlapped biclusters in a given matrix, some algorithms, e.g., CC, need to mask the discovered biclusters with random values which is not necessary for BiMine. ISA and OPSM are sensitive to overlapping biclusters. They use the normalization step in the first preprocessing step of their algorithms. With overlapping biclusters, the expression value range after normalization becomes narrower. Table 5 shows that BiMine is marginally affected by the implanted overlap biclusters. We can conclude that BiMine can extract all implanted biclusters unlike other algorithms that can extract only certain types of biclusters.
Real data
Data Sets
We applied our approach to the well-known yeast cell-cycle dataset. This dataset is publicly available from [36] and described in [37] and processed in [1]. It contains the expression profiles of more than 6000 yeast genes measured at 17 conditions over two complete cell cycles. In our experiments we use 2884 genes selected by [1].
Comparison Criteria
Two criteria are used. First, in order to evaluate the biological relevance of our proposed biclustering algorithm, we compute the p-values to indicate the quality of the extracted biclusters. Second, we identify the biological annotations for the extracted biclusters.
Protocol for Experiments
For our biclustering algorithm, we have fixed δ = 0.1 and threshold of ASR = 0.85. The parameter settings used for the different reference biclustering algorithms are the default settings as used in [12]. For the first experiment, we run all the algorithms and we compute the p-value for extracted biclusters. With BiMine (resp. Bimax), we have obtained more than 1800 (resp. 3700) biclusters. Since a biological analysis on 1800 (resp. 3700) biclusters was not feasible, only the 100 biggest biclusters with high ASR were selected for analysis like Christinat et al. [38]. Post-filtering was applied for all algorithms in order to eliminate insignificant biclusters like Cheng et al. [13]. With the others algorithms, we obtained 10 biclusters for CC, 45 biclusters for ISA and 14 biclusters for OPSM. For the second experiment, we use a well-known web-tool to search for the significant shared Gene Ontology terms of the groups of genes.
Biological relevance
Proportions of Biclusters significantly enriched by GO annotations.
p-value | 5% | 1% | 0.5% | 0.1% | 0.001% |
---|---|---|---|---|---|
Algorithms | |||||
BiMine | 100 | 100 | 93 | 82 | 51 |
OPSM | 100 | 100 | 86 | 36 | 22 |
Bimax | 100 | 100 | 89 | 79 | 64 |
ISA | 89 | 89 | 87 | 69 | 32 |
CC | 80 | 70 | 60 | 20 | 10 |
Furthermore, in order to identify the biological annotations for the extracted biclusters we use GOTermFinder http://db.yeastgenome.org/cgi-bin/GO/goTermFinder which is a tool available in the Saccharomyces Genome Database (SGD). GOTermFinder is designed to search for the significant shared GO terms of the groups of genes and provides users with the means to identify the characteristics that the genes may have in common.
Most significant shared GO terms (process, function, component) for two biclusters on Yeast data.
Bicluster volume (genes × conditions) | Process Ontology | Function Ontology | Component Ontology |
---|---|---|---|
(12 × 13) | cellular response to DNA damage stimulus (66.7%, 1.87e-08) response to DNA damage stimulus (66.7%, 6.30e-08) cellular response to stress(66.7%, 2.12e-07) cellular response to stimulus(66,7%, 3.25e-07) DNA repair(50%, 2.58e-05) response to stress(66.7%, 2.98e-05) | chromatin binding (25%,0.00037) | microtubule organizing center part(16.7%, 0.00742) |
(11 × 11) | cell cycle process (63.6%, 2.93e-05) cell cycle (63.6%, 6.85e-05) | GTPase activator activity (18.2%,0.00994) | microtubule cytoskeleton (45.5%, 6.33e-06) microtubule organizing center (36.4%,4.97e-05) spindle pole body (36.4%, 4.97e-05) spindle pole (36.4%, 6.77e-05) |
The values within parentheses after each GO term in Table 7, such as (66.7%, 1.87e-08) in the first bicluster, indicate the cluster frequency and the statistical significance. The cluster frequency (66.7%) shows that out of 12 genes in the first bicluster 8 belong to this process, and the statistical significance is provided by a p-value of 1.87e-08 (highly significant).
All these experiments show that for this dataset, the proposed approach is able to detect biologically significant and functionally enriched biclusters with low p-value. Furthermore, BiMine gives a good degree of homogeneity.
Discussion
BiMine algorithm has several interesting features. First, with BiMine, we avoid using a discretization of the data matrix. Indeed, classifying the gene expression values using intervals often leads to bad results [44]. Also, the discretization may limit the performance of an algorithm to discover a biological model because of noises which are inherent in most experiences of microarrays [31]. Thus, to discretize biological data we must have a good knowledge of these data to assign good values. However, this is not always possible.
Second, the BiMine algorithm can enumerate all possible cases of attributes while reducing the tree size. In fact, the parametric rule based on ASR threshold allows the enumeration process to prune tree branches that cannot lead to good biclusters.
Third, the BiMine algorithm provides naturally multiple biclusters of variable sizes. The number of the desired biclusters can be determined by tuning the ASR threshold. These multiple solutions of different sizes and different characteristics may be of interest for biological investigations.
Forth, the new ASR evaluation function can be applied by other biclustering algorithm in replacement of MSR or ACV. It can also be used as a complementary function to these previously ones.
Finally, in [45], it has been shown that Spearman's rank correlation is less sensitive to the presence of noise in the data. Since our evaluation function ASR is based on Spearman rank correlation, ASR would also be less sensitive to the presence of noise in the data.
Conclusions
In this paper, we described BiMine, a new algorithm for biclustering of DNA microarray data. Compared with existing biclustering algorithms, BiMine distinguishes itself by a number of original features. First, BiMine operates directly on the raw data matrix without resorting to a discretization of data, reducing thus the risk of loss of information. Second, with BiMine, it is not necessary to fix a minimum or maximum number of genes or conditions, enabling the generation of diversified biclusters. Third, using a convenient tree structure for representing biclusters with a parametric and effective branch pruning rule, BiMine is able to explore effectively the search space. Notice that ASR can also be used by other biclustering algorithm as an alternative evaluation function.
The performance of the BiMine algorithm is tested and assessed on a set of synthetic data as well as a real microarray data (yeast cell-cycle). Computational experiments showed highly competitive results of BiMine in comparison with four other popular biclustering algorithms for both types of datasets. In addition, a biological validation of the selected genes within the biclusters for yeast cell-cycle has been provided based on a publicly available Gene Ontology (GO) annotation tool. Notice that although we presented BiMine with the context of DNA microarray data analysis, it should be clear that the algorithm can be applied or adapted to other biclustering problems.
Finally, let us mention that the proposed algorithm is computational time expensive; one of our ongoing works aims to find new heuristics to speed up the enumeration process. In particular, it would be possible to define other heuristic rules to improve the branch pruning in order to further reduce the size of the explored search tree.
Declarations
Acknowledgements
The authors are grateful to Dr. Jason Moore and Dr. Federico Divina for their insightful comments and questions that helped us to improve the work.
Authors’ Affiliations
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