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Implementation of BiClusO and its comparison with other biclustering algorithms
Applied Network Science volumeÂ 4, ArticleÂ number:Â 79 (2019)
Abstract
This paper describes the implementation of biclustering algorithm BiClusO using graphical user interface and different parameters to generate overlapping biclusters from a binary sparse matrix. We compare our algorithm with several other biclustering algorithms in the context of two different types of biological datasets and four synthetic datasets with known embedded biclusters. Biclustering technique is widely used in different fields of studies for analyzing bipartite relationship dataset. Over the past decade, different biclustering algorithms have been proposed by researchers which are mainly used for biological data analysis. The performance of these algorithms differs depending on dataset size, pattern, and property. These issues create difficulties for a researcher to take the right decision for selecting a good biclustering algorithm. Two different scoring methods along with Gene Ontology(GO) term enrichment analysis have been used to measure and compare the performance of our algorithm. Our algorithm shows the best performance over some other wellknown biclustering algorithms.
Introduction
The rapid growth of advanced computeraided technology is creating a huge amount of high dimensional and heterogeneous data in different fields of study. Sometimes such data form a complex network which is very difficult to analyze by conventional database structure. In order to find the gist information from these data of complex structure, network analysis and graph clustering algorithms have evolved. A bipartite graph is a network of two disjoint sets of nodes where each edge connects a node from one set to a node from the other set. No edge is allowed within any single set. A bicluster is a high density (in terms of connected edges) subgraph of a bipartite graph. When a bicluster is fully connected by all possible edges then it is called a biclique. There are various applications of biclustering in different fields of study. In biology, gene expression under certain conditions forms a bipartite network which helps to understand the cellular response, disease diagnosis and pathway analysis (Beatriz et al. 2015; Andrew and Halappanavar 2015). Biological network analysis of the pairwise combinations of protein, miRNA, metabolite, conserved functional subsequences, and factor binding sites can predict or understand different cellular mechanisms (Gurkan and Yang 2007; Rui and Madeira 2014; Shu et al. 2007). In text mining, content related document searching is done using biclustering where rows represent different document types and columns represent frequently used keywords in the documents (de Castro et al. 2007; Arindam et al. 2007). In a social network, finding a common group of people with different shared interest are analyzed by biclustering (Dmitry et al. 2012; Qinghua 2011).
Some biclustering algorithms have been implemented which usually produce the textual presentation of output and are widely used mainly for analyzing biological data (Cheng and Church 2000; Lazzeroni and Owen 2002; Preli et al. 2006; Murali and Kasif 2002; Yuval et al. 2003; Guojun et al. 2009; Bergmann et al. 2003; Hochreiter et al. 2010; Tanay et al. 2002). Most of these implementations do not have overlapping controlling parameter or technique to separate bicliques from bicluster set. Moreover, only a few algorithms have been implemented with comprehensive visual presentations of biclusters (SantamarÃa et al. 2014; Streit et al. 2014; Heinrich et al. 2011; GonÃ§alves et al. 2009). Researchers have compared the performance of these biclustering algorithms by using artificial data and real biological data (Preli et al. 2006; Eren et al. 2012; Li et al. 2012). Due to performance variations mentioned in literature, it is very difficult to make the right decision to select an efficient biclustering algorithm. Optimal parameter selection for these algorithms is another important issue which needs persistent trial. We implement our algorithm using GUI and two different overlapping controlling parameters which allow the user to extract effective biclusters as well as bicliques. We selected five different biclustering algorithms BiMax, Spectral, CC, Plaid and xMOTIF for comparing with our algorithm. We selected these algorithms based on three criteria 1) whether the implementation is available in R 2) whether the algorithm can deal with binary dataset 3) whether the algorithm can produce some reasonable number of biclusters(at least two). Implementation of these algorithms is available within R package â€˜biclustâ€™ (Kaiser et al. 2018). We perform the comparison using three different types of datasets i.e. speciesVOC (volatile organic compound) relational data, geneexpression data of C. Cerevisiae and synthetic data. We created synthetic datasets of known embedded biclusters of varying size and overlapping properties. A partial portion of the current work was published in the conference of Complex Network 2018 (Karim et al. 2018). In the current work, we report the extended results of the experiment with another biological dataset and on the implementation of the algorithm using GUI.
Method of BiClusO
A bipartite graph also called a bigraph is a graph which consists of a finite number of vertices consisting of two disjoint sets. Each pair of vertices in the same set is nonadjacent. To denote a bipartite graph we use the notation G=(U,V,E) where U,V are two disjoint independent sets of nodes and E is the set of edges connecting the elements of U and V. By definition a bicluster is a subgraph induced by a pair of subsets U^{â€²}âŠ†U and V^{â€²}âŠ†V such that the size and edge density of the subgraph are maximal. The goal of the biclustering algorithm is to find a finite number of biclusters in a bipartite graph. If a node from U or V is included in more than one bicluster then it is called overlapping biclustering. Figure 1 illustrates the algorithm of BiClusO (Karim et al. 2019). Below we describe different steps to generate a bicluster set from a bipartite graph in the context of Fig. 1.
Matrix representation
A simple bipartite graph can be represented by a binary matrix (Fig. 1a). In a simple bipartite graph, edge weights are confined to 0,1. For a weighted graph, the edge weights can be converted to binary values by scaling the weights and defining a threshold limit for 0 and 1. In Fig. 1a, the row and column labels belong to the sets of U and V respectively. We place 1 to each cell of the matrix if there exists an edge between the corresponding row label and column label in the original graph.
Simple graph construction
Figure 1a denotes a binary matrix of a simple bipartite graph where U=7,V=7. We used data folding mechanism to generate two independent simple graphs involving the elements of U and V separately. In the weighted graph of Fig. 1b, the edge weights are determined based on common neighbors between corresponding nodes in the original bipartite graph which we call relation number. For example, node u1 has neighbor set {v1,v2,v3,v4} and u2 has neighbor set {v2,v3,v4,v5,v6}. So the relation number between u1 and u2 is N(u1)âˆ©N(u2)=3. From Fig. 1b some of the edges are filtered out using the relation number threshold <3 which generate the graph of Fig. 1c. Then Tanimoto coefficient is calculated for each edge. For example Tanimoto coefficient between node u1 and u2 is N(u1)âˆ©N(u2)/N(u1)âˆªN(u2)=0.5. Figure 1d shows the Tanimoto coefficients for all the edges. Tanimoto coefficient of an edge measures the robustness of the relation between the nodes it connects. From Fig. 1d some of the edges have been filtered out by setting the Tanimoto coefficient threshold >0.5, which makes the simple graph of Fig. 1e. By following the same way, the simple graph of Fig. 1j has been created involving the elements of set V.
Applying DPclusO
After applying both types of filtering the remaining graph shows a significant amount of the reduction of less important edges. DPClusO (AltafUlAmin et al. 2012; Karim et al. 2017), which was developed by extending the concepts of DPClus algorithm (AltafUlAmin et al. 2006; AltafUlAmin et al. 2006), can easily separate clusters from such simple graphs. The overlapping controlling parameter of DPClusO can be used to generate overlapping clusters to some extent. The dotted line in Fig. 1e and j shows the separable region of the clusters.
Second node attachment
After separating the clusters (u2,u3),(u5,u6,u7),(v5,v6) and (v2,v3,v4) as shown in Fig. 1e and j, the corresponding neighbor of each element of the clusters are joined by using their attachment probability. The attachment probability of each neighbor node in a cluster is calculated by dividing the number of corresponding cluster nodes attached to it by the total number of nodes in the cluster. After calculating the attachment probability, some of the neighbor nodes are pruned by using attachment probability threshold. We can extract the bicliques by setting the attachment probability to be 1. In this above example we consider the attachment probability threshold >0.5. After attachment we get four biclusters {(u2,u3),(v2,v3,v4,v5,v6)}, {(u5,u6,u7),(v5,v6,v7)}, {(v5,v6),(u2,u3,u5,u6,u7)}, and {(v2,v3,v4),(u1,u2,u3)} as shown in Fig. 1f and k.
Join cluster sets
Finally, four biclusters are listed in Fig. 1l by arranging the nodes from U and V as first and second set respectively. Sometimes finding biclusters from both sides create some duplicate biclusters or similar biclusters with big overlapping. This duplicity can be removed by keeping only one set from each duplicate pair. We also introduced the biclustering overlapping coefficient which can be used to join or filter out similar biclusters to some extent.
Implementation of BiClusO software
We implemented our algorithm using JAVA 2d graphics and swing control. We used JAVA because it helps to run the program independent of any operating system. Users from any field can easily run the jar file or executable file without any prior knowledge in a programming language. Users can input matrix data in excel or text format and can interact with the tool using different parameters. GUI implementation allows three different types of view 1) matrix view and 2) graph view 3) hierarchical relation view. Also, user can export the tabular format of cluster set arranged in descending order of their size. Clustering a big graph and rendering it with large canvas is a challenge when additional facilities like user interaction of rearranging nodes or edges are added. Due to this problem, we also added additional filtering based on size range and random choice to view the partial portion of a large graph. Figure 2 shows the BiClusO software with bicluster sets using tabular format (Fig. 2a), hierarchical view (Fig. 2b) and a single bicluster (Fig. 2c).
Overlapping parameters
We used here two different overlapping coefficients which are 1) simple graph cluster overlapping coefficient and 2) bicluster overlapping coefficient. The first overlapping coefficient measures the overlapping between two clusters while applying DPClusO to the simple graph generated from the bipartite graph. If A,B are two simple clusters and Aâˆ©B be the set of common number of nodes between them (Fig. 3a) then the simple cluster overlapping coefficient is
Bicluster overlapping coefficient measures the overlapping between two biclusters. This measurement considers each bicluster as a biclique and finds the common block matrix in terms of total block area occupied by them. If the node sets of two biclusters are denoted by A1,B1 and A2,B2 (Fig. 3b). The block area which denotes the number of edges in each bicluster is A1Ã—B1 and A2Ã—B2. The common block area between two biclusters is A1âˆ©A2Ã—B1âˆ©B2. The overlapping coefficient can be expressed by the following Equation
These parameters help the user to join and filter some of the biclusters and control overlapping property of the generated bicluster set.
Computational complexity
A bipartite graph is a graph that consists of two disjoint sets of nodes, U and V, such that each edge connects a node in U with a node in V, i.e., U and V are independent sets. Let U=n and V=m and therefore, the bipartite graph can be represented by a matrix of size nÃ—m. The detail of the algorithm of BiClusO is available in Karim et al. (2019). In BiClusO, calculating relation numbers between n(nâˆ’1)/2 pairs of the nodes of set U requires mn(nâˆ’1)/2 computations (Fig. 1b). Therefore, the complexity for this calculation is O(n^{2}m). Similarly, the complexity for calculating Tanimito coefficient is O(n^{2}m) (Fig. 1d). Filtering the edges in the weighted graph and construction of a simple graph using the threshold values of the Tanimoto coefficient and relation number need n(nâˆ’1)/2 or O(n^{2}) computations (Fig. 1c and d. The computational complexity for DPClusO algorithm is O(n^{3}) where n is the total number of nodes in a simple graph (AltafUlAmin et al. 2012; AltafUlAmin et al. 2006) (Fig. 1e). If the clusters generated by DPClusO on the simple graph are of sizes p1,p2...pc, then assigning the second node set to all the clusters needs p1m,p2m....pcm=(p1+p2....pc)m computations. The value of p1+p2....pc is roughly equal to n. Thus the computational complexity for assigning the second node by calculating the attachment probability is O(nm). So the total computational complexity of biclustering on the first node set comprises of 1) weight graph construction using relation number and Tanimoto coefficient 2) filtering and construction of simple graph 3) simple graph clustering and 4) second node attachment which is in total O(2n^{2}m+n^{2}+n^{3}+nm). Similarly, the computational complexity considering the second node set is O(2m^{2}n+m^{2}+m^{3}+mn).
Reference methods and dataset selection
This section illustrates the reference biclustering algorithms, data selection and data preparation for evaluating the performance of BiClusO.
Reference BiClustering algorithms
We have chosen five other wellknown biclustering algorithms namely BiMax, Plaid, xMOTIF, Spectral, And CC (Chang and Church) for performance comparison with BiClusO. These algorithms have been implemented in the R package â€™biclustâ€™ which we utilized for our experiments. BiMax recursively divides the binary data matrix using a different concentration of regions made by 1 and 0 and generates all maximal biclusters (Preli et al. 2006). It can only work with binary data matrix. Plaid algorithm models the data matrices with the sum of different layers and an assumed number of biclusters. It fits the model by iteratively updating the different parameters (Lazzeroni and Owen 2002). xMOTIF discretizes the data matrix by searching a set of rows following the same linear order under a set of columns to find motif (Murali and Kasif 2002). Spectral algorithm uses the singular value decomposition in eigenvectors to search the coherent value over row and column where the variance is lower than a given threshold value (Yuval et al. 2003). CC uses the deterministic greedy method based on the mean square residue of a submatrix where the score is lower than a threshold (Cheng and Church 2000).
Selection of dataset
We used two different types of datasets i.e. biological data and synthetic data. It is easy to compare the performance of different biclustering algorithms when the properties of the results are foreknown. Two different biological datasets are speciesVOC (volatile organic compound) data and microarray gene expression data of S. Cerevisiae. We evaluated the performance of different algorithms by measuring the statistical significance of the generated biclusters in terms of biological properties. We also created synthetic dataset by embedding biclusters with different physical properties. These physical properties help us to measure the quality of the extracted biclusters by different algorithms.
SpeciesVOC data
The speciesVOC bipartite dataset was collected from KNApSAcK (Nakamura et al. 2014; Afendi et al. 2011; Afendi et al. 2013; Abdullah et al. 2015) database. This dataset forms a sparse matrix of dimension 710 species vs. 1760 VOCs emitted by those species under different biological stress. We categorized these species under five different taxonomic levels which are kingdom, phylum, class, order, and family. Usually, similar species group under family level produce many distinct types of VOCs (Karim et al. 2018). However, there are some common VOCs among such species groups.
Gene expression data
This dataset is the gene expression data of the species S. Cerevisiae (Brown et al. 2000; Alvaro et al. 2002; LÃ¦greid et al. 2003; Raghava and Han 2005; AltafUlAmin et al. 2014) consisting of a matrix with dimension 2644 genes vs. 79 conditions. The data point in each cell represents the logarithmic ratio of the expression levels of a particular gene under two different experimental conditions. Each column represents n logtransformed expressionlevel ratios of n genes for a single chip. We digitized the data by columnwise transformation using the following formula.
Where avg_{j} and sd_{j} represent average and standard deviation of the j^{th} column. The threshold th is the value which determines the quantity of 1s to be set in the digital matrix. After accessing three different threshold values (th=0.5,1 and 1.5) we considered the th to be 1.5 which makes the digitized matrix as a sparse matrix.
Synthetic data
We created synthetic dataset of binary matrices using hypothetical relations between genes and conditions. We inserted a block of 1s as embedded bicluster to such matrix where 1 represents the differential expression of a gene under a condition. We changed the overlapping property and size of the biclusters and constructed four different types of such dataset: (1) equal size nonoverlapping, (ii) different size nonoverlapping, (iii) equal size overlapping and (iv) different size overlapping. Each dataset is a matrix of dimension 100 genes vs. 100 conditions. Each bicluster of equal size nonoverlapping set has 10 genes and 10 conditions (Fig. 4a). Different size nonoverlapping set has a varying number of genes between 5 to 10 and a varying number of conditions between 8 to 20 (Fig. 4c). Each bicluster of equal size with overlapping consists of 18 genes and 18 conditions where 9 genes and 9 conditions are overlapped between biclusters (Fig. 4b). Each bicluster of different size with overlapping consist of maximum 18 genes and 18 conditions with the overlapping region varying between 1 to 8 for genes and 2 to 10 for conditions (Fig. 4d). We introduced noise to the noncluster region of each dataset by randomly inserting â€™1â€™s. Total 36 datasets were created with four types of variation and nine different noise levels from 1% to 9%.
Parameter setting
We used the default parameter setting recommended by different papers and the authors of the biclustering algorithms (Preli et al. 2006; Eren et al. 2012; Li et al. 2012). In some cases, we slightly changed the parameters to force the algorithm to generate at least a minimum number of biclusters. For BiMax we set minimum row = 2, and minimum column=2. For Spectral we set minimum row =2, minimum column =2, maximum within variation =1, normalization = â€™logâ€™ and number of eigenvalue = 1. A high number of eigenvalue for this algorithm creates a performance issue on calculation time. The recommended value is 1. For xMOTIF we set sd=7, alpha=0.1, ns=10, nd=1000. For CC we set alpha=1.2, delta =.0000005. We took delta to such small value to create at least two biclusters. We set default parameter for Plaid algorithm of R BiClust package. For our algorithm we set CD(Cluster density)=0.5, CP=0.5(Cluster property), Tanimoto coefficient = 0.33, relation number = 5 and attachment probability = 0.5Cluster density and Cluster property are default parameters for DPClusO algorithm. From our previous experience on some actual dataset, the optimal setting for both CD and CP is 0.5 (Eguchi et al. 2018; Hossain et al. 2018) which produced good results. Also user can change the cluster density between 0.5 to 0.7. Most of the cases, the actual datasets are sparse by nature. To select the best threshold on sparse matrix user can start with relation number = 2 or 3 and Tanimoto coefficient = 0.33 (Karim et al. 2019). Tanimoto coefficient â‰¥0.33 allows more than 50% similarity between 1s of two binary vectors. Very high relation number and Tanimoto coefficient might exclude some nodes from the analysis (Karim et al. 2019). If the data is not sparse then Tanimoto coefficient threshold can be adjusted between 0.4 to 1.0 depending on whether the required number of nodes are included in the biclusters.
Results of performance evaluation
We adopted different scoring methods to evaluate the strength of biclustering algorithms. The results of comparison based on different biological data and synthetic data are summarized in Figs. 5, 6 and 7. In the following we discuss in detail.
Classification of species based on VOCs
Biclustering can be used to classify species based on the VOCs they emit. We applied the hypergeometric test to the bicluster set generated from speciesVOC relational data. The richness of same categorical species in a bicluster in terms of familylevel taxonomy was evaluated by using the following formula
Here X is the number of species in a bicluster. N is the number of species in the data. n is the maximum number of same categorical species belong to a specific family in a cluster. G is the total number of species of that category in the data. After selecting the statistically significant biclusters where pâˆ’value<=0.05, we combined the results generated by different algorithms. We created a list of clusters together with the algorithm name and their corresponding âˆ’log(pâˆ’value) in each row. We rearranged the list by descending order of âˆ’log(pâˆ’value) and therefore the top row on the list indicates the most significant bicluster and the bottom row indicates the least significant bicluster out of all the biclusters generated by all the algorithms. In order to compare the performance of different algorithms, we selected different sets of rows from the top of the list and counted the number of rows corresponding to the different algorithms within each set. The number of biclusters generated by BiClusO, BiMax, and Spectral is more compared to other algorithms as shown in Fig. 5a. Except some slightly overlapping, the BiClusO produces almost distinct biclusters. Plaid, CC and xMOTIF produce a very small number of biclusters which implies that these algorithms are not suitable for finding biclusters from sparse matrices (Miranda van et al. 2008). One of the limitations of BiMax is that it requires to specify the number of biclusters beforehand, otherwise produces 100 biclusters by default. Figure 5b shows that BiClusO has the highest share among top ranking clusters based on statistical significance. Spectral biclustering shows the second best performance. A moderate number of the biclusters generated by Spectral has identical sets of species over different sets of VOCs.
Richness of similar function genes
Biclustering of gene expression data is expected to accumulate similar function genes in individual biclusters. In order to find the richness of similar function genes in the gene side of each bicluster, we calculated hypergeometric pvalues corresponding to three different Gene Ontology (GO) terms i.e. Biological Process(BP), Cellular Component(CC), Molecular Function(MF). We used the GOstats package from R for calculating the pâˆ’values. The result of this analysis produces a series of pâˆ’value related to different GO terms. The GO term for which the pâˆ’value is the smallest is the most significant. For each bicluster, we selected the lowest pâˆ’values for certain term under each type of ontology. The number of generated biclusters by different algorithms is different. For the sake of fair comparison, we took up to the best 50 biclusters from each algorithm. For the algorithms that produced less than 50 clusters, we took all of them. We combined the selected clusters of all algorithms corresponding to a specific ontology and sorted the clusters according to their respective âˆ’log(pâˆ’value).
Scoring results with respect to three different GO terms are summarized in Fig. 6. Five biclustering algorithms including BiClusO produced some meaningful biclusters in terms of biological significance. We tried with the spectral algorithm by changing the reference parameter to produce some reasonable number of biclusters but failed. Only BiClusO and BiMax produced a significant number of biclusters with the small number of nodes in both gene and condition sides. Plaid, xMOTIF, and CC (algorithm) produced a small number of biclusters with a large number of nodes in both gene and condition sides. According to Fig. 6 in all three cases of BP, CC and MF, BiClusO produces most of the best ranking biclusters among the first 30 top biclusters. In the case of BP and MF (Fig. 6a and c), BiClusO clearly outperformed the other algorithms. Only in case of CC, BiMax shows almost similar performance like BiClusO (Fig. 6b). The CC (algorithm) shows the third position in all three ontology analysis. Plaid produces the smallest number of biclusters with large sizes compare with other algorithms which fail to produce good pvalue.
Comparison based on synthetic data
We evaluated the degree of similarity of bicluster sets generated by an algorithm from synthetic data by using the following matching score formula (Preli et al. 2006; Eren et al. 2012).
Here A_{1} denotes the actual bicluster set and B_{1} denotes the bicluster set generated by an algorithm. An element of the bicluster set A_{1} is denoted by (G_{1},C_{1}) where G_{1} is the set of genes and C_{1} is the set of conditions. Similarly (G_{2},C_{2}) denotes an element of the bicluster set B_{1}. The Jaccard index formula: âˆ£G_{1}âˆ©G_{2}âˆ£/âˆ£G_{1}âˆªG_{2}âˆ£ express the similarity between two biclusters in terms of gene side. According to the Jaccard index, identical biclusters produce maximum value 1 whereas disjoint biclusters produce minimum value 0. The above equation finds the average of maximum matching score of each gene set from A_{1} to B_{1}. Similarly, the matching score S_{c}(A_{1},B_{1}) was calculated considering the matching of condition sides of biclusters. Finally the formula
calculates the overall matching score from A_{1} to B_{1} considering both genes and conditions. If A_{1} be the actual bicluster set and B_{1} be the generated bicluster set by an algorithm then the matching score S(A_{1},B_{1}) represents average module recovery whereas S(B_{1},A_{1}) represents average cluster relevance (Preli et al. 2006). Average module recovery reflects the algorithmâ€™s ability to retrieve the actual biclusters. The best case value for this score is 1 which means that all of the actual biclusters are successfully discovered by the algorithm. The number of biclusters generated by the algorithm, in this case, must be greater than or equal to the number of actual biclusters. Average cluster relevance measures the similarity between generated bicluster and actual bicluster. The best case value for this score is 1 which means maximum similarity is achieved by the algorithm. The number of biclusters generated by the algorithm, in this case, must be less than or equal to the number of actual biclusters. If both scores are 1 then the algorithm successfully generates actual biclusters.
Performance on average module recovery
Figure 7 shows the performance of average module recovery for all algorithms. BiClusO outperforms over other biclustering algorithms on the synthetic dataset of equal size nonoverlapping (Fig. 7a) and variable size nonoverlapping (Fig. 7c). On both dataset maximum matching score, â€™1â€™ is achieved over different noise level. BiMax and Spectral achieve the second and third position. BiClusO shows good performance over BiMax and Spectral in Fig. 7b of equal size overlapping data. From Fig. 7d of variable size overlapping data, BiMax performs better matching score while the noise level is low but as the noise level increases the performance drastically degrades. Most of the cases of four types of synthetic dataset with a varied level of noises, BiClusO achieves the best results in terms of average module recovery.
Performance on average cluster relevance
Almost all of the cases of four different synthetic datasets BiClusO shows the best performance over other algorithms in terms of average cluster relevance (Fig. 8). Spectral and Plaid achieve the second highest position alternatively. BiMax, xMOTIF and CC show poor performance. The Performance of BiMax is deteriorated because it produces a large number of biclusters where a substantial portion of them are dissimilar to the original bicluster set. Most of the biclusters produced by BiClusO show maximum similarity to the original bicluster set.
Conclusion
In this study, we present a GUI based implementation of our algorithm BiClusO and compare its performance with different biclustering algorithms based on two types of biological data and four types of synthetic data. Biclusters generated based on biological dataset are analyzed on the basis of statistical significance determined by Hypergeometric test in the context of taxonomical classes and different GO terms. The varying properties of size and overlapping on synthetic datasets simulate a diverse pattern of genecondition bipartite dataset which helps to compare different algorithms effectively. The implementation allows us to control the overlapping between biclusters. Also the attachment probability parameter helps to extract biclusters as bicliques. The performance of our algorithm in terms of different scoring methods on real biological datasets expresses the robustness and effectiveness of BiClusO over other algorithms. The consistency of the performance of different algorithms to a certain degree in the case of synthetic datasets implies the correctness and significance of the comparison method we adopted in the present work. The implementation of our algorithm will be available on http://www.knapsackfamily.com/BiClusO/.
Availability of data and materials
Data is available on request from the corresponding author
Abbreviations
 BCov:

Biclustering overlapping coefficient
 BP:

Biological process
 CC (Algorithm):

Biclustering algorithm
 CC:

Cellular component
 CD:

Cluster density
 CP:

Cluster property
 GO:

Gene ontology
 MF:

Molecular function
 SCov:

Simple clustering overlapping coefficient
 VOC:

Volatile organic compound
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Acknowledgements
We are very thankful to Dr. Naoaki Ono for his cordial help and to the anonymous reviewers for their important comments.
Funding
This work was supported by the Ministry of Education, Culture, Sports, Science, and Technology of Japan (16K07223 and 17K00406) and NAIST Big Data Project and was partially supported by Platform Project for Supporting Drug Discovery and Life Science Research funded by Japan Agency for Medical Research (18am0101111) and Development and the National Bioscience Database Center in Japan.
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Md. AUA and MBK designed the research and conducted the experiments. SK guided the research with valuable comments. All authors have read and approved the final manuscript.
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Karim, M.B., Kanaya, S. & AltafUlAmin, M. Implementation of BiClusO and its comparison with other biclustering algorithms. Appl Netw Sci 4, 79 (2019). https://doi.org/10.1007/s411090190180x
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DOI: https://doi.org/10.1007/s411090190180x