QUBIC Tutorial

Gene expression data is very important in experimental molecular biology (Brazma and Vilo 2000), especially for cancer study (Fehrmann et al. 2015). The large-scale microarray data and RNA-seq data provide good opportunity to do the gene co-expression analyses and identify co-expressed gene modules; and the effective and efficient algorithms are needed to implement such analysis. Substantial efforts have been made in this field, such as Cheng and Church (2000), Plaid (Lazzeroni, Owen, et al. 2002), Bayesian Biclustering (BCC, Gu and Liu 2008), among them Cheng and Church and Plaid has the R package implementation. It is worth noting that our in-house biclustering algorithm, QUBIC (Li et al. 2009), is reviewed as one of the best programs in terms of their prediction performance on benchmark datasets. Most importantly, it is reviewed as the best one for large-scale real biological data (Eren et al. 2012).

Until now, QUBIC has been cited over 200 times (via Google Scholar) and its web server, QServer, was developed in 2012 to facilitate the users without comprehensive computational background (Zhou et al. 2012). In the past five years, the cost of RNA-sequencing decreased dramatically, and the amount of gene expression data keeps increasing. Upon requests from users and collaborators, we developed this R package of QUBIC to void submitting large data to a webserver.

The unique features of our R package (Zhang et al. 2017) include (1) updated and more stable back-end resource code (re-written by C++), which has better memory control and is more efficient than the one published in 2009. For an input dataset in Arabidopsis, with 25,698 genes and 208 samples, we observed more than 40% time saving; and (2) comprehensive functions and examples, including discretize function, heatmap drawing and network analysis.

How to cite

citation("QUBIC")
Please cite the QUBIC package in your work, whenever you use it:

  Yu Zhang, Juan Xie, Jinyu Yang, Anne Fennell, Chi Zhang, Qin Ma; QUBIC: a
  bioconductor package for qualitative biclustering analysis of gene co-expression
  data. Bioinformatics, 2017; 33 (3): 450-452. doi: 10.1093/bioinformatics/btw635

A BibTeX entry for LaTeX users is

  @Article{,
    title = {{QUBIC}: a bioconductor package for qualitative biclustering analysis of gene co-expression data},
    author = {Yu Zhang and Juan Xie and Jinyu Yang and Anne Fennell and Chi Zhang and Qin Ma},
    journal = {Bioinformatics},
    year = {2017},
    volume = {33},
    number = {3},
    pages = {450--452},
    doi = {10.1093/bioinformatics/btw635},
  }

Other languages

If R is not your thing, there is also a C version of QUBIC.

Help

If you are having trouble with this R package, contact the maintainer, Yu Zhang.

Install and load

Stable version from BioConductor

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("QUBIC")

Or development version from GitHub

install.packages("devtools")
devtools::install_github("zy26/QUBIC")

Load QUBIC

library("QUBIC")

Functions

There are nine functions provided by QUBIC package.

  • qudiscretize()creates a discrete matrix for a given gene expression matrix;
  • BCQU()performs a qualitative biclustering for real matrix;
  • BCQUD()performs a qualitative biclustering for discretized matrix;
  • quheatmap()can draw heatmap for singe bicluster or overlapped biclusters;
  • qunetwork() can automatically create co-expression networks based on the identified biclusters by QUBIC;
  • qunet2xml()can convert the constructed co-expression networks into XGMML format for further network analysis in Cytoscape, Biomax and JNets;
  • query-based biclustering allows users to input additional biological information to guide the biclustering progress;
  • bicluster expanding expands existing biclusters under specified consistency level;
  • biclusters comparison compares biclusters obtained via different algorithms or parameters.

The following examples illustrate how these functions work.

Example of a random matrix with two diferent embedded biclusters

library(QUBIC)
set.seed(1)
# Create a random matrix
test <- matrix(rnorm(10000), 100, 100)
colnames(test) <- paste("cond", 1:100, sep = "_")
rownames(test) <- paste("gene", 1:100, sep = "_")

# Discretization
matrix1 <- test[1:7, 1:4]
matrix1
##            cond_1      cond_2     cond_3      cond_4
## gene_1 -0.6264538 -0.62036668  0.4094018  0.89367370
## gene_2  0.1836433  0.04211587  1.6888733 -1.04729815
## gene_3 -0.8356286 -0.91092165  1.5865884  1.97133739
## gene_4  1.5952808  0.15802877 -0.3309078 -0.38363211
## gene_5  0.3295078 -0.65458464 -2.2852355  1.65414530
## gene_6 -0.8204684  1.76728727  2.4976616  1.51221269
## gene_7  0.4874291  0.71670748  0.6670662  0.08296573
matrix2 <- qudiscretize(matrix1)
matrix2
##        cond_1 cond_2 cond_3 cond_4
## gene_1     -1      0      0      1
## gene_2      0      0      1     -1
## gene_3      0     -1      0      1
## gene_4      1      0      0     -1
## gene_5      0      0     -1      1
## gene_6     -1      0      1      0
## gene_7      0      1      0     -1
# Fill bicluster blocks
t1 <- runif(10, 0.8, 1)
t2 <- runif(10, 0.8, 1) * (-1)
t3 <- runif(10, 0.8, 1) * sample(c(-1, 1), 10, replace = TRUE)
test[11:20, 11:20] <- t(rep(t1, 10) * rnorm(100, 3, 0.3))
test[31:40, 31:40] <- t(rep(t2, 10) * rnorm(100, 3, 0.3))
test[51:60, 51:60] <- t(rep(t3, 10) * rnorm(100, 3, 0.3))

# QUBIC
res <- biclust::biclust(test, method = BCQU())
summary(res)
## 
## An object of class Biclust 
## 
## call:
##  biclust::biclust(x = test, method = BCQU())
## 
## Number of Clusters found:  40 
## 
## Cluster sizes:
##                    BC 1 BC 2 BC 3 BC 4 BC 5 BC 6 BC 7 BC 8 BC 9 BC 10 BC 11 BC 12 BC 13 BC 14
## Number of Rows:      10   10    9    9    3    5    3    3    2     3     3     3     2     2
## Number of Columns:    9    9    8    7    5    3    5    5    6     4     4     4     6     6
##                    BC 15 BC 16 BC 17 BC 18 BC 19 BC 20 BC 21 BC 22 BC 23 BC 24 BC 25 BC 26 BC 27
## Number of Rows:        2     2     2     2     2     2     2     2     2     2     2     2     2
## Number of Columns:     6     6     6     6     6     6     6     5     5     5     5     5     5
##                    BC 28 BC 29 BC 30 BC 31 BC 32 BC 33 BC 34 BC 35 BC 36 BC 37 BC 38 BC 39 BC 40
## Number of Rows:        2     2     2     2     2     2     2     2     2     2     2     2     2
## Number of Columns:     5     5     5     5     5     5     5     5     5     5     5     5     5
# Show heatmap
hmcols <- colorRampPalette(rev(c("#D73027", "#FC8D59", "#FEE090", "#FFFFBF", 
    "#E0F3F8", "#91BFDB", "#4575B4")))(100)
# Specify colors

par(mar = c(4, 5, 3, 5) + 0.1)
quheatmap(test, res, number = c(1, 3), col = hmcols, showlabel = TRUE)
Heatmap for two overlapped biclusters in the simulated matrix

Heatmap for two overlapped biclusters in the simulated matrix

Example of Saccharomyces cerevisiae

library(QUBIC)
data(BicatYeast)

# Discretization
matrix1 <- BicatYeast[1:7, 1:4]
matrix1
##           cold_green_6h cold_green_24h cold_roots_6h cold_roots_24h
## 249364_at    -0.2759300     -0.5108508    1.74476670     2.12442300
## 253423_at    -0.9405282      3.2669048   -0.37776557     0.06860917
## 250327_at     1.6419950      1.4484175    0.33474782    -0.15095752
## 247474_at     0.6903505      1.6705408    0.04386528    -0.45295456
## 252661_at     1.7493315      0.9260773    2.05519100     2.11200260
## 258239_at     0.6110116     -0.6083303    0.60419910     0.43582130
## 248910_at     1.4501406     -0.3107802    0.16640233     0.37186486
matrix2 <- qudiscretize(matrix1)
matrix2
##           cold_green_6h cold_green_24h cold_roots_6h cold_roots_24h
## 249364_at             0             -1             0              1
## 253423_at            -1              1             0              0
## 250327_at             1              0             0             -1
## 247474_at             0              1             0             -1
## 252661_at             0             -1             0              1
## 258239_at             1             -1             0              0
## 248910_at             1             -1             0              0
# QUBIC
x <- BicatYeast
system.time(res <- biclust::biclust(x, method = BCQU()))
##    user  system elapsed 
##   0.079   0.000   0.044
summary(res)
## 
## An object of class Biclust 
## 
## call:
##  biclust::biclust(x = x, method = BCQU())
## 
## Number of Clusters found:  77 
## 
## Cluster sizes:
##                    BC 1 BC 2 BC 3 BC 4 BC 5 BC 6 BC 7 BC 8 BC 9 BC 10 BC 11 BC 12 BC 13 BC 14
## Number of Rows:      72   53   37   80   56   98   47   26   45    35    29    34    42    33
## Number of Columns:    4    5    7    3    4    2    4    7    4     5     6     5     4     5
##                    BC 15 BC 16 BC 17 BC 18 BC 19 BC 20 BC 21 BC 22 BC 23 BC 24 BC 25 BC 26 BC 27
## Number of Rows:       41    41    32    53    39    38    74    29    47    23    27    22    33
## Number of Columns:     4     4     5     3     4     4     2     5     3     6     5     6     4
##                    BC 28 BC 29 BC 30 BC 31 BC 32 BC 33 BC 34 BC 35 BC 36 BC 37 BC 38 BC 39 BC 40
## Number of Rows:       41    40    19    51    20    16    32    31    23    30    44    44    41
## Number of Columns:     3     3     6     2     5     6     3     3     4     3     2     2     2
##                    BC 41 BC 42 BC 43 BC 44 BC 45 BC 46 BC 47 BC 48 BC 49 BC 50 BC 51 BC 52 BC 53
## Number of Rows:       13    39    26    39    19    25    25    25    25     8    36     9     8
## Number of Columns:     6     2     3     2     4     3     3     3     3     9     2     8     9
##                    BC 54 BC 55 BC 56 BC 57 BC 58 BC 59 BC 60 BC 61 BC 62 BC 63 BC 64 BC 65 BC 66
## Number of Rows:       12    18    23    16    32    31    20    28    14    11    18    18    17
## Number of Columns:     6     4     3     4     2     2     3     2     4     5     3     3     3
##                    BC 67 BC 68 BC 69 BC 70 BC 71 BC 72 BC 73 BC 74 BC 75 BC 76 BC 77
## Number of Rows:       24    24     9    14    14    20    12     9    17     8     3
## Number of Columns:     2     2     5     3     3     2     3     4     2     3     5

We can draw heatmap for single bicluster.

# Draw heatmap for the second bicluster identified in Saccharomyces cerevisiae data

library(RColorBrewer)
paleta <- colorRampPalette(rev(brewer.pal(11, "RdYlBu")))(11)
par(mar = c(5, 4, 3, 5) + 0.1, mgp = c(0, 1, 0), cex.lab = 1.1, cex.axis = 0.5, 
    cex.main = 1.1)
quheatmap(x, res, number = 2, showlabel = TRUE, col = paleta)
Heatmap for the second bicluster identified in the *Saccharomyces cerevisiae* data. The bicluster consists of 53 genes and 5 conditions

Heatmap for the second bicluster identified in the Saccharomyces cerevisiae data. The bicluster consists of 53 genes and 5 conditions

We can draw heatmap for overlapped biclusters.

# Draw for the second and third biclusters identified in Saccharomyces cerevisiae data

par(mar = c(5, 5, 5, 5), cex.lab = 1.1, cex.axis = 0.5, cex.main = 1.1)
paleta <- colorRampPalette(rev(brewer.pal(11, "RdYlBu")))(11)
quheatmap(x, res, number = c(2, 3), showlabel = TRUE, col = paleta)
Heatmap for the second and third biclusters identified in the *Saccharomyces cerevisiae* data. Bicluster #2 (topleft) consists of 53 genes and 5 conditions, and bicluster #3 (bottom right) consists of 37 genes and 7 conditions.

Heatmap for the second and third biclusters identified in the Saccharomyces cerevisiae data. Bicluster #2 (topleft) consists of 53 genes and 5 conditions, and bicluster #3 (bottom right) consists of 37 genes and 7 conditions.

We can draw network for single bicluster.

# Construct the network for the second identified bicluster in Saccharomyces cerevisiae
net <- qunetwork(x, res, number = 2, group = 2, method = "spearman")
if (requireNamespace("qgraph", quietly = TRUE))
    qgraph::qgraph(net[[1]], groups = net[[2]], layout = "spring", minimum = 0.6, 
        color = cbind(rainbow(length(net[[2]]) -  1), "gray"), edge.label = FALSE)
## Warning in qgraph::qgraph(net[[1]], groups = net[[2]], layout = "spring", : The following
## arguments are not documented and likely not arguments of qgraph and thus ignored: edge.label
Network for the second bicluster identified in the *Saccharomyces cerevisiae* data.

Network for the second bicluster identified in the Saccharomyces cerevisiae data.

We can also draw network for overlapped biclusters.

net <- qunetwork(x, res, number = c(2, 3), group = c(2, 3), method = "spearman")
if (requireNamespace("qgraph", quietly = TRUE))
    qgraph::qgraph(net[[1]], groups = net[[2]], layout = "spring", minimum = 0.6, 
        legend.cex = 0.5, color = c("red", "blue", "gold", "gray"), edge.label = FALSE)
## Warning in qgraph::qgraph(net[[1]], groups = net[[2]], layout = "spring", : The following
## arguments are not documented and likely not arguments of qgraph and thus ignored: edge.label
Network for the second and third biclusters identified in the *Saccharomyces cerevisiae* data.

Network for the second and third biclusters identified in the Saccharomyces cerevisiae data. 

# Output overlapping heatmap XML, could be used in other software such
# as Cytoscape, Biomax or JNets
sink('tempnetworkresult.gr')
qunet2xml(net, minimum = 0.6, color = cbind(rainbow(length(net[[2]]) - 1), "gray"))
sink()
# We can use Cytoscape, Biomax or JNets open file named 'tempnetworkresult.gr'

Example of Escherichia coli data

The Escherichia coli data consists of 4,297 genes and 466 conditions.

library(QUBIC)
library(QUBICdata)
data("ecoli", package = "QUBICdata")

# Discretization
matrix1 <- ecoli[1:7, 1:4]
matrix1
##       dinI_U_N0025 dinP_U_N0025 lexA_U_N0025 lon_U_N0025
## b4634     9.077693     9.225537     9.138900    9.114353
## b3241     7.122300     7.195453     7.051193    7.124200
## b3240     7.184417     7.336610     7.283377    7.188263
## b3243     7.902090     7.963167     7.847747    7.943650
## b3242     6.801900     6.843213     6.795007    6.889897
## b2836     9.114207     9.133303     9.167487    9.189480
## b0885     9.057120     8.918723     8.985483    9.002663
matrix2 <- qudiscretize(matrix1)
matrix2
##       dinI_U_N0025 dinP_U_N0025 lexA_U_N0025 lon_U_N0025
## b4634           -1            1            0           0
## b3241            0            1           -1           0
## b3240           -1            1            0           0
## b3243            0            1           -1           0
## b3242            0            0           -1           1
## b2836           -1            0            0           1
## b0885            1           -1            0           0
# QUBIC
res <- biclust::biclust(ecoli, method = BCQU(), r = 1, q = 0.06, c = 0.95, o = 100, 
    f = 0.25, k = max(ncol(ecoli)%/%20, 2))
system.time(res <- biclust::biclust(ecoli, method = BCQU(), r = 1, q = 0.06, c = 0.95, 
    o = 100, f = 0.25, k = max(ncol(ecoli)%/%20, 2)))
##    user  system elapsed 
##  10.802   0.016   5.154
summary(res)
## 
## An object of class Biclust 
## 
## call:
##  biclust::biclust(x = ecoli, method = BCQU(), r = 1, q = 0.06, c = 0.95, o = 100, 
##      f = 0.25, k = max(ncol(ecoli)%/%20, 2))
## 
## Number of Clusters found:  20 
## 
## Cluster sizes:
##                    BC 1 BC 2 BC 3 BC 4 BC 5 BC 6 BC 7 BC 8 BC 9 BC 10 BC 11 BC 12 BC 13 BC 14
## Number of Rows:     437  121   51  108  103   65   41   26   27    20    25    23    17    18
## Number of Columns:   29   45   94   44   38   38   31   33   31    27    19    20    23    21
##                    BC 15 BC 16 BC 17 BC 18 BC 19 BC 20
## Number of Rows:       14    15    13    11     5     6
## Number of Columns:    26    20    22    25    32    23
# Draw heatmap for the 5th bicluster identified in Escherichia coli data

library(RColorBrewer)
paleta <- colorRampPalette(rev(brewer.pal(11, "RdYlBu")))(11)
par(mar = c(5, 4, 3, 5) + 0.1, mgp = c(0, 1, 0), cex.lab = 1.1, cex.axis = 0.5, 
    cex.main = 1.1)
quheatmap(ecoli, res, number = 5, showlabel = TRUE, col = paleta)
Heatmap for the fifth bicluster identified in the *Escherichia coli* data. The bicluster consists of 103 genes and 38 conditions

Heatmap for the fifth bicluster identified in the Escherichia coli data. The bicluster consists of 103 genes and 38 conditions 

library(RColorBrewer)
paleta <- colorRampPalette(rev(brewer.pal(11, "RdYlBu")))(11)
par(mar = c(5, 4, 3, 5), cex.lab = 1.1, cex.axis = 0.5, cex.main = 1.1)
quheatmap(ecoli, res, number = c(4, 8), showlabel = TRUE, col = paleta)
Heatmap for the fourth and eighth biclusters identified in the *Escherichia coli* data. Bicluster #4 (topleft) consists of 108 genes and 44 conditions, and bicluster #8 (bottom right) consists of 26 genes and 33 conditions

Heatmap for the fourth and eighth biclusters identified in the Escherichia coli data. Bicluster #4 (topleft) consists of 108 genes and 44 conditions, and bicluster #8 (bottom right) consists of 26 genes and 33 conditions 

# construct the network for the 5th identified bicluster in Escherichia coli data
net <- qunetwork(ecoli, res, number = 5, group = 5, method = "spearman")
if (requireNamespace("qgraph", quietly = TRUE))
    qgraph::qgraph(net[[1]], groups = net[[2]], layout = "spring", minimum = 0.6, 
        color = cbind(rainbow(length(net[[2]]) - 1), "gray"), edge.label = FALSE)
## Warning in qgraph::qgraph(net[[1]], groups = net[[2]], layout = "spring", : The following
## arguments are not documented and likely not arguments of qgraph and thus ignored: edge.label
Network for the fifth bicluster identified in the *Escherichia coli* data.

Network for the fifth bicluster identified in the Escherichia coli data. 

# construct the network for the 4th and 8th identified bicluster in Escherichia coli data
net <- qunetwork(ecoli, res, number = c(4, 8), group = c(4, 8), method = "spearman")
if (requireNamespace("qgraph", quietly = TRUE))
    qgraph::qgraph(net[[1]], groups = net[[2]], legend.cex = 0.5, layout = "spring", 
        minimum = 0.6, color = c("red", "blue", "gold", "gray"), edge.label = FALSE)
## Warning in qgraph::qgraph(net[[1]], groups = net[[2]], legend.cex = 0.5, : The following
## arguments are not documented and likely not arguments of qgraph and thus ignored: edge.label
Network for the fourth and eighth biclusters identified in the *Escherichia coli* data.

Network for the fourth and eighth biclusters identified in the Escherichia coli data.

Query-based biclustering

We can conduct a query-based biclustering by adding the weight parameter. In this example, the instance file “511145.protein.links.v10.txt” (Szklarczyk et al. 2014) was downloaded from string (http://string-db.org/download/protein.links.v10/511145.protein.links.v10.txt.gz) and decompressed and saved in working directory.

# Here is an example to download and extract the weight
library(igraph)
url <- "http://string-db.org/download/protein.links.v10/511145.protein.links.v10.txt.gz"
tmp <- tempfile()
download.file(url, tmp)
graph = read.graph(gzfile(tmp), format = "ncol")
unlink(tmp)
ecoli.weight <- get.adjacency(graph, attr = "weight")
library(QUBIC)
library(QUBICdata)
data("ecoli", package = "QUBICdata")
data("ecoli.weight", package = "QUBICdata")
res0 <- biclust(ecoli, method = BCQU(), verbose = FALSE)
res0
## 
## An object of class Biclust 
## 
## call:
##  biclust(x = ecoli, method = BCQU(), verbose = FALSE)
## 
## Number of Clusters found:  100 
## 
## First  5  Cluster sizes:
##                    BC 1 BC 2 BC 3 BC 4 BC 5
## Number of Rows:     437  519  178  180  169
## Number of Columns:   29   22   52   50   53
res4 <- biclust(ecoli, method = BCQU(), weight = ecoli.weight, verbose = FALSE)
res4
## 
## An object of class Biclust 
## 
## call:
##  biclust(x = ecoli, method = BCQU(), weight = ecoli.weight, verbose = FALSE)
## 
## Number of Clusters found:  100 
## 
## First  5  Cluster sizes:
##                    BC 1 BC 2 BC 3 BC 4 BC 5
## Number of Rows:     437  519  178  180  169
## Number of Columns:   29   22   52   50   53

Bicluster-expanding

we can expand existing biclustering results to recruite more genes according to certain consistency level:

res5 <- biclust(x = ecoli, method = BCQU(), seedbicluster = res, f = 0.25, verbose = FALSE)
summary(res5)
## 
## An object of class Biclust 
## 
## call:
##  biclust(x = ecoli, method = BCQU(), seedbicluster = res, f = 0.25, verbose = FALSE)
## 
## Number of Clusters found:  20 
## 
## Cluster sizes:
##                    BC 1 BC 2 BC 3 BC 4 BC 5 BC 6 BC 7 BC 8 BC 9 BC 10 BC 11 BC 12 BC 13 BC 14
## Number of Rows:     593  151   51  110  117   68   84   27   43    20    36    30    17    19
## Number of Columns:   29   45   94   44   38   38   31   33   31    27    19    20    23    21
##                    BC 15 BC 16 BC 17 BC 18 BC 19 BC 20
## Number of Rows:       14    16    16    11     5     6
## Number of Columns:    26    20    22    25    32    23

Biclusters comparison

We can compare the biclustering results obtained from different algorithms, or from a same algorithm with different combinations of parameter.

test <- ecoli[1:50,]
res6 <- biclust(test, method = BCQU(), verbose = FALSE)
res7 <- biclust (test, method = BCCC())
res8 <- biclust(test, method = BCBimax())
showinfo (test, c(res6, res7, res8))
## 1: Call and Parameter
## 2: number of detected biclusters
## 3: nrow of the first bicluster
## 4: ncol of the first bicluster
## 5: area of the first bicluster
## 6: ratio (nrow / ncol) of the first bicluster
## 7: ratio (nrow / ncol) of the matrix
## 8: max nrow and corresponding bicluster
## 9: max ncol and corresponding bicluster
## 10: max area and corresponding bicluster
## 11: union of rows, (# and %)
## 12: union of columns, (# and %)
## 13: overlap of first two biclusters (row, col, area)
## 
## 
## 1    2   3   4   5   6   7   8   9   10  11  12  13  
## biclust(x = test, method = BCQU(), verbose = FALSE)  59  11  25  275 0.44    0.1072961   11 1    42 12   275 1   49 98   283 60.72961    1 0 0
## biclust(x = test, method = BCCC())   1   50  466 23300   0.1072961   0.1072961   50 1    466 1   23300 1 50 100  1 100   
## biclust(x = test, method = BCBimax())    1   50  466 23300   0.1072961   0.1072961   50 1    466 1   23300 1 50 100  466 100

References

Brazma, Alvis, and Jaak Vilo. 2000. “Gene Expression Data Analysis.” FEBS Letters 480 (1): 17–24.
Cheng, Yizong, and George M Church. 2000. “Biclustering of Expression Data.” In Proceedings of the Eighth International Conference on Intelligent Systems for Molecular Biology, edited by Philip Bourne, Michael Gribskov, Russ Altman, Nancy Jensen, Debra Hope, Thomas Lengauer, Julie Mitchell, et al., 8:93–103. AAAI Press, Menlo Park, CA.
Eren, Kemal, Mehmet Deveci, Onur Küçüktunç, and Ümit V. Çatalyürek. 2012. “A Comparative Analysis of Biclustering Algorithms for Gene Expression Data.” Briefings in Bioinformatics. https://doi.org/10.1093/bib/bbs032.
Fehrmann, Rudolf SN, Juha M Karjalainen, Małgorzata Krajewska, Harm-Jan Westra, David Maloney, Anton Simeonov, Tune H Pers, et al. 2015. “Gene Expression Analysis Identifies Global Gene Dosage Sensitivity in Cancer.” Nature Genetics 47 (2): 115–25.
Gu, Jiajun, and Jun S Liu. 2008. “Bayesian Biclustering of Gene Expression Data.” BMC Genomics 9 (Suppl 1): S4.
Lazzeroni, Laura, Art Owen, et al. 2002. “Plaid Models for Gene Expression Data.” Statistica Sinica 12 (1): 61–86.
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