A mesh generation and machine learning framework for Drosophilagene expression pattern image analysis
 Wenlu Zhang^{1},
 Daming Feng^{1},
 Rongjian Li^{1},
 Andrey Chernikov^{1},
 Nikos Chrisochoides^{1},
 Christopher Osgood^{2},
 Charlotte Konikoff^{3},
 Stuart Newfeld^{4},
 Sudhir Kumar^{3, 4, 5} and
 Shuiwang Ji^{1}Email author
DOI: 10.1186/1471210514372
© Zhang et al.; licensee BioMed Central Ltd. 2013
Received: 29 July 2013
Accepted: 16 December 2013
Published: 28 December 2013
Abstract
Background
Multicellular organisms consist of cells of many different types that are established during development. Each type of cell is characterized by the unique combination of expressed gene products as a result of spatiotemporal gene regulation. Currently, a fundamental challenge in regulatory biology is to elucidate the gene expression controls that generate the complex body plans during development. Recent advances in highthroughput biotechnologies have generated spatiotemporal expression patterns for thousands of genes in the model organism fruit fly Drosophila melanogaster. Existing qualitative methods enhanced by a quantitative analysis based on computational tools we present in this paper would provide promising ways for addressing key scientific questions.
Results
We develop a set of computational methods and open source tools for identifying coexpressed embryonic domains and the associated genes simultaneously. To map the expression patterns of many genes into the same coordinate space and account for the embryonic shape variations, we develop a mesh generation method to deform a meshed generic ellipse to each individual embryo. We then develop a coclustering formulation to cluster the genes and the mesh elements, thereby identifying coexpressed embryonic domains and the associated genes simultaneously. Experimental results indicate that the gene and mesh coclusters can be correlated to key developmental events during the stages of embryogenesis we study. The open source software tool has been made available at http://compbio.cs.odu.edu/fly/.
Conclusions
Our mesh generation and machine learning methods and tools improve upon the flexibility, easeofuse and accuracy of existing methods.
Background
Advances in sequencing and geneprediction technologies have led to the discovery of virtually complete sets of proteincoding sequences in many model systems. In contrast, how these coding sequences are controlled by the regulatory sequences to transform a single cell, through cell division and differentiation, into a complex multicellular organism remains largely unknown. In multicellular organisms, one of the primary purposes of gene control is execution of the genomic regulatory code to generate complex body plans during development [1, 2]. This process critically depends on the right gene being activated in the right cell (spatially) at the right time (temporally). Thus, analysis of spatiotemporal gene expression patterns provides a promising way for investigating the gene regulatory networks governing development.
In developmental biology, the fruit fly Drosophila melanogaster has long been established as a canonical model organism [3, 4]. Recent advances in highthroughput in situ hybridization (ISH) technologies have allowed scientists to produce spatiotemporal expression patterns for thousands of genes in Drosophila [58]. This wealth of data creates opportunities for studying the developmental regulatory networks. However, the sheer volume and complexity of these data preclude the traditional practice of manual analysis and make automated methods essential [816].
In this work, we develop a set of ISH image computing and machine learning methods for the automated analysis of Drosophila gene expression pattern images. Specifically, we develop a mesh generation pipeline for mapping the expression patterns of many genes into the same geometric space [8]. This enables accurate comparative analysis of the spatial expression patterns of multiple genes and accounts for the differences in embryo morphology. We fit an ellipse to the boundary of each embryo using the least squares criterion. We then average the fitted ellipses for all images in the same stage range to obtain a generic ellipse. We automatically interpolate the boundary of this generic ellipse and use a Delaunay mesh method [1720] to generate a triangulated mesh on this ellipse.
We accurately capture the morphology of each embryo by employing a systematic procedure to deform the generic, meshed ellipse to each individual embryo. We first establish correspondences between vertices on the generic ellipse and those on the fitted ellipses. Then the vertices on the fitted ellipses are deformed to the embryo boundary using the minimum distance criterion. Finally, the coordinates of all the other vertices are computed by solving an elastic finite element problem.
The mesh generation scheme allows us to organize the expression pattern images of many genes into a data matrix in which one dimension corresponds to genes and the other dimension corresponds to mesh elements as in the GenomewideExpressionMaps (GEMs) [12, 21]. To identify coexpressed embryonic domains and the associated genes, we develop a coclustering formulation to cluster the mesh elements and the genes simultaneously. We formulate the coclustering problem using a maximum likelihood formalism and employ an expectationmaximization algorithm to perform the parameter estimation.
We apply the mesh generation and coclustering methods to a set of gene expression pattern images in the FlyExpress database [12]. Our results show that our methods generate coexpressed domains that overlap with many embryonic structures. In addition, these results show that the proposed methods yield gene clusters that are functionally more enriched than those discovered in prior studies. More importantly, we show that the mesh and gene coclusters correlate strongly with key developmental events during the stages of embryogenesis under investigation.
Methods
Mesh generation
Requirements
Let I_{1},…,I_{ m } be a list of embryo images. The goal of this module of the pipeline is to overlay each of the embryo images with a triangular mesh, such that all meshes have the same number of triangles and connectivity. For a given image, all triangles we create are of approximately the same size, in terms of their area. Let a stand for an upper bound on triangle area. Then all triangles in a single mesh which we construct have area slightly less than a. Let M_{ j }(a) be the mesh that we construct for image I_{ j } that depends on area bound a. For simplicity we will omit the parameter a below.
More precisely, let M_{ j } = (V_{ j },T_{ j }), where V_{ j } is the list of vertices and T_{ j } is the list of triangles. Each vertex is defined by its twodimensional coordinate, and each triangle is defined by a triple of vertex indices (p_{1},p_{2},p_{3}), 1 ≤ p_{1},p_{2},p_{3} ≤ V_{ j }. These meshes are expected to satisfy the following requirements:

All of the T_{ j } contain the same number of triangles, i.e., T_{ j } = T_{ i } for i,j = 1,…,m.

All of the T_{ j } contain the same triples of vertex indices in the corresponding positions. As a result, we can omit the subscript and use T for all meshes M_{ j }, j = 1,…,m.

All of the V_{ j } contain the same number of vertices: V_{ j } = V_{ i } for i,j = 1,…,m.

All vertices on the boundary of mesh M_{ j } lie on the boundary of the embryo of image I_{ j }.

Each triangle in M_{ j } = M_{ j }(a) has area approximately equal to a.

All vertices in V_{ j } are geometrically close to the vertices in the corresponding positions in V_{ i } for all i,j = 1,…,m, with respect to their location within an embryo.
Construction and meshing of the average ellipse
For each image I_{ j }, j = 1,…,m, we compute the parameters of the equation of the ellipse E_{ j } that realizes the best fit to the boundary of the embryo in this image. We compute the best fitted ellipse using the least squares criterion to the set of the embryo’s boundary pixels. Then we average the parameters of all ellipses to obtain the average ellipse E^{′}.
Given a value of a, we construct a mesh of E^{′}. First, we use linear interpolation to approximate the boundary of E^{′}, and then use a Delaunay mesh generator, Triangle [17], to mesh the interior of E^{′}. Delaunay refinement is our meshing method of choice since it is backed by proven theoretical guarantees [1820] that make it a pushbutton technology: its being able to guarantee termination with angle and area bounds allow for a guaranteed quality automatic pipeline.
We interpolate the boundary of E^{′} by performing the following steps. First, we calculate the side length ℓ of an equilateral triangle with area a. Then we use an iterative subdivision of the boundary of E^{′} with a set of vertices v_{1},…,v_{ s } = v_{0} until all segment lengths v_{i1}v_{ i }, i = 1,…,s are approximately equal to ℓ. In other words, this is a uniform distribution of vertices with respect to the lengths of segments. The union of all these segments is a piecewise linear interpolation of the boundary of E^{′}.
To tessellate the interior of E^{′}, we use Triangle with the following parameters:

A planar straight line graph (PSLG) composed of the segments and the points interpolating the boundary of E^{′} plus one point in the center of E^{′}. We instruct Triangle to preserve this PSLG and not to split the boundary segments, so that the discretization of the PSLG appears as a subgraph of the final mesh.

The area bound a instructing Triangle to produce all triangles with areas bounded from above by a. Triangle starts with a coarse mesh and iteratively splits triangles until their areas fall below a, and therefore this is an approximate target area.

An angle bound of 25° which instructs Triangle to enforce all angles in the final mesh to be 25° or above. Theoretically, Triangle guarantees only a minimum angle bound of 20.7° or below, however we found that in practice it can mesh an ellipse with a 25° angle bound, since it is a simple shape.
Let the mesh of the average ellipse be denoted as M^{′}, and the list of radial angles corresponding to the subdivision vertices as ${\theta}_{1}^{\prime},\dots ,{\theta}_{s}^{\prime}$.
Deformation of the mesh of the average ellipse
For each ellipse E_{ j }, we use the angles ${\theta}_{1}^{\prime},\dots ,{\theta}_{s}^{\prime}$ to find the vertices that discretize the boundary of E_{ j }. Then we project these vertices onto the closest points from the boundary of the embryo in image I_{ j }. We define closeness in terms of the Euclidean distance, and use the Matlab’s Euclidean distance transform function to find the nearest boundary pixels simultaneously for all pixels in the image. Using the result of this function, we determine the required projections.
For each image I_{ j }, we deform the mesh M^{′}, such that the boundary vertices of M^{′} assume the coordinates of the corresponding vertices (with respect to their radial ordering) on the boundary of the embryo in I_{ j }. The target coordinates of all the other vertices in V^{′} are computed by solving an elastic finite element problem [22]. As a result, the triangles of the generic mesh are deformed minimally and proportionally to their distance to the projected vertices on the boundary of the embryo in I_{ j } and to the amount of the displacement at these boundary vertices.
Simultaneous clustering of mesh elements and genes
For a mesh with n elements (triangles), we assume that the elements are numbered from 1 to n in an arbitrary but fixed order. Following [8], we extract the median of graylevel intensities from each mesh element and represent each image using an ndimensional vector in which the ith component contains the median of intensities from the ith mesh element. Then the expression patterns of m genes can be encoded into a data matrix $A\in {\mathbb{R}}^{m\times n}$, in which each row corresponds to a gene, and each column corresponds to a mesh element. Note that, to simplify the notation, we assume that the number of images and the number of genes are the same. When the expression pattern of a gene is captured by multiple images, we treat them separately.
In [8], two clustering methods are applied independently to identify clusters in the rows or the columns of the matrix A. In their case the rowwise (columnwise) clustering requires the rows (columns) in the same cluster to be similar with respect to all columns (rows). However, a set of genes might be coexpressed only at certain local domain of the embryo corresponding to a subset of mesh elements. To identify the coexpressed embryonic domains and the associated genes, we employ a coclustering method to cluster the rows and columns of the data matrix A simultaneously. This generates coclusters consisting of a subset of genes that are coexpressed at a subset of mesh elements. Note that entries of matrix A encode the expression intensities of genes and thus are nonnegative. An appealing property of our coclustering method is that it is based on a probabilistic model and thus preserves the nonnegativity in the estimated parameters. It has been shown in [23] that a variant of this model consistently outperforms other methods that do not preserve nonnegativity.
A coclustering formulation
where the vertices in one set is ordered before vertices in the other set.
which matches the structure of W in Eq. (1).
It can be shown [24] that maximizing the log likelihood in Eq. (6) is equivalent to minimizing the divergence loss of the approximation in Eq. (2).
An EM algorithm
The results are then normalized such that ∑_{i} P_{ ik } = 1 and ∑_{j} Q_{ jk } = 1. The Estep and Mstep are repeated until a locally optimal solution is obtained. Then the matrices P and Q can be used as row and column cocluster indicator matrices, respectively, to obtain soft coclustering results. A variant of this method has been shown to compare favorably with other approaches on a variety of data sets [23].
Related work
Our work is related to the seminal work in [25], where the Gaussian mixture models (GMM) were applied to generate coexpression domains for the purpose of image comparison. Our work is different from [25] in both its objectives and approaches. In [25], image pixels were considered directly as the basic elements of modeling while we use triangulated mesh to warp and discretize the embryos in order to account for the shape and morphological variations. It has been shown in prior work [8] that the use of mesh leads to biologically significant results. In addition, GMM was used to cluster the pixels in [25], while we use a coclustering method to cocluster the mesh elements and the genes simultaneously. Since each domain is expected to be defined by only a subset of genes in the genome, coclustering aims at identifying the domains and the associated genes simultaneously. As shown by our experimental results, coclustering leads to more significant results.
Results and discussion
We evaluate the proposed computational methods on a set of gene expression pattern images retrieved from the FlyExpress database [12]. This database contains genomewide, twodimensional, standardized images obtained from multiple sources, including the Berkeley Drosophila Genome Project [5]. Other databases provide threedimensional images with higher resolution, but the data are not on the genomescale [26]. Following [8], we focus on stages 46 and generate two data sets. The larger data set contains 2693 images capturing the expression patterns of 1881 genes, and the smaller one is a subset of 553 images corresponding to 365 genes with clearly defined expression boundaries. The images are preprocessed by a set of tools developed in [8] before they are tesselated with our mesh generation tools. We apply the proposed mesh generation method to convert a set of images into a data matrix in which the rows correspond to genes and the columns correspond to mesh elements. We apply the coclustering method to compute coclusters of genes and mesh elements. We first study the mesh clusters and gene clusters separately in Sections “Clustering of mesh elements” and “Clustering of genes”, respectively. We then correlate mesh and gene coclusters with developmental events in Section “Coclustering of mesh elements and genes.
Clustering of mesh elements
Figures 2 and 3 reveal the resulting clusters resemble the fate map of the developing embryo [27]. The clusters represent domains of high coexpression. They invariably form spatially contiguous regions, and are composed of rectangular shapes. Further, the cluster boundaries are largely parallel to the anterior/posterior (A/P) and dorsal/ventral (D/V) axes of the embryo. As the number of coclusters is increased (Figure 2), the rectangular cluster shape is often retained, with larger clusters subdivided into smaller ones. In our data set, this subdivision of clusters often occurred at the far A/P and D/V regions of the embryo. These increased subdivisions correlate with major developmental events during stages 46 of Drosophila embryogenesis [4, 27]. Signals along the A/P and D/V axes drive this pattern formation [3]. During Stage 6 gastrulation begins, and the ventral and cephalic furrows form. Looking back at the clusters, we see a greater proportion of subdivisions along where these furrows form in the developing embryo. The general clustering patterns remain the same while the cluster boundaries become smoother as the number of mesh elements increases (Figure 3).
Clustering of genes
Coclustering of the data matrix leads to clusters of genes. We use gene ontology (GO) [28] to evaluate the gene clusters and compare the results with those reported in [8]. Our gene clusters are the combined results of the mesh generation and coclustering methods. Hence, we evaluate the effects of these two methods separately.
The numbers of enriched gene ontology terms generated by the original (Original) and the proposed (New) mesh generation methods
Number of clusters  Biological process  Cellular component  Molecular function  

New  Original  New  Original  New  Original  
30  168  169  36  36  43  43 
31  168  169  36  36  43  43 
32  155  156  35  35  38  38 
33  174  175  30  30  40  40 
34  174  175  30  30  40  40 
35  169  170  30  30  38  38 
36  189  176  30  29  38  38 
37  189  176  30  29  38  38 
38  189  176  30  29  38  38 
39  192  177  32  31  38  38 
40  192  177  32  31  38  38 
41  222  209  19  18  24  22 
42  227  209  20  19  29  22 
43  232  209  21  19  28  22 
44  231  209  27  19  28  22 
45  234  209  28  19  33  22 
46  234  209  28  19  33  22 
47  234  209  28  19  33  22 
48  234  209  28  19  33  22 
49  234  209  28  19  33  22 
50  228  209  28  19  27  23 
51  228  210  28  20  27  23 
52  228  221  29  20  27  25 
53  228  217  29  20  27  25 
54  228  196  29  19  27  25 
55  228  196  29  19  27  25 
56  228  195  29  22  27  25 
57  228  203  29  22  27  21 
58  228  204  29  22  28  25 
59  228  204  29  23  28  25 
60  229  208  29  21  28  26 
The significantly different results might be due to the fundamentally different approaches taken by the two studies. Specifically, Frise et al. [8] used clustering method to group the genes into clusters based on all the mesh elements. In another word, clustering method measures the expression patterns of genes across the whole embryo. That is, for two genes to be in the same cluster, they need to have similar expression patterns over the entire embryo. In comparison, we propose to use a coclustering method, which identifies gene and mesh coclusters simultaneously. In our approach, two genes can be grouped into the same cluster if they share similar local expression patterns. Note that coclustering was mainly motivated from gene expression studies [30], and our results show that coclustering method yields statistically more significant results.
Coclustering of mesh elements and genes
We next evaluate the gene coclusters, and correlate the results with major developmental events occurring during stages 46. To accomplish this, we first apply our mesh generation and coclustering methods to the data set of 2693 images depicting gene expression in stage 46 laterally oriented embryos [8]. We set the number of coclusters to 39 as in [8]. Then, enriched GO terms (biological process) are computed (pvalue < 0.001). A onesided significance test is applied, and enriched terms with ≥90% significance were retained. Of the 39 clusters, 21 are enriched in at least one term.
Among the many genes showing expression in clusters located in the forming cephalic furrow, we find a subset of genes known to be involved in the mesodermal developmental network [31, 32] among the images in our data set. These include the transcription factors twist, snail, Mes2, brinker, and tinman. These genes exhibit high coexpression, and are expressed in the ventral region of the embryo during stages 46 [31, 32]. We obtain similar results when examining other clusters located in close proximity to each other, overall suggesting that the discovered gene and mesh coclusters correlate well with major developmental events associated with the stage range.
Lastly, we examine the 18 clusters showing no GO biological process enrichment in our stage 46 coclustering. These include clusters 3, 6, 14, 34, and 37. Looking back at the images, we find a lack of localized gene expression at these embryonic domains during stages 46. These clusters initially form a single cluster in the interior region of embryo when the number of clusters is small (Figure 2). These regions are involved in later developmental processes and are not involved in the major developmental events occurring during stages 46 of Drosophila embryogenesis.
Conclusion
In this study, we aim at identifying coexpressed embryonic domains and the associated genes simultaneously. We develop a mesh generation pipeline that maps the expression patterns of many genes into the same coordinate space. We then employ a coclustering formulation to cluster the mesh elements and the genes. This identifies coexpressed genes and spatial embryonic domains simultaneously. Experimental results show that the embryonic domains identified in this purely datadriven manner correspond to many embryonic structures. Results also show that the gene and mesh coclusters correlate with major developmental events during the stages we study.
In the current mesh generation method, we only consider the shapes of embryos when deforming the generic ellipse to each embryo. A more accurate deformation method would take the intensity and texture information of images into account. We will develop more advanced mesh generation method in the future. In this work, we focus on a particular time period of development. We will extend our analysis to multiple stages and employ timevarying analysis in the future [23].
Declarations
Acknowledgements
We thank Bernard Van Emden and Michael McCutchan for help with access to the image data, Dr. Erwin Frise for help with interpreting their data and results. This work is supported by research grants from National Science Foundation (DBI1147134 and CCF1139864), National Institutes of Health (HG00251609), Old Dominion University Office of Research, and the Richard T. Cheng Endowment.
Authors’ Affiliations
References
 Lodish H, Berk A, Kaiser CA, Krieger M, Scott MP, Bretscher A, Ploegh H, Matsudaira P: Molecular Cell Biology. 2007, New York: W. H. FreemanGoogle Scholar
 Davidson EH: The Regulatory Genome: Gene Regulatory Networks in Development and Evolution. 2006, Burlington: Academic PressGoogle Scholar
 Wolpert L, Smith J, Jessell T, Lawrence P, Robertson E, Meyerowitz E: Principles of Development. 2006, Oxford: Oxford University PressGoogle Scholar
 CamposOrtega JA, Hartenstein V: The Embryonic Development of Drosophila Melanogaster. 1997, New York: SpringerView ArticleGoogle Scholar
 Tomancak P, Beaton A, Weiszmann R, Kwan E, Shu S, Lewis SE, Richards S, Ashburner M, Hartenstein V, Celniker SE, Rubin GM: Systematic determination of patterns of gene expression during Drosophila embryogenesis. Genome Biol. 2002, 3 (12): research0088.10088.14..View ArticleGoogle Scholar
 Tomancak P, Berman B, Beaton A, Weiszmann R, Kwan E, Hartenstein V, Celniker S, Rubin G: Global analysis of patterns of gene expression during Drosophila embryogenesis. Genome Biol. 2007, 8 (7): R14510.1186/gb200787r145.PubMed CentralView ArticlePubMedGoogle Scholar
 Lécuyer E, Yoshida H, Parthasarathy N, Alm C, Babak T, Cerovina T, Hughes T, Tomancak P, Krause H: Global analysis of mRNA localization reveals a prominent role in organizing cellular Architecture and function. Cell. 2007, 131: 174187. 10.1016/j.cell.2007.08.003.View ArticlePubMedGoogle Scholar
 Frise E, Hammonds AS, Celniker SE: Systematic imagedriven analysis of the spatial Drosophila embryonic expression landscape. Mol Syst Biol. 2010, 6: 345PubMed CentralView ArticlePubMedGoogle Scholar
 Kumar S, Jayaraman K, Panchanathan S, Gurunathan R, MartiSubirana A, Newfeld SJ: BEST: a novel computational approach for comparing gene expression patterns from early stages of Drosophila melanogaster develeopment. Genetics. 2002, 169: 20372047.Google Scholar
 Lécuyer E, Tomancak P: Mapping the gene expression universe. Curr Opin Genet Dev. 2008, 18 (6): 506512. 10.1016/j.gde.2008.08.003.View ArticlePubMedGoogle Scholar
 Walter T, Shattuck DW, Baldock R, Bastin ME, Carpenter AE, Duce S, Ellenberg J, Fraser A, Hamilton N, Pieper S, Ragan MA, Schneider JE, Tomancak P, Hériché JK: Visualization of image data from cells to organisms. Nat Methods. 2010, 7: S26S41. 10.1038/nmeth.1431.PubMed CentralView ArticlePubMedGoogle Scholar
 Kumar S, Konikoff C, Van Emden B, Busick C, Davis KT, Ji S, Wu LW, Ramos H, Brody T, Panchanathan S, Ye J, Karr TL, Gerold K, McCutchan M, Newfeld SJ: FlyExpress: visual mining of spatiotemporal patterns for genes and publications in Drosophila embryogenesis. Bioinformatics. 2011, 27 (23): 33193320. 10.1093/bioinformatics/btr567.PubMed CentralView ArticlePubMedGoogle Scholar
 Peng H: Bioimage informatics: a new area of engineering biology. Bioinformatics. 2008, 24 (17): 18271836. 10.1093/bioinformatics/btn346.PubMed CentralView ArticlePubMedGoogle Scholar
 Ji S, Sun L, Jin R, Kumar S, Ye J: Automated annotation of Drosophila gene expression patterns using a controlled vocabulary. Bioinformatics. 2008, 24 (17): 18811888. 10.1093/bioinformatics/btn347.PubMed CentralView ArticlePubMedGoogle Scholar
 Yuan L, Woodard A, Ji S, Jiang Y, Zhou ZH, Kumar S, Ye J: Learning sparse representations for fruitfly gene expression pattern image annotation and retrieval. BMC Bioinformatics. 2012, 13: 10710.1186/1471210513107.PubMed CentralView ArticlePubMedGoogle Scholar
 Ji S, Li YX, Zhou ZH, Kumar S, Ye J: A bagofwords approach for Drosophila gene expression pattern annotation. BMC Bioinform. 2009, 10: 11910.1186/1471210510119.View ArticleGoogle Scholar
 Shewchuk JR: Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. Applied Computational Geometry: Towards Geometric Engineering, Volume 1148 of Lecture Notes in Computer Science. Edited by: Lin MC, Manocha D. 1996, Berlin: SpringerVerlag, 203222.Google Scholar
 Shewchuk JR: Delaunay refinement algorithms for triangular mesh generation. Comput Geometry: Theory Appl. 2002, 22 (13): 2174.View ArticleGoogle Scholar
 Foteinos P, Chernikov A, Chrisochoides N: Fully generalized 2D constrained Delaunay mesh refinement. SIAM J Sci Comput. 2010, 32: 26592686. 10.1137/090763226.View ArticleGoogle Scholar
 Chernikov A, Chrisochoides N: Generalized insertion region guides for Delaunay mesh refinement. SIAM J Sci Comput. 2012, 34: A1333A1350. 10.1137/100809076.View ArticleGoogle Scholar
 Goering LM, Hunt PK, Heighington C, Busick C, Pennings PS, Hermisson J, Kumar S, Gibson G: Association of orthodenticle with natural variation for early embryonic patterning in Drosophila melanogaster. J Exp Zool Part BMol Dev Evol. 2009, 312B: 841854. 10.1002/jez.b.21299.View ArticleGoogle Scholar
 Zienkiewicz OC, Taylor RL, Zhu JZ: The Finite Element Method: Its Basis and Fundamentals. 2005, Oxford: ButterworthHeinemannGoogle Scholar
 Zhang W, Ji S, Zhang R: Evolutionary Soft CoClustering. Proceedings of the 2013 SIAM International Conference on Data Mining. 2013, Philadelphia: Society for Industrial and Applied Mathematics, 121129.View ArticleGoogle Scholar
 Yu K, Yu S, Tresp V: Soft clustering on graphs. Advances in Neural Information Processing Systems 18. Edited by: Weiss Y, Schölkopf B, Platt J. 2006, Cambridge: MIT Press, 15531560.Google Scholar
 Peng H, Myers EW: Comparing in situ mRNA expression patterns of Drosophila embryos. Proceedings of the Eighth Annual International Conference on Resaerch in Computational Molecular Biology. 2004, New York: ACM, 157166.Google Scholar
 Fowlkes CC, Luengo Hendriks CL, Keränen SV, Weber GH, Rübel O, Huang MY, Chatoor S, DePace AH, Simirenko L, Henriquez C, Beaton A, Weiszmann R, Celniker S, Hamann B, Knowles DW, Biggin MD, Eisen MB, Malik J: A quantitative spatiotemporal atlas of gene expression in the Drosophila blastoderm. Cell. 2008, 133 (2): 364374. 10.1016/j.cell.2008.01.053.View ArticlePubMedGoogle Scholar
 Hartenstein V: Atlas of Drosophila Development. 1995, Cold Spring Harbor: Cold Spring Harbor Laboratory PressGoogle Scholar
 Ashburner M, Ball CA, Blake JA, Botstein D, Butler H, Cherry JM, Davis AP, Dolinski K, Dwight SS, Eppig JT, Harris MA, Hill DP, IsselTarver L, Kasarskis A, Lewis S, Matese JC, Richardson JE, Ringwald M, Rubin GM, Sherlock G: Gene ontology: tool for the unification of biology. Nat Genet. 2000, 25: 2529. 10.1038/75556.PubMed CentralView ArticlePubMedGoogle Scholar
 Boyle EI, Weng S, Gollub J, Jin H, Botstein D, Cherry JM, Sherlock G: GO::TermFinderopen source software for accessing gene ontology information and finding significantly enriched gene ontology terms associated with a list of genes. Bioinformatics. 2004, 20 (18): 37103715. 10.1093/bioinformatics/bth456.PubMed CentralView ArticlePubMedGoogle Scholar
 Cheng Y, Church GM: Biclustering of expression data. Proceedings of the Eighth International Conference on Intelligent Systems for Molecular Biology. 2000, Menlo Park: The AAAI Press, 93103.Google Scholar
 Stathopoulos A, Levine M: Genomic regulatory networks and animal development. Dev Cell. 2005, 9 (4): 449462. 10.1016/j.devcel.2005.09.005.View ArticlePubMedGoogle Scholar
 Sandmann T, Girardot C, Brehme M, Tongprasit W, Stolc V, Furlong EE: A core transcriptional network for early mesoderm development in Drosophila melanogaster. Genes Dev. 2007, 21 (4): 436449. 10.1101/gad.1509007.PubMed CentralView ArticlePubMedGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.