Analysis of tiling array expression studies with flexible designs in Bioconductor (waveTiling)
© De Beuf et al.; licensee BioMed Central Ltd. 2012
Received: 22 May 2012
Accepted: 5 September 2012
Published: 14 September 2012
Existing statistical methods for tiling array transcriptome data either focus on transcript discovery in one biological or experimental condition or on the detection of differential expression between two conditions. Increasingly often, however, biologists are interested in time-course studies, studies with more than two conditions or even multiple-factor studies. As these studies are currently analyzed with the traditional microarray analysis techniques, they do not exploit the genome-wide nature of tiling array data to its full potential.
We present an R Bioconductor package, waveTiling, which implements a wavelet-based model for analyzing transcriptome data and extends it towards more complex experimental designs. With waveTiling the user is able to discover (1) group-wise expressed regions, (2) differentially expressed regions between any two groups in single-factor studies and in (3) multifactorial designs. Moreover, for time-course experiments it is also possible to detect (4) linear time effects and (5) a circadian rhythm of transcripts. By considering the expression values of the individual tiling probes as a function of genomic position, effect regions can be detected regardless of existing annotation. Three case studies with different experimental set-ups illustrate the use and the flexibility of the model-based transcriptome analysis.
The waveTiling package provides the user with a convenient tool for the analysis of tiling array trancriptome data for a multitude of experimental set-ups. Regardless of the study design, the probe-wise analysis allows for the detection of transcriptional effects in both exonic, intronic and intergenic regions, without prior consultation of existing annotation.
In the last few years tiling microarrays have become a well-established tool for whole-genome transcriptome analysis. They have shown to be very useful for exploring and unraveling the complex genome-wide trancriptional landscape of higher organisms, in which not only protein coding genes, but also non-coding RNAs play an important role [1–4]. The methods that have been developed for transcriptome analysis with tiling arrays either focus on segmentation and transcript discovery within a single biological condition [5–8], or on the detection of differential expression between two distinct conditions [9, 10]. Recently, the focus in tiling array studies has shifted towards more complex experimental designs, such as studies with more than two conditions  and studies with several experimental factors . Furthermore, it is recognized that expression is a dynamic rather than a static phenomenon. Hence, more and more time-course experiments are designed to provide insights into the whole-genome transcript regulation of species during different developmental stages or external periodic changes in the environment [13, 14].
Currently, most tiling array transcriptome analysis pipelines start with summarization of the probe-level data. This can be done by constructing probesets from the groups of probes that map to known annotated genes, (e.g. [11, 15]). Hereby unannotated regions are disregarded. In [12, 13, 16] a sliding window-based approach is adopted, combined with a thresholding rule for selecting transcriptional units, whereas in  segments with piece-wise constant intensity levels are constructed first . After the summarization a statistical test or a more heuristic analysis technique is conducted on the summarized expression values of the transcriptional units. In current time-course and single-factor studies this is merely done by directly applying traditional microarray analysis methods, such as a pairwise moderated t-test (Limma)  conducted in  or a permutated t-test (SAM)  conducted in . Other studies adopt ad-hoc approaches to filter the genes or transcriptional units of interest. Transcriptional units in a time-course experiment, for example, can be filtered based on thresholding the amplitude of the signal . In an alternative approach the correlations between temporal expression patterns are explored and a clustering is performed of genomic regions based on expression profiles in different gene classes showing expression at different time-points . The tests reported in  and  on the other hand are less ad hoc, but very specific for the periodic time-course design apparent in these studies [22–24]. The aforementioned methods either lack flexibility by only focusing on one specific experimental design, or they first summarize probes to probesets based on existing annotation, hence not exploiting the genome-wide nature of the data to the full extent.
Here, we present waveTiling, a R Bioconductor package for transcriptome analysis of tiling arrays with flexible designs. The package is based on and provides an extension to a recently introduced wavelet-based functional model for transcriptome analysis . While the methodology in  was initially developed to conduct the simultaneous tasks of transcript discovery and detection of differential expression, their framework can be easily extended by adapting the model design matrix. After modeling the specific effect function of interest, probe-wise inference can be conducted for detecting affected regions. The probe-wise analysis allows for the detection of transcriptional units in both exonic, intronic and intergenic regions, without prior consultation of existing annotation. Currently, waveTiling provides a standard analysis flow for transcriptome analysis on single-factor experiments with two or more biological conditions, the detection of linear and quadratic effects and circadian rhythms in time-course experiments, and the analysis of two-factor experiments, while more experienced users can also specify customized designs. Furthermore, it generates along-genome plots and contains functions to easily extract the detected genes and unannotated regions. The Implementation section gives an overview of the main functionalities of the waveTiling package and describes the model for the different designs, as well as the associated inference procedures. In Results and Discussion we illustrate the use of the package and the model on three different case studies with very distinct experimental designs.
The waveTiling package is an add-on package to the Bioconductor project  written in the programming language and statistical environment R . It provides all the tools necessary to conduct a full analysis of tiling microarray experiments for flexible designs based on the recently introduced wavelet-based functional model for trancriptome analysis . The package uses the standard Bioconductor S4-class data structures making it fully compatible with existing packages. The data is imported with the aid of the oligo-package  and the resulting object inherits from TilingFeatureSet, which is specifically designed for representing tiling array data and in turn extends ExpressionSet. Existing instance methods from oligo and other Bioconductor packages supporting this structure are therefore applicable as well. Before starting the analysis the probes can be remapped to the existing annotation. Moreover, probes that contain duplicated sequences for perfect match and mismatch probes or for probes on different strands can be filtered because they are deemed unreliable due to cross-hybridization effects. The main transcriptome analysis consists of two consecutive steps: (1) fitting the wavelet-based functional model to the data, and (2) model-based inference to identify transcriptionally affected regions. The fitted model is stored in a WfmFit-class object. Depending on the design of the study a WfmFitFactor (factorial design), WfmFitTime (time-course design), WfmFitCircadian (circadian rhythm design) or WfmFitCustom (custom design) subclass is used. Part of the code for fitting the model is implemented in C to speed up computation. In the second step, different inference procedures can be conducted depending on the research question. The inference procedure that can be conducted depends on the WfmFit-subclass. The results are stored as a WfmInf -class object. There are 3 main subclasses: WfmInfCompare which contains the results of a pairwise comparison between two groups or time points; WfmInfMeans with the results of transcript discovery for each individual group or time point; and WfmInfEffects which contains results with linear or quadratic time effects for time-course designs and circadian rhythm effects for circadian designs. All transcriptionally affected regions can be extracted from the WfmInf -class objects and are stored as IRanges-class objects . The model fitting and inference steps are described in more detail in the Statistical Methods part.
The results can be visually explored by means of a general plot function. The implementation is based on the GenomeGraphs package . For any genomic region the fitted expression values and transcriptionally affected regions can be plotted along the genomic coordinate. Furthermore, two functions are available for further post-processing of the results. Provided a suitable annotation file is given, the transcriptionally affected regions are mapped against the existing annotation. The first function outputs the genes that are transcriptionally affected, while the second function provides a list of the detected unannotated regions. The output of both functions is a list of GRanges-class objects .
We start by presenting an overview of the basic model introduced by . Subsequently, we show how we accomodate for several sampling schemes in time-course experiments or other experiments with more flexible designs.
Basic wavelet-based model for transcriptome analysis
with i=1,.. N, Y i (t) the measured log2-transformed expression values for the probe with position t (t=1,.. T) on array i (i=1,.. N). T is the number of probes that are more or less equally spaced along the genomic position of the chromosome, and N=N1 + N2 is the number of tiling arrays in the experiment, with N1 the number for biological condition 1, say C1, and N2 the number for biological condition 2, say C2. Further, X1,i is a dummy variable which is 1 for C1and −1 for C2, and E i (t) is a zero mean error term. It is assumed that E i (1),.. E i (T) are jointly MVN(0 Σ ε ). Here, MVN(μ Σ) denotes the density function of a multivariate normal distribution with mean μ and variance-covariance matrix Σ.
where is the element of B∗ corresponding to scale j and location k and m=1,2. In (4) N(μσ2) denotes the density function of a normal distribution with mean μand variance σ2. The smoothing parameters τ m (j k) and the error variances are estimated by marginal maximum likelihood using a Gauss-Seidel algorithm. The estimated induce a regularization of the wavelet coefficients of the effect functions. When backtransforming the modified coefficients to the original data space, this leads to a denoised expression signal whereby the main features are retained. The method has proven to be very fast which is essential when analyzing large datasets. For more details, see .
Wavelet-based models for transcriptome analysis in more flexible designs
To extend the modeling framework reviewed in the previous section and to make it suitable for the analysis of tiling array data with more flexible designs, the design matrix X needs to be adapted in an appropriate way. Firstly, the adaptation must enable the model to answer the specific research questions provoked by the experimental design. Secondly, it must allow us to use the same fast algorithms introduced in . This second argument comes down to the preservation of the orthogonality of X. In the first part of this section we focus on general time-course designs and single-factor designs for more than 2 groups. The second part aims at specific time-course designs for assessing circadian rhythms in the transcriptome. The section concludes with looking specifically at non-orthogonal designs, typically encountered in multi-factor studies.
General time-course designs
In tiling array time-course experiments one is often interested in the detection of differentially expressed regions between any two different time points. An additional concern might be to detect significant effects of transcriptional activity in time, e.g. linearly increasing or decreasing transcriptional expression of certain regions. These two possible research aims can be dealt with by considering a functional relationship of the designed time points described by orthogonal polynomials. This approach has also been used in quantitative trait associated expression studies based on traditional microarrays . In that paper the functional relationship with phenotype is considered instead of with time.
Just like for the polynomials, the design matrix Xbased on Helmert contrasts still needs to be normalized if the same smoothing for all factor effects is desired.
Designs for circadian rhythms
To estimate a common smoothing parameter for inducing the same amount of smoothing for all effect functions, X can again be normalized as described previously.
Design matrices for two- or multiple-factor designs are typically non-orthogonal. Using these in the wavelet-based model would imply that the fast algorithms presented in  would have to be adapted. This would lead to undesirably increased computation time during parameter estimation. A solution to this problem is to apply the Gram-Schmidt process to orthogonalize Xand subsequently estimate the model parameters based on the orthogonalized design matrix. The Gram-Schmidt orthogonalization comes down to a QR-decomposition  of X into an upper-triangular matrix X tri and an orthogonal matrix X orth , which is now used to fit the model. Afterwards, the estimated parameters have to be transformed back to obtain the parameter values for the original X. This is possible by premultiplying them with . Similar to single-factor and time-course designs, the coding of the initial design matrix X still determines how the parameters can be interpreted, and may thus be constructed according to the specific research interest.
Statistical inference: detection of transcriptional effect regions
Based on the size of A(t) circadian effect regions can be detected. In the case of non-orthogonal designs in multiple-factor studies, there are several possibilities for the choice of , depending on the aim of the analysis. The idea remains the same, however.
Results and discussion
The use and flexibility of the waveTiling package is illustrated in three case studies for transcriptome analysis with different experimental set-ups.
Case study 1: Time-course experiment
The first data set consists of a tiling array expression study for identifying the molecular events associated with early leaf development of the plant species Arabisopsis thaliana. Unraveling the underlying mechanisms of on one hand the transition from cell division to cell expansion and on the other hand the transition from non-photosynthetic to photosynthetic leaves, was the focus of this study. Trancriptome analysis for six developmental time points (day 8 to day 13) was conducted with AGRONOMICS1 tiling arrays , with three biological replicates per time point. Primarily, the researchers were focusing on the detection of differentially expressed regions between any two pairs of developmental time points. This specific study design, however, also allows for the detection of expression regions that change linearly over time. The functions and code used for this case study are described in more detail in the package vignette (see Additional file 1).
We evaluate the regions detected by the wavelet-based analysis against the genes produced by the well-established and often used RMA method . This is done by comparing the results of a gene set enrichment analysis based on both methods. By mapping the genomic regions found by the wavelet-based method to the Arabidopsis thaliana TAIR9 annotation , a list of genes is created for this method. Only genes that showed an overlap of at least 15% with the detected regions were retained. The enrichment analysis as performed with Plaza  revealed a strong overlap in the processes detected by both methods. A total of 483 enrichments were identified using both genesets of which 360 common enrichments were shared. The RMA gene list had 75 specific enrichments, while the wavelet-based gene list had 48.
Region was not in or near an exon or promoter from an annotated gene.
Longer regions containing more differentially expressed probes were preferentially selected.
Regions showing homogeneous probe directionality (all probes going in the same direction) across the entire region of differential expression were preferentially selected.
Using these criteria 12 regions were selected and qRT-PCR analysis was performed (see Additional file 2: Table S1). Of the 12 regions, 11 could be confirmed to contain differentially expressed transcripts during the time-course analysis. Only 1 region had no detectable transcriptional products. Log fold changes were calculated for confirming the expression and differential expression, as well as the directionality of the differential expression. From this analysis 9 of the 11 regions showed the same log fold change directionality as previously identified from the tiling arrays, and 2 regions showed opposite log fold change directionality. However, these 2 regions had the lowest log fold changes in the wavelet-based analysis. More details about the methods of enrichment and qRT-PCR analysis can be found in Additional file 2.
Linear and quadratic time effects
Case study 2: Circadian rhythms
The second case study concerns an expression analysis to examine circadian rhythms in Arabisopsis thaliana. It is known that photosynthetic organisms anticipate changes in the daily environment with an internal oscillator, called the circadian clock. The aim of the study was to explore the genome-wide extent of the rhythmic expression patterns governed by this oscillator. In this experiment, 12 samples were collected from Arabidopsis thaliana seedlings that were placed under a 12 h light / 12 h dark cycles regime. Every 4 hours 2 samples were taken and hybridized to the Affymetrix AtTile 1.0F and 1.0R tiling arrays. More information about the experiment can be found in .
Circadian effect for 9 genes put forward in the Hazen study
FLAVIN-BINDING KELCH DFB PROTEIN1
PSEUDO RESPONSE REGULATOR9
CIRCADIAN CLOCK ASSOCIATED1
TIME FOR COFFEE
TIMING OF CAB2 EXPRESSION1
Case study 3: Non-orthogonal two-factor design
Two-factor model gene-wise effects
In this paper, we have described the R package waveTiling for model-based analysis of tiling array expression studies with flexible designs. It implements the recently proposed wavelet-based model for transcriptome analysis  and extends its applicability towards more complex experimental set-ups. Unlike most currently applied methods, transcriptional activity is modeled at probe-level instead of gene- or exon-level. This probe-wise analysis allows for the detection of transcriptional units in both exonic, intronic and intergenic regions, without prior consultation of existing annotation. By appropriate adaptations of the basic model design matrix it becomes possible to easily analyze the transcriptome for single-factor experiments with more than two biological conditions, to detect linear and quadratic time effects or a circadian rhythm effect in time-course experiments, and to even conduct two- or multiple-factor studies. The package’s use and flexibility are illustrated with three case studies on the reference plant Arabidopsis thaliana. These cases show the potential of the package and method to cope with a multitude of study designs and associated specific research questions and still provide reliable results. The waveTiling package will be freely available as part of the Bioconductor project.
Availability and requirements
Project name: waveTiling
Project home page: http://r-forge.r-project.org/projects/wavetiling/
Operating system(s): Platform independent
Programming language: R
Other requirements: R >= 2.14
License: GNU GPL
Any restrictions to use by non-academics: None
Part of this research was supported by IAP research network grant no. P6/03 of the Belgian government (Belgian Science Policy) and Ghent University (Multidisciplinary Research Partnership “Bioinformatics: from nucleotides to networks”).
- Yamada K, Lim J, Dale JM, Chen H, Shinn P, Palm CJ, Southwick AM, Wu HC, Kim C, Nguyen M, Pham P, Cheuk R, Karlin-Newmann G, Liu SX, Lam B, Sakano H, Wu T, Yu G, Miranda M, Quach HL, Tripp M, Chang CH, Lee JM, Toriumi M, Chan MMH, Tang CC, Onodera CS, Deng JM, Akiyama K, Ansari Y, Arakawa T, et al: Empirical analysis of transcriptional activity in the arabidopsis genome. Science. 2003, 302 (5646): 842-846.View ArticlePubMedGoogle Scholar
- Kampa D, Cheng J, Kapranov P, Yamanaka M, Brubaker S, Cawley S, Drenkow J, Piccolboni A, Bekiranov S, Helt G, Tammana H, Gingeras TR: Novel RNAs identified from an in-depth analysis of the transcriptome of human chromosomes 21 and 22. Genome Res. 2004, 14: 331-342.PubMed CentralView ArticlePubMedGoogle Scholar
- Schadt E, Edwards S, GuhaThakurta D, Holder D, Ying L, Svetnik V, Leonardson A, Hart K, Russell A, Li G, Cavet G, Castle J, McDonagh P, Kan Z, Chen R, Kasarskis A, Margarint M, Caceres R, Johnson J, Armour C, Garrett-Engele P, Tsinoremas N, Shoemaker D: A comprehensive transcript index of the human genome generated using microarrays and computational approaches. Genome Biol. 2004, 5 (10): R73.PubMed CentralView ArticlePubMedGoogle Scholar
- Stolc V, Samanta MP, Tongprasit W, Sethi H, Liang S, Nelson DC, Hegeman A, Nelson C, Rancour D, Bednarek S, Ulrich EL, Zhao Q, Wrobel RL, Newman CS, Fox BG, Phillips GN, Markley JL, Sussman MR: Identification of transcribed sequences in Arabidopsis thaliana by using high-resolution genome tiling arrays. Proc Nat Acad Sci U S A. 2005, 102 (12): 4453-4458.View ArticleGoogle Scholar
- Toyoda T, Shinozaki K: Tiling array-driven elucidation of transcriptional structures based on maximum-likelihood and Markov models. Plant J. 2005, 43 (4): 611-621.View ArticlePubMedGoogle Scholar
- Zeller G, Henz SR, Laubinger S, Weigel D, Rätsch G: Transcript normalization and segmentation of tiling array data. Pac Symp Biocomput. 2008, 13: 527-538.Google Scholar
- Nicolas P, Leduc A, Robin S, Rasmussen S, Jarmer H, Bessires P: Transcriptional landscape estimation from tiling array data using a model of signal shift and drift. Bioinformatics. 2009, 25 (18): 2341-2347.PubMed CentralView ArticlePubMedGoogle Scholar
- Munch K, Gardner P, Arctander P, Krogh A: A hidden Markov model approach for determining expression from genomic tiling micro arrays. BMC Bioinformatics. 2006, 7: 239.PubMed CentralView ArticlePubMedGoogle Scholar
- Piccolboni A: Multivariate segmentation in the analysis of transcription tiling array data. J comput biol: a j comput mol cell biol. 2008, 15 (7): 845-856.View ArticleGoogle Scholar
- Otto C, Reiche K, Hackermüller J: Detection of differentially expressed segments in tiling array data. Bioinformatics. 2012, 28: 1471-1479.View ArticlePubMedGoogle Scholar
- Andriankaja M, Dhondt S, De Bodt, Vanhaeren H, Coppens F, De Milde, Mühlenbock P, Skirycz A, Gonzalez N, Beemster GT, Inzé D: Exit from proliferation during leaf development in Arabidopsis thaliana: a not-so-gradual process. Dev Cell. 2012, 22: 64-78.View ArticlePubMedGoogle Scholar
- Okamoto M, Tatematsu K, Matsui A, Morosawa T, Ishida J, Tanaka M, Endo TA, Mochizuki Y, Toyoda T, Kamiya Y, Shinozaki K, Nambara E, Seki M: Genome-wide analysis of endogenous abscisic acid-mediated transcription in dry and imbibed seeds of Arabidopsis using tiling arrays. Plant J. 2010, 62: 39-51.View ArticlePubMedGoogle Scholar
- Hazen S, Naef F, Quisel T, Gendron J, Chen H, Ecker J, Borevitz J, Kay S: Exploring the transcriptional landscape of plant circadian rhythms using genome tiling arrays. Genome Biol. 2009, 10 (2): R17.PubMed CentralView ArticlePubMedGoogle Scholar
- Granovskaia M, Jensen L, Ritchie M, Toedling J, Ning Y, Bork P, Huber W, Steinmetz L: High-resolution transcription atlas of the mitotic cell cycle in budding yeast. Genome Biol. 2010, 11 (3): R24.PubMed CentralView ArticlePubMedGoogle Scholar
- Naouar N, Vandepoele K, Lammens T, Casneuf T, Zeller G, Van Hummelen, Weigel D, Rätsch G: Quantitative RNA expression analysis with Affymetrix Tiling 1.0R arrays identifies new E2F target genes. Plant J. 2009, 57: 184-194.View ArticlePubMedGoogle Scholar
- Matsui A, Ishida J, Morosawa T, Mochizuki Y, Kaminuma E, Endo TA, Okamoto M, Nambara E, Nakajima M, Kawashima M, Satou M, Kim JM, Kobayashi N, Toyoda T, Shinozaki K, Seki M: Arabidopsis transcriptome analysis under drought, cold, high-salinity and ABA treatment conditions using a tiling array. Plant and Cell Physiol. 2008, 49 (8): 1135-1149.View ArticleGoogle Scholar
- Huber W, Toedling J, Steinmetz LM: Transcript mapping with high-density oligonucleotide tiling arrays. Bioinformatics. 2006, 22 (16): 1963-1970.View ArticlePubMedGoogle Scholar
- Smyth GK: Linear models and empirical bayes methods for assessing differential expression in microarray experiments. Stat Appl Genet and Mol Biol. 2004, 3: Iss. 1, Article 3.Google Scholar
- Tusher VG, Tibshirani R, Chu G: Significance analysis of microarrays applied to the ionizing radiation response. Proc Nat Acad Sci. 2001, 98 (9): 5116-5121.PubMed CentralView ArticlePubMedGoogle Scholar
- Samanta MP, Tongprasit W, Sethi H, Chin CS, Stolc V: Global identification of noncoding RNAs in Saccharomyces cerevisiae by modulating an essential RNA processing pathway. Proc Nat Acad Sci U S A. 2006, 103 (11): 4192-4197.View ArticleGoogle Scholar
- Assarsson E, Greenbaum JA, Sundström M, Schaffer L, Hammond JA, Pasquetto V, Oseroff C, Hendrickson RC, Lefkowitz EJ, Tscharke DC, Sidney J, Grey HM, Head SR, Peters B, Sette A: Kinetic analysis of a complete poxvirus transcriptome reveals an immediate-early class of genes. Proc Nat Acad Sci. 2008, 105 (6): 2140-2145.PubMed CentralView ArticlePubMedGoogle Scholar
- Wijnen H, Naef F, Young MW: Molecular and statistical tools for circadian transcript profiling. Methods Enzymol. 2005, 393: 341-365.View ArticlePubMedGoogle Scholar
- Ahdesmaki M, Lahdesmaki H, Pearson R, Huttunen H, Yli-Harja O: Robust detection of periodic time series measured from biological systems. BMC Bioinf. 2005, 6: 117.View ArticleGoogle Scholar
- de Lichtenberg U, Jensen L, Fausboll A, Jensen T, Bork P, Brunak S: Comparison of computational methods for the identification of cell cycle-regulated genes. Bioinformatics. 2005, 21: 1164-1171.View ArticlePubMedGoogle Scholar
- Clement L, De Beuf K, Thas O, Vuylsteke M, Irizarry RA, Crainiceanu C: Fast wavelet based functional models for transcriptome analysis with tiling arrays. Stat Appl Genet and Mol Biol. 2012, 11: Iss. 1, Article 4.Google Scholar
- Gentleman RC, Carey VJ, Bates DM: Bioconductor: open software development for computational biology and bioinformatics. Genome Biol. 2004, 5: R80.PubMed CentralView ArticlePubMedGoogle Scholar
- R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. 2011, Austria: Vienna, http://www.R-project.org,Google Scholar
- Carvalho BS, Irizarry RA: A framework for oligonucleotide microarray preprocessing. Bioinformatics. 2010, 26: 2363-2367.PubMed CentralView ArticlePubMedGoogle Scholar
- Pages H, Aboyoun P, Lawrence M: IRanges: Infrastructure for manipulating intervals on sequences (R package version 1.12.5).
- Durinck S, Bullard J, Spellman P, Dudoit S: GenomeGraphs: integrated genomic data visualization with R. BMC Bioinf. 2009, 10: 2.View ArticleGoogle Scholar
- Aboyoun P, Pages H, Lawrence M: GenomicRanges: Representation and manipulation of genomic intervals (R package version 1.6.6).
- Qu Y, Xu S: Quantitative trait associated microarray gene expression data analysis. Mol Biol and Evol. 2006, 23 (8): 1558-1573.View ArticleGoogle Scholar
- Narula SC: Orthogonal polynomial regression. Int Stat Rev. 1979, 47: 31-36.View ArticleGoogle Scholar
- Golub GH, Loan CFV: Matrix Computations, third ed. 1996, Baltimore: The Johns Hopkins University PressGoogle Scholar
- Morris JS, Brown PJ, Herrick RC, Baggerly KA, Coombes KR: Bayesian analysis of mass spectrometry proteomic data using wavelet-based functional mixed models. Biometrics. 2008, 64 (2): 479-489.PubMed CentralView ArticlePubMedGoogle Scholar
- Rehrauer H, Aquino C, Gruissem W, Henz SR, Hilson P, Laubinger S, Naouar N, Patrignani A, Rombauts S, Shu H, Van de Peer Y, Vuylsteke M, Weigel D, Zeller G, Hennig L: AGRONOMICS1: a new resource for arabidopsis transcriptome profiling. Plant Physiol. 2010, 152 (2): 487-499.PubMed CentralView ArticlePubMedGoogle Scholar
- Irizarry RA, Hobbs B, Collin F, Beazer-Barclay YD, Antonellis KJ, Scherf U, Speed TP: Exploration, normalization, and summaries of high density oligonucleotide array probe level data. Biostatistics. 2003, 4 (2): 249-264.View ArticlePubMedGoogle Scholar
- Swarbreck D, Wilks C, Lamesch P, Berardini TZ, Garcia-Hernandez M, Foerster H, Li D, Meyer T, Muller R, Ploetz L, Radenbaugh A, Singh S, Swing V, Tissier C, Zhang P, Huala E: The Arabidopsis Information Resource (TAIR): gene structure and function annotation. Nucleic Acids Res. 2008, 36 (suppl 1): D1009-D1014.PubMed CentralPubMedGoogle Scholar
- Proost S, Van Bel M, Stercka L, Billiaua K, Van Parysa T, Van de Peer Y, Vandepoele K: PLAZA: a comparative genomics resource to study gene and genome evolution in plants. The Plant Cell. 2009, 21 (12): 3718-3731.PubMed CentralView ArticlePubMedGoogle Scholar
- Gardner MJ, Hubbard KE, Hotta CT, Dodd AN, Webb AAR: How plants tell the time. Biochem J. 2006, 397: 15-24.PubMed CentralView ArticlePubMedGoogle Scholar
- Ding Z, Millar AJ, Davis AM, Davis SJ: TIME FOR COFFEE encodes a nuclear regulator in the arabidopsis thaliana circadian clock. Plant Cell. 2007, 19 (5): 1522-1536.PubMed CentralView ArticlePubMedGoogle Scholar
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