- Open Access
NPEBseq: nonparametric empirical bayesian-based procedure for differential expression analysis of RNA-seq data
© Bi and Davuluri; licensee BioMed Central Ltd. 2013
- Received: 11 February 2013
- Accepted: 21 August 2013
- Published: 27 August 2013
RNA-seq, a massive parallel-sequencing-based transcriptome profiling method, provides digital data in the form of aligned sequence read counts. The comparative analyses of the data require appropriate statistical methods to estimate the differential expression of transcript variants across different cell/tissue types and disease conditions.
We developed a novel nonparametric empirical Bayesian-based approach (NPEBseq) to model the RNA-seq data. The prior distribution of the Bayesian model is empirically estimated from the data without any parametric assumption, and hence the method is “nonparametric” in nature. Based on this model, we proposed a method for detecting differentially expressed genes across different conditions. We also extended this method to detect differential usage of exons from RNA-seq data. The evaluation of NPEBseq on both simulated and publicly available RNA-seq datasets and comparison with three popular methods showed improved results for experiments with or without biological replicates.
NPEBseq can successfully detect differential expression between different conditions not only at gene level but also at exon level from RNA-seq datasets. In addition, NPEBSeq performs significantly better than current methods and can be applied to genome-wide RNA-seq datasets. Sample datasets and R package are available at http://bioinformatics.wistar.upenn.edu/NPEBseq.
- Posterior Distribution
- Prior Distribution
- Read Count
- Dispersion Parameter
- Differential Expression Analysis
The advent of massive parallel sequencing, popularly known as Next-Generation Sequencing (NGS), is allowing whole genomes and transcriptomes to be sequenced with extraordinary speed and accuracy, providing insights into the bewildering complexity of gene expression at both gene and isoform levels . With decreasing sequencing cost per base, RNA-Seq approach has become a desirable method to get a complete view of the transcriptome and to identify differentially expressed rare transcripts and isoforms . The RNA-seq assay provides sensitive and accurate digital counts for the exon regions of expressed transcripts in a given sample. The count of short sequence reads for each exon region is the sum of read counts belonging to the overlapping exon region of different transcript isoforms that are expressed in the sample. Therefore, estimating the transcript-level expression from the collection of counts of short read sequences that map to exons (or exon slices) and exon junctions is a computationally challenging problem, which has been recently attempted by us and others, in programs such as IsoformEx , rSeq , Cufflinks , RSEM , BASIS , and GPSeq . However, none of these methods showed good agreement with qRT-PCR measurements, a gold standard in measuring differential RNA abundance between samples . The statistical challenges in analyzing RNA-Seq data arise from many perspectives. While some sources of error are due to inherent problems with the technology, some are contributed at laboratory or experimental levels, leading to non-biological or technical variation across samples. Therefore, there is a critical need for investigation of other statistical methods for normalization and differential expression analysis of RNA-seq data across different conditions.
RNA-seq experiments are now frequently employed for identifying genes and alternatively spliced gene isoforms that are differentially expressed across distinct tissue/cell types and disease conditions . This amounts to comparing one condition, A, with another condition, B, and producing a ranked list of differentially expressed genes according to the statistical significance of observed expression difference or fold-change between A and B [10, 11]. Thus, proper normalization between samples is crucial before differential expression (DE) analysis and, to a certain degree, the two aspects are linked with each other. Normalization can be divided into within-sample normalization and between-samples normalization . DE analysis is the study of the difference in absolute gene expression levels between two conditions. However, similar to microarray technology, RNA-seq is a relative abundance measure technology and does not allow for the measurement of absolute transcript abundance. This is because molecules are sampled proportionately from a large pool of cells and the initial number of cells and other technical factors are usually difficult to estimate or unknown. The standard procedure for computing the proportion of sequence reads that map to a gene relative to the total number of reads obtained in that RNA-seq experiment and for comparing those proportions across different samples can lead to high false positive rate. For example, a common method for normalization is to divide the gene-wise read counts by corresponding gene length and the total number of mapped reads to the genome. Recent reports show that the latter method, based on the total count of mapped reads, is not a robust method  and several alternative methods have been proposed. For example, an empirical strategy that equates the overall expression levels of genes between samples under the assumption that the majority of them are not DE was proposed recently [10, 14, 15]. Alternatively, the widely used quartile normalization method in the microarray field was also adapted for between-sample normalization of RNA-seq . A recent review evaluated seven proposed normalization methods for the differential analysis of RNA-seq data by using a varied group of real and simulated datasets involving different species and experimental designs . They concluded that the methods proposed in DESeq  and edgeR  have the most relative satisfactory behaviour compared to the others.
Similarly, several tools have been developed for DE analysis of RNA-seq data. The Poisson model has been successfully used to account for technical variations in RNA-seq data . When biological replicates are available, the negative binomial distribution is commonly used to model the over-dispersion in the count data, such as DESeq and edgeR. There are also pure non-parametric methods, which do not assume any particular distribution for the data, e.g. NOISeq . Approaches within a Bayesian framework for differential expression in RNA-Seq data have also been developed by many researchers, such as baySeq , GFOLD , ShrinkSeq  and EBSeq . It is acknowledged that Bayesian approach can be used to obtain accurate and robust estimates by sharing information across all genes when sample size is small . In baySeq, the genes are ranked according to the posterior probabilities of differential expression between conditions, using an empirical Bayes framework. To infer the posterior probability, the gene expression prior factors are integrated out by an approximation method  with mean and dispersion parameters empirically and iteratively estimated from the entire set of genes through a quasi-likelihood method. GFOLD assumes uniform prior distribution (vague prior) of gene expression level for technical replicate model. For data with biological replicates, a hierarchical model with log-normal prior for the gene expression is used to account for the biological variation. The posterior distribution of fold change is obtained through sampling. ShrinkSeq, which also takes a Bayesian perspective, is presented in a framework of generalized linear model setting, which infers the DE coefficient in the GLM directly instead of inferring the gene expression level first. ShrinksSeq explores both parametric mixture prior and non-parametric prior for the DE coefficient and extends the INLAs (integrated nested Laplace approximation) method to infer the marginal posterior distribution under non-parametric prior and shows the superior performance of non-parametric prior than parametric prior. EBSeq, similar to baySeq, ranks the genes/isoforms by posterior probability of DE, but assumes a parametric form of the prior distribution for the gene/isoform expression with parameters estimated from the data by method of moments and EM. All the aforementioned methods do not provide the close-form of posterior distribution of fold change. Because sequencing of cDNA reads is basically a sampling procedure, it is important to note that a large number of genes are unseen in a typical RNA-seq sample due to low expression or the limited depth of the experiment. For example, only approximately 0.0013% of the total number of available molecules in a RNA library are sampled in one lane of a typical Solexa/Illumina GAIIx RNA-seq experiment . Further, the fact that a small number of highly expressed genes consume a significant fraction of the total sequence reads can also influence the statistical inference procedures. These limitations affect the estimation of DE or differences in relative transcript distribution between samples. For almost all currently developed RNA-seq DE methods, genes with low read counts are usually omitted from the analysis because of unreliable estimation. Another issue is that a zero read count in one condition leads to unrealistic estimation of fold change.
Here, we developed a novel method to model the RNA-seq data and detect differentially expressed genes and exons across different conditions. To mitigate the biases caused by the nature of sampling and reliably estimate the expression levels of those unseen and lowly expressed genes, we adopted a previously developed Poisson mixture model to empirically estimate the prior distributions of read counts completely from the data . We propose a nonparametric, empirical Bayesian-based approach to model the RNA-seq data. We prepared five datasets, three simulated and two publicly available RNA-seq datasets, for systematically evaluating the performance of the new method. Also, the novel method is compared with the other popular methods for RNA-seq DE analysis, both using simulated and real RNA-seq datasets.
A few of the earlier RNA-seq assessment studies have reported highly reproducible results with little technical variation [27, 28], suggesting that the inclusion of technical replicates in the experimental plan is usually not essential. Numerous RNA-Seq studies have used the Poisson model to perform testing for differential gene expression. The Poisson model assumes equality of mean and variance of read counts per gene across replicates. Therefore, pooling technical replicates together to give read counts for each biological replicate does not lead to loss of information. Thus, we first discuss one replicate per condition and then consider biological replicates.
Model for single replicate
Let γ be the expression level of one gene under one condition and x be the number of observed reads mapped to this gene. It is well known that x follows a binomial distribution and can be approximated well by a Poisson distribution with mean λ = γdl, where 1 is the gene length and d is the normalization constant reflecting the sequencing depth. Given a prior mixing distribution G (with probability density function g(λ)) on λ, the posterior distribution of λ is , where hG(x) = ∫ λ x /x ! e‒ λdG(λ) is a G-mixture of Poisson.
Let λA and λB denote the read counts that represent the true expression level of a gene and G A and G B denote the corresponding prior distributions, under conditions A and B, respectively. However, as mentioned previously, since NGS is like sequencing a set of sampled reads from a pool of expressed sequences of gene, the read counts that are obtained, say xA and xB, denote the corresponding reads counts obtained in conditions A and B. The posterior distribution of log , which is log fold change of expression level of a gene, has a closed-form formula and is easy to derive, because and follow probability distribution of discrete form.
Thus, we rank the genes by the values of first and then estimate d by using the genes falling in the (ε, 1 − ε) quantile of all those values. In this paper, we used ε=0.25. That is, we used half of the genes to estimate d.
The default value for Δ is log(2). We consider this as our own pre-defined p-value. The false discovery rate is controlled with Benjamini-Hochberg adjustment.
Model dealing with biological replicates
RNA-seq datasets with large numbers of biological replicates are increasingly generated by many laboratories and consortia, for example, HapMap , ENCODE , and TCGA projects . TCGA data consists of hundreds of RNA-seq biological replicates for each cancer condition. Dealing with the large number of biological replicate data is challenging. Recent studies have found that while the Poisson model is appropriate for technical replicates of the same RNA samples, it can be a poor fit for biological replicates. Here we propose a hierarchical Bayesian model to account for the over-dispersion in the read counts.
where xij is the number of reads for gene i and replicate j; eij is the expression index; λi is the expression level of gene i under this condition A; θ is the scale parameter of Gamma distribution; and dj is normalization constant for replicate j. The prior distribution G is inferred as before by using the sample that has the largest data depth under each condition.
The p-values and FDR can be computed by equation (4).
Similar to the estimation of G, θ is also estimated by using the sample that has the largest data depth under each condition.
Differential exon usage analysis from RNA-seq data
RNA-seq also provides information for the study of alternative splicing. DE analysis of individual transcripts is essential in many comparative studies because of isoform-level changes in gene expression between conditions . Recently, two tools, Cufflink  and BitSeq , have been proposed to identify differential expression of transcripts by first estimating the expression of the transcripts. The expression or abundance estimates may contain significant correlated uncertainties that reduce the power for inference of differential expression . Another tool, DEXSeq , proposed an exon-centric analysis to test for differential exon usage in RNA-seq data based on a generalized linear model. The input of DEXSeq is a table that contains read counts for each exon of every gene in each sample. Note that one exon may be cut into two or more parts if its boundary is not the same in all transcripts. The basic unit for counting the number of reads overlapped is called “counting bin” in this manner, similar to the definition of exon slice used in IsoformEx algorithm .
The posterior distribution of tAk|yAk and tBk|yBk is derived based on our model applied to counting bin read count data.
Test the fitness of the distribution against the null hypothesis: the proportion of the number of reads overlapping with a counting bin to that of all the reads overlapping with the gene does not change between conditions (same as in DEXSeq).
Finally, define the p-value as the probability of , where Δ denotes a user-defined cutoff that represents the extent of differential expression one wishes to identify between conditions. For example, if Δ is set as log(1/3), it will test if the exon inclusion level of an exon is less than 1/3 or more than a three-fold change between conditions. Such an extreme switch of exon usage between conditions is a strong indicator of functional alterative splicing events.
Within-condition quantile normalization
For cases without replicates, the normalization constant can be computed by equation (3). With biological replicates, dj can be computed by the method proposed in DESeq or the trimmed mean of log-folder change method . Here we propose within-condition quantile normalization based on the assumption that the distributions of read count data within conditions are common. The samples in the same condition are first quantile normalized relative to the one that has the largest data depth and then equation (3) is applied to do the normalization between conditions. Quantile normalization of read count data for samples coming from different conditions might not be proper due to the fact that longer genes have more reads and they might be differentially expressed between conditions, which can put the gene expression level in a different scale within one sample. We compared this within-condition quantile normalization to the method proposed in DESeq for the simulated data and no significant difference was present (results are not shown here). For rest of this paper, we adopt the within-condition quantile normalization procedure.
Datasets for simulation
We generated three simulation datasets to evaluate the proposed method for identifying differentially expressed genes. Dataset1 is generated with biological replicates by assuming different priors across conditions, which means G A and GB follow different statistical distributions. Dataset2 consists of no replicates and is generated by following the simulation scheme adopted by baySeq and edgeR. Dataset3 is generated by the same scheme as dataset2 but with biological replicates.
Performance of NBESeq on simulated datasets
To evaluate the proposed method for identifying differentially expressed genes, we first conducted a simulation study.
Simulation 1 - Simulation with different priors between conditions (dataset1)
To generate data similar to those produced by real RNA-seq experiments, we first applied the empirical Bayes method on publicly available RNA-seq datasets, which were generated to compare liver and kidney transcriptomes . The prior distributions of kidney and liver samples were first estimated and then the data was normalized based on the expected values. The corresponding dispersion parameter θ for each condition was also estimated.
Dataset1 consist of 20 independent simulations with seven samples each for two conditions. The library size of each sample is uniformly sampled from 300,000 to 900,000. Each sample was generated by a mixture of negative binomial model with both the prior distributions and dispersion parameters estimated from Marioni’s data. Each sample consists of 10,000 genes for computational efficiency.
We performed a comparative analysis of our method with four popular methods, DESeq, edgeR, baySeq and NOISeq, which are available as part of Bioconductor packages at http://bioconductor.org. The edgeR implements two ways to estimate the dispersion parameter in its model, common dispersion and tag-wise dispersion. Both of them are studied here. baySeq provides two choices of model (Poisson and negative binomial). We adopted the negative binomial model for dataset1. Both DESeq and edgeR provide p-values for ranking the genes. baySeq provides log posterior likelihood ratio for ranking the differential expression of genes. In the case of NPEBseq, we rank the genes by p-values as defined in equation (4). The purpose of this simulation is to compare the ability of these methods to rank the genes in order of differential expression. The true ranking order of the genes is based on the fold change of differential expression values between the two conditions.
Simulation 2 - simulation with the same priors between conditions (dataset2 and dataset3)
A simulation scheme similar to the one suggested by Robinson and Smyth  is applied here to generate dataset2 and dataset3. The library size of each sample was uniformly sampled from 300,000 to 900,000. The prior distribution of λ was assumed to be common between the two conditions and estimated from the liver RNA-seq data of Marioni.
Estimated fold change of 10 genes from one sample of simulated dataset2 using NPEBSeq, DESeq and edgeR
Poisson mean under condition A
Poisson mean under condition B
TRUE fold change
observed #reads under condition A
observed #reads under condition B
estimated fc by NPEBSeq
estimated fc by DESeq
estimated fc by edgeR
Real RNA-seq data analysis
To further evaluate our method, we tested it on two published RNA-seq datasets.
Real RNA-seq data 1-Comparison based on one MAQC dataset
We first applied NPEBSeq on the MicroArray quality control (MAQC) dataset [40, 41] and compared with DESeq, baySeq, and edgeR. MAQC datasets contain gene expression data from multiple platforms and are extensively used in evaluating different data processing methods. We downloaded the MAQC2 Illumina RNA-seq data from http://www.ncbi.nlm.nih.gov/sra, which contains seven technical replicates of brain reference RNA samples and seven technical replicates of UHR RNA samples. Tophat  was used for tag alignment and counts for each gene were computed by means of HTSeq Python package (http://www-huber.embl.de/users/anders/HTSeq/), using the annotation of the Ensembl genes and only reads that mapped to exons.
Note that these are not the standard definitions of TPR and FPR.
qRT-PCR data were annotated by RefSeq. The BioMart R package  was used to convert the RefSeq genes IDs for qRT-PCR to Ensembl genes ids.
Real RNA-seq data 2-Detecting differential usage of exons from RNA-seq data
We also analysed the data by Brook et al. , where the effect of the RNAi knockdown of “pasilla” was studied by RNA-seq in the Drosophila melanogaster cell line. The data was downloaded as part of DEXSeq package. The data consists of four control samples and three knockdown samples. The analysis at gene level by NPEBseq reported 107 differentially expressed genes, with nominal FDR control at 0.1 for the comparison of control and knockdown. To access the specificity of the NPEBseq method we performed in-condition comparison by making use of the fact that there are four biological replicates in the control group. We applied NPEBseq for the comparison of two control samples versus the other two. NPEBseq reported zero differentially expressed genes with FDR control at 0.1, which indicates that NPEBseq has a very high specificity.
In this paper we developed a novel empirical Bayesian-based approach to model the RNA-seq data. This method has been widely used in ecology to estimate species diversity . The nonparametric form of the prior distribution of the Bayesian model is empirically estimated from the data. The expression level of genes with low read counts are estimated by borrowing information from the gene expression in the whole sample. For data with biological replicates, we developed a hierarchal Bayesian model to account for the over-dispersion and proposed an empirical Bayesian method to estimate the dispersion parameter. We also extended the model to detect differential usage of exons from RNA-seq datasets. The closed-form formula of the posterior distribution makes the computation of any statistics very efficient. At the final step, we evaluated the performance of this method in detecting the differentially expressed genes by conducting simulation and real RNA-seq data analysis.
There are many challenges still present in the processing and analyses of RNA-seq data. For example, it has been empirically observed that quantification of expression depends on the length of the biological features under study (genes, transcripts, or exons), as longer features tend to have more significant statistics than shorter ones . Also, it was recently shown that there exists a sample-specific guanine-cytosine content (GC-content) effect and the studies proposed normalization methods by GC-strata to remove such effects [46, 47]. Incorporating those factors into our model could further improve the performance.
Delineating the gene expression at an alternative transcript-level from RNA-seq data is still a very challenging problem. Our recently published IsoformEx method , based on non-negative least square, is aimed to estimate transcript abundance. In future enhancements to the proposed method we will integrate NPEBseq with IsoformEx to identify DE at isoform-level.
NPEBseq can be applied to not only detect differential gene expressions from the RNA-seq dataset with technical and biological replicates for both studied conditions, but also to detect differential usage of exons. It is robust, since it requires no limited assumptions to be made about the prior distribution of the data. NPEBseq also provides the closed form of posterior distribution of the fold change, which is useful for further analysis.
Project name: NPEBseq
Project home page: http://bioinformatics.wistar.upenn.edu/NPEBseq
Operating system and R version: The R package is platform independent and is compatible with all the versions of R same as or higher than 2.15.1.
Other requirements: No.
License: GPL (≥ 2)
Any restrictions to use: It is available for free download.
The work in the Davuluri laboratory is supported by Commonwealth Universal Research Enhancement (CURE) Research Program, Department of Health, Pennsylvania. RD holds a Philadelphia Healthcare Trust Endowed Chair Position; Research reported in this publication was partially supported by the National Library Of Medicine of the National Institutes of Health under Award Number R01LM011297. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The use of resources in the Bioinformatics Shared Facility of Wistar Cancer Centre (grant # P30 CA010815) is gratefully acknowledged. We thank Dr. Sharmistha Pal and Dr. Murali Bashyam for reading through the manuscript.
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