The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel

Molecular Microbiology and Biotechnology Department, Tel-Aviv University, Tel Aviv 69978, Israel

International Computer Science Institute, Berkeley, CA, 94704, USA

Abstract

Background

RNA-Seq is a technique that uses Next Generation Sequencing to identify transcripts and estimate transcription levels. When applying this technique for quantification, one must contend with reads that align to multiple positions in the genome (multireads). Previous efforts to resolve multireads have shown that RNA-Seq expression estimation can be improved using probabilistic allocation of reads to genes. These methods use a probabilistic generative model for data generation and resolve ambiguity using likelihood-based approaches. In many instances, RNA-seq experiments are performed in the context of a population. The generative models of current methods do not take into account such population information, and it is an open question whether this information can improve quantification of the individual samples

Results

In order to explore the contribution of population level information in RNA-seq quantification, we apply a hierarchical probabilistic generative model, which assumes that expression levels of different individuals are sampled from a Dirichlet distribution with parameters specific to the population, and reads are sampled from the distribution of expression levels. We introduce an optimization procedure for the estimation of the model parameters, and use HapMap data and simulated data to demonstrate that the model yields a significant improvement in the accuracy of expression levels of paralogous genes.

Conclusions

We provide a proof of principal of the benefit of drawing on population commonalities to estimate expression. The results of our experiments demonstrate this approach can be beneficial, primarily for estimation at the gene level.

Introduction

With the rapid decline in the cost of sequencing, RNA-Seq has emerged as a legitimate competitor to mi-croarrays as a means of assessing global gene expression. Even as arrays currently enjoy a cost advantage, many new applications of information accessible only through sequencing further strengthen the case that sequencing may soon supplant arrays as the technology of choice for transcription analysis. One such application is fine-grained assessment of variation in expression and the sources for such variation, as exemplified by recent large-scale RNA-Seq studies

Unfortunately, with any new technology come its limitations, and several studies have pointed out that RNA-Seq is not immune to bias

Several methods have emerged to address the multiread problem for paralog and isoform estimation

In many applications, a set of samples is studied. For instance, one may be interested in comparing the expression levels in cases verses controls, or in tissues originating from different organs. In such cases, it is plausible that the commonality of expression patterns within each of the defined groups of studied samples may be used to improve the quantification results in each of the samples. We demonstrate that by analyzing expression profiles of a population together, one gets expression estimates more accurate than those obtained by estimating individual sample expression levels independently. We describe and implement a probabilistic model of the sequencing process that incorporates population commonalities, and demonstrate its advantages over existing methods in the population setting.

Methods

RNA-Seq multiread expression estimation

As we seek to extend the prevalent generative model of RNA-Seq _{1}, _{M}) be the set of M transcribed regions considered and _{1}, _{M}) be the proportions of RNA bases attributed to each transcript out of the total number of transcribed bases in a sequenced sample. Regions may be either genes or transcripts, depending on the level of transcription being investigated. We require P to satisfy ∑_{gϵG }_{g }= 1 and ∀_{g }≤ 1.

The model describes an RNA sequencing experiment where regions in G are randomly chosen according to the distribution P, start positions in these regions are chosen uniformly, and reads of length _{1},..., _{ρ}). Sequencing is assumed to be error prone, leading to a certain probability of error for each read base. Based on the repetitions present in the set of regions and errors in alignment, reads may fail to map to the region from which they originate or may map to additional locations. Thus, we assign a probability of obtaining read _{j }given that it originated from region _{j }to _{k }to afford us the best match position instead of summing over all possible starting positions. _{k }is the effective length of _{k }(i.e., the number of start positions from which a full length read can be derived) as defined in _{jk }is the number of mismatches in the best alignment of _{j }to _{k}.

This formulation leads to the likelihood of observing the data:

This likelihood function is used to estimate P given the read alignments. Typically, one will use expectation maximization to find the P for which the likelihood is maximized. It is assumed that _{j}|_{k}) is zero for all regions to which _{j }is not aligned.

Common population extension

To estimate expression levels in N individuals from a defined population, we modify the above model by assuming that samples are drawn from a common population. This is imposed by having _{11}, ...,_{1M}), .., (_{N1 }_{N M})] be probability densities drawn from a common Dirichlet distribution, defined by a set of hyper-parameters specific to the population: ∀**p**_{i }= (_{i1},..., _{iM}) ~ _{1},..., _{M}).

For sample i, we denote the set of reads as _{ij }is mapped to one or more regions in G. The output of a read alignment program defines the set of accepted regions for the read (in practice only alignments with up to 2 errors are accepted) and provides the number of errors in alignment for each read-region pair. This allows us to calculate _{ij}|_{k}) as done above for one sample. For convenience we denote _{ij}_{k}) = _{ijk }(taken to be zero for all regions not mapped to), which is independent of ** α **and

As before, our objective is to estimate P, but in this case we must optimize by estimating P and ** α **together. We begin by writing the likelihood function:

Since expression values are sampled from the Dirichlet distribution,

Where

and similar to (1) above,

This leads to

Taking the log, we get

Multi-Genome Multi-Read (MGMR) algorithm

We wish to estimate ** α **and

Although for EM-based estimation methods convexity guarantees an optimal solution will be obtained, here (as shall be seen below) we have no such guarantee. Thus, we confine updates to be local by performing only one update for P and one for ** α**. By one MGMR iteration, we refer to one EM-based P update followed by one

Estimating P given α

If we assume ** α **is given, we can write the EM steps to find

**E step **Letting ** Match **signify a matching between reads and regions, and

which leads to

where

**M step **Given that each _{ijk }is fixed, the above reduces to maximizing

Using Lagrange multipliers and differentiating, we see that this is maximized with

Estimating α given P

Given a new estimate for ^{(t)}, we can use a fixed point iteration

By using the known bound

where

We maximize this bound with a fixed point iteration similar to EM, noting that for fixed values of

Heuristics/Implementation

As we have found **α**) presented in equation (15) is non-convex even in 2 dimensions (Figure ^{-20 }for their values in MGMR to avoid taking the log of zero. For P updates (e.g., equation 14), we avoid potentially negative P values by adding one to each alpha (thus ignoring -1 in the numerator and denominator). We implemented the algorithm in MATLAB, where the inputs required are read-gene map files for each sample as in SEQEM

A mesh representation of F(α) [equation (15)] showing non-convex behavior

**A mesh representation of F(α) [equation (15)] showing non-convex behavior**. _{1 }= .06 on the range [0,50] on the right.

Results

Experimental setup

As in

Simulating data

To derive artificial reads, we first estimated expression on real biological samples using one method and then used the resulting distribution of expression values to generate simulated datasets for testing. Real samples were drawn from lymphoblastoid cells of the Yoruba in Ibadan (YRI) population

To derive the sequence set for the SEQEM comparison, we expanded upon the procedure used in

For the RSEM-A and RSEM-B read sets, the transcript set used was also obtained by filtering the HomoloGene database to avoid gene overlaps, but no length filtering was required: reads were now sampled directly from transcripts which all had effective lengths greater than the read length used. 524 transcripts corresponding to 265 genes survived this filtering. For these read sets, we produced 30 repetitions of 74 samples, where each consisted of 100 bp reads at a coverage level of 20. In all other respects the sampling process and read generation steps were identical to those performed for the SEQEM-A and SEQEM-B read sets.

Error measures

Accuracy was assessed by three error measures, the first two of which were applied in

Simulated data with priors based on real estimates - estimating paralogous gene expression

To test performance on paralogous gene estimates, we set out to compare independent sample SEQEM estimates with MGMR's common population estimates. Before applying SEQEM, we looked to address one criticism of it from

With this correction in place, we estimated expression levels on the SEQEM-A and SEQEM-B read sets, applying SEQEM and MGMR to each. Outputs were recorded at 1-10, 20, 30, 40, 50 and 100 iterations for MGMR and at 100 iterations for SEQEM. The results are shown in Figure

Relative error measured on SEQEM-A and SEQEM-B data sets

**Relative error measured on SEQEM-A and SEQEM-B data sets**. MGMR outputs on SEQEM-A and SEQEM-B initializations were compared with SEQEM up to 100 iterations. MGMR outputs were recorded at 1-10, 20, 30, 40, 50 and 100 iterations. The first few iterations have been trimmed to allow a compact presentation.

MGMR vs. SEQEM error at 100 iterations on SEQEM-A and SEQEM-B data sets

**SEQEM-A sampling**

**SEQEM-B sampling**

**SEQEM**

**MGMR**

**SEQEM**

**MGMR**

**Error**

**SD**

**Error**

**SD**

**Error**

**SD**

**Error**

**SD**

E

1.27

1 * 10^{-2}

1.03

0.14

1.50

0.70

0.82

6 * 10^{-3}

χ^{2}

0.66

2 * 10^{-3}

0.22

4 * 10^{-3}

0.69

0.05

0.27

1 * 10^{-4}

KL

0.29

7 * 10^{-4}

0.14

1 * 10^{-4}

0.18

2 * 10^{-4}

0.17

1 * 10^{-4}

These data sets were derived from SEQEM and MGMR(SEQEM) estimates, respectively, on 20 YRI samples. (E: relative error rate; χ^{2}: Chi-squared error; KL: Kullback-Liebler divergence; SD: standard deviation)

Simulated data with priors based on real samples - estimating transcript level expression

We also sought to examine whether MGMR can improve results in the more challenging setting of estimating transcript level expression. Here, we expect estimates to be noisier due to low expression values in the real samples, and we must contend with multiread mappings due to paralogous genes as well as to isoforms of particular genes sharing subsequences as a result of alternative splicing. In anticipation of this challenge, we used a larger set consisting of 74 sample of single-end YRI samples as the real data source and simulated 100 bp reads instead of 35 bp. This was expected to be a difficult case for estimation, as all genes in the set are paralogs and many have multiple isoforms, as described in the section "Simulating data."

Once expression estimation was performed on the YRI samples and read sets RSEM-A and RSEM-B were generated, we again performed expression estimates with RSEM and MGMR on each set. In this case, unfortunately, we found the results did not exhibit a consistent trend as before and overall appeared inconclusive. These results are summarized in Table

MGMR vs. RSEM error at 100 iterations on RSEM-A and RSEM-B data sets

**RSEM-A Sampling**

**RSEM-B Sampling**

**RSEM**

**MGMR**

**RSEM**

**MGMR**

**Error**

**SD**

**Error**

**SD**

**Error**

**SD**

**Error**

**SD**

E

0.1

1 * 10^{-3}

0.69

1 * 10^{-3}

1.0

1 * 10^{-4}

0.61

1 * 10^{-3}

χ^{2}

0.02

6 * 10^{-4}

1.25

0.01

0.02

9 * 10^{-4}

0.58

3 * 10^{-4}

KL

1.5

0.22

0.6

1 * 10^{-3}

0.8

0.11

0.38

6 * 10^{-4}

E: relative error rate; χ^{2}: Chi-squared error; KL: Kullback-Liebler divergence; SD: standard deviation

Conclusion

As shown by the 1000 Genomes and HapMap projects, one of the drives of modern genetics and bioinformatics research is to characterize variation in populations. Because of cost and time constraints, such projects have only recently become feasible. In addition to such studies assessing genomic variation and its relation to disease phenotypes based on DNA, it is anticipated that RNA-Seq population studies will also grow in popularity to more directly assign functional significance to variant loci by means of transcription measures. Thus, it becomes essential to accurately measure the expression levels from each individual to characterize such variation. Here, we have shown that for one common study design an unexpected benefit can arise. When individuals in these studies are drawn from the same population, the estimates made on each can be made more accurate because of the commonalities among population members.

A shortcoming of the MGMR approach is that since it assumes commonality among the samples, outlier samples will be attracted towards the common denominator, and thus appear more similar to the group profile than they really are. In particular, if the data are subject to differential expression analysis, MGMR may reduce the number of differentially expressed genes.

We have investigated the efficacy of MGMR in tackling two typical experimental settings - inferring expression levels of paralogs at the gene level, and of isoforms (also drawn from a difficult set of paralogs). Although substantial gains were obtained in the first, more inquiry is required to demonstrate a benefit in the latter. It is worth noting that in each case at least a quarter of the regions considered showed improvement, as shown in Table

Proportion of genes for which MGMR improves estimates on different data sets

**SEQEM-A**

**SEQEM-B**

**RSEM-A**

**RSEM-B**

Proportion

104/285

78/285

126/524

173/524

%

36.5

27.3

24.0

33.0

Proportions of regions (genes for SEQEM and transcripts for RSEM, respectively) for which MGMR has lower relative error on average than each method compared to.

List of abbreviations

E: relative error rate; ^{2}: Chi-squared error; KL: Kullback-Liebler divergence; SD: standard deviation; bp: base pair

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

RR and EH developed the method. RS and EH designed the experiments. RR implemented the method and performed experiments. RR, EH, and RS analyzed results and wrote the manuscript. All authors read and approved the final manuscript.

Acknowledgements

E.H. is a faculty fellow of the Edmond J. Safra Bioinformatics program at Tel-Aviv University. R.S. was supported in part by the European Community's Seventh Framework Programme (grant HEALTH-F4-2009-223575 for the TRIREME project) and by the Israel Science Foundation (grant 802/08).

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