You are currently browsing the monthly archive for August 2015.

[Update July 15, 2016: A preprint describing sleuth is available on BioRxiv]

Today my student Harold Pimentel released the beta version of his new RNA-Seq analysis method and software program called sleuth. A sleuth for RNA-Seq begins with the quantification of samples with kallisto, and together a sleuth of kallistos can be used to analyze RNA-Seq data rigorously and rapidly.

Why do we need another RNA-Seq program?

A major challenge in transcriptome analysis is to determine the transcripts that have changed in their abundance across conditions.  This challenge is not entirely trivial because the stochasticity in transcription both within and between cells (biological variation), and the randomness in the experiment (RNA-Seq) that is used to determine transcript abundances (technical variation), make it difficult to determine what constitutes “significant” change.

Technical variation can be assessed by performing technical replicates of experiments. In the case of RNA-Seq, this can be done by repeatedly sequencing from one cDNA library. Such replicates are fundamentally distinct from biological replicates designed to assess biological variation. Biological replicates are performed by sequencing different cDNA libraries that have been constructed from repeated biological experiments performed under the same (or in practice near-same) conditions. Because biological replicates require sequencing different cDNA libraries, a key point is that biological replicates include technical variation.

In the early days of RNA-Seq a few papers (e.g. Marioni et al. 2008, Bullard et al. 2010) described analyses of technical replicates and concluded that they were not really needed in practice, because technical variation could be predicted statistically from the properties of the Poisson distribution. The point is that in an idealized RNA-Seq experiment counts of reads are multinomial (according to abundances of the transcripts they originate from), and therefore approximately Poisson distributed. Their variance is therefore approximately equal to the mean, so that it is possible to predict the variance in counts across technical replicates based on the abundance of the transcripts they originate from. There is, however, one important subtlety here: “counts of reads” as used above refers to the number of reads originating from a transcript, but in many cases, especially in higher eukaryotes, reads are frequently ambiguous as to their transcript of origin because of the presence of multi-isoform genes and genes families. In other words, transcript counts cannot be precisely measured. However, the statement about the Poisson distribution of counts in technical replicates remain true when considering counts of reads by genomic features because then reads are no longer ambiguous.

This is why, in so-called “count-based methods” for RNA-Seq analysis, there is an analysis only at the gene level. Programs such as DESeq/DESeq2, edgeR and a literally dozens of other count-based methods first require counting reads across genome features using tools such as HTSeq or featureCounts. By utilizing read counts to genomic features, technical replicates are unnecessary in lieu of the statistical assumption that they would reveal Poisson distributed data, and instead the methods focus on modeling biological variation. The issue of how to model biological variation is non-trivial because typically very few biological replicates are performed in experiments. Thus, there is a need for pooling information across genes to obtain reliable variance estimates via a statistical process called shrinkage. How and what to shrink is a matter of extensive debate among statisticians engaged in the development of count-based RNA-Seq methods, but one theme that has emerged is that shrinkage approaches can be compatible with general and generalized linear models, thus allowing for the analysis of complex experimental designs.

Despite these accomplishments,  count-based methods for RNA-Seq have two major (related) drawbacks: first, the use of counts to gene features prevents inference about the transcription of isoforms, and therefore with most count-based methods it is impossible to identify splicing switches and other isoform changes between conditions. Some methods have tried to address this issue by restricting genomic features to specific exons or splice junctions (e.g. DEXSeq) but this requires throwing out a lot of data, thereby reducing power for identifying statistically significant differences between conditions. Second, because of the fact that in general $\frac{a}{b} + \frac{c}{d} \neq \frac{a+b}{c+d}$ it is mathematically incorrect to estimate gene abundances by adding up counts to their genomic region. One consequence of this, is that it is not possible to accurately measure fold change between conditions by using counts to gene features. In other words, count-based methods are problematic even at the gene-level and it is necessary to estimate transcript-level counts.

While reads might be ambiguous as to exactly which transcripts they originated from, it is possible to statistically infer an estimate of the number of reads from each transcript in an experiment. This kind of quantification has its origin in papers of Jiang and Wong, 2009 and Trapnell et al. 2010. However the process of estimating transcript-level counts introduces technical variation. That is to say, if multiple technical replicates were performed on a cDNA library and then transcript-level counts were to be inferred, those inferred counts would no longer be Poisson distributed. Thus, there appears to be a need for performing technical replicates after all. Furthermore, it becomes unclear how to work within the shrinkage frameworks of count-based methods.

There have been a handful of attempts to develop methods that combine the uncertainty of count estimates at the transcript level with biological variation in the assessment of statistically significant changes in transcript abundances between conditions. For example, the Cuffdiff2 method generalizes DESeq while the bitSeq method relies on a Bayesian framework to simultaneously quantify abundances at the transcript level while modeling biological variability. Despite showing improved performance over count-based methods, they also have significant shortcomings. For example the methods are not as flexible as those of general(ized) linear models, and bitSeq is slow partly because it requires MCMC sampling.

Thus, despite intensive research on both statistical and computational methods for RNA-Seq over the past years, there has been no solution for analysis of experiments that allows biologists to take full advantage of the power and resolution of RNA-Seq.

The sleuth model

The main contribution of sleuth is an intuitive yet powerful model for RNA-Seq that bridges the gap between count-based methods and quantification algorithms in a way that fully exploits the advantages of both.

$Y_t = X_t\beta_t + \epsilon_t$.

Here the subscript t refers to a specific transcript, $Y_t$ is a vector describing transcript abundances (of length equal to the number of samples), $X_t$ is a design matrix (of size number of samples x number of confounders), $\beta_t$ is a parameter vector (of size the number of confounders) and $\epsilon_t$ is a noise vector (of size the number of samples). In this model the abundances $Y_t$ are normally distributed. For the purposes of RNA-Seq data, the $Y_t$ may be assumed to be the logarithm of the counts (or normalized counts per million) from a transcript, and indeed this is the approach taken in a number of approaches to RNA-Seq modeling, e.g. in limma-voom. A common alternative to the general linear model is the generalized linear model, which postulates that some function of $Y_t$ has a distribution with mean equal to $g^{-1}(X_t \beta_t)$ where g is a link function, such as log, thereby allowing for distributions other than the normal to be used for the observed data. In the RNA-Seq context, where the negative binomial distribution may make sense because it is frequently a good distribution for modeling count data, the generalized model is sometimes preferred to the standard general model (e.g. by DESeq2). There is much debate about which approach is “better”.

In the sleuth model the $Y_t$ in the general linear model are modeled as unobserved. They can be thought of us the unobserved logarithms of true counts for each transcript across samples and are assumed to be normally distributed. The observed data $D_t$ is the logarithm of estimated counts for each transcript across samples, and is modeled as

$D_t = Y_t + \zeta_t$

where the $\zeta_t$ vector parameterizes a perturbation to the unobserved $Y_t$. This can be understood as the technical noise due to the random sequencing of fragments from a cDNA library and the uncertainty introduced in estimating transcript counts.

The sleuth model incorporates the assumptions that the response error is additive, i.e. if  the variance of transcript in sample is $V(D_{t,i})$ then $V(D_{t,i}) = \sigma^2_t + \tau^2_t$ where the variance $V(\epsilon_{t,i}|y_{t,i}) = \sigma^2_t$ and the variance $V(\zeta_{t,i}|y_{t,i}) = \tau^2_t$. Intuitively, sleuth teases apart the two sources of variance by examining both technical and biological replicates, and in doing so directly estimates “true” biological variance, i.e. the variance in biological replicates that is not technical.  In lieu of actual technical replicates, sleuth takes advantage of the bootstraps of kallisto which serve as accurate proxies.

In a test of sleuth on data simulated according to the DESeq2 model we found that sleuth significantly outperforms other methods:

In this simulation transcript counts were simulated according to a negative binomial distribution, following closely the protocol of the DESeq2 paper simulations. Reference parameters for the simulation were first estimated by running DESeq2 on a the female Finnish population from the GEUVADIS dataset (59 individuals). In the simulation above size factors were set to be equal in accordance with typical experiments being performed, but we also tested sleuth with size factors drawn at random with geometric mean of 1 in accordance with the DESeq2 protocol (yielding factors of 1, 0.33, 3, 3, 0.33 and 1) and sleuth still outperformed other methods.

There are many details in the implementation of sleuth that are crucial to its performance, e.g. the approach to shrinkage to estimate the biological variance $\sigma^2_t$. A forthcoming preprint, together with Nicolas Bray and Páll Melsted that also contributed to the project along with myself, will provide the details.

Exploratory data analysis with sleuth

One of the design goals of sleuth was to create a simple and efficient workflow in line with the principles of kallisto. Working with the Shiny web application framework we have designed an html interface that allows users to interact with sleuth plots allowing for real time exploratory data analysis.

The sleuth Shiny interface is much more than just a GUI for making plots of kallisto processed data. First, it allows for the exploration of the sleuth fitted models; users can explore the technical variation of each transcript, see where statistically significant differential transcripts appear in relationship to others in terms of abundance and variance and much more. Particularly useful are interactive features in the plots. For example, when examining an MA plot, users can highlight a region of points (dynamically created box in upper panel) and see their variance breakdown of the transcripts the points correspond to, and the list of the transcripts in a table below:

The web interface contains diagnostics, summaries of the data, “maps” showing low-dimensional representations of the data and tools for analysis of differential transcripts. The interactivity via Shiny can be especially useful for diagnostics; for example, in the diagnostics users can examine scatterplots of any two samples, and then select outliers to examine their variance, including the breakdown of technical variance. This allows for a determination of whether outliers represent high variance transcripts, or specific samples gone awry. Users can of course make figures showing transcript abundances in all samples, including boxplots displaying the extent of technical variation. Interested in the differential transcribed isoform ENST00000349155 of the TBX3 gene shown in Figure 5d of the Cuffdiff2 paper? It’s trivial to examine using the transcript viewer:

One can immediately see visually that differences between conditions completely dominate both the technical and biological variation within conditions. The sleuth q-value for this transcript is 3*10^(-68).

Among the maps, users can examine PCA projections onto any pair of components, allowing for rapid exploration of the structure of the data. Thus, with kallisto and sleuth raw RNA-Seq reads can be converted into a complete analysis in a matter of minutes. Experts will be able to generate plots and analyses in R using the sleuth library as they would with any R package. We plan numerous improvements and developments to the sleuth interface in the near future that will further facilitate data exploration; in the meantime we welcome feedback from users.

How to try out sleuth

Since sleuth requires the bootstraps and quantifications output by kallisto we recommend starting by running kallisto on your samples. The kallisto program is very fast, processing 30 million reads on a laptop in a matter of minutes. You will have to run kallisto with bootstraps- we have been using 100 bootstraps per sample but it should be possible to work with many fewer. We have yet to fully investigate the minimum number of bootstraps required for sleuth to be accurate.

To learn how to use kallisto start here. If you have already run kallisto you can proceed to the tutorial for sleuth. If you’re really eager to see sleuth without first learning kallisto, you can skip ahead and try it out using pre-computed kallisto runs of the Cuffdiff2 data- the tutorial explains where to obtain the data for trying out sleuth.

For questions, suggestions or help see the program websites and also the kallisto-sleuth user group. We hope you enjoy the tools!

I recently read a “brainiac” column in the Boston Globe titled “For 40 years, computer scientists looked for a solution that doesn’t exist” that caused me to facepalm so violently I now have pain in my right knee.

The article is about the recent arXiv posting:

Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false) by Arturs Backurs and Piotr Indyk (first posted in December 2014 and most recently updated in April 2015).

In the preprint, which was presented at STOC, Backurs and Indyk show that computing edit distance sub-quadratically is hard in the sense that if it were possible to compute the edit distance between two strings of length n in time $O(n^{2-\delta})$ for some constant $\delta > 0$ then it would be possible to solve the satisfiability of conjunctive normal form formulas in time that would violate the Strong Exponential Time Hypothesis (SETH)This type of “lower bound” reduction is common practice in theoretical computer science. In this case the preprint is of interest because (a) it sheds light on the complexity of a computational problem (edit distance) that has been extensively studied by theoretical computer scientists and (b) it extends the relevance of SETH and sheds more light on what the hypothesis implies.

In the introduction to the preprint Backurs and Indyk motivate their study by writing that “Unfortunately, that [standard dynamic programming] algorithm runs in quadratic time, which is prohibitive for long sequences”  and note that “considerable effort has been invested into designing faster algorithms, either by assuming that the edit distance is bounded, by considering the average case or by resorting to approximation”. They mention that the fastest known exact algorithm is faster than quadratic time running in $O(n^2/logn)$, but note that “only barely so”. This is all true, and such preamble is typical for theoretical computer science, setting the stage for the result. In one single parenthetical remark intended to relate their work to applications they note that “the human genome consists of roughly 3 billions base pairs” and that exact edit distance computation is currently prohibitive for such long sequences.  This is a true sentence, but also not very precise. More on that later in the post…

The parenthetical genome comment in the preprint was picked up on in an MIT press release describing the results and implications of the paper. While the press release does a very good job explaining the reduction of Backurs and Indyk in an accessible way, it also picks up on the parenthetical genome comment and goes a bit further. The word “genome” appears three times, and it is stated that “a computer running the existing algorithm would take 1,000 years to exhaustively compare two human genomes.” This is technically true, but two human genomes are very similar, so that exhaustive comparison is not necessary to compute the edit distance. In fact, two human genomes are 99.9% identical, and with strong prior upper bounds on the edit distance the computation of edit distance is tractable.

In terms of different species, the edit distance between complete genomes is completely irrelevant, because of the fact that organisms undergo large scale segmental duplications and rearrangements over timescales of speciation. The MIT press release did not claim the Backurs and Indyk result was relevant for comparing genomes of species, but… the Boston Globe column states that

“By calculating the edit distance between the genomes of two organisms, biologists can estimate how far back in evolutionary time the organisms diverged from each other.”

No.

Actually, even for closely related genomes edit distance is not the relevant statistic in biology, but rather the alignment of the genomes. This is why biological sequence analysis is based on what is called the Neeldeman-Wunsch algorithm and not the Wagner-Fischer algorithm mentioned in the MIT press release. Actually, because the length of sequences for which it is relevant to compute alignments is bounded and rather small (usually a few tens or hundreds of thousands of base pairs at most), the real computational issue for biological sequence analysis is not running time but memory. The Needleman-Wunsch and Wagner-Fischer algorithms require $O(n^2)$ space and the algorithmic advance that made edit distance and alignment computations for biological sequence comparison tractable was Hirschberg’s algorithm which requires only linear space at a cost of only a factor of 2 in time.

Thus, the parenthetical comment of Backurs and Indyk, intended to merely draw attention of the reader to the possible practical relevance of the edit distance problem, mutated into utter rubbish by the time the work was being discussed in the Boston Globe. If that was the only issue with the brainiac column I might have just hurt my hand when I facepalmed, but the following paragraph caused me the real pain:

Because it’s computationally infeasible to compute the edit distance between two human genomes, biologists have to rely on approximations. They’d love a faster method, but Indyk and his coauthor, MIT graduate student Arturs Backurs, recently demonstrated a faster method is impossible to create. And by impossible they don’t mean “very hard” or “our technology has to improve first.” They mean something like, “by the laws of the mathematics, it can’t be done.”

We’ve already gone through point #1 but amazingly nothing is true in this paragraph:

1. No, it is not computationally infeasible to compute the edit distance between two human genomes and no, biologists do not have to rely on approximations. But then there is also the fact that…
2. Backurs and Indyk did not, in fact, demonstrate that a “faster method is impossible to create”. What they showed is that reducing the exponent in the quadratic algorithm would require SETH to be false. It turns out there are some respectable computer scientists that believe that SETH is indeed false. So…
3. No, Backurs and Indyk, who understand mathematics quite well… most certainly do not believe that “by the laws of the mathematics, it can’t be done”. Actually,
4. I’d bet that Backurs and Indyk don’t really believe that mathematics has laws. Axioms yes.. but laws…

I find inaccurate reporting of science to be highly problematic. How can we expect the general public to believe scientists claims about serious issues, e.g. the prospects for climate change, when news reports are filled with hype and rubbish about less serious issues, e.g. the complexity of edit distance?

There is a another issue that this example highlights. While complexity theory is extremely important and fundamental for computer science, and while a lot of the work in complexity has practical applications, there has been a consistent hyping of implications for biology where complexity theory is simply not very relevant. When I started my career it was fashionable to talk about protein folding being NP-complete. But of course millions of proteins were probably being folded in my body while I was writing this blog post. Does that fact that I am alive imply that P=NP?

Of course, exploring the complexity of simplified models for biological systems, such as the HP-model for protein folding that the above reference links to, has great value. It helps to sharpen the questions, to better specify models, and to define the limits of computational tractability for computational biology problems. But it is worth keeping in mind that in biology n does not go to infinity.

### Blog Stats

• 2,853,560 views