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I recently published a paper on the bioRxiv together with Vasilis Ntranos, Lynn Yi and Páll Melsted on Identification of transcriptional signatures for cell types from single-cell RNA-Seq. The contributions of the paper can be summed up as:

1. The simple technique of logistic regression, by taking advantage of the large number of cells assayed in single-cell RNA-Seq experiments, is much more effective than current approaches at identifying marker genes for clusters of cells.
2. The simplest single-cell RNA-Seq data, namely 3′ single-end reads produced by technologies such as Drop-Seq or 10X, can distinguish isoforms of genes.
3. The simple idea of GDE provides a unified perspective on DGE, DTU and DTE.

These simple, simple and simple ideas are so obvious that of course anyone could have discovered them, and one might be tempted to go so far as to say that even if people didn’t explicitly write them down, they were basically already known. After all, logistic regression was published by David Cox in 1958, and who didn’t know that there are many 3′ unannotated UTRs in the human genome? As for DGE, DTU and DTE (and DTE->G and DTE+G) I mean who doesn’t get these basic concepts? Indeed, after reading our paper someone remarked that one of the key results “was already known“, presumably because the successful application of logistic regression as a gene differential expression method for single-cell RNA-Seq follows from the fact that Šidák aggregation fails for differential gene expression in bulk RNA-Seq.

The “was already known” comment reminded me of a recent blog post about the dirty secret of mathematics. In the post, the author begins with the following math problem: Without taking your pencil off the paper/screen, can you draw four straight lines that go through the middle of all of the dots?

The problem may not yield immediately (try it!) but the solution is obvious once presented. This is a case of the solution requiring a bit of out-of-the-box thinking, leading to a perspective on the problem that is obvious in retrospect. In the Ntranos, Yi et al. paper, the change in perspective was the realization that “Instead of the traditional approach of using the cell labels as covariates for gene expression, logistic regression incorporates transcript quantifications as covariates for cell labels”. It’s no surprise the “was already known” reaction reared it’s head in this case. It’s easy to convince oneself, after the fact, that the “obvious” idea was in one’s head all along.

The egg of Columbus is an apocryphal tale about ideas that seem trivial after the fact. The story originates from the book “History of the New World” by Girolamo Benzoni, who wrote that Columbus, upon upon being told that his journey to the West Indies was unremarkable and that Spain “would not have been devoid of a man who would have attempted the same” had he not undertaken the journey, replied

“Gentlemen, I will lay a wager with any of you, that you will not make this egg stand up as I will, naked and without anything at all.” They all tried, and no one succeeded in making it stand up. When the egg came round to the hands of Columbus, by beating it down on the table he fixed it, having thus crushed a little of one end”

The story makes a good point. Discovery of the Caribbean in the 6th millennium BC was certainly not a trivial accomplishment even if it was obvious after the fact. The egg trick, which Columbus would have learned from the Amerindians who first brought chickens to the Americas, is a good metaphor for the discovery.

There are many Amerindian eggs in mathematics, which has its own apocryphal story to make the point: A professor proving a theorem during a lecture pauses to remark that “it is obvious that…”, upon which she is interrupted by a student asking if that’s truly the case. The professor runs out of the classroom to a nearby office,  returning after several minutes with a notepad filled with equations to exclaim “Why yes, it is obvious!” But even first-rate mathematicians can struggle to accept Amerindian eggs as worthy contributions, frequently succumbing to the temptation of dismissing others’ work as obvious. One of my former graduate school mentors was G.W. Peck, a math professor who created a pseudonym for the express purpose of publishing his Ameridian eggs in a way that would reduce unintended embarrassment for those whose work he was improving on in in “trivial ways”. G.W. Peck has an impressive publication record.

Bioinformatics is not very different from mathematics; the literature is populated with many Amerindian eggs. My favorite example is the Smith-Waterman algorithm, an algorithm for local alignment published by Temple Smith and Michael Waterman in 1981. The Smith-Waterman algorithm is a simple modification of the Needleman-Wunsch algorithm:

The table above shows the differences. That’s it! This table made for a (highly cited) paper. Just initialize the Needleman-Wunsch algorithm with zeroes instead of a gap penalty, set negative scores to 0, trace back from the highest score. In fact, it’s such a minor modification that when I first learned the details of the algorithm I thought “This is obvious! After all, it’s just the Needleman-Wunsch algorithm. Why does it even have a name?! Smith and Waterman got a highly cited paper?! For this?!” My skepticism lasted only as long as it took me to discover and read Peter Sellers’ 1980 paper attempting to solve the same problem. It’s a lot more complicated, relying on the idea of “inductive steps”, and requires untangling mysterious diagrams such as:

The Smith-Waterman solution was clever, simple and obvious (after the fact). Such ideas are a hallmark of Michael Waterman’s distinguished career. Consider the Lander-Waterman model, which is a formula for the expected number of contigs in a shotgun sequencing experiment:

$E(contigs) = Ne^{-R}.$

Here N is the number of reads sequenced and R=NL/G is the “redundancy” (reads * fragment length / genome length). At first glance the Lander-Waterman “model” is just a formula arising from the Poisson distribution! It was obvious… immediately after they published it. The Pevzner-Tang-Waterman approach to DNA assembly is another good example. It is no coincidence that all of these foundational, important and impactful ideas have Waterman in their name.

Looking back at my own career, some of the most satisfying projects have been Amerindian eggs, projects where I was lucky to participate in collaborations leading to ideas that were obvious (after the fact). Nowadays I know I’ve hit the mark when I receive the most authentic of compliments: “your work is trivial!” or “was widely known in the field“, as I did recently after blogging about plagiarism of key ideas from kallisto. However I’m still waiting to hear the ultimate compliment: “everything you do is obvious and was already known!”

(Click “read the rest of this entry” to see the solution to the 9 dot problem.)

The development of microarray technology two decades ago heralded genome-wide comparative studies of gene expression in human, but it was the widespread adoption of RNA-Seq that has led to differential expression analysis becoming a staple of molecular biology studies. RNA-Seq provides measurements of transcript abundance, making possible not only gene-level analyses, but also differential analysis of isoforms of genes. As such, its use has necessitated refinements of the term “differential expression”, and new terms such as “differential transcript expression” have emerged alongside “differential gene expression”. A difficulty with these concepts is that they are used to describe biology, statistical hypotheses, and sometimes to describe types of methods. The aims of this post are to provide a unifying framework for thinking about the various concepts, to clarify their meaning, and to describe connections between them.

To illustrate the different concepts associated to differential expression, I’ll use the following example, consisting of a comparison of a single two-isoform gene in two conditions (the figure is Supplementary Figure 1 in Ntranos, Yi et al. Identification of transcriptional signatures for cell types from single-cell RNA-Seq, 2018):

The isoforms are labeled primary and secondary, and the two conditions are called “A” and “B”. The black dots labeled conditions A and B have x-coordinates $x_A$ and $x_B$ corresponding to the abundances of the primary isoform in the respective conditions, and y-coordinates $y_A$ and $y_B$ corresponding to the abundance of the secondary isoforms. In data from an experiment the black dots will represent the mean level of expression of the constituent isoforms as derived from replicates, and there will be uncertainty as to their exact location. In this example I’ll assume they represent the true abundances.

Biology

Below is a list of terms used to characterize changes in expression:

Differential transcript expression (DTE) is change in one of the isoforms. In the figure, this is represented (conceptually) by the two red lines along the x- and y-axes respectively. Algebraically, one might compute the change in the primary isoform by $x_B-x_A$ and the change in the secondary isoform by $y_B-y_A$. However the term DTE is used to denote not only the extent of change, but also the event that a single isoform of a gene changes between conditions, i.e. when the two points lie on a horizontal or vertical line. DTE can be understood to occur as a result of transcriptional regulation if an isoform has a unique transcription start site, or post-transcriptional regulation if it is determined by a unique splicing event.

Differential gene expression (DGE) is the change in the overall output of the gene. Change in the overall output of a gene is change in the direction of  the line $y=x$, and the extent of change can be understood geometrically to be the distance between the projections of the two points onto the line $y=x$ (blue line labeled DGE). The distance will depend on the metric used. For example, the change in expression could be defined to be the total expression in condition B ($x_B+y_B$) minus the change in expression in condition A ($x_A+y_A$), which is $|x_B-x_A+y_B-y_A|$.  This is just the length of the blue line labeled “DGE” given by the $L_1$ norm. Alternatively, one could consider “DGE” to be the length of the blue line in the $L_2$ norm. As with DTE, DGE can also refer to a specific type of change in gene expression between conditions, one in which every isoform changes (relatively) by the same amount so that the line joining the two points has a slope of 1 (i.e. is angled at 45°). DGE can be understood to be the result of transcriptional regulation, driving overall gene expression up or down.

Differential transcript usage (DTU) is the change in relative expression between the primary and secondary isoforms. This can be interpreted geometrically as the angle between the two points, or alternatively as the length (as given by some norm) of the green line labeled DTU. As with DTE and DGE, DTU is also a term used to describe a certain kind of difference in expression between two conditions, one in which the line joining the two points has a slope of -1. DTU events are most likely controlled by post-transcriptional regulation.

Gene differential expression (GDE) is represented by the red line. It is the amount of change in expression along in the direction of line joining the two points. GDE is a notion that, for reasons explained below, is not typically tested for, and there are few methods that consider it. However GDE is biologically meaningful, in that it generalizes the notions of DGE, DTU and DTE, allowing for change in any direction. A gene that exhibits some change in expression between conditions is GDE regardless of the direction of change. GDE can represent complex changes in expression driven by a combination of transcriptional and post-transcriptional regulation. Note that DGE, DTU and DTE are all special cases of GDE.

If the $L_2$ norm is used to measure length and $DTE_1,DTE_2$ denote DTE in the primary and secondary isoforms respectively, then it is clear that DGE, DTU, DTE and GDE satisfy the relationship

$GDE^2 = DGE^2 + DTU^2 = DTE_1^2 + DTE_2^2.$

Statistics

The terms DTE, DGE, DTU and GDE have an intuitive biological meaning, but they are also used in genomics as descriptors of certain null hypotheses for statistical testing of differential expression.

The differential transcript expression (DTE) null hypothesis for an isoform is that it did not change between conditions, i.e. $x_A=x_B$ for the primary isoform, or $y_A=y_B$ for the secondary isoform. In other words, in this example there are two DTE null hypotheses one could consider.

The differential gene expresión (DGE) null hypothesis is that there is no change in overall expression of the gene, i.e. $x_A+y_A = x_B+y_B$.

The differential transcript usage (DTU) null hypothesis is that there is no change in the difference in expression of isoforms, i.e. $x_A-y_A = x_B - y_B$.

The gene differential expression (GDE) null hypothesis is that there is no change in expression in any direction, i.e. for all constants $a,b$, $ax_A+by_A = ax_B+by_B$.

The union differential transcript expression (UDTE) null hypothesis is that there is no change in expression of any isoform. That is, that $x_A = y_A$ and $x_B = y_B$ (this null hypothesis is sometimes called DTE+G). The terminology is motivated by $\neg \cup_i DTE_i = \cap_i DTE_i$.

Not that $UDTE \Leftrightarrow GDE$, because if we assume GDE, and set $a=1,b=0$ we obtain DTE for the primary isoform and setting $a=0,b=1$ we obtain DTE for the secondary isoform. To be clear, by GDE or DTE in this case we mean the GDE (respectively DTE) null hypothesis. Furthermore, we have that

$UDTE,GDE \Rightarrow DTE,DGE,DTU$.

This is clear because if $x_A=y_A$ and $x_B=y_B$ then both DTE null hypotheses are satisfied by definition, and both DGE and DTU are trivially satisfied. However no other implications hold, i.e. $DTE \not \Rightarrow DGE,DTU$, similarly $DGE \not \Rightarrow DTE,DTU$, and $DTU \not \Rightarrow DGE, DTE$.

Methods

The terms DGE, DTE, DTU and GDE also used to describe methods for differential analysis.

A differential gene expression method is one whose goal is to identify changes in overall gene expression. Because DGE depends on the projection of the points (representing gene abundances) to the line y=x, DGE methods typically take as input gene counts or abundances computed by summing transcript abundances $x_A+y_A$ and $x_B+y_B$. Examples of early DGE methods for RNA-Seq were DESeq (now DESeq2) and edgeR. One problem with DGE methods is that it is problematic to estimate gene abundance by adding up counts of the constituent isoforms. This issue was discussed extensively in Trapnell et al. 2013. On the other hand, if the biology of a gene is DGE, i.e. changes in expression are the same (relatively) in all isoforms, then DGE methods will be optimal, and the issue of summed counts not representing gene abundances accurately is moot.

differential transcript expression method is one whose goal is to identify individual transcripts that have undergone DTE. Early methods for DTE were Cufflinks (now Cuffdiff2) and MISO, and more recently sleuth, which improves DTE accuracy by modeling uncertainty in transcript quantifications. A key issue with DTE is that there are many more transcripts than genes, so that rejecting DTE null hypotheses is harder than rejecting DGE null hypotheses. On the other hand, DTE provides differential analysis at the highest resolution possible, pinpointing specific isoforms that change and opening a window to study post-transcriptional regulation. A number of recent examples highlight the importance of DTE in biomedicine (see, e.g., Vitting-Seerup and Sandelin 2017). Unfortunately DTE results do not always translate to testable hypotheses, as it is difficult to knock out individual isoforms of genes.

differential transcript usage method is one whose goal is to identify genes whose overall expression is constant, but where isoform switching leads to changes in relative isoform abundances. Cufflinks implemented a DTU test using Jensen-Shannon divergence, and more recently RATs is a method specialized for DTU.

As discussed in the previous section, none of null hypotheses DGE, DTE and DTU imply any other, so users have to choose, prior to performing an analysis, which type of test they will perform. There are differing opinions on the “right” approach to choosing between DGE, DTU and DTE. Sonseson et al. 2016 suggest that while DTE and DTU may be appropriate in certain niche applications, generally it’s better to choose DGE, and they therefore advise not to bother with transcript-level analysis. In Trapnell et al. 2010, an argument was made for focusing on DTE and DTU, with the conclusion to the paper speculating that “differential RNA level isoform regulation…suggests functional specialization of the isoforms in many genes.” Van den Berge et al. 2017 advocate for a middle ground: performing a gene-level analysis but saving some “FDR budget” for identifying DTE in genes for which the UDTE null hypothesis has been rejected.

There are two alternatives that have been proposed to get around the difficulty of having to choose, prior to analysis, whether to perform DGE, DTU or DTE:

differential transcript expression aggregation (DTE->G) method is a method that first performs DTE on all isoforms of every gene, and then aggregates the resulting p-values (by gene) to obtain gene-level p-values. The “aggregation” relies on the observation that under the null hypothesis, p-values are uniformly distributed. There are a number of different tests (e.g. Fisher’s method) for testing whether (independent) p-values are uniformly distributed. Applying such tests to isoform p-values per gene provides gene-level p-values and the ability to reject UDTE. A DTE->G method was tested in Soneson et al. 2016 (based on Šidák aggregation) and the stageR method (Van den Berge et al. 2017) uses the same method as a first step. Unfortunately, naïve DTE->G methods perform poorly when genes change by DGE, as shown in Yi et al. 2017. The same paper shows that Lancaster aggregation is a DTE->G method that achieves the best of both the DGE and DTU worlds. One major drawback of DTE->G methods is that they are non-constructive, i.e. the rejection of UDTE by a DTE->G method provides no information about which transcripts were differential and how. The stageR method averts this problem but requires sacrificing some power to reject UDTE in favor of the interpretability provided by subsequent DTE.

gene differential expression method is a method for gene-level analysis that tests for differences in the direction of change identified between conditions. For a GDE method to be successful, it must be able to identify the direction of change, and that is not possible with bulk RNA-Seq data. This is because of the one in ten rule that states that approximately one predictive variable can be estimated from ten events. In bulk RNA-Seq, the number of replicates in standard experiments is three, and the number of isoforms in multi-isoform genes is at least two, and sometimes much more than that.

In Ntranos, Yi et al. 2018, it is shown that single-cell RNA-Seq provides enough “replicates” in the form of cells, that logistic regression can be used to predict condition based on expression, effectively identifying the direction of change. As such, it provides an alternative to DTE->G for rejecting UDTE. The Ntranos and Yi GDE methods is extremely powerful: by identifying the direction of change it is a DGE methods when the change is DGE, it is a DTU method when the change is DTU, and it is a DTE method when the change is DTE. Interpretability is provided in the prediction step: it is the estimated direction of change.

Remarks

The discussion in this post is based on an example consisting of a gene with two isoforms, however the concepts discussed are easy to generalize to multi-isoform genes with more than two transcripts. I have not discussed differential exon usage (DEU), which is the focus of the DEXSeq method because of the complexities arising in genes which don’t have well-defined shared exons. Nevertheless, the DEXSeq approach to rejecting UDTE is similar to DTE->G, with DTE replaced by DEU. There are many programs for DTE, DTU and (especially) DGE that I haven’t mentioned; the ones cited are intended merely to serve as illustrative examples. This is not a comprehensive review of RNA-Seq differential expression methods.

Acknowledgments

The blog post was motivated by questions of Charlotte Soneson and Mark Robinson arising from an initial draft of the Ntranos, Yi et al. 2018 paper. The exposition was developed with Vasilis Ntranos and Lynn Yi. Valentine Svensson provided valuable comments and feedback.