Continuous-time Markov chain models for DNA mutations on a phylogenetic tree (e.g. the Jukes-Cantor model, the Kimura models, and more generally models of the Felsenstein hierarchy) have the simple and convenient property of multiplicativity. Specifically, if Q is a rate matrix then the associated substitution matrices are multiplicative in the following sense:

$e^{Q(t_1+t_2)} = e^{Qt_1}e^{Qt_2}$.

This follows directly from the fact that the matrices $Qt_1$ and $Qt_2$ commute, because for any two commuting matrices A and B

$e^{A+B} = e^{A}e^{B}$.

This means that substitutions over a time period 2t are equivalently described as substitutions occurring over a time period t, followed by substitutions occurring afterwards over another time period t.

But what if over the course of time the rate matrix changes? For example, suppose that for a period of time t mutations proceed according to a rate matrix Q, and following that, for another period of time t,  mutations proceed according to a rate matrix R? Is it true that the substitutions after time 2t will behave as if mutations occurred for a time 2t according to the (average) rate matrix $\frac{Q+R}{2}$?

If Q and R commute the answer will be yes, as Qt and Rt will also be commutative and the multiplicativity property will hold. But what if Q and don’t commute? Is there any relationship at all between $e^{\frac{Q+R}{2}2t}$ and the matrices $e^{Qt}$ and $e^{Rt}$?

This week I visited Yale University to give a talk in the Center for Biomedical Data Science seminar series.  I was invited by Smita Krishnaswamy, who organized a wonderful visit that included many interesting conversations not only in computational biology, but also applied math, computer science and statistics (Yale has strong programs in applied mathematics, statistics and data science, computer science and biostatistics). At dinner I learned from Dan Spielman of the Golden-Thompson inequality which provides a beautiful answer to the question above in the case where Q and R are symmetric. The theorem is a trace inequality for Hermitian matrices A and B:

$tr(e^{A+B}) \leq tr(e^Ae^B)$.

This inequality is well known in statistical mechanics and random matrix theory but I don’t believe it is known in the phylogenetics community, hence this post. The phylogenetic interpretation of the pieces of the Golden-Thompson inequality (replacing A with Qt and B with Rt) is straightforward:

• The matrices $e^{Qt}$ and $e^{Rt}$ are substitution matrices for the rate matrices Q and R respectively.
• The product $e^{Qt}e^{Rt}$ is the substitution matrix corresponding to mutations occurring with rate matrix Q for time t followed by rate matrix R for time t.
• The matrix $e^{Qt+Rt} = e^{\frac{Q+R}{2} \cdot 2t}$ is the substitution matrix for mutations occurring with rate $\frac{Q+R}{2}$ for time 2t.
• Since the trace of a substitution matrix is the probability that there is no transition, or equivalently the probability that a change in nucleotide does not occur, the Golden-Thompson inequality states that for two symmetric rate matrices and R, the probability of a substitution after time 2t is higher when mutations occur first at rate Q for time t and then at rate R for time t, than if they occur at rate $\frac{Q+R}{2}$ for time 2t.

In other words, rate changes decrease the expected number of substitutions in comparison to what one would see if rates are constant

The Golden-Thompson inequality was discovered independently by Sidney Golden and Colin Thompson in 1965. A proof is explained in an expository blog post by Terence Tao who heard of the Golden-Thompson inequality only eight years ago, which makes me feel a little bit better about not having heard of it until this week! It would be nice if there was a really simple proof but that appears not to be the case (there is a purported one page proof in a paper titled Golden-Thompson from Davis, however what is proved there is the different inequality $tr(e^{A+B}) \leq tr(e^A)tr(e^B)$, which can be shown, by virtue of another matrix trace inequality, to be a weaker inequality).

There is considerable interest in evolutionary biology in models that allow for time-varying rates of mutation, as there is substantial evidence of such variation. The Golden-Thompson inequality provides an additional insight for how mutation rate changes over time can affect naïve estimates based on homogeneity assumptions.

The Felsenstein hierarchy (from Algebraic Statistics for Computational Biology).

Six years ago I received an email from a colleague in the mathematics department at UC Berkeley asking me whether he should participate in a study that involved “collecting DNA from the brightest minds in the fields of theoretical physics and mathematics.”  I later learned that the codename for the study was “Project Einstein“, an initiative of entrepreneur Jonathan Rothberg with the goal of finding the genetic basis for “math genius”. After replying to my colleague I received an inquiry from another professor in the department, and then another and another… All were clearly flattered that they were selected for their “brightest mind”, and curious to understand the genetic secret of their brilliance.

I counseled my colleagues not to participate in this ill-advised genome-wide association study. The phenotype was ill-defined and in any case the study would be underpowered (only 400 “geniuses” were solicited), but I believe many of them sent in their samples. As far as I know their DNA now languishes in one of Jonathan Rothberg’s freezers. No result has ever emerged from “Project Einstein”, and I’d pretty much forgotten about the ego-driven inquiries I had received years ago. Then, last week, I remembered them when reading a series of blog posts and associated commentary on evolutionary biology by some of the most distinguished mathematicians in the world.

1. Sir Timothy Gowers is blogging about evolutionary biology?

It turns out that mathematicians such as Timothy Gowers and Terence Tao are hosting discussions about evolutionary biology (see On the recently removed paper from the New York Journal of Mathematics, Has an uncomfortable truth been suppressed, Additional thoughts on the Ted Hill paper) because some mathematician wrote a paper titled “An Evolutionary Theory for the Variability Hypothesis“, and an ensuing publication kerfuffle has the mathematics community up in arms. I’ll get to that in a moment, but first I want to focus on the scientific discourse in these elite math blogs. If you scroll to the bottom of the blog posts you’ll see hundreds of comments, many written by eminent mathematicians who are engaged in pseudoscientific speculation littered with sexist tropes. The number of inane comments is astonishing. For example, in a comment on Timothy Gowers’ blog, Gabriel Nivasch, a lecturer at Ariel University writes

“It’s also ironic that what causes so much controversy is not humans having descended from apes, which since Darwin people sort-of managed to swallow, but rather the relatively minor issue of differences between the sexes.”

This person’s understanding of the theory of evolution is where the Victorian public was at in England ca. 1871:

In mathematics, just a year later in 1872, Karl Weierstrass published what at the time was considered another monstrosity, one that threw the entire mathematics community into disarray. The result was just as counterintuitive for mathematics as Darwin’s theory of evolution was for biology. Weierstrass had constructed a function that is uniformly continuous on the real line, but not differentiable on any interval:

$f(x) = \sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^ncos({11}^n\pi x)$.

Not only does this construction remain valid today as it was back then, but lots of mathematics has been developed in its wake. What is certain is that if one doesn’t understand the first thing about Weierstrass’ construction, e.g. one doesn’t know what a derivative is, one won’t be able to contribute meaningfully to modern research in analysis. With that in mind consider the level of ignorance of someone who does not even understand the notion of common ancestor in evolutionary biology, and who presumes that biologists have been idle and have learned nothing during the last 150 years. Imagine the hubris of mathematicians spewing incoherent theories about sexual selection when they literally don’t know anything about human genetics or evolutionary biology, and haven’t read any of the relevant scientific literature about the subject they are rambling about. You don’t have to imagine. Just go and read the Tao and Gowers blogs and the hundreds of comments they have accrued over the past few days.

2. Hijacking a journal

To understand what is going on requires an introduction to Igor Rivin, a professor of mathematics at Temple University and, of relevance in this mathematics matter, an editor of  the New York Journal of Mathematics (NYJM) [Update November 21, 2018: Igor Rivin is no longer an editor of NYJM]. Last year Rivin invited the author of a paper on the variability hypothesis to submit his work to NYJM. He solicited two reviews and published it in the journal. For a mathematics paper such a process is standard practice at NYJM,  but in this case the facts point to Igor Rivin hijacking the editorial process to advance a sexist agenda. To wit:

• The paper in question, “An Evolutionary Theory for the Variability Hypothesis” is not a mathematics or biology paper but rather a sexist opinion piece. As such it was not suitable for publication in any mathematics or biology journal, let alone in the NYJM which is a venue for publication of pure mathematics.
• Editor Igor Rivin did not understand the topic and therefore had no business soliciting or handling review of the paper.
• The “reviewers” of the paper were not experts in the relevant mathematics or biology.

To elaborate on these points I begin with a brief history of the variability hypothesis. Its origin is Darwin’s 1875 book on “The Descent of Man and Selection in Relation to Sex” which was ostensibly the beginning of the study of sexual selection. However as explained in Stephanie Shields’ excellent review, while the variability hypothesis started out as a hypothesis about variance in physical and intellectual traits, at the turn of 20th century it morphed to a specific statement about sex differences in intelligence. I will not, in this blog post, attempt to review the entire field of sexual selection nor will I discuss in detail the breadth of work on the variability hypothesis. But there are three important points to glean from the Shields review: 1. The variability hypothesis is about intellectual differences between men and women and in fact this is what “An evolutionary theory for the variability hypothesis” tries really hard to get across. Specifically, that the best mathematicians are males because of biology. 2. There has been dispute for over a century about the extent of differences, should they even exist, and 3. Naïve attempts at modeling sexual selection are seriously flawed and completely unrealistic. For example naïve models that assume the same genetic mechanism produces both high IQ and mental deficits are ignoring ample evidence to the contrary.

Insofar as modeling of sexual selection is concerned, there was already statistical work in the area by Karl Pearson in 1895 (see “Note on regression and inheritance in the case of two parents“). In the paper Pearson explicitly considers the sex-specific variance of traits and the relationship of said variance to heritability. However as with much of population genetics, it was Ronald Fisher, first in the 1930s (Fisher’s principle) and then later in important work from 1958 what is now referred to as Darwin-Fisher theory (see, e.g. Kirkpatrick, Price and Arnold 1990) who significantly advanced the theory of sexual selection. Amazingly, despite including 51 citations in the final arXiv version of “An Evolutionary Theory for the Variability Hypothesis”, there isn’t a single reference to prior work in the area. I believe the author was completely unaware of the 150 years of work by biologists, statisticians, and mathematical biologists in the field.

What is cited in “An Evolutionary Theory for the Variability Hypothesis”? There is an inordinate amount of cherry picking of quotes from papers to bolster the message the author is intent on getting across: that there are sex-differences in variance of intelligence (whatever that means), specifically males are more variable. The arXiv posting has undergone eight revisions, and somewhere among these revisions there is even a brief cameo by Lawrence Summers and a regurgitation of his infamous sexist remarks. One of the thorough papers reviewing evidence for such claims is “The science of sex differences in science and mathematics” by Halpern et al. 2007. The author cherry picks a quote from the abstract of that paper, namely that “the reasons why males are often more variable remain elusive.” and follows it with a question posed by statistician Howard Wainer that implicitly makes a claim: “Why was our genetic structure built to yield greater variation among males than females?” An actual reading of the Halpern et al. paper reveals that the excess of males in the top tail of the distribution of quantitative reasoning has dramatically decreased during the last few decades, an observation that cannot be explained by genetics. Furthermore, females have a greater variability in reading and writing than males. They point out that these findings “run counter to the usual conclusion that males are more variable in all cognitive ability domains”. The author of “An Evolutionary Theory for the Variability Hypothesis” conveniently omits this from a very short section titled “Primary Analyses Inconsistent with the Greater Male Variability Hypothesis.” This is serious amateur time.

One of the commenters on Terence Tao’s blog explained that the mathematical theory in “An Evolutionary Theory for the Variability Hypothesis” is “obviously true”, and explained its premise for the layman:

It’s assumed that women only pick the “best” – according to some quantity X percent of men as partners where X is (much) smaller than 50, let’s assume. On the contrary, men are OK to date women from the best Y percent where Y is above 50 or at least greater than X.

Let’s go with this for a second, but think about how this premise would have to change to be consistent with results for reading and writing (where variance is higher in females). Then we must go with the following premise for everything to work out:

It’s assumed that men only pick the “best” – according to some quantity X percent of women as partners where X is (much) smaller than 50, let’s assume. On the contrary, women are OK to date men from the best Y percent where Y is above 50 or at least greater than X.

Perhaps I should write up this up (citing only studies on reading and writing) and send it to Igor Rivin, editor at the New York Journal of Mathematics as my explanation for my greater variability hypothesis?

Actually, I hope that will not be possible. Igor Rivin should be immediately removed from the editorial board of the New York Journal of Mathematics. I looked up Rivin’s credentials in terms of handling a paper in mathematical biology. Rivin has an impressive publication list, mostly in geometry but also a handful of publications in other areas. He, and separately Mary Rees, are known for showing that the number of simple closed geodesics of length at most L grows polynomially in L (this result was the beginning of some of the impressive results of Maryam Mirzakhani who went much further and subsequently won the Fields Medal for her work). Nowhere among Rivin’s publications, or in many of his talks which are online, or in his extensive online writings (on Twitter, Facebook etc.) is there any evidence that he has a shred of knowledge about evolutionary biology. The fact that he accepted a paper that is completely untethered from the field in which it purports to make an advance is further evidence of his ignorance.

Ignorance is one thing but hijacking a journal for a sexist agenda is another. Last year I encountered a Facebook thread on which Rivin had commented in response to a BuzzFeed article titled A Former Student Says UC Berkeley’s Star Philosophy Professor Groped Her and Watched Porn at Work. It discussed a lawsuit alleging that John Searle had sexually harassed, assaulted and retaliated against a former student and employee. While working for Searle the student was paid $1,000 a month with an additional$3,000 for being his assistant. On the Facebook thread Igor Rivin wrote

Here is an editor of the NYJM suggesting that a student should have effectively known that if she was paid $36K/year for work as an assistant of a professor (not a high salary for such work), she ought to expect sexual harassment and sexual assault as part of her job. Her LinkedIn profile (which he linked to) showed her to have worked a summer in litigation. So he was essentially saying that this victim prostituted herself with the intent of benefiting financially via suing John Searle. Below is, thankfully, a quick and stern rebuke from a professor of mathematics at Indiana University: I mention this because it shows that Igor Rivin has a documented history of misogyny. Thus his acceptance of a paper providing a “theory” for “higher general intelligence” in males, a paper in an area he knows nothing about to a journal in pure mathematics is nothing other than hijacking the editorial process of the journal to further a sexist agenda. How did he actually do it? He solicited a paper that had been rejected elsewhere, and sent it out for review to two reviewers who turned it around in 3 weeks. I mentioned above that the “reviewers” of the paper were not experts in the relevant mathematics or biology. This is clear from an examination of the version of the paper that the NYJM accepted. The 51 references were reduced to 11 (one of them is to the author’s preprint). None of the remaining 10 references cite any relevant prior work in evolutionary biology on sexual selection. The fundamental flaws of the paper remain unaddressed. The entire content of the reviews was presumably something along the lines of “please tone down some of the blatant sexism in the paper by removing 40 gratuitous references”. In defending the three week turnaround Rivin wrote (on Gowers’ blog) “Three weeks: I assume you have read the paper, if so, you will have found that it is quite short and does not require a huge amount of background.” Since when does a mathematician judge the complexity of reviewing a paper by its length? I took a look at Rivin’s publications; many of them are very short. Consider for example “On geometry of convex ideal polyhedra in hyperbolic 3-space”. The paper is 5 pages with 3 references. It was received 15 October 1990 and in revised form 27 January 1992. Also excuse me, but if one thinks that a mathematical biology paper “does not require a huge amount of background” then one simply doesn’t know any mathematical biology. 3. Time for mathematicians to wet their paws The irony of mathematicians who believe they are in the high end tail of some ill-specified distribution of intelligence demonstrating en masse that they are idiots is not lost on those of us who actually work in mathematics and biology. Gian-Carlo Rota’s ghost can be heard screaming from Vigevano “The lack of real contact between mathematics and biology is either a tragedy, a scandal, or a challenge, it is hard to decide which!!” I’ve spent the past 15 years of my career focusing on Rota’s call to address the challenge of making more contacts between mathematics and biology. The two cultures are sometimes far apart but the potential for both fields, if there is real contact, is tremendous. Not only can mathematics lead to breakthroughs in biology, biology can also lead to new theorems in mathematics. In response to incoherent rambling about genetics on Gowers’ blog, Noah Snyder, a math professor at Indiana University gave sage advice: I really wish you wouldn’t do this. A bunch of mathematicians speculating about stuff they know nothing about is not a good way to get to the truth. If you really want to do some modeling of evolutionary biology, then find some experts to collaborate or at least spend a year learning some background. What he is saying is די קאַץ האָט ליב פֿיש אָבער זי װיל ניט די פֿיס אײַננעצן (the cat likes fish but she doesn’t want to wet her paws). If you’re a mathematician who is interested in questions of evolutionary biology, great! But first you must get your paws wet. If you refuse to do so then you can do real harm. It might be tempting to imagine that mathematics is divorced from reality and has no impact or influence on the world, but nothing could be farther from the truth. Mathematics matters. In the case discussed in this blog post, the underlying subtext is pervasive sexism and misogyny in the mathematics profession, and if this sham paper on the variance hypothesis had gotten the stamp of approval of a journal as respected as NYJM, real harm to women in mathematics and women who in the future may have chosen to study mathematics could have been done. It’s no different than the case of Andrew Wakefield‘s paper in The Lancet implying a link between vaccinations and autism. By the time of the retraction (twelve years after publication of the article, in 2010), the paper had significantly damaged public health, and even today its effects, namely death as a result of reduced vaccination, continue to be felt. It’s not good enough to say: “Once the rockets are up, who cares where they come down? That’s not my department,” says Wernher von Braun. Here are two IQ test questions for you: 1. Fill in the blank in the sequence 1, 4, 9, 16, 25, __ , 49, 64, 81. 2. What number comes next in the sequence 1, 1, 2, 3, 5, 8, 13, .. ? Please stop and think about these questions before proceeding. Spoiler alert: the blog post reveals the answers. Earlier this month I posted a new paper on the bioRxiv: Jase Gehring, Jeff Park, Sisi Chen, Matt Thomson, and Lior Pachter, Highly Multiplexed Single-Cell RNA-seq for Defining Cell Population and Transciptional Spaces, bioRxiv, 2018. The paper offers some insights into the benefits of multiplex single-cell RNA-Seq, a molecular implementation of information multiplexing. The paper also reflects the benefits of a multiplex lab, and the project came about thanks to Jase Gehring, a multiplex molecular biologist/computational biologist in my lab. mult·i·plex /`məltəˌpleks/ adjective – consisting of many elements in a complex relationship. – involving simultaneous transmission of several messages along a single channel of communication. Conceptually, Jase’s work presents a method for chemically labeling cells from multiple samples with DNA nucleotides so that samples can be pooled prior to single-cell RNA-Seq, yet cells can subsequently be associated with their samples of origin after sequencing. This is achieved by labeling all cells from a sample with DNA that is unique to that sample; in the figure below colors are used to represent the different DNA tags that are used for each sample: This is analogous to the barcoding of transcripts in single-cell RNA-Seq, that allows for transcripts from the same cell of origin to be associated with each other, yet in this framework there is an additional layer of barcoding of cells. The tagging mechanism is a click chemistry one-pot, two-step reaction in which cell samples are exposed to methyltetrazine-activated DNA (MTZ-DNA) oligos as well as the amine-reactive cross-linker NHS-trans-cyclooctene (NHS-TCO). The NHS functionalized oligos are formed in situ by reaction of methyltetrazine with trans-cyclooctene (the inverse-electron demand Diels-Alder (IEDDA) reaction). Nucleophilic amines present on all proteins, but not nucleic acids, attack the in situ-formed NHS-DNA, chemoprecipitating the functionalized oligos directly onto the cells: MTZ-DNAs are made by activating 5′-amine modified oligos with NHS-MTZ for the IEDDA reaction, and they are designed with a PCR primer, a cell tag (a unique “barcode” sequence) and a poly-A tract so that they can be captured by poly-T during single-cell RNA-Seq: Such oligos can be readily ordered from IDT. We are careful to refer to the identifying sequences in these oligos as cell tags rather than barcodes so as not to confuse them with cell barcodes which are used in single-cell RNA-Seq to associate transcripts with cells. The process of sample tagging for single-cell RNA-Seq is illustrated in the figure below. It shows how the tags, appearing as synthetic “transcripts” in cells, are captured during 3′ based microfluidic single-cell RNA-Seq and are subsequently deciphered by sequencing a tag library alongside the cDNA library: This significance of multiplexing is manifold. First, by labeling cells prior to performing single-cell RNA-Seq, multiplexing allows for controlling a trade off between the number of cells assayed per sample, and the total number of samples analyzed. This allows for leveraging the large number of cells that can be assayed with current technologies to enable complex experimental designs based on many samples. In our paper we demonstrate this by performing an experiment consisting of single-cell RNA-Seq of neural stem cells (NSCs) exposed to 96 different combinations of growth factors. The experiment was conducted in collaboration with the Thomson lab that is interested in performing large-scale perturbation experiments to understand cell fate decisions in response to developmental signals. We examined NSCs subjected to different concentrations of Scriptaid/Decitabine, epidermal growth factor/basic fibroblast growth factor, retinoid acid, and bone morphogenic protein 4. In other words, our experiment corresponded to a 4x4x6 table of conditions, and for each condition we performed a single-cell RNA-Seq experiment (in multiplex). This is one of the largest (in terms of samples) single-cell RNA-Seq experiments to date: a 100-fold decrease in the number of cells we collected per sample allowed us to perform an experiment with 100x more samples. Without multiplexing, an experiment that cost us ~$7,000 would cost a few hundred thousand dollars, well outside the scope of what is possible in a typical lab. We certainly would have not been able to perform the experiment without multiplexing. Although the cost tradeoff is impactful, there are many other important implications of multiplexing as well:

• Whereas simplex single-cell RNA-Seq is descriptive, focusing on what is in a single sample, multiplex single-cell RNA-Seq allows for asking how? For example how do cell states change in response to perturbations? How does disease affect cell state and type?
• Simplex single-cell RNA-Seq leads to systematics arguments about clustering: when do cells that cluster together constitute a “cell type”? How many clusters are real? How should clustering be performed? Multiplex single-cell RNA-Seq provides an approach to assigning significance to clusters via their association with samples. In our paper, we specifically utilized sample identification to determine the parameters/thresholds for the clustering algorithm:On the left hand side is a t-SNE plot labeled by different samples, and on the right hand side de novo clusters. The experiment allowed us to confirm the functional significance of a cluster as a cell state resulting from a specific range of perturbation conditions.
• Multiplexing reduces batch effect, and also makes possible the procurement of more replicates in experiments, an important aspect of single-cell RNA-Seq as noted by Hicks et al. 2017.
• Multiplexing has numerous other benefits, e.g. allowing for the detection of doublets and their removal prior to analysis. This useful observation of Stoeckius et al. makes possible higher-throughput single-cell RNA-Seq. We also found an intriguing relationship between tag abundance and cell size. Both of these phenomena are illustrated in one supplementary figure of our paper that I’m particularly fond of:

It shows a multiplexing experiment in which 8 different samples have been pooled together. Two of these samples are human-only samples, and two are mouse-only. The remaining four are samples in which human and mouse cells have been mixed together (with 2,3,4 and 5 tags being used for each sample respectively). The t-SNE plot is made from the tag counts, which is why the samples are neatly separated into 8 clusters. However in Panel b, the cells are colored by their cDNA content (human, mouse, or both). The pure samples are readily identifiable, as are the mixed samples. Cell doublets (purple) can be easily identified and therefore removed from analysis. The relationship between cell size and tag abundance is shown in Panel d. For a given sample with both human and mouse cells (bottom row), human cells give consistently higher sample tag counts. Along with all of this, the figure shows we are able to label a sample with 5 tags, which means that using only 20 oligos (this is how many we worked with for all of our experiments) it is possible to label ${20 \choose 5} = 15,504$ samples.

• Thinking about hundreds (and soon thousands) of single-cell experiments is going to be complicated. The cell-gene matrix that is the fundamental object of study in single-cell RNA-Seq extends to a cell-gene-sample tensor. While more complicated, there is an opportunity for novel analysis paradigms to be developed. A hint of this is evident in our visualization of the samples by projecting the sample-cluster matrix. Specifically, the matrix below shows which clusters are represented within each sample, and the matrix is quantitative in the sense that the magnitude of each entry represents the relative abundance of cells in a sample occupying a given cluster:
A three-dimensional PCA of this matrix reveals interesting structure in the experiment. Here each point is an entire sample, not a cell, and one can see how changes in factors move samples in “experiment space”:

As experiments become even more complicated, and single-cell assays become increasingly multimodal (including not only RNA-Seq but also protein measurements, methylation data, etc.) development of a coherent mathematical framework for single-cell genomics will be central to interpreting the data. As Dueck et al. 2015 point out, such analysis is likely to not only be mathematically interesting, but also functionally important.

We aren’t the only group thinking about sample multiplexing for single-cell RNA-Seq. The “demuxlet” method by Kang et al., 2017 is an in silico approach based on multiplexing from genomic variation. Kang et al. show that if pooled samples are genetically heterogeneous, genotype data can be used to separate samples providing an effective solution for multiplexing single-cell RNA-Seq in large human studies. However demuxlet has limitations, for example it cannot be used for samples from a homogenous genetic background. Two papers at the end of last year develop an epitope labeling strategy for multiplexing: Stoeckius et al. 2017 and Peterson et al. 2017. While epitope labeling provides additional information that can be of interest, our method is more universal in that it can be used to multiplex any kind of samples, even from different organisms (a point we make with the species mixing multiplex experiment I described above). The approaches are also not exclusive, epitope labeling could be coupled to a live cell DNA tagging multiplex experiment allowing for the same epitopes to be assayed together in different samples. Finally, our click chemistry approach is fast, cheap and convenient, immediately providing multiplex capability for thousands, or even hundreds of thousands of samples.

One interesting aspect of Jase’s multiplexing paper is that the project it describes was itself a multiplexing experiment of sorts. The origins of the experiment date to 2005 when I was awarded tenure in the mathematics department at UC Berkeley. As is customary after tenure trauma, I went on sabbatical for a year, and I used that time to ponder career related questions that one is typically too busy for. Questions I remember thinking about: Why exactly did I become a computational biologist? Was a mathematics department the ideal home for me? Should I be more deeply engaged with biologists? Were the computational biology papers I’d been writing meaningful? What is computational biology anyway?

In 2008, partly as a result of my sabbatical rumination but mostly thanks to the encouragement and support of Jasper Rine, I changed the structure of my appointment and joined the UC Berkeley Molecular and Cell Biology (MCB) department (50%). A year later, I responded to a call by then Dean Mark Schlissel and requested wet lab space in what was to become the Li Ka Shing Center at UC Berkeley. This was not a rash decision. After working with Cole Trapnell on RNA-Seq I’d come to the conclusion that a small wet lab would be ideal for our group to better learn the details of the technologies we were working on, and I felt that practicing them ourselves would ultimately be the best way to arrive at meaningful (computational) methods contributions. I’d also visited David Haussler‘s wet lab where I met Jason Underwood who was working on FragSeq at the time. I was impressed with his work and what I saw were important benefits of real contact between wet and dry, experiment and computation.

In 2011 I was delighted to move into my new wet lab. The decision to give me a few benches was a bold and unexpected one, spearheaded by Mark Schlissel, but also supported by a committee he formed to decide on the make up of the building. I am especially grateful to John Ngai, Art Reingold and Randy Scheckman for their help. However I was in a strange position starting a wet lab as a tenured professor. On the one hand the security of tenure provided some reassurance that a failure in the wet lab would not immediately translate to a failure of career. On the other hand, I had no startup funds to buy all the basic infrastructure necessary to run a lab. CIRM, Mark Schlissel, and later other senior faculty in Molecular & Cell Biology at UC Berkeley, stepped in to provide me with the basics: a -80 and -20, access to a shared cold room, a Bioanalyzer (to be shared with others in the building), and a thermocycler. I bought some other basic equipment but the most important piece was the recruitment of my first MCB graduate student: Shannon Hateley. Shannon and I agreed that she would set up the lab and also be lab manager, while I would supervise purchasing and other organization lab matters. I obtained informed consent from Shannon prior to her joining my lab, for what would be a monumental effort requested of her. We also agreed she would be co-advised by another molecular biologist “just in case”.

With Shannon’s work and then my second molecular biology student, Lorian Schaeffer, the lab officially became multiplexed. Jase, who initiated and developed not only the molecular biology but also the computational biology of Gehring et al. 2018 is the latest experimentalist to multiplex in our group. However some of the mathematicians now multiplex as well. This has been a boon to the research of the group and I see Jase’s paper as fruit that has grown from the diversity in the lab. Moving forward, I see increasing use of mathematics ideas in the development of novel molecular biology. For example, current single-cell RNA-Seq multiplexing is a form of information multiplexing that is trivial in comparison to the multiplexing ideas from information theory; the achievements are in the molecular implementations, but in the future I foresee much more of a blur between wet and dry and increasingly sophisticated mathematical ideas being implemented with molecular biology.

Hedy Lamarr, the mother of multiplexing.

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.

I have been fascinated with mini computers for some time, and have wondered when they will become suitable for bioinformatics. The 4273π project, which is an online course that is distributed as a 32Gb SD card image for the Raspberry Pi, has been around for a few years and demonstrated the utility of mini computers for training. The course is a proof of principle that bioinformatics software can work on a mini computer; the distributed software includes some comparative genomics and phylogenetics programs. However there is not much one can do with 1Gb RAM. The data in 4273π are small FASTA files, and while the Raspberry Pi is powerful enough to allow for experimentation and exploration of such datasets, even the new Raspberry Pi 3, with ten times the performance of the original 2012 model, still only has 1Gb of RAM and is not powerful enough for handling the current primary data type of genomics: high-throughput sequencing data.

Enter the Rock64.

The Rock64 is a new single-board computer from Pine64 that competes with the Raspberry Pi 3:

The Rock64 is evidence of the rapid and impressive development in single-board computers over the past few years, and Pine64 crosses a major threshold by offering a model with 4Gb RAM. The machine is also cheap. A 4Gb RAM Rock64, which is a 64-bit, quad core 1.5GHz machine, costs $44.95 (the 1Gb model is just$24.95). An enclosure is $7.95, a power supply$6.99, and a 64Gb SSD drive is only $31.95 (the 16Gb drive is$15.95). When my student Jase Gehring found out the specs of the machine last summer, he immediately realized that it was powerful enough to run kallisto for RNA-Seq analyses, and we preordered a handful of the boards for the lab. These arrived in the fall and we have been testing the machines for a while. One of them is hooked up to a monitor, and together with a bluetooth mouse and keyboard is serving as a general desktop computer in the wet lab. They are extraordinary versatile mini computers that, in my opinion, portend a future of mobile, low-cost, and light-weight computing for clinical and field genomics applications.

Unfortunately ARM is not an architecture known to most computational biologists, and my initial enthusiasm for the Rock64 was dampened when I found out that most genomics software does not work on ARM architecture. However I managed to install R, and Páll Melsted compiled kallisto on the Rock64 for the new release of version 0.44 (the release introduces an ARM binary, along with pseudobam for visualization of pseudoalignments). With these programs in place on Gibraltar (our first Rock64 with 4Gb of RAM, a 64Gb SSD drive, and a quad-core 1.5GHz processor), there was ample processing power to quantify RNA-Seq datasets.

For example, I was able to build the Saccharomyces cerevisae release 81 transcriptome index in one minute. A complete quantification of 6 samples from Ellahi, Thurtle and Rine, 2015 using two cores (with 30 bootstraps per sample) took 21 minutes. The quantification consisted of processing 47,744,312 paired-end reads. Amazingly, the Rock64 can quantify human RNA-Seq, which requires pseudoalignment of reads to a much larger transcriptome than yeast. A human 15,117,833 paired-end read sample (SRR493366) took less than 11 minutes to quantify using a single core. These results show that the Rock64 is not a toy; it can be used for the analysis of high-throughput sequencing data from substantial biological experiments.

It’s mind boggling to consider just how amazing it is to be able to quantify RNA-Seq on such a machine. When we developed kallisto we knew that the two orders of magnitude speedup was a game-changer, but I never thought we would literally be able to run it on what is not much more than a phone. We’re not going to switch over all of our RNA-Seq analyses to the Rock64s quite yet, but cluster assemblies such as the Pico5S have piqued my interest.

I imagine that it won’t be long before mini computers are even more powerful, and provide ultra low-cost portable alternatives to current server and cloud computing solutions. Having said that, I still miss my Commodore 64. Fortunately the mini revolution isn’t leaving me behind: a mini version of the C64 is slated for release early this year.

On April 11th 2016, I contacted the Office for Prevention of Harassment and Discrimination at UC Berkeley to report that Professor Terry Speed had sexually harassed a postdoctoral researcher in the UC Berkeley statistics department in the period 2000–2002. Two specific allegations were subsequently investigated:

Allegation One: Respondent, a professor in the Statistics Department, sexually harassed Complainant One, a post-doctoral student in the same department, from 2000-2002 by making sexual advances toward her, asking her for dates, telling her he had a “crush” on her, giving her hugs, and communicating with her, including by email, in an intimate or romantic manner, when such behavior was not welcome.

Allegation Two: Respondent, a professor in the Statistics Department, created a hostile work environment for Complainant Two, an Assistant Professor in the Mathematics Department, in 2002, through Respondent’s persistent discussions and emails regarding his romantic interest in Complainant One and by pressuring Complainant Two to persuade Complainant One to interact with Respondent.

The investigation took 14 months to complete, and the result was a 47 page report along with 89 pages of supporting evidence based on interviews, hundreds of pages of emails that I disclosed at the outset of the investigation, and letters and emails provided by Respondent. The report concludes as follows:

CONCLUSION
For the reasons stated above, I conclude that the preponderance of the evidence substantiates that Respondent violated the 1992 Sexual Harassment Policy in that he engaged in unwelcome conduct of a sexual nature that created a hostile environment for Complainant One and Complainant Two, and conditioned an academic or personnel decision on Complainant One’s submission to his conduct. This report is being submitted to the Vice Provost for Faculty for review under the Faculty Code of Conduct.

I have waited since June of last year to hear from the Vice Provost for Faculty at UC Berkeley what action the university will take in light of the findings, however despite multiple requests for information the university has yet to respond as to whether it will enact any sanctions on Respondent.

My close-up encounter with sexual harassment was devastating. I never expected, when I arrived in Berkeley in 1999, that Terry Speed, a senior professor in my field who I admired and thought of as a mentor would end up as Respondent and myself as Complainant Two. However much more serious and significant than my ordeal were the devastating consequences his sexual harassment had on the life and well being of Complainant One. The sexual harassment that took place was not an isolated event. Despite repeated verbal and written requests by Complainant One that Speed stop, his sexual harassment continued unabated for months. The case was not reported at the time the sexual harassment happened because of the structure of Title IX. Complainant One knew that Speed would be informed if a complaint was made, and Complainant One was terrified of reprisal. Her fear was not hypothetical; after months of asking Speed to stop sexually harassing her, he communicated to her that, unless she was willing to reconcile with him as he wished, she could not count on his recommendation.

Speed has been an advocate for women in academia in recent years. However no amount of advocacy on behalf of women can cancel out the physical and mental harm caused by prolonged sexual harassment. Speed’s self-proclamation that he is a “male feminist” rings hollow.

Update on June 6, 2018: Terry Speed is no longer listed as Professor Emeritus at UC Berkeley.

Update on June 22, 2018: This is the “notice of outcome” I received from UC Berkeley regarding the case:

The GTEx consortium has just published a collection of papers in a special issue of Nature that together provide an unprecedented view of the human transcriptome across dozens of tissues. The work is based on a large-scale RNA-Seq experiment of postmortem tissue from hundreds of human donors, illustrated in Figure 1 of the overview by Ward and Gilad 2017:

The data provide a powerful new opportunity for several analyses, highlighted (at least for me) by the discovery of 673 trans-eQTLs at 10% genome-wide FDR. Undoubtedly more discoveries will be published when the sequencing data, available via dbGAP, is analyzed in future studies. As a result, the GTEx project is likely to garner many citations, both for specific results, but also drive-by-citations that highlight the scope and innovation of the project. Hopefully, these citations will include the key GTEx paper:

Carithers, Latarsha J, Ardlie, Kristin, Barcus, Mary, Branton, Philip A, Britton, Angela, Buia, Stephen A, Compton, Carolyn C, DeLuca, David S, Peter-Demchok, Joanne, Gelfand, Ellen T, Guan, Ping, Korzeniewski, Greg E, Lockhart, Nicole C, Rabiner, Chana A, Rao, Abhi K, Robinson, Karna L, Roche, Nancy V, Sawyer, Sherilyn J, Segrè, Ayellet V, Shive, Charles E, Smith, Anna M, Sobin, Leslie H, Undale, Anita H, Valentino, Kimberly M, Vaught, Jim, Young, Taylor R, Moore, Helen M, on behalf of the GTEx consortium, A Novel Approach to High-Quality Postmortem Tissue Procurement: The GTEx Project, Biopreservation and Biobanking 13(5), 2015, p 311–319.

The paper by Latarsha Carithers et al. provides an overview of the consent and laboratory procedures that GTEx developed and applied to obtain tissues from hundreds of deceased donors. The monumental effort is, to my knowledge, unprecedented in scale and scope, and it relied on the kindness and generosity of hundreds of family members and next-of-kin of donors, who consented to donate their loved ones to science.

To develop effective and appropriate consent procedures, the GTEx project organized a sub-study to determine how best to approach, interact and explain the project to family members. Ultimately consent was obtained either in person or over the phone, and one can only imagine the courage of families to agree to donate, especially during times of grief and for a project whose goals could only be explained in terms of the long-term benefits of basic science.

The consent procedures for GTEx were complicated by a need to rapidly place tissue in preservative postmortem. RNA degrades rapidly after the time of death, and there is a window of only a few hours before expression can no longer be effectively measured. The RNA Integrity Number (RIN) measures the extent of degradation of RNA. It used to be measured with gel electrophoresis by examining the ratio of 28S:18S rRNA; more recently RIN is computed using more sophisticated analyses with, e.g. the Agilent bioanalyzer (see Schroeder et al. 2006 for details). GTEx conducted extensive studies to determine the correspondence between postmortem interval (time taken to preserve tissue) and RIN, and also examined the RIN necessary for effective RNA-Seq library construction.

The effect of ischemic time time on RIN values (Fig 6 from Carithers et al. 2015).

These studies were used to deploy standard operating procedures across multiple source sites (an obvious necessity given the number of donors needed). All of this research was not only crucial for GTEx, but will be extremely valuable for studies relying on postmortem RNA-Seq in the future.

The collection of specimens from each source site required training of individuals at that site, and one of GTEx’s achievements is the gathering of knowledge of how to orchestrate such a complex distributed sample collection and preparation enterprise. The workflow shown below (Figure 2 from Carithers et al. 2015) hints at the complexities involved (e.g. the need for separate treatment of brain due to the requirement of proper sectioning).

A meeting discussing the findings of Carithers et al. was held on May 20-21 2015 and I encourage all users of GTEx data/results to view the recording of it (Day 1, Day 2).

It is truly moving and humbling to consider the generosity of the hundreds of families, in many cases of donors in their twenties or thirties, who enabled GTEx. The scale of their contribution, and the suffering that preceded collection of the data cannot be captured in cartoons that describe the experiment. The families can also never be fully acknowledged, not matter how many times they are thanked in acknowledgment sections. But at a minimum, I think that reading Carithers et al. 2015 is the least one can do to honor them, and those who turned their good-will into science.

Acknowledgment: the idea for this blog post originated during a conversation with Roderic Guigó.

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