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A tutorial on Burrows-Wheeler indexing methods

There are many resources explaining how the Burrows-Wheeler transform works and how it can be used to index a text, but so far I have not found anything explaining what makes it so awesome. I figured I would write a tutorial like this for those who are not afraid of the detail.

The problem

Say we have a sequencing run with over 100 million reads. After processing, the reads are between 20 and 25 nucleotide long. We would like to know if these sequences are in the human genome, and if so where.

The first idea would be to use grep to find out. On my computer, looking for a 20-mer such as ACGTGTGACGTGATCTGAGC takes about 10 seconds. Nice, but querying 100 million sequences would take more than 30 years. Not using any search index, grep needs to scan the whole human genome, and this takes time when the file is large.

We could build an index to speed up the search. The simplest would be a dictionary that associates each 20 to 25-mer to its locations in the human genome. The nice thing about dictionaries is that the access time is fast and does...

Did Mendel fake his results?

You went to high school and you learned genetics. You heard about a certain Gregor Mendel who crossed peas and came up with the idea that there is a dominant and a recessive allele. You did not particularly like the guy because there would always be a question about peas with recessive and dominant alleles at the exam. But you grew up, became wiser and just as you started to like him, you heard from someone that he faked his data. You felt disoriented for a while, why annoy you with this stuff at school if it is wrong? But then you came to the conclusion that he just got lucky and that he was right for the wrong reasons. After all, he was just a monk on gardening duties, why would you expect him to understand anything about real science?

Gregor Mendel

Gregor Mendel was a monk, but he was also a trained scientist. He studied assiduously for twelve years (including about seven years on physics and mathematics), to then become a teacher of physics and natural sciences at the gymnasium of Brno. He prepared his most famous experiment for two years, meticulously checking and choosing his...

How to teach research?

One of my students once asked me how to become a better researcher. “Great question!” I thought. Instead of a straight and clear answer, I heard myself babbling, as happens when you have no idea what you are talking about. Somewhat disappointed by my non answer, I turned to my colleagues and asked how they teach research. The answer ranged from embarrassed silence to politely switching topic. Strange, I thought, it is our responsibility as principal investigators to educate the junior researchers of our group, so we’d better have some idea of what we are doing.

We have a fairly good idea of how to teach skills, like mathematics, molecular biology etc. But I would argue that research is not a skill. I would argue that it is an attitude. So the real question is how to teach an attitude. There is no foolproof recipe, but here goes a subjective view that I gathered from reading, from discussions and from personal experience.

1. Motivate the change

Most students and junior scientists enter the lab with an open mind, ready to learn as many new things as possible. But most of them focus on learning technical skills (at...

ENCODE data, Principal Components and racism

“Thinking is classifying” wrote Georges Clémenceau*. This tells, in simple words, everything about the obsession of the human mind to keep things tidy. No surprise we ask computers a little help here and there. Is this email spam? Is this online user human? Is this text written by that author? Training machines to put things into the boxes created by our human mind is called supervised learning and it can be very lucrative. But what about the more philosophical cases where machines make their own boxes? Can we reverse the process and put things in boxes created by computers? Unsupervised learning, as it is called, creates a lot of interesting problems where we, humans, are left wondering whether the boxes make any sense.

The mother of all classification techniques is undisputedly Principal Component Analysis (PCA). But let me reassure those who hate PCA and those who never heard of it: I will just touch the surface, and then very briefly. PCA automatically arranges similar items close to each other on a plane. The rest is up to you. Similarity, in particular, depends on a bunch of arbitrary features, size, height, number of legs... In a classical introductory...

Stick breaking and DNA alignment

A couple of months ago, I posted an approximate formula for the longest match in the problem of DNA alignment. I recently used it to calibrate a seeding heuristic to map Illumina reads and I was surprised to see that it was not just bad, but epic bad. Upon closer inspection, I realized that the main assumption does not hold when the error rate is small (which is typically the case for Illumina reads). The formula was based on longest runs in Bernoulli trials. This time I present more accurate results with an approach based on a stick breaking process.

Stick breaking (spacings)

Inserting $(k)$ mutations at random in a sequencing read will produce $(k+1)$ (possibly empty) subsequences without errors. The process is analogous to inserting $(k)$ breaks at random in a stick of length 1, and we can approximate the distribution of the longest subsequence without error by that of the longest fragment when breaking the stick.

The example above illustrates this concept graphically. A sequencing read of 60 nucleotides contains 2 mutations highlighted in red and the longest error-free stretch is the central subsequence of 28 nucleotides. If mutations occur uniformly on the read...

Bayesian networks and causation

The first thing you learn in statistics is that “correlation does not imply causation”. As obvious as it sounds, most human mistakes fall in this category, and not only in statistics. The major difficulty with this question is that it is fairly easy to define correlation, but it is much harder to define causation, let alone quantify it. No surprise many statisticians just avoid talking about causation to stay out of the danger zone.

However, for Judea Pearl, this is not a satisfactory answer. In his book Causality: Models, Reasoning, and Inference, he expresses his opinion vividly.

I see no greater impediment to scientific progress than the prevailing practice of focusing all our mathematical resources on probabilistic and statistical inferences while leaving causal considerations to the mercy of intuition and good judgment.

This book lays the foundation of the now popular Bayesian networks. The key idea is that you can distinguish correlation from causation if you can observe several independent causes. For instance, suppose that patients suffering from a certain type of cancer are often immunodeficient. You wonder whether immunodeficiency is a cause or a consequence of this cancer type.

Say that variable A is whether patients have...

What is bioinformatics about?

A brief note published a few weeks ago initiated a discussion on the blogosphere about who is a bioinformatician and who is not. According to Wikipedia

bioinformatics combines computer science, statistics, mathematics, and engineering to study and process biological data.

To see how the community defines itself, I downloaded the abstracts of Bioinformatics from 2014 (for a total around 1000 articles, extracted the most relevant keywords and put the top 100 terms in a word cloud where a word size shows its frequency*.

Obviously, bioinformatics is about data, mostly gene/protein sequences and expression. It is also good to know that bioinformatics likes genomes and networks, and that it has more affinities with structural biology than with evolution.

The favourite organism of bioinformaticians is Homo sapiens, actually it is the only one mentioned in the word cloud, and when bioinformaticians work on a disease, it is cancer.

When bioinformaticians describe their work, it is “novel” and “new”, and what they talk about is “biological”, “different”, “multiple” and “single” (the last two are usually followed by “sequence alignment” and “nucleotide polymorphism”). It is also “functional” and “available”, but somewhat less. I expected it to be “fast” and “accurate”, but...

Why do bioinformatics?

I never planned to do bioinformatics. It just happened because I liked the time in front of my computer. Still, as every sane individual, I sometimes think that I could do something else with my life, and I wonder whether I am doing the right thing. On this topic, I recently came across the famous farewell to bioinformatics by Frederick J. Ross, which is worth reading, and of which the most emblematic quote is definitely the following.

My attitude towards the subject after all my work in it can probably be best summarized thus: Fuck you, bioinformatics. Eat shit and die.

There is nothing to agree or disagree in this quote, but Frederick gives further detail about his point of view in the post. In short, bioinformaticians are bad programmers, and community-level obfuscation maintains the illusion.

By making the tools unusable, by inventing file format after file format, by seeking out the most brittle techniques and the slowest languages, by not publishing their algorithms and making their results impossible to replicate, the field managed to reduce its productivity by at least 90%, probably closer to 99%.

There are indeed many issues in the bioinformatics community and I...

If cars were made by bioinformaticians...

1. Cars would have nice names

Here is what an abstract describing a car would look like.

Transporting people to defined locations of interest is a challenge of significant economic importance. To achieve this goal, people usually use cars or public transport. However, these solutions are suboptimal in several conditions. For instance, when people are extremely close to their target location, both cars and public transport are inappropriate, which limits their practical use. Here we present CaЯ (vehiCle for chAnging geo-cooЯdinates), a fast and accurate tool as an alternative to existing vehicles.

2. Cars would be fast and accurate

Bioinformaticians develop fast and accurate software. Their cars would be just the same. Here is what a typical benchmark sections would look like.

To show that CaЯ is faster and more accurate than existing alternatives, we benchmarked CaЯ against Volvo XC90 and Ferrari F430. In the first series of tests, we measured the time to lower the front windows of the vehicles. The average run duration was 2.3 seconds for CaЯ, 3.1 seconds for Volvo XC90 and 3.9 seconds for Ferrari F430, which demonstrates that CaЯ is substantially faster than Volvo XC90 and Ferrari...

Longest runs and DNA alignments

Update: The key assumption of the approximation below is that $(nq)$ is a relatively big number. This happens if the read is very long ($(n)$ is large) and/or the eror rate is high ($(q)$ is large). While these assumptions are reasonable with PacBio or Oxford Nanopore technologies, they are not for Illumina reads. In this case, I recommend using the formula based on stick-breaking from this post (it also holds for PacBio and Oxford Nanopore by the way).

The problem of sequence alignment gets a lot of attention from bioinformaticians (the list of alignment software counts more than 200 entries). Yet, the statistical aspect of the problem is often neglected. In the post Once upon a BLAST, David Lipman explained that the breakthrough of BLAST was not a new algorithm, but the careful calibration of a heuristic by a sound statistical framework.

Inspired by this idea, I wanted to work out the probability of identifying best hits in the problem of long read alignments. Since this is a fairly general result and that it may be useful for many similar applications, I post it here for reference.

Longest runs of 1s

I start with generalities on...


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