Field of Science

Domains of Applicability (DOA) in top-down and bottom-up drug discovery

You don’t use a hammer to do impressionistic painting. And although you technically could, you won’t use a spoon for drinking beer. The domains of applicability of these tools are different, in terms of quality and quantity.

The idea of domains of applicability (DOA) is an idea that is somehow both blatantly simple as well as easily forgotten. As the examples above indicate, the definition is apparent; every tool, every idea, every protocol, has a certain reach. There are certain kinds of data for which it works well and certain others for which it fails miserably. Then there are the most interesting cases; pieces of data on the boundary between applicable and non-applicable. These often serve as real testing grounds for your tool or idea.

Often the DOA of a tool becomes clear only when it’s been used for a long time on enough number of test cases. Sometimes the DOA reveals itself accidentally, when you are trying to use the tool on data for which it’s not really designed. That way can lie much heartbreak. It’s better instead to be constantly aware of the DOA for your techniques and deliberately stress-test its range. The DOA can also inform you about the sensitivity of your model; for instance, for a certain model a small change from a methyl to a hydroxy might fall within its DOA, while for another it might exceed it.

The development and use of molecular docking, an important part of bottom-up drug discovery, makes the idea of DOA clear. By now there’s an extensive body of knowledge about docking, developed over at least twenty years, which makes it clear when docking works well and when you can trust it less. For example, docking works quite well in reproducing known crystal poses and generating new poses when the protein is well resolved and relatively rigid; when there are no large-scale conformational changes; when there are no unusual interactions in the binding site; when water molecules aren’t playing any weird or special role in the binding. On the other hand, if you are doing docking on a homology model built on sparse homology that features a highly flexible loop and several bridging water molecules as key binding elements, all bets are off. You have probably stepped way outside the DOA of docking. Then there are the intermediate and in many ways the most interesting cases; somewhat rigid proteins, just one or two water molecules, a good knowledge base around that protein that tells you what works. In these cases, one can be cautiously optimistic and make some testable hypotheses.

Fortunately there are ways to pressure-test the DOA of a favorite technique. If you suspect that the system under consideration does not fall within the DOA, there are simple tests you can run and questions you can ask. The first set of questions concerns the quality and quantity of data that is available. This data falls into two categories; data that was used for training the method and the data that you actually have in your test case. If the test data closely matches the training data then there’s a fair chance that your DOA is covered. If not, you ask the second important question: What’s the quickest way I can actually test the DOA? Usually the quickest way to test any hypothesis in early stage drug discovery is to propose a set of molecules that your model suggests as top candidates. As always, the easier these are to make, the faster you can test them and the better you can convince chemists to make them in the first place. It might also be a good idea to sneak in a molecule that your model says has no chance in hell of working. If neither of these predictions come true within a reasonable margin, you clearly have a problem, either with the data itself or with your DOA.

There are also ways to fix the DOA of a technique, but because that task involves generating more training data and tweaking the code accordingly, it’s not something that most end users can do. In case of docking for instance, a DOA failure might result from inadequate sampling or inadequate scoring. Both of these issues can be fixed in principle through better data and better force fields, but that’s really something only a methods developer can do.

When a technique is new it always struggles to establish its DOA. Unfortunately both technical users and management don’t understand this and can immediately start proclaiming the method as a cure for all your problems; they think that just because it has worked well on certain cases it will do so on most others. The lure of publicity, funding and career advancement can further encourage this behavior. That certainly happened with docking and other bottom-up drug design tools in the Wild West of the late 80s and early 90s. I believe that something similar is happening with machine learning and deep learning now.

For instance it’s well known that when it comes to problems like image recognition and natural language processing (NLP), machine learning can do extremely well. In that case one is clearly operating well within the DOA. But what about modeling traffic patterns or brain activity or social networks or SAR data for that matter? What is the DOA of machine learning in these areas? The honest answer is that we don’t know. Now some users and developers of machine learning acknowledge this and are actually trying to circumscribe the right DOA by pressure-testing the algorithms. Others unfortunately simply take it for granted that more data must translate to better accuracy; in other words, they assume that the DOA is purely dictated by data quantity. This is true only in a narrow sense. Yes, less data can certainly hamper your efforts, but more data is neither always necessary and certainly not sufficient. You can have as much data as possible, but your technique can still be operating in the wrong DOA. For example, the presence of a discontinuous landscape of molecular activity places limitations on using machine learning in medicinal chemistry. Would more data ameliorate this problem? We don’t know yet, but this kind of thinking would not be inconsistent with the new religion of “dataism” which says that data is everything.

There are many opportunities to test the DOA of top-down approaches like deep learning in drug discovery and beyond. But to do this, both scientists and management must have realistic goals about the efficacy of the techniques, and more importantly must honestly acknowledge that they don’t know the DOA in the first place. In other words, they need to honestly acknowledge that they don’t yet know whether the technique will work for their specific problem. Unfortunately these kinds of decisions and proclamations are severely subject to hype and the enticement of dollars and drama. Machine learning is seen as a technique with such an outsize potential impact on diverse areas of our lives, that many err on the side of wishful thinking. Companies have sunk billions of dollars into the technology; how many of them would be willing to admit that the investment was really based on hope rather than reality?

In this context, machine learning can draw some useful lessons from the cautionary tale of drug design in the 80s, when companies were throwing money from all directions at molecular modeling. Did that money result in important lessons learnt and egos burnt? Indeed it did, but one might argue that computational chemists are still suffering from the negative effects of that hype, both in accurately using their techniques and in communicating the true value of those techniques to what seem like perpetually skeptical Nervous Nellies and Debbie Downers. Machine learning could go down the same route and it would be a real tragedy, not only because the technique is promising but because it could potentially impact many other aspects of science, technology, engineering and business and not just pharmaceutical development. And it might all happen because we were unable or unwilling to acknowledge the DOA of our methods.

Whether it’s top-down or bottom-up approaches, we can all ultimately benefit from Feynman’s words: “For a successful technology, reality has to take precedence over public relations, for Nature cannot be fooled.” For starters, let’s try not to fool each other.

2017 Nobel Prize picks

The nice thing about Nobel Prizes is that it gets easier to predict them every year, simply because most of the people you nominate don't win and automatically become candidates for the next year (note however that I said "easier to predict", not "easier to correctly predict"). That's why every year you can carry over much of the same list of likely candidates as before.

Having said that, there is a Bayesian quality to the predictions since the previous year's prize does compel you to tweak your priors, even if ever so slightly. Recent developments and a better understanding of scientific history also might make you add or subtract from your choices. For instance, last year the chemistry prize was awarded for molecular machines and nanotechnology. This was widely considered a “pure chemistry” prize, so this year’s prize is unlikely to be in the same area. Knowing the recent history of recent prizes for chemistry, my bets are on biological chemistry or inorganic chemistry as leading contenders this year.

As in previous years, I have decided to separate the prizes into lifetime achievement awards and specific discoveries. There have been fewer of the former in Nobel history and I have only two in mind myself, although the ones that do stand out are no lightweights - for instance R B Woodward, E J Corey, Linus Pauling and Martin Karplus were all lifetime achievement awardees. If you had to place a bet though, then statistically speaking you would bet on specific discoveries since there have been many more of these. So here goes:

Lifetime achievement awards

Inorganic chemistry: Harry Gray and Steve Lippard: For their pioneering and foundational work in the field of bioinorganic chemistry;work which has illuminated the workings of untold number of enzymatic and biological processes including electron transfer.

Biological chemistry: Stuart Schreiber and Peter Schultz: For their founding of the field of modern chemical genetics and their impact on the various ramifications of this field in chemistry, biology and medicine. Schreiber has already received the Wolf Prize last year so that improves his chances for the Nobel. The only glitch with this kind of recognition is that a lot of people contributed to the founding of chemical biology in the 1980s and 90s, so it might be a bit controversial to single out Schreiber and Schultz. The Thomson-Reuters website has a Schreiber prediction, but for rapamycin and mTOR; in my opinion that contribution, while noteworthy, would be too narrow and probably not sufficient for a prize.

Specific awards

John Goodenough and Stanley Whittingham for lithium-ion batteries: This has been on my list for a very long time and it will remain so. Very few science-based innovations have revolutionized our basic standard of living the way lithium-ion batteries have, and I cannot think of anyone else who deserves a prize for this as much as Goodenough. Just a few months ago there was a book about the making of the iPhone which featured him and his outsize impact on enabling our modern electronics age. As this recent New York Times profile noted, Goodenough is 94 and is still going strong, but that’s no reason to delay a recognition for him.

Generally speaking, recognition for the invention of specific devices have been rather rare, with the charged-coupled device (CCD) and the integrated circuit being exceptions. More importantly, a device prize was given out just three years ago in physics (for blue light-emitting diodes) so based on the Bayesian argument stated above, it might make it a bit unlikely for another device-based invention to win this year. Nonetheless, a prize for lithium ion batteries more than most other inventions would conform to the line in Alfred Nobel's will about the discovery that has "conferred the greatest benefits on mankind."

Franz-Ulrich Hartl and Arthur Horwich for their discovery of chaperones: This is clearly a discovery which has had a huge impact on our understanding of both basic biological processes as well as their therapeutic relevance.

Barry Sharpless for click chemistry, Marvin Caruthers for DNA synthesis:
I am grouping these two together under the heading of organic synthesis. Sharpless’s click chemistry has seen widespread enough use since it was developed. However, it’s worth contrasting it with two other kinds of novel reactions which were awarded the prize – olefin metathesis and palladium-catalyzed couplings. One of the reasons these two were recognized was because they had a huge impact on industrial synthesis of drugs, polymers, agricultural chemicals etc. I don’t know enough to know whether click chemistry has also had such a practical impact – it may not have since it’s still rather new – but I assume that this practical aspect would certainly play a role in the decision.

Of the two, I think Caruthers deserves it even more; the technology he invented in the 1980s has been chugging along for thirty years now, quietly fueling the biotechnology revolution. While perhaps not as monumental as sequencing, it’s certainly a close second, and unlike click chemistry its practical applications are uncontested. If Sanger could get a prize for figuring out the basic chemistry of DNA sequencing, then I don’t see why Caruthers shouldn’t get one for figuring out the basic chemistry of DNA synthesis. Caruthers could also nicely split the prize with Leroy Hood (below), who really pioneered both commercial DNA sequencers as well as synthesizers.

The medicine prize

As is traditionally the case, several of the above discoveries and inventions can be contenders for the medicine prize. However we have left out what is potentially the biggest contender of all until now.

Jennifer Doudna, Emmanuelle Charpentier and Feng Zhang for CRISP-Cas9: I don't think there is a reasonable soul who thinks CRISPR-Cas9 does not deserve a Nobel Prize at some point in time. In terms of revolutionary impact and ubiquitous use it almost certainly belongs in the same shelf that houses PCR and Sanger sequencing.

There are two sets of questions I have about it though: Firstly, whether an award for it would still be rather premature. While there is no doubt as to the broad applicability of CRISPR, it also seems to me that it's rather hard right now to apply it with complete confidence to a wide variety of systems. I haven't seen numbers describing the percentage of times that CRISPR works reliably, and one would think that kind of statistics would be important for anyone wanting to reach an informed decision on the matter (I would be happy to have someone point me to such numbers). While that infamous Chinese embryo study that made the headlines last year was quite flawed, it also exposed the problems with efficacy and specificity that still bedevil CRISPR (these are problems similar to the two major problems for drugs). My personal take on it is that we might have to wait for just a few more years before the technique becomes robust and reliable enough to thoroughly enter the realm of reality from one of possibility.

The second question I have about it is the whole patent controversy, which if anything seems to have become even more acrimonious since last year, reaching worthy-of-optioning-movie-rights level of acrimonious in fact. Doudna also wrote a book on CRISPR this year which I reviewed here; while it’s generally fair and certainly well-written, it does downplay Church and Zhang’s role (and wisely omits any discussion of the patent controversy). Generally speaking Nobel Prizes try to stay clear of controversy, and one would think that the Nobel committee would be especially averse to sullying their hands with a commercial one. The lack of clear assignment of priority that is being played out in the courts right now not only tarnishes the intellectual purity of the discovery, but on a more practical level it also makes the decision to award the prize to all three major contenders (Doudna, Charpentier and Zhang) difficult.

Hopefully, as would be fitting for a good novel, the allure of a Nobel Prize would make the three protagonists reach an agreement to settle their differences over a few beers. But that could still take some time. A different way to look at the whole issue however is to say that the Nobel committee could actually heal the divisions by awarding the prize to the trio. Either way, a recognition of CRISPR is likely going to be one of the most publicly debated prizes of recent times.

The bottom line in my mind: CRISPR definitely deserves a prize, and its past results and tremendous future potential may very well tip the balance this year, but it could also happen that the lack of robust, public vindication of the method and the patent controversy could make the recognition seem premature and delay the actual award.

Mary-Claire King: For the discovery of the BRCA1 breast cancer gene. Not only did this discovery revolutionize the treatment and detection of breast cancer, but it really helped to solidify the connection between genetics and cancer.

Craig Venter, Francis Collins, Eric Lander, Leroy Hood and others for genomics and sequencing: The split here may be pretty hard here and they might have to rope in a few consortiums, but as incomplete and even misleading as the sequencing of the human genome might have been, there is little doubt that it was a signal scientific achievement deserving of a Nobel Prize.

Alec Jeffreys for DNA fingerprinting and assorted applications: Alec Jeffreys is another perpetual favorite on the list and one whose invention has had a huge societal impact. I have never really understood why he has never been awarded the prize; the societal impact of DNA fingerprinting is almost as great as the contraceptive pill (for which Carl Djerassi was unfortunately never recognized).

Ronald Evans and Pierre Chambon for nuclear receptors: After GPCRs, nuclear receptors are the biggest targets for drugs, and GPCRs have already been recognized a few years ago. The third discoverer of nuclear receptors, Elwood Jensen, sadly passed away in 2012.

Bert Vogelstein, Robert Weinberg and others for cancer genes: This again seems like a no-brainer to me. Several medicine prizes have been awarded to cancer genetics so this certainly wouldn't be a novel idea, and it's also clear that Vogelstein and Weinberg have done more than almost anyone else in identifying rogue cancer genes and their key roles in health and disease.

The physics prize: There should be zero doubt in anyone’s mind that this year's Nobel Prize in physics will be awarded to Kip Thorne and Rainer Weiss for their decades-long dogged leadership and work that culminated in last year's breakthrough discovery of gravitational waves by the LIGO observatory. I would both love and hate to be in their position right now. It's a dead ringer, and the only reason they missed it last year was because the discovery came after the nomination. Sadly Ron Drever died of dementia this year. For those wanting to know more about the kind of dedication and personality clashes these three men brought to the project, Janna Levin's book which came out earlier this year is a great source.

There is another recognition that I have always thought has been due: a recognition of the ATLAS-CMS collaboration at the LHC which discovered the Higgs boson. A prize for them would emphasize several things: it would put experiment at the center of this important scientific discovery (there would have been no 2013 Nobel Prize without the LHC) and it would herald a new and necessary tradition of awarding the prize to teams rather than individuals, reflecting the reality of contemporary science. The Nobel committee could also recognize the international, collaborative nature of science and actually award the prize to the entire LIGO team and not just to Thorne and Weiss, but that’s unlikely to happen.

It also seems to me that a Nobel Prize for chaos theory and the study of dynamical systems - a field that surprisingly has not been recognized yet - should include any number of pioneers featured for instance in James Gleick's amazing book "Chaos", most notably Mitchell Feigenbaum.


My interest in fiction has picked up again over the last few years so I am going to venture a few guesses here. Unlike the science awards, Nobel Prizes for literature are usually more of lifetime achievement awards rather than awards for specific books; in fact nobody has won the prize for writing just one book, no matter how transformational it might have seemed.

More accurately, the literature prize is usually given for writers who have consistently explored specific themes in their work. For instance, Naipaul and Coetzee were recognized for vividly exploring issues of post-colonial identity, Tony Morrison was recognized for exploring issues of black identity and Bertrand Russell was recognized for extolling the virtues of individual freedom and rationality. It thus makes sense to think in terms of themes when considering potential literature Nobel Laureates.

My personal favorite pick for the prize is Haruki Murakami. Interestingly, my introduction to him came not through his novels but through his amazing book on running which was a great driving force for my own running efforts. But whether it’s in that book or in his novels, Murakami is quite stunning at exploring existential angst, isolation and anxiety in a world where technology is supposed to function as a palliative that connects humans together. His prose is characteristically Japanese; spare, stark and straight to the point. More than most writers I know, I think Murakami deserves to be recognized for his substantial body of work with unifying themes.

Among other people who have traditionally been on nomination lists are Salman Rushdie, Milan Kundera, Philip Roth, Cormac McCarthy and Joan Didion. I think all of these are great writers, but Cormac McCarthy would top the list for me, again because he has produced a consistent body of work that investigates raw themes of Americana in devastatingly brief and searing prose. Rushdie’s “Midnight’s Children” is brilliant, but I honestly don’t think his other work really is up to Nobel caliber. Roth is also an eminent contender in my opinion, and I would be happy to see him receive the award. Joyce Carol Oates is another favorite, but frankly I haven’t read enough of her work to form an informed judgement. In any case, it’s now been twenty four years since an American won the prize, and there are certainly a few worthy contenders by this point.

Far and away, I personally think the most creative writer in English alive today is Richard Powers; he's one of those few novelists to whom I would apply the "genius" label. His sentence constructions and metaphors defy belief and for sheer imaginative prose I cannot think of his equal. Unfortunately, while he has a cult following, his novels are rather challenging to become widely read, and generally speaking the Nobel Committee does simple rather than complex.

So that's it from my side. Let the bloodbath games commence!

Heisenberg on Helgoland

The sun was setting on a cloudless sky, the gulls screeching in the distance. The air was bracing and clear. Land rose from the blue ocean, a vague apparition on the horizon.

He breathed the elixir of pure evening air in and heaved a sigh of relief. This would help the godforsaken hay fever which had plagued him like a demon for the last four days. It had necessitated a trip away from the mainland to this tiny outcrop of flaming red rock out in the North Sea. Here he could be free not just of the hay fever but of his mentor, Niels Bohr.

For the last several months, Bohr had followed him like a shadow, an affliction that seemed almost as bad as the hay fever. It had all started about a year earlier, but really, it started when he was a child. His father, an erudite scholar but unsparing disciplinarian, made his brother and him compete mercilessly with each other. Even now he was not on the best terms with his brother, but the cutthroat competition produced at least one happy outcome: a passion for mathematics and physics that continued to provide him with intense pleasure.

He remembered those war torn years when Germany seemed to be on the brink of collapse, when one revolution after another threatened to tear apart the fabric of society. Physics was the one refuge. It sustained him then, and it promised to sustain him now.

If only he could understand what Bohr wanted. Bohr was not his first mentor. That place of pride belonged to Arnold Sommerfeld in Munich. Sommerfeld, the man with the impeccably waxed mustache who his friend Pauli called a Hussar officer. Sommerfeld, who would immerse his students not only in the latest physics but in his own home, where discussions went on late into the night. Discussions in which physics, politics and philosophy co-existed. His own father was often distant; Sommerfeld was the father figure in his life. It was also in Sommerfeld’s classes that he met his first real friend – Wolfgang Pauli. Pauli was still having trouble attending classes in the morning when there were all those clubs and parties to frequent at night. He always enjoyed long discussions with Pauli, the ones during which his friend often complimented him by telling him he was not completely stupid. It was Pauli who had steered him away from relativity and toward the most exciting new field in physics – quantum theory.

Quantum theory was the brainchild of several people, but Bohr was its godfather, the man who everyone looked up to. It was Bohr who had first applied the notion of discontinuity to the interior of the atom. It was Bohr who had explained the behavior of the simplest of atoms, hydrogen. But much more than that, it was Bohr who had an almost demonic obsession both with the truths of quantum theory and the dissemination of its central tenets to young physicists like him.

Darkness was approaching as he descended the rock and started walking back to his inn. He smiled as he remembered his first meeting with Bohr. After the war, Germany was the world’s most hated nation. Nobody wanted to deal with her. The Versailles treaty had imposed draconian measures on her already devastated economy. How could they do this? Bohr was one of those very few who had extended a statesmanlike hand toward his country. War is war, Bohr had said, but science is science. Its purity cannot be violated by the failings of humanity. The University of Göttingen had invited Bohr to inaugurate a new scientific relationship between Germany and the rest of the world. The day was as clear in his memory as the air around him. The smell of roses wafting through the windows, the audience standing or sitting on the windowsills, the medieval churches chiming in the distance.

Bohr was explaining one of the finer points related to spectroscopy which his quantum theory explained. But there was clearly a mathematical error. Had anyone else seen it? The error was an elementary one, and it did not seem worthy of Bohr. Later as he found out, Bohr was a competent but not particularly noteworthy mathematician. Physical and philosophical intuition was his forte. The mathematics he left to lesser souls, to young men who he called scientific assistants. At Göttingen he pointed out the mistake from the back and offered some other comments. He was all of twenty. Bohr graciously admitted the mistake. After the talk, when he was leaving, Bohr caught up with him. Walk with me, said Bohr. Walk, and talk. It was what Bohr did best.

They climbed up the hill near the university, then discussed the problems of atomic physics in a nearby cafe. He felt he could pledge his soul to Bohr. After Munich he had been tempted to go to Copenhagen right away, but Sommerfeld had cautioned him otherwise. Bohr was an excellent physicist, Sommerfeld had said, but at this stage in his career he would be much better served by a more rigorous and mathematical immersion in atomic physics. The best man to mentor him in this regard was Max Born in Göttingen. Born was hesitant, sometimes too sensitive to perceived slights, often in undue awe of his own students, but there was no one else who combined physical insights with mathematical rigor the way he did. Born could acquaint him much better with the formal techniques; he could always spend time with the philosophical Niels. His friend Pauli had already served as Born’s assistant and had vouched for Born’s first-rate mentorship. However he had cautioned him about Born’s insistence on early morning meetings, an expectation that had been so hard for him to meet that Born had had to send a maid to wake him up.

The moonlight illuminated the path in front of him, but there were few other lights on the tiny island. This was what he liked best about it though. Very few people, very few lights, almost nobody to talk with, but plenty of opportunities for walking and swimming in the cool water. And the air, the air. Crystal clear and seemingly designed for clearing both his nasal passages and the cobwebs in his mind. His hay fever seemed almost gone already. He could read Goethe and think about physics as much as he wanted. When he arrived at the inn he greeted the innkeeper, who when he arrived four days ago, had seemed horrified at his swollen face. She had asked him if he had been in a brawl. Sadly, political brawls and beatings were not uncommon in Germany. After a light meal of sausages and potato dumplings, he retired to his room.

In Munich, for his doctoral dissertation, he had chosen an uncontroversial topic in fluid dynamics. The final exam had been a fiasco though, and he wrinkled his brow as he thought about it. One of the examiners, Wilhelm Wien, had asked him a question from elementary physics about the resolving power of a microscope. He had forgotten the formula and had gotten hopelessly entangled in trying to work it out. He was trying to solve problems at the forefront of quantum theory; why was he being asked to answer questions that were better suited to a second-rate undergraduate? Wien would not let up, however, and Sommerfeld finally had to step in, assuring the examiners that his student was certainly promising enough to be awarded his doctorate. He had still barely escaped with a passing grade. It still rankled.

He had packed his bags and gone straight to Göttingen from Munich. It was partly to start off on quantum theory right away, but also to escape the depressing pessimism that gripped German society. The past year had seen unprecedented inflation cripple his beloved country. At its height an American dollar had been worth a trillion marks. People were carrying entire carts full of money to trade for a load of bread or for some potatoes. They were using it as insulating wallpaper in their homes. Is this what his country really deserved? As he pondered the situation he felt a spring of resentment welling up inside him. If nothing else, he would show them that Germany was still not lacking in scientific talent.

After spending some time with Born and becoming familiar with the fundamental mathematical tools of atomic physics, he had finally made it to Copenhagen. The past few months there had been among the happiest of his life. Bohr had created an atmosphere whose spirit of camaraderie exceeded even Sommerfeld’s seminars. The days would be filled with deep scientific and philosophical discussions, long walks in the Faelledparken behind the institute and games of ping-pong. Evenings were spent in entertaining Bohr and his kind wife Margarethe with Beethoven and Schubert on the piano, which after physics had been his main passion. Even more than Sommerfeld Bohr had become a father figure to him. His avuncular nature, his obsession with quantum theory and his physical agility; all of these were impressive. He would take stairs two at a time, and it seemed nobody could beat him at ping-pong.

But he had also encountered aspects of Bohr’s personality that had not been apparent before. Bohr was very gentle in personal relations, but when it came to divining scientific truth he could be ferocious, unremittingly persistent, a fanatic without scruples. He had been arguing the validity of some rather well known facts of atomic physics, but Bohr’s relentless questioning of even the basic existence of the properties of electrons and photons - questioning that continued well into the night even after he had expressed his fatigue - had almost reduced him to tears. As if Bohr’s inquisition-style interrogation had not been enough, another hitherto unobserved particle had entered Bohr’s orbit since he last met him. His name was Hendrik Kramers. Kramers was Dutch, voluble, mathematically sophisticated, could speak four languages and could play both the piano and the cello. He had been struggling with Danish and English for some time and it was difficult not to be jealous of Kramers. A kind of sibling rivalry had developed between them, both vying for the attention of the father figure.

While he had been putting the finishing touches on his mundane dissertation on fluid dynamics, Bohr, Kramers, and a young American postdoctoral fellow named John Slater had created a compelling picture of electrons in the atom as a set of pendulum-like objects. The technical term for this was harmonic oscillators. The oscillators would vibrate with certain frequencies that would correspond to transitions of electrons between different states in the atoms. Bohr and Kramers were using these oscillators as convenient representations to picture what goes on inside an atom, but they were still concerned with the well-known basic properties of atoms like their positions and velocities. He had been asked to see what he could do with Bohr and Kramers’s model.

This was where the problems had started. He liked the idea of using oscillators to represent electrons. The oscillators expressed themselves in the form of a well-known mathematical device called a Fourier series. His time with Born had made him quite familiar with Fourier series. But when he had inserted formulas for the series into the basic equations of motion, single numbers had grotesquely multiplied into entire lists of numbers. Every time he got rid of certain numbers others would mushroom, like the heads of a Hydra. He had played algebraic games, filled tables upon tables with numerical legerdemain, had gotten not an inch closer to expressing any physical quantity. And then, suddenly, like a gale from the North Sea, he had been swept off his feet by the worst bout of hay fever he remembered. It kept him awake at night. It made him feel groggy during the day. It made the morass of numbers appear even bigger than what it was.

He had finally had enough. Time for resetting the mental gears, he had told himself. The little rocky outcrop with its very low pollen count had been a favored destination for sufferers. That’s where he would go, away from the stifling hay fever and the intellectual hothouse, to the ocean, mountains and clear air which he loved best. He had known this part of the country during expeditions with his youthful Pfadfinder classmates. There they had sung songs about the fatherland and had had fervent patriotic discussions about the spiritual and political revival of Germany. He felt at home there.

The light on the ceiling was flickering as he started thinking about oscillators, about frequencies, about electrons. How does one ever know what goes inside an atom? And that’s when it struck him. It seemed like a bolt out of the blue then, but later on he realized that it was part of a continuum of mental states, a flash of insight that only seemed discontinuous like the transitions of electrons. Once again, how does one ever know what goes on inside an atom? Nobody has seen an atom or electron; they are unobservable. And yet we know they are real because we observe their tangible effects. Unobservable entities have been part of science for a very long time. Nobody knew what went on inside the sun. But scientists – German scientists among them – had figured it out based on the frequencies of spectral lines that indicated the presence of certain elements. Spectroscopy had also been paramount in the development of atomic theory. Bohr himself had demonstrated the success of the theory by using it to explain spectral lines of hydrogen.

He took a step back, looked at the whole picture from a fresh viewpoint, saw the forest for the trees. What we see are spectral lines and nothing else but spectral lines. We do not see the electron’s position; we do not see its momentum. Position and momentum may have been the primary variables in classical physics, but that was because we could measure them. In case of atoms and electrons, all we see are the frequencies of the spectral lines. What we do not see we do not know. Then why pretend to use it? Why pretend to calculate it? The frequencies are the observables. Why not use them as the primary variables, with the positions and momenta as secondary quantities? He had always been a first-rate mathematician, but now he thought about the physics. It was a fundamental shift of a frame of reference, so memorably introduced by Einstein before. The problem was that representing the position and momenta as Fourier series and frequencies still led to a list of numbers rather than a single number obtained by multiplication. But here is where his physical intuition proved pivotal. One could know which numbers from the list to keep and which ones to discard based on whether they represented transitions between real energy states in atoms. That information was available and implicit in the frequency of the spectral lines. Nature could steady that tentative march of numbers.

It was finally time to use his strange calculus to calculate the energy of a real physical system. As his excitement mounted he kept on making mistakes and correcting them, but finally he had it. When he looked at it he was struck with joy and astonishment. Out of the dance of calculations emerged an answer for the energy of the system, but crucially, this energy could only exist in a restricted set of values. In one fell swoop he had rediscovered Max Planck’s original formulation of quantum theory without explicitly using Planck’s energy formula. An answer this correct must be true. An answer this elegant must be true.

It was almost three o’clock in the morning. The night outside seemed to deepen into a deep chasm. He had hardly talked to anyone during his four days on the island, and now it seemed that all that silence was culminating in a full-throated expression of revolutionary insight. The hand of nature and his own dexterous mind had cracked the puzzle in front of him, just as invisible writing is suddenly revealed by the application of the right chemical solution. But the sheer multiplicity of applications that he now foresaw was startling. At first he was deeply alarmed. He had the feeling that, through the surface of atomic phenomena he was looking at a strangely beautiful interior, and now had to probe this wealth of mathematical structures that nature had so generously spread before him.

But that could wait. He now knew that he had a general scheme of quantum theory that could be used to solve any number of old and new problems. Bohr would be pleased, although he would still insist on several modifications to his formulation when it was time to publish. And of course he would show it to his friend Pauli who would provide the most stringent test of the correctness of his theory.

His hay fever seemed to have disappeared. He felt strong again. There did not seem much point in trying to fall asleep at this very late hour. He put on his boots and set out. There was a distant rocky outcrop, the northernmost tip of the island that he had not explored yet. He walked in the predawn light. Not a gull cried around him, not a leaf seemed to tremble. An hour later he was at the base of the rock and scaled it without much effort. There he sat for a long time until he saw the first rays of the sun penetrate the darkness. Photons of light falling on his eyes, stimulating electron transitions in atoms of carbon, nitrogen and oxygen. And at that moment he was the sole human being on earth who knew how this was happening.

Note: This is my latest column for 3 Quarks Daily. It's a piece of historical fiction in which I imagine 24-year-old Werner Heisenberg inventing quantum mechanics on the small island of Helgoland in the North Sea. Heisenberg's formulation was not the easiest to use and was supplanted by Schrödinger's more familiar wave mechanics, but it inaugurated modern quantum theory and was by any reckoning one of the most important discoveries in the history of physics.