No doubt you’ve seen the cartoon depiction of the novel coronavirus: A big ball, usually red, with numerous projections sporting what seem like suction cups on their ends. Like one of those toy balls you can throw it at a wall and it sticks there.
I don’t know if this is at all an accurate representation of the virus, but maybe that doesn’t matter. What we do know is that the projections stand for so-called “spike proteins”, and it is these proteins through which the virus attacks and invades other cells. In a sense, they are the weapons that the virus carries to war, wielding them as it moves through its surroundings in search of vulnerabilites to exploit, thus people to infect. Think of battles you’ve heard of, in which an army finds and bursts through some weakness in an enemy formation. The virus operates similarly. It uses spike proteins to bind to “receptor” proteins on the cells it encounters. Once that happens, it’s like the virus has opened a door into the hapless cell. Now, troops pour through the door and they begin infecting this new host — in our case, that person you know or have read about who has “caught” the virus and needs treatment.
Too anthropomorphic a picture for you? Maybe so. But it helps us focus on what we need to do to battle this virus. For just as these spike proteins are the virus’s sweeper patrols, opening up new routes to infections, they also seem to corona researchers to be the virus’s Achilles heel. They figure that if they can find a way to stop the spike proteins from binding to receptor proteins, they will have neutralized the virus. That’s easier said than done, admittedly. But crucial to this effort is to understand the structure of the spike protein—what it looks like. Again anthropomorphising: If you want to stop an invasion, it helps to know the nature of the beast—the size of those patrols, their command structure, how they move and behave, etc.
In February, researchers were able to put together a “blueprint” of the spike proteins, meaning a 3D representation of the structure of the protein molecule. Now they knew, if they encountered such a molecule, where in it to look for each of its atoms, and how those atoms are connected to each other.
The task that remains, then, is to identify the weak points in this structure. We know that the proteins change shape substantially when they bond to receptors. Do those changes happen because of unstable or weak points in the molecular structure? Can we identify them? That is, which atoms, or which bonds between atoms, are vulnerable to attack by whatever drug we might invent for the purpose? It’s a good question to ask. Because if we do manage to hit the virus at those weak points, perhaps we will have damaged its ability to bond to receptor proteins.
In essence, this is the framework for the ongoing search for a vaccine against the virus.
In just the last few weeks, we’ve had reports of progress towards finding the weak spots from an unexpected angle: Mathematics. Robert Penner is a mathematician at France’s Institute of Advanced Scientific Studies. He decided to look for spots of high energy on the protein molecule. His rationale was that these “high free energy sites” are likely to be weaker than elsewhere on the molecule. (Anthropomorphising one last time: You have to enter a building guarded by two men. One is jumpy, nervous and talkative. The other is stolid and silent. Which one would you assume is more amenable to persuasion?)
But how do we locate these spots? Considerably simplified, it goes like this. Penner examined the bonds between atoms in the protein molecule. As the virus finds and binds to a receptor, the shape of the protein changes—and with this transformation, these bonds rotate around each other. How much they rotate depends on how much energy they have to begin with. So the search for high-energy spots is, in effect, a search for specific degrees of rotation. Now if while trying to find biological answers, we start looking at these rotations, we are really looking at geometry. We are in entirely mathematical territory. We’ve converted a problem in biology into one in mathematics, and that itself might produce answers we would not otherwise have come upon. This is often how science progresses, and this was Penner’s motivation.
In the 19th Century, the great German mathematician Carl-Friedrich Gauss gave us a way to describe a rotation: Specify its axis and the amount of the rotation. What does this mean? Let’s say you’re running laps around the local cricket stadium—that is, we might say you’re “rotating” around its centre. You call your husband to tell him what you’re doing. “I’m running on the track at Brabourne Stadium,” you say, and he immediately visualizes the loop of the track and thus its axis, which is an imaginary pole stuck in the centre. But he also wants to know exactly where on the track you are as you speak. “45 degrees,” you might say. That is, imagine a rod attached to the pole and stretching to the entrance, where you started your run. Rotate it anti-clockwise 45 degrees and it will point to precisely where you are as you pant sweet nothings to your husband. That’s what Gauss meant: Specify the axis and the angle, and you’ve described a given rotation precisely.
Take this a little further. The husband wants to know about any others doing laps like you. There are six more runners, at different spots on the track. Looking around, you tell your husband: “30, 44, 151, 190, 286, 319″. He’ll know exactly where each runner is (and that one is right behind you — which one?). Now imagine making notches on the pole corresponding to each of these angles. That is, the distance from the base of the pole to a given notch tells you how far that runner has “rotated” around the stadium. That pole perfectly describes where all seven runners are at that instant. In a sense, it is a snapshot of the rotations on display at that instant in Brabourne Stadium.
Penner set himself the task of finding just such snapshots of rotations in protein molecules. Working with a huge database of proteins, he was able to put together a large number of these poles — strictly theoretical, of course — pointing in all kinds of directions, with the requisite notches on them. Think of all those poles attached to a single point, and you will visualize the spiky ball — strictly theoretical, of course — they form. (Sort of like a spherical pincushion, sort of like the cartoon image of the virus itself).
What does this ball tell Penner? Some parts of it were less dense than others, meaning rotations in those regions are rarer than others. This is useful, because it is also known that the energy in these less frequently-occurring rotations is higher than elsewhere. (Remember, we are searching for spots with higher energy). As Penner notes in his paper (Conserved High Free Energy Sites in Human Coronavirus Spike Glycoprotein Backbones, Journal of Computational Biology, 13 May 2020), he was able to “pinpoint regions of high free energy … whose obstruction might interrupt function.” That is, using this ball that is just an abstraction, he identified parts of the protein molecule that are potential targets to attack — Penner called them “exotic” spots — in the effort to disable the virus’s function.
There’s evidence that Penner might be on to something with this geometrical approach: Some regions his spiky ball uncovered are already identified weak points in the protein molecule. So maybe the other exotic spots it has uncovered are worth focusing on as well.
Still, these are findings that need further research and validation. Penner’s paper identified three exotic sites on the virus. Are they accessible? Are there drugs that can target them? What happens if they are attacked? Are those drugs safe for human consumption? Lots of questions to be answered. Penner also sounds this caution: “Mother Nature, with her wider vocabulary of sites and compounds, has not succeeded in finding antibodies [for the virus], so how can we expect greater success?”
A sobering thought. But as Penner notes, there is a “pressing need for an effective and robust vaccine to treat COVID-19, for human society and humanity itself are under siege.”
Mathematics possibly rescuing us from that siege? I thoroughly like that idea.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun