Tony’s Take September 2022

This month’s topics:

Interview with Dennis Sullivan.

Dennis Sullivan, recipient of the 2022 Abel Prize, was interviewed by Mihai Andrei for the web newsletter ZME Science (posted on July 26, 2022). Dennis (my friend and colleague at Stony Brook; also at the CUNY Graduate Center) spends some time on generalities about being a mathematician. His passion is simplicity: everything in mathematics reduces to space and number. He tells Mihai that these are exactly the parts of our surroundings that every toddler starts to explore. But in school, mathematics becomes a chore, made even more onerous by the idea that you have to be clever and fast to succeed. Dennis tells a story from his own college days with the moral: “if you really want to understand something, you have to keep plugging at it.”

The interview gets into some detailed mathematics with a discussion of the limitations of calculus in describing the world, in particular fluid motion. The problem is that calculus relies on limits of ratios of lengths, as the lengths involved get closer and closer to zero. But in the real world, arbitrarily small lengths don’t exist. As Dennis puts it, “physics doesn’t make sense below thirty-three zeroes.” Mihai interprets for us: he’s referring to the Planck length, $10^{-33}$cm, below which measurements have no physical meaning. Back to general statements about the profession: You don’t have to be a math genius to succeed, but love of understanding is key. His last words: “So try to understand, don’t try to learn a lot, try to understand.”

“Math is the Great Secret” in the New York Times.

Alec Wilkinson is back, proclaiming his new-found fascination with mathematics to an even larger audience, on the Times
Op-Ed page
, September 25, 2022. Along with the story of how he struggled with mathematics as a teenager but reattacked at age 65, he brings in some new mathematical details: the prime numbers; as he reminds us, these are the numbers like 2, 3, 5, 7, 11 and 13 which have only 1 and themselves as divisors. The context is the age-old question about whether mathematics is discovered or invented. If we claim that humans invented numbers and counting, how can we account for the prime numbers [there are infinitely many of them, as we have known since Euclid], “that have attributes no one gave them?” Wilkinson alludes poetically to the realm where mathematics exists: “It is the timeless nowhere that never has and never will exist anywhere but that nevertheless is.”

Topology and mutations in the COVID spike protein.

Quenisha Baldwin (Tuskegee), Bobby Sumpter (Oak Ridge) and Eleni Panagiotou (UT Chattanooga, now at Arizona State) published “The Local Topological Free Energy of the SARS-CoV-2 Spike Protein” in Polymers, July 26, 2022. As they explain, the spike protein is part of the mechanism by which the COVID virus attaches itself to, and then penetrates, a target cell. Furthermore, many of the variants of COVID come along with mutations at sites along the backbone of this particular protein. (The CoSARS-CoV-2spike protein is a linear chain of 1273 amino acids (the residues), linked along a backbone of carbon atoms). But which sites? The authors give a characterization of the sites most likely to be loci of mutations, in terms of the nearby topology-geometry of the backbone, considered as a 1273-node polygonal curve in 3-space.

The criterion they define is the Local Topological Free Energy: using the free energy formalism from statistical mechanics, they measure the LTE by comparing the twistedness (in a specific sense—see below—borrowed from knot theory) of each 4-node subunit with the average twistedness of such subunits all along the molecule.

The authors refer to two different measures of twistedness, although in the main part of the article they concentrate on the first. Writhe, as they describe it, “is a measure of the number of times a chain winds around itself.” For closed, smooth curves in 3-space (most interestingly, knots) it can be defined in terms of numbers and signs of crossings in a planar projection of the curve (here is an example); recently the definition has been extended to open curves, including polygonal ones. Torsion (“describes how much [a chain] deviates from being planar”) was also initially defined for smooth curves. For polygonal curves, it is simpler to define than the writhe: if the curve has $n$ segments its torsion is $(1/2\pi)$ times the sum of the $n-2$ dihedral angles between the planes spanned by consecutive segments. (The dihedral angle between two intersecting planes is the angle between the lines they determine in a third plane perpendicular to their intersection line).

From left: a square with a blue line traced around 3 of its 4 sides. Writhe for this figure is 1/2, torsion 0. Next, a cube with a blue line traveling up one of the edges, then around two edges of the top face. Writhe is 1/6, torsion 1/4. Finally, three cubes stacked on top of one another. The line travels diagonally across the front face of the bottom cube, curves diagonally up the side of the middle cube, and finally travels diagonally up across the back face of the top cube. For this figure, writhe is 0.106 and torsion is 0.196.
Three 4-node curves for which the writhe (Wr) and torsion (T) can be calculated by hand. Values like those for (c) are reported to occur along the backbone of the COVID spike protein.

As the authors report, “most mutations of concern are either in or in the vicinity of high local topological free energy conformations, suggesting that high local topological free energy conformations could be targets for mutations with significant impact of protein function.” An example of the current relevance of this research: they mention that “the recently discovered omicron variant has two new mutations … at two conserved high LTE residues.”