This month’s topics:
Perplexanes, a new class of molecular topologies.
Molecular topology is becoming increasingly important in nano-technology. Molecular knots have been used to accelerate chemical reactions, and to act as delivery agents in cancer therapy. These and other applications are itemized by Fredrik Schaufelberger in a 2020 article in Nature Communications Chemistry, where he writes: “Molecular knots are evolving from academic curiosities to a practically useful class of mechanically interlocked molecules, capable of performing unique tasks at the nanoscale.”
A May paper in the Journal of the American Chemical Society shares an improved process for producing these topologically complex molecules. According to the authors of the study, attempts at synthesizing knots and links in molecules have been “straining the limits of synthetic chemistry,” because the building blocks are usually linear chains of subunits, whereas knots and links are essentially curvilinear, looped objects. To circumvent this problem, they used a new method of molecular synthesis using a tripodal (Y-shaped) building block. This allowed them to create tetrahedral-cage molecules in a process with a 66% yield rate. By comparison, they report that previous state-of-the-art methods generated a 0.8%-20% yield rate. When they enriched the synthesis with flexible peptides that allowed non-planar couplings, the reactions produced novel molecules with “an unusual combination of interlocking and interweaving.” These novel molecules are the perplexanes.
A challenge to topologists? The authors remark: “The higher complexity associated with more branching components should allow access to new subclasses of topology that have not yet been mathematically identified.” [Emphasis mine.]
Geometry and brain genetics.
In a June 13 report in Science Advances, a team of neuroscientists investigated how the human genome controls the shapes within the brain. These shapes are very irregular, so how did they quantify them? By borrowing a tool from a mathematical line of research that goes back to the 18th century: the study of vibrating membranes.
The central observation is that membranes of different shapes vibrate at different frequencies. This is one of the reasons for different musical instruments producing different sounds. A large drum sounds at a lower pitch than a small drum, because the small drum is only capable of vibrating at high frequencies, while a large drum can vibrate more slowly. The set of possible vibration frequencies for a surface (actually, their squares; see below) is called its spectrum.
Computationally, the spectrum of a surface is determined by a partial differential equation called the wave equation. For very simple or symmetrical surfaces, the wave equation and the resulting spectrum can be worked out on paper using geometry and calculus.
Here’s an example. A cord stretched between two pegs on a board (we can consider it a 1-dimensional “membrane”) will vibrate if it is plucked, producing in many cases an audible sound. To lighten notation, we work with a cord of length $\pi$. We choose one end to start, and represent by $x$ the distance along the cord. Let $f(x,t)$ represent how far the cord has been moved from its rest position at location $x$ and time $t$. In these terms, the wave equation is $$\frac{\partial ^2f}{\partial t^2}=c^2 \frac{\partial ^2f}{\partial x^2},$$where $c$ is the speed at which waves travel along the cord.
The equation states that at any point the vertical acceleration of a point on the cord is directly proportional to its concavity. This makes sense, since when the deformation at $x$ is concave downwards ($\frown$), elasticity will pull $x$ downward ($\downarrow$), whereas when it is concave upwards ($\smile$), elasticity will pull $x$ up ($\uparrow$).
The functions $$f_k(x,t)=\sin(k x) \cos(k ct),$$ for $k=1, 2, 3, \dots$ are solutions to the wave equation. Each of these elementary solutions, setting $t=0$, corresponds to a particular initial configuration of the cord. These are called the modes of vibration. By Fourier analysis, any initial configuration of the cord can be written as a linear combination of these modes. In the case of a plucked cord, where the initial vertical velocity is zero, the solution to the wave equation is the corresponding linear combination of the $f_k$.
The possible frequencies of the elementary solutions are all the integer multiples of $\frac{c}{2\pi}$, and the spectrum of the cord is the squares of those frequencies: $\frac{c^2}{4\pi^2}, \frac{c^2}{\pi^2}, \frac{9c^2}{4\pi^2},$ and so on. (In terms of tonality, if $\frac{c}{2\pi}$ is the pitch of a musical note, then $\frac{c}{\pi}$ is its octave and $\frac{3c}{2\pi}$ is a perfect fifth above the octave).
This method can be generalized to much more complex situations once we know the structure of the elementary solutions. Each is of the form $h(x)\cos (kct)$ (where $h$ is a function of $x$ alone). Writing out that the solution satisfies the wave equation
$$ \frac{\partial^2}{\partial x^2} (h(x)\cos (kct)) = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} (h(x)\cos (kct)) $$
gives
$$\frac{\partial^2 h}{\partial x^2}\cos (kct) = \frac{1}{c^2}h(x)\frac{\partial^2}{\partial t^2} \left( \cos (kct)\right),$$
which simplifies to$$\frac{\partial^2 h}{\partial x^2} = -k^2h.$$
This means that the operator $\frac{\partial^2}{\partial x^2}$ (which measures the concavity at each point) takes $h(x)$ to a multiple ($-k^2$) of itself. We say that $h$ is an eigenfunction of $\frac{\partial^2}{\partial x^2}$ with eigenvalue $-k^2$. In the example of the cord, $\sin (2x)$ is an eigenfunction of the operator $\frac{\partial^2}{\partial x^2}$ with eigenvalue $-4$.
For a curved surface, $\frac{\partial^2}{\partial x^2}$ has an analogue called the Laplace-Beltrami operator. This operator is usually denoted by $\Delta$ and incorporates information about the local geometry; the wave equation is now $$\frac{\partial^2 f}{\partial t^2} = c^2 \Delta f,$$ where $f$ is a function of the surface variables and time. The procedure for calculating the spectrum is the same as in the 1-dimensional case: If a function $h(x,y)$ defined on the surface is an eigenfunction for $\Delta$ with eigenvalue $-k^2$, then $h(x,y)\cos (kct)$ will solve the wave equation on that surface. Suppose we can identify enough eigenfunctions $h_1, h_2, \dots$ so that any initial configuration $f(x, y)$ of the surface can be written as a linear combination $a_1h_1 + a_2h_2 + \cdots$, in a process akin to Fourier analysis. Then if $-k_i^2$ is the eigenvalue associated with $h_i$, the sum
$$a_1h_1\cos (k_1ct) + a_2h_2\cos (k_2ct) + \cdots$$solves the wave equation with initial configuration $f(x,y)$.
The collection of eigenvalues (by convention, without their minus signs) $(k_1^2, k_2^2, \dots )$, written in increasing order, is the Laplace-Beltrami spectrum, or just “the spectrum,” of the surface. Note that the vibration frequencies of the various eigenfunctions are the square roots of the corresponding eigenvalues, multiplied by $c$.
To what extent does the spectrum of a surface determine its shape? Mark Kac popularized this question with his 1966 paper “Can one hear the shape of a drum?” One can’t, in general, but counterexamples are very hard to construct. For the vast majority of surfaces, if two have the same spectrum they must be isometric, i.e., geometrically identical.
In the recent paper, the authors assumed that the shape of a human hippocampus falls into this category, using the calculated spectrum as a numerical characterization of shape. The hippocampus is too complex for the wave equation to be solved exactly; instead, they used a computer to make a three-dimensional model out of tiny triangles, and calculated the spectrum using a discrete version of the wave equation.
As they write, “This multidimensional intrinsic shape representation, as an isometric invariant, is independent of rotation, translation, and scaling of the coordinate system, eliminating the need for error-prone interindividual image registration, and behaves continuously with any change in the [surface].”
Primus and collaborators worked with the Laplace-Beltrami spectrum, represented by its first 49 eigenvalues. They found the spectrum of 22 separate parts of the brain for 19,862 individuals, along with records of their genomes. The spectra were normalized to control for volume, since the authors were interested in shape independent of size. For each brain part, they ran a Multivariate Omnibus Statistical Test to search for correlations between spectrum and genome. They report 80 single-letter changes in DNA, of which 31 were unknown previously, that are “independently associated with the shape of at least one brain structure.” In particular, they found significant correlations between brain shape and genomic markers of susceptibility to hypertension, ischemic stroke and schizophrenia, “suggesting the potential use of the [Laplace-Beltrami spectrum] as an early disease biomarker.”
—Tony Phillips, Stony Brook University