Geometry of molecules

SUMMARY: The geometry of molecules, also called the structure of molecules, can be an important property, as in the shape of a protein molecule or the double-helix of DNA.

The physical structure of molecules is important in chemistry, biology, physics, and engineering. The precise structure can influence the chemical reactivity of a molecule as well as its response to other physical interactions, such as how it can absorb energy in the form of photons (light particles or X-ray particles). These interactions can have important implications in biology, medicine, health, and engineering. For instance, how proteins fold determines their function, and the shapes of certain protein molecules influence the existence of diseases. For example, shape is important in the normal function of the hemoglobin molecule, the molecule crucial for absorbing oxygen in red blood cells so that they can transport it throughout the body.

Hemoglobin consists of four protein subunits, associated with four heme subunits (ring-like structures containing an iron atom). As one oxygen molecule (O₂) binds to one of the heme units, the molecule distorts so as to allow another oxygen molecule to more readily bind in a cooperative way to another heme unit. This, in turn, distorts the molecule so that another O₂ finds it even more readily. Altogether, four O₂ molecules can ordinarily bind to one hemoglobin molecule. In sickle-cell anemia, two mutations in two of the four protein units distort the hemoglobin molecule so that the misshapen units form long chains. These, in turn, cause the red blood cell to become misshapen and lose its elasticity so that it can no longer readily move through small capillaries. Besides being painful, the misshapen red blood cells are destroyed by the spleen, resulting in anemia.

Another example of how the shape of a protein can cause disease is that of prions, which are misshapen proteins that enter (or “infect”) cells and cause the cells’ proteins to become misshapen. Prions are probably best known as the cause of bovine spongiform encephalopathy (commonly called “mad cow disease”) in cattle. Finally, protein folding is also implicated in Alzheimer’s disease. Thus, there is a natural interest in understanding how these molecules fold. Knowing precisely how any particular protein folds in a particular chemical environment generally requires intensive mathematical computations that implement various equations from the area of physics known as “quantum mechanics.” While supercomputers are commonly used for this work, many scientific articles have been written that instead relied on computations performed by harnessing millions of ordinary personal computers volunteered by millions of individuals.

DNA

Besides proteins, another important molecule studied extensively for its structure is DNA. While the double helix structure has been known for over 50 years, precisely how DNA is used in the cells of the body is still a source of research in the twenty-first century. In order to fit inside a cell nucleus, DNA must be very tightly coiled. How the appropriate sequence of DNA that a cell might need at a particular time can be rapidly located and then rapidly transcribed into messenger RNA for making a particular enzyme of interest is a complex process. Simply understanding how unknotting the knotted DNA takes place within the nucleus is nontrivial, and the mathematical discipline known as “topology” (and its subdiscipline, knot theory) has helped to elucidate how the cell handles the knotted DNA. One key equation to help understand the process of DNA supercoiling is Lk=Tw+Wr, where Lk is the linking number, Tw is the twist, and Wr is the writhe. This equation, attributed to G. Calugareanu, J. H. White, and F. B. Fuller, relates the linking number of the DNA (which essentially describes how the two backbones of the double-stranded DNA are linked) to the twist (the twisting of either backbone relative to the central axis of the DNA) and the writhe (which relates how the central axis of the DNA is oriented in three-dimensional space).

Other Structures

Besides proteins and DNA, molecular modeling is important in other areas. In the past, a scientist looking for a chemical that would have a certain effect in a certain situation, given a compound that reacts in a slightly different way in a slightly different situation, would likely have changed one part of the molecule and tested the new product; changed another feature and tested that product; and so on. Combinatorial chemistry is devoted to trying to automate the synthesis—and efficacy studies—of a huge number of different permutations of some basic chemical structure, somewhat in parallel. Interest in combinatorial chemistry is widespread among pharmaceutical companies.

Determining the molecular structure of molecules often relies on the general area of spectroscopy, which involves examining the spectrum that results when visible, ultraviolet, or infrared light or X-ray radiation, is applied to molecules. Mathematics that can categorize the different types of symmetry that molecules can assume can be used to help spectroscopy determine what shape the molecule must have. As one example, analysis of DNA in crystalline form by X-ray crystallography led to James D. Watson and Francis Crick’s determination of the double-helix structure of DNA in 1953.

In the late twentieth century, a form of pure carbon was found to be created from an electric arc between graphite electrodes (or from high-temperature burning of gaseous hydrocarbons). The carbon compounds created are known as “fullerenes,” which are cage-like in appearance. The first fullerene to be discovered and have the results scientifically published is now known as Buckminsterfullerene or C₆₀. Discovered in 1985 by Richard Buckminster “Bucky” Fuller, it was determined to essentially look like a soccer ball in appearance (a truncated icosahedron). How the precise polyhedral cagelike structure was determined from spectroscopy relied heavily on mathematics, specifically the area of abstract algebra known as “group theory,” applied to quantum mechanics. Whereas fullerenes like C₆₀ and C₇₀ are cage-like, other pure forms of carbon obtained from graphite that do not fully close up include nanotubes. While fullerenes and nanotubes may have health applications, they are also of interest purely as nanotechnological objects. Indeed, some nanotubes are extremely strong and one day may make superstrong fibers; some, when other atoms, such as potassium, are added, are superconductors. For instance, the orientation of carbon atoms in nanotubes affects electrical conductivity (whether the molecules are conducting or semiconducting).

Another approach to determining molecular structure, particularly to surfaces, is to use instrumentation such as the scanning tunneling microscope. This tool relies heavily on physics (quantum mechanical tunneling) principles.

Bibliography

Helmenstine, Anne M. "Molecular Geometry Introduction." ThoughtCo., 4 Nov. 2019, www.thoughtco.com/introduction-to-molecular-geometry-603800. Accessed 18 Dec. 2024.

Plunkett, Mathew J., and Jonathan A. Ellman. “Combinatorial Chemistry and New Drugs.” Scientific American, vol. 276, 1997, www.scientificamerican.com/article/combinatorial-chemistry-and-new-dru. Accessed 18 Dec. 2024.

Schlick, Tamar. Molecular Modeling and Simulation: An Interdisciplinary Guide. Springer Verlag, 2006.

Sumners, De Witt. “Lifting the Curtain: Using Topology to Probe the Hidden Action of Enzymes.” Notices of the American Mathematical Society, vol. 42, 1995, homepages.math.uic.edu/~kauffman/sumners.pdf. Accessed 18 Dec. 2024.