April 2003

# The Double Helix

Quick, what do you get when you double a helix?

The answer, as everyone knows, is a Nobel Prize. Exactly fifty years ago this month, on April 25, 1953, the molecular biologists James D. Watson and Francis H. C. Crick published their pivotal paper in Nature in which they described the geometric shape of DNA, the molecule of life. The molecule was, they said, in the form of a double helix - two helices that spiral around each other, connected by molecular bonds, to resemble nothing more than a rope ladder that has been repeatedly twisted along its length. Their Nobel Prizewinning discovery opened the door to a new understanding of life in general and genetics in particular, setting humanity on a path that in many quite literal ways would change life forever.

#### Watson and Crick with their model of DNA (1953), alongside a modern illustration of the now famous molecule

"This structure has novel features which are of considerable biological interest," they wrote. Well, duh. You're telling me it does. But does the structure have any mathematical interest? More generally, never mind the double helix, does the single helix offer the mathematician much of interest?

Given the neat way the two intertwined helices in DNA function in terms of genetic reproduction, you might think that the helix had important mathematical properties. But as far as I am aware, there's relatively little to catch the mathematician's attention.

The equation of the helix is quite unremarkable. In terms of a single parameter t, the equation is

x = a cos t, y = a sin t, z = b t
This is simply a circular locus in the xy-plane subjected to constant growth in the z-direction.

A deeper characterization of a helix is that it is the unique curve in 3-space for which the ratio of curvature to torsion is a constant, a result known as Lancret's Theorem.

Helices are common in the world around us. Various sea creatures have helical shells, like the ones shown here

#### Helical shaped shells

and climbing vines wind around supports to trace out a helix.

In the technological world of our own making, spiral staircases, corkscrews, drills, bedsprings, and telephone handset chords are helix-shaped.

#### A spiral staircase: where the helix leads to a higher things

The popular Slinky toy, pictured below, shows that the helix is capable of providing amusement for even the most non mathematical among us.

And what kind of a world would it be without the binding capacity the helix provides in the form of various kinds of screws and bolts.

#### Making a bolt for it: the helix in everyday use

One of the most ingenious uses of a helix was due to the ancient Greek mathematician Archimedes, who was born in Syracuse around 287 BC. Among his many inventions was an elegant device for pumping water uphill for irrigation purposes. Known nowadays as the Archimedes screw, it comprised a long, helix-shaped wooden screw encased in a wooden cylinder, like this:

#### The Archimedes screw

By turning the screw, the water is forced up the tube. The same device was also used to pump water out of the bilges of ships.

But when you look at each of these useful applications, you see that there is no deep mathematics involved. The reason the helix is so useful is that it is the shape you get when you trace out a circle at the same time as you move at a constant rate in the direction perpendicular to the plane of the circle. In other words, the usefulness of the helix comes down to that of the circle.

So where does that leave mathematicians as biologists celebrate the fiftieth anniversary of the discovery that the helix was fundamental to life? Well, if what you are looking for is a mathematical explanation of why nature chose a double helix for DNA, the answer is: on the sidelines. On this occasion, the mathematics of the structure simply does not appear to be significant.

On the other hand, that does not mean that Crick and Watson did not need mathematics to make their discovery. Quite the contrary. Crick's own work on the x-ray defraction pattern of a helix was a significant step in solving the structure of DNA, which involved significant applications of mathematics (Fourier transforms, Bessel functions, etc.). Based on these theoretical calculations, Watson quickly recognized the helical nature of DNA when he saw one of Rosalind Franklin's x-ray diffraction patterns. In particular, Watson and Crick looked for parameters that came from the discrete nature of the DNA helices.

Now, in the scientific advances that followed Crick and Watson's breakthrough, in particular the cracking of the DNA code, mathematics was much more to the fore. But that is another story. In the meantime, I hope I speak for all mathematicians when I wish the double-helix a very happy fiftieth birthday.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His most recent book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, published last fall by Basic Books.

Jeff Denny of the Department of Mathematics at Mercer University contributed to this month's column.