This is a lasercut version of the Bessel Functions, as a handy desk ornament:
The helical diffraction theory, (and hence the Bessel functions) were the major key to solving the structure of DNA.
In 1952 (well before the DNA structure was solved) Francis Crick & Bill Cochran wrote a paper explaining how the expected form of X-ray diffraction from a helix is the sum of various Bessel functions:
Quick bit of background. When you’re using X-rays & film to find the structure of something, what you get when the film is developed isn’t a picture of the structure. Instead, it’s (more or less) the Fourier Transform of the structure.
We can simulate this in python like so. Let’s say we have a simple helix, (which we’ll assume is smoothly continuous, and not made up of any yucky atoms). The pattern we get looks like this:
and then if we take a photo of a helix which is made up of a discrete atoms, we see a pattern like this: :
The way the maths works out is something like this; the ‘dotty helix‘ can be thought of as the (piece-wise) multiplication of two functions:
- H – a helix with constant radius
- K – a function for the ‘planes’, which is zero everywhere except at a plane every ‘p’ units
and the result in the ‘Reciprocal Space’ (i.e. what the X-ray picture will look like) can be neatly expressed as the convolution of the [Fourier transform of H] with [the Fourier transform of K].
In other words, the big ‘X’ is “stamped” on the image every where the red planes are. The result looks like a series of diamonds.
Let’s make a larger diagram. If we sketch out the expected pattern for a continuous helix, we’ll see an x-shaped pattern, roughly like:
And if the helix is made up of discrete units (atoms or rungs), then we’ll see the above pattern ‘stamped out’ multiple times on the image.
For example, if we have a helix which has 10 layer lines per twist (like real DNA), we’d expect to see a pattern like this:
That’s an amazingly good match for this (terrible quality) photo of the real thing :
You can see most of the characteristic features. The double diamond (4+ diamonds, really). Note that they meet up on the 10th line, indicating that every 10 rungs the helix makes one turn.
There’s a whole bunch more cool stuff covered in the Cochran/Crick paper, like:
- They explicitly consider cases where the number of rungs per turn isn’t a neat integer
- They do worked examples to show how it explains features in the Pauling’s recently discovered alpha helix
- They propose practical methods for analog computing via paper charts and movable masks in order for people to be able to quickly synthesize patterns for arbitrarily complex helicices in the future.
Side note: the mathematician Alexander Stokes had also worked out the helical diffraction theory at around the same time, but didn’t bother to publish it. He famously did the work on the train on the way home, and presented it to the lab the next morning. You can see the lovely sketch he did here:
Which Wilkins was so impressed with, that he stuck it on the lab notice board, with the name “Waves at Bessel-on-sea”.
It was after seeing Stoke’s picture, that I decided I wanted to make my own copy of Bessel-on-Sea for my coffee table:
Files here for anyone that wants to make their own: