Tube bundles in
physics

Mathematical aspects of tube bundles

A key consideration in the mathematical modelling
of tubes and tube bundles is their changing shape when they are
bent, stretched, twisted and folded. For
example, one can attempt to tighten a straightened rope by
twisting it (see Figure 2). However, no matter how hard you try it
is impossible to tighten a rope beyond a certain limit (try it if
you have a rope!). Eventually the rope would have to pass through
itself in order to wind any further, a physical impossibility. On the other hand we often have to
wrap and knot ropes, once again we use the fact that the rope
cannot pass through itself in order to allow the knot to catch. A
classic example of this is the formation of a plectoneme, as shown
in panels A-E to the left (Figures courtesy of Sebastien Neukirch).
The equations used to define the systems I discussed above do not
generally ensure this necessary physical constraint is actually
enforced. In order to do so we have
to define mathematical measurements of shape which remain
unchanged as long as the tubes do not pass through themselves, but
change when they do. Ensuring this quantity always stays fixed for
our system allows the mathematician to enforce this necessary
physical constraint. This is a particular branch of mathematics
called Topology (knot and braid theory in this case). The problem
is that it is hard to define examples of these measurements which
are simple to apply in practice for the mathematical modeller. It
is this topic which was the subject of my doctoral work, which
concerned the creation of a new topological measurement of the
linking of two open ended curves (published in J. Phys. A).
This
lead to the creation of an open analogue of the celebrated
Calugareanu theorem (based on an open curve definition of the
writhing). Recently I extended this open topological framework to
accounted form knotting and belt trick type deformations, in joint
work with Sebastien Neukirch (accepted in J Phys A, a pre-print is
here).
The code to calculate the writhing of a twisted ribbon/tube can be
found in the code section of this website.

Practical Application of Techniques

One of the most common methods currently used in identifying the structure of the hundreds of thousands of protein structures which exist in nature are diffraction techniques. Put roughly, electromagnetic radiation (such as x-rays) is fired at a sample of the protein of interest and the radiation is scattered by its atomic structure. The pattern formed after scattering is thus indicative of the protein's shape and the theory gives us the necessary mathematical tool to translate this data, the Fourier transform. If we can guess a structure whose Fourier transform fits this pattern then we probably have the right structure (in reality it’s a little more sophisticated than just guessing!). Unfortunately proteins can have spectacularly complex structure (just look at haemoglobin). Currently, defining the transform relies on a description of the molecule which depends on defining the hundreds of thousands to millions of orientation angles between the neighbouring molecules composing the structure. For medium resolution diffraction techniques, which are used for the more complex protein structures, this level of detail is unnecessary in order to determine its tertiary and quaternary structure (i.e., its inter-wound shape rather than the specific molecular sequence). With Alain Goriely and Andrew Hausrath I have developed a technique, based on modelling the protein as a continuum tubular (electron) density of arbitrary structural complexity, which can be used to describe protein structures with a vastly reduced number of parameters, but still having sufficient accuracy for the purpose of determining its shape (published in J. Phys. A). This work is now being used to develop efficient methods for identifying new protein structures with Dr Ehmke Pohl.

A common method of modelling physical tubes is elastic rod theory. Basically we describe a tube which bends stretches and twists under applied forces (compression and stretching) and moments (twisting and bending forces). The aim of this is to work out what shape such a tube will adopt under the application of a set of forces/moments. I am involved with modelling not single tubes but an ensemble of tubes which are inter-wound, a more complex task which involves some of the mathematical methods mentioned above. The model was used to produce the bent rope-like structure on my homepage (under an applied moment). This work will in future be used to model both biopolymer structures and wire ropes in specific physical scenarios.

Magnetic flux ropes in the Sun’s corona

The Corona is the Sun’s atmosphere directly above its surface. Rope-like structures composed of inter-wound tubes of magnetic field (magnetic flux) emerge though the surface into the corona. The coronal environment is very different from the interior of the Sun and the flux ropes begins to contort adopting coiled structures which appear as S or Z shapes from above (see x-ray images in Figure 4 and model simulations in Figure 5). Often these flux ropes will become unstable and be ejected into space. There is much interest in this process partly because it is vital to understanding what happens in the Sun's interior which, unlike the Corona, we cannot view directly using current observational techniques. In particular there is interest in measuring its helicity, a topological measure of the inter-winding of the flux tubes composing the rope. The results from my thesis (detailed in J. Phys. A) gave the correct method for measuring the helicity in this case for which the flux ropes are anchored at points on the Sun’s surface, rather than forming closed loops. Previously there had only been a measure of helicity of closed loop flux bundles. With Mitchell Berger I showed that this measurement actually indicated certain assumptions about how the helicity could be measured from observational data would not in general be correct (published in Solar Physics). A future aim will be to use this work to improve the techniques for estimating the amount of helicity of these flux ropes.

Recently, on a more technical note, Anthony Yeates and I have obtained some fundamental results regarding the interpretation of helicity of fields which allow flow through their boundary (Pre-Print). The helicity of such fields seems to be ill-defined as is is not Gauge invariant, however, we show that this is a function of mistake of defining the helicity though its vector potential, rather than basing its definition on the curves composing the field itself.