Structures which commonly occur in both man-made applications and
nature are bundles of inter-wound tubular shapes. We will all
familiar with ropes, used as a device to climb or pull and in
addition large steel ropes are a common engineering structure used
to stabilise bridges and oil-rigs. Tube bundles also proliferate in
the human body. For example, the famed DNA double helix can be thought of as a
pair of tubular electron densities, as depicted in the major Figure
on my Homepage.
More complex bundles from fibred molecules such as Fibronin, Keratin
and the Optic nerve (amongst many others). We also see rope-like
structures in the Sun, where bundles of magnetic flux are expelled
from the Sun's interior to its surface, they then contort in its
atmosphere (see Figure 1 to the right) and are often then expelled
explosively (a coronal mass ejection). One school of thought holds
that this ejection is vital to the so-called dynamo process, which
is theorised to generate the Sun’s large scale magnetic field.I have and am currently involved with the
mathematical modelling of bundle structures in all these areas and
have particular interest in their response to applied forces and
moments, as well as the analysing the complexity of structure they
can exhibit
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
Biopolymers.
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.
Elastic tubes
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.
can be thought of as a
pair of tubular electron densities, as depicted in the major Figure
on my Homepage.
More complex bundles from fibred molecules such as Fibronin, Keratin
and the Optic nerve (amongst many others). We also see rope-like
structures in the Sun, where bundles of magnetic flux are expelled
from the Sun's interior to its surface, they then contort in its
atmosphere (see Figure 1 to the right) and are often then expelled
explosively (a coronal mass ejection). One school of thought holds
that this ejection is vital to the so-called dynamo process, which
is theorised to generate the Sun’s large scale magnetic field.
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 im
possibility. 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.
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).