Introduction
Biological filament bundles are ubiquitous in nature. In high school a student would likely have first encountered them as muscle fibers, a bundle of smaller filaments collected together in a bundle (as shown below) which contract/bend (muscle contraction) to allow the body to move. On a much smaller scale they play a crucial role in the human body. Microtubule bundles are crucial to providing human cells their rigidity, Intermediate filament bundles play a crucial role in cell adhesion, and the optic nerve bundle is the name given to the organised fibrous grouping of neuron fibers which transmit visual signals to the brain.One crucial aspect of these bundles are their deformational modes of failure under applied pressures. This can be both good, the buckling of these bundles (see (a) below) is what allows for a supported release of tension in the cell interior. The pinching of microtubule bundles (b) and then localised collapse (c) is a crucial to of cell abscission (the splitting in two cells for replication). The local collapse (c) is what allows the cells to be easily split. However, it can also be catastrophic, the localised collapse of neuron bundles in the optic nerve (c), is linked to the formation of Glaucoma.
We do not yet have a full understanding of what properties of these bundles and the external forces acting on them lead to different modes of collapse (and why they react differently in different bodies). This project will aim to explore this issue using a biomechanical filament bundle model.
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Figure1: Examples of Biological filament bundles. Panel (a) is a buckled Actin filament, b) a pinched MEF Fibroblast. Left and right in (c) we have microtubule bundle during cell abscission (left) and an image of the optic nerve head (right). |
The basic model used for the individual filaments snake is the sol called slender elastic body model (the cosserat model). Visualised below
This model uses the differential geometry of frames to derive a system of partial differential equations which combine the internal mechanics (musculature) of the body and its external interaction (friction) to model its motion. It has been used to model DNA, spermatozoa, space cables, proteins, growing plants, the optic nerve and sea-shell growth amongst other
applications.
The bundle model is still yet more complex as it involves interactions between neighbouring filaments mediated by a set of interactive forces compressing and stretching the connective material which links the filaments (the distances d and areas A are sued to quantify this but he modelled behaviour can vary dramatically).
A recent variant of this model, aided
massively by a former Durham studnet, Hannah Tatman,
has put this model in 3D:
But there is much we are yet to understand about this 3d model.
Project aims
The student will specialise the model to a specific physical application. This could involved developing specific interaction forces, more exotic applied forcings (which would be required, for example, to specialise the model to cell abscission). In practice this will involved some analytic work and some numerical work (the exact balance can be chosen by the student as the project proceeds). The numerical work can be performed in Python, Julia, C++ or Java. I cannot stress enough that you do not need to be an experienced coder to perform this task. I have have many students tell me they are not confident with coding over the years, only to go ahead and create excellent projects which involve solving P.D.E's numerically.
But there is much we are yet to understand about this 3d model.
Project aims
The student will specialise the model to a specific physical application. This could involved developing specific interaction forces, more exotic applied forcings (which would be required, for example, to specialise the model to cell abscission). In practice this will involved some analytic work and some numerical work (the exact balance can be chosen by the student as the project proceeds). The numerical work can be performed in Python, Julia, C++ or Java. I cannot stress enough that you do not need to be an experienced coder to perform this task. I have have many students tell me they are not confident with coding over the years, only to go ahead and create excellent projects which involve solving P.D.E's numerically.
If you have any questions please contact me via email. I am happy to give your more details.
Prerequisites
None, even with a numerical approach my experience has been that students can develop sufficient skills during the project to treat complex systems. However, any of the following modules could be helpful: Mathematical Biology III, Differential geometry or fluid mechanics . It would match very well with the 4th year mathematical biology course!
Resources
An
introduction to the 2D bundle model
An introduction to
the Cosserat model.
Hannah's
project