MSE7: Hierarchical scaling, structure, and mechanics of filamentary assemblies

Project completed July 16, 2012

Background

Rope-like structures are prevalent in everyday life, for example in wire ropes and wool, as well as biopolymer structures, such as keratin, collagen and micro-tubules. These types of structure can be considered as intertwined tubular units, which allows them to be studied mathematically.

Researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) investigated structural identification techniques for large-scale biopolymer assemblies, including small angle X-ray scattering, and they also investigated the mechanical properties of rope-like structures.

Techniques and Challenges

Tubular density: Tubular densities (i.e. a description of density of a given field that surrounds a space curve), are a common way to represent a variety of physical systems. Using classical differential geometry, the researchers developed new techniques for understanding these structures. The structural identification technique uses the Fourier transform of a tubular density. A key issue in this study was the concept of density overlap (see Figure 1e) with the correct characterisation achieved by defining the density as a distribution and utilising the superposition principle.

Elastic birods: The project also considered the mechanics of intertwined elastic rods. Particular focus was given to defining, characterising and describing the geometry of the system. In addition, the researchers had to define the range of interaction mechanics the system permitted and relate this to the system’s geometry. A second, similar issue concerned the allowable range of boundary conditions for the system. These issues had not been well-studied in pre-existing literature.

Results

Tubular density: The researchers found general expressions for the Fourier transform of a tubular density, with an arbitrarily complex geometrical structure and cross-sectional density (see Figure 1). Simple expressions for the transform were developed which allow evaluation of the full three-dimensional transform as a single integral along the length of the tube’s axis.

In addition, expressions for including arbitrary rotations and translations of the density were defined, which enable complex structures to be constructed from a number of repeated basic units, generally helical for proteins. This functionality is important as macro-molecular complexes tend to be constructed like this.

Elastic birods: The researchers also developed a model of two contacting elastic rod structures, termed a birod. This model describes a comprehensive range of interaction mechanics and boundary conditions, and how these relate to the general geometrical description of the birod.

A linear model of a particular case of the birod, in which the contact pressure dominates, was solved. The physically interesting large length-to-width ratio limit of this birod was considered, and linear stiffness coefficients for the system were obtained. It was shown that these were independent of the boundary conditions subject to physically reasonable assumptions. In addition, this birod has been used to develop a simple model of the bacterial flagellar motor.

The researchers also completed a study of the stability of twisted elastic rods.

On a different note, this work was used in order to study the magnetic flux tubes in the sun’s atmosphere. The conclusions demonstrated that commonly made assumptions were actually incorrect.

The Future

The Fourier transform expressions will be used in combination with existing results to construct continuum polymer models from a finite number of helical units. By using the Fourier transform, the technique developed provides a means for searching experimentally for proposed polymer structures. This theoretical work will be compared with experimental results.

In terms of birods, the linear stiffness coefficient of the birod model will be used to study its stability. Also, the birod model will be extended to the dynamic setting in order to incorporate time-varying forces such as friction. The work will also be extended to rods embedded in a generic medium, modelled as a continuum. Finally, these results for two rods will be furthered to n rods, since many structures, such as wire ropes, have self-similar structure on several scales.

Related Publications

[12/03] Prior C.B., Goriely A.: The Fourier transform of tubular densities, Journal of Physics A-Mathematical and Theoretical

[11/66] Prior C.B., Berger M.A.: On the shape of force-free field lines in the solar corona, Solar Physics

[11/64] Majumdar A., Prior C.B., Goriely A.: Stability estimates for a twisted rod under terminal loads: a three dimensional study, Journal of Elasticity

Goriely A., Hausrath A, Neukirch S.: The differential geometry of proteins and its applications to structure determination, Biophysical Reviews and Letters, 3, 77-101, 2008

Neukirch S., Van der Heijden G.: Geometry and mechanics of uniform n-plies: from engineering ropes to biological filaments, Journal of Elasticity, 69, 41-72, 2002