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Journal of Elasticity
Published in: Journal of Elasticity 2/2014

01-04-2014

A Catalogue of Stable Equilibria of Planar Extensible or Inextensible Elastic Rods for All Possible Dirichlet Boundary Conditions

Author: Robert S. Manning

Published in: Journal of Elasticity | Issue 2/2014

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Abstract

We catalogue configurations that locally minimize energy for a planar elastic rod (extensible-shearable or inextensible-unshearable) subject to arbitrary Dirichlet boundary conditions in position and orientation. Via a combination of analysis and computation, we determine several bifurcation surfaces in the 3-parameter space of boundary conditions and explore how they depend on the rod material parameters, including in the inextensible limit. For each possible boundary condition, we find all stable equilibria with sufficiently low energy that they might be competitive within a Boltzmann distribution if the rod were used to model DNA with tens or hundreds of base pairs, the length-scale relevant for DNA looping. Depending on the boundary conditions, there are as many as three such equilibria.
next article Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals

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Footnotes
1
If the binding protein is rigid, then it is plausible to model the protein as Dirichlet boundary conditions. If one assumes substantial protein flexibility, then the problem is more complicated but the results for Dirichlet boundary conditions may still be exploited, as was done in [28].
 
3
The values A 1=2560, A 3=25600 are of an order of magnitude relevant to DNA looping. Nondimensionalization of N-basepair DNA illustrates that A 3/K∝N 2 and preliminary computations from the laboratory of John Maddocks [private communication] suggest a proportionality constant on the order of 10, so that for loops with 10≤N≤100, we would expect 1000≤A 3≤100000. The ratio A 3/A 1 is less well-studied in the DNA literature, but preliminary analysis of molecular dynamics simulations [John Maddocks, private communication] suggests a ratio on the order of 1/10; in Sect. 4.1 below, we show that the key features of our results are relatively insensitive to A 3/A 1 once we are close to the inextensible limit.
 
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Metadata
Title
A Catalogue of Stable Equilibria of Planar Extensible or Inextensible Elastic Rods for All Possible Dirichlet Boundary Conditions
Author
Robert S. Manning
Publication date
01-04-2014
Publisher
Springer Netherlands
Published in
Journal of Elasticity / Issue 2/2014
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-013-9449-y

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