The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on May 23, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(3):431-451; doi:10.1093/qjmam/hbn012
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Energy density functions for protein structures
( School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia )
ngamta{at}uow.edu.au
Received 27 March 2008. Accepted 18 April 2008.
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Abstract |
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In this paper, we adopt the calculus of variations to study the structure of protein with an energy functional 
dependent on the curvature, torsion and their derivatives with respect to the arc length of the protein backbone. Minimising this energy among smooth normal variations yields two Euler–Lagrange equations, which can be reduced to a single equation. This equation is identically satisfied for the special case when the free-energy density satisfies a certain linear condition on the partial derivatives. In the case when the energy depends only on the curvature and torsion, it can be shown that this condition is satisfied if the free-energy density is a homogeneous function of degree one. Another simple special solution for this case is shown to coincide with an energy density linear in curvature, which has been examined in detail by previous authors. The Euler–Lagrange equations are illustrated with reference to certain simple special cases of the energy density function, and a family of conical helices is examined in some detail.