Energy density functions for protein structures
In this paper, we adopt the calculus of variations to study the protein folding problem with an energy functional dependent on the curvature, torsion and the derivatives of both the curvature and torsion of the protein backbone. Minimizing this energy amongst smooth normal variations yields two Euler-Lagrange equations, which can be reduced to a single equation. In the case when the energy depends only on the curvature and torsion, it can be shown that the free energy density is the form of 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, which has not been studied previously, is examined in some detail.