Helices for mathematical modelling of proteins, nucleic acids and polymers

Helices occur in modelling a range of structures including ropes, fibers, polymers and biopolymers. In a recent paper with Thamwattana and Hill, the authors derived Euler–Lagrange equations for modelling protein structure, where the energy being minimised was assumed to depend only on the curvature and torsion of the protein backbone space curve and their first derivatives. Such a model is applicable to helices occurring in other scenarios and in this article the author considers more generally which energies will yield helices as solutions of their corresponding Euler–Lagrange equations. He finds in particular classes of energy for which all circular helices are solutions and an energy depending on curvature and its derivative which generates conical helices as solutions of the Euler–Lagrange equations. Also included are some new results for energies depending only on curvature, extending previous investigations by Feoli et al. [A. Feoli, V.V. Nesterenko, G. Scarpetta, Functionals linear in curvature and statistics of helical proteins, Nuclear Phys. B 705 (2005) 577–592].