Optimal kinematics of supercoiled filaments

New kinematics of supercoiling of closed filaments as solutions of the elastic energy minimization are proposed. The analysis is based on the thin rod approximation of linear elastic theory, under conservation of self-linking number with elastic energy evaluated by means of bending contribution, due to curvature effects and torsional influence due to torsion and intrinsic twist. The deformation energy of the system is required to be monotonically decreasing in time, favoring the folding process for fixed initial condition given by a critical twist value chosen to generate writhing instability [9]. Constraints to ensure the inextensibility of the filament are also included. Time evolution functions are described by means of piecewise polynomial transformations based on cubic B-spline and Hermite spline functions. We impose proper constraints so that the transformation is globally C2 and smoothing penalties on norm and curvature are also included. In contrast with traditional interpolation, values at grids points of the evolution functions are considered as the unknowns in a non-linear optimization problem. We show how the coiling process is associated with conversion of mean twist energy into bending energy through the passage by an inflexional configuration in relation to geometric characteristics of the filament evolution. These results shed new light on the folding mechanism and associated energy contents and find useful applications in the general context of structural genomics and proteomics.