Elastic line deformed on a surface by an external field: Intrinsic formulation and preliminary application to nucleosome energetics

If an elastically resilient thin rod (elastic line) is subjected to a static force field, it adopts a trajectory that minimizes the sum of its elastic energy and its energy of interaction with the field. Using ‘‘intrinsic’’ methods of differential geometry and variational calculus, we formulate and solve this problem for an elastic line confined to a general surface. The biological structure known as the nucleosome core particle provides motivation to apply the solution to an elastic line (DNA) subject to a centrosymmetric force emanating from a circle. In this case the surface is a plane (although the surface will be curved in a more accurate model). In preliminary numerical work we are able to represent the angular deviation of the elastic line to fifth order in powers of arc length. The result of greatest interest may be that the equilibrium trajectory appears not to approach the circle progressively with increasing strength of the attraction between line and circle. Instead, the line approaches a ‘‘barrier’’ trajectory and, as it does, becomes unstable relative to the circular arc.