Helical states of nonlocally interacting molecules and their linear stability: geometric approach

The equations for strands of rigid charge configurations interacting nonlocally are formulated on the special Euclidean group, SE(3), which naturally generates helical conformations. Helical stationary shapes are found by minimizing the energy for rigid charge configurations positioned along an infinitely long molecule with charges that are off-axis. The classical energy landscape for such a molecule is complex with a large number of energy minima, even when limited to helical shapes. The question of linear stability and selection of stationary shapes is studied using an SE(3) method that naturally accounts for the helical geometry. We investigate the linear stability of a general helical polymer that possesses torque-inducing non-local self-interactions and find the exact dispersion relation for the stability of the helical shapes with an arbitrary interaction potential. We explicitly determine the linearization operators and compute the numerical stability for the particular example of a linear polymer comprising a flexible rod with a repeated configuration of two equal and opposite off-axis charges, thereby showing that even in this simple case the non-local terms can induce instability that leads to the rod assuming helical shapes.Comment: 34 pages, 9 figure

Tops and Writhing DNA

The torsional elasticity of semiflexible polymers like DNA is of biological significance. A mathematical treatment of this problem was begun by Fuller using the relation between link, twist and writhe, but progress has been hindered by the non-local nature of the writhe. This stands in the way of an analytic statistical mechanical treatment, which takes into account thermal fluctuations, in computing the partition function. In this paper we use the well known analogy with the dynamics of tops to show that when subjected to stretch and twist, the polymer configurations which dominate the partition function admit a local writhe formulation in the spirit of Fuller and thus provide an underlying justification for the use of Fuller's "local writhe expression" which leads to considerable mathematical simplification in solving theoretical models of DNA and elucidating their predictions. Our result facilitates comparison of the theoretical models with single molecule micromanipulation experiments and computer simulations.Comment: 17 pages two figure

Localized Helix Configurations of Discrete Cosserat Rods

Cosserat rods are the prefered choice for modeling large spatial deformations of slender flexible structures at small local strains. Discrete Cosserat rod models based on geometric finite differences preserve essential properties of the continuum theory. In previous work kinetic aspects of discrete quaternionic Cosserat rods defined on a staggered grid were investigated. In particular it was shown that equilibrium configurations obtained by energy minimization correspond to solutions of finite difference type discrete balance equations for the sectional forces and moments in conservation form. The present contribution complements the numerical studies shown in by considering localized helix configurations of discrete Cosserat rods as more complex benchmark examples

Helices

Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical centerlines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance