Helices

Abstract

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.biomolecules ͉ differential geometry ͉ elasticity ͉ filaments ͉ rods S cientists have long held a fascination, sometimes bordering on mystical obsession, for helical structures in nature (1, 2). Helices arise in nanosprings, carbon nanotubes, ␣-helices, DNA double and collagen triple helices, lipid bilayers, bacterial flagella in Salmonella and Escherichia coli, aerial hyphae in actynomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, and helical staircases (3-13). Helical structures can be understood from a discrete point of view as a regular, periodic stacking of rigid blocks, such as the bases in DNA strands (14), the tail sheaths of bacteriophages (15), the packing of flagellin subunits (16), or simply the stairs in spiral staircases (1). However, it also can be beneficial to adopt a continuous description in which a filamentary structure is represented by a central space curve along with a frame that captures the orientation of the material cross sections at each point along the curve. Although this description neglects some fine features of deformations in the cross section, it is nevertheless suitable to describe the large-scale geometrical and physical properties of long, thin structures. If the deformations are small enough with respect to the appropriate characteristic length scales of the problem, the physical attributes of the filament such as the stresses acting across cross sections can be averaged and represented as a resultant force and moment acting on the centerline. We will refer to such a description of a filamentary structure as a rod model (see, for instance, refs 5, 7, 8, and 10-13 for examples of such rod theories).In continuum mechanics, a semi-inverse problem is generally taken to mean the study of a special class of solutions w...

The generic geometry of helices and their close-packed structures

“…It was shown by an entropy argument, that when the spheres are relatively small, the optimal shape of the polymer is again a helix of the shape originally found by Maritan et al [2]. The complete classification of the single helices has been discussed in the Kirchhoff frame by Chouaieb et al [7], and the close-packing of many rods has been studied by Starostin [8]. The work of Starostin generalizes the densest hexagonal packing of many infinite straight cylinders by applying a collective helical twist.…”

Tops and Writhing DNA

“…Configurations near the minimum will also contribute due to thermal fluctuations. We first do a purely classical elastic analysis taking only the energy into account, while ignoring the entropy [10,15,5,8]. This allows us to draw on physical intuition derived from the elasticity of beams, cables, telephone cords and ribbons and paves the way for a fuller treatment which incorporates thermal fluctuations around the classical solutions.…”

Position Vector of Spacelike Slant Helices in Minkowski 3-Space

“…This fact was published for the first time by Watson and Crick in 1953 [3]. They constructed a molecular model of DNA in which there were two complementary, antiparallel (side-by-side in opposite directions) strands of the bases guanine, adenine, thymine and cytosine, covalently linked through phosphodiesterase bonds [4,5]. In the fields of computer aided design and computer graphics, helices can be used for the tool path description, the simulation of kinematic motion or design of highways, etc.…”