Helices
- †Institut Préparatoire aux Etudes d‘Ingénieurs d’El Manar, 2092 El Manar, Tunisia;
- ‡Department of Mathematics and Program in Applied Mathematics, University of Arizona, Tucson, AZ 85721; and
- §Institute of Mathematics B, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
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Edited by Michael Levitt, Stanford University School of Medicine, Stanford, CA, and approved May 5, 2006 (received for review September 26, 2005)
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.
Footnotes
- ¶To whom correspondence should be addressed. E-mail: john.maddocks{at}epfl.ch
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Author contributions: N.C., A.G., and J.H.M. performed research and wrote the paper.
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Conflict of interest statement: No conflicts declared.
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This paper was submitted directly (Track II) to the PNAS office.
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↵ ‖ In general, nonisotropic rods have the property that if the uniform, i.e., s-independent, constitutive relations are rewritten with respect to another reference framing, then the transformed constitutive relations will be nonuniform unless the old and new reference frames are related by means of a simple constant rotation. Thus, for nonisotropic rods there is a family of preferred choices of û leading to uniform constitutive relations, and the notion of uniform helices introduced above is with respect to any of these preferred coordinate systems. However, for isotropic rods the constitutive relations will remain uniform for any choice of û 3, even though the different reference framings are not related through a constant rotation. Thus, for isotropic rods, the notion of which helical configurations are uniform and which are not, is coordinate system dependent and is accordingly not of physical importance. Because the set of helical configurations for isotropic rods is so well understood, the ambiguity in the isotropic case will turn out to be of no practical significance (see text below).
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↵ †† Further to footnote ‖, we remark that the precise value of the excess twist c 1 on a given helical equilibrium depends on the choice of reference framing that has been made, so that in particular for an isotropic rod the subset of uniform helical equilibria, i.e., those with c 1 = 0, depends on the coordinate system chosen.
- © 2006 by The National Academy of Sciences of the USA
