Differential geometry of polymer models: worm-like chains, ribbons and Fourier knots

Abstract

We analyze several continuum models of polymers: worm-like chains, ribbons and Fourier knots.We show that the torsion of worm-like chains diverges and conclude that such chains can not be described by the Frenet-Serret (FS) equation of space curves. While the same holds for ribbons as well, their rate of twist is finite and, therefore, they can be described by the generalized FS equation of stripes. Finally, Fourier knots have finite curvature and torsion and, therefore, are sufficiently smooth to be described by the FS equation of space curves.

Models of random knots

“…In the search of a more numerically accessible model for worm-like loops, Rappaport et al (2006) and Rappaport and Rabin (2007) suggested the following mathematical model. One way to construct a Brownian bridge in R 3 is by the following Fourier series, with w k ¼ 1.…”

Optimal Control of a Formula One Car on a Three-Dimensional Track—Part 1: Track Modeling and Identification

“…Both of these quantities will be defined in terms of a unit camber vector n(s), which lies in the plane of the ribbon and is orthogonal to the tangent vector t. We will represent the ribbon parametrically as R ¼ rðs; nÞ ¼ xðsÞ þ nðsÞn 2 R 3 : s 2 ½s 0 ; s f ; n 2 ½n l ðsÞ; n r ðsÞ È É (11) in which s is the arc length and n is a lateral offset in the ribbon plane. Lateral displacements on the ribbon surface are measured from the spine in the direction of n. The width of the ribbon at each s is given by jn r ðsÞ À n l ðsÞj.…”

“…1, which will later be returned to when making these ideas precise. Ribbon-based models have been used to describe DNA, polymers, and other chainlike molecules and some of their properties [8][9][10][11]. For our purposes, these papers play a useful role by describing some of the differential-geometric ideas that we will use here; the account given in Ref.…”

“…For our purposes, these papers play a useful role by describing some of the differential-geometric ideas that we will use here; the account given in Ref. [11] is particularly useful.…”

Conformational collapse of surface-bound helical filaments

“…Furthermore, classical models of polymers such as the worm-like chain model do not take anisotropies in elastic moduli or finite widths of the biopolymers into account. A number of recent efforts have therefore focused on the statistical mechanics of elastic "ribbons" with a finite aspect ratio as a more realistic model for typical biopolymers or other protein aggregates [18][19][20][21][22][23][24][25][26][27][28]. The ribbon-like nature of the polymers can lead to interesting effects including layering transitions in highly anisotropic condensates and the existence of an underlying helical structure even in the absence of twist [29,30].…”