Tops and Writhing DNA

Abstract

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.

Biopolymer elasticity: Mechanics and thermal fluctuations

“…The period of the orbit is P = 2s 0 . As was shown in [36], the configuration is a local minimum of the energy only if the length L of the polymer is equal to P (and not a multiple of it). It has been shown earlier [36] that the local minima of the energy are "good curves" [30] i.e they satisfy the conditions of Fuller's [16,15] theorem and so the writhe can be computed from a simple local formula 3 .…”

“…This problem is hard because of the appearance of the writhe W CW [x(s)], which is a non local function of the curve. However, we will make progress by noting that variations of the writhe are local [37,36]. We will compute the partition function assuming that for high stretch forces, the sum over curves is dominated by configurations near minima of the energy.…”

“…Since we have already established [36] that the writhing family are "good curves" [30] we may compute the writhe using the simpler "Fuller formula" W F , which (apart from normalization) measures the solid angle swept out by the unique shorter geodesic connecting the tangent vector to the north pole. We find by a straightforward calculation that the writhe of the writhing family is…”

Statistical mechanics of bent twisted ribbons

“…The constraint is enforced by the Lagrange multiplier τ , which has the physical interpretation of the torque imposed on the ribbon. (The writhe of an open space curve whose initial and final tangent vectors are fixed is defined [17] by extending the curve beyond its ends to infinity adding straight line segments with constant tangent vectors ti and tf .) We will work in the constant torque ensemble, where τ is held fixed.…”

“…The problem is formally similar to that of a symmetric top, (as was known to Kirchoff [18]) and this greatly aids the solution. Borrowing from classical mechanics, we can use variational techniques and identify "constants of the motion" The Euler-Lagrange (E-L) equations that one arrives at from the variational prob-lem are [17]:…”