Writhing Geometry of Open DNA

Motivated by recent experiments on DNA torsion-force-extension characteristics we consider the writhing geometry of open stiff molecules. We exhibit a cyclic motion which allows arbitrarily large twisting of the end of a molecule via an activated process. This process is suppressed for forces larger than femto-Newtons which allows us to show that experiments are sensitive to a generalization of the Calugareanu-White formula for the writhe. Using numerical methods we compare this formulation of the writhe with recent analytic calculations.Comment: 12 pages 10 figures. Revtex

Multi-plectoneme phase of double-stranded DNA under torsion

We use the worm-like chain model to study supercoiling of DNA under tension and torque. The model reproduces experimental data for a much broader range of forces, salt concentrations and contour lengths than previous approaches. Our theory shows, for the first time, how the behavior of the system is controlled by a multi-plectoneme phase in a wide range of parameters. This phase does not only affect turn-extension curves but also leads to a non-constant torque in the plectonemic phase. Shortcomings from previous models and inconsistencies between experimental data are resolved in our theory without the need of adjustable parameters.Comment: 4 pages, 6 figures, submitted, 2 typo's corrected, one reference adde

Denaturation of Circular DNA: Supercoils and Overtwist

The denaturation transition of circular DNA is studied within a Poland-Scheraga type approach, generalized to account for the fact that the total linking number (LK), which measures the number of windings of one strand around the other, is conserved. In the model the LK conservation is maintained by invoking both overtwisting and writhing (supercoiling) mechanisms. This generalizes previous studies which considered each mechanism separately. The phase diagram of the model is analyzed as a function of the temperature and the elastic constant $\kappa$ associated with the overtwisting energy for any given loop entropy exponent, $c$. As is the case where the two mechanisms apply separately, the model exhibits no denaturation transition for $c \le 2$. For $c>2$ and $\kappa=0$ we find that the model exhibits a first order transition. The transition becomes of higher order for any $\kappa>0$. We also calculate the contribution of the two mechanisms separately in maintaining the conservation of the linking number and find that it is weakly dependent on the loop exponent $c$.Comment: 10 pages, 6 figure

Total positive curvature of circular DNA

The interplay between global constraints and local material properties of chain molecules is a subject of emerging interest. Studies of molecules that are intrinsically chiral, such as double-stranded DNA, is one example. Their properties generally depend on the local geometry, i.e. on curvature and torsion, yet the paths of closed molecules are globally restricted by topology. Molecules that fulfill a twist neutrality condition, a zero sum rule for the incremental change in the rate of winding along the curve, will behave neutrally to strain. This has implications for plasmids. For small circular microDNAs it follows that there must exist a minimum length for these to be double-stranded. It also follows that all microDNAs longer than the minimum length must be concave. This counterintuitive result is consistent with the kink-like appearance which has been observed for circular DNA. A prediction for the total negative curvature of a circular microDNA is given as a function of its length.Comment: 6 pages, 1 figure; v2: references added;v3: a crucial mistake in Eq. (8) of v2 has been corrected, and the conclusions changed accordingl

Comment on "Elasticity Model of a Supercoiled DNA Molecule"

We perform simulations to numerically study the writhe distribution of a stiff polymer. We compare with analytic results of Bouchiat and Mezard (PRL 80 1556- (1998); cond-mat/9706050).Comment: 1 page, 1 figure revtex

Remarks on Legendrian Self-Linking

The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.Comment: 42 pages, many figures; v2: minor revisions, published versio

Writhe formulas and antipodal points in plectonemic DNA configurations

The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double-helix, these numbers are not always straightforward to compute, though a simplified formula exists. We examine the range of applicability of this widely-used simplified formula, and show that it cannot be employed for plectonemic DNA. We show that inapplicability is due to a hypothesis of Fuller theorem that is not met. The hypothesis seems to have been overlooked in many works.Comment: 20 pages, 7 figures, 47 reference

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

Dynamics of filaments and membranes in a viscous fluid

Motivated by the motion of biopolymers and membranes in solution, this article presents a formulation of the equations of motion for curves and surfaces in a viscous fluid. We focus on geometrical aspects and simple variational methods for calculating internal stresses and forces, and we derive the full nonlinear equations of motion. In the case of membranes, we pay particular attention to the formulation of the equations of hydrodynamics on a curved, deforming surface. The formalism is illustrated by two simple case studies: (1) the twirling instability of straight elastic rod rotating in a viscous fluid, and (2) the pearling and buckling instabilities of a tubular liposome or polymersome.Comment: 26 pages, 12 figures, to be published in Reviews of Modern Physic