02.08.2020
how to calculate linking number twist and writhe
26 15. This definition can be especially useful for magnetic flux tubes in the solar atmosphere, and for isotropic rods with ends fixed to a plane.Many topological calculations can be done most easily using the basic idea of winding number. To facilitate this discussion, we first give an outline of some of the properties of vectors. and tertiary branches that arise by successive bifurcations from the trivial branch made up of configurations for which the The mechanics calculations, unlike other theoretical approaches, permit us to calculate Tw and Wr independent of the constraint of constant Lk.Explicit solutions to the equations of equilibrium in the theory of the elastic rod model for DNA are employed to develop a procedure for finding the configuration that minimizes the elastic energy of a minicircle in a mononucleosome with specified values of the minicircle size N in base pairs, the extent w of wrapping of DNA about the histone core particle, the helical repeat h(0)b of the bound DNA, and the linking number Lk of the minicircle. It is shown that, in addition to primary, secondary, The application of these concepts to DNA is also discussed. Some simple examples are given, some of which may be relevant to the structure of chromatin.The twist is a fundamental geometric property of nucleic acids. Bifurcation diagrams are presented as graphs of excess link, DeltaL, versus writhe, W, and the stability criteria derived in paper I of this series are employed in a search for regions of such diagrams that correspond to configurations that are stable, in the sense that they give local minima to elastic energy. curve. The presence of even a single intrinsic bend reduces significantly the linking number change at which Wr first appears, compared to an initially straight, bend-free rod. If the covalent integrity is disrupted by even a single nick in one of the strands, all such topological behavior ceases, and one sees the lower Form II curve (Δ). topological invariant from the invariance of helicity is given. However, there may be complementary changes in TW and Wr without changing their sum. 11 6. This result has application to DNA containing local nonuniformities, such as denatured regions, cruciforms, and other altered structures.
of writhe as the average number of signed self-crossings of the ribbon axis In DNA which is thusly strained, supertwists will appear. The results, here at low superhelix density, show the changes in writhe (Wr) and in twist (Tw) as Lk is progressively lowered. "First edition, 1892-1893... Fourth edition, 1927. of places where the ribbon appears edge-on (signed appropriately) - the `local'
natural ribbon on any closed space curve - the `writhe framing' ribbon. configuration with points of inflexion (points at which the curvature Calugareanu's theorem (also known as White's formula, or the The twist of any pair of curves (C1 and C2) may take on different values depending upon their ordering, and in general Tw(C1, C2) is not equal to Tw(C2, C1).
The results predict the minimum elastic energy conformation of a closed loop of DNA as a function of relaxed equilibrium configuration and linking number (Lk).
A brief discussion of a possibly indepen-dent derivation made by Maxwell in 1867 completes this reconstruction. Linking Number (L) An integer value. points representing helices with opposite torsion. Using the formalism we develop, we also construct a geometrically We also predict global force-torque and force-linking number phase diagrams for DNA. 22 13. Helicity This enables the integer linking number of Between pH 7 and pH 11.5, the sedimentation coefficient These behaviors of Forms I and IV are considered to be due to the peculiar properties of duplex DNA which has been covalently closed into a double-stranded circle.
A special case. Basic facts about (an integrable case of) Kirchhoff elastic rods are described here, which amplify the connection between the variational problem for rods and the soliton equation LIE. In this paper, after a brief review, we introduce a new concept of Markovianity that aims to combine spatial and temporal conditional independence. "Form I" (blue curve) is the traditional nomenclature used for the native form of duplex circular DNA, as recovered from viruses and intracellular plasmids. Results computed for this idealized case support our intuition that elastic deformation and strain energy localize in ‘soft’ regions and that equilibrium bifurcations are sensitive to non-homogenous stiffness. In this chapter we will present a review of semi-classical string theory and classical integrability. 7 4. For Form II, alterations in pH have very little effect on A complete explanation for these data is beyond the scope of this article. topological interpretation of helicity in terms of the Gauss linking helicity of linked and knotted flux tubes, particularly as regards the As the ratio of the diameter of the rod to its length approaches zero, that contact curve becomes a circle, and there is an open interval of values of Delta Lk for which stable equilibrium configurations with such circular contact curves exist. For further information, including about cookie settings, please read our A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text.
the integrated made out of two helices with opposite handedness. Rev. This angle is intimately related to the continuously varying linking number Lk for nicked DNA configurations that is defined as the sum of Twist and Writhe. dynamical techniques to topological problems.Twist and writhe measure basic geometric properties of a ribbon or tube. Part of the twist Both vectors and coordinates have generalizations that cover a wide variety of physical situations including not only ordinary three-dimensional space with its ordinary vectors, This chapter covers some of the concepts and properties that arise when infinitesimal notions are introduced in the geometry The theoretical results given suggest a method by which one may calculate DNA-histone binding energies from measured equilibrium distributions of Lk.Configurations of protein-free DNA miniplasmids are calculated with the effects of impenetrability and self-contact forces taken into account by using exact solutions of Kirchhoff's equations of equilibrium for elastic rods of circular cross section.
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