General Helix Research Articles

Towards a general triple helix mediated DNA recognition scheme

This review describes new perspectives offered by the synthesis of non-natural nucleosides to overcome current limitations and extend the triplex-mediated DNA recognition scheme to any sequence. Alternate strand purine binding, direct pyrimidine recognition, and binding to the whole base-pair are described. The review highlights structural requirements to the design of modified nucleosides, as well as perturbing events such as tautomeric ambiguity and intercalation for the extended heterocyclic bases.

Curves of Constant Precession

1. INTRODUCTION. Given initial position and direction, the flight-path of a ship in Euclidean space is completely determined by how much it turns and how much it twists at each odometer reading. This is an intuitive interpretation of the Fundamental Theorem for Space Curves, which states that curvature K and torsion , as functions of arclength s, determine a space curve uniquely up to rigid motion. This statement of the Fundamental Theorem ([14], §1-8) should be tempered with the reservations expressed by Nomizu [12] and Wong & Lai [15]. Given a parametric space curve, there are well-known formulae for the arclength, curvature, and torsion (as functions of the parameter). Given two functions of one parameter (potentially curvature and torsion parametrized by arc-length) one might like to find a parametrized space curve for which the two functions are the curvature and torsion. This activity, called natural ([14], §1-10), is generally achieved by solving Riccati equations like dw/ds = -iz/2 iKW + i7W /2. Although the solution generally exists, it usually cannot be obtained explicitly. Euler [6] found explicit integral formulae for plane curves (where z - O) through direct geometric analysis. Hoppe [9] developed a general method for solving the natural equations for space curves by solving Riccati equations through a complicated sequence of integral transformations. He digressed to obtain formulae for the tangent, normal, and binormal indicatrices for general helices and essentially for curves of constant precession. Enneper [5] obtained explicit closed-form solutions for helices on revolved conic sections through direct geometric analysis. A curve of constant precession is defined by the property that as the curve is traversed with unit speed, its centrode revolves about a fixed axis with constant angle and constant speed. In this paper we obtain an arclength-parametrized closed-form solution of the natural equations for curves of constant precession through direct geometric analysis. As part of this analysis, we obtain a new theorem for curves of constant precession analogous with Lancret's Theorem for general helices. We provide the first rendering of a curve of constant precession. We also note for the first time that curves of constant precession lie on circular hyperboloids of one sheet and have closure conditions that are simply related to their arclength, curvature, and torsion. These are 3-type curves, except one family of closed 2-type curves (when Z = 4,u; see [2], [3], and [1]). Given a closed C3 curve in space, it is rather obvious that the curvature and torsion functions will be periodic functions of the arclength, with period equal the total arclength. This is a necessary condition but, as the circular helices (K and z both constant) show, not a sufficient condition that integral curves be closed. Efimov [4] and Fenchel [7] independently formulated The Closed Curve Problem. Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions K(S) and z(s) with the same period L, the integral curve is closed.

The Differential Geometry of the General Helix as Applied to Mechanical Springs

In order to improve the performance of helical springs, such as increasing the fatigue life and suppressing resonance, variable pitch angle and variable helix radius may be incorporated into the helical spring geometry. Employing the tool of differential geometry, new and complete formulae of curvature, torsion, and spring force are derived. It is shown that these formulae are more general and accurate than Kelvin’s curvature and torsion formulae, than commonly used force formulae (Wahl, 1963). Possible simplifications to the complete formulae and the corresponding errors introduced are both discussed and compared with experimental data.

Electronic absorption, polarised excitation, and circular dichroism spectra of [5]-helicene (dibenzo[c,g]phenanthrene)

The absorption and circular dichroism spectra of (+)-dibenzo[c,g]phenanthrene-9,10-dicarboxylic acid are reported, together with the polarised excitation and fluorescence spectra of the parent hydrocarbon, [5]-helicene (dibenzo[c,g]phenanthrene). The transition energies and dipole and rotational strengths of [5]-helicene have been calculated in the π–SCF approximation, using a general helix model and a model based on X-ray diffraction data for the molecular structure and a constant and a bond-length-dependent π-resonance integral. It is found that the theoretical spectroscopic quantities are not sensitive to the choice of structural model or of a constant or variable β-value, and that their relative values are in satisfactory agreement with experiment. It is concluded that (–)-[5]-helicene has the stereochemical form of a segment of a left-handed helix, and that the (+)-isomer of the derived 9,10-dicarboxylic acid has the enantiomorphic configuration.

Comparison of theories of the helix-coil transition in polypeptides.

Using Lifson's recent formalism of sequence-generating functions, several theories of the helix—coil transition in polypeptides are compared. The general helix—coil model, say as treated by Lifson and Roig, gives rise to a cubic secular equation. The largest root of this equation can be approximated closely by a quadratic equation, similar to the procedure of Zimm and Bragg, if isolated single helical states are assigned hydrogen bonds of positive (unfavorable) energy of formation. Equations are given to evaluate the effect of this artifice on the entropy of the random coil. It is pointed out that Peller's combinatorial theory, in its present form, significantly underestimates the combinatorial entropy due to the artifice of introducing a triplet of three amino acid residues (the span of a hydrogen bond) to define a unit. If the use of triplets as units is replaced by the use of amino acid residues, and if a positive energy of formation of a hydrogen bond is assigned to single helical states, then Peller's result is identical to the quadratic approximation.

Radiation field of an elliptical helical antenna

Rigorous expressions for the radiation field of a helical antenna of elliptical shape are derived on the assumption of a traveling-wave type of current distribution along the helix conductor. The analysis is valid for integral and nonintegral numbers of turns. These expressions for the general helix are employed to determine the fields for the limiting case of a circular helical antenna, and the results are essentially the same as those derived by both Knudsen and Kornhauser. Allowing the ellipse to degenerate to the other limiting case, a solution for the radiation field is obtained for the planar or commonly known zig-zag antenna. Therefore, it is possible to achieve a truly circularly polarized field with an elliptical helix of slight ellipticity.

(1) Elementary Aeronautical Science (2) Les Hélicoptères: Recherches expérimentales sur le fonctionnement le plus général des hélices: Études sur le miéanique de l'hélicoptère (3) Étude sur le ballon captif et les aéronefs matins (4) Sur la théorie des surfaces portantes

(1) An obvious need in aeronautical literature is a JL˜\ book giving a reasonably competent account of the principles of aviation, and not merely popular or spectacular on one hand, but not containing masses of technical detail of an engineering or mathematical character on the other. Messrs. Hart and Laidler's book is an attempt to provide for this need: it is intended primarily for young students among the aircraft apprentices in the Royal Air Force. The mathematical equipment of such readers is necessarily slender, and the authors do not use any more advanced mathematics than elementary algebra and trigonometry, with a few exceptions where the calculus notation has had to be employed. It may be said at once that the book seems at least to be a very good attempt in a new direction. It is pleasantly written, fairly full but not too big, comparatively cheap and nicely got up, with numerous interesting illustrations. (1) Elementary Aeronautical Science. By Ivor B. Hart W. Laidler. Pp. vi + 288. (Oxford: Clarendon Press; London: Oxford University Press, 1923.) 7s. 6d. net. (2) Les Helicopteres: Recherches experimentales sur le fonctionnement le plus general des helices: Etudes sur le mieanique de l'helicoptere. Par W. Margoulis. Pp. xi + 90. (Paris: Gauthier-Villars et Cie, 1922.) 10 francs. (3) Etude sur le ballon captif et les aeronefs matins. Par Le Commandant Charles Lafon. Pp. vi + 206. (Paris: Gauthier-Villars et Cie, 1922.) 20 francs. (4) Sur la theorie des surfaces portantes. Par Maurice Roy. (Collection “Scientia,” No. 39.) Pp. 130. (Paris: Gauthier-Villars et Cie, 1923.) 12 francs.