Conformational analysis of nucleic acids revisited: Curves+

We describe Curves+, a new nucleic acid conformational analysis program which is applicable to a wide range of nucleic acid structures, including those with up to four strands and with either canonical or modified bases and backbones. The program is algorithmically simpler and computationally much faster than the earlier Curves approach, although it still provides both helical and backbone parameters, including a curvilinear axis and parameters relating the position of the bases to this axis. It additionally provides a full analysis of groove widths and depths. Curves+ can also be used to analyse molecular dynamics trajectories. With the help of the accompanying program Canal, it is possible to produce a variety of graphical output including parameter variations along a given structure and time series or histograms of parameter variations during dynamics.     

General method for calculating helical parameters of polymer chains from bond lengths,bond angles,and internal-rotation angles

Helical conformations of infinite polymer chains may be described by the helical parameters, d and θ (the translation along the helix axis and the angle of rotation about the axis per repeat unit), _pi (the distance of the ith atom from the axis), d_ij, and d_ij (the translation along the axis and the angle of rotation, respectively, on passing from the ith atom to the jth). A general method has been worked out for calculating all those helical parameters from the bond lengths, bond angles, and internal-rotation angles. The positions of the main chain and side chain atoms with respect to the axis may also be calculated. All the equations are applicable to any helical polymer chain and are readily programmed for electronic computers. A method is also presented for calculating the partial derivatives of helical parameters with respect to molecular parameters.     

Z-curve: a computer program calculating DNA helical axis coordinates for three-dimensional graphic presentation of curvature.

In order to predict curvature of DNA fragments, we previously developed a computer program for simply calculating a vectorial sum of all individual roll, tilt and twist wedge angles between the nearest base pairs for a given DNA fragment [Lee et al., (1991)]. Now, a new program, called Z-curve, was developed to calculate three-dimensional coordinates of the helical center of each base pair along the DNA, using helical axis deviations from B-form DNA by wedge angles. The output file of the new program was designed to become an input file for a graphics program, Insight II. Thus, we were able to obtain three-dimensional graphic presentations of DNA helical axis curvatures of any length. It visualized spatial details of the DNA curvature, where and how much it curves, and to which direction. It also allowed calculation of the three-dimensional distance between two ends of a DNA fragment, which could provide a measure of its curvature. Here, three DNA fragments, both curved and straight, were subjected to the Z-curve and Insight II programs. The results showed that their curvature details could be visualized to the level of the base pair, whether the DNA fragments contained an oligo(A) track or not. Their estimated curvatures were consistent with the experimental results of permutation gel mobility assay.     

How dinoflagellates swim

Dinoflagellates possess two flagella; usually these are directed perpendicular to one another constituting a transversal flagellum and a longitudinal, trailing flagellum, respectively. The transversal flagellum causes the cell to rotate around its length axis. The trailing flagellum is responsible for the translation of the cell; due to its asymmetric insertion it also causes a rotation of the cell around an axis perpendicular to the longitudinal axis. Together, these two rotational components result in a helical swimming path. Cells can vary the two rotational components independently as well as the translational velocity. With these three degrees of freedom, cells can vary the parameters of their helical swimming paths for steering. Dinoflagellates use this mechanism for orientation in chemical concentration gradients (“helical klinotaxis”).     

Minimizing errors associated with calculating the location of the helical axis for spinal motions

One of the more common comparative tools used to quantify the motion of the vertebral joint is the orientation and position of the (finite) helical axis of motion as well as the amount of translation along, and rotation about, this axis. A survey of recent studies that utilize the helical axis of motion to compare motion before and after total disc replacement reveals a lack of concern for the relative errors associated with this metric. Indeed, intrinsic algorithmic and experimental errors that arise when interpreting motion tracking data can easily lead to a misinterpretation of the changes caused by replacement disc devices. While previous studies examining these errors exist, most have overlooked the errors associated with the determination of the location of the helical axis and its intersection with a chosen plane. The purpose of the study presented in this paper was to evaluate the sensitivity and reliability of the helical axis of motion as a comparative tool for kinematically evaluating spinal prostheses devices. To this end, we simulated a typical spine biomechanics testing experiment to investigate the accuracy of calculating the helical axis and its associated parameters using several popular algorithms. The resultant data motivated the development of a new algorithm that is a hybrid of two existing algorithms. The improved accuracy of this hybrid method made it possible to quantify some of the changes to the kinematics of a spinal unit that are induced by distinct placements of a total disc replacement.     

Quantum-mechanical theory of CD for infinite helical polymers

Quantum-mechanical equations are derived that are particularly well suited to actual computations of the CD for helical polymers. They make use of cyclic boundary conditions and helical symmetry, so that only two matrices with a size equal to the number of transitions considered need be diagonalized. The final equations are expressed directly in terms of monomer properties and helical parameters to invite the same input as earlier calculations, and are given as a rotational strength times a shape function for ease of comparison with the earlier work. The shape of the helix term is expressed as a derivative with respect to ω and depends on the distance between monomers along the helix axis. Other terms involving two electric transition dipoles depend on the distance from the helix axis to the transition center. These equations are directly comparable to the classical equations derived for cyclic boundary conditions and helical symmetry. We present an outline of the derivation and enough intermediate steps to clarify how the equations arise.     

The study of helical distortions due to environmental changes: choice of parameters

Parameters like interhelical angles, helical parameters, levels of distortions, etc., have been analysed to test their sensitivity to environmental changes using a method developed in this laboratory. This analysis was done on protein structures solved under different environmental conditions like temperature and pH, and ligand binding. The study reveals that the helical parameters are not sensitive enough to study the effect of environmental changes on protein helices. On the other hand the helical distortions as well as changes in the interhelical angles are more sensitive to these changes. The study also provides with additional information like the origin of distortions in a helix when a ligand binds to a protein, bending in helical axis, identification and extent of domain movements, etc.     

AUGUR: a program to predict, display and analyze the tertiary structure of B-DNA

A UGUR is a program to predict, display and analyze the three-dimensionalstructure of B-DNA. The user can choose one of six models topredict the helical parameters of a given sequence. These parametersare then used to generate the coordinates of the DNA model inthree-dimensional space (trajectory). The trajectory can bedisplayed and rotated on a graphics terminal The trajectoryand helical parameters can also be searched for bends and structuralhomologues. Received on August 17, 1987; accepted on December 31, 1987     

A finite helical axis as a landmark for kinematic reference of the knee

Reference coordinates based on the finite helical axis for flexion of the knee from 0 to 90 deg are proposed. Six degree-of-freedom tracking allows the use of such a helical axis as a kinematic landmark for knee motion representation. Data from five human subjects in vivo are presented as a path of finite helical axes for flexion of the knee from 20 to 80 deg. The finite helical axis rotates by an average of 11.4 deg, the centrode translates an average of 19.8 mm, and the total axial translation averages 0.1 mm during flexion from 20 to 80 deg. Error due to the transducer was measured on a fixed-pivot pendulum and found to be 1.0 deg and 1.9 mm rms for the helical axis orientation and position, respectively, and 0.1 mm for the axial translation. Reproducibility and soft tissue effects on the measurements were repeatable to 4.0 deg and 2.7 mm rms in orientation and position, respectively, and 0.1 mm for the axial translations. Soft tissue errors averaged 4.9 deg and 3.6 mm in position and orientation, and 0.3 mm in the axial translations.     

Orientation of silk III at the air-water interface.

A threefold helical crystal structure of Bombyx mori silk fibroin has been observed in films prepared from aqueous silk fibroin solutions using the Langmuir Blodgett (LB) technique. The films were studied using a combination of transmission electron microscopy and electron diffraction techniques. Films prepared at a surface pressure of 16.7 mN/m have a uniaxially oriented crystalline texture, with the helical axis oriented perpendicular to the plane of the LB film. Films obtained from the air-water interface without compression have a different orientation, with the helical axes lying roughly in the plane of the film. In both cases the d-spacings observed in electron diffraction are the same and match a threefold helical model crystal structure, silk III, described in previous publications. Differences in the relative intensities of the observed reflections in both types of oriented samples, as compared to unoriented samples, allows estimations of orientation distributions and the calculations of orientation parameters. The orientation of the fibroin chain axis in the plane of the interfacial film for uncompressed samples is consistent with the amphiphilic behavior previously postulated to drive the formation of the threefold helical silk III conformation.     

Real-space processing of helical filaments in SPARX

We present a major revision of the iterative helical real-space refinement (IHRSR) procedure and its implementation in the SPARX single particle image processing environment. We built on over a decade of experience with IHRSR helical structure determination and we took advantage of the flexible SPARX infrastructure to arrive at an implementation that offers ease of use, flexibility in designing helical structure determination strategy, and high computational efficiency. We introduced the 3D projection matching code which now is able to work with non-cubic volumes, the geometry better suited for long helical filaments, we enhanced procedures for establishing helical symmetry parameters, and we parallelized the code using distributed memory paradigm. Additional features include a graphical user interface that facilitates entering and editing of parameters controlling the structure determination strategy of the program. In addition, we present a novel approach to detect and evaluate structural heterogeneity due to conformer mixtures that takes advantage of helical structure redundancy.     

Protein folding: evaluation of some simple rules for the assembly of helices into tertiary structures with myoglobin as an example

A computer program designed to fold a peptide chain consisting solely of helical segments and connecting links of known length is described and evaluated. This study is a detailed extension of certain aspects of the earlier work of Ptitsyn &; Rashin (1975). Possible interaction sites on the helices are sequence dependent and are calculated as described by Richmond &; Richards (1978) using probable changes in solvent contact area as a guide. The helices are then paired according to the list of potential sites, with each helix being paired at least once. The lists of pairings are then examined geometrically, each site having a defined dihedral helix axis angle, a specified inter-helix axis distance, and defined rotations, when required, about each helix axis. Two simplified filters are used: (1) lengths of connecting links must be equal to or greater than the end-to-end distances of the helices; and (2) non-paired helices must not collide. With myoglobin as a test example and only six of the eight helices being considered, a conformation space consisting of more than 3 × 10⁸ structures was surveyed. The two filters reduced the acceptable structure list to 121. Slight readjustment of the parameters in the filters would have reduced this to 20 structures. Of these 20, one closely resembles the actual distribution of helices in myoglobin. The possible utility and pitfalls of this approach as part of an overall protein folding program are discussed.     

Polynucleotide models. II. Prediction of the radial position and tilt of the bases from the helical parameters

A simple geometrical analysis is applied to base-stacking interactions in helical polynucleotides. A base-pair or other H-bonded arrangement of bases is represented by a corresponding plane. In a helical structure, these planes will intersect each other if the bases are not perpendicular to the helix axis. The tilt of the bases and the position of intersection of these planes are geometrically related, and demonstrate the interrelation of the tilt of the bases with their radial position. This approach may be coupled with a constraint which predicts the radial position of the bases based solely upon the helical parameters. These two constraints taken together allow a simple means to predict the approximate position and tilt of a given H-bonded arrangement of bases placed according to specified helical parameters. For a number of polydeoxyribonucleotides, there is more than one base position and tilt consistent with these constraints, while there is only one base position and tilt indicated for each of the polyribonucleotides examined.     

Electron-spin-resonance studies of a chlorpromazine derivative bound to DNA fibres.

Solutions of DNA, spin-labelled with the radical cation of chlorpromazine, were used to produce oriented species of fibres pulled from a gel obtained by ultracentrifugation. The electron spin resonance spectra, recorded at X and Q band frequencies, are given for both gel and fibres; 14N hyperfine coupling parameters were obtained by computer fitting. The spectra are explained in terms of a strongly immobilized label having one principal hyperfine tensor axis parallel to the axis of the DNA helix, and the preferential orientation of the chlorpromazine ions with their planes perpendicular to the DNA helical axis.     

Comparison of analyses of DNA curvature

Popular programs for characterizing DNA structure include Curves 5.1 (Lavery, R. and Sklenar, H., J. Biomol. Struct. Dyn. 6, 63-91, 1988; Lavery, R. and Sklenar, H., J. Biomol. Struct. Dyn. 6, 655-67, 1989) and Freehelix98 (Dickerson, R. E., Nucleic Acids Res. 26, 1906-1926, 1998), along with the more recent 3DNA (X. J. Lu, Z. Shakked and W. K. Olson., J. Mol. Biol. 300, 819-840 (2000). Given input of structural coordinates, all of these programs return values of the local helical parameters, such as roll, tilt, twist, etc. The first two programs also provide characterization of global curvature. Madbend (Strahs, D. and Schlick, T., J. Mol. Biol. 301, 643-663, 2000), a program that computes global curvature from local roll, tilt, and twist parameters, can be applied to the output of all three structural programs. We have compared the curvature predicted by the three programs with and without the use of Madbend. Global bend magnitudes and directions as well as values of helical kinks were calculated for four high-resolution DNA structures and four model DNA helices. Global curvature determined by Curves 5.1 without Madbend was found to differ from values obtained using Freehelix98 with or without Madbend or 3DNA and Curves 5.1 with Madbend. Using model helices, this difference was attributed the fact that Curves 5.1 is the only program sensitive to changes in axial displacement, such as shift and slide. Madbend produced robust values of bend magnitude and direction, and displayed little sensitivity to axis displacement or the source of local helical parameters. Madbend also appears to be the method of choice for bending comparisons of high-resolution structures with results from cyclization kinetics, a method that measures DNA curvature as a vectorial sum of local roll and tilt angles.     

Orientation by helical motion—II. Changing the direction of the axis of motion

We analyse the helical motion of organisms, concentrating on the means by which organisms change the direction in space of the axis of the helical trajectory, which is the net direction of motion. We demonstrate that the direction of the axis is determined largely by the direction of the organism's rotational velocity. Changes in direction of the rotational velocity, with respect to the organism's body, change the direction in space of the axis of the helical trajectory. Conversely, changes in direction of the translational velocity, with respect to the body of the organism, have little effect on the direction in space of the axis of the trajectory. Because the axis of helical motion is the net direction of motion, it is likely that organisms that move in helices change direction by pointing their rotational velocity, not their translational velocity, in a new direction.