Nucleic acid structure analysis. Mathematics for local Cartesian and helical structure parameters that are truly comparable between structures.

Analyzing nucleic acid structures in a comparable manner has become increasingly important as the number of solved structures has increased. This paper presents the concepts, mathematics, theorems, and proofs that form the basis of a new program to analyze three-dimensional DNA and RNA structures. The approach taken here provides numerical data in accordance with guidelines set at a 1988 EMBO workshop. Mathematical definitions are provided for all local structural parameters described in the guidelines. The definitions satisfy the guideline requirements while preserving the original physical intuition of the parameters. In particular, the rotational parameters are true rotations based on a simple physical model (net rotation at constant angular velocity), not Euler angles or angles between vectors and planes as is the case with other approaches. As a result, the mathematical definitions are symmetrical with the property that a 5 degrees tilt is the same as a 5 degrees roll and a 5 degrees twist, except that the rotations take place about different axes. In other approaches, a 5 degrees tilt can mean a different amount of net rotation than a 5 degrees roll or a 5 degrees twist. A second unique feature of the mathematics is that it explicitly incorporates the concept of a pivot point, which is the point about which a base in a base-pair rotates as it buckles, propeller twists, and opens. Pivot points enable one to model the physical motion of bases more accurately. As a result, they greatly reduce and/or eliminate the statistical correlations between rotational and translational parameters that arise as mathematically induced artifacts in other approaches. This paper, together with the statistical analysis in the companion paper for determining the locations of the pivot points, provides everything needed to understand the output of the program as it relates to individual structures.

Nucleic acid structure analysis: a users guide to a collection of new analysis programs.

Common nomenclature describing the geometry of nucleic acid structures was established at a 1988 EMBO Workshop on DNA Curvature and Bending (Diekmann, S. (1988) J. Mol. Biol. 208, 787-791; Diekmann, S. (1989) The EMBO Journal 8, 1-4; Sarma, R.H. (1988) J. Biomol. Structure & Dynamics 6, 391-395; Dickerson, R.E. (1989) J. Biomol. Structure & Dynamics 6, 627-634; Dickerson, R.E. et al. (1989) Nuc. Acids Res. 17, 1979-1803). We have subsequently developed and incorporated sophisticated mathematics in a computer program to calculate the parameters described by the guidelines. The program calculates all the local parameters relating complementary bases and neighboring base and base pairs in both Cartesian and helical coordinate frames. In addition, the main mathematical property requested by the EMBO guidelines--that the magnitude of the parameters be independent of strand or direction of measurement--is accomplished without the use of a midway coordinate frame for the rotational parameters. The mathematics preserve the physical intuition used in defining the parameters; in particular, the rotational parameters are true rotations based on a simple physical model (rotation at constant angular velocity for a unit amount of time), not Euler angles or angles between vectors and planes as is the case with other approaches. As a result, the mathematical equations are symmetric with the property that a 5 degrees tilt is the same as a 5 degrees roll or a 5 degrees twist, except that the rotations take place about different axes. In other approaches, a 5 degrees tilt can mean a different amount of net rotation from a 5 degrees roll or a 5 degrees twist. In addition, a great deal of flexibility is built into the program so that the user has control over the analysis, including the input format, the coordinate frame used for the base pairing relationship, the point about which the rotations are performed, and which geometric relationships are analyzed. While there is a great deal of flexibility, the program is easy to use. Interactive queries and user accessible files make the options in the program very convenient to tailor to individual needs. In addition, there is also a program that calculates bond lengths, valence angles, and torsion angles along the nucleic acid backbone, and within the sugar and base rings. Another program 'learns' the identities of the bond lengths, valence angles, and torsion angles that the user would like to determine.(ABSTRACT TRUNCATED AT 400 WORDS)