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.