Comparison of rotation models for describing DNA conformations: application to static and polymorphic forms

A new method, based on a space-fixed rotation axis, or local helix axis, is proposed for the calculation of the relative orientation variables for a sequence of base pairs. With this method, orientation variables are determined through the rotation of a base pair about this axis. These variables uniquely determine a set of helical variables, similar to the roll, tilt, and twist, commonly used for a description of spatial orientations of internally rigid base pairs. The proposed identification of roll and tilt with the direction cosines of the space-fixed rotation axis agrees well with their customary definitions as the openings of the angles between adjoining base pairs toward the minor groove and toward the ascending (5' to 3') backbone strand, respectively. These new variables permit a more direct physical comprehension of DNA conformations and also the behavior of self-complementary sequences. These direction cosines, together with the rotation angle about the space-fixed axis, form a set of three independent orientation variables of the bases that afford some advantages over the variously defined twist, roll, and tilt angles, either for static or average forms. An example for the static form of these variables is shown through their use to interpret crystal coordinates. An example for the average of orientation variables is based on statistical calculations. In this example, the orientation variables, together with the translational variables that describe the relative displacements of a pair of adjacent base pairs, form a canonically distributed ensemble in phase space spanned by these variables. Two sets of conformational variables are generated by using two different methods for performing rotation operations on the sequences of base pairs. The first method is based on the new single rotation about a space-fixed axis of rotation. This space-fixed axis of rotation is, in fact, the local helical axis as constructed previously by others. The second method is based on three consecutive rotations by Euler angles. Because of large flexibilities and anisotropies along various conformational variables of DNA base pairs, the two sets of generated conformational variables, based on these two different methods of performing rotation operations, lead to slightly different sets of structurally different, but energetically equivalent, spatial arrangements of the base pairs.