Quaternion modeling of the helical path for analysis of the shape of the DNA molecule

The three­dimensional shape of a DNA molecule is a key property influencing its functional specificity and the nature of its molecular interactions. The characteristic shape into which a DNA molecule folds under certain conditions is a manifestation of its micromechanical and structural features, which are sequence­dependent. DNA shape­related properties can there fore be determined in a predictable manner. A number of models have been designed to describe intrinsic DNA curvature, incorporating a set of helical parameters which can be applied to operative three­dimensional reconstruction of the DNA structures. Alternatively, desired base pair parameters can be computed based on publicly available information about atomic DNA structures. Further, taking the base pairs as rigid bodies, their relative location in space can be estimated based on these parameters. Matrices are a common method to implement any rigid body transformations and are widely used in the modeling of DNA structures. Quaternions are the more straightforward and robust alternative for matrices. Unit quaternions can represent only a rotation, whereas dual quaternions combine rotation and translation into a single state. In the present guide, the algebra of unit and dual quaternions is applied for the first time to modeling of the DNA helical path, based on conformational parameters of the base pair steps. Although dual quaternions are preferable for  modeling of DNA structure in detail, the use of unit quaternions is sufficient to predict the DNA trajectory and all calculations of DNA shape features. In order to analyze DNA shape and chain sta ­ tistics, and predict the micromechanical properties of DNA molecules based on coordinates of the helical path, the widely used as well as original algorithms for computing DNA curvature, radius of gyration, persistence length and phasing of DNA bends are described. Taken together, these algorithms will be useful both in the in silico analysis of relatively short DNA fragments as well as in topological mapping of whole genomes.

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