The Application of 3D Zernike Moments for the Description of "Model-Free" Molecular Structure, Functional Motion, and Structural Reliability

Protein structures are not static entities consisting of equally well-determined atomic coordinates. Proteins undergo continuous motion, and as catalytic machines, these movements can be of high relevance for understanding function. In addition to this strong biological motivation for considering shape changes is the necessity to correctly capture different levels of detail and error in protein structures. Some parts of a structural model are often poorly defined, and the atomic displacement parameters provide an excellent means to characterize the confidence in an atom's spatial coordinates. A mathematical framework for studying these shape changes, and handling positional variance is therefore of high importance. We present an approach for capturing various protein structure properties in a concise mathematical framework that allows us to compare features in a highly efficient manner. We demonstrate how three-dimensional Zernike moments can be employed to describe functions, not only on the surface of a protein but throughout the entire molecule. A number of proof-of-principle examples are given which demonstrate how this approach may be used in practice for the representation of movement and uncertainty.