Quantitative symmetry and chirality--a fast computational algorithm for large structures: proteins, macromolecules, nanotubes, and unit cells

Symmetry is one of the most fundamental properties of nature and is used to understand and investigate physical properties. Classically, symmetry is treated as a binary qualitative property, although other physical properties are quantitative. Using the continuous symmetry measure (CSM) methodology one can quantify symmetry and correlate it quantitatively to physical, chemical, and biological properties. The exact analytical procedures for calculating the CSM are computationally expensive and the calculation time grows rapidly as the structure contains more atoms. In this article, we present a new method for calculating the CSM and the related continuous chirality measure (CCM) for large systems. The new method is much faster than the full analytical procedures and it reduces the calculation time dependency from N! to N(2), where N is the number of atoms in the structure. We evaluate the cost of the applied approximations, estimate the error of the method, and show that deviations from the analytical solutions are within an error of 2%, and in many cases even less. The method is applicable at the moment for the cyclic symmetry point groups- C(i), C(s), C(n), and S(n), and therefore it can be used also for chirality measures, which are the minimal of the S(n) measures. We demonstrate the application of the method for large structures across chemistry: proteins, macromolecules, nanotubes, and large unit cells of crystals.