Quantitative symmetry and chirality—A fast computational algorithm for large structures: Proteins, macromolecules, nanotubes, and unit cells

Abstract

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.

Chiral Dualism as an Instrument of Hierarchical Structure Formation in Molecular Biology

“…Perhaps this problem does not exist at all. The paradoxical outcome of this study may be a new look at an old hypothesis about the relationship between the chiral polarization of biomolecules and a spontaneous break of electroweak symmetry [55][56][57][58]. In other words, the initial cause of the chiral polarization of biomolecules could lie not in local symmetry breaking at the level of chemical compounds-probionts-but in the conjugate sequence of chiral symmetry breaking, leading into the depths of the microworld.…”

The near‐symmetry of proteins

“…Ref. describes in detail how this problem is solved, resulting calculation times which scale reasonably with N 2 . Typically, computing the S( G ) value of a protein of 70 KDa takes less than 5 min on an Intel Core i5–3470 @ 3.20 GHz processor.…”

“…; and so on. The use of quantitative descriptors of symmetry—the Continuous Symmetry Measure (CSM)—has already proven very useful in identifying and understanding a host of symmetry‐related phenomena, and some encouraging indications for its usefulness in protein structure analyses exist. For instance, Keinan et al .…”

“…The CSM quantitative symmetry analysis of proteins suffered for two decades from the magnitude of the problem: The large number of atoms with all of their potential permutations (a key step in the calculations) has drastically limited the ability to apply CSM analyses to proteins. The computational problem was recently solved, decreasing the dependence of the computation time on the number of atoms, N, from N! to N 2 ; this has opened the gates for a detailed symmetry analysis of proteins.…”

Continuous symmetry measures for complex symmetry group

“…We will start by describing the procedure for the complex groups and then we will generalize it for the complex groups. We start by choosing one of the cyclic subgroups, for example, G 1 , and calculate the best permutation and axis for it using the analytical procedures (or the fast approximation method for large structures). Now, we need to find the axis for the second cyclic subgroup, G 2 , with the constraint that the angle between it and the axis of the first cyclic subgroup equals to a known angle.…”