ABSTRACT
We present a comprehensive and updated Python-based open software to calculate continuous symmetry measures (CSMs) and their related continuous chirality measure (CCM) of molecules across chemistry. These descriptors are used to quantify distortion levels of molecular structures on a continuous scale and were proven insightful in numerous studies. The input information includes the coordinates of the molecular geometry and a desired cyclic symmetry point group (i.e., Cs, Ci, Cn, or Sn). The results include the coordinates of the nearest symmetric structure that belong to the desired symmetry point group, the permutation that defines the symmetry operation, the direction of the symmetry element in space, and a number, between zero and 100, representing the level of symmetry or chirality. Rather than treating symmetry as a binary property by which a structure is either symmetric or asymmetric, the CSM approach quantifies the level of gray between black and white and allows one to follow the course of change. The software can be downloaded from https://github.com/continuous-symmetry-measure/csm or used online at https://csm.ouproj.org.il.
ABSTRACT
Quantifying imperfect symmetry of molecules can help explore the sources, roles and extent of structural distortion. Based on the established methodology of continuous symmetry and chirality measures, we develop a set of three-dimensional molecular descriptors to estimate distortion of large structures. These three-dimensional geometrical descriptors quantify the gap between the desirable symmetry (or chirality) and the actual one. They are global parameters of the molecular geometry, intuitively defined, and have the ability to detect even minute structural changes of a given molecule across chemistry, including organic, inorganic, and biochemical systems. Application of these methods to large structures is challenging due to countless permutations that are involved in the symmetry operations and have to be accounted for. Our approach focuses on iteratively finding the approximate direction of the symmetry element in the three-dimensional space, and the relevant permutation. Major algorithmic improvements over previous versions are described, showing increased accuracy, reliability and structure preservation. The new algorithms are tested for three sets of molecular structures including pillar[5]arene complexes with Li+, C100 fullerenes, and large unit cells of metal organic frameworks. These developments complement our recent algorithms for calculating continuous symmetry and chirality measures for small molecules as well as protein homomers, and simplify the usage of the full set of measures for various research goals, in molecular modeling, QSAR and cheminformatics.
ABSTRACT
Symmetry of proteins, an important source of their elegant structure and unique functions, is not as perfect as it may seem. In the framework of continuous symmetry, in which symmetry is no longer a binary yes/no property, such imperfections can be quantified and used as a global descriptor of the three-dimensional structure. We present an improved algorithm for calculating the continuous symmetry measure for proteins that takes into account their complete set of atoms including all side chains. Our method takes advantage of the protein sequence and the division into peptides in order to improve the accuracy and efficiency of the calculation over previous methods. The Hungarian algorithm is applied to solve the assignment problem and find the permutation that defines the symmetry operation. Analysis of the symmetry of several sets of protein homomers, with various degrees of rotational symmetry is presented. The new methodology lays the foundations for accurate, efficient and reliable large scale symmetry analysis of protein structure and can be used as a collective variable that describes changes of the protein geometry along various processes, both at the backbone level and for the complete protein structure.
