CSM Software: Continuous Symmetry and Chirality Measures for Quantitative Structural Analysis.

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.

The SARS-CoV-2 spike protein structure: a symmetry tale on distortion trail.

A preliminary step in the SARS-CoV-2 human infection process is a conformational change of the receptor binding domain (RBD) of its spike protein, characterized by a significant loss of symmetry. During this process, the residues which later on bind to the human angiotensin converting enzyme 2 (ACE2) receptor, become exposed at the surface of the protein. Symmetry analysis of a data set of 33 protein structures from experimental measurements and 32 structures from molecular dynamics simulation, show that the initial state carries clear indications on the structure of the final state, with respect to the local distortion along the sequence. This surprising finding implies that this type of analysis predicts the mechanism of change. We further show that the level of local distortion at the initial state increases with variant's transmissibility, for the wild type (WT) along with past and present variants of concern (WT ∼ alpha < beta < delta < Omicron BA.1), in accordance with the trend of their evolutionary path. In other words, the initial structure of the variant which is most infectious is also the most distorted, making its path to the final state shorter. It has been claimed that the RBD migration of the spike protein is allosterically controlled. Our analysis provides a quantitative support to a major theorem in this respect - that information about an allosteric process is encoded in the structure itself, suggesting that the path of local distortion is related to an allosteric information network.

Symmetry-Binding Correlations of Crown Ether Complexes with Li+ and Na.

The gas-phase structure of 18-crown-6 in the presence of Li+ and Na+ cations is highly flexible and generally distorted. Using density functional theory calculations, natural bond orbital analysis, and symmetry measures, we reveal the driving forces behind the structural and energy trends of 18-crown-6 and its phenyl substituents. We show that the structural deviation from C 3-symmetry increases with the non-bonded interactions between the occupied spx orbitals of the crowns' oxygen atoms and the unoccupied 2s orbital of the cation. These orbital interactions are strongly correlated with the overall host-guest interaction energy. Our approach highlights the role of non-bonded interactions and paves the way for deeper understanding of structure-reactivity relations of flexible host-guest systems.

Side chain flexibility and the symmetry of protein homodimers.

A comprehensive analysis of crystallographic data of 565 high-resolution protein homodimers comprised of over 250,000 residues suggests that amino acids form two groups that differ in their tendency to distort or symmetrize the structure of protein homodimers. Residues of the first group tend to distort the protein homodimer and generally have long or polar side chains. These include: Lys, Gln, Glu, Arg, Asn, Met, Ser, Thr and Asp. Residues of the second group contribute to protein symmetry and are generally characterized by short or aromatic side chains. These include: Ile, Pro, His, Val, Cys, Leu, Trp, Tyr, Phe, Ala and Gly. The distributions of the continuous symmetry measures of the proteins and the continuous chirality measures of their building blocks highlight the role of side chain geometry and the interplay between entropy and symmetry in dictating the conformational flexibility of proteins.