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 (, , , or ).

Continuous symmetry and chirality measures: approximate algorithms for large molecular structures.

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.

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.

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 -symmetry increases with the non-bonded interactions between the occupied sp orbitals of the crowns' oxygen atoms and the unoccupied 2s orbital of the cation.

Head to Tail Distortion Wave Characterizes the Enantiomerization of Helicenes.

A fresh look on helicenes' enantiomerization process with a focus on ring conformation reveals that it can be described as a step-by-step mechanism in which maximal distortion is consecutively transferred along the helicene skeleton, head to tail. Density functional theory methods were used to compute the enantiomerization pathway, and continuous symmetry measures were applied to quantify the distortion of even-number helicenes with 8-14 rings. Our findings show that the distortion wave is additive-the process always starts from one edge of the helicene and progresses along the rings until the other edge is reached.

Binding Mode Multiplicity and Multiscale Chirality in the Supramolecular Assembly of DNA and a π-Conjugated Polymer.

Water-soluble π-conjugated polymers are increasingly considered for DNA biosensing. However, the conformational rearrangement, supramolecular organization and dynamics upon interaction with DNA have been overlooked, which prevents the rational design of such detection tools. To elucidate the binding of a cationic polythiophene (CPT) to DNA with atomistic resolution, we performed molecular simulations of their supramolecular assembly.

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.

Improved algorithms for quantifying the near symmetry of proteins: complete side chains analysis.

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.

Chiral Ramachandran Plots II: General Trends and Protein Chirality Spectra.

The degree of chirality of protein backbone residues is used to enrich the Ramachandran plot (RP) and create three-dimensional chiral RPs with much more structural information. Detailed comparative analysis of the four classical RPs (general, glycine, proline, and pre-proline) is provided, including statistical analysis of quantitative chirality distributions in the maps and in the secondary structures. Our results show that points with outlier chirality levels represent special transitional points in the folded protein such as α-helix kinks, twists of β-strands, and transition points between secondary structures.

Chiral Ramachandran Plots I: Glycine.

Ramachandran plots (RPs) map the wealth of conformations of the polypeptide backbone and are widely used to characterize protein structures. A limitation of the RPs is that they are based solely on two dihedral angles for each amino acid residue and provide therefore only a partial picture of the conformational richness of the protein. Here we extend the structural RP analysis of proteins from a two-dimensional (2D) map to a three-dimensional map by adding the quantitative degree of chirality-the continuous chirality measure (CCM)-of the amino acid residue at each point in the RP.

High handaxe symmetry at the beginning of the European Acheulian: The data from la Noira (France) in context.

In the last few decades, new discoveries have pushed the beginning of the biface-rich European Acheulian from 500 thousand years (ka) ago back to at least 700 ka, and possibly to 1 million years (Ma) ago. It remains, however, unclear to date if handaxes arrived in Europe as a fully developed technology or if they evolved locally from core-and-flake industries. This issue is also linked with another long-standing debate on the existence and behavioral, cognitive, and social meaning of a possibly chronological trend for increased handaxe symmetry throughout the Lower Paleolithic.

Effect of temperature and substitution on Cope rearrangement: a symmetry perspective.

Many reactions feature symmetry variation along the reaction path on the potential energy surface. The interconversion of the point group symmetry of the stationary points can be characteristic of these processes. Increasing the temperature, however, leads to the loss of symmetry in its traditional yes-no language.

Symmetry-enthalpy correlations in Diels-Alder reactions.

Woodward-Hoffmann (WH) rules provide strict symmetry selection rules: when they are obeyed, a reaction proceeds; when they are not obeyed, there is no reaction. However, the voluminous experimental literature provides ample evidence that strict compliance to symmetry requirements is not an obstacle for a concerted reaction to proceed, and therefore the idea has developed that it is enough to have a certain degree of the required symmetry to have reactivity. Here we provide quantitative evidence of that link, and show that as one deviates from the desired symmetry, the enthalpy of activation increases, that is, we show that concerted reactions slow down the further they are from the ideal symmetry.

Quantifying asymmetry in concerted reactions: solvents effect on a Diels-Alder cycloaddition.

We propose the notion that if asymmetry characterizes a concerted reaction, a quantitative treatment in terms of continuous symmetry can bridge the gap between the Woodward-Hoffmann (WH) rules, originally formulated for symmetry-idealized unsubstituted reactants, and the fact that these rules hold for a much wider scope of reactions. Instead of focusing on symmetry conservation along the minimum energy path, we suggest that the distortion with respect to the original expected symmetry must attain a certain minimal value, not necessarily zero. To demonstrate this approach we studied the effect of solvents on the symmetry and reactivity of the classical [4 + 2] Diels-Alder cycloaddition of (E,E)-1,4-dimethoxy-1,3-butadiene with tetracyanoethylene, revealing the predictive value of this approach.