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.

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. 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.

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.

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. As more rings are added to the helicene, extra steps are appended to the distortion wave. Implications of this fundamental process are discussed in light of similar natural phenomena from polymer dynamics to snake locomotion.

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. Comparison of replicated simulations show a multiplicity of CPT binding geometries that contribute to the wrapping of CPT around DNA. The different binding geometries are stabilized by both electrostatic interactions between CPT lateral cations and DNA phosphodiesters and van der Waals interactions between the CPT backbone and the DNA grooves. Simulated circular dichroism (CD) spectra show that the induced CD signal stems from a conserved geometrical feature across the replicated simulations, i. e. the presence of segments of syn configurations between thiophene units along the CPT chain. At the macromolecular scale, we inspected the different shapes related to the CPT binding modes around the DNA through symmetry metrics. Altogether, molecular dynamics (MD) simulations, model Hamiltonian calculations of the CD spectra, and symmetry indices provide insights into the origin of induced chirality from the atomic to the macromolecular scale. Our multidisciplinary approach points out the hierarchical aspect of CPT chiral organization induced by DNA.

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.

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. 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.

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 ß-strands, and transition points between secondary structures. A protein chirality spectrum in which the degree of chirality of each residue is plotted against the sequence number explores these special points. More than 65000 residues extracted from 200 high-quality proteins are used for this study, which shows that quantitative chirality is a general and useful structural parameter for protein conformational studies.

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. This measure encompasses all bond angles and bond lengths of an amino acid residue. We focus in this report on glycine (Gly) because, due to its flexibility, it occupies a large portion of the 2D map, thus allowing a detailed study of the chirality measure, and in order to evaluate the justification of classically labeling Gly as the only achiral amino acid. We have analyzed in detail 4366 Gly residues extracted from high resolution crystallographic data of 160 proteins. This analysis reveals not only that Gly is practically always conformationally chiral, but that upon comparing with the backbone of all amino acids, the quantitative chirality values of Gly are of similar magnitudes to those of the (chiral) amino acids. Structural trends and energetic considerations are discussed in detail. Generally we show that adding chirality to Ramachandran plots creates far more informative plots that highlight the sensitivity of the protein structure to minor conformational changes.

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. The newly discovered sites can provide a link between the much older Acheulian in Africa and the Levant and the well-known assemblages from the later European Acheulian, enabling a rigorous testing of these hypotheses using modern morphometric methods. Here we use the Continuous Symmetry Measure (CSM) method to quantify handaxe symmetry at la Noira, a newly excavated site in central France, which features two archaeological levels, respectively ca. 700 ka and 500 ka old. In order to provide a context for the new data, we use a large aggregate from the well-known 500 ka old site of Boxgrove, England. We show that handaxes from the oldest layer at la Noira, although on average less symmetric than both those from the younger layers at the same site and than those from Boxgrove, are nevertheless much more symmetric than other early Acheulian specimens evaluated using the CSM method. We also correlate trends in symmetry to degree of reduction, demonstrating that raw material availability and discard patterns may affect observed symmetry values. We conclude that it is likely that, by the time the Acheulian arrived in Europe, its makers were, from a cognitive and motor-control point of view, already capable of producing the symmetric variant of this technology.

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. We find that in such cases the instantaneous distance of the molecular structure from its symmetric counterpart is a suitable collective variable that can describe the reaction process. We show that this quantity, the continuous symmetry measure (CSM), has a positive linear relationship with temperature, implying that even highly symmetric molecules should be considered as asymmetric above 0 K. Using ab initio molecular dynamics, we simulate the temperature-induced Cope rearrangements of several fluxional molecules and employ different CSMs to follow the reaction progress. We use this methodology to demonstrate the validity of important concepts governing these reactions: Woodward-Hoffmann rules and TS aromaticity. Statistical analysis of the CSM distributions reveals that ligands connected to the carbon frame have profound effect on the reaction course. In particular, our results show that lower temperatures tend to enhance the differences between the TS-stabilizing effect of the substituents.

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. Specifically, we study the deviation from mirror symmetry (evaluated with the continuous symmetry measure (CSM)) of the [4+2] carbon skeleton of the transition state of a series of twelve Diels-Alder reactions in seven different solvents (and in the gas phase), in which the dienes are butadiene, cyclopentadiene, cyclohexadiene, and cycloheptadiene; the dienophiles are the 1-, 1,1-, and 1,1,2-cyanoethylene derivatives; the solvents were chosen to sample a range of dielectric constants from heptane to ethanol. These components provide twenty-four symmetry-enthalpy DFT-calculated correlation lines (out of which only one case is a relatively mild exception) that show the general trend of increase in enthalpy as symmetry decreases. The various combinations between the dienophiles, cyanoethylenes, and solvents provide all kinds of sources for symmetry deviations; it is therefore remarkable that although the enthalpy of activation is dictated by various parameters, symmetry emerges as a primary parameter. In our analysis we also bisected this overall picture into solvent effects and geometry variation effects to evaluate under which conditions the electronic effects are more dominant than symmetry effects.

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. Calculations of the enthalpy of activation and the charge separation at the transition state (TS) predict increased reactivity with the polarity of the solvent. The symmetry measure is in excellent correlation with the enthalpy of activation and the charge separation at the TS, indicating the higher reactivity of the more symmetric case, thus quantifying the main teaching of the WH rules. The advantages of using a global structural parameter that takes into account all geometrical parameters, i.e., the symmetry measure, over specific ones (e.g., asynchronicity) are discussed.