Inbal Tuvi-Arad

Continuous Symmetry and Chemistry Teachers: Learning Advanced Chemistry Content through Novel Visualization Tools

In this paper we describe the learning process of a group of experienced chemistry teachers in a specially designed workshop on molecular symmetry and continuous symmetry. The workshop was based on interactive visualization tools that allow molecules and their symmetry elements to be rotated in three dimensions. The topic of continuous symmetry is a new field of study that provides a quantitative description of the distance of a specific structure from perfect symmetry. Using novel online tools, teachers were able to perform these calculations with the emphasis on the chemistry, rather than on the mathematics of the calculations. Our results show that even a very basic knowledge of symmetry and continuous symmetry opens up new ways of thinking about and looking at molecules. The addition of visualization tools creates a deeper understanding of molecular structure. Moreover, even though molecular symmetry is not a mandatory part of the chemistry high-school curriculum in Israel, familiarity with concepts of symmetry can help teachers understand and explain other topics, such as chirality and the polarity of molecules. Our results indicate that highly advanced content can influence the way teachers think, understand and teach. This experience can shed light on curriculum choices for teachers’ education.

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.

New visualization tools for learning molecular symmetry: a preliminary evaluation

A Website that helps students visualize and locate symmetry elements on three- dimensional molecular structures was developed. It includes textual explanations, an interactive example window and a Microsoft-Excel based symmetry toolkit that enables students to draw symmetry elements in three dimensions. Preliminary qualitative research aimed at exploring how students learned with this tool was performed. It was found that the three-dimensional graphical capabilities of the toolkit (1) helped students overcome difficulties in three-dimensional visualization, (2) enabled students to find symmetry elements of complex molecules generally not accessible from drawings and (3) contributed to a deeper understanding of molecular structure and chemical symmetry. [Chem. Educ. Res. Pract., 2007, 8 (1), 61-72]

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.

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.

Improved algorithms for symmetry analysis: structure preserving permutations

We propose an improved algorithm for calculating Avnir’s continuous symmetry and chirality measures of molecules. These measures evaluate the deviation of a given structure from symmetry by calculating the distance between the structure and its nearest symmetric counterpart. Our new algorithm utilizes structural properties of the given molecule to increase the accuracy of the calculation and dramatically reduce the running time by up to tens orders of magnitude. Consequently, a wide variety of molecules of medium size with ca. 100 atoms and even more can be analyzed within seconds. Numerical evidence of the algorithm’s efficiency is presented for several families of molecules such as helicenes, porphyrins, dendrimers building blocks, fullerene and more. The ease and efficiency of the calculation make the continuous symmetry and chirality measures promising descriptors for integration in quantitative structure–activity relationship tools, as well as chemical databases and molecular visualization software.

Chiral Ramachandran Plots II: General trends and proteins 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. 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.