Continuous Symmetry and Chirality Measures
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Transcript of Continuous Symmetry and Chirality Measures
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Continuous Symmetry and Chirality Measures
David Avnir
Institute of ChemistryThe Hebrew University of JerusalemHarvard, Boston, January 28, 2013
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Near C2 symmetry: HIV Protease mutant V82A complexed with A77 inhibitorWhat, quantitatively, is the C2 symmetry content of that protein?
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Gradual changing chirality and C2-ness in aggregatesIs it possible to quantify these changes?
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Since achirality relates to symmetry, similar questions pop up also in the context of chirality:By how much is one molecule more chiral than the other?
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In fact, asymmetry and chirality are very common:
Given a sufficiently high resolution in space or time it is quite difficult to find a fully symmetric, achiral molecule.
Consider watching methane on a vibrational time-scale: Only one in zillion frames will show the following:
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Given a sufficiently high resolution in space or time it is quite difficult to find a fully symmetric, achiral molecule
Spatial resolutions: Often, symmetry is lost at the condensed phase:# An adsorbed molecule# A matrix-entrapped molecule# A molecule packed in the crystal# A molecule in the glassy state# A molecule within a cluster
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A methodology is needed in order to quantify the degree of symmetry and the degree of chirality:
# Comparing different molecules# Following changes within a single molecule
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The proposed methodology for a symmetry-measure design:
Find the minimal distance between the original structure, and the one obtained after the G point-group symmetry is operated on it.
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The continuous symmetry measure* The scale is 0 - 1 (0 - 100): The larger S(G) is, the higher is the deviation from G-symmetry
: The original structure: The symmetry-operated structureN : Number of verticesd : Size normalization factorH. Zabrodsky
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Measuring the degree of C3-ness (S(C3)) of a triangleCh. Dryzun
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All three triangles are superimposed. The set of 9 points is C3-symmetric. Its blues average is a C3-symmetric triangleThe measure is the collection of distances between the blue and the (original) red
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G: The achiral symmetry point group which minimizes S(G)Achiral molecule: S(G) = 0
The more chiral the molecule is, the higher is S(G) S(G) as a continuous chirality measure
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The Continuous Shape MeasureS. Alvarez, P. Alemany* The CSM estimates the distance to an a-priori unknown shape with the desired symmetry
* The Shape Measure estimates the minimal distance to a specific pre-selected shape (any shape)
* For ML6:# Shape: What is the degree of ML6-octahedricity (S(L6-Oh))?# Symmetry: What is the degree of Oh-ness (S(Oh))? D4h-ness (S(D4h)? And of S(D2h)?
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* The measure is a global structural parameter: It takes into account all bond angles and bond lengths
* A full profile of symmetry and chirality values is obtained
* All values are comparable either within the same molecule or between different ones
* The computational tools are efficient
* Analytical solutions have been obtained for many types of symmetry * The shape of the nearest symmetric object is an outcome
* The measure is well behaved, and its correlations with physical/chemical parameters agree with intuition
Some properties of the symmetry measure
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Planar square D4h
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The full scale of the CSM
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The most chiral monodentate complex
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Trends within families and classifications
Symmetry maps
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The symmetry map of 13,000 transition metal ML4 complexesS. Alvarez, P. Alemany, JACS 2004
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CuCl42-: The tetrahedral to planar-square symmetry map and pathwayS(Td)S(D4h)S. Keinan
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Several possible pathways for this transformationSpreadTwistCompression
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The tetrahedral to planar-square transformationCuCl42-S(Td)S(D4h)
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Minimal energy and minimal symmetry values coincideS(D4h)
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Tetracoordinated Bis-Chelate Metal ComplexesM(L-L')2: The [M(bipy)2] family L-M-L bond angles: # Spread From 90 to 109.4#Two Twist pathways: The bidentate nature is introduced by keeping the two opposite L-M-L bond angles constant at typical 82 and 73Twist
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We (mainly S. Alvarez) analyzed similarly all MLn families with n from 4 to 104Chem. Eur. J., 10, 190-207 (2004).5J. Chem. Soc., Dalton Trans., 3288-3303 (2000). 6New J. Chem., 26, 996-1009 (2002). 7Chem. Eur. J., 9, 1281-1295 (2003).8Chem. Eur. J., 11, 1479 (2005).9Inorg. Chem., 44, 6939-6948 (2005).10Work in progress
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Symmetry or chirality as reaction coordinates
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Stone-Wales Enantiomerizations in FullerenesY. Pinto, P. Fowler (Exeter)
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Hckel energy changes along the enantiomerization
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The sensitivity of energy/chirality dependence on the size of the fullerene
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Temperature and pressure effects on symmetry and chirality
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Temp (oK)S(Oh)Data: Wei, M. & Willett, R.D. Inorg. Chem. (1995) 34, 3780. Analysis: S. KeinanChanges in the degree of octahedricity with temperatureCuCl64-
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Low QuartzSiO2, P3221Temperature and pressure effects on the chirality and symmetry of extended materials: Quartz
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The building blocks of quartz SiO4Si(OSi)4SiSi4-O(SiO3)4-
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Combining temperature and pressure effects through symmetry analysisbS(C2) of a four tetrahedra unit: A measure of helicityA correlation between global and specific geometric parameters
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Predicting the high pressure symmetry behavior of quartz based on the isostrucutral GeO2D. Yogev-Einot , D. Avnir; Acta Cryst. (2004) B60 163-173
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The building blocks of quartz: All are chiral!SiO4Si(OSi)4SiSi4-O(SiO3)4-
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M. Pinsky et al, Statistical analysis of the estimation of distance measures J. Comput. Chem., 24, 786796 (2003)
How small can the measure be and still indicate chirality?The error bar
# Typical limit: In quartz, S(Chir) of SiO4 = 0.0007 # For S values near zero, the error bar is not symmetric: The + and - are different. # If the lower bound of S touches 0.00000, then the molecule is achiral.
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The optical rotation of quartz Le Chatelier, H. Com. Rend de I'Acad Sciences 1889, 109, 264.
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115 years later: Interpretation and exact match with quantitative chirality changesCrystallography: Kihara, 1990. Analysis: D. Yogev-EinotSiSi4
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Correlations between continuous symmetry and spectral properties
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S(Td)max d-d (cm-1)Jahn-Teller effects and symmetry:The d-d splitting in Cu complexesData: Halvorson, 1990. Analysis: S. Keinan
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Changes in transition probability as a function of octahedricityCuN4O2 Chromophores:S(Oh) [cm-1M-1]Data: P. Comba, 1999+2H2O
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Degree of allowedness of ESR transition as a function of the degree of tetrahedricity
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Maximal and minimal shielding in AB4 speciesSymmetry effects on NMR chemical shielding
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Current wisdom:But how does the shielding change when the symmetry changes continuously?
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350010203040CSA (ppm)S(D4h) deviation from planarityCSA vs. S(D4h)200 randomly distorted SiH4All 29Si NMR properties were calculated using Gaussian98, B3LYP/6-31G* and GIAOA. Steinberg, M. Karni
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Correlation between symmetry/chirality and chemical recognition
* Chromatography* Catalysis* Enzymatic activity
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The pioneering work of Gil-Av on chiral separations of helicenesE. Gil-Av, F. Mikes, G. Boshart, J. Chromatogr, 1976, 122, 205A pair of enantiomers of a [6]-heliceneSilica derivatized with a chiral silylating agent
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Enantioselectivity of a chiral chormatographic column towards helicenesIs there a relation between this behavior and the degree of chirality of helicenes?
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The chiral separation of helicenes on Gil-Avs column is dictated by their degree of chiralityO. KatzenelsonTetrahedron-Asymmetry, 11, 2695 (2000)Gil-AvQuantitative chirality
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Catalysis
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Catalytic Chiral Diels-Alder ReactionData: Davies, 1996. Analysis: Lipkowitz, Katzenelson
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The nearest symmetry plane of the catalystn = 1
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The enantiomeric excess of the product as a function of the degree of chirality of the catalystLipkowitz, JACS 123 6710 (2001)
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Which smallest fragment carries the essential chirality?S. Alvarez
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The smallest fragment which carries the essential chirality for catalysis
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Prediction 1: Replace the exocyclic ring with C=O or C=CH2 to get good homologue catalysts
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Prediction 2: Increase the twist angle
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Enzymatic activity
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Trypsin inhibitorsS. Keinan JACS 98
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Attempt to find a correlation between the inhibition constant and the chirality of the whole inhibitorNo correlation; but
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The correlation follows the degree of chirality but not the length of the alkyl chainCorrelation between inhibition and the chirality of the pharmacophor
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Inhibition of acetylcholine esterase by chiral organophosphates
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Ala82Asn83Ile84Gly50HIV protease complexed with A77 inhibitorHIV protease-drug complex C2-symmetric color map
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F: Native HIV-protease inhibitorsE: Native HIV-protease inhibitor A77J: V82A mutant HIV-protease inhibitor A77
Free energy of inhibitors binding vs. their C2-symmetry change
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Given a sufficiently high resolution in space or in time, nothing is symmetric, everything is chiral
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Our web-site (beta)http://chirality.ch.huji.ac.il/ or http://www.csm.huji.ac.il/
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The J. Am. Chem. Soc. Series:
114, 7843 (1992)115, 8278 (1993)117, 462 (1995)120, 6152 (1998)122, 4378 (2000)123, 6710 (2001)125, 4368 (2003)126, 1755 (2004)
LiteratureRecent:
A. Steinberg et al, "Continuous Symmetry Analysis of NMR Chemical Shielding Anisotropy, Chem. Eur. J., 12, 8534 8538 (2006)
D. Yogev-Einot et al, "The temperature-dependent optical activity of quartz: from Le Chatelier to chirality measures, Tetrahedron: Asymmetry 17, 2723 2725 (2006)
Mark Pinsky et al, "Symmetry operation measures, J. Comput. Chem., 2007
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