Continuous Symmetry and Chirality Measures

David Avnir

Institute of ChemistryThe Hebrew University of JerusalemHarvard, Boston, January 28, 2013

Near C2 symmetry: HIV Protease mutant V82A complexed with A77 inhibitorWhat, quantitatively, is the C2 symmetry content of that protein?

Gradual changing chirality and C2-ness in aggregatesIs it possible to quantify these changes?

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?

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:

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

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

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.

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

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

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

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)?

* 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

Planar square D4h

The full scale of the CSM

The most chiral monodentate complex

Trends within families and classifications

Symmetry maps

The symmetry map of 13,000 transition metal ML4 complexesS. Alvarez, P. Alemany, JACS 2004

CuCl42-: The tetrahedral to planar-square symmetry map and pathwayS(Td)S(D4h)S. Keinan

Several possible pathways for this transformationSpreadTwistCompression

The tetrahedral to planar-square transformationCuCl42-S(Td)S(D4h)

Minimal energy and minimal symmetry values coincideS(D4h)

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

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

Symmetry or chirality as reaction coordinates

Stone-Wales Enantiomerizations in FullerenesY. Pinto, P. Fowler (Exeter)

Hckel energy changes along the enantiomerization

The sensitivity of energy/chirality dependence on the size of the fullerene

Temperature and pressure effects on symmetry and chirality

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-

Low QuartzSiO2, P3221Temperature and pressure effects on the chirality and symmetry of extended materials: Quartz

The building blocks of quartz SiO4Si(OSi)4SiSi4-O(SiO3)4-

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

Predicting the high pressure symmetry behavior of quartz based on the isostrucutral GeO2D. Yogev-Einot , D. Avnir; Acta Cryst. (2004) B60 163-173

The building blocks of quartz: All are chiral!SiO4Si(OSi)4SiSi4-O(SiO3)4-

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.

The optical rotation of quartz Le Chatelier, H. Com. Rend de I'Acad Sciences 1889, 109, 264.

115 years later: Interpretation and exact match with quantitative chirality changesCrystallography: Kihara, 1990. Analysis: D. Yogev-EinotSiSi4

Correlations between continuous symmetry and spectral properties

S(Td)max d-d (cm-1)Jahn-Teller effects and symmetry:The d-d splitting in Cu complexesData: Halvorson, 1990. Analysis: S. Keinan

Changes in transition probability as a function of octahedricityCuN4O2 Chromophores:S(Oh) [cm-1M-1]Data: P. Comba, 1999+2H2O

Degree of allowedness of ESR transition as a function of the degree of tetrahedricity

Maximal and minimal shielding in AB4 speciesSymmetry effects on NMR chemical shielding

Current wisdom:But how does the shielding change when the symmetry changes continuously?

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

Correlation between symmetry/chirality and chemical recognition

* Chromatography* Catalysis* Enzymatic activity

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

Enantioselectivity of a chiral chormatographic column towards helicenesIs there a relation between this behavior and the degree of chirality of helicenes?

The chiral separation of helicenes on Gil-Avs column is dictated by their degree of chiralityO. KatzenelsonTetrahedron-Asymmetry, 11, 2695 (2000)Gil-AvQuantitative chirality

Catalysis

Catalytic Chiral Diels-Alder ReactionData: Davies, 1996. Analysis: Lipkowitz, Katzenelson

The nearest symmetry plane of the catalystn = 1

The enantiomeric excess of the product as a function of the degree of chirality of the catalystLipkowitz, JACS 123 6710 (2001)

Which smallest fragment carries the essential chirality?S. Alvarez

The smallest fragment which carries the essential chirality for catalysis

Prediction 1: Replace the exocyclic ring with C=O or C=CH2 to get good homologue catalysts

Prediction 2: Increase the twist angle

Enzymatic activity

Trypsin inhibitorsS. Keinan JACS 98

Attempt to find a correlation between the inhibition constant and the chirality of the whole inhibitorNo correlation; but

The correlation follows the degree of chirality but not the length of the alkyl chainCorrelation between inhibition and the chirality of the pharmacophor

Inhibition of acetylcholine esterase by chiral organophosphates

Ala82Asn83Ile84Gly50HIV protease complexed with A77 inhibitorHIV protease-drug complex C2-symmetric color map

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

Given a sufficiently high resolution in space or in time, nothing is symmetric, everything is chiral

Our web-site (beta)http://chirality.ch.huji.ac.il/ or http://www.csm.huji.ac.il/

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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