Continuous Symmetry and Chirality Measures David Avnir Institute of Chemistry The Hebrew University of Jerusalem Harvard, Boston, January 28, 2013

“Near” C 2 symmetry: HIV Protease mutant V82A complexed with A77 inhibitor What, quantitatively, is the C 2 symmetry content of that protein?

Gradual changing chirality and C 2 -ness in aggregates Is 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 ( ): The larger S(G) is, the higher is the deviation from G-symmetry : The original structure : The symmetry-operated structure N : Number of vertices d : Size normalization factor H. Zabrodsky

E C3C3 C32C32 Measuring the degree of C 3 -ness (S(C 3 )) of a triangle Ch. Dryzun

All three triangles are superimposed. The set of 9 points is C 3 -symmetric. Its blues average is a C 3 - symmetric triangle The 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 Measure S. 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 ML 6 : # Shape: What is the degree of ML 6 - octahedricity (S(L 6 -O h ))? # Symmetry: What is the degree of O h - ness (S(O h ))? D 4h -ness (S(D 4h )? And of S(D 2h )?

* 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 – D 4h The CSM values of an AB 4 species with respect to tetrahedricity and planar-squareness Distorted tetrahedron S(T d ) = 0 S(D 4h ) = 33.3 S(T d ) = 10.6 S(D 4h ) = 7.84 S(T d ) = 33.3 S(D 4h ) = 0 Perfect tetrahedron - T d

Td D4h C3v CvCv S(Td) 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 ML 4 complexes S. Alvarez, P. Alemany, JACS 2004

CuCl 4 2- : The tetrahedral to planar-square symmetry map and pathway S(T d ) S(D 4h ) S. Keinan

Several possible pathways for this transformation Spread Twist Compression 70 o 110 o

The tetrahedral to planar-square transformation Spread Twist Compression CuCl 4 2- S(T d ) S(D 4h )

d J S(D ) S(T ) (136.8 kcal/mol) (105.4 kcal/mol) (74.1 kcal/mol) (42.67 kcal/mol) (11.29 kcal/mol) J (0 Kcal/mol) Spread simulation Energy in Hartree(relative energy in kcal/mol) Minimal energy and minimal symmetry values coincide S(D 4h )

Tetracoordinated Bis-Chelate Metal Complexes M(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 73° 70 o 110 o Twist

We (mainly S. Alvarez) analyzed similarly all MLn families with n from 4 to 10 4Chem. Eur. J., 10, (2004). 5J. Chem. Soc., Dalton Trans., (2000). 6New J. Chem., 26, (2002). 7Chem. Eur. J., 9, (2003). 8Chem. Eur. J., 11, 1479 (2005). 9Inorg. Chem., 44, (2005). 10Work in progress

Symmetry or chirality as reaction coordinates

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

Hückel 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 ( o K) S(O h ) Data: Wei, M. & Willett, R.D. Inorg. Chem. (1995) 34, Analysis: S. Keinan Changes in the degree of octahedricity with temperature CuCl 6 4-

Low Quartz SiO 2, P Temperature and pressure effects on the chirality and symmetry of extended materials: Quartz

The building blocks of quartz SiO 4 Si(OSi) 4 SiSi 4 -O(SiO 3 ) 4 -

Combining temperature and pressure effects through symmetry analysis b S(C 2 ) of a four tetrahedra unit: A measure of helicity A correlation between global and specific geometric parameters

GeO SiO Predicting the high pressure symmetry behavior of quartz based on the isostrucutral GeO 2 D. Yogev-Einot, D. Avnir; Acta Cryst. (2004) B

The building blocks of quartz: All are chiral! SiO 4 Si(OSi) 4 SiSi 4 -O(SiO 3 ) 4 -

M. Pinsky et al, “Statistical analysis of the estimation of distance measures” J. Comput. Chem., 24, 786–796 (2003) How small can the measure be and still indicate chirality? The error bar # Typical limit: In quartz, S(Chir) of SiO 4 = # For S values near zero, the error bar is not symmetric: The + and - are different. # If the lower bound of S touches , then the molecule is achiral.

Temperature (°K)  Le Chatelier   t  The optical rotation of quartz Le Chatelier, H. Com. Rend de I'Acad Sciences 1889, 109, 264.

Temperature (°K)  Le Chatelier   t  Chirality, SiSi 4 Chirality  t   115 years later: Interpretation and exact match with quantitative chirality changes Crystallography: Kihara, Analysis: D. Yogev-Einot SiSi 4

Correlations between continuous symmetry and spectral properties

S(T d ) max d-d (cm -1 ) Jahn-Teller effects and symmetry: The d-d splitting in Cu complexes Data: Halvorson, Analysis: S. Keinan

Changes in transition probability as a function of octahedricity CuN 4 O 2 Chromophores: S(O h ) a=b=c=(CH 2 ) 3 a=b=c=(CH 2 ) 2 a=c=(CH 2 ) 3 ; b=(CH 2 ) 2 a=c=(CH 2 ) 2 ; b=(CH 2 ) 3  [cm -1 M -1 ] Data: P. Comba, H 2 O

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

z x y z x y Maximal and minimal shielding in AB 4 species Symmetry effects on NMR chemical shielding

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

CSA (ppm) S(D 4h ) – deviation from planarity CSA vs. S(D 4h ) 200 randomly distorted SiH 4 All 29 Si NMR properties were calculated using Gaussian98, B3LYP/6-31G * and GIAO A. Steinberg, M. Karni

Random Spread: Maximal de-shielding S(D 4h ) – deviation from planarity CSA (ppm) CSA vs. S(D 4h )

Correlation between symmetry/chirality and chemical recognition * Chromatography * Catalysis * Enzymatic activity

The pioneering work of Gil-Av on chiral separations of helicenes E. Gil-Av, F. Mikes, G. Boshart, J. Chromatogr, 1976, 122, 205 A pair of enantiomers of a [6]-helicene Silica derivatized with a chiral silylating agent

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

The chiral separation of helicenes on Gil-Av’s column is dictated by their degree of chirality O. Katzenelson Tetrahedron-Asymmetry, 11, 2695 (2000) Gil-Av Quantitative chirality

Catalysis

Catalytic Chiral Diels-Alder Reaction Data: Davies, Analysis: Lipkowitz, Katzenelson

The nearest symmetry plane of the catalyst n = 1

The enantiomeric excess of the product as a function of the degree of chirality of the catalyst Lipkowitz, JACS (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=CH 2 to get good homologue catalysts

Prediction 2: Increase the twist angle

Enzymatic activity

Trypsin inhibitors S. Keinan JACS 98

Attempt to find a correlation between the inhibition constant and the chirality of the whole inhibitor No correlation; but…

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

Inhibition of acetylcholine esterase by chiral organophosphates

Ala82 Asn83 Ile84 Gly50 HIV protease complexed with A77 inhibitor HIV protease-drug complex C 2 -symmetric color map

F: Native HIV-protease inhibitors E: Native HIV-protease inhibitor A77 J: V82A mutant HIV-protease inhibitor A77 Free energy of inhibitors binding vs. their C 2 -symmetry change

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

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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) Literature Recent: 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 Chaˆtelier to chirality measures”, Tetrahedron: Asymmetry 17, 2723 – 2725 (2006) Mark Pinsky et al, "Symmetry operation measures”, J. Comput. Chem., 2007