Continuous Symmetry and Chirality Measures

David Avnir

Institute of ChemistryThe Hebrew University of Jerusalem

Harvard, Boston, January 28, 2013

“Near” C2 symmetry: HIV Protease mutant V82A complexed with A77 inhibitor

What, quantitatively, is the C2 symmetry content of that protein?

Gradual changing chirality and C2-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 0 - 1 (0 - 100):

The larger S(G) is, the higher is the deviation from G-symmetry

N

1k

2

2ˆ1

min100 kk QQNd

)S(G

kQ

kQצ

: The original structure

: The symmetry-operated structure

N : Number of vertices

d : Size normalization factor

H. Zabrodsky

E

C3

C32

Measuring the degree of C3-ness (S(C3)) of a triangle

Ch. Dryzun

All three triangles are superimposed. The set of 9 points is C3-symmetric. Its blues average is a C3-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

N

1k

2

2ˆ1

min100 kk QQNd

)S(G

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 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 CSM values of an AB4 species

with respect to tetrahedricity and planar-squareness

Distorted tetrahedron

S(Td) = 0

S(D4h) = 33.3

S(Td) = 10.6

S(D4h) = 7.84

S(Td) = 33.3

S(D4h) = 0

Perfect tetrahedron - Td

0 10072.22

Td

D4h

C3v

Cv

33.33 65.73

0 1

S(Td)

The full scale of the CSM

S(TP)

[Ta(CCSitBu3)6]- [Ti2(-SMe)3(SMe)6]2-[Zr(SC6H4-4-OMe)6]2-

1.88

18.8°

1.67

8.27

5.51

1.34

33.3°

4.45

3.94

2.16

30.4°

5.09

S(chir)

S(Oh)

The most chiral monodentate complex

Trends within families and classifications

Symmetry maps

The symmetry map of 13,000 transition metal ML4 complexes

S. Alvarez, P. Alemany, JACS 2004

0

5

10

15

20

25

30

0 5 10 15 20 25 30

CuCl42-: The tetrahedral to planar-square symmetry map and pathway

S(Td)

S(D

4h)

S. Keinan

Several possible pathways for this transformation

Spread

Twist

Compression

70o

110o

0

5

10

15

20

25

30

0 5 10 15 20 25 30

The tetrahedral to planar-square transformation 

Spread

Twist

Compression

CuCl42-

S(Td)

S(D

4h)

30

25

20

15

10

5

035302520151050

-2033.20-2033.15-2033.10-2033.05-2033.00-2032.95

d

JS(D

)

S(T )

-2033.15-2033.10

-2033.05

-2033.00

(136.8 kcal/mol)(105.4 kcal/mol)(74.1 kcal/mol)(42.67 kcal/mol)(11.29 kcal/mol)

J -2033.168 (0 Kcal/mol)

Spread simulation

Energy in Hartree (relative energy in kcal/mol)

Minimal energy and minimal symmetry values coincide

•S

( D4 h

)

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°

70o

110o

Twist

We (mainly S. Alvarez) analyzed similarly all MLn families with n from 4 to 10

4 Chem. Eur. J., 10, 190-207 (2004).

5 J. Chem. Soc., Dalton Trans., 3288-3303 (2000).

6 New J. Chem., 26, 996-1009 (2002).

7 Chem. Eur. J., 9, 1281-1295 (2003).

8 Chem. Eur. J., 11, 1479 (2005).

9 Inorg. Chem., 44, 6939-6948 (2005).

10 Work 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

Cl

NH3+

150

200

250

300

0.34 0.35 0.36 0.37 0.38 0.39

Tem

p (o K

)

S(Oh)

Data: Wei, M. & Willett, R.D. Inorg. Chem. (1995) 34, 3780. Analysis: S. Keinan

Changes in the degree of octahedricity

with temperature

CuCl64-

Low QuartzSiO2, P3221

Temperature and pressure effects on the chirality and symmetry of extended materials:

Quartz

The building blocks of quartz

SiO4 Si(OSi)4

SiSi4-O(SiO3)4-

Combining temperature and pressure effects through symmetry analysis

b

0

0.5

1

1.5

2

2.5

3

3.5

4

120 130 140 150SiOSi angle

C2

A

B

C

D

T

S(C2) of a four tetrahedra unit:

A measure of helicity

A correlation between global and specific geometric parameters

0 5 10 15

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15Pressure (GPa)

Te

tra

he

dri

city

GeO4

SiO4

GeO4

SiO4

a

b

20 SiO2

GeO2

SiO2

GeO2

20

GeGe4

SiSi4

0 5 10 15

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15Pressure (GPa)

Te

tra

he

dri

city

GeO4

SiO4

GeO4

SiO4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15Pressure (GPa)

Te

tra

he

dri

city

GeO4

SiO4

GeO4

SiO4

GeO4

SiO4

a

b

20 SiO2

GeO2

20 SiO2

GeO2

SiO2

GeO2

20 SiO2

GeO2

20 SiO2

GeO2

20 SiO2

GeO2

20

GeGe4

SiSi4

GeO

SiO

4

4

4

4

4

4

Predicting the high pressure symmetry behavior of quartz based on the isostrucutral GeO2

D. Yogev-Einot , D. Avnir; Acta Cryst. (2004) B60 163-173

The building blocks of quartz: All are chiral!

SiO4 Si(OSi)4

SiSi4-O(SiO3)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 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.

0.97

1.02

1.07

1.12

1.17

98 298 498 698 898 1098

Temperature (°K)

Le

Cha

teli

er

t

The optical rotation of quartz

Le Chatelier, H. Com. Rend de I'Acad Sciences 1889, 109, 264.

0.97

1.02

1.07

1.12

1.17

98 298 498 698 898 1098

Temperature ( K)

0.54

0.56

0.58

0.6

0.62

0.64

Temperature (°K)

Le

Cha

teli

er

t

Ch

irality, SiSi4

Chirality t

115 years later: Interpretation and exact match with quantitative chirality changes

Crystallography: Kihara, 1990. Analysis: D. Yogev-Einot

SiSi4

Correlations between continuous symmetry and spectral properties

7000

8000

9000

10000

11000

12000

13000

14000

15000

0 5 10 15 20 25 30 35S(Td)

max d-d

(c

m-1)

Jahn-Teller effects and symmetry:

The d-d splitting in Cu complexes

Data: Halvorson, 1990. Analysis: S. Keinan

Changes in transition probability as a function of octahedricity

CuN4O2 Chromophores:

S(Oh)

N

Cu NN

N

O2N

H

HH

H

(a)

(c) (b)

50

100

150

200

250

1 2 3 4 5 6 7

a=b=c=(CH2)3

a=b=c=(CH2)2

a=c=(CH2)3; b=(CH2)2

a=c=(CH2)2; b=(CH2)3

[c

m-1M

-1]

Data: P. Comba, 1999

+2H2O

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 AB4 species

Symmetry effects on NMR chemical shielding

Current wisdom:

But how does the shielding change when the symmetry changes continuously?

350

0 10 20 30 40

0

50

100

150

200

250

300C

SA

(ppm

)

S(D4h) – deviation from planarity

CSA vs. S(D4h)200 randomly distorted SiH4

All 29Si NMR properties were calculated using Gaussian98, B3LYP/6-31G* and GIAO

A. Steinberg, M. Karni

0

50

100

150

200

250

300

350

RandomSpread: Maximal de-shielding

0 10 20 30 40

S(D4h) – deviation from planarity

CSA

(ppm

)

CSA vs. S(D4h)

If you de le te a

parag raphmark, the

follow ing

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

N

OO

CH2

Cu N

X X

ON

O O

O N

OO

n

X = OTf

1 n = 12 n = 23 n = 34 n = 4

1-4

5 6

Catalytic Chiral Diels-Alder Reaction

Data: Davies, 1996. 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 123 6710 (2001)

N

Cu

N

O O

C C

CC

SS

S1 S2

CC C

N

Cu

N

CC

O O

C

C

C

C

SO

O

CF

F F

SO

O

CF

F F

C C

CC

CC

N

Cu

N

O O

C

C

C

C

S

CF

F F

S

CF

F F

C

CC C

C

C

Sb Sg

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 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

Ala82Asn83

Ile84

Gly50

HIV protease complexed

with A77 inhibitor

HIV protease-drug complex C2-symmetric color map

F

FF

FFF

FF

FF

F

J

E

-16000

-15000

-14000

-13000

-12000

-11000

-10000

-9000

-8000

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0

G [

Kcal/

mol]

[S(C )]2

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 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)

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