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