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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Written in a clear and understandable manner, this book provides a comprehensive, yet non-mathematical, treatment of the topic, covering the basic principles of symmetry and the important spectroscopic techniques used to probe molecular structure.
The chapters are extensively illustrated and deal with such topics as symmetry elements, operations and descriptors, symmetry guidelines, high-fidelity pseudosymmetry, crystallographic symmetry, molecular gears, and experimental techniques, including X-ray crystallography and NMR spectroscopy. As an additional feature, 3D animations of most of the structures and molecules covered are available online at wiley.com. As a result, chemists learn how to understand and predict molecular structures and reactivity.
Authored by a renowned expert with numerous publications and an excellent track record in research and teaching, this is a useful source for graduate students and researchers working in the field of organic synthesis, physical chemistry, biochemistry, and crystallography, while equally serving as supplementary reading for courses on stereochemistry, organic synthesis, or crystallography.
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Seitenzahl: 494
Veröffentlichungsjahr: 2015
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Table of Contents
Cover
Title Page
Related Titles
Copyright
Dedication
From the Author's Desk
Chapter 1: Symmetry/Pseudosymmetry: Chirality in Molecules,in Nature, and in the Cosmos
1.1 Introduction
1.2 Rudimentary Group Theory, Isometry, and Symmetry
1.3 Asymmetric versus Chiral: The
I
-Symmetry of Viral Capsids
1.4 The Birth of Chirality as a Chemical Concept
1.5 Apparent Symmetry (High-Fidelity
Pseudo
symmetry) and the Quantification of Distortion from the Ideal
1.6 Chirality in Form and Architecture: Symmetry versus Broken Symmetry
1.7 Chirality in Nature: Tropical Storms, Gastropods (Shells), and Fish
1.8 Extraterrestrial Macroscale Chirality: Spiral Galaxies, Martian Sand Devils, Jovian Great Red Spot, Neptune's Great Dark Spot, and Venusian South-Pole Cloud Vortex
1.9 Analyses of Amino Acid Chirality in Extraterrestrial Samples with Gas–Liquid Chromatography Chiral Columns
Chapter 2: Enantiospecificity of Pheromones, Sweeteners, Fragrances, and Drugs
2.1 Enantiospecificity of Pheromones, Sweeteners, and Fragrances
2.2 The Importance of Chirality in Drug Therapy
Chapter 3: Bonding Parameters and the Effect of Local Environment on Molecular Structure
3.1 Symmetry Arguments and the Effect of the Environment on Molecular Structure
3.2 The Effect of Local Environment on Molecular Models and Molecular Structure
3.3 Torsion Angles and Molecular Conformation
3.4 Symmetry Considerations of Atomic Orbital Hybridization and Bonding Parameters
Chapter 4: Historical Development of Structural Chemistry: From Alchemy to Modern Structural Theory
4.1 Hemihedralism in Quartz Crystals: Setting the Stage for the Birth of Stereochemistry
4.2 Tartaric Acid and Alchemy
4.3 Hemihedralism in Crystalline Tartaric Acid Salts: The Birth of Molecular Chirality
4.4 Gift for Prelog's Retirement: A Matched Pair of
u′,x′
-Hemihedral Faced Right- and Left-Handed Quartz Crystals
4.5 Early Structural Representations of Organic Substances and the Development of Modern Structural Concepts
4.6 Fischer Projections to Determine α- and β-Anomeric Configurations
Chapter 5: Chiroptical Properties
5.1 The Language of Symmetry, Isomerism, and the Characterization of Symmetry Relationships within and between Molecules
5.2 Chiroptical Properties: Circular Birefringence, Optical Rotatory Dispersion, and Circular Dichroism
5.3 Miller Indices and Fractional Coordinates in Crystallography
5.4 Scanning Tunneling Microscopy
5.5 Direct Visualization of an Enantiomer's Absolute Configuration in the Gas Phase
Chapter 6: Symmetry Comparison of Molecular Subunits: Symmetry in Nuclear Magnetic Resonance Spectroscopy and in Dynamic NMR
6.1 Symmetry in NMR Spectroscopy
6.2 Symmetry Comparison of Molecular Subunits, Topicity Relationships
6.3 Dynamic Stereochemistry, Dynamic Nuclear Magnetic Resonance Spectroscopy (DNMR)
6.4 Use of Permutations in DNMR for Topomerization-, Enantiomerization-, and Diastereomerization-Exchange Processes
Chapter 7: Prochirality, Asymmetric Hydrogenation Reactions, and the Curtin–Hammett Principle
7.1 Prochirality of Enantiotopic Subunits
7.2 Homogeneous Hydrogenation by Rhodium
I
/Achiral Diphosphine Catalysts Differentiates the Diastereotopic Prochiral Faces of Olefins
7.3 Homogeneous Hydrogenation by Rhodium
I
/(Chiral Diphosphine) Catalysts Differentiates the Enantiotopic Prochiral Faces of Olefins: The Curtin–Hammett Principle
Chapter 8: Stereogenic Elements, Chirotopicity, Permutational Isomers, and Gear-Like Correlated Motion of Molecular Subunits
8.1 Stereogenicity, Stereogenic Elements, Chirotopicity, and the Ambiguity of Some Stereochemical Terms
8.2 Triarylamine Propellers
8.3 Dynamic Stereochemistry of Permutational Isomers: Correlated Motion in Triarylamines
8.4 Relative Stereochemical Descriptors:
Retro-Inverso
Isomers
Chapter 9: Symmetry in Extended Periodic Arrays of Molecular Crystals and the Relevance of Penrose Tiling Rules for Nonperiodic Quasicrystal Packing
9.1 Symmetry in Extended Arrays/Molecular Crystals
9.2 Achiral Periodic Arrays of Chiral Objects and Racemic Compound Crystal Lattices
9.3 Chiral Periodic Arrays
9.4 Occupancy of Special Positions in Periodic Arrays
9.5 The Bragg Law and X-Ray Diffraction
9.6 The Interferogram Phenomenon in Single-Crystal X-Ray Crystallography
9.7 X-Ray Fiber Diffraction
9.8 Penrose Tiling Matching Rules, Quasicrystal Packing, and Dodecahedrane
Chapter 10: Multiple Molecules in the Asymmetric Unit, “Faking It”; Pseudosymmetry Emulation of Achiral Higher Order Space Filling in Kryptoracemate Chiral Crystals
10.1 Multiple Molecules within an Asymmetric Unit
10.2 “Faking It”:
Pseudo
symmetry Emulation of Achiral Higher-Order Space Filling in Kryptoracemate Chiral Crystals
10.3 Desymmetrization of Platonic-Solid Geometries Resulting from Crystallographic Symmetry Constraints
10.4 Mobility of Cubane and Dodecahedrane (CH)
n
Spherical Molecules within a Crystal Lattice
Chapter 11: Platonic-Solid Geometry Molecules and Crystallographic Constraints upon Molecular Geometry, Symmetry Distortions from Ideality
11.1 Geometrical Considerations in High-Symmetry Molecules
11.2 Syntheses Strategies of High-Symmetry Chiral Molecules
11.3 Ethano-Bridge Enantiomerization of
T
-Symmetry Molecules
11.4 Self-Assembly of
T
-Symmetry Chiral Molecules
11.5 Enantiomerization of T-Symmetry Clusters
11.6 Tetradentate Edge-Linker Units Separated by a Spacer
11.7 Self-Assembly of
O
-Symmetry Chiral Molecules
11.8
O
-Symmetry Ferritin Protein Octahedral Shell
11.9 Desymmetrization Resulting from Symmetry and Chemical Constraints
Chapter 12: Solid-State NMR Spectroscopic/X-Ray Crystallographic Investigation of Conformational Polymorphism/Pseudopolymorphism in Crystalline Stable and Labile Hydrated Drugs
12.1 Divalent Anions Linking Conformationally Different Ammonium Cations
12.2 Cross Polarization/Magic Angle Spinning Solid-State NMR and X-Ray Crystallographic Studies on the Elusive “Trihydrate” Form of Scopolamine·Hydrobromide, an Anticholinergic Drug
Chapter 13: NMR Spectroscopic Differentiation of Diastereomeric Isomers Having Special Positions of Molecular Symmetry
13.1 NMR Anisochronism of Nuclei at Special Positions of Molecular Symmetry
13.2 Pattern Recognition: A Graphical Approach to Deciphering Multiplet Patterns
Chapter 14: Stereochemistry of Medium Ring Conformations
14.1 A Short Primer on Medium Ring Stereochemistry
14.2 Assignment of
Equatorial-/Axial
-Substituent Descriptors to Rings of Any Size
14.3 NMR Structure Determination of Medium-Ring Solution-State Conformations
14.4 Dynamic Disorder in Crystals
Chapter 15: The Pharmacophore Method for Computer Assisted Drug Design
15.1 The Pharmacophore, Neurotransmitters and Synapse
15.2 The Pharmacophore Method for Computer Assisted Drug Design
15.3 Determination of the Dopamine Reuptake Site Pharmacophore
15.4 Methylphenidate (Ritalin·HCl) and (−)-Cocaine·HCl
15.5 Ritalin versus Cocaine: Binding Affinity and Inhibitory Concentration
15.6 Second Generation Pharmacophore: The Orientation of the NH Proton
15.7 Avoidance of Adjacent Gauche
+
Gauche
−
Interactions
15.8 Static Disorder in
N
-Methyl Ritalin Crystals
15.9 Development of Specific Dopamine Reuptake Inhibitors (SDRI)
Chapter 16: The X-Ray Structure–Based Method of Rational Design
16.1 X-Ray Crystallographic Structure–Based Molecular Design
16.2 The Different Primary Ammonium and Quaternary Aminium Binding Modes
16.3 Search for Unused Binding Sites
16.4 Primary Ammonium and Quaternary Aminium Binding Modes in CB[7 and 8] Complexes of Diamantane-4,9-Substituted Guests
Chapter 17: Helical Stereochemistry
17.1 Helical Stereochemistry
17.2 2
n
n
-Symmetry Achiral Helical Pathways
17.3 “
La Coupe du Roi
”: Chiral Apple Halves Produced by a
4
2
-Bisection
17.4 Intermeshing Molecular Threefold Helices: Symmetry, Chemical, and Phase Considerations
17.5
X
-Ray Fiber versus Single-Crystal Diffraction Models
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Begin Reading
List of Illustrations
Chapter 4: Historical Development of Structural Chemistry: From Alchemy to Modern Structural Theory
Figure 4.1 The achiral symmetry element in
meso
-tartaric acid is an inversion center (
i
) within stable staggered conformer
85
and definitely not a σ-plane in the Fischer projection eclipsed transition state
84
.
Figure 4.3 Reorientation of rotamer
87
(middle left) of Figure 4.2 by a 180° out-of-plane rotation of the entire molecule is then followed by a 120° in-plane rotation to afford
87
in the enantiomorphous disposition of
86
.
Figure 4.2 The
dynamic stereochemistry
of
meso
-tartaric acid is exemplified by three stable nonidentical staggered conformations interconverted by rapid 120° flips the (
S
)-half of the molecule about the central C–C bond.
Figure 4.4 Both the configurational hydroxyl group of the vertically oriented linear form of l-galactose (
90
) (marked with an arrow) and the anomeric hydroxyl (upper most structure) are disposed toward the same left side of the drawings. Therefore, pyranose ring formation affords
91
(the more stable
1
C
4
-
chair
conformation of the two possible α-anomers).
Chapter 5: Chiroptical Properties
Figure 5.1 Photon locations (black dot distances from the left-handed edge) on parallel left and right
in-phase
helical pathways traversing an achiral medium are depicted at constant time intervals. Since the photons travel on enantiomorphous helical propagation pathways through an achiral medium at exactly identical velocities, the polarization plane never rotates since the vector-sum remains constant.
Figure 5.2 Photon locations (black dots) on parallel left- and right-handed helical pathways traversing an achiral medium are depicted at the same constant time interval. In this example, the photon's
faster
velocity on the left circularly polarized path enables it to travel farther along the top axis relative to the distance of its
slower
counterpart on the right circularly polarized bottom axis path and results in a −
α
° rotation of the vector-sum plane.
Figure 5.3 Constant time interval photon locations (black dots) on parallel left- and right-handed helical pathways traversing an achiral medium depict equal absorption and equal indices of refraction of right and left circularly polarized light energy by an achiral chromophore at .
Figure 5.4
Circular dichroism
is the unequal absorption of right and left circularly polarized light energy combined with .
Chapter 6: Symmetry Comparison of Molecular Subunits: Symmetry in Nuclear Magnetic Resonance Spectroscopy and in Dynamic NMR
Figure 6.1 Interconversion of magnetic environments of
enantiotopic
geminal protons
H1
,
H2
in rotamers
I–III
of achiral molecule
114
.
Figure 6.2 Interconversion of magnetic environments of
diastereotopic
geminal protons
H1
,
H2
in rotamers
I–III
of a chiral molecule
115
.
Chapter 7: Prochirality, Asymmetric Hydrogenation Reactions, and the Curtin–Hammett Principle
Figure 7.1 The catalytic cycle for the homogeneous
cis
-hydrogenation of the (
re-re
) enantiotopic face of a generic
trans
-olefin yields a
cis
-(
S
,
S
)-alkane-
d
2
via a [Rh
I
/(diphosphine)]
+
BF
4
−
complex.
Chapter 14: Stereochemistry of Medium Ring Conformations
Figure 14.1
2
1
Screw rotation-related nefopam methohalide quaternary ammonium ions separated by halide ions in crystal structures measured at 193 K. (a) methiodide, (b) methobromide, and (c) methochloride.
Chapter 15: The Pharmacophore Method for Computer Assisted Drug Design
Figure 15.1 R = CH(C=O)OMe. Ring inversion interconverts
332
(inside solid-rectangle) with its
ring invertomer
(topmost structure) while keeping the
N
-configuration invariant. The ring invertomer is then converted to
336
(within the dashed rectangle) via an overall
prototropic shift/nitrogen inversion
mechanism.
List of Tables
Chapter 14: Stereochemistry of Medium Ring Conformations
Table 14.1
J
exptl
and
J
calcd
for the CD
2
Cl
2
solution-state
twist-chair–boat
(type III) conformation of the
N
-desmethyl 2,6-benzoxazonine analogue of homonefopam; estimated standard deviation (esd) of the values is 0.2 Hz [124] (I. Ergaz and R. Glaser, unpublished data). J
calcd
comes from a DFT B3LYP/6-311 +g(2d,p) NMR=spin-spin calculation.
Table 14.2 Comparison of
θ
J
exptl
solution-state,
θ
X-ray
X-ray
crystallographic solid-state,
θ
J
DFTcalcd
, and
θ
DFT
dihedral angles
calculated for the
twist-chair–boat
(type III) conformation of
N
-desmethyl 2,6-benzoxazonine within different environments [124] (I. Ergaz and R. Glaser, unpublished data).
Chapter 15: The Pharmacophore Method for Computer Assisted Drug Design
Table 15.1 Different pharmacotherapeutic profiles of (±)-
threo
-ritalin and (−)-cocaine.
Robert Glaser
Symmetry, Spectroscopy, and Crystallography
The Structural Nexus
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The Author
Prof. Dr. Robert Glaser
Ben-Gurion University of the Negev
Department of Chemistry
84105 Beer-Sheva
Israel
About the companion website
This book is accompanied by a companion website:
3D animations showing most of the molecules contained in this book can be found under www.wiley.com/go/Glaser/Symmetry
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To Darling Yael: a woman of valor (Eshet Hayil).
From the Author's Desk
This book is the outgrowth of a graduate-level “Special Topics in Structural chemistry and Symmetry Course” taught concurrently in Hebrew at Ben-Gurion University of the Negev, Beer-Sheva, and in English at the Feinberg Graduate School, Weizmann Institute of Science, Rehovot, Israel. One educational goal of the course was to expose the participants to a path toward stereochemical knowledge different than the one ordinarily presented in standard organic stereochemistry lectures. In recent years, this course has become part of the advanced undergraduate curriculum of the Cross-Border Program in Biological Chemistry administered by the Johannes Kepler Universität Linz and the Jihočeská Univerzita v Českých Budějovicích, (University of South Bohemia, Budweis, Czech Republic) – two neighboring universities on either side of a common, and now peaceful, international frontier. It is hoped that similar binational programs integrating students in a common goal toward mutual cultural exchange and tolerance through advanced science education will one day be the norm between my own country and its Arab neighbors. Over the past three decades, this course has been given, in whole or in part, around the world (Australasia: Uni. Wollongong AU; Uni. Auckland NZ; Massey Uni., NZ; Uni. Kyoto JP; Uni. Mandalay, Myanmar MM; Europe: Uni. Twente NL; Uni. Zagreb HR; Serbian Acad. Sci., Belgrade RS; Boğaziçi Uni., Istanbul TR; and the Americas: UCLA; Uni. Syracuse; Nacional Autonom. Univ. Mexico MX; CINVESTAV – Nacional Polytech. Inst. Mexico MX; Uni. Nacional Costa Rica CR; Uni. Fed. Rio de Janeiro BR; Fluminense Fed. Uni. BR; Uni. National Córdoba AR; and Uni. Nacional Tucumán AR).
One important secondary goal of this book is to demonstrate a common interface between the aesthetic world of form and that of structural chemistry. Structural chemistry is unfortunately often not understood at the time of important career-making decisions in secondary school. Many labor under the misconception that chemistry is a daily regimen of titrations and/or balancing redox equations. Another objective was to stress the intimate impact of environment upon a molecule's conformation and structure: change the first and the other is not invariant. In a limited-size book, such as this, the various subjects that can be discussed can obviously not be comprehensive. They were chosen to be a limited sampling of those that exhibited basic principles and provided vehicles for the exposition of stereochemistry.
Without the patience, encouragement, and understanding of Yael Burko-Glaser, my dear wife and life partner of 53 wonderful years, this dream of writing a book could not have come to pass. To my immediate family: daughter Yardenna, son Gil, daughter-in-law Adva Almog-Glaser, grandsons Ron Zeev, Tal David, and granddaughters Or and Hadar, now an IDF commander, it is my hope that intellectual and scientific endeavor will make their world a better and more peaceful place than it was at their birth. To them all, this book is dedicated.
Finally, it is both fit and proper to remember the teachers who shaped our minds and events that changed our lives. George Noyes was my inspiring 1958 11th grade chemistry teacher at Great Neck North Senior High School (Long Island, NY) who enthralled his class with the wonders of chemistry and also told us of the introduction of electric light in his boyhood home. Madeleine M. Joullié (University of Pennsylvania, Philadelphia), my undergraduate professor, showed us the world of functional group organic chemistry. Upon graduation in 1964, I was employed as a Development Chemist at PPG Industries, Structural Adhesives Division, Bloomfield, NJ. The PPG Employee Education Program rejected my enrollment in the New York University Graduate School of Business Administration, despite my already approved admission. Instead, they encouraged me to upgrade my Penn Chemistry B.A. into a full B.Sc. degree and then undertake part-time evening M.Sc. studies at the Polytechnic Institute of Brooklyn (a center of excellence in polymer chemistry). Dr Lincoln Hawkins next affected my life by informing me that acceptance of the Bell Laboratories (Murray Hill, NJ) job offer, having only the M.Sc. Chemistry degree, meant that I could not reach the highest echelons of their Scientific Staff. This provided the impetus to return to full-time studies. The Soviet Union Space Exploration program's Sputnik success prompted the United States Congress to pass the National Defense Education Act (NDEA). This provided me, and others, with predoctoral fellowships in the physical sciences. While at Rutgers University School of Graduate Studies, the scientific acumen of Donald J. Cram and George S. Hammond, in their revolutionary text “Organic Chemistry,” Second Edition, McGraw-Hill: New York, 1959, did more than anything else to reveal that there was a wonderful logical chemical-intermediate-based foundation to organic chemistry rather than reliance on rote memory. The provisions of a 2-year US National Institutes of Health Postdoctoral Fellowship did not permit its use abroad, despite my prior arrangements with the Israeli biophysicist Ephraim Katchalski-Katzir to be a postdoctoral research associate at the Weizmann Institute of Science, Rehovot. In 1971, Prof. Katzir, who later became the fourth President of Israel, encouraged me not to take his job offer at WIS, but rather to start out at the new University of the Negev and to have the pleasure (“nachas” in Yiddish) of building, designing, and molding the nascent Beer-Sheva Department of Chemistry. This is not the time, nor the place, to go into the reasons for immigration to the Jewish State in 1971. Suffice it to say, the feared destruction of the Jewish people in Israel prior to the Six Day War did not come to pass. However, those tension-filled prewar days brought about a radical change of mind and heart, and I then desired to do something for my people, and for myself, rather than continue a comfortable existence in the United States of America, the land of my birth.
Special mention must be given to my very talented and inspiring scientific mentors/advisors. First and foremost, the late Edmond J. Gabbay (Rutgers University, Chemistry Department), my PhD mentor and friend, who opened the door to the intriguing world of nucleic acid/diammonium ion interactions. I remember coming home after my first meeting with him, bubbling over with excitement about the “scientific” research project that was proposed to me (after having performed routine polymer chemistry development work at PPG Industries for 3 years). At Princeton, I owe my thanks to Paul von Rague Schleyer, who introduced me to the exciting and aesthetic world of adamantane chemistry. The second year of my NIH-funded postdoctoral studies was in X-ray crystallography with Robert Langridge at the Biochemical Sciences Department.
I have left the most significant for last: a 1978 sabbatical with Kurt Mislow back in Princeton probably had more impact upon my scientific development than any other single year in my career. I remember a conversation with K.M. in his office to this very day. He said “Robert, you are working in asymmetric hydrogenation where the rate determining transition-state is ephemeral. There is so much to be learned from ground-state stereochemistry, where at least you know what the structures are.” It was sage advice from a learned scholar. It is thus both an honor and a pleasure to also dedicate this book to all my teachers. Finally, throughout the years, I have had the honor to tutor many research students. Without their sense of inquiry, determination, hard effort, and thirst for knowledge, this book could not have been written.
Much have I learned from my teachers … but even more from my students
Talmud Bavli, Ta'anit 7a'
Robert Glaser
Omer (Negev), Israel, February 2015
Chapter 1Symmetry/Pseudosymmetry: Chirality in Molecules,in Nature, and in the Cosmos
1.1 Introduction
Symmetry and emulation of symmetry (pseudosymmetry) play a major role in the world of esthetics and science. In our macro-surroundings, symmetry and pseudosymmetry are an ever-present source of “visual pleasure” whose origins may arise from our genes. At times, prior to the achievements of modern medical science, symmetrical appearance of a prospective mate may have symbolized physical well-being (health) – an essential attribute for both the child bearer/parent and the successful hunter/defender. Often, our perception of beauty and form is related to the observation of physical proportions whose ratio approximates the “golden ratio” φ (an irrational number (1 + √5)/2) where The golden ratio is derived from a Fibonacci sequence of numbers 0,1,1,2,3,5,8,13,21,…,a,b whereby .
Leonardo da Vinci's “Vitruvian Man” drawing (1) illustrates the beauty of ideal human proportions as described by Vitruvius, an architect in ancient Rome. The proportions of the circle's radius (centered at the navel) and the square's side (the human figure's height) are 1.659, which approximates the 1.618 value of the golden ratio. It is doubtful whether da Vinci and other great artists thought to themselves that they should paint figures and objects according to the dimensions specified by the golden ratio. Instead, they knew in their creative minds “what looks good” in their mind's eye in terms of proportions. They were probably gifted from birth (i.e., their genes) and did not have to learn about the importance of painting with the golden ratio as a novice students in art school.
Prehistoric man's predilection to symmetry can be seen in anthropological findings of stone axe-heads. Increasingly symmetric stone axe-heads were unearthed at sites populated by progressively more developed societies. As the society of early prehistoric man matured, these finds seem to suggest that the hunter-gatherer crafted increasingly more functional hand tools that were also more visually pleasing [1]. David Avnir and coworkers have developed algorithms to measure distortion from an ideal symmetry and applied them to a morphological study of stone axe-heads unearthed at Pleistocene Age sites in the Jordan Valley. This enabled a quantitative correlation between increasing stone axe-head mirror pseudosymmetry and the decreasing age of the site. Illustrations 2–4 depict axe-heads dated 1.84 million years ago (the oldest site, (2)) 0.6(2) million years ago (intermediate aged site, (3)), and 0.3(2) million years ago (the youngest site, (4)) [1].
The visual pleasure we receive from the “classic proportions” of the Municipal Arch (5, photo: Yael Glaser) in the Roman ruins of Glanum (Provence) is undoubtedly related to the golden ratio of its dimensions (8.8 m width and 5.5 m height). It is clear that its intimately related symmetry and esthetics were concepts well understood by talented architects in ancient times.
Our ancestors seem to have been greatly fascinated by objects of “high symmetry” (i.e., objects with more than one Cn rotation axis of order n ≥ 3, where n denotes the number of times the rotation is performed on a subunit in order to return it to its original orientation). Gray illustration 6 illustrates a late Neolithic/Bronze Age (about 4500–5200 years ago) elaborately carved regular tetrahedron stone specimen with three of its knobs decorated with spirals or dots and rings. It was unearthed at Towie in Aberdeenshire, North-East Scotland. Simple carved regular tetrahedron (7) and octahedron (8) geometry objects were also unearthed in Aberdeenshire (hedron means “face” in Greek). The esthetically pleasing five convex regular polyhedra (tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron) were a source of learning, contemplation, and discussion for Plato and his students. In modern times, these geometrical structures were the impetus for creative organic syntheses of high-symmetry hydrocarbon molecules [2, 3].
1.2 Rudimentary Group Theory, Isometry, and Symmetry
Symmetry is based upon mathematical transforms, and to understand it, a short introduction to simple Group Theory will be presented. The elements (members) of a mathematical set all share some common trait. For example, the elements of a symmetry set are all the symmetry operations that can be performed for a particular object. This set may be acted upon by a mathematical operation (multiplication or successive application, that is, performing one operation followed by another). The combination of a set and an operation defines a mathematical group. The identity (E) element must be present in the set of every group. Multiplication of two elements always affords a third element that must also be an element of the set. For example, in the symmetry group C2, the elements E and C2 ( rotation about an axis) are the symmetry operations of the set, and . Every element has its own inverse element whereby multiplication of the element by its inverse affords identity. The inverse element must also be an element of the set. In the case of C2, it is its own inverse. Symmetry groups that result in at least one point within the object remaining spatially invariant are called point groups. Mathematical group elements should not be confused with symmetry elements. Symmetry elements in an object are those single points, linear arrays of points (axes), or flat surfaces composed of points (planes) that remain spatially invariant after a symmetry operation is performed. Symmetry elements in an object (or between objects) are generated by calculating the array of midpoints between all pairs of symmetry-equivalent points.
Symmetry is a subset of isometry (isos in Greek means “equal,” metron in Greek means “measure”), where the distance matrix between a set of points in one object is preserved when the operation generates a second object. Two objects are said to be isometric when their distance matrices are identical. The spatial orientation of two isometric objects may differ relative to external axes of reference. Symmetry should have actually been written as synmetry (syn in Greek means “together”), but the letter “n” cannot precede the letter “m.” Symmetry is an isometry in which full objects (or molecules), or subunits of an object, are exchanged while keeping their spatial orientations invariant. All symmetry operations are mathematical transforms performed using symmetry operators. These act upon a set of [x, y, z] coordinates of points defining an object to generate a new exact copy of the original object. One method (but not the only one) to make an exact copy is to generate the mirror image. When the mirror-image copied object or subunit is superimposable upon the original, then the object or subunit lacks the property called “handedness.” In other words, the object is achiral. When the mirror-image object is nonsuperimposable on the original, then it is chiral (from the Greek χϵιρ, pronounced “cheir”, meaning “hand”). Objects that are chiral possess the property of (right- or left-) handedness or chirality.
Four of the eight known symmetry operations preserve an object's handedness. By this, we mean that after the operation is performed, a “right-handed” object/molecule remains “right,” and a “left-handed” one remains “left,” that is, they are congruent. These operations are denoted as being of the First Kind. Operations of the First Kind include identity (E), rotation around a “proper axis” (Cn, where “C” means cyclic and “n” is the order of the rotation operation, that is, the number of times the operation is performed on a subunit until it returns to its original orientation and position), translation (linear movement by a fixed distance), and nm screw-rotation or helical displacement (combination of translation by m/n-th of the distance required for a full-turn of a helix plus rotation by 360/n degrees). Note: the m/n translation distance is the inverse of the descriptor's symbol nm.
As noted earlier, when the mirror-image object or subunit is nonsuperimposable upon the original, then the object possesses the property of chirality or handedness. By this, we mean that the object's or subunit's “handedness” has been inverted by a symmetry operator, that is, a “right-handed” object/subunit becomes “left,” and a “left-handed” one becomes “right.”
The remaining four of the eight symmetry operations are those that invert the handedness of an object via symmetry transforms of the Second Kind. These are reflection (σ, where sigma comes from “Spiegel,” which is German for “mirror”), inversion (i), rotatory-reflection (Sn, a combination of reflection and rotation by 360/n around an “improper axis”), and glide-reflection (g, a combination of reflection plus translation by one-half of the repeating distance in a linear periodic array of objects or molecules). Three of the eight known symmetry operations involve translation and, thus, can leave no point invariant in space. These are pure translation, screw-rotation (translation + rotation), and glide-reflection (translation + reflection). These three operations are found only in the 230 different space groups (groups of operations that describe the symmetry of a periodic array of objects or molecules). The remaining five (E, Cn, σ, i, and Sn) of the eight operations are found in both point groups and space groups.
There are 32 point groups and just five have sets containing only operations of the First Kind (those that preserve handedness). These five are known as chiral point groups: Cn (cyclic), Dn (dihedral), and the “high symmetry” T (tetrahedral, tetra in Greek means “four” and hedron in Greek means “face”), O (octahedral, octa in Greek means “eight”), and I (icosahedral, icosa in Greek means “twenty”). The other 27 point groups are achiral as their sets contain operations of both the First and Second Kinds.
In periodic ordered arrays of molecules (i.e., those within a crystal lattice), there is a sideways (or laterally or diagonally) translational repeat unit called a unit cell that builds the entire extended mosaic. Its volume is determined from the cell parameters: a,b,c-axes lengths (Å) and the α,β,γ-angles (°), where α is the angle between the b,c-axes, and so on. For ease of usage, crystallography uses decimal fractions of a unit cell axis length rather than Cartesian coordinates.
A symmetry transform is the mathematical basis for performing a symmetry operation. In this procedure, the set of [x, y, z] coordinates defining all the atoms of a molecule or subunit is changed by a particular symmetry operator to generate the exact copy's new set of atoms. For example, the [−y, x, z] symmetry operator engenders a C4 rotational symmetry operation (Cn, where ). When applied to the set of a molecule's fractional atomic coordinates, it produces a new same-handedness superimposable copy that has been rotated by about the Z-axis. The operator for a +180° (clockwise) symmetry-equivalent position is [−y, −x, z], and for a +270° position, it is [y, −x, z]. What does this mean? It means that a +90° clockwise rotation about the +Z-axis of an atom with coordinates [x, y, z] will reposition that atom to , , and . For example, if for an initial atomic position, then a +90° rotated atom will be at [−0.2321(1),0.0047(1),0.0479(5)] based upon the [−y, x, z] symmetry operator. The number in parenthesis is the estimated standard deviation (esd) of the experimentally measured position, that is, its precision. Note: the rotational tropicity (tropos means “directionality” in Greek) of all symmetry operations is arbitrarily but consistently chosen to be always clockwise. Why? Answer: by historical convention.
The symmetry operation for a −90° rotation is (i.e., performing C4 three times, i.e., ). In a symmetry comparison, one compares only the initial and final objects. Thus, any intermediate steps leading up to (i.e., C4 and ) are neglected. It is clear that the final position of the rotated atom is identical whether it was rotated either by −90° or by . An [x, y, −z] symmetry operator acting upon all the original atoms of a molecule would afford a new exact copy that has been reflected through the xy-plane. Similarly, a [−x, −y, −z] operator would produce a new exact copy that was inverted through the unit cell origin, while a [−x, 1/2 + y, −z] operator would generate a new copy that has been 21 screw-rotated about the b-axis (combination of 180° rotation about the b-axis plus translation of 1/2 the unit cell repeating distance on that axis). The meaning of the 21 screw rotation operation will be explained later. The main purpose of this discourse is to emphasize that genuine symmetry is a mathematical ideality.
1.3 Asymmetric versus Chiral: The I-Symmetry of Viral Capsids
Asymmetryis a special case, since only objects exhibiting solely the identity operation are both asymmetric and chiral, that is, they have C1 point group symmetry. Thus, while all asymmetric objects are chiral, certainly not all chiral objects are asymmetric. “Asymmetric” is definitely not a synonym for “chiral.” The human adenovirus has an I-symmetry viral capsid (a protein shell without DNA or RNA; see 9 [4–6]. The cryo-electron microscopy and X-ray crystallography structure is presented in Protein Data Bank (PDB) entries 3IYN [7] and 1VSZ [8], respectively. I-symmetry is the highest chiral symmetry point group. The 120 Ih achiral group elements are: E, i, 6(C5, , S10; ; ; ); 10(C3, , S6; ) 15C2, and 15σh, that is, . The presence of only L-amino acids in the capsid removes the 60 group elements of the Second Kind, whereby the I-symmetry chiral group elements that remain are just E, 6(C5, ); 10(C3, ); 15C2, that is, . The 60 Cn-rotation operations, where , preserve the L-(levo)-handedness of all the capsid's amino acids.
The geometric considerations of the capsid's structure will be discussed next. The asymmetric unit of a unit cell is that minimal structural entity that will build the entire contents of the unit cell repeat unit using all of the symmetry operations of the point or space group (with the exception of translation for space groups). Molecules (or objects) that contain a symmetry element within themselves will have an asymmetric unit that is a fraction of the entity itself. The Hexon is the main structural protein of capsid 9, and it is organized as a trimer. Four hexon trimer units reside within the capsid's asymmetric unit – a triangular shaped wedge extending from the interior center to the surface where it fills one-third of an equilateral triangular face (see dashed line triangular segment in 9). Asymmetric units will be discussed in detail later on. Suffice it to say that the 12 hexons (3 hexons/trimer × 4 trimers/asymmetric unit) in this building-block unit will be duplicated by all the I-symmetry operations to generate the entire capsid. A threefold rotation axis through the face center copies the asymmetric unit's 12 hexon proteins 2 more times for a total of 36 hexon proteins/face (3 hexons/trimer × 4 trimers/asymmetric unit × 3 asymmetric units/face). Since there are 20 faces, this means a total of 720 hexon proteins reside within the capsid assembly. Obviously, this chiral supramolecular I-symmetry assembly, exhibiting 60 operations of the First Kind, can in no way be referred to as “asymmetric” [5].
Rod units (protein helical fibers with knobs at the very end) are located at each of the 20 locations of fivefold symmetry. The human adenovirus structure provides a useful example of how Nature sometimes modifies a symmetry motif in order to attain a particular functional advantage. Why do this? Symmetry causes structural constraints that sometimes need to be partially removed to facilitate a particular functional effect (in this case, it is a biological effect). For this particular virus, there is a symmetry mismatch between the capsid's fivefold axes of symmetry and the threefold symmetry within the knob domain. The reasons for this are yet to be discovered. Usually, the rods show fivefold symmetry.
Due to the mismatch, the adenovirus only shows I-pseudosymmetry both in the solid and solution states. The Flock House virus is an RNA virus isolated from insects and exhibits near-perfect I-symmetry. One wonders why Nature has chosen such high symmetry for many viral particles? One obvious reason is that icosahedral symmetry enables efficient packing of the viral constituents within the capsid. The genetic material (DNA or RNA), packed within, often also has I-symmetry geometry. However, perhaps the high symmetry of the capsid also imparts a statistical advantage so that its knobs can attain auspicious orientations required for effective binding to the cell surface no matter how it lands upon it.
David Goodsell [6] has discussed the function of the adenovirus in his Protein Database “Structure of the Month” webpage. The viral capsid's function is to locate a cell and deliver the viral genome inside. Most of the action occurs at the knobs located at each vertex. The purpose of the long knobs is to selectively bind to transmembrane receptors called integrins located on the cell's surface. Once the virus attacks this surface, it is drawn into vesicles (a small bubble within the cell's hydrophobic phospholipid bilayer membrane) by a process called endocytosis. Endocytosis is an energy-consuming process by which the cell absorbs molecules (i.e., large polar molecules) by engulfing them so that they can pass through the hydrophobic outer membrane. The next step involves breaking through the vesicle membrane and releasing the viral DNA into the cell. In the final stage, the adenovirus enters the cell's nucleus and builds thousands of replicas of new viruses. The ultimate result will cause the unfortunate recipient to suffer from respiratory illnesses, such as the common cold, conjunctivitis (eye infections), croup, bronchitis, and also pneumonia.
Note, for a lethal virus (e.g., ebola) to cause an epidemic, it has to kill the infected person slowly enough so that many people can come in contact with the dying victim (as with ebola). If the virus kills the sufferer too quickly, then a critical number of new victims will not arise to cause a chain reaction.
In the time of Louis Pasteur, rotationally symmetrical objects were called dissymmetric, while only achiral objects were referred to as symmetric. However, the prefix “dis” implies “a lack” of something, and thus, dissymmetric appears to be a synonym of asymmetric, which it is not. Since symmetry operations are both of the First and Second Kinds, there is no logical basis to restrict the general concept of “symmetry” to encompass only the second of the two types. On the other hand, the term chirality makes no distinction between asymmetric objects and those (like viral capsids) that exhibit high symmetry of the First Kind. Bottom line: all nonasymmetric objects are symmetrical by definition. Obviously, viral capsid 9 cannot be described as being asymmetric.
1.4 The Birth of Chirality as a Chemical Concept
The use of the term “chirality” in a scientific sense owes its conception to Lord Kelvin (Sir William Thomson), who stated, in his lecture to the Oxford University Junior Scientific Club (1893), “I call any geometrical figure, or group of points, chiral, and say that it has chirality if its image in a plane, ideally realized, cannot be brought to coincide with itself.” Its present use in a chemical context to mean the structural property giving rise to two nonsuperimposable mirror-image molecular structures, or right- and left-handedness, can be traced to one of the eminent scholars of stereochemistry, Kurt Mislow. For nonphysicists, such as the author, Mislow says that the phrase “ideally realized” refers to “geometrical chirality” coming from a “mathematical idealization” which is common in physics and science in general (K. Mislow, personal communication, January 13, 2014).
Kurt Mislow was the first to use the terms chiral and chirality in chemistry. To see how this came about, we should first set the stage and discuss some important aspects of stereochemistry in the 1950s. Carl Djerassi, Luis Miramontes, and George Rosenkranz were at the Mexico City laboratories of the Syntex Corporation and were performing research in the developing field of steroid chemistry. In 1951, they invented the first oral contraceptive drug, progestin norethindrone, which, unlike progesterone, remained effective when taken orally. The drug triggers a hormonal response that stops further egg production in the female body during the period of pregnancy. This discovery radically changed the manner in which men and women interact as they heed the call of primordial hormonal urges originally designed to ensure the continuation of our species. In 1978, at the Knesset (Parliament) in Jerusalem, Carl Djerassi was awarded the first Wolf Prize in Chemistry for his work in bioorganic chemistry and for the invention of the first oral contraceptive drug.
At the time of his work, a very useful empirical link was discovered between steroidal absolute configuration and the (+ or –) sign of the “Cotton Effect” (a chiroptical property based on Optical Rotatory Dispersion (ORD) of plane polarized light in the n → π transition of steroidal saturated ketones). This was the empirically based Octant Rule. While the “rule” seemed to work nicely for steroidal ketones (10), later on, it was found that different (+ or –) signs of the Cotton Effect, versus those predicted by the Octant Rule, were experimentally observed for optically active, known absolute configuration 8-alkyladamantan-2-ones (11). Since this phenomenon was then dubbed the Anti-Octant Rule, it is logically apparent that, alas, there is no rule. The ORD chirotopical technique and octant rule will be discussed later when details of its related subtopics circular birefringence and circular dichroism are presented as a separate section.
The principal investigators of a project involving steroidal α,β-unsaturated ketones (10) and the sign of their Cotton Effects were a group of distinguished chemists Carl Djerassi, Kurt Mislow (a stereochemist), and Albert Moscowitz (a chemical physicist). In a letter to Djerassi dated 7 October 1961, Mislow was discussing the Cotton Effect's sign for 1-methyl epimers of 1-methyl-19-norprogesterones (see wavy gray line in 10). Mislow wrote “In the α-isomer the 1-methyl group is equatorial, but in the β-isomer the 1-methyl is axial, and interaction with C11-methylene forces A (the 1-methylcyclohex-4-ene-3-one ring, A) into a boat and changes the sense of chirality.” The ketones in the letter, and other steroids, were then discussed at length in their paper where the following is written: “… Dreiding models point definitely to a given chirality …” [9] A copy of the original letter was published in an article on language and molecular chirality written by chemical historian Joseph Gal [10].
1.5 Apparent Symmetry (High-Fidelity Pseudosymmetry) and the Quantification of Distortion from the Ideal
It is common to use “symmetry” in a logically “fuzzy” manner, that is, a concept whose borders or limitations are not absolutely well defined, such as the usual about 95% probability of finding electron densities in atomic orbitals. The pleasurable symmetry that one observes in our macro-world obviously does not exist via mathematical transforms, and yet viewers often perceive nonmathematically symmetrical objects as being “symmetrical.” This is the realm of pseudosymmetry. High-fidelity pseudosymmetry (rather than genuine symmetry) is the norm in our macro-world. Genuine symmetry in our world manifests itself as reflections in a plane mirror of high quality or in a pristine lake's windless surface (see 12, photo courtesy of Shaul Barkai, Switzerland).
Aside from the aforementioned special cases of reflection symmetry, our senses do not, or cannot, note the small deviations from an ideal geometry when confronted with “almost symmetrical/nonmathematically symmetrical” objects. A second look at da Vinci's “Vitruvian Man” (1) shows that it does not have true mirror reflection symmetry. Illustration 13 shows a pseudo left-half (inverted right-hand) drawing next to a genuine right-half drawing, while 14 depicts a real left-half next to a pseudo right-half (inverted left-half) drawing. Since these differences between the right and left halves are clearly noticeable, one wonders if they were intentionally made by the artist for esthetic purposes.
Many objects “look symmetrical” since their pseudosymmetry fidelities are very high. At first glance, starfish 15 appears to have multiple symmetry elements (fivefold rotational symmetry and five mirror planes, each one passed through a leg), that is, C5v point group symmetry. But, careful inspection of 16 and 17 shows that the left leg of starfish 15 is very slightly higher than its right leg. Note the very small differences between 16 (inverted right next to a real right starfish) and 17 (genuine left next to inverted left) make 15–17 all appear to be almost identical.
Our genes have programmed us to overlook minor distortions from ideal symmetry. Avnir and his students have provided us with very useful mathematical algorithms to quantify the continuum of distortion from a genuine symmetry element within an object [11]. Genuine symmetrical geometry is only the origin-point terminus of an ever-decreasing sliding scale of pseudosymmetrical fidelity, that is, the less an object is distorted from the ideal symmetry, the more it shows a higher-fidelity pseudosymmetry. Therefore, pseudosymmetry is a sliding-scale variable, while true symmetry is an ideal state (a mathematical singleton, a set with only one element). Molecules either possess a particular symmetry, described by mathematical transforms, or do not. On the other hand, what is undisputable is that objects or molecules can readily exhibit a “continuum of distortion” from a particular ideal geometry. In other words, they can appear to exhibit various degrees of being “almost symmetrical.”
Avnir and his coworkers at the Hebrew University of Jerusalem have shown that distortions from ideal point group symmetry or from an ideal shape should be considered to be continuous properties. The Avnir algorithms are commonly referred to as “Continuous Symmetry Measures” (CSMs) [11–17]. The broad utility of the CSM tools has been extensively demonstrated for quantification of symmetry distortion within objects or molecules and has led to correlations of these values with a wide range of physiochemical phenomena [18–23]. Continuous Chirality [24] and Continuous Shape Measures (CShMs) have also been reported [25, 26]. The application of Symmetry Operation Measures to inorganic chemistry [27] and the use of CShM [28–31] measurements for analysis of complex polyhedral structures have also been reported by Alvarez and coworkers.
It is useful to take the time to understand (in a very simplified and general manner) some of the methodology of the important Avnir [18–23] symmetry distortion algorithms. The “S(X)” CSM numerical index calculated for an object or a molecule distorted from any generic ideal X-symmetry (18, where X is any point group) is a normalized root-mean-square distance function from the closest theoretical geometry object (19) that exhibits the ideal X-symmetry [11, 12, 14, 16]. The theoretical object is simply the closest geometrical construct exhibiting ideal point group X-symmetry that is calculated from the distorted input structure. Despite the fact that the construct was derived from a molecule's structure, its geometry is not real in terms of its bonding parameters (i.e., bond lengths, bond angles, and torsion angles).
For simplicity, the distance geometry algorithm is based on Eq. (1.1), where Pk = Cartesian coordinates of the actual shape (e.g., distorted trigonal bipyramid colored black in 18); Nk = coordinates of the nearest ideal C3-symmetry “geometrical construct” (colored gray in 19); n = number of points (1 to n); and D = a normalization factor so that two objects differing solely in size will afford the same S(C3) value [11, 16].
Any object with ideal generic X-symmetry will afford an S(X)-value equal to integer number zero, that is, it is a special case as it results from symmetry constraints upon the object's geometry. S(X)-values that are not integer zero are general cases (e.g., does NOT result from symmetry considerations). As the object is distorted more and more from ideal X-symmetry, the S(X) value will increase. The S(X) scale has been designed to range from 0 to 100, and users may refer to this scale as a continuous measure of the ability to perceive the existence of a particular generic X-pseudosymmetry element within an object. Ranges of S(X)-parameter numerical values have been interpreted in terms of either a negligible loss, small loss, moderate loss, or the perceivable absence of a generic symmetry element-X [11, 23]. Thus, one may paraphrase this interpretation by proposing a rule of thumb, which states that S-values of 0.01 or less indicate negligible distortion from an ideal symmetry (i.e., high-fidelity pseudosymmetry that is not perceivable by the naked eye) [11], whereas values up to 0.1 correspond to small deviations from the ideal, which may or may not be visibly perceivable (i.e., moderate-fidelity pseudosymmetry). Larger S-values up to 1.0 signify structurally significant divergences from the ideal (i.e., low-fidelity pseudosymmetry) [11, 23]. Values larger than 1.0 testify to important distortions that are large enough so that the absence of a particular pseudosymmetry element within an object is visually recognized. For example, an Avnir CSM quantification of mirror-plane distortion in the stone axe-heads (2–4) as a function of the archeological site's age was reported to be 1.84 S(σ) for the oldest specimen (2), 0.77 S(σ) for the intermediate-aged one (3), and 0.29 S(σ) for the youngest axe-head (4) [1]. Inspection of axe-heads shows 4 to exhibit the highest fidelity mirror pseudosymmetry and the lowest S(σ) value.
The closer the Avnir S(X)-value approaches integer zero (the object's distortion from an ideal generic symmetry), the harder it is for our eyes to differentiate between high-quality pseudosymmetric objects and those exhibiting genuine symmetry. In the world around us, it is only our knowledge that perceived symmetry is almost always nonmathematical that enables us to be cognizant that an object's shape or form is actually only a high-quality emulation of symmetry rather than being the real thing. When does our perception of pseudosymmetry end and our realization of gross distortion or a complete loss of pseudosymmetry begin? It is not simple, since the pseudosymmetry continuum's frontier with perceivable asymmetry is mathematically fuzzy, rather than being an easily recognized step-function.
Should experts in symmetry proclaim, to one and all, that only pseudosymmetry (and not mathematical symmetry) primarily exists in our macro-world? Definitely not! There is nothing wrong with using fuzziness in communication unless the distinction between pseudosymmetry and genuine symmetry is absolutely required to explain a particular phenomenon. Communication and language should be free and not hampered by awkward phraseology and unnecessary accuracy. As a corollary, does this mean that one should use the adjective “pseudosymmetrical” to describe objects in our world? Probably not a good idea, since many will not understand the term. There is nothing wrong in using the concept of symmetry in its fuzzy sense. However, as scientists studying molecular structure, we should be aware of the conceptual differences between pseudosymmetry and genuine symmetry.
The concept of chirality seems to be a continuum without an end, since the generation of achiral geometries is a very special case due to specific symmetry constraints. Avnir has elegantly shown that when one of the two reflection symmetry-related vertices of the isosceles triangle 20 (isos and skelos, respectively, mean “equal” and “leg” in Greek) is ever so slightly depressed, then the formerly achiral figure is transformed into scalene triangle 21 (a two-dimensional chiral object). It exhibits a “small amount of chirality” as the ideal σ-plane has ceased to exist. The amount of this chirality can increase as the vertex is moved further downward. It is thus clear that it is almost impossible to observe ideal achirality in our macro-world. The general case is that an object's geometry will have some finite degree of chirality. In other words, chirality is the “general rule” in our universe, while achirality is a “special case” since it requires structural and mathematical constraints.
