Lecture 2: Crystals and symmetry.
2.1 Crystallization of Biological Molecules
2.1.1 Physical Principles of crystallization
2.1.2 Methods for growing crystals
2.1.3 Identifying crystallization conditions
2.2 Some Fundamentals of Symmetry
2.2.1 What is symmetry and why is it important in structural biology ?
2.2.2 Symmetry operations in 2D
2.2.3 Symmetry Groups
2.2.4 A critical concept : The asymmetric unit
2.2.5 Symmetry operations in 3D
2.2.6 Enantiomorphism and the symmetries of biological specimens
2.3 Three-dimensional Symmetry Groups.
2.3.1 Point groups
2.3.2 Lattices and Translational Periodicity
2.3.3 Line Groups
2.3.4 Layer Groups
2.3.5 Space Groups
2.4 “Racemic crystallography”
2.5 “Non-crystallographic” symmetry
Thursday, 6 March 14
1
Physical principles of crystallization.
•The ability of protein crystals to diffract X-rays provides the experimental data
required to determine the three-dimensional structure of proteins at atomic
resolution.
•However a primary difficulty in the application of the technique is that it
depends on on obtaining well-ordered crystals. This is not trivial.
•We'll discuss crystallization of water soluble proteins. However, pretty much the
same considerations apply to the crystallization of membrane proteins and
nucleic acids, with some added complications. e.g.
★Detergents
proteins.
are required to shield the hydrophobic surfaces of membrane
★The
crystallization of nucleic acids often requires addition of compounds to
neutralize the charge on the phosphate backbone.
Thursday, 6 March 14
2
Some examples of protein crystals
From McPherson (1999)
Very pretty, and very useful, but how do we make them ?
Thursday, 6 March 14
3
Protein crystallization: The basic idea
•Protein crystals form in supersaturated solutions in which the protein
concentration exceeds it's equilibrium solubility.
•Hence all the physical techniques for crystallizing proteins involve bringing a
protein solution into the supersaturated state by alteration of some property of
the system.
•Typically this is accomplished by gradually increasing the concentration of
substances which serve to reduce protein solubility (protein precipitants), via a
diffusive process. Salts, simple organic compounds and long chain synthetic
polymers are all used as protein precipitants.
•From a supersaturated protein solution, equilibrium can be restored by phase
separation.
Thursday, 6 March 14
4
Standard crystallization techniques used for
inorganic compounds are generally not
applicable
Crystallization by Temperature
Manipulation
Copper Sulfate crystals grown by cooling a
saturated hot CuSO4 solution
Crystallization by
Uncontrolled Evaporation
Salt crystals covering rocks, Great Salt Lake, Utah
(Photo Credit: Eve Andersson)
Proteins commonly denature (irreversibly unfold) at extreme
temperatures, or under other harsh conditions
Thursday, 6 March 14
5
How do protein precipitants work?
Long chain synthetic polymers work by simple volume exclusion
Finet et al. Controlling biomolecular crystallization by understanding the distinct effects of PEGs and
salts on solubility. Meth. Enzymol. (2003) vol. 368 pp. 105-29
Around the protein there is a zone which water can readily penetrate, but is not accessible to the
polymer (the light gray layer in the diagram). The associated loss in configurational entropy raises
the Gibbs energy of the system. Protein-Protein interactions, which reduce the volume of the
exclusion zone and minimize the Gibbs Energy, are favored.
Thursday, 6 March 14
6
sion of
Precipitation
by
long-chain
synthetic
polymers
Proteins by Polyethylene
Glycols
reasing size of the
5 are given in Fig.
n of the solubility
emolecular weight
) to 20,000. There
h appears to ap-
I
1
1
1.6
Most commonly used polymer is
Polyethylene Glycol (PEG)
1.2
HO-CH2-(CH2-O-CH2-)n-CH2-OH
0.8
3
5
0.4
v)
-
Ln
0.0
-0.4
-0.8
I
0
10
20
30
% PEG-4000 IwlvJ
40
FIG. 4. Solubility of various proteins in PEG-4000. Measurements were made in 0.05 M potassium phosphate, pH 7.0, containing
0.1 M KC1. Human fibrinogen (X), 20 mg/ml; human y-globulin (D),
20 mg/ml; lysozyme (A), 50 mg/ml; a-lactalbumin (+), 20 mg/ml;
chymotrypsin (O),7.5 mg/ml; aldolase (A),13 mg/ml; thyroglobulin
(0),14 mg/ml; human serum albumin (O),20 mg/ml.
Downloaded from www.jbc.org at UNIVERSITY OF AUCKLAND on Ap
obtained by least
he corresponding
able 11. The value
eement with that
min under similar
intercepts are
conh small decreases
own in Fig. 3, a
ained by assuming
on point and that
pe. This is reasonn the type of PEG.
in, thyroglobulin,
lubility curves are
r these conditions,
e term in Equation
crease in the slope
m 400 to 4000. At
eater, the slope for
4.6. However, the
ions (not shown)
cess of 1/5indicatactions.
of several proteins
tions are shown in
aracteristic linear
tration. Lysozyme
ologous and have
r under these condiffer greatly (4.7
though the larger
and thyroglobulin)
ntrations than the
I
“PEG 400”
Average Molecular Mass 400 Da
Average n = 9
Atha and Ingham. Mechanism of
precipitation of proteins by polyethylene
glycols. Analysis in terms of excluded
volume. Journal of Biological Chemistry
(1981) vol. 256 (23) pp. 12108
"1 theory, protein solubility varies with polymer concentration in
As predicted from volume-exclusion
a fairly simple way
Thursday, 6 March 14
7
Precipitation by salts
Salts exhibit more complicated behavior, and no compete physical theory
exists to explain their ability to precipitate proteins.
Generally at low concentrations they serve to increase protein solubility
(“salting-in” behavior) whereas at higher concentrations they force proteins
from solution (“salting out” behavior).
From Blundell and Johnson (1976)
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Precipitation by salts
The ability of salts to precipitate proteins varies systemically, and
is predicted by the Hofmeister series. The physical origins of the
series are still being debated. The series predicts other properties
too, such as the ability to stabilize and destabilize proteins.
Precipitate proteins, Structure stabilizing
Thursday, 6 March 14
Solubilize proteins, Structure destabilizing
9
Protein Conecntration
Protein crystallization: A schematic phase diagram
Supersaturation
Protein crystals form
in supersaturated
solutions
Undersaturation
Solubility limit
Precipitant Concentration
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Nucleation. A critical step in crystal growth.
•Protein crystal growth is generally a nucleated process. The critical nucleus = smallest
stable crystal. Once a critical nucleus has formed, crystal growth will follow spontaneously. In
practice the critical nucleus may be 10 -100 molecules in size.
•This occurs because in small aggregates, where a large fraction of the molecules are at the
surface, too few favorable intermolecular contacts are created to compensate for the
entropic cost of a molecule joining the crystal.
•The idea is easy to illustrate with a helical assembly (which is in essence a 1D crystal).
• A subunit adding to a small assembly experiences only lateral interactions with its neighbors. A
subunit adding to a large assembly experiences both lateral and vertical interactions. We might
guess that, in this system, a single turn of the helix might be a critical nucleus ...
Thursday, 6 March 14
11
Nucleation. A critical step in crystal growth.
Weber, P.C. (1991) Adv Prot. Chem. 41, 1-36
Thursday, 6 March 14
12
Protein crystallization: A more detailed (but still
schematic) phase diagram
Crystals will dissolve
Crystals will grow but not nucleate
Crystals will nucleate and grow
Protein will precipitate
Protein Conecntration
PRECIPITATION ZONE
NUCLEATION ZONE
METASTABLE
ZONE
Precipitant Concentration
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The Batch Method for Crystallization
In the batch method we simply mix protein and precipitant
in fixed ratio (typically 1:1)
A real
example: Batch
crystallization
of lysozyme
Feher and Kam (1985) Meth. Enzymol 114.
NaCl concentration %(w/v)
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14
The Batch
Method for
Crystallization
Returning to a schematic phase
diagram, batch crystallization
should proceed according to
path labelled (i).
Chayen & Saridakis (2008) Nature Methods 5, 147-153.
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15
The Vapor Diffusion method for
crystallization
A slightly gentler way of approaching supersaturation is the
vapor diffusion method. This relies on water transport through
the vapor phase as depicted below.
From Crystallization of Biological Macromolecules, McPherson
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The Vapor Diffusion
method for
crystallization
Returning to a schematic
phase diagram, vapor
diffusion should proceed
according to path labelled
(ii).
Chayen & Saridakis (2008) Nature Methods 5, 147-153.
Vapor diffusion is the most widely used technique for preparing protein crystals. There are other physical
methods available (e.g. dialysis and free-interface diffusion). We will not discuss them here.
Thursday, 6 March 14
17
Identifying crystallization conditions
From a supersaturated protein solution a solid phase can result from:
The formation of disordered protein aggregates, leading to an amorphous precipitate or flocculate
(easy to do) ...
Images courtesy of Andrea Kurtz, SLAC
... or the formation of ordered aggregates leading to nucleation and growth of crystals (hard to
do)
Image courtesy of Andrea Kurtz, SLAC
For a water-soluble protein, a typical crystallization trial will involve at least 3 solution
components; the protein; a buffer to control the pH, and a precipitant to exclude the protein from
solution. What must be experimentally determined are an appropriate buffer
and precipitant together with the ranges of temperature, pH and
concentration of the solution components that will support crystal growth
(vs amorphous precipitation).
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18
Identifying crystallization conditions
Because many hundreds of trials may be required to identify protein crystallization conditions, the
trials are usually carried out in multi-well plates, and often using robotics.
Thursday, 6 March 14
19
Protein crystals are fragile
Protein crystals are formed by a sparse network of weak molecular interactions. Hence protein
crystals are generally fragile and easily damaged. We’ll discuss some of the practicalities of
handling them in lecture 5.
Rupp (2010)
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20
Some Fundamentals of Symmetry
Notion of symmetry depends on the existence of equivalent parts in a pattern or object
From Bernal, Hamilton & Ricci (1972)
If there are motions which carry or map any part of an object to the original position of
any other, while leaving the appearance of the object unchanged, then the object is
symmetrical. Such motions are called symmetry operations for the object.
Symmetry is a mathematical ideal and may be imperfectly realized in nature.
Thursday, 6 March 14
21
Why is symmetry important in structural
biology?
1. Many biological molecules assemble into symmetric or nearly symmetric structures …
this is critical to their function.
Thursday, 6 March 14
22
Why is symmetry important in structural
biology?
1. We take advantage of symmetry in scattering-based techniques (e.g. electron
microscopy and X-ray crystallography) … it massively increases the signal-to-noise ratio.
Understanding symmetry is critical to proper data analysis in both methods.
3D Protein Crystals
Electron micrograph of icosahedral virus particles (L)
and a derived 3D reconstruction (R).
Images courtesy Tuli Mukhopadhyay
Thursday, 6 March 14
23
The purpose of this lecture …
1. To introduce the basic concepts and language of symmetry.
2.To provide some tables and notation … reference material for the
subsequent lectures.
Thursday, 6 March 14
24
Symmetry operations and symmetry
elements
A symmetry operation may keep some points fixed in place; the set of
such points is called the symmetry element for the operation. In the
case of reflection in the plane, the symmetry element is a line.
Once upon a midnight dreary, while I pondered weak and weary,
Over many a quaint and curious volume of forgotten lore...
Symmetry
operation:
reflection in
the plane
Symmetry element
Thursday, 6 March 14
25
Symmetry in 2D
Rotations, Reflections and Translations
Rotation and
Reflection are
“Point symmetry
Operations”. They
leave at least one
point of the object
fixed
From Bernal, Hamilton & Ricci (1972)
Translation is a
“space symmetry
operations”. It
leaves no point of
the object fixed.
Symmetry operations are rigid motions, motions which do not alter the distances
between points. Rigid motions are also called isometries
Thursday, 6 March 14
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Some group theory ...
Consider this object:
From Bernal, Hamilton & Ricci (1972)
It has exactly 5 symmetries. We may rotate by 72, 144, 216
and 288° without changing its appearance. That makes 4. The
fifth symmetry is to do nothing at all (or equivalently, to rotate
through 360°). This is termed the identity
Thursday, 6 March 14
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Some group theory ...
From Bernal, Hamilton & Ricci (1972)
These symmetries can be combined. We may first rotate by
288° (4x72) and then by 144° (2x72). The net result would be
the same as rotating by 432° (6x72) - or more simply - by 72°
- which is one of our original 5 symmetries.
Thursday, 6 March 14
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Some group theory ...
From Bernal, Hamilton & Ricci (1972)
The identity is special:
• When we combine the identity with any other symmetry we
don’t change the result.
•Also every symmetry has an inverse that is its undoing.
Combining a symmetry with its inverse produces the identity
e.g. combine a rotation of 288° (4x72) with a rotation of 72° and we get a rotation of 360°
(The identity operation). A rotation of 72° is the inverse of a rotation of 288°
Thursday, 6 March 14
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Some group theory ...
From Bernal, Hamilton & Ricci (1972)
Combination of symmetries is associative. If we define “x” to
mean “followed by” then:
(72° x 144°) x 216° =
72° x (144° x 216°) = 72°
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30
This collection of movements forms a
mathematical group.
A group of movements is a set that satisfies the following conditions
•The product of any two movements in the set, or of any movement with itself, is
a member of the set.
•The identity is included as a movement.
•For every movement there is an inverse, a member of the set such that the
product of the movement and its inverse is the identity.
•For three successive movements the associative law applies
(M1*M2)*M3 = M1*(M2*M3)
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Our object is described by a symmetry group
The operations of the group
From Bernal, Hamilton & Ricci (1972)
Thursday, 6 March 14
In this context, “multiplication” means
“followed by”.
5 x 5 = A rotation of 72°, followed by a
rotation of 72°
32
A more complicated point group in the
plane.
From Bernal, Hamilton & Ricci (1972)
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33
The asymmetric unit / fundamental region of
a symmetric object.
From Vainshtein (1994)
The asymmetric unit of an object, described by a symmetry group,
is the smallest compact region from which the whole
object can be reconstructed through application of the
symmetry.
Thursday, 6 March 14
34
Symmetry operations in 3D
One useful way to classify symmetry operations is by the number of points
they keep fixed
Point
symmetry
operations
The identity
Fixes 4 non-coplanar points
Reflection
Fixes 3 non-colinear points
Rotation
Fixes 2 points
Rotary inversion/
Rotary reflection*
Fixes 1 point
Translation
Fixes no points
Glide Reflection
Fixes no points
Screw Rotation
Fixes no points
Space
symmetry
operations
* N.B Rotary inversion and Rotary reflection are mathematically equivalent. Straight inversion is a
special case
Thursday, 6 March 14
35
Symmetry operations in 3D: Reflection
From Vainshtein (1994)
Thursday, 6 March 14
36
Symmetry operations in 3D: Rotation
From Glusker & TrueBlood (1985)
Thursday, 6 March 14
37
Symmetry operations in 3D: Inversion and
Rotary Inversion (Rotary Reflection)
From Vainshtein (1994)
From Glusker & TrueBlood (1985)
From Blow (2002)
Thursday, 6 March 14
Anything that can be achieved by rotary reflection can
be done by rotary inversion. For example rotation by 0°
coupled with reflection is the same as rotation by 180°
coupled with inversion
38
Symmetry operations in 3D: Translation
From Glusker & TrueBlood (1985)
Thursday, 6 March 14
39
Symmetry operations in 3D: Glide reflection
From Vainshtein (1994)
From Glusker & TrueBlood (1985)
Thursday, 6 March 14
40
Symmetry operations in 3D: Screw
Rotation
From Glusker & TrueBlood (1985)
From Glusker & TrueBlood (1985)
An nm screw axis involves a rotation of 360/n° accompanied
by a translation of m/n of the fundamental repeat distance.
Thursday, 6 March 14
41
Symmetry operations in 3D: Screw
Rotation
From Glusker & TrueBlood (1985)
An nm screw axis involves a rotation of
360/n° accompanied by a translation of
m/n of the fundamental repeat distance.
An 21 screw axis involves a rotation of
360 / 2 = 180° accompanied by a
translation of 1/2 of the fundamental
repeat distance.
Thursday, 6 March 14
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Symmetry operations in 3D: Screw
Rotation
An nm screw axis involves a rotation of
360/n° accompanied by a translation of
m/n of the fundamental repeat distance.
An 41 screw axis involves a rotation of
360/4 = 90° accompanied by a translation
of 1/4 of the fundamental repeat distance.
An 43 screw axis involves a rotation of
360/4 = 90° accompanied by a translation
of 3/4 of the fundamental repeat distance.
Thursday, 6 March 14
43
Graphical representations of symmetry
operations
From Giacovazzo (1992)
Thursday, 6 March 14
44
ble pointgroup symmetries for biological macromolecules, the reader is refe
complete list of all possible pointgroup symmetries including those with m
Enantiomorphism
and
the
etry, to the International Tables of Crystallography, and to Wikipedia: “Li
symmetries
of biological specimens
cal symmetry
groups”.
Short story: There is no mirror symmetry in proteins
Image courtesy of Marin van Heel.
There is no mirror symmetry in Proteins!
Thursday, 6 March 14
45
Slightly longer story:
From Rupp (2010)
Many biological molecules are chiral. This includes the amino acids - the building blocks of proteins.
Natural proteins are made of L-amino acids. The enantiomers - D-amino acids - are used very rarely in
nature.
This means that for proteins and protein assemblies reflections and inversions are not “permitted” since
they switch the handedness of an object. They would convert a protein made up of L-amino acids into a
protein made up of D-amino acids.
Enantiomorphism therefore limits the symmetries we observe with biological specimens
Thursday, 6 March 14
46
While a biological assembly cannot
accommodate mirror planes and centers of
inversion, its diffraction pattern may …
Tetragonal crystals of lysozyme
From Fraser & Macrae (1969)
Thursday, 6 March 14
47
Symmetry groups in 3D
• We’ll deal first with the symmetry groups that don’t
include translational symmetry (Point groups)
• Then we’ll discuss translational symmetry and the concept
of the lattice.
• Finally we’ll deal with the symmetry groups that include
translational symmetry (Line groups, Layer Groups,
Space groups)
Thursday, 6 March 14
48
Point groups
The symmetries of non-periodic objects
•A
group of symmetry operations that leaves at least one point
invariant (in the same place) is known as a point group
•There
are three families of point groups that can accommodate
chiral molecules of fixed hand … the cyclic, dihedral and cubic
point groups. They involve only rotations (reflections and inversions
being disallowed).
•Unluckily for you, point groups containing reflections and inversions
are important for describing the symmetry of diffraction patterns, so
we can’t ignore them entirely.
Thursday, 6 March 14
49
A note on notation ...
To describe the point groups we are going to use the International
(Hermann-Mauguin) notation. This is the standard in crystallography ...
Carl Hermann (1898-1961)
A German crystallographer
http://www.staff.uni-marburg.de/
Charles-Victor Mauguin (1878-1958)
A French Mineralogist
http://www-int.impmc.upmc.fr/public/Associations/afc/
A alternative notation - The Schönflies system - is commonly used
for isolated molecules, particularly in chemistry and spectroscopy
Thursday, 6 March 14
50
A note on notation ...
The basics of Hermann-Mauguin notation.
Operation: Rotation of 360°/n
Symmetry element: A Rotation Axis
Denoted by n (1,2,3,4,5,6,7,...)
Operation: Reflection
Symmetry Element: A Mirror plane
Denoted by m
Operation: Rotary inversion (Rotation of 360°/n coupled with inversion)
Symmetry Elements:
Combination
of
a
Rotation
axis
and
a
point.
- - -- -- - Denoted by n (1,2,3,4,5,6,7,...)
Thursday, 6 March 14
51
“Biological” Point Groups part 1. Cyclic
point groups
From Vainshtein (1994)
From Blow (2002)
Thursday, 6 March 14
52
seen in negatively stained EM preparations. However, it is actu
distinguish C4 from some of the other pointgroups (like 22
proper symmetry analysis. Head-to-tail types of inter-subu
helical polymers rather than to closed Cn pointgroup assembl
“n” values. To form a Cn structure requires “n” identical units
A particularly common symmetry
for pore-forming proteins
polypeptide chains.
Cyclic point groups: examples
Top view
Side view
α-hemolysin; A heptameric pore-forming protein
with 7-fold rotational symmetry
Thursday, 6 March 14
The 3D reconstruction
of the
LTX oligomers. a, The top and
Image courtesy of Marin
van Heel
C4 tetramer. c, The side and cut-open views of the tetramer.
calculated
from 128 class averages
that included 1,900 ori
Spider α-latrotoxin:
The tetrameric
(E.V. Orlova, M.A. Rahman, B. Gowen, K.E. Volynski, F. Meunie
neurotoxin
from the
widow
spider
Ushkaryov: Structure
of αblack
-Latrotoxin:
Dimers
Assemble
into T
with
4-fold
rotational
Large,
Gated
Membrane
Pores. symmetry
Nature Struct. Biol. 7 (2000) 4853
“Biological” Point Groups part 2. Dihedral
point groups
From Vainshtein (1994)
From Blow (2002)
Thursday, 6 March 14
54
Dihedral point groups: examples
Top view
Side view
Glutamine synthetase:
Dodecamer with 622 point group symmetry
Thursday, 6 March 14
Three
orthogonal
views,
looking
down the 2fold rotation
axes of the
tetramer
Glucose fructose oxidoreductase:
Tetramer with 222 point group symmetry
55
density maps of aromatic residues are clearly
shown in the regions with higher correlation coefficients (~0.95). An electron density map of each
80.0% were in the most favorable region of the
Ramachandran plot (25), 19.7% in the allowed
region, and 0.3% in the disallowed region. Out of
Dihedral point groups: examples
Fig. 1. Overall structure of the vault shell. One molecule of MVP is colored in tan, and the others are
colored in purple. (Left) Side view
the mysterious
ribbon representation.
Thecomplex
whole vault shell comprises a 78Theofvery
rat liver vault
oligomer polymer of MVP molecules.
The size
is ~670 Å from the top to the bottom
78-mer
withof39the2 whole
point particle
group symmetry
and ~400 Å in maximum diameter. The particle has two protruding caps, two shoulders, and a body with
Tanaka et
The structure
liver vault
at 3.5 angstrom resolution.
Science
vol. 323
(5912) pp. 384-8
an invaginated waist. Two half-vaults are associated
atal.the
waistof rat
with
N-terminal
domains
of(2009)
MVP.
(Right)
Thursday,
March 14
Top 6view
of the ribbon representation. The maximum diameter of the cap is ~200 Å. The outer and the
56
23 … tetrahedral
Order of the group: 24
432 … octahedral
Order of the group: 48
“Biological” Point
Groups part 3.
Cubic point
groups
Images courtesy Dr Stefan Immel
http://csi.chemie.tu-darmstadt.de/ak/immell
532 … icosahedral
Order of the group: 60
Thursday, 6 March 14
57
Cubic point groups: examples
Ferritin particle
Cut-away view
Ferritin - an iron storage protein
432 (octahedral) symmetry
Image courtesy David Goodsell http://www.pdb.org/
Thursday, 6 March 14
58
Cubic point groups: examples
http://www.virology.wisc.edu/virusworld/
Human echovirus
532 icosahedral symmetry
Thursday, 6 March 14
Norovirus (“Norwalk virus”)
532 icosahedral symmetry
59
Summary: The Biological point
groups
CYCLIC
138
Schönflies
notation
J. Janin, R. P. Bahadur and P. Chakrabarti
...
1
2
3
4
5
Hermann-Mauguin notation
Adapated from Janin et al. Protein-protein interaction and quaternary structure. Q Rev Biophys (2008) vol. 41 (2) pp. 133-180
Thursday, 6 March 14
60
Summary: The Biological point
groups
138
J. Janin, R. P. Bahadur and P. Chakrabarti
DIHEDRAL
Schönflies notation
...
222
32
422
52
Hermann-Mauguin notation
Fig. 1. Symmetry of oligomeric proteins. An oligomeric protein with n identical subunits may have the
symmetries of the cyclic Cn point group (top row), one with 2n subunits, the symmetries of the dihedral Dn
point group (middle
row)
cubic
row)
requirestructure.
the protein
to have
12,vol.2441or
identical
Adapated
from; Janin
et al.symmetries
Protein-protein(bottom
interaction and
quaternary
Q Rev Biophys
(2008)
(2)60
pp. 133-180
subunits. Symmetry axes of different types are marked as dotted lines. Courtesy of E. Lévy (Cambridge,
Thursday, 6 March 14
UK).
61
Summary: The Biological point
groups
CUBIC
Schönflies notation
T (Tetrahedral)
O (Octahedral)
I (Icosahedral)
Fig. 1. Symmetry of oligomeric proteins. An oligomeric protein with n identical subunits may have the
(top row), one with 2n subunits, the 532
symmetries of the dihedral Dn
symmetries 23
of the cyclic Cn point group 432
point group (middle row) ; cubic symmetries (bottom row) require the protein to have 12, 24 or 60 identical
subunits. Symmetry axes of different types are marked as dotted lines. Courtesy of E. Lévy (Cambridge,
UK).
Hermann-Mauguin notation
Thursday, 6 March 14
(Fig. 1). Oligomers that display the symmetries of a cyclic Cn group have an n-fold axis : their
subunits are related by 360x/n rotations. The dihedral Dm groups require an even number of
subunits, n=2m ; they possess an m2-fold axis and m2-fold axes orthogonal to it. The T (tetrahedral) cubic point group has non-orthogonal twofold and threefold axes ; in addition, the
O (octahedral)
point
has fourfold
and the
I (icosahedral)
point
group,
fivefold axes.
Adapated from
Janin etgroup
al. Protein-protein
interactionaxes,
and quaternary
structure.
Q Rev Biophys (2008)
vol. 41
(2) pp. 133-180
Symmetry is a general property of oligomeric proteins (Goodsell & Olson, 2000). The most
62
Other Families of Point Groups, incorporating
the “forbidden” inversions and mirror planes
Images from Vainshtein (1994)
32
Thursday, 6 March 14
63
Important sets of point groups I. The crystallographic point groups, which
describe the rotational symmetries of crystals
Crystals (which we’ll come too shortly) can only accommodate rotation axes of a certain order. 1,2,3,4
and 6 fold symmetries are okay. 5 fold symmetry is out. So is 7-fold, 8-fold and all higher rotational
symmetries. There are consequently only 11 “crystallographic” point groups involving pure rotations.
Here they are …
From Blow (2002)
Thursday, 6 March 14
64
Important sets of point groups II. The Laue groups which describe the
point symmetry of X-ray diffraction patterns
Notice there are 11 crystallographic point groups and 11 Laue groups - and there is an obvious
correspondence between them. If you add a center of inversion to a crystallographic point group
you generate one of the Laue groups.
Images from Vainshtein (1994)
Thursday, 6 March 14
65
Lattices and Translational periodicity
• A lattice is a convenient representation of translational symmetry.
• Lattices are the framework on which much crystallographic theory
is built.
It’s all best illustrated by example …
Thursday, 6 March 14
66
A crystal is the convolution of an object and a
lattice
From Holmes and Blow (1965)
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A crystal is the convolution of an object and a
lattice
From Chiu, Schmid and Prasad (1993)
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Lattices in 1 and 2 dimensions
There is 1 unique 1-dimensional lattice …
a
There are 5 unique 2-dimensional lattices
A 1D lattice is
generated by a
single translation
A 2D lattice is
generated by 2
linearly
independent
translations
From Vainshtein (1994)
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The unit cell
The generators of a 2D lattice form two edges of a box … if we complete the
box we have a simple unit cell. This is a parallelogram with a lattice point at each
vertex and no lattice point anywhere else inside it or on its surface.
The orthorhombic primitive lattice
a
b
Conventional unit cell choice
The entire lattice can be constructed from a single unit cell by stacking these
boxes together. Since each lattice point is shared by four boxes and each box has
four vertices, this works out to be one lattice point per unit cell.
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The unit cell
The lattice generators are not unique, and the unit cell can be chosen in infinitely
many different ways. This is easy to see in two dimensions. However these
choices do not reflect the symmetry of the lattice. By convention we choose a
unit cell which does reflect this symmetry
The orthorhombic primitive lattice
Some alternate cell choices
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The unit cell
Following this line of reasoning crystallographers select a “non-primitive” or
“centered” unit cell for one of the 2D lattices This cell contains 2 lattice points
per cell. This choice drives mathematicians crazy*, but is actually quite sensible
and convenient for crystallography.
The orthorhombic centered lattice
a
a
b
A primitive unit cell
b
The Centered unit cell standard in crystallography.
* The cell edges no longer generate the lattice, which destroys the mathematical theory
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Lattices in 3 dimensions. The 14 Bravais Lattices
A 3D lattice is generated by 3 linearly independent
translations. There are exactly 14 of them, no more, no less.
Figures from
Woolfson (1970)
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The Unit Cell in 3 Dimensions
The generators of a 3D lattice form three edges of a box … if we
complete the box we have a simple unit cell. This is a parallelepiped
with a lattice point at each vertex and no lattice point anywhere else
inside it or on it’s surface.
From Drenth (2002)
The entire lattice can be constructed from a single unit cell by stacking
these boxes together. Since each lattice point is shared by eight boxes
and each box has eight vertices, this works out to be one lattice point per
unit cell.
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“Centered unit cells”
Just as in 2D, crystallographers choose the unit cell so that it
reflects the symmetry of the lattice. Sometimes this puts lattice
points at the center of a unit cell, or on its faces. Once again this
generates “non-primitive”, “centered” unit cells containing more than
one lattice point per cell.
From Drenth (2002)
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Symmetry of the lattice imposes constraints on
the unit cell dimensions
From Blow (2002)
Lattices are assigned to crystal
systems according to their
symmetry though mathematicians
and crystallographers do this
slightly differently. This time we’re
going to side with the
mathematicians (The argument
centers over where to place the
Rhombohedral Lattice).
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The seven crystal systems
System
Essential
rotational
symmetry
Conventional
Unit Cell
choice of unit cell restrictions
axes
Possible
Lattices
Triclinic
None
No constraints
None
P
Monoclinic
Two-fold axis
b parallel to 2-fold
α = γ = 90°
P,C
Orthorhombic
Three perpendicular
2-fold axes
a, b, c parallel to 2fold axes
α = β = γ = 90°
P,C,I,F
Trigonal/Hexagonal
3-fold or 6-fold axis
c parallel to 3-fold or
6-fold
a=b
α = β = 90°, γ = 120°
P
Rhombohedral
3-fold axis
a, b, c related by
three fold axis
a=b=c
α=β=γ
R
Tetragonal
4-fold axis
c parallel to 4-fold
a=b
α = β = γ = 90°
P,I
Cubic
4 3-fold axes
a, b, c related by
three fold axis
a=b=c
α = β = γ = 90°
P,I,F
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Symmetry groups involving translational
periodicity.
Skipping over the mathematical complexities, these groups are derived by
combining point group symmetry with a lattice. The lattice might be 1, 2 or
3 dimensional leading to …
The Line (or Rod) groups …
the symmetries of helices
The Layer (or Two-sided Plane) groups …
the symmetries of 2D crystals
The Space groups …
the symmetries of 3D crystals.
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Line Groups: The symmetries of 3D objects
periodic in 1 Dimension
These groups describe the symmetries of helices and rods. They are derived
by combination of a 3D point group and a 1D lattice.
Very important in X-ray Fiber diffraction and Helical image reconstruction. They are constructed by combination of
a point group and a 1D lattice. We do not consider them in detail in this course.
Tobacco Mosaic Virus … image from Don
Caspar
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Layer Groups (Two-sided plane groups): The
symmetries of 3D objects periodic in 2 Dimensions
These groups describe the symmetries of “2D” crystals. They are constructed through
combination of a 3D point group and a 2D lattice.
From Mosser (2001)
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Transmission electron microscopy … The relationship between the Layer
groups and the Plane groups
Electrons
Layer Group
Crystal
Projection
Plane Group
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81
Layer Groups
(Two-sided plane
groups)
The symmetries of 3D objects
periodic in 2 Dimensions
These groups describe the symmetries
of “2D” protein crystals. There are 80
Layer groups in total, but only 17 can
accommodate biological specimens.
From Engelhardt (1988)
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Plane Groups
The symmetries of 2D objects
periodic in 2 Dimensions
These groups describe the symmetries
of crystals in projection. They are
therefore of critical importance in
electron crystallography. Happily, the
17 “Biological” Layer groups each have
a unique corresponding Plane group.
From Engelhardt (1988)
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The 17 “Biological” Layer Groups
2D Bravais
Lattice
Oblique
3D point
group
Layer group
1
p1
2
p21
Primitive orthorhombic 2
p12, p121
222
p222, p2212, p22121
Centered
orthorhombic
2
c12
222
c222
Square
4
p4
422
p422, p4212
3
p3
32
p312, p321
6
p6
622
p622
Hexagonal
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A closer look at the Layer groups based on point group 222
From Engelhardt (1988)
Tshudy, D. K. & Litvin, D. B. VRML general position/symmetry diagrams
of the 80 layer groups. J Appl Cryst 31, 973–973 (1998).
There are three different ways to combine the point group 222 with the primitive orthorhombic lattice
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Space Groups: The symmetries of 3D objects
periodic in 3 Dimensions
These groups describe the symmetries of 3D crystals. They underpin
X-ray crystallography. They are constructed by combining a 3D point
group with a 3D lattice.
There are 230 Space groups in total. 65 of these do not involve
reflection or inversion, and can accommodate biological molecules
of a fixed hand.
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Space Groups: The symmetries of 3D objects
periodic in 3 Dimensions
The simplest space groups to understand are those that result from “stacking” the layer
groups (if you stack an object periodic in two dimensions, you get an object periodic in three
dimensions). To illustrate here’s a schematic of a protein molecule with P4 layer group
symmetry.
If we stacked these structures, one on top of
another, to generate a 3D crystal - this crystal
would have P4 space group symmetry.
Not too difficult too visualize.
Rupp (2010)
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Space Groups: The symmetries of 3D objects
periodic in 3 Dimensions
While we are here, let’s consider what the asymmetric unit of this crystal would be. Recall that
the asymmetric unit of a crystal is the region from which the entire crystal can be
generated through application of the symmetry operations.
Rupp (2010)
The asymmetric unit of layer group/space group P4 covers 1/4 of the unit cell. Above are two possible
asymmetric units. The choice on the left is ideal for visualizing the tiling, but the choice on the right is
much better suited for visualizing the molecule. The asymmetric unit needs to cover 1/4 of the cell - it
doesn’t matter mathematically which 1/4 you choose
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Another very simple space group …
P2
From Blow (2002)
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The asymmetric unit of P2
From Outline of Crystallography for
Biologists, Blow
The asymmetric unit of a crystal with P2 symmetry, covers 1/2 of the unit cell. In this case it
contains in totality, 1 duck. Note that again - the duck doesn’t fit neatly in the bounds of the
crystallographic cell.
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A unit cell’s worth of some other simple space groups
Alas, not all the 3D space
groups are so simply
related to the 2D plane
groups. In general, the
space groups can be quite
hard to visualize. The great
compendium of
information on the space
groups is the
International Tables
for Crystallography.
From Cantor and Schimmel (1980)
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International Tables:Space Group P2
space group
point group
crystal system
Symmetry of the
diffraction pattern
Projection diagrams
showing the
position of the
symmetry elements
in the unit cell
Bounds of the
asymmetric unit
Mathematical
operations which will
generate the cell
contents from the
asymmetric unit.
Specifies diffraction
data which will be
“systematically
absent” ... this is critical
for space group
determination
For Further information see Dauter, Z. & Jaskolski, M. How to read (and understand) Volume A of International Tables for Crystallography: An
introduction for nonspecialists. J Appl Cryst 43, 1150-1171 (2010).
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The 65 “Biological” Space Groups
Bravais Lattice
Possible space groups
Associated
point group
Primitive Cubic
P23 (195), P213 (198)
P432 (207), P4132 (213), P4232 (208), P4332 (212)
23
432
I centered Cubic
I23 (197), I213 (199)
I432 (211), I4132 (214)
23
432
F centered Cubic
F23 (196)
F432 (209),F4132 (210)
23
432
Rhombohedral
R3 (146)
R32 (155)
3
32
Primitive Hexagonal
P3 (143), P31 (144), P32 (145)
P312 (149), P3112 (151), P3212 (153), P321 (150), P3121 (152), P3221 (154)
P6 (168), P61 (169), P62 (171), P63 (173), P64 (172), P65 (170)
P622 (177), P6122 (178), P6222 (180), P6322 (182), P6422 (181), P6522 (179)
3
32
6
622
Primitive Tetragonal
P4 (75), P41 (76), P42 (77), P43 (78)
P422 (89), P4212 (90), P4122 (91), P41212 (92), P4222 (93), P42212 (94), P4322,
(95), P43212 (96)
4
422
I centred Tetragonal
I4 (79), I41 (80)
I422 (97), I4122 (98)
4
422
Primitive Orthorhombic
P222 (16), P2221 (17), P21212 (18), P212121 (19)
222
C Centered Orthorhombic
C222 (21), C2221 (20)
222
I Centered Orthorhombic
I222 (23), I212121 (24)
222
F Centered Orthorhombic
F222 (22)
222
Primitive Monoclinic
P2 (3), P21 (4)
2
C Centered Monoclinic
C2 (5)
2
Triclinic
P1 (1)
1
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“Racemic” crystallography
Having emphasized that space groups containing mirror planes and centers of inversion are not
relevant to the crystallography of biological molecules, we note some circumstances in which this is
not true.
Using total chemical synthesis one can produce
small proteins entirely from D- amino acids. This
may fold into the mirror image of the biologically
occurring L-protein
If you create a racemic mixture of L- and Dproteins, the 165 “disallowed” space groups can
be accessed.
L- and D- plectasin
crystallized in space
group P1, which
contains a center of
inversion.
It appears that racemic mixtures of proteins may
be easier to crystalize than the single biological
L-enantiomers. There are also other
crystallographic advantages which we will touch
on later.
The principal problem is that total chemical
synthesis of proteins, is time consuming, difficult,
and not applicable to large polypeptides.
Mandal, K. et al. Racemic crystallography of synthetic protein enantiomers used to determine the
X-ray structure of plectasin by direct methods. Protein Sci 18, 1146-1154 (2009).
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Non-crystallographic
symmetry
If there’s more than one equivalent copy
of a molecule in the asymmetric unit of a
crystal, then there will be some
symmetry operation(s) which relate the
molecules. These symmetry operations
are called non-crystallographic or “local”
because they extend only a finite
distance through the crystal.
The diagram on the right illustrates the
basic idea, in 2 dimensions.
Non-crystallographic symmetry occurs
frequently with protein cr ystals.
Depending on it’s nature, it can either
help or hinder structure determination.
We’ll return to this in the later lectures.
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