Syllabus

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Syllabus
1. Crystals Principles of crystal growth 2.
Symmetry Unit cells, Symmetry elements, point
groups and space groups  3. Diffraction Introduct
ion to diffraction of waves The reciprocal
lattice Diffraction by crystals Bragg
equation 4. Obtaining the diffraction
pattern Instruments Data collection strategies 5.
Deriving a trial structure - phase
problem Molecular Replacement (MR) Isomorphous
replacement Anomalous scattering (MAD methods) 6.
Refining the structure Fourier and least-squares
methods  7. Analysis of structural parameters

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Crystals and symmetry
Other good resources Outline of
Crystallography for Biologists, David Blow
(Oxford University Press) Introduction to
Macromolecular Crystallography, Alexander
McPherson (Wiley) Principles of Protein X-ray
Crystallography, Jan Drenth (Springer) Internation
al Tables for Crystallography, Volume A
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Growing crystals
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Proteins pack symmeterically within crystals
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Symmetry
An object is symmetrical if, after some operation
has been carried out, the result is
indistinguishable from the original object.
Symmetry operators (or elements)
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Importance of Symmetry

• A crystallographer needs to analyze the underling
  symmetry of a crystal at an EARLY stage
• Needed to decide on the appropriate STRATEGY for
  data collection
• Crystallographic results must satisfy the
  symmetry and are constrained by it
• Precise symmetry required to interpret scattering
  data and SOLVE STRUCTURE

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Types of Symmetry

• The types of symmetry operation for finite
  three-dimensional bodies are
• rotation
• reflection (mirror symmetry)
• inversion (centrosymmetry)

Only rotation can exist in biological
macromolecules, which lack a centre of symmetry
and are called chiral
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Mirror symmetry is not allowed in biological
macromolecules
Crystals of chiral molecules cannot contain
mirror planes (centers of inversion)
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symmetry
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Rotational symmetry of molecular oligomers

• Rotational symmetry operations must always be
  through an angle which is an integral fraction of
  360 degrees
• Many protein molecules are composed of several
  identical peptide chains in a symmetrical
  arrangement eg. 2-, 3-fold..etc
• 4-fold rare in protein tetramers, normally
  2-fold symmetry about each of the perpendicular
  directions x, y and z i.e 222 symmetry. Eg.
  Glyceraldehyde 3-phospate dehydrogenase
• The kinds of symmetry that can be possessed by a
  local assembly of objects are called the POINT
  GROUPS
• By creating a 2-fold symmetry axis perpindicular
  to any n-fold axis, a second kind of 2-fold axis
  is always generated. For every point group with
  n-fold symmetry, another exist with n22 symmetry

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Symmetry in chiral molecules
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Tetramer with 222 symmetry
This kind of tetramer (2-fold symmetry about each
of three perpendicular directions) is often seen
in proteins.
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Crystal Symmetry Crystals are regular periodic
arrays, i.e. they have long range translational
symmetry. Crystals are often considered to have
essentially infinite dimensions.
Unit cell The smallest volume from which the
entire crystal can be constructed by translation
only. All crystals have translational symmetry,
with the translational vectors equal to edges of
the unit cell.
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The unit cell in three dimensions. The unit cell
is defined by three edge vectors a, b, and c,
with ?, ?, ?, corresponding to the angles between
b-c, ac, and a-b, respectively.
Unit cells are usually defined in terms of the
lengths of the three vectors and the three
angles. For example, a94.2Å, b72.6Å, c30.1Å,
?90, ?102.1, ?90.
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The crystal lattice
Lattice translation a
An ideal crystal has lattice symmetry a 3-d
arrangement of imaginary points so that view in a
given direction from each point in the lattice is
identical with the view in the same direction
from any other lattice point.
Lattice is the network of points on which the
repeating unit (unit cell) may be imagined to be
laid down so that the regularly repeating
structure of the crystal is obtained NB We could
choose a unit cell whose lattice points dont
coincide with any atoms at all.
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Crystal symmetry
Rotational symmetry may be added to lattice
symmetry Crystals can only accommodate certain
kinds of symmetry because of constraints of the
crystal lattice i.e lattice translation Only
2-,3-,4-, and 6-fold allowed. Crystals do not
contain 5-fold rotations, or any rotation axis
that is incompatible with translational symmetry
- limited point groups
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There are 32 distinct combinations of
crystallographic symmetry operations relating to
finite groups 32 point groups For chiral
units, there are 11 point groups
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The point groups that can exist in protein
crystals
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Screw axis, nr
An n-fold screw axis results from the combination
of rotation (of 360/n) and translation parallel
to the axis by a fraction r/n of the identity
period along that axis.
2-fold screw axis, 21
The degree of translation is added as a subscript
showing how far along the axis the translation
is, as a portion of the parallel lattice vector.
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The unit cell
The unit cell is the parallelepiped repeating
unit in the crystal. Defined by 3 lengths
(a,b,c) and 3 angles (? - between b and c ? -
between b and c ? - between b and c)
Distances along a, b and c are referred to in
terms of x, y and z respectively.
The relationship between these 6 parameters yield
7 types of unit cell (and only 7) called crystal
systems a b c ? ? ? 90 cubic a b
c ? ? ? ? 90 trigonal a b c ? ?
90 ? 120 hexagonal a b ? c ? ? ?
90 tetragonal a ? b ? c ? ? ?
90 orthorhombic a ? b ? c ? ? 90 ? ?
monoclinic a ? b ? c ? ? ? ? ?
triclinic
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Choosing the unit cell
Convention is to choose unit cell whose shape
displays the full symmetry (rotational and
translational) of the crystal lattice and that
is most convenient (axial lengths may be shortest
possible and angles near as possible to 90)
There are 14 possible crystal lattices (Bravais
lattices) (combination of 7 crystal systems and 4
packing modes (P, I, F, C)
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Bravais lattices

• Not all lattice points need coincide with unit
  cell vertices. Primitive unit cells .
  Non-primitive unit cells, however, contain extra
  lattice points not at the corners.
• The 14 Bravais lattices are arrived at by
  combining one of the seven crystal systems (or
  axial systems) with one of the lattice
  centerings.
• end-centered an extra lattice point is
  centered in each of two opposing faces of the
  cell - eg. C centering
• face-centered an extra lattice point is
  centered in every face of the cell - F
• body-centered an extra lattice point is
  centered in the exact middle of the cell - I

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Asymmetric unit and unit cell
Unit cell The smallest volume from which the
entire crystal can be constructed by translation
only.
Asymmetric Unit The smallest volume from which
the unit cell can be constructed by application
of the crystallographic symmetry.
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Crystallographic Symmetry Symmetry operators,
such as rotation axes, that apply over the entire
crystal.
Non-crystallographic Symmetry (NCS) also called
local symmetry Symmetry operators that apply to
a local region of the crystal, but do not apply
over the entire crystal. For example, two
molecules in an asymmetric unit may be related to
each other by an NCS 2-fold, but the same
operation will not superimpose more distant parts
of the structure onto equivalent sites. NCS
elements can include rotation axis that are not
compatible with translational symmetry, such as
five-fold axes.
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Asymmetric Unit The smallest volume from which
the unit cell can be constructed by application
of the crystallographic symmetry.
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Symmetry Operators and Elements
Symmetry Operator an operation that leaves the
structure unchanged.
Apart from the identity and translational
symmetry, protein crystals can only contain the
following symmetry elements Proper rotation
Rotate by 360/n. Screw rotation Rotate by
360/n translate by d(m/n) d unit cell edge.
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Example of a 2-fold screw axis.
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Space Groups Because crystallographic symmetry
must be compatible with translational symmetry
(i.e. a crystal), symmetry elements can only
occur in certain combinations.
Combinations of symmetry elements that are
compatible with translational symmetry in three
dimensions are called space groups. The figure
illustrates plane group P2. Assuming that the
third unit cell axis was normal to the page, this
would be a projection of Space Group P2.
There are 230 space groups. Because protein and
nucleic acid molecules are chiral, there are only
65 biological space groups.
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Space groups
Space groups are listed in International Tables
for X-ray Crystallography (Vol. A)
Once space group is determined, only the
structure of the contents of the asymmetric unit
need to be determined. Centring of the lattice
or the presence screw symmetry elements can
result in Systematic absences in diffraction
pattern. Use to identify precise space group
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P1

• Simplest space group

• No symmetry except crystal lattice translations

• Triclinic

P1
Highest rotational symmetry 1-fold (360, ie.
no rot. sym.)
Primitive unit cell

• The whole unit cell forms the asymmetric unit.

• Origin may be placed wherever convenient

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P2

• 2-fold axis at the origin (0,y,0)
  creates 2 asymmetric units

• Monoclinic

• Operation of lattice creates 3 more 2-fold
  axes

• Convention calls cell axis // to 2-fold
  axis b.

P2
Highest rotational symmetry 2-foldtransforms
(x,y,z) to (-x, y, -z) equivalent positions
Primitive unit cell
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P222

• Orthorhombic

• asymmetric unit is ¼ unit cell

P222
3 perpendicular 2-fold axes
Primitive unit cell
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The Seven Crystal Systems The 230 space groups
can be grouped into seven crystal systems
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Fractional Coordinates. Positions in the unit
cell are often given in fractional coordinates,
i.e. the full length along the a edge corresponds
to x 1.0. The fractional distances along b and
c y and z.
Because of lattice (translational) symmetry, the
coordinates x 0.5, x 1.5, x -0.5, are
identical.
Final coordinate files, such as from the PDB, are
given in orthogonal Å. (Which have a defined
relationship to the unit cell)
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Summary - terms

• Basic building block of a crystal is the unit
  cell - box, defined by three lengths a, b, c
  (one for each edge of the box) and three angles
  ?, ?, ? (between the axes b-c, a-c, and a-b,
  respectively), collectively referred to as
  lattice constants.
• The unique part of the unit cell is called the
  asymmetric unit containing 1 or more molecules.
  Latter related by NCS
• In order to describe a crystal, several symmetry
  elements may be combined and a particular
  combination of symmetry elements is called a
  space group - 65 possible for protein crystals.

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Conclusions

• The most common space groups observed for protein
  crystals are P212121 (primitive orthorhombic) and
  P21 (primitive monoclinic).
• Diffraction of X-rays by a crystal results in a
  pattern which is mathematically related to the
  pattern of the crystal lattice.
• One of the first steps in analysing diffraction
  patterns is to assign the crystal to its specific
  space group in a Bravais lattice with maximum
  symmetry.
• This analysis also determines the shape and
  dimensions of the unit cell which are important
  parameters in calculation of structure from
  crystallographic data.