Syllabus

1. Crystals

Principles of crystal growth

2. Symmetry

Unit cells, Symmetry elements, point groups and space groups

3. Diffraction

Introduction 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

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) International Tables for Crystallography, Volume A

Growing crystals

Proteins pack symmeterically within crystals

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)

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

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

Mirror symmetry is not allowed in biological macromolecules

Crystals of chiral molecules cannot contain mirror planes (centers of inversion)

symmetry

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

Symmetry in chiral molecules

Tetramer with 222 symmetry

This kind of tetramer (2-fold symmetry about each of three perpendicular directions) is often seen in proteins.

Crystal Symmetry Crystals are regular periodic arrays, i.e. they have long range translational symmetry. Crystals are often considered to have essentially infinite dimensions. a b 1 Unit Cell 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.

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, a–c, and a-b, respectively.

b

  

a c

Unit cells are usually defined in terms of the lengths of the three vectors and the three angles. For example, a=94.2Å, b=72.6Å, c=30.1Å,  =90°,  =102.1°,  =90°.

The crystal lattice

Lattice translation

+a y

b a

x

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 don ’t coincide with any atoms at all.

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

There are 32 distinct combinations of crystallographic symmetry operations relating to finite groups = 32 point groups For chiral units, there are 11 point groups:

The point groups that can exist in protein crystals

Screw axis, n r

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, 2 1 :

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.

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) c a

  

b 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 a

=

b

=

c

=

c a

=

b a

=

b a



b

=

c



a



a



b b



c c

 

c c

      = = = = = =       =  = = 90 °   90 ° = 90;  = 120 ° = =        = 90 ° = 90 = 90 °   ° cubic trigonal hexagonal tetragonal orthorhombic monoclinic triclinic

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) systems and 4 packing modes (P, I, F, C) (combination of 7 crystal

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

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.

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.

NCS applies only here

Asymmetric Unit = The smallest volume from which the unit cell can be constructed by application of the crystallographic symmetry.

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.

Proper Rotations Symbol ( n ) Two-fold 2 Three-fold 3 Four-fold 4 Six-fold 6 Screw Rotations Symbol ( n m ) 2 1 3 1 , 3 2 4 1 , 4 2 , 4 3 6 1 , 6 2 , 6 3 , 6 4 , 6 5

Example of a 2-fold screw axis.

2 1

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.

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

P1

 Simplest space group  No symmetry except crystal lattice translations  Triclinic P1 Primitive unit cell Highest rotational symmetry : 1-fold (360 °, ie. no rot. sym.)  The whole unit cell forms the asymmetric unit.

 Origin may be placed wherever convenient

P2

 Monoclinic  Convention calls cell axis // to 2-fold axis

b.

P2  2-fold axis at the origin (0,

y

,0) creates 2 asymmetric units  Operation of lattice creates 3 more 2-fold axes Primitive unit cell Highest rotational symmetry :2-fold transforms (

x,y,z

) to (

-x, y, -z)

[equivalent positions]



P222

Orthorhombic  asymmetric unit is ¼ unit cell Primitive unit cell P222 3 perpendicular 2-fold axes

The Seven Crystal Systems The 230 space groups can be grouped into seven crystal systems

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.

a

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)

b

at x=0.5, y=0.5

at x=0.25, y=0.5

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 .

Conclusions

The most common space groups observed for protein crystals are P2 1 2 1 2 1 (primitive orthorhombic) and P2 1 (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.