1. Crystals Principles of crystal growth2. SymmetryUnit cells, Symmetry elements, point groups and space groups 3. DiffractionIntroduction to diffraction of wavesThe reciprocal latticeDiffraction by crystals; Bragg equation4. Obtaining the diffraction patternInstrumentsData collection strategies5. Deriving a trial structure - phase problemMolecular Replacement (MR)Isomorphous replacementAnomalous scattering (MAD methods)6. Refining the structureFourier and least-squares methods 7. Analysis of structural parameters

Syllabus

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

SymmetrySymmetry

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 SymmetryImportance 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 SymmetryTypes 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 Rotational symmetry of molecular oligomersmolecular 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.

1 Unit Cell

Crystal SymmetryCrystals 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.

ab

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.

a

b

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

ba x

y

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 don’t coincide with any atoms at all.

Crystal symmetryCrystal 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, 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.

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)

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

c

ba

Distances along a, b and c are referred to in terms of x, y and z respectively.

There are 14 possible crystal lattices (Bravais lattices) (combination of 7 crystal systems and 4 packing modes (P, I, F, C)

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°)

Bravais latticesBravais 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.

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.

Crystallographic Symmetry = Symmetry operators, such as rotation axes, that apply over the entire crystal.

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 ElementsSymmetry 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

Two-fold

Three-fold

Four-fold

Six-fold

Symbol (n)

2

3

4

6

Screw Rotations

Symbol (nm)

21

31, 32

41, 42, 43

61, 62, 63, 64, 65

Example of a 2-fold screw axis.

21

Space GroupsBecause 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

No symmetry except crystal lattice translations

P1

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

Triclinic

The whole unit cell forms the asymmetric unit.

Origin may be placed wherever convenient

Simplest space group

P2

P2

Primitive unit cell Highest rotational symmetry :2-foldtransforms (x,y,z) to (-x, y, -z) [equivalent positions]

Monoclinic

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

Operation of lattice creates 3 more 2-fold axes

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

P222

P222

Primitive unit cell 3 perpendicular 2-fold axes

Orthorhombic asymmetric unit is ¼ unit cell

The Seven Crystal SystemsThe 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

b

at x=0.5, y=0.5

at x=0.25, y=0.5

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)

Summary - terms 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.

ConclusionsConclusions

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.