Overview - Macromolecular Crystallography

Size: px

Start display at page:

Download "Overview - Macromolecular Crystallography"

Kristin Dickerson
5 years ago
Views:

1 Overview - Macromolecular Crystallography 1. Overexpression and crystallization 2. Crystal characterization and data collection 3. The diffraction experiment 4. Phase problem 1. MIR (Multiple Isomorphous Replacement) - No homologous structure known - Incorporate heavy atoms - Locate bound heavy atoms - Phase refinement 2. MAD (Multiwavelength Anomalous Diffraction) - No homologous structure known - Incorporate anomalous scatterers - Locate anomalous scatterers - Phase refinement 3. MR (Molecular Replacement) - Requires homologous structure - Rotation function (determine orientation of known molecule in unknown crystal) - Translation function (determine position of search molecule in unknown crystal) - Rigid body refinement 5. Phase improvement 6. Model building 7. Refinement

2 2a. Crystal Characterization 2b. Data Collection

3 Crystal Mounting Cold N 2 Gas Capillary-mounted crystal for room temperature data collection Loop-mounted crystal for cryogenic data collection

$Diffraction$ Visualization

4 Diffraction Photograph Visualization of the reciprocal lattice

5 Asymmetric Unit, Unit Cell and Crystal Asymmetric Units 2-fold Axis of Symmetry

6 N-fold Axis of Rotation 2 422

7 Icosahedral symmetry

8 Molecular Symmetry I C2 or 2 C3 or 3

9 Molecular Symmetry II D2 / 222 D3 / 32

10 2-Fold and 2 1 Screw Axis

11 Mirror Planes and Inversion Centers Mirror plane Inversion center

12 Point Groups I Molecular symmetries are described by point groups There are 32 point groups, which are relevant for crystallography (those containing 2-, 3-, 4- and 6-fold axes of symmetry) Additional point groups are possible (e.g. 52 or D 5 ), but not as crystallographic symmetry operations This does not mean that a pentamer with 5-fold symmetry cannot be crystallized, it only means that the five-fold axis of rotation cannot be a crystallographic symmetry element

13 Point Groups II n n-fold axis of rotation n n-fold axis of rotation and inversion center n/m nm nm n-fold axis of rotation and perpendicular mirror plane n-fold axis of rotation and mirror plane containing n-fold axis n-fold axis of rotation, mirror plane containing the n-fold axis and inversion center n2 n-fold axis of rotation and perpendicular 2-fold n/mm n-fold axis of rotation, perpendicular mirror plane and mirror plane containing n-fold axis

14 Cubic System

15 Simple Lattices I A 2-fold axis (1 2) and a lattice translation (2 3) generate a secondary 2-fold axis (1 3) Crystallographic symmetry operations are valid throughout the crystal (global symmetry)

16 Simple Lattices II A two-dimensional lattice with 2-fold symmetry axes (indicated by ) perpendicular to the plane of the figure and 2-fold screw axes (indicated by the half-arrows) in the plane.

17 Crystal Systems Crystal System Bravais Type Minimal Symmetry Properties Triclinic P None a b c, α β γ Monoclinic P, C One 2-fold axis β 90 Orthorhombic P, C, I, F Three perpendicular 2-fold axes α = β = γ = 90 Tetragonal P, I One 4-fold a = b c, α = β = γ = 90 Trigonal/Hexagonal P 3-fold or 6-fold a = b c, α = β = 90, γ = 120 Rhombohedral R 3-fold a = b = c, α = β = γ 90 Cubic P, I, F Four 3-folds a = b = c, α = β = γ = 90

18 Primitive and Centered Unit Cells c a b Primitive P B-centered B (could also be A- or C-centered) I-centered I F-centered F

19 Bravais Lattices 14 Bravais lattices Some crystal systems have only one lattice type (triclinic, trigonal/hexagonal and rhombohedral), some have two (monoclinic and tetragonal), cubic has three and orthorhombic four Rhombohedral Rhombohedral R can be set up either hexagonal or rhombohedral

20 Space Groups - Introduction Combination of 32 point groups with translational symmetry elements yields the 230 space groups All space groups are tabulated in the International Tables of Crystallography For chiral molecules (protein, DNA, RNA, etc.) only 65 space groups are possible, i.e. those space groups which do not contain mirror planes, glide planes or inversion centers Space group P is the most common space group found with proteins and will be discussed in more detail

21 Space groups - P P Hermann-Mauguin symbol D Point group Orthorhombic - Crystal system 19 - Space group number Space group diagrams - Projections onto the ab-, bc- and acplanes showing symmetry elements Cartoon diagram illustrating the symmetry operation acting on a molecule represented by the open circle(+, above plane/ -, below plane) Origin - Location of origin with respect to the symmetry elements Asymmetric unit - One choice of the asymmetric unit Symmetry operation - (1) Identity operation (2) 2 1 screw axis along z at a = 1/4 and b = 0

22 Space Group - P Symmetry operations: (1) x,y,z (2) -x+1/2, -y, z+1/2 (3) -x, y+1/2, -z+1/2 (4) x+1/2,-y+1/2,-z Reflection conditions: For reflections of type h00 only those with even h are observed. Reflections with odd h are forbidden by symmetry. Same for 0k0 and 00l

23 Asymmetric Unit, Unit Cell and Crystal

A Real Crystal Space group P2 1 2 1 2 1 4 asymmetric units Molecular boundaries Bovine pancreatic trypsin inhibitor (BPTI) Empty spaces

24 A Real Crystal Space group P asymmetric units Molecular boundaries Bovine pancreatic trypsin inhibitor (BPTI) Empty spaces between molecules: Macromolecular crystals consist of protein and solvent (30% - >80%). The Matthew s coefficient describes crystal packing.

25 Packing Analysis I Calculate unit cell volume: V = a ( b x c) = abc (1 - cos 2 α - cos 2 β - cos 2 γ + 2 cosα cosβ cosγ) 1/2 (if all angles = 90 : V = abc, if one angle (β) 90 : V= abc sin β) Calculate volume of asymmetric unit: V AU = V / n AU Calculate Matthew s coefficient: V M = V AU / M r The Matthew s coefficient is between Å 3 /Da Calculate solvent content percentage: ρ = / V M

26 Packing Analysis II The MogA protein (molecular mass 21,500 Da) crystallizes in the hexagonal space group P6 3 (six asymmetric units) with unit cell dimensions of a=b=66 Å, c=65 Å and γ=120. Calculate volume of unit cell: V= abc sin γ = Å 3 Calculate volume of asymmetric unit: V AU = V / 6 = Å 3 Calculate Matthew s coefficient: V M = V AU / M r = 1.9 Å 3 /Da Calculate solvent content percentage: ρ = / V M = 35% This is a tightly packed crystal

27 Packing Analysis III The MogA protein also crystallizes in orthorhombic space group P (4 asymmetric units) with unit cell dimensions of a=55 Å, b=71 Å and c=165 Å. The protein is present as a trimer in solution. Calculate volume of unit cell: V = abc = Å 3 Calculate volume of asymmetric unit: V AU = V / 4 = Å 3 Calculate Matthew s coefficient: 1 monomer: V M = 7.5 Å 3 /Da 2 monomers: V M = 3.75 Å 3 /Da 3 monomers: V M = 2.5 Å 3 /Da 4 monomers: V M = 1.9 Å 3 /Da 5 monomers: V M = 1.5 Å 3 /Da This crystal form contains a trimer in the asymmetric unit.

28 Packing Analysis IV How to reconcile the results? The protein is a trimer in solution and also in both crystal forms Hexagonal crystals: A monomer is present in the asymmetric unit A crystallographic 3-fold axis of symmetry generates the trimer Check the International Tables to find the 3-fold axis Orthorhombic crystals: A trimer is present in the asymmetric unit There is a local (non-crystallographic) 3-fold axis of symmetry The monomers within the trimer do not have to be identical The monomers within the trimer will be similar Non-crystallographic symmetry elements can be detected

29 2a. Crystal Characterization 2b. Data Collection

30 X-rays High-energy electromagnetic radiation (λ ~ 1 Å) Produced by x-ray generator or synchrotron In-house: rotating anode x-ray generator with copper anode (λ = Å) Synchrotron: Large scale research facilities DESY (Hamburg) BESSY (Berlin) ESRF (Grenoble) SLS (Villigen, Switzerland) Interatomic distances are on the order of 1 Å Interference effects

The structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures

The structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures Describing condensed phase structures Describing the structure of an isolated small molecule is easy to do Just specify the bond distances and angles How do we describe the structure of a condensed phase?