Molecular Biology Course 2006 Protein Crystallography Part I Tim Grüne University of Göttingen Dept. of Structural Chemistry November 2006 http://shelx.uni-ac.gwdg.de tg@shelx.uni-ac.gwdg.de
Overview Overview 1/43
Molecular Biology Course 2006 Protein Crystallography I Tim Grüne Macromolecular Crystallography The Unit Cell interpretation of electron density... A macromolecular model is... and Crystal 6 - measured crystals from Introduction derived from diffraction... 2/43
Examples of Crystals Ice Crystals Salt Crystals from inorganic and especially ionic compounds (like N acl) tend to be rather stable. They can be kept at room temperature, for a long time and in a dry environment. Quartz Crystal Artificial Monocrystal for a kj Laser 3/43
Protein Crystals Proteins are generally more sensitive than minerals. They are usually grown from solution and must be kept in a humid environment throughout the experiment. Protein Crystals grown in solution. If handled improperly, proteins quickly degenerate or aggregate 4/43
What are X rays? (Visible) light is composed of electromagnetic waves: every colour has a wavelength/energy X rays are the same, but with higher energy, i.e. shorter wavelength To distinguish objects which are a distance d apart, one must use light with a wavelength λ 2d. Typical bond lengths (C C) are about 1.5Å, so light with λ 3Å is required: X rays. The spectrum of electromagnetic waves wavelength λ Radio 4.1µeV Micro 1.25 ev Infrared 1.25 kev 125 kev 1.6 3.1eV visible UV X-rays γ-rays energy 10km 10 13 Å 30cm 3 10 9 Å 1mm 10 7 Å 800-400nm 8 4 10 3 Å 1nm 10Å 10pm 0.1Å Typical X-ray experiment: 1Å X rays diffraction 5/43
Principle of an X ray experiment X-ray beam λ 1Å (0.1nm) crystal (0.2mm) 3 10 13 unit cells Diffraction pattern T. Schneider The crystal diffracts in all directions. Since the detector cannot cover the full sphere around the crystal, the crystal is rotated during an experiment. One typical data set consists of tens to hundreds of images and the crystal is rotated for between 90 and 360. X rays diffraction 6/43
Molecular Biology Course 2006 Protein Crystallography I Tim Grüne X ray sources Rotating Anode Inhouse Equipment: The SMART 6000 The Unit Cell and Crystal Cryostream (100K) Detector Beam stop Rotating Anode Focussing unit Crystal hold (goniometer head) G. Sheldrick A beam of electrons hits a (rotating) Copper plate. This induces a transition of the inner shell electrons which in turn produces characteristic CuKα radiation (1.54 Å) X rays diffraction 7/43
X ray sources Synchrotron The ESRF (European Synchrotron Radiation Facility) Grenoble G. Sheldrick Bunches of electrons are circulating at high velocity and bent by strong magnets. This generates a continuous spectrum of X rays. At the hutch (where the experiment takes place), one particular wavelength is filtered out. Some set-ups allow to tune the wavelength (important for MAD and SAD phasing, see later in the lecture). X rays diffraction 8/43
Diffraction Pattern visible light 00 11 01 00 11 000 11100 01 111000 111000 00000 11111 111000 00000 11111 111000 00000 11111 111000 00000 11111 000 11100 00000 11111 1100 00000 11111 00 110 1 00000 11111 00000 11111 00 11 object (focussing) lense image Screen Light is scattered from an object in all directions. A lense (e.g., eye lense, camera, microscope, or telescope) collects some of the scattered light and focusses it on a screen (and eventually the retina). This creates an image of the object. X rays diffraction 9/43
Diffraction Pattern Screen X-rays object no lenses for X-rays no image - "blur" For X-rays, lenses do not exist, therefore one cannot create an X-ray image. With a normal object, no information would be collected on the screen. What is special about crystals so that one can retrieve information through X-rays from them? X rays diffraction 10/43
Diffraction Pattern X-ray source beam crystal recorded image When irradiated with X rays, crystals produce a typical pattern of localised spots. This is because of the regular structure of crystal. This does not replace a lense the molecule cannot be seen directly. But from the pattern one can deduce what s inside the crystal. X rays diffraction 11/43
Regular Composition of Crystals A crystal consists of a unit cell which is piled up like bricks (but in three directions) to compose the whole crystal. The unit cell is characterised by the three side lengths, a, b, c and angles α, β, γ. c γ α: angle between b and c β: angle between c and a γ: angle between a and b α β b a Irrespective of the lengths and angles a unit cell can always be piled without gaps. The Unit Cell 12/43
Unit Cell where is the molecule? b a The unit cell is only a theoretical concept the molecule is real, but there is no box in the crystal. However, if we start at one atom of one molecule and translate it by, say, axis a, we end up on the same atom of a second molecule. The Unit Cell 13/43
The Crystal In order to understand the diffraction pattern of crystals, one introduces the concept of crystal planes. The corner points of all unit cells create the crystal lattice. Three non linear lattice points (corners of different unit cells) define a plane. (2 5) (1 1) b a One set of planes is described by the number of section it divides each side of the unit cell into (in the figure, side a is divided into two parts by the green set, and b into five parts). These numbers are called the Miller-Indices (hkl) (three in three dimensions). I.e., the green set of planes has the Miller Indices (2 5 0), the purple one the Indices (110). The Unit Cell 14/43
Bragg Reflection and Bragg s Law X-rays are reflected at the lattice planes. For every set of parallel planes there is a certain angle θ at which constructive interference occurs, and a signal is emitted. At other angles, destructive interference ruins any detectable signal. X-rays θ One set of planes inside the crystal d The angle θ, the plane separation d and the wavelength λ of the incident X-rays are related by Bragg s law or the Bragg condition a : λ = 2d sin θ Every plane can give rise to one reflection on the detector. Therefore one observes the spotty pattern. a Remark: Bragg s law can be derived from the above picture by using the fact that the path difference between rays reflected from two adjacent planes must be an integer multiple of the wavelength. This leads to the actual law nλ = 2d sin θ, but higher order reflections (n > 1) are generally too weak to be detected. The Unit Cell 15/43
Indexing Given the unit cell dimensions a, b, c, α, β, γ and the orientation of the unit cell relative to the lab coordinate system, one can label each spot on the detector with the Miller index (hkl) that gave rise to the reflection. This assignment of one Miller index to every reflection on the detector is called indexing. (hkl) recorded image beam 2θ max crystal The distance d which can be calculated via θ from Bragg s law is called the resolution of that reflection. The resolution of the reflection with maximal θ max determines the resolution of the data set. The Unit Cell 16/43
The Meaning of Resolution The resolution of data collected by X-ray diffraction is a measure for how much detail can be seen. It is related with the plane distance d via Bragg s law by λ d = 2 sin θ max θ max is the maximum angle to which data (i.e. a reasonable number of reflections) could be collected. The final goal of data collection is to calculate an electron density map for the crystallised molecule. The resolution is closely related to the minimum distance between two atoms that can still be resolved in the electron density map. high resolution density map low resolution density map d d The Unit Cell 17/43
Examples for the Resolution of Electron Density Maps Low Resolution Medium Resolution High Resolution G. Sheldrick The images show three times the same region of a protein map at different resolutions. N.B.: Crystallographers usually speak of high resolution when the number is small (e.g. 1.2Å) and of low resolution when the number is large (e.g. 3Å). The Unit Cell 18/43
The Benefits of using Crystal Symmetry It often happens that there are several copies of the same molecules inside the box. To move one molecule to overlap with the next one, one can apply a symmetry operation. If this is the case, two advantages can be drawn from this: 1. One only needs to build one instead of, say, four identical molecules. 2. The symmetry also applies to the reflections measured from the crystal. This means that the data can be determined with higher accuracy. A major part of crystallography deals with symmetry and its effect on the reflections. 19/43
The Seven Types Seven different categories of boxes or unit cells can be distinguished. There is no other type that can be stacked without gaps. α = β = γ = 90 cubic tetragonal orthorhombic b γ c α β a a = b = c a = b c a b c hexagonal trigonal monoclinic triclinic a = b c α = β = 90 γ = 120 a = b = c α = β = 90 γ = 120 a b c α = γ = 90 β a b c α β γ Triclinic is the most general one (six numbers required for characterisation), whereas cubic is the most special one (only one number). 20/43
The Seven Types Type Restrictions Sides Restrictions Angles triclinic none none monoclinic none α = γ = 90 trigonal a = b = c α = β = 90, γ = 120 hexagonal a = b α = β = 90, γ = 120 orthorhombic none α = β = γ = 90 tetragonal a = b α = β = γ = 90 cubic a = b = c α = β = γ = 90 21/43
Symmetry Operations The composition of unit cells implies translational symmetry: If we were inside a crystal and were moved by a translation h a + k b + l c we would not see any difference. Most crystals show additional symmetry elements. Examples of such symmetry elements are: rotations ( only 2-, 3-, 4- and 6-fold axes possible) 2-fold 3-fold 4-fold 6-fold mirrors and inversion centres mirror plane centre of inversion 22/43
Combining s with Symmetry One can combine the symmetry element (i.e. symmetry operations) with each other. One can apply (but not arbitrarily) the symmetry operations to the lattices. The combination of one or several symmetry element with a lattice type is called a space group. Not every symmetry operation can be combined with every lattice type, e.g., the cube does not have a 6 fold rotational axis. Altogether, there are 230 space groups. macromolecules like proteins and DNA or RNA are chiral. Therefore, crystals from these molecules cannot possess an inversion centre or a mirror plane. This leave 65 chiral space groups for macromolecular crystals. 23/43
Combining s with Symmetry The screw axis is one example for the combination of symmetry operators. top view side view The figure shows an example of a 4 1 screw axis: A rotation by 1/4 360, i.e. 90, is combined with a translation of 1/4 of the length of the unit cell axis along which the screw axis runs. After four such screws, one comes to a point in the next unit cell which is the starting point translated by the cell axis. 24/43
The Asymmetric Unit The unit cell was sufficient to create the whole crystal (better: the whole crystal lattice) by translating it in all directions. The asymmetric unit is sufficient to create the whole lattice if we use both translations and symmetry operators. The asymmetric unit is always smaller than the unit cell: If a crystal has one single 4 fold axis, the volume of the asymmetric unit is 1/4th of the volume of the unit cell. 25/43
The Asymmetric Unit A real world example of a DNA oligo a that crystallised in space group P6 ( hexagonal, one 6 fold axis) with the cell a = b = 53.77Å c = 34.35Å (and α = β = 90, γ = 120 ). DNA often oligomerises with its axis parallel to a crystallographic axis. a d(m5cgggm5cg) + d(m5cgccm5cg) Z-DNA, PDB-ID 145D 26/43
Data Collection Crystals and their lattices are three dimensional objects, but detectors are only two dimensional. Therefore, data are recorded in frames. ω rotation 0 1 1 2 179 180 X-ray beam φ rotation Typical frame widths range from 0.2 1. For a 180 scan, this gives 180 720 images for each rotation axis. This is typical for proteins that diffract to moderate resolution. A more thorough data collection rotates the crystal about two axes. One easily ends up with a few thousand image. Data Collection, Processing, and Scaling 27/43
Data Processing Data Collection results in a (large) number of images. Each one representing a small wedge at the rotation of the crystal in the beam. Since the reflections are projected on a plane (alias the detector), they are distorted. Data Processing has to reconstitute the original, undistorted 3 dimensional lattice of reflections. The result usually is a list with one reflection per line: det. coord s H K L Intensity error x y z[ ] -3 0-3 4,162 153.7 1181.5 1235.6 107.4-3 -3 0 2,747 107.5 1110.9 1205.1 76.0-3 0 3 3,946 145.1 1156.2 1233.4 18.3 1 1-4 5,933 213.9 1215.0 1226.7 165.0 4 1-1 5,640 206.4 1209.5 1074.0 57.3 The x and y coordinates are coordinates on the detector. The z coordinate is the rotation angle of the crystal relative to its starting position. Data Collection, Processing, and Scaling 28/43
Data Scaling After data processing the data must be scaled. There are two types of scaling: 1. Scaling of reflections within one dataset this must always be done: The intensity of the beam might change from frame to frame especially at synchrotron sources The crystal might be larger in one direction than another spot intensity increases with the volume the beam traversed. Different regions of the detector have different sensitivity 2. Scaling of reflections of different datasets, e.g. from two different crystals or from two different detector positions. This is only necessary if two or more datasets exist. Both types of scaling take advantage of reflections which are symmetry related. Symmetry related reflections should have the same intensity. Data Collection, Processing, and Scaling 29/43
Friedel s Law Even in the simplest space group (P1) with no symmetries, scaling can be carried out because of Friedel s law: Reflections with negated indices, i.e., (h, k, l) and ( h, k, l) have the same intensity. They arise from reflection at the same set of planes, but on opposite sides. X-rays Before rotation of crystal, i.e. rotation of lattice leads to reflection (h k l) top bottom bottom After rotation of crystal, i.e. rotation of lattice X-rays top leads to reflection ( h k l) Data Collection, Processing, and Scaling 30/43
The Relationship between Reflections and Electron Density With a diffraction experiment one measures intensities of reflections The intensities I are related to the Structure Factors F a : I(hkl) = F (hkl) 2 The Structure Factors are related to the electron density ρ at the point (x, y, z) inside the crystal by a Fourier transform: ρ(x, y, z) = 1 V unit cell h,k,l F (hkl) e iφ(hkl) e 2πi(hx+ky+lz) The amplitudes of the structure factors, F (hkl), are the results of the experiment. To calculate the electron density, one also needs the phases φ(hkl). a NB: F is a complex number, i.e. generally F I 31/43
The intensities from a diffraction experiment yield the structure factor amplitudes F (hkl). The phases φ(hkl) cannot be extracted. This is known as the phase problem. yields F (h, k, l), but not φ(h, k, l) The loss of the phase can be compared with a projection on a plane wall: The eye may see a three dimensional object but which face points forward? 32/43
The Importance of Phases The phase of the structure factor contains the main information about the shape of the molecule. F (h, k, l), φ(h, k, l) inverse FT φ(h, k, l) FT inverse FT F (h, k, l) The phase φ of the duck determines the picture F (h, k, l), φ(h, k, l) pictures from http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html 33/43
Techniques for Retrieving Phases Overview The phase problem makes it necessary to recover the phase information by indirect means. It is one of the major efforts of macromolecular crystallography to determine good phases. 1. direct methods (small molecules and high resolution only) 2. molecular replacement 3. isomorphous replacement 4. anomalous dispersion 5. exploitation of radiation damage Phasing 34/43
Method 1: Direct Methods Brute Force With small molecules ( <1000 unique atoms) and high resolution data (better than 1.2Å), random starting phases are assigned to all reflections. The starting phases are optimised using the assumption that the structure consists of resolved atoms. This assumption imposes statistical restraints on the phase probability distribution. With more than 1000 atoms the problem cannot be solved in a reasonable amount of time. The Patterson Function Very small structures can also be solved by interpreting the Patterson function. The Patterson Function is a Fourier transform based on intensities rather than structure factors, i.e., it can be calculated from experimental data. A Peak of the Patterson function corresponds to a vector connecting two atoms in the structure. For too many atoms, the peaks of the Patterson function come too close to be interpreted. Phasing 35/43
Method 2: Molecular Replacement By November 2004, the PDB, the Protein Data Base (http://www.rcsb.org/pdb), held more than 28,000 structures, both from X-ray crystallography and NMR. Therefore only very few of newly deposited structures reveal a new fold. Sequence homology between two proteins normally also implies structural similarity, and therefore chances are good that a new structure is similar to an already determined one. For successful molecular replacement, sequence homology of at least 30% is required. The problem with molecular replacement is that one has to find the correct placement of the search model within the asymmetric unit. This is usually performed in two steps Rotational search The search model is rotated. At every rotation, the Patterson function calculated from the model is compared with the Patterson function calculated from the data. This exploits that the Patterson function is independent from the translational position of the model. Translational search For the best fitting rotation, one translates the model and compares the calculated reflection pattern with the measured one. Phasing 36/43
Method 3: Isomorphous Replacement Isomorphous Replacement is one of the oldest methods of phasing for protein structures. It is based on the idea that introduction of a small molecule into a protein or nucleic acid crystal does not or hardly alter the structure of the macromolecule. On the other hand, a few heavy metal atoms can contribute detectably to the structure factors and hence introduce changes in the reflection intensities. Common heavy metals are Hg (80e ), Pb (82e ), Au (79e ), Pt (78e ), or U (92e ). They can be incorporated by co-crystallisation or by soaking after the crystals have grown. The first protein structures like myoglobin or hemoglobin were solved by isomorphous replacement, making use of the iron in the heme cluster. G. Sheldrick Phasing 37/43
Method 3: Isomorphous Replacement In order to use the extra information, one needs at least two data sets: a native one (no heavy metal) and a derivative (with heavy metal). derivative: F T difference co-ordinates Harker- construction F H, φ H F T, φ T native: F P The co-ordinates of the heavy metal(s) can be derived via either direct methods or Patterson methods. From the co-ordinates one can calculate structure factors (amplitude and phase!) for the heavy atom(s). The phases for the derivative follow from the Harker construction. Phasing 38/43
The Harker Construction The Harker Construction solves the phases of a large structure from the phases and amplitudes of a small subset of the structure in a geometrical way. With one native dataset and one derivative, the Harker constructions results in a twofold ambiguity for the phases of the protein: 1. Draw a circle with radius F T 2. Draw the vector for the heavy atom, F H, φ H 3. From its endpoint, draw a circle with radius F P The two circles have two points of intersection from which one reads the two possible phases φ T for the derivative or ( drawing the vector from the endpoint of the heavy atom) the native structure φ P. F T Im F T, φ T F H, φ H F P Re With only one derivative, one speaks of SIR, single isomorphous replacement, with more than one, one speaks of MIR, multiple isomorphous replacement. MIR removes the ambiguity of SIR. The more derivatives, the better the phases (and their errors) can be determined. Phasing 39/43
Method 4: Anomalous Dispersion For a normal diffraction experiment, Friedel s law is valid, which states that the intensities of the reflection (h, k, l) and ( h, k, l) are equal and that the phases of the underlying structure factor have opposite signs, φ(h, k, l) = φ( h, k, l). In the case of anomalous dispersion, the wavelength of the experiment is chosen to cause a transition of electrons of heavy atoms present in the crystal. The absorption of X-rays by the heavy metals causes the break down of Friedel s Law. Mathematically this can be exploited in a very similar way to isomorphous replacement. Phasing 40/43
Method 4: Anomalous Dispersion MAD and SAD SAD or Single wavelength Anomalous Dispersion requires only one dataset, collected at one wavelength. MAD or Multi wavelength Anomalous Dispersion uses datasets collected at different wavelengths. The effect of anomalous dispersion varies strongly with wavelength. Like with SIR, SAD yields two possible solutions for the phases. The mean of the two is taken to calculate the electron density map. This seems like a very rough approximation, but is sufficient for a first approach. Phasing 41/43
Some exotic experimental phasing techniques RIP: Radiation Induced Phasing makes use of the formation of radicals due to radiation. They damage the molecule. Apart from rather random destruction, carboxyl groups are removed and disulphides destroyed. I.e. one dataset is collected before and after exposure to a high dose of X-rays. They serve as native and derivative dataset. Sulphur SAD exploitation of the very weak signal of native S (or P for nucleic acids). Requires very accurately measured data. Halide Soaking: The crystal is soaked in 1M KI or NaBr. The Iodide or Bromide ions go into solvent channels inside the crystal. This allows e.g. SAD or MAD phasing. Phasing 42/43
Examples for the Quality of Phases (Initial) centroid phases from SAD Resolved twofold ambiguity Final (refined) phases G.M. Sheldrick Phasing 43/43