Day I: Crystals and X-Ray Diffraction

Transcription

1 Macromolecular Structure Determination I Day I: Crystals and X-Ray Diffraction Tim Grüne Dept. of Structural Chemistry, University of Göttingen September tg@shelx.uni-ac.gwdg.de Tim Grüne 1/80

2 Macromolecule A macromolecule is a protein or nucleic acid compound bigger than a couple of kda, i.e. a protein consisting of 50 or more residues. The term macromolecular also includes complexes, e.g. between a protein and a ligand or DNA and an antibiotic. Tim Grüne Methods for Structure Determination 2/80

3 Structure Structure Determination means the description of how something looks like. This is a very vague description, because it depends on the applied technique. A microscopist may describe the compartments inside a bacterial cell, e.g in terms of colour and composition. For an electron microscopist, structural information of a macromolecule consists mostly of its shape. For a crystallographer or an NMR spectroscopist, structure means the determination of the coordinates of the atoms a molecule or complex consists of. Tim Grüne Methods for Structure Determination 3/80

4 Methods for Structure Determination Some of the common methods for macromolecular structure determination: Method Sample Information Remarks X-ray Crystallography Crystal atom positions Neutron Diffraction Crystal atom positions detects H-atoms Electron Diffraction Crystal atom positions often only 2D information Nuclear Magnetic Res- Solution atom positions size limits onance Electron Microscopy Solution shape large complexes only These methods are complementary, i.e. the information they provide add to one another (even though some might regard NMR and X-ray crystallography as competitive). This course concentrates on X-ray Crystallography. Tim Grüne Methods for Structure Determination 4/80

5 Outline of X-ray Structure Determination Crystal Data growth collection Electron density map Refinement Phasing Data & building Deposition Validation Tim Grüne X-ray Structure Determination 5/80

6 Wat is en Kristall? Definition of a Crystal Macromolecular Structure Determination I The International Union of Crystallography (IUCr) defines a crystal as a solid material with an essentially discrete diffraction pattern. For this course it is easier to think of a crystal as one motif the unit cell containing the molecule or molecules which is repeated in all three directions without any gaps, like building a house from bricks. The sides of the bricks can have arbitrary lengths and the sides can be inclined. But all (crystallographic) bricks must be identical to each other. Tim Grüne Crystals 6/80

7 Crystal Types Tim Grüne Crystal Types 7/80

8 Crystal Types All matter is held together by electrostatic interaction, i.e. because of the attraction of positive and negative charge, also crystals. There are different sub-types of interaction. Those which are important for crystals can be classified as: 1. ionic 2. metallic 3. covalent bonds 4. van-der-waals interactions The categories are not distinct": there are compounds which belong to inbetween two categories. Tim Grüne Crystal Types 8/80

9 Ionic Crystals Ionic crystals are composed of negatively charged anions and positively charged cations. The net-charge of an ionic crystal is always 0e, otherwise the crystal would fly apart. NaCl is the simplest example for an ionic crystals: Na passes its outer shell electron to Cl, leaving a positively charged Na + -ion and a negatively charged Cl -ion. The total energy gain by this transition is 6.4eV. Tim Grüne Crystal Types 9/80

10 Metals The valence electrons dis- Al 13e Al 13e Al 13e Al 13e Al 13e Al 13e Electron lake (3 electrons per Al atom) Al 13e Al 13e sociate from the atom and are shared amongst all ionic bodies. The valence elctrons create an electron lake. This explains the high conductivity and elasticity of metals. Tim Grüne Crystal Types 10/80

11 Covalent Bonds (Usually) two atoms share their covalent electrons to fill their outer electron shell. E.g. C or Si have four electrons in their outer shell and can therefore have up to four bonding partners. This results in a rather complicated network in crystalline carbon and the mechanical stability of diamonds. Crystal packing of C (diamond) or Si. Tim Grüne Crystal Types 11/80

12 van-der-waals Interaction van-der-waals interaction is the main interaction for macromolecules, not only in crystals but also e.g. in the formation of oligomers in solution. It is based on the random or accidental displacement of electrons which creates a temporary electric field which propagates through adjacent molecules. A snapshot of a charge distribution three putative, aligned molecules which induces a temporary dipole moment by which the molecules attract each other. One moment later the charge distribution might look different again. Tim Grüne Crystal Types 12/80

13 Macromolecular Structure Determination I Interaction between Macromolecules and their Environment Hydrophobic patches negatively charged patches positively charged patches Surface charge distribu- Schematic view of a protein tion of the nucleosome Macromolecules are much more likely to aggregate than to crystallise. Tim Grüne Crystal Types 13/80

14 Crystal Growth Tim Grüne Crystal Growth 14/80

15 Growing Crystals Metals Solid metals are generally crystalline, so e.g. cooling molten metal results in crystalline metal. Salts Drying salt dissolved in water often results in crystals because of the strong ionic force Proteins are difficult to crystallise. Their natural solid state is a disordered aggregate, because the intermolecular forces are relatively weak and the large surface of the molecule allows many mutual (irregular) orientations. Tim Grüne Crystal Growth 15/80

16 Crystallisation Methods Macromolecules are usually crystallised by driving them out of solution by competition with precipitants for solvent molecules. Example: Precipitation with salt Common precipitants are salts e.g. (NH 4 ) 2 SO 4, NaCl, KH 2 PO 4 organic polymers mostly polyethylen glycol (PEG) alcohols e.g. isopropanol protein solubility salting in salting out salt concentration good for purification good for crystallisation Tim Grüne Crystal Growth 16/80

17 Phase Diagram Protein vs. Precipitant Simplified Phase diagram between precipitant and protein concentration. protein concentration soluble protein meta stable (growth) precipitant concentration solid (precipitation) labile (nucleation) Crystal growth occurs in the labile and mostly the metastable zone. Nucleation, i.e. the formation of the initial crystal seed, occurs in the labile zone. At too high protein and/or precipitant concentration, proteins aggregate and precipitate without forming crystals. Tim Grüne Crystal Growth 17/80

18 Crystallisation Conditions The phase diagram depends on many factors, e.g. ph (buffer) ionic strength (salt concentration) type of salt additive compounds temperature. For many (most) precipitants and conditions, the labile and metastable zone are virtually non-existant. The art of crystal growth consists of finding the right precipitant under the right conditions. Tim Grüne Crystal Growth 18/80

19 Crystallisation Methods The most common crystallisation methods are 1. vapour diffusion 2. liquid phase diffusion Tim Grüne Crystal Growth 19/80

20 Vapour Diffusion Protein sample: c Prot = 20mg/ml 20mM Tris ph=8.0 50mM NaCl 1µl 1µl Sealed chamber Reservoir solution: 100mM Hepes ph=7.0 25% PEG mM CaCl 2 drop at setup: c Prot =10mg/ml c = 12.5% PEG after equilibration: cprot =20mg/ml c PEG = 25% Tim Grüne Crystal Growth 20/80

21 Vapour Diffusion Protein sample: c Prot = 20mg/ml 20mM Tris ph=8.0 50mM NaCl 1µl 1µl Sealed chamber Reservoir solution: 100mM Hepes ph=7.0 25% PEG mM CaCl 2 drop at setup: c Prot =10mg/ml c = 12.5% PEG after equilibration: cprot =20mg/ml c PEG = 25% Tim Grüne Crystal Growth 21/80

22 Vapour Diffusion Protein sample: c Prot = 20mg/ml 20mM Tris ph=8.0 50mM NaCl 1µl 1µl Sealed chamber Reservoir solution: 100mM Hepes ph=7.0 25% PEG mM CaCl 2 drop at setup: c Prot =10mg/ml c = 12.5% PEG after equilibration: cprot =20mg/ml c PEG = 25% Tim Grüne Crystal Growth 22/80

23 Vapour Diffusion It is usually impossible to predict the conditions that will result in crystals of the macromolecule. Therefore one tests a large number of random conditions (matrix screen). The vapour diffusion method is the most popular crystallisation method because it is easy and fast to set up and has even been automatised to a large extent (1000 conditions in 1hr; manually only about 50 conditions per 1hr). Tim Grüne Crystal Growth 23/80

24 Liquid Phase Diffusion Dialysis membrane The MWCO of the dialysis membrane Reservoir solution O ring (seal) Protein sample Dialysis button must be smaller than the protein size. By exchanging the reservoir, the conditions can be very finely tuned. Awkward to set up, requires large amounts ( 5µl) of sample. Dialysis buttons are well suited to improve/ fine-tune known crystallisation conditions. Tim Grüne Crystal Growth 24/80

25 Further Reading: Crystallisation of Macromolecules Drenth, Principles of Protein X-Ray Crystallography (Springer, 2007) Rupp, Biomolecular Crystallography: Principles, Practice, and Application to Structural Biology (Garland Science, 2009) Documentation at Documentation at Tim Grüne Crystal Growth 25/80

26 X-Rays Tim Grüne X-Rays 26/80

27 X-rays: Electromagnetic Waves Like visible light, UV-radiation, or radiowaves, X-rays are electromagnetic waves. wavelength λ Infrared UV X-rays Radio Micro γ-rays energy 10km 30cm 1mm nm 1nm=10Å 10pm =0.1Å According to the formula E = h c, a wave with a long wavelength λ has low energy E and vice versa. λ The energy of X-rays lies usually between Å. Physicists measure the energy of electromagnetic waves in electronvolt, ev. 1eV = energy of one electron (or proton) accelerated through 1V. Tim Grüne X-Rays 27/80

28 X-rays: Electromagnetic Waves Name wavelength λ Name wavelength λ Radio waves 30cm 10km Microwaves 1mm 30cm Infrared 800nm 1mm Visible Light 400nm 800nm Ultra violet 1nm 400nm X-rays 10pm 1nm γ-rays <1pm (1nm = 10 Å, 10pm = 0.1 Å) Tim Grüne X-Rays 28/80

29 Why X-Rays? Why do we use X-rays for structure determination? As a rule of thumb, light can only used to visualise objects greater than at least half the wavelength of that particular light, e.g. visible light/ light microscopy (λ > 400nm) can only be used to see objects greater than 200nm. The typical distance between atoms in (macro)molecules is about 1.5 Å - 2 Å. Therefore the wavelength to investigate molecules must be below 4 Å. Typically X-rays between 0.5 Å and 2 Å are used for X-ray experiments with macromolecular crystals. Tim Grüne X-Rays 29/80

30 Carrying out an X-ray Experiment Detector X ray source X ray waves Crystal (sample) The X-ray from an X-ray source are filtered to a single wave- (diffraction) length (monochromatic X-rays) and focussed as much as possible. beamstop The crystal diffracts the X-rays which are collected as spots on the detector. Tim Grüne X-Rays 30/80

31 Macromolecular Structure Determination I Result of a Diffraction Experiment The reflections (= spots) are the data we seek to measure: Their position and their intensity. The dark ring stems from scattering of solvent in the crystal. It always lies between about 3 and 4 Å and can be used as rough guideline for the resolution of a diffraction image. However it hampers the quality of the data and one tries to reduce its intensity. The spots are the result of the interaction of the X-rays with the periodic nature of the crystal. Tim Grüne X-Rays 31/80

32 Light vs. X-rays visible light object (focussing) lense Screen image Lenses allow us to build microscopes, telescopes, to actually see (with our own eyes lenses). Tim Grüne X-Rays 32/80

33 Light vs. X-rays We are forced to use X-rays (wavelength λ = Å) because bond distances are in the range to 1.5 Å. object X rays Screen no lense = no image, only "blur" Lenses for X-rays do not exist. Therefore, X-rays cannot be focussed as light can and there are not microscopes for X-rays. Otherwise, we could look at single molecules under a microscope. Tim Grüne X-Rays 33/80

34 Macromolecular Structure Determination I Crystals and X-rays The blur contains no useful information that could help us reconstruct the image of the tree. This changes in the case of crystals: Their periodic composition made up of myriads of unit cells causes spots (reflections) to appear on top of the blur. How this happens will be explained later during this course. Tim Grüne X-Rays 34/80

35 Generation of X-rays There are two main methods to generate X-rays for crystallographic purposes: Inhouse sources like rotating anodes. micro sources, or sealed tubes. A beam of electrons directed at a heavy metal anode initiates the transition of inner shell electrons. Synchrotrons Bending of Electron Beam: An electron beam forced by a magnetic field to drive a curve generate X-rays. This principle is exploited at Synchrotrons. Tim Grüne X-Rays 35/80

36 Rotating Anodes Hitting metal (Cu, Mo, Cr,... ) with electrons generates two types of radiation: 1. bremsstrahlung due to the deceleration of electrons 2. radiation due to shell transitions, usually from L to K. The metal is an anode because it is positively charged to attract the electrons. It is rotating because this facilitates cooling of the anode. Images courtesy of Jan-Olof Lill Tim Grüne X-Rays 36/80

37 Rotating Anodes tube Rh-spectrum The bremsstrahlung creates a broad spectrum at medium intensity. Intensity K β K α The shell transitions create sharp peaks at high intensity. The main peak is filtered from the rest and used for the measurement as monochromatic light. Wavelength [pm] Tim Grüne X-Rays 37/80

38 Macromolecular Structure Determination I Typical Inhouse Machine Tim Grüne X-Rays 38/80

39 Generation of X-rays: Rotating Anode The wavelength generated from rotating anodes is exact and fixed. It can only be modified by exchanging the type of heavy metal in use. Some common metals and their wavelengths: Metal wavelength λ Copper Cu Å high intensity Molybdenum Mo Å small molecules (higher resolution) Silver Ag Å charge density Tungsten W Å medical applications Tim Grüne X-rays 39/80

40 Generation of X-rays: Synchrotrons electrons e N N e S S e Magnets Vacuum tube e Beamlines "Light" (from Infrared to X Rays) Electrons are circled inside a vacuum tube. At bends they generate a wide spectrum of electro-magnetic radiation, from infrared to X-rays. The beamlines (experimental stations) the desired wavelength is selected. Tim Grüne X-rays 40/80

41 Synchrotron vs. Inhouse + Synchrotron radiation is much stronger than inhouse sources. A full data set can be collected in minutes as opposed to hours or days with an inhouse source. + Synchrotrons allow to select (tune) the wavelength. This is important for the phasing step. - Inhouse sources are often more stable and deliver more accurate data. - Inhouse sources often allow more advanced settings of crystal and detector with respect to each other, resulting in higher data quality (but not higher resolution data). Tim Grüne X-rays 41/80

42 Cryo-Crystallography Tim Grüne Cryo-Crystallography 42/80

43 Cryo-Crystallography The quality of data measured from X-ray crystallography has been greatly improved with the introduction of cryo-crystallography. The crystals are cooled to 100K (or less) during data collection. Tim Grüne Cryo-Crystallography 43/80

44 Room Temperature Measurement: Capillary Radiation damage by beam E. Garman & T.R. Schneider, Macromolecular Cryocrystallography, J. Appl. Cryst. (1997). 30, At room temperature the crystal must be kept in a humid atmosphere and is therefore mounted in a glass capillary. Tim Grüne Cryo-Crystallography 44/80

45 Reasons for Cryo-Crystallography Crystal with visible consequences of radiation damage after data collection at a synchrotron. From E. Garman, Radiation damage in macromolecular crystallography: what is it and why should we care?, Acta Cryst. D66 (2010), p. 339 Tim Grüne Cryo-Crystallography 45/80

46 Reasons for Cryo-Crystallography Radiation causes radiation damage, i.e. the breaking of covalent bonds and the generation of free radicals. This degrades the crystal. Radiation damage is not removed but at least greatly reduced at 100 K compared to room temperature. The thermal motion of the atoms is reduced. Thermal motion (vibration of the atoms) causes a smearing of the reflections, they can be measured less accurately better data quality. Weak reflections (at high resolution) are buried in the background noise if the more they are spread out higher resolution data. Sample preparation is actually easier when frozen than at room temperature at room temperature the crystal can slip more easily. Tim Grüne Cryo-Crystallography 46/80

47 Sample Preparation Macromolecular crystals always contain water. Water crystallises when it is frozen, and the ice crystal lattice would destroy the protein crystal (they are not compatible). Courtesy Stephen Curry, Imperial College London Therefore the formation of ice crystals must be prevented by the addition of a cryo-protectant. Tim Grüne Cryo-Crystallography 47/80

48 Sample Preparation 298K 120K, no cryo 120K, cryo Images from E. Garman & T.R. Schneider, Macromolecular Cryocrystallography, J. Appl. Cryst. (1997). 30, Tim Grüne Cryo-Crystallography 48/80

49 Sample Preparation Common cryo-protectants: glycerol PEG400 MPD sucrose 2,3-butanediol Na-malonate LiCl (2M) Required concentration ranges between 15% and 35%, depending on the composition of the mother liquor, and the minimum required amount should always be tested beforehand without a crystal. Tim Grüne Cryo-Crystallography 49/80

50 Further Reading: Freezing Crystals Rodgers, D.W., Practical Cryocrystallography, chapter 14 in Methods in Enzymology, Vol. 276A (1997) Garman, E.F. and Schneider, T.R., Macromolecular Cryocrystallography, J. Appl. (1997), 30, p. 211 Cryst. Tim Grüne Cryo-Crystallography 50/80

51 Diffraction Theory Tim Grüne Diffraction Theory 51/80

52 The Unit Cell The unit cell is the smallest unit from which we can form the crystal solely by translations (shifting). γ c β a α b The unit cell is characterised by the three side lengths, a, b, c and angles α, β, γ. α: angle between b and c β: angle between c and a γ: angle between a and b Tim Grüne Diffraction Theory 52/80

53 Unit Cell: an X-ray Amplifier We are going to see that the regular repetition of the unit cell causes a certain amplification of the X-rays that enables us to calculate the atom positions inside the unit cell. Tim Grüne Diffraction Theory 53/80

54 X-Ray meets Electron X ray source X ray waves electron The X-rays from the source are plane waves An ϑ electron in the crystal (sample) reacts to this incoming wave by emitting a spherical wave (travelling in all directions) of much weaker intensity. The wave intensity is distributed as 1 2 (1 + cos2 ϑ) around the electron, but this is not important for further understanding. Tim Grüne Diffraction Theory 54/80

55 Wave Emitted by the Electron The wave emitted by the electron is an electromagnetic wave. travels away from the electron. The electromagnetic field The description as wave is merely a mathematical trick to simplify the calculations. The observed intensity of the wave is the square of the amplitude. Therefore, a light-source does not flicker. Tim Grüne Diffraction Theory 55/80

56 Multiple Waves: Interference Multiple electrons emit one wave each. The resulting wave is again a wave, but this time it is more complicated. It is an interference pattern. In some directions the amplitude get stronger (constructive interference), but in some directions the amplitude stays 0 at all times (destructive interference). Note that the electrons are aligned in a regular pattern, just like the unit cells in a crystal. The more electrons there are the more destructive interference occurs and only certain directions remain where a signal can be detected. This is the origin of the distinct spots observed with an X-ray crystallography experiment. Tim Grüne Diffraction Theory 56/80

57 The Laue Conditions The Laue Conditions are the main tool to predict whether or not a crystal diffracts in a certain direction and are also the basis for the interpretation and measurement of diffraction data. Tim Grüne Diffraction Theory 57/80

58 The Laue Conditions Crystal and Unit Cell in some orientation Incoming X-rays at wavelength λ We want to find out if there is a reflection on the detector at the circled position: 1. Draw input vector with length 1/λ to centre of crystal incoming X rays in b a 2. Draw output vector with length 1/λ from centre of crystal to point on detector. 3. Scattering vector S = difference between out and in 4. The angle between input and output vector is called 2θ. θ is the scattering angle (the 2 is explained shortly). Detector Another point on the detector results in another scattering vector S. Tim Grüne Diffraction Theory 58/80

59 The Laue Conditions direction of observation Crystal and Unit Cell in some orientation Incoming X-rays at wavelength λ out (1/λ) 2θ We want to find out if there is a reflection on the detector at the circled position: 1. Draw input vector with length 1/λ to centre of crystal incoming X rays in(1/λ) b a 2. Draw output vector with length 1/λ from centre of crystal to point on detector. 3. Scattering vector S = difference between out and in 4. The angle between input and output vector is called 2θ. θ is the scattering angle (the 2 is explained shortly). Detector Another point on the detector results in another scattering vector S. Tim Grüne Diffraction Theory 59/80

60 The Laue Conditions direction of observation Crystal and Unit Cell in some orientation Incoming X-rays at wavelength λ We want to find out if there is a reflection on the detector at the circled position: 2θ 1. Draw input vector with length 1/λ to centre of crystal incoming X rays in b out a S 2. Draw output vector with length 1/λ from centre of crystal to point on detector. 3. Scattering vector S = difference between out and in 4. The angle between input and output vector is called 2θ. θ is the scattering angle (the 2 is explained shortly). Detector Another point on the detector results in another scattering vector S. Tim Grüne Diffraction Theory 60/80

61 The Laue Conditions direction of observation Crystal and Unit Cell in some orientation Incoming X-rays at wavelength λ 2θ We want to find out if there is a reflection on the detector at the circled position: out 1. Draw input vector with length 1/λ to centre of crystal incoming X rays in b a out S 2θ 2. Draw output vector with length 1/λ from centre of crystal to point on detector. 3. Scattering vector S = difference between out and in Detector another direction of observation 4. The angle between input and output vector is called 2θ. θ is the scattering angle (the 2 is explained shortly). Another point on the detector results in another scattering vector S. Tim Grüne Diffraction Theory 61/80

62 Macromolecular Structure Determination I The Laue Conditions ~ carries information about the direction of the incoming beam, the The scattering vector S wavelength λ and the position on the detector we are interested in. The unit cell vectors ~a, ~b, ~c define how the unit cell is oriented with respect to the incoming beam. There is a reflection spot on the detector at the ~ only position described by the scattering vector S if there are three integers h, k, l such that: ~ a cos( (S, ~ ~a)) = h 1. S ~ ~ ~b cos( (S, ~ ~b)) = k 2. S ~ c cos( (S, ~ ~c)) = l 3. S ~ Equations 1-3 are called the Laue Conditions. Tim Grüne Diffraction Theory 62/80

63 The Laue Conditions The Laue conditions are if-and-only-if conditions: There is a spot on the detector if the numbers h, k, l are all integers. Each integer triplet (h, k, l) corresponds to uniquely one reflection, because the vector S can be calculated from the Laue conditions. An integer triplet (h, k, l) is called the Miller index of the corresponding reflection. Tim Grüne Diffraction Theory 63/80

64 The origin of 2 in 2θ in b out 2θ θ θ in in out θ out S a By rotating the picture on the left by θ, the incoming and the outgoing wave vectors become much more symmetrical and the picture looks like a light-ray reflected by a mirror plane. Like in optics the θ in = θ out = θ. This also justifies the term reflection for the diffraction spots. Tim Grüne Diffraction Theory 64/80

65 Lattice Planes There is a connection between the aforementioned mirror plane and the Miller indices. Consider the crystal lattice with the unit cell highlighted in green: Pick one corner of the unit cell. Pick a corner from a second unit cell (in 3D, pick two other ones) Shift the line (plane) so that it hits all unit cell corners as long as it passes through the original unit cell. Tim Grüne Diffraction Theory 65/80

66 Lattice Planes There is a connection between the aforementioned mirror plane and the Miller indices. Consider the crystal lattice with the unit cell highlighted in green: Pick one corner of the unit cell. Pick a corner from a second unit cell (in 3D, pick two other ones) Shift the line (plane) so that it hits all unit cell corners as long as it passes through the original unit cell. Tim Grüne Diffraction Theory 66/80

67 Lattice Planes There is a connection between the aforementioned mirror plane and the Miller indices. Consider the crystal lattice with the unit cell highlighted in green: Pick one corner of the unit cell. Pick a corner from a second unit cell (in 3D: two other ones) Shift the line (plane) so that it hits all unit cell corners as long as it passes through the original unit cell. Tim Grüne Diffraction Theory 67/80

68 Lattice Planes There is a connection between the aforementioned mirror plane and the Miller indices. Consider the crystal lattice with the unit cell highlighted in green: Pick one corner of the unit cell. Pick a corner from a second unit cell (in 3D: two other ones) Shift the line (plane) so that it hits all unit cell corners as long as it passes through the original unit cell. Tim Grüne Diffraction Theory 68/80

69 Lattice Planes There is a connection between the aforementioned mirror plane and the Miller indices. Consider the crystal lattice with the unit cell highlighted in green: a b The planes divide the side a 1x, the b side 2x, and the c side 0x. The planes we thus constructed are the mirror planes for the reflection with the Miller index (1, 2, 0). From the incoming beam direction and the unit cell we could now predict the orientation of the crystal in the beam so that the reflection (1, 2, 0) can be collected. Tim Grüne Diffraction Theory 69/80

70 Lattice Planes For every such plane (which runs through three unit cell corners) there is a scattering vector S and integer Miller indices (hkl) which fulfill the Laue conditions. Any other plane never fulfills the Laue conditions. The construction also helps to understand the resolution limit of a realistic crystal (see later in this lecture). Tim Grüne Diffraction Theory 70/80

71 Bragg s Law Another important consequence from the Laue conditions is Bragg s Law: In order that the reflection that belongs to the purple lattice planes can be measured, the planes (and hence the crystal) must be oriented to the beam such that θ d λ = 2d sin θ a θ d : distance between two adjacent planes. called the resolution of the reflection. λ : wavelength of the X-rays It is a The exact law is nλ = 2d sin θ, but n > 1 corresponds to multiple refraction in the crystal and can usually be neglected. Tim Grüne Diffraction Theory 71/80

72 Spot Position and Intensity Bragg s law and the Laue conditions depend on the unit cell parameters a, b, c, but not the unit cell content, i.e. the molecule inside. The diffraction pattern tells us about the unit cell parameters a, b, c. The spot intensities tell us about what is inside the unit cell. Tim Grüne Diffraction Theory 72/80

73 Spot Position and Intensity Atoms A and its correspondinh atom A in the next B B d d unit cell are both on the plane (120) and contribute with their small waves to the spot (120). A A The shifted atoms B and B contribute to the same spot (the shift does not change the Laue conditions!). Depending on the small shift, the contribution interferes constructively or destructively and therefore changes the spot intensity: Its intensity changes depending on the number and positions of the atoms inside the unit cell, i.e. depending on the molecule in the unit cell. Tim Grüne Diffraction Theory 73/80

74 Resolution Limit: Theory and Practice Bragg s law λ = 2d sin θ sets a lower limit for the plane distance d that can be measured with a fixed wavelength λ: d = λ 2 sin θ λ 2 This assumes a perfectly ordered crystal. Unfortunately, the molecules inside the crystal do not know about crystallography and the concept of the unit cell (or they do and only want to tease you). Tim Grüne Diffraction Theory 74/80

75 Resolution Limit: Theory and Practice A small lattice distance d corresponds to a long-distance order of the unit cells. A realistic crystal, however, only as a limited order, and spots with a small lattice distance d are not formed beyond a certain limit, the resolution limit of the crystal. Tim Grüne Diffraction Theory 75/80

76 Resolution Limit: Theory and Practice A small lattice distance d corresponds to a long-distance order of the unit cells. A realistic crystal, however, only as a limited order, and spots with a small lattice distance d are not formed beyond a certain limit, the resolution limit of the crystal. Tim Grüne Diffraction Theory 76/80

77 Sample Images Resolution: 1.5 Å at edge Cell: a = 92.6Å, b = 92.6Å, c = 128.9Å, α = β = 90, γ = 120 sharp and small spots Some overloads (saturated counter) white bar: beam stop white lines: detector tiling Tim Grüne Diffraction Theory 77/80

78 Macromolecular Structure Determination I Sample Images Resolution: 2.5 Å at edge Cell: a = 111.7Å, b = 80.5Å, c = 70.3Å, α = γ = 90, β = 94.2 Smeared spots (very common) Ice rings (from cryo stream or poor freezing) Multiple lattices (twin) Tim Grüne Diffraction Theory 78/80

79 Sample Images Cell: a = 10.56Å, b = 11.64Å, c = 16.14Å, α = β = γ = 90 Small cell few (large) spots Tim Grüne Diffraction Theory 79/80

80 Further Reading: Diffraction Theory Drenth, Principles of Protein X-Ray Crystallography (Springer, 2007) T. L. Blundell & L. N. Johnson, Protein Crystallography (Academic Press London, 1976) Tim Grüne Diffraction Theory 80/80