1. Purify your protein 2. Crystallize protein 3. Collect diffraction data 4. Get experimental phases 5. Generate an electron density map 6. Build a model 7. Refine the model 8. Publish X-ray Crystallography BMB/Bi/Ch173 02/06/2017 Catalysis-dependent selenium incorporation and migration in the nitrogenase active site iron-molybdenum cofactor Spatzal, Perez, Howard & Rees elife 2015 Petsko and Ringe
Pauling s September 1953 Protein Conference in Pasadena Max Perutz, Vernon Shomaker, James Watson, Jack Dunitz, Julian Huxley, Francis Crick, Richard Marsh, Ken Trueblood, Maurice Huggins, Ray Pepinsky, Ken Palmer, John Rollet, Vitorio Luzzati, George Beadle, David Davies, Maurice Wilkins, John Kendrew, Alex Rich, Bea Magdoff, Maurry King, Linus Pauling, Robert Corey, David Harker, William Astbury, Richard Bear, William Bragg, Lindo Patterson, John Edsall, Francis O. Schmidt, John Randall, Barbara Low, I.F. Trotter
Satellite Tobacco Mosaic Virus (STMV) Crystal MacPherson
Useful texts and links Rhodes Crystallography made crystal clear Drenth Principles of protein X-ray crystallography Lattman & Loll Protein crystallography: a concise guide MacPherson Preparation and analysis of protein crystals Ed. Rossmann & Arnold Crystallography of biological macromolecules: volume F Bernhard Rupp Biomolecular Crystallography http://www.ruppweb.org/xray/101index.html
Nobel prizes related to X-ray Crystallography 1901 Physics (1 st ) Röntgen X-rays 1905 Physics von Lenard cathode rays 1914 Physics von Laue X-ray diffraction by crystals 1915 Physics Bragg & Bragg first crystal structure 1946 Chemistry Sumner First enzyme crystals 1962 Chemistry Perutz & Kendrew First protein structure 1962 Medicine Watson, Crick & Wilkins DNA structure 1964 Chemistry Hodgkin Protein crystallography 1976 Chemistry Lipscomb Borane structure (Rees mentor) 1982 Chemistry Klug Crystallographic EM 1985 Chemistry Hauptman & Karle Direct methods (Isabella credited) 1988 Chemistry Deisenhofer, Huber & Michel Photosynthetic reaction center 1997 Chemistry Agre & Walker F1 ATPase structure, aquaporins 2003 Chemistry MacKinnon Ion channel structures 2006 Chemistry Kornberg RNA polymerase structure 2009 Chemistry Ramakrishnan, Steitz and Yonath Ribosome structures 2012 Chemistry Leftkowitz & Kobilka - GPCRs
The international year of crystallography (2014)
X-ray crystallography Why X-rays? Right wavelength to resolve atoms Why crystal? Immobilize protein, enhance weak signal from scattering What is a protein crystal? Large solvent channels 20-80% solvent Same density as cytoplasm Are crystal structures valid compared with solution structures? Enzymes active in crystals Usually -- Compare NMR and x-ray structures Structures correlate with biological function Multiple crystal forms look same -- small effects of packing (flexible hinges can differ depending on packing)
Crystal lattices and symmetry y A crystal is a regular, 3-dimensional repeating array. The fundamental building block is the unit cell The crystal is built up by translations along x, y and z (which are not necessarily orthogonal) x z Note that x, y, z form a right-handed coordinate system
Unit cell Described by three vectors, a, b, and c, which are related by angles α, β and γ Lattice built up by translating the unit cell along each of the lattice vectors
Choice of unit cell Criteria for choosing a unit cell: There are many different cell choices for a lattice; I, II and III all constitute unit cells from which the entire lattice can be generated 1. Right-handed axis system 2. The basis vectors should coincide as much as possible with directions of highest symmetry 3. Cell should be smallest one that satisfies previous condition. This may mean the choice of a non-primitive unit cell. 4. a > b > c 5. Angles either all <90 or 90
Primitive and non-primitive unit cells Primitive cells have a single point in each unit cell (1/8 of each corner point) Non-primitive cells have more than one point in each unit cell
Contents of a unit cell A unit cell does not have to contain a full object but it must contain the sum of the repeating object The contents of a unit cell can have no intrinsic symmetry, or can contain objects related by symmetry operations.
Space filling repeats Forms closed lattice Can t fill 5-fold 2-fold 3-fold 7-fold 8-fold 4-fold 6-fold
Symmetry operations 12 1 1 (2 1 )screw axis m n The symmetry operator m: rotate 360 o /m along an axis to unit cell plane The screw axis m n rotate 360 o /m along an axis to unit cell plane translate n/m along unit cell Crystallographic symmetry operations describe the symmetry of the unit cell as well as of the entire crystal. Symmetry of 5 or >6 cannot be used to build a 3-dimensional lattice therefore they do not exist except in local symmetry
Examples of screw axis Rotate 360/m = 120 Translate n/m n/m unit cells = 1/3 360/m = 120 n/m unit cells = 2/3 360/m = 60 n/m unit cells = 1/6
Biological systems are limited in symmetry Mirror Plane Biological systems are chiral and can t generate mirror planes or inversion centers. Inversion center
Point groups in proteins
Space groups Crystal System Bravais Type Condition of geometry Triclinic P None 1 Minimum symmetry a complete description of a crystal lattice defines a unit cell type a set of symmetry operations 230 possible 3D space groups only 65 biological space groups due to chirality (no mirror or inversion centers) Crystal systems define classes of space groups Higher symmetry means less data needed for completeness but more molecules in the unit cell (larger cells ) Monoclinic P, C α=γ=90 1 2-fold parallel to b Orthrombic P, I, F α=β=γ=90 Three 2-folds Tetragonal P, I a=b; α=β=γ=90 1 4-fold parallel to c Trigonal P, R Hex: a=b; α=β=90; γ=120 Rhomb: a=b=c; α=β=γ 1 3-fold axis Hexagonal P a=b; α=β=90; γ=120 1 6-fold axis Cubic P, F, I a=b=c; α=β=γ=90 4 3-folds along diagonal Defined in excruciating detail in The International Tables for Crystallography volume A
65 biological space groups Lot s of possible space groups P2 1 2 1 2 1 and P2 1 most common Least restrictive packing Triclinic P 1 Monoclinic P 2 P 21 C 2 Orthorhombic P 2 2 2 P 2 2 21 P 21 21 2 P 21 21 21 C 2 2 21 C 2 2 2 F 2 2 2 I 2 2 2 I 21 21 21 Tetragonal P 4 P 41 P 42 P 43 I 4 I 41 P 4 2 2 P 4 21 2 P 41 2 2 P 41 21 2 P 42 2 2 P 42 21 2 P 43 2 2 P 43 21 2 I 4 2 2 I 41 2 2 Trigonal P 3 P 31 P 32 R 3 P 3 1 2 P 3 2 1 P 31 1 2 P 31 2 1 P 32 1 2 P 32 2 1 R 3 2 Hexagonal P 6 P 61 P 65 P 62 P 64 P 63 P 6 2 2 P 61 2 2 P 65 2 2 P 62 2 2 P 64 2 2 P 63 2 2 Cubic P 2 3 F 2 3 I 2 3 P 21 3 I 21 3 P 4 3 2 P 42 3 2 F 4 3 2 F 41 3 2 I 4 3 2 P 43 3 2 P 41 3 2 I 41 3 2 Wuckovic & Yeates (1995) NSB 2(12) 1062
Space groups Space groups define internal symmetry Given the space group and cell dimensions one can construct a unit cell Space group defines relationship of molecules P21 21 21 (19) Laue class mmm Orthrombic 4 Transformations X, Y, Z ½+X, ½-Y, -Z -X, ½+Y, ½-Z ½-X, -Y, ½+Z P31 2 1 (152) Laue class -3m1 Trigonal 6 Transformations X, Y, Z -Y, X-Y, ⅓+Z -X+Y, -X, ⅔+Z Y, X, -Z X-Y, -Y, ⅔-Z -X, -X+Y, ⅓-Z
The asymmetric unit P2 a a.u. 2-fold axis b a.u. The asymmetric unit is the smallest fraction of the unit cell lacking internal symmetry The unit cell can be built up by applying symmetry operations to the asymmetric unit Unit cell
MacPherson
Joseph Fourier Théorie analytique de la chaleur - 1822 The observation: a periodic function can be described as the sum of simple sine and cosine functions that have wavelengths as integrals of the function Also first predicted the Greenhouse Effect
Fourier Series A set of functions from a Fourier synthesis that can be summed to reproduce a periodic function f (x) = F cos2π(hx + α) s(t) cos2π(x) 1 cos2π(3x) 3 t 1 cos2π(5x) 5 Rhodes Crystallography Made Crystal Clear
Fourier Series of atoms f (x) = F cos2π(hx + α) A set of functions from a Fourier synthesis that can be summed to reproduce a periodic function C C O
How can we describe the diffraction pattern of a protein in a crystal? Because there is no lens to refocus x-rays, we have to understand reciprocal space. Diffraction: Scattering followed by interference
Diffraction by a wave Diffraction: deviation of light from rectilinear propagation, is a characteristic of wave phenomena which occurs when a portion of a wave front is obstructed in some way. When various portions of a wave front propagate past some obstacle, and interfere at a later point past the obstacle, the pattern formed is called a diffraction pattern. Obstruction (slit) is smaller than the wavelength
Diffraction pattern With 2 slits you now get patterns of interference. -Constructive when peaks or troughs intersect -Destructructive when peaks and troughs intersect.
Convolution theorem The FT of the convolution of two functions is the product of their FTs The diffraction pattern of a lattice is a lattice The diffraction pattern of a molecular crystal product of the transform of the molecule (molecular transform) the diffraction pattern of a lattice (reciprocal lattice) Sampling of molecular transform at reciprocal lattice points
The convolution of two functions
Convolution theorem Diffraction pattern of a set of lines is a row of dots perpendicular to the lines The separation between the dots is proportional to the inverse of the separation between the lines Form lattice by multiplying the two functions The diffraction pattern of the lattice is a convolution of the diffraction patterns of the two sets of lines
All of the lens contributes to each point
Fourier transforms
Laser fun Red, Blue, Green Wavelength Red>Green>Blue ROYGBIV Red 650nm, Green 530nm, Blue 405nm 1mW, 5mW, 10mW 50 µm vs 100 µm mesh
The transform of the convolution of two functions is the product of the transforms a b * = Fourier transform of this F.T. F.T. F.T. x = = this F.T.(a b)
Same molecular transform sampled by different lattices a) Molecular transform b) Lattice c) Convolution of lattice and transform d - f ) Same molecular transform sampled by different lattices Modified from Lipson & Taylor, 1964
Diffraction from an atom e - diffract X-rays Can be described by an approximation of the scattering by the electron shell An atom diffracts x- rays in all directions
When do we get a diffraction pattern? When we get constructive interference from two diffracted waves Bragg s Law nλ = 2d sinθ
Diffraction planes reflect X-rays λ d θ θ θ Bragg s Law nλ = 2dsinθ
λ Bragg s Law d θ θ λ θ d λ θ θ 2dsinθ The difference in travel is equal to a multiple of the wavelength. All of the atoms close to the Bragg plane contribute to the diffraction nλ = 2dsinθ
Lattice planes (2 1 0) plane (1 3 0) plane Plane defined by ( a h, b k, c l ) Can have negative integers
Getting to reciprocal space 1/d 1 1/d 2 θ 1 θ 2 d 1 d 2 The angle that satisfies Bragg s law is inversely proportional to the lattice spacing Reciprocal space can be defined by a vector normal to the real plane giving us the reciprocal lattice d = nλ 2sinθ
Constructing the reciprocal lattice J.D. Bernal (1926) On the interpretation of X-ray, Single Crystal, Rotation Photographs Proceedings of the Royal Society 113:117 When do planes diffract? Rhodes Crystallography Made Crystal Clear
Ewald Sphere Diffracted X-ray 1/λ 1/d θ 1/λ θ θ Reciprocal lattice origin Crystal origin A construction to indicate which Bragg planes diffract for a given orientation 1 2d = 1 λ sinθ