Introduction to Macromolecular X-ray Crystallography Biochem 300 Borden Lacy Print and online resources: Introduction to Macromolecular X-ray Crystallography, by Alexander McPherson Crystallography Made Crystal Clear, by Gale Rhodes Online tutorial with interactive applets and quizzes. Nice pictures demonstrating Fourier transforms Cool movies demonstrating key points about diffraction, resolution, data quality, and refinement.

Crystal -> Diffraction pattern -> Electron density -> Model Overview of X-ray Crystallography Resolution, Fourier transforms, the ‘phase problem’, B-factors, R-factors, R-free …

Diffraction: The interference caused by an object in the path of waves (sound, water, light, radio, electrons, neutron..) Observable when object size similar to wavelength. Object Visible light: nm X-rays: nm, 1-2 Å

Can we image a molecule with X-rays? 1) We do not have a lens to focus X-rays. 2) The X-ray scattering from a single molecule is weak. Not currently. Measure the direction and strength of the diffracted X-rays and calculate the image mathematically. Amplify the signal with a crystal - an array of ordered molecules in identical orientations.

Interference of two waves Wave 1 + Wave 2 Wave 1 Wave 2 In-phase Out -of-phase

Historical background 1895: X-rays discovered by Wilhelm Roentgen, Germany. 1910: Max von Laue suggested that X-rays might diffract when passed through a crystal – since wavelengths comparable to separation of lattice planes. 1912: Friedrich and Knipping use a crystal as a diffraction grating and prove the existence of lattices in crystals and the wave nature of X-rays.

1913: English physicists Sir W.H. Bragg and his son Sir W.L. Bragg explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence with Bragg’s Law. Historical background continued Sin  = AB/d AB = d sin  AB + BC = 2d sin  n = 2d sin 

Practically: Assign a coordinate (h, k, l) and intensity (I) to every spot in the diffraction pattern— Index and Integrate. I hkl,  hkl The intensity of each spot contains information about the entire molecule. The spacing of the spots is due to the size and symmetry of your lattice. Diffraction pattern

The wave nature of light f(x) = Fcos2π(hx +  ) f(x) = Fsin2π(hx +  ) F = amplitude h = frequency  = phase a)f(x) = cos 2πx b)f(x) = 3cos2πx c)f(x) = cos2π(3x) d)f(x) = cos2π(x + 1/4) x

Fourier Series: an expansion of a periodic function in terms of an infinite sum of sines and cosines. Approximate a square wave f 0 =1 f 1 =cos2π(x) f 2 =(-1/3)cos2π(3x) f 3 =(1/5)cos2π(5x)

Fourier transform: F(h)= ∫ f(x)e 2πi(hx) dx where units of h are reciprocals of the units of x Reversible! f(x)= ∫ F(h)e -2πi(hx) dh

Fourier series from the diffraction pattern Reverse Fourier transform Electron density function  x,y,z)

Fourier series from the diffraction pattern  hkl,  hkl f(x) = Fcos2π(hx +  ) f(x) = Fsin2π(hx +  ) F = amplitude h = frequency  = phase F hkl ~ √I hkl

Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods

Heavy Atom Methods (Isomorphous Replacement) The unknown phase of a wave of measurable amplitude can be determined by ‘beating’ it against a reference wave of known phase and amplitude. Combined Wave Unknown Reference

Generation of a reference wave: Max Perutz showed ~1950 that a reference wave could be created through the binding of heavy atoms. Heavy atoms are electron-rich. If you can specifically incorporate a heavy atom into your crystal without destroying it, you can use the resulting scatter as your reference wave. Crystals are ~50% solvent. Reactive heavy atom compounds can enter by diffusion. Derivatized crystals need to be isomorphous to the native.

Native F nat Heavy atom derivative F deriv

The steps of the isomorphous replacement method

Heavy Atom Methods (Isomorphous Replacement) The unknown phase of a wave of measurable amplitude can be determined by ‘beating’ it against a reference wave of known phase and amplitude. F PH FPFP F H and  H Can use the reference wave to infer  P. Will be either of two possibilities. To distinguish you need a second reference wave. Therefore, the technique is referred to as Multiple Isomorphous Replacement (MIR).

Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods

Anomalous scattering Incident X-rays can resonate with atomic electrons to result in absorption and re-emission of X-rays. Results in measurable differences in amplitude F hkl ≠ F -h-k-l

Advances for anomalous scattering methods Use of synchrotron radiation allows one to ‘tune’ the wavelength of the X-ray beam to the absorption edge of the heavy atom. Incorporation of seleno-methionine into protein crystals.

Anomalous scattering/dispersion in practice Anomalous differences can improve the phases in a MIR experiment (MIRAS) or resolve the phase ambiguity from a single derivative allowing for SIRAS. Measuring anomalous differences at 2 or more wavelengths around the absorption edge: Multiple-wavelength anomalous dispersion (MAD). Advantage: All data can be collected from a single crystal. Single-wavelength anomalous dispersion (SAD) methods can work if additional phase information can be obtained from density modification.

Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods

1- Compute the diffraction pattern for your model. 2- Use Patterson methods to compare the calculated and measured diffraction patterns. 3- Use the rotational and translational relationships to orient the model in your unit cell. 4- Use the coordinates to calculate phases for the measured amplitudes. 5- Cycles of model building and refinement to remove phase bias. Molecular Replacement If a model of your molecule (or a structural homolog) exists, initial phases can be calculated by putting the known model into the unit cell of your new molecule.

Direct Methods Ab initio methods for solving the phase problem either by finding mathematical relationships among certain phase combinations or by generating phases at random. Typically requires high resolution (~1 Å) and a small number of atoms. Can be helpful in locating large numbers of seleno-methionines for a MAD/SAD experiment.

Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods F = amplitude h = frequency  = phase  (x,y,z) electron density FT

Electron density maps

Duck intensities and cat phases Are phases important?

Does molecular replacement introduce model bias? Cat intensities with Manx phases

An iterative cycle of phase improvement Solvent flattening NCS averaging Building Refinement

Model building Interactive graphics programs allow for the creation of a ‘PDB’ file. Atom type, x, y, z, Occupancy, B-factor

ATOM 1 N GLU A ATOM 2 CA GLU A ATOM 3 C GLU A ATOM 4 O GLU A ATOM 5 CB GLU A ATOM 6 CG GLU A ATOM 7 CD GLU A ATOM 8 OE1 GLU A ATOM 9 OE2 GLU A ATOM 10 N ARG A ATOM 11 CA ARG A ATOM 12 C ARG A ATOM 13 O ARG A ATOM 14 CB ARG A ATOM 15 CG ARG A ATOM 16 CD ARG A ATOM 17 NE ARG A ATOM 18 CZ ARG A ATOM 19 NH1 ARG A ATOM 20 NH2 ARG A The PDB File:

Occupancy B-factor How much does the atom oscillate around the x,y,z position? What fraction of the molecules have an atom at this x,y,z position? Can refine for the whole molecule, individual sidechains, or individual atoms. With sufficient data anisotropic B-factors can be refined.

Refinement Least -squares refinement  =  w hkl (|F o | - |F c |) 2 hkl Apply constraints (ex. set occupancy = 1) and restraints (ex. specify a range of values for bond lengths and angles) Energetic refinements include restraints on conformational energies, H-bonds, etc. Refinement with molecular dynamics An energetic minimization in which the agreement between measured and calculated data is included as an energy term. Simulated annealing often increases the radius of convergence.

Monitoring refinement R =  | |Fobs| - |Fcalc| |  |Fobs| R free : an R-factor calculated from a test set that has not been used in refinement.