1 Diffraction by Atoms Fhkl = V∑∑∑ρxyz cos [2π(hx+ky+lz)] + ρxyz sin [2π(hx+ky+lz)] "x y z Fhkl = ∑ fi cos [2π(hx+ky+lz)] + fi sin [2π(hx+ky+lz)] N Atomic Scattering Factor Note: the volume (V) is used in the top equation to account for ρ being expressed in electrons per unit volume, whereas F is in electrons per unit cell. "Scattering by an Atom 2 dhkl sin θ = λ$For d = 2 Å, sin θ/ λ = 0.25$2 Scattering Factor Temperature Factor fi = fo e -B(sin2 θ) / λ2 Argand Diagram φ = tan-1(y/x) Ahkl = ∑ fi cos [2π(hxi+kyi+lzi)]; Bhkl = ∑ fi sin [2π(hxi+kyi+lzi)] |Fhkl| = √(Ahkl2 + Bhkl2) "αhkl = tan-1(Bhkl / Ahkl) i i3 1-D Example: 2 Carbon Atoms Atom 1, x = 1.8 Å (angstrom coordinates) = 0.18 (fractional coordinates) Atom 2, x = 8.2 Å (angstrom coordinates) = 0.82 (fractional coordinates) 0 10 8.2 1.8 6 e- 1-D Example: Structure Factors Atom 1, x = 1.8 Å Atom 2, x = 8.2 Å λ = 1.54 Å4 1-D Example: Questions Consider h = 5 • How many atoms in the asymmetric unit? • What is sin θ/λ? (Bragg’s Law: 2 dhkl sin θ = λ)$• What is fi ? (See individual scattering factor plot)"• What is F5 ? (Fourier transform)"Homework 1: repeat for h = 2 Answer: see Table 8.1 above Homework 2: prove to yourself that the sin component (Bhkl) = 0 Calculating Electron Density ρxyz = 1/V∑∑∑ Fhkl cos [2π(hx+ky+lz) + αhkl] " - Fhkl sin [2π(hx+ky+lz) + αhkl]"h k l5 Typical Electron Density Maps Fc, αc (calculated amplitude and phase) Fo, αc (observed amplitude and calc. phase) Fo, αexp (observed amplitude and exp. phase) Fo- Fc, αc (difference between crystal and model) Fo1- Fo2, αc (difference between 2 crystals) 2Fo- Fc, αc (best working map) Difference Maps Typically have: Fo, Fc Fc Fo Missing atoms αc6 Model Improvement through Refinement Approach: Least squares fit of model to data, such that model best fits the data. Similar to R-factor Model Improvement through Refinement Press et al., Numerical Recipes, 1989.7 Include X-ray and Stereochemistry Terms Drenth, Principles of Protein X-ray Crystallography, 1999 Radius of Convergence is Small Iteration is key – build, refine, examine for many cycles8 R-factors There are numerous residuals used in crystallography. The three most widely used are Rsym, Rcrys, Rfree. Rsym = Σh Σi |Ihi - <Ihi>| \ Σh Σi Ihi Rcrys = Σh ||Fo| - |Fc|| / Σh |Fo| Rfree same as Rcrys but using reflections excluded from refinement (usually 5%) Rsym should be ~0.05 overall, and < 0.40 at edge of diffraction. Redundancy should be 3-5 fold Rcrys should < 0.20 for a completed structure. Rfree should < 0.25 for a completed structure. Interpreting Errors and Disorder • Poorly fit atoms can arise from errors in the experiment (usually phases) or from multiple positions for the atoms (e.g. Q12 in thioredoxin) • Sometimes disorder can be modeled as two conformations (e.g. T9; note also nearby difference peak) ATOM 68 CA ATHR X 9 49.802 22.471 9.638 0.50 9.07 ATOM 69 CA BTHR X 9 49.739 22.639 9.629 0.50 8.49 ATOM 70 CB ATHR X 9 50.301 23.907 9.713 0.50 10.05 ATOM 71 CB BTHR X 9 50.022 24.189 9.746 0.50 8.47 ATOM 72 OG1ATHR X 9 49.173 24.767 9.810 0.50 11.99 ATOM 73 OG1BTHR X 9 50.614 24.670 8.527 0.50 11.159 Interpreting Errors and Disorder Anisotropic B factors: modeling non-spherical dynamics as ellipsoids (6-parameters, e.g. Wat 113). ATOM 1 N MET X 1 37.114 15.448 24.395 1.00 24.73 ANISOU 1 N MET X 1 3280 3113 3002 7 43 -35 ATOM 2 CA MET X 1 37.285 16.656 23.555 1.00 23.89 ANISOU 2 CA MET X 1 3102 3089 2884 30 45 -51 ATOM 4 CB MET X 1 38.236 17.672 24.173 1.00 26.07 ANISOU 4 CB MET X 1 3377 3350 3178 -68 -18 -28 Data : Parameter Ratio • Data: number of reflections • Parameters: number of variables (XYZ, B, etc.) • Ratio of ~ 10:1 for best least squares result • Rare in proteins, so restraints are used as additional