Phase problem: Determining an initial phase angle α hkl for each recorded reflection. 1 ρ(x,y,z) = F hkl cos 2π (hx+ky+ lz - α hkl ) V h k l

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1 Phase problem: Determining an initial phase angle α hkl for each recorded reflection 1 ρ(x,y,z) = F hkl cos 2π (hx+ky+ lz - α hkl ) V h k l Methods: Heavy atom methods (isomorphous replacement Hg, Pt) Halides soaks (Br, I salts) Anomalous scattering (SAD, MAD) Molecular replacement Direct Methods Computational maximum entropy methods Lecture 11 1

2 Molecular replacement: Requirement: a suitable phasing model (approximates your molecule to get a starting set of initial phases) How to get a model: Multiple sequence alignments (greater than 20% identity) Threading techniques (~70% of all protein folds known) Protein data base (+20,000 entries) Other biophysical techniques (NMR, solution scattering) Lecture 11 2

3 1) Rotation function: Determine the orientation of the crystal structure to be determined (θ 1, θ 2, θ 3 ) relative to a model unit cell (space group) Phasing model (X,Y, Z) c cθ origin a dimer bθ 2 aθ 1 1,1 1,2 1,3 X X 2,1 2,2 2,3 Y = Y 3,1 3,2 3,3 Z Z Crystal (X,Y.Z ) Rotation matrix Lecture 11 3

4 2) Translation function: Determine the position of the crystal structure to be determined (T x, T y, T z ) relative to a model. The search (phasing) model is orientated to be the same as that of the unknown molecule in the crystal unit cell. It is then translated in real space within the unit cell and structure factors F hcal and α hcal are calculated and compared with observed values of F hob. (grid search -imposing crystallographic symmetry) 1,1 1,2 1,3 X T X 2,1 2,2 2,3 Y + T = Y Concept: 3,1 3,2 3,3 Z T Z Minimize the difference between F obs and F cals (ie maximize the agreement between the search model and observed data and calculate initial phases. R-factor search Lecture 11 4

5 Determining an initial phase angle α hkl for each recorded reflection 1 ρ(x,y,z) = F hkl cos 2π (hx+ky+ lz - α hkl ) V h k l R work = Σ F hob -KF hcal h Σ F hob h F hob F hcal F hob = measured structure factor of reflection h F hcal = calculated structure factor of reflection h K = scaling factor α hcal Lecture 11 5

6 The heavy atom method Concept: Each atom in the unit cell contributes to every reflection in reciprocal space. The contribution of atom is greatest to the reflection whose indices correspond to lattice planes that intersect that atom. So a specific atom contributes to some reflections strongly and weakly to others. Hence one atom could contribute much more strongly than others (having a higher atomic number). A so-called heavy atom (an electron rich atom). Lecture 11 6

7 Concept: Collect native and then heavy atom soaked diffraction data set. The heavy atom ion will bind to one or a number of specific sites on the protein, without effecting the the protein conformation or crystal packing (isomorphous). The difference between the two data sets is only from the heavy atoms. Examples of binding sites: Cysteines Cysteines, histidines, methionines Hg Pt Specific binding must be found Tips try different ionic compounds at various phs Lecture 11 7

8 Calculation of the initial electron density map Once R work is minimized (~40% or lower) Dependent on resolution, model accuracy etc. The calculated α hcal can be directly used as the initial phases for the observed F hobs and an electron density map can be generated Lecture 11 8

9 Map interpretation: Model building and refinement 1 ρ(x,y,z) = F (hkl) cos 2π (hx+ky+ lz - α hkl ) V h k l REAL SPACE REPICROCAL SPACE Lecture 11 9

10 Least-squares model refinement: The model/improved electron density map can be used to calculate improved structure factors Goal to find a set of atom positions/parameters that give F hcal that minimize the difference to the observed Fobs minimize Φ = Σ w( F hob - F hcal ) 2 h F hobs = measured structure factor F hcal = calculated structure factor w = weighting (reliability of F hob ) (1/σ h ) 2 Lecture 11 10

11 Other considerations: Model Geometry: bond lengths bond angles Φ = Σ w( F hob - F hcal ) 2 + Σ w(l ideal -l model ) 2 + Σ w(a ideal -a model ) 2 h l a torsion angles + Σ w(t ideal -t model ) 2 t Always check rms deviation of model geometry Additional refinement parameters: F hcal = G n j f j e -2πi(hxj+kyj+lzj). e -Bj(sinθ/λ)2 j n j =occupancy, B j = temperature factor of atom j G overall scale factor of F hcal to F hob Lecture 11 11

12 R-Factor h = all reflections hkl R factor = Σ w ( F hob - K F hcal ) h Σ F hob h R-Free R free = Σ w ( F hob - K F hcal ) h T Σ F hob h T Where hkl T means reflections belonging to a test set T of unique reflections not used during the model s refinement Lecture 11 12

13 Data Collection (I hob F hob ) Space group Expression, Purification Crystallization Phase assignment ( α hcal ) F hob α hcal FT Electron density ρ(x,y,z) R-factor model / density Generate F hcal α hcal FT improvement Lecture 11 13

14 Resolution (Ǻ): Structure ONLY as good as the resolution of map! Lecture 11 14

Direct Method. Very few protein diffraction data meet the 2nd condition

$Direct Method. Very few protein diffraction data meet the 2nd condition$ Direct Method Two conditions: -atoms in the structure are equal-weighted -resolution of data are higher than the distance between the atoms in the structure Very few protein diffraction data meet the 2nd