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2

 Real space (x, y, z)
 Reciprocal space (h, k, l)
 Patterson space (u, v, w)
 u = x_{1} – x_{2}
 v = y_{1} – y_{2}
 w = z_{1} – z_{2}

3

 Electron density calculated by taking the inverse Fourier transform of
the structure factors
 Patterson function (map) is Fourier transform of Intensities (no phases)
 Requires measured data ONLY
 Gives map of vectors between atoms

4

 To explain in onedimension,
 Atom 1 at position x_{1}
 Atom 2 at position x_{2}
 Both have electron density, then…
 The Patterson map will have peaks at x_{1}  x_{2} and
at x_{2}  x_{1}
 Peaks heights proportional to the product of the peaks heights in the
electron density

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6

 Calculated from crystal data
 Shows presence of noncrystallographic symmetry within the unit cell
 Should be inspected if MR fails
 Provides information about how molecules are oriented to each other
(such as rotations)
 Some newer programs don’t use Patterson functions, but instead
“likelihood” functions which are more robust (Phaser)

7

 To run molecular replacement
 Need a “model”
 Need data from crystal
 Download structure of similar protein (called “model”) from www.pdb.org
 Should have at least ~40% sequence identity
 Can have no more than ~1Å RMS difference between structures
 Run molecular replacement program:
 Molrep (CCP4i)
 Amore (CCP4i)
 Beast (CCP4i)
 Phaser (Phenix, CCP4i)
 CNS
 EMPR

8

 Model – molecule is centered on origin by its center of mass
 Patterson map calculated from model
 Patterson map calculated from crystal data
 Once the peaks in the model Patterson map overlap the peaks in the
observed Patterson map, we have a solution
 Performed in 2 steps to simplify the calculation (N^{2}N peaks)

9

 To simply calculation, performed in 2 steps
 First, rotation function looks at intramolecular vectors (atoms within
same molecule)
 Depends on orientation of molecule, not position
 3 parameters
 Second, translation function looks at intermolecular vectors to find
position in cell

10

 Three angles needed to describe rotation of model
 Two typical ways to explain rotation

11

 Observed Patterson map calculated from crystal data
 Program rotates model on a grid according to the 3 angles described
previously and a model Patterson map is calculated
 For each point on the grid, all the overlapped Patterson peaks are
scored and printed as the RF peak list

12

 Calculated from crystal data collected
 Only need Intensities (I) to make Patterson map

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14

 Once rotation of molecule is known (peak in RF list), its rotation is
fixed and the molecule is translated throughout the cell on a grid
 Patterson map is calculated for each step on the grid and compared to
observed Patterson map
 A correlation coefficient and Rfactor are calculated for each grid
point and listed (TF peak list)

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16

 After translation function
 Get solution from the TF list (provides the rotation and translation of
model from the origin to the correct place in the cell)
 List usually ordered from highest CC to lowest
 Want to see top solution has highest CC and lowest Rfactor.
 CC is better to look at in this stage, compared to Rfactor
 Typical CC are 5070% for a correct solution
 Translation Function is repeated for all the possible space groups to
determine if screw axis is present
 Example. Data processed in P6, but space group could be P6, P6_{1},
P6_{2}, P6_{3}, P6_{4}, or P6_{5}. Can
only place molecules in the correct space group.

17

 Check molecular packing in modeling program for any overlapped
molecules or missing layers in the crystal packing indicating not all
molecules were found by MR
 If it looks good, START BUILDING!!!

18

 Meth. Enz. 277, pg. 26
 J. Denth, Prin. of Protein Xray Crystallography, Feb. 1999, pg.
130139.
