Molecular Replacement Phases may be calculated given a known structure. molecular replacement If the structure of the molecule is known approximately, then the phases can be calculated. R R R R R

R R R BUT. We need to know how the molecule is oriented. fraction patterns of these two crystals are not the We can use homology to infer structure Protein sequences tell us whether or not the protein structures are likely to be the same. If the sequence similarity is > 25%, then we say the sequences are "homologous", meaning they evolved

from the same common ancestor, and they therefore must have similar structures. How similar is not known until both structures are solved. Molecular replacement will not work if the structures are tooof different. It is used to If a homolog known structure solve structures of close homologs or even the exists, then can be used to do same molecule in ait

non-isomorphic crystal. molecular replacement 6-dimensional search space Every possible rigidbody transformation of a molecule can be described using 6 parameters. 3 angles of rotation (defining a matrix of 9 coefficients), and a vector of translation (3 values). i.e. x' = c11x + c21y + c31z + vx y' = c12x + c22y + c32z + vy

z' = c13x + c23y + c33z + vz Therefore, the position of our molecule in the crystal unit cell must be a 6D transformation of its current position. Molecular replacement is the method for finding the angles and vector that define Procedure for molecular replacement: (1)Calculate fake diffraction data for the model, using a large P1 unit cell. F c (2)Calculate the Patterson map. P c (3) Calculate the Patterson map of the observed crystal data (Pobs or Po). (4) Rotate one Patterson versus the other. Find

the rotation with the maximum correlation. (5) Re-calculate structure factors (F model) for a P1 unit cell of the same cell dimensions as the true crystal unit cell (isomorphous except for symmetry). (6) Translate the P1 cell origin to every position in the unit cell, then sum the 'syms' (hkl's related by space group symmetry) to get point-by-point (1) Calculate Fake diffraction data The |Fc|'s are used to calculate the Patterson map. A large P1 unit cell is used

because then the Patterson map (the part close to the origin) will have only intramolecular peaks. P1=no symmetry, not necessarilly the same cell Intramolecular versus intermolecular Patterson peaks Only the part of point-by-point the Patterson map within the shaded

region is used. Short vectors are blocked out because they contain little shape info. Long vectors are blocked out because they are all intermolecular (symmetry) and Patterson space therefore depend on Rotated Patterson map for Gly intermole cular

vectors Rotated Patterson map for Gly intermole cular vectors intramolecular vectors rotate intermolecular vectors are (2) The Patterson map of the crystal The Patterson map represents all atom-atom vectors, translated to the origin.

Included in this mess are vectors within molecules (this is what we want to detect), and vectors between symmetry-related molecules (these are considered noise to the Rotation Function). Both intramolecular and intermolecular vectors exist in Z copies, oriented according to the rotational symmetry within the cell. Z is the number of symmetry operators in the space group. If there is more than one molecule in the asymmetric unit, then there are n*Z copies of the intramolecular vectors. (3) The Rotation Function z Three angles () define all possible rigidbody rotations. The solution of the rotation function are the angles that

y give the highest Patterson correlation function. x Correlation, defined

The correlation between any two functions x and y is defined as: r= (xx)(yy) (xx) (yy) 2 2 x-bar means the average value of the function x If the correlation is perfect, r=1.000 If the anti-correlation is perfect, r=-1.000 If there is no correlation, r

is close to zero. Patterson correlation function r= (P (v) P )(P (v) P ) (P (v) P ) (P (v) P o o mod 2 o

o mod mod 2 ) mod The sums are generally done over v in a spherical shell of the Patterson map that excludes the huge self-peak (v < 4) and also excludes long (mostly intermolecular) vectors (v > 20).

So, 4 |v| 20, is a good range for the rotation function. Non-crystallographic symmetry can be detected using the Self Rotation Function If the native Patterson is rotated against itself and the correlation (r) is calculated, the result (call the Self Rotation Function) will have at a nonsymmetry-related position only if the asymmetric unit has NON-CRYSTALLOGRAPHIC SYMMETR (NCS). NCS means that an envelope of the asu exists for which: (r) = (Mncsr + vncs)

(4) The Translation Function The model is oriented correctly with respect to the cell axes, but it is still at the origin. The green vector translates the model to its position in the crystal unit cell. (4) The Translation Function Symmetry related positions for each atom are calculated as follows: x = Mx + v (M is the sym-op matrix and v is the sym-op vector) A translation of the coordinates is: x = x + t

Symmetry-related, translated coordinates are: What happens to the phases and amplitudes when we translate? Amplitudes dont change. Phases change depending on the dot product of the translation vector and the scattering vector S (alias hkl) New phase = old phase + 2(hva + kvb + lvc) note: v is in fractional coordinates Calculating Fcalc using all symmetry

all equivs atoms v v F (h) = f ( h)e i 2h( Z(rg ) ) calc Z=symops g=atoms . g where Z(r) = Mr + v. v

us define vFmod: Fmod(h) = Let f (h)e i2hrg g g=atoms Fcalc is, therefore, just Fmod summed over the symmetry operators Z. v v i 2h(Z(rg ))

Fcalc(h) = Fmod( Z(h))e Z . syms Reciprocal space symmetry Rotating atoms in real space, r = Mr then multiplying by h to get the phase, phase = 2(h(Mr)) is the same as rotating reciprocal space the other way. can prove this by h(Mr) = 2( MThr) You writing out the matrix multiplication.

Conclusions, summary Molecular replacement is the solution of the problem r=Mr+v where r are the model coordinates (from a homolog model) and r are the true crystallographic coordinates. The rotation function finds the rotation matrix M. The translation function finds the translation vector v. The rotation function is done in Patterson space. Problems with the MR method: Phase bias Molecular replacement solutions may be suspicious due to the possibility of phase bias. Parts of the model may be

wrong, but the map may not show this. We have already discussed ways to detect/correct this: real space R-factor omit maps B-factors