Macromolecular Crystallography

Section 7, Phasing: Isomorphous

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Phasing - Isomorphous Derivative Method

Add just one or a few heavy atoms to the crystal without disturbing the rest of the structure.
In general, anything that changes the effect scattering power in a place without significant change to the rest of the molecular structure.
This can be done either by:
   Substitution -- Ba++ for Ca++ , I for CH3
   or Addition -- PtCl4 in space between molecules
   or changing wavelength so a particular “heavy” atom scatters more or less -- a small effect but remarkably useful in special circumstances. (Anomalous scattering, MAD, etc. considered in another section.)


P.H.PH.png FPH = FP + FH
|FPH| ~ |FP| + |FH|
FP is from native protein crystals.
FPH is from derivative crystals.
FH is from an imaginary crystal containing only the heavy atoms.
(Traditionally the native protein and the derivatives are usually separate crystals.*)
   *Same crystal: changing wavelength, or adding heavy atoms to a crystal in a flow cell.
|FP| and |FPH| are measured quantities
so an approximate |FH| can always be calculated. |FH| ~ |FPH| - |FP|


We do a Patterson map using ≈ |FH|2, find the heavy atoms, and calculate φH. This works even though this approximation is only really good if FPH and FP are of nearly the same phase.

PhsIR.1.png

Fortunately,
   1) Sometimes the phases are constrained to be either 0° or 180°.
   2) There is a trick we can use: anomalous scattering by the heavy atoms reveals the difference of φH from φP.
   3) Simple subtraction actually works fairly well because
      a) Error can only reduce |FH|, so that term merely contributes less to the Patterson than it should have.
      b) We are using hundreds or thousands of measurements to determine a few heavy atom positions, so the Patterson peaks build-up in the right places.



Phasing -- Determination of φP

|FP| can be measured; we must find φP (for each h, k, l reflection).
|FPH| can be measured; |FH| and |φH| can be calculated for each h, k, l.

PhsIR.2a.png

Lay out circles of radii |FP| and |FPH| at the ends of the calculated vector FH (arrow in diagrams). The intersections of the circles give possible values for φP; but there is a twofold ambiguity!


So, get a second heavy-atom derivative: measure |FPH'|, calculate |FH'| and φP'.
Layout the circles for this second case also: *

PhsIR.2b.png

This case will give a different twofold ambiguity, and the alternative that agrees with one of the intersections from the first case should give the correct φP.


Do this for all reflections.


Using |FP(h, k, l)| and φP(h, k, l) , plug into the Fourier transform formula to calculate an electron density map.


* For each derivative, calculate a |FPH| - |FP| Patterson map and solve for those heavy atoms. Thus for each derivative |FH| and φH can be calculated for each h, k, l.
The heavy atom models are independent, and before they can be combined, they must be related to the same origin. There are various tricks that take advantage of symmetry and patterns in the diffraction from the native protein molecule to do this. However, the best way to relate the derivatives is to make a crystal with BOTH derivatives at once. The Patterson map from this this combined derivative will give the relative postions of the heavy atoms, even if the crystal is of poorer quality.


HeavyAtomIsomorphousReplacement #1

PhsIR.3a.png

Reference wave FH1
Amplitude calculated
Phase calculated from
known heavy atom
position


2 choices for wave FP
from all protein atoms,
Amplitude measured

What 2 phases for FP
will give the measured
|FH1|?

FH1 + FP = FPH1
Reference wave +
protein wave =
Amplitude measured
for FPH1
PhsIR.3b.png


HeavyAtomIsomorphousReplacement #2

PhsIR.4a.png

Reference wave FH2
Amplitude calculated
Phase calculated from
known heavy atom
position


2 choices for wave FP
from all protein atoms,
Amplitude measured

What 2 phases for FP
will give the measured
|FH2|?

FH2 + FP = FPH2
Reference wave +
protein wave =
Amplitude measured
for FPH2
PhsIR.4b.png


MIR Multiple Isomorphous Replacement

PhsIR.5a.png

Reference waves 1 & 2
Amplitude calculated
Phases calculated from
known heavy atom
positions

wave FP
from all protein atoms,
Amplitude measured

What phases for FP
will give the measured
|FH|'s ?

FH1 + FP = FPH1
FH2 + FP = FPH2
Amplitudes measured
for FPH1 and FPH2
PhsIR.5b.png


Download:   download arrow PS-PhaseTriangles.pdf(80KB)  Problem for this section


Phase Probability Distributions

Because the data are by no means perfect, one does not try to find a unique φP where for both derivatives the circles intersect. Instead, a probability of intersection is defined that is related to the distance between these circles (as measured along a FPH radius.)

   P = e(-dist/ε)2 where ε is an appropriate estimate of the error in the data.

Each probability distribution will be bimodal. They can be plotted on circular graphs, with radial distance out beyond a reference circle representing the probability of a given phase angle for φP. The probability distributions then can be multiplied together to give a new probability distribution that hopefully will have only one large peak at the true φP, as shown below for the first example of derivatives discussed above.

Often more than two derivatives are used in solving a protein structure. These probability distributions provide a convenient method of combining several sets of imperfect data.

PhsIR.6.png

prob. derivative H       prob. derivative H'        prob. combined


Triangles and Phase Probability Distributions

Probability distributions correlated with triangles. The FP circle is centered on the arrow head. The probability distributions are drawn around the circumference of the FP circle.

GOOD CASE

PhsIR.2amod.png PhsIR.2bmod.png PhsIR.2abmod.png
PhsIR.6mod.png

This is a good determination of the Phase! If the distribution as shown really measures the density on the circumference of the FP circle, then the center of mass of the resultant probability distribution is very close to the tip of the FP arrowhead. This reflection would have a figure of merit very close to 1.0.


Triangles and Phase Probability Distributions

Probability distributions correlated with triangles. The FP circle is centered on the arrow head. The probability distributions are drawn around the circumference of the FP circle.

NOT SO GOOD CASE

PhsIR.2amod.png PhsIR.2brot.png PhsIR.2abrot.png
PhsIR.6bad.png

This reflections has a less sure determination of its Phase! If the distribution as shown really measures the density on the circumference of the FP circle, then the fractional radius of the center of mass of the resultant probability distribution is the figure of merit. This reflection would have a figure of merit about 0.7.

[Note: There is now a more elaborate method called Maximum Likelihood, which explicitly takes into account estimates of possible uncertainty in both the amplitude measurement and in knowledge of the phase of all components.]


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