2. Finding Heavy Atom Sites
Calculation of a Differences Patterson Map enables you to find the locations of heavy atoms in a derivative crystal by comparing it to a native crystal.
We'll calculate both Isomorphous and Anomalous Difference Patterson maps (separately) for each derivative. This will give us two independent checks for the sites of each of the two derivatives.
First, let's calculate the four different patterson maps. We'll do this with the CCP4 fast fourier transform program fft, used in patterson mode. To do this properly, you need to include the cutoffs for unrealistically large differences that we found in our scaling output. I've put these in for you, but you should look at the files and double-check to make sure they are correct.
1. isomorphous patterson map 123
2. anomalous patterson map 123
3. isomorphous patterson map 124
4. anomalous patterson map 124
These calculations will give you four different maps. Notice for the anomalous difference patterson maps you only need derivative data. That is why they look cleaner. Now we have to interpret them. Let's start with isomorphous patterson map 124:
First, reformat the map using the CCP4 program npo. Here is my shell script for one of the above maps. You can use it as a template for the other three.
Next, we are going to display each of the cross sections in the (u,v) plane using the CCP4 program xplot84driver. To do this, type the command
% xplot84driver filename.plt &
and an X-window display will appear that allows you to display any cross section. Page through all of them to see how the map looks. I've chosen convenient contour levels for you, but in reality you would have to play with them to figure out how best to display the patterson maps.
As an example, let's look at the Isomorphous Differences Patterson map for derivative number 124. Use xplot84driver to display plots number 12, 16 and 28. They should look like these: plot12.pdf, plot16.pdf and plot28.pdf
If you want to save these as postscript files or print them, use the "control panel" button. You can use the ghostview command
% ps2pdf filename.ps
to make your postscript file a pdf file (which is what I did). Or you can pay $100+ for Adobe software that does the same in twice the time.
Now comes the fun part: Solving the Patterson maps. For the example above, cross-sections 12, 16 and 28 are three different Harker vectors. Section 28, which is 1/3 on the c axis, is a Harker Section and is independent of the z coordinate of the heavy atom, so all spacegroups have this Harker vector here on this section. Section 28 will allow you to find the x and y coordinates, and the other two will allow you to deduce the z coordinates. (Yes, you can do this computationally, but the point here is to learn.)
Here is a list of the six symmetry operators of the space group P3121:
X, Y, Z
-Y, X-Y, Z+1/3
Y-X, -X, Z+2/3
Y, X, -Z
X-Y, -Y, 2/3-Z
-X, Y-X 1/3-Z
You can obtain Harker vectors by subtracting any one of these from any other. Not all will be unique. In fact, few will be unique. Find the one that corresponds to w=1/3 (section 28) by subtracting the first from the second operator. Then find two more that give you x and z and y and z for z between 0 and 1/3. These will allow you to solve uniquely for the x,y,z coordinates of the Br in one of the asymmetric units of our crystal.
My answers, used to calculate phases, are given on the Next Page.
A description of the problem of correlating origins is here.
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