Difference Patterson Maps and Determination of Heavy Atom Sites
Structural Molecular Biology Laboratory, ChemM230D

P43212 crystal symmetry and its corresponding Patterson symmetry.
Suggested Reading Materials

1) Chapter 6 (page 101-127) from Crystallography Made Crystal Clearby Gale Rhodes. 

2) Chapter 12 (page 279-291) from X-ray Structure Determination by Stout & Jensen.  Copies handed out in lab.

Plane group p2 Patterson demo in powerpoint format.

Space group P43212 Patterson demo in powerpoint format.

Complete lecture notes in powerpoint format. Complete lecture notes in pdf format.

Lecture outline in powerpoint format.

Assignment & Procedures
2nd Assignment: Heavy Atom Site Determination
due week of, February 9, 2004
Objective: To determine the mercury position (x,y,z) from a difference Patterson map calculated with data from the PCMBS derivative crystal. 


1) Label each difference Patterson peak with its (u,v,w) coordinates read from the difference Patterson map. The coordinates of each peak corresponds to a vector difference between one pair of crystallographically related heavy atoms (i.e. atoms related by space group symmetry operators). The mathematical expression for the difference vector is given by taking the difference between each pair of symmetry operators in the space group. 

2) Assign each of the labeled peaks with the difference vector equation that describes it. There are 8 symmetry operators in space group P43212 so there are 82-8=56 possible difference vectors to choose from in your assignment. Difference vectors between each pair of  symmetry operators from the P43212 space group have been calculated for you (see table of Patterson difference vectors below). OK, actually, 28 difference vectors are calculated below; the remaining 28 are just the same except the signs are negated in the result  (i.e. symop#3-symop#1 is the negative of symop#1-symop#3).  To help you choose the appropriate difference vector for each peak, the difference vectors have been color coded. Difference vectors in the blue boxes produce peaks on the w=1/4  or w=3/4 section. Difference vectors in the yellow boxes produce peaks on the w=1/2 section. Difference vectors in the red boxes produce peaks on the u=1/2 or v=1/2 sections.  These sections are called Harker sections. Let's begin with a peak on the w=1/4 Harker section. This peak must have been produced by one of the pairs of symmetry operators shown in blue boxes.  Does it matter which of the blue boxed equations I use?  Well, yes and no.  Yes, it makes a difference in the numerical value obtained in the result. But, these values should all be equivalent by crystallographic symmetry. Remember, if there is a single heavy atom in the asymmetric unit, then there will be seven other symmetry equivalents in the unit cell, these are related by the crystallographic symmetry operators. By chosing a different blue boxed equation, you will be solving for a different symmetry equivalent.  It's OK. Just pick one.  Write the equation in the margin, next to the peak. (Technical note: actually there are more than just eight possible correct (x,y,z) answers you could obtain. There are 8*4=32 possible different correct answers. The 24 additional possible correct answers arise from the arbitrary choice of origin.  If you look at the symmetry diagram for space group P43212 shown above, you will see that by adding (1/2,1/2,0), or (0,0,1/2), or (1/2,1/2,/1,2) to x,y,z the atom would remain in the same symmetry environment.  The particular origin that is chosen will affect the numerical value of x, y, or z, but they do not its validity. The 32 symmetry operators that relate these alternative correct choices are called Cheshire symmetry operators and are given below.

3) Solve for the x,y coordinate of the heavy atom.  Plug u and v (coordinates of the Patterson peak) into the equation and solve for x and y (coordinates of the heavy atom). OK, that was easy; I have x and y. But, I still need z!

4) Solve for the z coordinate of the heavy atom. Chose a peak on the u=1/2 section. Find the corresponding difference vector equation.  Plug in the v,w coordinates to Solve for x and z. 

5) Check for consistency in the x coordinates derived from the steps 3 and 4. We would like to combine the x,y coordinates from step 3 with the z coordinate derived from step 4 in order to get a complete set of x,y,z coordinates for the heavy atom. But, before we do this, we must check that the x coordinate derived from both sections  agree. If they agree, then we have our x,y,z.  But, if the two values of x do not agree then you must find a symmetry operator that transforms x from step 3 to have the same value as the x from step 4, and then use this symmetry operator to transform y from step 3 so it is consistent with x and y from step 4.  After all this we should have a self consistent set of x,y,z. What symmetry operator do I use?  It can be any of the Cheshire symmetry operators for this space group. 

Cheshire Symmetry Operators for space group P43212

     X,     Y,     Z
-X, -Y, Z
-Y, X, 1/4+Z
Y, -X, 1/4+Z
Y, X, -Z
-Y, -X, -Z
X, -Y, 1/4-Z
-X, Y, 1/4-Z

1/2+X, 1/2+Y, Z
1/2-X, 1/2-Y, Z
1/2-Y, 1/2+X, 1/4+Z
1/2+Y, 1/2-X, 1/4+Z
1/2+Y, 1/2+X, -Z
1/2-Y, 1/2-X, -Z
1/2+X, 1/2-Y, 1/4-Z
1/2-X, 1/2+Y, 1/4-Z

X, Y, 1/2+Z
-X, -Y, 1/2+Z
-Y, X, 3/4+Z
Y, -X, 3/4+Z
Y, X, 1/2-Z
-Y, -X, 1/2-Z
X, -Y, 3/4-Z
-X, Y, 3/4-Z

1/2+X, 1/2+Y, 1/2+Z
1/2-X, 1/2-Y, 1/2+Z
1/2-Y, 1/2+X, 3/4+Z
1/2+Y, 1/2-X, 3/4+Z
1/2+Y, 1/2+X, 1/2-Z
1/2-Y, 1/2-X, 1/2-Z
1/2+X, 1/2-Y, 3/4-Z
1/2-X, 1/2+Y, 3/4-Z

6) Check that xyz can predict the position of a third Patterson peak, not lying on a Harker section.   At this point, the  x,y,z position you have calculated is still tentative. One ambiguity remains; both x,y,z and -x,y,z can satisfy the difference vector equations you have used so far.  Both can account for the Harker peaks you observed in the difference Patterson map. But, only one of these coordinates sets is correct and will lead you to a beautiful electron density map, the other is not correct and would lead to miserable phases and uninterpretable electron density.  There is a 50/50 chance that you got the sign of x correct. The best way to determine whether  x,y,z or -x,y,z is correct is to calculate the u,v,w coordinates for a Patterson peak not lying on a Harker section.  Use one of the difference vector equations in the non-colored box below. Check that this peak is present in the Patterson map (if the coordinates of the peak fall outside the asymmetric unit, you can use a Patterson symmetry operator to transform u, v, and w to values between 0.0 and 0.5).  If you found the peak, congratulations!  If not, then simply change the sign of y in your xyz coordinate describing the position of the heavy atom.  Now re-calculate u,v,w.  You should be able to find the peak now.  

7) Report one x,y,z position for the heavy atom site.  Make sure that you performed the check in step 6.  Submit your x,y,z coordinate to Mike or Duilio by the start of your next lab period. We will check your answer during the lab.

summary slide
step 1
Step 1.

step 2
Steps 2 and 3.

step 4 continued


Step 5.

Step 6

Part One:
Calculate Isomophous Difference Patterson Map
Objective: To calculate an isomorphous difference Patterson map.  Plot the Harker sections.

Background: When last we met, we converted the output from Scalepack (mydata.sca) into mtz format using a GUI (graphical user interface) from CCP4 (Crystallographic Computing Project 4)  Duilio and I have collected all the data sets and merged them into one file.  This file is called prok_2004.mtz.  It contains multiple columns of data: 

native data set 2

PCMBS derivative data set 1
(strong single site)

EuCl3  derivative data set 2 
(strong single site)

Iodide data set 3 
(mutliple sites)

PtCN4  derivative

PIP derivative

thimerosol derivative

SmCl3  derivative

PMSF data set

Where F means structure factor, SIGF means standard deviation of the structure factor, DANO means anomalous difference, SIGDANO means standard deviation of the anomalous difference.  The structure factors have all been scaled to the native 1 data set which is the strongest of the two native data sets.  The scaled data is in a file called prok_2004_scaleit1.mtz. This is the file we will use for the isomorphous difference Patterson map calculation.

 Calculate the difference Patterson map using the ccp4i GUI.  Choose the FFT for Pattersons button.  Calculate the isomorphous difference Patterson map for F_nat2-F_pcmbs1. Give appropriate names to the output map files.  View the map with mapslicer. Send the plot to the printer.  Print out the Patterson peak positions in the .ha files.  Do the same thing for another derivative of your choice. Check if the anomalous difference Patterson is of better quality than the isomorphous difference Patterson.  Why?


fft for patterson
FFT for Pattersons

Table of Patterson Difference Vectors

Patterson vectors for P43212

Table of all possible self-vectors in space group P43212.  Colored blocks appear on Harker sections. Blocks of the same color are related by Patterson symmetry operators.  The Patterson symmetry operators for space group P43212 belong are the same as space group P4/mmm

Patterson Symmetry Operators for P43212

  u, v, w;   -u,-v, w;   u, v,-w;   -u,-v,-w;
-u, v, w;    u,-v, w;   -u, v,-w;   u,-v,-w;
-v, u, w;    v,-u, w;   -v, u,-w;   v,-u,-w;
 v, u, w;    -v,-u, w;   v, u,-w;  -v,-u,-w.

Instructor's preparations

Back to CHEM M230D course syllabus 

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