In the previous sections we have seen how to process individual images of 2D crystals. From a single micrograph we generate a list of spots containing their amplitudes and phases in Fourier space. This section describes how to merge the projection maps from each micrograph of an non-tilted specimen in order to produce one common projection map. This is similar to creating class averages in single particle electron microscopy, where projections of particles with the same orientation are averaged together.

In contrast to single particle classification in the context of 2D crystals we already know the projection direction from the electron microscope, all under the assumption that the crystals are perfectly flat. The central section theorem gives us the mathematical backbone to consider each image created by a transmission electron microscope as a projection of the crystal. Thus all images of non-tilted 2D crystals represent a projection of the crystal in z direction. In the previous section we have seen how filtering the Fourier transform of 2D crystal results equals averaging the proteins in real space. The task of 2D merging is to average the Fourier components of the images of untitled crystals together. To be able to average the densities gained from each image one has to align the images first. They have to be in register with each other which is best done by aligning them to a common reference. As we see later in the practical part the reference is just one of the images chosen by the user. The alignment can also be understood as setting a common origin.

We again make use of the Fourier theory for this task. For single image processing we only considered at the amplitudes of the Fourier transform and the lattice it spans. This lattice always had a common origin and approximately the same dimensions, only the orientation varies across the images. This is due to that the magnitude of the Fourier Transform does not change even when the image is shifted. Therefore averaging the amplitudes is just averaging the diffraction spots with the same miller indices. The phases of the Fourier transform though determines the origin of the image in real space. Thus aligning all the images results in shifting the phases of each image to match with the reference. The last step of 2D merging is averaging. The simplest way would be to take the mean of the values for each unique reflection depicted by a Miller index. But the program AVRGAMPHS written by Richard Henderson weights each reflection based on its IQ value. Hence each single diffraction spot of every image contributes to the average according to its signal-to noise ratio. For the user this has the advantage that even images of bad quality can be merged without making the resulting 2D map worse, additionally one can still benefit from the few spots that have a good IQ value. The final merged 2D map is then the inverse Fourier transform of the list of averaged amplitude and phases.

Having processed all the images of non-tilted specimen we are ready to merge them to an average projection map. We will use the Merge Tool panel to do so.

1. Open the Merge Tool panel
2. Select the 2D merge scripts panel from the left bar.
3. We want to align the images to a common phase origin. Therefore we have to select one image as reference. Usually the image with the highest QVal and/or lowest phase residual error is used. The QVal can be read from the “QVal2” column in the project panel. To make it easier one can also sort the images by the this attribute by clicking on the title of the column.
4. When choosing the reference make sure that the phase origin of the image was chosen accordingly the present space group. If the image suffers from a wrong phase origin you should set the phase origin manually.
5. The appropriate reference should be selected with the tick mark from the “Project Library” panel. Make sure no other images are selected.
6. Before creating the reference one should to choose the resolution the data on which the alignment process can rely on. This resolution is specified by the parameter Resolution of the merged dataset for the reference. We set this value initially to 15 Å.
7. Make sure the Symmetry is set correctly in the processing data panel because the selected symmetry will be applied to the reference, i.e. correction of symmetry-restricted phase values and removal of symmetry-forbidden reflections.
8. Now run the script Merge Selected Images. This script merges the data of the selected image into a reference file. Several output files are created as listed in the Images panel. The essential ones are the merge.aph, which is the list of merged amplitude and phases, and the MTZ file, which is then used as reference named MTZ: Merged full reciproc. space 2D data for reference. Double clicking the MTZ file will show the content in your web browser. The file mainly consist of a reflection list as already seen in the APH file. Every reflection is depicted by index (H,K,L), where the L is set to 0 in the 2D merging, followed by amplitude and phase, which have been CTF corrected and symmetrized. The MTZ file also contains information about the crystal unit cell dimensions, the symmetry, the resolution range, etc.

Now that we have created a reference we can start aligning the non-tilted image data.

1. Uncheck the image you have selected to be the reference and select all other images recorded from non-tilted specimen that you want to align. Tip (Automatically selecting the images): The images from the “Project Library” can be automatically selected using the “Automatic Selection” tool located at the bottom right of the “Project Tools” panel. Choose the parameter to be used, provide the minimum and maximum values of the range. Finally hit “Change Selection” once you are done. This will automatically select all the images with selected parameter in provided range.
2. With the script Refine Once you can now align the selected images to the reference. All you need to do is to specify the range over which the phase origin search should be performed. Unless you have already manually selected the phase origin in all of the projection maps, we advise you to initially cover the entire crystal unit cell i.e. a search range of 360° phase angle. The search range is defined by the parameters Stepsize of the phase origin search and the Number of steps in the phase origin search in the Merging section. In a first phase origin search you should use a coarse step size of 6° which with a number of 60 steps will go from -180° to 174° . In later stages you should decrease the step size to a smaller value in order to perform a local refinement of the phase origin.
3. Run the Refine PhaseOrigins Once script. This will refine the phase origin for all selected images with the help of the MRC program origtilt. The last phase origin change for every image will be listed as phaori_last_change in the Results panel as well as in the PhaOri Change column.
4. Now we visually inspect the phase origin change for each projection map by running the Generate Image Maps script. Therewith the projection map are updated with the refined phase origin and is reflected in the Image Preview panel. You can scroll through the images in the Images panel or the Project panel. Tip (Limiting library to selected images): If you want to see only the selected images in the Project Library panel you can select the option “Show checked images only” from the “Library Tools” located on the right of the panel. The same menu option can be used to switch back to view all images.
5. Run Refine PhaseOrigins Once again with the current step size until the phase origin of all refined images does not improve anymore. Depending on the step size it might happen that the optimal phase origin is in-between the two considered phase origins and will jump back and forth. This is also a sign that you have reached the optimum for this step size and should decrease it for the next merge and refinement runs.
6. Besides the updated parameters in the results panel, the script also produces output files in the images panel. If you are curious how origtilt is called to refine every single image you should have a look at CSH: refinement script. The output of origtilt might be more interesting and is in the text file listed as LOG: origtilt B output. This file should be considered when evaluating the refinement process. It contains the list of reflections for all refined images. The most important value in this file though is the refined phase origin. It is followed by a table of resolution ranges with the associated phase residual for the reflections in that range. This allows the verification up to what resolution the refinement is trustworthy. One thing to keep in mind is that for the coarse refinement we limited the resolution of the reference. After the resolution table in LOG: origtilt B output a cross-correlation map is plotted through a matrix where each entry represents the normalized cross-correlation value between the refined image and the reference. It should contain a clear peak which should end up in the center after the last rounds of refinement.
7. After the first cycle of refinement we should update our reference with the refined data. Therefore include the image you initially used as a reference by adding it to the selection. Now run Merge Selected Images again to create a new reference.
8. If you set the Symmetry in the Processing Data panel and the symmetry is above space group P2 you will get a table named “Phase Residual In Resolution Ranges”. You should consider the phase residuals as measure to what resolution you can trust your data and adapt the Resolution of the merged dataset for the reference accordingly. The resolution ranges of the table list the phase residual up to the Upper Resolution limit plus 1Å further.
9. With the new reference at hand we can perform a finer refinement of the phase origin. Hence the Stepsize of the phase origin search should be decreased to e.g. 0.5Å. In any case make sure that the step size together with the number of steps at least span the range of the previous step size (in our case 6Å).
10. Now run Refine Once again for the refinement until the phase origin for each image does not change anymore i.e. PhaOri Change will be zero. The cross-correlation maps in LOG: origtilt B output should now show sharp and centered peaks.
11. Until now we have always executed the merge and refine process separately. But when you are at the point where the images merged to a reference are the same set as the images being refined i.e. the selection does not change, you can also use the Merge & Refine script and do merging and refining iteratively for the selected number of rounds.
12. When using Merge & Refine script one should have a look at the parameters phaori_last_change and MergePhaseResidual in the Results panel. As we have mentioned before the phase origin change should decrease with each iteration and ideally end up being zero. The measure of how good the refined phase origin is quantified MergePhaseResidual. This value will also decrease in the process of refinement. If the value remains around 90° then the alignment is not working.
13. For more in-depth knowledge about your merged data set there is the option List Reflections into logfile (ILIST). If this is enabled a separate log-file is created called LOG: reflections after origtiltk. This file is the complete list of reflections from every merged image. This means that for every H,K index the available values from the contributing images is listed. The values associated with each reflection besides the H,K index are the amplitude, phase, image number, IQ, film weight, background amplitude and CTF. Appending these values are dashed lines followed by the IQ value again. The number of dashed lines is inverse proportional to the IQ value i.e. a reflection with an IQ of 1 has the most dashed lines appended. This helps to identify the good reflections even in a large list. The reflections with IQ values of 1 to 4, even have the amplitude and phases repeated at the end. This should make it easier to detect reflections with data differing from the amplitude and phases of the same reflections stemming from other images. This deviation can be a sign for a wrong CTF correction in that image, therefore you should have a look if the defocus was detected correctly in this image.
14. If you did not add all the non-tilted images yet or have gained new images from the microscope you should use the existing reference to align the novel images in the same fashion by jumping back to step 1.

Once all the data of the non-tilted images are aligned the last step is to merge them all together to one.

1. As described above the final alignment should also be visually inspected by running the Generate Image Maps script again and then scrolling through the images.
2. Running the Final Merge script is similar to the previous merge steps, except that we will not use the result as a reference for further alignment, rather as resulting average 2D map. Hence it is of great importance to chose the Upper Resolution Limit (RESMAX) carefully. One should only trust the data up to a resolution where the phase residual are low. If you are looking at the table of phase residuals a good estimate if phase residual reflects meaningful data can be given by the following relation:phase residual 90 -(90/n), where n is the number of spots in a given resolution range. Final merge also creates two kind of 2D maps: one in form of an MRC image and the other in form of a contour plot. For our example data set where we set the symmetry to P4 and the files the images are named p4-symmetrized final 2D map and PS: p4-symmetrized final 2D map plot. Tip: (Saving and Loading selections): Selection of checked images can be saved using the button “Save checked list” in the right toolbar in the “Project Panel”. These selections can be later loaded using the button “Load checked list”.