The general goal of electron transmission imaging techniques such as Electron Crystallography, Tomography or Single Particle reconstruction is to gain a 3D computer model by integrating sets of 2D data recorded by an electron microscope. The common mathematical tool that all these techniques share is the Central Section Theorem. The theorem states that the projection along the z direction of a three dimensional volume contains the same information as the central slice in the reciprocal space (z*=0) of the 3D-Fourier transform of the volume. This means that we can gain a full sampling of the Fourier Transform of the object by the projections along all spatial directions. Accordingly, the 3D reconstruction become just a Inverse Fourier transform. The theory of Electron Optics allows to some approximation the use of images recorded by a Transfer Electron Microscope (TEM) as a projection of the sample. This means that with sufficient images of samples imaged in different orientation to the electron beam, one can simulate the sample’s three dimensional density distribution.

We have seen that the Fourier transform can be used to harvest the repeating signal of the 2D crystal. But since we are dealing with 2D crystals the periodicity is limited to the x,y-plane. The crystal is ideally only formed by one layer in the vertical direction. This results in continuously distributed Fourier component in vertical direction as illustrated. The diffraction spots we have seen in 2D with the index (h,k) now become so called lattice lines (h,k,z*) where h and k still correspond to the same Miller index, but z*defines the position along the lattice line. Projections of tilted 2D crystals now produce diffraction spots, where each reflection corresponds to the lattice line h,k and height z*. The height corresponds to the intersection of the tilted plane with the lattice line h, k and thanks to the central section theorem this can be calculated from the tilt geometry of the crystal. So each recorded image produces a set of amplitude and phases with the coordinates (h,k,z*). The goal is now to densely sample the lattice lines by integrating as much data as possible with varying orientation and tilt angle. This allows us to sample the Fourier transform of our crystal to a certain degree, but due to instrumental restrictions of the TEM, where can not record an image of a any specimen tilted higher than ±70 there will always be a missing cone.

3D merging in 2dx in follows the same workflow as for 2D merging. The problem is still merging data from different images into one common data set. Therefore finding the common phase origin is the task at hand. But in contrast to 2D data where each reflection h,k and height z*=0, the amplitude and phases of a tilted sample are usually associated with a reflection at a height that is not zero. These measurements belong to a tilted plane in the 3D Fourier space, where the tilt geometry is defined by the tilt angle (TANGL) and the orientation of the lattice on that plane (TAXA). If we would now try to refine the phase origin of the tilted image data with non-tilted dataset, then the data points available for comparison would only be the one that lie on the tilt axis. Those measurements are too few to give a sufficient alignment of the tilted data. To deal with the alignment problem of the first tilted data we need to increase the range in z* direction, so that a wider region for allowed comparisons between the non-tilted and the reference data exists as illustrated. In the following steps we will show how 2DX can merge data from 2D crystals with different tilt geometry to a 3D reconstruction of your protein structure.

Tip (Backup and Synchronize): We advise you to always backup your project before you start with 3D Merging, using the script Synchronize with Backup from “Project Tools”. This script allows you to submit your project data to a different directory.

Tip (Automatically selecting the images): Also save the merged map from 2D merging with help of the Copy Merged Dataset custom script. This script allows you to copy the last merging result to any of the given 10 registers.

1. Select the 3D merge scripts panel from the left bar.
2. Select all processed images that belong to the lowest tilt group (TANGL < 30), except the non-tilt data. Preferably, the selection should include ten or more projections with reliable measurements. The projections of sample with low tilt is determined by defocus gradient which can be unreliable to some extent, therefore it is essential that the handedness is correct.
3. We have mentioned the critical step of aligning the first tilted dataset to a 2D merged map. So before running Refine PhaseOrigins Once limit the selected tilted data set to 15 Å with the parameter RESMAX. In this first 3D refinement step it is indispensable to have sufficient reference points (>20 ) for the phase origin determination. This can be reached by increasing vertical z*tolerance along a lattice line, which is defined by the processing parameter 3D: zstarwin in the 3D Merging section. In the beginning of 3D merging set this parameter to 0.2 or something higher. This measure depends on the vertical thickness of your crystal defined by the parameter ALAT . When refining the highest tilt data zstarwin should be decreased to 12*ALAT. So for a 200 Å sample we eventually set zstarwin to 0.0025, just to give you a feeling for the value range.
4. As in 2D merging we start out with a step size of 6.0 (Stepsize of the phase origin search)and 60 steps (Number of steps in the phase origin search).
5. Run Refine PhaseOrigins Once until the phase origins do not essentially change anymore.
6. Examine the refinement of the tilted data to the non-tilted by looking at the origtilt output LOG: origtilt B output. You should be familiar with this file from 2D merging, but in contrast to 2D you find that the refined reflection now have non-zero z* values. Check the cross-correlation maps and if there is a distinct peak in the center. In that file you will also find a line like this “ORIGIN REFINEMENT DONE BETWEEN 48 OF THE NEW REFLECTIONS”, make sure the number of listed comparisons are above 20. Also have a look at the phase residual.
7. To visually inspect the refinement result run the Generate Image Maps script and scroll through the phase shifted projection maps.
8. Select the images for which the phase origin could be determined and the images which were used for the 2D reference. If you saved the selection file, just use that file to load those selection again.
9. Merge the selected images all to one common 3D reference through Merge Selected Images, but with a Resolution of the merged dataset for the reference set to 15 Å.
10. Now refine the alignment of the tilted images through the Refine PhaseOrigins Once script, but with a smaller Stepsize of the phase origin search while reducing zstarwin .
11. After the refinement have a look at the phase residual (MergePhaseRes) for every refined image. If the phase residual is high it could have its origin in the CTF correction with a false defocus. This can be detected by verifying the merged data set. Because the data from most images will be complete up to 15 Å one can profit from the redundancy in the number of reflections to identify errors in the CTF correction. The defocus in the images, where the errors were detected has to be adjusted before proceeding with the refinement.
12. If the cause of the bad phase residual was not the CTF correction, check if refining the tilt geometry can improve the alignment. This would be done in another Merge & Refine (Iterative) run while having the parameter Refine Tilt Geometry (Only in 3D mode) set to Yes. This should only be done if a thorough reference already exist. You will discover that if the reference is not sufficient most commonly the tilt refinement will cause the tilt angle of all the refined images to increase.
13. If you find that your image data has a resolution of 5 Å or higher you should refine the beam tilt at this stage. This can be achieved by setting enabling the option Refine Beam Tilt in the Merging Refinement section for the refinement.
14. Now you can include images of higher tilt and align them to your 3D reference with the Refine PhaseOrigins Once script. The procedure of the refinement is done as before, so you should redo all the steps from item 7 to here. Remember that you can now decrease the parameter zstarwin.
15. The benefit of data stemming from sample tilted more than 25° is that the tilt geometry is more trustworthy, since it was determined by lattice distortions through emtilt. This means when you have merged a fair amount of higher tilt data you can in turn refine the tilt geometry of the lower tilt data.
16. Once all image data has been included and all refinements have been done you can decide on the final resolution cut-off for the individual images (Upper Resolution Limit) and run Final Merge.
17. An important file that appears in the Images panel during merging is PS: TLTPLOT file . This shows you the completeness of your 3D data set and allows you to determine at what tilt angle more data is needed for an even sampling of the 3D Fourier space.
18. The final step in the 3D reconstruction of your structure is the calculation of the 3D map. This can be achieved with the Generate Merged Map script. This creates a volume in CCP4 format named MAP: Final 3D Volume. You can examine the result in Chimera by double-clicking it.
19. Once a reliable 3D-model has been generated on can re-process the already used images with the (Re-)Process all images script. Thereby one can improve the data from the individual image data by unbending with a reference created through back projection from the model. We call this procedure synthetic unbending.