So given that background information of what the transform of a two-dimensional crystal looks like in 3-D, let's now talk about the best strategies for data collection. How do we actually measure that transform. So let's go back to this picture which we've seen many times now, where there's a sample and electrons are scattered into all directions. That scattering is focused onto the back focal plane of the objective lens. And then sails on through and re-interferes with all of the other scattered waves to produce a magnified image. And as we've mentioned before, the object is in real space. And the wave function that exists here on the back focal plane is the Fourier transform of that real space object. And the wave function on the image plane which appears on the detector, is again, in real space. Because the lens does an effect in inverse Fourier transform. So in the case of a two-dimensional crystal, in real space, you can see all the different unit cells and see that it's a crystal. The Fourier transform of that crystal is a series of discrete spots as we've discussed. And so the wave function on the back focal plane of the objective lens will look something like this. And if we shrink all that down and move it up so we can look more carefully at what happens below. You'll reca, recall that at some level down below that first image plane, we have a detector. And there's a series of additional lenses here in the microscope. But depending on the strength of this final lens here, this plane can be made conjugate either to the back focal plane of the objective lens. In which case, what we get on the detector is a magnified representation of the diffraction pattern. We can actually image the diffraction pattern of the original crystal. Or if the detector is made conjugate instead to the image plane down here below the objective lens, then we see a real space picture of the original crystal. And so this is called diffraction mode where the wave function on the back focal plane of the objective lens is made conjugate to the detector. And this is called imaging mode [NOISE], because the detectors made conjugate to an image plane. And so what are these advantages and disadvantages of the two modes? Well, the first big difference is that if you act, if you record an actual image of your sample, you can then calculate the Fourier transform of that image. And recover both the amplitudes and the phases in the transform. However, if you only record a picture of the diffraction pattern, phase information is lost, and all you would get is the amplitudes. However, because these amplitudes are shift in variant. Meaning that if the crystal is drifting slightly in the microscope as it's imaged, that slight drift doesn't actually impact the amplitudes of the spots in the diffraction pattern. And so even though the crystal's drifting just a little bit, you can still get clean measurements of the amplitudes. So the best way to get good measurements of the amplitudes is through diffraction mode. So I'll write amplitudes are better recorded in the diffraction mode. Here in the imaging mode, if you're imaging a crystal that's drifting slightly during the image, then it blurs out the, the crystal. Just like all of the beam-induced motion that we've discussed in other contexts. So imaging mode gives you both amplitudes and phases, but the best way to measure accurate amplitudes is through diffraction mode. So the best way to proceed is to record a lot of images of crystals but also a lot of diffraction patterns directly. So for instance, in the aquaporin crystallography project, many images were recorded of these crystals. Here this, this is obviously an image. You can see the double-layered nature of that crystal. And from an image like this, a Fourier transform would reveal both amplitudes and phases along all the lattice lines. But in addition to the images, it's advantageous to record a large number of diffraction patterns as well. So here's an electron diffraction pattern recorded of a crystal just like this in an untilted condition. And you see all the discrete spots, and the intensity of each spot can be measured directly as its square root of the amplitude that's needed. And when you're recording diffraction patterns to protect the camera, we insert a beam stop to block the unscattered beam because it's so intense. Once a lot of diffraction patterns of an untilted crystal are recorded, the next step is to tilt lots of different crystals and record diffraction patterns in their tilted condition. And so here example is an electron diffraction pattern of one of the crystals tilted to 70 degrees. And you still see the discrete spots, but notice that the spacing between these spots is much greater here than the spacing between these spots. Because, in the tilted position, the, we're, we're sampling the last lines at an angle now. And so, the untitled images and diffraction patterns give us these amplitudes and phases here on the x-y plane in, in reciprocal space. And all the tilted images give us measurements of amplitude and phase along the lattice lines and some other position. And the next step is to consider each lattice line independently. So this lattice line would be, say, the 1, 2, 3, 0 [NOISE] lattice line. And this lattice line would be the 4, 0 lattice line. And we plot all the data that we have in a plot like this. So here is the z coordinate, meaning this is z equals 0, and this is positive z. And as the lattice line goes down below 0, that is negative z. So positive z. And so, we'll plot along that lattice line from 0, right here, to both positive z coordinates and negative z positions. And in this case, what is being plotted is the diffraction intensity, here in this plot. And then above it is being plotted the image phase. And so image, diffraction intensity is varying from essentially zero to high numbers. And so each time we have the measurement of the diffraction intensity, it's plotted on this, on this graph. And so, a central section cutting through here, say, might give us this value of amplitude at some, some higher z value. Let's say the z value here [NOISE] was equivalent to 0.15. Then the amplitude and phase that were measured in that tilted image, we could plot the amplitude here. And we could plot the phase up here. Phases you see vary from minus 180 to positive 180. And so each of these dots here, like this dot, and this dot, and this dot. Each of these dots here is an independent measurement of the amplitude and phase obtained from either an image of the crystal, or a diffraction pattern of the crystal. And this whole plot is all about just the lattice line 813. So 813 is too high resolution to be seen on this schematic. But all of this data is about just the lattice line 813. How its amplitude varies from low z to high z, and how the phase changes from low z to high z. And this procedure is repeated for each lattice line. So here's a separate plot for the lattice line 1212. These were the two examples shown in the paper that I'm using from Eva Nogales in Nature 1998 of tubulin 2-D crystals. And so here's the plot where the 1212 lattice line. Again, going in z from low z to high z. And plotting both the amplitudes and the phases. And once all the data has made plotted here, then a curve is found that connects these it, its curve fit. Both the amplitudes and the phases. Our curve fit. And then they can be sampled at regular intervals, can be sampled at zero. They can be sampled at regular intervals along these lattice lines,which essentially in this kind of a diagram, means that we're going to sample the amplitudes in phases on a regular lattice in z. And we do this for all the lattice lines. And once all the lattice lines are sampled, then we can do an inverse Fourier transform in 3-D [NOISE] to produce a three-dimensional reconstruction of the unit cell that made up the 2-D crystal. And in the beautiful case of the tubulin 2-D crystals that we're looking at here in this nature paper, that reconstruction had sufficient resolution to build an atomic model of the tubulin sub-units. Now, one of the challenges in 2-D crystallography is that because the proteins are only constrained within a single plane. Sometimes their packing is less precise and less ordered than it would be if it was a three-dimensional crystal with packing interactions in all three directions. So what has been observed is sometimes these 2-D crystals have distortions and bends within them. Fortunately, some of this can be corrected through image processing. This process is called crystal unbending. And here, I'll show a figure from Braun and Engel, that illustrates this. So given an image of a two-dimensional crystal, one can then calculate its Fourier transform. And if you look, you see the individual spots in the diffraction pattern. That transform can then be filtered by masking out only the discrete spots in its regular lattice, and zeroing all of the pixels in between those spots. Remember, the pixels in between the spots carry all the information about how each unit cell is subtly different than the next one. And the information in the spots tells you what the average unit cell looks like. So if we just mask out the regions around the spots, and then do an inverse Fourier transform, we get an image that contains an average unit cell in the crystal pattern. And so this is a, is a filtered image with a much clearer representation of the unit cell in the crystal. Now, if we cut a small part of that out as a reference, we can then cross-correlate that reference against the entire original image of the crystal. And so the cross-correlation map shows where in the original image unit cells appear. From that, we can plot a distortion map of how each unit cell, whether it is in exactly the place as expected or whether it's shifted a little bit from its predicted position. It shows you how the crystal has been squeezed or expanded in certain regions by the individual unit cells moving a little bit. And so here, the distortion vectors give an impression of how the crystal as, as a, as a sheet actually spread out, say, towards the edge of this image. And once that, those distortion vectors are known, we can create a unbent image of the crystal by moving parts of the image. Shifting them back to the locations to make a more regular lattice, re-interpolating it. And then, we see in the Fourier transform of that unbent image that the diffraction spots are much cleaner. So here is before unbending, and here is after unbending. And so from these unbent images, one can retrieve amplitudes and phases. And then if you mask those out and calculate an inverse for a transform back to real space, now you get a much cleaner picture with a lot more detail about the crystal. Once you have this, it can be used again as a reference to move back, to redo the cross-correlation map of how the crystal was bent. And it can be unbent, and this is an iterative loop, until the best result is obtained. Now, to finish our discussion of 2-D crystallography, let's enumerate the challenges. One of the main challenges is to get well-ordered crystals. And this is partly because if the proteins are only stuck to each other within a plane. In many cases, it's difficult for them to have enough contacts between each other to precisely lock into the regular lattice. And despite our ability to unbend the images after taking them, still, the difficulty of getting well-ordered crystals is a limitation. In addition, it can be hard to get flat crystals. So if we think about the grid surface that the crystals are settling on, in some cases, you have a continuous carbon film. But that carbon film may not be actually anatomically flat. It may, there may be bumps and ridges in it, especially if it was frozen and there was some cryo-crinkling involved. And so as the crystal rests on that surface, it also can get bent and crumpled. On the other hand, if we're thinking about a Quantifoil grid, where we have some carbon, but then holes in the carbon. Sometimes the water will bulge out of that hole, or there may be a ridge where the water dips into the hole. And so this can cause the crystals to be unflat. And unfortunately, because the crystals are just a single unit cell thick, they don't have much mechanical strength to prevent bending out of plane. As in other cases, charging and beam-induced movement can also blur the images. And finally, as we discussed, because we can't tilt crystals all the way to 90 degrees, there's a missing cone of amplitudes in phases in reciprocal space. That distort the final reconstructions and give them an elongation in the direction of the beam.
