Now that we've talked about data collection and reconstruction, and how to identify objects of interest in tomography, next we're going to talk about resolution and other limitations. The ultimate resolution limitation in tomography depends on what kind of sample we're imaging. For native samples, frozen hydrated material, the ultimate resolution limit is radiation damage. And remember I showed this picture earlier of a series of hemocyanin molecules, and after they were imaged with just tens of electrons per square angstroms, you can still see that they were barrel shaped, and you could even seen the inner ring of the barrel. You could see that they were stacks of six different layers. But then after applying 100 electrons per square angstrom or even 200, those details were lost. Nevertheless, the rough outlines of the particles were still present. And so the ultimate resolution limit is the fact that we simply can't fire enough electrons at the sample to get detailed images to high resolution before we've actually destroyed the sample. And all we're looking at is the remnants of the sample and a blurry object with lower resolution detail. Now, one way to overcome that resolution limitation is what we call sub-tomogram averaging. And to illustrate that, I'll play this movie. In one of our projects we were imaging a bacterial cell, and here you see it coiled up. And in this particular kind of bacteria there are two flagella. One flagellum emerges from this pole of the cell, and here it's being shown in gold, and then there's another flagellum that emerges from this pole of the cell. And each of this flagella have a motor that drives them. And so there's one motor over here in the reconstruction and there's another motor over here in the reconstruction. And these are the same macromolecular complex, or they're copies of the same macromolecular complex. Perhaps they're in a little bit different conformational states, but, nevertheless, what sub-tomographical averaging means is that we take the sub-tomograms that contain the object of interest, in this case, a flagellar motor, and we box them out of the reconstruction wherever they appear. And then we align them together in space and average them to produce a higher signal-to-noise ratio reconstruction of that object. And so here we average them to produce a much cleaner, higher signal to noise ratio reconstruction. And this can be done with two objects, or five objects, or ten, or 50, or 1,000 objects, as long as they are copies of each other. That should be averaged. And so, for instance, if, after recording hundreds of tomograms of a particular species, there are hundreds of example motors or some other object of interest within those tomograms, hundreds can be selected. And they can all be averaged together to produce a much higher signal to noise ratio reconstruction. An average reconstruction of that object. In this case, it's the flagellar motor from campylobacter jejuni. Now, for fixed and stained samples, the ultimate resolution limit is the fidelity of the stain. And to talk about that, let me bring up the image again here of insect sperm showing individual microtubules. Where we noticed that you could actually see the outlines of individual protofilaments in a microtubule in this tomogram, because the stain that was used coated each protofilament. Now, because this sample has been chemically fixed and dehydrated, and because what we're really looking at is the metal stains encasing the proteins, rather than the native material of the proteins themselves, we can record very high resolution tomograms. We can crank up the magnification and use very high dose and get very high signal to noise ratio images. But what we'll be seeing at the end of the day is the stain and how well it mimics the true structure that we're interested in underlying the stain. And so the resolution limit in this kind of a project is the fidelity to which the stain reveals the structure of the biological object of interest. Now another resolution limit in tomography is the tilt increment used. So here I've drawn reciprocal space. This is spatial frequency in the x direction and this is spatial frequency in the y direction. And let's suppose that our first tilted image gives us the amplitudes and phases of the 3D reconstruction along this plane. And our second tilted image gives us these measurements of amplitude and phase. Now in these images, of course, we have a series of pixels. And so we have measurements of amplitude and phase in a regularly spaced lattice. Okay, and in the next image again we have measurements of amplitude and phase along a regularly spaced lattice. And in order to calculate the three dimensional reconstruction, we're going to have to interpolate these measurements onto this Cartesian coordinate system. Marked, for instance, by these grid points, and then calculate an inverse 3D Fourier transform to get the real space reconstruction. Now clearly, at low resolution, we have lots of measurements for each value that we have to find. We have lots of amplitude and phase measurements in the vicinities of these lattice points where we need to estimate the amplitude and phase of the 3D Fourier transform. But a much higher resolution, say out here at this resolution, now we start to have more unknowns, where we need to estimate amplitude and phase. And we actually have measurements. And at this point it becomes poorly determined. And so the resolution that you can reach depends critically on the tilt increment that's used in between each image of the tilt series. And as a rule of thumb, it's been shown that the number of images that are needed is approximately equal to pi times the diameter of the object being imaged by tomography. For instance, of a cell, or of a virus, or the, whatever the object of interest. This is diameter. And this is spatial frequency in, in reciprocal units of length. And the number of images is approximately equal to pi times the diameter, times the spatial frequency that you can reach before your number of measurements is too sparse. And so, for example, let's say you're trying to record a tomagram of an, of a cell. And its diameter, just to make it a little easier, let's say its diameter's 180 nanometers. And let's say you were trying to reach a resolution of one over three nanometers in spatial frequency. Then the number of images that you would need, if we say that this is approximately three here, would be, you would need approximately 180 images. And because each image covers both sides of reciprocal space,. You really only need to cover 180 degrees. And so covering 180 degrees in 180 images, mean that you would pick a tilt increment of 1 degree, for example. Now the effects of tilt increment, and also the phenomenon of a missing wedge is well illustrated in this figure. We've seen this part of it before, illustrating the idea that tomography is recording projection images of an object and then taking those projection images and calculating the structure of the object that must have existed to give rise to those projections. And here, as an example it's shown what the reconstruction, at least of a two dimensional image, might look like if this series of images that's recorded, the series of projections, spans all the way from plus or minus 90 degrees. In other words the full set of projections is recorded. In this case, if the full set of projections are recorded at a fine step size like two degrees step size then the reconstruction looks very good. Obviously this is a picture of Einstein and you can clearly see lots of details, the, creases in his forehead, the shape of his hair. Many of the details here are present. However, if you limit the angular range through which the projections are recorded to say only plus or minus 60 degrees, rather than the full plus or minus 90 degrees,. So, if you're missing some of these projection angles here, then the reconstruction is obviously degraded. And it's specifically degraded in, in ways that we'll talk about more in a second. But you can see that the creases in the forehead here are missing in this image and that's related to which projections,have, have been left out. Here the step size is again 2 degrees. Now a further problem that we might have is what if our tilt increment is a large increment? So what would happen if we had the full tilt range from plus or minus 90 degrees, but our tilt increment was a full 5 degrees? In this case, you can see artifacts arising in the image. The image is not nearly as clear as it was before and that's because there's fewer images involved. The tilt increment is high, so the resolution here is lower. And, finally, if you have both a high tilt increment and a limited tilt range. Then there's both kinds of degradation in the final image. Now let me try to draw the situation in 3D. Let's let this represent the three dimensional Fourier transform of some object of interest. And what we record are projection images, which give us the amplitudes and phases of central slices through this three dimensional Fourier transform. So this might be the data from the un-tilted image. And the next image might give us the data through this plane. And the next image gives us the data through this plane and so forth. And if we're able to tilt to say, plus 60 degrees here, we find that we fill up this entire region of reciprocal space. And if we are then able to tilt say to minus 60 degrees, we would find that we could further. Fill up all of this region of reciprocal space. And what is missing is a wedge of information here and here. There's a missing wedge of information in the three dimensional Fourier transform. Now let's think about what that would mean for specific objects. Suppose we had a filament in our object of interest that was long, skinny, and parallel to the electron beam. The Fourier transform of a vertical rod is almost entirely zero except for amplitudes in phases in a plane that's perpendicular to that rod. And so here is the plane that would contain all of the most important information that characterizes the structure of a vertical rod. And as you can see, despite the missing wedge, we still collect all of that information in our tomogram. So this rod would be very well defined. And that makes sense intuitively. Imagine that you have a vertical rod and your electron beam is viewing down that rod. The very first image shows you clearly that it, it's confined in all these dimensions, and as soon as you begin to tilt it, you can see very clearly that it's, a single skinny rod. What if however, the object of interest was a skinny rod in this direction? Well, all of the most important amplitudes and phases that define a rod in this direction are found in this plane in reciprocal space. This vertical plane. And as you can see we're missing almost all of that data because of the missing wedge. There's just a single line here on this central slice that's actually measured. And so that kind of an object is almost invisible in tomograms. And that makes sense intuitively, because if the electron beam is coming this way, and one is tilting the object this way, you can see it's extent in this dimension from the first image. But you learn almost nothing more about it by tilting it, because it always looks the same, and you have no idea how, how large it is in this plane. You have no idea that it's confined to just a single rod in this direction. Finally, let's consider a long, skinny rod parallel to the tilt axis. In this case, the information about it is contained in this plane in reciprocal space. And what you can see is that we have this this part of the information. And we have this part of the information of that plane. But we're missing a wedge of data, both at the top and at the bottom. And so we partially define that rod, but not entirely. And that makes sense because if the rod is oriented this way, and the electron beam comes down, we can see immediately that it's finite in this dimension and then when we roll it to different angles during the tilt series, we also see that's it's limited in its height to just that amount. So we learn most of what there is to know about this object in this particular orientation. Now let me show a more practical example of the effects of the missing wedge. This figure comes from a project where we were imaging a bacterial cell, here's its boundaries on the grid. This is a projection image of that cell. And we recorded a tomogram of this cell. We are interested in, actually, the protein machinery that exists here at the cell division site. And you can see this cell is beginning to divide, and we're interested in the protein machinery here at the division plane. And incidentally, in this, species, there's an outer membrane along the cell and there's an inner membrane to the cell, and there's also a surface layer outside those membranes. And what we observed is that just inside the inner membrane here. There was a couple of dark spots that turned out to be filaments. These are FtsZ filaments that drive constriction of the cell. And the point is, that this picture is what we call an XY slice of the tomogram perpendicular to the direction of the electron beam. And features are very, are, are pretty well defined in an XY slice. The effects of the missing wedge are obscured. But if we take a cross section through the cell like this, and we lay it in this direction. So here, this could be an X, Z slice over here. Now we see the surface layer and the outer membrane, the inner membrane, and here we see those filaments, the FtsZ filaments. There's one over here on this side. There's one over here on this side. But what you notice is that the top and the bottom of the cell appear to be missing. Where's the membranes that are supposed to come around here and cover it? I can dare, guarantee you that the cell was not missing its membranes on the top and the bottom. They're just simply missing in the tomogram. And that's because the missing wedge obscures features that are perpendicular to the direction of the electron beam. And so these membranes, as they go over the top here, they are lost. They, they're not visible in the tomogram, because of the missing wedge. Along the sides, these features are well-defined, despite the missing wedge. That being said, it's important to remember the missing wedge shapes the point spread function that affects all the densities in the entire tomogram. And to a first approximation, it turns a single point in the object. The reconstruction is no longer a single point like it would be in a perfect microscope. But because of the missing wedge, the point spread function is actually shaped like an American football. And it's blurred in the Z direction because we don't get enough of the tilt to be able to see the extents of objects in the vertical direction. And so the point spread function looks like an American football. Now, think about what that would do to a membrane. If you had a cell membrane here encircling the cell, and that membrane consisted of a number of densities all around the cell. I mean, you can think of individual lipids or you can think of, you know, proteins in the membrane. They're just densities within the membrane. Each of those densities in the tomogram is converted from a spherical density to actually an American football. So imagine smearing all of these out into an American football shape. Now what you can see is that along the sides of the cell where one of these densities smears up and down. It compensates for where another density in that same object is being smeared up and down. And so a series of dots that are vertically related to each other, the point spread function is much less detrimental because the density loss from one is compensated by extra density coming from the one below. And so you see a nice clear membrane along the side of the cell. But up at the top of the cell, the densities instead of being nice clear dots, they're actually smeared vertically to the point where none of the density in any of the voxels is actually high enough to stand out above the noise. And it's not compensated by any other high densities that are contributing density to that position. So, the effect of the missing wedge is to shape the point spread function into something that looks like an American football. And this is convolved on every density on the tomogram. But because of the geometry of certain features, the missing wedge is more or less detrimental to how you can interpret that object. And what, the end result is that features that are vertical in the image still appear fine in a reconstruction. But features that are perpendicular to the electron beam are often lost. Another factor that can limit the resolution of tomograms is defocus. To illustrate this, I'm showing a picture. This is a single slice through the middle of a three dimensional reconstruction, a tomogram, of a bacterial cell. And in the surface of the cell, right here, I'm showing a slice through the chemo receptor ray. This is a regular lattice of molecules that senses signals in the environment and tells the cell what to do. You can show that there's a regular pattern of densities in this area by calculating the power spectrum. And here because we see a pattern of distinct spots, we know that this was periodic. And so because of that, we did sub-tomogram averaging like I've shown you before. And the result here is the sub-tomogram average. And you can see six major densities in a hexagonal lattice. And so we understood then, that the receptors were packed in this hexagonal lattice. Our principal resolution limitation here was the defocus. We were using a high defocus to increase contrast. And because of that, the first zero of our contrast transfer function was hitting at around three and a half or four nanometer resolution. And so, for these reconstructions we were imposing a low-pass filter with a cutoff around four nanometer resolution. And because of that, the sub-tomogram averages had no more detail past about four nanometer resolution. So one way to overcome this would be to record the pictures closer to focus. But another way is through CTF-correction. So we can take our original images that went in to produce this tomogram. And we can calculate their power spectra and then fit the intensities in the power spectra to curves. And so this is what I'm showing here. So this is a graph of spatial frequency from zero out to higher spatial frequencies. This is 0.2, or about 5 nanometer resolution. And this is 0.4, meaning 2.5 nanometer resolution. And the red curve here is the actual intensities that were seen in the power spectrum. So they have a strong bump there and then another bump and several bumps as you get to higher resolution. And eventually the signal is lost. The green curve is a theoretical contrast transfer function, the expected pattern of maxima and minima that you would get for a particular defocus. So you can see that there was plenty of bump pattern in the experimental data, the red curve, to cleanly fit a contrast transfer function curve to that. And so we were able to identify what defocus was being used for this tilt series. And once this is done, using the techniques of CTF-correction that we went through previously, we can phase flip. And also boost the amplitudes of those spatial frequencies that were flipped or diminished by the contrast transfer function. And then redo the three dimensional reconstruction and produce CTF-corrected tomograms. Then from the CTF-corrected tomograms. When we did another sub-tomogram average. Now, there was clearly three distinct densities seen in each of these hexagonally-related objects. And in fact, each of these dark spots turns out to be a bundle of four alpha helices. We know this because we know the crystal structure of the object that's being imaged there. And this sub-tomogram average was sufficient to allow us to, to fit in crystal structures and build a pseudo-atomic model of the chemoreceptor ray. Here, these are the receptor bundles and there's another ring of proteins here that links the bundles together. So one of the resolution limitation's in tomography can be the defocus. And that can be overcome by taking the pictures closer to focus, or through CTF correction. The next resolution limitation we'll talk about is the precision of image alignment. Now to illustrate this, let's look again at the tilt series of that Bdellovibrio cell. So here it is, the tilt series that we recorded. And you can see that we have added the gold fiducial markers into the media. And as the tilt series is recorded, we can use those gold fiducials to align the images. And as we described, there's a lot of elements in the alignment. First of all, we have to find the translational shift of each image with respect to its neighbors in the tilt series. Then we have to find what is the rotation. There can be a relative rotation of each image within the tilt series. Then we have to find if there has been any magnification shifts of each image with respect to the others in the tilt series. Then we refine where exactly was the tilt axis in each image, and by what angle was it tilted. So there's a lot of parameters that we need to deduce from tracking the positions of these gold beads throughout the tilt series. And obviously, they more precisely we can determine all of those parameters, the higher the resolution of the tomogram will be. Or perhaps more intuitively, you can think of, that if there are errors in the way the images are aligned. For instance, if one image is shifted with respect to another, obviously, features in the reconstruction will be blurred because the images weren't aligned. Or if there is a magnification error, one image is a thousandth higher magnification than the one before it. Then objects on the edge are going to be blurred, because they're going to be pushed further to the edge of the, of the image. And any of these shifts then translates to limitations in the resolution of the ultimate reconstruction. The next parameter that affects the resolution of the tomogram is magnification. Shannon's sampling theorem says that pixels should be at least two times smaller than the resolution target. But in practice, we should for three or even more times smaller. So for instance, if you want a reconstruction with, say, 4 nanometer final resolution, then you need to use a magnification high enough that your pixel size, each pixel in each image is no more than 2 nanometers. And it would be much better if it was only a single nanometer. And if for instance you are planning to do sub-tomogram averaging. And get to a 1 nanometer resolution-reconstruction, then you need to have a pixel size no bigger than five angstroms. And much better would be two or three angstroms. Smaller pixel sizes, however, result in smaller fields of view. So obviously, you don't want to exaggerate the magnification if you're never going to reach the resolution that that magnification supports anyway. You pick a magnification that's high enough to get the resolution you want, but no higher. because otherwise, you're just limiting the field of view that you'll capture in each image. And finally, to operate a direct detector in accounting mode, the electron hits must be separated in either space, because you used a higher magnification. In other words, you use a high enough magnification that the electron hits are spread on on the detector. Or you can separate the electron hits in time by reducing the dose rate of the beam and recording longer exposure times. In practice, let your resolution target dictate your magnification. And then increase the exposure time as necessary, so that your direct detector can count individual electron hits.