So, of the three basic approaches in 3D EM, namely, tomography, single particle analysis and 2D crystallography. We'll start with tomography. Now again, the idea of tomography is that if you have a single unique object and you record projection images from lots of different directions. You can then take this set of projection images and you can calculate what the structure of the object must have been in 3D to give rise to all those different projections. As an example, this is a picture of a bacterial cell. Here's the bacterial cell plunge frozen in vitreous ice over an EM grid. Now the black dots that you see all around, these are gold fiducial markers, usually ten nanometers in diameter that were added into the solution with the bacterial cells before they were plunge frozen into the liquid ethane and these gold fiducials just act as markers to help us align the images afterwards. After we insert the sample into the electron microscope, we then record what's called a tilt series. So here in the movie, each frame of the movie is one of the images that was recorded as the sample was being incrementally tilted about a tilt axis. Now, once the sample tilts to about 60 or 70 degrees, it becomes too thick for us to image through it. And so we just stop recording images once we get that high. Also, in the extreme tilt images, you can see the edges of the circular pattern of the quantifoil holes come into view. There are circular holes in the carbon film and these cells are spread in thin layers across the circular holes. And in the tilt series, you can see a different texture inside the cell than outside the cell. And that's because inside the cell, it's full of proteins and others macromolecules. Now remembering the projection theorem, which is that if you have a 3D object and you record a projection image, this is a 2D image and then calculate its Fourier transform, so this is the Fourier transform of that image. The amplitudes and phases here are the same as the amplitudes and phases on a central section through the three-dimensional Fourier transform of the original object. So remembering this, what we do next is calculate the Fourier transform of each of those individual images and then we merge those images in reciprocal space. So let's draw reciprocal space. Let's that, let that be spatial frequency in the x direction. Let's, let that be spatial frequency in the z direction and then let's let spatial frequency in the y direction be into the plane. Now let's suppose the very first image that we recorded as a projection of an untilted sample, then the amplitudes and phases in the transform of that image will be the amplitudes and phases on the special frequency in x and y planes, then we tilt the sample just a little bit and record another image. And so these will be the amplitudes and phases of say, the first tilted image and then we tilt the image again. We tilt the sample again and record a second image and its Fourier transform fills reciprocal space there. And as you can see, as we record images to higher and higher tilt angle, we will sample, measure the amplitudes and phases in reciprocal space. Now, in tomography, we typically can only get up to about 60, 65 maybe even 70 degrees tilt before the sample itself becomes too thick. Of course, at 90 degrees, it would be infinitely thick and so we can only record useful images up to 60 or 70 degrees. And so there's a missing region here of data that we don't record. Now let's suppose we started tilting our sample in the opposite direction and recorded images towards what you might call negative tilt angles and then we would sample these amplitudes and phases in the negative direction. And again, we couldn't tilt past about minus negative 60 or negative 70 degrees. Now, each of the images that we recorded is digital of course. And so when we calculate the Fourier transform, it's a series of equally spaces pixels, each pixel contains an amplitude and a phase. And so we might represent that by little circles representing each measurement of an amplitude and phase. And the first image gives us these measurements and I'll try to make them equally spaced. Okay. And likewise, on the first tilted image, we get evenly spaced measurements of amplitude and phase along that plane. And so we get a series of measurements like this. And of course, here as well. And the second image gives us equally spaced measurements of amplitude and face and so forth. But in order to do an inverse 3D Fourier transform to generate the three-dimensional reconstruction, what we need are the amplitudes and phases on a regular lattice. For instance, a Cartesian coordinate system. And so we need amplitudes in phases on the intersections of all of these points and so we have an interpolation problem. So for instance, if we need to find the amplitude in phase of the 3D Fourier transform right in that position. We have at our disposal a number of measurements in the vicinity and so there's different algorithms to merge the amplitudes and phases. More sophisticated algorithms use amplitudes and phases further and further away. But at the end of the day, they all make an estimate of what is the amplitude and phase of the 3D Fourier transform at a regular lattice point. And so we make these estimates as we go out into further and further into reciprocal space and then we calculate an inverse 3D Fourier transform to generate the reconstruction of our object. So, in the case of the tilt series that I showed you as an example of the Bdellovibrio bacterial cell, this is what the three-dimensional reconstruction looks like. And here in this movie, I'm showing it from the bottom of the cell up to the top of the cell from the perspective of the electron beam. And in the movie, it's shown slice by slice. And so by tomography, we've been able to resolve the interior features of that cell in 3D. And we call such a three-dimensional reconstruction from tomography, we call it a tomogram. And so, in order to better appreciate the information content of this tomogram of the cell, it's hard to see things in detail when we're moving slice by slice up and down through the cell. Here, I'm showing just a single slice through the cell and I'm showing it in duplicate. These are identical copies, so that I can draw on this one and you can still look at the other one without the drawing on it. In this tomogram, we can clearly see an outer membrane. We see an inner membrane of the cell, you can even see the peptidoglycan layer, the cell wall in between them. Do you see that there's a region here, I'll outline it, that really has a different texture inside the cell than the surround. These larger dark objects that are about 25 nanometers in size and they're darker. These are ribosomes inside the cell. They're darker, because they contain a lot of phosphorus and so they scatter more strongly. And we know from their size that you know, ribosomes are about 25 nanometers in diameter. And so do you see that in the middle of the cell here, we have a region that has a really different texture. There don't seem to be any ribosomes in here. And there some swirly patterns and some linear features. This is presumably the nuclide where the DNA inside this cell is being packed together. And finally, at the pole of this cell, there is a flagellum. So here's a flagellum and at the base of the flagellum is a flagellar motor. And to get a better appreciation of the detail present in such tomograms. Here is an enlarged view of that single slice of the flageller motor. Again, in duplicate, so I can draw on one, while you can still see the other one. And do you see this density here? And this density here, for instance? These two densities form a cup called the C ring. We're viewing that C ring in cross section, so you just see this side of it and this side of it, but it's actually a whole cup. And you see this density here and this density here, we've been able to show that that's the cytoplasmic domain of a protein called Fla-H. In addition, we can see protein layers here and here there's an important protein piece. There's some kind of a plate that exists in the periplasm here. There are, are clear densities as part of the M, the MS ring. And anchor points to the growing flagellum here as it emerges from the cell. And so there is present in these cryotomograms enough resolution to resolve, even the shapes of protein complexes within the cell in a near native state. Now because tomography produces three-dimensional reconstructions of our samples, we can then interrogate it in 3D. And this is illustrated by the following movie. This is another bacterial cell in a tomogram being presented from the back to the front, slice by slice, like the previous one. But here on the second pass, we've identified on the computer object's of interest. Here are filament and gold and some iron core, iron objects inside the cell called magnetosomes. And finally, we layer on the membranes, the outer membrane and the inner membrane of this cell. And because it's in 3D, we can then move into the cell and interrogate the structure from any angle that we'd like. Here, we were interested in the relationship between the vesicles and the inner membrane and so tomography gives you a 3D reconstruction. Now tomography can be done on very large objects and very small objects. Let's first talk about the very large. Suppose that we had a large block of tissue, for instance, that even contain many cells inside of it. We can use a microtome to section this block of tissue and then each section, if we were to lay it down, for instance, we might see a pattern of cells on it. And then the next section would have a different slice through those same cells. And finally, the next section would have yet another slice through those cells. And so we call this serial sectioning. Now, on the scale of a cell, the typical field of view of a tomogram might be very small, might be just a small piece of one of those cells. And in order to expand the area that we can record in a tomogram, it's possible to do what's called montage tomography. And for the purposes of illustration, I'm going to enlarge the, the kind of field of view you might get in a tomogram suppose it was this large. Well, you could record a tomogram of this region and then you could record another tilt series of the adjacent region and another tilt series of the adjacent region and so on. And you'd want to have a little bit of overlap, so that you could stitch them together later. And then you could record another tilt series of each of these frames. And finally, perhaps another row of frames here. And they're always overlapped with ones next to them and above and below them, so that you can stitch them together. And then if we did the same thing on the next slice, et cetera. And the next slice, et cetera, et cetera. Then we could combine all these tomograms and stitch them together in, in the final reconstruction. And this strategy is called serial section montage and then if you were to record tilt series and produce 3D reconstruction, montage tomography. The serial section refers to sectioning, you know, collecting serial sections through the cell in one dimension. The montage refers to the fact that instead of recording just a single tilt series of a small area on that cell, we would record a whole set of tilt series, overlapping tilt series. And then later, stitch them together in the reconstruction process. And this can be extended, so you can reconstruct even very large regions of a cell. Some of the most ambitious research projects have been to produce three-dimensional reconstructions of entire cells for instance, entire human cells by this serial section montage tomography. As an example of a serial section tomogram, I'm going to show you one through a Golgi complex that was recorded by Mark Ledinski. And to show you the serial sections, this is a light microscope image of five serial sections that mark cut of this object of interest. Now the object of interest is a very small part of a cell that's right here in the sections. And this section wasn't imaged completely, but these dark star shaped patterns were produced by imaging the section in the electron microscope in a dual axis tilt series. So, one axis tilt series produced one of these elliptical burn marks and the other axis produced this other elliptical burn mark. And so mark image this section and then the imaged the same feature of the cell in the next section and the next section. And then finally, a fourth section. Now, after three dimensional reconstructions were calculated from each of those dual axis tilt series, those four reconstructions were then stitched together to produce a much larger reconstruction of a bigger volume of the Golgi and it looks like this form the bottom to the top. So here, we're going slice by slice through all four of those sections and between the sections, you see a little blurriness, because the stitching doesn't work exactly right. But nevertheless, you can see certain Golgi cisternae progress all the way from the bottom to the top. There is incredible amount of detail in these reconstructions. You can follow all the membranes and the vesicles and the Golgi cisternae. Now, once that's done, typically we segment the objects of interest. And so here each of the cisternae and vesicles involved in the Golgi are colored in different colors, so that we can then dissect them later. Look at them individually, if we like. Look at the interrelationship between them. So after the object has been segmented, it can be viewed in all of its complexity, in 3D. So you can see the power of tomography can allow an investigator to visualize, the very detailed three-dimensional relationships in the structures of even large objects within the cell. So that was an example of a very large object that can be studied by tomography. Now I'm going to talk about a very small object that was studied by tomography. And this is the case of the pyruvate dehydrogenase multi-enzyme complex. Now, as far as macro-molecular complexes go, this is a rather large one. It contains three enzymes has been named E1, E2, and E3. And together these three enzymes catalyze five biochemical reactions. And while the structures of these individual enzymes, E1, E2, and E3, were fairly quickly crystalized and their structures were solved by x-ray crystallography, the architecture of the entire complex remained unclear. One model, so-called ordered model, was that the E2 enzyme formed a cubic core in the middle and the E1s and the E3s decorated its faces in an ordered pattern. But there was another model, which we'll call the flexible model, which posited that the E2 enzyme formed the core but the E1s and E3s were just flexibly tethered along the outside. We took a purified sample of this complex, and spread it into a very thin layer on an EM grid, plunge froze it, inserted it in the electron microscope, and recorded this tilt series. Now in this tilt series, you clearly see this circular hole of the carbon film. And inside that hole, the highest contract objects are the black dots. Those are the gold fiducials that we use to align the tilt series. Another high-contrast object is right here. This is a piece of contamination, ice contamination. Probably a, a very tiny water droplet that froze on to the grid while we were inserting it into the electron microscope. And those are not the objects of interest. The objects of interest are very faint low contrast objects but there's one right here. They're easiest to see right on the tilt axis which is vertical in this slide. And you can see very faintly a pattern of 10 or 20 small densities closely packed together. Each of those small dots is one domain of this multi-enzyme complex. And so there's, there's an example. Here's another example, right there. There's a lot of others in the field of view but the ones that are further from the tilt axis are harder to follow. In the three-dimensional reconstruction, that was calculated from that tilt series, if we take a single slice through that reconstruction and look carefully at one of the enzyme complexes, reconstructed here in the ice, we clearly see four dots, which are the cubic core built of the E2 enzyme. And then we see surrounding that core, other densities that don't seem to be well ordered, at any particular distance away from it. So, here's a beautiful example of the core decorated by other densities. And there's other cases where you can see a core from a different angle, for instance, this one, you see the cubic core in the middle, and density surrounding it. So this is one of the enzyme complexes, cutting cross section, this is another one, cutting cross section, etc. One of the ways to view three-dimensional reconstructions is called volume rendering. And volume rendering, each voxel is like it's own little light bulb. And the brightness and the color of that light can be very depending on the density of that voxel. And this allows you to essentially see through a 3D object and see all these voxels at the same time. Rather than just a single slice or perhaps an isosurface which shows you one surface. So, it turns out that volume rendering was an effective way to look at the reconstructions of individual PDMC enzyme complexes in this experiment. And here's an example of one of the complexes shown in A. The orange spots are the densities of protein. So, you see in the middle, the four dots representing the cubic core made of the E2 enzyme. We know that because the crystal structure of E2 fits very well into that pattern. And, then surrounding it, you see these other densities all around it. And those are the E1 and E3 enzymes. And they seem to be flexibly arranged. We can look at another example of a different enzyme complex. And here again, you see the cubic core of the E2 enzyme. And you see a different arrangement of its E1s and E3s, around the outside. And finally, as a third example, we show another one. Here's the core in the middle. And then it's decorated by E1s and E3s. And the conclusion from this work is, the flexible model is the correct one. Because while the E2 core has a regular structure, the E1s and E3s were in a different position for each of the complexes that were imaged. And, in fact, in the PDB right now, it was featured as one of the molecules of the month a while ago. Here is a recent drawing of the current understanding of the structure of the PDMC multi-enzyme complex. So this illustrates how tomography can be applied to small objects like individual macromolecular complexes. And it can be used to resolve even conformational states. Here we were able to resolve the key information that was missing in the crystal structures. Namely how are all of the individual components arranged in 3D. And we were able to visualize which ones are flexibly tethered and which ones had a regular structure. So tomography is used to produce three-dimensional reconstructions of unique objects in size, all the way from an organelle like a Golgi apparatus to an individual macromolecular complex.