One of the key steps in any single particle reconstruction is generating the initial model. And obviously, the initial model has a high impact on the final model because part of the process is really just an iterative refinement of that initial model. So the initial model can heavily bias the result and it's very important to then start with a reasonable initial model. The procedure I described of finding common lines in between the transforms of different class averages, is what I'm calling here the de novo using common lines method of generating an initial model. It comes straight from the first set of class averages. And so in that sense it's unbiased because it comes right from the experimental data. However in many single particle projects, the initial class averages are so noisy that the first generation of the initial model. It's very difficult to determine the common lines precisely and so the initial model has significant errors in it. And so there's other ways to generate initial models. Another one is to use a known partial structure. For instance, a lot of the single-particle reconstructions that have been done to date have been done of ribosomes. And, in that case, the previous structure can be used for the initial model. This works particularly well if the new structure trying to be determined is just the core ribosome with a different elongation factor, or something bound to it, then using a known structure of the core ribosome is very effective. Two other related techniques are called random conical tilt and orthogonal tilt. And these important techniques, we'll go through them in a second. But they also lead to an initial model. In addition, tomography can be used to generate an initial model of a single individual particle, or perhaps, tomographic reconstructions of a number of individual particles could be produced, and those could be averaged, as we described, to produce a sub-tomogram average. And that could have enough features in it to serve as a very good initial model. And finally, for some structures it's been shown that simply guessing a very low resolution model with maybe balls and sticks in different places can be sufficient to guide the very first classification of the images into initial class averages that then refines effectively into a true structure and so obviously each of these methods to generate an initial model has it's advantages and disadvantages and so for each particular case they should all be considered. But now let's look in more detail at this method called random conical tilt, using this figure from Andres Leschziner's web page. The idea of a random conical tilt is you take a field of particles that have frozen in the ice. And you first tilt it to 50 degrees or so and take a projection image through it. After you take that image, you rotate the field of particles into an un-tilted orientation and take another projection image. So that's what's being depicted here. This is a single particle in that field. And you first tilt the field up to 50 degrees and record an image. Then you un-tilt the field of particles and record a second image. Now then, if you take the whole set of particles from the untilted images, and you classify them. That's what's being depicted here. Suppose this is a single picture in the untilted orientation of one particle, and this is a picture of a second particle in the untilted orientation. And you may have thousands of pictures like this, but right now we're just considering two of them and from those images, you classify them and find sets of particles. They're all presenting the same view, except they may be related to each other by a simple in plane rotation. Well, if you take this set of particles that you know is presenting the same view in the untilted orientation, and you align them with their in plane rotation, you can then consider them a class. A class of particles all presenting the same view here. But now we can ask, if all of these images were presenting the same view, what is the relationship between all of their tilt pairs? The images that were first taken at a 50 degree angle and the answer is, okay, given all of these particles presenting the same view in the untilted image, their tilt pairs now present all of the views that you could get on a cone-shaped surface of perspectives. And that's why it's called a random conical tilt. It's because you get a cone of images of the particle in the 50 degree angle image. And you can take that set of images. And you know the relative orientation of each one because you know the relative orientation of un-tilted counterparts and you know the angle between each of these untilted images and the tilt pair. And so you can take that set of more precisely known orientations and produce a three dimensional reconstruction of a particle from each point of view that you obtain in these class averages. Now you might ask, why do this instead of just recording the untilted projection images, classifying all the particles, finding common lines, and producing a three dimensional reconstruction? Well the answer is that in many cases it's easier to determine which of a large number of images are actually presenting the same view of a particle. Which is what you have to do in order to form this class. It's easier to do this. You can be more successful and precise in doing this. Then you can of taking independent class averages and then novel detecting their common lines And orienting them in reciprocal space. And so, this procedure can be more successful in generating a reliable initial model of the particle. And so this is why during the data collection module, I showed that sometimes in single particle analysis. We record tilt pairs and that's to do the random conical tilt reconstruction method. You might first have an image of your field of particles at 45 degree tilt. Where here you see the individual particles and this is showing the tilt access here and then you record another picture at 0 degree tilt angle. And here is showing the tilt axis. And you can match a particle here in this image with a particle in that image. And then you can generate a set of images at 0 degrees and their tilt pair counter part, let say 45 degrees in this case. You take the 0 degree images and you align and classify them And pick out of all your 0 degree images which ones are really exhibiting, presenting the same point of view. And then you form your reconstructions out of their specific Tilt pair partners. Now, if you think about it, if you had an object of interest, and you used projection images to find by perspectives that map on the surface of a cone, like this random conical tilt method gives you. And each projection image gives you the amplitudes and faces of a central section through the 4A transform. If you were to look at the situation in reciprocal space and show each central section here as a line where you have data, then all of the central sections that you collect define a volume depicted here with a missing cone at the top of angles that you don't collect. And so here's a cross-section through this object, showing that there is a missing cone of data, that is not collected by this procedure. And because there is a missing cone, all of the features in your initial reconstruction would be smeared in the direction of the missing cone. Just like we explored in more detail in the case of tomography. It was this limitation that prompted the development of a related technique called the orthogonal tilt reconstruction method. It's very similar. It starts with taking that field of particles and tilting it to, say, forty five degrees, and recording a projection image. But then the second image is not just of an untilted sample, but you continue tilting it to a negative 45 degrees to collect the second sample. In other words, this first image is at, say, positive 45 degrees, then you tilt the sample all the way to negative 45 degrees to record the second image. So now the angular difference between your two images are a full 90 degrees. Now given a whole set of images, where you have tilt pairs at minus 45 and plus 45 degrees. You can take all of the images at one of those angles, say the minus 45 degrees, and identify particles that are all presenting the same point of view in that tilted image except for, perhaps, an in-plane rotation. And you'll align all of these to form a class. Then notice that the first images, each partner in those tilt pairs now populate the range of views that go all the way around a circle, a flat plane of views of a particle. And think now about the tomography technique. This allows you essentially to record a tilt series of your particle all the way around the full circle of possible angles and so these images can then be used to produce a three dimensional reconstruction of the object of interest with no missing cone at all, and fewer distortions. So what are the similarities and differences? Well both the random conical tilt method and the orthogonal tilt reconstruction method allow you to generate an initial model that in many times is going to be more accurate than an initial model from the common lines method because it relies on detecting particle images that are really presenting the same view and that is, it can just be more accurately found. A difference is that in the random conical tilt approach, your images are taken at a tilt angle, and then an untilted position, and as a result, there's a missing cone of angles. In the orthogonal tilt of reconstruction method, the images are recorded at minus 45 degrees and positive 45 degrees, and this allows a more isotropic resolution in the reconstruction. However, sometimes the images of untilted specimens are higher quality than any tilted image that you can get. For some of the reasons that we've described how particles, the ice flows and bulges in an image, all of these problems are less severe in the image of an untilted sample. Tilted samples of the eye starts bulging, particles move more drastically, and so, there can be an advantage in working with untilted images here. But the disadvantage is that you have a missing cone. Just to show an example of the data collection for orthogonal tilt reconstruction, here's a field of particles recorded at minus 45 degrees, and here's the tilt pair collected at plus 45 degrees. And enlarged inserts of sets of particles from both images. Now in addition to recording tilt pairs, it can also be advantageous to record images with different defocus values. Sometimes we even record focused pairs of images. And the reason we do this is that including particles recorded with different defoci can fill in gaps in the contrast transfer function. And again, let's return to this picture. These are images of a field of icosahedral viruses. Here's one virus particle, here's another virus particle, here's another virus particle. And this picture was taken far from focus where the contrast transfer function had its first maximum here closer to the origin. So, with low frequencies being emphasized and that's why you can see the particles. This is the focal pair. This image was recorded closer to focus, and because of that the contrast transfer function oscillates less rapidly, and the first maximum is pushed to higher resolution. And so this image contains high resolution details at the expense of it being more difficult to find and align the particles here. This image contains the low resolution details. It's easy to see the particles and see where they are, but the high-resolution details are scrambled and dampened because the image is taken far from focus. The net effect of taking many pictures at different defocused values is illustrated in this figure. So imagine this is the contrast transfer function of a single picture taken at a particular D focus. So here's contrast transfer as a function of spatial frequency, and it oscillates as expected, and there's an envelope function dampening it at high resolution. Now, let's suppose we took our next picture slightly further from focus. Which has a contrast transfer function here in the purple line. And so it's maximum is closer to the origin and it oscillates a little bit more rapidly. Now you can see that some of the spacial frequencies that are missing in the red curve, like this one right here, is strongly present in the other image, because it has a CTF maximum in that location. And suppose we took hundred and maybe even thousands of images, and each one had a different defocus value. Here if we draw each of the CTF curves in a different color and superimpose them and then add them them all up in here in panel D, we see that by adding lots of images with different defocus values, we start to collect at least some data at all of these different spacial frequencies Now this dampens out at higher resolution because of the envelopes that reduce the signal at high resolution. But nevertheless, this is a way to collect the signal at all the different spatial frequencies. Now let's talk about taking advantage of the new capability of direct detectors to record in what has been called movie mode. Where you can record sub-frames during an exposure. And so we're going to talk about taking advantage of sub-frames. If your detector is a CCD, CCDs generally take a very long time to read out, 5, 10, 20 seconds to read out. And so they deliver single images. And so let's imagine an image here where we have a single particle there, and there, and we have a couple over here. And in a typical scenario, you might invest 20 electrons per square angstrom into that image. And it might be delivered in a total of about one second, like I say, and this is on a CCD camera. If however, you have a direct detector that can deliver images in movie mode, you might instead collect a series of images. And you might collect an image every one 20th of a second. And each image then will contain the signal from one electron per square angstrom. This is on a direct detector. And let's suppose you record this movie mode image for a total of three seconds. And so this corresponds to a total dose delivered of 60 electrons per square angstrom throughout the whole movie. Then you're going to have a series of images 1, 2, 3, 4, up to 60 images in the entire movie. And each of these images will contain pictures of these single particles. And one can ask, what is going to be different about each of the frames, these are called sub-frames, of the movie? What is going to be different about each sub-frame? Well, one big difference between these frames is the radiation damage contained in the particles at the time the picture was recorded. These particles are absolutely fresh. And so this image is of a more nearly native state. And with each progressive sub-frame, the structure of the proteins is going to be damaged. And so by the 60th sub-frame, the high resolution information is going to be lost. These are more damaged particles. Also, there's going to be specimen movement between these sub-frames. So for instance, if the ice is doming during the movie, you might see one of these particles flowing and moving with respect to each other during the movie. But this allows us to do some really sophisticated corrections. So the first is that you can do motion correction by identifying a particular particle in each one of the sub-frames. Then using cross correlation, you can find and see how that particle is shifting throughout the movie and correct for those shifts computationally, to get a motion corrected, crisp image of that particular particle. This is illustrated in this beautiful figure from 2012. Here is an icosahedral virus, and another virus and another virus, and this was an image that was recorded with 60 sub-frames. And if those 60 sub-frames were simply averaged, this is the quality of the picture that resulted. But, when the motion between each sub-frame was estimated, and then each sub-frame was shifted to align with the others. And then those aligned sub-frames were averaged, then this picture resulted, of the viruses. And you can clearly see that there's much more high resolution detail. This is a much crisper image which will clearly go to much higher resolution in the reconstruction than the uncorrected image. Now after the relative orientation, and the centers of each of these particles are determined to high precision through the standard iterative three-dimensional reconstruction process. The acquisition of sub-frames allows one to then calculate full reconstructions using the alignment parameters obtained from the motion corrected images. But only include certain sub-frames in a later reconstruction. Another possibility that these movies allow is to have sub-frame selected reconstructions. What this means is that one could take the motion corrected composite image of each of the particles in the field of view and use them to determine the relative orientations of all of the particles. And produce a three dimensional reconstruction using all of the sub-frames. But, once the relative orientations of all the particles are known, one could just take the images from the first sub-frame. And using only the first sub-frame, produce a complete three-dimensional reconstruction. And this three-dimensional reconstruction may have a lower signal-to-noise ratio, because it's only the first electron per square angstrom of each of the images. However, the particles themselves are not radiation damaged yet, so they might show a high quality three-dimensional reconstruction. Whereas if you produce a three-dimensional reconstruction just from the last sub-frame, that could be a low resolution result because of radiation damage. But you could also produce a 3-D reconstruction of, say, just sub-frames two through four. You could have a two through four three-dimensional reconstruction. And in fact when people have done this, you can plot the resolution of each of these reconstructions that you might do. And we'll have, let's say, high resolution up here and low resolution down here, and we can plot it as a function of this sub-frame. So here would be data from the first sub-frame, and this is the second, and this is the third, and this is the fourth, etc. All the way to the 60th sub-frame, for instance. And what's been observed is that the first couple sub-frames are no good. And the best reconstruction comes from the third or fourth sub-frame. And then the resolution goes down from there as radiation damage takes over. And it brings up the very interesting question, why isn't the first sub-frame, why doesn't that produce the highest resolution reconstruction or even the second sub-frame? And nobody really knows the answer to that question yet. There's something about the very first image where, these images the image quality is just poor. And so it takes just a very small dose before you start getting high quality signal. And then from there, radiation damage deteriorates the information content of the later sub-frames. An even more Sophisticated way to exploit the sub-frames is to apply custom, Band pass filters, To each sub-frame in the movie so for instance in this last sub frame, One might apply a low past filter to this sub frame knowing that the high resolution information here has been lost because of radiation damaging. And you might want to throw that out and just focus on the low resolution information in this sub-frame to help align the rest of the particles. But you might want to have a generous band pass filter applied to the early sub-frames because they contain the high resolution information that you need in the reconstruction. Now this approach was very recently developed and demonstrated by Sjors Scheres, and here's a figure of his paper. On this axis is frequency, in reciprocal angstrom, from low frequency to high frequency. And on this axis, is plotted the relative weight pf each sub frame being involved. So in the data collection process there was 17 sub frames collected of each image and then the width here shows how much at each spatial frequency. Let's say take a particular low spatial frequency here, you see that if this is 10%, only about 5% of the average comes from the first sub-frame, and maybe 6% of the average comes from the second sub-frame, and more comes from the third sub-frame. The third sub-frame had good, high resolution information here. And so the width of this stripe shows how much of the total information in the ultimate reconstruction came from that particular sub-frame of the movie. And so looking out here first at high resolution you can see that the very first image contributed almost nothing at high resolution. And that's because the first images are always a poor quality. The second was also contributed little. The third there was filtered somewhat. The most influential sub frames at high resolution were 4, 5 and 6. These contain a lot of the information here came from those sub frames and very little from the later sub frames, in fact sub frame 17 hardly contribute anything, because radiation damage has already damaged the particles at that point. However, at low spacial frequency, the sub-frames all contribute significantly. Now, one of the most powerful advantages of single particle analysis is it's ability to deal with sample heterogeneity, and to illustrate that I'm going to use this figure from a very recent paper reporting the structure of the human mitochondrial ribosome. So here's an example image, kirlian image of purified human mitochondrial ribosomes. And after each particle was selected and classified and aligned, these class average among others were calculated. So here are six different class averages among many that were produced. And so far we've only talked about the idea that each different class average could represent a different view of the particle of interest. But in addition to that, maybe the sample is heterogeneous. And so some of the class averages may represent particles with more or less mass, or particles with different conformational states. And so in this case, using maximum likelihood methods from all of these particle images, there were six different 3D reconstructions produced. Presumably of different classes of particles that existed in the original data set. And it was recognized that this class of particles was just the large ribosomal sub-unit not with a small rivisobal sum unit, also with it. These particles and these particles were deemed lower quality because they didn't produce high resolution reconstruction and so they were removed from further analysis. But all the particles that produced this reconstruction and this reconstruction and this reconstruction were once again pooled together and iteratively classified, again using maximum likelihood methods to produce four new reconstructions. Now these two reconstructions only contained a small number of the original images, and these contained the major fraction of the images. These reconstructions arose from complete particles, but they were in two different ratcheting states of the ribosome. Now the ribosome has a large subunit and a small subunit, and during translation they ratchet back and forth with respect to each other. And so it was observed that these two reconstructions represented different ratcheting states of the ribosome. In order to push to the highest resolution possible given this image set, the authors then applied a mask around just the large ribosomal subunit in these images, and just the large ribosomal subunit in these images as well. So here's an illustration of the mask, here's a class average with an entire particle. But here is that same class average with the mask applied just around the signal containing the large, ribosomal sub unit. And so, all of these 108,000 particle images were re-aligned based only on the signal present within this mask, based only on the signal of the large ribosomal sub-unit. And by aligning them all only on that sub-unit, they then yielded a reconstruction with even higher resolution. In fact, in this case the final resolution was assessed to be 3.4 angstrom resolution, which allowed the generation of an atomic model. So the point is, that in this case, single particle analysis allowed two different kinds of heterogeneity to be sorted out. In this initial population of particles there were both full ribosomes and also just partial ribosomes. And so classification allowed these partial ribosomes to be ignored. In addition, this initial particle set had particles in two different conformational states and, again, single particle processing enabled just the common element in these two different confirmationally heterogeneous states, just the common element to be mutually aligned to produce a high resolution reconstruction. So let's try to summarize starting with a sample. Obviously, the first step is to record the images. The next step is to align and classify those images to produce class averages. And this step has sub-steps. First of all, one has to identify the particles. Next, if possible the particle images should be motion corrected. This is also a good time to CTF correct each image. And then each particle image needs to be aligned to the rest, and classified. And this results in class averages. The next step is to go from class averages to a model of the particle. And the substeps here are to first select the highest quality class averages, and by the way, within those class averages, one can look at the images that went into them, and usually a lot of the images are clearly belonging to that class and they're high quality images. But sometimes, there's some other images that classified with that set that aren't so clearly belonging, and so sometimes this also involves throwing out or selecting the best images within each class to make a cleaner class average. And once you have cleaner, high-quality class averages, to find their relative orientations, then one can fill 3D reciprocal space, interpolate. And inverse Fourier transform, to produce the model. Next, one takes the model and re-projects it in a range of directions to produce three projections, so the step here is to re-project in all directions. Using the re-projections, one then realigns and reclassifies the original experimental images to produce refined class averages. So here the original experimental images are realigned and reclassified. And again, the best class averages are selected and just the best particles within each class average. Those are selected to produce the refined class averages. And given the refined class averages, one then reinterpolates in 3D reciprocal space and then inverse Fourier transforms back to get a refined model. And this process is iterated over, and over, and over until the model converges to the final result. Now, in each iteration, as the model progresses to higher resolution, one of the parameters that can be changed along the way is that as you re-project the model in all directions, one changes the angular step size. Because in the initial case, earlier on, like the initial model, one might project that model at a very course angular step and compare the experimental images with those re-projections. But as the model reaches higher and higher resolution that won't be sufficient. And so one needs to project the model into very fine angular steps to represent all of the subtly different orientations contained within the experimental dataset. In addition, as the particles are realigned and reclassified, different band pass filters can be applied. Again, one might start with low pass filters to focus on the low resolution information for the initial models. But once the model reaches higher and higher resolutions, it's important to allow more of that high resolution information, or to focus on the high resolution information contained in the images, both to align the particles and also as they contribute to the average. Now, we also talked about several methods that one might use to generate the initial model, so let's call this an initial model. Now we mentioned that it might come from de novo, I'll say de novo using the common lines approach. But a more accurate initial model can also be obtained from random conical tilt or the orthogonal tilt reconstruction methods. In addition, one might have a known structure that can be used as a starting point. Maybe it's a lower resolution structure or maybe it's just a part of the particle that's being studied, but a known structure can be a great starting point. In addition, one might use tomography to generate an initial average, or one might simply build a model, say with balls and sticks, in some cases, that's sufficient. Now obviously, there's lots, and lots of variations on this very generic flowchart of the steps involved in 3D reconstruction. There's lots of different kinds of samples with all their own particular caveats, and so this is just meant to illustrate the kinds of operations that happen. But as you read different papers and as you start to do this yourself, you feel free to rearrange things in whatever order seems to make sense for that situation. Now one of the ways that single particle reconstruction can go terribly wrong is if referenced bias is introduced at any point in the process. Referenced bias is when any assumptions that you made during the process about the structure of the object end up influencing the final result in a wrong way, and it can be illustrated using this figure from a JSB paper in 2009. What these authors did is to generate 1000 images of random noise, and then they aligned each of those images as if they were experimental cryo-EM images to a reference and the reference they chose was something like this. I just pulled this off the internet, a picture of Einstein, but by aligning their noise images all to this, the alignments had the effect of enhancing any kind of feature that was present in the reference. So that when the 1000 noise images were then averaged together, it clearly recapitulated some of the features in this average of the original reference, and obviously there was no information about Einstein's face in the original images. It came out in the average, simply, as an artifact of reference bias of the image they used as a reference. Reference bias can creep up in many different steps in a cryo-EM reconstruction. For instance, even at the very beginning, identifying particles. If you use some known structure that you think your particles look like and you search the micrographs for objects that look like what you're looking for, you're at risk of identifying regions of perhaps even pure noise that happen to look a lot like your reference. Just like in the case of Einstein from noise that I just showed. So you have to be careful that the particles that you pick are true particles. Another step is here in the alignment procedure. You may pick images of true particles but if again if you align them using a known reference. You may bias the alignment of each of those images and so when they're averaged they'll look a lot more like the reference than they should. Another step is if you use a known structure as your initial model. Here in this step where you find the relative orientations if you take your known structure and align all of your particle images to that known structure there's a chance that you'll bias those alignments. So that the average again looks more like the known structure than it should. Sometimes investigators will take the set of re-projections at that model. And use it to search the images again to identify more or more accurately identify the particles in the images that originally existed. And if the re-projections have bias within them, they don't always do, but if you followed some kind of procedure that biases your re-projections to look like some assumed structure that really isn't there. It's possible that when you re-look for particles you'll find regions of noise that look a lot like that assumed structure. Now this doesn't mean that it's never okay to use some kind of known structure to either identify the particles, or align and classify them, or find their relative orientations. It just means you have to be careful. One of the keys to look for is that whatever structural details you find in the end of your project, if they are not at high resolution, and if they aren't reproducible and at higher resolution then the details that were present in the initial reference, then you need to be weary. For instance, if you use even a PDB structure, for instance of some component that you think is part of your particle, and follow this process and end up with reconstruction, that beautifully recapitulates the structure that you used as the reference. Well, then you need to be careful that you may not have discovered anything. You may have simply reproduced that reference. If however, you use a very low resolution reference, and you can reproducibly generate 3D reconstructions of much higher resolution, that have reproducible details. I mean, if you were to take your data set and split it in half and use this low resolution reference, and use it to generate a reconstruction that had much higher resolution details. And you did it with the other half of your data set as well, and you generated a reconstruction with much higher resolution details. And those high resolution details matched between the results, then, you've discovered something new. And just as there's lots of different variations on the image processing pathway to produce a three-dimensional reconstruction. So to there are lots of packages, software packages available now for the steps. So here is a Wikibooks webpage that I captured the other day, listing the different software tools for molecular microscopy or general packages for cryo Image processing. So there are now 17 different packages on the list, and that number keeps growing. And there's a lot of activity in this field, they're developing better and better algorithms to handle single particle reconstructions.
