1:19

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.

3:39

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.

5:57

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.

7:28

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.

8:41

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

9:32

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.

11:48

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.

14:38

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

16:33

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.

18:31

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.

23:21

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.

25:53

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

27:10

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.