This class covers the fundamental principles underlying cryo-electron microscopy (cryo-EM) starting with the basic anatomy of electron microscopes, an introduction to Fourier transforms, and the principles of image formation. Building upon that foundation, the class then covers the sample preparation issues, data collection strategies, and basic image processing workflows for all 3 basic modalities of modern cryo-EM: tomography, single particle analysis, and 2-D crystallography. Philosophy: The course emphasizes concepts rather than mathematical details, taught through numerous drawings and example images. It is meant for anyone interested in the burgeoning fields of cryo-EM and 3-D EM, including cell biologists or molecular biologists without extensive training in mathematics or imaging physics and practicing electron microscopists who want to broaden their understanding of the field. The class is perfect as a primer for anyone who is about to be trained as a cryo-electron microscopist, or for anyone who needs an introduction to the field to be able to understand the literature or the talks and conversations they will hear at cryo-EM meetings. Pre-requisites: The recommended prerequisites are college-freshman-level math, physics, and biochemistry. Pace: There are 14.5 hours of lecture videos total separated into 40 individual “modules” lasting on average 20 minutes each. Each module has at the end a list of “concept check” questions you can use to test your knowledge of what was presented. As the modules are grouped into seven major subjects, one reasonable plan would be to go through one major subject each day. That would mean watching a couple hours of lecture and spending another hour or so thinking through the concept check questions each day for a week. Another reasonable plan would be to go through one module each day for a little over a month, or even three modules a week (Monday, Wednesday, and Friday) for a 3-month term. It is likely that as you then move on to actually begin using a cryo-EM or otherwise engage in the field, you will want to repeat certain modules.
Envelopes
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来自 Caltech 的课程
冷冻电子显微镜（cryo-EM）入门
215 个评分
从本节课中
Image Formation
与讲师见面
Grant J. JensenProfessor of Biophysics and Biology; Investigator, Howard Hughes Medical Institute
Biology and Biological Engineering
Now, in addition to oscillating,
the contrast transfer function of an electron
microscope is damped at high spatial frequencies.
And so, if we plot the CTF as a function of spatial frequency.
We see it not only oscillating, but
also being damped at higher spatial frequencies.
That's a contrast of 1 and minus 1.
We see that it's damped and the shape of the damping.
We call an envelope function.
[SOUND] And now, I'd like to talk about the causes of this damping.
So let's imagine that our sample has a profile something like this.
Okay.
A high density, a peak and a subpeak next to it.
And then let's imagine that we image that with an electron microscope and
one electron is heading straight down the column in this direction.
And because of that, it will produce an image on the image plane.
That closely resembles the sample.
But what happens at the next electron that comes down the microscope,
comes down at a different angle than the former?
And so instead, it passes at an angle say, like that.
Then it's going to produce an image on the image plane that looks like that and
these two images together are going to look something like.
Now to understand what's going on here, let's think about.
If, if we were to a Fourier transform of this original sample,
there would be at least two important waves.
One, the, the major wave would have a maximum around the top
of this sample and it would have minimum at the extremities.
The next high resolution Fourier component would have three maximum here,
representing these three peaks and so it would be a higher frequency component.
Now, if we look at the images formed by these two electrons,
the first electron would carry its low frequency
component in just the right position.
The higher frequency component would also be in just the right position.
Now the second electron,
the low frequency component would be shifted
somewhat as would the higher frequency component.
And when we compare these two in detail,
we see that the low frequency components match thoroughly well.
There displacement is small compared to their wave length.
And so they match fairly well to give a major bump in the sum, but the high
frequency components are completely shifted with respect to each other.
This one and this one are completely shifted.
And so, it together,
those two high frequency components hardly contribute anything.
And as a result, the sum image, the net image that we record has the low frequency
components, but it's missing the high frequency components.
In other words, because the electrons were coming down
at subtly different angles, the image became blurred.
And when an image is blurred, it affects the higher frequency
components much more severely than the lower frequency components.
And so one of the reasons that the contrast transfer function
of an electron microscope is damped at higher spatial frequencies
is due to only partial spatial coherence of the electron gun.
Remember, spatial coherence was a measure of how uniformly each
electron that the gun produced, whether or
not it comes down the microscope in exactly the same direction.
If some of the electrons are coming down at a slightly different angle,
then the pictures that each one produces will be slightly shifted with
respect to each other and that's partial spatial coherence.
And because of that, we can actually plot the,
the impact of that shift as a function of spatial frequency.
And if we do,
we find that those small shifts don't matter much at low spatial frequency.
But as we get to higher and higher spatial frequency, they matter more and more and
more until the high frequency components are essentially eliminated from the image,
because each electron is not coming down in exactly the same direction.
Remember that each electron individually comes down the microscope and
contributes to the ultimate image and
so the next electron that comes, contributes its image.
And if these images aren't perfectly superimposed,
the sum becomes blurred and that dampens the high frequency components.
Now, in an analogous fashion, if the different electrons
coming down the microscope come with different energies,
so let's suppose the first electron comes this way.
And if we draw the objective lens,
if this first electron has energy e,
then the scattering that occurs at
the sample will be focused by the objective
lens to form an image at a particular plane.
Let's call it here.
[SOUND] The image plane.
If however, the next electron comes down the microscope with a different energy.
Let's say, e plus delta.
It is scattered from the sample again, just like the first electron.
But in this case, because the energy electron is different, if the energy
is higher, it will be, it will be less focused by the objective lens.
And so, it won't be focused until a plane that's lower.
Okay.
So this is the image plane for a higher energy electron.
Now of course, there is in a microscope, a detector further on down.
And the detector as we have discussed before is conjugate
to some plain up here where the image is being formed.
And what we see is that the, the first electron with energy e,
it's going to contribute in over-focused image to the detector.
But the second electron with higher energy is going to
contribute an under-focused image to the detector.
And so the net image is going to be the sum of different
images with different focus values.
And if you were to plot, let's plot
the CTF as a function of spatial frequency
again for the lower electron e.
If its contrast transfer function, looked like this.
The contrast transfer function of the higher energy electron,
which is now under focused in comparison is going to look like this.
And if we were to add these together to see what kind of net
contrast transfer function there was in an image that was
the sum of many images with different defocus,
we would see that the average CTF starts out fairly strong.
But at this point, it starts to be weakened by the fact
that these two curves are no longer in phase.
And essentially, the signal is damped at high spatial frequency,
because the individual electrons contributing to the image
are contributing images with different defocus value.
And so again, the result is an envelope that describes how
the signal is dampened at higher spatial frequency.
And this is the result of partial temporal coherence.
And so there are at least two envelope functions that
affect the CTF in the electron microscope.
First, there will be an envelope function that we'll call the envelope
due to partial spacial coherence, which reflects the fact that
not all the electrons are coming exactly the same direction down the microscope.
In addition, there will be another envelope that reflects
the fact that not all the electrons have the same energy and
this we will call the envelope due to partial temporal coherence.
In addition to this, there are other miscellaneous problems that cause
an electron microscope image to lose some of the high resolution detail and
each can be characterized as en envelope function.
The net envelope function is the product of all the individual envelope functions.
So the net envelope function is even more severe than any of the components.
So this we'll call the total envelope function.
And it's the total envelope function that defines
how the contrast transfer function is damped more and
more at high spatial frequency.
Now again, what the contract transfer function tells you is how much of
each spatial frequency is transferred, delivered into the final image.
So for instance, this spatial frequency, we might see that because
the contrast transfer function here is approximately negative 0.5,
half of the signal that is present in the sample,
half of the strength of that Fourier component that's present in
the sample is actually delivered into the image.
Here if the CTF reaches a value of say, negative 0.9,
90% of that signal is delivered into the image.
But then as it oscillates, we get to a point where at this spatial frequency,
none of it is delivered to the image.
And so on, as we progress.
And we see that the envelope functions also contribute to that damping.
For instance, if this, at this spatial frequency,
their total contrast transfer function due to oscillations and the damping.
Due to partial, spatial, and temporal coherence and
other factors causes the CTF to be only 0.2.
It means that only 20% of the signal present in the sample
at that spatial frequency is being transferred into the image.
And this is why, when you look at the power spectrum of an actual EM image,
you see that these oscillations of the CTF or the ton rings decrease in
intensity as you go to higher and higher spatial frequency.
Their intensity is lost, because if we were to plot the contrast transfer
function not only is it oscillating, but it's also being damped at higher and
higher spatial frequencies until the variations are almost undetectable.
Now these envelope functions also depend on defocus.
So going back to this picture,
a picture of viruses at high defocus delta z and
this picture at low defocus delta z.
If I were to plot the contrast transfer function for
this image as a function of spatial frequency,
the envelope functions that limit the contrast
transfer function would attenuate more in the high
defocus image than they would in the low defocus image.
In this case, the envelope functions would be more generous.
And as a result, the CTF here in addition
to oscillating more rapidly would also
attenuate more quickly at high spacial
frequency than in this case, closer to focus.
So the advantages of taking pictures far from focus is that
you have stronger, low resolution frequencies here.
So, it's easier to see your particles.
But the high resolution details are more difficult to recover for two reasons.
First of all, the contrast transfer function is oscillating rapidly.
And second of all,
the envelope functions dampen the signal at those high spatial frequencies.
In comparison, images at close to focus, the high resolution details are going
to be more easily recovered, because the oscillations are slower and
because the envelope functions are more generous.
But this comes at the cost of a loss of low spatial frequency signal,
which makes it difficult to see the particles.
