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Estimating microscope CTF parameters.
In cryo-EM imaging, protein samples frozen in ice are imaged using a transmission electron beam. The protein molecules are not substantially more opaque to electrons than the ice, so simply passing the beam through the sample would not result in significant contrast in the image. Instead, the microscope is operated in "phase contrast" mode, where it is not the brightness but actually the interference of the electron beam (with itself) going through the sample that creates contrast, and allows for imaging differences in density rather than opaqueness.
As electrons interact with the sample under study, there is a finite probability that they will scatter (i.e. their path will be changed). This scattering can be elastic (the electron keeps all of its energy) or inelastic (the electron loses some energy to the sample). Only the elastically scattered electrons contribute usefully to the resulting "phase contrast" image, so the following discussion considers elastically scattered electrons only.
In quantum mechanics, electrons are modeled as a "wavefunction" - an oscillatory function of both space and time, the squared magnitude of which corresponds to the probability of finding the electron at a given position/moment, if a measurement were to be made. When an electron is elastically scattered by a cryo-EM sample, this wavefunction is scattered in many directions simultaneously, and the amount of signal scattered at a particular angle is proportional to the spatial frequency spectrum of the sample. For example, if a sample contains a lot of signal at high spatial frequencies, then more of the electron wavefunction will be scattered at high angles. These scattered waves are subsequently focused by various lenses in the microscope to form an image, but because of their different scattering angles, they traverse a different total path length to arrive at the electron detector. These differences in path length manifest as phase shifts in the electron wavefunction, causing the scattered components of the wavefunction to interfere with each other. Depending on spatial frequency, this interference can be constructive, destructive, or inverting, and as a result, some spatial frequencies show up in the resulting image more strongly than others, and some are flipped in sign. This phenomenon is mathematically modelled by the microscope's Contrast Transfer Function (CTF). It is important to estimate and correct for the CTF during cryo-EM image processing, otherwise the achievable reconstruction will be of very limited resolution.
A simplified, one-dimensional model of the CTF can be expressed via the equation below:
is the wavelength of the incident electrons,
is spherical aberration, and
is spatial frequency.
represents a phase shift factor, which is important e.g. when a phase plate is used. Though the CTF function used by CryoSPARC is two-dimensional (to account for astigmatism, third, and fourth order aberrations), it reduces to the above expression in the simplest case where both defoci are equal and higher-order aberrations are absent. Notice that the cosine argument is a function of defocus, so defocus affects the relative strength of different spatial frequencies in the resulting image. In practice, low defocus tends to lead to less contrast at low spatial frequencies. Given the amount of noise present in cryo-EM images, this means that particles at low defocus are harder to see and pick out (both manually and algorithmically) than particles at high defocus. For an example of this phenomenon, see the figure below.
Example (synthetic) particle images at various defocus values.
One very important concept to note about the CTF is that the dominant term in the CTF is due to defocus. Defocus can be caused by changing the power of the lenses in the microscope so that the actual sample is no longer in focus. Conversely, physically moving the sample up and down in the microscope column moves it towards/away from the focal plane, causing exactly the same effect as defocus. This is important because it means that if a sample is not flat, the different parts will be at different levels of defocus.
It is well-known that the vast majority of cryo-EM samples are not "flat". Particles tend to concentrate near the air-water interfaces prior to the sample being frozen, and the ice surface itself is often nonplanar. Recalling that defocus affects the CTF, this means that a single image can contain particles with different defoci and therefore different CTFs. CryoSPARC provides a patch-based CTF estimator, which examines many different areas in the micrograph to compute a "defocus landscape", to combat this issue. The patch CTF estimator requires no prior information about particle locations within the micrograph, and can be used immediately after motion correction. It can even work on tilted samples without knowing about the tilt beforehand.
An important aspect of CTF estimation is the effect of aliasing. If you inspect the third column of the figure above carefully, you'll notice that the CTF curve appears to have oscillations with varying amplitude and at a frequency which seems to increase and then decrease. But how can this be the case, you may ask, if the CTF model dictates that the curve should oscillate between -1 and 1 with increasing frequency? This effect is an undesirable artifact known as CTF aliasing.
CTF aliasing can occur when we represent a continuous CTF curve with a discrete set of samples in the Fourier domain. Any discrete sampling can only represent a periodic function with certain frequencies -- namely, it can account for all frequencies up to the Nyquist rate. All other 'faster' oscillations are 'aliased' into the frequencies below the Nyquist rate (see figure below) and result in the complex oscillations we observe in the aforementioned plot.
Aliasing conflates frequency information. Here, when trying to sample a continuous signal with true frequency of
(shown in black dashes, and in grayscale in the top half of the image), the sampling rate is not high enough to faithfully recover the signal. Thus, the sampled points (black dots, shown in grayscale in the bottom half of the image) appear to show the signal oscillating at an apparent frequency of
, which is lower than the true frequency. This occurs because the true frequency of the continuous signal is higher than the Nyquist frequency of the sampling rate.
The Fourier domain represents spatial frequency (in units of 1 / Angstrom). Its range is from 0 (the DC component) to the real-space Nyquist rate
. For a typical pixel size of 1 Angstrom / pixel, the spatial frequency will range from 0 to 0.5 (as is the case in the above plots). For a particle box size of
, each Fourier pixel accounts for
in spatial frequency. This, in turn, defines a Nyquist rate (in the Fourier domain) of
. Any oscillations above this rate in the CTF curve will result in CTF aliasing.
CTF aliasing can corrupt higher frequencies and its effect will usually be most evident at higher defocus values. To remove its effect, we can either increase the particle box size
or the pixel size
. The former approach will reduce the spatial frequency extent of each Fourier pixel, and thus increase the maximum frequency content we can represent. The latter approach will remove the effect of high frequency oscillations by reducing the spatial frequency extent of the entire image and effectively 'zoom into' a portion of the Fourier domain. Both approaches are shown in the figure below.
Sampled CTF at different box and pixel sizes compared to the continuous 'unaliased' CTF model (see equation above).
Note that CTF aliasing is often best visualized via a CTF 'image' that depicts amplitudes as pixel intensities (see figure below). Using this new format, we can more clearly identify when aliasing begins and decide whether this is deleterious for the estimated defocus value, desired target resolution, and CTF fit. Importantly, the effect of CTF aliasing at higher resolutions and at high defocus may be made moot by the poor CTF fit in this domain.
CTF curves (same data as above) visualized as images with coloured intensities. Visualized this way, the effect of CTF aliasing can often be made clearer.
To see if CTF aliasing occurs for a given set of experimental parameters in 2D, consider using this very useful online script, written by Takanori Nakane at MRC-LMB, which produces a 2D CTF image similar to the 1D images above.
To help tie together all of these concepts, we've created a short animation (see below) to illustrate how a 2D CTF plot, like the one you might find in the diagnostics section of a Manually Curate Exposures job, is related to the 1D curves and images we present here. Note that we omit many details for clarity (e.g., astigmatism, equi-phase averaging, etc.).
Relationship between a 2D CTF fit (shown above the power spectrum of a micrograph) and 1D CTF curves.