Low frequencies may be recovered by phase plates introducing a phase shift to the
CTF and recovering low resolution signal [29]. Phase shift of Pi/2 in the (10.2)
changes the sine-function to a cosine amplifying low-resolution contrast. The
increased low resolution contrast allows recording data close to focus, thereby minimizing the envelope function due to defocusing and avoiding errors in defocus
determination (Sect. 10.3.2). Volta Phase plate (VPP) preserves the resolution at least
to 3.2 Å as shown by a single particle cryo-EM reconstruction of 20S proteasome
[30]. VPP was successfully used for cryo electron tomography [31] and subtomogram
averaging [32]. However, the full potential of VPP for cryo electron tomography is
still to be implemented.
10.3 Resolution of Tomographic Reconstructions
Electron tomography allows resolution on the order of nanometres and therefore is
typically performed on well-preserved samples that have been high pressure frozen,
freeze substituted, or otherwise cryo-preserved. Details of tomographic acquisition
and reconstruction are discussed in Chap. 9 of this book. There are two types of
resolution limits—defined by geometry of acquisition and by SNR of the actual
data in the projections. Geometrical limits of tomographic reconstructions were
quantitatively discussed previously [33, 34]. While the geometrical limits apply to
all the types of sample preparation, the SNR considerations are particularly
important for frozen hydrated samples where the resolution is not limited by sample
preservation.
10.3.1 Geometric Limits
Three-dimensional reconstruction from 2D projection is based on the central slice
theorem: inserting a Fourier transform of each 2D image into the 3D Fourier space
perpendicular to the projection direction followed by an inverse Fourier transform is
equivalent to back-projecting the slice into the 3D volume (Chap. 9). For the most
commonly used single-axis tomography around Y-axis this may be reduced to a
series of independent reconstructions for the x-z slices, therefore only one slice may
be considered in 2D (Fig. 10.3). The first geometric limitation of electron tomography is that the sample has non-negligible length in the direction perpendicular to
the electron axis Z (“slab-geometry”), therefore with tilting by an angle Theta it
becomes effectively thicker. The thickness increases as 1/cos(Theta) which is
twofold for Theta = 60, threefold for Theta = 70, fourfold for Theta = 75.5.
Higher sample thickness reduces the fraction of the elastically scattered electrons,
degrades the image quality and in practical terms limits the maximal used tilting
angle to 60°–70°. Missing information in Fourier space constitutes a “missing
wedge”, resulting in a resolution anisotropy in real space for a given tomogram.
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M. Kudryashev
CTF and recovering low resolution signal [29]. Phase shift of Pi/2 in the (10.2)
changes the sine-function to a cosine amplifying low-resolution contrast. The
increased low resolution contrast allows recording data close to focus, thereby minimizing the envelope function due to defocusing and avoiding errors in defocus
determination (Sect. 10.3.2). Volta Phase plate (VPP) preserves the resolution at least
to 3.2 Å as shown by a single particle cryo-EM reconstruction of 20S proteasome
[30]. VPP was successfully used for cryo electron tomography [31] and subtomogram
averaging [32]. However, the full potential of VPP for cryo electron tomography is
still to be implemented.
10.3 Resolution of Tomographic Reconstructions
Electron tomography allows resolution on the order of nanometres and therefore is
typically performed on well-preserved samples that have been high pressure frozen,
freeze substituted, or otherwise cryo-preserved. Details of tomographic acquisition
and reconstruction are discussed in Chap. 9 of this book. There are two types of
resolution limits—defined by geometry of acquisition and by SNR of the actual
data in the projections. Geometrical limits of tomographic reconstructions were
quantitatively discussed previously [33, 34]. While the geometrical limits apply to
all the types of sample preparation, the SNR considerations are particularly
important for frozen hydrated samples where the resolution is not limited by sample
preservation.
10.3.1 Geometric Limits
Three-dimensional reconstruction from 2D projection is based on the central slice
theorem: inserting a Fourier transform of each 2D image into the 3D Fourier space
perpendicular to the projection direction followed by an inverse Fourier transform is
equivalent to back-projecting the slice into the 3D volume (Chap. 9). For the most
commonly used single-axis tomography around Y-axis this may be reduced to a
series of independent reconstructions for the x-z slices, therefore only one slice may
be considered in 2D (Fig. 10.3). The first geometric limitation of electron tomography is that the sample has non-negligible length in the direction perpendicular to
the electron axis Z (“slab-geometry”), therefore with tilting by an angle Theta it
becomes effectively thicker. The thickness increases as 1/cos(Theta) which is
twofold for Theta = 60, threefold for Theta = 70, fourfold for Theta = 75.5.
Higher sample thickness reduces the fraction of the elastically scattered electrons,
degrades the image quality and in practical terms limits the maximal used tilting
angle to 60°–70°. Missing information in Fourier space constitutes a “missing
wedge”, resulting in a resolution anisotropy in real space for a given tomogram.
266
M. Kudryashev