In order to reduce the risk of aligning and enhancing noise (‘overfitting‘) or
getting caught in local minima during the alignment, an adaptive bandpass filter can
be applied to the reference during the iterative alignment procedure. In the
beginning of each iteration, this bandpass filter is set according to the resolution
determined by Fourier Shell Correlation (FSC) of two halves of the data averaged
according to the previous iteration [17]. A way to further reduce the influence of
noise on subtomogram alignment is to split the dataset into two halves that are
processed completely independently from each other (‘gold standard alignment’)
[16, 18]. By determining the adaptive bandpass filter that is applied to the reference
for alignment by FSC of the two completely independent halfsets of the data, the
resolution can be approximated essentially without contribution of artificially correlated noise [19]. Note that both independent averages still contain artificially
enhanced (‘overfitted’) noise, but it will not affect the FSC because these artefacts
are different in both subtomogram averages.
Typically, the translational search is carried out in Fourier space using the
convolution theorem, while rotational search can be performed either in real space
[17] or in spherical harmonics space using a generalized convolution theorem [19].
The latter approach to rotational sampling is also known as Fast Rotational
Matching (FRM). Since the rotational search using spherical harmonics is orders of
magnitude faster than in real space, it provides the opportunity to sample rotations
exhaustively in a sensible amount of time, while in real space, rotational search is
usually restricted around a predetermined starting angle (e.g. from template
matching). Sampling rotations globally allows a reference-free alignment approach,
in which subtomograms are initially aligned to a featureless sphere, instead of an
external reference structure. This strongly reduces template bias during the subtomogram alignment and allows determining the structure of unknown macromolecular complexes without using a reference.
9.4.3 Example Dataset
For demonstration purposes, we selected approximately 350 ribosome-containing
subtomograms from the tomogram shown in Fig. 9.2b and aligned them without
any prior knowledge about ribosome orientations using alternating translational and
global rotational search in Fourier space using FRM in PyTom. The subtomogram
average evolves from a featureless sphere to a defined ribosome within 10 alignment iterations (Fig. 9.3a). This approach can be chosen to minimize template bias
during subtomogram alignment or to align subtomograms extracted from manually
picked coordinates, if no prior structural knowledge about the macromolecule of
interest is available. From the example dataset depicting ER membrane-associated
ribosomes used for illustration, 17,500 subtomograms were selected and aligned
using simultaneous translational and restricted rotational search in real space following either the ‘conventional’ or the ‘gold standard’ alignment approach
implemented in PyTom. Both approaches yielded essentially identical densities
244
S. Pfeffer and F. Förster
getting caught in local minima during the alignment, an adaptive bandpass filter can
be applied to the reference during the iterative alignment procedure. In the
beginning of each iteration, this bandpass filter is set according to the resolution
determined by Fourier Shell Correlation (FSC) of two halves of the data averaged
according to the previous iteration [17]. A way to further reduce the influence of
noise on subtomogram alignment is to split the dataset into two halves that are
processed completely independently from each other (‘gold standard alignment’)
[16, 18]. By determining the adaptive bandpass filter that is applied to the reference
for alignment by FSC of the two completely independent halfsets of the data, the
resolution can be approximated essentially without contribution of artificially correlated noise [19]. Note that both independent averages still contain artificially
enhanced (‘overfitted’) noise, but it will not affect the FSC because these artefacts
are different in both subtomogram averages.
Typically, the translational search is carried out in Fourier space using the
convolution theorem, while rotational search can be performed either in real space
[17] or in spherical harmonics space using a generalized convolution theorem [19].
The latter approach to rotational sampling is also known as Fast Rotational
Matching (FRM). Since the rotational search using spherical harmonics is orders of
magnitude faster than in real space, it provides the opportunity to sample rotations
exhaustively in a sensible amount of time, while in real space, rotational search is
usually restricted around a predetermined starting angle (e.g. from template
matching). Sampling rotations globally allows a reference-free alignment approach,
in which subtomograms are initially aligned to a featureless sphere, instead of an
external reference structure. This strongly reduces template bias during the subtomogram alignment and allows determining the structure of unknown macromolecular complexes without using a reference.
9.4.3 Example Dataset
For demonstration purposes, we selected approximately 350 ribosome-containing
subtomograms from the tomogram shown in Fig. 9.2b and aligned them without
any prior knowledge about ribosome orientations using alternating translational and
global rotational search in Fourier space using FRM in PyTom. The subtomogram
average evolves from a featureless sphere to a defined ribosome within 10 alignment iterations (Fig. 9.3a). This approach can be chosen to minimize template bias
during subtomogram alignment or to align subtomograms extracted from manually
picked coordinates, if no prior structural knowledge about the macromolecule of
interest is available. From the example dataset depicting ER membrane-associated
ribosomes used for illustration, 17,500 subtomograms were selected and aligned
using simultaneous translational and restricted rotational search in real space following either the ‘conventional’ or the ‘gold standard’ alignment approach
implemented in PyTom. Both approaches yielded essentially identical densities
244
S. Pfeffer and F. Förster