Three-dimensional reconstruction from two-dimensional images
In a frozen-hydrated sample of homogeneous, identical particles, each
particle is embedded within the vitreous ice layer in a different
orientation. The image that is formed of such a layer of particles in a
micrograph, recorded on photographic film or a charged-couple device (CCD),
is a two-dimensional projection of each particle. The challenge is
thus to reconstruct a three-dimensional image of the particle from different
two-dimensional projections. The mathematical theory underlying three-
dimensional reconstruction, which goes back to Radon, was first put into
practice by DeRosier and Klug, who reconstructed a bacteriophage tail
from electron micrographs. However, unlike the reconstruction of the highly
ordered, helical bacteriophage tail, which only requires a single view, the
reconstruction of a totally asymmetric particle requires that a large number
of views are available, and that their relative orientation are known with
high accuracy.
For a homogeneous sample, the spatial relationship of the particles in the
vitreous ice layer can be described mathematically by a series of rigid-body
translations and rotations of a single object. Thus, if the angular
distribution of the particles is sufficiently uniform, i.e., if particles are
present in orientations that sample a large part of the angular space, then a
set of micrographs of the specimen, each showing projections of hundreds of
particles, contains all the information necessary to reconstruct the particle
in three dimensions.
The problem of determining the relative orientation among all projections is
challenging because of the low SNR of the data. If no prior information
about the structure of the particle is known, two ab initio methods can be
used for orientation determination: (i) In the ‘random-conical’ data
collection method, which involves the use of an additional tilted view
of the specimen, geometric relationships are established among a subset of
particles that face the grid in the same orientation. (ii) In the ‘method of
common lines’, particle images are first classified, then class
averages representing different views of the particle are related to one
another following the common lines principle first formulated by Crowther. The latter method cannot establish the handedness, however, and
requires an extra tilt for this purpose. From a set of projections whose
angles have been determined by either one of these ab initio methods, a
coarse reconstruction can then be computed, which is inaccurate yet contains
important shape information allowing higher resolution and accuracy to be
achieved through a succession of iterative steps, known as angular
refinement.
In the refinement, the first, coarse reconstruction serves as an initial
reference structure from which two-dimensional projections are computed,
which are then compared to the experimental projections, yielding refined
angles. With those angles, a new reconstruction is obtained, which is then
used as reference in the next cycle, etc. Refinement continues until no
improvement in the reconstructed three-dimensional model is observed, or the
orientation angles have stabilized. Currently, resolutions in the range between 10 and 13 Ă…
are routinely obtained, where detailed features of proteins and RNA are
observed at the level of protein domains or RNA backbone. Resolutions in the
range up to 7 Ă…, where structural motifs such as alpha-helices start to be
discernable, still require a large effort. Typically, more than
100,000 particle projections are required to meet this goal. Still larger
numbers are needed – an estimated 1,000,000 for the ribosome – to approach 3
Ă… resolution. However, while the achievement of atomic resolution still
awaits improvements in several instrumental and data processing aspects,
existing moderate-resolution cryo-EM density maps of a molecular assembly can
already be interpreted at the atomic level if X-ray or NMR structures of its
components are known.
See reference:
Mitra, K et al. (2006). Ribosome dynamics: insights from atomic structure modeling into cryo-electron microscopy maps. Annu. Rev. Biophys. Biomol. Struct. 35: 299
back to top
Protocols
Protocols
back to top
