186 research outputs found
3D ab initio modeling in cryo-EM by autocorrelation analysis
Single-Particle Reconstruction (SPR) in Cryo-Electron Microscopy (cryo-EM) is
the task of estimating the 3D structure of a molecule from a set of noisy 2D
projections, taken from unknown viewing directions. Many algorithms for SPR
start from an initial reference molecule, and alternate between refining the
estimated viewing angles given the molecule, and refining the molecule given
the viewing angles. This scheme is called iterative refinement. Reliance on an
initial, user-chosen reference introduces model bias, and poor initialization
can lead to slow convergence. Furthermore, since no ground truth is available
for an unsolved molecule, it is difficult to validate the obtained results.
This creates the need for high quality ab initio models that can be quickly
obtained from experimental data with minimal priors, and which can also be used
for validation. We propose a procedure to obtain such an ab initio model
directly from raw data using Kam's autocorrelation method. Kam's method has
been known since 1980, but it leads to an underdetermined system, with missing
orthogonal matrices. Until now, this system has been solved only for special
cases, such as highly symmetric molecules or molecules for which a homologous
structure was already available. In this paper, we show that knowledge of just
two clean projections is sufficient to guarantee a unique solution to the
system. This system is solved by an optimization-based heuristic. For the first
time, we are then able to obtain a low-resolution ab initio model of an
asymmetric molecule directly from raw data, without 2D class averaging and
without tilting. Numerical results are presented on both synthetic and
experimental data
Reconstructing continuous distributions of 3D protein structure from cryo-EM images
Cryo-electron microscopy (cryo-EM) is a powerful technique for determining
the structure of proteins and other macromolecular complexes at near-atomic
resolution. In single particle cryo-EM, the central problem is to reconstruct
the three-dimensional structure of a macromolecule from noisy and
randomly oriented two-dimensional projections. However, the imaged protein
complexes may exhibit structural variability, which complicates reconstruction
and is typically addressed using discrete clustering approaches that fail to
capture the full range of protein dynamics. Here, we introduce a novel method
for cryo-EM reconstruction that extends naturally to modeling continuous
generative factors of structural heterogeneity. This method encodes structures
in Fourier space using coordinate-based deep neural networks, and trains these
networks from unlabeled 2D cryo-EM images by combining exact inference over
image orientation with variational inference for structural heterogeneity. We
demonstrate that the proposed method, termed cryoDRGN, can perform ab initio
reconstruction of 3D protein complexes from simulated and real 2D cryo-EM image
data. To our knowledge, cryoDRGN is the first neural network-based approach for
cryo-EM reconstruction and the first end-to-end method for directly
reconstructing continuous ensembles of protein structures from cryo-EM images
Structural Variability from Noisy Tomographic Projections
In cryo-electron microscopy, the 3D electric potentials of an ensemble of
molecules are projected along arbitrary viewing directions to yield noisy 2D
images. The volume maps representing these potentials typically exhibit a great
deal of structural variability, which is described by their 3D covariance
matrix. Typically, this covariance matrix is approximately low-rank and can be
used to cluster the volumes or estimate the intrinsic geometry of the
conformation space. We formulate the estimation of this covariance matrix as a
linear inverse problem, yielding a consistent least-squares estimator. For
images of size -by- pixels, we propose an algorithm for calculating this
covariance estimator with computational complexity
, where the condition number
is empirically in the range --. Its efficiency relies on the
observation that the normal equations are equivalent to a deconvolution problem
in 6D. This is then solved by the conjugate gradient method with an appropriate
circulant preconditioner. The result is the first computationally efficient
algorithm for consistent estimation of 3D covariance from noisy projections. It
also compares favorably in runtime with respect to previously proposed
non-consistent estimators. Motivated by the recent success of eigenvalue
shrinkage procedures for high-dimensional covariance matrices, we introduce a
shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We
evaluate our methods on simulated datasets and achieve classification results
comparable to state-of-the-art methods in shorter running time. We also present
results on clustering volumes in an experimental dataset, illustrating the
power of the proposed algorithm for practical determination of structural
variability.Comment: 52 pages, 11 figure
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