Cryo-electron microscopy (cryo-EM) has emerged as a powerful technique for resolving the three-dimensional structures of macromolecules. A key challenge in cryo-EM is characterizing continuous heterogeneity, where molecules adopt a continuum of conformational states. Covariance-based methods offer a principled approach to modeling structural variability. However, estimating the covariance matrix efficiently remains a challenging computational task. In this paper, we present SOLVAR (Stochastic Optimization for Low-rank Variability Analysis), which leverages a low-rank assumption on the covariance matrix to provide a tractable estimator for its principal components, despite the apparently prohibitive large size of the covariance matrix. Under this low-rank assumption, our estimator can be formulated as an optimization problem that can be solved quickly and accurately. Moreover, our framework enables refinement of the poses of the input particle images, a capability absent from most heterogeneity-analysis methods, and all covariance-based methods. Numerical experiments on both synthetic and experimental datasets demonstrate that the algorithm accurately captures dominant components of variability while maintaining computational efficiency. SOLVAR achieves state-of-the-art performance across multiple datasets in a recent heterogeneity benchmark. The code of the algorithm is freely available at https://github.com/RoeyYadgar/SOLVAR.
Cryo-electron microscopy (cryo-EM) single-particle analysis is a method for determining the high-resolution structures of molecules, enabling biologists to analyze their function. In this method, copies of a molecule are suspended in a thin layer of vitrified ice, which preserves them in their native state, with each copy assuming a random orientation and position in the ice layer [4]. An electron microscope then captures two-dimensional noisy projection images of these molecules, resulting in up to several million particle images, which are the input to the subsequent data processing pipeline.
The classic simplified model of analyzing cryo-EM data often assumes that all the imaged particles are exact copies of the same molecular structure. However, in practice, biological molecules exhibit structural heterogeneity that is often crucial to the biological processes in which they participate. In such cases, the imaged molecules can no longer be considered to be identical copies. Traditional cryo-EM analysis software models the heterogeneity as a small number of discrete states; while this approach has been very successful in many applications, it cannot resolve more complex dynamics exhibited by many molecular complexes, nor achieve high-resolution reconstructions. It is therefore necessary to develop new techniques that can accurately recover the molecule’s structure while accommodating its inherent variability. This problem, known as the “continuous heterogeneity problem,” is an active and challenging research area.
In this paper, we introduce SOLVAR, a new method for analyzing structural heterogeneity in cryo-EM. Our method uses the covariance based approach to heterogeneity analysis; this approach has significant theoretical advantages [16] and has been shown to outperform other methods in recent benchmarks. Traditionally, covariance-based methods were limited in their resolution due to the sheer size of the covariance matrix they attempted to compute. More recent covariance-based methods [39,31,9] tackle this limitation by exploiting low dimensional structure of the covariance matrix and estimate the matrix’s principal components directly. Despite the advancements in resolution, all covariance-based methods (and most other methods) share a critical limitation: they all require the user to provide particle images’ poses, and they are inherently unable to refine these initial poses. In realistic pipelines, such poses are computed by software that ignores the heterogeneity in the data; these estimates are distorted by heterogeneity, and existing covariance-based methods are unable to correct the error. Building on this line of work, SOLVAR leverages the low rank of the covariance to reformulate the estimation into an optimization problem, which enables the direct estimation of principal components using gradient-based methods. Unlike prior methods, our framework readily accommodates particle image poses within the optimization problem. We show, using recent benchmarks, that our algorithm, SOLVAR, outperforms existing algorithms while remaining computationally efficient.
Cryo-EM particle images are tomographic projections of the electrostatic potential of the imaged biomolecules. For convenience, we consider the particle images and volumes in Fourier space. We will denote the Fourier transform of a particle image by Y i ∈ C N ×N , where N × N are the dimensions in pixels of each particle image. For brevity, we refer to the Fourier transform of a particle image also as the particle image. We will denote the Fourier transform of the electrostatic potential of the biomolecule by X i ∈ C N ×N ×N and refer to it as the volume or structure. The main advantage of the Fourier domain representation is that the complicated tomographic projection in real space can be described as a simple 2-d slice through the center of the 3-d space (the Fourier slice theorem [25]).
Using this notation, the imaging model under the usual weak phase assumption [34] can be written as
where P ϕ i is the 2-d slice operator corresponding to orientation ϕ i , T t i is the in-plane translation operator (in the Fourier domain) corresponding to the in-plane shift t i from the center of the particle, C i is the contrast transfer function (CTF), which is a filter related to the imaging procedure, a i is the per-image contrast scaling factor, and e i is additive Gaussian noise which we will assume throughout the paper to be white e i ∼ N (0, σ 2 I). In practice, a preprocessing step is performed to whiten the particle stack and estimate the variance σ 2 . For the sake of simplicity, since all the transformations applied to the volume X i are linear, we will merge them into a single operator P i = a i C i T i P ϕ i to which we will refer as the projection operator. Using this notation, the imaging model is given by
We refer to the pair (ϕ i , t i ) as the pose of the i-th image. The classic simplified model for analyzing cryo-EM data often as
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