Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem.

Area of Science:

• Structural biology
• Computational imaging
• Biophysics

Background:

• Cryo-electron microscopy (cryo-EM) generates 2D projections of molecules for 3D structure reconstruction.
• Structural variability (heterogeneity) in molecules presents a significant challenge for single particle reconstruction (SPR).
• Previous methods suggested using eigenvectors of the covariance matrix to address heterogeneity, but estimating this matrix from projections is difficult.

Purpose of the Study:

• To develop a general method for estimating the covariance matrix from noisy 2D projections of molecules.
• To address the challenge of mapping conformational states (heterogeneity) in cryo-EM data.
• To enable more accurate 3D structure reconstruction of molecules with structural variability.

Main Methods:

• Formulated a general problem of covariance estimation from noisy projections, linking it to matrix completion and high-dimensional principal component analysis.
• Proposed a consistent estimator for the covariance matrix.
• Developed a sparse basis representation for the 'projection covariance transform' to enable tractably inverting the linear operator.

Main Results:

• The proposed estimator for the covariance matrix is proven to be consistent.
• The spectrum of the estimated covariance matrix can reveal the number of conformational states when heterogeneity is finite.
• Numerical experiments on synthetic data demonstrate the algorithm's robustness to high noise levels.

Conclusions:

• The developed method provides a robust solution for covariance estimation from cryo-EM projections, crucial for tackling molecular heterogeneity.
• The approach facilitates the mapping of conformational states, advancing the field of single particle reconstruction.
• The technique offers a computationally tractable way to invert a complex linear operator, essential for tomographic reconstruction problems with structural variation.