Sparse approximations of protein structure from noisy random projections

TL;DR

This paper introduces a new method for reconstructing 3D protein structures from noisy, random projections without prior knowledge of projection angles, using sparse representations and LASSO regularization.

Contribution

It presents a novel approach that estimates low-resolution protein structures directly from noisy data, bypassing the need for initial angle estimates, leveraging sparsity and shape theory.

Findings

Successfully reconstructed the E. coli Klenow fragment structure.

Demonstrated robustness to noise and unknown projection angles.

Provided a computational framework for structure approximation without initial estimates.

Abstract

Single-particle electron microscopy is a modern technique that biophysicists employ to learn the structure of proteins. It yields data that consist of noisy random projections of the protein structure in random directions, with the added complication that the projection angles cannot be observed. In order to reconstruct a three-dimensional model, the projection directions need to be estimated by use of an ad-hoc starting estimate of the unknown particle. In this paper we propose a methodology that does not rely on knowledge of the projection angles, to construct an objective data-dependent low-resolution approximation of the unknown structure that can serve as such a starting estimate. The approach assumes that the protein admits a suitable sparse representation, and employs discrete $L^1$-regularization (LASSO) as well as notions from shape theory to tackle the peculiar challenges involved in the associated inverse problem. We illustrate the approach by application to the reconstruction of an E. coli protein component called the Klenow fragment.