Bayesian inference for three-dimensional helical reconstruction using a soft-body model

Estimation of the three-dimensional (3D) structure of a protein using cryo transmission electron microscopy (cryo-TEM) is an inverse problem, which aims to estimate the parameters of a specific physical process from observations. In general, we need to model the observation process to estimate a structure. However, the inconsistency between the model and a real observation process decreases the estimation accuracy. In cryo-TEM, the flexibility of a soft protein, including the bending of a helix, can lead to inconsistencies between the observations because of the assumption that there is a consistent 3D structure behind each observed image. In this paper, we propose a 3D reconstruction algorithm for helical structures using a parametric soft-body model that can represent continuous deformation. We performed an approximate Bayesian inference for unobservable (hidden) variables, such as the deformation parameters, projection angle, and two-dimensional origin offset (shift) of each protein in the 3D structure estimation problem. Our principled approach is not only beneficial to deal with the uncertainties in the estimation, but also beneficial to make the optimization algorithm convergent and efficient. Reconstructions with artificial molecules validated the advantage of the proposed method, particularly, when deformed helices were imaged under a low signal-to-noise ratio condition. Moreover, we confirmed that the proposed method successfully reconstructed a 3D structure from cryo-TEM images of the tobacco mosaic virus.