A fully automatic 3D reconstruction method using simulated annealing enables accurate posterioric angular assignment of protein projections

Single-particle analysis is a structure determining method using electron microscopic (EM) images, which does not require protein crystal. In this method, projections are picked up and used to reconstruct a three-dimensional (3D) structure. When the conical tilting method is not available, the particle images are usually classified and averaged to improve the signal-to-noise ratio. The Euler angles of these average images must be posteriorically assigned to create a primary 3D model. We developed a new, fully automatic unsupervised Euler angle assignment method, which does not require an initial 3D reference and which is applicable to asymmetric molecules. In this method, the Euler angle of each average image is initially set randomly and then automatically corrected in relation to those of the other averages by iterated optimizations using the Simulated Annealing (SA) algorithm. At each iteration, the 3D structure is reconstructed based on the current Euler angles and reprojected back in the average-input directions. A modified cross-correlation between each reprojection and its corresponding original average is then calculated. The correlations are summed as a total 3D echo-correlation score to evaluate the Euler angles at this iteration. Then, one of the projections is selected, its Euler angle is changed randomly, and the score is also calculated. Based on the score change, judgment of whether to accept or reject the new angle is made using the SA algorithm, which is introduced to overcome the local minimums. After a certain number of iterations of this process, the angles of all averages converge so as to create a reliable primary 3D model. This echo-correlated 3D reconstruction with simulated annealing also has potential for wide application to general 3D reconstruction from various types of 2D images.