Non-uniform refinement: Adaptive regularization improves single particle cryo-EM reconstruction

Ali Punjani; Haowei Zhang; David J. Fleet

In the cryo-EM refinement problem, like many latent variable inverse problems, there is an additional interplay between regularization, noise buildup, and the estimation of latent variables. Retained noise due to under-regularization will contaminate the estimation of latent variables. This contamination is propagated to subsequent iterations and causes over-fitting. In Fig. 1 , with the TRPA1 membrane protein [21] , over-fitting occurs in the disordered micelle and in the dynamic tail at the bottom of the protein. Conversely, over-regularization attenuates useful signal structure, further degrading latent variable estimates. With TRPA1, signal loss occurs in the central region where the structure is not as well resolved as it otherwise could be. In general, the amount of regularization (or smoothing) depends on a set of regularization parameters, the tuning of which is often critical to successful outcomes. This paper reconsiders the task of regularization based on the observation that common iterative refinement algorithms often systematically under-fit and over-fit different regions of a 3D structure, simultaneously. This causes a loss of resolvable detail in some parts of a structure, and accumulation of noise in others. The reason stems from the use of frequency-space filtering as a form of regularization. Some programs, like cisTEM [11] , use a strict resolution cutoff, beyond which Fourier amplitudes are set to zero before alignment of particle images to the current 3D structure. In RELION [27] , regularization was initially formulated with an explicit Gaussian prior on Fourier amplitudes of the 3D structure, with a hand-tuned parameter that controls Fourier amplitude shrinkage. Later versions of RELION [28] and cryoSPARC's homogeneous (uniform) refinement [22] use a transfer function (or Wiener filter) determined by Fourier Shell Correlation (FSC) computed between two independent half-maps (e.g., [3, 13, 25] ).