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

Ali Punjani; Haowei Zhang; David J. Fleet

The difference between uniform and non-uniform refinement is in the regularization step. First, in non-uniform refinement, regularization is performed independently in the two half-maps. As such, the 1 Two key factors directly influence the choice of ρ(x) for estimating θ(x) reliably. The first is noise in the raw reconstruction. Aggregating squared error measures the residual variance (the residual power); larger windows provide more accurate variance estimates. The second is residual signal structure. When over-regularized, some high-frequency signal is lost, in which case signal residual is expected to have significant amplitudes near the wavelength θ. (Higher frequencies with insignificant amplitudes are dominated by noise, while lower frequencies are left unchanged by the regularizer and thus cancel the signal in the opposite halfmap.) Integrating a squared band-pass signal provides a phase-independent measure of the local signal power; in our case the minimum window ρ(x) is determined by pass-band of the residual signal, which is expected to be close to θ. Thus the minimum ρ(x) is naturally constrained to be small multiple of θ. The same reconstruction is shown after uniform isotropic filtering (based on FSC between half-maps). Right: The same half-map reconstruction after non-uniform regularization with the optimal CV regularizer. Non-uniform regularization removes noise from the solvent background and nanodisc region, while preserving the high-resolution structure needed for particle alignments in the well-ordered protein region. It is particularly effective due to its implicitly small number of degrees of freedom (Eqn. 6) and capacity to model sharp transitions between ordered and disordered regions.