A Hierarchical Robust Linear Model for Cryo-EM Map Analysis

Abstract

Cryo-electron microscopy (cryo-EM) has become a central tool for determining the atomic structures of biological macromolecules, producing three-dimensional reconstruction maps that guide atomic model building. Current quantitative analyses of cryo-EM maps largely rely on simplified Gaussian signal models with fixed width parameters, limiting their ability to capture atom-specific signal characteristics directly from the maps. We propose a robust hierarchical linear (RHL) model for the statistical analysis of paired cryo-EM maps and atomic structures deposited in the EMDB and PDB. Within this framework, the logarithms of local voxel intensities surrounding each atom are modeled using a linearized Gaussian form, and atom-specific amplitude and width parameters are estimated through a hierarchical structure that pools information across atoms of the same type. The primary objective is to provide a statistically grounded framework for extracting quantitative atomic signal features from cryo-EM maps without imposing rigid physical assumptions on these parameters. To address contamination arising from overlapping atomic signals, spatially varying resolution, and experimental noise, we incorporate a data-adaptive weighting scheme based on minimum density power divergence estimation (MDPDE). This formulation preserves computational efficiency and ensures stable parameter estimation, while naturally facilitating the identification of deviated atomic profiles that may stem from model misspecification or local structural heterogeneity. Simulation studies demonstrate that the proposed method yields stable group-level parameter estimates even under substantial contamination. Applications to cryo-EM maps at multiple resolutions show that an atomic resolution (1.25 Angstrom) apoferritin map exhibits systematic atom-type differences in Gaussian amplitude and width, whereas a near-atomic-resolution map (2.20 Angstrom) shows closer agreement with conventional uniform-width assumptions. Beyondstructural biology, the proposed RHL framework provides a general statistical methodology for hierarchical data integration under heterogeneous noise, with potential applications in multi-center studies and federated learning.