LieMap: A Fast and Transparent Gradient-Based Method for Cryo-EM Structure Fitting

Fitting high-resolution atomic models into lower-resolution cryo-electron microscopy (cryo-EM) maps is a critical yet challenging step in structural biology. We present LieMap, a straightforward, fast, and transparent gradient-based fitting protocol designed for practical applications. Our method parameterizes rigid-body motion on the Lie algebra se(3) to provide a minimal and singularity-free representation. The optimization objective is to directly maximize a real-space cross-correlation score, calculated as the mean density value sampled from the map at the atomic coordinates. This score is computed differentially using trilinear interpolation and optimized with the Adam optimizer in a multi-stage fashion. We demonstrate the effectiveness of LieMap by fitting the apo-state structure of E. coli GroEL (PDB: 1aon) into its ATP-bound state cryo-EM map (EMD-1046). The protocol successfully navigates the large conformational change, achieving a inal RMSD of 3.23 Angstrom. LieMap's simplicity, computational efficiency, and transparent optimization process make it a valuable and practical tool for structural biologists.