LieOTMap: A Differentiable Approach to Cryo-EM Fitting via Lie-Theoretic Optimal Transport

Motivation The integration of high-resolution atomic models with lower-resolution cryo-electron microscopy (cryo-EM) maps is a fundamental task in structural biology. However, this rigid-body fitting problem is challenged by complex scoring landscapes and the non-differentiable nature of standard structural similarity metrics like TM-score, precluding their direct use in modern gradient-based optimization pipelines. Method We present LieOTMap, a novel, fully differentiable framework for cryo-EM fitting. Our approach introduces two key innovations. First, we parameterize rigid-body transformations on the Lie algebra se (3), which provides a minimal, singularity-free representation of the motion. Second, we formulate a loss function based on the Sinkhorn divergence, a differentiable proxy for the Optimal Transport (Wasserstein) distance. This loss function serves as a robust, geometrically meaningful surrogate for non-differentiable scores, allowing us to leverage powerful gradient descent optimizers to navigate the search space. Results We demonstrate the effectiveness of LieOTMap by fitting the apo-state structure of E. coli GroEL (PDB: 1aon) into the ATP-bound state cryo-EM map (EMD-1046). Our method successfully navigates a large-scale conformational change, achieving a highly accurate final RMSD of 3.08 Å with respect to the ground-truth structure (PDB: 1GRU). This result showcases the power of combining Lie-theoretic representations with differentiable geometric loss functions for complex structural alignment tasks. Availability The source code is available at https://github.com/YueHuLab/LieOTMap .