LieOTMap-FFT: A Differentiable Fitting Framework Combining Lie Algebra and FFT-accelerated Optimal Transport for Cryo-EM Maps

Interpreting cryo-electron microscopy (cryo-EM) density maps often requires the accurate placement of atomic models, a challenging high-dimensional fitting problem. We introduce LieOTMap-FFT , a novel, fully differentiable framework for rigid body map-to-map alignment. Our framework synergistically combines four key mathematical and computational techniques. First, we parameterize the SE(3) rigid body transformation using Lie algebra, ensuring a continuous and singularity-free optimization landscape suitable for gradient descent. Second, we treat the mobile and target density maps as probability distributions on a 3D grid and measure their similarity using an Optimal Transport (OT) score based on the Sinkhorn algorithm. Third, to make this computationally feasible for large maps, we leverage the Fast Fourier Transform (FFT) to accelerate the core convolution operations within the Sinkhorn iterations from O ( N ² ) to O ( N log N ). Finally, a TM-align-inspired similarity kernel provides a robust score landscape for global search. We demonstrate the power of LieOTMap-FFT by fitting a large ribosomal subunit (PDB: 1AON) into a cryo-EM map (EMD-1046), refining a random initial placement to a final RMSD of 4.81 Å. This result showcases the framework’s ability to achieve high-accuracy global convergence by effectively integrating these four core technologies.