Lie-RMSD: A Gradient-Based Framework for Protein Structural Alignment using Lie Algebra

The comparison of protein structures is a fundamental task in computational biology, crucial for understanding protein function, evolution, and for drug design. While analytical methods like the Kabsch algorithm provide an exact, closed-form solution for minimizing the Root Mean Square Deviation (RMSD) between two sets of corresponding atoms, their application is limited to this specific metric. The rise of deep learning and automatic differentiation frameworks offers a new, more flexible paradigm for such optimization problems. We present Lie-RMSD, a novel, fully differentiable framework for protein structural alignment. Our method represents the rigid-body transformation (rotation and translation) as a 6-dimensional vector in the Lie algebra se(3) of the special Euclidean group SE(3). This representation allows the RMSD to be formulated as a loss function that can be directly minimized by modern gradient-based optimizers. We benchmarked our framework by aligning two allosteric conformations of Adenylate Kinase (PDB IDs: 4AKE and 1AKE). We demonstrate that a suite of standard optimizers (SGD, Adam, AdamW, and Sophia) can robustly converge to the global minimum, achieving precision effectively identical to the analytical Kabsch algorithm. This work validates the accuracy of the Lie algebra-based gradient descent approach and establishes a robust foundation for its extension to more sophisticated and biologically relevant scoring functions where no analytical solutions exist.