FoldDoF: Utilizing the Primary Degrees of Freedom of Protein Backbone for Geometric Modeling and Generation.

Robust modeling of protein structures through internal coordinates requires modeling a mixture of bond angles and torsion angles. Smoothing out bond angles can lead to significant error accumulation during structure reconstruction. We propose using a concise yet precise representation for modeling and generating protein structures, which views the protein backbone structure as a sequence of 3D rotations of peptide units, unifying bond angles and torsion angles on a single rotation manifold. We therefore present differentiable algorithms for efficient conversion between such internal coordinates and Cartesian coordinates. We demonstrate that the relative 3D rotations between consecutive peptide units describe the primary degrees of freedom of protein backbone conformation and thus guarantee the fidelity of the reconstructed full-atomic backbone structures in an optimization-free manner. Given the advantage of this representation that allows for straightforward geometric reasoning and optimization in 3D space, we incorporated this peptide-unit-centric formulation into protein backbone generative models, resulting in FrameFlow variants with improved in silico design performance in terms of diversity, novelty, and length generalizability. The enhanced efficacy of probabilistic modeling of backbone conformations demonstrates the superiority of this backbone representation and lays the foundation for developing more effective generative models for both protein design and structure prediction.