Hierarchical Hybrid-Coordinate Flow Matching for Accurate and Physically Plausible Apo-to-Holo Protein Generation

Predicting holo-like protein conformations from apo structures is essential in structure-based drug design. Most existing apo-to-holo models operate in Cartesian space and achieve strong global accuracy. However, high global accuracy does not ensure physically valid local backbone geometry. Internal-coordinate models offer local control and validity, but regressive generation can amplify angular errors—the lever-arm effect. We introduce CrypticFlow, a hierarchical flow-matching model that combines these two complementary representations. An ESM-3-conditioned E(3)-equivariant graph network first generates sparse Cα anchors in Cartesian space. A conditional midpoint Transformer then recursively fills the gaps in local internal coordinates. During recursive generation, the decoder constrains the distance between each midpoint and any sequence-adjacent parent anchor to the corresponding apo Cα–Cα distance. On the D3PM test split, CrypticFlow achieves state-of-the-art median and mean Cα RMSD. Across five independent training models, the median and mean direct RMSD are 1.333 ± 0.056 Å and 2.766 ± 0.041 Å, respectively. The quality of local geometry also improves substantially. Our model achieves 99.58 ± 0.02% adjacent-distance fidelity and 95.87 ± 0.25% pseudo-angle fidelity, measured within 0.2 Å and 20° of the holo reference. Then, we reconstruct the predicted Cα-only traces into all-atom structures using the cg2all coarse-to-all-atom backmapping method and evaluate their physical and geometric quality with MolProbity. Our model achieves a MolProbity score of 3.034 ± 0.054, outperforming the evaluated baselines. These results show that a hierarchical flow-matching model with hybrid coordinates can improve global conformational accuracy and structural fidelity together.