Element interaction manifolds for comparative analysis of geometric representations in protein-ligand recognition

Accurate modeling of protein–ligand interactions requires representations that capture both molecular geometry and chemically specific interactions. Existing approaches encode molecular recognition through several geometric formalisms, including differential geometry and spherical harmonic surface descriptors such as three-dimensional Zernike descriptors (3DZD). However, the relative roles of local curvature, surface area, volume, global shape, and chemical partitioning in protein–ligand recognition remain insufficiently understood. In this work, we present a unified geometric study centered on the Element Interaction Manifold (EIM) framework. We integrate element-interactive curvature, surface-area, and volume descriptors into a common representation and comparatively evaluate their roles in binding-affinity prediction and ligand-aware protein–ligand complex recognition. To place differential geometry in context, we compare EIM with standard 3DZD and a chemically partitioned extension termed element-pair 3DZD (EP-3DZD). Across PDBbind/CASF-2016 affinity prediction and unsupervised ligand-aware recognition tasks, chemically resolved interaction geometry provides strong predictive performance. Chemical partitioning improves the Zernike-based representation relative to standard global 3DZD, whereas combining Zernike descriptors with Hybrid EIM produces only small numerical gains, indicating limited additional complementarity under the evaluated protocols. Additional robustness and control analyses support the stability of the affinity-prediction results and clarify the contribution of shared ligand identity to the ligand-aware recognition task. Overall, these results support chemically resolved geometric information as an interpretable and informative representation for protein–ligand modeling.