Decoupling Size from Shape: Cellular Sheaf Laplacians as Ligand Geometry Descriptors for Binding Affinity Prediction

Binding affinity prediction in computational drug discovery is confounded by trivial correlations between molecular size and measured potency. We introduce cellular sheaf Laplacians as descriptors of ligand molecular geometry that quantify geometric frustration independent of system size. Sheaves are constructed over molecular graphs by assigning three-dimensional coordinate spaces to atoms and projection operators encoding ideal bonding geometry to edges; eigendecomposition of the resulting Laplacian yields spectral features measuring inconsistencies between local geometric constraints and global topology. Applied to 14,050 protein-ligand complexes from the PDBbind v2020 refined set, MW-residualized Sheaf features capture a statistically significant geometric signal (rpartial = 0.171, p<10−70) that is orthogonal to the Wiener index (r=0.013) and persists after controlling for both molecular weight and classical graph-theoretic descriptors (rpartial = 0.390, p<10−9). Sheaf spectral features alone achieve predictive performance (R2=0.403) approaching that of fourteen classical cheminformatics descriptors (R2=0.446), and their combination yields consistent improvements across the binding affinity spectrum (RMSE =1.43pKd). Permutation importance analysis confirms the Sheaf Frobenius norm as the second most influential descriptor after molecular weight. We introduce Topological Binding Efficiency as a size-normalized quality metric identifying ligands that achieve potent binding through geometric complementarity rather than molecular bulk. Gaussian mixture analysis of the maximum eigenvalue distribution among strong binders reveals two distinct spectral modes corresponding to planar aromatic and three-dimensional sp3-rich scaffolds, confirmed by significant differences in fraction of sp3 carbons and aromatic ring counts (p<10−8). As an intentionally ligand-centric framework, our approach complements rather than replaces protein-aware co-modelling architectures. This work establishes cellular sheaf theory as a principled framework for encoding molecular topology with statistically significant associations with binding affinity, providing interpretable geometric insights that are inaccessible to conventional molecular descriptors.