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

Area of Science:

• Computational chemistry and drug discovery
• Mathematical physics and topology
• Structural biology and cheminformatics

Background:

• Predicting binding affinity is crucial for drug discovery but often confounded by molecular size.
• Existing descriptors struggle to capture complex geometric information independent of molecular weight.
• Need for novel descriptors that quantify ligand geometry and its relation to binding potency.

Purpose of the Study:

• Introduce cellular sheaf Laplacians as novel descriptors for ligand molecular geometry.
• Quantify geometric frustration independent of system size for improved binding affinity prediction.
• Develop a size-normalized metric (Topological Binding Efficiency) for ligand quality.

Main Methods:

• Constructed sheaves over molecular graphs using 3D atomic coordinates and ideal bonding geometry.
• Eigendecomposition of the cellular sheaf Laplacian to extract spectral features.
• Applied features to 14,050 protein-ligand complexes from PDBbind v2020, performing residualization and correlation analyses.

Main Results:

• Sheaf features capture a statistically significant geometric signal (rpartial = 0.171, p<10-70) orthogonal to molecular weight and Wiener index.
• Sheaf spectral features alone achieve predictive performance (R2=0.403), approaching classical descriptors (R2=0.446).
• Topological Binding Efficiency metric reveals distinct spectral modes for planar aromatic and 3D sp3-rich scaffolds.

Conclusions:

• Cellular sheaf theory provides a principled framework for encoding molecular topology relevant to binding affinity.
• Sheaf features offer interpretable geometric insights inaccessible to conventional descriptors.
• This ligand-centric approach complements protein-aware co-modelling strategies in drug discovery.