End-to-End Modeling of Reaction Field Energy Using Data-Driven Geometric Graph Neural Networks

Abstract

Electrostatic interactions are fundamental to the structure, dynamics, and function of biomolecules, with broad applications in protein–ligand binding, enzymatic catalysis, and nucleic acid regulation. The Poisson–Boltzmann (PB) equation provides a physically grounded framework for modeling these interactions. However, solving the PB equation for large and complex biomolecular systems remains computationally expensive as traditional numerical solvers scale poorly with system size. While the Generalized Born (GB) model offers a more computationally efficient approximation, it does so at the cost of reduced accuracy relative to full PB solutions. To overcome these limitations, we propose PBGNN, a novel end-to-end framework that uses data-driven geometric graph neural networks to directly approximate PB electrostatic energies without relying on the GB approximation. PBGNN incorporates sinusoidal embeddings of atomic charges and a message-passing architecture to efficiently capture long-range interactions in large biomolecules. To address training instability caused by high variance in atomic electrostatic potentials, we introduce a charge-weighted mean squared error (CMSE) optimization objective that improves convergence. We benchmark PBGNN on the AMBER PBSA suite and PBSMALL, a new dataset designed for rapid evaluation of small-molecule electrostatics in drug discovery contexts. The results demonstrate that PBGNN consistently achieves high accuracy in predicting the PB energy with linear computational complexity. Furthermore, it provides reliable and precise PB free energy predictions for both large biomolecular complexes and small-molecule datasets, showcasing its strong generalizability, scalability, and potential utility in drug discovery tasks requiring accurate electrostatic modeling of small molecules. Comprehensive ablation studies further reveal the impact of architectural components, such as geometric representation, objective design, and cutoff strategy, informing future research directions. Finally, we release PBGNN as an open-source, self-contained codebase, along with preprocessed datasets and a complete training and evaluation pipeline, to support scalable and accurate electrostatic analysis that facilitates future research in associated areas.