Gaussian Process Regression for Mapping Free EnergyLandscape of Mg²⁺-Cl^- Ion Pairing in Aqueous Solution: Molecular...

Free energy landscapes are pivotal for understanding molecular interactions in solution, yet their reconstruction in complex systems remains computationally demanding. In this study, we integrated Gaussian process regression (GPR) with well-tempered metadynamics (WT-MTD) to efficiently map the free energy landscape of the Mg²⁺-Cl^- ion pairing in an aqueous solution, a system central to biological processes such as magnesium hydration and ligand exchange. We compared traditional umbrella sampling (WHAM) with WT-MTD-derived free energy profiles, identifying critical discrepancies attributed to insufficient sampling in barrier regions. WT-MTD captures two distinct minima corresponding to the contact ion pair (CIP: 0.23 nm) and solvent-separated ion pair (SSIP: 0.47 nm) configurations, consistent with previous computational and experimental studies. GPR, trained on free energy gradients from WT-MTD trajectories, reconstructs smooth landscapes with small datasets (5000 points) while reducing computational costs via grid sparsification. Our results demonstrate that GPR hyperparameters can be optimized based on the insights from WT-MTD simulations, enabling accurate reconstructions even in sparse data regimes. This approach bridges computational efficiency with molecular-level resolution, offering a robust framework for studying ion solvation dynamics and hydration effects in complex systems, where this work is the first application of GPR in ionic solvation environments. The methodology's scalability to multidimensional landscapes further underscores its potential for advancing molecular simulations in biochemistry and material science.