Automated bias potential optimization via position and variance control in umbrella sampling

Free-energy landscapes (FELs) play a crucial role in understanding molecular processes via molecular dynamics (MD) simulations. However, standard umbrella sampling (US), a common technique for enhancing FEL sampling efficiency, struggles with adequately sampling high-free-energy regions and controlling the distributions of windows. We previously introduced an optimization-based approach that adaptively adjusts window positions through bias potential optimization. Here, we significantly refine this approach by explicitly controlling both the positions and variances of the sampling distributions. Our optimization method employs target Gaussian distributions with imposed upper bounds on variance, preventing excessive broadening and ensuring stable, unimodal sampling within each window. We demonstrate our method's efficacy using Langevin dynamics simulations of the two-dimensional π/4-rotated Wolfe-Quapp potential and MD simulations of alanine dipeptide in water. For the Wolfe-Quapp potential, in the non-optimized simulations, increasing bias potential strength improved accuracy, but even the best case yielded only 95% agreement. In contrast, all optimized simulations exhibited superior convergence compared to the nonoptimized simulations. Compared to the standard US, our optimized method yields significantly improved accuracy and faster convergence in reconstructing FELs, particularly near saddle points and steep free-energy gradients. This improved optimization framework provides a robust and generalizable strategy for automated tuning of bias potentials, facilitating accurate FEL reconstruction in diverse and complex molecular systems. The method is openly accessible via GitHub (https://github.com/YukiMitsuta/plumed_USopt) and fully integrates with the widely used PLUMED package. This method can be easily extended to multiple dimensions, making it possible to extend it to more complex biomolecular systems.