Bayesian umbrella quadrature accelerates free-energy calculations across diverse molecular systems and processes

TL;DR

This work introduces Bayesian Umbrella Quadrature (BUQ), a framework that fuses umbrella integration with Bayesian quadrature to accelerate free-energy calculations in molecular simulations. By modeling the free-energy gradient $-oldsymbol{F}(oldsymbol{s})$ with a Gaussian Process and actively selecting bias centers via acquisition functions, BUQ dramatically reduces the number of biased MD windows required to converge $A(oldsymbol{s})$ while providing uncertainty quantification. Demonstrations across alanine dipeptide conformational changes, water–ice nucleation, and an S$_{\mathrm{N2}}$ reaction show 1.6×–2.8× speedups and accurate barrier predictions, with robustness to kernel choice and hyperparameters. The approach is implemented in an open-source pipeline interfaced with PLUMED, GROMACS, LAMMPS, and ASE, enabling broad adoption for automated, generalizable free-energy estimates. These results establish BUQ as a versatile, automated alternative to traditional UI/US pipelines, with potential extensions to multi-fidelity and alchemical calculations.

Abstract

Biased sampling in molecular dynamics simulations overcomes timescale limitations and delivers free-energy landscapes, essential to understand complex atomistic phenomena. However, when applied across diverse systems and processes, biasing protocols often require time- and resource-consuming fine-tuning. In search for robustness, we boost a prominent biasing method, Umbrella Sampling. To estimate the value of an integral, i.e., the free energy, our Bayesian Umbrella Quadrature (BUQ) method iteratively selects gradient samples, i.e., bias locations, that most reduce the posterior integral variance based on a noise-tolerant Gaussian process model, which also effectively interpolates between samples. We validate the method for a conformational change in a small peptide, a water-to-ice phase transition, and a substitution chemical reaction; obtaining excellent accuracies and speedups. To ease adoption of this more automated and universal free-energy method, we interface BUQ with wide-spread simulation packages and share hyperparametrization guidelines.

Figures (8)

• Figure 1: Schematic representation of the Bayesian Umbrella Quadrature (BUQ) loop applied to free-energy calculations. (1) The Gaussian Process (GP) gradient model is initialized considering a variance $\sigma$ and a few initial gradient samples from umbrella-biased molecular dynamics (MD) simulations. (2) Based on the GP gradient model, the variance of the integral, i.e., the free energy, is estimated. (3) Using an acquisition function, BUQ selects the next umbrella center at which sampling the gradient will maximally reduce the variance of the free-energy estimate. (4) A new biased MD simulation is run at the proposed umbrella center, generating a new gradient sample. This new observation is incorporated into the GP model, and the loop is repeated until convergence. This adaptive strategy allows BUQ to prioritize the most informative sampling regions, reducing the computational cost compared to standard grid-based umbrella integration.
• Figure 2: Bayesian Umbrella Quadrature (BUQ) sampling of the conformational free-energy landscape of alanine dipeptide. a, Representation of the alanine dipeptide molecule with relevant collective variables (CVs), the dihedral angles $\phi$ and $\psi$. b, Conformational free-energy landscape of alanine dipeptide in vacuo obtained by BUQ (blue) compared with a standard umbrella integration (UI) (orange) benchmark. The distribution of umbrella centers shows how BUQ adaptively places samples along the contours of valleys and barriers on the free-energy landscape. c, Root-mean-squared deviation (RMSD) of BUQ and UI with respect to the ground truth as a function of the number of samples, or total simulation time.
• Figure 3: Bayesian Umbrella Quadrature (BUQ) sampling of the nucleation free-energy profile of the phase transition from water to ice Ih. Free energy as a function of the number of ice-like water molecules, $N_\text{ice}$, obtained by BUQ (blue) compared against the reference profile from variational enhanced sampling (black) taken from Ref. Piaggi2020. BUQ samples are indicated by red dots.
• Figure 4: Bayesian Umbrella Quadrature (BUQ) sampling of the S$_\text{N}2$ reaction free-energy landscape between ethyl chloride and a fluoride anion. a, The free-energy surface is represented as a function of the collective variables $d_1$ and $d_2$, defined as the distances between the carbon atom and the fluorine and chlorine atoms in Å, respectively. Sampling points selected by BUQ are shown as white markers. The Minimum Free-Energy Path (MFEP), obtained via the string method, is shown in white. b, free energy along the MFEP and the structures of reactants, products, and transition state, indicating the reaction coordinates $d_1$ and $d_2$.
• Figure 5: Block-averaging convergence analysis of the benchmark metadynamics free energy surface (FES). FES estimations were averaged over blocks of 500 (1 ns) and 1000 (2 ns) Gaussian depositions, aligned to zero at their minimum, and compared to the final block via RMSD.