Estimating Solvation Free Energies with Boltzmann Generators

TL;DR

The paper tackles the challenge of calculating solvation free energies by addressing poor phase-space overlap between gas‑phase and solvated states. It introduces Boltzmann Generators, a subset of normalizing flows, to learn invertible mappings that transform solvent configurations and enhance overlap, enabling learned free energy perturbation estimations. Through Lennard‑Jones solvent tests, it demonstrates that BG can reproduce MBAR results for challenging transformations like solute growth and varying solute separation, with meaningful RDF changes and entropy trends reflected. Overall, the work presents a promising direction for accelerating solvation free-energy calculations, highlighting both efficiency gains in certain transformations and limitations related to system size, flow expressivity, and the need for validation on more realistic, correlated solvent environments.

Abstract

Accurate calculations of solvation free energies remain a central challenge in molecular simulations, often requiring extensive sampling and numerous alchemical intermediates to ensure sufficient overlap between phase-space distributions of a solute in the gas phase and in solution. Here, we introduce a computational framework based on normalizing flows that directly maps solvent configurations between solutes of different sizes, and compare the accuracy and efficiency to conventional free energy estimates. For a Lennard-Jones solvent, we demonstrate that this approach yields acceptable accuracy in estimating free energy differences for challenging transformations, such as solute growth or increased solute-solute separation, which typically demand multiple intermediate simulation steps along the transformation. Analysis of radial distribution functions indicates that the flow generates physically meaningful solvent rearrangements, substantially enhancing configurational overlap between states in configuration space. These results suggest flow-based models as a promising alternative to traditional free energy estimation methods.

Figures (3)

• Figure 1: (a) Illustration of the solvation experiment showing solvents (blue) and solutes $A$ and $B$ (red), with particle $B$ varying in radius across the simulations. (b) Radial distribution function $g_{B-\mathrm{sol}}(r)$ between solute $B$ and the solvent particles at 100 K. The orange line (mapped) is obtained by applying the flow to solvent configurations from the $\sigma_B = 2.0~\text{\AA}$ simulation before inserting a $\sigma_B = 4.0~\text{\AA}$ solute, while the red dashed line (unmapped) uses the same solvent configurations without transformation prior to solute insertion. (c) Reduced free energy differences relative to the base state with $\sigma_B=3.0~\text{\AA}$ and $T=100~\text{K}$ (marked with a star symbol). (d) Change in $\Delta F$ for $\sigma_B=2.0~$Å as a function of temperature together with a linear regression of the data points. (e) Entropy obtained as the slope of the linear fits in (d) for various $\sigma_B$-values.
• Figure 2: (a) Illustration of the experiment in which the solute-solute-distance is varied. Solute and solvent particles are depicted as red and blue spheres, respectively. (b) RDF $g_{B-\mathrm{sol}}(r)$ between solute $B$ and solvent particles, showing both the unmapped and mapped RDFs compared to the reference MD RDF at a distance $d_{AB} = 4~$Å. For the unmapped RDF, solvents configurations from the base state $d^0_{AB}=2.0~$Å were combined with the solutes placed at the target distance $d_{AB}=4.0~$Å. (c) $\Delta f$ as a function of particle separation $d_{AB}$ relative to the base state $d^0_{AB}=2$ Å, marked by a star symbol, estimated using both the flow method and MD+MBAR.
• Figure 3: (a) $\Delta f_{\text{sol}}$ between states with $\sigma_B = 2.0$ Å and 4.0 Å at $T = 100$ K as obtained from MD+MBAR shown as a function of the spacing between intermediate states $\sigma_{\rm step}$. (b) $\Delta f_\mathrm{sol}$ as a function of the temperature step size between states with $\sigma_B =3.0$ Å at $T=100$ K and 140 K. (c) $\Delta f_\mathrm{sol}$ as a function of the displacement step size for transitions from $d_{AB} = 0$ Å to $d_{AB} = 4$ Å at $T=100$ K.