Expanded ensemble predictions of absolute binding free energies in the SAMPL9 host-guest challenge

Abstract

As part of the SAMPL9 community-wide blind host-guest challenge, we implemented an expanded ensemble workflow to predict absolute binding free energies for 13 small molecules against pillar[6]arene. Notable features of our protocol include consideration of a variety of protonation and enantiomeric states for both host and guests, optimization of alchemical intermediates, and analysis of free energy estimates and their uncertainty using large numbers of simulation replicates performed using distributed computing. Our predictions of absolute binding free energies resulted in a mean absolute error of 2.29 kcal mol^-1 and an R² of 0.54. Overall, results show that expanded ensemble calculations using all-atom molecular dynamics simulations are a valuable and efficient computational tool in predicting absolute binding free energies.

Fig. 1

Molecular structures of the pillar[6]arene host (WP6) and guests (G1–G13).

Fig. 2

An example of $\lambda$-value optimization using pylambdaopt. The top panel shows ${\lambda }_i$ values before optimization. The bottom panel shows the optimized ${\lambda }_i^{\ast }$ values. Since the vdW decoupling transformation occurs independently and subsequent to the Coulomb decoupling transformation, pylambdaopt treats the transformation using a single $\lambda =0\to 2$, where $\lambda =0\to 1$ represents Coulomb decoupling and $\lambda =1\to 2$ represents vdW decoupling.

Fig. 3

(a) Predicted free energies of host–guest binding using single microstates $i$ (${\Delta G}_{\text{binding}}={\Delta G}_{\text{rest}}+{\Delta G}_{L,i}-{\Delta G}_{RL,i}$) between host microstates and guest microstates. Error bars are colored by guest (see panel b) and data markers are colored by microstate index (see legend). (b) ${\Delta G}_{\text{binding}}$ in kcal mol⁻¹ for each guest to WP6 (colored by guest) calculated according to Equation (3)

Fig. 4

(a) Comparisons of predicted (ranked submission) vs. experimental binding free energies for all host–guest systems. Annotations report the correlation coefficient ($R^2$), mean absolute error (MAE), mean standard error (MSE), root-mean squared error (RMSE). (b) Comparisons for the “Voelz RL8” submission. (c) Comparisons for the “Voelz all” submission.

Fig. 5

Predictions of host–guest binding free energies submitted by all SAMPL9 participants. The color map and numerical values inside the cells are the absolute difference in predictions against experiment (in kcal mol⁻¹). The last row is the mean absolute error (MAE) over all host–guest predictions for each group.

Fig. 6

Convergence of expanded ensemble (EE) estimates of guest-only decoupling free energies ${\Delta G}_L$ for G13. Superimposed are (up to) fifty independent trajectories (distinguished by color). (a) The Wang-Landau (WL) increment vs. simulation time, with a dotted line denoting our 0.02 convergence threshold. (b) EE estimates of ${\Delta G}_L$ vs. simulation time, with a dotted line denoting the final estimate.

Fig. 7

Convergence of expanded ensemble (EE) estimates of host–guest decoupling free energies ${\Delta G}_{RL}$ for G13. Superimposed are (up to) fifty independent trajectories (distinguished by color). (a) The Wang-Landau (WL) increment vs. simulation time, with a dotted line denoting our 0.02 convergence threshold. (b) EE estimates of ${\Delta G}_{RL}$ vs. simulation time, with a dotted line denoting the final estimate.

Fig. 8

Convergence of expanded ensemble (EE) estimates of guest-only decoupling free energies ${\Delta G}_L$ for G6. Superimposed are (up to) fifty independent trajectories (distinguished by color). (a) The Wang-Landau (WL) increment vs. simulation time, with a dotted line denoting our 0.02 convergence threshold. (b) EE estimates of ${\Delta G}_L$ vs. simulation time, with a dotted line denoting the final estimate.

Fig. 9

Convergence of expanded ensemble (EE) estimates of host–guest decoupling free energies ${\Delta G}_{RL}$ for G6. Superimposed are (up to) fifty independent trajectories (distinguished by color). (a) The Wang-Landau (WL) increment vs. simulation time, with a dotted line denoting our 0.02 convergence threshold. (b) EE estimates of ${\Delta G}_{RL}$ vs. simulation time, with a dotted line denoting the final estimate.

Fig. 10

Estimated correction terms to our submitted free energy predictions for each guest, to correctly account for restraint biases.

Fig. 11

Transitions between a number of host-bound states are shown for G5. (a) A trace of the $z$-axis displacement of the guest center-of-mass (COM) with respect to the host COM shows free energy minima at both +/− edges of the host, in addition to those near the center of mass. (b,c, and d) depict Conformations representative of binding for G5 are shown for poses near the (b) top, (c) middle, and (d) bottom of the host.

Fig. 12

Changes to our submitted predicted before (red) and after (green) corrections are applied to correctly account for restraint biases. Left: “Voelz RL_8” submissions vs. experiment. Right: “Voelz ranked” submissions vs. experiment.

Fig. 13

Comparison of the convergence of free energies from expanded ensemble simulations, calculated using the EE biases versus the MBAR free energy estimator. Ten independent EE trajectories for the decoupling of guest G13 from the WP6 (−12) host (see Figure 7) were used for these tests. (a) Free energies ${\Delta G}_{RL}$ over time for each trajectory estimated from the EE biases. Traces change from transparent to solid lines as each trajectory reached the $\delta <0.02$ convergence criterion. (b) Free energies ${\Delta G}_{RL}$ over time for each trajectory estimated from MBAR. The appearance of each trace over time occurs when both the convergence criterion is met, and there are samples from each thermodynamic ensemble. Mean free energy estimates averaged over each trajectory, standard deviations ${\sigma }_{\Delta G}$ across trajectories, and standard errors of the mean, shown over time for the EE bias estimator (c,e) and MBAR estimator (d,f).

Fig. 14

Spearman rank correlation coefficients $r_s$ for participants in the SAMPL9 host–guest challenge, in comparison with the null distribution $P(r_s)$ constructed from the nonparametric sampling of randomly permuted ranks.