The difference between analytically calculated free energy ΔG ij, g and free energy calculated using MBAR with mapping ΔG ij, ℓ should be consistent with the difference in the solvation free energies calculated using MBAR without mapping for thermodynamic cycle (a) according to Eq. (6) for the three transformations. ΔG ij, ℓ, ΔG jk, ℓ, and ΔG ki, ℓ should add up to zero in thermodynamic cycle and (b) according to Eq. (7) .
Free energy differences along the transformation coordinate are plotted for the three water transformations. Three intermediate states are sufficient to accurately estimate the free energy difference, or a single intermediate state if less precision is required. Uncertainties are the same size as or smaller than the symbols.
Free energy estimates converge with just five intermediate states for all the transformations.
ΔΔG hyd for SPC/E–TIP4P estimated using MBAR with mapping with different number of states (2, 3, 5, 11, 21) and number of samples per state (10, 100, 1000, 10 000) has 2–3 orders lower uncertainty compared with ΔΔG hyd for SPC/E–TIP4P estimated using MBAR without mapping with 21 states and 20 000 samples per state.
Ratio of samples required for MBAR without remapping vs. with remapping to achieve a target statistical uncertainty. Mapping and reweighting approaches require 102–105 times fewer samples compared to MBAR without mapping to achieve the same precision in ΔΔG hyd .
The free energy differences between dipoles of different equilibrium lengths are calculated using two approaches: (a) MBAR without mapping and (b) MBAR with mapping. Different number of states (2, 3, 5, 11) and number of samples in each state (10, 100, 1000, 10 000) are used to estimate the free energies. MBAR with mapping estimates converged free energies with low uncertainty using only 2 states and 100 to 1000 samples per state whereas MBAR without mapping gives unconverged free energy estimate with high uncertainty unless many intermediates are used. In subplot (c) we see that the uncertainty in free energy using MBAR with mapping is an order lower compared to uncertainty in free energy estimated using MBAR without mapping using the same amount of sampling.
Log ratio of samples required for MBAR without remapping vs. with remapping to achieve a target statistical uncertainty for the free energy difference between dipoles of different lengths. Phase space remapping combined with MBAR requires 3–300 times fewer samples compared to MBAR without mapping to achieve the same precision in estimating free energies between dipoles of different length.
Force field parameters for the water models used in this study.
Solvation free energies, ΔG hyd , and enthalpy of vaporization H vap . H vap estimated using MBAR without mapping (column 3) has half the error as H vap using the standard method of energy averages at the endpoint alone (column 4), as it uses information from nearby intermediate states.
ΔG ij, g (column 1) is calculated analytically and ΔG ij, ℓ (column 2) is calculated using MBAR with mapping. The differences ΔΔG hyd (decoupling) are calculated using direct subtraction and error propagation using data from Table II . ΔΔG hyd (mapping) − ΔΔG hyd (decoupling) are zero within two and three standard deviations according to the thermodynamic cycle in Eq. (6) for all transformations. Subscripts a, b, and c refer to the transformations labeled a, b, and c in the first three rows. The result of decoupling cycle in column 5 of the last row is constrained to be identically zero numerically because it consists of quantities (i − j) + (j − k) + (k − i). The result of the mapping cycle in column 4 of the last row is not constrained to be identically zero numerically as it consists of three independent calculations using different simulations; instead, it is in such close agreement with the thermodynamic cycle in Eq. (7) because of the high statistical precision of the approach. All free energies are in kJ/mol.
All enthalpies are in kJ/mol and all entropies are in J/mol/K. Decoupling ΔH vap s are calculated using direct subtraction and error propagation using data from Table II . The differences ΔS ij, ℓ are calculated using the relationship ΔS = (ΔH − ΔG)/T for both decoupling and mapping cases, with uncertainties propagated using the covariance calculated with MBAR. Subscripts a, b, and c refer to the transformations labeled a, b, and c in the first three rows. The thermodynamic cycle in Eq. (7) is satisfied for enthalpy H and entropy S. Enthalpies and entropies calculated with decoupling method agree with the ones calculated using mapping formalism within statistical error.
MBAR with mapping requires just 2 states and 1000 samples per state to estimate converged and precise free energy difference between dipoles of very different equilibrium lengths.
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