The use of the most accurate (i.e., QM or QM/MM) levels of theory for free energy simulations (FES) is typically not possible. Primarily, this is because the computational cost associated with the ...extensive configurational sampling needed for converging FES is prohibitive. To ensure the feasibility of QM-based FES, the “indirect” approach is generally taken, necessitating a free energy calculation between the MM and QM/MM potential energy surfaces. Ideally, this step is performed with standard free energy perturbation (Zwanzig’s equation) as it only requires simulations be carried out at the low level of theory; however, work from several groups over the past few years has conclusively shown that Zwanzig’s equation is ill-suited to this task. As such, many approximations have arisen to mitigate difficulties with Zwanzig’s equation. One particularly popular notion is that the convergence of Zwanzig’s equation can be improved by using interaction energy differences instead of total energy differences. Although problematic numerical fluctuations (a major problem when using Zwanzig’s equation) are indeed reduced, our results and analysis demonstrate that this “interaction energy approximation” (IEA) is theoretically incorrect, and the implicit approximation invoked is spurious at best. Herein, we demonstrate this via solvation free energy calculations using IEA from two different low levels of theory to the same target high level. Results from this proof-of-concept consistently yield the wrong results, deviating by ∼1.5 kcal/mol from the rigorously obtained value.
The reliability of free energy simulations (FES) is limited by two factors: (a) the need for correct sampling and (b) the accuracy of the computational method employed. Classical methods (e.g., force ...fields) are typically used for FES and present a myriad of challenges, with parametrization being a principle one. On the other hand, parameter-free quantum mechanical (QM) methods tend to be too computationally expensive for adequate sampling. One widely used approach is a combination of methods, where the free energy difference between the two end states is computed by, e.g., molecular mechanics (MM), and the end states are corrected by more accurate methods, such as QM or hybrid QM/MM techniques. Here we report two new approaches that significantly improve the aforementioned scheme; with a focus on how to compute corrections between, e.g., the MM and the more accurate QM calculations. First, a molecular dynamics trajectory that properly samples relevant conformational degrees of freedom is generated. Next, potential energies of each trajectory frame are generated with a QM or QM/MM Hamiltonian. Free energy differences are then calculated based on the QM or QM/MM energies using either a non-Boltzmann Bennett approach (QM-NBB) or non-Boltzmann free energy perturbation (NB-FEP). Both approaches are applied to calculate relative and absolute solvation free energies in explicit and implicit solvent environments. Solvation free energy differences (relative and absolute) between ethane and methanol in explicit solvent are used as the initial test case for QM-NBB. Next, implicit solvent methods are employed in conjunction with both QM-NBB and NB-FEP to compute absolute solvation free energies for 21 compounds. These compounds range from small molecules such as ethane and methanol to fairly large, flexible solutes, such as triacetyl glycerol. Several technical aspects were investigated. Ultimately some best practices are suggested for improving methods that seek to connect MM to QM (or QM/MM) levels of theory in FES.
The calculation of free energy differences between levels of theory has numerous potential pitfalls. Chief among them is the lack of overlap, i.e., ensembles generated at one level of theory (e.g., ...“low”) not being good approximations of ensembles at the other (e.g., “high”). Numerous strategies have been devised to mitigate this issue. However, the most straightforward approach is to ensure that the “low” level ensemble more closely resembles that of the “high”. Ideally, this is done without increasing computational cost. Herein, we demonstrate that by reparametrizing classical intramolecular potentials to reproduce high level forces (i.e., force matching) configurational overlap between a “low” (i.e., classical) and “high” (i.e., quantum) level can be significantly improved. This procedure is validated on two test cases and results in vastly improved convergence of free energy simulations.
Carrying out free energy simulations (FES) using quantum mechanical (QM) Hamiltonians remains an attractive, albeit elusive goal. Renewed efforts in this area have focused on using “indirect” ...thermodynamic cycles to connect “low level” simulation results to “high level” free energies. The main obstacle to computing converged free energy results between molecular mechanical (MM) and QM (ΔA MM→QM), as recently demonstrated by us and others, is differences in the so-called “stiff” degrees of freedom (e.g., bond stretching) between the respective energy surfaces. Herein, we demonstrate that this problem can be efficiently circumvented using nonequilibrium work (NEW) techniques, i.e., Jarzynski’s and Crooks’ equations. Initial applications of computing ΔA NEW MM→QM, for blocked amino acids alanine and serine as well as to generate butane’s potentials of mean force via the indirect QM/MM FES method, showed marked improvement over traditional FES approaches.
Conventional quantum mechanical–molecular mechanics (QM/MM) simulation approaches for modeling enzyme reactions often assume that there is one dominant reaction pathway and that this pathway can be ...sampled starting from an X-ray structure of the enzyme. These assumptions reduce computational cost; however, their validity has not been extensively tested. This is due in part to the lack of a rigorous formalism for integrating disparate pathway information from dynamical QM/MM calculations. Here, we present a way to model ensembles of reaction pathways efficiently using a divide-and-conquer strategy through Hierarchical Markov State Modeling (Hi-MSM). This approach allows information on multiple, distinct pathways to be incorporated into a chemical kinetic model, and it allows us to test these two assumptions. Applying Hi-MSM to the reaction carried out by dihydrofolate reductase (DHFR) we find (i) there are multiple, distinct pathways significantly contributing to the overall flux of the reaction that the conventional approach does not identify and (ii) that the conventional approach does not identify the dominant reaction pathway. Thus, both assumptions underpinning the conventional approach are violated. Since DHFR is a relatively small enzyme, and configuration space scales exponentially with protein size, accounting for multiple reaction pathways is likely to be necessary for most enzymes.
Use of quantum mechanical/molecular mechanical (QM/MM) methods in binding free energy calculations, particularly in the SAMPL challenge, often fail to achieve improvement over standard additive (MM) ...force fields. Frequently, the implementation is through use of reference potentials, or the so-called “indirect approach”, and inherently relies on sufficient overlap existing between MM and QM/MM configurational spaces. This overlap is generally poor, particularly for the use of free energy perturbation to perform the MM to QM/MM free energy correction at the end states of interest (e.g., bound and unbound states). However, by utilizing MM parameters that best reproduce forces obtained at the desired QM level of theory, it is possible to lessen the configurational disparity between MM and QM/MM. To this end, we sought to use force matching to generate MM parameters for the SAMPL6 CB8 host–guest binding challenge, classically compute binding free energies, and apply energetic end state corrections to obtain QM/MM binding free energy differences. For the standard set of 11 molecules and the bonus set (including three additional challenge molecules), error statistics, such as the root mean square deviation (RMSE) were moderately poor (5.5 and 5.4 kcal/mol). Correlation statistics, however, were in the top two for both standard and bonus set submissions (
R
2
of 0.42 and 0.26,
τ
of 0.64 and 0.47 respectively). High RMSE and moderate correlation strongly indicated the presence of systematic error. Identifiable issues were ameliorated for two of the guest molecules, resulting in a reduction of error and pointing to strong prospects for the future use of this methodology.
This study reports the results of binding free energy calculations for CB8 host–guest systems in the SAMPL6 blind challenge (receipt ID 3z83m). Force-field parameters were developed specific for each ...of host and guest molecules to improve configurational sampling. We used quantum mechanical (QM) implicit solvent calculations and QM force matching to determine non-bonded (partial atomic charges) and bonded terms, respectively. Free energy calculations were carried out using the double-decoupling method (DDM) combined with Hamiltonian replica exchange method (HREM) and Bennett acceptance ratio (BAR) method. The root mean square error (RMSE) of the predicted values using DDM with respect to the experimental results was 4.32 kcal/mol. The coefficient of determination (R
2
) and Kendall rank coefficient (
τ
) were 0.49 and 0.52, respectively, highest of all submissions. In addition, these were compared to the results obtained by umbrella sampling (US) and weighted histogram analysis method (WHAM). Overall, DDM achieved a higher prediction accuracy than the US method. Results are discussed in terms of parameterization and free energy simulations.
Accurately computing partition coefficients is a pivotal part of drug discovery. Specifically, octanol–water partition coefficients can provide information into hydrophobicity of drug-like molecules, ...as well as a de facto representation of membrane permeability. However, one challenge facing the computation of partition coefficients is the need to encapsulate various microscopic environments. These include areas of largely bulk solvent (i.e., either water or octanol) or regions where octanol is saturated with water or areas of higher salt concentration. Also, tautomeric effects require consideration. Thus, we present a Boltzmann weighting approach that incorporates transfer free energies across varying microscopic media, as well as varying tautomeric state, to compute partition coefficients in the SAMPL6 challenge.
Accurately predicting free energy differences is essential in realizing the full potential of rational drug design. Unfortunately, high levels of accuracy often require computationally expensive ...QM/MM Hamiltonians. Fortuitously, the cost of employing QM/MM approaches in rigorous free energy simulation can be reduced through the use of the so-called “indirect” approach to QM/MM free energies, in which the need for QM/MM simulations is avoided via a QM/MM “correction” at the classical endpoints of interest. Herein, we focus on the computation of QM/MM binding free energies in the context of the SAMPL8 Drugs of Abuse host–guest challenge. Of the 5 QM/MM correction coupled with force-matching submissions, PM6-D3H4/MM ranked submission proved the best overall QM/MM entry, with an RMSE from experimental results of 2.43 kcal/mol (best in ranked submissions), a Pearson’s correlation of 0.78 (second-best in ranked submissions), and a Kendall
τ
correlation of 0.52 (best in ranked submissions).
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