Development of a quantum mechanics-based free-energy perturbation method: use in the calculation of relative solvation free energies

Free-energy perturbation (FEP) is considered the most accurate computational method for calculating relative solvation and binding free-energy differences. Despite some success in applying FEP methods to both drug design and lead optimization, FEP calculations are rarely used in the pharmaceutical industry. One factor limiting the use of FEP is its low throughput, which is attributed in part to the dependence of conventional methods on the user's ability to develop accurate molecular mechanics (MM) force field parameters for individual drug candidates and the time required to complete the process. In an attempt to find an FEP method that could eventually be automated, we developed a method that uses quantum mechanics (QM) for treating the solute, MM for treating the solute surroundings, and the FEP method for computing free-energy differences. The thread technique was used in all transformations and proved to be essential for the successful completion of the calculations. Relative solvation free energies for 10 structurally diverse molecular pairs were calculated, and the results were in close agreement with both the calculated results generated by conventional FEP methods and the experimentally derived values. While considerably more CPU demanding than conventional FEP methods, this method (QM/MM-based FEP) alleviates the need for development of molecule-specific MM force field parameters and therefore may enable future automation of FEP-based calculations. Moreover, calculation accuracy should be improved over conventional methods, especially for calculations reliant on MM parameters derived in the absence of experimental data.     

Efficient calculation of temperature dependence of solid-phase free energies by overlap sampling coupled with harmonically targeted perturbation

We examine a method for computing the change in free energy with temperature of a crystalline solid. In the method, the free-energy difference between nearby temperatures is calculated via overlap-sampling free-energy perturbation with Bennett's optimization. Coupled to this is a harmonically targeted perturbation that displaces the atoms in a manner consistent with the temperature change, such that for a harmonic system, the free-energy difference would be recovered with no error. A series of such perturbations can be assembled to bridge larger gaps in temperature. We test this harmonically targeted temperature perturbation (HTTP) method through the application to the inverse-power soft potential, u(r)=ε(σ/r)(n), over a range of temperatures up to the melting condition. Three exponent values (n=12, 9, and 6) for the potential are studied with different crystal structures, specifically face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal close packing. Absolute free energies (classical only) for each system are obtained by implementing the series to near-zero temperature, where the harmonic model becomes very accurate. The HTTP method is shown to provide very precise results, with errors in the free energy smaller than two parts in 10(5). An analysis of the thermodynamic stability of the various structures in the infinite-system limit confirms previous findings. In particular, for n=12 and 9, the fcc structure is stable for all temperatures up to melting, and for n=6, the bcc crystal becomes stable relative to fcc for temperatures above kT/ε=0.802±0.001. The effects of vacancies and other defects are not considered in the analysis.     

Use of a QM/MM-based FEP method to evaluate the anomalous hydration behavior of simple alkyl amines and amides: application to the design of FBPase inhibitors for the treatment of type-2 diabetes

Standard molecular mechanics (MM) force fields predict a nearly linear decrease in hydration free energy with each successive addition of a methyl group to ammonia or acetamide, whereas a nonadditive relationship is observed experimentally. In contrast, the non-additive hydration behavior is reproduced directly using a quantum mechanics (QM)/MM-based free-energy perturbation (FEP) method wherein the solute partial atomic charges are updated at every window. Decomposing the free energies into electrostatic and van der Waals contributions and comparing the results with the corresponding free energies obtained using a conventional FEP method and a QM/MM method wherein the charges are not updated suggests that inaccuracies in the electrostatic free energies are the primary reason for the inability of the conventional FEP method to predict the experimental findings. The QM/MM-based FEP method was subsequently used to evaluate inhibitors of the diabetes drug target fructose-1,6-bisphosphatase adenosine 5'-monophosphate and 6-methylamino purine riboside 5'-monophosphate. The predicted relative binding free energy was consistent with the experimental findings, whereas the relative binding free energy predicted using the conventional FEP method differed from the experimental finding by an amount consistent with the overestimated relative solvation free energies calculated for alkylamines. Accordingly, the QM/MM-based FEP method offers potential advantages over conventional FEP methods, including greater accuracy and reduced user input. Moreover, since drug candidates often contain either functionality that is inadequately treated by MM (e.g., simple alkylamines and alkylamides) or new molecular scaffolds that require time-consuming development of MM parameters, these advantages could enable future automation of FEP calculations as well as greatly increase the use and impact of FEP calculations in drug discovery.     

Improved precision and efficiency of free energy calculations for small systems using lambda-scaled atomic masses and separating conformational and transformational sampling

We present results showing the importance of appropriate treatment of atomic masses in molecular dynamics (MD)-based single topology free-energy perturbations (FEPs) on small molecule systems. The reversibility of gas phase simulations is significantly improved by scaling the atomic mass of mutated atoms with the lambda variable normally used for the scaling of energy terms. Because this effect is less pronounced for solvated systems, it will not cancel in estimates of the relative hydration free energy difference. The advantage of mass scaling is demonstrated by a null mutation of ethane to ethane and the calculation of the relative hydration free energy difference between ethane and n-propane. Furthermore, it is found that the simulation time necessary for converged MD/FEPs is prohibitively large for relative hydration free energy calculations on cyclic alkanes. Therefore, we explore an alternative free energy pathway including strongly constrained conformations to improve convergence in FEP simulations of flexible molecules.     

Understanding free-energy perturbation calculations through a model of harmonic oscillators: theory and implications to improve the sampling efficiency by molecular simulation

Free-energy perturbation calculation is frequently used to calculate free-energy differences because it is easy to implement and the computation is fast. However, the calculation is subject to large inaccuracies in some circumstances due to the insufficient sampling of the relevant tails of the energy-difference distributions. Here we expand this knowledge of insufficient sampling into a two-dimensional (2D) energy space using a model of harmonic oscillators. We show analytically the relation between the energies of the sampling system and those of the desired target energy spaces, which provide the basis to understand the difficulties in free-energy perturbation calculations. We clarify the reasons of the inaccurate calculation in the different harmonic cases that stem from the spatial separations of the reference and the target energy pairs located in the two-dimensional energy space. The potential-energy space introduced into this 2D energy-space model provides additional clues to improve the sampling efficiency. Based on this understanding, we propose two ways to calculate the free-energy differences using the two schemes of the distribution method. We show that the distribution method implemented in the appropriate energy space--the energy-difference space and the potential-energy space, respectively--can improve the calculation of free energies in different circumstances. This analysis implies that the sampling can be improved if it is directed toward the appropriate region in the potential-energy space, which is easily implemented in various types of free-energy calculations. To test this, we calculate the free-energy surface of alanine dipeptide in gas phase and in aqueous phase, respectively. We demonstrate that the free-energy surface calculation is improved when the biased sampling of the potential energy is integrated into the sampling scheme.     

Efficient calculation of α- and β-nitrogen free energies and coexistence conditions via overlap sampling with targeted perturbation

A recently introduced solid-phase free-energy calculation method that is based upon overlap sampling with targeted free-energy perturbation is further developed and extended to systems with orientational degrees of freedom. Specifically we calculate the absolute free energy of the linear-molecular nitrogen model of Etter et al., examining both the low-temperature low-pressure α-N(2) structure and the orientationally disordered β-N(2) phase. In each perturbation (for the α-N(2) phase) to determine the free-energy difference between systems at adjacent temperatures, harmonic coordinate scaling is applied to both the translational and rotational degrees of freedom in the nitrogen molecule to increase the phase-space overlap of the two perturbing systems and consequently, improve the free-energy difference results. For the plastic β-N(2) phase, a novel method that requires several perturbation paths is introduced to calculate its absolute free energy. Through these methods, the absolute free energies for both the α-N(2) and β-N(2) phase can be accurately and precisely determined. We find again that the anharmonic contribution to the free energy has weak dependence on system size. The transition properties for the α-N(2) and β-N(2) phase are also investigated. The α-β phase transition for the model at atmospheric pressure (0.1 MPa) is found to occur at 40.35 ± 0.01 K with volumetric and entropy changes of 0.44 ± 0.01 cm(3)/mol and 1.99 ± 0.01 cal/mol.K respectively.     

Contributions of conformational compression and preferential transition state stabilization to the rate enhancement by chorismate mutase

The rate enhancement provided by the chorismate mutase (CM) enzyme for the Claisen rearrangement of chorismate to prephenate has been investigated by application of the concept of near attack conformations (NACs). Using a combined QM/MM Monte Carlo/free-energy perturbation (MC/FEP) method, 82% and 100% of chorismate conformers were found to be NAC structures in water and in the CM active site, respectively. Consequently, the conversion of non-NACs to NACs does not contribute to the free energy of activation from preorganization of the substrate into NACs. The FEP calculations yielded differences in free energies of activation that well reproduce the experimental data. Additional calculations indicate that the rate enhancement by CM over the aqueous phase results primarily from conformational compression of NACs by the enzyme and that this process is enthalpically controlled. This suggests that preferential stabilization of the transition state in the enzyme environment relative to water plays a secondary role in the catalysis by CM.     

One-dimensional free-energy profiles of complex systems: progress variables that preserve the barriers

We show that the balanced minimum-cut procedure introduced in PNAS 2004, 101, 14766 can be reinterpreted as a method for solving the constrained optimization problem of finding the minimum cut among the cuts with a particular value of an additive function of the nodes on either side of the cut. Such an additive function (e.g., the partition function of the reactant region) can be used as a progress coordinate to determine a one-dimensional profile (FEP) of the free-energy surface of the protein-folding reaction as well as other complex reactions. The algorithm is based on the network (obtained from an equilibrium molecular dynamics simulation) that represents the calculated reaction behavior. The resulting FEP gives the exact values of the free energy as a function of the progress coordinate; i.e., at each value of the progress coordinate, the profile is obtained from the surface with the minimal partition function among the surfaces that divide the full free-energy surface between two chosen end points. In many cases, the balanced minimum-cut procedure gives results for only a limited set of points. An approximate method based on p(fold) is shown to provide the profile for a more complete set of values of the progress coordinate. Applications of the approach to model problems and to realistic systems (beta-hairpin of protein G, LJ38 cluster) are presented.     

Calculation of relative binding free energy differences for fructose 1,6-bisphosphatase inhibitors using the thermodynamic cycle perturbation approach.

An iterative, computer-assisted, drug design strategy that combines molecular design, molecular mechanics, molecular dynamics (MD), and free energy perturbation (FEP) calculations with compound synthesis, biochemical testing of inhibitors, and crystallographic structure determination of protein-inhibitor complexes was successfully used to predict the rank order of a series of nucleoside monophosphate analogues as fructose 1,6-bisphosphatase (FBPase) inhibitors. The X-ray structure of FBPase complexed with 5-aminoimidazole-4-carboxamide-1-beta-D-ribofuranosyl 5'-monophosphate (ZMP) provided structural information used for subsequent analogue design and free energy calculations. The FEP protocol was validated by calculating the free energy differences for the mutation of ZMP (1) to AMP (2). The calculated results showed a net gain of 1.7 kcal/mol, which agreed with the experimental result of 1.3 kcal/mol. FEP calculations were performed for 18 other AMP analogues. Inhibition constants were determined for over half of these analogues, usually after completion of the calculation, and were consistent with the predictions. Solvation free energy differences between AMP and various AMP analogues proved to be an important factor in binding free energies, suggesting that increased desolvation costs associated with the addition of polar groups to an inhibitor must be overcome by stronger ligand-protein interactions if the structural modification is to enhance inhibitor potency. The results indicate that FEP calculations predict relative binding affinities with high accuracy and provide valuable insight into the factors that influence inhibitor binding and therefore should greatly aid efforts to optimize initial lead compounds and reduce the time required for the discovery of new drug candidates.     

Can free energy calculations be fast and accurate at the same time? Binding of low-affinity,non-peptide inhibitors to the SH2 domain of the src protein

The usefulness of free-energy calculations in non-academic environments, in general, and in the pharmaceutical industry, in particular, is a long-time debated issue, often considered from the angle of cost/performance criteria. In the context of the rational drug design of low-affinity, non-peptide inhibitors to the SH2 domain of the (pp60)src tyrosine kinase, the continuing difficulties encountered in an attempt to obtain accurate free-energy estimates are addressed. free-energy calculations can provide a convincing answer, assuming that two key-requirements are fulfilled: (i) thorough sampling of the configurational space is necessary to minimize the statistical error, hence raising the question: to which extent can we sacrifice the computational effort, yet without jeopardizing the precision of the free-energy calculation? (ii) the sensitivity of binding free-energies to the parameters utilized imposes an appropriate parametrization of the potential energy function, especially for non-peptide molecules that are usually poorly described by multipurpose macromolecular force fields. Employing the free-energy perturbation method, accurate ranking, within +/-0.7 kcal/mol, is obtained in the case of four non-peptide mimes of a sequence recognized by the (pp60)src SH2 domain.     

Binding free energies in the SAMPL6 octa-acid host–guest challenge calculated with MM and QM methods

We have estimated free energies for the binding of eight carboxylate ligands to two variants of the octa-acid deep-cavity host in the SAMPL6 blind-test challenge (with or without endo methyl groups on the four upper-rim benzoate groups, OAM and OAH, respectively). We employed free-energy perturbation (FEP) for relative binding energies at the molecular mechanics (MM) and the combined quantum mechanical (QM) and MM (QM/MM) levels, the latter obtained with the reference-potential approach with QM/MM sampling for the MM → QM/MM FEP. The semiempirical QM method PM6-DH+ was employed for the ligand in the latter calculations. Moreover, binding free energies were also estimated from QM/MM optimised structures, combined with COSMO-RS estimates of the solvation energy and thermostatistical corrections from MM frequencies. They were performed at the PM6-DH+ level of theory with the full host and guest molecule in the QM system (and also four water molecules in the geometry optimisations) for 10–20 snapshots from molecular dynamics simulations of the complex. Finally, the structure with the lowest free energy was recalculated using the dispersion-corrected density-functional theory method TPSS-D3, for both the structure and the energy. The two FEP approaches gave similar results (PM6-DH+/MM slightly better for OAM), which were among the five submissions with the best performance in the challenge and gave the best results without any fit to data from the SAMPL5 challenge, with mean absolute deviations (MAD) of 2.4–5.2 kJ/mol and a correlation coefficient (R²) of 0.77–0.93. This is the first time QM/MM approaches give binding free energies that are competitive to those obtained with MM for the octa-acid host. The QM/MM-optimised structures gave somewhat worse performance (MAD?=?3–8 kJ/mol and R²?=?0.1–0.9), but the results were improved compared to previous studies of this system with similar methods.     

Achieving fast convergence of ab initio free energy perturbation calculations with the adaptive force-matching method

This paper studies the possibility of improving the convergence of ab initio free energy perturbation (FEP) calculations by developing customized force fields with the adaptive force-matching (AFM) method. The ab initio FEP method relies on a molecular mechanics (MM) potential to sample configuration space. If the Boltzmann weight of the MM sampling is close to that of the ab initio method, the efficiency of ab initio FEP will be optimal. The difference in the Boltzmann weights can be quantified by the relative energy difference distribution (REDD). The force field developed through AFM significantly improves the REDD when compared with standard MM models, thus improving the convergence of the ab initio FEP calculation. The static dielectric constant ε_s of ice-Ih was studied with PW-91 through ab initio FEP. With a customized force field developed through AFM, we were able to converge ε_s to 80 ± 4 with 3,600 configurations. A similar ab initio FEP calculation with the TIP4P model would require 220 times more configurations to achieve the same accuracy. Our study indicates that the PW-91 functional underestimates ice-Ih ε_s by about 20%.     

Efficiency of alchemical free energy simulations. I. A practical comparison of the exponential formula, thermodynamic integration, and Bennett's acceptance ratio method

We investigate the relative efficiency of thermodynamic integration, three variants of the exponential formula, also referred to as thermodynamic perturbation, and Bennett's acceptance ratio method to compute relative and absolute solvation free energy differences. Our primary goal is the development of efficient protocols that are robust in practice. We focus on minimizing the number of unphysical intermediate states (λ-states) required for the computation of accurate and precise free energy differences. Several indicators are presented which help decide when additional λ-states are necessary. In all tests Bennett's acceptance ratio method required the least number of λ-states, closely followed by the "double-wide" variant of the exponential formula. Use of the exponential formula in only strict "forward" or "backward" mode was not found to be competitive. Similarly, the performance of thermodynamic integration in terms of efficiency was rather poor. We show that this is caused by the use of the trapezoidal rule as method of numerical quadrature. A systematic study focusing on the optimization of thermodynamic integration is presented in a companion paper.     

Phase-space overlap measures. I. Fail-safe bias detection in free energies calculated by molecular simulation

We consider ways to quantify the overlap of the parts of phase space important to two systems, labeled A and B. Of interest is how much of the A-important phase space lies in that important to B, and how much of B lies in A. Two measures are proposed. The first considers four total-energy distributions, formed from all combinations made by tabulating either the A-system or the B-system energy when sampling either the A or B system. Measures for A in B and B in A are given by two overlap integrals defined on pairs of these distributions. The second measure is based on information theory, and defines two relative entropies which are conveniently expressed in terms of the dissipated work for free-energy perturbation (FEP) calculations in the A-->B and B-->A directions, respectively. Phase-space overlap is an important consideration in the performance of free-energy calculations. To demonstrate this connection, we examine bias in FEP calculations applied to a system of independent particles in a harmonic potential. Systems are selected to represent a range of overlap situations, including extreme subset, subset, partial overlap, and nonoverlap. The magnitude and symmetry of the bias (A-->B vs B-->A) are shown to correlate well with the overlap, and consequently with the overlap measures. The relative entropies are used to scale the amount of sampling to obtain a universal bias curve. This result leads to develop a simple heuristic that can be applied to determine whether a work-based free-energy measurement is free of bias. The heuristic is based in part on the measured free energy, but we argue that it is fail-safe inasmuch as any bias in the measurement will not promote a false indication of accuracy.     

Ab initio quantum mechanics-based free energy perturbation method for calculating relative solvation free energies

A free energy perturbation (FEP) method was developed that uses ab initio quantum mechanics (QM) for treating the solute molecules and molecular mechanics (MM) for treating the surroundings. Like our earlier results using AM1 semi empirical QMs, the ab initio QM/MM-based FEP method was shown to accurately calculate relative solvation free energies for a diverse set of small molecules that differ significantly in structure, aromaticity, hydrogen bonding potential, and electron density. Accuracy was similar to or better than conventional FEP methods. The QM/MM-based methods eliminate the need for time-consuming development of MM force field parameters, which are frequently required for drug-like molecules containing structural motifs not adequately described by MM. Future automation of the method and parallelization of the code for Linux 128/256/512 clusters is expected to enhance the speed and increase its use for drug design and lead optimization.     

Relative energies of binding for antibody-carbohydrate-antigen complexes computed from free-energy simulations

Free-energy perturbation (FEP) simulations have been applied to a series of analogues of the natural trisaccharide epitope of Salmonella serotype B bound to a fragment of the monoclonal anti-Salmonella antibody Se155-4. This system was selected in order to assess the ability of free-energy perturbation (FEP) simulations to predict carbohydrate-protein interaction energies. The ultimate goal is to use FEP simulations to aid in the design of synthetic high affinity ligands for carbohydrate-binding proteins. The molecular dynamics (MD) simulations were performed in the explicit presence of water molecules, at room temperature. The AMBER force field, with the GLYCAM parameter set for oligosaccharides, was employed. In contrast to many modeling protocols, FEP simulations are capable of including the effects of entropy, arising from differential ligand flexibilities and solvation properties. The experimental binding affinities are all close in value, resulting in small relative free energies of binding. Many of the DeltaDeltaG values are on the order of 0-1 kcal mol(-1), making their accurate calculation particularly challenging. The simulations were shown to reasonably reproduce the known geometries of the ligands and the ligand-protein complexes. A model for the conformational behavior of the unbound antigen is proposed that is consistent with the reported NMR data. The best agreement with experiment was obtained when histidine 97H was treated as fully protonated, for which the relative binding energies were predicted to well within 1 kcal mol(-1). To our knowledge this is the first report of FEP simulations applied to an oligosaccharide-protein complex.     

Use of quantum mechanics/molecular mechanics-based FEP method for calculating relative binding affinities of FBPase inhibitors for type-2 diabetes

A quantum mechanics (QM)/molecular mechanics (MM)-based free energy perturbation (FEP) method, developed recently, provides most accurate estimation of binding affinities. The validity of the method was evaluated for a large set of diverse inhibitors for fructose 1,6-bisphosphatase (FBPase), a target enzyme for type-II diabetes mellitus. The validation set comprises of 22 important structurally different mutations. The calculated relative binding free energies using the QM/MM-based FEP method reproduce the experimental values with exceptional precision of less than ±0.5 kcal/mol. The CPU requirements for QM/MM-based FEP are about fivefold greater than conventional FEP methods, but it is superior in accuracy of predictions. In addition, the QM/MM-based FEP method eliminates the need for time-consuming development of MM force field parameters, which are frequently required for novel inhibitors described by MM. Future automation of the method and parallelization of the code for 128/256/512 cluster computers is expected to enhance the speed and increase its use for drug design and lead optimization. The present application of QM/MM-based FEP method for structurally diverse set of analogs serves to enhance the scope of FEP method and demonstrate the utility of QM/MM-based FEP method for its potential in drug discovery.     

Do all pieces make a whole? Thiele cumulants and the free energy decomposition

The partitioning of the free energy into additive contributions originating from different groups of atoms or force field terms has the potential to provide relationship between structure and biological activity of molecules. In this article, the theoretical foundation for the free energy decomposition in the free energy perturbation (FEP) methodology is formulated using Thiele cumulants, a powerful tool from the arsenal of probability theory and mathematical statistics. We establish that rigorous decomposition of the free energy into its components is precluded by the presence of mixed potential energy terms in Thiele cumulants of second and higher orders. However, we also show that the resultant nonadditivity error can be reduced to an arbitrary value by increasing the number of FEP steps. Consequently, the whole system can be in the limit of small perturbation steps adequately represented by the sum of its constituent parts.     

Evaluating polarizable potentials on distributed memory parallel computers: Program development and applications

The efficient evaluation of polarizable molecular mechanics potentials on distributed memory parallel computers is discussed. The program executes at 7–10 Mflops/node on a 32-node CM-5 partition and is 19 times faster than comparable code running on a single-processor HP 9000/735. On the parallel computer, matrix inversion becomes a practical alternative to the commonly used iterative method for the calculation of induced dipole moments. The former method is useful in cases such as free-energy perturbation (FEP) simulations, which require highly accurate induced dipole moments. Matrix inversion is performed 110 times faster on the CM-5 than on the HP. We show that the accuracy which is needed for FEP calculations with polarization can be obtained by either matrix inversion or by performing a large number of iteration cycles to satisfy convergence tolerances that are less than 10^?6 D. © 1995 by John Wiley & Sons, Inc.