An efficient method for the calculation of quantum mechanics/molecular mechanics free energies

The combination of quantum mechanics (QM) with molecular mechanics (MM) offers a route to improved accuracy in the study of biological systems, and there is now significant research effort being spent to develop QM/MM methods that can be applied to the calculation of relative free energies. Currently, the computational expense of the QM part of the calculation means that there is no single method that achieves both efficiency and rigor; either the QM/MM free energy method is rigorous and computationally expensive, or the method introduces efficiency-led assumptions that can lead to errors in the result, or a lack of generality of application. In this paper we demonstrate a combined approach to form a single, efficient, and, in principle, exact QM/MM free energy method. We demonstrate the application of this method by using it to explore the difference in hydration of water and methane. We demonstrate that it is possible to calculate highly converged QM/MM relative free energies at the MP2/aug-cc-pVDZ/OPLS level within just two days of computation, using commodity processors, and show how the method allows consistent, high-quality sampling of complex solvent configurational change, both when perturbing hydrophilic water into hydrophobic methane, and also when moving from a MM Hamiltonian to a QM/MM Hamiltonian. The results demonstrate the validity and power of this methodology, and raise important questions regarding the compatibility of MM and QM/MM forcefields, and offer a potential route to improved compatibility.     

Calculation of ligand binding free energies from molecular dynamics simulations

A recently developed method for predicting binding affinities in ligand–receptor complexes, based on interaction energy averaging and conformational sampling by molecular dynamics simulation, is presented. Polar and nonpolar contributions to the binding free energy are approximated by a linear scaling of the corresponding terms in the average intermolecular interaction energy for the bound and free states of the ligand. While the method originally assumed the validity of electrostatic linear response, we show that incorporation of systematic deviations from linear response derived from free energy perturbation calculations enhances the accuracy of the approach. The method is applied to complexes of wild-type and mutant human dihydrofolate reductases with 2,4-diaminopteridine and 2,4-diaminoquinazoline inhibitors. It is shown that a binding energy accuracy of about 1 kcal/mol is attainable even for multiply ionized compounds, such as methotrexate, for which electrostatic interactions energies are very large. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 77–88, 1998     

Evaluation of molecular mechanics calculated binding energies for isolated and monolayer organic molecules on graphite

The calculated molecule-surface binding energy, E(cal)( *), for physical adsorption was determined using molecular mechanics MM2 parameters for a model graphite surface and various organic molecules. The results for E(cal)( *) were compared to published experimental binding energy values, E( *), from gas chromatography (GC) or thermal desorption (TD). The binding energies from GC were for isolated molecules in the Henry's law region of adsorption, and the binding energies from TD were for molecules in monolayer coverage on a highly oriented pyrolytic graphite (HOPG). A simple desorption model was used to allow the calculation of monolayer coverage to include both molecule-surface and molecule-molecule interactions and then the results were compared to experimental values. For the 14 TD organic adsorbates (polyaromatic hydrocarbons, alcohols, benzene, substituted benzenes, methane, chloroalkanes, N,N-dimethylformamide, and C(60) Buckyball), the experimental versus calculated binding energies were E( *)=1.1193E(cal)( *) and r(2)=0.967. The GC E( *) values were also well correlated by calculated E(cal)( *) values for a set of 11 benzene and methyl substituted benzenes and for another set of 10 alkanes and haloalkanes. The TD E(cal)( *) mechanics computation provides a useful comparison to the one for GC data since adsorbate-adsorbate interactions as well as adsorbate-surface must be considered.     

A comparison of conformational energies calculated by several molecular mechanics methods

Several commonly used molecular mechanics force fields have been tested for accuracy in conformational energy calculations. Differences in performance between the force fields are discussed for different classes of structures. MMFF93 and force fields based on the MM2 or MM3 functional form are found to perform significantly better than other force fields in the test, with average conformational energy errors around 0.5 kcal/mol. CFF91 also reaches this accuracy for the subset in which fully determined parameters are used, but it doubles the overall error due to use of estimated parameters. Harmonic force fields generally have average errors exceeding 1 kcal/mol. Factors influencing accuracy are identified and discussed. © 1996 by John Wiley & Son s, Inc.     

Coupled semiempirical quantum mechanics and molecular mechanics (QM/MM) calculations on the aqueous solvation free energies of ionized molecules

The aqueous solvation free energies of ionized molecules were computed using a coupled quantum mechanical and molecular mechanical (QM/MM) model based on the AM1, MNDO, and PM3 semiempirical molecular orbital methods for the solute molecule and the TIP3P molecular mechanics model for liquid water. The present work is an extension of our model for neutral solutes where we assumed that the total free energy is the sum of components derived from the electrostatic/polarization terms in the Hamiltonian plus an empirical “nonpolar” term. The electrostatic/polarization contributions to the solvation free energies were computed using molecular dynamics (MD) simulation and thermodynamic integration techniques, while the nonpolar contributions were taken from the literature. The contribution to the electrostatic/polarization component of the free energy due to nonbonded interactions outside the cutoff radii used in the MD simulations was approximated by a Born solvation term. The experimental free energies were reproduced satisfactorily using variational parameters from the vdW terms as in the original model, in addition to a parameter from the one-electron integral terms. The new one-electron parameter was required to account for the short-range effects of overlapping atomic charge densities. The radial distribution functions obtained from the MD simulations showed the expected H-bonded structures between the ionized solute molecule and solvent molecules. We also obtained satisfactory results by neglecting both the empirical nonpolar term and the electronic polarization of the solute, i.e., by implementing a nonpolarization model. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1028–1038, 1999     

PDB ligand conformational energies calculated quantum-mechanically

We present here a greatly updated version of an earlier study on the conformational energies of protein-ligand complexes in the Protein Data Bank (PDB) [Nicklaus et al. Bioorg. Med. Chem. 1995, 3, 411-428], with the goal of improving on all possible aspects such as number and selection of ligand instances, energy calculations performed, and additional analyses conducted. Starting from about 357,000 ligand instances deposited in the 2008 version of the Ligand Expo database of the experimental 3D coordinates of all small-molecule instances in the PDB, we created a "high-quality" subset of ligand instances by various filtering steps including application of crystallographic quality criteria and structural unambiguousness. Submission of 640 Gaussian 03 jobs yielded a set of about 415 successfully concluded runs. We used a stepwise optimization of internal degrees of freedom at the DFT level of theory with the B3LYP/6-31G(d) basis set and a single-point energy calculation at B3LYP/6-311++G(3df,2p) after each round of (partial) optimization to separate energy changes due to bond length stretches vs bond angle changes vs torsion changes. Even for the most "conservative" choice of all the possible conformational energies-the energy difference between the conformation in which all internal degrees of freedom except torsions have been optimized and the fully optimized conformer-significant energy values were found. The range of 0 to ~25 kcal/mol was populated quite evenly and independently of the crystallographic resolution. A smaller number of "outliers" of yet higher energies were seen only at resolutions above 1.3 ?. The energies showed some correlation with molecular size and flexibility but not with crystallographic quality metrics such as the Cruickshank diffraction-component precision index (DPI) and R(free)-R, or with the ligand instance-specific metrics such as occupancy-weighted B-factor (OWAB), real-space R factor (RSR), and real-space correlation coefficient (RSCC). We repeated these calculations with the solvent model IEFPCM, which yielded energy differences that were generally somewhat lower than the corresponding vacuum results but did not produce a qualitatively different picture. Torsional sampling around the crystal conformation at the molecular mechanics level using the MMFF94s force field typically led to an increase in energy.     

Atom equivalents for converting DFT energies calculated on molecular mechanics structures to formation enthalpies

New atom equivalents are introduced to convert BP/DN**//MMFF energies into formation enthalpies. As a result of using molecular mechanics structures, poor results are obtained for compounds outside the scope of the force field, such as those bearing  NF₂ groups or some nitrogenous systems. Notwithstanding these limitations, present procedures compare well with the results of previous atom equivalents schemes. Indeed, rms deviations from experiment are below 9 kJ/mol for hydrocarbons, and close to 16 kJ/mol for a variety of compounds reasonably well described by MMFF. The explicit inclusion of thermal and vibrational contributions, using calculated frequencies, does not improve the results. This study demonstrates that cost‐effective approaches to formation enthalpies may be developed on the basis of a combination of DFT with a suitable molecular mechanics force field. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 367–379, 2000     

Precision of free energies calculated by molecular dynamics simulations of peptides in solution

Errors in free energies for molecular replacement and for conformation change of a small model peptide have been determined empirically by repeated simulations from different starting points. All calculations have been done using thermodynamic integration, in which the system's potential energy is coupled to a parameter λ, that is increased or decreased by a small amount at each step of the simulation. The effects of several factors that may alter the precision are evaluated. These factors include: the length of the simulation, the dependence of the potential energy on λ, the use of conformational restraints, and their magnitude and form. The methods used for restraint and conformational forcing are described in detail. The free energy change, calculated as the mean from several successive simulations with alternately increasing and decreasing λ, is found to be independent of the length of the simulations. As expected, longer simulations produce more precise results. The variation of the calculated free energies is found to consist of two parts, a random error and a systematic hysteresis, i.e., a dependence on the direction in which λ changes. The hysteresis varies as the inverse of the length of the simulation and the random error as the inverse square root The advantage of the use of a different (nonlinear) dependence of the attractive and repulsive parts of the nonbonded potential energy on the coupling parameter when “creating” particles in solution is found to be very large. This nonlinear coupling was found to be superior to the use of linear coupling and a nonlinear change of the coupling parameter with the simulation time. The hysteresis in conformational free energy calculations is found to increase markedly if too weak a forcing restraint is chosen. It is shown how to deconvolute the contribution of a torsional restraint from the dependence of the free energy on a torsion angle.     

Alchemical absolute protein–ligand binding free energies for drug design

The recent advances in relative protein–ligand binding free energy calculations have shown the value of alchemical methods in drug discovery. Accurately assessing absolute binding free energies, although highly desired, remains a challenging endeavour, mostly limited to small model cases. Here, we demonstrate accurate first principles based absolute binding free energy estimates for 128 pharmaceutically relevant targets. We use a novel rigorous method to generate protein–ligand ensembles for the ligand in its decoupled state. Not only do the calculations deliver accurate protein–ligand binding affinity estimates, but they also provide detailed physical insight into the structural determinants of binding. We identify subtle rotamer rearrangements between apo and holo states of a protein that are crucial for binding. When compared to relative binding free energy calculations, obtaining absolute binding free energies is considerably more challenging in large part due to the need to explicitly account for the protein in its apo state. In this work we present several approaches to obtain apo state ensembles for accurate absolute ΔG calculations, thus outlining protocols for prospective application of the methods for drug discovery.

Molecular dynamics based absolute protein–ligand binding free energies can be calculated accurately and at large scale to facilitate drug discovery.     

计算受体和配体结合自由能的新方法

利用ABEEMσπ浮动电荷力场与连续介质模型相结合的方法,计算了受体和配体的结合自由能.将结合自由能分解为真空中的力场作用项、溶剂化能量以及熵效应.由于ABEEMσπ/MM方法充分考虑了外界环境发生变化引起的体系中各个位点之间的电荷极化,因而极大地提高了结合自由能的计算精度.利用该方法计算的2个复合物的结合自由能与实验值的偏差均小于0.5kJ/mol.     

Quantum mechanics and mixed quantum mechanics/molecular mechanics simulations of model nerve agents with acetylcholinesterase

 The accurate modeling of biological processes presents major computational difficulties owing to the inherent complexity of the macromolecular systems of interest. Simulations of biochemical reactivity tend to require highly computationally intensive quantum mechanical methods, but localized chemical effects tend to depend significantly on properties of the extended biological environment – a regime far more readily examined with lower-level classical empirical models. Mixed quantum/classical techniques are gaining in popularity as a means of bridging these competing requirements. Here we present results comparing two quantum mechanics/molecular mechanics implementations (the SIMOMM technique of Gordon et al. as implemented in GAMESS, and the ONIOM technique of Morokuma et al. found in Gaussian 98) as performed on the enzyme acetylcholinesterase and model nerve agents. This work represents part of the initial phase of a DoD HPCMP Challenge project in which we are attempting to reliably characterize the biochemical processes responsible for nerve agent activity and inhibition, thereby allowing predictions on compounds unrelated to those already studied. Received: 10 October 2001 / Accepted: 13 November 2002 / Published online: 1 April 2003 Contribution to the Proceedings of the Symposium on Combined QM/MM Methods at the 222nd National Meeting of the American Chemical Society, 2001 Correspondence to: M. M. Hurley e-mail: hurley@arl.army.mil     

Overcoming dissipation in the calculation of standard binding free energies by ligand extraction

This article addresses calculations of the standard free energy of binding from molecular simulations in which a bound ligand is extracted from its binding site by steered molecular dynamics (MD) simulations or equilibrium umbrella sampling (US). Host–guest systems are used as test beds to examine the requirements for obtaining the reversible work of ligand extraction. We find that, for both steered MD and US, marked irreversibilities can occur when the guest molecule crosses an energy barrier and suddenly jumps to a new position, causing dissipation of energy stored in the stretched molecule(s). For flexible molecules, this occurs even when a stiff pulling spring is used, and it is difficult to suppress in calculations where the spring is attached to the molecules by single, fixed attachment points. We, therefore, introduce and test a method, fluctuation‐guided pulling, which adaptively adjusts the spring's attachment points based on the guest's atomic fluctuations relative to the host. This adaptive approach is found to substantially improve the reversibility of both steered MD and US calculations for the present systems. The results are then used to estimate standard binding free energies within a comprehensive framework, termed attach‐pull‐release, which recognizes that the standard free energy of binding must include not only the pulling work itself, but also the work of attaching and then releasing the spring, where the release work includes an accounting of the standard concentration to which the ligand is discharged. © 2013 Wiley Periodicals, Inc.     

MoBioTools: A toolkit to setup quantum mechanics/molecular mechanics calculations

We present a toolkit that allows for the preparation of QM/MM input files from a conformational ensemble of molecular geometries. The package is currently compatible with trajectory and topology files in Amber, CHARMM, GROMACS and NAMD formats, and has the possibility to generate QM/MM input files for Gaussian (09 and 16), Orca (≥4.0), NWChem and (Open)Molcas. The toolkit can be used in command line, so that no programming experience is required, although it presents some features that can also be employed as a python application programming interface. We apply the toolkit in four situations in which different electronic-structure properties of organic molecules in the presence of a solvent or a complex biological environment are computed: the reduction potential of the nucleobases in acetonitrile, an energy decomposition analysis of tyrosine interacting with water, the absorption spectrum of an azobenzene derivative integrated into a voltage-gated ion channel, and the absorption and emission spectra of the luciferine/luciferase complex. These examples show that the toolkit can be employed in a manifold of situations for both the electronic ground state and electronically excited states. It also allows for the automatic correction of the active space in the case of CASSCF calculations on an ensemble of geometries, as it is shown for the azobenzene derivative photoswitch case.     

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.     

Integrating quantum and molecular mechanics

A computer algorithm is developed for integrating density functional quantum mechanics into a molecular mechanics program. The computationally infeasible aspects of the standard LCAO-MO approach (the iterative calculation of eigenvectors and the requirement of orthogonal expansions for the orbitals) are replaced with an efficient use of optimization via the trace theorem of linear algebra. The construction of a basis is also described for expanding the electron density that transforms with the molecular geometry. The combination of the trace method and the basis allow the solution for one configuration of atoms and electrons to be tracked over a wide range of internal conformations. The approach is readily adaptable to being used in the context of an imposed classical field that allows it to be used on part of a macromolecular complex. The initial implementation in the program AMMP is described. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1618–1633, 1999     

Solvation free energies of amino acid side chain analogs for common molecular mechanics water models

Quantitative free energy computation involves both using a model that is sufficiently faithful to the experimental system under study (accuracy) and establishing statistically meaningful measures of the uncertainties resulting from finite sampling (precision). In order to examine the accuracy of a range of common water models used for protein simulation for their solute/solvent properties, we calculate the free energy of hydration of 15 amino acid side chain analogs derived from the OPLS-AA parameter set with the TIP3P, TIP4P, SPC, SPC/E, TIP3P-MOD, and TIP4P-Ew water models. We achieve a high degree of statistical precision in our simulations, obtaining uncertainties for the free energy of hydration of 0.02-0.06 kcal/mol, equivalent to that obtained in experimental hydration free energy measurements of the same molecules. We find that TIP3P-MOD, a model designed to give improved free energy of hydration for methane, gives uniformly the closest match to experiment; we also find that the ability to accurately model pure water properties does not necessarily predict ability to predict solute/solvent behavior. We also evaluate the free energies of a number of novel modifications of TIP3P designed as a proof of concept that it is possible to obtain much better solute/solvent free energetic behavior without substantially negatively affecting pure water properties. We decrease the average error to zero while reducing the root mean square error below that of any of the published water models, with measured liquid water properties remaining almost constant with respect to our perturbations. This demonstrates there is still both room for improvement within current fixed-charge biomolecular force fields and significant parameter flexibility to make these improvements. Recent research in computational efficiency of free energy methods allows us to perform simulations on a local cluster that previously required large scale distributed computing, performing four times as much computational work in approximately a tenth of the computer time as a similar study a year ago.     

Structural forms of green fluorescent protein by quantum mechanics/molecular mechanics calculations

The molecular modeling of structural forms of the green fluorescent protein (GFP) with the Ser65Thr single-site mutation was performed by the quantum mechanics/molecular mechanics (QM/MM) method. Two model systems were constructed based on the crystallographic structure from the Protein Data Bank (PDB entry code 1EMA.) The model systems differ in the initial protonation state of the side chain of the amino acid residue Glu222 near the chromophore. The atomic coordinates of the protein macromolecule corresponding to the equilibrium geometric configurations were determined by total energy minimization using the QM/MM method within the density functional theory approximation PBE0/cc-pVDZ for the quantum subsystem that consists of the chromophore, a water molecule, and the side chains of Arg96, Glu222, and Ser205, and with the parameters of the AMBER force field for the molecular mechanics subsystem. In the analysis of the results, particular attention was given to the hydrogen bond redistribution in the chromophore-containing region of the protein caused by a change in the protonation state of the chromophore. The results obtained from the model containing the initially protonated side chain of Glu222 suggest a new interpretation of the photophysical processes in the green fluorescent protein.