Multidimensional free-energy calculations using the weighted histogram analysis method

The recently formulated weighted histogram analysis method (WHAM)¹ is an extension of Ferrenberg and Swendsen's multiple histogram technique for free-energy and potential of mean force calculations. As an illustration of the method, we have calculated the two-dimensional potential of mean force surface of the dihedrals gamma and chi in deoxyadenosine with Monte Carlo simulations using the all-atom and united-atom representation of the AMBER force fields. This also demonstrates one of the major advantages of WHAM over umbrella sampling techniques. The method also provides an analysis of the statistical accuracy of the potential of mean force as well as a guide to the most efficient use of additional simulations to minimize errors. © 1995 John Wiley & Sons, Inc.     

Multidimensional potentials of mean force from biased experiments along a single coordinate

External biasing forces are often applied to enhance sampling in regions of phase space which would otherwise be rarely observed. While the typical goal of these experiments is to calculate the potential of mean force (PMF) along the biasing coordinate, here I present a method to construct PMFs in multiple dimensions and along arbitary alternative degrees of freedom. A protocol for multidimensional PMF reconstruction from nonequilibrium single-molecule pulling experiments is introduced and tested on a series of two-dimensional potential surfaces with varying levels of correlation. Reconstruction accuracy and convergence from several methods--this new protocol, equilibrium umbrella sampling, and free diffusion--are compared, and nonequilibrium pulling is found to be the most efficient. To facilitate the use of this method, the source code for this analysis is made freely available.     

Convergence and error estimation in free energy calculations using the weighted histogram analysis method

The weighted histogram analysis method (WHAM) has become the standard technique for the analysis of umbrella sampling simulations. In this article, we address the challenges (1) of obtaining fast and accurate solutions of the coupled nonlinear WHAM equations, (2) of quantifying the statistical errors of the resulting free energies, (3) of diagnosing possible systematic errors, and (4) of optimally allocating of the computational resources. Traditionally, the WHAM equations are solved by a fixed-point direct iteration method, despite poor convergence and possible numerical inaccuracies in the solutions. Here, we instead solve the mathematically equivalent problem of maximizing a target likelihood function, by using superlinear numerical optimization algorithms with a significantly faster convergence rate. To estimate the statistical errors in one-dimensional free energy profiles obtained from WHAM, we note that for densely spaced umbrella windows with harmonic biasing potentials, the WHAM free energy profile can be approximated by a coarse-grained free energy obtained by integrating the mean restraining forces. The statistical errors of the coarse-grained free energies can be estimated straightforwardly and then used for the WHAM results. A generalization to multidimensional WHAM is described. We also propose two simple statistical criteria to test the consistency between the histograms of adjacent umbrella windows, which help identify inadequate sampling and hysteresis in the degrees of freedom orthogonal to the reaction coordinate. Together, the estimates of the statistical errors and the diagnostics of inconsistencies in the potentials of mean force provide a basis for the efficient allocation of computational resources in free energy simulations.     

Dynamic information for cardiotoxin protein desorption from a methyl-terminated self-assembled monolayer using steered molecular dynamics simulation

Dynamic information, such as force, structural change, interaction energy, and potential of mean force (PMF), about the desorption of a single cardiotoxin (CTX) protein from a methyl-terminated self-assembled monolayer (SAM) surface was investigated by means of steered molecular dynamics (SMD) simulations. The simulation results indicated that Loop I is the first loop to depart from the SAM surface, which is in good agreement with the results of the nuclear magnetic resonance spectroscopy experiment. The free energy landscape and the thermodynamic force of the CTX desorption process was represented by the PMF and by the derivative of PMF with respect to distance, respectively. By applying Jarzynski's equality, the PMF can be reconstructed from the SMD simulation. The PMFs, calculated by different estimators based upon Jarzynski's equality, were compared with the conventional umbrella sampling method. The best estimation was obtained by using the fluctuation-dissipation estimator with a pulling velocity of v = 0.25 nm/ns for the present study.     

Accurate calculations of free-energy differences by the distribution method

We employ the strategy used in the successive umbrella sampling method [P. Virnau and M. Muller, J. Chem. Phys. 120, 10925 (2004)] to obtain the energy-difference distribution over its desired range. This is very helpful in calculating free-energy differences, where the source of the error is well recognized as the insufficient sampling over the relevant tail region in the energy-difference distribution. The distribution method proposed here employs the idea of restricting the sampling within an appropriate energy range, as was presented by Shing and Gubbins in their restricted umbrella sampling method [Mol. Phys. 46, 1109 (1982)]. We demonstrate the efficiency of the distribution method by calculating the free-energy difference of a model of harmonic oscillators where the systems exhibit nonoverlap features in their important phase spaces through the original Metropolis sampling. For this particular case, we show that the distribution method outperforms the free-energy perturbation method and even the Bennett's acceptance ratio method [J. Comput. Phys. 22, 245 (1976)] with the fastest convergence and the smallest relative errors. We further demonstrate the application of the distribution method with a simple point charge water model.     

An algorithm to find minimum free-energy paths using umbrella integration

The calculation of free-energy barriers by umbrella sampling and many other methods is hampered by the necessity for an a priori choice of the reaction coordinate along which to sample. We avoid this problem by providing a method to search for saddle points on the free-energy surface in many coordinates. The necessary gradients and Hessians of the free energy are obtained by multidimensional umbrella integration. We construct the minimum free-energy path by following the gradient down to minima on the free-energy surface. The change of free energy along the path is obtained by integrating out all coordinates orthogonal to the path. While we expect the method to be applicable to large systems, we test it on the alanine dipeptide in vacuum. The minima, transition states, and free-energy barriers agree well with those obtained previously with other methods.     

Potential of mean force calculations of ligand binding to ion channels from Jarzynski's equality and umbrella sampling

Potential of mean force (PMF) calculations provide a reliable method for determination of the absolute binding free energies for protein-ligand systems. The common method used for this purpose -- umbrella sampling with weighted histogram analysis -- is computationally very laborious, which limits its applications. Recently, a much simpler alternative for PMF calculations has become available, namely, using Jarzynski's equality in steered molecular dynamics simulations. So far, there have been a few comparisons of the two methods and mostly in simple systems that do not reflect the complexities of protein-ligand systems. Here, we use both methods to calculate the PMF for ion permeation and ligand binding to ion channels. Comparison of results indicate that Jarzynski's method suffers from relaxation problems in complex systems and would require much longer simulation times to yield reliable PMFs for protein-ligand systems.     

Alternative approaches to potential of mean force calculations: Free energy perturbation versus thermodynamic integration. Case study of some representative nonpolar interactions

We investigated the convergence behavior of potential of mean force (PMF) calculations using free energy perturbation (FEP), thermodynamic integration (TI), and “slow growth” (SG) techniques. The critical comparison of these alternative approaches is illustrated by the study of three different systems: two tagged argon atoms in a periodic box of argon, two methane molecules, and two benzene molecules maintained in a “T-shaped” conformation, both dimers embedded in a periodic box of water. The complete PMF simulations were carried out considering several protocols, in which the number of intermediate “λ” states, together with the amount of sampling per individual state, were varied. In most cases, as much as 1 ns of molecular dynamics (MD) sampling was used to derive each free energy profile. For the different systems examined, we find that FEP and TI unquestionably constitute robust computational methods leading to results of comparable accuracy. We also show that proper convergence of the free energy calculations, and further quantitative interpretation of the PMFs, requires total simulation times much higher than has been hitherto estimated. In some circumstances, the free energy profiles derived from FEP calculations tend to be slightly poorer than those obtained with TI, as a probable consequence of the greater sensitivity of FEP to the window spacing δλ. In the context of TI, and to a lesser extent FEP, simulations, it appears preferable to employ a limited number of “λ” points of the integrand involving extensive sampling, rather than numerous points with fewer samplings. Finally, we note that, at least in the case of nonpolar interactions, PMFs of reasonable quality can be generated using SG, and at a substantially lower cost than with either FEP or TI. © 1996 by John Wiley & Sons, Inc.     

Efficient calculation of two-dimensional adiabatic and free energy maps: Application to the isomerization of the C13C14 and C15N16 bonds in the retinal of bacteriorhodopsin

Accurate calculation of potential energy and free-energy profiles along reaction coordinates of biological processes such as enzymatic reactions or conformational changes is fundamental to the obtention of theoretical insight into protein function. We describe here the practical implementation of the Automatic Map Refinement Procedure (AMRP) and two-dimensional Weighted Histogram Analysis Method (WHAM) for efficient computation of adiabatic potential energy and free-energy maps, respectively. Methods for efficiently sampling configuration space with high-energy barriers and for removing hysteresis in the case of periodic reaction coordinates are presented. The application of these techniques to the isomerization of the C13C14 and C15N16 bonds in the retinal of bacteriorhodopsin is described. In dark-adapted bacteriorhodopsin (bR), the retinal moiety exists in two conformers, all-trans and (13,15)cis, with the latter making ≃67% of the population. This experimental free energy difference is reproduced here to within k_BT. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1644–1658, 1999     

A comparison of methods to compute the potential of mean force.

Most processes occurring in a system are determined by the relative free energy between two or more states because the free energy is a measure of the probability of finding the system in a given state. When the two states of interest are connected by a pathway, usually called reaction coordinate, along which the free-energy profile is determined, this profile or potential of mean force (PMF) will also yield the relative free energy of the two states. Twelve different methods to compute a PMF are reviewed and compared, with regard to their precision, for a system consisting of a pair of methane molecules in aqueous solution. We analyze all combinations of the type of sampling (unbiased, umbrella-biased or constraint-biased), how to compute free energies (from density of states or force averaging) and the type of coordinate system (internal or Cartesian) used for the PMF degree of freedom. The method of choice is constraint-bias simulation combined with force averaging for either an internal or a Cartesian PMF degree of freedom.     

Potentials of mean force between ionizable amino acid side chains in water

Potentials of mean force (PMF) between all possible ionizable amino acid side chain pairs in various protonation states were calculated using explicit solvent molecular dynamics simulations with umbrella sampling and the weighted histogram analysis method. The side chains were constrained in various orientations inside a spherical cluster of 200 water molecules. Beglov and Roux's Spherical Solvent Boundary Potential was used to account for the solvent outside this sphere. This approach was first validated by calculating PMFs between monatomic ions (K(+), Na(+), Cl(-)) and comparing them to results from the literature and results obtained using Ewald summation. The strongest interaction (-4.5 kcal/mol) was found for the coaxial Arg(+).Glu(-) pair. Many like-charged side chains display a remarkable lack of repulsion, and occasionally a weak attraction. The PMFs are compared to effective energy curves obtained with common implicit solvation models, namely Generalized Born (GB), EEF1, and uniform dielectric of 80. Overall, the EEF1 curves are too attractive, whereas the GB curves in most cases match the minima of the PMF curves quite well. The uniform dielectric model, despite some fortuitous successes, is grossly inadequate.     

Generalized gradient-augmented harmonic Fourier beads method with multiple atomic and/or center-of-mass positional restraints

We describe a generalization of the gradient-augmented harmonic Fourier beads method for finding minimum free-energy transition path ensembles and similarly minimum potential energy paths to allow positional restraints on the centers of mass of selected atoms. The generalized gradient-augmented harmonic Fourier beads (ggaHFB) method further extends the scope of the HFB methodology to studying molecule transport across various mobile phases such as lipid membranes. Furthermore, the new implementation improves the applicability of the HFB method to studies of ligand binding, protein folding, and enzyme catalysis as well as modeling equilibrium pulling experiments. Like its predecessor, the ggaHFB method provides accurate energy profiles along the specified paths and in certain simple cases avoids the need for path optimization. The utility of the ggaHFB method is demonstrated with an application to the water permeation through a single-wall (5,5) carbon nanotube with a diameter of 6.78 A and length of 16.0 A. We provide a simple rationale as to why water enters the hydrophobic nanotube and why it does so in pulses and in wire assembly.     

Cell-Penetrating Peptides: Correlation between Peptide-Lipid Interaction and Penetration Efficiency

Cell-penetrating peptides are used in the delivery of peptides and biologics, with some cell-penetrating peptides found to be more efficient than others. The exact mechanism of how they interact with the cell membrane and penetrate it, however, remains unclear. This study attempts to investigate the difference in free energy profiles of three cell-penetrating peptides (TAT, CPP1 and CPP9) with a model lipid bilayer (DOPC) using molecular dynamics pulling simulations with umbrella sampling. Potential mean force (PMF) and free energy barrier between the peptides and DOPC are determined using WHAM analysis and MM-PBSA analysis, respectively. CPP9 is found to have the smallest PMF value, followed by CPP1 and TAT, consistent with the experimental data. YDEGE peptide, however, does not give the highest PMF value, although it is a non-cell-permeable peptide. YDEGE is also found to form water pores, alongside with TAT and CPP9, suggesting that it is difficult to distinguish true water pore formation from artefacts arising from pulling simulations. On the contrary, free energy analysis of the peptide-DOPC complex at the lipid-water interface with MM-PBSA provides results consistent with experimental data with CPP9 having the least interaction with DOPC and lowest free energy barrier, followed by CPP1, TAT and YDEGE. These findings suggest that peptide-lipid interaction at the lipid-water interface has a direct correlation with the penetration efficiency of peptides across the lipid bilayer.     

Potential of mean force of hydrophobic association: dependence on solute size

The potentials of mean force (PMFs) were determined for systems involving formation of nonpolar dimers composed of methane, ethane, propane, isobutane, and neopentane, respectively, in water, using the TIP3P water model, and in vacuo. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field was carried out for each pair in either water or in vacuo. The PMFs were calculated by using the weighted histogram analysis method (WHAM). The shape of the PMFs for dimers of all five nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule, that is, for larger molecules they shift toward larger distances. The PMF of the neopentane dimer is similar to those of other small nonpolar molecules studied in this work, and hence the neopentane dimer is too small to be treated as a nanoscale hydrophobic object. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from the PMF in explicit solvent. The molecular surface area model correctly describes the solvent contribution to the PMF together with the changes of the height and positions of the desolvation barrier for all dimers investigated. The water molecules in the first solvation sphere of the dimer are more ordered compared to bulk water, with their dipole moments pointing away from the surface of the dimer. The average number of hydrogen bonds per water molecule in this first hydration shell is smaller compared to that in bulk water, which can be explained by coordination of water molecules to the hydrocarbon surface. In the second hydration shell, the average number of hydrogen bonds is greater compared to bulk water, which can be explained by increased ordering of water from the first hydration shell; the net effect is more efficient hydrogen bonding between the water molecules in the first and second hydration shells.     

Theoretical calculations of homoconjugation equilibrium constants in systems modeling acid-base interactions in side chains of biomolecules using the potential of mean force

The potentials of mean force (PMFs) were determined for systems forming cationic and anionic homocomplexes composed of acetic acid, phenol, isopropylamine, n-butylamine, imidazole, and 4(5)-methylimidazole, and their conjugated bases or acids, respectively, in three solvents with different polarity and hydrogen-bonding propensity: acetonitrile (AN), dimethyl sulfoxide (DMSO), and water (H(2)O). For each pair and each solvent a series of umbrella-sampling molecular dynamics simulations with the AMBER force field, explicit solvent, and counterions added to maintain a zero net charge of a system were carried out and the PMF was calculated by using the Weighted Histogram Analysis Method (WHAM). Subsequently, homoconjugation-equilibrium constants were calculated by numerical integration of the respective PMF profiles. In all cases but imidazole stable homocomplexes were found to form in solution, which was manifested as the presence of contact minima corresponding to hydrogen-bonded species in the PMF curves. The calculated homoconjugation constants were found to be greater for complexes with the OHO bridge (acetic acid and phenol) than with the NHN bridge and they were found to decrease with increasing polarity and hydrogen-bonding propensity of the solvent (i.e., in the series AN > DMSO > H(2)O), both facts being in agreement with the available experimental data. It was also found that interactions with counterions are manifested as the broadening of the contact minimum or appearance of additional minima in the PMF profiles of the acetic acid-acetate, phenol/phenolate system in acetonitrile, and the 4(5)-methylimidazole/4(5)-methylimidzole cation conjugated base system in dimethyl sulfoxide.     

Quantum mechanics/molecular mechanics minimum free-energy path for accurate reaction energetics in solution and enzymes: sequential sampling and optimization on the potential of mean force surface

To accurately determine the reaction path and its energetics for enzymatic and solution-phase reactions, we present a sequential sampling and optimization approach that greatly enhances the efficiency of the ab initio quantum mechanics/molecular mechanics minimum free-energy path (QM/MM-MFEP) method. In the QM/MM-MFEP method, the thermodynamics of a complex reaction system is described by the potential of mean force (PMF) surface of the quantum mechanical (QM) subsystem with a small number of degrees of freedom, somewhat like describing a reaction process in the gas phase. The main computational cost of the QM/MM-MFEP method comes from the statistical sampling of conformations of the molecular mechanical (MM) subsystem required for the calculation of the QM PMF and its gradient. In our new sequential sampling and optimization approach, we aim to reduce the amount of MM sampling while still retaining the accuracy of the results by first carrying out MM phase-space sampling and then optimizing the QM subsystem in the fixed-size ensemble of MM conformations. The resulting QM optimized structures are then used to obtain more accurate sampling of the MM subsystem. This process of sequential MM sampling and QM optimization is iterated until convergence. The use of a fixed-size, finite MM conformational ensemble enables the precise evaluation of the QM potential of mean force and its gradient within the ensemble, thus circumventing the challenges associated with statistical averaging and significantly speeding up the convergence of the optimization process. To further improve the accuracy of the QM/MM-MFEP method, the reaction path potential method developed by Lu and Yang [Z. Lu and W. Yang, J. Chem. Phys. 121, 89 (2004)] is employed to describe the QM/MM electrostatic interactions in an approximate yet accurate way with a computational cost that is comparable to classical MM simulations. The new method was successfully applied to two example reaction processes, the classical SN2 reaction of Cl-+CH3Cl in solution and the second proton transfer step of the reaction catalyzed by the enzyme 4-oxalocrotonate tautomerase. The activation free energies calculated with this new sequential sampling and optimization approach to the QM/MM-MFEP method agree well with results from other simulation approaches such as the umbrella sampling technique with direct QM/MM dynamics sampling, demonstrating the accuracy of the iterative QM/MM-MFEP method.     

Enhanced sampling simulations of DNA step parameters

A novel approach for the selection of step parameters as reaction coordinates in enhanced sampling simulations of DNA is presented. The method uses three atoms per base and does not require coordinate overlays or idealized base pairs. This allowed for a highly efficient implementation of the calculation of all step parameters and their Cartesian derivatives in molecular dynamics simulations. Good correlation between the calculated and actual twist, roll, tilt, shift, and slide parameters is obtained, while the correlation with rise is modest. The method is illustrated by its application to the methylated and unmethylated 5′‐CATGTGACGTCACATG‐3′ double stranded DNA sequence. One‐dimensional umbrella simulations indicate that the flexibility of the central CG step is only marginally affected by methylation. © 2014 Wiley Periodicals, Inc.     

On the mechanism of surfactant adsorption on solid surfaces: free-energy investigations

The adsorption free-energy of surfactant on solid surfaces has been calculated by molecular dynamics (MD) simulation for a model surfactant/solvent system. The umbrella-sampling with the weight histogram analysis method (WHAM) was applied. The entropic and enthalpic contributions to the full potential of mean force (PMF) were obtained to evaluate the detailed thermodynamics of surfactant adsorption in solid/liquid interfaces. Although we observed that this surfactant adsorption process is driven mainly by a favorable enthalpy change, a highly unfavorable entropic contribution still existed. By decomposing the free energy (including its entropic and enthalpic components) into the solvent-induced contribution and the surfactant-wall term, the effect of surface and solvent on the adsorption free-energy has been distinguished. The contribution to the PMF from the surface effect is thermodynamically favorable, whereas the solvent term displays an obviously unfavorable component with a monotonic increase as the surfactant approaches to the surface. The impact of various interactions from the surfaces (both solvent-philic and solvent-phobic) and the solvent on the adsorption PMF of surfactant has been compared and discussed. Compared to the solvent-philic surface, the solvent-phobic surface generates more stable site for the surfactant adsorption. However, the full PMF profile for the solvent-phobic system shows a clear positive maximum value at the bulk-interface transition region, which leads to a considerable long-range free-energy barrier to the surfactant adsorption. These results have been analyzed in terms of the local interfacial structures. In summary, this comprehensive study is expected to reveal the microscopic interaction mechanisms determining the surfactant adsorption on solid surfaces.     

Rapid free energy calculation of peptide self-assembly by REMD umbrella sampling

We extend umbrella sampling with replica exchange steps to calculate free energies that are important in the self-assembly of peptides. This leads to a more than 10-fold speed up over conventional umbrella sampling, thereby providing a practical method to calculate these free-energy differences. This approach can also observe first-order phase transitions and pinpoint the location of the concomitant boundary. When conformational changes are involved, this method can handle peptides up to approximately 7 residues, providing a rapid and accurate assessment of the thermodynamic properties of model systems, and can thus be used to answer fundamental questions about peptide self-assembly. When no major conformational changes are involved, we expect the size limit to be equal to that of standard molecular dynamics.     

Through the channel and around the channel: Validating and comparing microscopic approaches for the evaluation of free energy profiles for ion penetration through ion channels

Microscopic calculations of free energy profiles for ion transport through biological ion channels present a very serious challenge to modern simulation approaches. The main problem is due to the major convergence problems associated with the heterogeneous landscape of the electrostatic environment in ion channels and with the need to evaluate the profile associated with the transfer of the ion from bulk water to the channel environment. This problem is compounded by the lack of reliable and relevant benchmarks that can discriminate between alternative approaches. The present study is aimed at reducing the above problems by defining benchmarks that are directly relevant to ion channels and can also give converging results. This is done by constructing a series of models of a truncated gramicidin channel with different numbers of water molecules and by comparing the profiles for going around the channel and through the channel. These discriminating models are then used to validate and compare the adiabatic charging free energy perturbation (FEP) approach combined with an umbrella sampling approach (Warshel, A. J. Phys. Chem. 1982, 86, 2218) and the potential of mean force (PMF) approach used frequently in studies of ion channels. It is found that both approaches work quite well until one moves to the case of the fully solvated channel. In this limit, the PMF approach may give different results for the overall work of going through the channel and around the channel, while the FEP approach gives physically consistent results. The present benchmark also indicates that the weighted histogram analysis method (WHAM) approach does not offer a significant advantage over earlier approaches at least as much as studies of ion channels are concerned. Finally, it is concluded that the FEP approach may be more useful in evaluating the overall barrier for moving ions from water to ion channels and that in some cases it might be beneficial to use the FEP approach for selective points along the channel and then to connect these points by PMF calculations.