Comparison of thermodynamic integration and Bennett acceptance ratio for calculating relative protein‐ligand binding free energies

The performances of Bennett's acceptance ratio method and thermodynamic integration (TI) for the calculation of free energy differences in protein simulations are compared. For the latter, the standard trapezoidal rule, Simpson's rule, and Clenshaw‐Curtis integration are used as numerical integration methods. We evaluate the influence of the number and definition of intermediate states on the precision, accuracy, and efficiency of the free energy calculations. Our results show that non‐equidistantly spaced intermediate states are in some cases beneficial for the TI methods. Using several combinations of softness parameters and the λ power dependence, it is shown that these benefits are strongly dependent on the shape of the integrand. Although TI is more user‐friendly due to its simplicity, it was found that Bennett's acceptance ratio method is the more efficient method. It is also the least dependent on the choice of the intermediate states, making it more robust than TI. © 2013 Wiley Periodicals, Inc.     

Computational Characterization of Binding of Small Molecule Inhibitors to HIV‐1 gp41

Developing orally available small molecule inhibitors of HIV‐1 fusion has attracted significant interest over many years. Frey had recently reported several synthetic compounds which are experimentally shown to inhibit cell‐cell fusion in the low micromolar range. We carried out computational study to help identify possible binding modes by docking these compounds onto the hydrophobic pocket on gp41 and to characterize structures of binding complexes. The detailed gp41‐molecule binding interactions and free energies of binding are obtained through molecular dynamics simulation and MM‐PBSA calculation. Specific molecular interactions in the gp41‐inhibitor complexes are identified. The present computational study complements the corresponding experimental investigation and helps establish a good starting point for further refinement of small molecular gp41 inhibitors.     

Minimum MD simulation length required to achieve reliable results in free energy perturbation calculations: Case study of relative binding free energies of fructose‐1,6‐bisphosphatase inhibitors

In an attempt to establish the criteria for the length of simulation to achieve the desired convergence of free energy calculations, two studies were carried out on chosen complexes of FBPase‐AMP mimics. Calculations were performed for varied length of simulations and for different starting configurations using both conventional‐ and QM/MM‐FEP methods. The results demonstrate that for small perturbations, 1248 ps simulation time could be regarded a reasonable yardstick to achieve convergence of the results. As the simulation time is extended, the errors associated with free energy calculations also gradually tapers off. Moreover, when starting the simulation from different initial configurations of the systems, the results are not changed significantly, when performed for 1248 ps. This study carried on FBPase‐AMP mimics corroborates well with our previous successful demonstration of requirement of simulation time for solvation studies, both by conventional and ab initio FEP. The establishment of aforementioned criteria of simulation length serves a useful benchmark in drug design efforts using FEP methodologies, to draw a meaningful and unequivocal conclusion. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Net charge changes in the calculation of relative ligand‐binding free energies via classical atomistic molecular dynamics simulation

The calculation of binding free energies of charged species to a target molecule is a frequently encountered problem in molecular dynamics studies of (bio‐)chemical thermodynamics. Many important endogenous receptor‐binding molecules, enzyme substrates, or drug molecules have a nonzero net charge. Absolute binding free energies, as well as binding free energies relative to another molecule with a different net charge will be affected by artifacts due to the used effective electrostatic interaction function and associated parameters (e.g., size of the computational box). In the present study, charging contributions to binding free energies of small oligoatomic ions to a series of model host cavities functionalized with different chemical groups are calculated with classical atomistic molecular dynamics simulation. Electrostatic interactions are treated using a lattice‐summation scheme or a cutoff‐truncation scheme with Barker–Watts reaction‐field correction, and the simulations are conducted in boxes of different edge lengths. It is illustrated that the charging free energies of the guest molecules in water and in the host strongly depend on the applied methodology and that neglect of correction terms for the artifacts introduced by the finite size of the simulated system and the use of an effective electrostatic interaction function considerably impairs the thermodynamic interpretation of guest‐host interactions. Application of correction terms for the various artifacts yields consistent results for the charging contribution to binding free energies and is thus a prerequisite for the valid interpretation or prediction of experimental data via molecular dynamics simulation. Analysis and correction of electrostatic artifacts according to the scheme proposed in the present study should therefore be considered an integral part of careful free‐energy calculation studies if changes in the net charge are involved. © 2013 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Molecular dynamics simulations and free energy calculations on the enzyme 4‐hydroxyphenylpyruvate dioxygenase

4‐Hydroxyphenylpyruvate dioxygenase is a relevant target in both pharmaceutical and agricultural research. We report on molecular dynamics simulations and free energy calculations on this enzyme, in complex with 12 inhibitors for which experimental affinities were determined. We applied the thermodynamic integration approach and the more efficient one‐step perturbation. Even though simulations seem well converged and both methods show excellent agreement between them, the correlation with the experimental values remains poor. We investigate the effect of slight modifications on the charge distribution of these highly conjugated systems and find that accurate models can be obtained when using improved force field parameters. This study gives insight into the applicability of free energy methods and current limitations in force field parameterization. © 2011 Wiley Periodicals, Inc. J Comput Chem 2011     

Three pillars for achieving quantum mechanical molecular dynamics simulations of huge systems: Divide‐and‐conquer,density‐functional tight‐binding,and massively parallel computation

The linear‐scaling divide‐and‐conquer (DC) quantum chemical methodology is applied to the density‐functional tight‐binding (DFTB) theory to develop a massively parallel program that achieves on‐the‐fly molecular reaction dynamics simulations of huge systems from scratch. The functions to perform large scale geometry optimization and molecular dynamics with DC‐DFTB potential energy surface are implemented to the program called DC‐DFTB‐K. A novel interpolation‐based algorithm is developed for parallelizing the determination of the Fermi level in the DC method. The performance of the DC‐DFTB‐K program is assessed using a laboratory computer and the K computer. Numerical tests show the high efficiency of the DC‐DFTB‐K program, a single‐point energy gradient calculation of a one‐million‐atom system is completed within 60 s using 7290 nodes of the K computer. © 2016 Wiley Periodicals, Inc.     

On the Relative Merits of Equilibrium and Non‐Equilibrium Simulations for the Estimation of Free‐Energy Differences

The possibility of estimating equilibrium free‐energy profiles from multiple non‐equilibrium simulations using the fluctuation–dissipation theory or the relation proposed by Jarzynski has attracted much attention. Although the Jarzynski estimator has poor convergence properties for simulations far from equilibrium, corrections have been derived for cases in which the work is Gaussian distributed. Here, we examine the utility of corrections proposed by Gore and collaborators using a simple dissipative system as a test case. The system consists of a single methane‐like particle in explicit water. The Jarzynski equality is used to estimate the change in free energy associated with pulling the methane particle a distance of 3.9 nm at rates ranging from ～0.1 to 100 m s^?1. It is shown that although the corrections proposed by Gore and collaborators have excellent numerical performance, the profiles still converge slowly. Even when the corrections are applied in an ideal case where the work distribution is necessarily Gaussian, performing simulations under quasi‐equilibrium conditions is still most efficient. Furthermore, it is shown that even for a single methane molecule in water, pulling rates as low as 1 m s^?1 can be problematic. The implications of this finding for studies in which small molecules or even large biomolecules are pulled through inhomogeneous environments at similar pulling rates are discussed.     

Computational gibberellin‐binding channel discovery unraveling the unexpected perception mechanism of hormone signal by gibberellin receptor

Gibberellins (GAs) are phytohormones essential for many developmental processes in plants. In this work, fundamental mechanism of hormone perception by receptor GID1 has been studied by performing computational simulations, revealing a new GA‐binding channel of GID1 and a novel hormone perception mechanism involving only one conformational state of GID1. The novel hormone perception mechanism demonstrated here is remarkably different from the previously proposed/speculated mechanism [Murase et al., Nature 2008 , 456, 459] involving two conformational states (“OPEN” and “CLOSED”) of GID1. According to the new perception mechanism, GA acts as a “conformational stabilizer,” rather than the previously speculated “allosteric inducer,” to induce the recognition of protein DELLA by GID1. The novel mechanistic insights obtained in this study provide a new starting point for further studies on the detailed molecular mechanisms of GID1 interacting with DELLA and various hormones and for mechanism‐based rational design of novel, potent growth regulators that target crops and ornamental plants. © 2013 Wiley Periodicals, Inc.     

Automated Search of Minimum Free‐Energy Path by Umbrella Integration

The free‐energy landscape is an important factor for understanding the conformational equilibria of chemical reactions, and many techniques have been developed to calculate the potential of the mean force. Unfortunately, these methods require a previous knowledge of the system for calculations because the results depend on the reaction coordinates. In this study, we combine the scaled hypersphere search method with the umbrella integration method to obtain the transition states on free‐energy landscapes and minimum‐free‐energy paths (MFEPs). With this approach, the MFEP connections between known and unknown equilibrium points are constructed without the prior knowledge of the free‐energy landscape. The problem of reaction coordinates can be solved by using a multidimensional, fully automated interrogation of MFEPs for acquiring the potential of mean force. The efficiency of the proposed method is demonstrated by applying it to alanine dipeptide and alanine tripeptide. © 2018 Wiley Periodicals, Inc.     

Effect of sampling on BACE‐1 ligands binding free energy predictions via MM‐PBSA calculations

The BACE‐1 enzyme is a prime target to find a cure to Alzheimer's disease. In this article, we used the MM‐PBSA approach to compute the binding free energies of 46 reported ligands to this enzyme. After showing that the most probable protonation state of the catalytic dyad is mono‐protonated (on ASP32), we performed a thorough analysis of the parameters influencing the sampling of the conformational space (in total, more than 35 μs of simulations were performed). We show that ten simulations of 2 ns gives better results than one of 50 ns. We also investigated the influence of the protein force field, the water model, the periodic boundary conditions artifacts (box size), as well as the ionic strength. Amber03 with TIP3P, a minimal distance of 1.0 nm between the protein and the box edges and a ionic strength of I = 0.2 M provides the optimal correlation with experiments. Overall, when using these parameters, a Pearson correlation coefficient of R = 0.84 (R ² = 0.71) is obtained for the 46 ligands, spanning eight orders of magnitude of K _d (from 0.017 nm to 2000 μM, i.e., from −14.7 to −3.7 kcal/mol), with a ligand size from 22 to 136 atoms (from 138 to 937 g/mol). After a two‐parameter fit of the binding affinities for 12 of the ligands, an error of RMSD = 1.7 kcal/mol was obtained for the remaining ligands. © 2017 Wiley Periodicals, Inc.     

Atomistic detailed free‐energy landscape of intrinsically disordered protein studied by multi‐scale divide‐and‐conquer molecular dynamics simulation

Calcineurin (CaN) is a eukaryotic serine/threonine protein phosphatase activated by both Ca²⁺ and calmodulin (CaM), including intrinsically disordered region (IDR). The region undergoes folding into an α‐helix form in the presence Ca²⁺‐loaded CaM. To sample the ordered structure of the IDR by conventional all atom model (AAM) molecular dynamics (MD) simulation, the IDR and Ca²⁺‐loaded CaM must be simultaneously treated. However, it is time‐consuming task because the coupled folding and binding should include repeated binding and dissociation. Then, in this study, we propose novel multi‐scale divide‐and‐conquer MD (MSDC‐MD), which combines AAM‐MD and coarse‐grained model MD (CGM‐MD). To speed up the conformation sampling, MSDC‐MD simulation first treats the IDR by CGM to sample conformations from wide conformation space; then, multiple AAM‐MD in a limited area is initiated using the resultant CGM conformation, which is reconstructed by homology modeling method. To investigate performance, we sampled the ordered conformation of the IDR using MSDC‐MD; the root‐mean‐square distance (RMSD) with respect to the experimental structure was 2.23 Å.     

Possible mechanism of BN fullerene formation from a boron cluster: Density‐functional tight‐binding molecular dynamics simulations

We simulate the formation of a BN fullerene from an amorphous B cluster at 2000 K by quantum mechanical molecular dynamics based on the density‐functional tight‐binding method. We run 30 trajectories 200 ps in length, where N atoms are supplied around the target cluster, which is initially an amorphous B₃₆ cluster. Most of the incident N atoms are promptly incorporated into the target cluster to form B‐N‐B bridges or NB₃ pyramidal local substructures. BN fullerene formation is initiated by alternating BN ring condensation. Spontaneous atomic rearrangement and N₂ dissociation lead to the construction of an sp² single‐shelled structure, during which the BN cluster undergoes a transition from a liquid‐like to a solid‐like state. Continual atomic rearrangement and sporadic N₂ dissociation decrease the number of defective rings in the BN cluster and increase the number of six‐membered rings, forming a more regular shell structure. The number of four‐membered rings tends to remain constant, and contributes to more ordered isolated‐tetragon‐rule ring placement. © 2016 Wiley Periodicals, Inc.     

Fast and accurate determination of the relative binding affinities of small compounds to HIV‐1 protease using non‐equilibrium work

The fast pulling ligand (FPL) out of binding cavity using non‐equilibrium molecular dynamics (MD) simulations was demonstrated to be a rapid, accurate and low CPU demand method for the determination of the relative binding affinities of a large number of HIV‐1 protease (PR) inhibitors. In this approach, the ligand is pulled out of the binding cavity of the protein using external harmonic forces, and the work of pulling force corresponds to the relative binding affinity of HIV‐1 PR inhibitor. The correlation coefficient between the pulling work and the experimental binding free energy of shows that FPL results are in good agreement with experiment. It is thus easier to rank the binding affinities of HIV‐1 PR inhibitors, that have similar binding affinities because the mean error bar of pulling work amounts to . The nature of binding is discovered using the FPL approach. © 2016 Wiley Periodicals, Inc.     

Free energy perturbation and molecular dynamics calculations of copper binding to azurin

Free energy perturbation/molecular dynamics simulations have been carried out on copper/azurin systems calculating the binding affinities of copper (II) ion to azurin either in the native or in the unfolded state. In order to test the validity of the strategy adopted for the calculations and to establish what force field is suitable for these kinds of calculations, three different force fields, AMBER, CVFF, and CFF, have been alternatively used for the calculations and the results have been compared with experimental data obtained by spectroscopic titrations of copper (II)/azurin solutions and denaturation experiments. Our findings have pointed out that only CFF gives satisfactory results, thus providing a reliable tool for copper binding simulations in copper protein.     

Interaction identification of Zif268 and TATAZF proteins with GC‐/AT‐rich DNA sequence: A theoretical study

Molecular dynamics (MD) simulations for Zif268 (a zinc‐finger‐protein binding specifically to the GC‐rich DNA)‐d(A₁G₂C₃G₄T₅G₆G₇G₈C₉A₁₀C₁₁)₂ and TATA_ZF (a zinc‐finger‐protein recognizing the AT‐rich DNA)‐d(A₁C₂G₃C₄T₅A₆T₇A₈A₉A₁₀A₁₁G₁₂G₁₃)₂ complexes have been performed for investigating the DNA binding affinities and specific recognitions of zinc fingers to GC‐rich and AT‐rich DNA sequences. The binding free energies for the two systems have been further analyzed by using the molecular mechanics Poisson‐Boltzmann surface area (MM‐PBSA) method. The calculations of the binding free energies reveal that the affinity energy of Zif268‐DNA complex is larger than that of TATA_ZF‐DNA one. The affinity between the zinc‐finger‐protein and DNA is mainly driven by more favorable van‐der‐Waals and nonpolar/solvation interactions in both complexes. However, the affinity energy difference of the two binding systems is mainly caused by the difference of van‐der‐Waals interactions and entropy components. The decomposition analysis of MM‐PBSA free energies on each residue of the proteins predicts that the interactions between the residues with the positive charges and DNA favor the binding process; while the interactions between the residues with the negative charges and DNA behave in the opposite way. The interhydrogen‐bonds at the protein‐DNA interface and the induced intrafinger hydrogen bonds between the residues of protein for the Zif268‐DNA complex have been identified at some key contact sites. However, only the interhydrogen‐bonds between the residues of protein and DNA for TATA_ZF‐DNA complex have been found. The interactions of hydrogen‐bonds, electrostatistics and van‐der‐Waals type at some new contact sites have been identified. Moreover, the recognition characteristics of the two studied zinc‐finger‐proteins have also been discussed. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Computation of the binding affinities of catechol‐O‐methyltransferase inhibitors: Multisubstate relative free energy calculations

Alchemical free energy simulations are amongst the most accurate techniques for the computation of the free energy changes associated with noncovalent protein–ligand interactions. A procedure is presented to estimate the relative binding free energies of several ligands to the same protein target where multiple, low‐energy configurational substates might coexist, as opposed to one unique structure. The contributions of all individual substates were estimated, explicitly, with the free energy perturbation method, and combined in a rigorous fashion to compute the overall relative binding free energies and dissociation constants. It is shown that, unless the most stable bound forms are known a priori, inaccurate results may be obtained if the contributions of multiple substates are ignored. The method was applied to study the complex formed between human catechol‐O‐methyltransferase and BIA 9‐1067, a newly developed tight‐binding inhibitor that is currently under clinical evaluation for the therapy of Parkinson's disease. Our results reveal an exceptionally high‐binding affinity (K_d in subpicomolar range) and provide insightful clues on the interactions and mechanism of inhibition. The inhibitor is, itself, a slowly reacting substrate of the target enzyme and is released from the complex in the form of O‐methylated product. By comparing the experimental catalytic rate (k_cat) and the estimated dissociation rate (k_off) constants of the enzyme‐inhibitor complex, one can conclude that the observed inhibition potency (K_i) is primarily dependent on the catalytic rate constant of the inhibitor's O‐methylation, rather than the rate constant of dissociation of the complex. © 2012 Wiley Periodicals, Inc.     

Hot spot computational identification: Application to the complex formed between the hen egg white lysozyme (HEL) and the antibody HyHEL‐10

The definition and comprehension of the hot spots in an interface is a subject of primary interest for a variety of fields, including structure‐based drug design. Therefore, to achieve an alanine mutagenesis computational approach that is at the same time accurate and predictive, capable of reproducing the experimental mutagenesis values is a major challenge in the computational biochemistry field. Antibody/protein antigen complexes provide one of the greatest models to study protein–protein recognition process because they have three fundamentally features: specificity, high complementary association and a small epitope restricted to the diminutive complementary determining regions (CDR) region, while the remainder of the antibody is largely invariant. Thus, we apply a computational mutational methodological approach to the study of the antigen–antibody complex formed between the hen egg white lysozyme (HEL) and the antibody HyHEL‐10. A critical evaluation that focuses essentially on the limitations and advantages between different computational methods for hot spot determination, as well as between experimental and computational methodological approaches, is presented. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Fast and accurate predictions of binding free energies using MM‐PBSA and MM‐GBSA

In the drug discovery process, accurate methods of computing the affinity of small molecules with a biological target are strongly needed. This is particularly true for molecular docking and virtual screening methods, which use approximated scoring functions and struggle in estimating binding energies in correlation with experimental values. Among the various methods, MM‐PBSA and MM‐GBSA are emerging as useful and effective approaches. Although these methods are typically applied to large collections of equilibrated structures of protein‐ligand complexes sampled during molecular dynamics in water, the possibility to reliably estimate ligand affinity using a single energy‐minimized structure and implicit solvation models has not been explored in sufficient detail. Herein, we thoroughly investigate this hypothesis by comparing different methods for the generation of protein‐ligand complexes and diverse methods for free energy prediction for their ability to correlate with experimental values. The methods were tested on a series of structurally diverse inhibitors of Plasmodium falciparum DHFR with known binding mode and measured affinities. The results showed that correlations between MM‐PBSA or MM‐GBSA binding free energies with experimental affinities were in most cases excellent. Importantly, we found that correlations obtained with the use of a single protein‐ligand minimized structure and with implicit solvation models were similar to those obtained after averaging over multiple MD snapshots with explicit water molecules, with consequent save of computing time without loss of accuracy. When applied to a virtual screening experiment, such an approach proved to discriminate between true binders and decoy molecules and yielded significantly better enrichment curves. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Coarse‐grained molecular dynamics simulations of protein–ligand binding

Coarse‐grained molecular dynamics (CGMD) simulations with the MARTINI force field were performed to reproduce the protein–ligand binding processes. We chose two protein–ligand systems, the levansucrase–sugar (glucose or sucrose), and LinB–1,2‐dichloroethane systems, as target systems that differ in terms of the size and shape of the ligand‐binding pocket and the physicochemical properties of the pocket and the ligand. Spatial distributions of the Coarse‐grained (CG) ligand molecules revealed potential ligand‐binding sites on the protein surfaces other than the real ligand‐binding sites. The ligands bound most strongly to the real ligand‐binding sites. The binding and unbinding rate constants obtained from the CGMD simulation of the levansucrase–sucrose system were approximately 10 times greater than the experimental values; this is mainly due to faster diffusion of the CG ligand in the CG water model. We could obtain dissociation constants close to the experimental values for both systems. Analysis of the ligand fluxes demonstrated that the CG ligand molecules entered the ligand‐binding pockets through specific pathways. The ligands tended to move through grooves on the protein surface. Thus, the CGMD simulations produced reasonable results for the two different systems overall and are useful for studying the protein–ligand binding processes. © 2014 Wiley Periodicals, Inc.     

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.