Accurate determination of the binding free energy for KcsA-charybdotoxin complex from the potential of mean force calculations with restraints

Free energy calculations for protein-ligand dissociation have been tested and validated for small ligands (50 atoms or less), but there has been a paucity of studies for larger, peptide-size ligands due to computational limitations. Previously we have studied the energetics of dissociation in a potassium channel-charybdotoxin complex by using umbrella sampling molecular-dynamics simulations, and established the need for carefully chosen coordinates and restraints to maintain the physiological ligand conformation. Here we address the ligand integrity problem further by constructing additional potential of mean forces for dissociation of charybdotoxin using restraints. We show that the large discrepancies in binding free energy arising from simulation artifacts can be avoided by using appropriate restraints on the ligand, which enables determination of the binding free energy within the chemical accuracy. We make several suggestions for optimal choices of harmonic potential parameters and restraints to be used in binding studies of large ligands.     

Sparse sampling of water density fluctuations near liquid-vapor coexistence

Abstract

The free energetics of water density fluctuations in bulk water, at interfaces, and in hydrophobic confinement inform the hydration of hydrophobic solutes as well as their interactions and assembly. The characterisation of such free energetics is typically performed using enhanced sampling techniques such as umbrella sampling. In umbrella sampling, order parameter distributions obtained from adjacent biased simulations must overlap in order to estimate free energy differences between biased ensembles. Many biased simulations are typically required to ensure such overlap, which exacts a steep computational cost. We recently introduced a sparse sampling method, which circumvents the overlap requirement by using thermodynamic integration to estimate free energy differences between biased ensembles. Here we build upon and generalise sparse sampling for characterising the free energetics of water density fluctuations in systems near liquid-vapor coexistence. We also introduce sensible heuristics for choosing the biasing potential parameters and strategies for adaptively refining them, which facilitate the estimation of such free energetics accurately and efficiently. We illustrate the method by characterising the free energetics of cavitation in a large volume in bulk water. We also use sparse sampling to characterise the free energetics of capillary evaporation for water confined between two hydrophobic plates. In both cases, sparse sampling is nearly two orders of magnitude faster than umbrella sampling. Given its efficiency, the sparse sampling method is particularly well suited for characterising free energy landscapes for systems wherein umbrella sampling is prohibitively expensive.     

Free energy surface for rotamers of cis-enol malonaldehyde in aqueous solution studied by molecular dynamics calculations

Two-dimensional free energy surfaces for four rotamers of cis-enol malonaldehyde in water have been investigated by umbrella sampling molecular dynamics (MD) calculations. Biasing potential used in the umbrella sampling calculation was adopted to be the minus of conformational free energy preliminary obtained by the thermodynamic integration MD calculations for the rigid malonaldehyde whose stretching and bending were all fixed. The calculated free energy surface shows that, in water, a rotamer that has an intramolecular hydrogen bond is most stable among the rotamers. This is the same as that in vacuum, while order of relative stability of the other three rotamers is different in water and in vacuum. Inclusion of intramolecular vibrations changed the free energy surface little, i.e. at most 2.6 kJ/mol, which is much smaller than the solvation free energy. Free energy barriers from the most stable intramolecular hydrogen bonded rotamer to the others are lowered by hydration but they are still very high, >50 kJ/mol, such that the malonaldehyde molecule spends most of its time in water taking this conformation. Thus, reaction coordinate for intramolecular proton transfer reaction in water may be constructed assuming this rotamer.     

Dimerization of a flocculent protein from Moringa oleifera: experimental evidence and in silico interpretation

Many proteins exist in dimeric and other oligomeric forms to gain stability and functional advantages. In this study, the dimerization property of a coagulant protein (MO_2.1) from Moringa oleifera seeds was addressed through laboratory experiments, protein–protein docking studies and binding free energy calculations. The structure of MO_2.1 was predicted by homology modelling, while binding free energy and residues-distance profile analyses provided insight into the energetics and structural factors for dimer formation. Since the coagulation activities of the monomeric and dimeric forms of MO_2.1 were comparable, it was concluded that oligomerization does not affect the biological activity of the protein.     

QM/MM free energy simulations: recent progress and challenges

Due to the higher computational cost relative to pure molecular mechanical (MM) simulations, hybrid quantum mechanical/molecular mechanical (QM/MM) free energy simulations particularly require a careful consideration of balancing computational cost and accuracy. Here, we review several recent developments in free energy methods most relevant to QM/MM simulations and discuss several topics motivated by these developments using simple but informative examples that involve processes in water. For chemical reactions, we highlight the value of invoking enhanced sampling technique (e.g. replica-exchange) in umbrella sampling calculations and the value of including collective environmental variables (e.g. hydration level) in metadynamics simulations; we also illustrate the sensitivity of string calculations, especially free energy along the path, to various parameters in the computation. Alchemical free energy simulations with a specific thermodynamic cycle are used to probe the effect of including the first solvation shell into the QM region when computing solvation free energies. For cases where high-level QM/MM potential functions are needed, we analyse two different approaches: the QM/MM–MFEP method of Yang and co-workers and perturbative correction to low-level QM/MM free energy results. For the examples analysed here, both approaches seem productive although care needs to be exercised when analysing the perturbative corrections.     

Exhaustive mutagenesis in silico: multicoordinate free energy calculations on proteins and peptides

We have extended and applied a multicoordinate free energy method, chemical Monte Carlo/Molecular Dynamics (CMC/MD), to calculate the relative free energies of different amino acid side-chains. CMC/MD allows the calculation of the relative free energies for many chemical species from a single free energy calculation. We have previously shown its utility in host:guest chemistry (Pitera and Kollman, J Am Chem Soc 1998;120:7557-7567)1 and ligand design (Eriksson et al., J Med Chem 1999;42:868-881)2, and here demonstrate its utility in calculations of amino acid properties and protein stability. We first study the relative solvation free energies of N-methylated and acetylated alanine, valine, and serine amino acids. With careful inclusion of rotameric states, internal energies, and both the solution and vacuum states of the calculation, we calculate relative solvation free energies in good agreement with thermodynamic integration (TI) calculations. Interestingly, we find that a significant amount of the unfavorable solvation of valine seen in prior work (Sun et al., J Am Chem Soc 1992;114:6798-6801)3 is caused by restraining the backbone in an extended conformation. In contrast, the solvation free energy of serine is calculated to be less favorable than expected from experiment, due to the formation of a favorable intramolecular hydrogen bond in the vacuum state. These monomer calculations emphasize the need to accurately consider all significant conformations of flexible molecules in free energy calculations. This development of the CMC/MD method paves the way for computations of protein stability analogous to the biochemical technique of "exhaustive mutagenesis." We have carried out just such a calculation at position 133 of T4 lysozyme, where we use CMC/MD to calculate the relative stability of eight different side-chain mutants in a single free energy calculation. Our T4 calculations show good agreement with the prior free energy calculations of Veenstra et al. (Prot Eng 1997;10:789-807)4 and excellent agreement with the experiments of Mendel et al. (Science 1992;256:1798-1802).     

Molecular Dynamics Simulations of some Small Organic Molecules

Abstract

Free energy differences between different conformers of D-ribofuranose, L-malic acid and meso-tartaric acid in solution were calculated using Molecular Dynamics simulations. In case of ribose the α → β transition was studied. For the acids attention was focussed on the transitions between the three possible staggered conformers with respect to the central C-C bond. In all cases a thermodynamic integration method was employed to evaluate the free energy difference. The use of an alternative technique, umbrella sampling, for ribose did not give promising results.

It was shown that one needs a fairly accurate picture of the accessible conformational space in case of flexible molecules like the ones considered here before one can determine meaningful free energy differences. Large hysteresis effects between forward and reverse simulated transitions were observed, but contrary to the general belief they are no direct measure of the accuracy of the calculated ΔG values. In all cases the ΔG values resulting from the simulations and from NMR experiments agree within the, considerable, error limits and for the different forms of D-ribose, L-malic acid and L-tartaric acid the relative order of their populations is also correctly reproduced.     

The Influence of Starting Coordinates in Free Energy Simulations of Ligand Binding to Dihydrofolate Reductase

Abstract

Molecular dynamics (MD) simulation combined with free energy perturbation (FEP) methods have been used to study the key structural differences and relative free energies for the binding of 6-methyl-N5-deazapterin (N8 protonated) and the 8-substituted compound, 6,8-dimethyl-N5-deazapterin (N3 protonated), to dihydrofolate reductase (DHFR). The free energy changes have been calculated using a variety of initial X-ray coordinates derived from bacterial and vertebrate (including human) DHFRs, and both with and without the reduced cofactor nicotinamide adenine dinucleotide (NADPH) bound. Given a sufficiently long simulation time for the FEP calculations (ca. 200 ps), all structures obtained after mutating 6,8-methyl-N5-deazapterin to 6-methyl-N5-deazapterin exhibited hydrogen bond formation between a backbone carbonyl group of DHFR and H(N8) of 6-methyl-N5-deazapterin, analogous to that found in the X-ray crystal structure of N5-deazafolate(N8 protonated) bound to human DHFR. However, both simulation and experiment suggest this additional H-bonding does not greatly enhance thermodynamic stability, with experiment indicating at most a factor of 2 difference in the relative affinities of the two ligand cations for vertebrate DHFR. Moreover, a binding differential of 10 in favour of the protonated 8-substituted compound is found experimentally for bacterial DHFR. The MD/FEP calculations suggest that the relative cost of ligand desolvation may largely cancel the lowering of free energy obtained in the active site, resulting in predicted binding differences within the range indicated by the vertebrate and bacterial DHFR experiments. However, the theoretical free energy changes could not be obtained with the accuracy required for the rationalization of the observed species dependence. While sampling difficulties are known to be inherent in MD simulation methodologies, these studies with several initial coordinate sets have demonstrated the contribution of coordinate choice to this problem. The results indicate that for demanding protein-ligand binding problems such as this one, the accuracy of the method may be no better than ± 2 kcal/mol.     

Free energy simulations: Applications to the study of liquid water,hydrophobic interactions and solvent effects on conformational stability

Free energy simulations using the Metropolis Monte Carlo method and the coupling parameter approach with umbrella sampling are described for several problems of interest in structural biochemistry; the liquid water, the hydrophobic interaction of alkyl and phenyl groups in water and solvent effects on the conformational stability of the alanine dipeptide and the dimethyl phosphate anion in water. Proximity analysis of results is employed to identify stabilizing factors. Implications of result with respect to the structural chemistry of proteins and nucleic acids is considered.     

Evaluation of the Adaptive Umbrella Sampling Method

Abstract

The adaptive umbrella sampling technique, introduced recently to improve the probability ratio method and found to perform more reliably than the customary harmonic umbrella sampling, is tested and compared with other free energy methods. One of the tests applies the method to a transition involving a chemical change: calculation of the hydration free energy difference between acetone and dimethylamine and the other test calculates the conformational free energy difference between the C ₇ and α_R conformations of the alanide dipeptide. The dipeptide problem is also treated by two types of thermodynamic integrations and by the perturbation method. The result for the acetone-dimethylamine problem is compared with previous calculations on the same system using the perturbation method, overlap ratio method and finite difference thermodynamic integration. Enhancements to the adaptive umbrella sampling method are also presented.     

X-ray structural and simulation analysis of a protein mutant: the value of a combined approach

The effect of the mutation Arg 96 to His on the stability of bacteriophage T4 lysozyme has been previously studied by calorimetric experiments, X-ray crystallography, and free energy simulation techniques. The experimental and calculated values for the difference between the free energy of denaturation of the mutant and the wild type are in reasonable agreement. However, the two approaches led to different explanations for the loss in stability. To analyze the differences, a series of refinements based on the crystallographic data were performed, a number of aspects of the simulations were reexamined, and continuum electrostatic calculations were done to complement the latter. The results of those comparisons provide a better understanding of the origin of the free energy difference in this mutant. Furthermore, they show the importance of the combined use of simulations and crystallography for interpreting the effects of mutations on the energetics of the system.     

Determination of the structural ensemble of the molten globule state of a protein by computer simulations

We have used computer simulations to investigate the structural nature of the molten globule (MG) state of canine milk lysozyme. To sample the conformational space efficiently, we performed replica-exchange umbrella sampling simulations with the radius of gyration as a reaction coordinate. We applied the Weighted Histogram Analysis Method to the trajectory of the simulations to obtain the potential of mean force, from which we identified representative structures corresponding to local minima in the free energy surface. The representative structures obtained in this way are in accord with the characteristics of the MG state reported previously by experimental studies. We conjecture that the MG state comprises a series of partially structured states undergoing relatively fast conformational interchange.     

The N-Terminal Segment of the Voltage-Dependent Anion Channel: A Possible Membrane-Bound Intermediate in Pore Unbinding

The voltage-dependent anion channel (VDAC) resides in the outer mitochondrial membrane and can adopt a closed or open configuration, most likely depending on whether the N-terminal segment (NTS) occupies the pore or protrudes into the cytoplasm. In this study, we calculate the free energy of releasing the NTS from the pore using molecular dynamics simulation. This is complicated by the flexible nature of the NTS, in particular its disordered structure in aqueous solution compared to the pore lumen. We carried out potential of mean force calculations using enhanced sampling or conformational restraints to address the conformational sampling problem. For the binding to the VDAC pore, two systems were considered, featuring either the native VDAC system or a modified system where the NTS is detached from the pore, that is, noncovalently bound in the pore lumen. The calculated free energies required to translocate the NTS from the pore into the solvent moiety are 83.8 or 74.3 kJ mol⁻¹, respectively. The dissociation pathway in VDAC presents two in-pore minima, separated by a low free energy barrier and a membrane-bound intermediate state. Since we observe small changes in pore shape along the NTS dissociation pathway, we suggest that rigidification of the VDAC pore might impair NTS dissociation. The stability of the membrane-bound state of the VDAC NTS is confirmed by independent molecular dynamics simulations showing spontaneous membrane binding of a NTS-derived peptide as well as nuclear magnetic resonance experiments where chemical shift perturbations of the NTS-derived peptide evidence binding to phospholipid nanodiscs.     

Mechanism and Energetics of Charybdotoxin Unbinding from a Potassium Channel from Molecular Dynamics Simulations

Ion channel-toxin complexes are ideal systems for computational studies of protein-ligand interactions, because, in most cases, the channel axis provides a natural reaction coordinate for unbinding of a ligand and a wealth of physiological data is available to check the computational results. We use a recently determined structure of a potassium channel-charybdotoxin complex in molecular dynamics simulations to investigate the mechanism and energetics of unbinding. Pairs of residues on the channel protein and charybdotoxin that are involved in the binding are identified, and their behavior is traced during umbrella-sampling simulations as charybdotoxin is moved away from the binding site. The potential of mean force for the unbinding of charybdotoxin is constructed from the umbrella sampling simulations using the weighted histogram analysis method, and barriers observed are correlated with specific breaking of interactions and influx of water molecules into the binding site. Charybdotoxin is found to undergo conformational changes as a result of the reaction coordinate choice—a nontrivial decision for larger ligands—which we explore in detail, and for which we propose solutions. Agreement between the calculated and the experimental binding energies is obtained once the energetic consequences of these conformational changes are included in the calculations.     

Predicting absolute ligand binding free energies to a simple model site

A central challenge in structure-based ligand design is the accurate prediction of binding free energies. Here we apply alchemical free energy calculations in explicit solvent to predict ligand binding in a model cavity in T4 lysozyme. Even in this simple site, there are challenges. We made systematic improvements, beginning with single poses from docking, then including multiple poses, additional protein conformational changes, and using an improved charge model. Computed absolute binding free energies had an RMS error of 1.9 kcal/mol relative to previously determined experimental values. In blind prospective tests, the methods correctly discriminated between several true ligands and decoys in a set of putative binders identified by docking. In these prospective tests, the RMS error in predicted binding free energies relative to those subsequently determined experimentally was only 0.6 kcal/mol. X-ray crystal structures of the new ligands bound in the cavity corresponded closely to predictions from the free energy calculations, but sometimes differed from those predicted by docking. Finally, we examined the impact of holding the protein rigid, as in docking, with a view to learning how approximations made in docking affect accuracy and how they may be improved.     

Interactions of liquid crystal-forming molecules with phospholipid bilayers studied by molecular dynamics simulations

Recent experiments have shown that liquid crystals can be used to image mammalian cell membranes and to amplify structural reorganization in phospholipid-laden liquid crystal-aqueous interfaces. In this work, molecular dynamics simulations were employed to explore the interactions between commonly used liquid crystal-forming molecules and phospholipid bilayers. In particular, umbrella sampling was used to obtain the potential of mean force of 4-cyano-4'-pentylbiphenyl (5CB) and 4'-(3,4-difluor-phenyl)-4-pentyl-bicylohexyl (5CF) molecules partitioning into a dipalmitoylphosphatidylcholine bilayer. In addition, results of simulations are presented for systems consisting of a fully hydrated bilayer with 5CB or 5CF molecules at the lowest (4.5 mol %) and highest (20 mol %) concentrations used in recent laboratory experiments. It is found that mesogens preferentially partition from the aqueous phase into the membrane; the potential of mean force exhibits highly favorable free energy differences for partitioning (-18 k(B)T for 5CB and -26 k(B)T for 5CF). The location and orientation of mesogens associated with the most stable free energies in umbrella sampling simulations of dilute systems were found to be consistent with those observed in liquid-crystal-rich bilayers. It is found that the presence of mesogens in the bilayer enhances the order of lipid acyl tails, and changes the spatial and orientational arrangement of lipid headgroup atoms. These effects are more pronounced at higher liquid-crystal concentrations. In comparing the behavior of 5CB and 5CF, a stronger spatial correlation (i.e., possibly leading to aggregation) is observed between 5CB molecules within a bilayer than between 5CF molecules. Also, the range of molecular orientations and positions along the bilayer normal is larger for 5CB molecules. At the same time, 5CF molecules were found to bind more strongly to lipid headgroups, thereby slowing the lateral motion of lipid molecules.     

Thermodynamics of Micelle Formation and Membrane Fusion Modulate Antimicrobial Lipopeptide Activity

Antimicrobial lipopeptides (AMLPs) are antimicrobial drug candidates that preferentially target microbial membranes. One class of AMLPs, composed of cationic tetrapeptides attached to an acyl chain, have minimal inhibitory concentrations in the micromolar range against a range of bacteria and fungi. Previously, we used coarse-grained molecular dynamics simulations and free energy methods to study the thermodynamics of their interaction with membranes in their monomeric state. Here, we extended the study to the biologically relevant micellar state, using, to our knowledge, a novel reaction coordinate based on hydrophobic contacts. Using umbrella sampling along this reaction coordinate, we identified the critical transition states when micelles insert into membranes. The results indicate that the binding of these AMLP micelles to membranes is thermodynamically favorable, but in contrast to the monomeric case, there are significant free energy barriers. The height of these free energy barriers depends on the membrane composition, suggesting that the AMLPs’ ability to selectively target bacterial membranes may be as much kinetic as thermodynamic. This mechanism highlights the importance of considering oligomeric state in solution as criterion when optimizing peptides or lipopeptides as antibiotic leads.