A Comparative Study on the Ability of Two Implicit Solvent Lipid Models to Predict Transmembrane Helix Tilt Angles

Free-energy profiles describing the relative orientation of membrane proteins along predefined coordinates can be efficiently calculated by means of umbrella simulations. Such simulations generate reliable orientational distributions but are difficult to converge because of the very long equilibration times of the solvent and the lipid bilayer in explicit representation. Two implicit lipid membrane models are here applied in combination with the umbrella sampling strategy to the simulation of the transmembrane (TM) helical segment from virus protein U (Vpu). The models are used to study both orientation and energetics of this α-helical peptide as a function of hydrophobic mismatch. We observe that increasing the degree of positive hydrophobic mismatch increased the tilt angle of Vpu. These findings agree well with experimental data and as such validate the solvation models used in this study.     

Exploration of the conformational space of myosin recovery stroke via molecular dynamics

Muscle contractions are driven by cyclic conformational changes of myosin, whose molecular mechanisms of operation are being elucidated by recent advances in crystallographic studies and single molecule experiments. To complement such structural studies and consider the energetics of the conformational changes of myosin head, umbrella sampling molecular dynamics (MD) simulations were performed with the all-atom model of the scallop myosin sub-fragment 1 (S1) with a bound ATP in solution in explicit water using the crystallographic near-rigor and transition state conformations as two references. The constraints on RMSD reaction coordinates used for the umbrella sampling were found to steer the conformational changes efficiently, and relatively close correlations have been observed between the set of characteristic structural changes including the lever arm rotation and the closing of the nucleotide binding pocket. The lever arm angle and key residue interaction distances in the nucleotide binding pocket and the relay helix show gradual changes along the recovery stroke reaction coordinate, consistent with previous crystallographic and computational minimum energy studies. Thermal fluctuations, however, appear to make the switch-2 coordination of ATP more flexible than suggested by crystal structures. The local solvation environment of the fluorescence probe, Trp 507 (scallop numbering), also appears highly mobile in the presence of thermal fluctuations.     

The energetics of transmembrane helix insertion into a lipid bilayer

Free energy profiles for insertion of a hydrophobic transmembrane protein α-helix (M2 from CFTR) into a lipid bilayer have been calculated using coarse-grained molecular dynamics simulations and umbrella sampling to yield potentials of mean force along a reaction path corresponding to translation of a helix across a lipid bilayer. The calculated free energy of insertion is smaller when a bilayer with a thinner hydrophobic region is used. The free energies of insertion from the potentials of mean force are compared with those derived from a number of hydrophobicity scales and with those derived from translocon-mediated insertion. This comparison supports recent models of translocon-mediated insertion and in particular suggests that: 1), helices in an about-to-be-inserted state may be located in a hydrophobic region somewhat thinner than the core of a lipid bilayer; and/or 2), helices in a not-to-be-inserted state may experience an environment more akin (e.g., in polarity/hydrophobicity) to the bilayer/water interface than to bulk water.     

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.     

Free energy simulations for protein ligand binding and stability

Abstract

We summarize several computational techniques to determine relative free energies for condensed-phase systems. The focus is on practical considerations which are capable of making direct contact with experiments. Particular applications include the thermodynamic stability of apo- and holo-myoglobin, insulin dimerization free energy, ligand binding in lysozyme, and ligand diffusion in globular proteins. In addition to provide differential free energies between neighboring states, converged umbrella sampling simulations provide insight into migration barriers and ligand dissociation barriers and analysis of the trajectories yield additional insight into the structural dynamics of fundamental processes. Also, such simulations are useful tools to quantify relative stability changes for situations where experiments are difficult. This is illustrated for NO-bound myoglobin. For the dissociation of benzonitrile from lysozyme it is found that long umbrella sampling simulations are required to approximately converge the free energy profile. Then, however, the resulting differential free energy between the bound and unbound state is in good agreement with estimates from molecular mechanics with generalized Born surface area simulations. Furthermore, comparing the barrier height for ligand escape suggests that ligand dissociation contains a non-equilibrium component.     

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.     

Potential of mean force for separation of the repeating units in cellulose and hemicellulose

As a first step toward understanding the energetics of removal of cello-oligomers from the cellulose surface, we have performed umbrella sampling calculations to determine the free energy required for separation of repeating units of cellulose and hemicellulose from each other. Molecular dynamics (MD) simulations were performed for both the stacked and non-stacked arrangements of the cellobiose pair system and the xylobiose pair system. These stacked and non-stacked arrangements were taken as representative systems for the crystalline and amorphous domains in cellulose and hemicellulose. In addition, similar calculations were also carried out to determine the energetics involved in the separation of the cellobiose–xylobiose molecule pair in the non-stacked arrangement. The potential of mean force profiles exhibit a single minimum in all cases and are qualitatively similar. Our results show that the location of the minimum as well as the depth of the well can be correlated with the size of the disaccharide molecules.     

Thermodynamic stability of water molecules in the bacteriorhodopsin proton channel: a molecular dynamics free energy perturbation study.

The proton transfer activity of the light-driven proton pump, bacteriorhodopsin (bR) in the photochemical cycle might imply internal water molecules. The free energy of inserting water molecules in specific sites along the bR transmembrane channel has been calculated using molecular dynamics simulations based on a microscopic model. The existence of internal hydration is related to the free energy change on transfer of a water molecule from bulk solvent into a specific binding site. Thermodynamic integration and perturbation methods were used to calculate free energies of hydration for each hydrated model from molecular dynamics simulations of the creation of water molecules into specific protein-binding sites. A rigorous statistical mechanical formulation allowing the calculation of the free energy of transfer of water molecules from the bulk to a protein cavity is used to estimate the probabilities of occupancy in the putative bR proton channel. The channel contains a region lined primarily by nonpolar side-chains. Nevertheless, the results indicate that the transfer of four water molecules from bulk water to this apparently hydrophobic region is thermodynamically permitted. The column forms a continuous hydrogen-bonded chain over 12 A between a proton donor, Asp 96, and the retinal Schiff base acceptor. The presence of two water molecules in direct hydrogen-bonding association with the Schiff base is found to be strongly favorable thermodynamically. The implications of these results for the mechanism of proton transfer in bR are discussed.     

Elucidation of Ligand-Dependent Modulation of Disorder-Order Transitions in the Oncoprotein MDM2

Numerous biomolecular interactions involve unstructured protein regions, but how to exploit such interactions to enhance the affinity of a lead molecule in the context of rational drug design remains uncertain. Here clarification was sought for cases where interactions of different ligands with the same disordered protein region yield qualitatively different results. Specifically, conformational ensembles for the disordered lid region of the N-terminal domain of the oncoprotein MDM2 in the presence of different ligands were computed by means of a novel combination of accelerated molecular dynamics, umbrella sampling, and variational free energy profile methodologies. The resulting conformational ensembles for MDM2, free and bound to p53 TAD (17-29) peptide identify lid states compatible with previous NMR measurements. Remarkably, the MDM2 lid region is shown to adopt distinct conformational states in the presence of different small-molecule ligands. Detailed analyses of small-molecule bound ensembles reveal that the ca. 25-fold affinity improvement of the piperidinone family of inhibitors for MDM2 constructs that include the full lid correlates with interactions between ligand hydrophobic groups and the C-terminal lid region that is already partially ordered in apo MDM2. By contrast, Nutlin or benzodiazepinedione inhibitors, that bind with similar affinity to full lid and lid-truncated MDM2 constructs, interact additionally through their solubilizing groups with N-terminal lid residues that are more disordered in apo MDM2.     

Simulations of skin barrier function: free energies of hydrophobic and hydrophilic transmembrane pores in ceramide bilayers

Transmembrane pore formation is central to many biological processes such as ion transport, cell fusion, and viral infection. Furthermore, pore formation in the ceramide bilayers of the stratum corneum may be an important mechanism by which penetration enhancers such as dimethylsulfoxide (DMSO) weaken the barrier function of the skin. We have used the potential of mean constraint force (PMCF) method to calculate the free energy of pore formation in ceramide bilayers in both the innate gel phase and in the DMSO-induced fluidized state. Our simulations show that the fluid phase bilayers form archetypal water-filled hydrophilic pores similar to those observed in phospholipid bilayers. In contrast, the rigid gel-phase bilayers develop hydrophobic pores. At the relatively small pore diameters studied here, the hydrophobic pores are empty rather than filled with bulk water, suggesting that they do not compromise the barrier function of ceramide membranes. A phenomenological analysis suggests that these vapor pores are stable, below a critical radius, because the penalty of creating water-vapor and tail-vapor interfaces is lower than that of directly exposing the strongly hydrophobic tails to water. The PMCF free energy profile of the vapor pore supports this analysis. The simulations indicate that high DMSO concentrations drastically impair the barrier function of the skin by strongly reducing the free energy required for pore opening.     

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.     

Vectorial insertion of a β-helical peptide into membrane: a theoretical study on polytheonamide B

Spontaneous unidirectional, or vectorial, insertion of transmembrane peptides is a fundamental biophysical process for toxin and viral actions. Polytheonamide B (pTB) is a potent cytotoxic peptide with a β^6.3-helical structure. Previous experimental studies revealed that the pTB inserts into the membrane in a vectorial fashion and forms a channel with its single molecular length long enough to span the membrane. Also, molecular dynamics simulation studies demonstrated that the pTB is prefolded in aqueous solution. These are unique features of pTB because most of the peptide toxins form channels through oligomerization of transmembrane helices. Here, we performed all-atom molecular dynamics simulations to examine the dynamic mechanism of the vectorial insertion of pTB, providing underlying elementary processes of the membrane insertion of a prefolded single transmembrane peptide. We find that the insertion of pTB proceeds with only the local lateral compression of the membrane in three successive phases: “landing,” “penetration,” and “equilibration” phases. The free energy calculations using the replica-exchange umbrella sampling simulations present an energy cost of 4.3 kcal/mol at the membrane surface for the membrane insertion of pTB from bulk water. The trajectories of membrane insertion revealed that the insertion process can occur in two possible pathways, namely “trapped” and “untrapped” insertions; in some cases, pTB is trapped in the upper leaflet during the penetration phase. Our simulations demonstrated the importance of membrane anchoring by the hydrophobic N-terminal blocking group in the landing phase, leading to subsequent vectorial insertion.     

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.     

Energetics of K+ permeability through Gramicidin A by forward-reverse steered molecular dynamics

The estimation of ion channel permeability poses a considerable challenge for computer simulations because of the significant free energy barriers involved, but also offers valuable molecular information on the ion permeation process not directly available from experiments. In this article we determine the equilibrium free energy barrier for potassium ion permeability in Gramicidin A in an efficient way by atomistic forward-reverse non-equilibrium steered molecular dynamics simulations, opening the way for its use in more complex biochemical systems. Our results indicate that the tent-shaped energetics of translocation of K+ ions in Gramicidin A is dictated by the different polarization responses to the ion of the external bulk water and the less polar environment of the membrane.     

Extracting hydrophobic free energies from experimental data: relationship to protein folding and theoretical models

Solubility and vapor pressure measurements of hydrocarbons in water are generally thought to provide estimates of the strength of the hydrophobic effect in the range 20-30 cal/(mol.A2). Our reassessment of the solubility data on the basis of new developments in solution thermodynamics suggests that the hydrophobic surface free energy for hydrocarbon solutes is 46-47 cal/(mol.A2), although the actual value depends strongly on curvature effects [Nicholls et al. (1991) Proteins (in press); Sharp et al. (1991) Science 252, 106-109]. The arguments to support such a significant increase in the estimate of the hydrophobic effect stem partly from theoretical considerations and partly from the experimental results of De Young and Dill [(1990) J. Phys. Chem. 94, 801-809] on benzene partition between water and alkane solvents. Previous estimates of the hydrophobic effect derive from an analysis of solute partition data, which does not fully account for changes in volume entropy. We show here how the ideal gas equations, combined with experimental molar volumes, can account for such changes. Revised solubility scales for the 20 amino acids, based on cyclohexane to water and octanol to water transfer energies, are derived. The agreement between these scales, particularly the octanol scale, and mutant protein stability measurements from Kellis et al. [(1989) Biochemistry 28, 4914-4922] and Shortle et al. [(1990) Biochemistry 29, 8033-8041] is good. The increased strength of the hydrophobic interaction has implications for the energetics of protein folding, substrate binding, and nucleic acid base stacking and the interpretation of computer simulations.     

Coarse-grained molecular dynamics simulations of the energetics of helix insertion into a lipid bilayer

Experimental and computational studies have indicated that hydrophobicity plays a key role in driving the insertion of transmembrane alpha-helices into lipid bilayers. Molecular dynamics simulations allow exploration of the nature of the interactions of transmembrane alpha-helices with their lipid bilayer environment. In particular, coarse-grained simulations have considerable potential for studying many aspects of membrane proteins, ranging from their self-assembly to the relation between their structure and function. However, there is a need to evaluate the accuracy of coarse-grained estimates of the energetics of transmembrane helix insertion. Here, three levels of complexity of model system have been explored to enable such an evaluation. First, calculated free energies of partitioning of amino acid side chains between water and alkane yielded an excellent correlation with experiment. Second, free energy profiles for transfer of amino acid side chains along the normal to a phosphatidylcholine bilayer were in good agreement with experimental and atomistic simulation studies. Third, estimation of the free energy profile for transfer of an arginine residue, embedded within a hydrophobic alpha-helix, to the center of a lipid bilayer gave a barrier of approximately 15 kT. Hence, there is a substantial barrier to membrane insertion for charged amino acids, but the coarse-grained model still underestimates the corresponding free energy estimate (approximately 29 kT) from atomistic simulations (Dorairaj, S., and Allen, T. W. (2007) Proc. Natl. Acad. Sci. U.S.A. 104, 4943-4948). Coarse-grained simulations were then used to predict the free energy profile for transfer of a simple model transmembrane alpha-helix (WALP23) across a lipid bilayer. The results indicated that a transmembrane orientation was favored by about -70 kT.     

Ion transport through membrane-spanning nanopores studied by molecular dynamics simulations and continuum electrostatics calculations

Narrow hydrophobic regions are a common feature of biological channels, with possible roles in ion-channel gating. We study the principles that govern ion transport through narrow hydrophobic membrane pores by molecular dynamics simulation of model membranes formed of hexagonally packed carbon nanotubes. We focus on the factors that determine the energetics of ion translocation through such nonpolar nanopores and compare the resulting free-energy barriers for pores with different diameters corresponding to the gating regions in closed and open forms of potassium channels. Our model system also allows us to compare the results from molecular dynamics simulations directly to continuum electrostatics calculations. Both simulations and continuum calculations show that subnanometer wide pores pose a huge free-energy barrier for ions, but a small increase in the pore diameter to approximately 1 nm nearly eliminates that barrier. We also find that in those wider channels the ion mobility is comparable to that in the bulk phase. By calculating local electrostatic potentials, we show that the long range Coulomb interactions of ions are strongly screened in the wide water-filled channels. Whereas continuum calculations capture the overall energetics reasonably well, the local water structure, which is not accounted for in this model, leads to interesting effects such as the preference of hydrated ions to move along the pore wall rather than through the center of the pore.     

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.     

Free energetics of rigid body association of ubiquitin binding domains: A biochemical model for binding mediated by hydrophobic interaction

Weak intermolecular interactions, such as hydrophobic associations, underlie numerous biomolecular recognition processes. Ubiquitin is a small protein that represents a biochemical model for exploring thermodynamic signatures of hydrophobic association as it is widely held that a major component of ubiquitin's binding to numerous partners is mediated by hydrophobic regions on both partners. Here, we use atomistic molecular dynamics simulations in conjunction with the Adaptive Biasing Force sampling method to compute potentials of mean force (the reversible work, or free energy, associated with the binding process) to investigate the thermodynamic signature of complexation in this well‐studied biochemical model of hydrophobic association. We observe that much like in the case of a purely hydrophobic solute (i.e., graphene, carbon nanotubes), association is favored by entropic contributions from release of water from the interprotein regions. Moreover, association is disfavored by loss of enthalpic interactions, but unlike in the case of purely hydrophobic solutes, in this case protein‐water interactions are lost and not compensated for by additional water‐water interactions generated upon release of interprotein and moreso, hydration, water. We further find that relative orientations of the proteins that mutually present hydrophobic regions of each protein to its partner are favored over those that do not. In fact, the free energy minimum as predicted by a force field based method recapitulates the experimental NMR solution structure of the complex. Proteins 2014; 82:1453–1468. © 2014 Wiley Periodicals, Inc.     

End-to-end attraction of duplex DNA

Recent experiments [Nakata, M. et al., End-to-end stacking and liquid crystal condensation of 6 to 20 basepair DNA duplexes. Science 2007; 318:1276-1279] have demonstrated spontaneous end-to-end association of short duplex DNA fragments into long rod-like structures. By means of extensive all-atom molecular dynamic simulations, we characterized end-to-end interactions of duplex DNA, quantitatively describing the forces, free energy and kinetics of the end-to-end association process. We found short DNA duplexes to spontaneously aggregate end-to-end when axially aligned in a small volume of monovalent electrolyte. It was observed that electrostatic repulsion of 5'-phosphoryl groups promoted the formation of aggregates in a conformation similar to the B-form DNA double helix. Application of an external force revealed that rupture of the end-to-end assembly occurs by the shearing of the terminal base pairs. The standard binding free energy and the kinetic rates of end-to-end association and dissociation processes were estimated using two complementary methods: umbrella sampling simulations of two DNA fragments and direct observation of the aggregation process in a system containing 458 DNA fragments. We found the end-to-end force to be short range, attractive, hydrophobic and only weakly dependent on the ion concentration. The relation between the stacking free energy and end-to-end attraction is discussed as well as possible roles of the end-to-end interaction in biological and nanotechnological systems.