On the Improvement of Free-Energy Calculation from Steered Molecular Dynamics Simulations Using Adaptive Stochastic Perturbation Protocols

The potential of mean force (PMF) calculation in single molecule manipulation experiments performed via the steered molecular dynamics (SMD) technique is a computationally very demanding task because the analyzed system has to be perturbed very slowly to be kept close to equilibrium. Faster perturbations, far from equilibrium, increase dissipation and move the average work away from the underlying free energy profile, and thus introduce a bias into the PMF estimate. The Jarzynski equality offers a way to overcome the bias problem by being able to produce an exact estimate of the free energy difference, regardless of the perturbation regime. However, with a limited number of samples and high dissipation the Jarzynski equality also introduces a bias. In our previous work, based on the Brownian motion formalism, we introduced three stochastic perturbation protocols aimed at improving the PMF calculation with the Jarzynski equality in single molecule manipulation experiments and analogous computer simulations. This paper describes the PMF reconstruction results based on full-atom molecular dynamics simulations, obtained with those three protocols. We also want to show that the protocols are applicable with the second-order cumulant expansion formula. Our protocols offer a very noticeable improvement over the simple constant velocity pulling. They are able to produce an acceptable estimate of PMF with a significantly reduced bias, even with very fast perturbation regimes. Therefore, the protocols can be adopted as practical and efficient tools for the analysis of mechanical properties of biological molecules.     

Theoretical and computational models of biological ion channels

The goal of this review is to establish a broad and rigorous theoretical framework to describe ion permeation through biological channels. This framework is developed in the context of atomic models on the basis of the statistical mechanical projection-operator formalism of Mori and Zwanzig. The review is divided into two main parts. The first part introduces the fundamental concepts needed to construct a hierarchy of dynamical models at different level of approximation. In particular, the potential of mean force (PMF) as a configuration-dependent free energy is introduced, and its significance concerning equilibrium and non-equilibrium phenomena is discussed. In addition, fundamental aspects of membrane electrostatics, with a particular emphasis on the influence of the transmembrane potential, as well as important computational techniques for extracting essential information from all-atom molecular dynamics (MD) simulations are described and discussed. The first part of the review provides a theoretical formalism to 'translate' the information from the atomic structure into the familiar language of phenomenological models of ion permeation. The second part is aimed at reviewing and contrasting results obtained in recent computational studies of three very different channels: the gramicidin A (gA) channel, which is a narrow one-ion pore (at moderate concentration), the KcsA channel from Streptomyces lividans, which is a narrow multi-ion pore, and the outer membrane matrix porin F (OmpF) from Escherichia coli, which is a trimer of three beta-barrel subunits each forming wide aqueous multi-ion pores. Comparison with experiments demonstrates that current computational models are approaching semi-quantitative accuracy and are able to provide significant insight into the microscopic mechanisms of ion conduction and selectivity. We conclude that all-atom MD with explicit water molecules can represent important structural features of complex biological channels accurately, including such features as the location of ion-binding sites along the permeation pathway. We finally discuss the broader issue of the validity of ion permeation models and an outlook to the future.     

Molecular dynamics simulation of water permeation through the alpha-hemolysin channel

The alpha-hemolysin (AHL) nanochannel is a non-selective channel that allows for uncontrolled transport of small molecules across membranes leading to cell death. Although it is a bacterial toxin, it has promising applications, ranging from drug delivery systems to nano-sensing devices. This study focuses on the transport of water molecules through an AHL nanochannel using molecular dynamics (MD) simulations. Our results show that AHL can quickly transport water across membranes. The first-passage time approach was used to estimate the diffusion coefficient and the mean exit time. To study the energetics of transport, the potential of mean force (PMF) of a water molecule along the AHL nanochannel was calculated. The results show that the energy barriers of water permeation across a nanopore are always positive along the channel and the values are close to thermal energy (k_BT). These findings suggest that the observed quick permeation of water is due to small energy barriers and a hydrophobic inner channel surface resulting in smaller friction. We speculate that these physical mechanisms are important in how AHL causes cell death.     

Free energy and kinetics of cAMP permeation through connexin26 via applied voltage and milestoning

The connexin family is a diverse group of highly regulated wide-pore channels permeable to biological signaling molecules. Despite the critical roles of connexins in mediating selective molecular signaling in health and disease, the basis of molecular permeation through these pores remains unclear. Here, we report the thermodynamics and kinetics of binding and transport of a second messenger, adenosine-3′,5′-cyclophosphate (cAMP), through a connexin26 hemichannel (Cx26). First, inward and outward fluxes of cAMP molecules solvated in KCl solution were obtained from 4 μs of ± 200 mV simulations. These fluxes data yielded a single-channel permeability of cAMP and cAMP/K⁺ permeability ratio consistent with experimentally measured values. The results from voltage simulations were then compared with the potential of mean force (PMF) and the mean first passage times (MFPTs) of a single cAMP without voltage, obtained from a total of 16.5 μs of Voronoi-tessellated Markovian milestoning simulations. Both the voltage simulations and the milestoning simulations revealed two cAMP-binding sites, for which the binding constants K_D and dissociation rates k_off were computed from PMF and MFPTs. The protein dipole inside the pore produces an asymmetric PMF, reflected in unequal cAMP MFPTs in each direction once within the pore. The free energy profiles under opposite voltages were derived from the milestoning PMF and revealed the interplay between voltage and channel polarity on the total free energy. In addition, we show how these factors influence the cAMP dipole vector during permeation, and how cAMP affects the local and nonlocal pore diameter in a position-dependent manner.     

Ion permeation through a narrow channel: using gramicidin to ascertain all-atom molecular dynamics potential of mean force methodology and biomolecular force fields

We investigate methods for extracting the potential of mean force (PMF) governing ion permeation from molecular dynamics simulations (MD) using gramicidin A as a prototypical narrow ion channel. It is possible to obtain well-converged meaningful PMFs using all-atom MD, which predict experimental observables within order-of-magnitude agreement with experimental results. This was possible by careful attention to issues of statistical convergence of the PMF, finite size effects, and lipid hydrocarbon chain polarizability. When comparing the modern all-atom force fields of CHARMM27 and AMBER94, we found that a fairly consistent picture emerges, and that both AMBER94 and CHARMM27 predict observables that are in semiquantitative agreement with both the experimental conductance and dissociation coefficient. Even small changes in the force field, however, result in significant changes in permeation energetics. Furthermore, the full two-dimensional free-energy surface describing permeation reveals the location and magnitude of the central barrier and the location of two binding sites for K(+) ion permeation near the channel entrance--i.e., an inner site on-axis and an outer site off-axis. We conclude that the MD-PMF approach is a powerful tool for understanding and predicting the function of narrow ion channels in a manner that is consistent with the atomic and thermally fluctuating nature of proteins.     

The influence of cholesterol on interactions and dynamics of ibuprofen in a lipid bilayer

In this work, molecular dynamics (MD) simulations with atomistic details were performed to examine the influence of the cholesterol on the interactions and the partitioning of the hydrophobic drug ibuprofen in a fully hydrated 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) bilayer. Analysis of MD simulations indicated that ibuprofen molecules prefer to be located in the hydrophobic acyl chain region of DMPC/cholesterol bilayers. This distribution decreases the lateral motion of lipid molecules. The presence of ibuprofen molecules in the bilayers with 0 and 25 mol% cholesterol increases the ordering of hydrocarbon tails of lipids whereas for the bilayers with 50 mol% cholesterol, ibuprofen molecules perturb the flexible chains of DMPC lipids which leads to the reduction of the acyl chain order parameter. The potential of the mean force (PMF) method was used to calculate the free energy profile for the transferring of an ibuprofen molecule from the bulk water into the DMPC/cholesterol membranes. The PMF studies indicated that the presence of 50 mol% cholesterol in the bilayers increases the free energy barrier and slows down the permeation of the ibuprofen drug across the DMPC bilayer. This can be due to the condensing and ordering effects of the cholesterol on the bilayer.     

Water permeation dynamics of AqpZ: a tale of two states

Molecular dynamics simulations of aquaporin Z homotetramer which is a membrane protein facilitating rapid water movement through the plasma membrane of Escherichia coli were performed. Initial configurations were taken from the open and closed states of crystal structures separately. The resulting water osmotic permeability (pf) and diffusive permeability (pd) displayed distinct features. Consistent with previous studies, the side chain conformation of arginine189 was found to mediate the water permeability. A potential of mean force (PMF) as a function of the distance between NH1 of R189 and carbonyl oxygen of A117 was constructed based on the umbrella sampling technique. There are multiple local minima and transition states on the PMF. The assignment of the open or closed state was supported by the permeability pf, calculated within trajectories in umbrella sampling simulations. Our study disclosed a detailed mechanism of the gated water transport.     

Simulating the free energy of passive membrane permeation for small molecules

Abstract

Computer simulations of passive membrane permeation provide important microscopic insights into the molecular mechanism of this important biological process that are complementary to experimental data. Our review focuses on the main approaches for calculating the free energy, or potential of mean force, for permeation of small molecules through lipid bilayers. The theoretical background for most currently used methods for potential of mean force calculation is described, including particle insertion, thermodynamic integration, umbrella sampling, metadynamics, adaptive biasing force and milestoning. A brief comparison of strengths and weaknesses of the competing approaches is presented. This is followed by a survey of results obtained by the different methods, with special attention to describing the mechanistic insights generated by modelling and illustrating capabilities of the different techniques. We conclude with a discussion of recent advances and future directions in modelling membrane permeation, including latest methodological enhancements, consideration of multiple slow variables and memory effects.     

Exploring gas permeability of cellular membranes and membrane channels with molecular dynamics

Aquaporins are a family of membrane proteins specialized in rapid water conduction across biological membranes. Whether these channels also conduct gas molecules and the physiological significance of this potential function have not been well understood. Here we report 140 ns of molecular dynamics simulations of membrane-embedded AQP1 and of a pure POPE bilayer addressing these questions. The permeability of AQP1 to two types of gas molecules, O2 and CO2, was investigated using two complementary methods, namely, explicit gas diffusion simulation and implicit ligand sampling. The simulations show that the central (tetrameric) pore of AQP1 can be readily used by either gas molecule to permeate the channel. The two approaches produced similar free energy profiles associated with gas permeation through the central pore: a -0.4 to -1.7 kcal/mol energy well in the middle, and a 3.6-4.6 kcal/mol energy barrier in the periplasmic vestibule. The barrier appears to be mainly due to a dense cluster of water molecules anchored in the periplasmic mouth of the central pore by four aspartate residues. Water pores show a very low permeability to O2, but may contribute to the overall permeation of CO2 due to its more hydrophilic nature. Although the central pore of AQP1 is found to be gas permeable, the pure POPE bilayer provides a much larger cross-sectional area, thus exhibiting a much lower free energy barrier for CO2 and O2 permeation. As such, gas conduction through AQP1 may only be physiologically relevant either in membranes of low gas permeability, or in cells where a major fraction of the cellular membrane is occupied by AQPs.     

Biomembrane simulations of 12 lipid types using the general amber force field in a tensionless ensemble

The AMBER family of force fields is one of the most commonly used alternatives to describe proteins and drug-like molecules in molecular dynamics simulations. However, the absence of a specific set of parameters for lipids has been limiting the widespread application of this force field in biomembrane simulations, including membrane protein simulations and drug-membrane simulations. Here, we report the systematic parameterization of 12 common lipid types consistent with the General Amber Force Field (GAFF), with charge-parameters determined with RESP at the HF/6–31G(d) level of theory, to be consistent with AMBER. The accuracy of the scheme was evaluated by comparing predicted and experimental values for structural lipid properties in MD simulations in an NPT ensemble with explicit solvent in 100:100 bilayer systems. Globally, a consistent agreement with experimental reference data on membrane structures was achieved for some lipid types when using the typical MD conditions normally employed when handling membrane proteins and drug-membrane simulations (a tensionless NPT ensemble, 310?K), without the application of any of the constraints often used in other biomembrane simulations (such as the surface tension and the total simulation box area). The present set of parameters and the universal approach used in the parameterization of all the lipid types described here, as well as the consistency with the AMBER force field family, together with the tensionless NPT ensemble used, opens the door to systematic studies combining lipid components with small drug-like molecules or membrane proteins and show the potential of GAFF in dealing with biomembranes.     

Computer simulation of small molecule permeation across a lipid bilayer: dependence on bilayer properties and solute volume, size, and cross-sectional area

Cell membrane permeation is required for most drugs to reach their biological target, and understanding this process is therefore crucial for rational drug design. Recent molecular dynamics simulations have studied the permeation of eight small molecules through a phospholipid bilayer. Unlike experiments, atomistic simulations allow the direct calculation of diffusion and partition coefficients of solutes at different depths inside a lipid membrane. Further analyses of the simulations suggest that solute diffusion is less size-dependent and solute partitioning more size-dependent than was commonly thought.     

Computational active site analysis of molecular pathways to improve functional classification of enzymes

This study describes a method to computationally assess the function of homologous enzymes through small molecule binding interaction energy. Three experimentally determined X-ray structures and four enzyme models from ornithine cyclo-deaminase, alanine dehydrogenase, and mu-crystallin were used in combination with nine small molecules to derive a function score (FS) for each enzyme-model combination. While energy values varied for a single molecule-enzyme combination due to differences in the active sites, we observe that the binding energies for the entire pathway were proportional for each set of small molecules investigated. This proportionality of energies for a reaction pathway appears to be dependent on the amino acids in the active site and their direct interactions with the small molecules, which allows a function score (FS) to be calculated to assess the specificity of each enzyme. Potential of mean force (PMF) calculations were used to obtain the energies, and the resulting FS values demonstrate that a measurement of function may be obtained using differences between these PMF values. Additionally, limitations of this method are discussed based on: (a) larger substrates with significant conformational flexibility; (b) low homology enzymes; and (c) open active sites. This method should be useful in accurately predicting specificity for single enzymes that have multiple steps in their reactions and in high throughput computational methods to accurately annotate uncharacterized proteins based on active site interaction analysis.     

Effective potentials between gold nano crystals – functional dependence on temperature

A method is presented that allows to combine the effective potential between two nano crystals (NC), the potential of mean force (PMF), as obtained from all-atomistic molecular dynamics simulations with perturbation theory. In this way, a functional dependence of the PMF on temperature is derived, which enables the prediction of the PMF in a wide temperature range. We applied the method to systems of capped gold NCs of different size. They show very good agreement with data from atomistic simulations.     

Molecular dynamics methods to predict peptide locations in membranes: LAH4 as a stringent test case

Determining the structure of membrane-active peptides inside lipid bilayers is essential to understand their mechanism of action. Molecular dynamics simulations can easily provide atomistic details, but need experimental validation. We assessed the reliability of self-assembling (or “minimum-bias”) and potential of mean force (PMF) approaches, using all-atom (AA) and coarse-grained (CG) force-fields. The LAH4 peptide was selected as a stringent test case, since it is known to attain different orientations depending on the protonation state of its four histidine residues.In all simulations the histidine side-chains inserted in the membrane when neutral, while they interacted with phospholipid headgroups in their charged state. This led to transmembrane orientations for neutral-His LAH4 in all minimum-bias AA simulations and in most CG trajectories. By contrast, the charged-His peptide stabilized membrane defects in AA simulations, whereas it was located at the membrane surface in some CG trajectories, and interacted with both lipid leaflets in others. This behavior is consistent with the higher antimicrobial activity and membrane-permeabilizing behavior of the charged-His LAH4. In addition, good agreement with solid-state NMR orientational data was observed in AA simulations.PMF calculations correctly predicted a higher membrane affinity for the neutral-His peptide. Interestingly, the structures and relative populations of PMF local free-energy minima corresponded to those determined in the less computationally demanding minimum-bias simulations.These data provide an indication about the possible membrane-perturbation mechanism of the charged-His LAH4 peptide: by interacting with lipid headgroups of both leaflets through its cationic side-chains, it could favor membrane defects and facilitate translocation across the bilayer.     

Determination of drug permeability and prediction of drug absorption in Caco-2 monolayers

Permeability coefficients across monolayers of the human colon carcinoma cell line Caco-2, cultured on permeable supports, are commonly used to predict the absorption of orally administered drugs and other xenobiotics. This protocol describes our method for the cultivation, characterization and determination of permeability coefficients of xenobiotics (which are, typically, drug-like compounds) in the Caco-2 model. A few modifications that have been introduced over the years are incorporated in the protocol. The method can be used to trace the permeability of a test compound in two directions, from the apical to the basolateral side or vice versa, and both passive and active transport processes can be studied. The permeability assay can be completed within one working day, provided that the Caco-2 monolayers have been cultured and differentiated on the permeable supports 3 weeks in advance.     

Proton transfer in carbonic anhydrase is controlled by electrostatics rather than the orientation of the acceptor

Combined quantum mechanical/molecular mechanical (QM/MM) simulations are carried out to analyze factors that dictate the proton transfer in carbonic anhydrase II (CAII), an enzyme that has been used as a prototypical example of long-range proton transfers in biomolecules. In contrast to the long-held conjecture in the experimental literature, the computed potentials of mean force (PMF) suggest that the proton transfer in CAII is not very sensitive to the orientation of the acceptor group (His 64) and, therefore, the number of water molecules that bridge the donor (zinc-water) and acceptor groups. Perturbative analysis indicates that a series of polar and charged residues close to the transfer pathways make the dominant contribution to the barrier and exothermicity of the proton transfer reaction, thus supporting the proposal from previous studies of Warshel and co-workers using a somewhat simpler QM/MM model that electrostatic interactions play a major role in the proton transfer in CAII. The PMF results are in striking contrast to previous analysis using the same QM/MM method but an ensemble of minimum energy path (MEP) calculations, which found a steep dependence of the barrier height on the number of bridging water molecules. Analysis of the configurations sampled in the PMF and MEP simulations suggests that this difference arises because the PMF simulations sample a largely stepwise mechanism while the local MEP calculations artificially favored concerted transfers due to the specific protocol used to generate the initial configurations. Therefore, this study presents a compelling argument for carrying out proper conformational sampling in the study of long-range proton transfers. Finally, we illustrate that Phi analysis, which has been widely used in protein folding studies, can potentially generate new mechanistic information for long-range proton transfers regarding the sequence of events. The results of the perturbation analysis and the Phi analysis provide opportunities for experimentally testing the mechanistic proposals from this study and our recent work in which a stepwise "proton hole" transfer pathway has been proposed.     

How to predict diffusion of medium-sized molecules in polymer matrices. From atomistic to coarse grain simulations

The normal diffusion regime of many small and medium-sized molecules occurs on a time scale that is too long to be studied by atomistic simulations. Coarse-grained (CG) molecular simulations allow to investigate length and time scales that are orders of magnitude larger compared to classical molecular dynamics simulations, hence providing a valuable approach to span time and length scales where normal diffusion occurs. Here we develop a novel multi-scale method for the prediction of diffusivity in polymer matrices which combines classical and CG molecular simulations. We applied an atomistic-based method in order to parameterize the CG MARTINI force field, providing an extension for the study of diffusion behavior of penetrant molecules in polymer matrices. As a case study, we found the parameters for benzene (as medium sized penetrant molecule whose diffusivity cannot be determined through atomistic models) and Poly (vinyl alcohol) (PVA) as polymer matrix. We validated our extended MARTINI force field determining the self diffusion coefficient of benzene (2.27·10⁻⁹ m² s⁻¹) and the diffusion coefficient of benzene in PVA (0.263·10⁻¹² m² s⁻¹). The obtained diffusion coefficients are in remarkable agreement with experimental data (2.20·10⁻⁹ m² s⁻¹ and 0.25·10⁻¹² m² s⁻¹, respectively). We believe that this method can extend the application range of computational modeling, providing modeling tools to study the diffusion of larger molecules and complex polymeric materials.     

Velocity-dependent mechanical unfolding of bacteriorhodopsin is governed by a dynamic interaction network

Bacteriorhodopsin is a model system for membrane proteins. This seven transmembrane helical protein is embedded within a membrane structure called purple membrane. Its structural stability against mechanical stress was recently investigated by atomic force microscopy experiments, in which single proteins were extracted from the purple membrane. Here, we study this process by all-atom molecular dynamics simulations, in which single bacteriorhodopsin molecules were extracted and unfolded from an atomistic purple membrane model. In our simulations, key features from the experiments like force profiles and location of key residues that resist mechanical unfolding were reproduced. These key residues were seen to be stabilized by a dynamic network of intramolecular interactions. Further, the unfolding pathway was found to be velocity-dependent. Simulations in which the mechanical stress was released during unfolding revealed relaxation motions that allowed characterization of the nonequilibrium processes during fast extraction.     

Behaviour of small solutes and large drugs in a lipid bilayer from computer simulations

To reach their biological target, drugs have to cross cell membranes, and understanding passive membrane permeation is therefore crucial for rational drug design. Molecular dynamics simulations offer a powerful way of studying permeation at the single molecule level. Starting from a computer model proven to be able to reproduce the physical properties of a biological membrane, the behaviour of small solutes and large drugs in a lipid bilayer has been studied. Analysis of dihedral angles shows that a few nano seconds are sufficient for the simulations to converge towards common values for those angles, even if the starting structures belong to different conformations. Results clearly show that, despite their difference in size, small solutes and large drugs tend to lie parallel to the bilayer normal and that, when moving from water solution into biomembranes, permeants lose degrees of freedom. This explains the experimental observation that partitioning and permeation are highly affected by entropic effects and are size-dependent. Tilted orientations, however, occur when they make possible the formation of hydrogen bonds. This helps to understand the reason why hydrogen bonding possibilities are an important parameter in cruder approaches which predict drug absorption after administration. Interestingly, hydration is found to occur even in the membrane core, which is usually considered an almost hydrophobic region. Simulations suggest the possibility for highly polar compounds like acetic acid to cross biological membranes while hydrated. These simulations prove useful for drug design in rationalising experimental observations and predicting solute behaviour in biomembranes.