Modeling kinetics and equilibrium of membranes with fields: milestoning analysis and implication to permeation

A methodology of quantifying membrane permeability based on returning probability theory and molecular dynamics simulation

We propose a theoretical approach to estimate the permeability coefficients of substrates (permeants) for crossing membranes from donor (D) phase to acceptor (A) phase by means of molecular dynamics (MD) simulation. A fundamental aspect of our approach involves reformulating the returning probability (RP) theory, a rigorous bimolecular reaction theory, to describe permeation phenomena. This reformulation relies on the parallelism between permeation and bimolecular reaction processes. In the present method, the permeability coefficient is represented in terms of the thermodynamic and kinetic quantities for the reactive (R) phase that exists within the inner region of a membrane. One can evaluate these quantities using multiple MD trajectories starting from phase R. We apply the RP theory to the permeation of ethanol and methylamine at different concentrations (infinitely dilute and 1 mol % conditions of permeants). Under the 1 mol% condition, the present method yields a larger permeability coefficient for ethanol (0.12 ± 0.01 cm s-1) than for methylamine (0.069 ± 0.006 cm s-1), while the values of the permeability coefficient are satisfactorily close to those obtained from the brute-force MD simulations (0.18 ± 0.03 and 0.052 ± 0.005 cm s-1 for ethanol and methylamine, respectively). Moreover, upon analyzing the thermodynamic and kinetic contributions to the permeability, we clarify that a higher concentration dependency of permeability for ethanol, as compared to methylamine, arises from the sensitive nature of ethanol's free-energy barrier within the inner region of the membrane against ethanol concentration.

Predictions from First-Principles of Membrane Permeability to Small Molecules: How Useful Are They in Practice?

Predicting from first-principles the rate of passive permeation of small molecules across the biological membrane represents a promising strategy for screening lead compounds upstream in the drug-discovery and development pipeline. One popular avenue for the estimation of permeation rates rests on computer simulations in conjunction with the inhomogeneous solubility-diffusion model, which requires the determination of the free-energy change and position-dependent diffusivity of the substrate along the translocation pathway through the lipid bilayer. In this Perspective, we will clarify the physical meaning of the membrane permeability inferred from such computer simulations, and how theoretical predictions actually relate to what is commonly measured experimentally. We will also examine why these calculations remain both technically challenging and overly computationally expensive, which has hitherto precluded their routine use in nonacademic settings. We finally synopsize possible research directions to meet these challenges, increase the predictive power of physics-based rates of passive permeation, and, by ricochet, improve their practical usefulness.

Free-Energy Profiles for Membrane Permeation of Compounds Calculated Using Rare-Event Sampling Methods

The free-energy profile of a compound is an essential measurement in evaluating the membrane permeation process by means of theoretical methods. Computationally, molecular dynamics (MD) simulation allows the free-energy profile calculation. However, MD simulations frequently fail to sample membrane permeation because they are rare events induced in longer timescales than the accessible timescale of MD, leading to an insufficient conformational search to calculate an incorrect free-energy profile. To achieve a sufficient conformational search, several enhanced sampling methods have been developed and elucidated the membrane permeation process. In addition to these enhanced sampling methods, we proposed a simple yet powerful free-energy calculation of a compound for the membrane permeation process based on originally rare-event sampling methods developed by us. Our methods have a weak dependency on external biases and their optimizations to promote the membrane permeation process. Based on distributed computing, our methods only require the selection of initial structures and their conformational resampling, whereas the enhanced sampling methods may be required to adjust external biases. Furthermore, our methods efficiently search membrane permeation processes with simple scripts without modifying any MD program. As demonstrations, we calculated the free-energy profiles of seven linear compounds for their membrane permeation based on a hybrid conformational search using two rare-event sampling methods, that is, (1) parallel cascade selection MD (PaCS-MD) and (2) outlier flooding method (OFLOOD), combined with a Markov state model (MSM) construction. In the first step, PaCS-MD generated initial membrane permeation paths of a compound. In the second step, OFLOOD expanded the unsearched conformational area around the initial paths, allowing for a broad conformational search. Finally, the trajectories were employed to construct reliable MSMs, enabling correct free-energy profile calculations. Furthermore, we estimated the membrane permeability coefficients of all compounds by constructing the reliable MSMs for their membrane permeation. In conclusion, the calculated coefficients were qualitatively correlated with the experimental measurements (correlation coefficient (R²) = 0.8689), indicating that the hybrid conformational search successfully calculated the free-energy profiles and membrane permeability coefficients of the seven compounds.

Mechanistic Insights into Passive Membrane Permeability of Drug-like Molecules from a Weighted Ensemble of Trajectories

Passive permeability of a drug-like molecule is a critical property assayed early in a drug discovery campaign that informs a medicinal chemist how well a compound can traverse biological membranes, such as gastrointestinal epithelial or restrictive organ barriers, so it can perform a specific therapeutic function. However, the challenge that remains is the development of a method, experimental or computational, which can both determine the permeation rate and provide mechanistic insights into the transport process to help with the rational design of any given molecule. Typically, one of the following three methods are used to measure the membrane permeability: (1) experimental permeation assays acting on either artificial or natural membranes; (2) quantitative structure–permeability relationship models that rely on experimental values of permeability or related pharmacokinetic properties of a range of molecules to infer those for new molecules; and (3) estimation of permeability from the Smoluchowski equation, where free energy and diffusion profiles along the membrane normal are taken as input from large-scale molecular dynamics simulations. While all these methods provide estimates of permeation coefficients, they provide very little information for guiding rational drug design. In this study, we employ a highly parallelizable weighted ensemble (WE) path sampling strategy, empowered by cloud computing techniques, to generate unbiased permeation pathways and permeability coefficients for a set of drug-like molecules across a neat 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine membrane bilayer. Our WE method predicts permeability coefficients that compare well to experimental values from an MDCK-LE cell line and PAMPA assays for a set of drug-like amines of varying size, shape, and flexibility. Our method also yields a series of continuous permeation pathways weighted and ranked by their associated probabilities. Taken together, the ensemble of reactive permeation pathways, along with the estimate of the permeability coefficient, provides a clearer picture of the microscopic underpinnings of small-molecule membrane permeation.

Dissecting Multiple Pathways in the Relaxation Dynamics of Helix <==> Coil Transitions with Optimum Dimensionality Reduction

Fast kinetic experiments with dramatically improved time resolution have contributed significantly to understanding the fundamental processes in protein folding pathways involving the formation of a-helices and b-hairpin, contact formation, and overall collapse of the peptide chain. Interpretation of experimental results through application of a simple statistical mechanical model was key to this understanding. Atomistic description of all events observed in the experimental findings was challenging. Recent advancements in theory, more sophisticated algorithms, and a true long-term trajectory made way for an atomically detailed description of kinetics, examining folding pathways, validating experimental results, and reporting new findings for a wide range of molecular processes in biophysical chemistry. This review describes how optimum dimensionality reduction theory can construct a simplified coarse-grained model with low dimensionality involving a kinetic matrix that captures novel insights into folding pathways. A set of metastable states derived from molecular dynamics analysis generate an optimally reduced dimensionality rate matrix following transition pathway analysis. Analysis of the actual long-term simulation trajectory extracts a relaxation time directly comparable to the experimental results and confirms the validity of the combined approach. The application of the theory is discussed and illustrated using several examples of helix <==> coil transition pathways. This paper focuses primarily on a combined approach of time-resolved experiments and long-term molecular dynamics simulation from our ongoing work.

Milestoning: An Efficient Approach for Atomically Detailed Simulations of Kinetics in Biophysics

Recent advances in theory and algorithms for atomically detailed simulations open the way to the study of the kinetics of a wide range of molecular processes in biophysics. The theories propose a shift from the traditionally very long molecular dynamic trajectories, which are exact but may not be efficient in the study of kinetics, to the use of a large number of short trajectories. The short trajectories exploit a mapping to a mesh in coarse space and allow for efficient calculations of kinetics and thermodynamics. In this review, I focus on one theory: Milestoning is a theory and an algorithm that offers a hierarchical calculation of properties of interest, such as the free energy profile and the mean first passage time. Approximations to the true long-time dynamics can be computed efficiently and assessed at different steps of the investigation. The theory is discussed and illustrated using two biophysical examples: ion permeation through a phospholipid membrane and protein translocation through a channel.

Unassisted N-acetyl-phenylalanine-amide transport across membrane with varying lipid size and composition: kinetic measurements and atomistic molecular dynamics simulation

Biological membranes are essential to preserve structural integrity and regulate functional properties through the permeability of nutrients, pharmaceutical drugs, and neurotransmitters of a living cell. The movement of acetylated and amidated phenylalanine (NAFA) across 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) membrane bilayers is investigated to probe physical transport. The rate of transport is measured experimentally applying parallel artificial membrane permeation assay (PAMPA). At the physiological temperature, 310 K, the measured time constants in the neutral pH were ∼6 h in DOPC and ∼3 h in POPC, while in a more acidic condition, at a pH 4.8, the time constants were ∼8 h in both lipids. Computationally, we have expanded our transport study of three aromatic dipeptides across a bilayer composed of DOPC¹⁸. In this study, we have examined the effects of lipid composition and bilayer size on the passive transport of NAFA by simulating the dipeptide in three different bilayers, with 50 DOPC lipids, 50 POPC lipids, and 40 POPC molecules. Specifically, atomistic molecular dynamics simulations with umbrella sampling were used to calculate the potential of mean force for the passive permeation of NAFA across the bilayers. Diffusion constants were then calculated by numerically solving the Smoluchowski equation. Permeability coefficients and mean first passage times were then calculated. Structural properties - Ramachandran plots, sidechain torsions, peptide insertion angles, radial distribution functions, and proximal peptide water molecules - were also examined to determine the effect of system size and lipid type. In terms of systems size, we observed a small decrease in the highest barrier of the potential of mean force and fewer sampled sidechain dihedral angle conformations with 40 versus 50 POPC lipids due to weaker membrane deformations within a smaller lipid bilayer. In terms of lipid type, DOPC contains two monounsaturated acyl chains compared to only one such acyl chain in POPC; therefore, DOPC bilayers are less ordered and more easily deformed, as seen by a much broader potential of mean force profile. The NAFA in DOPC lipid also transitioned to an internally hydrogen-bonded backbone conformation at lower membrane depths than in POPC. Similarly, as for other aromatic dipeptides, NAFA tends to insert into the membrane sidechain-first, remains mostly desolvated in the membrane center, and exhibits slow reorientations within the bilayer in both DOPC and POPC. With a joint experimental and computational study we have gained a new insight into the rate of transport and the underlying microscopic mechanism in different lipid bilayer conditions of the simplest hydrophobic aromatic dipeptide.Communicated by Ramaswamy H. Sarma.

Exploring the Reaction Mechanism of HIV Reverse Transcriptase with a Nucleotide Substrate

Enzymatic reactions consist of several steps: (i) a weak binding event of the substrate to the enzyme, (ii) an induced fit or a protein conformational transition upon ligand binding, (iii) the chemical reaction, and (iv) the release of the product. Here we focus on step iii of the reaction of a DNA polymerase, HIV RT, with a nucleotide. We determine the rate and the free energy profile for the addition of a nucleotide to a DNA strand using a combination of a QM/MM model, the string method, and exact Milestoning. The barrier height and the time scale of the reaction are consistent with experiment. We show that the observables (free energies and mean first passage time) converge rapidly, as a function of the Milestoning iteration number. We also consider the substitution of an oxygen of the incoming nucleotide by a nonbridging sulfur atom and its impact on the enzymatic reaction. This substitution has been suggested in the past as a tool to examine the influence of the chemical step on the overall rate. Our joint computational and experimental study suggests that the impact of the substitution is small. Computationally, the differences between the two are within the estimated error bars. Experiments suggest a small difference. Finally, we examine step i, the weak binding of the nucleotide to the protein surface. We suggest that this step has only a small contribution to the selectivity of the enzyme. Comments are made on the impact of these steps on the overall mechanism.

One-Step Assembly of TiO2-Liposomes Based on Interfacial Sol-Gel Process within Lipid Bilayer

There is a growing interest in the use of hybrid liposomes for various biochemical and biomedical applications. In this study, we report the first preparation and characterization of a class of TiO₂-reinforced liposomes by a one-step assembly approach. The amphiphilic natural structure of lipids is exploited to localize a hydrophobic molecule, namely, precursor tetrabutyl titanate (TBOT), in the mid-plane of the liposomal bilayer assemblies in the aqueous phase. In situ TiO₂ nanoshell formation is driven by subsequent interfacial hydrolysis of TBOT and the ensuing condensation within the hydrophobic interstices of the lipid bilayer. The core-shell structure, like cell and wall, is demonstrated by means of scanning electron microscopy and transmission electron microscopy images, and the formation of the TiO₂ shell is confirmed using energy-dispersive X-ray spectroscopy, thermogravimetric analysis, and differential scanning calorimetry. To study the structural evolution of the hybrid liposomes during titania formation, fluorescence probe technique and surface pressure versus molecular area (π- A) isotherms are designed. The results demonstrate that the incorporation of TBOT into the mid-membrane of the lipid and titania in the core of the membrane strengthened the assembly of the lipid bilayer. We further demonstrate that titania shell improved the stability and release property of liposomes. We expect that the reported new TiO₂-coated liposomes by co-assembly will be valuable in designing hybrid liposomes, exhibiting integrative capacity for drug encapsulation, compartment reaction, and photocatalysis.

Molecular Dynamics Simulations of Membrane Permeability

This Review illustrates the evaluation of permeability of lipid membranes from molecular dynamics (MD) simulation primarily using water and oxygen as examples. Membrane entrance, translocation, and exit of these simple permeants (one hydrophilic and one hydrophobic) can be simulated by conventional MD, and permeabilities can be evaluated directly by Fick's First Law, transition rates, and a global Bayesian analysis of the inhomogeneous solubility-diffusion model. The assorted results, many of which are applicable to simulations of nonbiological membranes, highlight the limitations of the homogeneous solubility diffusion model; support the utility of inhomogeneous solubility diffusion and compartmental models; underscore the need for comparison with experiment for both simple solvent systems (such as water/hexadecane) and well-characterized membranes; and demonstrate the need for microsecond simulations for even simple permeants like water and oxygen. Undulations, subdiffusion, fractional viscosity dependence, periodic boundary conditions, and recent developments in the field are also discussed. Last, while enhanced sampling methods and increasingly sophisticated treatments of diffusion add substantially to the repertoire of simulation-based approaches, they do not address directly the critical need for force fields with polarizability and multipoles, and constant pH methods.

Partition of Positively and Negatively Charged Tryptophan Ions in Membranes with Inverted Phospholipid Heads: Simulations and Experiments

A joint experimental and computational study illustrates that the partitioning of positively and negatively charged tryptophan in a phospholipid bilayer is significantly altered by a reversal in the head group dipole arrangement. Experiments were conducted using tryptophan as a fluorescent reporter of its local environment. Based on the experimental design in a recent publication ( Anderson , C. M. ; Cardenas , A. ; Elber , R. ; Webb , L. J. J. Phys. Chem. B 2018 , 123 , 170 - 179 ), we were able to determine that the arrangement of the head group dipole altered the degree of partitioning of charged tryptophan in the lipid bilayer. In parallel, atomically detailed simulations were performed for the two membrane systems. The simulation results are in accord with the experimental findings and support a simple molecular partition mechanism of electrostatic interactions with the head groups, glycerol linkers, and interfacial water dipoles.

Link between Membrane Composition and Permeability to Drugs

Prediction of membrane permeability to small molecules represents an important aspect of drug discovery. First-principles calculations of this quantity require an accurate description of both the thermodynamics and kinetics that underlie translocation of the permeant across the lipid bilayer. In this contribution, the membrane permeability to three drugs, or drug-like molecules, namely, 9-anthroic acid (ANA), 2',3'-dideoxyadenosine (DDA), and hydrocortisone (HYL), are estimated in a pure 1-palmitoyl-2-oleoylphosphatidylcholine (POPC) and in a POPC:cholesterol (2:1) mixture. On the basis of independent 2-5-μs free-energy calculations combined with a time-fractional Smoluchowski determination of the diffusivity, the estimated membrane permeabilities to these chemically diverse permeants fall within an order of magnitude from the experimental values obtained in egg-lecithin bilayers, with the exception of HYL in pure POPC. This exception is particularly interesting because the calculated permeability of the sterol-rich bilayer to HYL, in close agreement with the experimental value, is about 600 times lower than that of the pure POPC bilayer to HYL. In contrast, the permeabilities to ANA and DDA differ by less than a factor of 10 between the pure POPC and POPC:cholesterol bilayers. The unusual behavior of HYL, a large, amphiphilic compound, may be linked with the longer range perturbation of the lipid bilayer it induces, compared to ANA and DDA, suggestive of a possibly different translocation mechanism. We find that the tendency of lower permeabilities of the POPC:cholesterol bilayer relative to those of the pure POPC one is a consequence of increased free-energy barriers. Beyond reporting accurate estimates of the membrane permeability, the present contribution also demonstrates that rigorous free-energy calculations and a fractional-diffusion model are key in revealing the molecular phenomena linking the composition of a membrane to its permeability to drugs.

A new paradigm for atomically detailed simulations of kinetics in biophysical systems

The kinetics of biochemical and biophysical events determined the course of life processes and attracted considerable interest and research. For example, modeling of biological networks and cellular responses relies on the availability of information on rate coefficients. Atomically detailed simulations hold the promise of supplementing experimental data to obtain a more complete kinetic picture. However, simulations at biological time scales are challenging. Typical computer resources are insufficient to provide the ensemble of trajectories at the correct length that is required for straightforward calculations of time scales. In the last years, new technologies emerged that make atomically detailed simulations of rate coefficients possible. Instead of computing complete trajectories from reactants to products, these approaches launch a large number of short trajectories at different positions. Since the trajectories are short, they are computed trivially in parallel on modern computer architecture. The starting and termination positions of the short trajectories are chosen, following statistical mechanics theory, to enhance efficiency. These trajectories are analyzed. The analysis produces accurate estimates of time scales as long as hours. The theory of Milestoning that exploits the use of short trajectories is discussed, and several applications are described.

Permeability of a Fluid Lipid Bilayer to Short-Chain Alcohols from First Principles

Computational prediction of membrane permeability to small molecules requires accurate description of both the thermodynamics and kinetics underlying translocation across the lipid bilayer. In this contribution, well-converged, microsecond-long free-energy calculations are combined with a recently developed subdiffusive kinetics framework to describe the membrane permeation of a homologous series of short-tail alcohols, from methanol to 1-butanol, with unprecedented fidelity to the underlying molecular models. While the free-energy profiles exhibit barriers for passage through the center of the bilayer in all cases, the height of these barriers decreases with the length of the aliphatic chain of the alcohol, in quantitative agreement with experimentally determined differential solvation free energies in water and oil. A unique aspect of the subdiffusive model employed herein, which was developed in a previous article, is the determination of a position-dependent fractional order which quantifies the degree to which the motion of the alcohol deviates from classical diffusion along the thickness of the membrane. In the aqueous medium far from the bilayer, this quantity approaches 1.0, the asymptotic limit for purely classical diffusion, whereas it dips below 0.75 near the center of the membrane irrespective of the permeant. Remarkably, the fractional diffusivity near the center of membrane, where its influence on the permeability is the greatest, is similar among the four permeants despite the large difference in molecular weight and lipophilicity between methanol and 1-butanol. The relative permeabilities, which are estimated from the free-energy and fractional diffusivity profiles, are therefore determined predominantly by differences in the former rather than the latter. The predicted relative permeabilities are highly correlated with existing experimental results, albeit they do not agree quantitatively with them. On the other hand, quite unexpectedly, the reported experimental values for the short-tail alcohols are nearly three orders of magnitude lower than the available experimental measurement for water. Plausible explanations for this apparent disagreement between theory and experiment are considered in detail.

Subdiffusion in Membrane Permeation of Small Molecules

Within the solubility-diffusion model of passive membrane permeation of small molecules, translocation of the permeant across the biological membrane is traditionally assumed to obey the Smoluchowski diffusion equation, which is germane for classical diffusion on an inhomogeneous free-energy and diffusivity landscape. This equation, however, cannot accommodate subdiffusive regimes, which have long been recognized in lipid bilayer dynamics, notably in the lateral diffusion of individual lipids. Through extensive biased and unbiased molecular dynamics simulations, we show that one-dimensional translocation of methanol across a pure lipid membrane remains subdiffusive on timescales approaching typical permeation times. Analysis of permeant motion within the lipid bilayer reveals that, in the absence of a net force, the mean squared displacement depends on time as t0.7, in stark contrast with the conventional model, which assumes a strictly linear dependence. We further show that an alternate model using a fractional-derivative generalization of the Smoluchowski equation provides a rigorous framework for describing the motion of the permeant molecule on the pico- to nanosecond timescale. The observed subdiffusive behavior appears to emerge from a crossover between small-scale rattling of the permeant around its present position in the membrane and larger-scale displacements precipitated by the formation of transient voids.

Perspective: Computer simulations of long time dynamics

Atomically detailed computer simulations of complex molecular events attracted the imagination of many researchers in the field as providing comprehensive information on chemical, biological, and physical processes. However, one of the greatest limitations of these simulations is of time scales. The physical time scales accessible to straightforward simulations are too short to address many interesting and important molecular events. In the last decade significant advances were made in different directions (theory, software, and hardware) that significantly expand the capabilities and accuracies of these techniques. This perspective describes and critically examines some of these advances.

Simulating Current-Voltage Relationships for a Narrow Ion Channel Using the Weighted Ensemble Method

Ion channels are responsible for a myriad of fundamental biological processes via their role in controlling the flow of ions through water-filled membrane-spanning pores in response to environmental cues. Molecular simulation has played an important role in elucidating the mechanism of ion conduction, but connecting atomistically detailed structural models of the protein to electrophysiological measurements remains a broad challenge due to the computational cost of reaching the necessary time scales. Here, we introduce an enhanced sampling method for simulating the conduction properties of narrow ion channels using the Weighted ensemble (WE) sampling approach. We demonstrate the application of this method to calculate the current–voltage relationship as well as the nonequilibrium ion distribution at steady-state of a simple model ion channel. By direct comparisons with long brute force simulations, we show that the WE simulations rigorously reproduce the correct long-time scale kinetics of the system and are capable of determining these quantities using significantly less aggregate simulation time under conditions where permeation events are rare.

Two Relations to Estimate Membrane Permeability Using Milestoning

Prediction of passive permeation rates of solutes across lipid bilayers is important to drug design, toxicology, and other biological processes such as signaling. The inhomogeneous solubility-diffusion (ISD) equation is traditionally used to relate the position-dependent potential of mean force and diffusivity to the permeability coefficient. The ISD equation is derived via the Smoluchowski equation and assumes overdamped system dynamics. It has been suggested that the complex membrane environment may exhibit more complicated damping conditions. Here we derive a variant of the inhomogeneous solubility diffusion equation as a function of the mean first passage time (MFPT) and show how milestoning, a method that can estimate kinetic quantities of interest, can be used to estimate the MFPT of membrane crossing and, by extension, the permeability coefficient. We further describe a second scheme, agnostic to the damping condition, to estimate the permeability coefficient from milestoning results or other methods that compute a probability of membrane crossing. The derived relationships are tested using a one-dimensional Langevin dynamics toy system confirming that the presented theoretical methods can be used to estimate permeabilities given simulation and milestoning results.

Simulation-Based Approaches for Determining Membrane Permeability of Small Compounds

Predicting the rate of nonfacilitated permeation of solutes across lipid bilayers is important to drug design, toxicology, and signaling. These rates can be estimated using molecular dynamics simulations combined with the inhomogeneous solubility-diffusion model, which requires calculation of the potential of mean force and position-dependent diffusivity of the solute along the transmembrane axis. In this paper, we assess the efficiency and accuracy of several methods for the calculation of the permeability of a model DMPC bilayer to urea, benzoic acid, and codeine. We compare umbrella sampling, replica exchange umbrella sampling, adaptive biasing force, and multiple-walker adaptive biasing force for the calculation of the transmembrane PMF. No definitive advantage for any of these methods in their ability to predict the membrane permeability coefficient Pm was found, provided that a sufficiently long equilibration is performed. For diffusivities, a Bayesian inference method was compared to a generalized Langevin method, both being sensitive to chosen parameters and the slow relaxation of membrane defects. Agreement within 1.5 log units of the computed Pm with experiment is found for all permeants and methods. Remaining discrepancies can likely be attributed to limitations of the force field as well as slowly relaxing collective movements within the lipid environment. Numerical calculations based on model profiles show that Pm can be reliably estimated from only a few data points, leading to recommendations for calculating Pm from simulations.

Permeability across lipid membranes

Molecular permeation through lipid membranes is a fundamental biological process that is important for small neutral molecules and drug molecules. Precise characterization of free energy surface and diffusion coefficients along the permeation pathway is required in order to predict molecular permeability and elucidate the molecular mechanisms of permeation. Several recent technical developments, including improved molecular models and efficient sampling schemes, are illustrated in this review. For larger penetrants, explicit consideration of multiple collective variables, including orientational, conformational degrees of freedom, are required to be considered in addition to the distance from the membrane center along the membrane normal. Although computationally demanding, this method can provide significant insights into the molecular mechanisms of permeation for molecules of medical and pharmaceutical importance. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

Markovian and Non-Markovian Modeling of Membrane Dynamics with Milestoning

We exploit atomically detailed simulations and the milestoning theory to extract coarse grained models of membrane kinetics and thermodynamics. Non-Markovian and Markovian theories for the phosphate group displacements are used to coarsely represent membrane motions. The construction of the two models makes it possible to examine their consistency and accuracy. The equilibrium and fluctuations of the phosphate groups along the normal to the membrane plane are estimated, and milestoning equations are constructed and solved. An optimal Markovian model is constructed that reproduces exactly the equilibrium and mean first passage time (MFPT) of the non-Markovian model. The equilibrium solution of both models is favorably compared to distributions obtained from straightforward molecular dynamics simulations. The picture for the kinetics is complex. Multiple local relaxation times of the mass density are illustrated emphasizing the non-Markovian characteristics of the process. In Markovian modeling, only a single relaxation time is assumed for a state. Mapping of particle dynamics to the dynamics of a field density offers a new way of coarse graining complex systems as membranes that may bridge between atomically detailed models and phenomenological descriptions of macroscopic membranes.

A Stochastic Algorithm for the Isobaric-Isothermal Ensemble with Ewald Summations for All Long Range Forces

We present an algorithm termed COMPEL (COnstant Molecular Pressure with Ewald sum for Long range forces) to conduct simulations in the NPT ensemble. The algorithm combines novel features recently proposed in the literature to obtain a highly efficient and accurate numerical integrator. COMPEL exploits the concepts of molecular pressure, rapid stochastic relaxation to equilibrium, exact calculation of the contribution to the pressure of long-range nonbonded forces with Ewald summation, and the use of Trotter expansion to generate a robust, highly stable, symmetric, and accurate algorithm. Explicit implementation in the MOIL program and illustrative numerical examples are discussed.

Exact milestoning

A new theory and an exact computer algorithm for calculating kinetics and thermodynamic properties of a particle system are described. The algorithm avoids trapping in metastable states, which are typical challenges for Molecular Dynamics (MD) simulations on rough energy landscapes. It is based on the division of the full space into Voronoi cells. Prior knowledge or coarse sampling of space points provides the centers of the Voronoi cells. Short time trajectories are computed between the boundaries of the cells that we call milestones and are used to determine fluxes at the milestones. The flux function, an essential component of the new theory, provides a complete description of the statistical mechanics of the system at the resolution of the milestones. We illustrate the accuracy and efficiency of the exact Milestoning approach by comparing numerical results obtained on a model system using exact Milestoning with the results of long trajectories and with a solution of the corresponding Fokker-Planck equation. The theory uses an equation that resembles the approximate Milestoning method that was introduced in 2004 [A. K. Faradjian and R. Elber, J. Chem. Phys. 120(23), 10880-10889 (2004)]. However, the current formulation is exact and is still significantly more efficient than straightforward MD simulations on the system studied.

Simulations of thermodynamics and kinetics on rough energy landscapes with milestoning

We investigated by computational means the kinetics and stationary behavior of stochastic dynamics on an ensemble of rough two-dimensional energy landscapes. There are no obvious separations of temporal scales in these systems, which constitute a simple model for the behavior of glasses and some biomaterials. Even though there are significant computational challenges present in these systems due to the large number of metastable states, the Milestoning method is able to compute their kinetic and thermodynamic properties exactly. We observe two clearly distinguished regimes in the overall kinetics: one in which diffusive behavior dominates and another that follows an Arrhenius law (despite the absence of a dominant barrier). We compare our results with those obtained with an exactly-solvable one-dimensional model, and with the results from the rough one-dimensional energy model introduced by Zwanzig. © 2015 Wiley Periodicals, Inc.