A multiscale model for kinetics of formation and disintegration of spherical micelles

Micellar Composition Affects Lipid Accretion Kinetics in Molecular Dynamics Simulations: Support for Lipid Network Reproduction

Mixed lipid micelles were proposed to facilitate life through their documented growth dynamics and catalytic properties. Our previous research predicted that micellar self-reproduction involves catalyzed accretion of lipid molecules by the residing lipids, leading to compositional homeostasis. Here, we employ atomistic Molecular Dynamics simulations, beginning with 54 lipid monomers, tracking an entire course of micellar accretion. This was done to examine the self-assembly of variegated lipid clusters, allowing us to measure entry and exit rates of monomeric lipids into pre-micelles with different compositions and sizes. We observe considerable rate-modifications that depend on the assembly composition and scrutinize the underlying mechanisms as well as the energy contributions. Lastly, we describe the measured potential for compositional homeostasis in our simulated mixed micelles. This affirms the basis for micellar self-reproduction, with implications for the study of the origin of life.

Free energy of micellization of dodecyl phosphocholine (DPC) from molecular simulation: Hybrid PEACH-BAR method

A new method to extract the free energy of aggregation versus aggregate size from molecular simulation data is proposed and applied to a united atom model of the zwitterionic surfactant dodecyl phosphocholine in water. This system's slow dissociation rate and low critical micelle concentration (CMC of approximately 1-2 mM) make extraction of cluster free energies directly from simulation results using the "partition-enabled analysis of cluster histogram" (PEACH) method impractical. The new approach applies PEACH to a model with weakened attractions between aggregants, which allows sampling of a continuous range of cluster sizes, then recovers the free energy of aggregation under the original fully-attractive force field using the BAR free energy difference method. PEACH-BAR results are compared with free energy differences calculated via umbrella sampling, and are used to make predictions of CMC, average cluster size, and SAXS scattering profiles that are in fair agreement with experiment.

Aerosol-OT Surfactant Forms Stable Reverse Micelles in Apolar Solvent in the Absence of Water

Normal micelle aggregates of amphiphilic surfactant in aqueous solvents are formed by a process of entropically driven self-assembly. The self-assembly of reverse micelles from amphiphilic surfactant in a nonpolar solvent in the presence of water is considered to be an enthalpically driven process. Although the formation of normal and reverse surfactant micelles has been well characterized in theory and experiment, the nature of dry micelle formation, from amphiphilic surfactant in a nonpolar solvent in the absence of water, is poorly understood. In this study, a theory of dry reverse micelle formation is developed. Variation in free energy during micelle assembly is derived for the specific case of aerosol-OT surfactant in isooctane solvent using atomistic molecular dynamics simulation analyzed using the energy representation method. The existence and thermodynamic stability of dry reverse micelles of limited size are confirmed. The abrupt occurrence of monodisperse aggregates is a clear signature of a critical micelle concentration, commonly observed in the formation of normal surfactant micelles. The morphology of large dry micelles provides insight into the nature of the thermodynamic driving forces stabilizing the formation of the surfactant aggregates. Overall, this study provides detailed insight into the structure and stability of dry reverse micelles assembly in a nonpolar solvent.

Molecular Modeling of Surfactant Micellization Using Solvent-Accessible Surface Area

We report a new implicit solvent simulation model for studying the self-assembly of surfactants, where the hydrophobic interactions were captured by calculating the relative changes of the solvent-accessible surface area (SASA) of the hydrophobic domains. Using histogram-reweighting grand canonical Monte Carlo simulations, we demonstrate that this approach allows us to match both the experimental critical micelle concentrations (cmc) and micellar aggregation numbers simultaneously with a single phenomenological surface tension γ_SASA for the poly(oxyethylene) monoalkyl ether (C _mE _n) surfactants in aqueous solutions. Excellent transferability is observed: the same model can accurately predict the experimental cmc and aggregation numbers for the C _mE _n surfactants with the alkyl lengths m between 6 and 12 and the poly(oxyethylene) lengths n between 1 and 9. The SASA-based implicit solvent model put forward in this work is general and may be applied to study more complex amphiphilic systems such as surfactants with branched alkyl chains or surfactant-hydrocarbon mixtures.

Extracting Aggregation Free Energies of Mixed Clusters from Simulations of Small Systems: Application to Ionic Surfactant Micelles

Micelle cluster distributions from molecular dynamics simulations of a solvent-free coarse-grained model of sodium octyl sulfate (SOS) were analyzed using an improved method to extract equilibrium association constants from small-system simulations containing one or two micelle clusters at equilibrium with free surfactants and counterions. The statistical-thermodynamic and mathematical foundations of this partition-enabled analysis of cluster histograms (PEACH) approach are presented. A dramatic reduction in computational time for analysis was achieved through a strategy similar to the selector variable method to circumvent the need for exhaustive enumeration of the possible partitions of surfactants and counterions into clusters. Using statistics from a set of small-system (up to 60 SOS molecules) simulations as input, equilibrium association constants for micelle clusters were obtained as a function of both number of surfactants and number of associated counterions through a global fitting procedure. The resulting free energies were able to accurately predict micelle size and charge distributions in a large (560 molecule) system. The evolution of micelle size and charge with SOS concentration as predicted by the PEACH-derived free energies and by a phenomenological four-parameter model fit, along with the sensitivity of these predictions to variations in cluster definitions, are analyzed and discussed.

Logarithmic Exchange Kinetics in Monodisperse Copolymeric Micelles

Experimental measurements of the relaxation kinetics of copolymeric surfactant exchange for micellar systems unexpectedly show a peculiar logarithmic decay. Several authors use polydispersity as an explanation for this behavior. However, in coarse-grained simulations that preserve microscopic details of the surfactants, we find evidence of the same logarithmic behavior. Since we use a strictly monodisperse distribution of chain lengths such a relaxation process cannot be attributed to polydispersity, but has to be caused by an inherent physical process characteristic of this type of system. This is supported by the fact that the decay is specifically logarithmic and not a power law with an exponent inherited from the particular polydispersity distribution of the sample. We suggest that the degeneracy of the energy states of the hydrophobic block in the core, which is broken on leaving the micelle, can qualitatively explain the broad distribution of energy barriers, which gives rise to the observed nonexponential relaxation.

Micelle response to changes in solvent properties

The dynamics of co-polymer systems play an important role in the preparation and stability of formulations, as well as on their capability to function in drug delivery systems. Micelle inversion can occur as a result of a change in concentration when a solvent is very volatile and evaporates, or as a result of a change in solvent quality upon addition of another solvent to the original solution, or upon changes in pH. In this work, dissipative particle dynamics (DPD) is used to examine the dynamics of micelle inversion in concentrated systems of diblock and triblock amphiphiles, where interactions between neighboring aggregates are observed. Significant differences were observed in the inversion process of different amphiphilic molecules, with a large amount of co-polymer exchange between inverting aggregates made of diblock amphiphiles, and practically no exchange of molecules between aggregates during the inversion of triblock copolymer aggregates. Fundamental mechanisms of inversion are revealed that provide information which can be used to help design micelles for targeted drug release and allow understanding of history dependant formulations.

Kinetics of formation of bile salt micelles from coarse-grained Langevin dynamics simulations

We examine the mechanism of formation of micelles of dihydroxy bile salts using a coarse-grained, implicit solvent model and Langevin dynamics simulations. We find that bile salt micelles primarily form via addition and removal of monomers, similarly to surfactants with typical head-tail molecular structures, and not via a two-stage mechanism - involving formation of oligomers and their subsequent aggregation to form larger micelles - originally proposed for bile salts. The free energy barrier to removal of single bile monomers from micelles is ≈2kBT, much less than what has been observed for head-tail surfactants. Such a low barrier may be biologically relevant: it allows for rapid release of bile monomers into the intestine, possibly enabling the coverage of fat droplets by bile salt monomers and subsequent release of micelles containing fats and bile salts - a mechanism that is not possible for ionic head-tail surfactants of similar critical micellar concentrations.

Multi-scale times and modes of fast and slow relaxation in solutions with coexisting spherical and cylindrical micelles according to the difference Becker-Döring kinetic equations

The eigenvalues and eigenvectors of the matrix of coefficients of the linearized kinetic equations applied to aggregation in surfactant solution determine the full spectrum of characteristic times and specific modes of micellar relaxation. The dependence of these relaxation times and modes on the total surfactant concentration has been analyzed for concentrations in the vicinity and well above the second critical micelle concentration (cmc2) for systems with coexisting spherical and cylindrical micelles. The analysis has been done on the basis of a discrete form of the Becker-Döring kinetic equations employing the Smoluchowsky diffusion model for the attachment rates of surfactant monomers to surfactant aggregates with matching the rates for spherical aggregates and the rates for large cylindrical micelles. The equilibrium distribution of surfactant aggregates in solution has been modeled as having one maximum for monomers, another maximum for spherical micelles and wide slowly descending branch for cylindrical micelles. The results of computations have been compared with the analytical ones known in the limiting cases from solutions of the continuous Becker-Döring kinetic equation. They demonstrated a fair agreement even in the vicinity of the cmc2 where the analytical theory looses formally its applicability.

Kinetic modeling of self-aggregation in solutions with coexisting spherical and cylindrical micelles at arbitrary initial conditions

The whole picture of evolution of coexisting spherical and cylindrical micelles has been described for initial states far from equilibrium.

Direct observation of the formation of surfactant micelles under nonisothermal conditions by synchrotron SAXS

Self-assembly of amphiphilic molecules into micelles occurs on very short times scales of typically some milliseconds, and the structural evolution is therefore very challenging to observe experimentally. While rate constants of surfactant micelle kinetics have been accessed by spectroscopic techniques for decades, so far no experiments providing detailed information on the structural evolution of surfactant micelles during their formation process have been reported. In this work we show that by applying synchrotron small-angle X-ray scattering (SAXS) in combination with the stopped-flow mixing technique, the entire micelle formation process from single surfactants to equilibrium micelles can be followed in situ. Using a sugar-based surfactant system of dodecyl maltoside (DDM) in dimethylformamide (DMF), micelle formation can be induced simply by adding water, and this can be followed in situ by SAXS. Mixing of water and DMF is an exothermic process where the micelle formation process occurs under nonisothermal conditions with a temperature gradient relaxing from about 40 to 20 °C. A kinetic nucleation and growth mechanism model describing micelle formation by insertion/expulsion of single molecules under nonisothermal conditions was developed and shown to describe the data very well.

A Wang-Landau study of a lattice model for lipid bilayer self-assembly

The Wang-Landau (WL) Monte Carlo method has been applied to simulate the self-assembly of a lipid bilayer on a 3D lattice. The WL method differs from conventional Monte Carlo methods in that a complete density of states is obtained directly for the system, from which properties, such as the free energy, can be derived. Furthermore, from a single WL simulation, continuous curves of the average energy and heat capacity can be determined, which provide a complete picture of the phase behavior. The lipid model studied consists of 3 or 5 coarse-grained segments on lattices of varying sizes, with the empty lattice sites representing water. A bilayer structure is found to form at low temperatures, with phase transitions to clusters as temperature increases. For 3-segment chains, varying lattice sizes were studied, with the observation that the ratio of chain number to lattice area (i.e., area per lipid) affects the phase transition temperature. At small ratios, only one phase transition occurs between the bilayer and cluster phases, while at high lipid ratios the phase transition occurs in a two-step process with a stable intermediate phase. This second phase transition was not observed in conventional Metropolis Monte Carlo simulations on the same model, demonstrating the advantage of being able to perform a complete scan of the whole temperature range with the WL method. For longer 5-segment chains similar phase transitions are also observed with changes in temperature. In the WL method, due to the extensive nature of the energy, the number of energy bins required to represent the density of states increases as the system size increases and so limits its practical application to larger systems. To improve this, an extension of the WL algorithm, the statistical-temperature Monte Carlo method that allows simulations with larger energy bin sizes, has recently been proposed and is implemented in this work for the 3-segment lattice model. The results obtained are in good agreement with the original WL method and appear to be independent of the energy bin size used.

Molecular transport through surfactant-covered oil-water interfaces: role of physical properties of solutes and surfactants in creating energy barriers for transport

Mechanisms of molecular transport across oil-water interfaces covered by nonionic surfactants are investigated using coarse-grained molecular dynamics simulations. Resistance of the surfactant monolayer to the solute transport is shown to be controlled by dense regions in the monolayer. The dense regions are formed on both sides of the dividing surface and the barrier to the solute transport is created by those of them experiencing unfavorable interactions with the solute. Resistance to the transport of a hydrophobic (hydrophilic) solute increases with the excess density of the head (tail) group region of the monolayer, which in turn increases with the length of the surfactant head (tail) group. Barriers for solute transport through surfactant monolayers are also influenced by the solute size. However, the extent of this influence is determined by the monolayer thickness and the solute structure and composition. For example, it is shown that resistance offered by thin monolayers to transport of linear oligomers is relatively insensitive to the solute length. The barrier sensitivity to the length of these solutes increases with the monolayer thickness. In addition to the static barriers, the solute transport is shown to be affected by dynamic barriers due to a nonadiabatic coupling of the monolayer surface with the solute position and configuration. This coupling leads to deviations of the system dynamics from the minimum energy path. The deviations are most significant in the neighborhood of the static energy barrier, which effectively leads to an increase of the barrier for the solute transport.

Principal role of the stepwise aggregation mechanism in ionic surfactant solutions near the critical micelle concentration. Molecular dynamics study

The validity of the assumption on the predominant contribution of the stepwise processes to the ionic micelle formation/destruction in the vicinity of critical micelle concentration was investigated by molecular dynamics simulation. A coarse-grained model was used to describe the surfactant/water mixture. The cluster size distribution was estimated directly from molecular dynamics simulations or obtained from a reduced set of kinetic equations. The good agreement between two approaches shows that the neglect of the terms responsible for cluster fusion/fission is fully justified and that such processes are less important than stepwise aggregation.

Molecular transport across fluid interfaces: coupling between solute dynamics and interface fluctuations

We investigate the transport mechanism of a small hydrophobic solute molecule across two types of fluid interfaces, (i) an interface between two immiscible liquids and (ii) a surfactant-covered liquid-liquid interface. These systems are modeled by coarse-grained molecular dynamics simulations. It is demonstrated that the dynamics of the solute molecule near the interface significantly deviates from Markovian Brownian motion. Specifically, the correlation time of the random force acting on the solute strongly depends on the distance between the solute and the interface and increases by two orders of magnitude within a very narrow (less than 1 nm wide) region near the interface. The slow fluctuations of the random force in this narrow region are caused by capillary waves. The region location and width are determined by interface protrusions caused by attraction between the solute and the hydrophobic phase. We use results of molecular dynamics simulations to develop a stochastic model for the coupled solute-interface dynamics and estimate the rate of the solute transport across the interface. The observed phenomenon appears to be a general feature of mass transport across fluid or flexible membranes. The coupling between the solute transport and the interface fluctuations is the strongest in areas corresponding to a large free energy gradient or near a free energy barrier for the solute transport. This suggests a strong influence of the coupled solute-interface dynamics on the rate of mass transfer across interfaces.

Studying the unfolding kinetics of proteins under pressure using long molecular dynamic simulation runs

The usefulness of computational methods such as molecular dynamics simulation has been extensively established for studying systems in equilibrium. Nevertheless, its application to complex non-equilibrium biological processes such as protein unfolding has been generally regarded as producing results which cannot be interpreted straightforwardly. In the present study, we present results for the kinetics of unfolding of apomyoglobin, based on the analysis of long simulation runs of this protein in solution at 3 kbar (1 atm = 1.01325, bar = 101,325 Pa). We hereby demonstrate that the analysis of the data collected within a simulated time span of 0.18 mus suffices for producing results, which coincide remarkably with the available unfolding kinetics experimental data. This not only validates molecular dynamics simulation as a valuable alternative for studying non-equilibrium processes, but also enables a detailed analysis of the actual structural mechanism which underlies the unfolding process of proteins under elusive denaturing conditions such as high pressure.