Statistical Thermodynamics of Amphiphile Self-Assembly: Structure and Phase Transitions in Micellar Solutions. In: Gelbart WM, Ben-shaul A, Roux D, editors. Micelles, Membranes, Microemulsions, and Monolayers. New York: Springer; 1994. pp. 1-104. [DOI: 10.1007/978-1-4613-8389-5_1]

Mechanism of the Initial Tubulin Nucleation Phase

Tubulin nucleation is a highly frequent event in microtubule (MT) dynamics but is poorly understood. In this work, we characterized the structural changes during the initial nucleation phase of dynamic tubulin. Using size-exclusion chromatography-eluted tubulin dimers in an assembly buffer solution free of glycerol and tubulin aggregates enabled us to start from a well-defined initial thermodynamic ensemble of isolated dynamic tubulin dimers and short oligomers. Following a temperature increase, time-resolved X-ray scattering and cryo-transmission electron microscopy during the initial nucleation phase revealed an isodesmic assembly mechanism of one-dimensional (1D) tubulin oligomers (where dimers were added and/or removed one at a time), leading to sufficiently stable two-dimensional (2D) dynamic nanostructures, required for MT assembly. A substantial amount of tubulin octamers accumulated before two-dimensional lattices appeared. Under subcritical assembly conditions, we observed a slower isodesmic assembly mechanism, but the concentration of 1D oligomers was insufficient to form the multistranded 2D nucleus required for MT formation.

Multiscale Exploration of Concentration-Dependent Amyloid-β(16-21) Amyloid Nucleation

Atomic descriptions of peptide aggregation nucleation remain lacking due to the difficulty of exploring complex configurational spaces on long time scales. To elucidate this process, we develop a multiscale approach combining a metadynamics-based method with cluster statistical mechanics to derive concentration-dependent free energy surfaces of nucleation at near-atomic resolution. A kinetic transition network of nucleation is then constructed and employed to systematically explore nucleation pathways and kinetics through stochastic simulations. This approach is applied to describe Aβ_16-21 amyloid nucleation, revealing a two-step mechanism involving disordered aggregates at millimolar concentration, and an unexpected mechanism at submillimolar concentrations that exhibits kinetics reminiscent of classical nucleation but atypical pathways involving growing clusters with structured cores wrapped by disordered surface. When this atypical mechanism is operative, critical nucleus size can be reflected by the nucleation reaction order. Collectively, our approach paves the way for a more quantitative and detailed understanding of peptide aggregation nucleation.

Mechanism of Tubulin Oligomers and Single-Ring Disassembly Catastrophe

Cold tubulin dimers coexist with tubulin oligomers and single rings. These structures are involved in microtubule assembly; however, their dynamics are poorly understood. Using state-of-the-art solution synchrotron time-resolved small-angle X-ray scattering, we discovered a disassembly catastrophe (half-life of ∼0.1 s) of tubulin rings and oligomers upon dilution or addition of guanosine triphosphate. A slower disassembly (half-life of ∼38 s) was observed following an increase in temperature. Our analysis showed that the assembly and disassembly processes were consistent with an isodesmic mechanism, involving a sequence of reversible reactions in which dimers were rapidly added or removed one at a time, terminated by a 2 order-of-magnitude slower ring-closing/opening step. We revealed how assembly conditions varied the mass fraction of tubulin in each of the coexisting structures, the rate constants, and the standard Helmholtz free energies for closing a ring and for longitudinal dimer-dimer associations.

Microemulsions: Breakthrough Electrolytes for Redox Flow Batteries

Aqueous and non-aqueous redox flow batteries (RFBs) have limited energy and current densities, respectively, due to the nature of the electrolytes. New approaches to electrolyte design are needed to improve the performance of RFBs. In this work, we combined a highly conductive aqueous phase and an organic redox-active phase in a microemulsion to formulate a novel RFB electrolyte. As a proof-of-concept, we demonstrate an RFB using this microemulsion electrolyte with maximum current density of 17.5 mA·cm⁻² with a 0.19 M posolyte and 0.09 M negolyte at a flow rate of only ∼2.5 ml·min⁻¹, comparable to early vanadium electrolyte RFBs at similar flow rates on a per molar basis. The novel active negolyte component is an inexpensive oil-soluble vitamin (K₃). By combining aqueous and organic phases, the solvent potential window and energy density may be increased without sacrificing current density and new redox couples may be accessed. Microemulsion electrolytes show great promise for improved performance and increased energy densities in aqueous RFBs but the path forward is complex. We end with discussion of areas that need work to achieve the potential of these electrolytes.

The time complexity of self-assembly

Time efficiency of self-assembly is crucial for many biological processes. Moreover, with the advances of nanotechnology, time efficiency in artificial self-assembly becomes ever more important. While structural determinants and the final assembly yield are increasingly well understood, kinetic aspects concerning the time efficiency, however, remain much more elusive. In computer science, the concept of time complexity is used to characterize the efficiency of an algorithm and describes how the algorithm's runtime depends on the size of the input data. Here we characterize the time complexity of nonequilibrium self-assembly processes by exploring how the time required to realize a certain, substantial yield of a given target structure scales with its size. We identify distinct classes of assembly scenarios, i.e., "algorithms" to accomplish this task, and show that they exhibit drastically different degrees of complexity. Our analysis enables us to identify optimal control strategies for nonequilibrium self-assembly processes. Furthermore, we suggest an efficient irreversible scheme for the artificial self-assembly of nanostructures, which complements the state-of-the-art approach using reversible binding reactions and requires no fine-tuning of binding energies.

Structure and Energetics of GTP- and GDP-Tubulin Isodesmic Self-Association

Tubulin self-association is a critical process in microtubule dynamics. The early intermediate structures, energetics, and their regulation by fluxes of chemical energy, associated with guanosine triphosphate (GTP) hydrolysis, are poorly understood. We reconstituted an in vitro minimal model system, mimicking the key elements of the nontemplated tubulin assembly. To resolve the distribution of GTP- and guanosine diphosphate (GDP)-tubulin structures, at low temperatures (∼10 °C) and below the critical concentration for the microtubule assembly, we analyzed in-line size-exclusion chromatography-small-angle X-ray scattering (SEC-SAXS) chromatograms of GTP- and GDP-tubulin solutions. Both solutions rapidly attained steady state. The SEC-SAXS data were consistent with an isodesmic thermodynamic model of longitudinal tubulin self-association into 1D oligomers, terminated by the formation of tubulin single rings. The analysis showed that free dimers coexisted with tetramers and hexamers. Tubulin monomers and lateral association between dimers were not detected. The dimer-dimer longitudinal self-association standard Helmholtz free energies were -14.2 ± 0.4 k_BT (-8.0 ± 0.2 kcal mol^-1) and -13.1 ± 0.5 k_BT (-7.4 ± 0.3 kcal mol^-1) for GDP- and GTP-tubulin, respectively. We then determined the mass fractions of dimers, tetramers, and hexamers as a function of the total tubulin concentration. A small fraction of stable tubulin single rings, with a radius of 19.2 ± 0.2 nm, was detected in the GDP-tubulin solution. In the GTP-tubulin solution, this fraction was significantly lower. Cryo-TEM images and SEC-multiangle light-scattering analysis corroborated these findings. Our analyses provide an accurate structure-stability description of cold tubulin solutions.

Origin of the Hierarchical Structure of Dendritic Fibrous Nanosilica: A Small-Angle X-ray Scattering Perspective

The discovery of dendritic fibrous nanosilica (DFNS) has attracted great attention to the field of catalysis, CO₂ capture, drug delivery due to its distinct morphology, and pore size distribution. Despite extensive research, the understanding of the DFNS formation process and its internal structure remains incomplete as microscopy and gas sorption techniques were not able to provide necessary in-depth structural information due to their inherent limitations. In the current work, we present a structural model of DFNS derived using small-angle X-ray scattering (SAXS) supported by ¹²⁹Xe nuclear magnetic resonance (NMR), which provided intricate details of DFNS and its internal structure. Mechanistic understanding of the DFNS formation and growth process was achieved by performing time-resolved SAXS measurements during the synthesis of DFNS, which unveils the evolution of two levels of a bicontinuous microemulsion structure responsible for intricate DFNS morphology. The validity and the accuracy of the SAXS method and the model were successfully established through a direct correlation among the functionality of the DFNS scattering profile and its pore size distribution, as well as results obtained from the ¹²⁹Xe NMR studies. It has been established that the DFNS structure originates from direct modulation of the bicontinuous structure controlled by a surfactant, a co-surfactant, and the silicate species formed during hydrolysis and the condensation reaction of the silica precursor.

Scaling relationships for the elastic moduli and viscosity of mixed lipid membranes

The elastic and viscous properties of biological membranes play a vital role in controlling cell functions that require local reorganization of the membrane components as well as dramatic shape changes such as endocytosis, vesicular trafficking, and cell division. These properties are widely acknowledged to depend on the unique composition of lipids within the membrane, yet the effects of lipid mixing on the membrane biophysical properties remain poorly understood. Here, we present a comprehensive characterization of the structural, elastic, and viscous properties of fluid membranes composed of binary mixtures of lipids with different tail lengths. We show that the mixed lipid membrane properties are not simply additive quantities of the single-component analogs. Instead, the mixed membranes are more dynamic than either of their constituents, quantified as a decrease in their bending modulus, area compressibility modulus, and viscosity. While the enhanced dynamics are seemingly unexpected, we show that the measured moduli and viscosity for both the mixed and single-component bilayers all scale with the area per lipid and collapse onto respective master curves. This scaling links the increase in dynamics to mixing-induced changes in the lipid packing and membrane structure. More importantly, the results show that the membrane properties can be manipulated through lipid composition the same way bimodal blends of surfactants, liquid crystals, and polymers are used to engineer the mechanical properties of soft materials, with broad implications for understanding how lipid diversity relates to biomembrane function.

Self-Assembly of Cations in Aqueous Solutions of Multiheaded Cationic Surfactants: All Atom Molecular Dynamics Simulation Studies

Self-assembly of multiheaded surfactants in aqueous solutions has been investigated using atomistic molecular dynamics simulations. The model multiheaded surfactants contain multiple head groups ranging from one to four for a single tail group. Increase in the number of charged head groups has substantial consequences in the aggregation properties of surfactants in their aqueous solutions. Polydisperse aggregates of surfactants are formed in the aqueous solution. The shape and size of the aggregates are dictated by the number of charged head groups present in the surfactant. Our simulations demonstrate that with the increase in the number of charged head groups on the surfactants, the aggregation number decreases, which corroborates previous experimental and theoretical studies. Though experimental studies on the surfactant with four head groups is yet to be performed, we have included the surfactant having four head groups in our studies and compared the results with previous coarse-grained computational study involving four head groups.

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.

Line Tension Assists Membrane Permeation at the Transition Temperature in Mixed-Phase Lipid Bilayers

The umbrella sampling method has been used to evaluate the free energy profile for a large permeant moving through a lipid bilayer, represented using a coarse-grained simulation model, at and below its gel-fluid transition temperature. At the lipid transition temperature, determined to be 302 K for the MARTINI 2.0 model of DPPC, the permeation barrier for passage through an enclosed fluid domain embedded in a patch of gel was significantly lower than that for passage through a fluid stripe domain. In contrast, permeation through a fluid domain in a stripe geometry produced a free energy profile nearly identical to that of a gel-free fluid bilayer. In both cases, insertion of the permeant into a fluid domain coexisting with the gel phase led to a shift in phase composition, as lipids transitioned from fluid to gel to accommodate the area occupied by the permeant. In the case of the enclosed fluid domain, this transition produced a decrease in the length of the fluid-gel interface as the approximately circular fluid domain shrank. The observed decrease in the apparent permeation barrier, combined with an approximation for the change in interfacial length, enabled estimation of the interfacial line tension to be between 10 and 13 pN for this model. The permeation barrier was shown to drop even further in simulations performed at temperatures below the transition temperature. The results suggest a mechanism to explain the experimentally observed anomalous peak in the temperature-dependent permeability of lipid bilayers near their transition temperatures. The contribution of this mechanism toward the permeability of a gel phase containing a thermal distribution of fluid-phase domains is estimated using a simple statistical thermodynamic model.

Bending elasticity of nonionic surfactant layers

A novel approach to evaluate the bending elasticity of monolayers formed by nonionic surfactants with a rigid head group is introduced by means of considering head group repulsion as derived from the free energy of mixing rigid hydrophilic head groups with surrounding solvent molecules as well as contributions related to the hydrophobic tails. Explicit expressions for the spontaneous curvature (H0), bending rigidity (kc) and saddle-splay constant (kc) have been derived for the constraint of constant chemical potential of free surfactant (thermodynamically open layers) as well as the constraint of constant aggregation number (thermodynamically closed layers). Most interestingly, it is demonstrated that kc for thermodynamically open layers formed by a nonionic surfactant with rigid tail and head group always must be zero. However, kc for surfactants with a flexible tail as a function of the head group-to-tail volume ratio is found to go through a maximum at some large, positive value of kc and H0 approximately 0. Eventually, kc falls below zero as the head group volume increases above a certain value. Hence, we may conclude that nonionic surfactants with a rigid head group may form thermodynamically stable fluid layers or aggregates only insofar the hydrophobic part is flexible with respect to chain conformational degrees of freedom and the head group is not too voluminous. It is found that the head group repulsion contribution to kcH0 is always positive whereas the corresponding contribution to kc may be positive or negative depending on whether the hydrophobic layer of the film is thicker or thinner than the hydrophilic layer.

Molecular recognition of poly(A) by small ligands: an alternative method of analysis reveals nanomolar, cooperative and shape-selective binding

A few drug-like molecules have recently been found to bind poly(A) and induce a stable secondary structure (T(m) approximately 60 degrees C), even though this RNA homopolymer is single-stranded in the absence of a ligand. Here, we report results from experiments specifically designed to explore the association of small molecules with poly(A). We demonstrate that coralyne, the first small molecule discovered to bind poly(dA), binds with unexpectedly high affinity (K(a) >10(7) M(-1)), and that the crescent shape of coralyne appears necessary for poly(A) binding. We also show that the binding of similar ligands to poly(A) can be highly cooperative. For one particular ligand, at least six ligand molecules are required to stabilize the poly(A) self-structure at room temperature. This highly cooperative binding produces very sharp transitions between unstructured and structured poly(A) as a function of ligand concentration. Given the fact that junctions between Watson-Crick and A.A duplexes are tolerated, we propose that poly(A) sequence elements and appropriate ligands could be used to reversibly drive transitions in DNA and RNA-based molecular structures by simply diluting/concentrating a sample about the poly(A)-ligand 'critical concentration'. The ligands described here may also find biological or medicinal applications, owing to the 3'-polyadenylation of mRNA in living cells.

On the concentration-induced growth of nonionic wormlike micelles

We present a comprehensive study of the concentration-induced growth of nonionic wormlike micelles in dilute solutions, below the overlap concentration, where we combine static and dynamic light scattering and NMR self-diffusion. The data are analyzed in detail in terms of the number-averaged contour length, assuming an exponential size distribution, as predicted by theory, and that the micellar flexibility can be described by the wormlike chain model with a certain persistence length. A very good agreement between the different experimental methods is obtained. The number-averaged contour length increases with increasing micelle volume fraction. The data are consistent with a power law, where the power law exponent is in the range 0.5-0.8, depending on the system. The result is in reasonable agreement with the theory of living polymers.

Simulation and theory of self-assembly and network formation in reversibly cross-linked equilibrium polymers

A simulation model of hard spheres capable of reversible assembly into chains, which then may reversibly cross-link into networks, has been studied through grand canonical Monte Carlo simulation. Effects of varying intra- and interchain bond strengths, chain flexibilities, and restrictions on cross-linking angle were investigated. Observations including chain-length distributions and phase separation could be captured in most cases using a simple model theory. The coupling of chain growth to cross-linking was shown to be highly sensitive to the treatment of cross-linking by chain ends. In some systems, ladderlike domains of several cross-links joining two chains were common, resulting from cooperativity in the cross-linking. Extended to account for this phenomenon, the model theory predicts that such cooperativity will suppress phase separation in weakly polymerizing chains and at high cross-link concentration. In the present model, cross-linking stabilizes the isotropic phase with respect to the nematic phase, causing a shift in the isotropic-nematic transition to higher monomer concentration than in simple equilibrium polymers.

Simulation and theory of flexible equilibrium polymers under poor solvent conditions

Grand canonical Monte Carlo simulation and simple statistical thermodynamic theory are used to model the aggregation and phase separation of systems of reversibly polymerizing monomers, capable of forming chains with or without the ability to cyclize into rings, with isotropic square-well attractions between nonbonded pairs of monomers. The general trend observed in simulation of chain-only systems, as predicted in a number of published theoretical works, is that the critical temperature for phase separation increases and the critical monomer density decreases with rising polymer bond strength. Introduction of the equilibrium between chains and rings into the theory lowers the predicted critical temperature and increases the predicted critical density. While the chain-only theories predict a vanishing critical density in the limit of complete polymerization, when ring formation is taken into account the predicted critical density in the same limit approaches the density of the onset of the ring-chain transition. The theoretically predicted effect of cyclization on chemical potential is in good qualitative agreement with a subset of simulation results in which chain-only systems were compared with equilibrium mixtures of rings and chains. The influence of attractions on the aggregation number and radius of gyration of chains and rings observed in simulations is also discussed.

Can Linear Micelles Bridge between Two Surfaces?

Surfactants are extensively used as stabilizers of colloidal particles, even though the use of high surfactant concentrations can induce a loss of the stability of the dispersion. The depletion mechanism is believed to be responsible for this instability. In this paper, we show that there exists an alternative interpretation, namely that wormlike micelles can bridge between two surfaces. Such a stalk-like object connecting two adsorbed bilayers is (in first order) stable when the endcap (free) energy of the wormlike micelle (in solution) is higher than the connection (free) energy of the stalk with the surface layer. As an example, we consider an aqueous solution of nonionic C(12)E(6) surfactants and use a molecularly realistic self-consistent field approach to evaluate the free-energy of bridge formation. It appears impossible to connect linear micelles to hydrophobic surfaces onto which a monolayer of surfactants exists, and stalks only occur with an exponentially low probability for very hydrophilic surfaces. However, at a wide regime of moderately hydrophilic surfaces the stalks are thermodynamically stable. In this regime, the adsorbed bilayers are typically only marginally stable. We identify a range of parameters for which such adsorbed bilayer ruptures around the stalk and then the wormlike micelle essentially connects (head-on) to the bare surface. The strength of interaction is of the order of the endcap energy which easily exceeds 10 k(B)T. The range of interactions is expected to be large as it is set by the characteristic size of the linear micelles in solution. The regime of moderately hydrophilic surfaces is relevant experimentally, and we conclude that surfactant-induced flocculation may well be the result of stalks. The depletion mechanism is only expected for systems with extremely hydrophobic and with very hydrophilic particles.

Thermodynamics of membrane elasticity--a molecular level approach to one- and two-component fluid amphiphilic membranes, part II: applications

The theoretical framework developed in the accompanying publication is applied to a number of experimentally relevant amphiphilic systems. These include the influence of thermodynamic conditions and non-ideal mixing on bending elasticity, ellipsoidal modes of microemulsions and vesicles, hydrocarbon chain coupling in bilayers and the effect of osmotic and hydrostatic pressure on inverse hexagonal (H(II)) phases. It is found that the bending moduli at constant surface tension and constant chemical potentials are markedly different only for two-component membranes and non-ideal mixing with a tendency towards phase separation. The results indicate that non-ideal mixing is the main reason behind the experimentally observed strong compositional dependence of membrane elasticity. It is generally recommended to prefer the bending elastic moduli at constant chemical potentials to those at constant surface tension. A comparison between the area-difference-elasticity (ADE) model and explicit free energy calculations using a molecular model shows a good qualitative agreement for the sphere-to-ellipsoid transition of vesicles. Results for different free energy models of the hydrocarbon chains of amphiphilic molecules suggest that monolayer-monolayer chain coupling is responsible for the relatively higher bending stiffness of bilayers compared to single monolayers. For H(II)-phases an instability at negative pressure differences is predicted.

Perturbation of a lipid membrane by amphipathic peptides and its role in pore formation

We study the structural and energetic consequences of (alpha-helical) amphipathic peptide adsorption onto a lipid membrane and the subsequent formation of a transmembrane peptide pore. Initially, each peptide binds to the membrane surface, with the hydrophobic face of its cylinder-like body inserted into the hydrocarbon core. Pore formation results from subsequent peptide crowding, oligomerization, and eventually reorientation along the membrane normal. We have theoretically analyzed three peptide-membrane association states: interfacially-adsorbed monomeric and dimeric peptides, and the multi-peptide transmembrane pore state. Our molecular-level model for the lipid bilayer is based on a combination of detailed chain packing theory and a phenomenological description of the headgroup region. We show that the membrane perturbation free energy depends critically on peptide orientation: in the transmembrane pore state the lipid perturbation energy, per peptide, is smaller than in the adsorbed state. This suggests that the gain in conformational freedom of the lipid chains is a central driving force for pore formation. We also find a weak, lipid-mediated, gain in membrane perturbation free energy upon dimerization of interfacially-adsorbed peptides. Although the results pertain mainly to weakly-charged peptides, they reveal general properties of the interaction of amphipathic peptides with lipid membranes.

Molecular-dynamics simulation of amphiphilic bilayer membranes and wormlike micelles: a multi-scale modelling approach to the design of viscoelastic surfactant solutions

Bilayer membranes and wormlike micelles have been studied using molecular-dynamics simulations. The structure of the worm is analysed in terms of radial density distribution functions, and mechanical properties such as the elastic modulus are calculated. From an analysis of the fluctuation spectra of the tensionless states, we have calculated bending rigidities. Micelles consisting of coarse-grained (CG) model surfactants are studied in order to map the properties of the atomistic micelle. We optimize the CG model with respect to the structure factor of the atomistic micelle. The mechanical properties thus obtained will be used as input for a mesoscopic model of wormlike micelles where the persistence length is the smallest length-scale.

Solvent extraction for separating micellar-solubilized contaminants and anionic surfactants

Decontamination of contaminant-laden surfactant solutions is critical to successful implementation of surfactant-enhanced aquifer remediation (SEAR). Solvent extraction was studied for removing micellar-solubilized contaminants having low equivalent alkyl carbon numbers (EACNs) from surfactant solutions. Factors influencing the solvent extraction of micellar-solubilized contaminant were studied, including surfactant concentration, solution salinity, solvent solubilization, and solvent/solution volumetric ratio. A model was developed to quantify the impacts of these factors on contaminant removal. The good agreement between experimental results and model predictions corroborates assumptions made in the model development. From these results, it is concluded that extracting solvents must have much higher EACNs than that of the contaminant to reduce the micellar solubilization of the solvents, which can significantly reduce contaminant removal efficiency. However, the highest EACN solvent is not necessarily the best one for contaminant removal due to other constraints (e.g., molecular weight and viscosity). Increasing the total surfactant concentration or salinity of an anionic surfactant solution increases its contaminant solubilization capacity but reduces the contaminant removal efficiency by solvent extraction. Continuous column extraction operated at a low column surface loading rate allowed contaminant partitioning to approach equilibrium conditions, and multistage column extraction was able to improve the contaminant removal efficiency while minimizing solvent requirement.