Hierarchical approximations to the nucleation work in the entire range of metastability.

The work W to form a nucleus (also known as the critical nucleus) is a key quantity in the description of nucleation phenomena because of its exponentially strong effect on the nucleation rate. The present study provides a general approximate expression for W, which comprises a hierarchy of approximations to the dependence of W on the experimentally controlled overpressure Δp of a nucleating multicomponent phase. This general expression is used to derive explicit formulas for the lowest-order members of the W(Δp) hierarchy as well as for the respective lowest-order approximations to the Δp dependences of the nucleus surface tension, the nucleus radius, the Gibbs-Tolman length, and the stationary nucleation rate. The second-order and the third-order approximations to the W(Δp) dependence are confronted with available W(Δp) data, and the latter is found to agree very well with the data. The results obtained are applicable to homogeneous single-component or multicomponent nucleation from the binodal to the spinodal of the old phase, i.e., in the entire range of the old-phase metastability.

Nucleation work, surface tension, and Gibbs-Tolman length for nucleus of any size.

In the framework of the Gibbs approach to nucleation thermodynamics, expressions are derived for the nucleation work, nucleus size, surface tension, and Gibbs-Tolman length in homogeneous single-component nucleation at a fixed temperature. These expressions are in terms of the experimentally controlled overpressure of the nucleating phase and are valid for the entire overpressure range, i.e., for nucleus of any size. Analysis of available data for bubble and droplet nucleation in Lennard-Jones fluid shows that the theory describes well the data by means of a single free parameter, the Gibbs-Tolman length of the planar liquid/vapor interface. It is found that this length is about one-tenth of the Lennard-Jones molecular-diameter parameter and that it is positive for the bubble nucleus and negative for the droplet nucleus. In a sufficiently narrow temperature range, the nucleation work, nucleus radius, scaled surface tension, and Gibbs-Tolman length are apparently universal functions of scaled overpressure.

Growth probability and formation time of the individual Oosawa-Kasai protein fibril.

Protein fibrils are currently of great academic and practical interest because of their involvement in scores of severe human diseases and their promising use in various high-technology devices. The Oosawa-Kasai (OK) model of protein self-assembly into fibrils has been widely used to gain mechanistic insight into the process of fibril formation and growth. Here this model is employed to obtain exact and mathematically simple expressions for the probability P_{n} of an individual fibril of n protein monomers to grow to a macroscopically large size and for the mean time τ_{n} that such a fibril needs for its formation. These expressions quantify the increase of P_{n} and the decrease of τ_{n} with increasing the concentration of monomeric protein in the solution. When used for analysis of experimental P_{n} and τ_{n} data, they make it possible to determine the parameters characterizing fibril nucleation and growth in the framework of the OK model. Finally, an expression is found for the mean time of the first appearance of an n-sized fibril in the protein solution. The results obtained are applicable to the formation of other aggregates corresponding to the OK fibrils, such as the one-dimensional Kossel-Stranski crystals and Ising ferromagnets.

Modeling the Effect of Monomer Conformational Change on the Early Stage of Protein Self-Assembly into Fibrils.

Filamentous self-assembly of proteins is an important process implicated in a plethora of human diseases and of interest for nanotechnology. Using rate equations, we analyze the early stage of the process in solutions that initially contain fibrillation-passive protein monomers and in which the nascent fibrils are practically insoluble. The analysis is based on a model accounting for the conformational and/or other changes the passive monomers experience to transform themselves into fibrillation-active monomers and thus become fibril nuclei. The model allows exact, comprehensive, and simple mathematical description of the early stage of fibrillation, which reveals the usually neglected role of the nucleation nonstationarity in this stage of fibrillation. We obtain exact and user-friendly expressions for experimentally accessible quantities such as the size distribution of fibrils, their number and mass concentrations, the rate and nonstationary period of fibril nucleation, and the delay time of fibril formation. Analyzing available experimental data, we find that the theory successfully describes the fibrillation time course of pathological and nonpathological ataxin-3, a protein involved in the neurodegenerative disorder spinocerebellar ataxia type-3. The analysis provides mechanistic insight into the reason for the higher fibril nucleation and elongation rates of the pathological ataxin-3.

Protein Polymerization into Fibrils from the Viewpoint of Nucleation Theory.

The assembly of various proteins into fibrillar aggregates is an important phenomenon with wide implications ranging from human disease to nanoscience. Using general kinetic results of nucleation theory, we analyze the polymerization of protein into linear or helical fibrils in the framework of the Oosawa-Kasai (OK) model. We show that while within the original OK model of linear polymerization the process does not involve nucleation, within a modified OK model it is nucleation-mediated. Expressions are derived for the size of the fibril nucleus, the work for fibril formation, the nucleation barrier, the equilibrium and stationary fibril size distributions, and the stationary fibril nucleation rate. Under otherwise equal conditions, this rate decreases considerably when the short (subnucleus) fibrils lose monomers much more frequently than the long (supernucleus) fibrils, a feature that should be born in mind when designing a strategy for stymying or stimulating fibril nucleation. The obtained dependence of the nucleation rate on the concentration of monomeric protein is convenient for experimental verification and for use in rate equations accounting for nucleation-mediated fibril formation. The analysis and the results obtained for linear fibrils are fully applicable to helical fibrils whose formation is describable by a simplified OK model.

Kinetics of protein fibrillation controlled by fibril elongation.

Numerous proteins have the ability to assemble into fibrillar aggregates which are of great interest, because they feature in scores of human diseases and many technological products. In the present work, we analyze the kinetics of protein fibrillation when the process is governed solely by elongation of initially appeared fibrils in the protein solution. We derive exact expressions for the time dependences of the fibrillation degree, the concentration of monomeric protein in the solution, and the average fibril size. Furthermore, we present formulas for the initial fibrillation rate and the half-fibrillation time in terms of experimentally controllable quantities. The results obtained provide a mechanistic insight into the kinetics of protein fibrillation mediated by fibril elongation. We confront theory with experiment and find that it allows a good description of available experimental data for fibrillation of the Alzheimer's disease-associated protein Aß(1-40) and the yeast prion protein Sup35.

Protein fibrillation due to elongation and fragmentation of initially appeared fibrils: a simple kinetic model.

The assembly of various proteins into fibrillar aggregates is an important phenomenon with wide implications ranging from human disease to nanoscience. Employing a new model, we analyze the kinetics of protein fibrillation in the case when the process occurs by elongation of initially appeared fibrils which multiply solely by fragmentation, because fibril nucleation is negligible. Owing to its simplicity, our model leads to mathematically friendly and physically clear formulas for the time dependence of the fibrillation degree and for a number of experimental observables such as the maximum fibrillation rate, the fibrillation lag time, and the half-fibrillation time. These formulas provide a mechanistic insight into the kinetics of fragmentation-affected fibrillation of proteins. We confront theory with experiment and find that our model allows a good global description of a large dataset [W.-F. Xue, S. W. Homans, and S. E. Radford, Proc. Natl. Acad. Sci. U.S.A. 105, 8926 (2008)] for the fibrillation kinetics of beta-2 microglobulin. Our analysis leads to new methods for experimental determination of the fibril solubility, elongation rate constant, and nucleation rate from data for the time course of protein fibrillation.

Confounding the paradigm: peculiarities of amyloid fibril nucleation.

Fibrils of amyloid proteins are currently of great interest because of their involvement in various amyloid-related diseases and nanotechnological products. In a recent kinetic Monte Carlo simulation study (Cabriolu, R.; Kashchiev, D.; Auer, S. J. Chem. Phys.2012, 137, 204903), we found that our simulation data for the rate of amyloid fibril nucleation occurring by direct polymerization of monomeric protein could not be described adequately by nucleation theory. It turned out that the process occurred in a peculiar way, thus confounding the nucleation paradigm and demanding a new theoretical treatment. In the present study, we reconsider the theoretical approach to nucleation of amyloid fibrils and derive new expressions for the stationary rate of the process. As these expressions provide a remarkably good description of the simulation data, by using them we propose a theoretical dependence of the amyloid-ß(40) fibril nucleation rate on the concentration of monomeric protein in the solution. This dependence reveals the existence of a threshold concentration below which the fibril nucleation in small enough solution volumes is practically arrested, and above which the process occurs vigorously, because then each monomeric protein in the solution acts as fibril nucleus. The presented expressions for the threshold concentration and for the dependence of the fibril nucleation rate on the concentration of monomeric protein can be a valuable guide in designing new therapeutic and/or technological strategies for prevention or stimulation of amyloid fibril formation.

Breakdown of nucleation theory for crystals with strongly anisotropic interactions between molecules.

We study the nucleation of model two-dimensional crystals in order to gain insight into the effect of anisotropic interactions between molecules on the stationary nucleation rate J. With the aid of kinetic Monte Carlo simulations, we determine J as a function of the supersaturation s. It turns out that with increasing degree of interaction anisotropy the dependence of ln J on s becomes step-like, with jumps at certain s values. We show that this J(s) dependence cannot be described by the classical and atomistic nucleation theories. A formula that predicts the identified J(s) behavior is yet to be derived and verified, and the present study provides the necessary data and understanding for doing that.

Two-step nucleation of amyloid fibrils: omnipresent or not?

Amyloid protein fibrils feature in various diseases and nanotechnological products. Currently, it is debated whether they nucleate in one step (i.e., directly from the protein solution) or in two steps (step one being the appearance of nonfibrillar oligomers in the solution and step two being the oligomer conversion into fibrils). We employ nucleation theory to gain insight into the idiosyncrasy of two-step fibril nucleation and to determine the conditions under which this process can take place. Presenting an expression for the rate of two-step fibril nucleation, we use it to qualitatively describe experimental data for two-step nucleated amyloid-ß fibrils. Our analysis helps in understanding why, in some experiments, oligomers rather than fibrils form and remain structurally unchanged and why, in others, the oligomers convert into fibrils.

Size distribution of amyloid nanofibrils.

We consider the size distribution of amyloid nanofibrils (protofilaments) in nucleating protein solutions when the nucleation process occurs by the mechanism of direct polymerization of ß-strands (extended peptides or protein segments) into ß-sheets. Employing the atomistic nucleation theory, we derive a general expression for the stationary size distribution of amyloid nanofibrils constituted of successively layered ß-sheets. The application of this expression to amyloid ß(1-40) (Aß(40)) fibrils allows us to determine the nanofibril size distribution as a function of the protein concentration and temperature. The distribution is most remarkable with its exhibiting a series of peaks positioned at "magic" nanofibril sizes (or lengths), which are due to deep local minima in the work for fibril formation. This finding of magic sizes or lengths is consistent with experimental results for the size distribution of aggregates in solutions of Aß(40) proteins. Also, our approach makes it possible to gain insight into the effect of point mutations on the nanofibril size distribution, an effect that may play a role in experimentally observed substantial differences in the fibrillation lag-time of wild-type and point-mutated amyloid-ß proteins.

Atomistic theory of amyloid fibril nucleation.

We consider the nucleation of amyloid fibrils at the molecular level when the process takes place by a direct polymerization of peptides or protein segments into ß-sheets. Employing the atomistic nucleation theory (ANT), we derive a general expression for the work to form a nanosized amyloid fibril (protofilament) composed of successively layered ß-sheets. The application of this expression to a recently studied peptide system allows us to determine the size of the fibril nucleus, the fibril nucleation work, and the fibril nucleation rate as functions of the supersaturation of the protein solution. Our analysis illustrates the unique feature of ANT that the size of the fibril nucleus is a constant integer in a given supersaturation range. We obtain the ANT nucleation rate and compare it with the rates determined previously in the scope of the classical nucleation theory (CNT) and the corrected classical nucleation theory (CCNT). We find that while the CNT nucleation rate is orders of magnitude greater than the ANT one, the CCNT and ANT nucleation rates are in very good quantitative agreement. The results obtained are applicable to homogeneous nucleation, which occurs when the protein solution is sufficiently pure and/or strongly supersaturated.

Insight into the correlation between lag time and aggregation rate in the kinetics of protein aggregation.

Under favorable conditions, many proteins can assemble into macroscopically large aggregates such as the amyloid fibrils that are associated with Alzheimer's, Parkinson's, and other neurological and systemic diseases. The overall process of protein aggregation is characterized by initial lag time during which no detectable aggregation occurs in the solution and by maximal aggregation rate at which the dissolved protein converts into aggregates. In this study, the correlation between the lag time and the maximal rate of protein aggregation is analyzed. It is found that the product of these two quantities depends on a single numerical parameter, the kinetic index of the curve quantifying the time evolution of the fraction of protein aggregated. As this index depends relatively little on the conditions and/or system studied, our finding provides insight into why for many experiments the values of the product of the lag time and the maximal aggregation rate are often equal or quite close to each other. It is shown how the kinetic index is related to a basic kinetic parameter of a recently proposed theory of protein aggregation.

Nucleation of amyloid fibrils.

We consider nucleation of amyloid fibrils in the case when the process occurs by the mechanism of direct polymerization of practically fully extended protein segments, i.e., beta-strands, into beta-sheets. Applying the classical nucleation theory, we derive a general expression for the work to form a nanosized amyloid fibril (protofilament) constituted of successively layered beta-sheets. Analysis of this expression reveals that with increasing its size, the fibril transforms from one-dimensional to two-dimensional aggregate in order to preserve the equilibrium shape corresponding to minimal formation work. We determine the size of the fibril nucleus, the fibril nucleation work, and the fibril nucleation rate as explicit functions of the concentration and temperature of the protein solution. The results obtained are applicable to homogeneous nucleation, which occurs when the solution is sufficiently pure and/or strongly supersaturated.

Phase diagram of alpha-helical and beta-sheet forming peptides.

The intrinsic property of proteins to form structural motifs such as alpha helices and beta sheets leads to a complex phase behavior in which proteins can assemble into various types of aggregates including crystals, liquidlike phases of unfolded or natively folded proteins, and amyloid fibrils. Here we use a coarse-grained protein model that enables us to perform Monte Carlo simulations for determining the phase diagram of natively folded alpha-helical and unfolded beta-sheet forming peptides. The simulations reveal the existence of various metastable peptide phases. The liquidlike phases are metastable with respect to the fibrillar phases, and there is a hierarchy of metastability.

Dependence of the critical undercooling for crystallization on the cooling rate.

Crystallization in solutions under steady cooling is considered in the case of instantaneous nucleation, in which all crystallites appear at once in the solution and grow in the absence of crystallites born subsequently. Expressions are obtained for the total crystallite volume as a function of the steadily increasing undercooling. These expressions are employed for determining the dependence of the relative critical undercooling u(c) for crystallization on the cooling rate q. The resulting u(c)(q) formula reveals the physical meaning of the parameters in the linear relationship, often reported experimentally, between u(c) and q in double logarithmic coordinates. The results obtained are also directly applicable to overall crystallization of melts at sufficiently small undercoolings.

Effect of surfactant adsorption time on the observation of Newton black film in foam film.

Observation of Newton black film (NBF) in foam film is possible only with a certain probability W which depends on the concentration C of surfactant in the solution and on the time t(a) during which adsorption of surfactant at the solution/air interface has taken place. In the paper, the W(C,t(a)) dependence is derived and used to analyze the effect of t(a) on the critical surfactant concentration C(c) below which NBF in foam film practically cannot be observed. An expression for the C(c)(t(a)) function is obtained which reveals that C(c) decreases substantially with increasing t(a). This expression is found to describe well experimental C(c)(t(a)) data for foam films obtained from aqueous solution of the therapeutic surfactant INFASURF.

Toward a better description of the nucleation rate of crystals and crystalline monolayers.

The ability of the classical nucleation theory (CNT) and atomistic nucleation theory (ANT) to predict the stationary nucleation rate J of single-component crystals and crystalline monolayers is verified with the aid of numerical and computer simulation data obtained in the scope of the Kossel crystal model. It is found that in both cases CNT significantly overestimates J because it does not account for the work needed to attach an atom to the periphery of the two-dimensional nucleus or to form such a nucleus on the surface of the three-dimensional one. In contrast, ANT is successful in providing a good quantitative description of J, especially for high enough effective binding energy between nearest-neighbor atoms in the crystal and in capturing the existence of extended, nearly linear portions in the dependence of ln J on the supersaturation s when the values of both s and the binding energy are sufficiently great. However, the ANT prediction about broken linear ln J versus s dependence is not confirmed by the numerical and simulation results presented. General formulas for the nucleation work, the nucleus size, and the nucleation rate are proposed which are applicable to nucleation of single-component crystals and crystalline monolayers in vapors, solutions, or melts and which correct the respective CNT formulas. The proposed J(s) formula provides a good description of the numerical and simulation data and can justifiably be used up to the supersaturation at which the nucleus becomes monomer. When experimental data for the J(s) dependence are available and the nucleus specific edge and surface energies are unknown parameters, the proposed J(s) formula can be employed for estimation of these energies even if the nucleus is constituted of a few atoms only.

Rate of two-dimensional nucleation: verifying classical and atomistic theories by Monte Carlo simulation.

We present kinetic Monte Carlo simulation data for the stationary rate J of two-dimensional nucleation of monolayer-thick crystal clusters in the growth of an atomically smooth (100) face of single-component Kossel crystal. The data are over a wide range of supersaturations s and effective broken-bond energies omega of nearest-neighbor atoms, and the J values span about 15 orders of magnitude. The simulation reveals that, in the s,omega range studied, the ln J vs s curve is smooth but with nearly linear portions connected with rather sharply curved segments when the omega value is sufficiently great. The simulation J(s) data are used for verification of the classical (CNT) and atomistic (ANT) nucleation theories without free parameters. It turns out that whereas J is overestimated by CNT, it is underestimated by ANT. The disagreement between theory and simulation is much greater for CNT than for ANT and, with increasing omega, it increases for the former but almost disappears for the latter. However, the ANT prediction about broken linear ln J vs s dependence is not confirmed by the simulation in the s,omega range studied.