Liquid-liquid separation in solutions of normal and sickle cell hemoglobin

Free heme and the polymerization of sickle cell hemoglobin

In search of novel control parameters for the polymerization of sickle cell hemoglobin (HbS), the primary pathogenic event of sickle cell anemia, we explore the role of free heme, which may be excessively released in sickle erythrocytes. We show that the concentration of free heme in HbS solutions typically used in the laboratory is 0.02-0.04 mole heme/mole HbS. We show that dialysis of small molecules out of HbS solutions arrests HbS polymerization. The addition of 100-260 μM of free heme to dialyzed HbS solutions leads to rates of nucleation and polymer fiber growth faster by two orders of magnitude than before dialysis. Toward an understanding of the mechanism of nucleation enhancement by heme, we show that free heme at a concentration of 66 μM increases by two orders of magnitude the volume of the metastable clusters of dense HbS liquid, the locations where HbS polymer nuclei form. These results suggest that spikes of the free heme concentration in the erythrocytes of sickle cell anemia patients may be a significant factor in the complexity of the clinical manifestations of sickle cell anemia. The prevention of free heme accumulation in the erythrocyte cytosol may be a novel avenue to sickle cell therapy.

Nucleation

Crystallization starts with nucleation and control of nucleation is crucial for the control of the number, size, perfection, polymorphism and other characteristics of crystalline materials. This is particularly true for crystallization in solution, which is an essential part of processes in the chemical and pharmaceutical industries and a major step in physiological and pathological phenomena. There have been significant recent advances in the understanding of the mechanism of nucleation of crystals in solution. The foremost of these are the two-step mechanism of nucleation and the notion of the solution-crystal spinodal. According to the two-step mechanism, the crystalline nucleus appears inside pre-existing metastable clusters of size several hundred nanometers, which consist of dense liquid and are suspended in the solution. While initially proposed for protein crystals, the applicability of this mechanism has been demonstrated for small molecule organic materials, colloids, polymers, and biominerals. This mechanism helps to explain several long-standing puzzles of crystal nucleation in solution: nucleation rates which are many orders of magnitude lower than theoretical predictions, the significance of the dense protein liquid, and others. At high supersaturations typical of most crystallizing systems, the generation of crystal embryos occurs in the spinodal regime, where the nucleation barrier is negligible. The solution-crystal spinodal helps to understand the role of heterogeneous substrates in nucleation and the selection of crystalline polymorphs. Importantly, these ideas provide powerful tools for control of the nucleation process by varying the solution thermodynamic parameters.

The two-step mechanism of nucleation of crystals in solution

The formation of crystalline nanoparticles starts with nucleation and control of nucleation is crucial for the control of the number, size, perfection, polymorph modification and other characteristics of particles. Recently, there have been significant advances in the understanding of the mechanism of nucleation of crystals in solution. The most significant of these is the two-step mechanism of nucleation, according to which the crystalline nucleus appears inside pre-existing metastable clusters of size several hundred nanometers, which consist of dense liquid and are suspended in the solution. While initially proposed for protein crystals, the applicability of this mechanism has been demonstrated for small-molecule organic and inorganic materials, colloids, and biominerals. This mechanism helps to explain several long-standing puzzles of crystal nucleation in solution: nucleation rates which are many orders of magnitude lower than theoretical predictions, nucleation kinetic dependencies with steady or receding parts at increasing supersaturation, the role of heterogeneous substrates for polymorph selection, the significance of the dense protein liquid, and others. More importantly, this mechanism provides powerful tools for control of the nucleation process by varying the solution thermodynamic parameters so that the volume occupied by the dense liquid shrinks or expands.

The intranuclear environment

Many of the chapters in this volume are concerned with processes or structures inside the nucleus, and it is relevant to consider the properties of their environment, or rather of the multiple different and specific environments that must exist in local regions of the highly heterogeneous intranuclear space. Relatively little is known about the fundamental physical properties of these environments, and theoretical treatments of phenomena in such concentrated mixtures of charged macromolecules are complex and as yet poorly developed. Some of the phenomena that occur at the molecular level are unexpected and counterintuitive for biologists, although well known to colloid and polymer scientists; for example, the existence of short-range attractive forces between macromolecules or structures with like charges. As a background for the chapters that follow, we consider here some of the particular features of intranuclear environments, how they may influence processes and structures in the nucleus, and their implications for working with nuclei.

Self-catalyzed growth of S layers via an amorphous-to-crystalline transition limited by folding kinetics

The importance of nonclassical, multistage crystallization pathways is increasingly evident from theoretical studies on colloidal systems and experimental investigations of proteins and biomineral phases. Although theoretical predictions suggest that proteins follow these pathways as a result of fluctuations that create unstable dense-liquid states, microscopic studies indicate these states are long-lived. Using in situ atomic force microscopy to follow 2D assembly of S-layer proteins on supported lipid bilayers, we have obtained a molecular-scale picture of multistage protein crystallization that reveals the importance of conformational transformations in directing the pathway of assembly. We find that monomers with an extended conformation first form a mobile adsorbed phase, from which they condense into amorphous clusters. These clusters undergo a phase transition through S-layer folding into crystalline clusters composed of compact tetramers. Growth then proceeds by formation of new tetramers exclusively at cluster edges, implying tetramer formation is autocatalytic. Analysis of the growth kinetics leads to a quantitative model in which tetramer creation is rate limiting. However, the estimated barrier is much smaller than expected for folding of isolated S-layer proteins, suggesting an energetic rationale for this multistage pathway.

Nucleation of crystals from solution: classical and two-step models

Crystallization is vital to many processes occurring in nature and in the chemical, pharmaceutical, and food industries. Notably, crystallization is an attractive isolation step for manufacturing because this single process combines both particle formation and purification. Almost all of the products based on fine chemicals, such as dyes, explosives, and photographic materials, require crystallization in their manufacture, and more than 90% of all pharmaceutical products contain bioactive drug substances and excipients in the crystalline solid state. Hence control over the crystallization process allows manufacturers to obtain products with desired and reproducible properties. We judge the quality of a crystalline product based on four main properties: size, purity, morphology, and crystal structure. The pharmaceutical industry in particular requires production of the desired crystal form (polymorph) to assure the bioavailability and stability of the drug substance. In solution crystallization, nucleation plays a decisive role in determining the crystal structure and size distribution. Therefore, understanding the fundamentals of nucleation is crucial to achieve control over these properties. Because of its analytical simplicity, researchers have widely applied classical nucleation theory to solution crystallization. However, a number of differences between theoretical predictions and experimental results suggest that nucleation of solids from solution does not proceed via the classical pathway but follows more complex routes. In this Account, we discuss the shortcomings of classical nucleation theory and review studies contributing to the development of the modern two-step model. In the two-step model that was initially proposed for protein crystallization, a sufficient-sized cluster of solute molecules forms first, followed by reorganization of that cluster into an ordered structure. In recent experimental and theoretical studies, we and other researchers have demonstrated the applicability of the two-step mechanism to both macromolecules and small organic molecules, suggesting that this mechanism may underlie most crystallization processes from solutions. Because we have observed an increase in the organization time of appropriate lattice structures with greater molecular complexity, we propose that organization is the rate-determining step. Further development of a clearer picture of nucleation may help determine the optimum conditions necessary for the effective organization within the clusters. In addition, greater understanding of these processes may lead to the design of auxiliaries that can increase the rate of nucleation and avoid the formation of undesired solid forms, allowing researchers to obtain the final product in a timely and reproducible manner.

NMR study of the cataract-linked P23T mutant of human gammaD-crystallin shows minor changes in hydrophobic patches that reflect its retrograde solubility

The Pro23 to Thr (P23T) mutation in human gammaD-crystallin (HGD) shows several cataract phenotypes. We found earlier [A. Pande, O. Annunziata, N. Asherie, O. Ogun, G.B. Benedek, J. Pande, Decrease in protein solubility and cataract formation caused by the Pro23 to Thr mutation in human gamma D-crystallin, Biochemistry 44 (2005) 2491-2500] that the mutation dramatically lowers the solubility of P23T but the overall protein fold is maintained. Recently we observed that solutions of P23T showed liquid-liquid phase transition behavior similar to that of HGD but the liquid-protein crystal phase transition was altered, suggesting an asymmetric distribution of "sticky" patches on the protein surface [J.J. McManus, A. Lomakin, O. Ogun, A. Pande, M. Basan, J. Pande, G.B. Benedek, Altered phase diagram due to a single point mutation in human gammaD-crystallin, Proc. Natl. Acad. Sci. USA 104 (2007) 16856-16861]. Here we present high-resolution NMR studies of HGD and P23T in which we have made nearly complete backbone assignments. The data provide a structural basis for explaining the retrograde solubility of P23T by (a) identifying possible "sticky" patches on the surface of P23T and (b) highlighting their asymmetric distribution.

Phase separation and crystallization of hemoglobin C in transgenic mouse and human erythrocytes

Individuals expressing hemoglobin C (beta6 Glu-->Lys) present red blood cells (RBC) with intraerythrocytic crystals that form when hemoglobin (Hb) is oxygenated. Our earlier in vitro liquid-liquid (L-L) phase separation studies demonstrated that liganded HbC exhibits a stronger net intermolecular attraction with a longer range than liganded HbS or HbA, and that L-L phase separation preceded and enhanced crystallization. We now present evidence for the role of phase separation in HbC crystallization in the RBC, and the role of the RBC membrane as a nucleation center. RBC obtained from both human homozygous HbC patients and transgenic mice expressing only human HbC were studied by bright-field and differential interference contrast video-enhanced microscopy. RBC were exposed to hypertonic NaCl solution (1.5-3%) to induce crystallization within an appropriate experimental time frame. L-L phase separation occurred inside the RBC, which in turn enhanced the formation of intraerythrocytic crystals. RBC L-L phase separation and crystallization comply with the thermodynamic and kinetics laws established through in vitro studies of phase transformations. This is the first report, to the best of our knowledge, to capture a temporal view of intraerythrocytic HbC phase separation, crystal formation, and dissolution.

Role of the beta4Thr-beta73Asp hydrogen bond in HbS polymer and domain formation from multinucleate-containing clusters

Fiber formation and domain formation from deoxy-HbS as well as from beta4 and beta73 HbS variants were investigated after temperature jump using DIC microscopy to gain a basic understanding of the determinants involved. Oversaturated deoxy-HbS generated numerous 14-stranded fibers and formed ovoid-shaped, multispherulitic domains. Domain number increased linearly as a function of time. Oversaturated deoxy-alpha2beta2(E6V,T4S) also generated time-dependent, ovoid-shaped spherulitic domains like HbS and alpha 2beta2(E6V,D73H) in the deoxy form. In contrast, alpha 2beta2(E6V,T4Y) and HbC-Harlem (alpha2beta2(E6V,D73N)) in the deoxy form generated time-dependent, ball-shaped domains containing many straight, crystalline-like fibers without evidence of branching. Some of these domains formed large needlelike crystals after overnight incubation. The inhibitory effect on polymer formation by beta4Tyr in HbS was stronger than that by beta4Ser but weaker than that by beta73Asn or beta73Leu. In contrast, both deoxy- and oxy-alpha2beta2(E6V,T4V) promoted formation of tiny, disordered amorphous aggregates without a delay time like oxy-HbS, which is in contrast to formation after a delay time of needlelike fibers for alpha 2beta2(E6V,D73L). Solubilities for both deoxy- and oxy-alpha 2beta2(E6V,T4V) were similar to that of deoxy-alpha 2beta2(E6V,D73H) but approximately 10-fold lower than that of deoxy-HbS. These results suggest that the strength of the hydrogen bond between beta4Thr and beta73Asp and the balance between the hydrogen bond and beta6Val hydrophobic interactions in deoxy-HbS polymers control formation of different types of fibers in a single domain or lead to formation of disordered, non-nucleated amorphous aggregates. These results also lead to a model in which multinucleation rather than a single-nucleation event occurs in a single cluster to generate numerous fibers growing from a single domain.

Protein-protein interaction on lysozyme crystallization revealed by rotational diffusion analysis

Intermolecular interactions between protein molecules diffusing in various environments underlie many biological processes as well as control protein crystallization, which is a crucial step in x-ray protein structure determinations. Protein interactions were investigated through protein rotational diffusion analysis. First, it was confirmed that tetragonal lysozyme crystals containing fluorescein-tagged lysozyme were successfully formed with the same morphology as that of native protein. Using this nondisruptive fluorescent tracer system, we characterized the effects of sodium chloride and ammonium sulfate concentrations on lysozyme-lysozyme interactions by steady-state and time-resolved fluorescence anisotropy measurements and the introduction of a novel interaction parameter, k(rot). The results suggested that the specific attractive interaction, which was reflected in the retardation of the protein rotational diffusion, was induced depending on the salt type and its concentration. The change in the attractive interactions also correlated with the crystallization/precipitation behavior of lysozyme. Moreover, we discuss the validity of our rotational diffusion analysis through comparison with the osmotic second virial coefficient, B(22), previously reported for lysozyme and those estimated from k(rot).

Altered phase diagram due to a single point mutation in human gammaD-crystallin

The P23T mutant of human gammaD-crystallin (HGD) is associated with cataract. We have previously investigated the solution properties of this mutant, as well as those of the closely related P23V and P23S mutants, and shown that although mutations at site 23 of HGD do not produce a significant structural change in the protein, they nevertheless profoundly alter the solubility of the protein. Remarkably, the solubility of the mutants decreases with increasing temperature, in sharp contrast to the behavior of the native protein. This inverted solubility corresponds to a strong increase in the binding energy with temperature. Here we have investigated the liquid-liquid coexistence curve and the diffusivity of the P23V mutant and find that these solution properties are unaffected by the mutation. This means that the chemical potentials in the solution phase are essentially unaltered. The apparent discrepancy between the interaction energies in the solution phase, as compared with the solid phase, is explicable in terms of highly anisotropic interprotein interactions, which are averaged out in the solution phase but are fully engaged in the solid phase.

Sickle-cell haemoglobin polymerization: is it the primary pathogenic event of sickle-cell anaemia?

Sickle cell anaemia is associated with a mutant haemoglobin, HbS, which forms polymers in the red blood cells of patients. The primary role of the HbS polymerization for the pathophysiology has been questioned: observations in patients and model organisms contradict deterministic scenarios of sickling crises triggered by polymerization. However, results with knock-out sickle-cell mice, which were cured by delaying HbS polymerization, reconfirm polymerization's primary role. To reconcile the contradictory observations, this article reviews recent findings on two steps in polymerization: homogeneous nucleation of fibres, and their growth. The fibre growth is faster by far than for any other protein ordered structure. This is due to a negligible free-energy barrier for incorporation into a fibre, determined by an entropy gain, stemming from the release of water molecules structured around HbS. The kinetics of fibre nucleation have shown that the formation of the polymer nucleus is preceded by a metastable droplet of a dense liquid. The properties of the dense liquid are sensitive functions of solution composition, including components in micro- and nanomolar amounts. This mechanism allows low-concentration solution components to strongly affect the nucleation kinetics, accounting for the high variability of the disease. These insights can potentially be utilized for control of HbS polymerization and treatment of the disease.

Protein phase behavior and crystallization: Effect of glycerol

Glycerol is widely used as an additive to stabilize proteins in aqueous solution. We have studied the effect of up to 40 wt % glycerol on the crystallization of lysozyme from brine. As the glycerol concentration increased, progressively larger amounts of salt were needed to crystallize the protein. Like previous authors, we interpret this as evidence for glycerol changing the interaction between lysozyme molecules. We quantitatively model the interprotein interaction using a Derjaguin-Landau-Verwey-Overbeek potential. We find that the effect of glycerol can be entirely accounted for by the way it modifies the dielectric constant and refractive index of the solvent. Quantifying the interprotein interaction by the second virial coefficient, B(2), we find a universal crystallization boundary for all glycerol concentrations.

Influence of Poly(ethylene glycol) Molecular Mass on Separation and Ordering in Solutions of CiEj Nonionic Surfactants: Depletion Interactions and Steric Effects

We study ternary mixtures of nonionic surfactants C(i)E(j) (i = 12; j = 5, 6, 8) and poly(ethylene glycol) (PEG) in water. For sufficiently large molecular mass of PEG (M >M(sep) approximately 600), we observe a lowering of phase separation temperature with an increase in polymer concentration. The value of M(sep) is consistent with the analysis based on depletion interactions between micelles induced by polymer chains. We also demonstrate that there is another critical molecular mass of PEG (M = M* approximately 2000) necessary to induce ordering in the surfactant-rich phase. This critical molecular mass follows from two requirements: (a) PEG has to reduce the separation temperature below a temperature of hexagonal-isotropic phase transition in a binary surfactant-water mixture and (b) the PEG radius of gyration has to be larger than the size of the water channels in the hexagonal phase.

Two-step mechanism of homogeneous nucleation of sickle cell hemoglobin polymers

Sickle cell anemia is a debilitating genetic disease that affects hundreds of thousands of babies born each year worldwide. Its primary pathogenic event is the polymerization of a mutant, sickle cell, hemoglobin (HbS); and this is one of a line of diseases (Alzheimer's, Huntington's, prion, etc.) in which nucleation initiates pathophysiology. We show that the homogeneous nucleation of HbS polymers follows a two-step mechanism with metastable dense liquid clusters serving as precursor to the ordered nuclei of the HbS polymer. The evidence comes from data on the rates of fiber nucleation and growth and nucleation delay times, the interaction of fibers with polarized light, and mesoscopic metastable HbS clusters in solution. The presence of a precursor in the HbS nucleation mechanism potentially allows low-concentration solution components to strongly affect the nucleation kinetics. The variations of these concentrations in patients might account for the high variability of the disease in genetically identical patients. In addition, these components can potentially be utilized for control of HbS polymerization and treatment of the disease.

Metastable Liquid Clusters in Super- and Undersaturated Protein Solutions

Dense liquid phases, metastable with respect to a solid phase, but stable with respect to the solution, have been known to form in solutions of proteins and small-molecule substances. Here, with the protein lumazine synthase as a test system, using dynamic and static light scattering and atomic force microscopy, we demonstrate submicron size clusters of dense liquid. In contrast to the macroscopic dense liquid, these clusters are metastable not only with respect to the crystals, but also with respect to the low-concentration solution: the characteristic cluster lifetime is limited to approximately 10 s, after which they decay. The cluster population is detectable only if they occupy >10(-6) of the solution volume and have a number density >105 cm-3 for 3 to 11% of the monitored time. The cluster volume fraction varies within wide limits and reaches up to 10(-3). Increasing protein concentration increases the frequency of cluster detection but does not affect the ranges of the cluster sizes, suggesting that a preferred cluster size exists. A simple Monte Carlo model with protein-like potentials reproduces the metastable clusters of dense liquid with limited lifetimes and variable sizes and suggests that the mean cluster size is determined by the kinetics of growth and decay and not by thermodynamics.

Association Kinetics of Wild- and Mutant-Type Ynd1p in Relation to Quality of Grown Crystals

The intermolecular interaction and association dynamics of the Ynd1p protein were investigated using dynamic and time-resolved static light scattering measurements. The mutual diffusion coefficients of wild- and mutant-type (a single amino acid substitution) Ynd1p monomer were measured in 50 mM 2-(N-morpholino)ethanesulfonic acid (MES) buffer with 5 mM MnCl2 and 7.5% (v/v) ethylene glycol. Both translational diffusion coefficients at a zero protein concentration were (40.3 +/- 0.2) x 10(-12) m2/s at 20 degrees C and a pH of 7.0, so the hydrodynamic radius of the monomers was 4.1 +/- 0.1 nm. The measured intermolecular interaction between monomers, however, showed that the mutant-type Ynd1p had a stronger attractive force. Time-resolved static light scattering measurements showed that the association of mutant-type Ynd1p yielded a larger number of aggregates than that of wild-type Ynd1p. The time dependence of aggregate gyration radius differed between the two types. Fractal dimension analysis using scattering intensity data suggested that the inner structure of the aggregates changed from loose to rigid with time. Although this phenomenon is common for wild and mutant types, the differences in the number of aggregates yielded in the initial stages and in the intermolecular interaction affected the quality of the final grown crystals. That is, single crystals of Ynd1p grew in the mutant-type protein solution and polycrystals of Ynd1p grew in the wild-type protein solution.

Metastable mesoscopic clusters in solutions of sickle-cell hemoglobin

Sickle cell hemoglobin (HbS) is a mutant, whose polymerization while in deoxy state in the venous circulation underlies the debilitating sickle cell anemia. It has been suggested that the nucleation of the HbS polymers occurs within clusters of dense liquid, existing in HbS solutions. We use dynamic light scattering with solutions of deoxy-HbS, and, for comparison, of oxy-HbS and oxy-normal adult hemoglobin, HbA. We show that solutions of all three Hb variants contain clusters of dense liquid, several hundred nanometers in size, which are metastable with respect to the Hb solutions. The clusters form within a few seconds after solution preparation and their sizes and numbers remain relatively steady for up to 3 h. The lower bound of the cluster lifetime is 15 ms. The clusters exist in broad temperature and Hb concentration ranges, and occupy 10(-5)-10(-2) of the solution volume. The results on the cluster properties can serve as test data for a potential future microscopic theory of cluster stability and kinetics. More importantly, if the clusters are a part of the nucleation mechanism of HbS polymers, the rate of HbS polymerization can be controlled by varying the cluster properties.

Multiple extrema in the intermolecular potential and the phase diagram of protein solutions

Recent experiments have revealed several surprising features of the phase equilibria in protein solutions: liquid-liquid phase separation which is, in some cases, metastable with respect to the liquid-solid equilibrium, and in others-unobservable; widely varying crystallization enthalpies, including completely athermal crystallization; the co-existence of several crystalline polymorphs; and others. Other studies have shown that the solvent molecules at the hydrophobic and polar patches on the protein molecular surfaces are structured, introducing repulsive forces at surface separations equal to several water molecule sizes. In search of a causal link between the latter and former findings, we apply Monte Carlo simulation techniques in the investigation of phase diagrams associated with globular biological molecules in solution. We account for the solvent structuring via short-range isotropic two-body intermolecular potentials exhibiting multiple extrema. We show that the introduction of a repulsive maximum or a secondary attractive minimum at separations longer than the primary attractive minimum has dramatic effects on the phase diagram: liquid-liquid separation curves are driven to lower or higher temperatures, the sensitivity of the solubility curve (liquidus) to temperature, i.e., the enthalpy of crystallization, is significantly reduced or enhanced, metastable liquid-liquid separation may become stable and vice versa, and both low- and high-density crystalline phases are observed. The similarity of these features of the simulated phase behavior to those observed experimentally suggests that at least some of the mysteries of the protein phase equilibria may be due to the structuring of the solvent around the protein molecular surfaces. Another conclusion is that at least some of the dense liquids seen in protein solutions may be stable and not metastable with respect to a solid phase.

Spinodal for the solution-to-crystal phase transformation

The formation of crystalline nuclei from solution has been shown for many systems to occur in two steps: the formation of quasidroplets of a disordered intermediate, followed by the nucleation of ordered crystalline embryos within these droplets. The rate of each step depends on a respective free-energy barrier and on the growth rate of its near-critical clusters. We address experimentally the relative significance of the free-energy barriers and the kinetic factors for the nucleation of crystals from solution using a model protein system. We show that crystal nucleation is 8-10 orders of magnitude slower than the nucleation of dense liquid droplets, i.e., the second step is rate determining. We show that at supersaturations of three or four k(B)T units, crystal nuclei of five, four, or three molecules transform into single-molecule nuclei, i.e., the significant nucleation barrier vanishes below the thermal energy of the molecules. We show that the main factor, which determines the rate of crystal nucleation, is the slow growth of the near-critical ordered clusters within the quasidroplets of the disordered intermediate. Analogous to the spinodal in supersaturated fluids, we define a solution-to-crystal spinodal from the transition to single-molecule crystalline nuclei. We show that heterogeneous nucleation centers accelerate nucleation not only because of the wettinglike effects that lower the nucleation barrier, as envisioned by classical theory, but by helping the kinetics of growth of the ordered crystalline embryos.

Growth and form of spherulites

Many structural materials (metal alloys, polymers, minerals, etc.) are formed by quenching liquids into crystalline solids. This highly nonequilibrium process often leads to polycrystalline growth patterns that are broadly termed "spherulites" because of their large-scale average spherical shape. Despite the prevalence and practical importance of spherulite formation, only rather qualitative concepts of this phenomenon exist. It is established that phase field methods naturally account for diffusional instabilities that are responsible for dendritic single-crystal growth. However, a generalization of this model is required to describe spherulitic growth patterns, and in the present paper we propose a minimal model of this fundamental crystal growth process. Our calculations indicate that the diversity of spherulitic growth morphologies arises from a competition between the ordering effect of discrete local crystallographic symmetries and the randomization of the local crystallographic orientation that accompanies crystal grain nucleation at the growth front [growth front nucleation (GFN)]. This randomization in the orientation accounts for the isotropy of spherulitic growth at large length scales and long times. In practice, many mechanisms can give rise to GFN, and the present work describes and explores three physically prevalent sources of disorder that lead to this kind of growth. While previous phase field modeling elucidated two of these mechanisms--disorder created by particulate impurities or other static disorder or by the dynamic heterogeneities that spontaneously form in supercooled liquids (even pure ones)--the present paper considers an additional mechanism, crystalline branching induced by a misorientation-dependent grain boundary energy, which can significantly affect spherulite morphology. We find the entire range of observed spherulite morphologies can be reproduced by this generalized phase field model of polycrystalline growth.

Kinetics of two-step nucleation of crystals

When the nucleation of a stable crystalline phase directly in a supersaturated old phase is greatly retarded, the crystal nuclei might nucleate within faster-forming particles of an intermediate phase. Here we present a theoretical investigation of the kinetics of this two-step nucleation of crystals and derive general expressions for the time dependence of the number of crystals nucleated within the particles of the intermediate phase. The results reveal that crystal nucleation can be strongly delayed by the slow growth of the particles and/or by the slow nucleation of the crystals in them. Furthermore, the linear part of the time dependence of the number of nucleated crystals is determined by the formation rate of the intermediate particles. This is in contrast with the one-step nucleation of crystals when this linear part is determined by the rate of crystal nucleation directly in the old phase. Criteria are proposed for distinction between the one- and two-step nucleation mechanisms, based on the supersaturation dependence of the delay time for nucleation. The application of the theoretical approach to the analysis of experimental data on the nucleation of crystals and other ordered aggregates of protein and other soluble materials is discussed.

Nucleation of ordered solid phases of proteins via a disordered high-density state: Phenomenological approach

Nucleation of ordered solid phases of proteins triggers numerous phenomena in laboratory, industry, and in healthy and sick organisms. Recent simulations and experiments with protein crystals suggest that the formation of an ordered crystalline nucleus is preceded by a disordered high-density cluster, akin to a droplet of high-density liquid that has been observed with some proteins; this mechanism allowed a qualitative explanation of recorded complex nucleation kinetics curves. Here, we present a simple phenomenological theory that takes into account intermediate high-density metastable states in the nucleation process. Nucleation rate data at varying temperature and protein concentration are reproduced with high fidelity using literature values of the thermodynamic and kinetic parameters of the system. Our calculations show that the growth rate of the near-critical and supercritical ordered clusters within the dense intermediate is a major factor for the overall nucleation rate. This highlights the role of viscosity within the dense intermediate for the formation of the ordered nucleus. The model provides an understanding of the action of additives that delay or accelerate nucleation and presents a framework within which the nucleation of other ordered protein solid phases, e.g., the sickle cell hemoglobin polymers, can be analyzed.

A Metastable Prerequisite for the Growth of Lumazine Synthase Crystals

Dense liquid phases, metastable with respect to a solid phase, form in solutions of proteins and small-molecule materials. They have been shown to serve as a prerequisite for the nucleation of crystals and other ordered solid phases. Here, using crystals of the protein lumazine synthase from Bacillus subtilis, which grow by the generation and spreading of layers, we demonstrate that within a range of supersaturations the only mechanism of generation of growth layers involves the association of submicrometer-size droplets of the dense liquid to the crystal surface. The dense liquid is metastable not only with respect to the crystals, but also with respect to the low-concentration solution: dynamic light scattering reveals that the droplets' lifetime is limited to several seconds, after which they decay into the low-concentration solution. The short lifetime does not allow growth to detectable dimensions so that liquid-liquid phase separation is not observed within a range of conditions broader than the one used for crystallization. If during their lifetime the droplets encounter a crystal surface, they lower their free energy not by decay, but by transformation into crystalline matter, ensuring perfect registry with the substrate. These observations illustrate two novel features of phase transformations in solutions: the existence of doubly metastable, short-lifetime dense phases and their crucial role for the growth of an ordered solid phase.

Liquid-liquid phase separation in hemoglobins: distinct aggregation mechanisms of the beta6 mutants

Reversible liquid-liquid (L-L) phase separation in the form of high concentration hemoglobin (Hb) solution droplets is favored in an equilibrium with a low-concentration Hb solution when induced by inositol-hexaphosphate in the presence of polyethylene glycol 4000 at pH 6.35 HEPES (50 mM). The L-L phase separation of Hb serves as a model to elucidate intermolecular interactions that may give rise to accelerated nucleation kinetics of liganded HbC (beta6 Lys) compared to HbS (beta6 Val) and HbA (beta6 Glu). Under conditions of low pH (pH 6.35) in the presence of inositol-hexaphosphate, COHb assumes an altered R-state. The phase lines for the three Hb variants in concentration and temperature coordinates indicate that liganded HbC exhibits a stronger net intermolecular attraction with a longer range than liganded HbS and HbA. Over time, L-L phase separation gives rise to amorphous aggregation and subsequent formation of crystals of different kinetics and habits, unique to the individual Hb. The composite of R- and T-like solution aggregation behavior indicates that this is a conformationally driven event. These results indicate that specific contact sites, thermodynamics, and kinetics all play a role in L-L phase separation and differ for the beta6 mutant hemoglobins compared to HbA. In addition, the dense liquid droplet interface or aggregate interface noticeably participates in crystal nucleation.

Smooth transition from metastability to instability in phase separating protein solutions

For insight into the structure and dynamics of phases emerging upon crossing the metastability/instability boundary we monitor with optical microscopy, in real time and in real space, the generation of a dense liquid phase in high-concentration solutions of the protein lysozyme after temperature quenches into thermodynamically defined metastable and unstable regions. We show with this system, which is a poor fit to mean-field assumptions, that the evolution of the structure factor during nucleation is similar to that during spinodal decomposition and reveals no singularity predicted upon crossing the metastability boundary. We introduce two kinetic definitions of the metastability/instability boundary that yield values within approximately 1.5 K, i.e., the boundary appears as an area rather than a line, which is near and above the thermodynamic prediction. Delay times for the appearance of the new phase in the unstable regime are significant, i.e., new-phase growth is hindered by kinetic barriers. While our results agree with predictions of the non-mean-field theories of phase transformations, the experimentally observed behavior is richer than the one envisioned by theory.

Aggregation of normal and sickle hemoglobin in high concentration phosphate buffer

Sickle cell disease is caused by a mutant form of hemoglobin, hemoglobin S, that polymerizes under hypoxic conditions. The extent and mechanism of polymerization are thus the subject of many studies of the pathophysiology of the disease and potential treatment strategies. To facilitate such studies, a model system using high concentration phosphate buffer (1.5 M-1.8 M) has been developed. To properly interpret results from studies using this model it is important to understand the similarities and differences in hemoglobin S polymerization in the model compared to polymerization under physiological conditions. In this article, we show that hemoglobin S and normal adult hemoglobin, hemoglobin A, aggregate in high concentration phosphate buffer even when the concentration of hemoglobin is below the solubility defined for polymerization. This phenomenon was not observed using 0.05 M phosphate buffer or in another model system we studied that uses dextran to enhance polymerization. We have used static light scattering, dynamic light scattering, and differential interference contrast microscopy to confirm aggregation of deoxygenated and oxygenated hemoglobins below their solubility and have shown that this aggregation is not observable using turbidity measurements, a common technique for assessing polymerization. We have also shown that the aggregation increases with increasing temperature in the range of 15 degrees -37 degrees C and that it increases as the concentration of phosphate increases. These studies contribute to the working knowledge of how to properly apply studies of hemoglobin S polymerization that are conducted using the high phosphate model.

A finite-size scaling study of a model of globular proteins

Grand canonical Monte Carlo simulations are used to explore the metastable fluid-fluid coexistence curve of the modified Lennard-Jones model of globular proteins of ten Wolde and Frenkel [Science, 277, 1975 (1997)]. Using both mixed-field finite-size scaling and histogram-reweighting methods, the joint distribution of density and energy fluctuations is analyzed at coexistence to accurately determine the critical-point parameters. The subcritical coexistence region is explored using the recently developed hyper parallel tempering Monte Carlo simulation method along with histogram reweighting to obtain the density distributions. The phase diagram for the metastable fluid-fluid coexistence curve is calculated in close proximity to the critical point, a region previously unattained by simulations.

Nucleation of protein crystals

This paper introduces nucleation theory applied to crystallizing protein solutions. It is shown that the classical approach explains the available nucleation data under most conditions used for growing protein crystals for structural studies and for industrial crystallization. However, it fails to explain most experimental data on the structure of the critical clusters. It is also shown that for open systems working out of equilibrium, such as hanging-drop and counterdiffusion techniques, the geometry of the Ostwald-Myers protein solubility diagram and the number, size, and quality of the forming crystals depend not only on supersaturation but also on the rate of development of supersaturation.

Mechanisms of Homogeneous Nucleation of Polymers of Sickle Cell Anemia Hemoglobin in Deoxy State

The primary pathogenic event of sickle cell anemia is the polymerization of the mutant hemoglobin (Hb) S within the red blood cells, occurring when HbS is in deoxy state in the venous circulation. Polymerization is known to start with nucleation of individual polymer fibers, followed by growth and branching via secondary nucleation, yet the mechanisms of nucleation of the primary fibers have never been subjected to dedicated tests. We implement a technique for direct determination of rates and induction times of primary nucleation of HbS fibers, based on detection of emerging HbS polymers using optical differential interference contrast microscopy after laser photolysis of CO-HbS. We show that: (i). nucleation throughout these determinations occurs homogeneously and not on foreign substrates; (ii). individual nucleation events are independent of each other; (iii). the nucleation rates are of the order of 10(6)-10(8)cm(-3)s(-1); (iv). nucleation induction times agree with an a priori prediction based on Zeldovich's theory; (v). in the probed parameter space, the nucleus contains 11 or 12 molecules. The nucleation rate values are comparable to those leading to erythrocyte sickling in vivo and suggest that the mechanisms deduced from in vitro experiments might provide physiologically relevant insights. While the statistics and dynamics of nucleation suggest mechanisms akin to those for small-molecule and protein crystals, the nucleation rate values are nine to ten orders of magnitude higher than those known for protein crystals. These high values cannot be rationalized within the current understanding of the nucleation processes.

Sequence variability of proteins evolutionarily constrained by solution-thermodynamic function

Focusing on silk fibroin and hemoglobin molecules as templates, we model protein homolog dispersal across sequence-fitness landscapes determined by solution thermodynamics. Landscapes are constructed by inspecting an idealized theoretical phase topology associated with sequence length and hydrophobic-polar composition, comprising liquid-liquid phase separation, gelation and liquid crystalline self-assembly. We then calculate the distribution of homologs in sequence space as steady states of a simple mutation-selection dynamics. The results are consistent with Swiss-Prot bioinformatic data.

Liquid-liquid phase separation of a surfactant-solubilized membrane protein

Solubilization of membrane proteins requires surfactants, whose structural properties play a crucial role in determining the protein phase behavior. We show that ionization of a pH-sensitive surfactant, lauryldymethylamino-N-oxide, bound to the bacterial photosynthetic Reaction Center, induces protein phase segregation in micrometric "droplets." Liquid-liquid phase separation takes place in a narrow pH range, is promoted by increasing temperature, and vanishes by adding salt. After a fast initial droplet growth, the nearly arrested kinetics at a later stage leaves the system in a finely divided, long-lasting emulsified state.

Atomic-level observation of macromolecular crowding effects: escape of a protein from the GroEL cage

Experimental work has demonstrated that the efficient operation of the GroEL-GroES chaperonin machinery is sensitive to the presence of macromolecular crowding agents. Here, I describe atomically detailed computer simulations that provide a microscopic view of how crowding effects are exerted. Simulations were performed to compute the free energy required to extract the protein rhodanese from the central cavity of GroEL into solutions containing a range of crowder concentrations. The computed energetics allow the total yield of folded protein to be predicted; the calculated yields show a nonlinear dependence on the concentration of crowding agent identical to that observed experimentally. The close correspondence between simulation and experiment prompts the use of the former in a truly predictive setting: simulations are used to suggest that more effective crowding agents might be designed by exploiting an "agoraphobic effect."

Observation of liquid-liquid phase separation for eye lens gammaS-crystallin

gammaS-crystallin (gammaS) is an important human and bovine eye lens protein involved in maintaining the transparency of the eye. By adding small amounts of polyethylene glycol (PEG) to the binary aqueous bovine gammaS solutions, we have observed liquid-liquid phase separation (LLPS) at -8 degrees C and revealed that, in the binary gammaS-water system, this phase transition would occur at -28 degrees C. We have measured both the effect of PEG concentration on the LLPS temperature and proteinPEG partitioning between the two liquid coexisting phases. We use our measurements of proteinPEG partitioning to determine the nature and the magnitude of the gammaS-PEG interactions and to quantitatively assess the effectiveness of PEG as a crystallizing agent for gammaS. We use our measurements of LLPS temperature as a function of protein and PEG concentration to successfully determine the location of the critical point for the binary gammaS-water system. This phase transition cannot be observed in the absence of PEG because it is inaccessible due to the freezing of the system. Our findings indicate that the effective interactions between gammaS molecules in the binary gammaS-water system are attractive. We compare the magnitude of the attraction found for gammaS with the results obtained for the other gamma-crystallins for which the critical temperature is located above the freezing point of the system. This work suggests that PEG can be used to reveal the existence of LLPS for a much wider range of binary protein-water systems than known previously.

Effect of polyethylene glycol on the liquid-liquid phase transition in aqueous protein solutions

We have studied the effect of polyethylene glycol (PEG) on the liquid-liquid phase separation (LLPS) of aqueous solutions of bovine gammaD-crystallin (gammaD), a protein in the eye lens. We observe that the phase separation temperature increases with both PEG concentration and PEG molecular weight. PEG partitioning, which is the difference between the PEG concentration in the two coexisting phases, has been measured experimentally and observed to increase with PEG molecular weight. The measurements of both LLPS temperature and PEG partitioning in the ternary gammaD-PEG-water systems are used to successfully predict the location of the liquid-liquid phase boundary of the binary gammaD-water system. We show that our LLPS measurements can be also used to estimate the protein solubility as a function of the concentration of crystallizing agents. Moreover, the slope of the tie-lines and the dependence of LLPS temperature on polymer concentration provide a powerful and sensitive check of the validity of excluded volume models. Finally, we show that the increase of the LLPS temperature with PEG concentration is due to attractive protein-protein interactions.