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.

Interactions of hemin with model erythrocyte membranes.

To address the interactions of hemin with phospholipid bilayers, we introduce hemin to a solution of dimyristoylphosphatidylcholine (DMPC), a long chain phospholipid, and 3-(cholamidopropyl)(dimethylammonio)-2-hydroxy-1-propanesulfonate (CHAPSO), a detergent, in which vesicles form at around 37 degrees C. We show that vesicles composed of DMPC/CHAPSO form and grow, following a mechanism that does not trap solution and excludes larger solutes, such as hemin, from the vesicle interior. The existence of a limited number of patches of likely 2D crystalline hemin embedded in the phospholipid bilayer suggests that this layer is saturated with hemin molecules. We show that despite this saturation, even after prolonged contact with hemin-containing solution outside the vesicles, hemin is not released on the other side of the membrane; i.e., the phospholipid bilayer is impermeable to hemin. Comparison of the properties of the model membrane to those of the erythrocyte membrane suggests that the latter might also be impermeable to hemin and, given the absence of pores suitable to hemin in the erythrocyte membranes, that hemin might accumulate in erythrocytes after its release due to hemoglobin instability.

Localized generation of attoliter protein solution droplets by electrofocused liquid-liquid separation.

We develop a simple technique for the generation of droplets of a high-concentration protein solution of attoliter volume and a diameter of >or=500 nm, containing several thousand protein molecules, and their accurate deposition on micron-size electrodes, standalone or in an array. The technique is based on liquid-liquid phase separation in the homogeneous region of the phase diagram, induced by a nonuniform weak electric field, which also accurately directs the generated droplets toward the electrode. We show that the protein molecules are not chemically modified and retain their integrity, conformation, and activity post deposition. The technique is applicable to the manufacture of various types of protein/microelectrode arrays and is tunable, scalable, and applicable to a broad range of proteins.

Viscoelasticity in homogeneous protein solutions.

We probe the transport properties in protein solutions stable with respect to any, solid or liquid, phase separation as a step in the understanding of transport in the cytosol of live cells. We determine the mean-squared displacement of probe particles in the time range 10;{-3}-10 s in solutions of a model protein. The tested solutions exhibit significant elasticity at high frequencies, while at low frequencies, they are purely viscous. We attribute this viscoelasticity to a dense network of weakly-bound chains of protein molecules with characteristic lifetime of 10-100 ms. The found intrinsic viscoelasticity of protein solutions should be considered in biochemical kinetics models.

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.

Determination of the transition-state entropy for aggregation suggests how the growth of sickle cell hemoglobin polymers can be slowed.

Sickle cell anemia is associated with the mutant hemoglobin HbS, which forms polymers in red blood cells of patients. The growth rate of the polymers is several micrometers per second, ensuring that a polymer fiber reaches the walls of an erythrocyte (which has a 7-microm diameter) within a few seconds after its nucleation. To understand the factors that determine this unusually fast rate, we analyze data on the growth rate of the polymer fibers. We show that the fiber growth follows a first-order Kramers-type kinetics model. The entropy of the transition state for incorporation into a fiber is 95 J mol(-1) K(-1), very close to the known entropy of polymerization. This agrees with a recent theoretical estimate for the hydrophobic interaction and suggests that the gain of entropy in the transition state is due to the release of the last layer of water molecules structured around contact sites on the surface of the HbS molecules. As a result of this entropy gain, the free-energy barrier for incorporation of HbS molecules into a fiber is negligible and fiber growth is unprecedentedly fast. This finding suggests that fiber growth can be slowed by components of the red cell cytosol, native or intentionally introduced, which restructure the hydration layer around the HbS molecules and thus lower the transition state entropy for incorporation of an incoming molecule into the growing fiber.

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.

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.

The kinetics of nucleation and growth of sickle cell hemoglobin fibers.

Polymerization of sickle cell hemoglobin (HbS) in deoxy state is one of the basic events in the pathophysiology of sickle cell anemia. For insight into the polymerization process, we monitor the kinetics of nucleation and growth of the HbS polymer fibers. We define a technique for the determination of the rates J and delay times theta of nucleation and the fiber growth rates R of deoxy-HbS fibers, based on photolysis of CO-HbS by laser illumination. We solve numerically time-dependent equations of heat conductance and CO transport, coupled with respective photo-chemical processes, during kinetics experiments under continuous illumination. After calibration with experimentally determined values, we define a regime of illumination ensuring uniform temperature and deoxy-HbS concentration, and fast (within <1 s) egress to steady conditions. With these procedures, data on the nucleation and growth kinetics have relative errors of <5% and are reproducible within 10% in independent experiments. The nucleation rates and delay times have steep, exponential dependencies on temperature. In contrast, the average fiber growth rates only weakly depend on temperature. The individual growth rates vary by up to 40% under identical conditions. These variations are attributed to instability of the coupled kinetics and diffusion towards the growing end of a fiber. The activation energy for incorporation of HbS molecules into a polymer is E(A)=50 kJ mol(-1), a low value indicating the significance of the hydrophobic contacts in the HbS polymer. More importantly, the contrast between the strong theta(T) and weak R(T) dependencies suggests that the homogenous nucleation of HbS polymers occurs within clusters of a precursor phase. This conclusion may have significant consequences for the understanding of the pathophysiology of sickle cell anemia and should be tested in further work.

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.

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.

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.

Thermodynamics of the hydrophobicity in crystallization of insulin.

For insight into the solvent structure around protein molecules and its role in phase transformations, we investigate the thermodynamics of crystallization of the rhombohedral form of porcine insulin crystals. We determine the temperature dependence of the solubility at varying concentration of the co-solvent acetone, Cac=0%, 5%, 10%, 15%, and 20%, and find that, as a rule, the solubility of insulin increases as temperature increases. The enthalpy of crystallization, undergoes a stepwise shift from approximately -20 kJ mol(-1) at Cac=0%, 5%, and 10% to approximately -55 kJ mol(-1) at Cac=15% and 20%. The entropy change upon crystallization is approximately 35 J mol(-1) K(-1) for the first three acetone concentrations, and drops to approximately -110 J mol(-1) K(-1) at Cac=15% and 20%. DeltaS degrees cryst>0 indicates release of solvent, mostly water, molecules structured around the hydrophobic patches on the insulin molecules' surface in the solution. As Cac increases to 15% and above, unstructured acetone molecules apparently displace the waters and their contribution to DeltaS degrees cryst is minimal. This shifts DeltaS degrees cryst to a negative value close to the value expected for tying up of one insulin molecule from the solution. The accompanying increase in DeltaH degrees cryst suggests that the water structured around the hydrophobic surface moieties has a minimal enthalpy effect, likely due to the small size of these moieties. These findings provide values of the parameters needed to better control insulin crystallization, elucidate the role of organic additives in the crystallization of proteins, and help us to understand the thermodynamics of the hydrophobicity of protein molecules and other large molecules.

Intermolecular interactions, nucleation, and thermodynamics of crystallization of hemoglobin C.

The mutated hemoglobin HbC (beta 6 Glu-->Lys), in the oxygenated (R) liganded state, forms crystals inside red blood cells of patients with CC and SC diseases. Static and dynamic light scattering characterization of the interactions between the R-state (CO) HbC, HbA, and HbS molecules in low-ionic-strength solutions showed that electrostatics is unimportant and that the interactions are dominated by the specific binding of solutions' ions to the proteins. Microscopic observations and determinations of the nucleation statistics showed that the crystals of HbC nucleate and grow by the attachment of native molecules from the solution and that concurrent amorphous phases, spherulites, and microfibers are not building blocks for the crystal. Using a novel miniaturized light-scintillation technique, we quantified a strong retrograde solubility dependence on temperature. Thermodynamic analyses of HbC crystallization yielded a high positive enthalpy of 155 kJ mol(-1), i.e., the specific interactions favor HbC molecules in the solute state. Then, HbC crystallization is only possible because of the huge entropy gain of 610 J mol(-1) K(-1), likely stemming from the release of up to 10 water molecules per protein intermolecular contact-hydrophobic interaction. Thus, the higher crystallization propensity of R-state HbC is attributable to increased hydrophobicity resulting from the conformational changes that accompany the HbC beta 6 mutation.

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

We show that in solutions of human hemoglobin (Hb)--oxy- and deoxy-Hb A or S--of near-physiological pH, ionic strength, and Hb concentration, liquid-liquid phase separation occurs reversibly and reproducibly at temperatures between 35 and 40 degrees C. In solutions of deoxy-HbS, we demonstrate that the dense liquid droplets facilitate the nucleation of HbS polymers, whose formation is the primary pathogenic event for sickle cell anemia. In view of recent results that shifts of the liquid-liquid separation phase boundary can be achieved by nontoxic additives at molar concentrations up to 30 times lower than the protein concentrations, these findings open new avenues for the inhibition of the HbS polymerization.

Phase-locking for maps of a torus: a computer assisted study.

In the description of bifurcations in a family of maps of an n-torus it is natural to consider phase-locked regions in the parameter space that correspond approximately to the sets of parameter values for which the maps have invariant tori. The extreme case of phase-locking is resonance, where the torus map has a periodic orbit. We study a family of maps of an n-torus that only differ from a family of torus translations by a small nonlinear perturbation. The widths of the phase-locked regions for this family generally increase linearly with the perturbation amplitude. However, this growth varies to a higher power law for families of maps that are given by trigonometric polynomials (the so-called Mathieu-type maps). The exponent of the asymptotic power law can be found by simple arithmetic calculations that relate the spectrum of the trigonometric polynomial to the unperturbed translation. Perturbation theory and these calculations predict that typical resonance regions for the family of Mathieu-type maps are narrow elliptical annuli. All these results are illustrated in a number of numerical examples.