Homogeneous nucleation in sickle hemoglobin: stochastic measurements with a parallel method

An α-chain modification rivals the effect of fetal hemoglobin in retarding the rate of sickle cell fiber formation

Adults with sickle cell disease bear a mutation in the β-globin gene, leading to the expression of sickle hemoglobin (HbS; α2βS2). Adults also possess the gene for γ-globin, which is a component of fetal hemoglobin (HbF, α2γ2); however, γ-chain expression normally ceases after birth. As HbF does not form the fibers that cause the disease, pharmacological and gene-modifying interventions have attempted to either reactivate expression of the γ chain or introduce a gene encoding a modified β chain having γ-like character. Here, we show that a single-site modification on the α chain, αPro114Arg, retards fiber formation as effectively as HbF. Because this addition to the repertoire of anti-sickling approaches acts independently of other modifications, it could be coupled with other therapies to significantly enhance their effectiveness.

Entropy of Charge Inversion in DNA including One-Loop Fluctuations

The entropy and charge distributions have been calculated for a simple model of polyelectrolytes attached to the surface of DNA using a field-theoretic method that includes fluctuations to the lowest one-loop order beyond mean-field theory. Experiments have revealed correlation-driven behavior of DNA in charged solutions, including charge inversion and condensation. In our model, the condensed polyelectrolytes are taken to be doubly charged dimers of length comparable to the distance between sites along the phosphate chains. Within this lattice gas model, each adsorption site is assumed to have either a vacancy or a positively charged dimer attached with the dimer oriented either parallel or perpendicular to the double-helix DNA chain. We find that the inclusion of the fluctuation terms decreases the entropy by ∼50% in the weak-binding regime. There, the bound dimer concentration is low because the dimers are repelled from the DNA molecule, which competes with the chemical potential driving them from the solution to the DNA surface. Surprisingly, this decrease in entropy due to correlations is so significant that it overcompensates for the entropy increase at the mean-field level, so that the total entropy is even lower than in the absence of interactions between lattice sites. As a bonus, we present a transparent exposition of the methods used that could be useful to students and others wishing to use this formulation to extend this calculation to more realistic models.

Voxelotor does not inhibit sickle hemoglobin fiber formation upon complete deoxygenation

The drug voxelotor (commercially known as Oxbryta) has been approved by the US Food and Drug Administration for the treatment of sickle cell disease. It is known to reduce disease-causing sickling by inhibiting the transformation of the non-polymerizing, high-oxygen-affinity R quaternary structure of sickle hemoglobin into its polymerizing, low-affinity T quaternary structure. It has not been established whether the binding of the drug has anti-sickling effects beyond restricting the change of quaternary structure. By using a laser photolysis method that employs microscope optics, we have determined that fully deoxygenated sickle hemoglobin will assume the T structure. We show that the nucleation rates essential to generate the sickle fibers are not significantly affected by voxelotor. The method employed here should be useful for determining the mechanism of sickling inhibition for proposed drugs.

The Sickle-Cell Fiber Revisited

Sickle cell disease is the consequence of a single point mutation on the surface of the β chains of the hemoglobin molecule leading to the formation of rigid polymers that disrupt circulation. It has long been established that the polymers are comprised of seven pairs of double strands that are twisted replicas of the double strands found in crystals. Here, we review several newer developments that elaborate on that simple model and provide deeper insights into the process.

Temperature-dependent Self assembly of biofilaments during red blood cell sickling

Molecular self-assembly plays vital role in various biological functions. However, when aberrant molecules self-assemble to form large aggregates, it can give rise to various diseases. For example, the sickle cell disease andAlzheimer's disease are caused by self-assembled hemoglobin fibers and amyloid plaques, respectively. Here we studythe assembly kinetics of such fibers using kinetic Monte- Carlo simulation. We focus on the initial lag time of thesehighly stochastic processes, during which self-assembly is very slow. The lag time distributions turn out to be similarfor two very different regimes of polymerization, namely, a) when polymerization is slow and depolymerization is fast,and b) the opposite case, when polymerization is fast and depolymerization is slow. Using temperature dependent on-and off-rates for hemoglobin fiber growth, reported in recent in-vitro experiments, we show that the mean lag time canexhibit non-monotonic behavior with respect to change in temperature.

The role of surfaces on amyloid formation

Interfaces can strongly accelerate or inhibit protein aggregation, destabilizing proteins that are stable in solution or, conversely, stabilizing proteins that are aggregation-prone. Although this behaviour is well-known, our understanding of the molecular mechanisms underlying surface-induced protein aggregation is still largely incomplete. A major challenge is represented by the high number of physico-chemical parameters involved, which are highly specific to the considered combination of protein, surface properties, and solution conditions. The key aspect determining the role of interfaces is the relative propensity of the protein to aggregate at the surface with respect to bulk. In this review, we discuss the multiple molecular determinants that regulate this balance. We summarize current experimental techniques aimed at characterizing protein aggregation at interfaces, and highlight the need to complement experimental analysis with theoretical modelling. In particular, we illustrate how chemical kinetic analysis can be combined with experimental methods to provide insights into the molecular mechanisms underlying surface-induced protein aggregation, under both stagnant and agitation conditions. We summarize recent progress in the study of important amyloids systems, focusing on selected relevant interfaces.

Rational Drug Design of Peptide-Based Therapies for Sickle Cell Disease

Sickle cell disease (SCD) is a group of inherited disorders affecting red blood cells, which is caused by a single mutation that results in substitution of the amino acid valine for glutamic acid in the sixth position of the β-globin chain of hemoglobin. These mutant hemoglobin molecules, called hemoglobin S, can polymerize upon deoxygenation, causing erythrocytes to adopt a sickled form and to suffer hemolysis and vaso-occlusion. Until recently, only two drug therapies for SCD, which do not even fully address the manifestations of SCD, were approved by the United States (US) Food and Drug Administration. A third treatment was newly approved, while a monoclonal antibody preventing vaso-occlusive crises is also now available. The complex nature of SCD manifestations provides multiple critical points where drug discovery efforts can be and have been directed. These notwithstanding, the need for new therapeutic approaches remains high and one of the recent efforts includes developments aimed at inhibiting the polymerization of hemoglobin S. This review focuses on anti-sickling approaches using peptide-based inhibitors, ranging from individual amino acid dipeptides investigated 30-40 years ago up to more promising 12- and 15-mers under consideration in recent years.

Rapid and inefficient kinetics of sickle hemoglobin fiber growth

In sickle cell disease, the aberrant assembly of hemoglobin fibers induces changes in red blood cell morphology and stiffness, which leads to downstream symptoms of the disease. Therefore, understanding of this assembly process will be important for the treatment of sickle cell disease. By performing the highest spatiotemporal resolution measurements (55 nm at 1 Hz) of single sickle hemoglobin fiber assembly to date and combining them with a model that accounts for the multistranded structure of the fibers, we show that the rates of sickle hemoglobin addition and loss have been underestimated in the literature by at least an order of magnitude. These results reveal that the sickle hemoglobin self-assembly process is very rapid and inefficient (4% efficient versus 96% efficient based on previous analyses), where net growth is the small difference between over a million addition-loss events occurring every second.

Sensitivity analysis of the variability of amyloid aggregation profiles

The variability of amyloid aggregation profiles is linearly proportional to the duration of the aggregation process, and arises from a perturbation of one or more of the initial conditions.

Water, Ions, and Hemoglobin: Effects on Allostery and Polymerization

Proteins that function in aqueous solution can be perturbed by the solvent. Here we present experimental studies on two such interactions in the hemoglobin molecule. (1) Hemoglobin's oxygen binding is altered by introduction of crowding species or osmoticants, such as sucrose, through the linked binding of ions such as Cl or CO₂, but not otherwise. This rules out a significant role of buried surface in the allosteric energetics. (2) Sickle hemoglobin (HbS) polymerizes more readily in high concentrations of phosphate buffer. Such polymerization is analyzed quantitatively here for the first time in terms of the double nucleation mechanism. The changes in solubility are found to account for the increase in monomer addition rates and nucleation rates without requiring additional parameter adjustments. In the analysis, we also show how the analytical formulation of HbS nucleation may be adapted to include water that occupies the interstices between the assembled molecules. While such a "correction" has been applied to the equilibrium process, it has not previously been applied to the nucleation process.

Stochastic Formation of Fibrillar and Amorphous Superoxide Dismutase Oligomers Linked to Amyotrophic Lateral Sclerosis

Recent reports suggest that the nucleation and propagation of oligomeric superoxide dismutase-1 (SOD1) is effectively stochastic in vivo and in vitro. This perplexing kinetic variability-observed for other proteins and frequently attributed to experimental error-plagues attempts to discern how SOD1 mutations and post-translational modifications linked to amyotrophic lateral sclerosis (ALS) affect SOD1 aggregation. This study used microplate fluorescence spectroscopy and dynamic light scattering to measure rates of fibrillar and amorphous SOD1 aggregation at high iteration (ntotal = 1.2 × 10(3)). Rates of oligomerization were intrinsically irreproducible and populated continuous probability distributions. Modifying reaction conditions to mimic random and systematic experimental error could not account for kinetic outliers in standard assays, suggesting that stochasticity is not an experimental artifact, rather an intrinsic property of SOD1 oligomerization (presumably caused by competing pathways of oligomerization). Moreover, mean rates of fibrillar and amorphous nucleation were not uniformly increased by mutations that cause ALS; however, mutations did increase kinetic noise (variation) associated with nucleation and propagation. The stochastic aggregation of SOD1 provides a plausible statistical framework to rationalize how a pathogenic mutation can increase the probability of oligomer nucleation within a single cell, without increasing the mean rate of nucleation across an entire population of cells.

Universality of supersaturation in protein-fiber formation

The thermodynamics and kinetics of the aggregation of sickle-cell hemoglobin into fibers have been studied in great detail under a wide range of solution conditions. The stability of the fiber is measured by the solubility; the kinetics is characterized by a delay before the appearance of fibers. A review of data in the literature shows that there is no correlation of the delay time with fiber stability and only a weak correlation with the initial protein concentration. There is, however, a striking collapse of all the data onto a single universal curve when the delay time is plotted versus the supersaturation, which is the ratio of the initial protein concentration to the solubility, expressed as activities. Collapse onto the same universal curve is also obtained when using delay times calculated from the double-nucleation theoretical model.

Calibrating Sickle Cell Disease

Sickle cell disease is fundamentally a kinetic disorder, in which cells containing the mutated hemoglobin (hemoglobin S; HbS) will cause occlusion if they sickle in the microvasculature, but have minimal (or no) consequences if they sickle in the venous return. Physiologically, sickling always occurs when some ligands are present; nonetheless, the kinetics in the presence of ligands are virtually unstudied. Sickling arises from nucleation-controlled polymer formation, triggered when the HbS loses ligands (e.g., oxygen). Thus, understanding how nucleation responds to the presence of oxygen is the key to understanding how sickling proceeds in a physiological context. We have measured the rate of nucleus formation in HbS partially liganded with NO or CO, which we find have equivalent effects in reducing the nucleation rates. We find that hemoglobin must be in the T (tense) quaternary structure for nucleation, but the presence of ligands inhibits nucleus formation even when the correct quaternary structure is present. From these results, we can predict the fraction of cells that will sickle at any given partial ligand saturations. The ability to make such predictions may prove especially useful in designing future therapies, particularly those where the oxygen affinity is perturbed.

The delay time in sickle cell disease after 40 years: A paradigm assessed

Sickle hemoglobin polymerization commences with a striking latency period, called a "delay time" followed by abrupt polymer formation. The delay time is exceedingly concentration dependent. This discovery (40 years ago) led to the "kinetic hypothesis," that is, that the pathophysiology was related to the relationship between the delay time and the capillary transit. The delay time is well described by a double-nucleation mechanism of polymer formation. In macroscopic volumes, the delay time is highly reproducible, but in small volumes such as erythrocytes, under certain conditions, the intrinsic delay time can be augmented by a stochastic delay owing to random waiting times for the first nucleus to form. This lengthens the average delay and adds further protection from vaso-occlusion. When oxygen removal is not sudden, the growth of polymers after the delay time is limited by the rate of oxygen removal, further lengthening the time before occlusion may occur. This is important if some polymers have remained in the cell after pulmonary transit as their presence otherwise would obliterate any delay. The difficulty of deforming a cell once polymerized rationalizes the "two-step" model of vaso-occlusion in which a postcapillary adhesion event is followed by a sickling logjam. The delay time that is required is therefore generalized to be the delay time for an erythrocyte to move beyond regions in the venuoles where adherent cells have reduced the available lumen. The measurements of delay times correlate well with the severity of sickling syndromes. They also correlate with the improvements owing to the administration of hydroxyurea.

Dissecting the energies that stabilize sickle hemoglobin polymers

Sickle hemoglobin forms long, multistranded polymers that account for the pathophysiology of the disease. The molecules in these polymers make significant contacts along the polymer axis (i.e., axial contacts) as well as making diagonally directed contacts (i.e., lateral contacts). The axial contacts do not engage the mutant β6 Val and its nonmutant receptor region on an adjacent molecule, in contrast to the lateral contacts which do involve the mutation site. We have studied the association process by elastic light scattering measurements as a function of temperature, concentration, and primary and quaternary structure, employing an instrument of our own construction. Even well below the solubility for polymer formation, we find a difference between the association behavior of deoxy sickle hemoglobin molecules (HbS), which can polymerize at higher concentration, in comparison to COHbS, COHbA, or deoxygenated Hemoglobin A (HbA), none of which have the capacity to form polymers. The nonpolymerizable species are all quite similar to one another, and show much less association than deoxy HbS. We conclude that axial contacts are significantly weaker than the lateral ones. All the associations are entropically favored, and enthalpically disfavored, typical of hydrophobic interactions. For nonpolymerizable Hemoglobin, ΔH(o) was 35 ± 4 kcal/mol, and ΔS was 102.7 ± 0.5 cal/(mol-K). For deoxyHbS, ΔH(o) was 19 ± 2 kcal/mol, and ΔS was 56.9 ± 0.5 cal/(mol-K). The results are quantitatively consistent with the thermodynamics of polymer assembly, suggesting that the dimer contacts and polymer contacts are very similar, and they explain a previously documented significant anisotropy between bending and torsional moduli. Unexpectedly, the results also imply that a substantial fraction of the hemoglobin has associated into dimeric species at physiological conditions.

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.

Statistical mechanical treatments of protein amyloid formation

Protein aggregation is an important field of investigation because it is closely related to the problem of neurodegenerative diseases, to the development of biomaterials, and to the growth of cellular structures such as cyto-skeleton. Self-aggregation of protein amyloids, for example, is a complicated process involving many species and levels of structures. This complexity, however, can be dealt with using statistical mechanical tools, such as free energies, partition functions, and transfer matrices. In this article, we review general strategies for studying protein aggregation using statistical mechanical approaches and show that canonical and grand canonical ensembles can be used in such approaches. The grand canonical approach is particularly convenient since competing pathways of assembly and dis-assembly can be considered simultaneously. Another advantage of using statistical mechanics is that numerically exact solutions can be obtained for all of the thermodynamic properties of fibrils, such as the amount of fibrils formed, as a function of initial protein concentration. Furthermore, statistical mechanics models can be used to fit experimental data when they are available for comparison.

Hydrodynamics of hemostasis in sickle-cell disease

Vaso-occlusion, the stoppage of blood flow in sickle-cell disease, is a complex dynamical process spanning multiple time and length scales. Motivated by recent ex vivo microfluidic measurements of hemostasis using blood from sickle-cell patients, we develop a multiphase model that couples the kinetics and hydrodynamics of a flowing suspension of normal and sickled cells in a fluid. We use the model to derive expressions for the cell velocities and concentrations that quantify the hydrodynamics of hemostasis, and provide simple criteria as well as a phase diagram for occlusion, consistent with our simulations and earlier observations.

Nucleation of sickle hemoglobin mixed with hemoglobin A: experimental and theoretical studies of hybrid-forming mixtures

Sickle hemoglobin (HbS) is a point mutation of the two β subunits in normal Hb (HbA) that leads to nucleated polymerization and accompanying pathology. We measured the rates of homogeneous and heterogeneous nucleation of HbS in the presence of up to 50% HbA under conditions in which hybrid HbAS molecules will also form. The replacement of 50% of HbS by HbA slows polymerization by factors of ∼100 in the physiological range, which is substantially less than previously thought. To provide a theoretical description of these data, we extended the double nucleation model for HbS polymerization to conditions in which hybridized mixtures are present. Measurements of homogeneous nucleation and the theory agree only when at least one of the molecules in the nucleus is not a hybrid. We attribute this to the necessary presence in the nucleus of a molecule that utilizes both β-subunit mutation sites in intermolecular contacts, whereas the remaining molecules engage only one of the mutation sites. Heterogeneous nucleation appears to require an even greater number of nonhybrid molecules, presumably because of the need for the nucleus to attach to the polymer as well as to form internal bonds. These results also provide insights into the pathophysiology of sickle cell disease, including the occasional severe events that strike persons in whom both HbS and HbA are expressed, a condition known as sickle trait. The studies reported here are necessary for understanding physiologically relevant polymerization in the presence of ligands as well as therapeutically relevant copolymerizing inhibitors.

Band 3 catalyzes sickle hemoglobin polymerization

We have measured homogeneous and heterogeneous nucleation rates of sickle hemoglobin (HbS) in the presence of a strongly binding deletion mutant of the cytoplasmic domain of band 3 (cdb3), a membrane protein known to form dimers and to bind 2 HbS molecules to such a dimer, and we find that it accelerated both rates by a factor of 2. A weakly binding mutant, in contrast showed no impact on nucleation rates, contrary to naïve expectations of a slight enhancement based on the molecular crowding of the solution by the mutant. We find we can explain these phenomena by a model of HbS-cdb3 interaction in which the strong binding mutant, by stabilizing an HbS dimer, catalyzes the nucleation process, while the weak mutant binds only 1 HbS molecule, effectively inactivating it and thereby compensating for the crowding of the solution by the cdb3. The catalytic behavior we observe could play a role in intracellular processes.

The double nucleation model for sickle cell haemoglobin polymerization: full integration and comparison with experimental data

Sickle cell haemoglobin (HbS) polymerization reduces erythrocyte deformability, causing deleterous vaso-occlusions. The double-nucleation model states that polymers grow from HbS aggregates, the nuclei, (i) in solution (homogeneous nucleation), (ii) onto existing polymers (heterogeneous nucleation). When linearized at initial HbS concentration, this model predicts early polymerization and its characteristic delay-time (Ferrone et al. J Mol Biol 183(4):591-610, 611-631, 1985). Addressing its relevance for describing complete polymerization, we constructed the full, non-linearized model (Simulink), The MathWorks). Here, we compare the simulated outputs to experimental progress curves (n = 6-8 different [HbS], 3-6 mM range, from Ferrone's group). Within 10% from start, average root mean square (rms) deviation between simulated and experimental curves is 0.04 +/- 0.01 (25 degrees C, n = 8; mean +/- standard error). Conversely, for complete progress curves, averaged rms is 0.48 +/- 0.04. This figure is improved to 0.13 +/- 0.01 by adjusting heterogeneous pathway parameters (p < 0.01): the nucleus stability (sigma(2) micro( cc ): + 40%), and the fraction of polymer surface available for nucleation (phi), from 5e(-7), (3 mM) to 13 (6 mM). Similar results are obtained at 37 degrees C. We conclude that the physico-chemical description of heterogeneous nucleation warrants refinements in order to capture the whole HbS polymerization process.

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 polymerization of sickle hemoglobin in droplets

Sickle cell disease arises from a genetic mutation of one amino acid in each of the two hemoglobin beta chains, leading to the polymerization of hemoglobin in the red cell upon deoxygenation, and is characterized by vascular crises and tissue damage due to the obstruction of small vessels by sickled cells. It has been an untested assumption that, in red cells that sickle, the growing polymer mass would consume monomers until the thermodynamically well-described monomer solubility was reached. By photolysing droplets of sickle hemoglobin suspended in oil we find that polymerization does not exhaust the available store of monomers, but stops prematurely, leaving the solutions in a supersaturated, metastable state typically 20% above solubility at 37 degrees C, though the particular values depend on the details of the experiment. We propose that polymer growth stops because the growing ends reach the droplet edge, whereas new polymer formation is thwarted by long nucleation times, since the concentration of hemoglobin is lowered by depletion of monomers into the polymers that have formed. This finding suggests a new aspect to the pathophysiology of sickle cell disease; namely, that cells deoxygenated in the microcirculation are not merely undeformable, but will actively wedge themselves tightly against the walls of the microvasculature by a ratchet-like mechanism driven by the supersaturated solution.

Nucleation: the connections between equilibrium and kinetic behavior

This chapter describes the thermodynamics that govern the formation of nuclei, the least stable species in the reaction path of large, linear aggregates. In the approach described here, parameters are used that have direct molecular interpretations, such as contact energies of the molecular species. The extensive work on sickle hemoglobin is used as a model. An important result is that the nucleus size is expected to vary with initial conditions, such as the initial monomer concentration. Another unexpected result of some generality is that motion of the molecules within the nucleus recovers significant amounts of entropy that would be lost on complete immobilization.

The Hb A variant (beta73 Asp-->Leu) disrupts Hb S polymerization by a novel mechanism

Polymerization of a 1:1 mixture of hemoglobin S (Hb S) and the artificial mutant HbAbeta73Leu produces a dramatic morphological change in the polymer domains in 1.0 M phosphate buffer that are a characteristic feature of polymer formation. Instead of feathery domains with quasi 2-fold symmetry that characterize polymerization of Hb S and all previously known mixtures such as Hb A/S and Hb F/S mixtures, these domains are compact structures of quasi-spherical symmetry. Solubility of Hb S/Abeta73Leu mixtures was similar to that of Hb S/F mixtures. Kinetics of polymerization indicated that homogeneous nucleation rates of Hb S/Abeta73Leu mixtures were the same as those of Hb S/F mixtures, while exponential polymer growth (B) of Hb S/Abeta73Leu mixtures were about three times slower than those of Hb S/F mixtures. Differential interference contrast (DIC) image analysis also showed that fibers in the mixture appear to elongate between three and five times more slowly than in equivalent Hb S/F mixtures by direct measurements of exponential growth of mass of polymer in a domain. We propose that these results of Hb S/Abeta73Leu mixtures arise from a non-productive binding of the hybrid species of this mixture to the end of the growing polymer. This "cap" prohibits growth of polymers, but by nature is temporary, so that the net effect is a lowered growth rate of polymers. Such a cap is consistent with known features of the structure of the Hb S polymer. Domains would be more spherulitic because slower growth provides more opportunity for fiber bending to spread domains from their initial 2-fold symmetry. Moreover, since monomer depletion proceeds more slowly in this mixture, more homogeneous nucleation events occur, and the resulting gel has a far more granular character than normally seen in mixtures of non-polymerizing hemoglobins with Hb S. This mixture is likely to be less stiff than polymerized mixtures of other hybrids such as Hb S with HbF, potentially providing a novel approach to therapy.

Effect of T-R conformational change on sickle-cell hemoglobin interactions and aggregation

We compare the role of a conformational switch and that of a point mutation in the thermodynamic stability of a protein solution and in the consequent propensity toward aggregation. We study sickle-cell hemoglobin (HbS), the beta6 Glu-Val point mutant of adult human hemoglobin (HbA), in its R (CO-liganded) conformation, and compare its aggregation properties to those of both HbS and HbA in their T (unliganded) conformation. Static and dynamic light scattering measurements performed for various hemoglobin concentrations showed critical divergences with mean field exponents as temperature was increased. This allowed determining spinodal data points T(S)(c) by extrapolation. These points were fitted to theoretical expressions of the T(S)(c) spinodal line, which delimits the region where the homogeneous solution becomes thermodynamically unstable against demixing in two sets of denser and dilute mesoscopic domains, while remaining still liquid. Fitting provided model-free numerical values of enthalpy and entropy parameters measuring the stability of solutions against demixing, namely, 93.2 kJ/mol and 314 J/ degrees K-mol, respectively. Aggregation was observed also for R-HbS, but in amorphous form and above physiological temperatures close to the spinodal, consistent with the role played in nucleation by anomalous fluctuations governed by the parameter epsilon = (T - T(S))/T(S). Fourier transform infrared (FTIR) and optical spectroscopy showed that aggregation is neither preceded nor followed by denaturation. Transient multiple interprotein contacts occur in the denser liquid domains for R-HbS, T-HbS, and T-HbA. The distinct effects of their specific nature and configurations, and those of desolvation on the demixing and aggregation thermodynamics, and on the aggregate structure are highlighted.

Stochastic kinetics of intracellular huntingtin aggregate formation

Neurodegeneration in Huntington disease is described by neuronal loss in which the probability of cell death remains constant with time. However, the quantitative connection between the kinetics of cell death and the molecular mechanism initiating neurodegeneration remains unclear. One hypothesis is that nucleation of protein aggregates containing exon I fragments of the mutant huntingtin protein (mhttex1), which contains an expanded polyglutamine region in patients with the disease, is the explanation for the infrequent but steady occurrence of neuronal death, resulting in adult onset of the disease. Recent in vitro evidence suggests that sufficiently long polyglutamine peptides undergo a unimolecular conformational change to form a nucleus that seeds aggregation. Here we use this nucleation mechanism as the basis to derive a stochastic mathematical model describing the probability of aggregate formation in cells as a function of time and mhttex1 protein concentration, and validate the model experimentally. These findings suggest that therapeutic strategies for Huntington disease predicated on reducing the rate of mhttex1 aggregation need only make modest reductions in huntingtin expression level to substantially increase the delay time until aggregate formation.

Symmetry, equivalence, and molecular self-assembly

Molecular self-assembly at equilibrium is fundamental to the fields of biological self-organization, the development of novel environmentally responsive polymeric materials, and nanofabrication. Our approach to understanding the principles governing this process is inspired by existing models and measurements for the self-assembly of actin, tubulin, and the ubiquitous icosahedral shell structures of viral capsids. We introduce a family of simple potentials that give rise to the self-assembly of linear polymeric, random surface ("membrane"), tubular ("nanotube"), and hollow icosahedral structures that are similar in many respects to their biological counterparts. The potentials involve equivalent particles and an interplay between directional (dipolar, multipolar) and short-range (van der Waals) interactions. Specifically, we find that the dipolar potential, having a continuous rotational symmetry about the dipolar axis, gives rise to chain formation, while particles with multipolar potentials, having discrete rotational symmetries (square quadrupole or triangular ring of dipoles or "hexapole"), lead to the self-assembly of open sheet, nanotube, and hollow icosahedral geometries. These changes in the geometry of self-assembly are accompanied by significant changes in the kinetics of the organization.

Heterogeneous nucleation in sickle hemoglobin: experimental validation of a structural mechanism

Sickle hemoglobin polymerizes by two types of nucleation: homogeneous nucleation of aggregates in solution, and heterogeneous nucleation on preexisting polymers. It has been proposed that the same contact that is made in the interior of the polymer between the mutant site beta6 and its receptor pocket on an adjacent molecule is the primary contact site for the heterogeneous nucleus. We have constructed cross-linked hybrid molecules in which one beta-subunit is from HbA with Glu at beta6, and the other is from HbS with a Val at beta6. We measured solubility (using sedimentation) and polymerization kinetics (using laser photolysis) on cross-linked hybrids, and cross-linked HbS as controls. We find approximately 4000 times less heterogeneous nucleation in the cross-linked AS molecules than in cross-linked HbS, in strong confirmation of the proposal. In addition, changes in stability of the nucleus support a further proposal that more than one beta6 contact is involved in the homogeneous nucleus.

Molecular Crowding Limits the Role of Fetal Hemoglobin in Therapy for Sickle Cell Disease

The dominant assumption central to most treatments for sickle cell anemia has been that replacement of sickle hemoglobin (HbS) by fetal hemoglobin (HbF) would have major clinical benefit. Using laser photolysis, we have measured polymerization kinetics including rates of homogeneous and heterogeneous nucleation on mixtures of 20% and 30% HbF with HbS. We find that the present model for polymerization, including molecular crowding, can accurately predict the rates of such mixtures, by using the single assumption that no significant amount of HbF enters the polymer. The effects of replacing HbS by HbF on the rates of polymer formation are found to be significantly lower than previous measurements appeared to indicate because the impact of the replacement is also highly dependent on the total hemoglobin concentration. This is because the molecular crowding of non-polymerizing HbF offsets substantially the effects of decreasing the concentration of HbS concentration, an effect that increases with concentration. Most strikingly, the demonstrated benefit of hydroxyurea therapy in slowing the kinetics of intracellular polymerization cannot be primarily due to enhanced HbF, but must have some other origin, which could itself represent a promising therapeutic approach.

The effects of erythrocyte membranes on the nucleation of sickle hemoglobin

Pathology in sickle cell disease begins with nucleation-dependent polymerization of deoxyhemoglobin S into stiff, rodlike fibers that deform and rigidify red cells. We have measured the effect of erythrocyte membranes on the rate of homogeneous nucleation in sickle hemoglobin, using preparations of open ghosts (OGs) with intact cytoskeletons from sickle (SS) and normal adult (AA) red cells. Nucleation rates were measured by inducing polymerization by laser photolysis of carboxy sickle hemoglobin and observing stochastic variation of replicate experiments of the time for the scattering signals to reach 10% of their respective maxima. By optical imaging of membrane fragments added to a hemoglobin solution we contrast the rate of nucleation immediately adjacent to membrane fragments with nucleation in a region of the same solution but devoid of membranes. From analysis of 29,272 kinetic curves obtained, we conclude that the effect of AA OGs is negligible (10% enhancement of nucleation rates +/-20%), whereas SS OGs caused 80% enhancement (+/-20%). In red cells, where more membrane surface is available to Hb, this implies enhancement of nucleation by a factor of 6. These experiments represent a 10-fold improvement in precision over previous approaches and are the first direct, quantitative measure of the impact of erythrocyte membranes on the homogeneous nucleation process that is responsible for polymer initiation in sickle cell disease.

Understanding the shape of sickled red cells

To understand the physical basis of the wide variety of shapes of deoxygenated red cells from patients with sickle cell anemia, we have measured the formation rate and volume distribution of the birefringent domains of hemoglobin S fibers. We find that the domain formation rate depends on the approximately 80th power of the protein concentration, compared to approximately 40th power for the concentration dependence of the reciprocal of the delay time that precedes fiber formation. These remarkably high concentration dependences, as well as the exponential distribution of domain volumes, can be explained by the previously proposed double nucleation model in which homogeneous nucleation of a single fiber triggers the formation of an entire domain via heterogeneous nucleation and growth. The enormous sensitivity of the domain formation rate to intracellular hemoglobin S concentration explains the variable cell morphology and why rapid polymerization results in cells that do not appear sickled at all.

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.

Time scale of protein aggregation dictated by liquid-liquid demixing

The growing impact of protein aggregation pathologies, together with the current high need for extensive information on protein structures are focusing much interest on the physics underlying the nucleation and growth of protein aggregates and crystals. Sickle Cell Hemoglobin (HbS), a point-mutant form of normal human Hemoglobin (HbA), is the first recognized and best-studied case of pathologically aggregating protein. Here we reanalyze kinetic data on nucleation of deoxy-HbS aggregates by referring them to the (concentration-dependent) temperature T(s) characterizing the occurrence of the phase transition of liquid-liquid demixing (LLD) of the solution. In this way, and by appropriate scaling of kinetic data at different concentrations, so as to normalize their spans, the apparently disparate sets of data are seen to fall on a master curve. Expressing the master curve vs. the parameter epsilon = (T - T(s)) / T(s), familiar from phase transition theory, allows eliciting the role of anomalously large concentration fluctuations associated with the LLD phase transition and also allows decoupling quantitatively the role of such fluctuations from that of microscopic, inter-protein interactions leading to nucleation. Referring to epsilon shows how in a narrow temperature span, that is at T - T(s), nucleation kinetics can undergo orders-of-magnitude changes, unexpected in terms of ordinary chemical kinetics. The same is true for similarly small changes of other parameters (pH, salts, precipitants), capable of altering T(s) and consequently epsilon. This offers the rationale for understanding how apparently minor changes of parameters can dramatically affect protein aggregation and related diseases.

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.

Sickle hemoglobin polymer stability probed by triple and quadruple mutant hybrids

As part of an effort to understand the interactions in HbS polymerization, we have produced and studied a recombinant triple mutant, D6A(alpha)/D75Y(alpha)/E121R(beta), and a quadruple mutant comprising the preceding mutation plus the natural genetic mutation of sickle hemoglobin, E6V(beta). These recombinant hemoglobins expressed in yeast were extensively characterized, and their structure and oxygen binding cooperativity were found to be normal. Their tetramer-dimer dissociation constants were within a factor of 2 of HbA and HbS. Polymerization of these mutants mixed with HbS was investigated by a micromethod based on volume exclusion by dextran. The elevated solubility of mixtures of HbS with HbA and HbF in dextran could be accurately predicted without any variable parameters. Relative to HbS, the copolymerization probability of the quadruple mutant/HbS hybrid was found to be 6.2, and the copolymerization probability for the triple mutant/HbS hybrid was 0.52. The pure quadruple mutant had a solubility slightly above that of its hybrid with HbS. One way to explain these results is to require significant cis-trans differences in the polymer and that HbA assemble above 42.5 g/dl. A second way to explain these data is by the modification of motional freedom, thereby changing vibrational entropy in the polymer.

Heterogeneous nucleation and crowding in sickle hemoglobin: an analytic approach

Sickle hemoglobin nucleation occurs in solution as a homogeneous process or on existing polymers in a heterogeneous process. We have developed an analytic formulation to describe the solution crowding and large nonideality that affects the heterogeneous nucleation of sickle hemoglobin by using convex particle theory. The formulation successfully fits the concentration and temperature dependence of the heterogeneous nucleation process over 14 orders of magnitude. Unlike previous approaches, however, the new formulation can also accurately describe the effects of adding nonpolymerizing agents to the solution. Without additional adjustable parameters, the model now describes the data of M. Ivanova, R. Jasuja, S. Kwong, R. W. Briehl, and F. A. Ferrone, (Biophys. J. 2000, 79:1016-1022), in which up to 50% of the sickle hemoglobin is substituted by cross-linked hemoglobin A, which does not polymerize, and which substitution causes the rates to decrease by 10(5). The success of this approach provides insight into the polymerization process: from the size-dependence of the contact energy deduced here, it also appears that various contacts of unknown origin are energetically significant in the heterogeneous nucleation process.

Flexibility and nucleation in sickle hemoglobin

We have studied the self-assembly of Hemoglobin C-Harlem (HbC-Harlem), a double mutant of hemoglobin that possesses the beta6 Glu-->Val mutation of sickle hemoglobin (HbS) plus beta73 Asp-->Asn. By electron microscopy we find it forms crystals, rather than the wrapped multistranded fibers seen in HbS. Fourier transforms of the crystals yield unit cell parameters indistinguishable from crystals of HbS. Differential interference contrast (DIC) microscopy and birefringence also show crystal formation rather than the polymers or domains seen for HbS, while the growth patterns showed radiating crystal structures rather than simple linear crystalline forms. The solubility of the assembly was measured using a photolytic micromethod over a temperature range of 17-31 degrees C in 0.15 M phosphate buffer and found to be essentially the same as that of fibers of HbS. The assembly kinetics were observed by photolysis of the carbon monoxide derivative, and the mass of assembled hemoglobin was found to grow exponentially, with onset times that were stochastically distributed for small volumes. The stochastic onset of assembly showed strong concentration dependence, similar to but slightly greater than that seen in sickle hemoglobin nucleation. These observations suggest that like HbS, HbC-Harlem assembly proceeds by a homogeneous nucleation process, followed by heterogeneous nucleation. However, relative to HbS, both homogeneous and heterogeneous nucleation are suppressed by almost 11 orders of magnitude. The slowness of nucleation can be reconciled with the similarity of the solubility to HbS by an increase in contact energy coupled with a decrease in vibrational entropy recovered on assembly. This also explains the linearity of the double-strands, and agrees with the chemical nature of the structural replacement.

Nonideality and the nucleation of sickle hemoglobin

The homogeneous and heterogeneous nucleation kinetics of sickle hemoglobin (HbS) have been studied for various degrees of solution crowding by substitution of cross-linked hemoglobin A, amounting to 50% of the total hemoglobin. By cross-linking hemoglobin A, hybrid formation between hemoglobin A and hemoglobin S was prevented, thus simplifying the analysis of the results. Polymerization was induced by laser photolysis, and homogeneous nucleation kinetics were determined by observation of the stochastic behavior of the onset of light scattering. Heterogeneous nucleation was determined by observing the exponential growth of the progress curves, monitored by light scattering. At concentrations between 4 and 5 mM tetramer (i.e., approximately 30 g/dl), the substitution of 50% HbA for HbS slows the reaction by a factor of 10(3) to 10(4). Using scaled particle theory to account for the crowding of HbA, the observed decrease in the homogeneous nucleation rate was accurately predicted, with no variation of parameters required. Heterogeneous nucleation, on the other hand, is not well described in the present formulation, and the theory for this process appears to require modification of the way in which nonideality is introduced. Nonetheless, the accuracy of the homogeneous nucleation description suggests that such an approach may be useful for other assembly processes that occur in a crowded intracellular milieu.

A model for the sickle hemoglobin fiber using both mutation sites

The standard molecular model of the fiber of the sickle hemoglobin (HbS: beta6 Glu-->Val) has been revised to allow both beta6 mutation sites to participate in intermolecular contacts, rather than only one beta6 site as previously thought, for four molecules per 14-molecule fiber cross section. This structure accurately predicts the copolymerization of hybridized mixtures of HbS with HbA or HbC (beta6 Glu-->Lys), which could not be reconciled with prior models in which only half the beta6 sites were required for assembly. This model suggests new contacts within the fiber and raises the question of whether these cross-linked double strands could possess added stability important in such processes as nucleation.

Analysis of protein aggregation kinetics

Given a set of kinetic data, then, the preceding discussions suggest the following approach to its analysis. 1. For purposes of establishing the reaction, ignore the final stages and concentrate on the initial 10-20% of the reaction at first. A globally optimized model may be based on a faulty assumption for the initial steps. Thus, although the whole data set may look reasonably well fit, the reaction could be misrepresented, and thus the fit unhelpful if accuracy at the later stages has come at the expense of the initial phase of the reaction. 2. What is the time course of the initial reaction? (A) Is the reaction exponential? Exponential growth gives dramatic lag times (see Fig. 3), whereas nonexponential "lag times" have a visible signal from time 0 (i.e., Fig. 2). If the data set shows the abrupt appearance of signals after a period of quiescence, the chances are excellent that the time course is exponential. High sensitivity measurement of the signal at times during the lag phase should be used to confirm the exponential nature quantitatively. Exponential reactions mean a secondary pathway is operative. (a) A cascade (tn) can look similar to an exponential, but may proceed from a multistep single-path reaction. Thus the exponential needs to be ascertained with some accuracy. (b) It is possible that some or all of the lag results from a stochastic process, i.e., formation of a single nucleus being observed. This, however, is likely to be accompanied by a secondary process, as few techniques are sensitive enough to detect a single polymer at a time, and having one nucleus form many polymers is a hallmark of a secondary process. Thus, the reproducibility of the kinetics must be established to rule out stochastics. If data show wide variation, stochastic methods as described earlier may be employed. (c) Given a secondary process, one must separate the primary nucleation process from the secondary process (by stochastic means or by use of the product B2A, as described earlier). (B) If the reaction does not begin with an exponential, is it parabolic? If so, it falls in the general class of linear polymerizations. 3. What is the concentration dependence of the reaction(s)? This will separate nucleation processes from growth, and so on. 4. If the initial reaction is neither exponential nor parabolic, a reaction mechanism needs to be proposed and evaluated. Solving the resulting equations is best done by linearization, which has the best chance of giving equations whose solutions and their sensitivity to parameters are readily understood. If this proves fruitful, full numeric solutions may be useful. 5. At this point, the full reaction may be considered to completion. 6. The physical basis of the description (sizes of parameters and their dependencies) needs to be finally considered to ensure that the mathematical success of the description rests on tenable physical grounds.

Excitation-contraction coupling in gastrointestinal and other smooth muscles

The main contributors to increases in [Ca2+]i and tension are the entry of Ca2+ through voltage-dependent channels opened by depolarization or during action potential (AP) or slow-wave discharge, and Ca2+ release from store sites in the cell by the action of IP3 or by Ca(2+)-induced Ca(2+)-release (CICR). The entry of Ca2+ during an AP triggers CICR from up to 20 or more subplasmalemmal store sites (seen as hot spots, using fluorescent indicators); Ca2+ waves then spread from these hot spots, which results in a rise in [Ca2+]i throughout the cell. Spontaneous transient releases of store Ca2+, previously detected as spontaneous transient outward currents (STOCs), are seen as sparks when fluorescent indicators are used. Sparks occur at certain preferred locations--frequent discharge sites (FDSs)--and these and hot spots may represent aggregations of sarcoplasmic reticulum scattered throughout the cytoplasm. Activation of receptors for excitatory signal molecules generally depolarizes the cell while it increases the production of IP3 (causing calcium store release) and diacylglycerols (which activate protein kinases). Activation of receptors for inhibitory signal molecules increases the activity of protein kinases through increases in cAMP or cGMP and often hyperpolarizes the cell. Other receptors link to tyrosine kinases, which trigger signal cascades interacting with trimeric G-protein systems.

Thyroid Na+/I- symporter. Mechanism, stoichiometry, and specificity

The rat thyroid Na+/I- symporter (NIS) was expressed in Xenopus laevis oocytes and characterized using electrophysiological, tracer uptake, and electron microscopic methods. NIS activity was found to be electrogenic and Na+-dependent (Na+ >> Li+ >> H+). The apparent affinity constants for Na+ and I- were 28 +/- 3 mM and 33 +/- 9 microM, respectively. Stoichiometry of Na+/anion cotransport was 2:1. NIS was capable of transporting a wide variety of anions (I-, ClO3-, SCN-, SeCN-, NO3-, Br-, BF4-, IO4-, BrO3-, but perchlorate (ClO4-) was not transported. In the absence of anion substrate, NIS exhibited a Na+-dependent leak current (approximately 35% of maximum substrate-induced current) with an apparent Na+ affinity of 74 +/- 14 mM and a Hill coefficient (n) of 1. In response to step voltage changes, NIS exhibited current transients that relaxed with a time constant of 8-14 ms. Presteady-state charge movements (integral of the current transients) versus voltage relations obey a Boltzmann relation. The voltage for half-maximal charge translocation (V0.5) was -15 +/- 3 mV, and the apparent valence of the movable charge was 1. Total charge was insensitive to [Na+]o, but V0.5 shifted to more negative potentials as [Na+]o was reduced. NIS charge movements are attributed to the conformational changes of the empty transporter within the membrane electric field. The turnover rate of NIS was >/=22 s-1 in the Na+ uniport mode and >/=36 s-1 in the Na+/I- cotransport mode. Transporter density in the plasma membrane was determined using freeze-fracture electron microscopy. Expression of NIS in oocytes led to a approximately 2. 5-fold increase in the density of plasma membrane protoplasmic face intramembrane particles. On the basis of the kinetic results, we propose an ordered simultaneous transport mechanism in which the binding of Na+ to NIS occurs first.

Nucleation and polymerization of sickle hemoglobin with Leu beta 88 substituted by Ala

We have measured the solubility, and the rates of homogeneous and heterogeneous nucleation on sickle hemoglobin (HbS beta 6 Glu-->Val) additionally modified by site-directed mutagenesis to possess Ala rather than Leu at beta 88, which forms part of the receptor site for beta 6 Val in the sickle polymer. The solubility of the hemoglobin is increased at all temperatures, and is about 29 g/dl at 25 degrees C. Polymerization kinetics, induced by laser photolysis and observed by light-scattering intensity, showed exponential growth with rates about 300 times slower than experiments done on similar concentrations of HbS. When polymerization is carried out in small volumes, the time of measurable light-scattering signal to reach one-tenth of its final value (denoted as the tenth time) showed stochastic fluctuations, as is seen in pure HbS. Homogeneous nucleation rates were measured by observing distributions of tenth times and these rates were slowed by the mutation by almost 1000-fold relative to pure HbS. The kinetics, including the exponential progress curves and shape of the tenth time distributions, are well described by the double nucleation mechanism for polymerization. Analysis of the homogeneous nucleation rates leads to the surprising conclusion that the mutation has scarcely changed the energy of the intermolecular contacts despite the increase in solubility of the double mutant. This conclusion is supported by the stereochemistry of the modified contact site, in which the amount of exposed hydrophobic surface appears to be unchanged by the mutation. The increased solubility must therefore result from decreased motional freedom of molecules within the polymer, which could arise from tighter packing into the enlarged receptor pocket. This points up the ability of kinetic analysis to reveal important thermodynamic properties of assembly, and underlines the importance of the vibrational degrees of freedom in setting the final equilibrium constant. Chemical modifications to restrict vibrations and enhance the cost of polymerization may prove useful in constructing compounds to act as inhibitors of sickle cell gelation.