Kinetics of sickle hemoglobin polymerization. III. Nucleation rates determined from stochastic fluctuations in polymerization progress curves

The polymerization kinetics of sickle cell hemoglobin are found to exhibit stochastic variations when observed in very small volumes (approximately 10(-10) cm3). The distribution of progress curves has been measured at several temperatures for a 4.50 mM-hemoglobin S sample using a laser-photolysis, light-scattering technique. The progress curves at a given temperature are superimposable when translated along the time axis, showing that the variability of the kinetic progress curves results primarily from fluctuations in the time at which polymerization is initiated. The shapes of the initial part of the progress curves are well-fitted using the functional form I(t) = Io + As exp (Bt), derived from a dual nucleation model. When the distribution of the measured tenth times is broad, the rate of homogeneous nucleation can be obtained by fitting the exponential tail of the distribution. As the distribution sharpen, the rate of homogeneous nucleation can be estimated by modelling the width of the distribution function using a simple Monte-Carlo simulation of the polymerization kinetics. Using the rates of homogeneous nucleation obtained from the distributions, the rates of heterogeneous nucleation and polymer growth can be obtained from the experimental parameters As and B. The resulting nucleation rates are roughly 1000 times greater than those obtained from an analysis of bulk kinetic data. The results provide strong support for the dual-nucleation mechanism and show that the distribution of progress curves provides a powerful independent method for measuring the rate of homogeneous nucleation and thereby obtaining values for the other principal rates of the mechanism.     

Kinetics of nucleation-controlled polymerization. A perturbation treatment for use with a secondary pathway

We present a perturbation method for analyzing nucleation-controlled polymerization augmented by a secondary pathway for polymer growth. With this method, the solution to the kinetic equations assumes a simple analytic closed form that can easily be used in fitting data. So long as the formation of polymers by the secondary pathway depends linearly on the concentration of monomers polymerized, the form of the solutions is the same. This permits the analysis of augmented growth models with a minimum number of modeling assumptions, and thus makes it readily possible to distinguish between a variety of secondary processes (heterogeneous nucleation, lateral growth, and fragmentation). In addition, the parameters of the homogeneous process, such as the homogeneous nucleus size, can be determined independent of the nature of the secondary mechanism. We describe applications of this method to the polymerization of actin, collagen, and sickle hemoglobin. We present an extensive analysis of data on actin polymerization (Wegner, A., and P. Savko, 1982, Biochemistry, 21:1909-1913) to illustrate the use of the method. Although our conclusions generally agree with theirs, we find that lateral growth describes the secondary pathway better than the fragmentation model originally proposed. We also show how this method can be used to study the degree of polymerization, the parentage of polymers, and the behavior of polymers in cycling experiments.     

Kinetic studies on photolysis-induced gelation of sickle cell hemoglobin suggest a new mechanism.

The kinetics of deoxyhemoglobin S gelation have been investigated using photolytic dissociation of the carbon monoxide complex to initiate the process. Measurements over a wide range of times, 10(-3)-10(4) show that both the concentration dependence of the tenth-time (i.e., the time required to complete one-tenth the reaction) and the time dependence of the process decrease as gelation speeds up. In slowly gelling samples, where single domains of polymers are formed in the small sample volumes employed with this technique (1-2 x 10(-9) cm3), there is a marked increase in the variability of the tenth-times. These results are explained by a mechanism in which gelation is initiated by homogeneous nucleation of polymers in the bulk solution phase, followed by heterogeneous nucleation on the surface of existing polymers. At the lowest concentrations, homogeneous nucleation is so improbable that stochastic behavior is observed in the small sample volumes, and heterogeneous nucleation is the dominant pathway for polymer formation, thereby accounting for the high time dependence. At the highest concentrations homogeneous nucleation becomes much more probable, and the time dependence decreases. The decrease in concentration dependence of the tenth-time with increasing concentration results from a decrease in size of both the homogeneous and heterogeneous critical nuclei. The model rationalizes the major observations on the kinetics of gelation of deoxyhemoglobin S, and is readily testable by further experiments.     

Kinetics of sickle hemoglobin polymerization. I. Studies using temperature-jump and laser photolysis techniques

Using a combination of laser photolysis and temperature-jump techniques, the kinetics of hemoglobin S polymerization have been studied over a wide range of delay times (10(-3) to 10(5)s), concentrations (0.2 to 0.4 g/cm3) and temperatures (5 to 50 degrees C). A slow temperature-jump technique was used to induce polymerization in samples with delay times between 10(2) seconds and 10(5) seconds by heating a solution of completely deoxygenated hemoglobin S. For samples with shorter delay times, polymerization was induced by photodissociating the carbon monoxide complex in small volumes (10(-9) cm3) using a microspectrophotometer equipped with a cw argon ion laser. The photolysis technique is described in some detail because of its importance in studying hemoglobin S polymerization at physiological concentrations and temperatures. In order, to establish conditions for complete photodissociation with minimal laser heating, a series of control experiments on normal human hemoglobin was performed and theoretically modeled. The concentration dependence of the tenth time is found to decrease with increasing hemoglobin S concentration. In the range 0.2 to 0.3 g/cm3, the tenth time varies as the 36th power of the hemoglobin S concentration, while in the range 0.3 to 0.4 g/cm3 it decreases to 16th power. As the tenth times become shorter, the progress curves broaden, with the onset of polymerization becoming less abrupt. For tenth times greater than about 30 seconds, measurements with the laser photolysis technique on small volumes yield highly irreproducible tenth times, but superimposable progress curves, indicating stochastic behavior. The initial part of the progress curves from both temperature-jump and laser photolysis experiments is well fit with an equation for the concentration of polymerized monomer, delta (t) = A[cosh (Bt) -1], which results from integration of the linearized rate equations for the double nucleation mechanism described in the accompanying paper (Ferrone et al., 1985). The dependence of the parameters A and B on temperature and concentration is obtained from fitting over 300 progress curves. The rate B has a large concentration dependence, varying at 25 degrees C from about 10(-4) S-1 at 0.2 g/cm3 to about 100 s-1 at 0.4 g/cm3.     

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.     

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 10⁵. 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.     

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

The homogeneous nucleation rate for sickle hemoglobin polymerization has been measured for concentrations from 3.9 to 4.9 mM and temperatures from 13 degrees C to 35 degrees C by observing the stochastic fluctuations of the time to complete 10% of the reaction after photolysis of the carboxy derivative. To allow efficient data collection, a mesh was used to divide the photolysis beam into an array of smaller beams, which allowed parallel observation of about 100 different regions. Nucleation rates measured here are consistent with more restricted previously published data and, when combined with directly measured monomer addition rates, are consistent with previous analysis of progress curves. By describing these rates with equilibrium nucleation theory, the concentration of nuclei and hence their stability can be ascertained. Consequently, the chemical potential by which a monomer is attached to the polymer is determined. This attachment energy ranges from -6.6 to -8.0 kcal/mol between 15 degrees C and 35 degrees C. The enthalpic part of that chemical potential is found to be equal to the enthalpy determined by solubility measurements, as expected from thermodynamic considerations. The entropic portion of the contact chemical potential contributes from -21.4 to -8.7 kcal/mol. The vibrational chemical potential of monomers in the polymer ranges from -25.7 to -27.4 kcal/mol over the same temperatures.     

Length distributions of hemoglobin S fibers

Electron microscopy of sickle cell hemoglobin fibers fixed at different times during gelation shows an exponential distribution of fiber lengths, with many short fibers and few long ones. The distribution does not change significantly with time as polymerization progresses. If this distribution of lengths reflects kinetic mechanism of fiber assembly, it complements information from studies of the progress of average properties of the polymers and, as has been done for other rod-like polymerizing systems, permits testing of models for the mechanism of fiber assembly. In this case, the results are consistent with the double nucleation model of Ferrone et al. or with a related alternative model based on fiber breakage. However, other possible causes of this microheterogeneity exist, including: breakage due to solution shearing of the long, rod-like, fibers; the presence of residual nuclei; equilibrium relations governing polymerization; and breakage of solid-like but weak gels that develop early and adhere to the grid. The arguments against the first three of these possibilities suggest that they are not responsible. However, breakage of entanglements or cross-links in a solid-like and adherent gel is consistent with the distributions.     

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.     

Non-native aggregation of alpha-chymotrypsinogen occurs through nucleation and growth with competing nucleus sizes and negative activation energies

The kinetics and structural transitions of non-native aggregation of alpha-chymotrypsinogen (aCgn) were investigated over a wide range of temperature and initial protein concentration at pH 3.5, where high molecular weight aggregates remained soluble throughout the reaction. A comparison of thermodynamic, kinetic, and spectroscopic data shows that aggregation under non-native-favoring conditions proceeds through a molten globule unfolded monomer state, with a nucleation and growth mechanism. Formation of irreversible aggregates and conversion to beta-sheet secondary structures occur simultaneously without detectable intermediates, suggesting that beta-sheet formation may be a commitment step during the nucleation and growth stages. Analysis of the kinetics using a Lumry-Eyring with nucleated polymerization (LENP) model provides the predominant nucleus size and the product of the intrinsic nucleation and intrinsic growth time scales at each state point. We find that the nucleus size depends on both temperature and protein concentration, and in some cases there is competition between two distinct nucleus sizes. The observed rate coefficient (kobs) for aggregation displays a maximum as a function of temperature because of the competition between folding-unfolding thermodynamics and the intrinsic growth and nucleation rates; the latter contribution has a large, negative activation enthalpy that dominates kobs at elevated temperatures. Temperature-jump experiments reveal that aggregates depolymerize at high temperatures, indicating that they are lower in enthalpy than the free monomer. Overall, the results suggest more generally that non-native aggregation may proceed through more than one nucleus size and that intrinsic kinetics of nucleation and growth may have significant entropic barriers.     

Cooperative polymerization reactions. Analytical approximations, numerical examples, and experimental strategy

How does one obtain kinetic rate constants from the time course of a reversible and cooperative polymerization reaction? We examine a simple version of the homogeneous nucleation-elongation model with both analytical and numerical techniques to test some common assumptions and develop an experimental strategy. The assumption of irreversible polymer formation is found to be a useful and adequate approximation for the numerical determination of monomer disappearance. The assumption of early "pre-equilibrium" between monomer and seed, however, is shown numerically and analytically to produce significant errors over a wide range of parameters, particularly for small seed lengths. We exhibit numerical solutions for many different parameters, and also discuss analytical techniques that allow approximate solutions for several conditions: the high-concentration limit; the long-time limit; and the long-seed-length, lows concentration limit. The overall reaction simplifies when the monomer concentration is large. An experimental strategy for elucidating the seed size and the rate constants for polymerization and depolymerization is presented.     

Solubility of fluoromethemoglobin S: effect of phosphate and temperature on polymerization

The polymerization properties of the fully liganded fluoromet derivative of hemoglobin S (FmetHb S) were investigated by electron microscopy and absorption spectroscopy. Polymerization progress curves, as measured by increasing sample turbidity at 700 nm, exhibit a delay time (t(d)) consistent with the double nucleation mechanism. The pattern of fiber growth, as monitored by electron microscopy, is also indicative of a heterogeneous nucleation process, and dimensions of the fibers were found to be comparable to that of deoxyHb S. The polymerization rate constant (1/t(d)) depends exponentially on Hb S concentration, and the size of the homogeneous and heterogeneous nuclei also depend on FmetHb S concentration. As for deoxyHb S, higher concentrations of protein and phosphate favor fiber formation, while lower temperatures inhibit polymerization. Solubility experiments reveal, however, that eight times more FmetHb S is required for polymerization. The current studies further show that reaction order is independent of phosphate concentration if Hb S activity and not concentration is considered. The allosteric effector, inositol hexaphosphate (IHP), promotes fiber formation, and temperature-dependent reaggregation of FmetHb S suggests that IHP stabilizes pregelation aggregates. These studies show that FmetHb S resembles deoxyHb S in many of its polymerization properties; however, IHP-bound FmetHb S potentially provides a unique avenue for future studies of the early stages of Hb S polymerization and the effect of tertiary and quaternary protein structure on the polymerization process.     

Kinetic analysis of cooperativity in tubulin polymerization in the presence of guanosine di- or triphosphate nucleotides

In vitro polymerization of pig brain tubulin, highly purified and deprived of microtubule-associated proteins, was followed by turbidimetry. Treatment of the data yielded the relation existing between the observed turbidity and the amount of polymer formed. This allowed a kinetic analysis, according to Oosawa's theories, of the polymerization process, which consisted of a slow spontaneous nucleation followed by the growth process. The apparent elongation rate constant was closely related to the nucleation process and exhibited a highly cooperative variation with tubulin concentration. The cooperativity was indicative of the size of the nucleus which appears to remain the same whether sheets or microtubules are formed. Magnesium ions appear to play a role in the polymorphism of tubulin polymers, the proportion of microtubules to sheets increasing with magnesium ion concentration. From kinetic experiments evidence was provided for GDP binding in competition with GTP, with a sixfold lower affinity. The tubulin-GDP complex could participate in microtubules elongation, but was not able to form nuclei. The critical concentration of tubulin in the presence of GDP was roughly twice as high as in the presence of GTP.     

Fluctuations in the polymerization of sickle hemoglobin. A simple analytic model

Using the stochastic theory of chemical reactions and the theory of first passage times, a simple analytic expression is derived for the distribution of delay times that has been observed in studies of the polymerization kinetics of sickle hemoglobin under conditions where the polymerization progress curves exhibit stochastic variation. The rate of homogeneous nucleation can be readily extracted from such experiments using this expression. This work constitutes a significant addition to the rather limited number of examples where contact can be successfully made between the stochastic theory of chemical kinetics and experiment.     

Nucleation limits the lengths of actin filaments assembled by formin

Formins stimulate actin polymerization by promoting both filament nucleation and elongation. Because nucleation and elongation draw upon a common pool of actin monomers, the rate at which each reaction proceeds influences the other. This interdependent mechanism determines the number of filaments assembled over the course of a polymerization reaction, as well as their equilibrium lengths. In this study, we used kinetic modeling and in vitro polymerization reactions to dissect the contributions of filament nucleation and elongation to the process of formin-mediated actin assembly. We found that the rates of nucleation and elongation evolve over the course of a polymerization reaction. The period over which each process occurs is a key determinant of the total number of filaments that are assembled, as well as their average lengths at equilibrium. Inclusion of formin in polymerization reactions speeds filament nucleation, thus increasing the number and shortening the lengths of filaments that are assembled over the course of the reaction. Modulation of the elongation rate produces modest changes in the equilibrium lengths of formin-bound filaments. However, the dependence of filament length on the elongation rate is limited by the number of filament ends generated via formin’s nucleation activity. Sustained elongation of small numbers of formin-bound filaments, therefore, requires inhibition of nucleation via monomer sequestration and a low concentration of activated formin. Our results underscore the mechanistic advantage for keeping formin’s nucleation efficiency relatively low in cells, where unregulated actin assembly would produce deleterious effects on cytoskeletal dynamics. Under these conditions, differences in the elongation rates mediated by formin isoforms are most likely to impact the kinetics of actin assembly.     

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.     

Dynamics of polymerization shed light on the mechanisms that lead to multiple amyloid structures of the prion protein

It is generally accepted that spongiform encephalopathies result from the aggregation into amyloid of a ubiquitous protein, the so-called prion protein. As a consequence, the dynamics of amyloid formation should explain the characteristics of the prion diseases: infectivity as well as sporadic and genetic occurrence, long incubation time, species barriers and strain specificities. The success of this amyloid hypothesis is due to the good qualitative agreement of this hypothesis with the observations. However, a number of difficulties appeared when comparing quantitatively the in vitro experimental results with the theoretical models, suggesting that some differences should hide important discrepancies. We used well defined quantitative models to analyze the experimental results obtained by in vitro polymerization of the recombinant hamster prion protein. Although the dynamics of polymerization resembles a simple nucleus-dependent fibrillogenesis, neither the initial concentration dependence nor off-pathway hypothesis fit with experimental results. Furthermore, seeded polymerization starts after a long time delay suggesting the existence of a specific mechanism that takes place before nucleus formation. On the other hand, polymerization dynamics reveals a highly stochastic mechanism, the origin of which appears to be caused by nucleation heterogeneity. Moreover, the specific structures generated during nucleation are maintained during successive seeding although a clear improvement of the dynamics parameters (polymerization rate and lag time) is observed. We propose that an additional on-pathway reaction takes place before nucleation and it is responsible for the heterogeneity of structures produced during prion protein polymerization in vitro. These amyloid structures behave like prion strains. A model is proposed to explain the genesis of heterogeneity among prion amyloid.