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

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.

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.

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.

Feedback control of protein aggregation

The self-assembly of peptides and proteins into amyloid fibrils plays a causative role in a wide range of increasingly common and currently incurable diseases. The molecular mechanisms underlying this process have recently been discovered, prompting the development of drugs that inhibit specific reaction steps as possible treatments for some of these disorders. A crucial part of treatment design is to determine how much drug to give and when to give it, informed by its efficacy and intrinsic toxicity. Since amyloid formation does not proceed at the same pace in different individuals, it is also important that treatment design is informed by local measurements of the extent of protein aggregation. Here, we use stochastic optimal control theory to determine treatment regimens for inhibitory drugs targeting several key reaction steps in protein aggregation, explicitly taking into account variability in the reaction kinetics. We demonstrate how these regimens may be updated "on the fly" as new measurements of the protein aggregate concentration become available, in principle, enabling treatments to be tailored to the individual. We find that treatment timing, duration, and drug dosage all depend strongly on the particular reaction step being targeted. Moreover, for some kinds of inhibitory drugs, the optimal regimen exhibits high sensitivity to stochastic fluctuations. Feedback controls tailored to the individual may therefore substantially increase the effectiveness of future treatments.

The kinetics of islet amyloid polypeptide phase-separated system and hydrogel formation are critically influenced by macromolecular crowding

Many protein misfolding diseases (e.g. type II diabetes and Alzheimer's disease) are characterised by amyloid deposition. Human islet amyloid polypeptide (hIAPP, involved in type II diabetes) spontaneously undergoes liquid-liquid phase separation (LLPS) and a kinetically complex hydrogelation, both catalysed by hydrophobic-hydrophilic interfaces (e.g. air-water interface and/or phospholipids-water interfaces). Gelation of hIAPP phase-separated liquid droplets initiates amyloid aggregation and the formation of clusters of interconnected aggregates, which grow and fuse to eventually percolate the whole system. Droplet maturation into irreversible hydrogels via amyloid aggregation is thought to be behind the pathology of several diseases. Biological fluids contain a high volume fraction of macromolecules, leading to macromolecular crowding. Despite crowding agent addition in in vitro studies playing a significant role in changing protein phase diagrams, the mechanism underlying enhanced LLPS, and the effect(s) on stages beyond LLPS remain poorly or not characterised.We investigated the effect of macromolecular crowding and increased viscosity on the kinetics of hIAPP hydrogelation using rheology and the evolution of the system beyond LLPS by microscopy. We demonstrate that increased viscosity exacerbated the kinetic variability of hydrogelation and of the phase separated-aggregated system, whereas macromolecular crowding abolished heterogeneity. Increased viscosity also strengthened the gel meshwork and accelerated aggregate cluster fusion. In contrast, crowding either delayed cluster fusion onset (dextran) or promoted it (Ficoll). Our study highlights that an in vivo crowded environment would critically influence amyloid stages beyond LLPS and pathogenesis.

Impact of hemoglobin biophysical studies on molecular pathogenesis and drug therapy for sickle cell disease

Basic research on hemoglobin has been essential for understanding the origin and treatment of many hematological disorders due to abnormal hemoglobins. The most important of the hemoglobinopathies is sickle cell disease - Linus Pauling's "molecular disease" that gave birth to molecular medicine. In this review, I will describe the contributions of basic biophysical research on normal and sickle cell hemoglobin (HbS) to understanding the molecular pathogenesis of the disease and providing the conceptual basis for the various approaches to drug therapy that target HbS polymerization. Most prominent among these are the experimental results on the solubility of HbS as a function of oxygen saturation explained by the allosteric model of Monod, Wyman, and Changeux and the Gill-Wyman thermodynamic linkage relation between solubility and oxygen binding, the solubility of mixtures of HbS with normal or fetal hemoglobin explained by Minton's thermodynamic model, and the highly unusual kinetics of HbS polymerization explained by a novel double nucleation mechanism that also accounts for the aggregation kinetics of the Alzheimer's peptide. The HbS polymerization kinetics are of great importance to understanding the pathophysiology and clinical course, as well as guiding drug development for treating this common and severe disease. The article focuses primarily on experimental and theoretical results from my lab, so it is not a comprehensive review of the subject.

Stochastic Kinetic Treatment of Protein Aggregation and the Effects of Macromolecular Crowding

Investigation of protein self-assembly processes is important for understanding the growth processes of functional proteins as well as disease-causing amyloids. Inside cells, intrinsic molecular fluctuations are so high that they cast doubt on the validity of the deterministic rate-equation approach. Furthermore, the protein environments inside cells are often crowded with other macromolecules, with volume fractions of the crowders as high as 40%. We have developed a stochastic kinetic framework using Gillespie's algorithm for general systems undergoing particle self-assembly, including particularly protein aggregation at the cellular level. The effects of macromolecular crowding are investigated using models built on scaled-particle and transition-state theories. The stochastic kinetic method can be formulated to provide information on the dominating aggregation mechanisms in a method called reaction frequency (or propensity) analysis. This method reveals that the change of scaling laws related to the lag time can be directly related to the change in the frequencies of reaction mechanisms. Further examination of the time evolution of the fibril mass and length quantities unveils that maximal fluctuations occur in the periods of rapid fibril growth and the fluctuations of both quantities can be sensitive functions of rate constants. The presence of crowders often amplifies the roles of primary and secondary nucleation and causes shifting in the relative importance of elongation, shrinking, fragmentation, and coagulation of linear aggregates. We also show a dual effect of changing volume on the halftime of aggregation for ApoC2 which is reduced in the presence of crowders. A comparison of the results of stochastic simulations with those of rate equations gives us information on the convergence relation between them and how the roles of reaction mechanisms change as the system volume is varied.

Hemoglobin S polymerization and sickle cell disease: A retrospective on the occasion of the 70th anniversary of Pauling's Science paper

70 years ago, Linus Pauling, the legendary genius of 20^th century chemistry, published his famous work on the molecular cause of sickle cell disease, a paper that gave birth to what is now called molecular medicine. In this paper, Pauling left important questions unanswered that have motivated an enormous amount of scientific and clinical research since then. This retrospective discusses the basic science studies that have answered those questions directly related to the kinetics and thermodynamics of hemoglobin S polymerization.

Vaso-Occlusion in Sickle Cell Disease: Is Autonomic Dysregulation of the Microvasculature the Trigger?

Sickle cell disease (SCD) is an inherited hemoglobinopathy characterized by polymerization of hemoglobin S upon deoxygenation that results in the formation of rigid sickled-shaped red blood cells that can occlude the microvasculature, which leads to sudden onsets of pain. The severity of vaso-occlusive crises (VOC) is quite variable among patients, which is not fully explained by their genetic and biological profiles. The mechanism that initiates the transition from steady state to VOC remains unknown, as is the role of clinically reported triggers such as stress, cold and pain. The rate of hemoglobin S polymerization after deoxygenation is an important determinant of vaso-occlusion. Similarly, the microvascular blood flow rate plays a critical role as fast-moving red blood cells are better able to escape the microvasculature before polymerization of deoxy-hemoglobin S causes the red cells to become rigid and lodge in small vessels. The role of the autonomic nervous system (ANS) activity in VOC initiation and propagation has been underestimated considering that the ANS is the major regulator of microvascular blood flow and that most triggers of VOC can alter the autonomic balance. Here, we will briefly review the evidence supporting the presence of ANS dysfunction in SCD, its implications in the onset of VOC, and how differences in autonomic vasoreactivity might potentially contribute to variability in VOC severity.

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.

Statistical Mechanics of Globular Oligomer Formation by Protein Molecules

The misfolding and aggregation of proteins into linear fibrils is widespread in human biology, for example, in connection with amyloid formation and the pathology of neurodegenerative disorders such as Alzheimer's and Parkinson's diseases. The oligomeric species that are formed in the early stages of protein aggregation are of great interest, having been linked with the cellular toxicity associated with these conditions. However, these species are not characterized in any detail experimentally, and their properties are not well understood. Many of these species have been found to have approximately spherical morphology and to be held together by hydrophobic interactions. We present here an analytical statistical mechanical model of globular oligomer formation from simple idealized amphiphilic protein monomers and show that this correlates well with Monte Carlo simulations of oligomer formation. We identify the controlling parameters of the model, which are closely related to simple quantities that may be fitted directly from experiment. We predict that globular oligomers are unlikely to form at equilibrium in many polypeptide systems but instead form transiently in the early stages of amyloid formation. We contrast the globular model of oligomer formation to a well-established model of linear oligomer formation, highlighting how the differing ensemble properties of linear and globular oligomers offer a potential strategy for characterizing oligomers from experimental measurements.

Individuals with sickle cell disease have a significantly greater vasoconstriction response to thermal pain than controls and have significant vasoconstriction in response to anticipation of pain

The painful vaso-occlusive crises (VOC) that characterize sickle cell disease (SCD) progress over hours from the asymptomatic steady-state. SCD patients report that VOC can be triggered by stress, cold exposure, and, pain itself. We anticipated that pain could cause neural-mediated vasoconstriction, decreasing regional blood flow and promoting entrapment of sickle cells in the microvasculature. Therefore, we measured microvascular blood flow in the fingers of both hands using plethysmography and laser-Doppler flowmetry while applying a series of painful thermal stimuli on the right forearm in 23 SCD patients and 25 controls. Heat pain applied to one arm caused bilateral decrease in microvascular perfusion. The vasoconstriction response started before administration of the thermal pain stimulus in all subjects, suggesting that pain anticipation also causes significant vasoconstriction. The time delay between thermal pain application and global vasoconstriction ranged from 5 to 15.5 seconds and increased with age (P < .01). Although subjective measures, pain threshold and pain tolerance were not different between SCD subjects and controls, but the vaso-reactivity index characterizing the microvascular blood flow response to painful stimuli was significantly higher in SCD patients (P = .0028). This global vasoconstriction increases microvascular transit time, and may promote entrapment of sickle cells in the microvasculature, making vaso-occlusion more likely. The rapidity of the global vasoconstriction response indicates a neural origin that may play a part in the transition from steady-state to VOC, and may also contribute to the variability in VOC frequency observed in SCD patients.

Identification of Aptamers That Bind to Sickle Hemoglobin and Inhibit Its Polymerization

The pathophysiology of sickle cell disease (SCD) is dependent on the polymerization of deoxygenated sickle hemoglobin (HbS), leading to erythrocyte deformation (sickling) and vaso-occlusion within the microvasculature. Following deoxygenation, there is a delay time before polymerization is initiated, during which nucleation of HbS monomers occurs. An agent with the ability to extend this delay time or slow polymerization would therefore hold a therapeutic, possibly curative, potential. We used the Systematic Evolution of Ligands by Exponential Enrichment (SELEX) method to screen for HbS-binding RNA aptamers modified with nuclease-resistant 2'-fluoropyrimidines. Polymerization assays were employed to identify aptamers with polymerization-inhibitory properties. Two noncompeting aptamers, DE3A and OX3B, were found to bind hemoglobin, significantly increase the delay time, and reduce the rate of polymerization of HbS. These modifiable, nuclease-resistant aptamers are potential new therapeutic agents for SCD.

Physical principles of filamentous protein self-assembly kinetics

The polymerization of proteins and peptides into filamentous supramolecular structures is an elementary form of self-organization of key importance to the functioning biological systems, as in the case of actin biofilaments that compose the cellular cytoskeleton. Aberrant filamentous protein self-assembly, however, is associated with undesired effects and severe clinical disorders, such as Alzheimer's and Parkinson's diseases, which, at the molecular level, are associated with the formation of certain forms of filamentous protein aggregates known as amyloids. Moreover, due to their unique physicochemical properties, protein filaments are finding extensive applications as biomaterials for nanotechnology. With all these different factors at play, the field of filamentous protein self-assembly has experienced tremendous activity in recent years. A key question in this area has been to elucidate the microscopic mechanisms through which filamentous aggregates emerge from dispersed proteins with the goal of uncovering the underlying physical principles. With the latest developments in the mathematical modeling of protein aggregation kinetics as well as the improvement of the available experimental techniques it is now possible to tackle many of these complex systems and carry out detailed analyses of the underlying microscopic steps involved in protein filament formation. In this paper, we review some classical and modern kinetic theories of protein filament formation, highlighting their use as a general strategy for quantifying the molecular-level mechanisms and transition states involved in these processes.

Fluctuations in the Kinetics of Linear Protein Self-Assembly

Biological systems are characterized by compartmentalization from the subcellular to the tissue level, and thus reactions in small volumes are ubiquitous in living systems. Under such conditions, statistical number fluctuations, which are commonly negligible in bulk reactions, can become dominant and lead to stochastic behavior. We present here a stochastic model of protein filament formation in small volumes. We show that two principal regimes emerge for the system behavior, a small fluctuation regime close to bulk behavior and a large fluctuation regime characterized by single rare events. Our analysis shows that in both regimes the reaction lag-time scales inversely with the system volume, unlike in bulk. Finally, we use our stochastic model to connect data from small-volume microdroplet experiments of amyloid formation to bulk aggregation rates, and show that digital analysis of an ensemble of protein aggregation reactions taking place under microconfinement provides an accurate measure of the rate of primary nucleation of protein aggregates, a process that has been challenging to quantify from conventional bulk experiments.

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.

Inherent variability in the kinetics of autocatalytic protein self-assembly

In small volumes, the kinetics of filamentous protein self-assembly is expected to show significant variability, arising from intrinsic molecular noise. This is not accounted for in existing deterministic models. We introduce a simple stochastic model including nucleation and autocatalytic growth via elongation and fragmentation, which allows us to predict the effects of molecular noise on the kinetics of autocatalytic self-assembly. We derive an analytic expression for the lag-time distribution, which agrees well with experimental results for the fibrillation of bovine insulin. Our expression decomposes the lag-time variability into contributions from primary nucleation and autocatalytic growth and reveals how each of these scales with the key kinetic parameters. Our analysis shows that significant lag-time variability can arise from both primary nucleation and from autocatalytic growth and should provide a way to extract mechanistic information on early-stage aggregation from small-volume experiments.

Nucleation-conversion-polymerization reactions of biological macromolecules with prenucleation clusters

The self-assembly of biomolecules, such as peptides and proteins, into filaments is conventionally understood as a nucleated polymerization reaction. However, detailed analysis of experimental observation has revealed recently that nucleation pathways generate growth-competent nuclei via a cascade of metastable intermediate species, which are omitted in conventional models of filamentous growth based on classical nucleation theory. Here we take an analytical approach to generalizing the classical theory of nucleated polymerization to include the formation of these prenucleation clusters, providing a quantitative general classification of the behavior exhibited by these nucleation-conversion-polymerization reactions. A phase diagram is constructed, and analytical predictions are derived for key experimental observables. Using this approach, we delineate the characteristic time scales that determine the nature of biopolymer growth phenomena.

Effect of agitation on the peptide fibrillization: Alzheimer's amyloid-β peptide 1-42 but not amylin and insulin fibrils can grow under quiescent conditions

Many peptides and proteins can form fibrillar aggregates in vitro, but only a limited number of them are forming pathological amyloid structures in vivo. We studied the fibrillization of four peptides--Alzheimer's amyloid-β (Aβ) 1-40 and 1-42, amylin and insulin. In all cases, intensive mechanical agitation of the solution initiated fast fibrillization. However, when the mixing was stopped during the fibril growth phase, the fibrillization of amylin and insulin was practically stopped, and the rate for Aβ40 substantially decreased, whereas the fibrillization of Aβ42 peptide continued to proceed with almost the same rate as in the agitated conditions. The reason for the different sensitivity of the in vitro fibrillization of these peptides towards agitation in the fibril growth phase remains elusive.

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.

Off-pathway aggregation can inhibit fibrillation at high protein concentrations

Ribosomal protein S6 fibrillates readily at slightly elevated temperatures and acidic pH. We find that S6 fibrillation is retarded rather than favored when the protein concentration is increased above a threshold concentration of around 3.5mg/mL. We name this threshold concentration C(FR), the concentration at which fibrillation is retarded. Our data are consistent with a model in which this inhibition is due to the formation of an off-pathway oligomeric species with native-like secondary structure. The oligomeric species dominates at high protein concentrations but exists in dynamic equilibrium with the monomer so that seeding with fibrils can overrule oligomer formation and favors fibrillation under C(FR) conditions. Thus, fibrillation competes with formation of off-pathway oligomers, probably due to a monomeric conversion step that is required to commit the protein to the fibrillation pathway. The S6 oligomer is resistant to pepsin digestion. We also report that S6 forms different types of fibrils dependent on protein concentration. Our observations highlight the multitude of conformational states available to proteins under destabilizing conditions.

The physical foundation of vasoocclusion in sickle cell disease

The pathology of sickle cell disease arises from the occlusion of small blood vessels because of polymerization of the sickle hemoglobin within the red cells. We present measurements using a microfluidic method we have developed to determine the pressure required to eject individual red cells from a capillary-sized channel after the cell has sickled. We find that the maximum pressure is only ∼100 Pa, much smaller than typically found in the microcirculation. This explains why experiments using animal models have not observed occlusion beginning in capillaries. The magnitude of the pressure and its dependence on intracellular concentration are both well described as consequences of sickle hemoglobin polymerization acting as a Brownian ratchet. Given the recently determined stiffness of sickle hemoglobin gels, the observed obstruction seen in sickle cell disease as mediated by adherent cells can now be rationalized, and surprisingly suggests a window of maximum vulnerability during circulation of sickle cells.

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.

Nucleated polymerization with secondary pathways. III. Equilibrium behavior and oligomer populations

We explore the long-time behavior and equilibrium properties of a system of linear filaments growing through nucleated polymerisation. We show that the length distribution for breakable filaments evolves through two well defined limiting cases: first, a steady state distribution determined by the balance of breakage and elongation is reached; upon monomer depletion at the end of the growth phase, an equilibrium length distribution biased towards smaller filament fragments emerges. We furthermore compute the time evolution of the concentration of small oligomeric filament fragments. For frangible filaments, oligomers are present both at early times and at equilibrium, whereas in the absence of fragmentation, oligomers are only present in significant quantities at the beginning of the polymerisation reaction. Finally, we discuss the significance of these results for the biological consequences of filamentous protein aggregation.

Nucleated polymerization with secondary pathways. I. Time evolution of the principal moments

Self-assembly processes resulting in linear structures are often observed in molecular biology, and include the formation of functional filaments such as actin and tubulin, as well as generally dysfunctional ones such as amyloid aggregates. Although the basic kinetic equations describing these phenomena are well-established, it has proved to be challenging, due to their non-linear nature, to derive solutions to these equations except for special cases. The availability of general analytical solutions provides a route for determining the rates of molecular level processes from the analysis of macroscopic experimental measurements of the growth kinetics, in addition to the phenomenological parameters, such as lag times and maximal growth rates that are already obtainable from standard fitting procedures. We describe here an analytical approach based on fixed-point analysis, which provides self-consistent solutions for the growth of filamentous structures that can, in addition to elongation, undergo internal fracturing and monomer-dependent nucleation as mechanisms for generating new free ends acting as growth sites. Our results generalise the analytical expression for sigmoidal growth kinetics from the Oosawa theory for nucleated polymerisation to the case of fragmenting filaments. We determine the corresponding growth laws in closed form and derive from first principles a number of relationships which have been empirically established for the kinetics of the self-assembly of amyloid fibrils.

Size distribution of amyloid nanofibrils

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

Atomistic theory of amyloid fibril nucleation

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

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.

Tracking the heterogeneous distribution of amyloid spherulites and their population balance with free fibrils

The analysis of amyloidogenic systems reveals the appearance of distinct states of aggregation for amyloid fibrils. For different proteins and under specific experimental conditions, amyloid spherulites are recognized as a significant component occurring in several protein model systems used for in vitro fibrillation studies. In this work we have developed an approach to characterize solutions containing a mixture of amyloid spherulites and individual fibrils. Using bovine insulin as the model system, sedimentation kinetics for the amyloid aggregates were followed using a combination of UV-Vis spectroscopy and cross-polarized optical microscopy. Spherulites were identified as the species undergoing sedimentation. A simple mathematical approach allows the description of the kinetics in terms of decay time/rate distribution. Moreover, based on the sedimentation kinetics, a rough estimate of the balance between amyloid spherulites and individual fibrils can be provided. Fitting the experimental data with the proposed physico-chemical approach shows self-consistent results in reasonable agreement with quantitative imaging analysis previously reported. Our results provide new physical insights into the analysis of amyloidogenic systems, providing a method to characterize the heterogeneous distribution of amyloid spherulites and simultaneously distinguish spherulites and free fibril populations. Importantly, the method can be generally applied to the characterization of polydisperse solutions containing optically traceable spherical particles in the micrometric range.

The molecular chaperone alpha-crystallin as an excipient in an insulin formulation

Purpose

To investigate insulin fibrillation under accelerated stress conditions in the presence of a novel excipient, the molecular chaperone α-crystallin, in comparison with common excipients.

Methods

To induce fibrillation, recombinant human insulin (0.58 mg ml⁻¹) formulations without excipient or with bovine α-crystallin (0.01–0.2 mg ml⁻¹), human serum albumin (1–5 mg ml⁻¹), sucrose (10–100 mg ml⁻¹) or polysorbate 80 (0.075–0.3 mg ml⁻¹) were subjected to stirring stress in a fluorescence well plate reader and formulation vials. Protein fibrillation was monitored by thioflavin T. The formulations were further characterized by size-exclusion chromatography, light obscuration, UV/Vis and circular dichroism spectroscopy.

Results

In both methods, insulin formed thioflavin T-binding species, most likely fibrils. Addition of α-crystallin in the well plate assay greatly improved insulin’s resistance to fibrillation, measured as a 6-fold increase in fibrillation lag time for the lowest and 26-fold for the highest concentration used, whereas all other excipients showed only a marginal increase in lag time. The stabilizing effect of α-crystallin was shown by all characterization techniques used.

Conclusions

The effect of α-crystallin on insulin’s physical stability outperforms that of commonly used excipients. α-Crystallin is proposed to bind specifically to pre-fibrillation species, thereby inhibiting fibrillation. This makes α-crystallin an interesting excipient for proteins with propensity to fibrillate.

Self-organization pathways and spatial heterogeneity in insulin amyloid fibril formation

At high temperature and low pH, the protein hormone insulin is highly prone to form amyloid fibrils, and for this reason it is widely used as a model system to study fibril formation mechanisms. In this work, we focused on insulin aggregation mechanisms occurring in HCl solutions (pH 1.6) at 60 degrees C. By means of in situ Thioflavin T (ThT) staining, the kinetics profiles were characterized as a function of the protein concentration, and two concurrent aggregation pathways were pointed out, being concentration dependent. In correspondence to these pathways, different morphologies of self-assembled protein molecules were detected by atomic force microscopy images also evidencing the presence of secondary nucleation processes as a peculiar mechanism for insulin fibrillation. Moreover, combining ThT fluorescence and light scattering, the early stages of the process were analyzed in the low concentration regime, pointing out a pronounced spatial heterogeneity in the formation of the first stable fibrils in solution and the onset of the secondary nucleation pathways.

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.

Fiber depolymerization: fracture, fragments, vanishing times, and stochastics in sickle hemoglobin

The well-characterized rates, mechanisms, and stochastics of nucleation-dependent polymerization of deoxyhemoglobin S (HbS) are important in governing whether or not vaso-occlusive sickle cell crises will occur. The less well studied kinetics of depolymerization may also be important, for example in achieving full dissolution of polymers in the lungs, in resolution of crises and/or in minimizing gelation-induced cellular damage. We examine depolymerization by microscopic observations on depolymerizing HbS fibers, by Monte Carlo simulations and by analytical characterization of the mechanisms. We show that fibers fracture. Experimental scatter of rates is consistent with stochastic features of the analytical model and Monte Carlo results. We derive a model for the distribution of vanishing times and also show the distribution of fracture-dependent fiber fragment lengths and its time dependence. We describe differences between depolymerization of single fibers and bundles and propose models for bundle dissolution. Our basic model can be extended to dissolution of gels containing many fibers and is also applicable to other reversible linear polymers that dissolve by random fracture and end-depolymerization. Under the model, conditions in which residual HbS polymers exist and facilitate repolymerization and thus pathology can be defined; whereas for normal polymers requiring cyclic polymerization and depolymerization for function, conditions for rapid cycling due to residual aggregates can be identified.

Systematic analysis of nucleation-dependent polymerization reveals new insights into the mechanism of amyloid self-assembly

Self-assembly of misfolded proteins into ordered fibrillar aggregates known as amyloid results in numerous human diseases. Despite an increasing number of proteins and peptide fragments being recognised as amyloidogenic, how these amyloid aggregates assemble remains unclear. In particular, the identity of the nucleating species, an ephemeral entity that defines the rate of fibril formation, remains a key outstanding question. Here, we propose a new strategy for analyzing the self-assembly of amyloid fibrils involving global analysis of a large number of reaction progress curves and the subsequent systematic testing and ranking of a large number of possible assembly mechanisms. Using this approach, we have characterized the mechanism of the nucleation-dependent formation of beta(2)-microglobulin (beta(2)m) amyloid fibrils. We show, by defining nucleation in the context of both structural and thermodynamic aspects, that a model involving a structural nucleus size approximately the size of a hexamer is consistent with the relatively small concentration dependence of the rate of fibril formation, contrary to expectations based on simpler theories of nucleated assembly. We also demonstrate that fibril fragmentation is the dominant secondary process that produces higher apparent cooperatively in fibril formation than predicted by nucleated assembly theories alone. The model developed is able to explain and predict the behavior of beta(2)m fibril formation and provides a rationale for explaining generic properties observed in other amyloid systems, such as fibril growth acceleration and pathway shifts under agitation.

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.

Kinetic studies of protein L aggregation and disaggregation

We have investigated the aggregation of protein L in 25% (vol/vol) TFE and 10 mM HCl. Under both conditions, aggregates adopt a fibrillar structure and bind dyes Congo Red and Thioflavin T consistent with the presence of amyloid fibrils. The kinetics of aggregation in 25% TFE suggest a linear-elongation mechanism with critical nucleus size of either two or three monomers. Aggregation kinetics in 10 mM HCl show a prolonged lag phase prior to a rapid increase in aggregation. The lag phase is time-dependent, but the time dependence can be eliminated by the addition of pre-formed seeds. Disaggregation studies show that for aggregates formed in TFE, aggregate stability is a strong function of aggregate age. For example, after 200 min of aggregation, 40% of the aggregation reaction is irreversible, while after 3 days over 60% is irreversible. When the final concentration of the denaturant, TFE, is reduced from 5% to 0, the amount of reversible aggregation doubles. Disaggregation studies of aggregates formed in TFE and 10 mM HCl reveal a complicated effect of pH on aggregate stability.

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.

The kinetic behavior of insulin fibrillation is determined by heterogeneous nucleation pathways

When subjected to acidic conditions and high temperature, insulin is known to produce fibrils that display the common properties of disease amyloids. Thus, clarifying the mechanisms of insulin fibrillation can help the general understanding of amyloidal aggregation. Insulin fibrillation exhibits a very sharp time dependence, with a pronounced lag phase and subsequent explosive growth of amyloidal aggregates. Here we show that the initial stages of this process can be well described by exponential growth of the fibrillated proteins. This indicates that the process is mainly controlled by a secondary nucleation pathway.