Asymptotic solutions of the Oosawa model for the length distribution of biofilaments

Molecular mechanism of α-synuclein aggregation on lipid membranes revealed

The central hallmark of Parkinson's disease pathology is the aggregation of the α-synuclein protein, which, in its healthy form, is associated with lipid membranes. Purified monomeric α-synuclein is relatively stable in vitro, but its aggregation can be triggered by the presence of lipid vesicles. Despite this central importance of lipids in the context of α-synuclein aggregation, their detailed mechanistic role in this process has not been established to date. Here, we use chemical kinetics to develop a mechanistic model that is able to globally describe the aggregation behaviour of α-synuclein in the presence of DMPS lipid vesicles, across a range of lipid and protein concentrations. Through the application of our kinetic model to experimental data, we find that the reaction is a co-aggregation process involving both protein and lipids and that lipids promote aggregation as much by enabling fibril elongation as by enabling their initial formation. Moreover, we find that the primary nucleation of lipid-protein co-aggregates takes place not on the surface of lipid vesicles in bulk solution but at the air-water and/or plate interfaces, where lipids and proteins are likely adsorbed. Our model forms the basis for mechanistic insights, also in other lipid-protein co-aggregation systems, which will be crucial in the rational design of drugs that inhibit aggregate formation and act at the key points in the α-synuclein aggregation cascade.

Stability matters, too - the thermodynamics of amyloid fibril formation

Amyloid fibrils are supramolecular homopolymers of proteins that play important roles in biological functions and disease. These objects have received an exponential increase in attention during the last few decades, due to their role in the aetiology of a range of severe disorders, most notably some of a neurodegenerative nature. While an overwhelming number of experimental studies exist that investigate how, and how fast, amyloid fibrils form and how their formation can be inhibited, a much more limited body of experimental work attempts to answer the question as to why these types of structures form (i.e. the thermodynamic driving force) and how stable they actually are. In this review, I attempt to give an overview of the types of experiments that have been performed to-date to answer these questions, and to summarise our current understanding of amyloid thermodynamics.

Modeling Protein Aggregation Kinetics: The Method of Second Stochasticization

The nucleation of protein aggregates and their growth are important in determining the structure of the cell's membraneless organelles as well as the pathogenesis of many diseases. The large number of molecular types of such aggregates along with the intrinsically stochastic nature of aggregation challenges our theoretical and computational abilities. Kinetic Monte Carlo simulation using the Gillespie algorithm is a powerful tool for modeling stochastic kinetics, but it is computationally demanding when a large number of diverse species is involved. To explore the mechanisms and statistics of aggregation more efficiently, we introduce a new approach to model stochastic aggregation kinetics which introduces noise into already statistically averaged equations obtained using mathematical moment closure schemes. Stochastic moment equations summarize succinctly the dynamics of the large diversity of species with different molecularity involved in aggregation but still take into account the stochastic fluctuations that accompany not only primary and secondary nucleation but also aggregate elongation, dissociation, and fragmentation. This method of "second stochasticization" works well where the fluctuations are modest in magnitude as is often encountered in vivo where the number of protein copies in some computations can be in the hundreds to thousands. Simulations using second stochasticization reveal a scaling law that correlates the size of the fluctuations in aggregate size and number with the total number of monomers. This scaling law is confirmed using experimental data. We believe second stochasticization schemes will prove valuable for bridging the gap between in vivo cell biology and detailed modeling. (The code is released on https://github.com/MYTLab/stoch-agg.).

Directionality of growth and kinetics of branched fibril formation

The self-assembly of fibrils is a subject of intense interest, primarily due to its relevance to the formation of pathological structures. Some fibrils develop branches via the so-called secondary nucleation. In this paper, we use the master equation approach to model the kinetics of formation of branched fibrils. In our model, a branched fibril consists of one mother branch and several daughter branches. We consider five basic processes of fibril formation, namely, nucleation, elongation, branching, fragmentation, and dissociation of the primary nucleus of fibrils into free monomers. Our main focus is on the effect of the directionality of growth on the kinetics of fibril formation. We consider several cases. At first, the mother branch may elongate from one or from both ends, while the daughter branch elongates only from one end. We also study the case of branched fibrils with bidirectionally growing daughter branches, tangentially to the main stem, which resembles the intertwining process. We derive a set of ordinary differential equations for the moments of the number concentration of fibrils, which can be solved numerically. Assuming that the primary nucleus of fibrils dissociates with the fragmentation rate, in the limit of the zero branching rate, our model reproduces the results of a previous model that considers only the three basic processes of nucleation, elongation, and fragmentation. We also use the experimental parameters for the fibril formation of Huntingtin fragments to investigate the effect of unidirectional vs bidirectional elongation of the filaments on the kinetics of fibrillogenesis.

Stochastic master equation for early protein aggregation in the transthyretin amyloid disease

It is significant to understand the earliest molecular events occurring in the nucleation of the amyloid aggregation cascade for the prevention of amyloid related diseases such as transthyretin amyloid disease. We develop chemical master equation for the aggregation of monomers into oligomers using reaction rate law in chemical kinetics. For this stochastic model, lognormal moment closure method is applied to track the evolution of relevant statistical moments and its high accuracy is confirmed by the results obtained from Gillespie's stochastic simulation algorithm. Our results show that the formation of oligomers is highly dependent on the number of monomers. Furthermore, the misfolding rate also has an important impact on the process of oligomers formation. The quantitative investigation should be helpful for shedding more light on the mechanism of amyloid fibril nucleation.

A two-step biopolymer nucleation model shows a nonequilibrium critical point

Biopolymer self-assembly pathways are complicated by the ability of their monomeric subunits to adopt different conformational states. This means nucleation often involves a two-step mechanism where the monomers first condense to form a metastable intermediate, which then converts to a stable polymer by conformational rearrangement of constituent monomers. Nucleation intermediates play a causative role in amyloid diseases such as Alzheimer's and Parkinson's. While existing mathematical models neglect the conversion dynamics, experiments show that conversion events frequently occur on comparable timescales to the condensation of intermediates and growth of mature polymers and thus cannot be ignored. We present a model that explicitly accounts for simultaneous assembly and conversion. To describe conversion, we propose an experimentally motivated initiation-propagation mechanism in which the stable phase arises locally within the intermediate and then spreads by nearest-neighbor interactions, in a manner analogous to one-dimensional Glauber dynamics. Our analysis shows that the competing timescales of assembly and conversion result in a nonequilibrium critical point, separating a regime where intermediates are kinetically unstable from one where conformationally mixed intermediates accumulate. This strongly affects the accumulation rate of the stable biopolymer phase. Our model is uniquely able to explain experimental phenomena such as the formation of mixed intermediates and abrupt changes in the scaling exponent γ, which relates the total monomer concentration to the accumulation rate of the stable phase. This provides a first step toward a general model of two-step biopolymer nucleation, which can quantitatively predict the concentration and composition of biologically crucial intermediates.

Kinetic diversity of amyloid oligomers

The spontaneous assembly of proteins into amyloid fibrils is a phenomenon central to many increasingly common and currently incurable human disorders, including Alzheimer's and Parkinson's diseases. Oligomeric species form transiently during this process and not only act as essential intermediates in the assembly of new filaments but also represent major pathogenic agents in these diseases. While amyloid fibrils possess a common, defining set of physicochemical features, oligomers, by contrast, appear much more diverse, and their commonalities and differences have hitherto remained largely unexplored. Here, we use the framework of chemical kinetics to investigate their dynamical properties. By fitting experimental data for several unrelated amyloidogenic systems to newly derived mechanistic models, we find that oligomers present with a remarkably wide range of kinetic and thermodynamic stabilities but that they possess two properties that are generic: they are overwhelmingly nonfibrillar, and they predominantly dissociate back to monomers rather than maturing into fibrillar species. These discoveries change our understanding of the relationship between amyloid oligomers and amyloid fibrils and have important implications for the nature of their cellular toxicity.

Early mechanisms of amyloid fibril nucleation in model and disease-related proteins

Protein amyloid aggregation is a hallmark in neuropathologies and other diseases of tremendous impact such as Alzheimer's or Parkinson's diseases. During the last decade, it has become increasingly evident that neuronal death is mainly induced by proteinaceous oligomers rather than the mature amyloid fibrils. Therefore, the earliest molecular events occurring during the amyloid aggregation cascade represent a growing interest of study. Important breakthroughs have been achieved using experimental data from different proteins, used as models, as well as systems related to diseases. Here, we summarize the structural properties of amyloid oligomeric and fibrillar aggregates and review the recent advances on how biophysical techniques can be combined with quantitative kinetic analysis and theoretical models to study the detailed mechanism of oligomer formation and nucleation of fibrils. These insights into the mechanism of early oligomerization and amyloid nucleation are of relevant interest in drug discovery and in the design of preventive strategies against neurodegenerative diseases.

Chemical Kinetics for Bridging Molecular Mechanisms and Macroscopic Measurements of Amyloid Fibril Formation

Understanding how normally soluble peptides and proteins aggregate to form amyloid fibrils is central to many areas of modern biomolecular science, ranging from the development of functional biomaterials to the design of rational therapeutic strategies against increasingly prevalent medical conditions such as Alzheimer's and Parkinson's diseases. As such, there is a great need to develop models to mechanistically describe how amyloid fibrils are formed from precursor peptides and proteins. Here we review and discuss how ideas and concepts from chemical reaction kinetics can help to achieve this objective. In particular, we show how a combination of theory, experiments, and computer simulations, based on chemical kinetics, provides a general formalism for uncovering, at the molecular level, the mechanistic steps that underlie the phenomenon of amyloid fibril formation.

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.

Kinetic stability analysis of protein assembly on the center manifold around the critical point

Background

Non-linear kinetic analysis is a useful method for illustration of the dynamic behavior of cellular biological systems. To date, center manifold theory (CMT) has not been sufficiently applied for stability analysis of biological systems. The aim of this study is to demonstrate the application of CMT to kinetic analysis of protein assembly and disassembly, and to propose a novel framework for nonlinear multi-parametric analysis. We propose a protein assembly model with nonlinear kinetics provided by the fluctuation in monomer concentrations during their diffusion.

Results

When the diffusion process of a monomer is self-limited to give kinetics non-linearity, numerical simulations suggest the probability that the assembly and disassembly oscillate near the critical point. We applied CMT to kinetic analysis of the center manifold around the critical point in detail, and successfully demonstrated bifurcation around the critical point, which explained the observed oscillation.

Conclusions

The stability kinetics of the present model based on CMT illustrates a unique feature of protein assembly, namely non-linear behavior. Our findings are expected to provide methodology for analysis of biological systems.

Modeling the Effect of Monomer Conformational Change on the Early Stage of Protein Self-Assembly into Fibrils

Filamentous self-assembly of proteins is an important process implicated in a plethora of human diseases and of interest for nanotechnology. Using rate equations, we analyze the early stage of the process in solutions that initially contain fibrillation-passive protein monomers and in which the nascent fibrils are practically insoluble. The analysis is based on a model accounting for the conformational and/or other changes the passive monomers experience to transform themselves into fibrillation-active monomers and thus become fibril nuclei. The model allows exact, comprehensive, and simple mathematical description of the early stage of fibrillation, which reveals the usually neglected role of the nucleation nonstationarity in this stage of fibrillation. We obtain exact and user-friendly expressions for experimentally accessible quantities such as the size distribution of fibrils, their number and mass concentrations, the rate and nonstationary period of fibril nucleation, and the delay time of fibril formation. Analyzing available experimental data, we find that the theory successfully describes the fibrillation time course of pathological and nonpathological ataxin-3, a protein involved in the neurodegenerative disorder spinocerebellar ataxia type-3. The analysis provides mechanistic insight into the reason for the higher fibril nucleation and elongation rates of the pathological ataxin-3.

Protein Polymerization into Fibrils from the Viewpoint of Nucleation Theory

The assembly of various proteins into fibrillar aggregates is an important phenomenon with wide implications ranging from human disease to nanoscience. Using general kinetic results of nucleation theory, we analyze the polymerization of protein into linear or helical fibrils in the framework of the Oosawa-Kasai (OK) model. We show that while within the original OK model of linear polymerization the process does not involve nucleation, within a modified OK model it is nucleation-mediated. Expressions are derived for the size of the fibril nucleus, the work for fibril formation, the nucleation barrier, the equilibrium and stationary fibril size distributions, and the stationary fibril nucleation rate. Under otherwise equal conditions, this rate decreases considerably when the short (subnucleus) fibrils lose monomers much more frequently than the long (supernucleus) fibrils, a feature that should be born in mind when designing a strategy for stymying or stimulating fibril nucleation. The obtained dependence of the nucleation rate on the concentration of monomeric protein is convenient for experimental verification and for use in rate equations accounting for nucleation-mediated fibril formation. The analysis and the results obtained for linear fibrils are fully applicable to helical fibrils whose formation is describable by a simplified OK model.

Dynamics of protein aggregation and oligomer formation governed by secondary nucleation

The formation of aggregates in many protein systems can be significantly accelerated by secondary nucleation, a process where existing assemblies catalyse the nucleation of new species. In particular, secondary nucleation has emerged as a central process controlling the proliferation of many filamentous protein structures, including molecular species related to diseases such as sickle cell anemia and a range of neurodegenerative conditions. Increasing evidence suggests that the physical size of protein filaments plays a key role in determining their potential for deleterious interactions with living cells, with smaller aggregates of misfolded proteins, oligomers, being particularly toxic. It is thus crucial to progress towards an understanding of the factors that control the sizes of protein aggregates. However, the influence of secondary nucleation on the time evolution of aggregate size distributions has been challenging to quantify. This difficulty originates in large part from the fact that secondary nucleation couples the dynamics of species distant in size space. Here, we approach this problem by presenting an analytical treatment of the master equation describing the growth kinetics of linear protein structures proliferating through secondary nucleation and provide closed-form expressions for the temporal evolution of the resulting aggregate size distribution. We show how the availability of analytical solutions for the full filament distribution allows us to identify the key physical parameters that control the sizes of growing protein filaments. Furthermore, we use these results to probe the dynamics of the populations of small oligomeric species as they are formed through secondary nucleation and discuss the implications of our work for understanding the factors that promote or curtail the production of these species with a potentially high deleterious biological activity.

Analysis of the length distribution of amyloid fibrils by centrifugal sedimentation

The aggregation of normally soluble peptides and proteins into amyloid fibrils is a process associated with a wide range of pathological conditions, including Alzheimer's and Parkinson's diseases. It has become apparent that aggregates of different sizes possess markedly different biological effects, with aggregates of lower relative molecular weight being associated with stronger neurotoxicity. Yet, although many approaches exist to measure the total mass concentration of aggregates, the ability to probe the length distribution of growing aggregates in solution has remained more elusive. In this work, we applied a differential centrifugation technique to measure the sedimentation coefficients of amyloid fibrils produced during the aggregation process of the amyloid β (M1-42) peptide (Aβ42). The centrifugal method has the advantage of providing structural information on the fibril distribution directly in solution and affording a short analysis time with respect to alternative imaging and analytical centrifugation approaches. We show that under quiescent conditions interactions between Aβ42 fibrils lead to lateral association and to the formation of entangled clusters. By contrast, aggregation under shaking generates a population of filaments characterized by shorter lengths. The results, which have been validated by cryogenic transmission electron microscopy (cryo-TEM) analysis, highlight the important role that fibril-fibril assembly can play in the deposition of aggregation-prone peptides.

Role of filament annealing in the kinetics and thermodynamics of nucleated polymerization

The formation of nanoscale protein filaments from soluble precursor molecules through nucleated polymerization is a common form of supra-molecular assembly phenomenon. This process underlies the generation of a range of both functional and pathological structures in nature. Filament breakage has emerged as a key process controlling the kinetics of the growth reaction since it increases the number of filament ends in the system that can act as growth sites. In order to ensure microscopic reversibility, however, the inverse process of fragmentation, end-to-end annealing of filaments, is a necessary component of a consistent description of such systems. Here, we combine Smoluchowski kinetics with nucleated polymerization models to generate a master equation description of protein fibrillization, where filamentous structures can undergo end-to-end association, in addition to elongation, fragmentation, and nucleation processes. We obtain self-consistent closed-form expressions for the growth kinetics and discuss the key physics that emerges from considering filament fusion relative to current fragmentation only models. Furthermore, we study the key time scales that describe relaxation to equilibrium.