Fluctuations in the Kinetics of Linear Protein Self-Assembly

Mechanisms and pathology of protein misfolding and aggregation

Despite advances in machine learning-based protein structure prediction, we are still far from fully understanding how proteins fold into their native conformation. The conventional notion that polypeptides fold spontaneously to their biologically active states has gradually been replaced by our understanding that cellular protein folding often requires context-dependent guidance from molecular chaperones in order to avoid misfolding. Misfolded proteins can aggregate into larger structures, such as amyloid fibrils, which perpetuate the misfolding process, creating a self-reinforcing cascade. A surge in amyloid fibril structures has deepened our comprehension of how a single polypeptide sequence can exhibit multiple amyloid conformations, known as polymorphism. The assembly of these polymorphs is not a random process but is influenced by the specific conditions and tissues in which they originate. This observation suggests that, similar to the folding of native proteins, the kinetics of pathological amyloid assembly are modulated by interactions specific to cells and tissues. Here, we review the current understanding of how intrinsic protein conformational propensities are modulated by physiological and pathological interactions in the cell to shape protein misfolding and aggregation pathology.

Modified Smoluchowski Rate Equations for Aggregation and Fragmentation in Finite Systems

Protein self-assembly into supramolecular structures is important for cell biology. Theoretical methods employed to investigate protein aggregation and analogous processes include molecular dynamics simulations, stochastic models, and deterministic rate equations based on the mass-action law. In molecular dynamics simulations, the computation cost limits the system size, simulation length, and number of simulation repeats. Therefore, it is of practical interest to develop new methods for the kinetic analysis of simulations. In this work we consider the Smoluchowski rate equations modified to account for reversible aggregation in finite systems. We present several examples and argue that the modified Smoluchowski equations combined with Monte Carlo simulations of the corresponding master equation provide an effective tool for developing kinetic models of peptide aggregation in molecular dynamics simulations.

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.

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.

Kinetic analysis reveals that independent nucleation events determine the progression of polyglutamine aggregation in C. elegans

Protein aggregation is associated with a wide range of degenerative human diseases with devastating consequences, as exemplified by Alzheimer's, Parkinson's, and Huntington's diseases. In vitro kinetic studies have provided a mechanistic understanding of the aggregation process at the molecular level. However, it has so far remained largely unclear to what extent the biophysical principles of amyloid formation learned in vitro translate to the complex environment of living organisms. Here, we take advantage of the unique properties of a Caenorhabditis elegans model expressing a fluorescently tagged polyglutamine (polyQ) protein, which aggregates into discrete micrometer-sized inclusions that can be directly visualized in real time. We provide a quantitative analysis of protein aggregation in this system and show that the data are described by a molecular model where stochastic nucleation occurs independently in each cell, followed by rapid aggregate growth. Global fitting of the image-based aggregation kinetics reveals a nucleation rate corresponding to 0.01 h^-1 per cell at 1 mM intracellular protein concentration, and shows that the intrinsic molecular stochasticity of nucleation accounts for a significant fraction of the observed animal-to-animal variation. Our results highlight how independent, stochastic nucleation events in individual cells control the overall progression of polyQ aggregation in a living animal. The key finding that the biophysical principles associated with protein aggregation in small volumes remain the governing factors, even in the complex environment of a living organism, will be critical for the interpretation of in vivo data from a wide range of protein aggregation diseases.

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.).

MIRRAGGE - Minimum Information Required for Reproducible AGGregation Experiments

Reports on phase separation and amyloid formation for multiple proteins and aggregation-prone peptides are recurrently used to explore the molecular mechanisms associated with several human diseases. The information conveyed by these reports can be used directly in translational investigation, e.g., for the design of better drug screening strategies, or be compiled in databases for benchmarking novel aggregation-predicting algorithms. Given that minute protocol variations determine different outcomes of protein aggregation assays, there is a strong urge for standardized descriptions of the different types of aggregates and the detailed methods used in their production. In an attempt to address this need, we assembled the Minimum Information Required for Reproducible Aggregation Experiments (MIRRAGGE) guidelines, considering first-principles and the established literature on protein self-assembly and aggregation. This consensus information aims to cover the major and subtle determinants of experimental reproducibility while avoiding excessive technical details that are of limited practical interest for non-specialized users. The MIRRAGGE table (template available in Supplementary Information) is useful as a guide for the design of new studies and as a checklist during submission of experimental reports for publication. Full disclosure of relevant information also enables other researchers to reproduce results correctly and facilitates systematic data deposition into curated databases.

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.

Stochastic yield catastrophes and robustness in self-assembly

A guiding principle in self-assembly is that, for high production yield, nucleation of structures must be significantly slower than their growth. However, details of the mechanism that impedes nucleation are broadly considered irrelevant. Here, we analyze self-assembly into finite-sized target structures employing mathematical modeling. We investigate two key scenarios to delay nucleation: (i) by introducing a slow activation step for the assembling constituents and, (ii) by decreasing the dimerization rate. These scenarios have widely different characteristics. While the dimerization scenario exhibits robust behavior, the activation scenario is highly sensitive to demographic fluctuations. These demographic fluctuations ultimately disfavor growth compared to nucleation and can suppress yield completely. The occurrence of this stochastic yield catastrophe does not depend on model details but is generic as soon as number fluctuations between constituents are taken into account. On a broader perspective, our results reveal that stochasticity is an important limiting factor for self-assembly and that the specific implementation of the nucleation process plays a significant role in determining the yield.

The self-assembly of a large biological molecule from small building blocks is like finishing a puzzle of magnetic pieces by shaking the box. Even though each piece of the puzzle is attracted to its correct neighbours, the limited control makes it very hard to finish the puzzle in a short amount of time.

The problem becomes even more difficult if several copies of the same puzzle are assembled in one box. If several puzzles start at the same time, the different parts might steal pieces from each other, making it impossible to successfully complete any of the puzzles. This is called a depletion trap. If the box is only shaken and there is no real control over individual pieces, these traps occur at random.

Overcoming these random depletion traps is an important challenge when assembling nanostructures and other artificial molecules designed by humans without wasting many, potentially expensive, components. Previous studies have shown that when multiple copies of the same structure are assembled simultaneously, slowing the rate of initiation increases the yield of correctly-made structures. This prevents new structures from stealing pieces from existing structures before they are fully completed.

Now, Gartner, Graf, Wilke et al. have used a mathematical model to show that changing the way initiation is delayed leads to different yields. This was especially true for small systems where fluctuations in the availability of the different pieces strongly enhanced the initiation of new structures. In these cases, the self-assembly process terminated undesirably with many incomplete structures.

Nanostructures have various applications ranging from drug delivery to robotics. These findings suggest that in order to efficiently assemble biological molecules, the concentrations of the different building blocks need to be tightly controlled. A question for further research is to investigate strategies that reduce fluctuations in the availability of the building blocks to develop more efficient assembly protocols.

Molecular mechanisms of 33-mer gliadin peptide oligomerisation

The 33-mer gliadin peptide oligomerizes driven by its non-ionic polar character, flexible PPII secondary structure and stable glutamine H-bonds.

Sensitivity analysis of the variability of amyloid aggregation profiles

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

Influence of Zeolites on Amyloid-β Aggregation

Aggregation of Aβ plays a key role in the progression of Alzheimer's disease. Unfortunately, the Aβ aggregation mechanism is complex, leading to a structurally diverse population of oligomers and amyloid fibrils. Heterogeneous interfaces have been shown to influence the rate of fibrilization and may be useful tools to bias amyloid formation toward specific structures. In order to better understand how exogenous materials influence Aβ aggregation, Aβ_1-40 was exposed to zeolite Y containing different metal cations, including Na⁺, Mg²⁺, Fe³⁺, Zn²⁺, and Cu²⁺. NaY, MgY, and FeY, all accelerated the kinetics of fibrilization by increasing the primary nucleation rate, while CuY and ZnY inhibited fibrilization. These kinetic effects were supported through binding affinity measurements, in which ZnY and CuY showed higher association constants than the other zeolites. In addition to influencing the kinetics of fibrilization, the zeolites also affected the intermediate structures along the pathway. Western blots confirmed that Aβ_1-40 was arrested at the oligomeric stage in the presence of ZnY and CuY, while continuing to the fibrillary state in the presence of other zeolites. Seeding studies showed that NaY and FeY form on-pathway oligomers, while ZnY formed off-pathway oligomers. Overall, our results show that zeolites can impact the aggregation and speciation of amyloids.

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.

Kinetic Analysis of Amyloid Formation

The formation of amyloid fibrils is a central phenomenon in the progressive pathology of many neurodegenerative diseases, as well as in the fabrication of functional materials. Several different molecular processes acting in concert are responsible for the formation of amyloid fibrils from monomeric protein in solution. Here, we describe a method to determine which microscopic processes drive the overall formation of fibrils by using chemical kinetics in combination with systematic experimental datasets analysed in a global manner. We outline general concepts for obtaining suitable kinetic data and detail the key stages of data analysis, from quality control to the verification of a specific mechanism of aggregation.

Thermodynamic properties of amyloid fibrils in equilibrium

In this manuscript we use a two-dimensional coarse-grained model to study how amyloid fibrils grow towards an equilibrium state where they coexist with proteins dissolved in a solution. Free-energies to dissociate proteins from fibrils are estimated from the residual concentration of dissolved proteins. Consistent with experiments, the concentration of proteins in solution affects the growth rate of fibrils but not their equilibrium state. Also, studies of the temperature dependence of the equilibrium state can be used to estimate thermodynamic quantities, e.g., heat capacity and entropy.

CREIM: Coffee Ring Effect Imaging Model for Monitoring Protein Self-Assembly in Situ

Protein self-assembly is fundamental to nanotechnology. Self-assembling structures are produced under static in vitro conditions typically forming over hours. In contrast, hydrodynamic intracellular environments employ far shorter time scales to compartmentalize highly concentrated protein solutions. Herein, we exploit the radial capillary flow within a drying sessile droplet (the coffee ring effect) to emulate dynamic native environments and monitor an archetypal protein assembly in situ using high-speed super-resolution imaging. We demonstrate that the assembly can be empirically driven to completion within minutes to seconds without apparent changes in supramolecular morphology. The model offers a reliable tool for the diagnosis and engineering of self-assembling systems under nonequilibrium conditions.

Kinetic constraints on self-assembly into closed supramolecular structures

Many biological and synthetic systems exploit self-assembly to generate highly intricate closed supramolecular architectures, ranging from self-assembling cages to viral capsids. The fundamental design principles that control the structural determinants of the resulting assemblies are increasingly well-understood, but much less is known about the kinetics of such assembly phenomena and it remains a key challenge to elucidate how these systems can be engineered to assemble in an efficient manner and avoid kinetic trapping. We show here that simple scaling laws emerge from a set of kinetic equations describing the self-assembly of identical building blocks into closed supramolecular structures and that this scaling behavior provides general rules that determine efficient assembly in these systems. Using this framework, we uncover the existence of a narrow range of parameter space that supports efficient self-assembly and reveal that nature capitalizes on this behavior to direct the reliable assembly of viral capsids on biologically relevant timescales.

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.