Inherent variability in the kinetics of autocatalytic protein self-assembly

A computational model for single cell Lamin-A structural organization after microfluidic compression

In recent years, nuclear mechanobiology gained a lot of attention for the study of cell responses to external cues like adhesive forces, applied compression, and/or shear-stresses. In details, the Lamin-A protein-as major constituent of the cell nucleus structure-plays a crucial role in the overall nucleus mechanobiological response. However, modeling and analysis of Lamin-A protein organization upon rapid compression conditions in microfluidics are still difficult to be performed. Here, we introduce the possibility to control an applied microfluidic compression on single cells, deforming them up to the nucleus level. In a wide range of stresses (~1-102 kPa) applied on healthy and cancer cells, we report increasing Lamin-A intensities which scale as a power law with the applied compression. Then, an increase up to two times of the nuclear viscosity is measured in healthy cells, due to the modified Lamin-A organization. This is ascribable to the increasing assembly of Lamin-A filament-like branches which increment both in number and elongation (up to branches four-time longer). Moreover, the solution of a computational model of differential equations is presented as a powerful tool for a single cell prediction of the Lamin-A assembly as a function of the applied compression.

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.

Gaining Structural Control by Modification of Polymerization Rate in Ring-Opening Polymerization-Induced Crystallization-Driven Self-Assembly

Polymerization-induced self-assembly (PISA) has become an important one pot method for the preparation of well-defined block copolymer nanoparticles. In PISA, morphology is typically controlled by changing molecular architecture and polymer concentration. However, several computational and experimental studies have suggested that changes in polymerization rate can lead to morphological differences. Here, we demonstrate that catalyst selection can be used to control morphology independent of polymer structure and concentration in ring-opening polymerization-induced crystallization-driven self-assembly (ROPI-CDSA). Slower rates of polymerization give rise to slower rates of self-assembly, resulting in denser lamellae and more 3D structures when compared to faster rates of polymerization. Our explanation for this is that the fast samples transiently exist in a nonequilibrium state as self-assembly starts at a higher solvophobic block length when compared to the slow polymerization. We expect that subsequent examples of rate variation in PISA will allow for greater control over morphological outcome.

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.

Cellular organization in lab-evolved and extant multicellular species obeys a maximum entropy law

The prevalence of multicellular organisms is due in part to their ability to form complex structures. How cells pack in these structures is a fundamental biophysical issue, underlying their functional properties. However, much remains unknown about how cell packing geometries arise, and how they are affected by random noise during growth - especially absent developmental programs. Here, we quantify the statistics of cellular neighborhoods of two different multicellular eukaryotes: lab-evolved ‘snowflake’ yeast and the green alga Volvox carteri. We find that despite large differences in cellular organization, the free space associated with individual cells in both organisms closely fits a modified gamma distribution, consistent with maximum entropy predictions originally developed for granular materials. This ‘entropic’ cellular packing ensures a degree of predictability despite noise, facilitating parent-offspring fidelity even in the absence of developmental regulation. Together with simulations of diverse growth morphologies, these results suggest that gamma-distributed cell neighborhood sizes are a general feature of multicellularity, arising from conserved statistics of cellular packing.

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.

Kinetics of Ferritin Self-Assembly by Laser Light Scattering: Impact of Subunit Concentration, pH, and Ionic Strength

Ferritins, the cellular iron repositories, are self-assembled, hollow spherical nanocage proteins composed of 24 subunits. The self-assembly process in ferritin generates the electrostatic gradient to rapidly sequester Fe(II) ions, thereby minimizing its toxicity (Fenton reaction). Although the factors that drive self-assembly and control its kinetics are little investigated, its inherent reversibility has been utilized for cellular imaging and targeted drug delivery. The current work tracks the kinetics of ferritin self-assembly by laser light scattering and investigates the factors that influence the process. The formation of partially structured subunit-monomers/dimers, at pH ≤ 1.5, serves as the starting material for the self-assembly, which upon increasing the pH exhibits biphasic behavior (a rapid assembly process coupled with subunit folding followed by a slower reassembly/reorganization process) and completes within 10 min. The ferritin self-assembly accelerated with subunit concentration and ionic strength (t_1/2 decreases in both the cases) but slowed down with the pH of the medium from 5.5 to 7.5 (t_1/2 increases). These findings would help to regulate the ferritin self-assembly to enhance the loading/unloading of drugs/nanomaterials for exploiting it as a nanocarrier and nanoreactor.

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.).

Observation of molecular self-assembly events in massively parallel microdroplet arrays

The self-assembly of peptide and protein molecules into nanoscale filaments is a process associated with both biological function and malfunction. Microfluidic techniques can provide powerful tools in the study of such aggregation phenomena while providing access to exploring the role of molecular interactions in disease development. Yet, a common challenge encountered in the study of protein aggregation is the difficulty in achieving spatial and temporal control of the underlying processes. Here, we present a planar (2-D) device allowing for both the generation and confinement of 10 000 monodisperse water-in-oil droplets in an array of chambers with a trapping efficiency of 99%. Due to the specific geometry of the device, droplets can be formed and immediately trapped on the same chip, without the need for continuous flow of the oil phase. Furthermore, we demonstrate the capability of this device as a platform to study the aggregation kinetics and determine stochastic molecular nanoscale self-assembly events in a highly parallel manner for the aggregation of the dipeptide, diphenylalanine, the core recognition motif of the Aβ-42 peptide associated with Alzheimer's disease. The ability to reproducibly generate and confine monodisperse water-in-oil droplets with an extremely high trapping efficiency while maintaining entrapment under zero-flow conditions, on timescales compatible with observing molecular self-assembly events, renders it promising for numerous potential further applications in the biological and biophysical fields.

Exploiting limited valence patchy particles to understand autocatalytic kinetics

Autocatalysis, i.e., the speeding up of a reaction through the very same molecule which is produced, is common in chemistry, biophysics, and material science. Rate-equation-based approaches are often used to model the time dependence of products, but the key physical mechanisms behind the reaction cannot be properly recognized. Here, we develop a patchy particle model inspired by a bicomponent reactive mixture and endowed with adjustable autocatalytic ability. Such a coarse-grained model captures all general features of an autocatalytic aggregation process that takes place under controlled and realistic conditions, including crowded environments. Simulation reveals that a full understanding of the kinetics involves an unexpected effect that eludes the chemistry of the reaction, and which is crucially related to the presence of an activation barrier. The resulting analytical description can be exported to real systems, as confirmed by experimental data on epoxy-amine polymerizations, solving a long-standing issue in their mechanistic description.

Dynamics of proteins aggregation. II. Dynamic scaling in confined media

In this paper, the second in a series devoted to molecular modeling of protein aggregation, a mesoscale model of proteins together with extensive discontinuous molecular dynamics simulation is used to study the phenomenon in a confined medium. The medium, as a model of a crowded cellular environment, is represented by a spherical cavity, as well as cylindrical tubes with two aspect ratios. The aggregation process leads to the formation of β sheets and eventually fibrils, whose deposition on biological tissues is believed to be a major factor contributing to many neuro-degenerative diseases, such as Alzheimer's, Parkinson's, and amyotrophic lateral sclerosis diseases. Several important properties of the aggregation process, including dynamic evolution of the total number of the aggregates, the mean aggregate size, and the number of peptides that contribute to the formation of the β sheets, have been computed. We show, similar to the unconfined media studied in Paper I [S. Zheng et al., J. Chem. Phys. 145, 134306 (2016)], that the computed properties follow dynamic scaling, characterized by power laws. The existence of such dynamic scaling in unconfined media was recently confirmed by experiments. The exponents that characterize the power-law dependence on time of the properties of the aggregation process in spherical cavities are shown to agree with those in unbounded fluids at the same protein density, while the exponents for aggregation in the cylindrical tubes exhibit sensitivity to the geometry of the system. The effects of the number of amino acids in the protein, as well as the size of the confined media, have also been studied. Similarities and differences between aggregation in confined and unconfined media are described, including the possibility of no fibril formation, if confinement is severe.

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.

Measurement of amyloid formation by turbidity assay-seeing through the cloud

Detection of amyloid growth is commonly carried out by measurement of solution turbidity, a low-cost assay procedure based on the intrinsic light scattering properties of the protein aggregate. Here, we review the biophysical chemistry associated with the turbidimetric assay methodology, exploring the reviewed literature using a series of pedagogical kinetic simulations. In turn, these simulations are used to interrogate the literature concerned with in vitro drug screening and the assessment of amyloid aggregation mechanisms.

Recognizing and analyzing variability in amyloid formation kinetics: Simulation and statistical methods

We examine the phenomenon of variability in the kinetics of amyloid formation and detail methods for its simulation, identification and analysis. Simulated data, reflecting intrinsic variability, were produced using rate constants, randomly sampled from a pre-defined distribution, as parameters in an irreversible nucleation-growth kinetic model. Simulated kinetic traces were reduced in complexity through description in terms of three characteristic parameters. Practical methods for assessing convergence of the reduced parameter distributions were introduced and a bootstrap procedure was applied to determine convergence for different levels of intrinsic variation. Statistical methods for assessing the significance of shifts in parameter distributions, relating to either change in parameter mean or distribution shape, were tested. Robust methods for analyzing and interpreting kinetic data possessing significant intrinsic variance will allow greater scrutiny of the effects of anti-amyloid compounds in drug trials.

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.

Critical Nucleus Structure and Aggregation Mechanism of the C-terminal Fragment of Copper-Zinc Superoxide Dismutase Protein

The aggregation of the copper-zinc superoxide dismutase (SOD1) protein is linked to familial amyotrophic lateral sclerosis, a progressive neurodegenerative disease. A recent experimental study has shown that the (147)GVIGIAQ(153) SOD1 C-terminal segment not only forms amyloid fibrils in isolation but also accelerates the aggregation of full-length SOD1, while substitution of isoleucine at site 149 by proline blocks its fibril formation. Amyloid formation is a nucleation-polymerization process. In this study, we investigated the oligomerization and the nucleus structure of this heptapeptide. By performing extensive replica-exchange molecular dynamics (REMD) simulations and conventional MD simulations, we found that the GVIGIAQ hexamers can adopt highly ordered bilayer β-sheets and β-barrels. In contrast, substitution of I149 by proline significantly reduces the β-sheet probability and results in the disappearance of bilayer β-sheet structures and the increase of disordered hexamers. We identified mixed parallel-antiparallel bilayer β-sheets in both REMD and conventional MD simulations and provided the conformational transition from the experimentally observed parallel bilayer sheets to the mixed parallel-antiparallel bilayer β-sheets. Our simulations suggest that the critical nucleus consists of six peptide chains and two additional peptide chains strongly stabilize this critical nucleus. The stabilized octamer is able to recruit additional random peptides into the β-sheet. Therefore, our simulations provide insights into the critical nucleus formation and the smallest stable nucleus of the (147)GVIGIAQ(153) peptide.

What Can the Kinetics of Amyloid Fibril Formation Tell about Off-pathway Aggregation?

Some of the most prevalent neurodegenerative diseases are characterized by the accumulation of amyloid fibrils in organs and tissues. Although the pathogenic role of these fibrils has not been completely established, increasing evidence suggests off-pathway aggregation as a source of toxic/detoxicating deposits that still remains to be targeted. The present work is a step toward the development of off-pathway modulators using the same amyloid-specific dyes as those conventionally employed to screen amyloid inhibitors. We identified a series of kinetic signatures revealing the quantitative importance of off-pathway aggregation relative to amyloid fibrillization; these include non-linear semilog plots of amyloid progress curves, highly variable end point signals, and half-life coordinates weakly influenced by concentration. Molecules that attenuate/intensify the magnitude of these signals are considered promising off-pathway inhibitors/promoters. An illustrative example shows that amyloid deposits of lysozyme are only the tip of an iceberg hiding a crowd of insoluble aggregates. Thoroughly validated using advanced microscopy techniques and complementary measurements of dynamic light scattering, CD, and soluble protein depletion, the new analytical tools are compatible with the high-throughput methods currently employed in drug discovery.

Competition between primary nucleation and autocatalysis in amyloid fibril self-assembly

Kinetic measurements of the self-assembly of proteins into amyloid fibrils are often used to make inferences about molecular mechanisms. In particular, the lag time--the quiescent period before aggregates are detected--is often found to scale with the protein concentration as a power law, whose exponent has been used to infer the presence or absence of autocatalytic growth processes such as fibril fragmentation. Here we show that experimental data for lag time versus protein concentration can show signs of kinks: clear changes in scaling exponent, indicating changes in the dominant molecular mechanism determining the lag time. Classical models for the kinetics of fibril assembly suggest that at least two mechanisms are at play during the lag time: primary nucleation and autocatalytic growth. Using computer simulations and theoretical calculations, we investigate whether the competition between these two processes can account for the kinks which we observe in our and others' experimental data. We derive theoretical conditions for the crossover between nucleation-dominated and growth-dominated regimes, and analyze their dependence on system volume and autocatalysis mechanism. Comparing these predictions to the data, we find that the experimentally observed kinks cannot be explained by a simple crossover between nucleation-dominated and autocatalytic growth regimes. Our results show that existing kinetic models fail to explain detailed features of lag time versus concentration curves, suggesting that new mechanistic understanding is needed. More broadly, our work demonstrates that care is needed in interpreting lag-time scaling exponents from protein assembly data.

Thermal breakage of a semiflexible polymer: breakage profile and rate

Understanding fluctuation-induced breakages in polymers has important implications for basic and applied sciences. Here I present for the first time an analytical treatment of the thermal breakage problem of a semi-flexible polymer model that is asymptotically exact in the low temperature and high friction limits. Specifically, I provide analytical expressions for the breakage propensity and rate, and discuss the generalities of the results and their relevance to biopolymers. This work is fundamental to our understanding of the kinetics of living polymerisation.

Approximate probability distributions of the master equation

Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.