Scaling and dimensionality in the chemical kinetics of protein filament formation

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 catalytic nature of protein aggregation

The formation of amyloid fibrils from soluble peptide is a hallmark of many neurodegenerative diseases such as Alzheimer’s and Parkinson’s diseases. Characterization of the microscopic reaction processes that underlie these phenomena have yielded insights into the progression of such diseases and may inform rational approaches for the design of drugs to halt them. Experimental evidence suggests that most of these reaction processes are intrinsically catalytic in nature and may display enzymelike saturation effects under conditions typical of biological systems, yet a unified modeling framework accounting for these saturation effects is still lacking. In this paper, we therefore present a universal kinetic model for biofilament formation in which every fundamental process in the reaction network can be catalytic. The single closed-form expression derived is capable of describing with high accuracy a wide range of mechanisms of biofilament formation and providing the first integrated rate law of a system in which multiple reaction processes are saturated. Moreover, its unprecedented mathematical simplicity permits us to very clearly interpret the effects of increasing saturation on the overall kinetics. The effectiveness of the model is illustrated by fitting it to the data of in vitro Aβ40 aggregation. Remarkably, we find that primary nucleation becomes saturated, demonstrating that it must be heterogeneous, occurring at interfaces and not in solution.

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.

Polymer-Peptide Conjugates Convert Amyloid into Protein Nanobundles through Fragmentation and Lateral Association

The assembly of proteins into amyloid fibrils has become linked not only with the progression of myriad human diseases, but also important biological functions. Understanding and controlling the formation, structure, and stability of amyloid fibrils is therefore a major scientific goal. Here we utilize electron microscopy-based approaches combined with quantitative statistical analysis to show how recently developed kind of amyloid modulators-multivalent polymer-peptide conjugates (mPPCs)-can be applied to control the structure and stability of amyloid fibrils. In doing so, we demonstrate that mPPCs are able to convert 40-residue amyloid beta fibrils into ordered nanostructures through a combination of fragmentation and bundling. Fragmentation is shown to be consistent with a model where the rate constant of fibril breakage is independent of the fibril length, suggesting a local and specific interaction between fibrils and mPPCs. Subsequent bundling, which was previously not observed, leads to the formation of sheet-like nanostructures which are surprisingly much more uniform than the starting fibrils. These nanostructures have dimensions independent of the molecular weight of the mPPC and retain the molecular-level ordering of the starting amyloid fibrils. Collectively, we reveal quantitative and nanoscopic understanding of how mPPCs can be applied to control amyloid structure and stability, and demonstrate approaches to elucidate nanoscale amyloid phase behavior in the presence of functional macromolecules and other modulators.

Mathematical Modeling of Protein Misfolding Mechanisms in Neurological Diseases: A Historical Overview

Protein misfolding refers to a process where proteins become structurally abnormal and lose their specific 3-dimensional spatial configuration. The histopathological presence of misfolded protein (MP) aggregates has been associated as the primary evidence of multiple neurological diseases, including Prion diseases, Alzheimer's disease, Parkinson's disease, and Creutzfeldt-Jacob disease. However, the exact mechanisms of MP aggregation and propagation, as well as their impact in the long-term patient's clinical condition are still not well understood. With this aim, a variety of mathematical models has been proposed for a better insight into the kinetic rate laws that govern the microscopic processes of protein aggregation. Complementary, another class of large-scale models rely on modern molecular imaging techniques for describing the phenomenological effects of MP propagation over the whole brain. Unfortunately, those neuroimaging-based studies do not take full advantage of the tremendous capabilities offered by the chemical kinetics modeling approach. Actually, it has been barely acknowledged that the vast majority of large-scale models have foundations on previous mathematical approaches that describe the chemical kinetics of protein replication and propagation. The purpose of the current manuscript is to present a historical review about the development of mathematical models for describing both microscopic processes that occur during the MP aggregation and large-scale events that characterize the progression of neurodegenerative MP-mediated diseases.

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.