Stability analysis of 4-species A$\beta$ aggregation model: A novel approach to obtaining physically meaningful rate constants

Abstract

Protein misfolding and concomitant aggregation towards amyloid formation is the underlying biochemical commonality among a wide range of human pathologies. Amyloid formation involves the conversion of proteins from their native monomeric states (intrinsically disordered or globular) to well-organized, fibrillar aggregates in a nucleation-dependent manner. Understanding the mechanism of aggregation is important not only to gain better insight into amyloid pathology but also to simulate and predict molecular pathways. One of the main impediments in doing so is the stochastic nature of interactions that impedes thorough experimental characterization and the development of meaningful insights. In this study, we have utilized a well-known intermediate state along the amyloid-$\beta$ peptide aggregation pathway called protofibrils as a model system to investigate the molecular mechanisms by which they form fibrils using stability and perturbation analysis. Investigation of protofibril aggregation mechanism limits both the number of species to be modeled (monomers, and protofibrils), as well as the reactions to two (elongation by monomer addition, and protofibril–protofibril lateral association). Our new model is a reduced order four species model grounded in mass action kinetics. Our prior study required 3200 reactions, which makes determining the reaction parameters prohibitively difficult. Using this model, along with a linear perturbation argument, we rigorously determine stable ranges of rate constants for the reactions and ensure they are physically meaningful. This was accomplished by finding the ranges in which the perturbations die-out in a five-parameter sweep, which includes the monomer and protofibril equilibrium concentrations and three of the rate constants. The results presented are a proof-of-concept method in determining meaningful rate constants that can be used as a bonafide way for determining accurate rate constants for other models involving complex biological reactions such as amyloid aggregation.

Introduction

Protein aggregation is now being recognized as one of the fundamental processes in cell biology that seem to play a role in both cell toxicity and survival. More commonly known for their pathogenicity in neurodegenerative diseases, amyloid aggregates are also seen to take part in functional roles [6]. One of the most widely investigated amlyloid proteins is the amyloid-$\beta$ (A$\beta$) peptide that is implicated in Alzheimer’s disease (AD). The intrinsically disordered monomeric A$\beta$ peptide aggregates to form large molecular weight, insoluble fibrils that deposit as senile plaques in the brains of AD patients [19]. The process of A$\beta$ aggregation, as well as other amyloidogenic proteins, is highly stochastic but follows a nucleation-dependent mechanism in which a specific structural re-organization and concomitant self-assembly is a prerequisite for the aggregation process to occur. The nucleation-dependent mechanism displays a characteristic sigmoidal growth curve containing a lag-phase, where the nucleation occurs, followed by rapid growth and saturation (Fig. 1; inset). Stochasticity in this process can be appreciated by the fact that it involves multiple scales of the reactions (temporal and spatial) that can give raise to multiple nucleation events leading to heterogeneous assembly, depending on the experimental conditions.

In our previous study, we have demonstrated the temporal modeling of A$\beta$ aggregation using a top-down approach by systematically dissecting the sigmoidal growth into experimentally verifiable segments [7]. In the same article, we specifically described the post-nucleation event involving protofibril elongation and association using ODE-based simulations, and derived the rate constants involved in such processes. In the current paper, we have taken the biophysically and computationally well characterized processes of A$\beta$ protofibril elongation and association as a model interactions, to perform the perturbation analysis, to demonstrate and distinguish between the kinetically- and thermodynamically-stable products. More specifically, this paper demonstrates a novel method of selecting appropriate rate constants, when there is no clear way of identifying them, which render the system of equations physically meaningful by incorporation of kinetic and thermodynamic features.

In this work, we model the A$\beta$ aggregation reactions highlighted in Fig. 1. In the reduced-order model developed and employed here, the monomer to protofibril pathway (which includes nucleation) is combined into a single reaction step and the two potential pathways for elongation are conserved. The rate constants in this system of equations are unknown thereby making the solvability of the system impossible. Parametric fitting of the system to experimental data is very difficult due to the complexity of the problem and the abundance of species. Therefore, it is proposed that perturbation arguments and thermodynamic stability can be used to simplify the process of determining the rate constants. The corresponding differential equations are then used to derive a set of first-order perturbed differential equations. The two forward rate constants with which the present work is concerned are $k_{\mathit{pe}}$, for protofibril to elongation reactions, and $k_{\mathit{pa}}$, for protofibril to association reaction rate constant, which are systematically varied to determine which pairs of solutions produce stable solutions for the perturbed system. The pairs of solutions that are allowed are then subjected to thermodynamic constraints in an effort to further reduce the allowable pairs of solutions.

In Sections 2 Experimental materials and methods, 3 Experimental results: protofibril elongation and association, we discuss the experimental methodology and results. The following Section 4 is devoted to the development of the reduced order model and also to the introduction and implementation of the new methodology of determining rate constants. These new rate constants are then introduced into the model system and solved numerically. We also discuss the relevance and justification of this process and compare the results of our theory with those obtained from experiments.