Numerical and analytical simulation of the growth of amyloid-β plaques

Abstract

Numerical and analytical solutions to calculate the radius of an amyloid-β (Aβ) plaque over time were obtained. To my knowledge, this is the first model simulating the growth of Aβ plaques. It is shown that the plaque can reach the diameter of 50 μm after 20 years of growth only if degradation machinery responsible for degradation of Aβ monomers is malfunctioning. A mathematical model incorporates a nucleation and an autocatalytic growth using the Finke-Watzky model. The obtained system of ordinary differential equations was solved numerically, and for the simplified case of infinitely long Aβ monomer half-life, an analytical solution was found. Assuming that Aβ aggregates stick together, and using the distance between the plaques as an input parameter of the model, it was possible to calculate the plaque radius from the concentration of Aβ aggregates. This led to the formulation of the "cubic root hypothesis," which posits that Aβ plaque size increases as the cubic root of time. This hypothesis helps explain why larger plaques grow more slowly. Furthermore, the obtained results suggest that the plaque size is independent of the kinetic constants governing Aβ plaque agglomeration. This indicates that the kinetics of Aβ plaque agglomeration is not a limiting factor for the plaque growth. Instead, the plaque growth rate is limited by the rates of Aβ monomer production and degradation.

Numerical modeling of senile plaque development under conditions of limited diffusivity of amyloid-β monomers

“…The numerical solution is then compared with the approximate solution of Eqs. ( 8)- (12) obtained in [21] The growth of the Aβ plaque (Fig. 1) is simulated employing the following method.…”

“…The approximate and exact solutions for Eqs. ( 8)-( 12) were derived in [21]. The procedure is summarized below.…”

“…The numerical solution is then compared with the approximate solution of Eqs. (8)-(12) obtained in [21]. The approximate solution, detailed in section S1 of the Supplemental Materials, is derived under the assumption of dysfunctional protein degradation machinery, implying infinitely large half-lives for Aβ monomers and aggregates ( and ).…”

“…The proteolytic degradation of Aβ monomers and aggregates was simulated by assuming finite half-lives for these components. This paper aims to expand upon my previous investigations into the effect of Aβ monomer diffusivity [20] and studies based on the lumped capacitance approximation of Aβ plaque growth [21]. The objective is to develop a method for predicting the growth of Aβ plaques in scenarios where the concentrations of Aβ monomers and aggregates vary across the control volume (CV), thereby not adhering to the lumped capacitance assumption.…”

Simulating growth of TDP-43 cytosolic inclusion bodies in neuron soma

“…This builds upon my previous research, which explored the growth of amyloid-β plaques [18] and Lewy bodies [19,20]. The goal is to apply the methodology developed in [18,20] to simulate the growth of TDP-43 inclusion bodies.…”