Kinetic theory of fibrillogenesis of amyloid beta-protein

Abstract

Prior quasielastic light scattering (QLS) studies of fibrillogenesis of synthetic amyloid beta-protein (Abeta)-(1-40) at low pH have suggested a kinetic model in which: (i) fibrillogenesis requires a nucleation step; (ii) nuclei are produced by Abeta micelles in addition to seeds initially present; and (iii) fibril elongation occurs by irreversible binding of Abeta monomers to the fibril ends. Here we present the full mathematical formulation of this model. We describe the temporal evolution of the concentrations of Abeta monomers and micelles as well as the concentration and size distribution of fibrils. This formulation enables deduction of the fundamental parameters of the model-e.g., the nucleation and elongation rate constants kn and ke-from the time dependency of the apparent diffusion coefficient measured by QLS. The theory accurately represents the experimental observations for Abeta concentrations both below and above c*, the critical concentration for Abeta micelle formation. We suggest that the method of QLS in combination with this theory can serve as a powerful tool for understanding the molecular factors that control Abeta plaque formation.

Figure 1

Schematic representation of the kinetic model for Aβ fibrillogenesis. Fibrillization of Aβ protein is nucleation dependent. Two pathways of fibril nucleation are proposed. One is fibril nucleation on seeds. The second is nucleation within micelles, whose presence is postulated provided that the peptide concentration exceeds the critical micellar concentration c*. Micelles are in fast equilibrium with free monomers at concentration c*. Nuclei are spontaneously formed out of micelles with rate constant k_n. Fibrils grow by binding monomers to fibril ends with the rate proportional to the concentration of free monomers. The corresponding rate constant is k_e.

Figure 2

Calibration curve relating the hydrodynamic radius R_H of a monodisperse solution of rigid rods as a function of the rod length L, or the number of Aβ monomers p. The results are shown for three different rod diameters, d, and for scattering vectors q = 11.8/λ₀, corresponding to 90° scattering in aqueous solution for light having wavelength λ₀ of 633, 514, and 488 nm.

Figure 3

Temporal evolution of R̄_H(t) (dashed lines) for a polydisperse distribution of fibrils, as computed numerically, and the appropriate R̃_H(t) (solid lines) as calculated using the simple analytic theory for p̄(t), for the following values of the total monomer concentration: curves a, C = 5c*; curves b, C = 0.5c*; and curves c, C = 0.1c*. The parameters of the model used were N₀ = 0.001C, k_n = 2.4 × 10⁻⁶ sec⁻¹, k_e = 90 M⁻¹⋅sec⁻¹, m₀ = 25, c* = 0.1 mM, and n₀ = 10.

Figure 4

Comparison between temporal evolution of the samples with Aβ concentration 1.16 mM (A) and 0.47 mM (B) observed experimentally in 0.1 M HCl (+) and calculated using the simple analytic theory (solid curves) with the following parameters: k_n = 2.4 × 10⁻⁶ sec⁻¹, k_e = 90 M⁻¹⋅sec⁻¹, m₀ = 25, c* = 0.1 mM, and n₀ = 10. One percent of the protein was assumed to be in the form of seeds: n₀N₀ = 0.01C.