The kinetics of nucleated polymerizations at high concentrations: amyloid fibril formation near and above the "supercritical concentration"

Abstract

The formation of amyloid and other types of protein fibrils is thought to proceed by a nucleated polymerization mechanism. One of the most important features commonly associated with nucleated polymerizations is a strong dependence of the rate on the concentration. However, the dependence of fibril formation rates on concentration can weaken and nearly disappear as the concentration increases. Using numerical solutions to the rate equations for nucleated polymerization and analytical solutions to some limiting cases, we examine this phenomenon and show that it is caused by the concentration approaching and then exceeding the equilibrium constant for dissociation of monomers from species smaller than the nucleus, a quantity we have named the "supercritical concentration". When the concentration exceeds the supercritical concentration, the monomer, not the nucleus, is the highest-energy species on the fibril formation pathway, and the fibril formation reaction behaves initially like an irreversible polymerization. We also derive a relation that can be used in a straightforward method for determining the nucleus size and the supercritical concentration from experimental measurements of fibril formation rates.

FIGURE 1

Nucleated polymerization mechanism of protein fibril formation. (A) The sequence of reactions in a nucleated polymerization. Aggregates are assumed to grow by monomer addition. The association and dissociation rate constants are shown above and below the arrows, respectively. The rate constants shown in parentheses arise from the assumptions in the text. The n-mer, X_n, is known as the nucleus. Smaller species are called oligomers, whereas larger species are called fibrils. (B) Plots of free energy (relative to the monomer) versus aggregate size for the formation of a helical polymer with a nucleus size of 4. Addition of a monomer to a monomer or an oligomer creates one new interaction (in the drawings below the plot, the ovals overlap in one place for X₂, X₃, and X₄). Addition of a monomer to the nucleus or a fibril creates two new interactions (the ovals overlap in two places for X₅ and all larger species). Plots are shown for total protein concentrations i), below the critical concentration; ii), between the critical and supercritical concentrations; or iii), above the supercritical concentration (where the total protein concentration is [X]_tot, the critical concentration is K_c = c/a, and the supercritical concentration is K_s = b/a; see text). Below the critical concentration, neither oligomers nor fibrils are stable relative to the monomer. Fibril formation therefore does not occur when [X]_tot < K_c. Between the critical and supercritical concentrations, oligomers are less stable than the monomer, the nucleus is the highest-energy species on the fibril formation pathway, and fibrils become stable relative to the monomer when they are large enough. Above the supercritical concentration, both oligomers and fibrils are stable relative to the monomer. Curve ii corresponds to the classical picture of a nucleated polymerization. Note that the nucleus size is independent of the concentration.

FIGURE 2

(A) Time courses of the weight fractions of the monomer (X₁), dimer (X₂), trimer (X₃), tetramer (X₄), pentamer (X₅), and hexamer (X₆) in an irreversible polymerization. The plots were made using Eqs. 15 and 17. The quantity τx_tot is used for the time variable because τ is inversely proportional to x_tot (see Eq. 17), so using τx_tot for the independent variable enables the weight fraction plots to be independent of x_tot. (B) A plot of weight fraction versus species size at the end of an irreversible polymerization.

FIGURE 3

(A) Plots of the fraction completion versus rescaled time (τ) on a logarithmic scale for the test case with n = 6, σ = 1000, and selected values of x_tot. The fraction completion is defined as m/m_final, where m_final is the value of m at the near-steady-state point (see Eq. 10). Fraction completion was calculated using the numerical solutions of Eqs. 3–9. Each curve is labeled with its corresponding value of x_tot. As mentioned in the text, the curves for x_tot = 10⁵–10⁶ are nearly identical to each other. (B) A plot of the rescaled time required for a fibril formation reaction to reach 50% completion (τ₅₀) against the total protein concentration (x_tot), with both variables on a logarithmic scale for the test case with n = 6 and σ = 1000. The solid circles represent the τ₅₀ values obtained from the numerical solutions of Eqs. 3–9, the solid line represents the τ₅₀ values expected for a classical nucleated polymerization (Eq. 14), and the dashed curve represents the τ₅₀ values expected for a classical nucleated polymerization after correcting for oligomer formation (Eq. 21).

FIGURE 4

(A) Plots of monomer, oligomeric protein, and fibril mass concentrations versus rescaled time (τ) on a logarithmic scale for the test case with n = 6 and σ = 1000 in the low-concentration regime (x_tot = 10; the oligomeric protein concentration is defined as 2x₂ + 3x₃ + … + nx_n). The solid lines represent the time courses from the numerical solutions (NS) to the rate equations. The dashed lines represent the time courses expected for a classical nucleated polymerization (CNP). The three phases of the fibril formation reaction, preequilibration, nucleation, and conversion, are marked above the plots. (B) As in A, except that the plots are for the medium-concentration regime (x_tot = 100). (C) Plots of monomer, oligomeric protein, and fibril mass concentrations versus rescaled time (τ) on a logarithmic scale for the test case with n = 6 and σ = 1000 in the high-concentration regime (x_tot = 10⁵). The solid lines represent the time courses from the numerical solutions (NS) to the rate equations. The dashed lines represent the time courses expected for an irreversible polymerization (IP). (Inset) An expansion of the monomer concentration time course between 10⁻⁵ < τ < 10⁻².

FIGURE 5

Log-log plots of the rescaled time required for a fibril formation reaction to reach 50% completion (τ₅₀) against the total protein concentration (x_tot). Data for several values of n and σ are shown. The solid circles represent the τ₅₀ values obtained from the numerical solutions of the rate equations, the solid lines represent the τ₅₀ values expected from a classical nucleated polymerization (classical NP), and the dashed curves represent the τ₅₀ values expected from a classical nucleated polymerization after correcting for oligomer formation (corrected NP). The colors of the solid circles, solid lines, and dashed curves correspond to the values of σ as shown in the key in the lower right, and each graph shows data for a single value of n. (A) n = 3; (B) n = 6; and (C) n = 9.