The catalytic nature of protein aggregation

Abstract

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.

FIG. 1.

The catalytic nature of key reaction processes in biofilament assembly. (a) Catalytic conversion of substrate S to product O by an enzyme E, featuring an intermediate enzyme-substrate complex. (b) In secondary nucleation, the fibril surface acts as a catalyst. (c) In elongation, the growing fibril ends act as a catalyst. Although the chemical species (a shorter fibril) is not regenerated, the pseudospecies (the fibril end) is. (d) In heterogeneous primary nucleation, any surface or interface (we denote the total concentration of binding sites on the interface as ${\left[I\right]}_0$) present in the reaction vessel may act as a catalyst. In all cases, where the concentration is high enough, the surface may become completely saturated with monomers; at this point, further increases in concentration do not affect the rate, which is then given simply by k_cat · [catalyst]_t=0. The 50% binding concentration K_x ($x=P,E,S$) is given by setting the intermediate bound state to steady-state, and in the usual case that $k_d\gg k_{\mathrm{c}\mathrm{a}\mathrm{t}},K_x^{n_x}$ is approximately the dissociation constant for the corresponding dissociation reaction. We may thus interpret K_x as the geometric mean of the dissociation constants for each fundamental step in the dissociation reaction.

FIG. 2.

(a) Plot of the rate, scaled by the maximal rate $v_{\text{max}}$, vs the monomer concentration m, scaled by the half-saturation concentration K_M, for an elongation reaction (cyan) and for nucleation reactions with orders 2 (green) and 4 (magenta). Elongation obeys Michaelis-Menten kinetics precisely, with a sublinear dependence of the rate on monomer concentration, whereas the higher-order nucleation reactions obey Hill kinetics with the rate exhibiting a sigmoidal monomer dependence. (b)–(d) investigating the effect of saturation in elongation, secondary nucleation, and primary nucleation, respectively, on aggregation curves. Aβ40 rate constants employed with m(0) = 35 µM. Solid lines: K_M = 1M, i.e., no saturation. Dashed lines: K_M = 35, 20, or 10 µM. Shorter dashed lines correspond to lower saturation concentrations. Saturation in elongation and secondary nucleation mainly reduces the aggregation rate, whereas the sole effect of saturation in primary nucleation is to increase the lag time. Due to the logarithmic dependence of the half time on primary nucleation, saturation in the latter has the smallest effect on aggregation kinetics. Saturation in secondary nucleation has the largest overall impact, despite increasing ε, due to the higher reaction order of secondary nucleation compared to elongation.

FIG. 3.

Aggregation of Aβ40 monomers into filaments; fractional aggregation monitored by ThT fluorescence vs time. Initial monomer concentrations ranging from 3.5 µM to 70 µM (only 9.1–70 µM shown here). Data taken from Ref. . (a) The fit to the model by Meisl et al. is only able to reproduce aggregation curves below 35 µM. (b) Equation (10a) with K_P set to an arbitrarily large value fits the data significantly better, yielding K_E = 103 µM. (c) When fully unconstrained, Eq. (10a) fits the data better still, yielding a larger value of K_E and a moderate value of K_P, suggesting that saturation in primary nucleation is more important than in elongation for this system. However, the improvement in fit quality is too small to reach a firm conclusion and more analysis is needed.

FIG. 4.

Initial rates of aggregation of Aβ40 monomers into filaments in the presence of 21 µM seeds; initial monomer concentrations ranging from 3.5 µM to 70 µM. Data taken from Ref. . The initial rates in the presence of such high seeds should depend only on the elongation rate; if saturation effects are present in the elongation reaction, the initial aggregation should have a sublinear dependence on initial monomer concentration [see the cyan curve in Fig. 2(a)]. Instead, an approximately linear dependence on the monomer concentration is observed, demonstrating the absence of significant saturation in elongation at the monomer concentrations studied. This supports the tentative conclusion from Fig. 3 that the saturation effects additional to those from secondary nucleation, visible at the upper end of the concentration range studied, originate from primary nucleation and not from elongation.

FIG. 5.

The degree of saturation of the reaction processes of Aβ40 kinetics: blue for secondary nucleation; purple for elongation, and green for primary nucleation. (a) The concentrations at which the different reaction processes are 50% saturated. At these concentrations, saturation effects reduce the overall rates by 50%. (b) The fractional occupation of the catalytic surface for each reaction process over the range of monomer concentrations studied here. This makes a suitable definition for the degree of saturation of each process. Secondary nucleation is essentially fully saturated over all concentrations of interest, whereas elongation is largely unsaturated. Primary nucleation is fully saturated at the higher end of the range of concentrations studied. Where the fractional occupation is small, dissociation dominates over binding; where it is large, binding dominates over dissociation.

FIG. 6.

(a) The Richards solution (in blue) captures the dynamics of a system with secondary nucleation (n₂ = 4) even more accurately than the Hamiltonian solution (in red). (b) The kinetics of fragmenting filament assembly are almost exactly captured by the new c = 3 Richards solution, significantly more accurately than the current standard; the Gompertz solution (in red) and the numerical solution (in black) to the moment equation for fragmenting filament assembly. ε = 0.01.

FIG. 7.

The kinetics of filament assembly with secondary nucleation and saturating elongation are almost exactly captured by Eq. (10a), which is a significant improvement upon an earlier analytical solution based on the Lambert-W function. However, the Richards solution offers little advantage over the Lambert solution for fragmenting systems. Black: numerical solution to the moment equations. Red: old (Lambert) analytical approximate solution. Blue: new (Richards) analytical approximate solution. ε = 0.01. (a) $n_2=2,{\mathcal{K}}_E=2$. (b) $n_2=2,{\mathcal{K}}_E=0.4$. (c) $n_2=0,{\mathcal{K}}_E=2$. (d) $n_2=0,{\mathcal{K}}_E=0.4$.

FIG. 8.

(a) and (b): using rate constants for Aβ40 aggregation taken from Ref. , we calculate kinetic curves for initial monomer concentrations of 3 µM (a) and 70 µM (b). The curve calculated according to the solution developed in this paper [blue, Eq. (10a)] matches the numerical solutions (black) almost precisely; even better than the already highly accurate analytical solution developed in Ref. (red). (c) Raw Ab40 data from Ref. at a range of concentrations 35–3.5 μM (d) Scaling by κ′, and addition of $t_0=\ln \left({\epsilon }^{\prime }\right)/{\kappa }^{\prime }$, effectively collapses the data onto a universal curve, since experimental errors in these data are a greater source of variation from universality than the small differences in $n_2^{\prime }$ for each individual curve.