Thermodynamic and kinetic design principles for amyloid-aggregation inhibitors

Abstract

Understanding the mechanism of action of compounds capable of inhibiting amyloid-fibril formation is critical to the development of potential therapeutics against protein-misfolding diseases. A fundamental challenge for progress is the range of possible target species and the disparate timescales involved, since the aggregating proteins are simultaneously the reactants, products, intermediates, and catalysts of the reaction. It is a complex problem, therefore, to choose the states of the aggregating proteins that should be bound by the compounds to achieve the most potent inhibition. We present here a comprehensive kinetic theory of amyloid-aggregation inhibition that reveals the fundamental thermodynamic and kinetic signatures characterizing effective inhibitors by identifying quantitative relationships between the aggregation and binding rate constants. These results provide general physical laws to guide the design and optimization of inhibitors of amyloid-fibril formation, revealing in particular the important role of on-rates in the binding of the inhibitors.

Keywords: amyloid; drug discovery; inhibition; mathematical model; molecular mechanism.

Fig. 1.

Microscopic mechanisms of protein aggregation and possible inhibition pathways. (A) Schematic representation of the microscopic steps of protein aggregation into fibrillar structures. (B, Left) Potential target species during protein aggregation and associated binding rate constants. (B, Right) Schematic diagrams showing the microscopic steps that are targeted by the inhibitor.

Fig. 2.

Integrated rate laws for protein aggregation in the presence of inhibitors. Characteristic kinetic profiles for free species (A), bound species (B), and fibril mass concentration (C) in the presence of an inhibitor that binds free monomers (left column), fibrils ends (middle column), or fibril surfaces (right column). Dashed lines are the analytical integrated rate laws (see SI Appendix, section S2 for explicit expressions), which are in excellent agreement with numerical simulations of Eq. 3 (solid lines). The curves were generated using typical rate parameters for amyloid-forming systems; here, the parameters correspond to experimentally measured rates for in vitro aggregation of the A$\beta$42 peptide of Alzheimer’s disease (43): $m_{\mathrm{t}\mathrm{o}\mathrm{t}}=3\mu$M, $k_{+}=3\times 10^6{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_1=10^{-4}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_2=8\times 10^3{\mathrm{M}}^{-2}{\mathrm{s}}^{-1}$, $n_1=n_2=2$, $K_{\times }=0.3\mu {\mathrm{M}}^{-1}$, $k_{\times }^{\mathrm{on}}=1.3\times 10^{-2}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$; and $C_i=0.06,0.2,0.4\mu$M (A); $C_i=0.2,0.6,1.3\mu$M (B); and $C_i=0.2,0.8,3\mu$M (C). Dark grey curves are without inhibitor.

Fig. 3.

Thermodynamic and kinetic design principles for protein aggregation inhibitors. (A and B) Effective rates of aggregation in the presence of an inhibitor. (A) The parameter ${\lambda }_{\mathrm{eff}}/\lambda$ describes the extent of inhibition on the primary aggregation pathways. (B) The parameter ${\kappa }_{\mathrm{eff}}/\kappa$ describes the effect on the secondary pathways. These effective rates depend in characteristic ways on two combined parameters: a dimensionless binding rate $k_{\times }^{\mathrm{on}}{C}_i/\kappa$ and dimensionless binding constant $K_{\times }{C}_i$. Dashed and solid lines in A and B indicate sample inhibitor-response curves as a function of $k_{\times }^{\mathrm{on}}{C}_i/\kappa$ at constant $K_{\times }{C}_i$. (C) Depending on the rate of inhibitor binding to its target, $k_{\times }^{\mathrm{on}}{C}_i$, compared with the overall rate of aggregation ($\kappa$), we distinguish different inhibition regimes: inactive (I) ($k_{\times }^{\mathrm{on}}{C}_i\ll \kappa$), nonequilibrium inhibition (NE) ($k_{\times }^{\mathrm{on}}{C}_i\simeq \kappa$), and equilibrium inhibition (E) ($k_{\times }^{\mathrm{on}}{C}_i\gg \kappa$). For fixed $K_{\times }$, maximal inhibition is obtained in the equilibrium regime. The extent of maximal inhibition depends solely on $K_{\times }{C}_i$. The plots in C correspond to the dashed and solid lines shown in A and B, respectively. (D) Schematic phase diagram summarizing possible inhibition regimes for an inhibitor that binds to monomers, fibril ends, or fibril surfaces. These phase diagrams are top views of the plots in A and B, i.e., contour plots of ${\lambda }_{\mathrm{eff}}/\lambda$ (blue) and ${\kappa }_{\mathrm{eff}}/\kappa$ (red) (SI Appendix, Fig. S1). Contour lines in A–C are shown in intervals of 0.1. Plots of ${\kappa }_{\mathrm{eff}}/\kappa$ and ${\lambda }_{\mathrm{eff}}/\lambda$ are generated using the expressions in SI Appendix, Table S2 for the same parameters as in Fig. 2.

Fig. 4.

Application to the inhibition of A$\beta$42 aggregation. (A) Experimental data for the aggregation of 3 $\mu$M A$\beta$42 in the presence of increasing concentrations of Brichos, a molecular chaperone that binds amyloid fibrils (30) ($C_i=0.1,0.15,0.22,0.35,0.5$ A$\beta$42 molar equivalents). The experimental data are compared with the theoretical prediction of our integrated rate law (solid lines). The theoretical prediction for inhibited curves has no fitting parameters: the rate constants of aggregation are extracted from a fit of the aggregation curve in the absence of inhibitor ($k_{+}=3\times 10^6\hspace{0.17em}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_2=8.2\times 10^4\hspace{0.17em}{\mathrm{M}}^{-2}{\mathrm{s}}^{-1}$, $k_1=1.1\times 10^{-4}\hspace{0.17em}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$), and the effect of the inhibitor is predicted using the experimentally measured binding and dissociation rates ($k_{\mathrm{s}}^{\mathrm{on}}=5.1\times 10^3\hspace{0.17em}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_{\mathrm{s}}^{\mathrm{off}}=2.1\times 10^{-4}\hspace{0.17em}{\mathrm{s}}^{-1}$) (30). (B) Experimental data are compared with the theoretical prediction assuming equilibrium binding (Eq. 5) of Brichos with binding constant $K_{\mathrm{s}}=k_{\mathrm{s}}^{\mathrm{on}}/k_{\mathrm{s}}^{\mathrm{off}}=2.4\times 10^7\hspace{0.17em}{\mathrm{M}}^{-1}$. (C) Effective ${\kappa }_{\mathrm{eff}}/\kappa$ as a function of Brichos concentration and comparison with our theoretical prediction (SI Appendix, Table S2) (solid line). (D) Location of Brichos (30) in the phase diagram of possible inhibition regimes for an inhibitor binding fibril surface sites. The points correspond to the following concentrations of Brichos: $C_i=0.1,0.15,0.22,0.35,0.5$ molar equivalents. (E) Location of 10074-G5, a small molecule that binds A$\beta$42 monomers (56), in the phase diagram of possible inhibition regimes. The points correspond to the following inhibitor concentrations: $C_i=\mathrm{1,2,6,10,15,20}\mu$M for $1\mu$M A$\beta$42.