Probing the Occurrence of Soluble Oligomers through Amyloid Aggregation Scaling Laws

Abstract

Drug discovery frequently relies on the kinetic analysis of physicochemical reactions that are at the origin of the disease state. Amyloid fibril formation has been extensively investigated in relation to prevalent and rare neurodegenerative diseases, but thus far no therapeutic solution has directly arisen from this knowledge. Other aggregation pathways producing smaller, hard-to-detect soluble oligomers are increasingly appointed as the main reason for cell toxicity and cell-to-cell transmissibility. Here we show that amyloid fibrillation kinetics can be used to unveil the protein oligomerization state. This is illustrated for human insulin and ataxin-3, two model proteins for which the amyloidogenic and oligomeric pathways are well characterized. Aggregation curves measured by the standard thioflavin-T (ThT) fluorescence assay are shown to reflect the relative composition of protein monomers and soluble oligomers measured by nuclear magnetic resonance (NMR) for human insulin, and by dynamic light scattering (DLS) for ataxin-3. Unconventional scaling laws of kinetic measurables were explained using a single set of model parameters consisting of two rate constants, and in the case of ataxin-3, an additional order-of-reaction. The same fitted parameters were used in a discretized population balance that adequately describes time-course measurements of fibril size distributions. Our results provide the opportunity to study oligomeric targets using simple, high-throughput compatible, biophysical assays.

Keywords: amyloid; kinetic analysis; nucleation; protein aggregation; soluble oligomers.

Figure 1

Case study examples of human insulin and ataxin-3 aggregation. Transmission electron microscopy (TEM) micrographs of negatively stained fibrils of (A) 5 mg/mL human insulin and (B) $5$ μM (0.218 mg/mL) ataxin-3 captured after $6$ h and $65$ h incubation, respectively (scale bars, 100 nm). (C) Schematic amyloid fibrillation curves representing the progress of normalized thioflavin-T (ThT) fluorescence ($F/F_F$ ) during the aggregation of human insulin and ataxin-3 in the range of protein concentrations studied by Foderà et al. [26] and Silva et al. [17], respectively. The half-life coordinates $t_{50}$ and $v_{50}$ are indicated by the arrows and by the slopes of dashed lines, respectively.

Figure 2

Aggregation pathways of human insulin investigated through amyloid fibrillation kinetics. (A) Reaction steps and corresponding rate constants participating in the amyloid pathway. Green glows represent an increase in the mass of fibrils. This variation is detected by amyloid binding assays and can be used to estimate two parameters, $k_a$ and $k_b$, consisting of combinations of the other rate constants (see text for details). (B) Oligomeric equilibrium of insulin as determined by Bocian et al. [21] using 2D and pulsed field gradient spin echo (PFGSE) nuclear magnetic resonance (NMR) ($K_{12}=4.9\times {10}^5$, $K_{24}=5.0\times {10}^4$, $K_{46}=2.7\times {10}^3$ and $K_{\mathrm{iso}}=1.35\times {10}^4$ ). (C) Concentration of insulin monomers ($C_{1-\mathrm{mer}}$ ) predicted by the oligomeric equilibrium (B) for the values of total protein concentration ($C_T$ ) used in (D–F) (symbols). Pink line: polynomial fit to the data. (D–F) Reaction scaling laws measured by Foderà et al. [26] (symbols) and predicted by the model equations shown in blue for the monomer concentrations estimated in (C) (solid blue lines). (D) The final ThT fluorescence ($F_F$) is a direct proportion of supersaturation $\Delta C=C_{1-\mathrm{mer}}-C^{*}$ (proportionality constant $cnst=1.95\times {10}^5$ ) for an inferred solubility value of $C^{*}=0.029$ mg/mL. Pink line: The polynomial fit in (C) is used to estimate $F_F$ without the solubility correction ($cnst=1.15\times {10}^5$ and $C^{*}=0)$. (E) Double-logarithmic plot of half-life coordinate $t_{50}$ as a function of $C_T$. Red lines: Limit scaling exponents $\left|\gamma \right|$ of 1 (dashed line) and 0.5 (solid line) are still too high to represent the measured trend. (E,F) Both $k_a$ and $k_b$ are considered first-order dependent on $\Delta C$ (fitted values: $k_a=1.34\times {10}^2\Delta C$ h⁻¹ and $k_b=2.41\times {10}^{-7}\Delta C$ ). Measured data were adapted with permission from Foderà et al. [26]. Copyright 2017 American Chemical Society.

Figure 3

Aggregation pathways of ataxin-3 investigated through amyloid fibrillation kinetics. (A) The oligomeric and amyloid pathways take place simultaneously. The rate constants of oligomer formation/dissociation were previously determined (${\kappa }_{1+}=7.99\times {10}^{-4}$ μM⁻¹ h⁻¹, ${\kappa }_{1-}=9.73$ h⁻¹, ${\kappa }_{n+}=0.167$ μM⁻¹ h⁻¹, and ${\kappa }_{n-}=0.775$ h⁻¹) [17]. The steps of amyloid fibril formation are the same as in Figure 2A. The mass of amyloid fibrils is a function of only $k_a$ and $k_b$, whereas the number of filaments is also influenced by fibril breakage and by the critical size of fibrils formed by primary and secondary nucleation ($R^{*}$ and $R_2^{*}$, respectively). (B) Symbols: ThT fluorescence increase measured for ataxin-3 concentrations of (from top to bottom) $C_T=10$ μM, $7$ μM, $5$ μM, $4$ μM and $2$ μM [17]. Lines: individual (black) and global (blue) fittings of the experimental data by Equations (1) and (S7), respectively. Fitting statistics given in Figure S1A. Global fitting: $k_a=0.364C_T^{n_2}$ h⁻¹, $k_b=2.91\times {10}^{-10}{C}_T^{2-n_2}$ and $n_2=0.160$ ). (C–E) Reaction scaling laws corresponding to the kinetic measurements (symbols) and global fitting (blue lines) shown in (B). (C) Double-logarithmic plot. Red lines: Limit scaling exponents $\left|\gamma \right|$ of 1 (dashed line) and 0.5 (solid line) are still too high to represent the measured trend. (D) Red-shadowed area: typically, $v_{50}$ is positively correlated with $C_T$ (and with $t_{50}^{-1}$ ) [29]. Measured data were adapted with permission from Silva et al. [17]. Copyright 2018 John Wiley and Sons.

Figure 4

Time-course DLS analysis of human insulin aggregation—differences and common aspects with ataxin-3. (A–C) Symbols connected by lines: intensity-based size distributions measured at different time points as indicated by the color bar in (A). Larger symbols: values of the hydrodynamic radius ($R_h$) used as estimates of the mean size (${\overline{R}}_h$ ) of insulin fibrils. Vertical dashed lines: visual reference of the first ${\overline{R}}_h$ value of each panel. (D) Measured (symbols) and simulated (lines) time evolution of ${\overline{R}}_h$. Dashed lines: representations of Equation (4) using values of $k_a=4.13$ h⁻¹ and $k_b=6.90\times {10}^{-9}$ fitted beforehand to amyloid aggregation scaling laws (Figure 2), and $R^{*}=5.7$ nm, $k_{+}\approx k_a$ and $k_2\approx 0$; lines from top to bottom $k_a=4.13\times 1.2$ h⁻¹, $k_a=4.13\times 1.1$ h⁻¹ and $k_a=4.13$ h⁻¹. Solid line: solution of the discretized population balance taking into account the presence of pre-assembled clusters (Appendix B, Section B.2). (E) Measured (symbols) and simulated (lines) evolution of ${\overline{R}}_h$ during ataxin-3 aggregation (adapted from Silva et al. [17]). Lines: representations of Equation (4) using previously fitted values of $k_a$ and $k_b$, and $R^{*}=91$ nm, $R_2^{*}=15$ nm, $k_2\approx k_a$, $k_{+}\approx 0$ and (from top to bottom) $k_a/8$, $k_a/4$ and $k_a$ [17].

Figure 5

Equilibrium scaling laws used to unveil the oligomerization pathway. The initial distribution of monomer and total protein (left side) influences the end-point amyloid signal (right side); the correspondence is direct in the cases of (A) no oligomerization pathway and (B) irreversible oligomerization, and indirect in the case of (C) fully reversible oligomerization. Green lines represent cases of protein solubility values $C^{*}=0$ (solid lines) and $C^{*}>0$ (dashed lines).

Figure 6

Impact of oligomerization on amyloid fibrillation kinetics when the dominant autocatalytic step is either fibril elongation (top) or secondary nucleation (bottom). (A–D) If fibril elongation is prevalent, then $k_a\approx k_{+}$ (A) and the scaling laws of $t_{50}$ (B, double-logarithmic plot) and $v_{50}$ (C) can change from linear to markedly nonlinear depending on the rate of oligomer dissociation (D). (C) Inset: in the case of irreversible oligomerization (orange lines), the scaling laws of $v_{50}$ reflect the (effective) initial concentration of monomeric protein. (E–H) If secondary nucleation is prevalent, then $k_a\approx k_2$ (E) and the concentration dependences of $t_{50}$ (F, double-logarithmic plot) and $v_{50}$ (G) are either poorly defined or markedly nonlinear according to the rate of oligomer dissociation (H).