Probing the Occurrence of Soluble Oligomers through Amyloid Aggregation Scaling Laws

biomolecules Article Probing the Occurrence of Soluble Oligomers through Amyloid Aggregation Scaling Laws Alexandra Silva 1,2,† , Zsuzsa Sárkány 1,2,†,‡ , Joana S. Fraga 1,2,3 , Pablo Taboada 4,5, Sandra Macedo-Ribeiro 1,2 and Pedro M. Martins 1,2,3,* 1IBMC—Instituto de Biologia Molecular e Celular, Universidade do Porto, 4200-135 Porto, Portugal; [email protected] (A.S.); [email protected] (Z.S.); [email protected] (J.S.F.); [email protected] (S.M.-R.) 2Instituto de Investigação e Inovação em Saúde, Universidade do Porto, 4200-135 Porto, Portugal 3ICBAS—Instituto de Ciências Biomédicas Abel Salazar, Universidade do Porto, 4050-313 Porto, Portugal 4Área de Física de la Materia Condensada, Facultad de Física, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain; [email protected] 5Instituto de Investigación Sanitaria (IDIS), 15706 de Santiago de Compostela, Spain *Correspondence: [email protected]; Tel.: +351-220-408-800 † These authors contributed equally to this work. ‡ Present Address: LEPABE—Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto. Received: 4 September 2018; Accepted: 1 October 2018; Published: 4 October 2018   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: protein aggregation; amyloid; soluble oligomers; kinetic analysis; nucleation 1. Introduction The deposition of amyloid fibrils in the brain is a pathological hallmark of several different neurodegenerative disorders, yet the pathogenic role of these insoluble aggregates is not fully understood [ 1 ]. On the other hand, there is now substantial in vivo evidence of amyloidogenic proteins also forming small soluble oligomers that spread to neighboring cells and induce downstream processes associated with neurodegeneration [ 1 , 2 ]. Chemical kinetics, a classical cornerstone for drug discovery [ 3 ], is hardly applicable to the study of this new and pre-eminent target [ 4 – 6 ], in part due to the lack of straightforward methods to monitor the formation of a highly heterogeneous group of Biomolecules 2018,8, 108; doi:10.3390/biom8040108 www.mdpi.com/journal/biomolecules Biomolecules 2018,8, 108 2 of 20 species ranging from protein dimers to complex n-mers [ 7 , 8 ]. In contrast, extensive research has been devoted to protein aggregation kinetics based on the characteristic tinctorial properties of amyloid fibrils [9]. An important step towards the kinetic quantification of off-pathway aggregation was taken after the observation of protein precipitation occurring in parallel with the formation of amyloid fibrils of lysozyme [ 10 ]. Before, kinetic analysis of amyloid aggregation of the islet amyloid polypeptide (IAPP) suggested the formation of intermediate on- and off-pathway phases during IAPP fibrillogenesis [ 11 ]. The presence of non-amyloidogenic species produces perceptible deviations from the time evolution of the amyloid signal expected for the generic nucleation and growth processes of the phase transition: α=1−1 kb[exp(kat)−1]+1(1) where α is the normalized amyloid conversion, and ka and kb are combinations of elementary rate constants [ 12 ]. One of the kinetic signatures found to be associated with off-pathway aggregation was the unusually weak dependence of the lag phase duration on the initial concentration of lysozyme [ 10 ]. Similar behaviors observed with other protein models have provided the basis for varied interpretations of the amyloid aggregation mechanism encompassing, for example, Michaelis-Menten-like saturation of the elongation step [ 13 ], complex sub-steps of nucleation and growth [ 14 ], stochastic fluctuations in the nucleation time [ 15 ], and the suppression of fibril fragmentation at high fibril concentrations [ 16 ]. Unlike these possible explanations for the underperforming scaling laws, off-pathway aggregation can be directly investigated by analytical and microscopic techniques such as those used to identify insoluble aggregates of lysozyme [ 10 ], and later on, soluble oligomers of ataxin-3 [17], and metastable oligomers of Aβ40 and Aβ42 peptides [18,19]. Because the formation of soluble and insoluble assemblies is fed by a common pool of protein monomers, we propose that amyloid fibrillation kinetics can be used to reveal the presence of the parallel oligomeric pathway. To test this hypothesis, we chose two systems, human insulin and ataxin-3, for which the fibrillation kinetics have been measured under conditions of known oligomeric composition. Insulin is a protein hormone existing in solution in a thermodynamic equilibrium of monomers, dimers, tetramers, hexamers and higher-order oligomers [ 20 , 21 ]. Changes in the protein molecular structure induced by low pH, high temperature or the presence of organic solvents lead to the formation of amyloid fibrils through the direct association of insulin monomers [ 20 ], or by the assembly of intermediate on-pathway oligomers [ 22 ]. Ataxin-3 is a multi-domain protein with a globular Josephin domain and a C-terminal flexible tail containing a polyglutamine (polyQ) repeat whose expansion ultimately causes Machado–Joseph disease. Ataxin-3 aggregation involves an initial step mediated by the Josephin domain, and a second step dependent on the expanded polyQ tract that accelerates protein aggregation and promotes the formation of mature amyloid fibers [ 23 , 24 ]. The analysis of the thioflavin-T (ThT) binding assay run at different concentrations of human insulin and ataxin-3 uncovers mechanistic aspects of the oligomeric and fibrillar pathways. The distinct aggregation mechanisms predicted for each protein are experimentally validated by time-course dynamic light scattering (DLS) measurements. 2. Materials and Methods 2.1. Protein Preparation Human insulin purchased from Sigma-Aldrich (Saint Louis, MO, USA) (I2643) was dissolved without further purification in 20% acetic acid 0.5 M NaCl (pH 1.8) to a final concentration of 5 mg/mL. Before incubation, samples were filtered with 0.22 µ m syringe filter units (Millex-GV, Millipore, Cork, Ireland). Non-expanded ataxin-3 was expressed and purified as previously described [17]. Biomolecules 2018,8, 108 3 of 20 2.2. Transmission Electron Microscopy Transmission electron microscopy (TEM) visualization of insulin and ataxin-3 fibrils was performed using a TEM JEM-1400 (JEOL, Tokyo, Japan) at an accelerating voltage of 80 kV. 100 µ L samples of 5 mg/mL insulin were incubated for 6 h at 45 ◦ C without mechanical shaking in 1.5 mL eppendorf tubes (DNA LoBind, Eppendorf AG, Hamburg, Germany). 700 µ L samples of 5 µ M (0.218 mg/mL) ataxin-3 in 20 mM sodium phosphate pH 7.5, 150 mM NaCl, 1 mM dithiothreitol (DTT) were incubated in 1.5 mL eppendorf tubes (DNA LoBind, Eppendorf AG, Hamburg, Germany) for 65 h at 37 ◦ C without mechanical shaking. Protein samples were diluted in water (1:20 for insulin and 1:10 for ataxin-3), adsorbed to carbon-coated 200 mesh nickel grids (FCF300-NI, Electron Microscopy Sciences, Hatfield, PA, USA), negatively stained with 2% (w/v) uranyl acetate, dried and observed at a magnification of 80,000–100,000×. 2.3. Dynamic Light Scattering Dynamic light scattering measurements were performed using an ALV/DLS/ SLS-5000F, SP-86 goniometer system (ALV-GmbH, Langen, Germany) equipped with a CW diode-pumped Nd:YAG solid-state Compass-DPSS laser with a symmetrizer (Coherent Inc., Santa Clara, CA, USA). The laser operates at 488 nm with an output power of 400 mW. The intensity scale was calibrated against scattering from toluene. 700 µ L samples of 5 mg/mL insulin were incubated in glass cuvettes at 45 ◦ C without mechanical shaking and periodically analyzed at a scattering angle 90 ◦ to the incident beam. Hydrodynamic radii of the particles in solution were estimated from the diffusion coefficient(s) delivered from CONTIN analysis [ 25 ]. Discontinuous auto-correlation functions were not considered for CONTIN analysis. 3. Results and Discussion For the experimental conditions adopted in each model protein, human insulin and ataxin-3 produce fibrillar species with distinct morphologies (Figure 1A,B) and at markedly different aggregation rates (Figure 1C). Long, straight filaments of human insulin are formed much faster than the small, worm-like fibrils of ataxin-3, thereby suggesting that phase transition mechanisms are differently affected by the fibril elongation step. Chemical kinetic analysis pinpoints these differences, and reveals how the presence of soluble oligomers influences each type of protein aggregation curves. The quantitative methods proposed here are expected to contribute to the identification of mechanistic changes provoked, e.g., by the presence of aggregation modulators or by different conditions of temperature, pH, ionic strength, etc. 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 µ M (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/FF ) during the aggregation of human insulin and ataxin-3 in the range of protein concentrations studied by Foderàet al. [ 26 ] and Silva et al. [ 17 ], respectively. The half-life coordinates t50 and v50 are indicated by the arrows and by the slopes of dashed lines, respectively. Biomolecules 2018,8, 108 4 of 20 3.1. Mechanistic Analysis of Insulin Aggregation In the simplified mechanism represented in Figure 2A, the elementary intermediate steps participating in the primary nucleation, secondary nucleation and elongation of insulin fibrils are summed up into the overall rate constants kn , k2 , and k+ , respectively [ 17 ]. The sigmoidal (rather than hyperbolic) progress curve of insulin fibrillation (Figure 1C) points to low values of the parameter kb=kn/ka , which gives the relative weight of primary nucleation over the autocatalytic steps of secondary nucleation and elongation ( ka=k2+k+ ) [ 10 ]. The fast elongation rates suggested by the morphology of insulin fibrils (Figure 1A) are confirmed by the high value of ka associated to the steep burst phase (and high v50 value) in Figure 1C. The specific weight of secondary nucleation and elongation in determining the value of ka cannot be distinguished from single progress curve analysis because these steps follow similar rate laws [ 17 ]. Moreover, since fibril breakage (rate constant k− ) does not change the total mass of ThT-positive filaments but only their number [ 17 ], complementary measurements of fibril size distributions are required to directly assess the role of the breakage step. Prior knowledge of the protein oligomerization state is required before we can move into the deeper levels of the different aggregation pathways [ 27 ]. The oligomerization equilibrium of human insulin (Figure 2B) has been characterized by Bocian et al. [ 21 ] using 2D and pulsed field gradient spin echo (PFGSE) nuclear magnetic resonance (NMR). It is, therefore, possible to estimate the availability of insulin monomers under conditions of total protein concentration, presence of zinc, and acidic pH that are similar to those adopted by Foderàet al. [ 26 ] while measuring amyloid fibrillation kinetics. Based on the knowledge of the values of the monomer concentration C1−mer (Figure 2C), peculiar scaling laws of equilibrium (Figure 2D) and kinetic (Figure 2E,F) parameters can be explained using a number of fitted parameters commensurate with the number of independent observations. As an indicator of the amount of amyloid fibrils produced, the final ThT fluorescence intensity ( FF ) (pink line in Figure 2D) is not directly determined by the total protein available (closed circles in Figure 2D) or even by the monomer concentration alone. Since protein aggregation takes place until the monomer concentration C1−mer equals the thermodynamic solubility C∗ , the value of FF reflects the difference (C1−mer −C∗) otherwise known as supersaturation ( ∆C ) [ 12 ]. This is illustrated in Figure 2D (blue line) with no other fitting parameters than the fluorescence proportionality constant (in arbitrary units) and the insulin solubility, which is a measurable quantity. Besides confirming amyloid fibrillation as a phase transition process driven by supersaturation, the FF scaling law is consistent with a mechanism of monomer addition admitting no supplementary contribution from pre-existing soluble oligomers to the final ThT fluorescence signal. Consequently, the amyloid pathway (Figure 2A) and the oligomeric equilibrium (Figure 2B) are found to take place over distinct timescales, with insulin monomers being consumed by the first process at much faster rates than they are produced by the second. Biomolecules 2018,8, 108 5 of 20 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, ka and kb , 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) ( K12 = 4.9 × 10 5 , K24 = 5.0 × 10 4 , K46 = 2.7 × 10 3 and Kiso = 1.35 × 10 4 ). ( C ) Concentration of insulin monomers ( C1−mer ) predicted by the oligomeric equilibrium ( B ) for the values of total protein concentration ( CT ) used in ( D – F ) (symbols). Pink line: polynomial fit to the data. ( D – F ) Reaction scaling laws measured by Foderàet 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 ( FF ) is a direct proportion of supersaturation ∆C=C1−mer −C∗ (proportionality constant cnst = 1.95 × 10 5 ) for an inferred solubility value of C∗= 0.029 mg/mL. Pink line: The polynomial fit in ( C ) is used to estimate FF without the solubility correction ( cnst = 1.15 × 10 5 and C∗= 0 ) . ( E ) Double-logarithmic plot of half-life coordinate t50 as a function of CT . Red lines: Limit scaling exponents |γ| of 1 (dashed line) and 0.5 (solid line) are still too high to represent the measured trend. ( E , F ) Both ka and kb are considered first-order dependent on ∆C (fitted values: ka= 1.34 × 10 2∆C h −1 and kb= 2.41 × 10 −7∆C ). Measured data were adapted with permission from Foderàet al. [ 26 ]. Copyright 2017 American Chemical Society. The separation of timescales simplifies the application of analytic model equations that were originally derived by assuming the soluble protein fully dissociated [ 12 ]. Theoretical curves of t50 and v50 vs. protein concentration can be computed using the equations in Figure 2E,F (see Appendix Afor details), after expressing ka and kb as a function of ∆C (and of C1−mer ). If, as it seems to be the case of insulin, fibril elongation predominates over secondary nucleation (i.e., ka≈k+ and kb≈kn/k+ ), then both kaand kbare proportional to the initial supersaturation (∝ ∆C) considering that [10,12]: (k+∝ ∆C kn∝ ∆C2(2) Biomolecules 2018,8, 108 6 of 20 These simple premises and two model parameters are sufficient to elucidate the unconventionally weak CT -dependence of t50 (Figure 2E) as being the result of the lower molar fractions of insulin monomer observed for higher protein concentrations (Figure 2C). If the associated states of soluble insulin were ignored, the lower limits usually admitted for the absolute scaling factor |γ| would be too high to reproduce the measured trend in Figure 2E (red lines). Remarkably, the set of parameters, ka and kb fitted to the lag-time scaling data in Figure 2E are the same as those that describe the aggregation rate data in Figure 2F (Appendix B). In both cases, the used value of C∗ is the one resulting from the interpretation of Figure 2C. Far from being redundant, the confirmation of kinetic predictions by different and independent measurements provides unequivocal evidence that the present theoretical framework, with only two model parameters, is indeed valid. 3.2. Mechanistic Analysis of Ataxin-3 Aggregation The study of ataxin-3 aggregation follows the same underlying principle that was adopted for human insulin, and has a similar purpose: to show how traditional kinetics can be markedly distorted by the presence of soluble oligomers. As in the case of other polyQ-repeat proteins [ 28 ], the formation of ataxin-3 fibrils and the dissociation of ataxin-3 oligomers occur simultaneously (Figure 3A), and thus, timescale separation cannot be assumed as a simplifying hypothesis. Supported by DLS, size-exclusion chromatography and TEM data, a detailed account of the different steps shown in Figure 3A was recently provided [ 17 ], including quantitative estimations of the rate constants κ1+ , κ1− , κn+ , κn− characterizing the elementary steps of ataxin-3 oligomerization. The worm-like fibrils shown in Figure 1B are predominantly formed by secondary nucleation ( ka≈k2 ) and primary nucleation (kb≈kn/k2) , with minor contributions from the fibril elongation ( k+≈ 0) and fibril breakage (k−≈0) steps [ 17 ]. The measured effect of protein concentration on the ThT fluorescence progress curves (Figure 3B, symbols) is not fully assessed if the oligomeric pathway is not taken into account; on the whole, the black lines in Figure 3B are indicative of good numerical fits, yet they are based on Equation (1), which ignores the occurrence of the parallel reactions of soluble oligomer formation/dissociation. Regardless of how elaborated the theoretical model can be, the fitted parameters are, in this limited scenario, comparable to semi-empirical coefficients showing no evident fundamental meaning. In the illustrative case of Figure 3B, the empirically determined values of ka and kb would follow a proportional relationship with protein concentration, which is not reconcilable with established theories (Figure S1). Biomolecules 2018,8, 108 7 of 20 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 ( κ1+= 7.99 × 10 −4µ M −1 h −1 , κ1−= 9.73 h −1 , κn+= 0.167 µ M −1 h −1 , and κn−= 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 ka and kb , 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) CT= 10 µ M, 7 µ M, 5 µ M, 4 µ M and 2 µ M [ 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: ka= 0.364 Cn2 T h −1 , kb= 2.91 × 10 −10C2−n2 T and n2= 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 |γ| of 1 (dashed line) and 0.5 (solid line) are still too high to represent the measured trend. ( D ) Red-shadowed area: typically, v50 is positively correlated with CT (and with t−1 50 ) [ 29 ]. Measured data were adapted with permission from Silva et al. [17]. Copyright 2018 John Wiley and Sons. Instead of using the amyloid fibrillation model in its closed form solution, the original differential equation, was solved simultaneously with the oligomerization rate equilibrium, Equations (A6) and (A7) (Appendix A), and then fitted to the ThT fluorescence progress curves (Figure 3B, blue lines). Although computationally more demanding than the approach followed with human insulin, the number of degrees of freedom remains unusually low as regards to complex biophysical problems: three independent scaling laws of t50 (Figure 3C), v50 (Figure 3D) and FF (Figure 3E) are used to estimate no other unknowns but the scaling constants associated to ka and kb . Unlike the case of insulin, the value of ataxin-3 solubility is known beforehand to be very low ( C∗≈ 0) as evidenced by values of monomer concentration lower than the detection limits under equilibrium conditions [ 17 ]. In contrast, since the autocatalytic rate constant of ataxin-3 is determined by the secondary nucleation Biomolecules 2018,8, 108 8 of 20 step ( ka≈k2 ), a scaling exponent n2 is now introduced to account for the poorly understood k2 vs. ∆Crelationship: (k2∝ ∆Cn2 kn∝ ∆C2(3) In practice, different fitted parameters are provided by the individualized analysis of each progress curve in Figure 3B (black lines), whereas the global fit (blue lines) requires a single set of rate constants ka and kb to model both the aggregation assay and its scaling laws (Figure 3C–E). The better goodness-of-fit statistics of the former procedure (Figure S1) is not surprising since, as in the case of overparameterized problems, the individual numerical analysis is not cross-validated and tends to overfit the experimental error, therefore, compromising the model’s predictive power [17,30–32] . The global fitting confirms that pre-determined oligomerization constants can be integrated in aggregation reaction networks to explain highly peculiar kinetics, such as the very weak CT -dependence of t50 (Figure 3C), and notably, the negative CT -dependence of v50 (Figure 3D). Although a more conventional result in the absence of quenching phenomena [ 33 , 34 ], the linear scaling law of the end-point ThT fluorescence (Figure 3E) is explained by the dissociation of ataxin-3 oligomers occurring in the same time scale as amyloid fibrillation. The observed straight line crossing the origin also indicates that the soluble protein was converted into amyloid-like fibrils without the occurrence of significant monomer degradation during incubation [ 17 ]. The complex, yet self-consistent behaviors of half-life and end-point readings cross-validate the molecular-level implications arising from the definition of the secondary nucleation rate constant ( k2 ), and particularly, from the obtained value of the scaling exponent n2 close to 0. A direct comparison with the fibril elongation step would suggest a first-order dependence of k2 on the initial supersaturation ∆C since both rates linearly increase with the instantaneous values of supersaturation and fibril mass [ 10 , 17 ]. However, more than just a collisional rate coefficient, k2 is an overall rate constant accounting for the rate-limiting steps leading to the formation of secondary nuclei [ 35 ]. According to classical nucleation theory [ 36 ], the nucleation promoting effect elicited by higher supersaturation levels (and lower energetic barriers for phase transition) can be, in part, counteracted by the concomitant decrease in the critical sizes of the primary ( n∗ 1 ) and secondary ( n∗ 2 ) nucleus. This extra contribution, which is not evident for primary nucleation of amyloid fibrils [ 12 ], seems relevant for the secondary nucleation of ataxin-3. Somewhat undervalued in regard to induction time measurements, half-life aggregation rates v50 (or, equivalently, maximum aggregation rates) offer the opportunity to identify the predominant autocatalytic process. Whilst the scaling of t50 is greatly influenced by primary nucleation, the scaling of v50 is determined by the balance between elongation and secondary nucleation rates, with the effect of CT getting weaker as secondary nucleation becomes more important. Therefore, and similarly to what was concluded for insulin, the measured scaling laws of ataxin-3 aggregation are determined by the fibrillation mechanism itself and by the presence of thermodynamically stable, soluble aggregates that further deplete the concentration of free monomer in solution. 3.3. Model Predictions Are Further Confirmed by Size Distribution Analysis of Insulin Aggregation The previous models present a detailed picture of the different steps affecting the formation of the insoluble filaments that can be further tested using DLS measurements of particle size distributions (PSDs). Contrary to what was observed for ataxin-3 [ 17 ], the size of insulin fibrils tends to increase over time until reaching hydrodynamic radii ( Rh ) above the micrometer scale—Figure 4A–D (insulin) and Figure 4E (ataxin-3). This is not surprising taking into account the TEM images obtained at the end of each aggregation assay (Figure 1A,B), and the negligible role of the elongation step during ataxin-3 fibrillation. Another obvious difference to ataxin-3 is the persisting dominance of the left-side peak ( Rh< 10 nm) up to the end of the aggregation reaction (Figure 4A–C). To a certain extent, this is explained by the value of protein solubility ( C∗ ), which as already discussed, is much higher in the case of human insulin. While the final concentration of soluble ataxin-3 was too low to be Biomolecules 2018,8, 108 9 of 20 detected by DLS [ 17 ], the C∗ value of insulin is responsible for the population of soluble protein to continue predominating, even after large insoluble aggregates are formed (Figure 4C). Another reason explaining the modest increase in the intensity of scattered light of larger particles is associated with the dispersion of sizes and consequential broadening of PSDs provoked by the continuous elongation of old and newly-formed insulin fibrils, as opposed to the formation of ataxin-3 filaments with the constant dimension characteristic of the ataxin-3 secondary nucleus. In common with ataxin-3, fibril breakage has a minor role in determining the time variation of the PSD in quiescent insulin solutions: in the case of ataxin-3, the shape of these distributions did not change significantly during the burst and plateau phases of aggregation despite the increased relative importance of the population of ataxin-3 fibrils [ 17 ]. In the case of insulin, the elongation-dominated mechanism can be discerned from the expected fibril size increase during the burst phase (Figure 4A,B,D), whereas, after ∼ 4.5 h incubation, the mean aggregate size stabilizes at a constant value of Rh≈ 1100 nm without any visible signs of fibril fragmentation (Figure 4C,D). 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 ( Rh ) used as estimates of the mean size ( Rh ) of insulin fibrils. Vertical dashed lines: visual reference of the first Rhvalue of each panel. (D) Measured (symbols) and simulated (lines) time evolution of Rh. Biomolecules 2018,8, 108 16 of 20 n∞= 9 × 10 6 monomers was adopted as the maximum dimension of soluble ataxin-3 oligomers, and the condition dCn/dt= 0 was imposed for n≥n∞ . The initial conditions were set assuming no fibrillar aggregates present in solution ( M(0)= 0) and that the fractional compositions of monomers and n -mers correspond to those extracted from DLS measurements [ 17 ]. Since the predominant autocatalytic step during ataxin-3 fibrillation is secondary nucleation, ka≈k2 and kb≈kn/k2 . Owing to the low solubility of ataxin-3 ( C∗≈ 0), the initial supersaturation is a direct proportion of CT and the CLM parameters are given as ka=k0 aCn2 T and kb=k0 bC2−n2 T . The proportionality constants k0 2 and k0 n , and the order-of-reaction n2 were estimated by minimizing the absolute error between predicted and measured ThT fluorescence ( F ) progress curves using the experimentally determined calibration curve F(a.u.)= 0.70 ×M(µM) and the known oligomerization rate constants κ1+= 7.99 × 10 −4µ M −1 h −1 , κ1−=9.73 h−1,κn+=0.167 µM−1h−1, and κn−=0.775 h−1[17]. To simulate the t50 and v50 scaling laws in Figure 6, Equations (A6) and (A7) were numerically solved as described for ataxin-3 using illustrative values of C∗= 0, ka= 0.4 Cn2 T h −1 and kb= 5 × 10 −4(CT/5)2−n2 . Two limit situations corresponding to predominant elongation step ( n2= 1) or predominant secondary nucleation ( n2= 0) processes were considered. Soluble protein was admitted to occur either as a monomer or as a dimer ( κn+= 0 and κn−= 0) with the distribution of initial species given by the exemplar function C1= 10 ×1−e−CT/10 for CT values comprised between 0 and 20 in arbitrary concentration units. The relative weight of the oligomer dissociation rate was investigated by changing the value of κ1− between 0 and 5 h −1 while keeping a fixed value of κ1+= 0. The normalized amyloid signal ( M/MF ) was computed over time and the corresponding half-life coordinates, t50 and v50 were represented as a function of CT(Figure 6). Appendix B.2. Discretized Population Balance The time evolution of the size distribution of insulin fibrils was simulated using the discrete population balance derived for general phase transition processes comprising the steps of primary nucleation, secondary nucleation, growth/elongation and breakage [ 12 , 17 ]. As previously described for ataxin-3, the concentration of filaments composed by jmonomers (fj) is given as [17]: dfj dt=1 jhkn(1−α)2M∞δj,n∗+k2(1−α)αM∞δj,n∗ 2i+ +k+(1−α)(j−1)fj−1−k+(1−α)j fj+ +1 j"−k−fj(j−n∗ 2)H(j−2n∗−1)+ ∞ ∑ i=j+1 k−fiH(i−2n∗−1)#(A8) The Kronecker delta functions in Equation (A8) set the sizes of the primary and secondary nuclei to fixed values of n∗ and n∗ 2 , respectively, with the latter being adopted as the smallest possible filament size ( j≥n∗ 2 ). The Heaviside function establishes a minimum fibril size of 2 n∗+ 1 molecules above which fragmentation starts to occur [ 46 ]. In the case of insulin, this equation is simplified since secondary nucleation and fibril breakage take place to a negligible extent ( k2≈ 0 and k−≈ 0). The previous Equation (A2) can be obtained from Equation (A8) by extending the sum of j×dfj/dt to all filaments [17]: dM dt= ∞ ∑ j=n∗ 2 jdfj dt(A9) A population of pre-existing clusters with hydrodynamic radii Rh> 100 nm was identified in the analyzed insulin solutions (Figure S2). The DLS intensity-peak corresponding to this population gradually vanished from the measured size distributions as fibril elongation took place (Figure 4A–D). To simulate this behavior, a simple particle adhesion mechanism is proposed in which pre-existing clusters composed by k monomers are considered to join to the available j -mer fibrils and form heterogeneous agglomerates composed by l=k+j monomers. During the initial phases of the reaction, a number of pre-existing clusters remains isolated because there are fewer fibrils than clusters. Biomolecules 2018,8, 108 17 of 20 This scenario is then reversed as newly formed fibrils outnumber the initial clusters. Therefore, the less abundant species are the ones dictating the number of aggregates that will participate in the adhesion process (Pa): Pa=min P= ∞ ∑ j fj,Pc(0)= ∞ ∑ k fk(0)!(A10) where P is the concentration of insulin fibrils, Pc(0)≈ 5.5 × 10 −16 M is the total concentration of clusters and fk(0) is the concentration of k -mer clusters— Pc(0) and fk(0) are estimated from the initial DLS measurements. At a given instant, the number of k -mer clusters joining to j -mer fibrils is a function of the relative amounts of each type of aggregates, fk→j=Pa×fj P×fk(0) Pc(0), (A11) meaning that the concentrations of isolated fibrils (f0 j) and isolated clusters (f0 k) are f0 j=fj− ∞ ∑ k fk→j, (A12) f0 k=fk(0)− ∞ ∑ j fk→j, (A13) and that the concentration of heterogeneous agglomerates composed by a total lmonomers ( f1 l) is f1 l= ∞ ∑ k ∞ ∑ j fk→jδk+j,l. (A14) Here, the Kronecker delta function is used to limit the sum terms to the possible adhesion contacts producing l-mer agglomerates. Finally, the concentration of all l-mer aggregates (fl) is given as: fl=f0 j=l+f0 k=l+f1 l, (A15) which can be used to compute the theoretical size distributions. It follows from the proposed one-to-one adhesion mechanism that Pa= ∞ ∑ l f1 l, (A16) and that the total number of scattering particles corresponds to the concentration of the most abundant species: ∞ ∑ l fl=max(P,Pc(0)). (A17) The following additional approximations were adopted during the derivation of this model: (i) fibril elongation is unaffected by the presence of joined clusters, (ii) complex adhesion pathways are not considered and (iii) the presence of isolated fibrils and isolated clusters is a necessary and sufficient condition for productive adhesion. Other mechanisms explaining the observed decline of the cluster population are admissible, e.g., by admitting progressive cluster dissociation. 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