Modeling Protein Aggregation Kinetics from NMR Data

Abstract

We provide an overview of the practical aspects of using NMR spectroscopy to follow the time course of protein fibril formation (aggregation) and quantitatively model the kinetics of aggregation processes. Following a brief survey of the theoretical foundations of the kinetics of protein aggregation and its inhibition, the modeling of aggregation kinetics, from data acquired by a series of fast two-dimensional ${}^1\text{H-}{}^{15}\text{N}$ correlation NMR spectra, is described. Examples are drawn from our recent NMR-based studies of (1) the aggregation kinetics of a pathogenic huntingtin exon-1 protein whose fibrillization in neurons is responsible for Huntington’s disease, and (2) the kinetics of amyloid $\beta 42$ fibril formation and the mechanism of its inhibition by the chaperone Hsp104.

Introduction

Investigations of the processes of protein fibril formation and growth, which involve attachment of elementary units to the ends of growing filamentous structures, have a long history. Central to the kinetic theory of fibril assembly are the physical concepts of nucleation and growth (chain elongation) [1–4]. Early work on the kinetics of fibril formation in proteins focused on homogeneous nucleation events followed by linear chain polymerization [5,6]. In the early 1960s, Oosawa presented solutions to the kinetic equations for irreversible growth of non-fragmenting (infrangible) protein fibrils [5,6]. These solutions described a sigmoidal law for fibril growth defined by the microscopic rate constants of homogeneous (primary) nucleation $\left(k_n\right)$ and linear chain elongation $\left(k_{+}\right)$, and proved very successful in characterizing the polymerization of actin and tubulin [5–10], as well as amyloid fibril formation by a variety of different proteins [11–14].

Later studies in the 1970–1980s led to the extension of the Oosawa model to include secondary (heterogeneous) nucleation pathways [15–20], which may involve interactions of protein monomers or short oligomers with mature fibrils (e.g. Langmuir-type adsorption of monomers/oligomers on the fibril surface) or breakage (fragmentation) of fibrils, thereby enhancing the rate of homogeneous nucleation. Specifically, Eaton and coworkers showed that monomer-dependent secondary nucleation is of crucial importance in the formation of sickle hemoglobin fibrils [16,20–22], while fibril fragmentation has been shown to play a key role in the growth of amyloid fibrils [14,23–25] and fibrilization of mammalian prions [26–30]. Building on these earlier developments, Dobson, Knowles and co-workers significantly extended the repertoire of kinetic models available for analysis of protein aggregation processes [31–37], including the development of approximate analytical methods for integration of kinetic rate equations [31,34,38–40], as well as the models of inhibition of fibril growth [41–43].

The process of fibril formation is commonly monitored experimentally by fluorescence of Thioflavin T (ThT), a benzothiazole dye that exhibits enhanced fluorescence upon binding to $\beta$-sheet rich structures such as amyloid fibrils. The ThT assay is ‘invasive’ in the sense that direct interactions of the fluorescent dye with fibrils may perturb the rates of chain elongation. Here we review the practical aspects of using NMR spectroscopy to follow the time course of protein aggregation. Although much higher protein concentrations are needed for NMR analysis than for ThT assays, the monitoring of the disappearance of monomer signals by NMR does not involve addition of fluorescent or chromophoric probes to the aggregating system and is always performed under quiescent, non-perturbative conditions.

Monitoring the time course of protein aggregation by solution NMR has been reported on several occasions previously. Luchinat and co-workers used one-dimensional methyl ${}^1\text{H}$ NMR spectra to track and model the aggregation of the amyloid peptide $\mathrm{A}\beta 40$ [44]. Pauwels et al. [45] and Bax and co-workers [46] monitored the aggregation of $\mathrm{A}\beta 40$ and $\mathrm{A}\beta 42$ by two-dimensional heteronuclear NMR and ThT fluorescence but did not analyze the kinetics of the amyloidogenic process. Ramamoorthy and co-workers detected several oligomeric species during the time course of $\mathrm{A}\beta 40$ aggregation using ${}^{19}\text{F}$ NMR [47,48]. Oligomerization kinetics was monitored by Bertini and coworkers [49] by sedimentation NMR [50], trapping high molecular weight oligomeric species by magic-angle-spinning induced sedimentation. In none of these studies, however, was a systematic and comprehensive analysis of aggregation inhibition performed.

This review is structured as follows. First, we present a concise survey of the theoretical foundations of protein aggregation kinetics and the mechanisms of inhibition by ligands or proteins. This is followed by a brief discussion of the practical aspects of using NMR spectroscopy to monitor and model the processes of protein aggregation. We draw examples from our recent NMR-based studies of (1) the aggregation kinetics of a pathogenic exon-1-derived huntingtin protein [51], whose accumulation in neurons in fibrillized form is responsible for Huntington’s disease, and (2) the kinetics of $\mathrm{A}\beta 42$ fibril formation, an etiological component in the development of Alzheimer’s disease, and the mechanism of its inhibition by the chaperone Hsp104 [52].

Theoretical foundations of protein aggregation kinetics

The kinetic rate law

The basic kinetic model of irreversible aggregation that is commonly used for the analysis of protein aggregation profiles acquired under quiescent conditions, includes heterogeneous (secondary) nucleation, and can be described by the following system of differential equations [32,34,38–40],

$$\frac{\mathit{\text{dP}}(t)}{\mathit{\text{dt}}}=k_{n}m(t{)}^{\mathit{\text{nc}}}+k_{s}m(t{)}^{n2}M(t)$$ (1.1)

$$\frac{\mathit{\text{dM}}(t)}{\mathit{\text{dt}}}=-\frac{\mathit{\text{dm}}(t)}{\mathit{\text{dt}}}=2k_{+}m(t)P(t)$$ (1.2)

where $P(t)$ is the number concentration of fibrils proportional to the number concentration of growth competent (extendable) ends at which the aggregates can elongate (termed aggregation ‘nuclei’ in the following); $M(t)$ is the mass concentration of fibrils in monomer units (i.e. the concentration of monomers incorporated into fibrils); $m(t)$, the concentration of monomers, with $m(t)=m_{\mathit{\text{tot}}}-M(t)$, where $m_{\mathit{\text{tot}}}$ is the total monomer concentration; $\mathit{\text{nc}}$, the order of the primary nucleation; $n2$, the order of secondary nucleation; $k_n$, the primary nucleation rate constant (in units of $\left.{\text{hr}}^{-1}{\mathrm{M}}^{(1-\mathit{\text{nc}})}\right);k_s$, secondary nucleation rate constant (in units of ${\text{hr}}^{-1}{\mathrm{M}}^{-n2}$); and $k_{+}$, the rate constant of chain elongation $\left({\text{hr}}^{-1}{\mathrm{M}}^{-1}\right)$. The first and second terms in Eq. (1.1) describe the rates of primary $\left(k_n\right)$ and secondary $\left(k_s\right)$ nucleation, respectively – namely, the rates of production of the number of extendable ends $P$. The rate of secondary nucleation is dependent on the concentrations of both the monomers $m$ and mature fibrils $M$, and involves association of monomers or short oligomers with the fibril surface [33] (e.g. Langmuir-type adsorption) with subsequent release of the species possessing the ends $P$. The rate of chain elongation $\left(k_{+}\right)$ is described by Eq. (1.2) and is proportional to the number of ends $P$. Of note, the factor of ‘2’ in Eq. (1.2) reflects that fibrils may undergo elongation at each of the two available ends. In the case of ‘unseeded’ aggregation, $P\left(0\right)=0$ at the start of the aggregation process $(t=0)$. Note that the model in Eq. (1) assumes that the process of fibril formation is irreversible (no dissociation of monomers from fibrils taking place) and that the fibrils cannot disintegrate into smaller fragments producing new extendable ends $P$ in the process of fibril growth (no fragmentation occurring).

The important property of the kinetic law in Eq. (1) is that nucleation (Eq. (1.1)) constitutes the rate limiting step for fibril growth, with the rate of production of the ends $P$ invariably much lower (usually, by several orders of magnitude) than that of chain elongation (Eq. (1.2)). This separation of rates ensures that the ratio $M/P$ can serve as an approximate measure of average fibril length $L$ in monomer units, which by the end of the aggregation process typically reaches a value from several hundred to several thousand. It is worth noting that in the case of unseeded aggregation, (where $P\left(0\right)=0$), the rate constants of both primary and secondary nucleation ($k_n$ and $k_s$) are strongly correlated with the elongation rate constant ($k_{+}$), so that only the products, $k_{+}{k}_n$ and $k_{+}{k}_s$, may be determined with confidence [34]. In this case, the ratio $M/P$ obtained at the end of the aggregation process $(t\to \mathrm{\infty }),M_{\mathrm{\infty }}/P_{\mathrm{\infty }}$, serves as an important benchmark for the physical meaningfulness and validity of the individual rate constants obtained from least-squares fits of the data using the kinetic model described by Eq. (1).

Inhibition of protein aggregation

Kinetic mechanisms of inhibition may be differentiated according to the component of the aggregating system ($m,P$, or $M$) interacting with the inhibitor [42]. Three mechanisms of inhibition may be distinguished:

1. Inhibitor $I$ binds to monomers. In this case, the equilibrium, $m_{\mathit{\text{free}}}+I\rightleftharpoons m_{\mathit{\text{bound}}}$, has to considered, and Eq. (1) or variations thereof can be integrated with substitution of $m(t)$ for $m_f(t)=m_{\mathit{\text{tot}}}-M(t)-m_b(t)$, where $m_f(t)$ and $m_b(t)$ are the concentrations of free and bound monomers, respectively. Binding to monomers entails changes in all the three rates in Eq. (1), where all the terms are dependent on $m(t)$.

2. Inhibitor I is adsorbed on the surface of mature fibrils $M$. In this case, the equilibrium, $M_{\mathit{\text{free}}}+I\rightleftharpoons M$, has to considered, and the concentration of free fibrils in monomer units, $M_f(t)$, has to be used in the second term of Eq. (1.1). As $M_f(t)<M(t)$, this will entail a decrease in the rate of secondary nucleation. We note that this scenario is the most difficult to model since the stoichiometry and spatial characteristics of the inhibitor-fibril surface interactions are generally not known. Typically, $M_f(t)$ may be expressed as, $M_f(t)=\lambda M(t)-\theta M_b(t)$, where $M_b(t)$ is the concentration of bound fibrils, and the parameters $\lambda$ and $\theta$ account for the number of binding sites on the surface of fibrils per monomer and the number of monomers of fibrils occupied (occluded) by adsorption of inhibitor to a single binding site, respectively [41]. The parameters $\lambda$ and $\theta$ can then be used as global parameters when fitting aggregation profiles.

3. Inhibitor $I$ binds to extendable ends $P$. Two coupled binding equilibria have to be considered in this case: $P_{\mathit{\text{free}}}+I\stackrel{2k_{\mathit{\text{on}},1}}{\underset{k_{\mathit{\text{off}},1}}{\rightleftharpoons }}{P}_{\mathit{\text{bound}},1}$, and $P_{\mathit{\text{bound}},1}+I\stackrel{k_{\mathit{\text{on}},2}}{\underset{2k_{\mathit{\text{off}},2}}{\rightleftharpoons }}{P}_{\mathit{\text{bound}},2}$, where $P_{\mathit{\text{free}}}$ denotes the ends belonging to the fibrils free of inhibitor (unoccupied at both ends); $P_{\mathit{\text{bound}},1}$ and $P_{\mathit{\text{bound}},2}$ are the ends belonging to the fibrils bound to inhibitor at one and both ends, respectively; ${(k}_{on,1};k_{on,2})$ and $(k_{off,1};k_{off,2})$ are the association and dissociation rate constants of the corresponding binding events, respectively. The factors of 2 before $k_{on,1}$ and $k_{\mathit{\text{off}},2}$ account for the multiplicity of binding sites: there are two ways for binding to occur to $P_{\mathit{\text{free}}}$ and, likewise, two ways of dissociation from $P_{\mathit{\text{bound}},2}$. Eq. (1.2) describing fibril elongation is then altered to,

   $$\frac{\mathit{\text{dM}}(t)}{\mathit{\text{dt}}}=2k_{+}m(t)P_f(t)+k_{+}m(t)P_b^{*}(t)$$ (2)

   where $P_f(t)$ and $P_b^{*}(t)$ are the number concentrations of $P_{\mathit{\text{free}}}$ and $P_{\mathit{\text{bound}},1}$, respectively. Eq. (2) reflects the fact that fibrils bound to inhibitor only at a single end $\left(P_{\mathit{\text{bound}},1}\right)$ can still elongate from the free (unoccupied) end, while fibrils bound to inhibitor at both ends $\left(P_{\mathit{\text{bound}},2}\right)$ cannot be elongated further. Binding to the fibril ends therefore changes the rate of chain elongation.

Although the equilibria describing the binding of inhibitors to any component of the aggregating system ($m,P$, or $M$), may be explicitly introduced into Eq. (1) through association and dissociation rates constants, the rates of association/dissociation are usually not known and are difficult to measure in aggregating systems. Considering that the time-scale of inhibitor binding events (μs to ms) is typically much faster than the time-scale of aggregation (hours to days), the incorporation of dynamic binding equilibria into the aggregation kinetics results in a set of very stiff differential equations. To avoid using assumptions about the rates of binding on the one hand, and solving the sets of stiff differential equations on the other, we assume that inhibitor binding occurs ‘infinitely’ fast on the time-scale of the aggregation process, i.e. that the binding equilibrium is established effectively ‘instantaneously’ at each sampling point of the aggregation profile. Thus, in the cases (1) and (2), the concentrations of bound species ($m_b(t)$ and $M_b(t)$) may be re-calculated at each time-point of the aggregation profile using simple quadratic expressions. For example, for the partitioning of monomers onto the ‘free’ and ‘bound’ species (case 1),

$$m_b(t)=\frac{1}{2}\left[\left(m(t)+[I{]}_T+K_D\right)-\sqrt{{\left(m(t)+[I{]}_T+K_D\right)}^2-4[I{]}_{T}m(t)}\right]$$ (3)

where $[I{]}_T$ is the total concentration of the inhibitor added, and $K_D$ is the equilibrium dissociation constant of the monomer-inhibitor interaction. The concentration of monomers participating in nucleation and elongation (‘free’ monomers) is then given by $m_f(t)=m_{tot}-M(t)-m_b(t)$.

When interactions occur with the fibrils end $P$ (case 3), and the inhibitor is present in large excess relative to $P$ (i.e. $[I{]}_T\gg P$, where $P$ is typically in the low-to-middle nanomolar concentration range), the following expressions may be used for the partitioning of the total concentration $P$ onto the concentration of ‘free’, $P_f(t)$, ‘singly-bound’, $P_b^{*}(t)$, and ‘doubly-bound’, $P_b^{**}(t)$, parts,

$$P_f\left(t\right)=\frac{K_D^2}{[I{]}^2+2K_D[I]+K_D^2}P\left(t\right)$$ (4.1)

$$P_b^{*}(t)=\frac{2K_D\left[I\right]}{[I{]}^2+2K_D[I]+K_D^2}P(t)$$ (4.2)

$$P_b^{**}(t)=\frac{[I{]}^2}{[I{]}^2+2K_D[I]+K_D^2}P(t)$$ (4.3)

where the same equilibrium dissociation constant, $K_D=k_{off,1}/k_{on,1}=k_{off,2}/k_{on,2}$, is assumed for inhibitor binding to the ends of free fibrils and singly-bound fibrils (the two equilibria of case 3).

Of note, the macroscopic equilibrium constants for the two coupled equilibria of case 3 are related to each other through: $K_{d,1}=P_f\left[I\right]/P_b^{*}=K_D/2$; and $K_{d,2}=P_b^{*}\left[I\right]/P_b^{**}=2K_D$. When excess of inhibitor cannot be ensured, the concentration of free inhibitor $\left[I\right]$ is expressed through $P_b^{*}$ and $[I{]}_T$,

$$\left[I\right]=\frac{K_D\left([I{]}_T-P_b^{*}\right)}{K_D+P_b^{*}}$$ (5)

Solution of a quadratic equation for the fractional population of $P_{\mathit{\text{bound}},1},p_b=P_b^{*}/P$,

$$\alpha {\left(p_b\right)}^2+\beta \left(p_b\right)+\gamma =0$$ (6)

where $\alpha =2P^2,\beta ={\left([{I]}_T+K_D\right)}^2+2P\left(K_D-[{I]}_T\right)$, and $\gamma =-2K_D[I{]}_T$, provides $P_b^{*}$, and the concentration $P_f$ and $P_b^{**}$ can be calculated by inserting $\left[I\right]$ obtained from Eq. (5) into Eqs. (4.1) and (4.3), respectively. We note that when more than one of the inhibition mechanisms is operative at the same time, the material balance for inhibitor has to also be maintained in the integration of Eq. (1).

Tracking protein aggregation by NMR spectroscopy

In contrast to the ThT fluorescence assay that follows the course of formation of mature fibrils $M$, solution NMR tracks the decay of the signal arising from the monomeric species $m$. The time course of aggregation can be readily followed by recording a series of 2D ${}^1\text{H-}{}^{15}\text{N}$ selective optimized flip angle short transient (SOFAST)-heteronuclear multiple-quantum correlation (HMQC) spectra [53,54], with each spectrum recorded with reasonable signal-to-noise ratio over a period of several minutes. The SOFAST-HMQC experiment permits high repetition rates with short inter-scan delays by combining a small number of radiofrequency pulses, Ernst-angle excitation and optimization of longitudinal relaxation [53]. The intensities or volumes of cross-peaks showing similar aggregation decay profiles are averaged to obtain a single profile for each protein concentration $m_{\mathit{\text{tot}}}$ acquired.

The normalized NMR signal intensity, $I_{\mathit{\text{calc}}}$, as a function of time $t$, are calculated from the relationship,

$$I_{\mathit{\text{calc}}}(t)=A_0\left\{1-\alpha M(t)/m_{\mathit{\text{tot}}}\right\}$$ (7)

where $A_0$ is a global scaling factor that accounts for inaccuracies in the normalization of peak heights ($A_0$ typically optimizes to $~0.99);\alpha$ is a scaling factor that takes into account the inaccuracies in the measurement of very low (close to zero) NMR peak intensities and/or some residual NMR signal intensities in the aggregated state (typically, $0.9<\alpha <1$); and $M(t)$ is obtained by integration of the system of differential equations given by Eq. (1) or variations thereof (see examples below). The key premise of the analysis of NMR aggregation profiles according to Eq. (1) is that the decay in monomer concentration can be fully accounted for by the formation of mature fibrils (aggregates). The best-fit to the experimental data is performed by the minimization of the target function,

$$F=\sum _i\sum _j{\left[\frac{\left(I_{\mathit{\text{exp}}}-I_{\mathit{\text{calc}}}\right)}{\sigma }\right]}^2$$ (8)

where $I_{\mathit{\text{exp}}}$ and $I_{\mathit{\text{calc}}}$ are experimental and calculated normalized intensities, respectively; $\sigma$ is standard error (usually assumed to be equal to one standard deviation of the distribution of peak intensities for all sites included in analysis); the index $j$ in the inner sum of Eq. (8) runs over all time-points obtained, and the index $i$ runs over all protein concentrations, $m_{\mathit{\text{tot}}}$, analyzed.

Aggregation of huntingtin exon-1 protein constructs

Huntingtin (htt) exon-1 derived proteins, $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_n$, comprise a 16-residue N-terminal amphiphilic domain (NT), a poly-glutamine (polyQ) repeat of variable length $n$, and a proline rich domain (PRD) comprising two poly-proline repeats separated by a 17-residue linker [55]. ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_n$ proteins with $n>35$ play a central role in the etiology of Huntington’s disease [55–57] – a fatal neurodegenerative disorder whose age of onset is correlated with the length of the polyQ repeat [58–60]. Typically, the propensity of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_n$ proteins towards aggregation depends on the length of the polyQ stretch, with constructs having 10 or more glutamines aggregating on a time-scale of hours-to-days, while those with shorter polyQ stretches, such as $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_7$, remain stable in solution for several weeks before eventually forming fibrils [60]. In a series of exchange-based NMR studies on various constructs of ${\text{ht}\mathrm{t}}^{\text{ex}1}$ that aggregate very slowly (on a time-scale of several weeks), we showed that $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_7$ as well as a shorter, minimalistic construct, $\text{ht}{\mathrm{t}}^{\mathrm{N}\mathrm{T}}{\mathrm{Q}}_7$, which lacks the PRD domain, undergo transient tetramerization of the NT region on a time-scale of $~50-70\mu \mathrm{s}$ to form a sparsely-populated tetramer comprised of a dimer of dimers – a four helix bundle with $D_2$ symmetry [61–63].

For the pathogenic, fast-aggregating $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ construct, however, NMR signals decay too rapidly to permit data collection for time-intensive exchanged-based NMR experiments. In our study of $\text{ht}{\mathrm{t}}^{\text{ex}}{\mathrm{Q}}_{35}$, we showed that quantitative analysis of the kinetics of both pre-nucleation tetramerization and aggregation can be obtained simultaneously from a series of 2D ${}^1\text{H-}{}^{15}\text{N}$ SOFAST-HMQC correlation spectra [51]. Figure 1A shows the time courses of normalized cross-peak intensities (red circles) and volumes (blue circles) for residues in the NT, polyQ and PRD domains of ${{\text{ht}\mathrm{t}}^{\text{ex}1}\mathrm{Q}}_{35}$. While the tracked cross-peak intensities $\left(I\right)$ and volumes $\left(V\right)$ coincide for the polyQ and PRD domains (lower row of panels, Figure 1A, and the correlation between average cross-peak volumes and intensities for the PRD domain, Figure 1B), the decay of cross-peak volumes is contrasted with initial growth and subsequent decay of cross-peak intensities as a function of time in the NT domain (upper row of panels, Figure 1A). This phenomenon can be attributed to chemical exchange line broadening arising from the submillisecond tetramerization process, $m\rightleftharpoons D\rightleftharpoons T$, that affects the cross-peaks of the NT domain. The fractional population of tetramer $(T)$ is proportional to the third power of the monomer $\left(m\right)$ concentration. As the concentration of monomers decreases, exchange line broadening $\left(R_{ex}\right)$ is reduced, and the concomitant increase in cross-peak intensity due to line narrowing competes with the decrease in intensity resulting from the reduction in monomer concentration. In contrast, cross-peak volumes remain ‘immune’ to the chemical exchange line broadening/narrowing. This phenomenon can be interpreted in a semiquantitative manner by calculating the ratio of volumes to intensities $(V/I)$ and relating these ratios to (i) NMR spin relaxation parameters, (ii) acquisition parameters of the NMR experiments, and (iii) the kinetic parameters of concentration-dependent chemical exchange [51].

Figure 1.

Aggregation profiles of $750\mu \mathrm{M}$ ${}^{15}\text{N}$-labeled ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ monitored by a series of ${}^1\text{H-}{}^{15}\text{N}$ SOFAST-HMQC spectra (800 MHz; 5 °C). (A) Time courses of normalized cross-peak intensities (red circles) and volumes (blue circles) for residues in the NT, polyQ and PRD domains of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$. The data are normalized to the first (reference) experiment recorded immediately after NMR sample preparation $(t~0\mathrm{h})$. (B) Correlation between the normalized average ${}^1\text{H-}{}^{15}\text{N}$ cross-peak intensities (x axis) and the average normalized ${}^1\text{H-}{}^{15}\text{N}$ cross-peak volumes (y axis) for residues within the PRD domain. Adapted from Ceccon et al. [51] published in Proc. Natl. Acad. Sci. USA while the authors were U.S. Government employees at the National Institutes of Health.

Under conditions where $e^{-R_{2,\mathrm{H}}^{\mathit{\text{eff}}}{t}_2^{\mathit{\text{max}}}}\ll 1$ and $R_{2,\text{DQ}/\text{ZQ}}^{\mathit{\text{eff}}}{t}_1^{\mathit{\text{max}}}/2\ll 1$, and, hence $(1-e^{{-R}_{2,\text{DQ}/\text{ZQ}}^{\mathit{\text{eff}}}{t}_1^{\mathit{\text{max}}}/2})\approx R_{2,\text{DQ}/\text{ZQ}}^{\mathit{\text{eff}}}{t}_1^{\mathit{\text{max}}}/2$, the volume/intensity ratio $(V/I)$ of cross-peaks in a 2D ${}^1\text{H-}{}^{15}\text{N}$.

SOFAST-HMQC experiment, processed with no apodization of the time-domain data, can be expressed as (see Supplementary Information of Ref. [51] for derivation),

$$\frac{V}{I}\approx {\alpha }^{\prime }\left[\frac{R_{2,\mathrm{H}}^{\mathit{\text{eff}}}}{2+e^{-R_{2,\text{DQ}}^{\mathit{\text{eff}}}{t}_1^{\mathit{\text{max}}}}+e^{-R_{2,\text{ZQ}}^{\mathit{\text{eff}}}{t}_1^{\mathit{\text{max}}}}}\right]$$ (9)

where $t_1^{\mathit{\text{max}}}$ and $t_2^{\mathit{\text{max}}}$ are the acquisition times in the indirect (${}^{15}\text{N}$) and direct (${}^1\text{H}$) dimensions, respectively; the effective transverse relaxation rates $R_{2,\text{DQ}}^{\mathit{\text{eff}}},R_{2,\text{ZQ}}^{\mathit{\text{eff}}}$ and $R_{2,\mathrm{H}}^{\mathit{\text{eff}}}$ are equal to $\left(R_{2,\text{DQ}}\right.\left.+R_{ex,\text{DQ}}\right),\left(R_{2,\text{ZQ}}+R_{ex,\text{ZQ}}\right)$ and $\left(R_{2,\mathrm{H}}+R_{ex,\mathrm{H}}\right)$, respectively; $R_{2,\text{DQ}},R_{2,\text{ZQ}}$ and $R_{2,\mathrm{H}}$ are ${}^{15}\text{N-}{{}^1\text{H}}_{\text{N}}$ double-quantum (DQ), zero-quantum (ZQ) and amide proton (H) single-quantum (SQ) transverse relaxation rates, respectively; $R_{\text{ex},n}$ are the contributions to the relaxation rate from chemical exchange of each type of coherence $n(n\in ‘\text{DQ}’,‘\text{ZQ}’,‘\text{H}’)$, calculated from the smallest (real) eigenvalue of the appropriate NMR Liouvillian given by the Bloch-McConnell equations; and ${\alpha }^{\prime }$ is a fitted proportionality factor. Note that the relaxation rates of ${}^1\text{H-}{}^{15}\text{N}$ DQ and ZQ coherences are approximated in this analysis by SQ ${}^{15}\text{N}$ relaxation rates obtained from ${}^{15}\text{N}$ $R_{1\rho }$ measurements.

When combined with the time-dependence of the ${}^{15}\text{N}$ chemical shifts $\left({\delta }_{ex}\right)$, the analysis of the $V/I$ ratios of the $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ NT domain allows approximate estimation of the kinetic parameters of the tetramerization process. Figure 2 summarizes the quantitative analysis of prenucleation tetramerization of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ from a global fit of concentration-dependent ${}^{15}\text{N-}{\delta }_{ex}$ (Figure 2A) and ${}^1\text{H-}{}^{15}\text{N}$ $V/I$ ratios (Figure 2B). The derived kinetic characteristics of the tetramerization scheme, $m\rightleftharpoons D\rightleftharpoons T$, are shown in Figure 2C. Note a similar curvature of the plots of ${}^{15}\text{N-}{\delta }_{ex}$ and $V/I$ ratios as a function of the concentration of monomeric ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ in Figures 2A–B, arising from the approximately cubic dependence of the fractional population of tetramers on the concentration of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ monomers in both cases. Although of no particular consequence for kinetic parameters of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ tetramerization in this study, if needed, concentration-dependent line broadening during the ${}^1\text{H-}{}^{15}\text{N}$ and ${}^{15}\text{N-}{}^1\text{H}$ coherence transfer periods of the HMQC experiment, may be taken into account by including in Eq. (9) a factor, exp $\left(2R_{2,\mathrm{H}}^{\mathit{\text{eff}}}{T}_{\text{HC}}\right)$, where $R_{2,\mathrm{H}}^{\mathit{\text{eff}}}$ is the effective transverse relaxation rate of ${}^1\text{H}$ spins and $T_{\text{HC}}$ is the duration of ${}^1\text{H-}{}^{15}\text{N}$ and ${}^{15}\text{N-}{}^1\text{H}$ coherence transfer periods (~5 ms).

Figure 2.

Quantitative analysis of pre-nucleation tetramerization of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ from a global fit to concentrationdependent ${}^{15}\text{N}$ exchange induced shifts (${}^{15}\text{N-}{\delta }_{ex}$) and ${}^1\text{H-}{}^{15}\text{N}$ cross-peak volume/intensity (V/I) ratios. (A) ${}^{15}\text{N-}{\delta }_{ex}$ and (B) ${}^1\text{H-}{}^{15}\text{N}$ V/I ratios as a function of the concentration of monomeric ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ at 800 MHz (5 °C). The concentrations are derived from the time-dependence of the average cross-peak intensity for residues within the PRD domain. The total sample concentration is $750\mu \mathrm{M}$. The experimental data are shown as circles and the best fits to the tetramerization scheme are represented by the continuous solid lines. The ${}^{15}\text{N-}{\delta }_{ex}$ data report only on the monomer $\rightleftharpoons$ dimer and dimer $\rightleftharpoons$ tetramer equilibria, while the V/I ratios also report on the dissociation rate constant from dimer to monomer $\left(k_{-1}\right)$ and provide a lower limit on the dissociation rate constant from tetramer to dimer $\left(k_{-2}\right)$. (C) Kinetic scheme with the values of the fitted rate constants. The populations listed above each species correspond to a ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ concentration of 1 mM (at $t=0$). Adapted from Ceccon et al. [51] published in Proc. Natl. Acad. Sci. USA while the authors were U.S. Government employees at the National Institutes of Health.

The decay of ${}^1\text{H-}{}^{15}\text{N}$ cross-peak intensities for the PRD domain of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ over time (aggregation profiles) obtained at several starting concentrations of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ monomers $\left(m_{tot}\right)$ were first best-fit to the basic kinetic model of aggregation in Eq. (1). It was established that the optimal orders of the primary $\left(\mathit{\text{nc}}\right)$ and secondary $\left(n_2\right)$ nucleation processes are 4 and 1, respectively, implying that the ends $P$ produced by the two nucleation processes, belong to the tetrameric and monomeric species, respectively. In the following, we term the species produced by the two nucleation processes as primary or secondary ‘nuclei’. The tetrameric nature of the primary nuclei $(\mathit{\text{nc}}=4)$ is of particular importance, as it strongly implies that transient tetramerization of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ is a prerequisite for primary nucleation [51].

More accurate analysis of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation profiles is compounded by the presence of small amounts of oxidized material (where the sidechain of ${\text{Met}}^7$ is oxidized to a sulfoxide, ${\text{Met}}^7\mathrm{O}$) in this particular set of NMR samples – from ~7% to 8% in low concentrated samples to ~20% at the highest concentration of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ $\left(750\mu \mathrm{M}\right)$. Fully ${\text{Met}}^7\mathrm{O}$ oxidized ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ does not aggregate and does not undergo transient tetramerization, even in the presence of the native reduced form [51]. Since transient tetramerization was posited as a necessary condition for formation of primary nuclei [51], ${\text{Met}}^7\mathrm{O}$ oxidized ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ cannot be involved in primary nucleation. Nevertheless, the oxidized species can be incorporated into fibrils formed via nucleation of the reduced form and can be (at least, in principle) involved in secondary nucleation. The kinetic scheme that accounts for the presence of both reduced and oxidized forms of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ therefore involves primary nucleation for the reduced form only, secondary nucleation and elongation for all (reduced and oxidized) species, and is described by a system of three coupled ordinary differential equations,

$$\frac{\mathit{\text{dP}}}{\mathit{\text{dt}}}=k_n{\left[f{m}_{\mathit{\text{tot}}}-M_{\mathit{\text{red}}}\left(t\right)\right]}^{\mathit{\text{nc}}}+\left\{k_s^{\mathit{\text{red}}}{\left[f{m}_{\mathit{\text{tot}}}-M_{\mathit{\text{red}}}\left(t\right)\right]}^{n2}+\right.\left.k_s^{\mathit{\text{oxi}}}{\left[\left(1-f\right)m_{\mathit{\text{tot}}}-M_{\mathit{\text{oxi}}}\left(t\right)\right]}^{n2}\right\}\left[M_{\mathit{\text{red}}}\left(t\right)+M_{\mathit{\text{oxi}}}\left(t\right)\right]$$ (10.1)

$$\frac{d{M}_{\mathit{\text{red}}}}{\mathit{\text{dt}}}=2k_{+}^{\mathit{\text{red}}}\left[f{m}_{\mathit{\text{tot}}}-M_{\mathit{\text{red}}}(t)\right]P(t)$$ (10.2)

$$\frac{d{M}_{\mathit{\text{oxi}}}}{\mathit{\text{dt}}}=2k_{+}^{\mathit{\text{oxi}}}\left[(1-f)m_{\mathit{\text{tot}}}-M_{\mathit{\text{oxi}}}(t)\right]P(t)$$ (10.3)

where $f$ is the fraction of reduced species, and $(1-f)$ is the fraction of oxidized species in the NMR samples; the super-/subscripts ‘red’ and ‘oxi’ pertain to reduced and oxidized forms of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$, respectively. The secondary nucleation process $(n_2=1)$ involves reduced $\left(k_s^{\mathit{\text{red}}}\right)$ and oxidized $\left(k_s^{\mathit{\text{oxi}}}\right)$ species; and Eqs. (10.2) and (10.3) describe chain elongation by incorporation of reduced and oxidized subunits, respectively. NMR signal intensities of the reduced $\left(I_{\mathit{\text{calc}}}^{\mathit{\text{red}}}\right)$ and ${\text{Met}}^7\mathrm{O}$ oxidized ${(I}_{\mathit{\text{calc}}}^{\mathit{\text{oxi}}})$ forms of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ are calculated separately as,

$$I_{\mathit{\text{calc}}}^{\mathit{\text{red}}}=A_0\left[1-\alpha M_{\mathit{\text{red}}}(t)/f{m}_{\mathit{\text{tot}}}\right]$$ (11.1)

$$I_{\mathit{\text{calc}}}^{\mathit{\text{oxi}}}=A_0\left[1-\alpha M_{\mathit{\text{oxi}}}(t)/(1-f)m_{\mathit{\text{tot}}}\right]$$ (11.2)

in the instances where the cross-peaks corresponding to the reduced and oxidized forms are separated in the 2D spectra, and as the sum of Eqs. (11.1) and (11.2) when the cross-peaks of the two forms are completely superimposed (which is the case for all the cross-peaks in the PRD domain).

Figure 3A (top) shows the time course of the average ${}^1\text{H-}{}^{15}\text{N}$ cross-peak intensities for residues in the PRD domain of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$, globally best-fit to the kinetic scheme described by the set of differential Eq. (10). A slight but unavoidable seeding of NMR samples of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ ($P\left(0\right)$ values used in the fits varied from 2.5 nM for the lowest concentration of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ to 60 nM for the highest one) allowed us to determine the individual rate constants of the aggregation model for the reduced and oxidized forms: $k_n=3.4(\pm 0.4)\times {10}^5-{\mathrm{M}}^{-3}{\mathrm{h}}^{-1};k_s=0.50(\pm 0.02){\mathrm{M}}^{-1}{\mathrm{h}}^{-1};k_{+}=7.8(\pm 0.3)\times {10}^5{\mathrm{M}}^{-1}{\mathrm{h}}^{-1}$ for the reduced form, and $k_{+}=5.7(\pm 0.3)\times {10}^5{\mathrm{M}}^{-1}{\mathrm{h}}^{-1};k_s=0$ (optimized to a small, ill-determined value and therefore subsequently set to 0) for the oxidized form. The slightly slower incorporation of oxidized species into fibrils at the elongation stage $\left(k_{+}\right)$ compared to their reduced counterparts, is illustrated by the plots of ${}^1\text{H-}{}^{15}\text{N}$ cross-peak volume and intensity of Ser¹² in the reduced and oxidized forms (cross-peaks of the two forms separated in the 2D spectra), respectively (Figure 3A, bottom). Figure 3B shows simulated time-dependencies of all the components of the kinetic model: concentration of mature fibrils in monomer units ($M_{\mathit{\text{red}}}$ and $M_{\mathit{\text{oxi}}}$), the concentration of the ends $\left(P\right)$, and the ratio of total fibril concentration $(M_{\mathit{\text{red}}}+M_{\mathit{\text{oxi}}})$ to extendable ends $\left(P\right)$. The ratio, $\left(M_{\mathit{\text{oxi}}}+M_{\mathit{\text{red}}}\right)/P$, is an approximate measure of the average fibril length. Assuming a separation of ~4.2 Å (the distance between strands in a $\beta$-sheet) between monomer units along the length of the fibril, the fibril length reaches a maximum of ~850 nm, consistent with the negative stain electron microscopy image of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ fibrils shown in the lower right panel of Figure 3B.

Figure 3.

Quantitative analysis of the mechanism and kinetics of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation probed by serially acquired ${}^1\text{H-}{}^{15}\text{N}$ SOFAST-HMQC spectra. (A) Time-dependence of the average ${}^1\text{H-}{}^{15}\text{N}$ cross-peak intensities for residues in the PRD domain (top), and of the ${}^1\text{H-}{}^{15}\text{N}$ cross-peak volume and intensity of ${\mathrm{S}\mathrm{e}\mathrm{r}}^{12}$ in the native reduced and ${\text{Met}}^7\mathrm{O}$ oxidized forms, respectively, of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$. The total concentrations of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ at $t=0\left({\mathrm{m}}_{\mathrm{t}\mathrm{o}\mathrm{t}}\right)$ are indicated. The experimental data (shown as circles) were recorded at 800 MHz(5 °C) and normalized to the first time point (at $t~0$). The best-fit curves are shown as black continuous lines and were obtained from a global fit to the kinetic scheme described by the set of differential equations in Eq. (10) (B) Simulated time dependence of the components of the model of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ fibrillization at 5 °C calculated using the optimized values of the rate constants (see text). The inset in the lower right panel shows the negative stain electron microscopy image of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ taken after 70 h incubation of $40\mu \mathrm{M}{\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ at 5 °C. Adapted from Ceccon et al. [51] published in Proc. Natl. Acad. Sci. USA while the authors were U.S. Government employees at the National Institutes of Health.

Even though transient, reversible tetramerization of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ was posited as a prerequisite for primary nucleation in the kinetic models of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation described above based on solid evidence – the optimal order of primary nucleation $\mathit{\text{nc}}=4$, as well as the observation that the species that cannot form tetramers (e.g. ${\text{Met}}^7\mathrm{O}$ oxidized ones), cannot at the same time undergo primary nucleation – this treatment still lacks a direct link between the fast (microsecond time-scale) tetramerization and much slower (hours-to-days time-scale) irreversible formation of fibrils. The lack of such a connection is best illustrated by the observation that, at the onset of aggregation, the concentration of tetramers (typically, in the low micromolar range) exceeds that of nascent nuclei (typically, in the low nanomolar range) by 2-to-3 orders of magnitude (cf. Figures 2C and 3B, lower left panel). Consequently, the total pool of the ends supplied by $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ tetramers cannot quantitatively correspond to the pool of available extendable ends $P$. To resolve this contradiction, we recently developed a ‘unified’ kinetic model of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation that establishes a direct coupling between the fast, reversible tetramerization and slow, irreversible fibrillization [64].

The unified kinetic scheme is shown in Figure 4. The monomers of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ $\left(m\right)$ interconvert on a timescale of $~30\mu \mathrm{s}$ with sparsely populated dimeric species $D$, which, in turn, interconvert with sparsely-populated tetramers $T$ on a timescale of less than $100\mu \mathrm{s}$. These two reversible oligomerization reactions are characterized by equilibrium constants $K_{\mathrm{e}\mathrm{q},1}$ and $K_{\mathrm{e}\mathrm{q},2}$, respectively. The tetramer $T$ is slowly and irreversibly converted (with a rate constant $k_c$) in a monomer-independent manner into elongation-capable, ‘active’ primary nuclei, followed by elongation $\left(k_{+}\right)$ to produce mature fibrils $M$. Surface-mediated secondary nucleation ($k_s$ in Eqs. (1) and (10)) involves interactions of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ monomers $m$ with fibrils $M$ to produce secondary nuclei that, although physically distinct from the primary ones (i.e. converted tetramers), add to the total pool of nuclei, characterized by the number concentration of extendable ends, $P$. The time course of aggregation can then be described by two ordinary differential equations,

$$\begin{array}{rr}& \frac{\mathit{\text{dP}}}{\mathit{\text{dt}}}=k_{c}f[m(t)]+k_{s}m(t{)}^{n2}M(t)\end{array}$$ (12.1)

$$\frac{\mathit{\text{dM}}}{\mathit{\text{dt}}}=2k_{+}m(t)P(t)$$ (12.2)

where $f[m(t)]$ represents the time-dependent concentration of tetramers; $k_{\mathrm{c}}$ is the rate constant for the conversion of tetramers to ‘active’ nuclei in units of ${\mathrm{h}}^{-1}$; and the rest of the symbols are defined in Eq. (1). Assuming highly skewed fractional populations of the oligomeric states at equilibrium (i.e. $p_D,p_T\ll p_m),f[m(t)]$ can be expressed through the equilibrium and rate constants as,

$$\begin{gathered} f\left[m\left(t\right)\right]={\left(K_{\mathit{\text{eq}},1}\right)}^2{K}_{\mathit{\text{eq}},2}m{\left(t\right)}^4={\left(\frac{k_1}{k_{-1}}\right)}^2\left(\frac{k_2}{k_{-2}}\right)m{\left(t\right)}^4 \\ =K_{T}m{\left(t\right)}^4 \end{gathered}$$ (13)

Figure 4.

Schematic representation of the unified kinetic model of fast pre-nucleation tetramerization coupled with slow monomer-independent conversion, chain elongation and fibril surface-mediated secondary nucleation steps of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ fibrillization. $k_1$ and $k_2$ are association rate constants from monomer to dimer and from dimer to tetramer, respectively, $K_{eq,1}$ and $K_{eq,2}$ are the dimerization and tetramerization equilibrium constants, respectively, $k_c$ is the first order rate constant for the unimolecular conversion of pre-nucleation transient tetramer $T$ to elongation-competent species (nuclei) $P$, and $k_{+}$ and $k_s$ are the rate constants for fibril elongation and surface-mediated secondary nucleation, respectively. Adapted from Torricella et al. [65] published in J. Phys. Chem. Lett. while the authors were U.S. Government employees at the National Institutes of Health.

The first term on the right-hand side of Eq. (12.1), $k_{c}f[m(t)]$, describing the monomer-independent conversion of all pre-equilibrated tetramers $T$ to elongation-capable tetramers via conformational rearrangement, is related to the conventional rate constant ${(k}_n)$ of primary nucleation of order 4 (i.e. the rate, $k_{n}m(t{)}^4$) through the simple expression, $k_n={\left(k_1/k_{-1}\right)}^2\left(k_2/k_{-2}\right)k_{\mathrm{c}}$. Note that due to the large separation between the time-scales of tetramerization and aggregation processes, the equilibrium, $m\rightleftharpoons D\rightleftharpoons T$, is assumed to be established instantaneously at each time point sampled in the course of the aggregation decay.

Using the unified model described by Eqs. (12) and (13) (see also Figure 4), we revisited the kinetic analysis of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation – in this instance, under modified experimental conditions, where the ${\text{Met}}^7\mathrm{O}$ oxidized species are completely removed from the solution of aggregating ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ [64]. Figures 5A and 5B show the simulated time-dependence of the concentration of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ tetramers $T$ (in monomer units) and the time course of the average ${}^1\text{H-}{}^{15}\text{N}$ cross-peak intensities for the residues of the PRD domain, respectively. The use of slightly seeded initial conditions ($P\left(0\right)=\mathrm{2,4}$, and 6 nM for 160, 250 and $420\mu \mathrm{M}$ samples of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$) allowed the determination of individual aggregation rate constants: $k_c=0.07\pm 0.01{\mathrm{h}}^{-1};k_s=0.3\pm 0.04{\mathrm{M}}^{-1}{\mathrm{h}}^{-1}$; and $k_{+}=6.4(\pm 0.6)\times {10}^5{\mathrm{M}}^{-1}{\mathrm{h}}^{-1}$. Figure 5C shows the simulated time dependence of the $M,P$, and $M/P$ ratios during the course of $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation. Note that the $M/P$ ratio is approximately preserved between different models (cf. the plots in Figure 3B, lower right panel and Figure 5C, lower panel).

Figure 5.

Quantitative analysis of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation profiles. (A) Simulated time-dependence of the concentration of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ tetramers $T$ (in monomer units) for ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ samples with total monomer concentrations $m_{\mathit{\text{tot}}}$ of 160, 250 and $420\mu \mathrm{M}$ calculated using the values of $K_{\text{eq},1}$ and $K_{\text{eq},2}$. (B) Time-dependence of the average ${}^1\text{H-}{}^{15}\text{N}$ cross-peak intensities for residues in the PRD domain of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$. The experimental data (shown as circles) were recorded at 800 MHz (5 °C) and normalized to the first time point. The best-fit curves are shown as black continuous lines and were obtained from a global fit to the kinetic scheme in Figure 4. (C) Simulated time-dependence of $M,P$, and the $M/P$ ratios during the course of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation calculated using the optimized rate constants (see text). Adapted from Torricella et al. [64] published in Adv. Sci. (Weinh.) while the authors were U.S. Government employees at the National Institutes of Health.

The unified model described by Eqs. (12) and (13) bears similarities with the kinetic model developed by Dobson, Knowles and co-workers for aggregation of amyloid $\mathrm{A}\beta 42$ [37], where oligomers of $\mathrm{A}\beta 42$ are converted to elongation capable fibril ends. Direct comparison of the conversion rate constants $k_c$ between the two models is problematic since the conversion is independent of monomer concentration in the unified model, while it is proportional to the monomer concentration, $m(t{)}^{\mathit{\text{nconv}}}$, where nconv is the conversion order, for $\mathrm{A}\beta 42$ $(\mathit{\text{nc}}onv=2.7)$ [37]. Nevertheless, we note that for the largest monomer concentration used in the study of $A\beta 42$ aggregation $\left(5\mu \mathrm{M}\right)$, the rate of $\mathrm{A}\beta 42$ oligomer conversion $(~0.03{\mathrm{h}}^{-1})$ is estimated to be ~2-fold slower than the value of $k_c$ in the unified model $(~0.07\pm 0.01{\mathrm{h}}^{-1})$. The unified model proved to be robust and highly instrumental for interpretation of the effects exerted on $\text{ht}{\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation kinetics by macromolecular cosolutes [65], as well as for an ongoing study of ${\text{ht}\mathrm{t}}^{\text{ex}1}{\mathrm{Q}}_{35}$ aggregation in the presence of synthetic nanoparticles.

Modeling the aggregation kinetics of $\mathrm{A}\beta 42$ and its inhibition by the chaperone Hsp104 from NMR data

Fibrils formed by amyloid-$\beta$ ($\text{A}\beta$) peptides [66,67] are the main component of amyloid plaques found in the brains of individuals with Alzheimer’s disease [68,69]. Among several $\mathrm{A}\beta$ peptides produced by cleavage of the amyloid precursor protein, $\mathrm{A}\beta 42$ is the most susceptible to aggregation [70,71]. The kinetics of $\mathrm{A}\beta 42$ aggregation [32,36,37] and its inhibition by a number of chaperones [41,42,72–76] were studied earlier – typically, by ThT fluorescence assays using starting protein concentrations, $m_{\mathit{\text{tot}}}\le 6\mu \mathrm{M}$. Recently, we reported a NMR kinetic study of $\mathrm{A}\beta 42$ aggregation [52] in the absence and presence of substoichiometric quantities of the chaperone Hsp104 [77] – a ~600 kDa hexamer of identical subunits that enclose a large channel through which substrates are translocated [77–79].

The inherently low sensitivity of NMR detection requires the use of relatively high starting protein concentrations. We observed that substantial modifications of the kinetic laws of aggregation in Eq. (1), are required to satisfy the experimental data for concentrations of $\mathrm{A}\beta 42$ higher than $~20\mu \mathrm{M}$. The aggregation kinetics of $50\mu \mathrm{M}$ $\mathrm{A}\beta 42$ is governed by the following system of three ordinary differential equations,

$$\frac{\mathit{\text{dP}}(t)}{\mathit{\text{dt}}}=k_{n}m(t{)}^{\mathit{\text{nc}}}+k_{s}m(t{)}^{n2}M(t)$$ (14.1)

$$\frac{\mathit{\text{dM}}(t)}{\mathit{\text{dt}}}=2k_{+}\frac{m(t)}{1+m(t)/K_r}P(t)$$ (14.2)

$$\frac{d{O}^{\prime }(t)}{\mathit{\text{dt}}}=k_{p}m(t{)}^{\mathit{\text{np}}}-k_d{O}^{\prime }(t)$$ (14.3)

where the definitions of $m(t),P(t),M(t),k_n,k_{\mathrm{s}},k_{+},\mathit{\text{nc}}$ are $n_2$ are the same as in Eq. (1); $O^{\prime }(t)$ is the concentration of additional unobservable species that cannot convert to fibrils (‘off-pathway’ oligomers); $m(t)=m_{\mathit{\text{tot}}}-M(t)-O^{\prime }(t);K_r$ is a Michaelis constant for saturating elongation (in units of concentration, M) [34]; $k_p$ is the rate constant describing the production of the unobservable oligomers $O^{\prime }$ (in units of ${\text{hr}}^{-1}{\mathrm{M}}^{-1});k_d$ is the dissociation rate constant of oligomers $O^{\prime }$ back to monomers $\left({\text{hr}}^{-1}\right)$; $\mathit{\text{np}}$ is the order for the formation of unobservable oligomers $O^{\prime };\mathit{\text{nc}}=\mathit{\text{np}}=2$; and $n2=1$. Figure 6 depicts a schematic representation of $\mathrm{A}\beta 42$ aggregation at high concentrations.

Figure 6.

Schematic representation of the kinetic mechanism of $\mathrm{A}\beta 42$ aggregation at starting concentrations $m_{\mathit{\text{tot}}}>~20\mu \mathrm{M}$ described by Eq. (14). The model of $\mathrm{A}\beta 42$ aggregation comprises irreversible on-pathway fibril formation (primary nucleation, secondary nucleation, and saturating elongation, shown in blue) and reversible ‘off-pathway’ oligomerization (shown in red). Adapted from Ghosh et al. [52] published in Proc. Natl. Acad. Sci. USA while the authors were U.S. Government employees at the National Institutes of Health.

The model in Eq. (14) is different from that employed previously for $\mathrm{A}\beta 42$ at much lower concentrations by Dobson, Knowles and coworkers [32], in three respects listed below in order of their importance: (a) the ad hoc introduction of a reversible process describing the formation of ‘offpathway’ oligomers $O^{\prime }$ that cannot convert to fibrils. Note that Eq. (14.3) is coupled to Eqs. (14.1) and (14.2) only through the reduction in monomer concentration $m(t)$. Such ‘off-pathway’ oligomerization is formally akin to non-productive nucleation. It does not obey the tenets of classical nucleation theory (i.e. the reaction order of this process, $\mathit{\text{np}}$, is not necessarily related to the physical size of oligomers $O^{\prime }$), and may be viewed as a ‘coarse-grained’ representation of a mechanistically more complicated, multi-step process [37]; (b) retardation of the elongation stage by the denominator of Eq. (14.2) following the law of ‘saturating’ elongation; and (c) lower order of the secondary nucleation step $(n2=1)$.

Figure 7 illustrates the relative importance of these modifications on the best-fits of the aggregation profile of $50\mu \mathrm{M}\mathrm{A}\beta 42$ monitored by 2D cross peak intensities obtained from a series of ${}^1\text{H-}{}^{15}\text{N}$ SOFAST-HMQC spectra (at 20 °C). The normalized ${\chi }^2$ of the best-fit to the complete model described in Eq. (14) is 0.8 (black curve). Note that the profile loses its sigmoidal character at short times (up to ~15 hr), with the decrease in the concentration of the monomer surpassing by far that predicted from the standard model described by Eq. (1) (Figure 7). Setting $K_r$ to a very high value (i.e., eliminating retardation of the elongation stage, Eq. (14.2)) increases the normalized ${\chi }^2$ to 1.7 (green curve); removing the ‘off-pathway’ oligomerization (red curve) by setting $k_p=0$ increases the ${\chi }^2$ even further to 9.1; and, finally, setting $K_r$ to a very high value and $k_p$ to 0 increases the ${\chi }^2$ to 20.3 (blue curve).

Figure 7.

Best-fits to the time course of monomer disappearance of $50\mu \mathrm{M}$ ${}^{15}\text{N}$-labeled $\mathrm{A}\beta 42$ monitored by the normalized ${{}^1\text{H}}_{\text{N}}\text{/}{}^{15}\text{N}$ cross-peak intensity averaged over 21 well-resolved residues obtained from a series of ${}^1\text{H-}{}^{15}\text{N}$ SOFAST-HMQC spectra (at 20 °C). The experimental data are shown as black circles. The best-fit to the complete model in Eq. (14) is shown with black curve. The best-fit to the same model without saturating elongation (the denominator in Eq. (14.2) set to 1) or without production of ‘off-pathway’ oligomers (Eq. (14.3)) are shown by the green and red curves, respectively. The blue curve shows the best-fits obtained with exclusion of both saturating elongation and the ‘off-pathway’ oligomer production (the model in Eq. (1)). Regions of the time course labeled “1” and “2” are expanded for better visualization of systematic deviations between the experimental data and the calculated green, red and blue curves. Adapted from Ghosh et al. [52] published in Proc. Natl. Acad. Sci. USA while the authors were U.S. Government employees at the National Institutes of Health.

The chaperone Hsp104 inhibits aggregation of $\mathrm{A}\beta 42$ in an apparently sub-stoichiometric manner with respect to the concentration of monomeric $\mathrm{A}\beta 42$ [72,80–82] – addition of Hsp104 in the midnanomolar concentration range to $50\mu \mathrm{M}\mathrm{A}\beta 42$ results in practically complete abrogation of fibril formation. However, this inhibition cannot be viewed as truly sub-stoichiometric when we consider that Hsp104 interacts not with monomers $m$, but with the fibril ends $P$, which are present in solution in low nanomolar quantities. In our original publication, we presented the analysis of $P$ – Hsp104 interactions for the limiting case when the binding of Hsp104 can occur only to both $\mathrm{A}\beta 42$ fibril ends (producing the species $P_{\mathit{\text{bound}}}^{**}$; see the section ‘Inhibition of protein aggregation’). In this case, the concentration of the ‘doubly-bound’ ends, $P_b(t)$, may be calculated by analogy with Eq. (3),

$$P_b(t)=\frac{1}{2}\left\{[\mathit{\text{Hsp}}{]}_T+P(t)+K_D^{\mathit{\text{app}}}-\sqrt{{\left\{[\mathit{\text{Hsp}}{]}_T+P(t)+K_D^{\mathit{\text{app}}}\right\}}^2-4P(t)[\mathit{\text{Hsp}}{]}_T}\right\}$$ (15)

where $P(t)$ is the total concentration of ends obtained by numerical integration of the system of coupled differential Eq. (14); $[\mathit{\text{Hsp}}{]}_T$ is the total concentration of Hsp104 expressed in subunits; and $K_D^{\mathit{\text{app}}}$ is the ‘apparent’ equilibrium dissociation constant for the binding of Hsp104 to the ends $P$. The concentration of ‘free’ (available for elongation) ends is then calculated as, $P_f(t)=P(t)-P_b(t)$, and $P_f(t)$ is substituted for $P(t)$ in Eq. (14.2) at each integration time-step. Note that we refer to the equilibrium dissociation constant as ‘apparent’ because the stoichiometry and other properties of the $P$ – Hsp104 interaction are not known. This approach, while fully satisfying the experimental aggregation profiles of $\mathrm{A}\beta 42$ in the presence and absence of Hsp104, results in (i) the need for introducing two elongation rate constants as variable parameters in the fit, namely $k_{+}$ with and without Hsp104, and (ii) an ever so slight (~15%) overestimation of $K_D^{\mathit{\text{app}}}$.

A more rigorous treatment involves the possibility that Hsp104 also binds at one end of $\mathrm{A}\beta 42$ fibrils – as described in detail in the section ‘Inhibition of protein aggregation’. The concentrations of the free $\left(P_f\right)$, singly-bound $\left(P_b^{*}\right)$, and doubly-bound $\left(P_b^{**}\right)$ ends are calculated using Eqs. (4)–(6), with $\left[I\right]$ substituted for $\left[\mathit{\text{Hsp}}\right]$, and the appropriate law for the rate of fibril elongation, by analogy with Eq. (2), is given by,

$$\frac{\mathit{\text{dM}}(t)}{\mathit{\text{dt}}}=2k_{+}\frac{m(t)}{1+m(t)/K_r}{P}_f(t)+k_{+}\frac{m(t)}{1+m(t)/K_r}{P}_b^{*}(t)$$ (16)

Eq. (16) has to be used instead of Eq. (14.2) in the set of differential equations describing the kinetics of inhibition of $\mathrm{A}\beta 42$ aggregation by Hsp104. This treatment yields $K_D^{\mathit{\text{app}}}=12\pm 0.5\mathrm{n}\mathrm{M}$ for the $P$ – Hsp104 interaction, and allows for a single rate of elongation, $k_{+}$, in the absence and presence of Hsp104, to be determined. Note that Hsp104 cannot be assumed to be in large excess of $P$ in this case (i.e. that $[\mathit{\text{Hsp}}{]}_T\gg P$, and $\left.\left[\mathit{\text{Hsp}}\right]\approx [\mathit{\text{Hsp}}{]}_T\right)$, and the concentrations $P_f(t)$ and $P_b^{*}(t)$ have to be obtained from the solution of the quadratic Eq. (6) at each integration time-step.

Figure 8 shows the global best-fit of $A\beta 42$ aggregation profiles in the presence of varying amounts of Hsp104, to the model in Eq. (14) using Eqs. (4)–(6) for calculation of $P_f$ and $P_b^{*}$, and Eq. (16) for the rate of fibril elongation, with a different rate constant $k_p$ assumed for each Hsp104 concentration. The Michaelis constant that ensures saturation of elongation, $K_r$, has to still be fit separately for the data with and without Hsp104: $K_r=17.6\pm 1.5$ and $7.0\pm 0.3\mu \mathrm{M}$ with and without Hsp104, respectively, implying that transient Michaelis complexes formed between monomers and fibril ends in the course of the elongation process, have slightly different stabilities in the absence and presence of Hsp104. The aggregation of $\mathrm{A}\beta 42$ is virtually abrogated at $[\mathit{\text{Hsp}}{]}_T$ approaching $~1\mu \mathrm{M}$. The decay of $\mathrm{A}\beta 42$ monomer signal with time at higher concentrations of Hsp104 (Figure 8, right panel), is predicated on the reversible formation of the unobservable ‘off-pathway’ oligomers $O^{\prime }$ (Eq. (14.3)), with the equilibrium $m\rightleftharpoons O^{\prime }$ shifting to the right at higher $[\mathit{\text{Hsp}}{]}_T$ (via an unknown mechanism) by virtue of the increase of the rate constant $k_p$ (see inset in the left panel of Figure 8).

Figure 8.

Global best-fit to the experimental aggregation profiles of ${}^{15}\text{N}$-labeled $\mathrm{A}\beta 42$ in the presence of varying amounts of Hsp104, monitored by the disappearance of monomeric $\mathrm{A}\beta 42$ in serially acquired 2D ${}^1\text{H-}{}^{15}\text{N}$ SOFAST-HMQC experiments (20 °C). The experimental data, obtained by averaging the normalized time courses for 21 residues with well resolved ${}^1\text{H-}{}^{15}\text{N}$ cross-peaks are shown as circles (with the bars representing 1 standard error). All the experimental time courses were fit simultaneously by numerical solution of Eq. (14), with Eq. (14.2) substituted for Eq. (16), where $P_f$ and $P_b^{*}$ are calculated using Eqs. (4)–(6). The differential equations were integrated numerically with the following initial conditions: $P\left(0\right)=0;M\left(0\right)=0;O^{\prime }\left(0\right)=0$. The best-fits are shown by the continuous black curves. The normalized ${\chi }^2$ of the global fit is 0.8. The values of the optimized parameters are as follows: $k_n\left(\mathit{\text{nc}}=2\right)=0.19{\mathrm{M}}^{-1}{\mathrm{h}}^{-1};k_{+}=6.7\times {10}^6{\mathrm{M}}^{-1}{\mathrm{h}}^{-1};k_s\left(n_2=1\right)=7.5{\mathrm{M}}^{-1}{\mathrm{h}}^{-1};k_d=5.1\left(\pm 0.3\right)\times {10}^{-3}{\mathrm{h}}^{-1}$ (see text for the rest of the fitted parameters). The inset in the left panel shows the variation of the rate constant $k_p$ $(\mathit{\text{np}}=2)$ on the concentration of Hsp104. The uncertainties in the rate constants are not reported because of strong correlations between $k_{+}$ and $k_n$ and between $k_{+}$ and $k_2$ inherent in the model (the variability of the optimized parameters from the fits with different starting values are typically higher than those obtained from the covariance matrix of the fit). Adapted from Ghosh et al. [52] published in Proc. Natl. Acad. Sci. USA while the authors were U.S. Government employees at the National Institutes of Health.

Summary and perspectives

In this review we have provided an overview of the practical aspects of using NMR spectroscopy to monitor the time course of protein aggregation/fibril formation. Quantitative modeling of the kinetics of protein aggregation from NMR data acquired by a series of 2D ${}^1\text{H-}{}^{15}\text{N}$ SOFAST-HMQC NMR spectra, is described for two examples drawn from our recent studies: (1) aggregation kinetics of a pathogenic huntingtin exon-1 protein, and (2) the kinetics of fibril formation by $\mathrm{A}\beta 42$ and the mechanism of its inhibition by the chaperone Hsp104.

The principal limitation of using NMR for analysis of protein aggregation kinetics is associated with the need for much higher protein concentrations for NMR detection. Nevertheless, NMR spectroscopy affords a number of advantages for tracking protein aggregation over more conventional techniques. First, NMR detection is ‘non-invasive’, i.e. no fluorescent or chromophoric probes have to be added to the aggregating system – specific interactions of added probes with fibrils and non-specific interactions with monomers or short oligomers that may compromise quantitative analysis of fibril growth are therefore avoided. Second, no external perturbation is effected on the aggregating system, such as mixing or stirring that may lead to fragmentation of fibrils. Further, NMR aggregation profiles have better reproducibility than ThT fluorescence assays on the one hand, and are expected to be more accurate than turbidity measurements (nephelometry or turbidimetry) on the other. We anticipate that fast acquisition NMR methods will find more widespread use for tracking and modeling of protein aggregation in the near future.