General integrated rate law for complex self-assembly reactions reveals the mechanism of amyloid-beta co-aggregation

TL;DR

The paper tackles the challenge of extracting mechanistic insight from kinetic data of complex protein self-assembly involving multiple alloforms. It develops a general integrated rate law for a broad class of linear self-assembly reactions using asymptotic Lie symmetry, and applies it to Abeta42 co-aggregation with Aβ40/38/37. The analysis shows that Aβ42 fibril surfaces catalyze co-oligomer formation, accelerating Axx fibril formation while inhibiting Aβ42 secondary nucleation, and provides a universal solution formula that fits both unseeded and seeded data. The general solution extends to other filamentous aggregation systems and saturating or co-aggregation contexts, enabling rigorous kinetic deconvolution and mechanistic dissection. This method thus offers a practical, broadly applicable framework for understanding complex self-assembly in biological and therapeutic contexts.

Abstract

Analyzing kinetic experiments on protein aggregation using integrated rate laws has led to numerous advances in our understanding of the fundamental chemical mechanisms behind amyloidogenic disorders such as Alzheimer's and Parkinson's diseases. However, the description of biologically relevant processes may require rate equations that are too complex to solve using existing methods, hindering mechanistic insights into these processes. An example of significance is co-aggregation in environments containing multiple amyloid-beta (Abeta) peptide alloforms, which may play a crucial role in the biochemistry of Alzheimer's disease but whose mechanism is still poorly understood. Here, we use the mathematics of symmetry to derive a general integrated rate law valid for most plausible linear self-assembly reactions. We use it in conjunction with experimental data to determine the mechanism of co-aggregation of the most physiologically abundant Abeta alloforms: Abeta42, Abeta40, Abeta38 and Abeta37 peptides. We find that Abeta42 fibril surfaces catalyze the formation of co-oligomers, which accelerate new Abeta40, Abeta38 and Abeta37 fibril formation whilst inhibiting secondary nucleation of new Abeta42 fibrils. The simplicity, accuracy and broad applicability of our general integrated rate law will enable kinetic analysis of more complex filamentous self-assembly reactions, both with and without co-aggregation.

Figures (12)

• Figure 1: Previously established mechanistic features of A42 co-aggregation with A40/38/37 (A xx), illustrated using typical kinetic curves for these reactions.a: A42 and A xx co-aggregation at pH 7.4 shows separate sigmoidal increases in fibril mass, with the first corresponding to pure A42 fibril formation, and the second to pure A xx fibril formation. Thus, no significant cross-elongation occurs. Representative kinetic curves (black) are generated from the later-determined integrated rate laws for A alloform co-aggregation (Eq. \ref{['fullmodelf']}) using typical parameter values (see Table \ref{['Table1']}). b: Monomeric A xx has a clear inhibitory effect on A42 fibril formation, whereas monomeric A42 accelerates A xx fibril formation. (Addition of pure A42 fibrils to monomeric A xx was found in Cukalevski2015Braun2022 not to accelerate or "cross-seed" nucleation of new A xx fibrils.) The detailed mechanism of these inhibitory and accelerating effects was heretofore unknown and is a key focus of the present study. The red and blue curves are generated from published analytical solutions for A40 and A42 aggregation in isolation Dear2020JCP, using the same parameter values as in a (see Table \ref{['Table1']}).
• Figure 2: Kinetic analysis of first sigmoid of coaggregation data reveals molecular mechanism of A42 aggregation inhibition by A xx. Monomeric A42 (3 µ M) was aggregated with various initial A 40 (i), A 38 (ii) or A37 (iii) monomer concentrations. a: Global misfits of model in which A xx inhibits primary nucleation (Eqs. \ref{['Masoln']} with $K_E(ba)^{-1}=K_S(ba)^{-1}=0$). Mean residual errors (MREs) are $7.9\times10^{-3}$ (i), $4.9\times10^{-3}$ (ii), $1.4\times10^{-2}$ (iii). b: Global misfits of model in which A xx inhibits elongation (Eqs. \ref{['Masoln']} with $K_P(ba)^{-1}=K_S(ba)^{-1}=0$). MREs are $4.9\times10^{-3}$ (i), $3.7\times10^{-3}$ (ii), $9.4\times10^{-3}$ (iii). c: Global fits of model in which A xx inhibits secondary nucleation (Eqs. \ref{['Masoln']} with $K_E(ba)^{-1}=K_{P}(ba)^{-1}=0$). MREs are $1.8\times10^{-3}$ (i), $1.9\times10^{-3}$ (ii), $5.2\times10^{-3}$ (iii). Fitted parameter values are summarized in Tables S1-S3. Individually for each A xx alloform, the improvement in fit quality from b to c is arguably insufficient to eliminate the elongation inhibition mechanism with high confidence. (Brackets around the misfit "X" symbol indicate when the MREs are slightly less than double those achieved with the model used in c.) However, collectively they provide strong evidence in favour of secondary nucleation inhibition being the dominant cause of overall inhibition.
• Figure 3: Kinetic analysis of second sigmoid of seeded coaggregation data reveals molecular mechanism of A xx aggregation acceleration by A 42.i-ii: Kinetic data from Fig. 7 of Cukalevski2015, showing co-aggregation of 1.5 µ M each of monomeric A42 and A40 with several concentrations of preformed A42 fibril seeds, was additionally processed to suppress noise (see Methods \ref{['sec:processing']}). This reveals a clear trend of decreasing second sigmoid half-time with increasing A42 seed concentration. iii: We confirm this trend by performing a similar experiment but using different monomer concentrations (2 µ M A42 + 4 µ M A40; seed concentrations in legend). Only the second sigmoid is shown here; full timecourse is shown in Fig. S5. a: Global misfits to full kinetic curves for A42-A40 coaggregation using model in which only primary co-nucleation occurs (Eq. \ref{['fullmodelf']} with $k_2(ab)=0$). b: Global fits to full dataset for A42-A40 coaggregation using model in which only secondary co-nucleation occurs (Eq. \ref{['fullmodelf']} with $k_n(ab)=0$; fitted parameter values are summarized in Tables S4-S5).
• Figure 4: Schematic of unified co-aggregation model including all key states and reaction steps. A xx monomers inhibit pure A42 secondary nucleation by competing with A42 monomers for catalytic sites on A42 fibrils. Co-oligomers therefore form at these sites instead of pure A42 clusters. The co-oligomers undergo structural rearrangement into new growth-competent A xx fibrils, faster than they can form via primary nucleation. Any conversion of these co-oligomers into growth-competent A 42 fibrils is slow enough that A42 secondary nucleation is still inhibited overall. Note, formation of larger heterogeneous on-pathway nucleation intermediates such as protofibrils, rather than co-oligomers, would be equally consistent with the experimental findings, although co-oligomers are known to form in these reactions Iljina2016b.
• Figure 5: Unified co-aggregation model can successfully describe full kinetic curves for unseeded aggregation reactions using multiple initial concentrations of monomeric A xx.a: Global fit to full timecourse for A42-A40 coaggregation using unified model (Eq. \ref{['fullmodelf']}); fitted parameter values are summarized in Table S6. b: Global fit to full timecourse for A42-A38 coaggregation using unified model (Eq. \ref{['fullmodelf']}); fitted parameter values are summarized in Table S2.