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

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 coaggregation in environments containing multiple amyloid-beta (Aβ) 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 Lie 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 coaggregation of the most physiologically abundant Aβ alloforms: Aβ42, Aβ40, Aβ38 and Aβ37 peptides. We find that Aβ42 fibril surfaces catalyze the formation of co-oligomers, which accelerate new Aβ40, Aβ38 and Aβ37 fibril formation whilst inhibiting secondary nucleation of new Aβ42 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 coaggregation.