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The development of solutions to the kinetics of homomolecular self-assemblyinto amyloid fibrils using fixed-point methods, and their subsequentapplication to the analysis of in vitro kinetic experiments, has led tonumerous advances in our understanding of the fundamental chemical mechanismsbehind amyloidogenic disorders such as Alzheimer's and Parkinson's diseases.However, as our understanding becomes more detailed and new data becomeavailable, kinetic models need to increase in complexity. The resulting rateequations are no longer amenable to extant solution methods, hindering ongoingefforts to elucidate the mechanistic determinants of aggregation in livingsystems. Here, we demonstrate that most linear self-assembly reactions aredescribed by the same unusual class of singularly perturbed rate equations,that cannot be solved by normal singular perturbation techniques such asrenormalization group. We instead develop a new method based on Lie symmetrythat can reliably solve this class of equations, and use it in conjunction withexperimental data to determine the kinetics of co-aggregation of theAlzheimer's disease-associated Abeta42, Abeta40 and Abeta38 peptides. Ourmethod also rationalizes several successful earlier solutions for homomolecularself-assembly kinetics whose mathematical justification was previously unclear.Alongside its generality and mathematical clarity, its much greater accuracyand simplicity compared to extant methods will enable its rapid and widespreadadoption by researchers modelling filamentous self-assembly kinetics.

Developing integrated rate laws of complex self-assembly reactions using Lie symmetry: Kinetics of Abeta42, Abeta40 and Abeta38 co-aggregation