  The development of solutions to the kinetics of homomolecular self-assembly
into amyloid fibrils using fixed-point methods, and their subsequent
application to the analysis of in vitro kinetic experiments, 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, as our understanding becomes more detailed and new data become
available, kinetic models need to increase in complexity. The resulting rate
equations are no longer amenable to extant solution methods, hindering ongoing
efforts to elucidate the mechanistic determinants of aggregation in living
systems. Here, we demonstrate that most linear self-assembly reactions are
described by the same unusual class of singularly perturbed rate equations,
that cannot be solved by normal singular perturbation techniques such as
renormalization group. We instead develop a new method based on Lie symmetry
that can reliably solve this class of equations, and use it in conjunction with
experimental data to determine the kinetics of co-aggregation of the
Alzheimer's disease-associated Abeta42, Abeta40 and Abeta38 peptides. Our
method also rationalizes several successful earlier solutions for homomolecular
self-assembly kinetics whose mathematical justification was previously unclear.
Alongside its generality and mathematical clarity, its much greater accuracy
and simplicity compared to extant methods will enable its rapid and widespread
adoption by researchers modelling filamentous self-assembly kinetics.
