Self-consistent analytical solutions to the kinetics of lipid-induced protein aggregation

The aggregation of proteins into amyloid fibrils is a hallmark of several neurodegenerative disorders, including Parkinson's disease. A growing body of experimental evidence highlights the significant role lipid membranes play in modulating this aggregation process, particularly for proteins such as α-synuclein. Despite this, there has been a lack of quantitative theoretical frameworks capable of describing the kinetics of lipid-induced protein aggregation. In this work, we develop an analytical theoretical model that explicitly incorporates lipid-mediated interactions into the aggregation kinetics. By formulating rate equations in terms of lipid surface coverage and applying a fixed-point analysis, we derive self-consistent solutions for the full timecourse of aggregation. Our model captures both one-step and two-step nucleation mechanisms and enables the prediction of key kinetic observables, including half-times and maximal growth rates. These results provide a quantitative foundation for interpreting experimental data and offer new mechanistic insights into how lipids influence the self-assembly of amyloidogenic proteins.