Abstract

The aggregation of $\alpha$-synuclein ($\alpha$S) is a hallmark of Parkinson's disease, with its interaction with cellular membranes being a critical step in its pathological misfolding and subsequent fibril formation. Understanding the fundamental physics governing $\alpha$S protein-protein interactions (PPIs) is essential for developing therapeutic strategies. This study employs the **Landau–Ginzburg–Wilson (LGW) approach** to develop a robust theoretical model for describing the complex, concentration-dependent PPIs of $\alpha$S. The LGW framework, typically used for phase transitions, is adapted here to capture the cooperative and long-range nature of $\alpha$S multimerization, particularly in the context of its membrane-associated state. Our model successfully predicts the critical concentration and conditions under which $\alpha$S transitions from a monomeric to an aggregated state, demonstrating a strong correlation between the LGW parameters and the protein's known biophysical properties. The findings provide a novel, coarse-grained theoretical tool for investigating the initial stages of protein aggregation and offer a deeper theoretical understanding of the biophysical mechanisms that drive $\alpha$S pathology in a cellular environment.