# Stability Analysis of a Bulk-Surface Reaction Model for Membrane-Protein Clustering

## TL;DR

This paper investigates the stability and pattern formation of protein clusters on cell membranes using a reaction-diffusion model, revealing how ligand recruitment can lead to stable, heterogeneous protein distributions.

## Contribution

It provides a theoretical analysis of membrane protein clustering through a reaction-diffusion model, including diffusion-driven instability estimates and a threshold for pattern formation.

## Key findings

- Ligand recruitment promotes stable protein cluster formation.
- A threshold exists for ligand recruitment to induce pattern formation.
- Single-patch steady states emerge under certain conditions.

## Abstract

Protein aggregation on the plasma membrane (PM) is of critical importance to many cellular processes such as cell adhesion, endocytosis, fibrillar conformation, and vesicle transport. Lateral diffusion of protein aggregates or clusters on the surface of the PM plays an important role in governing their heterogeneous surface distribution. However, the stability behavior of the surface distribution of protein aggregates remains poorly understood. Therefore, understanding the spatial patterns that can emerge on the PM solely through protein-protein interaction, lateral diffusion, and feedback is an important step towards a complete description of the mechanisms behind protein clustering on the cell surface. In this work, we investigate the pattern formation of a reaction-diffusion model that describes the dynamics of a system of ligand-receptor complexes. The purely diffusive ligand in the cytosol can bind receptors in the PM, and the resultant ligand-receptor complexes not only diffuse laterally but can also form clusters resulting in different oligomers. Finally, the largest oligomers recruit ligands from the cytosol in a positive feedback. From a methodological viewpoint, we provide theoretical estimates for diffusion-driven instabilities of the protein aggregates based on the Turing mechanism. Our main result is a threshold phenomenon, in which a sufficiently high recruitment of ligands promotes the input of new monomeric components and consequently drives the formation of a single-patch spatially heterogeneous steady-state.