A Becker-D\"oring type model for cell polarization

TL;DR

This paper introduces a Becker-Döring type model for cell polarization, analyzing its long-term behavior, mass loss, and self-similar evolution of large clusters under various coagulation and fragmentation rates.

Contribution

It extends the Becker-Döring equations to model cell polarization, providing convergence results and analyzing large cluster dynamics with power-law and linear rates.

Findings

Convergence to equilibrium for power-law rates.

Mass loss depending on initial conditions.

Self-similar evolution of large clusters with linear rates.

Abstract

We propose a model for cell polarization based on the Becker-D\"oring equations with the first coagulation coefficient equal to zero. We show convergence to equilibrium for power-law coagulation and fragmentation rates and obtain a loss of mass in the limit $t \rightarrow \infty$ depending on the initial mass and the relative strengths of the coagulation and fragmentation processes. In the case of linear rates, we further show that large clusters evolve in a self-similar manner at large times by comparing limits of appropriately rescaled solutions in different spaces.