Quantifying random collisions between particles inside and outside a circle

Abstract

Random collisions of particles occur in various biophysical and physical systems. Inspired by the binding of receptor and ligand on the cell membrane, we devised a method based on stochastic dynamical modeling to quantify the likelihood of two random particles colliding on a circle. We consider the dynamics of a receptor binding to a ligand on the cell membrane, where the receptor and ligand perform different motions and are thus modeled by stochastic differential equations with non-Gaussian noise. We use neural networks based on the Onsager–Machlup function to compute the probability P1 of an unbounded receptor diffusing to the cell membrane. Meanwhile, we compute the probability P2 of the extracellular ligand arriving at the cell membrane by solving the associated nonlocal Fokker–Planck equation. We can then calculate the most probable binding probability by combining P1 and P2. In this way, we conclude with some indication of how the receptors could distribute on the membrane, as well as where the ligand will most probably encounter the receptor, contributing to a better understanding of the cell’s response to external stimuli and communication with other cells.