Quantifying random collisions between particles inside and outside a circle

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 $P_1$ of an unbounded receptor diffusing to the cell membrane. Meanwhile, we compute the probability $P_2$ 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 $P_1$ and $P_2$. 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.