A Stochastic Analysis of Particle Systems with Pairing

TL;DR

This paper models and analyzes the stochastic dynamics of particle pairing in biological systems, providing limit theorems and an averaging principle for systems with fixed or dynamic agent populations.

Contribution

It introduces a Markovian model for particle-agent interactions and establishes new limit theorems and an averaging principle for both fixed and dynamic agent scenarios.

Findings

First order limit theorems for particle pairing dynamics

An averaging principle for systems with dynamic agents

Limit theorems for fluctuations in fixed-agent systems

Abstract

Motivated by a general principle governing regulation mechanisms in biological cells, we investigate a general interaction scheme between different populations of particles and specific particles, referred to as agents. Assuming that each particle follows a random path in the medium, when a particle and an agent meet, they may bind and form a pair which has some specific functional properties. Such a pair is also subject to random events and it splits after some random amount of time. In a stochastic context, using a Markovian model for the vector of the number of paired particles, and by taking the total number of particles as a scaling parameter, we study the asymptotic behavior of the time evolution of the number of paired particles. Two scenarios are investigated: one with a large but fixed number of agents, and the other one, the dynamic case, when agents are created at a bounded rate and may die after some time when they are not paired. A first order limit theorem is established for the time evolution of the system in both cases. The proof of an averaging principle of the dynamic case is one of the main contributions of the paper. Limit theorems for fluctuations are obtained in the case of a fixed number agents. The impact of dynamical arrivals of agents on the level of pairing of the system is discussed.