Stochastic coagulation-fragmentation processes with a finite number of particles and applications

TL;DR

This paper develops a stochastic model for finite particle coagulation-fragmentation processes, deriving explicit distributions and moments, and explores applications in cellular biology.

Contribution

It introduces a new probabilistic framework with explicit formulas for cluster distributions and mean times, applicable to biological systems.

Findings

Derived steady-state distributions for cluster sizes.

Obtained explicit formulas using hypergeometric functions.

Analyzed mean times for particle interactions.

Abstract

Coagulation-fragmentation processes describe the stochastic association and dissociation of particles in clusters. Cluster dynamics with cluster-cluster interactions for a finite number of particles has recently attracted attention especially in stochastic analysis and statistical physics of cellular biology, as novel experimental data is now available, but their interpretation remains challenging.} We derive here probability distribution functions for clusters that can either aggregate upon binding to form clusters of arbitrary sizes or a single cluster can dissociate into two sub-clusters. Using combinatorics properties and Markov chain representation, we compute steady-state distributions and moments for the number of particles per cluster in the case where the coagulation and fragmentation rates follow a detailed balance condition. We obtain explicit and asymptotic formulas for the cluster size and the number of clusters in terms of hypergeometric functions. To further characterize clustering, we introduce and discuss two mean times: one is time two particles spend together before they separate and other is the time they spend separated before they meet again for the first time. Finally we discuss applications of the present stochastic coagulation-fragmentation framework in cell biology.