Coagulation-fragmentation for a finite number of particles and application to telomere clustering in the yeast nucleus

We develop a coagulation-fragmentation model to study a system composed of a small number of stochastic objects moving in a confined domain, that can aggregate upon binding to form local clusters of arbitrary sizes. A cluster can also dissociate into two subclusters with a uniform probability. To study the statistics of clusters, we combine a Markov chain analysis with a partition number approach. Interestingly, we obtain explicit formulas for the size and the number of clusters in terms of hypergeometric functions. Finally, we apply our analysis to study the statistical physics of telomeres (ends of chromosomes) clustering in the yeast nucleus and show that the diffusion-coagulation-fragmentation process can predict the organization of telomeres.

💡 Research Summary

The paper presents a mathematically rigorous coagulation‑fragmentation framework tailored for systems that contain only a modest number of stochastic particles confined within a finite domain. Unlike classical approaches that assume an infinite or continuum population, the authors treat each possible configuration of particles as a discrete state defined by a partition of the total particle number N into cluster sizes. This partition‑based representation naturally leads to a finite‑state Markov chain whose transition rates encode two elementary processes: (i) binary coagulation, where two distinct clusters of sizes n_i and n_j merge with a rate proportional to λ_c · n_i · n_j, reflecting diffusion‑limited encounter probability; and (ii) binary fragmentation, where a cluster of size n splits into two subclusters (i, n‑i) with a uniform probability over all admissible i, at a rate λ_f · n · (n‑1)/2. The master equation for the probability P(π,t) of each partition π is derived, and a generating‑function formalism is introduced to convert the high‑dimensional system into a set of partial differential equations in the auxiliary variables z_1,…,z_N.

A key analytical advance is the exploitation of integer‑partition combinatorics. The number of ways to partition N particles into k clusters, denoted p(N,k), satisfies a well‑known recursion and allows the authors to express stationary probabilities in closed form. By imposing detailed balance under the assumption that the ratio r = λ_c/λ_f is constant, the stationary distribution over the number of clusters k is shown to be proportional to a hypergeometric term:

P_k = C · r^k · (N‑k)! /