A Probabilistic Approach to Growth Networks

TL;DR

This paper introduces a probabilistic methodology to analyze stochastic growth networks, specifically cellular filaments, providing explicit filament length distributions without computationally infeasible partition function calculations.

Contribution

It develops a novel probabilistic approach using large-deviations inequalities to derive explicit marginal distributions for filament lengths in growth networks.

Findings

Explicit filament length distributions derived

Method provides exact order-one probabilities

Closed-form expressions for distribution parameters

Abstract

Widely used closed product-form networks have emerged recently as a primary model of stochastic growth of sub-cellular structures, e.g., cellular filaments. In the baseline model, homogeneous monomers attach and detach stochastically to individual filaments from a common pool of monomers, resulting in seemingly explicit product-form solutions. However, due to the large-scale nature of such networks, computing the partition functions for these solutions is numerically infeasible. To this end, we develop a novel methodology, based on a probabilistic representation of product-form solutions and large-deviations concentration inequalities, that yields explicit expressions for the marginal distributions of filament lengths. The parameters of the derived distributions can be computed from equations involving large-deviations rate functions, often admitting closed-form algebraic expressions. From a methodological perspective, a fundamental feature of our approach is that it provides exact results for order-one probabilities, even though our analysis involves large-deviations rate functions, which characterize only vanishing probabilities on a logarithmic scale.