Universality in stochastic exponential growth

TL;DR

This paper introduces the stochastic Hinshelwood cycle model, a microscopic framework explaining universal scaling behaviors in stochastic exponential growth observed in bacterial size and division-time distributions.

Contribution

The paper develops an exact analytical model, the stochastic Hinshelwood cycle, that captures universal scaling laws in stochastic exponential growth, extending understanding beyond existing models.

Findings

Universal size and division-time distribution scaling observed in bacteria.

Exact solutions reveal fundamental fluctuation signatures in exponential growth.

Model applicable to diverse systems exhibiting stochastic exponential growth.

Abstract

Recent imaging data for single bacterial cells reveal that their mean sizes grow exponentially in time and that their size distributions collapse to a single curve when rescaled by their means. An analogous result holds for the division-time distributions. A model is needed to delineate the minimal requirements for these scaling behaviors. We formulate a microscopic theory of stochastic exponential growth as a Master Equation that accounts for these observations, in contrast to existing quantitative models of stochastic exponential growth (e.g., the Black-Scholes equation or geometric Brownian motion). Our model, the stochastic Hinshelwood cycle (SHC), is an autocatalytic reaction cycle in which each molecular species catalyzes the production of the next. By finding exact analytical solutions to the SHC and the corresponding first passage time problem, we uncover universal signatures of fluctuations in exponential growth and division. The model makes minimal assumptions, and we describe how more complex reaction networks can reduce to such a cycle. We thus expect similar scalings to be discovered in stochastic processes resulting in exponential growth that appear in diverse contexts such as cosmology, finance, technology, and population growth.