Exactly Solvable Population Model with Square-Root Growth Noise and Cell-Size Regulation

TL;DR

This paper presents an exactly solvable model of cell population growth with square-root noise, revealing that growth rate remains unaffected by noise and providing explicit formulas for population size distribution and fluctuations.

Contribution

It introduces a novel size-structured branching process model with exact solutions, highlighting the neutral effect of square-root growth noise on long-term fitness.

Findings

Population growth rate equals mean single-cell growth rate.

Steady-state size distribution is an inverse-square law convolved with an exponential.

Long-term size fluctuations follow a stationary compound Poisson-exponential distribution.

Abstract

We analyze a size-structured branching process in which individual cells grow exponentially according to a Feller square-root process and divide under general size-control mechanisms. We obtain exact expressions for the asymptotic population growth rate, the steady-state snapshot distribution of cell sizes, and the fluctuations of the total cell number. Our first result is that the population growth rate is exactly equal to the mean single-cell growth rate, for all noise strengths and for all division and size-regulation schemes that maintain size homeostasis. Thus square-root growth noise is neutral with respect to long-term fitness, in sharp contrast to models with size-independent stochastic growth rates. Second, we show that the steady-state population cell-size distribution is obtained from the deterministic inverse-square-law solution by a one-sided exponential convolution with kernel width set by the strength of growth fluctuations. Third, the mean-rescaled population size $N_t/\left\langle N_t\right\rangle$ converges to a stationary compound Poisson-exponential distribution that depends only on growth noise. This distribution, and hence the long-time shape of population-size fluctuations, is unchanged by division-size noise or asymmetric partitioning. These results identify Feller-type exponential growth with square-root noise as an exactly solvable benchmark for stochastic growth in size-controlled populations and provide concrete signatures that distinguish it from models with size-independent growth-rate noise.