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

Farshid Jafarpour

TL;DR

The paper analyzes a size-structured population where cells grow via a Feller square-root process and divide under general size-control rules, deriving exact results for long-time growth, size distributions, and population fluctuations.A key finding is that the asymptotic population growth rate equals the mean single-cell growth rate $k$, independent of growth-noise strength and division details, making square-root noise a neutral fitness factor.The steady-state cell-size distribution is obtained as a one-sided exponential convolution of the deterministic inverse-square-law solution, with universal $\sigma^2$-dependent corrections; the total-population fluctuations have a closed-form stationary distribution independent of division mechanism.These results provide a solvable benchmark distinguishing square-root growth noise from size-independent noise and offer concrete, testable signatures for single-cell data and future extensions to colored noise or alternative growth models.

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.

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

TL;DR

The paper analyzes a size-structured population where cells grow via a Feller square-root process and divide under general size-control rules, deriving exact results for long-time growth, size distributions, and population fluctuations.A key finding is that the asymptotic population growth rate equals the mean single-cell growth rate $k$, independent of growth-noise strength and division details, making square-root noise a neutral fitness factor.The steady-state cell-size distribution is obtained as a one-sided exponential convolution of the deterministic inverse-square-law solution, with universal $\sigma^2$-dependent corrections; the total-population fluctuations have a closed-form stationary distribution independent of division mechanism.These results provide a solvable benchmark distinguishing square-root growth noise from size-independent noise and offer concrete, testable signatures for single-cell data and future extensions to colored noise or alternative growth models.

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 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.

Paper Structure

This paper contains 15 sections, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Population Growth Rate
  3. Deterministic Division
  4. Asymptotic Size Distribution
  5. Bellman-Harris Mapping and Generation Time Distribution
  6. Distribution of Total Numbers of Cells
  7. Noise in Division Size & Cell-Size Regulation
  8. Population Division & Birth Rates
  9. Population cell-size distribution
  10. Example: narrow Gamma distributed birth sizes
  11. Partitioning Noise & Asymmetric Division
  12. Discussion & Future Directions
  13. Moments recursion for binary Bellman-Harris
  14. Green's function method for cell-size distribution
  15. Asymptotic expansion of cell-size distribution for small $\sigma$

Figures (2)

  • Figure 1: Analytically calculated steady-state cell-size distribution from Eq. \ref{['eq:det_size_dist']} for $\sigma\in\{0.1,0.2,0.3\}$ (solid lines) compared to the cell-size distribution for deterministic growth (dashed line).
  • Figure 2: Steady-state population snapshot cell-size distribution for different variability $\sigma_s$ in lineage birth sizes. The left plot has close to zero growth noise $\sigma=0.01$, while the right plot has growth noise $\sigma=0.2$. The noise in growth effectively changes the cell-size distribution with an exponential moving average with the length scale $\sigma^2$.