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

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

Problem Statement
The paper addresses a central unresolved problem in quantitative cell physiology: how microscopic fluctuations in single-cell growth rates shape population-level fitness and size statistics. While single-cell technologies have enabled detailed characterization of division statistics and size-control strategies, the temporal statistics of single-cell growth rates remain poorly constrained. Existing theoretical models typically assume that the instantaneous exponential growth rate $\lambda(t)$ fluctuates independently of cell size $s$. However, from a microscopic perspective, intrinsic fluctuations in biomass production are expected to scale with the square root of cell size (due to the extensive nature of production noise), leading to a Feller-type diffusion process. The authors aim to bridge this gap by introducing an exactly solvable population model incorporating size-dependent (square-root) growth noise to determine its impact on population growth rates, steady-state size distributions, and population-size fluctuations.

Methodology
The authors analyze a size-structured population where individual cells grow according to a stochastic differential equation (SDE) representing Feller square-root diffusion:
$$\frac{ds}{dt} = k s + \sigma \sqrt{2k \bar{s}_b s} \, \xi(t)$$
where $k$ is the mean growth rate, $\bar{s}_b$ is the mean birth size, $\sigma$ is the dimensionless noise strength, and $\xi(t)$ is Gaussian white noise. Division and size regulation are modeled generally: cells divide at a size $s_d$ (which may be a random variable dependent on birth size $s_b$) and partition into two daughters with sizes $r s_d$ and $(1-r)s_d$.

The analysis proceeds through several analytical steps:

1. Population Mass Dynamics: The authors exploit the additivity of the square-root noise to show that the total population mass $M_t$ obeys the same SDE as a single cell, allowing for exact solutions regarding total mass and population size.
2. Steady-State Size Distribution: For a simplified case of constant division size, the authors solve the associated Fokker-Planck equation (a partial differential equation with drift and diffusion terms) to derive the exact steady-state density of cell sizes.
3. Branching Process Mapping: The model is mapped to a Bellman-Harris branching process. The authors derive the generation time distribution as the first-passage time of the Feller process and utilize its Laplace transform to compute moments of the population size.
4. Generalization: The results are extended to include noisy division sizes and general size-regulation mechanisms (e.g., adder, sizer, timer models) by decoupling the growth dynamics from the division statistics.

Key Contributions and Results

1. Exact Population Growth Rate:
   The primary result is that the asymptotic population growth rate $\Lambda$ is exactly equal to the mean single-cell growth rate $k$ ($\Lambda = k$). This holds for all noise strengths $\sigma$ and all division/size-regulation schemes that maintain size homeostasis.

   • Contrast: This stands in sharp contrast to models with size-independent growth-rate noise, where $\Lambda$ depends sensitively on the variance and correlation time of the noise.
   • Mechanism: This "strong decoupling" arises specifically from the additive square-root noise structure, which is not a generic property of stochastic growth models.

2. Steady-State Cell-Size Distribution:
   The authors derive an exact analytical solution for the steady-state distribution of cell sizes.

   • Convolution Structure: The solution for $\sigma > 0$ is obtained by a one-sided exponential convolution of the deterministic ($\sigma=0$) solution with a kernel of width $\sigma^2$.
   • Moments: The population mean cell size shifts by $-\sigma^2$ relative to the deterministic case, while the variance is modified only at order $\sigma^4$.
   • Decoupling: The shape of the distribution depends on size regulation only through the stationary lineage birth-size distribution; growth noise introduces a division-independent smoothing effect.

3. Population-Size Fluctuations:
   The distribution of the mean-rescaled population size $W = \lim_{t\to\infty} N_t / \langle N_t \rangle$ is derived in closed form.

   • Distribution Type: $W$ follows a compound Poisson-exponential distribution that depends only on the growth noise parameter $\sigma$.
   • Independence: This distribution is invariant to division-size noise, asymmetric partitioning, or specific size-regulation mechanisms. These factors only affect the prefactor relating total mass to cell number, not the shape of the fluctuations.
   • Variance: The long-time relative variance is given by $\text{Var}(N_t) / \langle N_t \rangle^2 = 2\sigma^2$.
   • Solvability: The authors note that, beyond the Yule process (exponential generation times), this is a rare example of a nontrivial branching model where the full stationary distribution of the rescaled population size is known in closed form.

Significance and Claims
The paper identifies stochastic exponential growth with square-root noise as an "exactly solvable benchmark" for studying growth fluctuations in size-controlled populations. The authors claim that their results reveal a hierarchy of decouplings:

1. Population growth rate is independent of growth noise and division details.
2. Snapshot cell-size distributions depend on regulation only via the lineage birth-size distribution, modified by a specific convolution.
3. The shape of population-size fluctuations is entirely determined by growth noise, independent of division mechanisms.

The authors assert that these predictions differ qualitatively from models based on size-independent growth-rate noise. Consequently, the derived signatures (specifically the independence of the population growth rate from noise strength and the specific form of population-size fluctuations) provide concrete, testable criteria to distinguish between competing microscopic models of growth using existing and future single-cell data. The work does not propose new experimental protocols but offers a theoretical framework to interpret empirical data.