Estimation of the lifetime distribution from fluctuations in Bellman-Harris processes

TL;DR

This work tackles the inverse problem of inferring the two-parameter Gamma lifetime distribution in Bellman-Harris processes from population-size data. It shows that the long-time fluctuations of the residual process $R_t^{\delta}=N_{t+\delta}-e^{\delta\alpha}N_t$ exhibit either Gaussian behavior or oscillations, depending on the Gamma shape parameter via a critical threshold $k_c\approx 57.24$. The authors derive explicit asymptotics for the Gaussian components and for the oscillatory regime, and they propose a practical estimation pipeline that first estimates the Malthusian rate $\alpha$, then detects the regime, and finally infers $(k,\theta)$ (or equivalently $(\mu,\sigma/\mu)$) from time-resolved population data. The approach is validated by simulations and applied to two real datasets, yielding plausible lifetime variability estimates without tracking individuals. The framework highlights the importance of regime-aware inference and provides a route to extract single-cell heterogeneity from population-level monitoring.

Abstract

The growth of a population is often modeled as branching process where each individual at the end of its life is replaced by a certain number of offspring. An example of these branching models is the Bellman-Harris process, where the lifetime of individuals is assumed to be independent and identically distributed. Here, we are interested in the estimation of the parameters of the Bellman-Harris model, motivated by the estimation of cell division time. Lifetimes are distributed according a Gamma distribution and we follow a population that starts from a small number of individuals by performing time-resolved measurements of the population size. The exponential growth of the population size at the beginning offers an easy estimation of the mean of the lifetime. Going farther and describing lifetime variability is a challenging task however, due to the complexity of the fluctuations of non-Markovian branching processes. Using fine and recent results on these fluctuations, we describe two time-asymptotic regimes and explain how to estimate the parameters. Then, we both consider simulations and biological data to validate and discuss our method. The results described here provide a method to determine single-cell parameters from time-resolved measurements of populations without the need to track each individual or to know the details of the initial condition.

Figures (15)

• Figure 1: Curve of $\lambda - \frac{\alpha}{2}$$= \frac{2^{\frac{1}{k}}\left(2\cos\left(\frac{2\pi}{k}\right) -1\right) -1}{2\theta}$ versus the parameter $k$, for $\theta = 1$.
• Figure 2: Illustration of the two regimes, for parameters with the same Malthusian coefficient.
• Figure 3: Comparison of $\overline{\sigma}_{\delta}^2$ with the variance of $\frac{R_{t,\delta}}{\sqrt{N_t}}$ in different cases. We use the grid of parameters $\mathbb{G}_{1/20}$, and we take $\delta$ as explained in Section \ref{['subsubsect:step_identifiability']}. We also estimate the variance of $\frac{R_{t,\delta}}{\sqrt{N_t}}$ using the empirical estimator of the variance with $2000$ simulations of Bellman-Harris dynamics. To simplify the computation of $\overline{\sigma}_{\delta}^2$, we do the approximation that $\sigma_{\delta}^2$ does not depends on $\alpha$ (or equivalently $\theta$), see Figure \ref{['fig:psi_independence_alpha']} and the end of Section \ref{['subsubsect:step_identifiability']}.
• Figure 4: Illustration of the estimation of $\alpha$ and $e^{\alpha\delta}$ for different parameters. Due to oscillations, see Figures \ref{['fig:mean_cells_small_amplitude']} and \ref{['fig:mean_cells_large_amplitude']}, the errors slightly increase when $k$ increases.
• Figure 5: Curve of $\left|e^{(\lambda+i\tau)\delta} - e^{\alpha\delta}\right|$ versus $\delta$, when $(k,\theta) = (80.2,4)$. Red lines represent multiples of $\log(2)/\alpha$.

Theorems & Definitions (15)

• Theorem 1
• Lemma 1
• proof
• Proposition 1
• Remark 1
• proof
• Lemma 2
• proof
• Proposition 2
• proof