A Time-Varying Branching Process Approach to Model Self-Renewing Cells

TL;DR

The paper addresses modeling stem cell proliferation when division probabilities evolve over time. It introduces a continuous-time, time-dependent multi-type branching process with three cell types and a rate parameter, deriving closed-form expressions for the mean, variance, and autocovariance of viable stem cells. It develops likelihood-based inference, including a forward algorithm to handle partially observed data, and demonstrates through simulations and an MEP clonal expansion case study that time-varying division probabilities and division rates can be accurately recovered. The approach reduces data requirements by enabling parameter estimation without full lineage tracking, with broad applicability to proliferative processes across biology.

Abstract

Stem cells, through their ability to produce daughter stem cells and differentiate into specialized cells, are essential in the growth, maintenance, and repair of biological tissues. Understanding the dynamics of cell populations in the proliferation process not only uncovers proliferative properties of stem cells, but also offers insight into tissue development under both normal conditions and pathological disruption. In this paper, we develop a continuous time branching process model with time-dependent offspring distribution to characterize stem cell proliferation process. We derive analytical expressions for mean, variance, and autocovariance of the stem cell counts, and develop likelihood-based inference procedures to estimate model parameters. Particularly, we construct a forward algorithm likelihood to handle situations when some cell types cannot be directly observed. Simulation results demonstrate that our estimation method recovers the time-dependent division probabilities with good accuracy.

Figures (5)

• Figure 1: Left: Fifty realizations of simulated stem cell count trajectories with initial count of 200, division rate of 0.2, and division probabilities $p_1(t) = \frac{0.55}{1 + 0.005(t-4)^2}$, $p_2(t) = \frac{0.15}{1 + (t-12)^2}$, $p_4(t) = \frac{0.20}{1+(t-20)^2}$, and $p_3(t) = 1-p_1(t)-p_2(t)-p_4(t)$. The blue curve shows the expected cell count and the shaded band indicates $\pm$2 standard deviations. Right: Distribution of stopping time, i.e. the time when all the viable stem cells are exhausted from 100 realizations of the simulated stem cell count trajectories with the same parameters, with the minimum expected stopping times is $\tau_{\min} = 26.491$.
• Figure 2: Time-dependent probability functions for division and differentiation events under three Lorentzian parameter configurations. Each panel displays the four probabilities $p_1(t), p_2(t), p_3(t),$ and $p_4(t)$ defined in equation (\ref{['lorentzian']}).
• Figure 3: Ratios of observed-to-expected cell counts. The predicted cell counts are generated by using the estimated parameters (in table \ref{['tab:par_est_forward']}) for partially observed data using forward algorithm likelihood.
• Figure 4: Observed and estimated cell counts calculated from the estimated parameters for the experimental MEP clonal expansion and differentiation data.
• Figure 5: Diagnostic plots assessing the fit of the inverse Gaussian distribution to the stimulated stopping times. The fitted distribution is based on parameters estimated using maximum likelihood.

Theorems & Definitions (5)

• Proposition 1: Moments of Number of Stem Cells
• proof
• Remark 1
• Proposition 2
• proof