Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data

Abstract

We consider an age-maturity structured model arising from a blood cell proliferation problem. This model is "hybrid", i.e., continuous in time and age but the maturity variable is discrete. This is due to the fact that we include the cell division marker carboxyfluorescein diacetate succinimidyl ester. We use our mathematical analysis in conjunction with experimental data taken from the division analysis of primitive murine bone marrow cells to characterize the maturation/proliferation process. Cell cycle parameters such as proliferative rate beta, cell cycle duration tau, apoptosis rate gamma, and loss rate micro can be evaluated from CarboxyFluorescein diacetate Succinimidyl Ester + cell tracking experiments. Our results indicate that after three days in vitro, primitive murine bone marrow cells have parameters beta = 2.2 day(-1), tau = 0.3 day, gamma = 0.3 day(-1), and micro = 0.05 day(-1).

FIGURE 1

A schematic representation of the G₀ stem cell model. Proliferating phase cells include those cells in G₁ (the first gap), S (DNA synthesis), G₂ (the second gap), and M (mitosis) while the resting phase cells are in the G₀ phase. μ is the loss rate of resting phase (G₀ cells due to death or differentiation, while γ represents a loss of proliferating phase cells due to apoptosis. β is the rate of cell reentry from G₀ into the proliferative phase, and τ is the duration of the proliferative phase (see Burns and Tannock, 1970; Mackey, 1978, ,, , for further details).

FIGURE 2

Comparison of CFSE fluorescence between the experimental data and theoretical results. This figure represents the CFSE profile after 3 days of culture (two days after isolating CFSE + cells). The first bar (black) represents the model predicted number of proliferating cells, the second one (dark) is for the predicted resting phase cells, the third bar (light) is the total cell population in the cell compartment and the fourth bar (white) is the experimental data. For comparison between the data and the model one should concentrate on the total population. Parameters: β = 2.24 day⁻¹, τ = 0.307 day, γ = 0.30 day⁻¹, and μ = 0.05 day⁻¹. The initial proportion of cells in resting phase is 0.65. Experimental data taken from Oostendorp et al., (2000) (Fig. 1, panel 3).

FIGURE 3

Representation of two subpopulations with the same parameters: τ = 0.25 day, γ = 0.90 day⁻¹, and μ = 0.05 day⁻¹, but a different reentry rate β. (Top) β = 0.08 day⁻¹, which corresponds to a slowly cycling population of cells. (Bottom) β = 2.30 day⁻¹ which corresponds to a rapidly cycling population. The experimental data come from Oostendorp et al. (2000) (Fig. 2, bottom). Bars as in Fig. 2.

FIGURE 4

Approximation of the experimental data after four days in culture (3 days after isolating CFSE + cells) from Oostendorp et al. (2000) (Fig. 2, bottom). Two subpopulations are represented in the present figure: one corresponding to the slowly cycling population (β = 0.08 day⁻¹) and the other one corresponding to the rapidly cycling population (β = 2.30 day⁻¹) parameters: β = 0.08 and 2.30 day⁻¹, τ = 0.25 day, γ = 0.90 day⁻¹, and μ = 0.05 day⁻¹. The initial proportion of cells in resting phase was 0.90; the slowly cycling population constituting 0.40 of the total and the rapidly cycling one 0.60 of the total initial population. This figure represents the weighted sum of subpanels in Fig. 3. Bars as in Fig. 2.

FIGURE 5

Model predicted numbers of proliferating and resting phase cells with respect to division number at t = 3 days based on the same parameter as in Fig. 2: β = 2.24 day⁻¹, τ = 0.307 day, γ = 0.30 day⁻¹, μ = 0.05 day⁻¹, and an initial proportion of resting cells of 0.65. The CFSE fluorescence profile is shown in the top panel. In the lower panel, the proportion of cells in the two different compartments for each generation is given.

FIGURE 6

Model predicted total number of proliferating and resting phase cells as a function of time for the same parameters as in Fig. 2, β = 2.24 day⁻¹, τ = 0.307 day, γ = 0.30 day⁻¹, μ = 0.05 day⁻¹, and an initial proportion of resting cells of 1. The evolution curves are compared to other ones with a smaller reentry rate β = 1.6 day⁻¹. As expected the proportion of resting phase cells gets larger as β decreases. The transient is due to the fact that proliferating cells take a time τ to divide and reenter in the resting phase. After time t = τ the curves stabilize rapidly.

FIGURE 7

Predicted CFSE fluorescence of labeled cells between 8 h and 72 h based on our analysis with β = 2.24 d⁻¹, τ = 0.307 d, γ = 0.30 d⁻¹, μ = 0.05 d⁻¹, and initial proportion of resting cells of 0.65. The peaks in each panel represent the total relative number of cells of each generation for different times. After two days (48 h) the CFSE profile corresponds to that of Fig. 2.