Stochastic modeling of stem-cell dynamics with control

Abstract

Tissue development and homeostasis are thought to be regulated endogenously by control loops that ensure that the numbers of stem cells and daughter cells are maintained at desired levels, and that the cell dynamics are robust to perturbations. In this paper we consider several classes of stochastic models that describe stem/daughter cell dynamics in a population of constant size, which are generalizations of the Moran process that include negative control loops that affect differentiation probabilities for stem cells. We present analytical solutions for the steady-state expectations and variances of the numbers of stem and daughter cells; these results remain valid for non-constant cell populations. We show that in the absence of differentiation/proliferation control, the number of stem cells is subject to extinction or overflow. In the presence of linear control, a steady state may be maintained but no tunable parameters are available to control the mean and the spread of the cell population sizes. Two types of nonlinear control considered here incorporate tunable parameters that allow specification of the expected number of stem cells and also provide control over the size of the standard deviation. We show that under a hyperbolic control law, there is a trade-off between minimizing standard deviations and maintaining the system robustness against external perturbations. For the Hill-type control, the standard deviation is inversely proportional to the Hill coefficient of the control loop. Biologically this means that ultrasensitive response that is observed in a number of regulatory loops may have evolved in order to reduce fluctuations while maintaining the desired population levels.

Figure 1

Typical trajectories corresponding to different probabilities: (a) the constant model, p_i = 0.5 and (b) the linear model, p_i = i/N, with N = 70.

Figure 2

The hyperbolic law: typical runs showing the number of stem cells as a function of time. (a) The value of β is fixed to be β = 0.1, and h varies. (b) The values of β and h are given by formulas (28) and (29), with n₀ = 35 and a taking the values {3, 5, 10, 20}; the runs corresponding to larger values of a show a larger standard deviation. These are the runs that were used to calculate the means and standard deviations in figure 3(b).

Figure 3

The hyperbolic law: variance as a function of control parameters. (a) Variance as a function of h for values of β given by equation (28), with n₀ = 35. (b) The numerical mean and standard deviation for the number of stem cells with β given by formula (28) with n₀ = 35 and h given by formula (29), with different values of a, the robustness parameter. The dotted lines show the theoretical estimates for the variance. The vertical bars show the standard deviations calculated from a single run, 5, 000 time-steps. The time-series of those runs are given in figure 2(b).

Figure 4

Hill-type control law. (a) Individual trajectories corresponding to the probability $p_i=\frac{i^{\alpha }}{k^{\alpha }+i^{\alpha }}$ are plotted, with k = 30, and N = 70. Different values of α are indicated in the figure. (b) The variance of the number of stem cells is computed numerically (the dots) and compared with formula (27) (solid line) for different values of α. The computations are performed for individuals runs with N = 400, k = 200, over 40, 000 time-steps.

Figure 5

The hyperbolic law: extension to variable total populations. Typical runs showing the number of stem and differentiated cells as well as the total populations as a function of time. (a) The total number if cells is regulated endogenously by a negative control loop, where the probability for a stem cell to divide is given by a/(1+bN), where N is the total population size, a = 2.5, and b = 0.02. (b) The total number of cells fluctuates according to a symmetric Markov walk with the probability to divide given by 1/2. In both plots, the proliferation/differentiation decisions are controlled by the hyperbolic law, formula (1), with β = 0.1 and h = 0.005.