Stem Cell Differentiation Dynamics Model

The present modeling approach is based on the dynamics of stem cell populations stratified with respect to cell differentiation. Cell differentiation is defined through the variable α∈[0,1] with α = 0 for pure stem cells and α = 1 for fully differentiated cells. The model assumes cell differentiation to be subject to random changes defined by the conditional probability density function (cpdf) for given α and the randomization rate R(α) quantifying the number of random events per time (Equations (2) and (3)). The cpdf is assumed to be Gaussian centered at α with standard deviation (noise amplitude) σ(α). It is renormalized to unity for each α to account for the truncation to the interval [0,1]. The noise amplitude is specified as a sum of piecewise linear or quadratic functions q_i(α) = u₀+u₁(α−α_iq)+u₂(α−α_iq)² localized by tanh-type radial basis functions with b_i(α) = 1/2 tanh[(r_i−α+α_ib)/s_i]+1/2 tanh[(r_i+α−α_ib)/s_i], in which α_iq and α_ib denote the offset of the polynomial and the radial base, respectively, whereas r_i specifies the characteristic radius, and s_i the transfer width of the base. Cells are assumed to proliferate according to the growth rate r(α) and the time-dependent apoptosis rate a(t) = a₁t irrespective of generation. The two-dimensional rate equation for the average number of cells is numerically solved by the explicit Euler Forward Method on a 2D-grid of discrete differentiation values α_i, i = 1,…,n_a, and generation-specific cell cycle phases k = 1,…,n_p according to in which n_c denotes the number of cell cycle phases per generation, ρ(k) = 2 if k≡1(mod n_c) to account for cell doubling and ρ(k) = 1 otherwise. Furthermore, ϕ(k) = 0 if k = n_p and ϕ(k) = 1 otherwise. It is understood that M(a_i, k−1) = 0 for k = 1. The cell cycle terms in the second row of Equation (4) implement the continuous cell cycle model of León et al. [50] (link) without G₀ phase arrest. The number of cells in generation l is calculated by summing over its cell cycle phases . The dynamics of the marginal relative frequencies associated with cell differentiation can easily be derived from .
Truncation of the cpdf to the unit interval generally results in non-symmetric scattering and thus in a non-vanishing drift term A(α) as defined for the Fokker-Planck equation [47] . This term mimics a deterministic dynamic component corresponding to f(α) in the Langevin equation (1). The results of the present study were checked against either using the numerically determined non-zero A(α) or setting A(α) = 0 in the equilibrated distributions. We found no notable difference except for the S1 to D1 transition shown in Figure 3, for which, however, also the Fokker-Planck approximation to the master equation breaks down.