Fig 1.
Symmetric and asymmetric SC divisions.
In the asymmetric division model, a SC produces one differentiated cell and one SC. In the symmetric division model, a SC produces two differentiated cells or two SCs.
Fig 2.
An example of endogenous control loops regulating SC decisions with the all symmetric division mode.
Division events are negatively regulated by daughter cells and differentiation decisions are positively regulated by SCs.
Table 1.
Notations used in the models.
Fig 3.
Classification of minimal control systems in two-compartment models.
Symbol “div” refers to the rate of symmetric stem cell divisions (both proliferations and differentiations). Symbol “diff” refers to the probability of differentiation; the probability of proliferation is 1-Prob(diff). Models #1–2 are the two-control systems. Models #3–5 are three-control systems. Division and differentiation decisions can be positively or negatively controlled by the population sizes of SCs or differentiated cells, as indicated by arch-like arrows that originate at the relevant cell population and point toward the process that this population controls. The rightmost column indicates how cell number variances depend on the symmetry of divisions, as obtained from the analysis of the Methods Section.
Fig 4.
Typical numerical simulations of cell dynamics.
(a) System Eq (1) with ϵ = 0.005 and S* = S = 0.5; (b) system Eq (6) with ϵ = 0.005 and S* = S = 0.8. Simulations are run for 2 ⋅ 105 time steps.
Fig 5.
The behavior of the means and the variances of the cell population described by Eq (1).
The analytical results given by Eqs (2–5) (solid line) are compared with the values obtained by numerical simulations (stars), for different values of ϵ with the fixed value of S: S = 0.5. The choice of S should satisfy: S = S* < Sc, where Sc is given by Eq (29). (‘T’) stands for the theoretical results, and (‘N’) stands for the numerical results.
Fig 6.
The behavior of the variances of the cell population described by Eq (1) with ϵ = 0.005, for different values of S.
The analytical results given by Eqs (2–5) (solid line) are compared with the values obtained by numerical simulations (stars). (‘T’) stands for the theoretical results, and (‘N’) stands for the numerical results.
Fig 7.
The behavior of the means and the variances of the cell population described by Eq (6).
The analytical results given by Eqs (7–10) (solid line) are compared with the values obtained by numerical simulations (stars), for different values of ϵ with the fixed value of S: S = 0.5. The choice of S should satisfy: S > Sc = 0 in this case, where Sc is given by Eq (29). (‘T’) stands for the theoretical results, and (‘N’) stands for the numerical results.
Fig 8.
The behavior of the variances of the cell population described by Eq (6) with ϵ = 0.005 and time steps = 2 ⋅ 106, for different values of S.
The analytical results given by Eqs (7–10) (solid line) are compared with the values obtained by numerical simulations (stars). (‘T’) stands for the theoretical results, and (‘N’) stands for the numerical results.
Fig 9.
The behavior of the two systems described by Eqs (1) and (6) with S = 0.1 and S = 1.
The top two diagrams, (a) and (b)m correspond to the first example, and the bottom two diagrams, (c) and (d), correspond to the second example. In Panels (a) and (c), S = 0.1 (mostly asymmetric divisions). In (b) and (d), S = 1 (symmetric divisions).
Fig 10.
Modeling micro-injuries by introducing a certain percent (1%, or 10%, or 20%) decrease in the number of differentiated cells in 0.01% of temporal updates (randomly chosen).
In (a,b) the numbers of SCs and differentiated cells is plotted as functions of time, for a typical run. In (a), most divisions are asymmetric (S = 0.1), and in (b), all divisions are symmetric (S = 1). (c,d) The variance and the relative standard deviation of the number of differentiated cells as a function of the percentage of symmetric divisions. The percent decrease of the number of differentiated cells is marked above each line. Eq (6) with ϵ = 0.002 was used.
Fig 11.
Modeling the possible role of hair follicles in epidermis turnover.
The numbers of differentiated and SCs are plotted as functions of time, for a typical run. (a) S = 0.1, in the presence of follicles; (b) S = 1, in the presence of follicles; (c) S = 0.1, in the absence of follicles; (d) S = 1, in the absence of follicles. Eqs (11 and 12) were used with parameters h = 0.3, ϵ = 0.05.
Fig 12.
The role of hair follicles in epidermis turnover.
The relative standard deviation of the number of differentiated cells () is plotted against the fraction of symmetric divisions, in the absence and in the presence of hair follicles. Parameters are as in Fig 11.