Mathematical models of hierarchically structured cell populations under equilibrium with application to the epidermis

Abstract

There are three categories of keratinocytes in the germinative compartment of the epidermis - stem, transit-amplifying and post-mitotic. Their population structure is hierarchical. This means that stem cells differentiate into transit-amplifying cells which, after a few rounds of division, become post-mitotic cells. The cell processes of birth, differentiation, death and migration affect the composition and proliferation rate of the germinative compartment. These phenomena are quantified by various cell kinetic parameters. In this paper we derive equations that relate these parameters for different models of hierarchically structured cell populations in equilibrium. We include in the models asymmetric and symmetric division, variations in cell-cycle times, apoptosis and variation in the number of transit generations. We conclude that variation in cell-cycle times need only be considered if apoptosis is not negligible. If it is negligible, then only average cell-cycle times are needed. Unfortunately, it is impossible to predict the importance of apoptosis from the available experimental data. However, the strength of its effect is determined by the other parameters, especially the fraction of cycling stem cells. We show that variation in the number of transit generations can have a potentially large effect on cell birth rate. We also show that cell birth rate does not directly depend on the mean transit-amplifying cell-cycle time, only on the mean stem cell-cycle time. We argue that 'homogeneous cell population' equations should not be used to study hierarchical cell populations as has been done in the past. Finally we argue that stem cell parameters and transit-amplifying cell parameters should not be lumped together.

Figure 1

Proposed hierarchical population structures for the germinative compartment of the epidermis. On average, each stem cell division produces another stem cell (s) and a transit‐amplifying cell (t). Transit‐amplifying cells can either undergo asymmetric (a) or symmetric (b) division for a limited number of divisions. Transit‐amplifying cells finally become post‐mitotic cells (p) that migrate from the germinative compartment.

Figure 2

An example density function of stem cell age ( s ( a ), equation 3) . The cell‐cycle times are gamma distributed with a mean of 16 h and a standard deviation of 4 h. The stem cell age distribution ( s ( a )δ a ) is the density of cells with ages between a and a  + δ a , where δ a is a small amount of age. The area under the curve gives the density of cycling stem cells f / N _s , which in this case has been normalized to one.

Figure 3

An example showing the calculation of Q _i from the frequencies of all possible transit‐amplifying cell lineages for asymmetric and symmetric division and g  = 3. Note that l _A  +  l _B  +  l _C  = 1 and l _A  +  l _B  +  l _C  +  l _D  = 1 for asymmetric and symmetric division, respectively. Q _i is calculated by summing, over all lineages, those i th generation cells that give rise to two post‐mitotic cells and dividing by the total number of i th generation cells.

Figure 4

The effect of apoptosis on the cell birth rates per unit area of skin surface. The curves are the ratios of birth rate with apoptosis to birth rate without apoptosis. There is little difference between asymmetric and symmetric division, compare (a) with (b) and (c) with (d). (a,b) T̄ _s  = T̄ _t  = 100 h and (c,d) T̄ _s  = 200 h and T̄ _t  = 20 h, with β _s  = β _t  = β’ g  = 3, φ _s  = φ _t  = 0.1 and gamma distributed cell‐cycle times. The two vertical lines represent the apoptotic rates in normal (right) and psoriatic (left) skin.

Figure 5

The effect of variation in the number of transit generations on cell birth rate per unit area of skin surface for a maximum of three generations, without apoptosis and with f  = 1, N _s  = 4000 cells mm ⁻² and T̄ = 100h for (a) asymmetric and (b) symmetric division. Birth rate is given by fN _s /T̄ _s  = (4 − 2 Q ₁  −  Q ₂  +  Q ₁ Q ₂ ) and fN _s /T̄ _s  = (8 − 6 Q ₁  − 4 Q ₂  + 4 Q ₁ Q ₂ )for asymmetric and symmetric division, respectively. When g  = 1 or g  = 2 the birth rate can also be read off from these graphs by setting Q ₁  =  Q ₂ = 1 or Q ₂  = 1, respectively. Also shown are points on the planes that correspond to specific values of the lineage frequencies from Fig. 4, i.e. l _A  = 1, l _B  = 1, l _C  = 1, l _D  = 1 and l _A  =  l _B  =  l _C  =  l  = 1/3 for asymmetric division and l _A  =  l _B  =  l _C  =  l _D  =  l  = 1/4 for symmetric division.

Figure 6

The fraction of cycling stem cells versus the ratio of the mean transit‐amplifying to the mean stem cell‐cycle times for different growth fractions for (a) asymmetric ( equation 75) and (b) symmetric ( equation 76) division. It is commonly thought that stem cells should have a longer cell‐cycle time than transit‐amplifying cells.