Intestinal crypt properties fit a model that incorporates replicative ageing and deep and proximate stem cells - PubMed (original) (raw)
Intestinal crypt properties fit a model that incorporates replicative ageing and deep and proximate stem cells
P N Lobachevsky et al. Cell Prolif. 2006 Oct.
Abstract
A model of intestinal crypt organization is suggested based on the assumption that stem cells have a finite replicative life span. The model assumes the existence in a crypt of a quiescent ('deep') stem cell and a few more actively cycling ('proximate') stem cells. Monte Carlo computer simulation of published intestinal crypt mutagenesis data is used to test the model. The results of the simulation indicate that stabilization of the crypt mutant phenotype following treatment with external mutagen is consistent with a stem cell replicative life span of about 40 divisions for mouse colon and 90-100 divisions for mouse small intestine, corresponding to a deep stem cell cycle time of about 3.9 and 8.5 weeks for colon and small intestine, respectively. Simulation of the data obtained for human colorectal crypts suggests that the proximate stem cell cycle time is about 80 h, assuming a replicative life span of 50-150 divisions, and that the deep stem cell divides approximately every 30 weeks.
Figures
Figure 1
A schematic representation of the model assumptions. The intestinal crypt is populated by one deep stem cell and a few (four in the diagram) proximate stem cells. The deep stem cell undergoes deterministic divisions that result in self‐renewal and the production of a proximate stem cell that fills a vacancy in the proliferative niche. Proximate stem cells normally undergo asymmetric divisions that result in self‐renewal and the production of a transit‐amplifying cell. In certain circumstances, they divide symmetrically, resulting in the production of two proximate stem cells, one of which fills a vacancy. A vacancy is formed when a proximate stem cell approaches the end of its replicative life span and leaves the niche (i.e. undergoes apoptosis or divides symmetrically to produce two transit‐amplifying cells). The counter on each stem cell shows its replicative age and represents the assumption that each stem cell is restricted to a finite number of divisions.
Figure 2
The kinetics of appearance of partially (open symbol) and wholly (closed symbol) mutant crypts following treatment with mutagen. Experimental data from Park et al. (1995) are for mouse small intestine (a) and colon (b). Mice were injected with ethyl nitrosourea at 6 weeks. The lines represent the results of simulation (dashed line is partially mutant crypts and solid line is wholly mutant crypts). The values of the model parameters used for the simulation were: proximate stem cell cycle time (T p) = 18 h; number of proximate stem cells (N p) = 4; and the limiting number of divisions (M 1) = 46–138 for small intestine and 20–60 for colon. The simulated values of the clonal half‐stabilization time (T 50) = 5.75 weeks for small intestine and 3.44 weeks for colon; the deep stem cell cycle time (T d) = 7.61 weeks for small intestine and 3.86 weeks for colon.
Figure 3
The relationship between stem cell parameters and the kinetics of appearance of wholly mutant crypts. T d is the deep stem cell cycle time, T c is the time between deep stem cell division and crypt conversion, and T 50 is the time after mutagen treatment at which the number of wholly mutant crypts reaches 50% of the maximum value.
Figure A1
The probabilities that a proximate stem cell leaves the niche and forms a vacancy (P_v(M_1, k_1, N_div)) (solid line) or divides symmetrically, renewing itself and filling the vacancy (Ps(M_2, k_2_, N_div)) (dashed line), as a function of the number of divisions it has completed (_N_div). Probabilities have been calculated for the following parameter values: M 1 = 60, M 2 = 20, k 1 = k 2 = 10.0. The probability that a stem cell divides symmetrically producing two daughter stem cells is also conditional upon the existence of a vacancy in the niche. The dash‐dot and dot lines represent the P v and P s probabilities, respectively, for the case of a threshold dependency, which is the limiting case for increasing k 1 and k 2.
Figure A2
Graphic representation of the solution of Equation A9 for mouse colon (bottom lines, _T_50 = 3.53 weeks) and small intestine (top lines, _T_50 = 5.91 weeks). The solution is obtained assuming the cycle time of stem cells (T p) = 16 h, and the number of proximate stem cells (N p) = 4 (solid lines), 8 (dot lines) or 16 (dash lines).
Figure A3
Graphic representation of the solution of (A9), (A10) for mouse colon (solid and dash‐dot lines, _T_50 = 3.53 weeks, _T_w = 5.01 weeks) and small intestine (dash and dot lines, _T_50 = 5.91 weeks, _T_w = 12.2 weeks). Solid and dash lines show values of T d calculated for various M 2 values according to expression A7, assuming the number of proximate stem cells (N p) = 4 and the cycle time of stem cells (T p) = 16 weeks. Dash‐dot and dot lines represent experimental values of T w (Table 1).
Figure 4
The kinetics of appearance of partially (open symbol) and wholly (closed symbol) mutant crypts in mouse small intestine following treatment with mutagen. Experimental data are from Winton & Ponder (1990). The wholly mutant crypt frequency includes ‘Paneth‐cell crypts’. The lines represent the results of simulation (dashed line is partially mutant crypts and solid line is wholly mutant crypts). The values of the model parameters used for the simulation were: proximate stem cell cycle time (T p) = 18 h; number of proximate stem cells (N p) = 4; mutagen treatment time (T m) = 6 weeks; and the limiting number of divisions (M 1) = 55–165. The simulated values of the clonal half‐stabilization time (T 50) = 6.70 weeks, and the deep stem cell cycle time (T d) = 9.56 weeks.
Figure 5
The kinetics of appearance of partially (open symbol) and wholly (closed symbol) mutant crypts in human colorectal specimens at various time intervals after therapeutic irradiation. Data are from Campbell et al. (1996). The lines represent the results of simulation (dashed line is partially mutant crypts and solid line is wholly mutant crypts). The values of the model parameters used for the simulation were: proximate stem cell cycle time (T p) = 80 h; number of proximate stem cells (N p) = 4; mutagen treatment time (T m) = 50 weeks; and the limiting number of divisions (M 1) = 50–150. The simulated values of the clonal half‐stabilization time (T 50) = 30.8 weeks, and the deep stem cell cycle time (T d) = 25.3 weeks.
Figure A4
Graphic representation of the solution of Equation 7 (_T_50= 21 weeks, _Np_= 4). The relationship between the cycle time and division limit of proximate stem cells, as predicted by the model, based on the human colorectal data of Campbell et al. (1996).
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