Figure 1.
Schematic representation of our model.
(A) Stochastic model for stem cell division. A stem cell can produce zero, one, or two stem cells with probabilities ,
, and
, respectively. The mean number of offspring is given by
. (B) Stem cells serve as an input to the proliferating progenitor population, and the progenitor population feeds back to the stem cell pool via dedifferentiation. (C) Mutation occurs with rate
during division and can affect cells both in the stem cell and progenitor pools. Blue circles represent wild-type cells, red circles cells with one mutation, and green circles cells with two mutations. (D) The sequence of models explored in this paper.
Table 1.
Parameter values used in numerical simulations.
Figure 2.
Steady-state progenitor distributions in the absence of stem cell mutation.
(A, B) Fraction of two-mutation cells as a function of mutation rate and proliferative advantage
for (A) unlimited growth, (B) logistic growth for each subpopulation. (C, D): Corresponding total population sizes for (C) unlimited growth, (D) logistic growth. Birth/death rates of progenitor cells are given by Eq. (5) with constant death rate
and sigmoidal birth rate with maximal growth rate
,
for
. In (B) the carrying capacity used is
Other parameters are as in Table 1. For two-mutation cells to reach appreciable levels in this scenario, both the mutation rate and the proliferative advantage must be unreasonably large.
Figure 3.
Times to fixation without dedifferentiation.
(A) Typical simulation trajectory for constant stem cell population size and mutation rate . The numbers of zero-, one-, and two-mutation stem cells are shown in blue, red, and green, respectively. The proportion of two-mutation cells in the progenitor population is shown in black. (B) Times to fixation for constant and variable stem cell population size models. Histograms of waiting times to fixation of two-mutation cells for constant (blue) and variable stem cell population size with high fluctuations (green,
) and low fluctuations (red,
). The semi-analytic distribution of waiting times calculated from Eq. (12) is shown in black. In both panels the mutation rate
.
Figure 4.
Times to fixation with dedifferentiation for constant stem cell population size.
(A) Typical simulation trajectory with dedifferentiation () for the same random number seed as Fig. 3A. Blue: zero-mutation stem cells, Red: one-mutation stem cells, Green: two-mutation stem cells. Black: proportion of two-mutation cells in the progenitor population. (B) Distributions of times to fixation of two-mutation cells under strict stem cell homeostasis. Normalized histograms (dots) and analytical approximations (solid lines) are shown for
for zero dedifferentiation (red;
) and non-zero dedifferentiation (black;
). (C) Median times to fixation of two-mutation cells (solid lines) and inter-quantile ranges (shaded regions) versus dedifferentiation rate
and mutation rate
. (D) Mean times to emergence of a successful two-mutation stem cell (solid lines, Eq. (9)) and fixation of such cells (dotted lines, Eq. (12)) in Moran models with selection coefficient
. Black curve indicates first appearance of two-mutant cell.
Figure 5.
Exponential growth given varying stem cell population size and dedifferentiation.
Total number of stem cells (A, C, E) and corresponding final progenitor age distributions (B, D, F) are shown in black. Also shown are best exponential fits of the growth rate (blue) and our semi-analytic solution given by Eq. (18) (green). In all panels the probability of asymmetric stem cell division and the mean reversion parameter is
.
Figure 6.
Fixation and exponential growth of two-mutation cells with dedifferentiation for variable stem cell population size.
A: Observed growth rate of the stem cell population (black curve) and the semi-analytic approximation Eq. (18) (green) for
,
, and
. The vertical line denotes
. B: Analytically predicted critical dedifferentiation rate
as a function of asymmetric division probability
and the growth advantage
of the two-mutation progenitor population. Exponential growth occurs for
. C: Normalized histogram (stars) of waiting times for exponential growth of the stem cell population with stochastic homeostasis and dedifferentiation for
,
. For comparison the histogram (red and black dots) as well as the analytical distributions of times to fixation given strict homeostasis for
and
are also shown. D: The median and inter-quantile range of times to first occurrence of
two-mutation stem cells, given stochastic homeostasis and a range of dedifferentiation rates
. For comparison, the waiting times to fixation for
given strict homeostasis (shaded areas) for the equivalent value of
are also shown. E: The probability that the first two-mutation stem cell arose from mutation in the stem cell compartment, rather than dedifferentiation. Vertical line denotes
. Parameters for all simulations given in Table 1.