Figure 1.
Symmetric and asymmetric stem cell divisions.
In the asymmetric division model, a stem cell produces one differentiated cell and one stem cell. In the symmetric division model, a stem cell produces two differentiated cells or two stem cells.
Table 1.
Model parameters.
Figure 2.
The six different approximation regimes (Table 2) for solutions of system (2–3).
Plotted is the quantity (a) and (b)
as a function of the frequency of symmetric divisions,
, for three different values of
(solid lines), together with the approximations given by the formulas in Table 2. Approximations
,
, and
are best demonstrated in panel (a), where the quantity
is plotted. Approximations
,
, and
are best demonstrated in panel (b), where the quantity
is plotted. The other parameters are
,
.
Table 2.
Important limiting cases for the tunneling rate (formula (1)).
Figure 3.
The reduction in the rate of double mutant production in stem cells with symmetric divisions compared to stem cells with asymmetric divisions only.
Plotted is the quantity in formula (4) as a function of the mutation rate, . The percentage of the stem cells in the whole population (
) is marked next to the lines. The other parameters are
,
.
Figure 4.
The probability of double-hit mutant generation as a function of , the probability of symmetric stem cell divisions.
The results of numerical simulations are presented as points connected with dotted lines (standard deviations are included). Analytical results are given by solid lines (formula (11). The horizontal line represents the calculations for the homogeneous model. We ran batches of
runs. The parameters are
,
,
,
.
Figure 5.
The probability of double-mutant generation as a function of , the ratio of TA cells to the total number of cells.
As in Figure 4, the results of numerical simulations are presented as points connected with dotted lines (standard deviations are included), and the analytical results are given by solid lines (formula (11)). The horizontal lines represent the calculations for the homogeneous model. We ran batches of
runs. Plotted is the probability of double-mutant generation as a function of
, for purely symmetric (
) and purely asymmetric (
) models, for three different values of
. The parameters are
.
Figure 6.
The probability of double-hit mutant generation in the symmetric division model.
The case of symmetrically dividing stem cells, same as in Figure 5.
Figure 7.
The threshold fraction of stem cells corresponding to stem and TA cells contributing equally to double-hit mutant production.
The quantity , which corresponds to
, is plotted as a function of the mutation rate,
, for three different values of
, and
. For the fraction of stem cells above these values, stem cells have a higher contribution to the rate of double-mutant production compared to the non-stem cells. Thin dashed lines show the approximations of equation (8).
Figure 8.
The immortal DNA strand hypothesis.
The probability of double-hit mutant generation is calculated for a particular set of parameters as a function of (the probability of symmetric divisions), according to formula (21. For
the minimum corresponds to at
(asymmetric divisions only), for
and
we have an intermediate minimum at
and
respectively, and for higher values of
the minimum is reached for
(symmetric divisions). Here,
,
,
,
,
, and the parameter
varies from
to
in increments of
.
Figure 9.
Why are symmetrically dividing stem cells produce mutants slower?
The weight of a typical symmetrically dividing mutant stem cell lineage, , relative to the weight of an asymmetrically dividing mutant stem cell lineage,
, is plotted as a function of the number of stem cell divisions,
. Here,
,
, and 20 batches of
simulations were performed to calculate the mean and the standard deviation.
Figure 10.
Stem cell division decision trees for the numerical algorithm.
(a) Divisions of wild-type stem cells. (b) Divisions of mutant stem cells. Stem cells are denoted by light circles with an “S” and TA cells by shaded circles with a “D”. One-hit mutants are marked with a star.