A graphical representation of stability conditions(4–5).

<p>For fixed values of controls and , we identify the region of the space corresponding to stability of the stem cell system. The borders of this region are given by lines and . (a) Negative division controls: , . (b) Positive division controls: , . The parameter .</p

Stability and robustness of two-compartment control systems.

<p>(a) For a wide range of positive and negative controls of the division, and , we show how robust the stability of the system is with respect to the choice of the controls of differentiation. The maximum robustness is (meaning that if we choose controls of differentiation and randomly, with probability we will get a stable solution). This corresponds to the lightest region in the south-west part of the diagram. The minimum robustness is zero, such that no choice of controls of differentiation will yield a stable solution. This corresponds to the darkest region in north-east part of the diagram. (b) Under the assumption of the connection between division and differentiation decisions, the same contour-plot is shown with the restriction , . The parameter .</p

Minimal regulatory networks for a three-compartment system with three control loops.

<p>(a) Left: the 7 networks containing (modifications of) the first pattern of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0072847#pone-0072847-g002" target="_blank">figure 2(a)</a>. Right: the 7 networks containing (modifications of) the second pattern of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0072847#pone-0072847-g002" target="_blank">figure 2(a)</a>. (b) The remaining 6 networks. Notations are similar to those of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0072847#pone-0072847-g002" target="_blank">figure 2</a>, with “I” denoting the intermediate cell type.</p

The threshold fraction of stem cells corresponding to stem and TA cells contributing equally to double-hit mutant production.

<p>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 <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076195#pone.0076195.e165" target="_blank">equation (8)</a>.</p

The reduction in the rate of double mutant production in stem cells with symmetric divisions compared to stem cells with asymmetric divisions only.

<p>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 , .</p

The immortal DNA strand hypothesis.

<p>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 .</p

Important limiting cases for the tunneling rate (formula (1)).

<p>The notations for the six different regimes refer to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076195#pone-0076195-g002" target="_blank">Figure 2</a>.</p

The probability of double-hit mutant generation in the symmetric division model.

<p>The case of symmetrically dividing stem cells, same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076195#pone-0076195-g005" target="_blank">Figure 5</a>.</p

Symmetric and asymmetric stem cell divisions.

<p>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.</p

Model parameters.

<p>Notations used in the text and their brief description.</p