Fig 1.
Schematics showing key concepts of the paper.
(a) Two types of symmetric divisions: self-renewals and differentiations. Each circle represents a cell, and i denotes the ith compartment, while i + 1 denotes the (i + 1)th compartment. Panels (b) and (c) demonstrate the division chains that replenish 8 differentiated cells eliminated from the top compartment. Dead cells are denoted by X’s and arrows show divisions. Cells are arranged in horizontal layers corresponding to compartments. Only the dividing cells are shown (for example, there may be more than 4 cells in the second to top compartment in panel (b)). In (b), the dead cells are replaced by a longer division tree, and in (c) by four shorter division trees.
Fig 2.
Properties of division trees as a function of v, the probability of self-renewal in compartments 1, …, n − 1.
(a) Mean tree length, defined as the mean number of compartments involved in the division trees. (b) Mean division position. For an individual tree, sequence (1), this is defined as ; plotted is the expectation of this quantity across all the trees. In this example, n = 4.
Fig 3.
Comparison between the average number of mutants from 100 stochastic simulations without replacement and the number of mutants predicted by the ODE approximation.
(a) Comparing the number of mutants for a high value of the proliferation probability, v = 0.9. (b) Comparing the number of mutants when the proliferation probability is small, v = 0.1. Solid lines correspond to the number of mutants in compartment Ci, i = 1, …, 3, predicted by the ODE approximation and dashed lines correspond to the mean number of mutants in compartment Ci from the stochastic simulations. We assume n = 3, a mutation rate of u = 10−3, and the compartment sizes are N0 = 1063, N1 = 104, N2 = 105, N3 = 106.
Fig 4.
The role of the self-renewal rate on mutant generation and mutant dynamics (the analytical approach).
(a) The probability of generating a mutant in each of the compartments for 6 different values of v: v = 0, 0.05, 0.1, 0.4, 0.7, 1. (b) The expected number of mutants produced from a single mutant cell in the absence of further de-novo mutations, plotted as a function of time for three different values of v. (c,d) The expected dynamics of mutants generated in different compartments at t = 0, in the absence of new mutations, for (c) v = 0.1 and (d) v = 0.5. Other parameters are: n = 4, and the compartment sizes are, from C0 to Cn: 40, 80, 120, 160, 200.
Fig 5.
Mean number of mutants from 1000 stochastic simulations.
We compare two arrangements of the compartment sizes for n = 3: constant from C0 to C3 (N0 = 65, N1 = 65, N2 = 65, N3 = 65) and increasing from C0 to C3 (N0 = 20, N1 = 40, N2 = 80, N3 = 120). (a) The mean number of mutants produced in compartments C1, C2 and C3 for v = 0.9. Note that no mutants were produced in compartment C0 over the time scale shown. (b) The mean number of mutants produced in compartments C0, C1, C2 and C3 for v = 0.1. (c) The mean of the total number of mutants comparing both architectures for v = 0.9 (blue line) and v = 0.1 (green line). For all panels, solid lines correspond to constant architecture and dashed lines to increasing architecture. In these simulations u = 0.001.
Fig 6.
Distribution of the generation times to second mutation from 5000 stochastic simulations.
We consider two arrangements of the compartment sizes for n = 3: constant from C0 to C3 (N0 = 65, N1 = 65, N2 = 65, N3 = 65) and increasing from C0 to C3 (N0 = 20, N1 = 40, N2 = 80, N3 = 120). (a) The time to observe a second mutant for both architectures and a high value of the self-renewal probability, v = 0.9. The mean time for constant and increasing architectures is 3.33 and 3.39 respectively; the p-value obtained by two-tailed t-test is p = 0.078, indicating that the means are different only at the 10% level (size effect is 0.04). (b) The time to observe a second mutant for both architectures and a small value of the self-renewal probability, v = 0.1. The mean time for constant and increasing architectures is 3.67 and 3.60 respectively; the p-value is p ≪ 0.001, indicating that the means are different; the effect size is 0.16. (c) and (d) The time to observe a second mutant for a constant and increasing architecture, respectively, for small and high v. In these simulations u = 0.001, and the effect size is 0.68 and 0.46 respectively.
Fig 7.
Distribution of the generation times to second mutation, obtained from stochastic simulations, in the case where v depends on the compartment sizes.
We perform 5000 stochastic simulations and compare two arrangements of the compartment sizes: (a) constant compartment size (red bars) and increasing (blue bars); (b) decreasing (red bars) and increasing (blue bars). Compartment sizes are as in Fig 6. The self-renewal probabilities are v1 = 1, v3 = 0, , i = 1, 2. We assume n = 3 and u = 0.001. In part (a), the mean time to a two-hit mutant for constant and increasing architecture is 3.65 and 3.59 respectively; the p value obtained by a two-tailed t-test is p ≪ 0.001, the effect size is 0.12. In part (b), the mean time to a two-hit mutant for decreasing architecture is 3.73; p ≪ 0.001; the effect size is 0.30.