Figure 1.
SC Replication Can Give Rise to Daughter Cells That Retain the Properties of the SCs or Become Committed to Differentiate
If the two daughter cells share the same fate, the preceding division is considered to be symmetric; while if one daughter cell retains SC capabilities and the other differentiates, the preceding mitotic event is considered to be asymmetric. Normal SCs are in blue and differentiated cells are depicted in red.
Figure 2.
Representation of the Stochastic Model of Mitotic SC Replication
The model is based on the Moran process, which maintains a constant population size (A) and in which cells are chosen for reproduction proportional to their fitness.
(B) Normal (blue) and mutant (red) SCs may divide asymmetrically with probability p, while they divide symmetrically with probability q to produce two differentiated cells. SCs also can divide symmetrically to produce two daughter SCs with probability 1 − p − q.
Figure 3.
(A) In the absence of mutant SCs, normal SC homeostasis is maintained in the stochastic process, and the abundance of normal SCs, n = 1,000, remains constant. Transition probabilities are pb = qb = 1/3.
(B) If a neutral mutant SC appears (i.e., a cell with the same reproductive fitness as a normal SC), random drift may allow the population to expand and take over the SC compartment (red) (pa = pb = qa = qb = 1/3, a0 = 1, b0 = 999, r = 1, averaged over 1,000 runs and the percentage plotted). As the probability for self-renewal of the mutant SC increases, a higher fraction of compartments are taken over by the mutant SC (green, pa = qa = 0.3; magenta, pa = qa = 0.25; black, pa = qa = 0.2; blue, pa = qa = 0.1; yellow, pa = qa = 0).
(C) A mutation that increases both the probability of self-renewal and the fitness is fixed in a large fraction of runs (red, pa = pb = 0.25, qa = qb = 1/3, a0 = 1, b0 = 999, r = 1.1; blue, pa = pb = 0.15, r = 1.3 averaged over 1,000 runs and the percentage plotted).
Figure 4.
The Impact of Compartment Size n on the Average Time to Fixation of a Mutant Cell Based on Equation 9
(A) As the compartment size increases, the average time to fixation increases. Parameter values are n = 10 (black), n = 50 (red), n = 100 (green), n = 200 (blue), and n = 1,000 (cyan). For all curves pa = qa = 0.25; pb = qb = 1/3; r = 1.02.
(B) The reproductive fitness of the mutant SCs compared with the normal SC pool plays a determining role in the evolutionary trajectory of the mutant population. A mutant with a low reproductive fitness (black) (pa = pb = 0.3; qa = qb = 1/3, r = 0.8, n = 1,000) has a low prevalence in the population, while the prevalence increases as the fitness increases (red, r = 1.0; green, r = 1.2; blue, r = 1.4; cyan, r = 1.6; yellow, r = 1.8). If the mutant cells have a higher fitness and increased probability for self-renewal, the probability of fixation is even higher. Each result is the summary of 1,000 simulations and the percentage plotted.
Figure 5.
Average Fixation Time of Mutant SC
The average fixation time of mutant SC depends on the relative fitness (A) (red, r = 1.02; blue, r = 1.05), population size (B), and probability of self-renewal (C) (Equation 9). The fixation time for a neutral mutant is always the longest. Although mutants with reduced fitness can reach fixation quickly, this is an improbable event. In all figures, n = 10 (red), n = 50 (green), n = 100 (blue), and n = 400 (black).