Figure 6.
Control of final state systems. (A, B) Shown are populations of terminal cells over time for renewal control (A) and fate control (B) for a two-stage lineage. The inset shows the corresponding stem cell populations. Large feedback gains (n) do not significantly alter development time. The initial population consists of 10 stem cells of type x 1 and no other cell types. (C, D) Shown is the error (deviation from desired final concentration divided by the desired final concentration) in development as a function of n in response to constant loss rates ρ = 0.1 and ρ = 0.2 of stem cells for the renewal control (C) and fate control (D). For renewal control, the benefits of feedback are modest. For fate control, sufficiently high gain can suppress the effects of stem cell loss nearly completely. (E, F) Shown are populations of the different terminal cell types x 2, x 4, x 6 in the multi-layered topology. Solid lines show population trajectories when there is no disturbance, and the dashed lines the trajectories when there is loss of all stem-like cells x 1, x 3, x 5 at rate ρ = 0.1. For renewal control (E), stem cell loss does not affect development time, but it results in smaller concentration of terminal cells (smaller tissue/organ) at every feedback level. For fate control (F), stem cell loss causes a delay in development time, but for high feedback gain the concentration of terminal cells is preserved. For panels A and C, p r = 1/(0.5 + 0.5(x 2/b n)n), b n = 23859, 25726, 27816, 29505, 29867, 29971, and 29985 for n = 1, 2, 4, 10, 20, 50, and 100 respectively. For panels B and D, p r = 1/(0.5 + 0.5(x 2/b n)n), p d = 0.5p r, b n = 29990, 34636, 40121, 47206, 51526, 55462, and 57294 for n = 1, 2, 4, 10, 20, 50, and 100 respectively. For parameter values used in panels E and F see Additional file 1: Table S2.