Fig. 1
a Pictorial description of the model involving neoblasts (N), non-mitotic progenitor cells (P), three classes of differentiated cells (D i ), and a resource pool. The maximum rates of asymmetric renewal (N→N, N), symmetric differentiation (N→N, P), and asymmetric differentiation (N→P, P) are p 1,p 2, and p 3 respectively. As explained in the Section “Methods”, a steady state exists only if p 1<p 3. Progenitor cells either complete differentiation or return to the resource pool. In the absence of external injury or death fully differentiated cells apoptose and return to the resource pool. These transitions have a maximum rate that is modified by feedback control. b Positive feedback controls superimposed upon the transitions. Neoblasts exert positive feedback f N (N) on the N→P, P transition in the sense that as the number of neoblasts increases, f N (N) increases, bounded by 0 and 1. The resource pool Q also has positive feedback on all transitions, in the sense that with larger resource pools the rate of transition is higher. However, we assume that the resource pool operates differentially on the different transitions, so use a 1(N,D,Q),a 2(N,D,Q),a 3(N,D,Q) where D is the vector (D 1,D 2,D 3) to indicate the feedback control of the resource pool on the transitions N→N,N,N→N, P, and N→P, P respectively. c Differentiated cells exert negative feedback control on the transitions, in the sense that as the number of differentiated cells increases, the rates of transitions decline, sharing a common feedback control f D (D). Here we assume the that the differentiated cell in shortest supply sets the feedback control. Absent an external source of mortality, the only transition for differentiated cells is D i →Q through cell death, which occurs for cell type i at rate M i (N,D,Q). In addition, progenitor cells may either fully differentiate or return to the resource pool through apoptosis. We assume that the rate of the former is determined by a function f Q (Q) that increases as the size of the resource pool increases