Dynamics of crypts with or without BMP. (A) With BMP, starting from a uniform distribution of progenitor cells localized in a small local area of the tissue (T = 0), a single crypt is formed near the steady state (T = 100) (μ1 = 2.5 × 10−4s−1μM). (B) The corresponding patterns of Wnt and BMP near the steady state in the crypt system described in (A). (C) If BMP is removed, starting from the single crypt (T = 100 in (A)), more small crypts are generated (T = 40) and eventually give rise to five identical crypts (T = 100). (D) Dynamics of crypt multiplication using a different energy function E without BMP (see Eq. S1.5 in Additional file 1). (E) A phase diagram of crypts in terms of γ
the strengths of feedback for Wnt and BMP, respectively. “Crypt extinction” corresponds to zero density of progenitor cells at the steady state; “Stable crypt number” means the number of crypts at the steady state is equal to the number of the initially localized spots of progenitor cells (e.g. se T = 100 in (A)); “Crypt multiplication” stands for the number of steady crypts increases from “stable crypt number” (e.g. T = 100 in (C) and (D)); “Unstable progenitor pool” represents the region where the density of progenitor cells in multiple crypts increases without bounds. Other parameters used are listed in Additional file 1: Table S2.