Figure 8

Exogenous Wnt added at a localized spatial region. (A) The exogenous Wnt is added at a fixed spatial region of a stable single crypt along the crypt direction $[p-\frac{\omega }{2},p+\frac{\omega }{2}]$ with a constant rate θ. The parameters used for the wild-type single crypt are listed in Additional file 1: Table S2. (B) A new stable pattern of a single crypt at a low level of exogenous Wnt: θ = 4 × 10⁻⁴s⁻¹μM. (C) A new stable pattern of two-crypt pattern at a high level of exogenous Wnt: θ = 4 × 10⁻³s⁻¹μM. (D) Removing the exogenous Wnt (i.e. θ = 0s⁻¹μM) in steady state of (C) leads to formation of two new stable crypts. (E) Wnt patterns at the steady states for (A), (B), (C), and (D). The other parameters are p = 0.75L_max, ω = 0.05L_max (L_max is the crypt length in (A)). The temporal dynamics of the three cases (B), (C), (D) are provided in Additional file 1: Figure S6