A reaction-diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts - PubMed (original) (raw)

A reaction-diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts

Lei Zhang et al. BMC Syst Biol. 2012.

Abstract

Background: Colon crypts, a single sheet of epithelia cells, consist of a periodic pattern of stem cells, transit-amplifying cells, and terminally differentiated cells that constantly renew and turnover. Experimental evidence suggests that Wnt signaling promotes and regulates stem cell division, differentiation, and possible cell migrations while intestinal BMP signaling inhibits stem cell self-renewal and repression in crypt formation. As more molecular details on Wnt and BMP in crypts are being discovered, little is still known about how complex interactions among Wnt, BMP, and different types of cells, and surrounding environments may lead to de novo formation of multiple crypts or how such interactions affect regeneration and stability of crypts.

Results: We present a mathematical model that contains Wnt and BMP, a cell lineage, and their feedback regulations to study formation, regeneration, and stability of multiple crypts. The computational explorations and linear stability analysis of the model suggest a reaction-diffusion mechanism, which exhibits a short-range activation of Wnt plus a long-range inhibition with modulation of BMP signals in a growing tissue of cell lineage, can account for spontaneous formation of multiple crypts with the spatial and temporal pattern observed in experiments. Through this mechanism, the model can recapitulate some distinctive and important experimental findings such as crypt regeneration and crypt multiplication. BMP is important in maintaining stability of crypts and loss of BMP usually leads to crypt multiplication with a fingering pattern.

Conclusions: The study provides a mechanism for de novo formation of multiple intestinal crypts and demonstrates a synergetic role of Wnt and BMP in regeneration and stability of intestinal crypts. The proposed model presents a robust framework for studying spatial and temporal dynamics of cell lineages in growing tissues driven by multiple signaling molecules.

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Figures

Figure 1

A schematic diagram of a two-stage cell lineage model. (A) A cartoon of one colonic crypt and the relative location of cells at different lineage stage adapted from [7,24]. (B) A diagram of an un-branched cell lineage that starts with progenitor cells, regarded as a combination of stem cells and transient amplifying cells, leading to terminally differentiated (TD) cells that may undergo death. Replication of progenitor cells may be enhanced by Wnt signaling and be inhibited by BMP signaling. Wnt and BMP are produced by progenitor cells and TD cells, respectively. Wnt has an autoregulation feedback and produces Wnt inhibitor which in turn represses Wnt.

Figure 2

Spatial distributions of progenitor cells and the crypt growth are driven by a Turing pattern of Wnt signaling. (A) Dynamics of the crypt growth colored in progenitor cell density at three different times. The solution is close to the steady state at T = 100. (B) Concentrations of Wnt, Wnt Inhibitor, BMP, progenitor cells, and TD cells along the crypt direction of s near the steady state at T = 100 (Wnt – magenta solid line, Wnt Inhibitor – black dash line). (C) Time evolution of the progenitor cells. Parameters used are listed in Additional file 1: Table S1 and Table S2.

Figure 3

The number of crypts at the steady state depends on removal rate of Wnt. (A) The number of crypts at steady state as the removal rate of Wnt is varied for two different initial distributions of Wnt (blue dash line and red solid line). For example, at the same Wnt removal rate of d W = 0.02_s_−1, there are five crypts (B) or six crypts (C) at steady state starting from two different initial conditions (see Additional file 1: Figure S2 for initial conditions). Other parameters used for the simulation are in Additional file 1: Table S2.

Figure 4

Maximal density and total amount of progenitor cells on a crypt as functions of removal rate of BMP or maximal death rate of TD cells. (A) Blue: removal rate of BMP versus maximal density of progenitor cells (max (_C_0)); green: removal rate of BMP versus total amount of progenitor cells. (B) Blue: maximal death rate of TD cells (d¯1) versus maximal density of progenitor cells (max (_C_0)); green: maximal death rate of TD cells versus the total amount of progenitor cells. (C) Spatial distribution of progenitor cells and the spatial crypt pattern at d¯1=0.02 and 0.2. Other parameters used for the simulation are in Additional file 1: Table S2.

Figure 5

Loss of progenitor cells or a crypt can be regenerated through Wnt signaling. (A) At T = 0, one half of progenitor cells in Crypt I are removed from the wild-type steady crypt (T = 100 in Figure 2A). With the regulation of Wnt signaling, progenitor cells in Crypt I start to repopulate (T = 10). Eventually, the original pattern of progenitor cells in Crypt I is regenerated (T = 100). (B) The maximal density of progenitor cells in Crypt I (blue dash line) and Crypt II (red solid line) as functions of time. Cell velocity, V, (C) and progenitor cell flux, _VC_0, (D) are plotted at the time T = 0 (blue dash curve), T = 10 (black dot curve), and T = 100 (red solid curve).

Figure 6

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) (μ_1 = 2.5 × 10−4_s_−1_μM). (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 B and γ W 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.

Figure 7

Spatial distributions of progenitor cells in a growing domain. (A) Crypt dynamics of progenitor cell density at three different times. A small amount of progenitor cells (C0 = 0.1) is initially placed in the center of the domain (1/10 of the length of the entire domain). The solution is close to the steady state at T = 100. (B) Comparison of the progenitor cell density along the crypt direction s in the fixed domain (blue dash curve) and the growing domain (red solid curve). (C) Time evolution of the progenitor cells in single crypt starting with three different initial levels of progenitor cells. The progenitor cells are initially placed at the center of the domain with _C_0 = 0.1, 0.5, and 1.0. Parameters in the simulations are listed in Additional file 1: Table S2.

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−ω2,p+ω2] 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 × 10−4_s_−1_μM_. (C) A new stable pattern of two-crypt pattern at a high level of exogenous Wnt: θ = 4 × 10−3_s_−1_μM_. (D) Removing the exogenous Wnt (i.e. θ = 0_s_−1_μM_) 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 = 0.75_L_max, ω = 0.05_L_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

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A reaction-diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts - PubMed (original) (raw)