An important aspect of colon crypt development and formation is its spatial patterning of cells at different lineage stages within intestinal tissues. Morphogens, signals induced by secreted molecules released from cells, their regulations on cell proliferation and differentiation, and the intimate coupling among signals and cells are generally believed to be the main factors governing growth, dynamics, and maintenance of spatial heterogeneity in crypts.
Here, we focus on spontaneous formation and stability of multiple crypts driven by Wnt and BMP signals that are regulated by cells in the intestinal tissue. Through a model based on a continuum description of two different types of cells in cell lineage with Wnt and BMP molecules produced at different rates by different types of cells, we have demonstrated that Wnt patterning driven by a reaction–diffusion mechanism consisting of short-range activation of Wnt and long-range inhibition by the Wnt inhibitor with additional modulation from BMP could spontaneously result in formation of multiple crypts that are observed experimentally. Unlike typical Turing patterning mechanisms, Wnt signaling in this case is intimately coupled with cell proliferation and expressed in only a portion of cell lineage within the growing tissue.
Our model can recapitulate some distinctive and important experimental facts. First, progenitor cells in crypts can be regenerated during an intestinal injury because Wnt signaling tightly mediates the self-renewal property of progenitor cells, leading to repopulate the original intestinal crypt; in particular, the observation in which new crypts first arise from the existing crypts is captured in the model through a suitable energy functional that mimics the overall mechanistic effects from the surrounding tissues on the crypts. Second, loss of BMP signal leads to crypt multiplication in a form of development of more stable crypts; however, with an additional increase in Wnt signaling during crypt multiplication, uncontrollable growth of crypts is then found in the model simulation – a signature of cancer initiation. The simulations have also shown that crypts exhibit fingering dynamics during crypt multiplications, an interesting pattern consistent with experiments. Model exploration has suggested that Wnt signaling pattern dictates crypt patterning while BMP is mostly responsible for crypt stability. The model presented herein has been developed mainly for studying the role of Wnt and BMP in growth of multiple crypts. In the current model, Wnt positively regulates the progenitor cell replication probability that is also being negatively regulated by BMP. As more molecular details are revealed on specific functions of Wnt and BMP in cell differentiation and proliferation, the model can be extended and refined to incorporate those details.
Because the diffusive molecules, Wnt, Wnt inhibitor, and BMP, may move into the lamina propria, further modulating spatial and temporal patterning of Wnt signaling, hence, affecting crypt organization [35
], one may need to add other spatial dimensions beyond the growth direction along the single layer of cells in the model. For the two- or three-dimensional models, the energy function like Eq. (4) needs to be refined to include transport aspects of cells and Wnt and BMP molecules in other spatial directions or different equations for growth of crypts are required based on other mechanistic or phenomenological descriptions.
On the other hand, more intermediate states and cellular types in one or multiple branching cell lineages, which is the case for human colonic crypts, can be added in the current model in a straightforward fashion. With inclusion of more cell types, it would be interesting to investigate effects of symmetric division versus asymmetric division and their interactions with Wnt and BMP signals on crypt patterning. Of course, downstream and feedback regulatory networks of Wnt may be added in the model as well to study functions of target genes of Wnt and BMP (e.g. c-Myc) and their role on cell proliferation or differentiation during growth of colonic crypts, as improperly regulated Wnt signaling results in constitutive renewal and limitless expansion of stem cells or confer stem cell behavior on the progenitor cells, leading to formation of cancerous tissues [5
Besides Wnt and BMP signaling, the model can be naturally extended to include other important signaling pathways, such as Notch signaling. Notch signaling is expressed in intestinal crypts [36
]. Wnt and Notch are found to jointly maintain stem cells with a fact that either Wnt or Notch signaling is insufficient to keep all progenitor cells in a proliferating state [7
]. Another avenue of research is why Wnt pathway mutations frequently give rise to intestinal cancers, whereas Notch pathway mutations have not been found to do so. This observation suggests that the stem cells are likely respond to Wnt and Notch signaling differently, for example, with a different regulation in the replication probability or different interactions with BMP or Wnt inhibitors in the model presented in this paper. Another possibility, which can be tested by the model, is that active Wnt signaling may switch on Notch activity, but not vice versa [7
The cell lineage framework in which cell population is described continuously has its limitation when single cell and small number of cells with strong cell migration and heterogeneity dictate dynamics of the crypt system, which requires the development of a discrete cell lineage model where one needs to trace single stem cell. A recent study by Murray et al.
] has shown that discrete models can be coarsed-grained in a continuum limit to generate a continuum model for more generalized description of the intestinal crypt. In such models, the incompressibility assumption on tissue growth is not required and the tissue compression may be accounted for as well.
In this paper, we have used a phenomenological approach with a simple energy functional describing the morphological change of crypts based on the experimental observation of the shape of crypts. Although the energy functional may be convenient in including certain simple mechanistic effects, incorporating both biochemical and physical mechanisms leading to the complex shape of the crypts requires further development of the continuum models presented in this paper. This more challenging problem is beyond the scope of our current paper and it will be pursued in our future study.
A recent modeling study suggests that the periodic pattern of crypt could be formed by the buckling instability caused by the negative tension produced by the dividing cells [37
]. In this model, the epithelia monolayer of cells is assumed to be lying on top of an elastic stroma, and the different patterns of villi and crypts are affected by a coupling of cell division and local curvature. It would be interesting to compare this mechanical mechanism with the chemical mechanism studied in this paper. Because the Wnt patterning is closely related to cell replication probability for cell division, these two mechanisms may be similar in certain aspects in driving the crypt pattern. One possible experimental approach in studying these two mechanisms is to utilize different culture conditions to examine morphological differences of intestinal crypt–villus units built from single stem cell [33
]. A mechanical modification of the laminin-rich Matrigel that supports intestinal epithelial growth [33
] would lead to pattern changes if the mechanics is the dominant mechanism for crypt patterning. On the other hand, a pulse of exogenous Wnt occurring at different spatially locations [34
] would affect Wnt patterns, resulting in different crypt patterning if the reaction–diffusion mechanism presented in this paper is dominant.