Department of Mathematics, University of California, Irvine, CA, 92697, USA

Department of Development and Cell Biology, University of California, Irvine, CA, 92697, USA

Center for Complex Biological Systems, University of California, Irvine, CA, 92697, USA

Center for Mathematical and Computational Biology, University of California, Irvine, CA, 92697, USA

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.

Background

The colonic crypt, a basic functional unit of the intestine, is made up of a single sheet of columnar epithelial cells, which form finger-like invaginations into the underlying connective tissue of lamina propria

Wnt signaling controls stem behaviors, maintains stem cell habitus, and regulates cell migration and differentiation

Another important player in regulating stem cells and crypt dynamics is bone morphogenetic proteins (BMPs), which are part of the transforming growth factor

Mathematical models for populations of different cell types and their interactions that do not consider spatial effects have been used to explain premalignant growths and the long lag phases in tumor growth

In the existing models, crypts are usually regarded as prescribed and fixed units, and focuses of study have mainly been on dynamics of different types of cells and their mutants within single crypt. Interesting questions at hand include what minimal regulatory interactions and components may give rise to spontaneous formation of multiple crypts with a periodic pattern of stem cells, TA cells, and TD cells along the crypt direction as observed in experiments? What emerging crypt dynamics may arise due to Wnt signaling with regulation from BMP? Because emanation of multiple crypts is directly related to crypt homeostasis

Here, we present a mathematical model that couples a two-stage cell lineage of progenitor cells, which are considered as a sum of stem cells and various types of TA cells, and TD cells with Wnt and BMP signals to study spontaneous formation of multiple crypts. We investigate pattern formation of Wnt signaling arising through interactions among Wnt and a possible Wnt inhibitor (for example, Dickkopf proteins

Our computational exploration of the model suggests that Wnt patterning driven by the mechanism of short-range activation and long-range inhibition from the Wnt inhibitor with additional modulation from BMP can spontaneously result in formation of multiple crypts. Once a stable multiple-crypt system is established, the loss of crypt or progenitor cells can be regenerated through the regulations and components presented in the model without invoking additional mechanisms. Our computation and analysis also indicate that a loss of BMP may result in formation of more crypts (i.e. crypt multiplication) with a stable configuration; however, stronger Wnt signaling for a system without BMP may weaken such stability, leading to unbounded growth of multiple crypts as seen in experiments

Results and discussion

A multiple-crypt model that couples a cell lineage with Wnt and BMP activities

The epithelial cells lining the crypts include stem cells, TA cells and TD cells. Take C_{0} (_{1} (_{0} and _{1} are governed by

A schematic diagram of a two-stage cell lineage model.

**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 **(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.

Here, _{0} represents the replication probability of progenitor cells, and parameter _{0} is the reciprocal of the cell cycle length scaled by ln 2; _{1} is the removal rate for TD cells and it may be spatially regulated (see Section I in Additional file _{0} + _{1} = 1, up to a normalization

**A Reaction–diffusion Mechanism Influences Cell Lineage Progression as a Basis for Formation, Regeneration, and Stability of Intestinal Crypts.** Equations for models, numerical methods, stability analysis, tables of parameters, and supplementary figures.

Click here for file

In colonic crypts, epithelial cells at different stages of the lineage may be exposed to different levels of signaling. The spatial heterogeneity of cell dynamics arises from the fact that the replication probability of progenitor cells is spatially and temporally regulated by secreted molecules such as Wnt and BMP, which may provide robust controls for homeostasis and the spatial arrangement of cells. Experimental evidence suggests that the proliferation rate of progenitor cells is enhanced by Wnt activity

Here, [Wnt] and [BMP] represent the concentrations of molecules Wnt and BMP, respectively; _{w} and _{B} are reciprocals of the corresponding half maximal effective concentrations (EC50), and

Because the spatial distribution of progenitor and TD cells from the basal to the apical surface may intimately depend on the spatial distribution of the diffusive molecules Wnt and/or BMP, we consider spatial and temporal dynamics of Wnt and BMP as well as a possible Wnt inhibitor

Here, each type of molecule assumes individual effective diffusion rates _{W}, _{I}, _{B}. The removal of molecules due to degradation or binding with other molecules is assumed to be linearly proportional to the concentration of each molecule, with a rate constant _{W}, _{I}, _{B}. The synthesis of Wnt and BMP is assumed to be proportional to the density of the cell types that produce Wnt and BMP with rates _{0}, _{1}. The coefficient _{W} is used to describe the rate of self-enhanced Wnt activity and _{W} is the synthesis rate of Wnt inhibitor. _{W}_{I}, and _{W} are reciprocals of the EC50 which reflect the strength of feedbacks of Wnt and Wnt inhibitor. _{W}, _{W}, and _{I} are Hill coefficients. The choice of the biochemical parameters, such as diffusion coefficients, is drawn from previous cell lineage models of similar molecules and from experimental approximations (for instance,

Typically, the time scale of the cell cycle length is days, whereas that of the molecule interactions is hours

Because the connected multiple crypts in the intestine typically exhibit periodic behavior if the interests of study is within a domain containing a small number of crypts

Here (

where _{0}) is the diameter of crypt, which is chosen as a scale of integration of the progenitor cells along a crypt; _{0} represents the sharpness of the crypt shape; and

Because the dynamics of crypt morphology changes according to Eq. (4), the total length of the crypt may vary over time, affecting patterns of cells and molecules. To investigate the effect of a growing domain, we include the dynamic length of the domain in the model _{max}(

in which the crypt(_{max}(

A Turing pattern of Wnt signaling determines spatial distribution of progenitor and TD cells

We first study how the interaction between the short range of Wnt activation and the long range of Wnt inhibition occurring in a domain of proliferating cells may generate a Turing pattern

Starting with uniform distributions of progenitor and TD cells along with only low and fluctuated levels of Wnt signals without any initial presence of BMP and Wnt inhibitor, the system can achieve a steady state spatial pattern of cells with progenitor cells localizing in the bottom of crypt and the TD cells residing in the top and majority of the crypt (Figure

Spatial distributions of progenitor cells and the crypt growth are driven by a Turing pattern of Wnt signaling.

**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 **(B)** Concentrations of Wnt, Wnt Inhibitor, BMP, progenitor cells, and TD cells along the crypt direction of **(C)** Time evolution of the progenitor cells. Parameters used are listed in Additional file

Linear stability analysis suggests that Wnt inhibitor must adapt rapidly to dynamics of Wnt in order for the system to form a stable heterogeneous pattern in Wnt and Wnt inhibitor (see Section II in Additional file _{I} for Wnt inhibitor and _{W} for Wnt, i.e.,

Because Wnt and BMP regulate cell proliferation, the removal rate of the secreted molecules due to degradation or binding with other molecules may significantly affect dynamics of homeostasis and spatial pattern of cells in the intestine. Simulations show that increasing of the equal removal rate of Wnt and Wnt inhibitor usually leads to creation of more crypts (Figure

The number of crypts at the steady state depends on removal rate of Wnt.

**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 _{W} = 0.02^{−1}, there are five crypts **(B)** or six crypts **(C)** at steady state starting from two different initial conditions (see Additional file

The effects of the BMP removal rate are found to be different from those of the Wnt/Wnt inhibitor removal rate. Typically, increasing the BMP removal rate leads to an increase in the maximum density of progenitor cells (denoted by max (_{0})) and the total amount of progenitor cells (Figure _{B} < 1.1 × 10^{−3}^{−1} in Figure _{B} > 1.75 × 10^{−3}^{−1} in Figure

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.

**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 (_{0})); green: removal rate of BMP versus total amount of progenitor cells. **(B)** Blue: maximal death rate of TD cells (_{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

Since homeostasis of tissue growth requires a balance between cell proliferation and cell death, the death rate of TD cells may also significantly affect dynamics and the spatial distribution of crypts. Specifically, increasing the death rate of TD cells results in a decrease in maximum density of progenitor cells and loss of the total amount of progenitor cells (Figure

Loss of crypts or progenitor cells can be regenerated through Wnt signaling

Experimental evidence has shown Wnt signaling plays a critical role in regulating progenitor cells, suggesting the possibility that the self-renewal property of progenitor cells might be tightly mediated by Wnt signaling

First, we replace half of the progenitor cells in one of the two crypts (denoted as Crypt I) from a two-crypt steady state system (as shown in Figure

Loss of progenitor cells or a crypt can be regenerated through Wnt signaling.

**Loss of progenitor cells or a crypt can be regenerated through Wnt signaling. (A)** At **(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, **(C)** and progenitor cell flux, _{0}, **(D)** are plotted at the time

Early dynamics show that the density of progenitor cells in Crypt I increases while Crypt I grows in height, because the replication probability of progenitor cells regulated by Wnt and BMP remains high, leading to fast proliferation of progenitor cells. The velocity of cells and the flux of progenitor cells in Crypt I gradually increase (Figure

To further investigate the potential mechanism of the regeneration of progenitor cells, we study the case in which progenitor cells in Crypt I are totally removed and replaced by TD cells from the two-crypt steady state (Additional file

BMP maintains stability of crypts and loss of BMP may result in crypt multiplication

Experimental data shows that a loss of BMP signaling (through knockout of BMP receptors) in the intestine leads to multiplication and fission of crypt units

With BMP, starting from a uniform distribution of progenitor cells localized in a small local area of the tissue (Figure

Dynamics of crypts with or without BMP.

**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 (_{1} = 2.5 × 10^{−4}^{−1}**(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 (**(D)** Dynamics of crypt multiplication using a different energy function **(E)** A phase diagram of crypts in terms of _{B} and _{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

Starting with this steady configuration of crypts, however, removing BMP from the system as would be the case in a knockout of BMP, more crypts immediately emerge (Figure

If initially localized spots of the progenitor cells coincide with the Wnt pattern (Figure

To further study this behavior, we systematically explore the system by varying _{w} and _{B}, which are reciprocal of the EC50s, a measurement of strength of feedback for Wnt and BMP (Figure

In particular, starting from a single steady crypt formed from a system with BMP and then removing BMP (i.e. setting _{B} = 0 in Figure

Localized stem cells and exogenous Wnt

Experiments in culture have suggested that single stem cell can generate crypt-villus structure

Spatial distributions of progenitor cells in a growing domain.

**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 (C_{0} = 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 **(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 _{0} = 0.1, 0.5, and 1.0. Parameters in the simulations are listed in Additional file

To study how sensitive the final pattern of the crypt depends on the initial density of progenitor cells, we place three different levels of progenitor cells in the same localized spatial region (Figure _{0} = 1 shows formation of a single crypt without initial TD cells, which is consistent with the experimental observation

We next study how local exogenous Wnt may affect the Wnt patterning and formation of crypts. Starting with a stable single crypt, we add exogenous Wnt at a localized region of the crypt (Figure

Exogenous Wnt added at a localized spatial region.

**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 **(B)** A new stable pattern of a single crypt at a low level of exogenous Wnt: ^{−4}^{−1}**(C)** A new stable pattern of two-crypt pattern at a high level of exogenous Wnt: ^{−3}^{−1}**(D)** Removing the exogenous Wnt (i.e. ^{−1}**(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 _{max}, _{max} (_{max} is the crypt length in (A)). The temporal dynamics of the three cases **(B)**, **(C)**, **(D)** are provided in Additional file

Conclusions

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

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

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

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

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

Methods

To solve the system of Eqs. (14), we apply Fourier spectral method to the spatial discretization, and a semi-implicit temporal scheme to the temporal discretization ^{−4}. Numerical tests have been conducted to ensure sufficient spatial and temporal resolutions for convergence of the numerical solution (see more details in Additional file

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LZ carried out model development, simulations, analysis, and wrote the manuscript. AL, and QN helped with model development, data analysis, and manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by NIH grants R01GM67247 and P50GM76516, and NSF grant DMS0917492. We thank Jeremy Ovadia for reading and comments of the manuscript. We also thank the anonymous reviewers for their valuable suggestions and references.