Mathematical modelling identifies conditions for maintaining and escaping feedback control in the intestinal epithelium

Abstract

The intestinal epithelium is one of the fastest renewing tissues in mammals. It shows a hierarchical organisation, where intestinal stem cells at the base of crypts give rise to rapidly dividing transit amplifying cells that in turn renew the pool of short-lived differentiated cells. Upon injury and stem-cell loss, cells can also de-differentiate. Tissue homeostasis requires a tightly regulated balance of differentiation and stem cell proliferation, and failure can lead to tissue extinction or to unbounded growth and cancerous lesions. Here, we present a two-compartment mathematical model of intestinal epithelium population dynamics that includes a known feedback inhibition of stem cell differentiation by differentiated cells. The model shows that feedback regulation stabilises the number of differentiated cells as these become invariant to changes in their apoptosis rate. Stability of the system is largely independent of feedback strength and shape, but specific thresholds exist which if bypassed cause unbounded growth. When dedifferentiation is added to the model, we find that the system can recover faster after certain external perturbations. However, dedifferentiation makes the system more prone to losing homeostasis. Taken together, our mathematical model shows how a feedback-controlled hierarchical tissue can maintain homeostasis and can be robust to many external perturbations.

Figure 1

The basic colon epithelium model. (a) Schematic sketch of the model. (b) The feedback function $\delta$ (upper row) determines the qualitative behaviour of the model (lower row). First column: Stable steady state. Second column: unbounded growth. Third column: sustained oscillations. The little arrows represent changes to the feedback function compared to the first column. The dotted line in the second and third column shows the feedback function from the first column for easier comparison. (c) In case of explosive growth, the system converges to a stable ratio of stem to differentiated cells. Exemplary numerical integration. (d) Bifurcation diagram of the system with linear feedback function $\delta$. Standard parametrisation: $\beta =1.0,\omega =0.1,{\delta }_0=0.9,{\delta }_{\mathit{slope}}={10}^{-4},{\delta }_{\mathit{max}}=2.0$. Note the switch from a stable steady state to unbounded growth for $\beta >{\delta }_{\mathit{max}}$ (upper-left panel ). Also note the invariance of $\overline{D}$ to changes in $\omega$ (upper-right panel). Both properties hold regardless of how we chose $\delta$. Finally, notice that ${\delta }_0>0,{\delta }_{\mathit{slope}}>0$ can be made arbitrarily small without switching to unbounded growth; however, if ${\delta }_0>\beta$, the system will go extinct (lower two panels). (e) A steeper feedback functions causes stronger and longer oscillations. Exemplary numerical simulations with linear function $\delta$ for varying ${\delta }_{\mathit{slope}}$; steady state fixed at $\overline{S}=100,\overline{D}=1000$.

Figure 2

All four one-looped model topologies that can show homeostasis. First column: schematic sketches; second column: exemplary numerical simulations; third column: model equations, as well as Jacobian at the non-trivial steady state and its eigensystem for the case of a linear feedback function $\beta (x)={\beta }_0+{\beta }_{\mathit{slope}}x$ or $\delta (x)={\delta }_0+{\delta }_{\mathit{slope}}x$, respectively.

Figure 3

(a) Illustration of our model comparison: After a perturbation at $t=0$, the model relaxes to its steady state (continuous blue and orange lines, depicting differentiated and stem cells, respectively). We use a first-order approximation (continuous red line) to not have to rely on costly numerical solutions of the system, and compute the ’defects’ $\chi$ of the models we want to compare (shaded red area). (b) The two models we compare in this figure. Top: model 1, where stem cell differentiation is stimulated by the differentiated cell compartment; bottom: model 2, where the stem cell compartment stimulates its own differentiation. Note that model 1 is equivalent to our basic colon epithelium model derived earlier. (c) Difference in defects of model 2 and 1 throughout the parameter space. Areas shaded in red depict regions where model 1 shows a bigger defect than the alternative model; blue areas indicate the opposite. Columns represent different cases of stem cell fraction at steady state (1, 10, and 25% respectively), rows represent the three different kinds of perturbations (removing differentiated cells, removing stem cells, and removing both, respectively).

Figure 4

The colon epithelium model with dedifferentiation and its most important properties. (a) Schematic sketch of the model. (b) Exemplary numerical simulations of the system with piecewise linear differentiation rate function $\delta$ and different dedifferentiation rate functions $\varrho$: no dedifferentiation (first column), linear dedifferentiation with ${\varrho }_0=0.5,{\varrho }_{\mathit{slope}}=-0.01$ (second column) and a faster linear dedifferentiation ${\varrho }_0=0.9,{\varrho }_{\mathit{slope}}=-0.01$ (third column). System parameters are $\beta =1,{\delta }_0=0.9,{\delta }_{\mathit{slope}}={10}^{-4},\omega =0.1$. Note how adding dedifferentiation, as well as increasing the higher maximum dedifferentiation rate ${\varrho }_0$ makes the system recover faster and with smaller oscillations after removing stem cells. (c) Adding dedifferentiation opens up a second way the system can lose homeostatic stability. First column shows the case of homeostasis. Equilibrium stem cell pool size $\overline{S}$ is given by solving $\beta \varrho (\overline{S})/\omega -\delta (\beta \overline{S}/\omega )=\beta$. Second column: Sufficient decrease of differentiation rate destroys the non-trivial steady state and unbounded growth occurs. Third column: Sufficient increase of dedifferentiation rate has the same consequence.