Modeling Crypt Geometry and Wnt Signaling

In our model we assume that the Wnt-activity of the individual cells is determined by the local curvature of the basal membrane. Thus, the crypt geometry impacts the lineage specification and differentiation and consequently the crypt turnover. In order to study these interrelations we set the shape parameter λ₁ to zero and varied the crypt length and width. By assigning the thresholds TP_WNT and TD_WNT fixed Gaussian curvatures 4×10⁻⁴/µm² and 0/µm², respectively, the shape changes resulted in a shifted position of these thresholds along the crypt axis. We found that the shape changes did result in quantitative changes of the systems behavior only. Selected results can be found in the Table S1. In all further simulations we considered a defined cell shape that agrees with experimental data on crypt geometry [8] (link). The crypt with this defined shape is called ‘reference crypt’ in the following. The parameters of the reference crypt are listed in Table 1.
In a series of simulations we varied the threshold TP_Wnt for a reference crypt (see Table S2). In this case the total number of cells remains approximately fixed. The changes result in changes of the size of the Paneth cell compartment (undifferentiated and Paneth cells) which are balanced by changes of the number of enteroytes and Goblet cells. A decrease of the size of the Paneth cell compartment increases the number of proliferative cells in the crypt and thus decreases the turnover time. We selected the position of the threshold TP_Wnt such that cell number of the Paneth cell compartment is about 40 [40] (link).
For a given position of TP_Wnt the steady state cell production of a crypt still depends on cell interaction parameters as well as internal parameters regulating fate decisions. For example the turnover is decreased as a result of an increase of the cell-cell interaction strength ε_c, an increase of the sensitivity to contact inhibition V_p or a decrease of the Wnt-activity threshold TD_Wnt. We used TD_Wnt, together with F_A^other and η_BM, to fit the turnover the results of the BrdU labelling experiments.
This was most efficient provided that the average apoptosis rate in the crypt was smaller than about 5% per day. Such low apoptosis rates were ensured assuming a high cell-knot interaction constant ε_k>5 nNm for all cells. Note that a migration force F_A^other>0 was required to fit the BrdU labelling data.
Steady state cell patterning also underlies a complex regulation as seen from the organisation of the Paneth cell population. For a given crypt geometry and Wnt-activity threshold TP_Wnt the sum of undifferentiated and Paneth cells is approximately fixed. Thereby, the number of Paneth cells depends sensitively on the cell-knot interaction strength ε_k^paneth, the migration force for Paneth cells F_A^Paneth and the Notch-activation strength LP^Paneth. Stable Paneth cell adhesion to the BM over their life time t_p required ε_k^paneth≥35 nN defining a constraint to this parameter. Moreover, a minimum ‘migration force’ F_A^Paneth of about 7nN is required to ensure that Paneth cells remain confined at the crypt bottom. Thus, we adjusted the number of Paneth cells using LP^Paneth (Fig. 5).