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Conceived and designed the experiments: NB HC. Performed the experiments: NB GA. Analyzed the data: PB JG. Wrote the paper: JG ML. Performed all simulations: PB. Designed the model: JG. Provided essential ideas regarding the model design: HC ML.

The authors have declared that no competing interests exist.

We introduce a novel dynamic model of stem cell and tissue organisation in murine intestinal crypts. Integrating the molecular, cellular and tissue level of description, this model links a broad spectrum of experimental observations encompassing spatially confined cell proliferation, directed cell migration, multiple cell lineage decisions and clonal competition.

Using computational simulations we demonstrate that the model is capable of quantitatively describing and predicting the dynamic behaviour of the intestinal tissue during steady state as well as after cell damage and following selective gain or loss of gene function manipulations affecting Wnt- and Notch-signalling. Our simulation results suggest that reversibility and flexibility of cellular decisions are key elements of robust tissue organisation of the intestine. We predict that the tissue should be able to fully recover after complete elimination of cellular subpopulations including subpopulations deemed to be functional stem cells. This challenges current views of tissue stem cell organisation.

In the murine small intestine there are more than a million organized groups of proliferating cells, the crypts, each of which contains about 250–300 cells. About 60% of these cells are in rapid cycle. The functional stem cells of this tissue have been demonstrated to reside at defined positions at the lower third of the crypt and to give rise to four different cell types. Considering this simple structure the murine intestine is an ideal system to study general aspects of tissue organization. Here, we introduce a comprehensive and predictive computer model of the spatio-temporal organization of the murine intestine which describes how cell production and cell fate decisions could be organized in steady state as well as under perturbations. The model is based on single cells acting as individual agents, updating their status within a certain set of options governed by some active rules and on signals received from the environment. This kind of self-organization enables effective tissue regeneration without assuming an explicit stem cell population that maintains itself by asymmetric division. Thus, the model offers a novel systems biological view on crypt stem cell and tissue organisation.

The epithelium of the small intestine is the most rapidly regenerating tissue of adult mammals. Cell production starts near the crypt base, producing numerous progeny which move up the crypt-villus axis. Cells moving up the crypt continue proliferating while in parallel becoming committed either to an absorptive or a secretory fate. Cells stop proliferating and differentiate while approaching the crypt-villus junction. Upon reaching the villus tip a few days later cells are shed into the lumen of the intestine. As an exception, cells that become committed to the secretory Paneth lineage move down the crypt-villus axis whilst differentiating until they occupy their final position at the very bottom of the crypt. These cells have a life time of up to 8 weeks

a) Histological section. Expression of the functional stem cell marker Lgr5-LacZ (blue) is mainly restricted to a few cells at the crypt bottom _{p} and x_{d} (see text). c) Snapshot of a crypt simulation. Undifferentiated cells (red) and Paneth cells (green) are found intermingled at the crypt bottom, progenitors of enterocytes (blue) and Goblet cells (yellow) move upwards along the crypt axis. d) Steady state cell numbers over time. Colour code as in c). Black line denotes the total number of cells.

Tissue and stem cell organisation of the adult small intestine has been studied extensively

The nature of the microenvironment that harbours and possibly conditions the functional stem cells is also not fully elucidated. Activation of the Wnt- and Notch- pathway was demonstrated to be essential for stem cell maintenance as well as proliferation and differentiation

The classical ‘pedigree concept’ of hierarchical tissue organization regards ‘stemness’ as a cellular property essentially fixed intrinsically to specified cells called stem cells annotating them as a specific cell type. These cells are assumed to divide asymmetrically and to give rise to a new stem cell and to a non-stem progenitor cell. Subsequently, the progenitor cell undergoes transient amplifying divisions before it differentiates terminally. In this pedigree model lineage specification is linked to cell stages in the cellular differentiation hierarchy. It is one of our objectives here to demonstrate that the pedigree related assumptions on stem cell populations are not required in order to provide a comprehensive explanation of the tissue self organisation and that an alternative concept can be more powerful.

Our approach is based on concepts of self-organizing systems assigning a greater emphasis to the interaction between cells and their environment. Moreover, they enable reversible developments for individual cells, allowing the system to flexibly react to changing demands

Cells are represented by elastic objects which can move, grow and divide, form contacts with other cells and the basal membrane (BM) and can communicate with one another. The extent by which these properties are expressed depends on the internal state vector _{Wnt} and I_{Notch}, respectively.

Wnt-signalling in intestinal crypts has been extensively studied

Notch-signalling is mediated via transmembrane proteins. Thus, it requires cell-cell contacts. It is activated in Notch-receptor expressing cells if their neighbour cells express Notch-ligands such as Jagged and Delta

In our model, lineage specification into enterocytes (absorptive lineage) and Paneth- and Goblet cells (secretory lineages) is assumed to depend on Wnt- and Notch-signalling

Cells with high Wnt- and high Notch-signalling, i.e. with I_{Wnt} and I_{Notch} above certain thresholds TP_{Wnt} and TP_{Notch}, respectively, are considered as undifferentiated (_{Wnt} become primed for switching on secretory properties if I_{Notch} drops below TP_{Notch}. This occurs if their neighbour cells do not express sufficient Notch-ligands. High Wnt-signalling is required for Paneth cell differentiation _{Wnt} become primed for switching on Paneth properties.

Enterocytes are characterised by a low Wnt-activity compared to undifferentiated cells _{Notch} becomes primed for switching on enterocyte properties if I_{Wnt} drops below the threshold TP_{Wnt}. This occurs if the cell reaches a position above x_{p} (_{Notch} falls below TP_{Notch}. In accordance with the concept of self-organizing systems

A primed cell can subsequently develop towards and may eventually reach an irreversible (terminal) differentiation state. Enterocyte and Goblet progenitors migrate out of the crypts and turn into differentiated cells if they reach a position near the crypt-villus junction _{Wnt} falls below a threshold TD_{Wnt,} which happens if the cells reach positions above x_{d} (

In our model the activity state of a cell is determined by its local environment (see above). As a result each conformational change of a cell changes its activity state and those of its neighbours. Accordingly lineage specification and differentiation strongly depend on cell migration, cell adhesion and cell elasticity which affect the spatio-temporal organisation of the crypt. These biomechanical features are modelled using an established individual cell-based model of epithelia

We assume identical biomechanical properties in all differentiation states, except for the migration properties. These properties were suggested to be controlled by the expression of Eph2/Eph3 receptors and their ligand ephrin-B1

Besides the cell properties, properties of the basal membrane (BM) of the crypt also impact the spatio-temporal organisation process. We explicitly model the BM by an artificial fibre network that interacts with each individual cell. The shape and the size of the network representing the basal membrane was chosen in order to fit experimental data on crypt geometry

Whether a cell proliferates depends on its differentiation state. Undifferentiated cells in the crypt are capable of proliferation

If a cell starts cycling, the cycle will be finished independently of changes in the differentiation state. Cell growth is modelled assuming stochastic growth steps. This leads to Γ-distributed growth times, where a cell doubles its volume. In our simulations the average value was adapted to about 16 hours. If the cell reaches twice the initial (minimal) volume it divides into two daughter cells of equal volume. Cells that are sufficiently compressed by their neighbour cells stop volume growth due to contact inhibition of growth

The system dynamics is described by equations of motion for each individual cell. Thereby, each update of the position and size of the cells potentially changes the internal state vector

Our crypt model is capable of quantitatively reproducing steady state cell production (

a,b) Positional BrdU label index obtained a) 2h and b) 24h after labelling. Experimental data: red, Simulation data: black. Bottom: Snapshots of simulated crypts. Colour code as in

Symbol | Value | Parameter | Reference |

V_{0} |
4/3π (5µm)^{3} |
Minimal volume of an isolated cell | Estimated |

τ | 14 h | cell growth time | results in an effective cell cycle time ∼24h |

E | 1kPa | Young modulus | |

ν | 1/3 | Poisson ratio | ‘’ |

ε_{c} |
200 µN/m | cell-cell anchorage | ‘’ |

V_{p} |
0.88 V_{0} |
threshold volume of contact inhibition | Set |

z_{0} |
150 µm | length of the crypt | Set, according to measured properties of the crypt shape |

r_{0} |
60 µm | crypt radius at the crypt-villus junction | ‘’ |

λ_{1} |
0.25 | shape parameter 1 | ‘’ |

λ_{2} |
0.1 | shape parameter 2 | ‘’ |

λ_{MAX} |
1.25 µm | maximum in-radius of a network triangle | Set (technical) |

Ω | 0.95 | threshold ratio | Set |

ε_{K}^{Paneth} |
35 10^{−12} Nm |
maximum cell-knot interaction energy of Paneth cells | ensuring apoptosis rates <5% |

ε_{K}^{other} |
5.5 10^{−12} Nm |
maximum cell-knot interaction energy of all other cells | |

η_{c} |
5×10^{10} Ns/m^{3} |
friction constant for cell-cell friction | |

η_{BM} |
3.2 Ns/m | friction coefficient for cell-BM friction | Fit: turnover |

η_{VO} |
400 Ns/m | friction coefficient regarding volume changes | |

F_{A}^{Paneth} |
7.5 nN | absolute value of the migration force of Paneth cells | Fit: Distribution of Paneth cells |

F_{A}^{other} |
4.5 nN | absolute value of the migration force of all other cells | Fit: turnover and Brdu data |

z_{p} |
−125 µm | position of the Wnt- threshold TP_{Wnt} for priming |
Fit: size of the Paneth cell compartment |

z_{d} |
−87.5 µm | position of the Wnt- threshold TD_{Wnt} for differentiation |
Fit: turnover and Brdu data |

LP^{Paneth} |
0.35 | Notch activation through Paneth cells | Fit: cell ratios |

LP^{Goblet} |
1.00 | Notch activation through Goblet cells | Set: maximum |

TD_{Notch} |
1 | Notch-threshold | Set |

t_{P} |
57 days | lifetime of a Paneth cell |

Steady state cell patterning in the model crypt results as a self-organized feature referring to the assumed modes of Notch- and Wnt-signalling. Undifferentiated functional stem cells, i.e. cells with high Wnt- and Notch-signalling are intermingled among the Paneth cells and appear up to cell position 4 and 5. Thus, they match the distribution of LGR-5 positive cells reported by Barker et al.

The model generates a Paneth cell population at the crypt bottom whose spatial distribution fits the data observed by Chwanlinski et al.

Simulating steady state conditions, we studied the dynamics of clonal expansion and conversion to monoclonality. Technically, in the model all cells were labelled at one instance with a clonal marker that is inherited by all offspring. The number of coexisting clones was followed over time. In the long run, few clones survive and only one clone eventually populates the entire crypt. The life-span time of clones in the crypt depends on the initial position of the labelled cell. While clones of functional stem cells initially located at the bottom of the crypt are more likely to persist for a long time (

a),b) Snapshots of simulated cell clones (pink) at labelling initiation (t_{0}) and 7days later (t_{1}) for clones derived from a) an undifferentiated functional stem cell and b) an enterocyte progenitor. Colour code as in _{1} and τ_{2} (red). A video showing an example of a simulation of clonal conversion in a crypt can be found in

Our model also permits a better understanding of the prompt effects of conditionally de-regulated Wnt- and Notch- signalling. A constitutive activation of the Wnt-signalling was implemented in the model by increasing the Wnt-activity over TP_{Wnt} in all cells along the crypt-villus axis. This results in a rapid expansion of the functional stem cell and Paneth cell population at the expense of the enterocyte and Goblet cell population (_{Wnt} in all cells (

Simulation results for crypt organisation following disturbed signalling. Colour code as in

Constitutive activation of Notch-signalling in all cells above TP_{Notch} resulted in a rapid Paneth- and Goblet- cell depletion and an increased cell turnover (

An essential model feature is the reversibility and flexibility of cell fate decisions in cells not yet terminally differentiated. In principle we give all enterocyte progenitors the capability of acting as functional stem cells if they enter the spatial Wnt-niche at the crypt bottom. Likewise, cells primed to become Paneth or Goblet cells may revert their status depending on the local Notch and Wnt-signals. Although possible in principle, such cell fate reversions are only occasional events in steady state (e.g. less than 0.01 transitions per enterocyte progenitor into the undifferentiated state per day). They can, however, become more relevant in states of perturbation.

This concept implies that the loss of a single cell or of a few cells is immediately compensated for by neighbouring cells which can rapidly adapt due to local signalling and thereby provide robustness against tissue perturbations. We analysed the model robustness to a sudden elimination of selected cell populations from steady state conditions. In independent simulations we separately deleted i) all undifferentiated cells, ii) all cells committed to the Paneth- cell lineage and iii) to the Goblet-cell lineage and finally iv) all enterocyte progenitors. A complete deletion of the undifferentiated cells at the crypt bottom only transiently affects the crypt system. This is in agreement with observations following the loss of functional Lgr5 stem cells upon conditional deletion of the stem cell-specific transcription factor ASCL2

The spatio-temporal organisation of the intestinal epithelium has been modelled using several approaches

The model is based on single cells acting as individual agents, updating their status within a certain set of options governed by some active rules and on signals received from the environment. Thereby, it accounts for and requires the 3-D spatial structure of the crypt. The model describes how cell production and cell fate decisions could be organized in steady state as well as under perturbations. Thus, the model offers a novel systems biological view on crypt stem cell and tissue organisation.

Recently, Lgr-5 and Bmi-1 have been identified as markers linked to functional stem cells in the small intestine _{Wnt}. We found that the behaviour of all these cells is in full accordance with the functional definition of stem cells

Our simulation results suggest that any single subpopulation of the crypt could be deleted at a certain time point without any long term consequence for crypt organisation. While this prediction remains to be validated experimentally, it does raise additional questions regarding the origin of this kind of robustness. In our model, robust organisation of the intestine depends on the assumption of i) reversible and flexible fate decisions of stem cells and ii) an ‘externally’ defined Wnt-activity gradient.

There is increasing evidence of reversible and flexible fate decisions from other tissue modelling studies

Our assumption of the dependence of the Wnt-activity on local curvature of the tissue rises the question on the underlying molecular regulation. A possible link between surface curvature, β-catenin stabilisation and enforced Wnt-signalling could be provided for example by integrin-linked kinase activity

A recent study by van Leeuwen et al.

Our model predicts that many more cells than the actual functional stem cells at the crypt base can be clonogenic but that the probability a certain clone overtakes the entire crypt depends on the position of its initiation. In the model, this probability is directly related to the probability that the progeny of the clone reaches the crypt bottom. This prediction also remains valid for systems with de-regulated signalling. Clonal expansion of individual APC-mutant cells was recently shown to be effective for Lgr-5 positive cells restricted to the crypt base

Long living clones were suggested to profit from specific environmental interactions such as interaction with enteroendocrine cells expressing growth inhibitory peptides

The model proposed in this study comprehensively explains numerous experimental observations regarding spatial patterns of proliferation, clonal dynamics, cell lineage specification and differentiation under both normal steady state and disturbed regulation. Thereby, it combines features of the molecular, cellular and tissue levels, providing a simplified but consistent picture of the dynamic organisation of small intestinal crypts. We expect that the model can be specified according to the specific organisation of duodenum, jejunum, ileum and colon crypts by adapting specific parameter sets. The predictions provided in this study can be validated experimentally. Thus, we expect that our novel approach will provoke further discussion about somatic stem cell organisation and will stimulate future experimental and modelling research in the field.

In the following we provide some details regarding the model, the fitting of experimental data, and experimental setups. In section A1) and A2) we describe the cell-cell and the cell-basal membrane (BM) interaction model, respectively. In section A3) the equations of motion are given and the update procedure of the internal state vector of the cells is explained. In section A4) the fitting strategy is described and some results on the effects of parameter variations are discussed. Finally, in section A5) material and methods of the BrdU labelling experiments and details of its simulation are provided.

An isolated cell is represented by an elastic sphere of radius R and volume V(R). If a cell i gets into contact with another cell j the cells adhere. Their adhesive interaction energy is approximated by:_{i}, R_{j}, and on the distance d_{i,j} between them. As a result of contact formation the shapes of the cells change by flattening at the contact area. Assuming that cells can be described by an isotropic homogenous elastic solid, the deformation energy for the contact is calculated using the Hertz model:

In our approach the BM is modelled by a triangulated fiber network. This network is represented by its knots. These knots are assumed to be located at the crypt surface, which is defined by the following equation for the local crypt radius:_{0} is the length of the crypt and r_{0} the radius of the crypt at the crypt-villus junction (z = 0). The parameters λ_{1} and λ_{2} are shape parameters. A further parameter of the network is the maximum inradius (mesh size) of its triangles λ_{MAX}. It was set narrow enough to avoid that cells can cross the network. In all simulations presented we used λ_{MAX} = 1.25 µm, corresponding to about 30.000 knots within the network of one crypt. On one hand this setting ensures low local variance of the network structure in terms of the coordination number. On the other hand it keeps the system computational tractable.

If the distance between a cell i and a knot k of the BM network d_{i,k} is smaller then the radius of the cell R_{i} they are assumed to interact. The interaction energy is modeled by:^{knot}_{ij} larger then _{i} the interaction is weakly adhesive, it becomes strongly repulsive for d_{ij} smaller then this threshold distance. The interaction energy is scaled by the number of knots

The generalized forces acting on cell i can be derived from the partial derivative of the interaction energies described above:_{ij} is given by d_{ij} = |_{ij}| = |_{i}−_{j}| where _{i} and _{j} are the position vectors of cell i and j, respectively. In the same way r_{i,k} is the distance between cell i and the knot k. _{i,j} = _{ij}/|r_{ij}| and _{i,k} = _{i,k}/|r_{i,k}|.

These forces organize the contacts between the cells by changing their distance or their radii. The resulting cell motion can be modeled using Langevin equations for each cell _{i} and the radius change dR_{i} of cell i are given by:_{BM} and η_{VO} describe friction between a cell and the BM and in course of volume changes, respectively. The friction between two cells is described by the coefficient

During each time step of a simulation position and radius of all cells are updated in parallel according to equation (A7). Thereby, a variable time step is used in order to avoid artificial cell interpenetration. Each update of position and radius potentially changes the internal state vector _{Wnt}, I_{Notch}) of the individual cells. Thus, _{Wnt}, I_{Notch} may cross one or more threshold values (TP_{Wnt}, TD_{Wnt}, TD_{Notch}). In this case the phenotype of the cells changes and all properties of the new phenotype - including the lineage characteristics, as well as migration and adhesion properties - are assigned to the cell.

In our model Wnt-activity is assumed to be a function _{Wnt} and TD_{Wnt} equal to _{p}) and _{d}) for lineage priming and terminal differentiation, respectively. Accordingly, changes of the cell fate occur if a cell crosses the position z_{p} or z_{d}. We used these threshold values as fit parameters to adjust the systems behavior (see A4 and _{p} cover the regions of high curvature. Moreover, at positions z_{p}<z<z_{d} both types of curvature are positive and nearly constant and at positions z>z_{d} the Gaussian curvature falls below zero. Thus, the fitting results are in full accordance with our assumptions of a correlation between Wnt-activity and positive curvature.

The Notch-activity is calculated via cell-cell contact analysis. A cell is Notch-activated by all cells being in direct contact with it and expressing Notch-ligands:_{i}+R_{j}>d_{ij} (condition of direct contact) otherwise it is zero. The degree of activation by a single cell (LP) depends on the cell type. LP is assumed to be larger than zero for Paneth and Goblet cells and zero for all other cells. In order to reproduce the correct cell patterning Paneth cells are required to induce weaker activation than Goblet cells (see A4 and _{Notch} (see also

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 λ_{1} to zero and varied the crypt length and width. By assigning the thresholds TP_{WNT} and TD_{WNT} fixed Gaussian curvatures 4×10^{−4}/µm^{2} and 0/µm^{2}, 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

In a series of simulations we varied the threshold TP_{Wnt} for a reference crypt (see _{Wnt} such that cell number of the Paneth cell compartment is about 40

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} (

a) Local fraction of Paneth cells in dependence of their migration force F_{A}^{Paneth}. b) Local fraction of Paneth cells in dependence of their Notch-activation strength LP^{Paneth}. All other parameters of the model are fix (see _{A}^{Paneth} (pink squares, compare a)) and changing LP^{Paneth} (cyan squares, compare b)).

To examine proliferating cells 2 mg/ml BrdU (Sigma-Aldrich, Deisenhofen, Germany)/PBS was injected i.p. into mice (50mg/kg b.w.). Mice were sacrificed 2 and 24 hrs after injection. The bowel segments were fixed in 4% paraformaldehyd/PBS and paraffin embedded. In 4µm sections proliferating cells were detected after blocking endogenous peroxidase activity in 3% H_{2}O_{2}/PBS for 10 min, 2N HCl DNA denaturation for 30 min and enzymatic pretreatment with 0.1% (w/v) Trypsin (SIGMA) for 20 min by incubating with an anti-BrdU monoclonal antibody (Sigma) for 2 hrs (all steps at 37°C) followed by the Vectastain® ABC kit (Vector Laboratories, Burlingame, US). Finally, sections were counter-stained with hematoxylin, dehydrated and mounted. Fifty half-crypts per mouse were scored on a cell positional basis according to whether or not cells were BrdU positive.

These experiments were simulated assuming that 70% of the proliferating cells were marked. This number was chosen somewhat larger than the fraction of the cell cycle time belonging to the S-phase (50–60%^{37}), accounting for an extended labeling time. Labels were inherited to the entire progeny. Sections of 4.5µm were analyzed.

Simulation results on the impact of the crypt shape on the systems behaviour. For a simple crypt shape (λ_{1} = 0) the length and width of the crypt was changed, by changing the parameter z_{0} and r_{0}. The thresholds TP_{Wnt} and TD_{Wnt} were set to the positions of Gaussian curvature 4×10^{−4}/µm^{2} and 0/µm^{2}, respectively (see black lines). Increasing length increases the number of cells leaving the crypt thereby the turnover time remains approximately constant. Increasing width increases the turnover time, i.e. the outgrowth is less efficient.

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Simulation results on the impact of the position of the threshold TP_{Wnt} on the systems behavior. Moving down the position of TP_{Wnt} (black lines) to the crypt bottom leads to a faster turnover. This refers to a decreasing number of Paneth cells which is mainly balanced by proliferative enterocyte progenitors. For positions x_{p}>x_{0} the system resembles the situation of a Wnt− system discussed in the text (

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Simulation results on the impact of the threshold TP_{Notch} on the systems behaviour. Increasing the threshold leads to an increased number of secretory cells in the crypt at the expense of undifferentiated cells and enterocyte progenitors. Note that the number of Goblet cell increases only if TP_{Notch} becomes larger than 1 due to discrete numbers of neighbour cells. At a certain value of TP_{Notch} stimulation by the neighbour cell is no longer sufficient and all cells will turn on secretory fates. In this case the system resembles the situation of a Notch− system discussed in the text (

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Example of a simulation of a steady state crypt. Undifferentiated cells (red) and Paneth cells (green) are found intermingled at the crypt bottom, progenitors of enterocytes (blue) and Goblet cells (yellow) move upwards the crypt axis. One second of the video represents 1.25 days.

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Example of simulated clonal conversion in a crypt. A labelled cell clone (pink), expands in the crypt. It originates from an undifferentiated functional stem cell. Colour code of crypt cells as in

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We like to thank Owen Sansom and Johan van Es for providing experimental data on mutation experiments.