Model Hypotheses Underlying Cell Proliferation and Differentiation in the Crypt

Asymmetric crypt and villus structures containing heterogeneous cell types are generated under uniform extracellular signalling environments. Therefore, differential responsiveness to signalling rather than differential exposure to extracellular signal governs cell proliferation and differentiation in the crypts [6]. We hypothesised that signals and responses are determined by interactions between neighbouring cells and coupled with cell lineage commitment. Figure 1 summarises our hypotheses to model cell differentiation and proliferation.

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Figure 1. Model for cellular differentiation and proliferation in the crypt of mouse small intestine.

A) Fate of stem cells progeny. Stem cells at the crypt bottom and at position +4 replenish each other and give rise to different progeny according to the strength of Wnt signalling. Wnt signalling is quantified according to the local cell composition as described in Equation 1 with a maximum value of 1 in the crypt base. Under the highest levels of Wnt signalling at the crypt bottom (Wnt  = 1), stem cells progeny consists of secretory cells and stem cells. At lower levels of Wnt signalling (Wnt <0) stem cells give rise to secretory cells and proliferative absorptive progenitors. In both cases the decision between these two fates, secretory and stem/absorptive, depends on Notch signalling. Differentiation into the secretory cellular lineage is inhibited if Notch is activated (Notch^⊕) as a result of 50% or more of adjacent cells belonging to the secretory lineage. B) Secretory cell specification. Secretory cells differentiate into goblet Paneth cell common progenitors with a probability estimated as q  =  Wnt signalling (quantified from Equation 1), otherwise secretory cells become either Tuft or enteroendocrine cells with equal probability. Goblet-Paneth cell common progenitors differentiate into Paneth cells with probability p  =  Wnt signalling (Equation 1), otherwise they differentiate into goblet cells. C) A snapshot of the simulated crypt in steady state shows how all cell types locate on the three dimensional spiral.

https://doi.org/10.1371/journal.pone.0037115.g001

Empirical observations demonstrate that only Paneth and stem cells are located in the crypt bottom cap where the highest levels of Wnt signals are expressed [2]. In our model, we have assumed that stem cells in the crypt base that are exposed to the highest levels of Wnt signalling give rise to stem cells and secretory cells. The size of the crypt base or crypt bottom cap is determined according to the crypt dimensions (see Methods section). By contrast, stem cells located above the crypt base or exposed to low Wnt stimuli generate proliferative progenitors of absorptive epithelial cells and secretory cells (Fig. 1). The decision as to whether stem cells give rise to secretory or to stem/absorptive lineages depends on Notch signalling.

1) Wnt signalling.

We have assumed that cells generated in the crypt bottom cap under the influence of the highest level of Wnt signalling are committed to becoming either stem cells or common goblet-Paneth cell progenitors and that these cell lineages simultaneously generate high Wnt signalling environments. Published results support the production of Wnt signals by Paneth cells and proliferative cells [19], [28]. Wnt signalling is quantified as follows:

For any cell in the crypt, Wnt signalling is modelled according to the local cell composition as:(1)Where

(2)N_Div is the number of divisions already undertaken by proliferative absorptive progenitors; I is the set of cells in the rings located immediately above and below the cell in question, cell_j; card(I) is the number of cells in those rings.

The Wnt signalling activity quantified for each cell varied from 0 to 1, with 0 reflecting cells that are not near Paneth and/or proliferative cells and 1 (or 100% in Fig. 2A) representing cells surrounded by Paneth and/or stem cells. From Equation 1, it follows that Wnt signalling is equal to 1 for all cells located in the crypt bottom (Fig. 1). A reduction in the levels of Wnt signalling is a consequence of the diminishing number or absence of Paneth and stem cells resulting in progressive cell differentiation as the cell ascends the crypt. These Wnt signalling predictions are consistent with independent published observations [19], [28] (Fig. 2A).

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Figure 2. Modelling Wnt and Notch signalling in Paneth cells intermingled with Lgr5^hi stem cells at the crypt base.

A) Wnt signalling activity (grey circles) as simulated by our model and measured experimentally (solid line) [28]. A Wnt activity level of 1 (Equation 1) is equivalent to 100%. The results shown were obtained from 30 simulated crypts. B) Number of Paneth or Paneth progenitor and stem cells at the bottom of the crypt under high Wnt signalling activity (100%) assuming that a newly generated cell differentiates into a common goblet Paneth cell progenitor if in contact with 1 or more stem cell; otherwise secretory fate is inhibited by Notch signalling (hypothesis I). Similarly, hypotheses II, III, IV, V and VI assume that newly generated cell differentiate into a Paneth cells if in contact with a number of stem cells equal or greater than 2, 3, 4, 5 and 6, respectively. The total number of cells in contact is 6. Hypothesis III, which states that secretory fate of a stem cell is inhibited by Notch signalling if it is in contact with more than 50% of cells that belong to the secretory lineage, which best fits empirical observations. The results shown are from 30 simulated crypts. Coefficient of variation for the number of cells (CV  =  standard deviation/mean) varied from 0.015 to 0.04 in all cases. C) Upper view of the distribution of Paneth and stem cells in the crypt base at day 20 according to the six hypotheses.

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Slow Dynamics and Cell Motion

We have used a Montecarlo model to describe crypt reorganization as a slow structural relaxation phenomenon. At each interval time of the simulation, the three dimensional spiral and helix are re-built according to the size of the cells and division or deletion events. Growing cells expand homogenously in all directions. The size of the rings of the spiral changes dynamically to accommodate cell size increases in the horizontal direction, perpendicular to the crypt-villus axis (Fig. S2A). The increase in cell size in the vertical direction creates a force translated mostly in upward migration. Figure S2B shows that the predicted ascending velocities are close to zero at the crypt bottom but increase progressively from position +4 to 12 where cells migrate at a velocity of ∼1 cell diameter per hour. These results agree with empirically determined migration velocities [3]. Thus, the force derived from cell growth is sufficient to explain cell dynamics in the crypt. The trajectories of Paneth cells that remain at the bottom of the crypt, and those of stem cells located at the crypt bottom, the progeny of which migrate upwards and differentiate into secretory and absorptive progenitors, are shown in Figure S2C. For cells located beyond position +7 in the crypt the model generates vertical trajectories towards the villi.

Our approach excludes active forces to account for the downward migration of goblet/Paneth progenitors and/or mature Paneth cells to the crypt bottom. The localisation of Paneth cells or their progenitors intermingling with stem cells at the crypt bottom is driven by Notch signalling as described above (Fig. 2C), and by the fact that Paneth cells are non-proliferating and therefore non-migratory cells. It can be demonstrated that in a mixed system of proliferating and non-proliferating cells restricted to the bottom of a cylindrical shape, active forces are not required to confine non-proliferating cells to the bottom of the cylinder. Actively dividing cells will eventually gain positions above non-proliferating cells (see Fig. S3).

Modelling Crypt Generation from One Stem Cell

The final number of cells in a crypt is the only input parameter needed to build the crypt from a single stem cell. The length and diameter of the crypt and the size of the crypt bottom cap is estimated from the final number of cells. Relationships between the number of cells and crypt morphometric measurements were estimated from published data [32], [44] (see Methods section).

The crypts are initially modelled as closed spheroids with a central lumen that expand radially through cell proliferation. By 4–5 days when the crypt reaches its final size of 235 cells (Fig. 3A–D), the spheroid shape progressively elongates to form a large open-ended cylinder. Of interest, individual stem cells from the small intestine or colon initially give rise to closed spheroid-like organoids that then undergo multiple crypt fission events to generate villus like epithelial domains when grown in culture [6], [45]. Initially, only proliferative cells are generated during the expansion phase of the crypt with mature epithelial cells being generated after the crypt reaches its final size (Fig. 3A–D). The predicted number of cells during crypt expansion is in agreement with that observed in cultured organoids generated from a single cell [6] (Fig. 3E). There is a lack of data regarding the timing and order of appearance of differentiated cell types. We have therefore assumed that differentiation starts simultaneously for all cell lineages, once absorptive epithelial proliferating progenitors complete their committed cellular divisions and the first enterocytes are generated (Fig. 3B and D). It has been reported that all cell lineages are observed in cultured crypts within two weeks of seeding with single cells [6]. In our simulated crypts, cell differentiation starts by day 6, with a steady state structure obtained after day 9–10 (Fig. 3 F and G). A video of the generation of a crypt from a single stem cell and its development over 40 days is included in the supplementary material (Video S1).

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Figure 3. Modelling crypt development from a single stem cell.

A–D) The number of cells during the development of a crypt from a single stem cell. The total number of cells, 235, is reached in about 5 days with a structural steady state reached by day 11. The simulation is for 100 days. E) Comparison between simulated (circles) and observed (columns) numbers of cells in a crypt developing from one stem cell [6]. Data shown is from 30 simulated crypts. F) Cellular crypt composition and Wnt signalling by day 4 when the growing crypt reaches its final number of cells. G) Cellular crypt composition and Wnt signalling by day 10 when the crypt reaches a structural steady state.

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Clonal Evolution of Equipotent Lgr5^hi Stem Cells is Consistent with Neutral Drift Dynamics

Asymmetric DNA segregation during stem cell division is regarded as a protective DNA mechanism in which stem cells undergoing mitosis retain the original DNA strand and transfer the newly synthesised strand to their descendents [46], [47]. However, the random segregation of chromosomes in intestinal stem cells has been recently confirmed by genetic lineage-tracing experiments [48]. In addition clonal expansion has been used to demonstrate that stem cells are equipotent and divide symmetrical symmetrically following a neutral drift dynamics in which clones are lost or expand at random until one takes over the crypt [31], [49]. In our model stem cells are equipotent but do not always divide symmetrically. The cellular fate is decided according to neighbouring cells. With our hypothesis we estimate that 53% of the stem cells divisions are symmetric and 43% are asymmetric giving a stem cell and goblet-Paneth cell progenitor with the remaining 4% producing common goblet-Paneth cell progenitors. To demonstrate that our model is consistent with empirically determined neutral drift dynamics [31], we followed the clonal evolution of 14 equipotent Lgr5^hi stem cells in a simulated crypt comprising all cell lineages (Video S2). We observed rapid clonal expansion during the first 7–10 days (Fig. S4A–B) during which the number of clones was reduced from 14 to 2–3. Analysis of clonal evolution during the first week was consistent with a pattern of neutral drift dynamics. We also observed that the cumulative clone size distribution converged rapidly into scaling behaviour, characteristic of equipotent stem cells dividing symmetrically [31], [49] (Fig. S4C). The scaling function was estimated as previously described [31] using a cell loss of 0.94/day, which was estimated from the average clone size simulated during the first week (Fig. S4D) and was similar to the expected replacement of stem cells according to the division cycle of stem cells (21.5 h). The generation of non-proliferating Paneth cells did not affect the neutral drift towards clonality.

After the first period of rapid clonal expansion the crypts remained in an oligoclonal state for relatively long and variable periods of time (Fig. S4A); the time required to observe the first monoclonal crypts varied from 15 to >200 days. Experimentally, the first monochromatic crypts are observed 2 weeks after labelling of stem cells with 75% of crypts becoming monoclonal after 2 months with some oligoclonal crypts still present after 30 weeks [31].

Altering Signalling Pathways

Our model is based upon reported observations [28] in which the absence of Wnt signalling halts stem cell proliferation forcing their differentiation into the enterocytic cell lineage (Fig. 1). In addition, we have modelled the effect of the inhibition of the Wnt signalling components Frizzle 5 receptor and Sox 9 and Math 1 transcriptional factors, by inhibiting death of Paneth cells migrating up the crypt, the maturation of progenitors into Paneth cells, and the differentiation of cells into the secretory cell lineage, respectively, as seen experimentally [16], [17], [50], [51]. Following inhibition of the Wnt signalling pathway, our model shows a complete block in proliferation by day 2 (Fig. 4A), which is the time needed for the proliferative absorptive progenitors to undertake committed divisions and differentiate into enterocytes. Thus, non-proliferative enterocytic cells were detected all along the simulated crypt including at the stem cell position with some Paneth cells mislocated along the crypt. After the first 2 days a progressive increase in apoptosis of the differentiated cells was detected leading to a marked reduction in the number of cells in the crypt by days 4–6. By day 8 the simulated crypts had less than 50 cells. These simulated results are consistent with the published experiments [28] (Fig. 4A) and the supplementary material includes a video (Video S3) of the simulation.

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Figure 4. Blocking Wnt and Notch signalling in a steady state crypt.

A) Blocking the Wnt signalling pathway halts cell proliferation and differentiation. Between days 2–3 all cells in the crypt differentiate into the enterocyte lineage with some aberrant Paneth cells scattered along the crypt. The colour code is as in Figure 3 with light orange lines representing aberrant mislocated Paneth cells in the crypt. Insert shows secretory and stem cells. B) Deletion of the Sox9 transcriptional factor of the Wnt signalling pathway blocks Paneth cell maturation which is coincident with a decline in the number of stem cells that co-localize with residual Paneth cells; C) Five or six days after blocking the Notch signalling pathway the entire proliferative absorptive progenitors compartment is populated exclusively by secretory cells, and while all secretory cell compartments expand, the number of stem cells decreases. Insert shows secretory and stem cells.

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Paneth cells are important for maintaining the stem cell niche [19], [31]. In our model, we assumed that suppression of Sox9 blocked the maturation of the common goblet/Paneth cell progenitors into Paneth cells (Fig. 1) resulting in the complete disappearance of Paneth cells and their replacement by epithelial proliferative cells at the crypt base as previously reported [50]. The decrease in Paneth cells causes a local reduction of the predicted levels of Wnt signals and therefore a reduction in the number of stem cells which co-localize with the Paneth cells (Fig. 4B). These results are consistent with experimental observations [19], [31].

In our simulated crypt when Notch is blocked we assumed that the inhibition of the secretory fate requires contact with more than 80% of secretory cells, instead of the 50% required according to hypothesis III derived from the intermingled distribution of Paneth and stem cells in the crypt base (Figs. 2B, 2C). We also assumed that the maturation time of the common goblet-Paneth cell progenitor increases by a factor of three. Consequently the number of stem cells is markedly reduced while maximum Wnt signalling levels are maintained such that proliferative absorptive progenitors are not generated (Fig. 1). Consistent with recent findings [23], in the simulated crypt the proliferating cell compartment is replaced by secretory cells within 5 days, which are mostly goblet cells and common goblet-Paneth cell progenitors. The number of Paneth, enteroendocrine and Tuft cells also increases, while the number of stem cells decreases (Fig. 4C and video S4).

Model Predictions

(1) Physical Interactions.

We have predicted the probability of physical interactions between cell types in the crypt by randomly sampling 500 times a simulated crypt in steady state for 80 days (Fig. 5A). In the simulated crypt, ∼70% of the cells in contact with stem cells were Paneth cells or their progenitors, while ∼20% were stem cells with the remaining cells being secretory cells and proliferative absorptive progenitors. These proportions of cells result from a non-random mixing process that is regulated by Notch signalling and increases the contact area between the two cell populations. It has been experimentally observed that 80% of the surface of stem cells is in contact with Paneth cells, with this physical interaction being important for the maintenance of the stem cells niche [19]. Goblet, enteroendocrine and Tuft cells as well as proliferative absorptive progenitors generated in the lower part of the crypt body eventually contact all the other cell types with a probability associated with the size of the crypt compartment. Mature enterocytes generated in the upper part of the crypt are unlikely to interact with stem or Paneth cells (Fig. 5A).

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Figure 5. Model Predictions.

A) Percentage of contact time between cell types in a mature crypt observed for 80 days. Colour code as in Figure 3. B) The chance of successful propagation of mutations affecting stem cells is ∼50% when the mutated cell is on the floor of the crypt and <2% for locations above. C) The time for the propagation of a mutation to the stem cell compartment is shorter when the mutation decreases the duration of the division cycle. Black columns show time for the propagation of mutated cells with 10-fold shorter division cycle, grey columns for 2-fold shorter, and white columns when division cycle is not affected. D) Time to replenish the crypt with the initial number of cells (•) and to recover the cellular structure of mature crypts after random killing of cells in the crypt (○). E) The numerical recovery rate of the injured crypt is 0.78 cells/day. The maturation rate is 0.34 cells/day. The time required to reach the original number of cells for an injured crypt that has lost half of its cells is approximately 0.9 days while the time required to reach a structural steady state is approximately 2 days. F) Time to replenish the bottom of the crypt with 10 Lgr5^hi cells after random killing of stem cells.

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Cell Growth

Cells are modelled as regular convex shapes in two dimensions. For proliferating cells, the cell surface, S, grows at a constant rate, k =  S_ini/D_c, reaching twice its initial size, S_ini, during the duration of the division cycle, D_c. For cells undertaking differentiation, the cell surface grows to reach a maximum cellular size, S_max, according to:(3)

where k is the growth rate determined at birth as described above.

The time to reach S_max is the maturation time. During maturation time goblet-Paneth cell common progenitors (Fig. 1) remain undifferentiated. Cells reaching the end of their lifespan undergo apoptosis and progressively detach from the mesenchyme. During apoptosis the crypt surface covered by the cell decreases at a constant rate estimated as h  =  S_max/(0.1*Lifespan).

The cell surface area is assumed to have a normal distribution. Estimations of ca. 10 and 13 µm for the mean surface area of proliferating and non-proliferating cells, respectively, and of 0.05 for their coefficient of variation (CV  =  standard deviation/mean) were empirically determined. Both S_ini and S_max are randomly generated at birth time.

Cell Division Cycle and Lifespan

The duration of the division cycle, D_c, has been reported to decrease as cells escalate up the crypt [32]. An average cell cycle length equal to 21.5 h has been observed for Lgr5^hi stem cells [13] while +4 stem cells have been reported to divide once a day [58], [59] and the measured shortest cycle length of a proliferative progenitor is of 9.7 hours [32]. We assumed that D_c decreases as proliferative absorptive progenitors undertake successive divisions during their migration up the crypt. D_c is assumed to have a gamma distribution and its parameters are estimated by the method of moments from the mean value and CV. The estimation of CV was equal to 0.144 according to published data and no association between the value of CV and the duration of the cell cycle has been observed [32]. In order to find the relationship between the mean value, , and the number of undertaken divisions, N_d, we ran our model to simulate N_d values at each crypt location and correlated them with the mean value for the duration of the cell cycle, , at each crypt location as previously observe [13], [32]. The relationship was as follows (Fig. S5A):(4)

For stem cells, N_d is fixed to 0. Data shows a lack of dependence for proliferative progenitors that undertake more than 2 division cycles; hence the maximum value of N_d is fixed to 2. Observed and estimated cell cycle durations according to cell position in the crypt were consistent (Fig. S5B).

Lifespan was modelled as a gamma distributed random variable for all cell lineages. Parameters of the distributions were estimated from empirically derived mean values and CV. Longevity of Paneth cells had a expected value equal to 54 days based on published studies [60]. The mean values for the lifespan of goblet and endocrine cells were equal to 5 and 23 days, respectively, as previously reported [32] while for enterocytes the mean value was estimated as 5 days from published results on the loss of all intestinal epithelial cells after blocking Wnt signalling [28] The CV of these variables was assumed to be equal to 0.144 as reported by Wright [32] for other cell cycle times.

Number of Divisions Undertaken by Proliferative Absorptive Progenitors

As previously proposed [61], the total number of proliferating cells is the sum of the expected number of cells in each proliferative compartment. The first proliferative compartment contains Lgr5^hi stem cells, named Lgr5^hi. The second compartment contains +4 stem cells, named Stem4, that receive an influx of cells from the previous compartment and provides an efflux of cells to the following compartment, T₁, of proliferative progenitors in first division, the descendents of these are proliferative progenitors in second division, T₂, and so on until the n-th division, T_n. The flux of cells between the first three compartments cannot be calculated straightforwardly from generation rates because part of the newly generated cells in these compartments will be non-proliferative Paneth, goblet and enteroendocrine cells with their replacement rate depending on the velocity of cell migration and rate of apoptosis at the bottom of the crypt. The expected number of cells in the compartments of proliferative absorptive progenitors, T_i for i >1, can be estimated as the efflux of cells coming from the previous compartment(5)

Where λ_i−1 is the growth rate of the cell population in the previous compartment estimated as:(6)

D_c estimated from equation (4).

The following expression describes the number of proliferating cells, N_prolif, in the crypt:(7)

As explained above λ_i  =  λ₂ for i >1, thus the number of committed divisions, n, to be undertaken by proliferative absorptive progenitors can be estimated from equation (7) as follows:

Then(8)

From published data [32], [44], it can be estimated that about 73% of the cells in a mature crypt are proliferating cells (Fig. S5C). We simulated crypts with sizes between 100 and 400 cells and solved equation (8) with the values obtained for the number of cells in the Lgr5^hi, +4 stem cells and T₁ compartments (Fig. S5D). The estimated number of divisions to be undertaken by proliferative progenitors, n, varied from ca. 5.4 to 6.7 (Fig. S5D), with a mean value of 6, which is the value used in the model.

Organization of the Three Dimensional Crypt

The bottom of the crypt is modelled as a three dimensional spiral followed by the crypt body, which is a three dimensional helix, constructed from single cells organized in a one-dimensional chain. The position of any cell in the spiral or helix is determined by the polar coordinates r and θ and the vertical coordinate z.

The value for the polar coordinate, r_i, of the i-th cell is estimated as(9)

The value for θ_i, the polar angle of the i-th cell is equal to(10)where the square root of S_i represents the width of the i-th cell.

The third coordinate z_i is calculated as follows:(11)where l is the index of the closest cell to the i-th cell from the ring immediately below.

The initial values for , k and n_spiral in equations (9), (10) and (11) are estimated according to the relationships between morphometric measurements of the crypt, cell sizes, number of proliferative cells and total cells in the crypt from published data of small and large intestinal mucosa of humans [44] and mice [32]. Based on these relationships, the total number of cells, ncells, is the only input parameter needed to build the crypt.

The relationship between the total number of cells and the circumference of the crypt is as follows (Fig. S5E):(12)From the same dataset we could estimate that 73% of the cells in a mature crypt are proliferating cells (Fig. S5C). Hence the initial value for the radius of any ring in the helix, i.e. for in equations (9) and (11), is equal to:(13)where Smax is the surface area of mature Paneth cells and (1.5 Sini) is the average surface area of proliferating cells.

Assuming that the bottom of the crypt is a semi-sphere, its surface can be estimated as:(14)

Knowing that there are aproximately 20 Paneth cells [32] and 14 Lgr5+^hi stem cells [31] at the base of the crypt, and assuming that this proportion holds for any number of cells in the crypt, the initial estimation of the number of cells covering the surface of the spiral will be:(15)

The first n_spiral cells of the one dimensional chain of cells will be located on the spiral and the initial value for the length of the spiral can be approximated as(16)

Since the polar coordinate r for the last cell in the spiral should be equal to the initial radius of the helix, the initial value for k in equations (9), (10) and (11) is equal to:(17)

Cells divide, grow, migrate and undergo apoptosis. These processes affect the cell size. The rings of the crypt are flexible to accommodate changes in cell size (Fig. S2A). To achieve this, the values for k and from equations (9), (10) and (11) are calculated for the m-th time step of a simulation as follows:(18)and(19)where {g1 .. gn} is the set of index in the g-th ring of the helix.

In this way changes in cell size affects the radius of the spiral and helix. The rings of the crypt expand and shrink to accommodate these changes while the angular position of the cells is virtually unaffected, which minimize lateral swings of cells (Fig. S2A,C).

We assumed that cells located on the spiral form the crypt bottom cap, which is characterized by the highest levels of Wnt signalling and contains Lgr5^hi stem cells and Paneth cells (Fig. 1). Figure S5F shows the number of Lgr5^hi stem cells, Paneth cells, and proliferative absorptive progenitors generated a day and the diameter and length of simulated crypts of several sizes between 100 and 400 cells.

Spatial Organization of Organoids During Development from One Single Cell

When modelling crypt development from a single cell, crypts are initially spheroids with a central lumen. The radius of the sphere containing n cells is estimated as:(20)where

(21)When the number of cells, n, reaches the target crypt size, the elongation of the crypt is modelled by the progressive decrease of the radius of the sphere that becomes a closed cylinder. The radius of the sphere decreases to the value of r_helix in equation (13) in about one day time. During the elongation time the crypt structure is still closed and the parameters for the bottom and top spirals are estimated according to equations (15–17).

Cell Dynamics

Growing cells expand in all directions and this expansion affects its position and the position of the other cells when the crypt is reorganized at each time interval. The size of the rings of the spiral changes dynamically to accommodate cell size increases in the horizontal direction, i.e. perpendicular to the crypt-villus axis. The increase in cell size in the parallel direction to the crypt-villus axis creates a force translated mostly as upward migration because the crypt bottom is a rigid boundary and we do not consider the possibility of cell compression. Downward motion could eventually be observed involving very short trajectories when cells located in lower positions undergo apoptosis (Fig. S2C). Therefore, the functional form for the instantaneous vertical velocity of cell j, v_i:(22)

Where I is the set of indices of those cells located in the vertical axis below cell j and L_i is the length in the vertical axis of cell i. Figure S2B shows the average vertical cell velocities calculated by simulation for each ring in the crypt.

After division a new cell has to be located in the neighbourhood of the original cell. The division event does not imply any increase of cell surface and thus its impact on cell dynamics can be neglected. In practice, after division the one dimensional chain used to build the three dimensional helix has to be reorganized to assign an index to the new cell. The new cell takes either the index of any apoptotic cell in contact, which is deleted, or otherwise it takes the index of the closest cell in the ring immediately above, position at which the new cell is already protruding. The vertical adjustment of indices spreads upwards along the crypt until the cell in the last ring, which is usually protruding out of the crypt, moves from the crypt to the adjacent villus. Newly generated cells in the first spiral ring forming the floor of the crypt are located laterally or vertically with equal probability. Therefore the reassignment of indices locates cells in their actual ring position after displacements due to growth.

Simulation

A Visual Basic Excel add-in has been written in Visual Basic to simulate the crypt. At simulation times, Wnt activity, size, position and age of all cells are updated. Time steps are variable.