Figure 1.
A schematic illustration of the crypts of Lieberkühn.
Neighbouring crypts are closely packed, and each is composed of a monolayer of columnar epithelial cells (nuclei shown in blue). The apical surface of each cell faces the lumen (the brush border of the cells is indicated in red), and the basal surface is in contact with the basement membrane (green). The arrows in the cells indicate the changing mitotic spindle alignment moving up the crypt axis, illustrating the switch from mostly asymmetric to symmetric cell division. Lastly, the surrounding pericryptal fibroblasts are shown in pink, myofibroblasts that provide chemical and mechanical factors for normal crypt structure. The surrounding musculature forms a basket beneath the base of each crypt.
Figure 2.
Musculature of the mouse intestinal wall.
(A) A frozen section of fluorescently labelled small intestine visualised by widefield fluorescence microscopy (red shows F-actin, green shows laminin, blue shows nuclei). (B–E) Fluorescently stained wholemounts of small and large intestine, visualised using confocal fluorescence microscopy (red shows F-actin, blue shows nuclei). (A) A section through the wall of the small intestine. Nuclei (blue) are stained with DAPI, F-actin (red) highlights the smooth muscle cells and apical surface of the gut epithelium. The gut epithelium is continuous from the crypts (C) and over the villi (V). Basement membranes (green) are stained with an anti-laminin antibody, and the basal surfaces of gut epithelial cells are directly attached to the basement membrane (arrowhead). The muscularis externa is composed of outer longitudinal and inner circularly oriented smooth muscle fibres. The muscularis mucosae (just above the dashed line) closely follows the crypt bases, with some fibres extending up into the villi (dashed arrow). Pericryptal fibroblasts surround the crypt epithelium (arrows). (B) A longitudinal section through the small intestine shows the base of crypts (outlined by dashed circles). Smooth muscle fibres of the muscularis mucosae are oriented parallel (arrows) to the longitudinal muscle. (C) A transverse section of small intestine shows the muscularis mucosae (arrow). It is comprised of a single cell layer that forms an incomplete layer or meshwork beneath crypts, just above the circular muscle (CM). (D) A longitudinal section through the colon shows the outer (solid arrow) and inner (dashed arrow) more disorganised layers of the muscularis mucosae. The outer layer is oriented parallel to the longitudinal muscle and the inner layer parallel with the circular muscle, as indicated by double headed arrows. (E) A transverse section through the colon shows the muscularis mucosae (arrow) at the base of the crypts, which is much thicker than that found in the small intestine. Scale bars = .
Table 1.
Model parameters.
Figure 3.
The simple geometry used to investigate the basement membrane force defined by Equation (8).
The top row of yellow cells are the proliferating epithelial cells, whilst the green cells are the non-proliferating stromal cells. The region of the basement membrane highlighted by the red rectangle is assigned a non-zero spontaneous curvature, , whilst the remainder outside this region is assigned a zero spontaneous curvature.
Figure 4.
The evolution of the epithelial layer, where the -coordinates of each epithelial cell are plotted after 60 hours, and the boundaries between the zero and non-zero spontaneous curvature regions are indicated by the dashed lines.
(a) , increasing
(indicated by the arrow); (b)
for increasing
(indicated by the arrows); (c) the total number of epithelial cells in the monolayer for
and increasing
.
Figure 5.
Simulation snapshots illustrating the deformation of the epithelial monolayer for increasing .
(a) ,
; (b)
,
. The layers have deformed from the initial state shown in Fig. 3, and snapshots are taken after 60 cell hours.
Figure 6.
Increasing the spring constant between epithelial cells.
(a) Plotting the -coordinates of the epithelial cells after 60 hours, where the arrow indicates the direction of increasing
. (b) Plotting the number of epithelial cells in the layer over time, averaged over 50 simulations. In both plots
(black), 30 (red), and
(blue).
Figure 7.
The spatial distribution of anoikis and division events, and epithelial cell locations.
Here ,
, and anoikis is the only mechanism of cell removal. (a) Anoikis events, (b) division events, (c) epithelial cell locations at the final timestep, (d) a plot of the
-coordinates of epithelial cells in the layer at the final timestep. Arc length represents the distance along the curve of epithelial cell centres, measured from the centre of the layer.
Figure 8.
Typical cell migration in the epithelial layer when anoikis is the only mechanism of cell removal.
Each colour corresponds to a different cell and tracks the and
-coordinates at each hour. In each case, the cells maintain a roughly constant position, showing that they do not migrate upwards: (a)
-coordinates, (b)
-coordinates. Results are shown from
hours to allow the layer sufficient time to equilibrate.
Figure 9.
The spatial distribution of anoikis and division events that occur in the epithelial layer when density-dependent inhibition of mitosis is implemented.
(a) Anoikis events, (b) division events, (c) epithelial cell locations at the final timestep, (d) six typical migratory tracks for epithelial cells in the monolayer (showing the -coordinates only).
Figure 10.
The spatial distribution of anoikis and division events that occur in the epithelial layer when density-dependent inhibition of mitosis and random cell death are implemented.
(a) Anoikis events, (b) division events, (c) epithelial cell locations at the final timestep, (d) six typical migratory tracks for epithelial cells in the monolayer (showing the -coordinates only).
Figure 11.
A comparison of the final state of the epithelial layer for the three cases considered.
The direction of the arrows indicate the extension of the epithelial layer when random cell death is included at the edges (blue curve), compared to the two cases where this second form of apoptosis is not present.
Figure 12.
Deforming a flat epithelial layer to adopt the test-tube shape of the crypt, as viewed in a cross-section.
To achieve this, a non-zero spontaneous curvature region is defined in the centre of the monolayer, equal to 20% of the width of the tissue block. It is also necessary to increase the size of the stromal cell compartment, as well as define a sufficiently wide epithelial monolayer. Each image from (a) to (h) is a snapshot in time, and the layer is fully deformed after approximately 100 cell hours.
Figure 13.
The cross-sectional crypt configuration.
(a) A haematoxylin and eosin stained wax section of a murine colonic crypt. Scale bar = . (b) The cross-sectional model geometry, plotted against the axes used to mark cell
-coordinates for the results presented. A linearly decreasing gradient of Wnt factors is imposed in the cross-sectional model, which is normalised to 1 at the base of the crypt, and 0 at the crypt collar. A threshold is prescribed such that cells in the region of insufficient Wnt factors are defined to be terminally differentiated. Thus, proliferating and non-proliferating epithelial cells are indicated in yellow and pink respectively, whilst non-proliferating stromal cells are indicated in green.
Figure 14.
A comparison of model results with cell area data for three healthy murine colonic epithelium samples.
(a) Experimental data obtained from three wildtype murine tissue samples, taken from midway down the length of the colon. (b) A histogram of cross-sectional area data collected from the simulation of 1000 hours. These data have been dimensionalised, scaling 1 cell width to , and the frequency has been averaged over total cell number to show the proportion of cells in each bin.
Figure 15.
Cell turnover statistics for the cross-sectional model.
(a) The total number of epithelial cells in the crypt over time. (b) Typical migration tracks for cells that are born at different positions in the epithelial monolayer (each coloured curve tracks the -coordinate of a specific epithelial cell, each born at different locations in the crypt).
Figure 16.
Cell division and death statistics for the cross-sectional model.
(a) The spatial distribution of anoikis events along the -axis, (c) the spatial distribution of division events along the
-axis.
Figure 17.
Defining the location of the basement membrane.
The red dashed line indicates the location of the basement membrane, which is defined to pass through the midpoints of the springs connecting neighbouring epithelial and stromal cell centres. The local discrete curvature is calculated for each epithelial and stromal node pair, and the midpoints of the neighbouring springs are taken to form a piecewise linear curve defined by three points. An example is indicated by the three points marked by black circles.