A Two-Dimensional Model of the Colonic Crypt Accounting for the Role of the Basement Membrane and Pericryptal Fibroblast Sheath
[['Colorectal cancer ( CRC ) is one of the leading causes of cancer-related death worldwide , demanding a response from scientists and clinicians to understand its aetiology and develop effective treatment .', 'CRC is thought to originate via genetic alterations that cause disruption to the cellular dynamics of the crypts of Lieberkühn , test-tube shaped glands located in the small and large intestine , which are lined with a monolayer of epithelial cells ( see Fig . 1 ) .', 'A delicate balance of cell division , migration and death is coordinated in the crypts to renew the epithelial layer every few days [1] , [2] .', 'The regular upward migration and removal of cells from the crypt provides a frontline defense mechanism against potential damage from mutated cells , which are prevented from remaining in the crypt long enough to do significant damage .', 'However , if cells accumulate genetic mutations that alter migration velocity or provide resistance to apoptosis cues , then such cells acquire the ability to persist and multiply in the crypts .', 'This alone can increase stress on the walls of the crypts , but the problem will be aggravated if such cells acquire additional mutations that increase proliferation , or alter cell-cell adhesion .', 'In turn , the increased stress can cause the walls of the crypt to buckle .', 'Dysplastic crypts allow the formation of a benign adenoma if mutated cells do not leave the crypt as they should , but rather persist and proliferate in a localised area .', 'Over time and via accumulated mutations , these growths can progress to a malignant lesion that can break through to the underlying tissue stroma , and so aid metastasis .', 'The dynamic cell properties that are required to initiate crypt buckling are poorly understood , as it is difficult for biologists to observe experimentally , either in vivo or in vitro , the initial changes in this sequence of events .', 'For example , the organoids grown in culture by Sato et al . [3] , while recapitulating the crypt geometry , have not yet been compared in detail with crypts in situ .', 'The organoids lack some of the forces that are present in vivo , and the cells themselves do not migrate .', 'However , performing in silico experiments using a computational model of the crypt in situ could highlight the conditions required for buckling to occur , and so provide crucial insight into the tissue-level effects of genetic mutations that lead to CRC .', 'To achieve reliable predictions of the breakdown of the crypt structure that occurs at the onset of carcinogenesis , such a theoretical model of the crypt must link processes occurring at the subcellular , cellular and tissue levels .', 'The model must also take into account the tissue structure and geometry .', 'While a fully comprehensive model is not yet realised , this work concerns a key step in the development of a predictive , computational model of the crypt which defines structural components in accordance with the tissue architecture that is observed experimentally , and reported here .', 'These elements are incorporated into a crypt model which also addresses the coordination of cell division , polarity , differentiation and apoptosis .', 'As depicted in Fig . 1 , individual crypts are closely packed , surrounded and separated by connective tissue .', 'Each crypt is lined with an epithelial monolayer that consists of contiguous cells separated from the connective tissue and musculature by the basement membrane , the primary contact site for epithelial cells to the extracellular matrix .', 'Below the basement membrane are myofibroblasts that provide chemical and mechanical factors for normal crypt structure .', 'There is an established proliferative hierarchy of cells within the epithelial layer: stem cells reside at the base and divide to produce transit amplifying cells , which migrate up the crypt and perform several symmetric divisions before terminally differentiating .', 'The polarised epithelial cells are oriented with the apical membrane facing the crypt lumen and , during symmetric division , mitotic spindles align parallel to the tissue layer [4] , [5] .', 'Consequently a cell places its daughter cell next to it within the plane and the monolayer is maintained throughout growth .', 'Asymmetric division occurs as a consequence of the perpendicular alignment of the mitotic spindle .', 'Differentiated epithelial cells , upon having reached the crypt collar , undergo apoptosis and/or are shed into the lumen [6] , [7] .', 'This process permits the renewal of the epithelial layer every few days .', 'In addition to this , a form of programmed cell death , anoikis , is triggered when there is inadequate adhesion of the epithelial cells to the extracellular matrix [8] , with detachment inducing apoptosis [9] .', 'Functioning correctly , this maintains tissue homeostasis by restricting proliferation to the monolayer , thereby averting dysplasia , and by preventing cells from reattaching in another location and resuming growth .', 'The Wingless/Int ( Wnt ) signalling pathway is involved in the control of cell proliferation , migration , differentiation and adhesion in the crypts [10] , [11] .', 'The Wnt signalling pathway is required to maintain the stem cell compartment in the crypt , and so is crucial to stem cell renewal and differentiation [12] .', 'Moreover , it has been observed that there is a spatial gradient of extracellular Wnt signalling factors along the vertical crypt axis , which suggests a localised source of diffusible Wnt factors in the stroma that surrounds the crypt base , and leads to the hypothesis that a Wnt gradient may be responsible for the observed proliferative hierarchy [13] .', 'As described in Van Leeuwen et al . [10] , cells in the presence of high concentrations of Wnt cycle for longer than those exposed to low Wnt and hence cells at the base of the crypt are expected to remain proliferative .', 'A number of mathematical models exist that aim to describe specific aspects of crypt behaviour , from Wnt dependent ordinary differential equation ( ODE ) cell cycle models that govern mitosis of individual cells [14] , [15] , to cellular automata and lattice-free mechanical models of cell proliferation and migration [16]–[19] .', 'These ideas have been combined in a multiscale model that has been used to investigate clonal expansion and the disruption of crypt homeostasis that forms the first step in colorectal carcinogenesis [15] .', 'However , these models restrict the domain of investigation by prescribing a rigid , cylindrical geometry to the crypt , and are limited by simplifying the tissue structure without considering the basement membrane and surrounding stroma .', 'This prevents such models from realistically examining the tissue-level effects of abnormal cell behaviour .', 'There also exist models that seek to describe crypt buckling .', 'Edwards and Chapman ( 2007 ) [20] present a continuum representation of the crypt , modelled as a growing beam , while Drasdo and Loeffler ( 2001 ) [21] apply an off-lattice overlapping spheres model to describe a two-dimensional ( 2D ) chain of deformable circles such as occurs during blastula formation , and then restrict this to a U-shape for modelling the crypt .', 'These models commonly assign a bending stiffness to the layer , and predict that buckling will occur if growth by cell division is not adequately matched by this force .', 'Edwards and Chapman generalise cell division events and so do not implement a specific cell cycle model to govern mitosis , though possible in this framework , and none of these examples take into account the deformation of the surrounding tissue stroma .', 'More recently , Nelson et al . ( 2010 ) [22] extended the continuum model due to Edwards and Chapman [20] to investigate how growth of an epithelial monolayer constrained to a flexible substrate can recapitulate the geometry of the crypt , and Hannezo et al . ( 2011 ) [23] present a model of the intestinal crypt-villus architecture arising from a buckling instability in a proliferating epithelial monolayer lying on an elastic substrate .', 'A three-dimensional ( 3D ) agent-based crypt model was proposed by Buske et al . ( 2011 ) [24] , which defines lineage specification and differentiation according to threshold-dependent rules that correspond to the effects of Wnt- and Notch- signalling .', 'This model addresses the pedigree concept of cell stemness , and reproduces the spatio-temporal organisation experimentally observed in the crypt without assuming an explicit stem cell population .', 'For this purpose , the authors model the basement membrane as a fiber network with a defined local radius for each cell position , which thereby defines a fixed crypt geometry .', 'Consequently , it is not possible to follow any deformation of the crypt structure , and the authors do not include more sophisticated subcellular pathways that determine cell division or fate .', 'Also relevant to the work presented here are those cell-based models which consider , for example , generic epithelial monolayers .', 'In particular , Galle et al . ( 2005 ) [25] present a 3D overlapping spheres model to examine growth regulation in epithelial layers , where deformation of the cells is calculated using the Hertz force law .', 'This model considers the role of anoikis and density-dependent inhibition of cell division , and how failure of the former can be prevented from corrupting the monolayer if contact-mediated growth inhibition is applied and there is sufficiently strong cell-substrate anchorage .', 'Schaller and Meyer-Hermann ( 2005 ) [26] propose a 3D model to investigate the growth of tumour spheroids , and while cell shapes are again defined as deformable spheres , the neighbour interactions are instead determined by a weighted Delaunay triangulation between cell centres .', 'The dual Voronoi tessellation is applied to provide a more realistic definition of the contact surface between neighbouring cells , which is subsequently used throughout the calculations instead of the sphere contact surface .', 'Drasdo et al . ( 2007 ) [27] also consider the growth of monolayers on a substrate and multi-cellular spheroids , and revisit single-layered tissues such as the blastula during development ( considered in [21] ) to examine the mechanical influence of contact inhibition on the growth of the cell population .', 'Such examples demonstrate the usefulness of individual-based models to investigate the growth dynamics of epithelial cell populations .', 'Dunn et al . ( 2011 ) [28] define a discrete off-lattice cell centre multiscale model that focuses on the role of the basement membrane beneath a growing epithelial monolayer in a simplified 2D geometry: a single layer of proliferating epithelial cells constrained to lie on a rectangular bed of stromal cells , which approximate the connective tissue .', 'Spatial connectivity is determined by a Delaunay triangulation of cell centres , and interactive forces are modelled as springs that act along the edges of this triangulation .', 'An additional force is applied to model the role of the basement membrane , which acts in proportion to the local curvature of the epithelial layer , and to maintain a zero spontaneous curvature .', 'Results from this simple geometry show that a large enough basement membrane force successfully maintains a stable , flat monolayer throughout successive division events , and that increasing the strength of this force favours horizontal migration along the layer , reducing the incidence of epithelial cell detachment from the layer ( whereupon cells are removed by anoikis ) .', 'This work presents the foundation of a realistic representation of epithelial cell growth and migration in a deformable environment , and is extended here to model a specific case in a realistic 2D geometry – the cross section of the crypt .', 'Given the coupling that exists between events at the genetic level and the tissue level , it is necessary to extend the scope of theoretical modelling to address both the role of the crypt geometry and subcellular events .', 'In addition , such a multiscale model should include a mechanical description of migration , cell-cell and cell-matrix adhesion; in so doing , the model can more fully describe all of the processes inherent in crypt dynamics and homeostasis .', 'The remainder of this paper is composed as follows .', 'Firstly , experimental results are discussed that examine the tissue structure of the crypt .', 'These results identify the composition of the connective tissue and surrounding musculature , and how the components relate to crypt shape and function .', 'These findings are incorporated into a new crypt model which assumes the basement membrane force proposed by Dunn et al . ( 2011 ) [28] , and investigations are conducted firstly using a simple rectangular geometry , to determine appropriate parameter balances and investigate the migration of epithelial cells out from the crypt base region .', 'Conclusions from this modelling step inform parameter choices for a complete 2D cross-sectional geometry which is subsequently defined , and the behaviour of the extended model is demonstrated .', 'The results and future work are discussed , where the advantages as well as the restrictions of the model are highlighted , and experiments to investigate model hypotheses are suggested .', 'The direction for future work is outlined , centred on an extension of the 2D cross-sectional model to a realistic 3D geometry .'], ['Immediately beneath crypts lies a thin layer of smooth muscle , known as the muscularis mucosae ( MM ) that forms the boundary between the mucosa and submucosa , as shown in Fig . 2 .', 'In the small intestine of the mouse , the MM is one or two cells thick and forms a network that follows the contours of the crypt bases ( Figs . 1 and 2 ) .', 'By examining intact mouse tissue in three dimensions , we found that , contrary to the reported structure of human gut tissue , the smooth muscle cells of the small intestinal MM are oriented mostly parallel to the longitudinal muscle layer of the muscularis externa ( ME ) ( Fig .', '2 ( A ) , ( B ) , ( C ) ) .', 'In the small intestine , the smooth muscle fibres of the MM extend up into the villi .', 'It is thought that the role of the MM is to constantly agitate the epithelium gently to help expel secretions from crypts and enhance contact between epithelium and luminal contents [29] .', 'When viewed in transverse section ( Fig .', '2 ( A ) ) , the MM appears to follow closely the outline of the base of each crypt .', 'When viewed in longitudinal section ( Fig .', '2 ( B ) ) the MM appears to form individual baskets beneath each crypt , analogous to an eggbox that contains each crypt base as a single egg .', 'The MM of the colon is composed of two distinct layers of smooth muscle fibres , the outer orientated parallel with the longitudinal ME , the inner layer more disorganized , but generally oriented parallel with the circular ME ( Fig .', '2 ( D ) , ( E ) ) .', 'Other components of the mucosa are a laminin-rich basement membrane that is directly attached to the basal surface of gut epithelial cells ( Fig .', '2 ( A ) ) and , just below , surrounding each crypt , a pericryptal fibroblast sheath ( PCFS ) , comprising a highly organized system of fibroblasts , collagen and mucopolysaccharide ground substance [30] .', 'There are 38 PCFS cells per mouse small intestinal crypt and 124 per colonic crypt [31] .', 'PCFS cells produce signaling factors involved in the growth and maintenance of the crypt .', 'Beneath the MM lies the submucosa ( SM ) , which consists of loose connective tissue rich in collagen and elastic fibres .', 'Embedded in this material are larger blood vessels , lymphatics and nerves .', 'The SM is enclosed by the muscular wall of the gut , called the muscularis .', 'It consists of outer longitudinal and inner circular layers of smooth muscle .', 'The muscularis is responsible for peristalsis , the contractile movements involved in advancing intestinal contents .', 'A discrete off-lattice cell centre model is defined , in which spatial connectivity is determined by a Delaunay triangulation of cell centres , and cell shapes are prescribed by the Voronoi tessellation of these centres .', 'Interactive forces are modelled as springs that act along the edges of this triangulation , as described in the Materials and Methods section .', 'Individual model components are now summarised .', 'All parameters are given in Table 1 .', 'Firstly , in silico experiments were run to demonstrate the effect of increasing the spontaneous curvature in the central region , , and the strength of the basement membrane force as governed by the parameter , which characterises the strength of adhesion of the epithelial layer to the basement membrane and the stiffness of the membrane itself .', 'Figs .', '4', '( a ) and', '( b ) illustrate the change in behaviour of the monolayer by plotting the -coordinates of all epithelial cells at the final timestep for typical simulations .', 'In', '( a ) , the arrows on these plots indicate the direction of increasing , while in', '( b ) the arrows indicate increasing .', 'Also marked are the boundaries between the non-zero and zero target curvature regions .', 'As these plots are generated from typical simulations , the curves are not symmetric due to recent division events .', 'Simulations reveal that as the spontaneous curvature increases , the epithelial monolayer is pushed further down into the tissue stroma as the central portion of the monolayer bends , behaviour that is demonstrated clearly in Fig . 4', '( a ) , where .', 'It is also observed that increasing decreases the radius of the circle that can be extrapolated from the arc length of the layer – this is as expected .', 'Simulation snapshots are shown in Fig . 5 , taken after 60 hours , to illustrate the difference in deformation of the layer for and .', 'As the basement membrane force increases , a stronger force acts on the outer edges to maintain a zero curvature , preventing these regions from bending to compensate the deformation of the region of non-zero curvature .', 'This is emphasised in Fig . 4', '( b ) , which fixes .', 'This plot shows that as increases , the outer edges flatten and are pushed further down into the stroma .', 'Accordingly , there is less distinction with the crypt base region , and the central portion of monolayer is not pushed down as much .', 'Fig . 4', '( c ) plots the total number of epithelial cells in the layer at the final timestep for and increasing , averaged over fifty simulations .', 'This reveals the trend that the number of epithelial cells in the layer decreases as increases .', 'As seen in Fig . 4', '( b ) , as increases the deformation of the epithelial layer decreases .', 'Correspondingly , the arc length of the layer decreases and fewer cells are held within the monolayer .', 'To relate this to the biology of the layer , it is necessary to know more about variability in the rigidity of , and adhesion of epithelial cells to , the basement membrane .', 'The simple geometric framework employed for the investigations thus far is a limiting factor preventing realistic modelling of the colonic epithelium , and so the next step is to incorporate the crypt geometry .', 'As shown in Fig . 12 , it is possible to deform an initially flat epithelial monolayer to adopt a test-tube crypt shape by suitable application of the basement membrane force ( see supplementary video S2 ) .', 'However , in order to do so it is necessary to define the initial rectangular geometry to be sufficiently wide , which in turn increases the width of the tissue stroma surrounding the crypt once the layer has fully deformed .', 'This is unrealistic , as the stroma between neighbouring crypts is only 2–3 cells thick .', 'The starting point for the following simulations is instead an initial geometry that approximates the shape of the crypt , described below .', 'Distinct proliferative compartments can be defined as dependent on an imposed Wnt gradient , and this also has the advantage of eliminating the time required to fully deform the flat layer .', 'From the approximate geometry , the basement membrane force acts to maintain the test-tube shape within the tissue through local calculation of the discrete curvature .', 'This is a key feature of the model , as the test-tube geometry emerges due to the action of the forces , rather than being fixed and imposed as in most earlier models .', 'The results found for the simple rectangular geometry are now translated to the cross-sectional geometry .', 'The conditions required for homeostasis are sought , which present a balance between the basement membrane force and the adhesion and repulsion between neighbouring cells , to allow constant upward migration that is matched by cell removal at the collar .', 'Thus , the number of epithelial cells in the crypt should fluctuate only slightly around a constant value , the cells should not be overly compressed , and the structure should not buckle .'], ['The work described in this paper thus far constitutes the foundation of a realistic , theoretical representation of growth in a deformable environment within the colonic crypt .', 'The usefulness and the need for mathematical modelling as a tool to guide and inform biological experimentation is becoming increasingly recognised [40] .', 'In particular , the field of oncology lacks a comprehensive model to which existing data can be applied , nonetheless such a model is required to identify key system parameters [41] .', 'The collaboration that resulted in this paper is intended to produce qualitative results that identify parameter balances and mechanisms that describe the behaviour of the system and which cannot be obtained by alternative methods for ethical , financial or viability reasons .', 'Such results will inform experimental work , and identify areas for future investigation .', 'A simple approach is adopted to describe the evolution of the tissue structure while linking subcellular processes ( Wnt signalling , cell cycle control , cell adhesion , cell differentiation ) with cellular mechanics that control division , migration and apoptosis .', 'The role of the basement membrane is defined by an additional force which takes into account the structural support provided by the surrounding connective tissue , including the PCFS .', 'Evidence for this is based on experimental observation of the tissue structure of the crypt in situ , reported here , which identifies key components that contribute towards crypt shape and function .', 'Although only the colonic crypt is considered here , extension to consider the small intestinal crypt would be possible based on this work , but would require the definition of the villus and the incorporation of paneth cells .', 'This is a possible direction for future work .', 'In addition , one could examine the effect of possible gradients of adhesion along the crypt axis , via a spatially-dependent basement membrane force parameter , and monitor any subsequent change in cell migration or cell death .', 'Simulations of the cross-sectional crypt model demonstrate that dynamic homeostasis can be achieved , in which repeated mitotic events evolve to force consistent epithelial cell migration towards the crypt collar , without compromising the overall structure and architecture .', 'This is characterised by a steady , constant turnover of cells , achieved in the presence of known constraints on the number of dividing cells , and applying the two known mechanisms of cell death .', 'It is by the application of the basement membrane force that the shape of the crypt evolves , and will allow the structure to deform , rather than imposing a fixed geometry .', 'This is a key aspect of this work , given that all previous crypt models , with the exception of Drasdo and Loeffler ( 2001 ) [21] , have assumed a fixed geometry [15] , [16] , [18] , [24] , and permits investigation of the destabilisation of the crypt structure that occurs at the onset of carcinogenesis , and indeed which can aggravate the growth of a pre-cancerous adenoma .', 'Moreover , two insights are proposed regarding mechanisms of cell dynamics within the crypts .', 'Firstly , in the absence of sufficient apoptosis at the crypt collar and intercrypt table , epithelial cells reach a state of confluence , do not divide and migration is inhibited .', 'This suggests that cell migration in the crypt may not be due solely to proliferative pressure from below , but that a feedback mechanism exists between cell birth and cell death , such that the epithelial cells move into the space created by cell death at the collar .', 'Subsequently , cells below are able to grow , divide and migrate upwards , which has the secondary effect of maintaining barrier function .', 'This is a theory that is in line with the extrusion process that occurs for apoptotic cells in epithelial layers [42] , [43] and is known as the negative pressure hypothesis [37] .', 'In the model , apoptosis has been defined to occur randomly , given that the cause of cell removal at the crypt collar is currently unknown .', 'It is likely that programmed cell death acts in combination with anoikis events that may be induced by the crowding of cells at the crypt collar , shown in Fig . 16 ( a ) , where the layer has a negative curvature , to remove cells and ultimately enable cell migration .', 'To test this hypothesis in a wet lab , it is suggested that apoptosis could be induced uniformly along the crypt-villus axis , and any alteration to the typical migratory pattern subsequently monitored .', 'Secondly , the model demonstrates a high incidence of anoikis events at the crypt base .', 'This is not commonly observed by experimentalists [34] .', 'However , when these results are considered in light of the process of asymmetric division in stem cells , whereby one nuclei is positioned apically ( towards the lumen ) before reinserting basally by an as yet unknown mechanism , it is suggested that the crypt shape may play a role in forcing the alignment of the mitotic spindle for the compressed cells at the base .', 'In the model , the compression of cells at the base forces one of the daughter cell centres to lose contact with the basement membrane , whereupon it is removed and this registers as an anoikis event .', 'That a high incidence of anoikis events happen following division at the base of the crypt therefore indicates a mechanosensory cause for asymmetric division in the stem cells at the base of the crypt .', 'This hypothesis is supported by experimental results which demonstrate that cells do respond to their mechanical environment , and moreover that cytokinesis is a mechanical process [44] .', 'It has also been suggested that the shape of cells and tissues can influence cell division via cortical tension heterogeneity which guides spindle orientation [45] .', 'The simulation results also show that anoikis events occur at the curve of the crypt collar .', 'As cells in this region are now differentiated , it is hypothesised that such events arise due to the negative curvature of the layer at this point , rendering cells vulnerable to extrusion .', 'To test this , one could grow epithelial cells on curved substrates to examine the incidence of extrusion on negatively curved regions .', 'Alternatively , cells could initially be grown on a flat substrate , which is bent once the cells reach confluence .', 'The experiments suggested above identify ways in which model development and wet lab experiments can enter a feedback loop to advance understanding of the system .', 'Support or invalidation of the hypotheses proposed will guide future model iterations , generally advancing the understanding of the system .', 'Ideally , the most useful experiments would be those imaging live tissue using appropriate markers , so as to measure individual cell behaviours over time .', 'This would be another way to parametrise the model out of imaging data ( in addition to that used to generate the data shown in Fig . 14 ) .', 'It has been demonstrated that crypt-like organoids can be grown in matrigel in the presence of growth factors distributed uniformly throughout the gel , which appear similar to crypts in tissue .', 'That additional forces do play a significant role in vivo is illustrated by the fact that crypts which are mutant in Apc form apparently normal shapes in whole tissue , but do not in matrigel [3] .', 'Therefore , at this stage , it is not yet possible to state decisively how crypt organoids could be used to test hypotheses generated by theoretical models .', 'When constructing a mathematical model of a biological system , it is wise to keep the model as simple as possible , focussing accurately on the key components and understanding the outcome of crucial interactions without over-complicating the description and analysis required [46] .', 'However , the computational framework within which the crypt model has been developed ( Chaste ) makes it extensible and amenable to additional complexity , should it be required .', 'For example , more detailed cell cycle models can be applied , and many currently exist within the Chaste framework .', 'At present , however , there are limiting factors that prevent full examination of the destabilisation of the crypt structure that occurs in the progression from a healthy system to the growth of a malignant tumour .', 'As the cross-sectional model consists of a 1D chain of cells , it is only possible for epithelial cells to move vertically or to displace the surrounding stroma .', 'Consequently , the introduction of a mutant cell that migrates aberrantly , e . g . more slowly , will always affect those cells directly beneath it in the chain .', 'In reality , these cells would be able to move around a blockage by moving laterally across the inner surface of the crypt .', 'To correct this , it would not be sufficient to simply apply radial symmetry to the model , a method that would produce unrealistic results due to the imposed symmetry , causing mutations to spread uniformly upwards as cells migrate .', 'In contrast , a full deformable 3D model which permits lateral movement could eliminate false positives observed in the 2D model .', 'For example , a detailed response of the system to the introduction of mutant cell populations could be obtained , and meaningful experiments to investigate the “top-down” [47] and “bottom-up” [48] , [49] theories of mutant cell invasion could be conducted .', 'In addition , a full 3D model makes it possible to define a localised stem cell compartment within the crypt that is distinguished from the transit-amplifying cell compartment by specific proliferative properties .', '( In 2D , the cell chain would force stem cells out of the base of the crypt . )', 'This would enable investigation of stem cell number in the crypt , an open question within the field , and how this affects cell dynamics .', 'The additional degrees of freedom associated with cell movement introduced by a deformable 3D model will increase the scope for accurately modelling the response of the system to different cellular events .', 'For example , the merging or rearrangement of cell columns , causing the lumen to narrow or widen should the number of cells decrease or increase .', 'This will have bearing on the persistence of mutations in the crypt , as well as on the incidence of anoikis events , which are likely to decrease as cells can exploit movement and growth in more directions .', 'However , it is necessary to re-evaluate the structure of forces in 3D , firstly in light of the additional complexity of the Delaunay triangulation , and also to consider the effect of shear forces that may contribute to the stability of the structure .', 'This work is underway .'], ['Mouse gut tissue was prepared for imaging following the methods described in Appleton et al . ( 2009 ) [50] .', 'Sectioned gut was imaged on a Leica DMIRB fluorescent microscope and the wholemounts imaged on a Zeiss 710 confocal microscope .', '3D images of fixed whole-mount tissue stained with DAPI and rhodamine-phalloidin were acquired using multiphoton fluorescence microscopy [50] .', 'The outer surface of the crypt is defined by the basement membrane on which the epithelial cells sit , and the outer crypt area and crypt lumen area were measured in a cross section half way along the crypt length using Volocity image ana
['Introduction', 'Results', 'Discussion', 'Materials and Methods', 'Notes Added in Proof']
['The role of the basement membrane is vital in maintaining the integrity and structure of an epithelial layer , acting as both a mechanical support and forming the physical interface between epithelial cells and the surrounding connective tissue .', 'The function of this membrane is explored here in the context of the epithelial monolayer that lines the colonic crypt , test-tube shaped invaginations that punctuate the lining of the intestine and coordinate a regular turnover of cells to replenish the epithelial layer every few days .', 'To investigate the consequence of genetic mutations that perturb the system dynamics and can lead to colorectal cancer , it must be possible to track the emerging tissue level changes that arise in the crypt .', 'To that end , a theoretical crypt model with a realistic , deformable geometry is required .', 'A new discrete crypt model is presented , which focuses on the interaction between cell- and tissue-level behaviour , while incorporating key subcellular components .', 'The model contains a novel description of the role of the surrounding tissue and musculature , based upon experimental observations of the tissue structure of the crypt , which are also reported .', 'A two-dimensional ( 2D ) cross-sectional geometry is considered , and the shape of the crypt is allowed to evolve and deform .', 'Simulation results reveal how the shape of the crypt may contribute mechanically to the asymmetric division events typically associated with the stem cells at the base .', 'The model predicts that epithelial cell migration may arise due to feedback between cell loss at the crypt collar and density-dependent cell division , an hypothesis which can be investigated in a wet lab .', 'This work forms the basis for investigation of the deformation of the crypt structure that can occur due to proliferation of cells exhibiting mutant phenotypes , experiments that would not be possible in vivo or in vitro .']
['At the onset of colorectal carcinogenesis , marked changes can be observed in the structure and dynamics of the crypts of Lieberkühn .', 'These test tube shaped glands regularly punctuate the surface of the gut and are lined with a monolayer of epithelial cells which divide and migrate upwards to renew the intestinal surface every few days .', 'The process by which the crypt structures breakdown , and the compliant environment that can be subsequently provided to mutated cells to allow the formation of adenomatous growths , is not yet well characterised .', 'A limiting factor in the understanding of this process is the ability to observe easily the initial changes that occur , and which are necessary to disrupt the normal behaviour of the system .', 'However , a predictive , theoretical model of the crypt that mimics the geometry and the tissue architecture can be used to perform in silico experiments and further such understanding .', 'A model is introduced here that addresses the tissue structure of the crypt , and the stability it provides to the epithelial layer , while remaining deformable and without imposing a fixed geometry .']
['histology', 'biophysic al simulations', 'biology', 'computational biology']