Modelling Spatially Regulated β-Catenin Dynamics and Invasion in Intestinal Crypts

Experimental data (e.g., genetic lineage and cell population studies) on intestinal crypts reveal that regulatory features of crypt behavior, such as control via morphogen gradients, are remarkably well conserved among numerous organisms (e.g., from mouse and rat to human) and throughout the different regions of the small and large intestines. In this article, we construct a partial differential equation model of a single colonic crypt that describes the spatial distribution of Wnt pathway proteins along the crypt axis. The novelty of our continuum model is that it is based upon assumptions that can be directly related to processes at the cellular and subcellular scales. We use the model to predict how the distributions of Wnt pathway proteins are affected by mutations. The model is then extended to investigate how mutant cell populations can invade neighboring crypts. The model simulations suggest that cell crowding caused by increased proliferation and decreased cell loss may be sufficient for a mutant cell population to colonize a neighboring healthy crypt.     

Stimulation of colonic mucosal growth associated with oxidized redox status in rats

Limited data in animal models suggest that colonic mucosa undergoes adaptive growth following massive small bowel resection (SBR). In vitro data suggest that intestinal cell growth is regulated by reactive oxygen species and redox couples [e.g., glutathione (GSH)/glutathione disulfide (GSSG) and cysteine (Cys)/cystine (CySS) redox]. We investigated the effects of SBR and alterations in redox on colonic growth indexes in rats after either small bowel transection (TX) or 80% midjejunoileal resection (RX). Rats were pair fed +/- blockade of endogenous GSH synthesis with buthionine sulfoximine (BSO). Indexes of colonic growth, proliferation, and apoptosis and GSH/GSSG and Cys/CySS redox potentials (E(h)) were determined. RX significantly increased colonic crypt depth, number of cells per crypt, and epithelial cell proliferation [crypt cell bromodeoxyuridine (BrdU) incorporation]. Administration of BSO markedly decreased colonic mucosal GSH, GSSG, and Cys concentrations in both TX and RX groups, with a resultant oxidation of GSH/GSSG and Cys/CySS E(h). BSO did not alter colonic crypt cell apoptosis but significantly increased all colonic mucosal growth indexes (crypt depth, cells/crypt, and BrdU incorporation) in both TX and RX groups in a time- and dose-dependent manner. BSO significantly decreased plasma GSH and GSSG, oxidized GSH/GSSG E(h), and increased plasma Cys and CySS concentrations. Collectively, these data provide in vivo evidence indicating that oxidized colonic mucosal redox status stimulates colonic mucosal growth in rats. The data also suggest that GSH is required to maintain normal colonic and plasma Cys/CySS homeostasis in these animal models.     

Patterns of Proliferative Activity in the Colonic Crypt Determine Crypt Stability and Rates of Somatic Evolution

Epithelial cells in the colon are arranged in cylindrical structures called crypts in which cellular proliferation and migration are tightly regulated. We hypothesized that the proliferation patterns of cells may determine the stability of crypts as well as the rates of somatic evolution towards colorectal tumorigenesis. Here, we propose a linear process model of colonic epithelial cells that explicitly takes into account the proliferation kinetics of cells as a function of cell position within the crypt. Our results indicate that proliferation kinetics has significant influence on the speed of cell movement, kinetics of mutation propagation, and sensitivity of the system to selective effects of mutated cells. We found that, of all proliferation curves tested, those with mitotic activities concentrated near the stem cell, including the actual proliferation kinetics determined in in vivo labeling experiments, have a greater ability of delaying the rate of mutation accumulation in colonic stem cells compared to hypothetical proliferation curves with mitotic activities focused near the top of the crypt column. Our model can be used to investigate the dynamics of proliferation and mutation accumulation in spatially arranged tissues.     

An enteroendocrine cell-based model for a quiescent intestinal stem cell niche

We have shown that the kinetics of conversion of intestinal crypt cell populations to a partially or wholly mutant phenotype are consistent with a model in which each crypt contains an infrequently dividing 'deep' stem cell that is the progenitor of several more frequently dividing 'proximate' stem cells. An assumption of our model is that each deep stem cell exists in a growth inhibitory niche. We have used information from the literature to develop a model for a quiescent intestinal stem cell niche. This niche is postulated to be primarily defined by an enteroendocrine cell type that maintains stem cell quiescence by secretion of growth inhibitory peptides such as somatostatin and guanylin/uroguanylin. Consistent with this model, there is evidence that the proteins postulated as defining a growth-inhibitory stem cell niche can act as intestinal tumour suppressors. Confirmation that a growth-inhibitory niche does exist would have important implications for our understanding of intestinal homeostasis and tumorigenesis.     

Programmed cell death, proliferating cell nuclear antigen and p53 expression in mouse colon mucosa during diet-induced tumorigenesis.

Western-style diets (WDs) trigger and sustain the early phases of tumorigenesis in mouse colon, and when continued throughout the life span lead to the development of dysplastic crypts. In order to evaluate the roles both of cell proliferation and programmed cell death (PCD) in WD-induced tumorigenesis, immunohistochemical detection of proliferating nuclear antigen (PCNA), in situ end labeling (TUNEL) of DNA breaks, and p53 protein were carried out in mouse colonic mucosa during prolonged feeding of two WDs. PCNA Labeling Index of colonic crypts was significantly higher in WD-treated animals than in controls only at the beginning of the nutritional study, the gap rapidly bridged by increased cell proliferation spontaneously occurring in the colonic mucosa during aging. A transient early homeostatic activation of PCD at the base of the crypt also was observed in WD groups. No changes in PCD were seen in the upper third of the crypt or in surface epithelium throughout the study, indicating that PCD in that colonic crypt segment produces a constant flux of cell loss, uninfluenced by homeostatic fluctuations. A major finding was an irreversible, progressive, age-related decline of PCD at the crypt base in both control and treated animals that occurred during the second half of the rodents' life span. p53 protein was not immunohistochemically detected, suggesting that neither overexpression of wild-type nor mutated forms of the protein are involved in the above mentioned changes.     

Emergent Stem Cell Homeostasis in the C.?elegans Germline Is Revealed by Hybrid Modeling

The establishment of homeostasis among cell growth, differentiation, and apoptosis is of key importance for organogenesis. Stem cells respond to temporally and spatially regulated signals by switching from mitotic proliferation to asymmetric cell division and differentiation. Executable computer models of signaling pathways can accurately reproduce a wide range of biological phenomena by reducing detailed chemical kinetics to a discrete, finite form. Moreover, coordinated cell movements and physical cell-cell interactions are required for the formation of three-dimensional structures that are the building blocks of organs. To capture all these aspects, we have developed a hybrid executable/physical model describing stem cell proliferation, differentiation, and homeostasis in the Caenorhabditis elegans germline. Using this hybrid model, we are able to track cell lineages and dynamic cell movements during germ cell differentiation. We further show how apoptosis regulates germ cell homeostasis in the gonad, and propose a role for intercellular pressure in developmental control. Finally, we use the model to demonstrate how an executable model can be developed from the hybrid system, identifying a mechanism that ensures invariance in fate patterns in the presence of instability.     

Protein Kinase-zeta inhibits collagen I-dependent and anchorage-independent growth and enhances apoptosis of human Caco-2 cells

Colonic carcinogenesis is accompanied by abnormalities in multiple signal transduction components, including alterations in protein kinase C (PKC). The expression level of PKC-zeta, an atypical PKC isoform, increases from the crypt base to the luminal surface and parallels crypt cell differentiation in normal colon. In prior studies in the azoxymethane model of colon cancer, we showed that PKC-zeta was down-regulated in rat colonic tumors. In this study, we showed that PKC-zeta is expressed predominantly in colonic epithelial and not stromal cells, and loss of PKC-zeta occurs as early as the adenoma stage in human colonic carcinogenesis. To assess the regulation of growth and differentiation by PKC-zeta, we altered this isoform in human Caco-2 colon cancer cells using stable constitutive or inducible expression vectors, specific peptide inhibitors or small interfering RNA. In ecdysone-regulated transfectants grown on collagen I, ponasterone A significantly induced PKC-zeta expression to 135% of empty vector cells, but did not alter nontargeted PKC isoforms. This up-regulation was accompanied by a 2-fold increase in basal and 4-fold increase in insulin-stimulated PKC-zeta biochemical activity. Furthermore, PKC-zeta up-regulation caused >50% inhibition of cell proliferation on collagen I (P < 0.05). Increased PKC-zeta also significantly enhanced Caco-2 cell differentiation, nearly doubling alkaline phosphatase activity, while inducing a 3-fold increase in the rate of apoptosis (P < 0.05). In contrast, knockdown of this isoform by small interfering RNA or kinase inhibition by myristoylated pseudosubstrate significantly and dose-dependently increased Caco-2 cell growth on collagen I. In transformation assays, constitutively up-regulated wild-type PKC-zeta significantly inhibited Caco-2 cell growth in soft agar, whereas a kinase-dead mutant caused a 3-fold increase in soft agar growth (P < 0.05). Taken together, these studies indicate that PKC-zeta inhibits colon cancer cell growth and enhances differentiation and apoptosis, while inhibiting the transformed phenotype of these cells. The observed down-regulation of this growth-suppressing PKC isoform in colonic carcinogenesis would be predicted to contribute to tumorigenesis.     

A Mathematical Model of Cancer Stem Cell Driven Tumor Initiation: Implications of Niche Size and Loss of Homeostatic Regulatory Mechanisms

Hierarchical organized tissue structures, with stem cell driven cell differentiation, are critical to the homeostatic maintenance of most tissues, and this underlying cellular architecture is potentially a critical player in the development of a many cancers. Here, we develop a mathematical model of mutation acquisition to investigate how deregulation of the mechanisms preserving stem cell homeostasis contributes to tumor initiation. A novel feature of the model is the inclusion of both extrinsic and intrinsic chemical signaling and interaction with the niche to control stem cell self-renewal. We use the model to simulate the effects of a variety of types and sequences of mutations and then compare and contrast all mutation pathways in order to determine which ones generate cancer cells fastest. The model predicts that the sequence in which mutations occur significantly affects the pace of tumorigenesis. In addition, tumor composition varies for different mutation pathways, so that some sequences generate tumors that are dominated by cancerous cells with all possible mutations, while others are primarily comprised of cells that more closely resemble normal cells with only one or two mutations. We are also able to show that, under certain circumstances, healthy stem cells diminish due to the displacement by mutated cells that have a competitive advantage in the niche. Finally, in the event that all homeostatic regulation is lost, exponential growth of the cancer population occurs in addition to the depletion of normal cells. This model helps to advance our understanding of how mutation acquisition affects mechanisms that influence cell-fate decisions and leads to the initiation of cancers.     

Homeostasis of peripheral immune effectors

In this paper, we use both mathematical modeling and simulation to explore homeostasis of peripheral immune system effector cells, particularly alveolar macrophages. Our interest is in the distributed control mechanisms that allow such a population to maintain itself. We introduce a multi-purpose simulator designed to study individual cell responses to local molecular signals and their effects on population dynamics. We use the simulator to develop a model of growth factor regulation of macrophage proliferation and survival. We examine the effects of this form of regulation in the context of two competing hypotheses regarding the source of new alveolar macrophages. In one model, local cells divide to replenish the population; in the other, only cells migrating from circulation divide. We find that either scenario is plausible, although the influx-driven system is inherently more stable. The proliferation-driven system requires lower cell death and efflux rates than the influx-driven system.     

Epithelial cell kinetics in the descending colon of the rat

Epithelial cell loss was induced in the descending colon of the rat by temporary ischaemia to investigate whether this would lead to an increase in crypt cell proliferation. Shortly after the temporary ischaemia the number of cells per crypt was markedly reduced, and it was shown that the cell loss occurred mainly from the non-proliferating upper half of the crypt. The number of cells per crypt reached control values again after 24-48 h. There was a marked increase in proliferative activity, as reflected by the labelling index after 3HTdR and by the mitotic index, with peak values at 16 and 24 h after ischaemia. After 48 h the proliferative indices were normal again. The increase in crypt cell proliferation was characterized by an increase in the labelling index as well as in the mitotic index per crypt cell position. No enlargement of the proliferative cell compartment in the crypt was observed. It is most likely then that the increase in crypt cell proliferation was brought about by a shortening of the cell cycle, since the growth fraction in the lower half of the crypt approaches 1.0. The possible implications of the present data for the control of colonic cell proliferation and colonic carcinogenesis are discussed.     

A stochastic branching model with formation of subunits applied to the growth of intestinal crypts

The intestinal epithelium is one of the most rapidly regenerating tissues in mammals. Cell production takes place in the intestinal crypts which contain about 250 cells. Only a minority of 1-60 proliferating cells are able to maintain a crypt over a long period of time. However, so far attempts to identify these stem cells were unsuccessful. Therefore, little is known about their cellular growth and selfmaintenance properties. On the other hand, the crypts appear to exhibit a life cycle which starts by fission of existing crypts and ends by fission or extinction. Data on these processes have recently become available. Here, we demonstrate how these data on the life cycle of the macroscopic crypt structure can be used to derive a quantitative model of the microscopic process of stem cell growth. The model assumptions are: (1) stem cells undergo a time independent supracritical Markovian branching process (Galton-Watson process); (2) a crypt divides if the number of stem cells exceeds a given threshold and the stem cells are distributed to both daughter crypts according to binomial statistics; (3) the size of the crypt is proportional to the stem cell number. This model combining two different stochastic branching processes describes a new class of processes whose stationary stability and asymptotic behavior are examined. This model should be applicable to various growth processes with formation of subunits (e.g. population growth with formation of colonies in biology, ecology and sociology). Comparison with crypt data shows that intestinal stem cells have a probability of over 0.8 of dividing asymmetrically and that the threshold number should be 8 or larger.     

Repair and regeneration: opportunities for carcinogenesis from tissue stem cells

This review will discuss the mechanisms of repair and regeneration in various tissue types and how dysregulation of these mechanisms may lead to cancer. Normal tissue homeostasis involves a careful balance between cell loss and cell renewal. Stem and progenitor cells perform these biologic processes as the functional units of regeneration during both tissue homeostasis and repair. The concept of tissue stem cells capable of giving rise to all differentiated cells within a given tissue led to the concept of a cellular hierarchy in tissues and in tumors. Thus, only a few cells may be necessary and sufficient for tissue repair or tumor regeneration. This is known as the hierarchical model of tumorigenesis. This report will compare this model with the stochastic model of tumorigenesis. Under normal circumstances, the processes of tissue regeneration or homeostasis are tightly regulated by several morphogen pathways to prevent excessive or inappropriate cell growth. This review presents the recent evidence that dysregulation of these processes may provide opportunities for carcinogenesis for the long-lived, highly proliferative tissue stem cell population. New findings of cancer initiating tissue stem cells identified in several solid and circulating cancers including breast, brain and hematopoietic tumors will also be reviewed. Finally, this report reviews the cellular biology of cancer and its relevance to the development of more effective cancer treatment protocols.     

The crypt cycle. Crypt and villus production in the adult intestinal epithelium.

We propose a model for the growth of individual crypts that is able to account for the observed changes in the number of cells in crypts under normal conditions, after irradiation, and after 30% resection. Parameter values for this model are estimated both for mouse and man, and detailed predictions of crypt growth rates are made. This model does not predict a steady-state crypt size; rather it suggests that crypts grow until they bifurcate. We therefore propose a crypt cycle (analogous to the cell cycle) and present evidence that most if not all crypts in the adult mouse are cycling asynchronously and independently. This evidence consists of four experiments that indicate that branching crypts are randomly distributed over the intestinal epithelium, that the plane of bifurcation of branching crypts is randomly oriented with respect to the villus base, and that the size distribution of crypts is consistent with an expanding crypt population. We also report for the first time evidence of villus production in the adult mouse intestinal epithelium. We conclude that the crypt and villus populations in the adult mouse are not in a steady state.     

Growth-induced buckling of an epithelial layer

We use a proof-of-concept experiment and two mathematical models to explore growth-induced tissue buckling, as may occur in colorectal crypt formation. Our experiment reveals how growth of a cultured epithelial monolayer on a thin flexible substrate can cause out-of-plane substrate deflections. We describe this system theoretically using a ‘bilayer’ model in which a growing cell layer adheres to a thin compressible elastic beam. We compare this with the ‘supported-monolayer’ model due to Edwards and Chapman (Bull Math Biol 69:1927–1942, 2007) for an incompressible expanding beam (representing crypt epithelium), which incorporates viscoelastic tethering to underlying stroma. We show that the bilayer model can exhibit buckling via parametric growth (in which the system passes through a sequence of equilibrium states, parameterised by the total beam length); in this case, non-uniformities in cell growth and variations in cell–substrate adhesion are predicted to have minimal effect on the shape of resulting buckled states. The supported-monolayer model reveals how competition between lateral supports and stromal adhesion influences the wavelength of buckled states (in parametric growth), and how non-equilibrium relaxation of tethering forces influences post-buckled shapes. This model also predicts that non-uniformities in growth patterns have a much weaker influence on buckled shapes than non-uniformities in material properties. Together, the experiment and models support the concept of patterning by growth-induced buckling and suggest that targeted softening of a growing cell layer provides greater control in shaping tissues than non-uniform growth.     

The Interplay between Wnt Mediated Expansion and Negative Regulation of Growth Promotes Robust Intestinal Crypt Structure and Homeostasis

The epithelium of the small intestinal crypt, which has a vital role in protecting the underlying tissue from the harsh intestinal environment, is completely renewed every 4–5 days by a small pool of stem cells at the base of each crypt. How is this renewal controlled and homeostasis maintained, particularly given the rapid nature of this process? Here, based on the recent observations from in vitro “mini gut” studies, we use a hybrid stochastic model of the crypt to investigate how exogenous niche signaling (from Wnt and BMP) combines with auto-regulation to promote homeostasis. This model builds on the sub-cellular element method to account for the three-dimensional structure of the crypt, external regulation by Wnt and BMP, internal regulation by Notch signaling, as well as regulation by internally generated diffusible signals. Results show that Paneth cell derived Wnt signals, which have been observed experimentally to sustain crypts in cultured organs, have a dramatically different influence on niche dynamics than does mesenchyme derived Wnt. While this signaling can indeed act as a redundant backup to the exogenous gradient, it introduces a positive feedback that destabilizes the niche and causes its uncontrolled expansion. We find that in this setting, BMP has a critical role in constraining this expansion, consistent with observations that its removal leads to crypt fission. Further results also point to a new hypothesis for the role of Ephrin mediated motility of Paneth cells, specifically that it is required to constrain niche expansion and maintain the crypt’s spatial structure. Combined, these provide an alternative view of crypt homeostasis where the niche is in a constant state of expansion and the spatial structure of the crypt arises as a balance between this expansion and the action of various sources of negative regulation that hold it in check.