The Growth and Life of A Monoclonal Crypt

Abstract. A computer simulation of a highly dynamic model for the birth, growth and adult life of a monoclonal crypt in the intestine was developed starting with a single precursor stem cell. the intestinal epithelial system was studied and observed in analogy to ' in vivo ' experiments. the model output, e.g. the geometric shape of a crypt, mitotic index, labelling index and the crypt length distribution in adult state, was compared with experimental data. There was evidence from the simulation that a certain steady state in the adult life could be reached regardless of some harmless influences in post-natal life, e.g. the influence of being weaned or non-weaned. the model is based on our hypothesis of the generation-controlled proliferation mechanism and at the same time is a confirmation of it.     

Intestinal crypt properties fit a model that incorporates replicative ageing and deep and proximate stem cells

A model of intestinal crypt organization is suggested based on the assumption that stem cells have a finite replicative life span. The model assumes the existence in a crypt of a quiescent ('deep') stem cell and a few more actively cycling ('proximate') stem cells. Monte Carlo computer simulation of published intestinal crypt mutagenesis data is used to test the model. The results of the simulation indicate that stabilization of the crypt mutant phenotype following treatment with external mutagen is consistent with a stem cell replicative life span of about 40 divisions for mouse colon and 90-100 divisions for mouse small intestine, corresponding to a deep stem cell cycle time of about 3.9 and 8.5 weeks for colon and small intestine, respectively. Simulation of the data obtained for human colorectal crypts suggests that the proximate stem cell cycle time is about 80 h, assuming a replicative life span of 50-150 divisions, and that the deep stem cell divides approximately every 30 weeks.     

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.     

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.     

Intestinal crypt proliferation. II. Computer modelling of mitotic index data provides further evidence for lateral and vertical cell migration in the absence of mitotic activity

The position-dependent mitotic index before, and 1, 2 and 3 h after vincristine was scored. The accumulation of cells in mitosis leads to an increase in the mitotic index from 0.06 to 0.34 at crypt positions 8-12. Surprisingly, the leading edge of the position-related mitotic index distribution moves to higher crypt positions although cell division was stopped. In addition, the vertical clustering of mitotic figures in sections was recorded. The data were examined using a previously described computer crypt model. We conclude: the average mitotic phase duration is about 0.7 h (40 min) and varies little with cell position; the geometrical correction factor for overscoring mitoses in crypt sections is about 0.6-0.7 and adjacent cell columns can merge. Lateral cell displacement after mitosis, as predicted in a previous model analysis, would be a mechanism to counteract other forces that tend to reduce the crypt circumference. In the normal steady state merging and expansion processes would just balance each other. This would not follow if one mechanism was blocked. Thus we propose a new concept in which the crypt geometry would be dynamically determined by cell proliferative activity in connection with lateral positioning of new cells on one hand and contracting forces on the other hand.     

The effect of T-lymphocytes on epithelial proliferation of the jejunal mucosa in thymectomized mice

Following 1.5 month period after thymectomy of adult mice the crypt 3H-thymidine labelling index is decreased and the proliferative zone is reduced. Analysis of labeled cell distributions in the crypt revealed that thymectomy resulted in the initiation of enterocyte entering a maturation phase in the site shifted to the crypt base. Syngeneic T-lymphocytes injected intravenously to thymectomized mice increased the labeling index, but exerted no effect on the size of proliferative zone. The proliferative changes described were not due to changes in the number of divisions of rapidly proliferating enterocytes. The role of T-lymphocytes in modulation of intestinal epithelium proliferation and steady state maintenance is discussed.     

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.     

Intestinal Crypt Proliferation. Ii. Computer Modelling of Mitotic Index Data Provides Further Evidence For Lateral and Vertical Cell Migration In the Absence of Mitotic Activity

Abstract. The position-dependent mitotic index before, and 1, 2 and 3 h after vincristine was scored. the accumulation of cells in mitosis leads to an increase in the mitotic index from 0.06 to 0.34 at crypt positions 8-12. Surprisingly, the leading edge of the position-related mitotic index distribution moves to higher crypt positions although cell division was stopped. In addition, the vertical clustering of mitotic figures in sections was recorded. the data were examined using a previously described computer crypt model. We conclude: the average mitotic phase duration is about 0.7 h (40 min) and varies little with cell position; the geometrical correction factor for overscoring mitoses in crypt sections is about 0.6-0.7 and adjacent cell columns can merge. Lateral cell displacement after mitosis, as predicted in a previous model analysis, would be a mechanism to counteract other forces that tend to reduce the crypt circumference. In the normal steady state merging and expansion processes would just balance each other. This would not follow if one mechanism was blocked. Thus we propose a new concept in which the crypt geometry would be dynamically determined by cell proliferative activity in connection with lateral positioning of new cells on one hand and contracting forces on the other hand.     

SpatialDemography: a spatially explicit,stage‐structured,metacommunity model

The responses of species and populations to changes in the environment (e.g. changes in climate and land use) are often complex and difficult to predict. We have created the SpatialDemography model (R package: spatialdemography). The model is a spatially explicit, stage‐structured, matrix‐based metacommunity model, with the potential for modeling species’ and populations’ potential responses to environmental heterogeneity and change. The SpatialDemography model assumes a cellular landscape populated by organisms with four life stages: a mobile dispersing stage, two sessile non‐reproductive stages, and a reproductive adult stage. Individuals are assumed to originate at the center of a given cell and disperse according to a specified dispersal kernel (e.g. log‐normal). All adult individuals are capable of producing offspring. The model approach and framework are described in the context of a hypothetical example with multiple competing species in a four cell landscape. In this example simulation, both spatial location and species interactions were important for understanding population dynamics. SpatialDemography can be applied to questions where an understanding of transient and long‐term demographic responses to spatiotemporal changes is desired. It is primarily applicable to metapopulations and metacommunities of organisms with early dispersal and sessile adults (i.e. modular organisms such as plants and some marine organisms). SpatialDemography differs from other population models in that it is spatially explicit, can incorporate biotic interactions, and is implemented in R.     

A model of the control of cellular regeneration in the intestinal crypt after perturbation based solely on local stem cell regulation

Abstract. The control mechanisms involved in regeneration of murine intestinal crypts after perturbations are presently not well understood. The existence of some feedback signals from the cells on the villus to the cells in the crypt has been suggested. However, some recent experimental data point to the fact that regeneration in the crypt starts very early after perturbation, at a time when the villus cell population has hardly changed. In particular, this early cell proliferative activity is seen specifically at the bottom of the crypt, i.e. in the presumed stem cell zone and furthest from the villus.
The objective of this study was to investigate whether a new concept of regulation operating solely at the stem cell level could explain the present mass of accumulated data on the post-irradiation recovery, which is an extensively studied perturbation from the experimental point of view. In order to check its validity, the new concept was formalized as a mathematical simulation model thus enabling comparison with experimental data. The model describes the cellular development from stem cells to the mature villus cells. As a basic feature it is assumed that the self-maintenance and the cell cycle activity of the stem cells are controlled by the number of these cells in an autoregulatory fashion. The essential features of the experimental data (i.e. the recovery with time and the consistency between different types of measurements) can be very well reproduced by simulations using a range of model parameters. Thus, we conclude that stem cell autoregulation is a valid concept which could replace the villus crypt feedback concept in explaining the early changes after irradiation when the damage primarily affects the crypt. The question of the detailed nature of the control process requires further investigation.     

Intestinal cell proliferation. I. A comprehensive model of steady-state proliferation in the crypt

Cell replacement in the crypt of the murine small intestine has been studied and modelled mathematically under steady-state conditions. A great deal of information is available for this system, e.g. cell cycle times, S phase durations, the rate of daily cell production, the Paneth cell distribution etc. The purpose of the present work was to consider simultaneously as much of these data as possible and to formulate a model based upon the behaviour of individual cells which adequately accounted for them. A simple mathematical representation of the crypt has been developed. This consists of sixteen stem cells per crypt (TC = 16 hr, TS = 9 hr), and four subsequent transit cell divisions (TC = 11 to 12 hr, TS = 8 hr) before maturation. Experimental data considered to test the modelling were LI and data on the number of vertical runs of similarly labelled cells. All data were obtained from the ileum after 25 microCi [3H]TdR given at 09:00 hours. A number of alternative assumptions have been considered and either accepted or rejected. Two alternative model concepts of cell displacement explain the data equally well. One is dependent upon strong local cell generation age determinance while the other could accommodate any weak local cell displacement process in conjunction with an environmental cut-off determinant at the middle of the crypt. Both models provide new interpretations of the data, e.g. certain rates of lateral cell exchange between neighbouring columns (250 to 350 per crypt per day out of a total of 420 cell divisions per day) can be concluded from run data, while LI data provide information about the mechanisms involved in maintaining a position-related age order in the crypt.     

Intestinal Cell Proliferation. I. A Comprehensive Model of Steady-State Proliferation In the Crypt

Abstract. Cell replacement in the crypt of the murine small intestine has been studied and modelled mathematically under steady-state conditions. A great deal of information is available for this system, e.g. cell cycle times, S phase durations, the rate of daily cell production, the Paneth cell distribution etc. the purpose of the present work was to consider simultaneously as much of these data as possible and to formulate a model based upon the behaviour of individual cells which adequately accounted for them. A simple mathematical representation of the crypt has been developed. This consists of sixteen stem cells per crypt (T_c= 16 hr, T_s= 9 hr), and four subsequent transit cell divisions (T_c= 11 to 12 hr, T_s= 8 hr) before maturation. Experimental data considered to test the modelling were LI and data on the number of vertical runs of similarly labelled cells. All data were obtained from the ileum after 25 μCi [³H]TdR given at 09.00 hours. A number of alternative assumptions have been considered and either accepted or rejected. Two alternative model concepts of cell displacement explain the data equally well. One is dependent upon strong local cell generation age determinance while the other could accommodate any weak local cell displacement process in conjunction with an environmental cut-off determinant at the middle of the crypt. Both models provide new interpretations of the data, e.g. certain rates of lateral cell exchange between neighbouring columns (250 to 350 per crypt per day out of a total of 420 cell divisions per day) can be concluded from run data, while LI data provide information about the mechanisms involved in maintaining a position-related age order in the crypt.     

Colon Cryptogenesis: Asymmetric Budding

The process of crypt formation and the roles of Wnt and cell-cell adhesion signaling in cryptogenesis are not well described; but are important to the understanding of both normal and cancer colon crypt biology. A quantitative 3D-microscopy and image analysis technique is used to study the frequency, morphology and molecular topography associated with crypt formation. Measurements along the colon reveal the details of crypt formation and some key underlying biochemical signals regulating normal colon biology. Our measurements revealed an asymmetrical crypt budding process, contrary to the previously reported symmetrical fission of crypts. 3D immunofluorescence analyses reveals heterogeneity in the subcellular distribution of E-cadherin and β-catenin in distinct crypt populations. This heterogeneity was also found in asymmetrical budding crypts. Singular crypt formation (i.e. no multiple new crypts forming from one parent crypt) were observed in crypts isolated from the normal colon mucosa, suggestive of a singular constraint mechanism to prevent aberrant crypt production. The technique presented improves our understanding of cryptogenesis and suggests that excess colon crypt formation occurs when Wnt signaling is perturbed (e.g. by truncation of adenomatous polyposis coli, APC protein) in most colon cancers.     

Development of the pattern of cell renewal in the crypt-villus unit of chimaeric mouse small intestine

We have previously shown that the epithelium of each adult intestinal crypt in chimaeric mice is derived from a single progenitor cell. Whether the crypts are monoclonal from the outset-that is, are formed by the proliferation of a single cell-or whether their formation is initiated by several cells was not known. Here we report that many crypts contain cells of both chimaeric genotypes in the neonatal period indicating a polyclonal origin at this stage of morphogenesis. The cellular organization of the early neonatal crypt is therefore different from that of the adult crypt, which includes a zone of 'anchored' stem cells above the crypt base. Within 2 weeks, however, the crypt progenitor cell and its descendants displace all other cells from the crypt and the crypt attains monoclonality. The distribution of enterocytes on chimaeric villi in the neonate shows a mottled pattern of mosaicism which is progressively replaced by coherent sheets of cells from the crypts, and within two weeks the orderly adult clonal pattern is established.     

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.     

Regulation of intestinal stem cell fate specification

The remarkable ability of rapid self-renewal makes the intestinal epithelium an ideal model for the study of adult stem cells. The intestinal epithelium is organized into villus and crypt, and a group of intestinal stem cells located at the base of crypt are responsible for this constant self-renewal throughout the life. Identification of the intestinal stem cell marker Lgr5, isolation and in vitro culture of Lgr5+ intestinal stem cells and the use of transgenic mouse models have significantly facilitated the studies of intestinal stem cell homeostasis and differentiation, therefore greatly expanding our knowledge of the regulatory mechanisms underlying the intestinal stem cell fate determination. In this review, we summarize the current understanding of how signals of Wnt, BMP, Notch and EGF in the stem cell niche modulate the intestinal stem cell fate.     

CELL POPULATION KINETICS IN THE RAT JEJUNAL CRYPT

Cell kinetics in the jejunal crypt of the male Wistar rat were studied using autoradiographic techniques with tritiated thymidine and a stathmokinetic technique with vincristine. The migration rate measured by following the movement of the 50 % peak on the labelling index distribution curve with time after injection of tritiated thymidine gave a value of 1.43 ± 0.14 (SE) cell positions per hour, compared with a value from a cumulative birth rate of 1.78 cell positions per hour. The crypt column length was 32.9 ± 0.2 cells and the column count was 22.3 ± 0.2. This measurement gave a total crypt population of 734 cells, compared with an estimate of 650 ± 6 from direct observation of squashed, microdissected crypts. In each crypt 22.5 ± 0.5 mitoses were present, and the crypt cell production rate was 32 cells per crypt per hour; this latter value was confirmed using two independent techniques. The crypt growth fraction calculated from the durations of phases of the cell cycle and the labelling index was 0.62. A value of 0.61 was found from the labelling index distribution curve. As assessed from crypt squashes, there were 403 proliferating cells per crypt.     

Variation in crypt size and its influence on the analysis of epithelial cell proliferation in the intestinal crypt.

The standard model of epithelial cell renewal in the intestine proposes a gradual transition between the region of the crypt containing actively proliferating cells and that containing solely terminally differentiating cells (Cairnie, Lamerton and Steel, 1965 a, b). The experimental justification for this conclusion was the gradual decrease towards the crypt top of the measured labeling and mitotic indices. Recently, however, we have proposed that intestinal crypts normally undergo a replicative cycle so that at any time in any region of the intestine, crypts will be found to have a wide range of sizes. We show here that if this intrinsic size variation is taken into account, then a sharp transition between the proliferative and nonproliferative compartments of individual intestinal crypts is consistent with the labeling and mitotic index distributions of mouse and rat jejunal crypts. Thus there is no need to invoke the region of gradual transition from proliferating to nonproliferating cells as is done in the standard model. The position of this sharp transition is estimated for both the mouse and rat. Experiments to further test our model are suggested and the significance of the results discussed.     

Cell population kinetics in the rat jejunal crypt.

Cell kinetics in the jejunal crypt of the male Wistar rat were studied using autoradiographic techniques with tritiated thymidine and a stathmokinetic technique with vincristine. The migration rate measured by following the movement of the 50% peak on the labelling index distribution curve with time after injection of tritiated thymidine gave a value of 1-43 +/- 0-14 (SE) cell positions per hour, compared with a value from a cumulative birth rate of 1-78 cell positions per hour. Tht crypt column length was 32-9 +/- 0-2 cells and the column count was 22-3 +/- 0-2. This measurement gave a total crypt population of 734 cells, compared with an estimate of 650 +/- l from direct observation of squashed, microdissected crypts. In each crypt 22-5 +/- 0-5 mitoses were present, and the crypt cell production rate was 32 cells per crypt per hour; this latter value was confirmed using two independent techniques. The crypt growth fraction calculated from the durations of phases of the cell cycle and the labelling index was 0-62. A value of 0-61 was found from the labelling index distribution curve. As assessed from crypt squashes, there were 403 proliferating cells per crypt.     

Phenology simulation model of Scotinophara lurida (Hemiptera: Pentatomidae)

A phenology simulation model was developed for Scotinophara lurida (Burmeister). The components for the model were a degree-day immigration flight model of overwintered adults, temperature-dependent developmental models of each stage, survival rates of each stage, and an adult oviposition model. A degree-day model for immigration flight of overwintered adults was developed with blacklight trap catch data by a Weibull function. Laboratory experiments using seven constant temperature regimens were conducted to determine the effect of temperature on the development of immature stages. Developmental rates of each immature stage fit well to a linear model. Distribution of developmental time for each immature stage was successfully modeled against physiological age by a Weibull function. To determine the temperature effect on longevity, fecundity, and survival of female adults, laboratory and greenhouse experiments were conducted. The adult developmental rate (1/median longevity) was described by a linear model. The oviposition model was developed incorporating the three components of average total fecundity, cumulative oviposition rate function, and survival rate function. The simulation model predicted the time of peak occurrences of life stages of S. lurida well.