Colorectal cancer is initiated in colonic crypts. A succession of genetic mutations or epigenetic changes can lead to homeostasis in the crypt being overcome, and subsequent unbounded growth. We consider the dynamics of a single colorectal crypt by using a compartmental approach [Tomlinson IPM, Bodmer WF (1995) Proc Natl Acad Sci USA 92:11130 -11134], which accounts for populations of stem cells, differentiated cells, and transit cells. That original model made the simplifying assumptions that each cell population divides synchronously, but we relax these assumptions by adopting an age-structured approach that models asynchronous cell division, and by using a continuum model. We discuss two mechanisms that could regulate the growth of cell numbers and maintain the equilibrium that is normally observed in the crypt. The first will always maintain an equilibrium for all parameter values, whereas the second can allow unbounded proliferation if the net per capita growth rates are large enough. Results show that an increase in cell renewal, which is equivalent to a failure of programmed cell death or of differentiation, can lead to the growth of cancers. The second model can be used to explain the long lag phases in tumor growth, during which new, higher equilibria are reached, before unlimited growth in cell numbers ensues.age-structure ͉ feedback ͉ mutations ͉ structural stability T he large intestine is one of the most frequent sites of carcinogenesis due, at least in part, to its continual selfrenewal and the large numbers of daily cell divisions (1). There are millions of invaginations in the lining of the colon, called crypts, and it is widely believed that colorectal cancer is initiated when mutations or relatively stable epigenetic changes occur in the single layer of epithelial cells that line the crypt. Consequently, much work has been directed toward understanding the mechanisms involved in the dynamics of the cells in healthy and neoplastic (abnormally growing) crypts.Stem cells are believed to reside near the bottom of the colorectal crypt (2), and these are capable of producing a variety of cell types that are required for tissue renewal and regeneration after injury (3). The stem cells divide to produce transit cells that migrate up the crypt wall toward the luminal surface. As the cells proceed up the crypt they differentiate into colonocytes, enteroendocrine cells, and Goblet cells (1). Once at the top, the cells either undergo apoptosis and͞or are shed into the lumen and transported away (4, 5).In the murine small intestine, the journey of the cells from the base of the crypt to its apex has been estimated to take between 2 and 3 days (6), and all the cells in the crypt, apart from the stem cells, will be renewed over this period. The stem cells are assumed to have a cycle time of between 12 and 32 h with an average of 24 h (7, 8). The transit cell population has an estimated cycle time of Ϸ11-12 h (4, 9).The crypt is homeostatic with an equilibrium maintained between cell proliferation and cell los...
Multiscale modeling is emerging as one of the key challenges in mathematical biology. However, the recent rapid increase in the number of modeling methodologies being used to describe cell populations has raised a number of interesting questions. For example, at the cellular scale, how can the appropriate discrete cell-level model be identified in a given context? Additionally, how can the many phenomenological assumptions used in the derivation of models at the continuum scale be related to individual cell behavior? In order to begin to address such questions, we consider a discrete one-dimensional cell-based model in which cells are assumed to interact via linear springs. From the discrete equations of motion, the continuous Rouse [P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)] model is obtained. This formalism readily allows the definition of a cell number density for which a nonlinear "fast" diffusion equation is derived. Excellent agreement is demonstrated between the continuum and discrete models. Subsequently, via the incorporation of cell division, we demonstrate that the derived nonlinear diffusion model is robust to the inclusion of more realistic biological detail. In the limit of stiff springs, where cells can be considered to be incompressible, we show that cell velocity can be directly related to cell production. This assumption is frequently made in the literature but our derivation places limits on its validity. Finally, the model is compared with a model of a similar form recently derived for a different discrete cell-based model and it is shown how the different diffusion coefficients can be understood in terms of the underlying assumptions about cell behavior in the respective discrete models.
Epithelial cell layers on soft elastic substrates or pillar arrays are commonly used as model systems for investigating the role of force in tissue growth, maintenance and repair. Here we show analytically that the experimentally observed localization of traction forces to the periphery of the cell layers does not necessarily imply increased local cell activity, but follows naturally from the elastic problem of a finite-sized contractile layer coupled to an elastic foundation. For homogeneous contractility, the force localization is determined by one dimensionless parameter interpolating between linear and exponential force profiles for the extreme cases of very soft and very stiff substrates, respectively. If contractility is sufficiently increased at the periphery, outward directed displacements can occur at intermediate positions. We also show that anisotropic extracellular stiffness can lead to force localization in the stiffer direction, as observed experimentally.PACS numbers: 87.10.+e, 87.16.Ln, 87.17.Rt Actively generated forces have emerged as a central element in the way tissue cells interact with their environment, for example during development and tissue maintenance [1]. It is becoming increasingly clear that cells use physical force to probe the mechanical properties of their environments and to create a structurally coherent state within the cell and tissue. Experimentally, however, it is challenging to measure cellular forces in a physiological context. Therefore their effect is usually assessed in an indirect manner, for example by mechanical relaxation after laser cutting [2].In order to directly measure the forces that cells exert on their environments, two complementary techniques have been established over the last decade. Embedding marker beads into soft elastic gels enables the measurement of the displacement field generated by cell traction forces applied to the surface of the gel, which can then be estimated using elasticity theory [3]. Alternatively, microfabrication techniques are used to create an array of elastomeric pillars, which for small displacements each act as a linear spring [4].In order to address the role of forces for larger cell clusters, both soft elastic substrates and pillar assays have been used for confluent layers of epithelial cells. Placing such cells on a pillar array, it was found that cellular traction forces are localized to the edge of the cell layer [5,6]. A qualitatively similar effect has also been observed for cell sheets migrating over soft elastic substrates [7], although in this case the length scales over which the stresses decayed were greater. It is tempting to assume that force localization to the edge reflects increased mechanical activity of the cells at the edge, for example, of putative leader cells [8]. However, a more detailed analysis requires a mechanical analysis of the cell sheet coupled to the elastic foundation.In this Letter, we analytically solve the problem of a contracting cell layer coupled to a layer of springs. For homogeneous...
In syncytial embryos nuclei undergo cycles of division and rearrangement within a common cytoplasm. It is presently unclear to what degree and how the nuclear array maintains positional order in the face of rapid cell divisions. Here we establish a quantitative assay, based on image processing, for analysing the dynamics of the nuclear array. By tracking nuclear trajectories in Drosophila melanogaster embryos, we are able to define and evaluate local and time-dependent measures for the level of geometrical order in the array. We find that after division, order is re-established in a biphasic manner, indicating the competition of different ordering processes. Using mutants and drug injections, we show that the order of the nuclear array depends on cytoskeletal networks organised by centrosomes. While both f-actin and microtubules are required for re-establishing order after mitosis, only f-actin is required to maintain the stability of this arrangement. Furthermore, f-actin function relies on myosin-independent non-contractile filaments that suppress individual nuclear mobility, whereas microtubules promote mobility and attract adjacent nuclei. Actin caps are shown to act to prevent nuclear incorporation into adjacent microtubule baskets. Our data demonstrate that two principal ordering mechanisms thus simultaneously contribute: (1) a passive crowding mechanism in which nuclei and actin caps act as spacers and (2) an active self-organisation mechanism based on a microtubule network.
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