Establishment of epidermal cell columns in mammalian skin: computer simulation

A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity

Most cancers in humans are large, measuring centimetres in diameter, and composed of many billions of cells. An equivalent mass of normal cells would be highly heterogeneous as a result of the mutations that occur during each cell division. What is remarkable about cancers is that virtually every neoplastic cell within a large tumour often contains the same core set of genetic alterations, with heterogeneity confined to mutations that emerge late during tumour growth. How such alterations expand within the spatially constrained three-dimensional architecture of a tumour, and come to dominate a large, pre-existing lesion, has been unclear. Here we describe a model for tumour evolution that shows how short-range dispersal and cell turnover can account for rapid cell mixing inside the tumour. We show that even a small selective advantage of a single cell within a large tumour allows the descendants of that cell to replace the precursor mass in a clinically relevant time frame. We also demonstrate that the same mechanisms can be responsible for the rapid onset of resistance to chemotherapy. Our model not only provides insights into spatial and temporal aspects of tumour growth, but also suggests that targeting short-range cellular migratory activity could have marked effects on tumour growth rates.

Cell models lead to understanding of multi-cellular morphogenesis consisting of successive self-construction of cells

Morphogenesis of multi-cellular organisms occurs through cell behaviours within a cell aggregate. Cell behaviours have been described using cell models involving equations of motion for cells. Cells in cell models construct shapes of the cell aggregate by themselves. Here, a history of cell models, the cell centre model and the vertex cell model, which we have constructed, are described. Furthermore, the application of these cell models is explained in detail. These cell models have been applied to transformation of cell aggregates to become spherical, formation of mammalian blastocysts and cell intercalation in elongating tissues. These are all elemental processes of morphogenesis and take place in succession during the whole developmental process. A chain of successive elemental processes leads to morphogenesis. Finally, we highlight that cell models are indispensable to understand the process whereby genes direct biological shapes.

Study on multicellular systems using a phase field model

A model of multicellular systems with several types of cells is developed from the phase field model. The model is presented as a set of partial differential equations of the field variables, each of which expresses the shape of one cell. The dynamics of each cell is based on the criteria for minimizing the surface area and retaining a certain volume. The effects of cell adhesion and excluded volume are also taken into account. The proposed model can be used to find the position of the membrane and/or the cortex of each cell without the need to adopt extra variables. This model is suitable for numerical simulations of a system having a large number of cells. The two-dimensional results of cell division, cell adhesion, rearrangement of a cell cluster, chemotaxis, and cell sorting as well as the three-dimensional results of cell clusters on the substrate are presented.

Growth based morphogenesis of vertebrate limb bud

Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea "growth-based morphogenesis." We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs.

A multicellular systems biology model predicts epidermal morphology, kinetics and Ca2+ flow

MOTIVATION

Systems biology is currently focused on integrating intracellular networks, although clinically, diseases are largely defined by their histological features. For example, no computational model can simulate today the formation of a horizontally layered epidermis. Since the epidermis is the most complex structured epithelial tissue, systems biology models could yield important insights in epithelial tissue, in which most of all human cancers arise.

RESULTS

We describe the algorithms of a system, capable of simulating the tissue homeostasis in human epidermis leading to a horizontally layered tissue with cells of different differentiation stages. The system predicts epidermal morphology, tissue kinetics and 2D flow of Ca2+ ions. Predicted properties of an epidermis with a healthy and a disturbed barrier are compared with the literature. The system closely mimics the respecting physiological situations.

AVAILABILITY

Additional information and films of the simulation are available at the website. Source code is available on request. http://www.zbh.uni-hamburg.de/research/ESB/index.php

CONTACT

grabe@zbh.uni-hamburg.de

Mathematical modeling of vascular endothelial layer maintenance: the role of endothelial cell division, progenitor cell homing, and telomere shortening

Maintenance of the endothelial cell (EC) layer of the vessel wall is essential for proper functioning of the vessel and prevention of vascular disorders. Replacement of damaged ECs could occur through division of surrounding ECs. Furthermore, EC progenitor cells (EPCs), derived from the bone marrow and circulating in the bloodstream, can differentiate into ECs. Therefore, these cells might also play a role in maintenance of the endothelial layer in the vascular system. The proliferative potential of both cell types is limited by shortening of telomeric DNA. Accelerated telomere shortening might lead to senescent vascular wall cells and eventually to the inability of the endothelium to maintain a continuous monolayer. The aim of this study was to describe the dynamics of EC damage and repair and telomere shortening by a mathematical model. In the model, ECs were integrated in a two-dimensional structure resembling the endothelium in a large artery. Telomere shortening was described as a stochastic process with oxidative damage as the main cause of attrition. Simulating the model illustrated that increased cellular turnover or elevated levels of oxidative stress could lead to critical telomere shortening and senescence at an age of 65 yr. The model predicted that under those conditions the EC layer could display defects, which could initiate severe vascular wall damage in reality. Furthermore, simulations showed that 5% progenitor cell homing/yr can significantly delay the EC layer defects. This stresses the potential importance of EPC number and function to the maintenance of vascular wall integrity during the human life span.

A three-dimensional vertex dynamics cell model of space-filling polyhedra simulating cell behavior in a cell aggregate

We developed a three-dimensional (3D) cell model of a multicellular aggregate consisting of several polyhedral cells to investigate the deformation and rearrangement of cells under the influence of external forces. The polyhedral cells fill the space in the aggregate without gaps or overlaps, consist of contracting interfaces and maintain their volumes. The interfaces and volumes were expressed by 3D vertex coordinates. Vertex movements obey equations of motion that rearrange the cells to minimize total free energy, and undergo an elementary process that exchanges vertex pair connections when vertices approach each other. The total free energy includes the interface energy of cells and the compression or expansion energy of cells. Computer simulations provided the following results: An aggregate of cells becomes spherical to minimize individual cell surface areas; Polygonal interfaces of cells remain flat; Cells within the 3D cell aggregate can move and rearrange despite the absence of free space. We examined cell rearrangement to elucidate the viscoelastic properties of the aggregate, e.g. when an external force flattens a cell aggregate (e.g. under centrifugation) its component cells quickly flatten. Under a continuous external force, the cells slowly rearrange to recover their original shape although the cell aggregate remains flat. The deformation and rearrangement of individual cells is a two-step process with a time lag. Our results showed that morphological and viscoelastic properties of the cell aggregate with long relaxation time are based on component cells where minimization of interfacial energy of cells provides a motive force for cell movement.

Spontaneous architectural organization of mammalian epidermis from random cell packing

The cells of the stratum corneum in the epidermis of some mammalian species are precisely stacked in columns in a honeycomb fashion. The epidermis constantly loses surface cells, which are replaced by basal cells that have differentiated during migration to the surface. The path of this migration is seen as precisely defined columns of cells that are in compressed Kelvin's tetrakaidecahedral form. We present a computer simulation of this architectural organization based on the assumption that the cells that migrate upward occupy less crowded regions. The simulation not only explained the mechanism by which the architecture is maintained during the process of cell replacement, but also showed that the architecture was spontaneously organized from initial cells supplied at random. Living organisms consist of self-organizing systems at various levels; however, self-organizing systems have been investigated mostly at the molecular level. The present computer simulation clarified the self-organizing system at the cellular level.

A computer simulation of cell stacking for even thickness in mammalian epidermis

The method of even stacking of epidermal cells in mammalian skin was studied by computer simulation. The epidermis consists of neat vertical columns of stacked, flattened, tetrakaidecahedral cells. Cells which have been proliferated in a basal layer migrate upwards, occupy the bottom regions of vertical columns, and become members constituting columns. Computer simulations demonstrated that the column height becomes considerably varied if the cells are randomly supplied from the basal layer. In contrast, if the cells are assumed to have an ability to find the uppermost region among the column's bases consisting of one base where the cell has reached and its neighbouring bases, the cells stack into columns whose heights are remarkably uniform even if the cells are randomly supplied. The results indicated that an epidermal structure consisting of the flattened polyhedral cells could itself function as a control mechanism of the epidermal thickness.

A computer simulation of geometrical configurations during cell division

A process of cell division in the blastular wall of the starfish, Asterina pectinifera, was observed, and an attempt was made to model with a computer simulation the way in which cell number increases in such a tissue. Dividing cells at stages between the 2(11)-cell and the beginning of rotation were observed to shift these positions to the outer surface of the cell sheet by rounding up, after which they divide and slip back into the sheet as two columnar daughter cells. The change of a polygonal pattern of the blastular wall by cell division was simulated by making use of geometrical models of polygonal cells and the rule of the direction of cell division which was confirmed by observation. The simulation proves valid for describing changes of polygonal patterns of cell sheets including dividing cells.

Cell movements in a living mammalian tissue: long-term observation of individual cells in wounded corneal endothelia of cats

Although the cells in tissues are known to be motile under special conditions (e.g., during tissue turnover or wound healing), there are not many reports that polygonal cells covering an area without leaving any gaps are also capable of movement. In the present study, cell movements (cell shifting and rearrangement) in a living mammalian eye tissue were documented by identifying and locating individual cells over intervals as long as 100 days. Cat corneal endothelium, a monolayered cell sheet, was wounded by removing a small number (about 180) of endothelial cells from the internal lining of the cornea. Healing of the wounded tissue was observed with a wide-view specular microscope applied to the outer surface of the cornea, enabling us to identify individual cells for as long as two to three months. Cells surrounding the wound underwent areal enlargement, elongated toward the wound, and shifted to cover the wound surface. During days 4-7, cells became rearranged by changing neighbors in such a way that they retained their enlarged size but recovered their non-elongated, original shape. This pattern of cell rearrangement was interpreted by a computer simulation which assumed that cells shorten their boundary length while maintaining contacts with contiguous cells. After day 7, the enlarged cells adjacent to the wounded area gradually contracted and pulled surrounding cells toward the wounded area. These movements were followed by a temporary halt in cell shifting, then by a recovery of shifting and cell elongation. These movements are interpreted as a result of the contractility of endothelial cell microfilaments.