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REVIEW ARTICLES

Computational modeling of cell sorting, tissue engulfment, and related phenomena: A review

[+] Author and Article Information
G Wayne Brodland

Department of Civil Engineering, University of Waterloo, Waterloo, ON N2L 3G1 Canada; brodland@uwaterloo.ca

Appl. Mech. Rev 57(1), 47-76 (Feb 10, 2004) (30 pages) doi:10.1115/1.1583758 History: Online February 10, 2004
Copyright © 2004 by ASME
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References

Figures

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Classifications for homotypic systems of cells: Under suitable conditions, dissociated cells join together to form an aggregate which typically reshapes into a spherical mass. If suitable changes are made to the medium (see 12), the cells on the surface of the mass will dissociate from the rest of the mass until, ultimately, all of the cells become dissociated from each other.
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Classifications for heterotypic systems of cells: Many states are possible when two cell types are present. Under suitable conditions, two separate homotypic masses can fuse together, with one partially or totally engulfing the other. Alternatively, isolated cells of these same two types can aggregate, sort and produce a final state that is similar to that produced by engulfment. Other possibilities include the formation of checkerboard patterns. Reprinted from 13.
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Schematic of cells in a 3D aggregate: The forward-most cell has been sectioned to reveal the nucleus and control biochemicals, some of which it responds to and others of which it produces. The upper-most section through the same cell shows primary structural components, including microfilaments along the inner surface of the cell, and microtubules and intermediate filaments that are present in the cytoplasm that fills the cell volume. Also shown are the extracellular matrix (ECM) and cell adhesion molecules (CAMs) that act between the cells.
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Schematic of a close-up view of the triple junction between cells: Three cells (shown lightly shaded) contact each other. The bilayer membranes along the edges of the cells are shown, as are two kinds of cell-cell adhesion systems and a cell-cell junction.
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Functional model of the cell: The cell is understood to take certain “input” and to respond by producing certain output.
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Contact of two isolated cells: Two isolated cells A and B, as shown in part a of the figure, are brought together b. During this process, surface regions of approximate area AAB on each cell come into contact with each other to form a new interface, also approximately of area AAB. As each new unit area of A-B contact is produced, approximately a unit area of A-M (medium) interface and a unit area of B-M interface are lost.
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Mechanics of a triple junction: a) A generic triple junction, a surface tension γ arises along each interface, and superscripts are used to identify the constituents of the interface; b) A triple junction between two types of cells. Depending on the relative strengths of the interfacial tensions, the junction may move. Movement to the left would produce mixing of the cell types, while movement to the right would produce sorting. c) An interface between two cell types and the medium. As noted in the text, a number of different outcomes are possible, depending on the relative values of the three interfacial tensions.
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Possible outcomes at a triple junction between two cell types: When two cell types interact at a triple junction, the value of γLD relative to γLL and γDD determines whether the cells will mix, sort, or form a checkerboard pattern. The value of γDD is assumed to be larger than γLL. The arrows along the top of the figure indicate where the simulations shown in Fig. 27 fall along the γLD continuum. Reprinted from 13.
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Possible outcomes at a triple junction between two cell types and the medium: When two cell types and the medium interact at a triple junction, the value of γLD relative to γLM and γDM determines whether the cells will totally engulf each other, separate or, if neither of these, partially engulf each other. In this figure, γDM is assumed to be smaller than γLM, and as a result, type D cells would tend to engulf those of type L. The arrow labeled A indicates the value of γLD in the simulation reported in Fig. 29, a simulation that produces total engulfment, as predicted by the theory. Arrow B corresponds to an unpublished simulation that produced separation of the two tissue masses.
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Possible outcomes at a triple junction between one cell type and the medium: When one cell type interacts with the medium at a triple junction, the value of γLM relative to γLL determines whether the cells dissociate, adhere to each other or round up. Arrows A and B correspond to unpublished simulations that confirm the predictions of the theory.
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Spherical mass of cells: A jagged fragment of liver cells was isolated from five-day-old chick embryos. After two days in liquid medium at 37°C, they formed the spherical mass shown. Reprinted from 12 with permission of the American Association for the Advancement of Science.
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Partial tissue engulfment: Pigmented retinal tissue (darkly pigmented cells) and heart tissue (unpigmented cells) from ten-day-old chick embryos were brought into contact with each other. After being kept in organ culture conditions for two days, the retinal tissue partially engulfed the heart tissue. Reprinted from 26 with permission of CRC Press.
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Cell sorting: Pigmented retinal tissue (darkly pigmented cells) and neural retina (unpigmented cells) from seven-day-old chick embryos were maintained in stirred suspension culture. a) After five hours, the cells are highly disordered, and the perimeter of the aggregate is convoluted. b) After 19 hours, the pigmented retinal cells have left the surface of the aggregate and have formed homotypic clumps, and the surface of the aggregate has become smooth. c) By two days, the clumps and the mass as a whole have become substantially rounded. Reprinted from 26 with permission of CRC Press.
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Checkerboard pattern: A tracing of the luminal surface of a mature quail oviduct epithelium showing the arrangement of the C-cells (shown stippled) and the G-cells. Reprinted from 45 with permission of the Company of Biologists Ltd.
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Cell exchange model: In this model, each biological cell occupies one cell of the grid. Square or hexagonal grids can be used. Here, light cells represent one cell type while dark cells represent another. The basic algorithm involves the exchange of entire cells, as has happened between parts a and b of the figure.
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Simulation of cell sorting by cell exchange: In this simulation, the original authors set the parameters such that D-D interfaces are preferred over L-L and L-D interfaces. The result is that the cells sort: a) the initial configuration; b) an intermediate state; c) the final state of the system. Sorting is nearly complete and is aided by the migration of small clumps that are an artifact of the algorithm used. The time scale was not reported in the original article. Reprinted from 40 with permission of Harcourt Inc.
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Cell-centric model: In this model, the boundaries of the cells are determined by a Voronoi (or Delauney) tessellation of the plane based on the positions of the forming points. If the tessellation points are placed entirely at random, irregular cell shapes result and cell sizes vary widely. The wide lines indicate a tessellation based on the forming points shown as filled-in circles. The arrow indicates a small displacement of one of the forming points, and the narrow lines show the modified tessellation that results. The lengths of several boundaries are changed and the conformation of the mesh is altered.
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Simulation of checkerboard pattern formation using a cell-centric model: When it is energetically preferable for a cell to contact a cell of a different type than to contact one of its own type, a checkerboard pattern results. In this model, the Voronoi boundaries have been augmented with circular boundaries of specified radius so that the cells need not remain contiguous with each other. Reprinted from 134 with permission of Harcourt Inc.
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Simulation of cell sorting using a cell-centric model: a) initial configuration; b) and c) intermediate states, and d) final configuration. The boundary parameters have been chosen so that it is energetically preferable for the cells to form a D-D boundary rather than an L-D boundary and to form either of these rather than an L-L boundary. The parametric values for the D-M and L-M boundaries are set equal to each other. The resulting sorting is explicitly predicted by theory, but the engulfment of the darker cells by the lighter ones is not. Reprinted from 134 with permission of Harcourt Inc.
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Vertex Model: In this model, each cell is defined by the positions of its vertices. The algorithm moves adjacent vertices in pairs (from PQ to PQ, for example) in such a way that the areas of all of the cells are unchanged. Changes in boundary length result, and the configuration that minimizes the energy of the five affected boundaries is chosen. Reprinted from 45 with permission of the Company of Biologists Ltd.
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Standard cell-rearrangment algorithm: a) Two cells A and C are initially in contact; b) If the length of the A-C boundary shortens sufficiently, the configuration shown will result; c) increasing contact between cells B and D will produce a new finite-length interface between these cells. The usual numerical implementation of the algorithm involves a direct change from a configuration in which the length of the A-C interface ceases to be longer than M to a new configuration in which the A-C interface is exchanged for a new B-D interface, which is assigned a length somewhat greater than M. A slight change in cell areas occurs, and this change may or may not be corrected during subsequent steps of the algorithm. Reprinted from 46.
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Sub-cellular lattice model: a) In a sub-cellular lattice model, each biological cell (demarcated by shading) occupies multiple lattice cells (demarcated by the rectangular grid). Normally, the lattice sites from which a single cell is formed are simply connected. b) The algorithm can change the type of a lattice site to the type of an adjoining lattice site, as has happened in the case of the site marked with an asterisk. The usual consequence of this change is to modify the position of the boundary between adjacent cells.
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Simulation of cell sorting using a sub-cellular lattice model: a) Initial configuration. Surface energies, which correspond directly with surface tensions, have been assigned values of γDD=2,γLD=11,γLL=14 and γLMDM=16,b) through e) Intermediate states showing the characteristic components of sorting. f) The final configuration. MSC is a pseudo-time parameter based on the number of Monte Carlo steps used. Sorting is explicitly predicted by the theory, but engulfment of the dark cells by the light cells is not. Reprinted from 135 with permission.
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Simulation of multiple interactions using a sub-cellular lattice model: The final state shown demonstrates the power of this approach to model multiple complex interactions. Here γDD=4,γLD=11,γLL=14,γLM=2, and γDM=16, and dark cells remain compact and round, while light cells in contact with the medium dissociate. (Reprinted from 27 with permission.)
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Schematic cell cross-section and its corresponding finite element model: a) A cross-section of a typical cell. b) the corresponding finite element model.
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Simulation of cell sorting using a finite element model: a) initial configuration, with γDD=5,γLD=30,γLL=15,γLM=40, and γDM=60,b) and c) intermediate states showing the characteristic components of sorting; d) the final configuration.
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Simulations showing the effect of γLD:a) Initial configuration with γLL=12 and γDD=20;b) When γLD=3, a final state demonstrating mixing, and partial checkerboarding results; c) When γLD=14, partial mixing occurs. d) When γLD=22, partial sorting occurs. e) When γLD=40, there is a strong tendency toward sorting, but its effect is limited by the boundary conditions. f) When the configuration shown in e) is used as a starting configuration, γLLDD=0 and γLD=40, the system of cells behaves like a system of immiscible fluids, and the boundaries between different cell types become smooth like the boundary between immiscible fluids. Reprinted from 13.
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Simulation of dissolution using the finite element method: When γLL=30 and γLM=10, it is energetically preferable for cells to be in contact with the medium rather than with each other. As a result, the triple junctions between pairs of cells and the medium are drawn into the mass pulling the medium in as they move. The result is that the medium infiltrates the cell-cell interfaces, causing the surface cells to dissociate from the mass. If the process were allowed to continue, the newly exposed surface cells would also dissociate from the mass until all of the cells become separated from each other.
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Simulation of total engulfment using the finite element method: a) Initial configuration with γLLDD=5,γLD=25,γLM=150,γDM=70;b) The dark cells are drawn over the light cells; c) Total engulfment of the light cells by the dark cells has occurred.
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Fundamental difference between two- and three-dimensional sorting. When cells form 3D aggregates, groups of like cells, B, in any one plane may be connected to other similar groups, such as A and C, by bridges that involve the third dimension. Interfacial tensions would act along the surfaces of these bridges and would either draw the isolated clumps together (A and B) or would draw cells out of the plane as the bridges shorten (C). A 2D model would not benefit from these effects, and cell sorting would be less complete in the sense that isolated groups of cells in the plane would remain isolated. Reprinted from 13.

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