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Computational Modelling of the Biomechanics of Epithelial and Mesenchymal Cell Interactions During Morphological Development
Abstract
Computational modelling of morphological development of tissues based on complex systems and cellular automata can be decomposed into three interdependent processes. Those three crucial parts are mechanical response of tissues, diffusion of signalling molecules, and gene regulatory network. It is shown that development of an adequate mechanical model of living tissues provides the morphological model with sufficient flexibility necessary to achieve expected morphological development scenarios. In this contribution, the attention is focussed to development of mesenchymal and epithelial tissues which, e.g., creates the
basic mechanism of tooth development. The future development of the model is discussed with emphasis on open questions.
Link to the book chapter:
http://link.springer.com/chapter/10.1007%2F978-3-540-75767-2_13
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We show that differential adhesion with fluctuations is sufficient to explain a wide variety of cell rearrangement, by using the extended large-Q Potts model with differential adhesivity to simulate different biological phenomena. Different values of relative surface energies correspond to different biological cases, including complete and partial cell sorting, checkerboard, position reversal, and dispersal. We examine the convergence and temperature dependence of the simulation and distinguish spontaneous, neutral, and activated processes by performing simulations at different temperatures. We discuss the biological and physical implications of our quantitative results.
The cellular automaton computer simulation technique has been applied to the problem of dynamic recrystallization. The model is composed of simple microstructural mechanisms - such as evolution of dislocation density, grain boundary movement and nucleation - that acting together result in complex macrostructural response expressed by evolution of flow stress and grain size. A strong sensitivity of the flow stress curves to the initial grain size and nucleation rate is observed. Simulations lead to the conclusion that model of a dynamically recrystallizing material has a strong structural memory resulting in wide variety of experimentally observed responses.
A model performing mesh partitioning into computationally equivalent mesh parts on regular square lattices using a diffusion controlled cellular automaton (DCCA) is proposed and studied in this article. Every processor has assigned a domain seed at the beginning of a simulation. Algorithm works with growth of seeds and migration of domain borders, the later is triggered when the difference of diffusive agents on both sides of a border exceeds a given threshold. The model is built using self-organization principles ensuring convergence. Solutions are dynamically stable configurations achieved from any initial configuration.
Developmental biology describes how tissues, organs, and bodies are made from living cells. There exists a large body of biological
data about developmental processes but there is still not ultimate understanding of how the whole orchestra of all involved
processes is working. It is the place where mathematical modelling could help to create biologically relevant models of morphological
development. The morphological development could be mathematically decomposed into three distinct but mutually interconnected
parts, namely to mechanical response of tissues, signalling by chemicals, and switching of cells into different types by a
gene regulatory network. This paper is focussed to the part dealing with mechanical interaction of growing mesenchyme and
epithelium within a living tissue modelled by a set of nodes interconnected by deformable bars as in tensegrity models.
A model of morphological development of living tissues could be split into three mutually entangled parts: a mechanical model of tissues, diffusion of signaling molecules, and a gene regulatory network. This paper propose a cellular automaton mechanical model of mesenchyme including its growth caused by proliferation of cells. The main attention is focussed to the mechanical interaction of mesenchyme and epithelium, to mesenchymal growth, and to the influence of growth rate along with epithelial forces on the final shape of the tissues. Future development of the model is briefly outlined.
Generation of morphological diversity remains a challenge for evolutionary biologists because it is unclear how an ultimately finite number of genes involved in initial pattern formation integrates with morphogenesis. Ideally, models used to search for the simplest developmental principles on how genes produce form should account for both developmental process and evolutionary change. Here we present a model reproducing the morphology of mammalian teeth by integrating experimental data on gene interactions and growth into a morphodynamic mechanism in which developing morphology has a causal role in patterning. The model predicts the course of tooth-shape development in different mammalian species and also reproduces key transitions in evolution. Furthermore, we reproduce the known expression patterns of several genes involved in tooth development and their dynamics over developmental time. Large morphological effects frequently can be achieved by small changes, according to this model, and similar morphologies can be produced by different changes. This finding may be consistent with why predicting the morphological outcomes of molecular experiments is challenging. Nevertheless, models incorporating morphology and gene activity show promise for linking genotypes to phenotypes.
We present a classification of developmental mechanisms that have been shown experimentally to generate pattern and form in metazoan organisms. We propose that all such mechanisms can be organized into three basic categories and that two of these may act as composite mechanisms in two different ways. The simple categories are cell autonomous mechanisms in which cells enter into specific arrangements ('patterns') without interacting, inductive mechanisms in which cell communication leads to changes in pattern by reciprocal or hierarchical alteration of cell phenotypes ('states') and morphogenetic mechanisms in which pattern changes by means of cell interactions that do not change cell states. The latter two types of mechanism can be combined either morphostatically, in which case inductive mechanisms act first, followed by the morphogenetic mechanism, or morphodynamically, in which case both types of mechanisms interact continuously to modify each other's dynamics. We propose that this previously unexplored distinction in the operation of composite developmental mechanisms provides insight into the dynamics of many developmental processes. In particular, morphostatic and morphodynamic mechanisms respond to small changes in their genetic and microenvironmental components in dramatically different ways. We suggest that these differences in 'variational properties' lead to morphostatic and morphodynamic mechanisms being represented to different extents in early and late stages of development and to their contributing in distinct ways to morphological transitions in evolution.
In this paper we present the foundation of a unified, object-oriented, three-dimensional biomodelling environment, which allows us to integrate multiple submodels at scales from subcellular to those of tissues and organs. Our current implementation combines a modified discrete model from statistical mechanics, the Cellular Potts Model, with a continuum reaction-diffusion model and a state automaton with well-defined conditions for cell differentiation transitions to model genetic regulation. This environment allows us to rapidly and compactly create computational models of a class of complex-developmental phenomena. To illustrate model development, we simulate a simplified version of the formation of the skeletal pattern in a growing embryonic vertebrate limb.
The feasibility of the use of Cellular Automata (CA) techniques for the design of two-dimensional structural problems, such as trusses and continuum structures under static loading, is investigated. The study implements an integrated analysis and design approach using the methods of CA to achieve an optimal structural configuration. The paper summarizes the basic features of the CA and demonstrates a formulation for the design of two-dimensional truss structures that exhibit linear and geometrically nonlinear response characteristics. The solution variables chosen for the approach include the in-plane displacements of the cells, which represent nodal points of the truss, and the cross sectional areas of the members (hat connect the neighboring cells, Equations derived from local equilibrium are used as the rules that govern the cell displacements, and simple rules that are based on fully stressed design conditions are used for sizing of the members. In addition to the theoretical formulation, an investigation into the numerical implementation of the CA was performed. Initially the Mathematica® programming language was used for demonstration purposes. For large-scale problems, an efficient and flexible Fortran 90 computer code capable of modeling complex two-dimensional geometries was developed. Examples demonstrating the capabilities of the design tools are included.
1. The Foundations for a New Kind of Science 2. The Crucial Experiment 3. The World of Simple Programs 4. Systems Based on Numbers 5. Two Dimensions and Beyond 6. Starting from Randomness 7. Mechanisms in Programs and Nature 8. Implications for Everyday Systems 9. Fundamental Physics 10. Processes of Perception and Analysis 11. The Notion of Computation 12. The Principle of Computational Equivalence
The crown pattern of the tooth is essentially that of the surface of the dentine (dentine-enamel junction), modified by the deposition of enamel which may be uneven in thickness. The dentine-enamel junction preserves in the completed tooth the form of the membrana praeformativa, the basement membrane of the inner enamel epithelium of the enamel organ. Folding of this membrane creates the crown pattern.
The inner enamel epithelium is subjected to pressure from both sides. On the basal side there is the rapidly growing mesenchyme of the dental papilla, and on the occlusal side there is the stellate reticulum, which swells by the accumulation of fluid. The stellate reticulum prevents distortion of the epithelium by growth of the papilla, and thus ensures that folding of the epithelium is due to its intrinsic growth pattern. This makes for more accurate control of the crown pattern, the details of which are of importance in the function of chewing.
The enamel knot is a region of the inner enamel epithelium from which cells are contributed to the stellate reticulum. It represents the tip of the primary cusp. The enamel cord (‘enamel septum’) which consists of cells which are in process of transforming into stellate reticulum, has been confused with two other structures that develop later: a cleavage septum, preparatory to the formation of crown cementum, and an epithelial septum, found in marsupials and crocodilians. The epithelial nodules of monotremes are probably degenerate relics of an epithelial septum.
The inner enamel epithelium is a diaphragm passing across the interior of the dental follicle, and folding to adapt its increased area to a confined space. The cement organ, which in some mammals develops from the follicle, probably plays no part in the deformation of the epithelium, but the follicle as a whole may be subject to compression by adjacent follicles.
A cusp is a centre of precocious maturation of the cells of the inner enamel epithelium. Here growth ceases (perhaps after a transitory burst of mitosis) and eventually the hard tissues are deposited. The process starts at the tip of the cusp and extends basally, so that growth continues longest in the valleys, intensifying the crown relief. Dens in dente is due to retarded maturation of an area of the enamel epithelium.
Throughout the development of the crown there is a marginal zona cingularis, where growth continues. The crown pattern depends upon the position and the stage of growth in which cusps are differentiated from the zona cingularis, by accelerated maturation of groups of cells. Cusps which appear late in development stand low on the crown and frequently form part of a cingulum. Changes in the timing of cusp formation play an important part in serial modifications of pattern, as well as in phylogeny.
Ridges are probably produced by tensions set up in the epithelium by the relative movement of cusps, owing to unequal growth or to changes in the shape of the follicle. They form in areas where growth has slowed down but the apposition of hard tissues has not begun.
The lobes of the basal outline of the tooth are produced by growth centres in the papilla, which cause evagination of the follicle. Each growth centre is supplied by a bundle of blood vessels. The roots form in relation to these blood vessels, and so reflect the organization of the papilla. Hertwig's epithelial sheath grows between the follicle and the base of the papilla, the direction of its growth being controlled by the surrounding tissues owing to the absence of stellate reticulum on the basal portion of the tooth.
There is no constant relation between cusps and roots, although marginal cusps frequently arise in association with lobes of the basal outline. The crown pattern results from an interaction between the growth pattern of the inner enamel epithelium and that of the papilla, the latter controlling the shape of the margin which limits the folding epithelium.
The morphogenesis and cell differentiation in developing teeth is governed by interactions between the oral epithelium and neural crest-derived ectomesenchyme. The fibroblast growth factors FGF-4, -8, and -9 have been implicated as epithelial signals regulating mesenchymal gene expression and cell proliferation during tooth initiation and later during epithelial folding morphogenesis and the establishment of tooth shape. To further evaluate the roles of FGFs in tooth development, we analyzed the roles of FGF-3, FGF-7, and FGF-10 in developing mouse teeth. In situ hybridization analysis showed developmentally regulated expression during tooth formation for Fgf-3 and Fgf-10 that was mainly restricted to the dental papilla mesenchymal cells. Fgf-7 transcripts were restricted to the developing bone surrounding the developing tooth germ. Fgf-10 expression was observed in the presumptive dental epithelium and mesenchyme during tooth initiation, whereas Fgf-3 expression appeared in the dental mesenchyme at the late bud stage. During the cap and bell stage, both Fgf-3 and Fgf-10 were intensely expressed in the dental papilla mesenchymal cells both in incisors and molars. It is of interest that Fgf-3 expression was also observed in the primary enamel knot, a putative signaling center of the tooth, whereas no transcripts were seen in the secondary enamel knots that appear in the tips of future cusps of the bell stage tooth germs. Down-regulation of Fgf-3 and Fgf-10 expression in postmitotic odontoblasts correlated with the terminal differentiation of the odontoblasts and the neighboring ameloblasts. In the incisors, mesenchymal cells of the cervical loop area showed partially overlapping expression patterns for all studied Fgfs. In vitro analyses showed that expression of Fgf-3 and Fgf-10 in the dental mesenchyme was dependent on dental epithelium and that epithelially expressed FGFs, FGF-4 and -8 induced Fgf-3 but not Fgf-10 expression in the isolated dental mesenchyme. Beads soaked in Shh, BMP-2, and TGF-β1 protein did not induce either Fgf-3 or Fgf-10 expression. Cells expressing Wnt-6 did not induce Fgf-10 expression. Furthermore, FGF-10 protein stimulated cell proliferation in the dental epithelium but not in the mesenchyme. These results suggest that FGF-3 and FGF-10 have redundant functions as mesenchymal signals regulating epithelial morphogenesis of the tooth and that their expressions appear to be differentially regulated. In addition, FGF-3 may participate in signaling functions of the primary enamel knot. The dynamic expression patterns of different Fgfs in dental epithelium and mesenchyme and their interactions suggest existence of regulatory signaling cascades between epithelial and mesenchymal FGFs during tooth development. © 2000 Wiley-Liss, Inc.
This paper presents a shape and topology optimization scheme for structures by using the concept of a cellular automaton
(CA). A design domain domain is divided into small square cells and then the thicknesses of the individual cells are taken
as the design variables. Considering the cells as the finite elements, the stress analysis is performed by the finite element
method. The design variables are modified by applying a local rule to the stress states of the cell and its neighbouring cells.
The present scheme is applied to a two-dimensional elastic problem in order to confirm its validity.
There has been recent interest in exploring alternative computational models for structural analysis that are better suited
for a design environment requiring repetitive analysis. The need for such models is brought about by significant increases
in computer processing speeds, realized primarily through parallel processing. To take full advantage of such parallel machines,
however, the computational approach itself must be revisited from a totally different perspective; parallelization of inherently
serial paradigms is subject to limitations introduced by a requirement of information coordination. The cellular automata
(CA) model of decentralized computations provides one such approach which is ideally tailored for parallel computers. The
proposed paper examines the applicability of the cellular automata model in problems of 2-D elasticity. The focus of the paper
is in the use of a genetic algorithm based optimization process to derive the rules for local interaction required in evolving
the cellular automata.
Cellular automata are dynamical systems where space, time, and variables are discrete. They are shown on two-dimensional examples to be capable of non-numerical simulations of physics. They are useful for faithful parallel processing of lattice models. At another level, they exhibit behaviours and illustrate concepts that are unmistakably physical, such as non-ergodicity and order parameters, frustration, relaxation to chaos through period doublings, a conspicuous arrow of time in reversible microscopic dynamics, causality and light-cone, and non-separability. In general, they constitute exactly computable models for complex phenomena and large-scale correlations that result from very simple short-range interactions. We study their space, time, and intrinsic symmetries and the corresponding conservation laws, with an emphasis on the conservation of information obeyed by reversible cellular automata.
Cellular automata are models of distributed dynamical systems whose structure is particularly well suited to ultrafast, exact numerical simulation. On the other hand, they constitute a radical departure from the traditional partial-differential-equation approach to distributed dynamics. Here we discuss the problem of encoding the state-variables and evolution laws of a physical system into this new setting, and of giving suitable correspondence rules for interpreting the model's behavior.
Mammalian dentition consists of teeth that develop as discrete organs. From anterior to posterior, the dentition is divided into regions of incisor, canine, premolar and molar tooth types. Particularly teeth in the molar region are very diverse in shape. The development of individual teeth involves epithelial-mesenchymal interactions that are mediated by signals shared with other organs. Parts of the molecular details of signaling networks have been established, particularly in the signal families BMP, FGF, Hh and Wnt, mostly by the analysis of gene expression and signaling responses in knockout mice with arrested tooth development. Recent evidence suggests that largely the same signaling cascade is used reiteratively throughout tooth development. The successional determination of tooth region, tooth type, tooth crown base and individual cusps involves signals that regulate tissue growth and differentiation. Tooth type appears to be determined by epithelial signals and to involve differential activation of homeobox genes in the mesenchyme. This differential signaling could have allowed the evolutionary divergence of tooth shapes among the four tooth types. The advancing tooth morphogenesis is punctuated by transient signaling centers in the epithelium corresponding to the initiation of tooth buds, tooth crowns and individual cusps. The latter two signaling centers, the primary enamel knot and the secondary enamel knot, have been well characterized and are thought to direct the differential growth and subsequent folding of the dental epithelium. Several members of the FGF signal family have been implicated in the control of cell proliferation around the non-dividing enamel knots. Spatiotemporal induction of the secondary enamel knots determines the cusp patterns of individual teeth and is likely to involve repeated activation and inhibition of signaling as suggested for patterning of other epithelial organs.
NASA
The role of tensegrity in the architecture of organic structures is examined. Topics include a definition of tensegrity, principles of tensegrity applied to the skeleton and cytoskeleton, mechanics in biochemistry, self-assembly of organic structures, geodesic forms in cellular structure, and the universality of the geodesic form.
Development of glandular organs such as the kidney, lung, and prostate involves the process of branching morphogenesis. The developing organ begins as an epithelial bud that invades the surrounding mesenchyme, projecting dividing epithelial cords or tubes away from the site of initiation. This is a tightly regulated process that requires complex epithelial-mesenchymal interactions, resulting in a three-dimensional treelike structure. We propose that activins are key growth and differentiation factors during this process. The purpose of this review is to examine the direct, indirect, and correlative lines of evidence to support this hypothesis. The expression of activins is reviewed together with the effect of activins and follistatins in the development of branched organs. We demonstrate that activin has both negative and positive effects on cell growth during branching morphogenesis, highlighting the complex nature of activin in the regulation of proliferation and differentiation. We propose potential mechanisms for the way in which activins modify branching and address the issue of whether activin is a regulator of branching morphogenesis.
Growth factors and other paracrine signal molecules regulate communication between cells in all developing organs. During tooth morphogenesis, molecules in several conserved signal families mediate interactions both between and within the epithelial and mesenchymal tissue layers. The same molecules are used repeatedly during advancing development, and several growth factors are coexpressed in epithelial signaling centers. The enamel knots are signaling centers that regulate the patterning of teeth and are associated with foldings of the epithelial sheet. Different signaling pathways form networks and are integrated at many levels. Many targets of the growth factors have been identified, and mutations in several genes within the signaling networks cause defective tooth formation in both humans and mice.
Teeth develop as ectodermal appendages in vertebrate embryos, and their early development resembles morphologically as well as molecularly other organs such as hairs and glands. Interactions between the ectoderm and underlying mesenchyme constitute a central mechanism regulating the morphogenesis of
Here we review studies of the physical, material properties of animal cells and their cytoskeleton, such as elastic stiffness and fluid viscosity, that determine how they respond to, and are shaped by, forces inside and out. Currently and historically, most such studies have reported a single value for a cell property and/or propose a single broad structural model based on nonliving materials. We believe that such physical studies would be of more interest to most cell biologists if greater emphasis were placed on the well-established regional differences within a cell and the ability of the cell to quickly change its mechanical behaviors
During development, embryonic tissues are shaped in a species-specific manner. Yet, across species, general classes of tissue remodeling events occur, such as tissue infolding and tissue elongation. The spatiotemporal control of these morphogenetic processes is responsible for the organization of different body plans, as well as for organogenesis. Cell morphogenesis in a mesenchyme contributes to the shaping of embryonic tissues. Epithelial cells, despite that they need to maintain an apicobasal organization, play an equally important role during morphogenesis. Moving from apical to basal, we review compartmentalized cellular rearrangements underlying tissue remodeling in Drosophila and compare them with those found in other organisms. Contractile activity at the apical surface triggers tissue folding and invagination. The regulation of adhesion at adherens junctions controls polarized neighbor exchange during intercalation and tissue elongation. Basolateral protrusive activity underlies other cases of intercalation. These localized cell shape changes are spatially regulated by developmental signals. Some signals define a local change in cell behavior (e.g., apical constriction), others orient a dynamic process in the plane of the tissue (e.g., junction remodeling).
This review centers on the role of the mesenchymal cell in development. The creation of this cell is a remarkable process, one where a tightly knit, impervious epithelium suddenly extends filopodia from its basal surface and gives rise to migrating cells. The ensuing process of epithelial-mesenchymal transformation (EMT) creates the mechanism that makes it possible for the mesenchymal cell to become mobile, so as to leave the epithelium and move through the extracellular matrix. EMT is now recognized as a very important mechanism for the remodeling of embryonic tissues, with the power to turn an epithelial somite into sclerotome mesenchyme, and the neural crest into mesenchyme that migrates to many targets. Thus, the time has come for serious study of the underlying mechanisms and the signaling pathways that are used to form the mesenchymal cell in the embryo. In this review, I discuss EMT centers in the embryo that are ready for such serious study and review our current understanding of the mechanisms used for EMT in vitro, as well as those that have been implicated in EMT in vivo. The purpose of this review is not to describe every study published in this rapidly expanding field but rather to stimulate the interest of the reader in the study of the role of the mesenchymal cell in the embryo, where it plays profound roles in development. In the adult, mesenchymal cells may give rise to metastatic tumor cells and other pathological conditions that we will touch on at the end of the review.
StarLogo - programmable environment for exploring decentralized systems flocks, traffic jams, termite and ant colonies
- M Resnick
Cellular Automata Theory
- T Toffoli
- N Margolus