Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory

This paper outlines the framework of a porous flow mixture theory for the mathematical modelling of in vitro tissue growth, and gives an application of this theory to an aspect of tissue engineering. The problem is formulated as a set of partial differential equations governing the space and time dependence of the amounts of each component of the tissue (phase), together with the physical stresses in each component. The theory requires constitutive relations to specify the material properties of each phase, and also requires relations to specify the stresses developed due to mechanical interactions, both within each phase and between different phases. An application of the theory is given to the study of the mobility and aggregation of a population of cells seeded into an artificial polymeric scaffold. Stability analysis techniques show that the interplay of the forces between the tissue constituents results in two different regimes: either the cells form aggregates or disperse through the scaffold.     

An individual based model of rippling movement in a myxobacteria population

Migrating cells of Myxococcus xanthus (MX) in the early stages of starvation-induced development exhibit elaborate patterns of propagating waves. These so-called rippling patterns are formed by two sets of waves travelling in opposite directions. It has been experimentally shown that formation of these waves is mediated by cell-cell contact signalling (C-signalling). Here, we develop an individual-based model to study the formation of rippling patterns in MX populations. Following the work of Igoshin et al. (Proc. Natl. Acad. Sci. 98 (2001) 14913) we consider each moving cell to have an internal clock which controls its turning behaviour and sensitivity to C-signal. Specifically, we examine the effects of changing: C-signal strength, sensitivity/refractoriness, cell density, and noise upon the formation and structure of the rippling patterns. We also consider three modified models that have no explicit refractory period and examine their ability to produce rippling patterns.     

A mathematical model of cleavage

In the present paper, we propose a mathematical model of cleavage. Cleavage is a process during the early stages of development in which the fertile egg undergoes repeated division keeping the cluster size almost constant. During the cleavage process individual cells repeat cell division in an orderly manner to form a blastula, however, the mechanism which achieves such a coordination is still not very clear. In the present research, we took sea urchin as an example and focused on the diffusion of chemical substances from the animal and vegetal pole. By considering chemotactic motion of the centrosomes, we constructed a mathematical model that describes the processes in the early stages of cleavage.     

A model of gradient interpretation based on morphogen binding

We propose a model in which pattern formation is controlled by several concentration gradients of “morphogens” and by allosteric proteins which bind them. In this model, each protein can bind up to two molecules of each morphogen and has an “active state” when one molecule of each morphogen is bound. The concentration of the active state of such a “morphogen binding protein” varies with position in a way that depends on the values given the binding constants. In a contour map of the active state concentration, the contours can have a variety of simple shapes.Simply-shaped regions of cell differentiation can be defined directly by concentration contours of a morphogen binding protein using a threshold-sensing mechanism. More complex shapes may be generated using several proteins and a “winner-take-all” rule according to which each protein specifies some particular sort of cell differentiation and the differentiation of cells in any position is governed by the protein with the highest active state concentration.We present an application of our model to the vertebrate limb skeleton; we use the “winner-take-all” mechanism and thirteen morphogen binding proteins, eleven of which specify cartilage formation. In this model we use one morphogen binding protein to specify the shaft of a typical long bone and one for each epiphysis. Our model is reasonably successful in imitating the in vivo positions and orientations of developing bones and in generating simple, plausible-looking articular surfaces.In addition to the morphogen-binding model we propose a mechanism which could transform morphogen-binding patterns into high-amplitude patterns capable of controlling the activity of structural genes. This “amplifying mechanism” can account for two previously unexplained features of limb skeletal development: the early formation of the diffusely-bounded “scleroblastema” in the limb bud and the center-to-edge gradations in cartilage formation rate which are later seen within individual chondrification foci.A simple modification of the morphogen-binding model provides an explanation for the general anatomical phenomenon of metamerism: The model can account for the formation of inexactly repeating patterns (such as the pattern of the vertebral column) and suggests a mechanism by which such patterns could (1) evolve from exactly repeating patterns, and (2) acquire, in further evolution, a high degree of specialization of the individual repeating units.The most promising approach for testing the morphogen-binding model would appear to involve experiments in which cytoplasm is transferred between cells at various stages of pattern development. Support for the model could also come from the discovery of certain kinds of hereditary limb defects.     

A model of growth restraints to explain the development and evolution of tooth shapes in mammals

The problem investigated here is control of the development of tooth shape. Cells at the growing soft tissue interface between the ectoderm and mesoderm in a tooth anlage are observed to buckle and fold into a template for the shape of the tooth crown. The final shape is created by enamel secreted onto the folds. The pattern in which the folds develop is generally explained as a response to the pattern in which genes are locally expressed at the interface. This congruence leaves the problem of control unanswered because it does not explain how either pattern is controlled. Obviously, cells are subject to Newton's laws of motion so that mechanical forces and constraints must ultimately cause the movements of cells during tooth morphogenesis. A computer model is used to test the hypothesis that directional resistances to growth of the epithelial part of the interface could account for the shape into which the interface folds. The model starts with a single epithelial cell whose growth is constrained by 4 constant directional resistances (anterior, posterior, medial and lateral). The constraints force the growing epithelium to buckle and fold. By entering into the model different values for these constraints the modeled epithelium is induced to buckle and fold into the different shapes associated with the evolution of a human upper molar from that of a reptilian ancestor. The patterns and sizes of cusps and the sequences in which they develop are all correctly reproduced. The model predicts the changes in the 4 directional constraints necessary to develop and evolve from one tooth shape into another. I conclude more generally expressed genes that control directional resistances to growth, not locally expressed genes, may provide the information for the shape into which a tooth develops.     

A Turing model with correlated random walk

 If in the classical Turing model the diffusion process (Brownian motion) is replaced by a more general correlated random walk, then the parameters describing spatial spread are the particle speeds and the rates of change in direction. As in the Turing model, a spatially constant equilibrium can become unstable if the different species have different turning rates and different speeds. Furthermore, a Hopf bifurcation can be found if the reproduction rate of the activator is greater than its rate of change of direction, and oscillating patterns are possible. Received 24 February 1995; received in revised form 6 September 1995     

Spots and stripes: Pleomorphic patterning of stem cells via p-ERK-dependent cell chemotaxis shown by feather morphogenesis and mathematical simulation

A key issue in stem cell biology is the differentiation of homogeneous stem cells towards different fates which are also organized into desired configurations. Little is known about the mechanisms underlying the process of periodic patterning. Feather explants offer a fundamental and testable model in which multi-potential cells are organized into hexagonally arranged primordia and the spacing between primordia. Previous work explored roles of a Turing reaction-diffusion mechanism in establishing chemical patterns. Here we show that a continuum of feather patterns, ranging from stripes to spots, can be obtained when the level of p-ERK activity is adjusted with chemical inhibitors. The patterns are dose-dependent, tissue stage-dependent, and irreversible. Analyses show that ERK activity-dependent mesenchymal cell chemotaxis is essential for converting micro-signaling centers into stable feather primordia. A mathematical model based on short-range activation, long-range inhibition, and cell chemotaxis is developed and shown to simulate observed experimental results. This generic cell behavior model can be applied to model stem cell patterning behavior at large.     

Numerical simulations of the effects of retinoids on pattern formation in a hydroid

Summary Retinoids have been shown to influence pattern formation in hydroid polyps (Hydractinia echinata) at various levels. These effects are counteracted by an inhibitor isolated from Hydra. The present study provides a theoretical attempt to elucidate the role of retinoids by computer simulations based on a simple model of pattern formation in Hydractinia. The elements of this model are morphogens of the Gierer-Meinhardt type, namely a long-range inhibitor and a short-range activator. From the calcualtions, a reducing effect of retinoids on the propagation of the inhibitor seems most probable.     

Pollen exine development precedes microtubule rearrangement inVigna unguiculata (Fabaceae): A model for pollen wall patterning

Summary Although patterns on pollen exines are highly conserved, heritable traits, there is no prevailing explanation for control of pattern development. InVigna unguiculata (Fabaceae), the exine reticulum is unusually coarse so that development of the reticulum can be followed by light microscopy. Developing exine patterns were compared with the arrangement of microtubules in the microspore using immunofluorescence labeling of microtubules. Exine pattern developed in microspores at stages with a regular microtubule pattern. At later stages of exine formation, microtubules were arranged in patches under the lumina of the reticulum. We conclude that microtubules do not determine exine pattern. The developing exine appears to rearrange microtubules. We interpret this as evidence for the selfpatterning of exine based on intrinsic properties of the matrix between the microspore and the callose wall.Abbreviations DIC differential interference contrast - ECM(s) extracellular matrix(ces) - MT(s) microtubule(s) - PBS phosphate buffered saline - SEM scanning electron microscopy     

Pattern formation in a spatial plant-wrack model with tide effect on the wrack

Spatial patterns are a subfield of spatial ecology, and these patterns modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Localized ecological interactions can generate striking large-scale spatial patterns in ecosystems through spatial self-organization. Possible mechanisms include oscillating consumer–resource interactions, localized disturbance–recovery processes, and scale-dependent feedback. However, in this paper, our main aim is to study the effect of tide on the pattern formation of a spatial plant-wrack model. We discuss the changes of the wavelength, wave speed, and the conditions of the spatial pattern formation, according to the dispersion relation formula. Both the mathematical analysis and numerical simulations reveal that the tide has great influence on the spatial pattern. More specifically, typical traveling spatial patterns can be obtained. Our obtained results are consistent with the previous observation that wracks exhibit traveling patterns, which is useful to help us better understand the dynamics of the real ecosystems.     

Multiple roles of the gene zinc finger homeodomain-2 in the development of the Drosophila wing

The gene zfh2 and its human homolog Atbf1 encode huge molecules with several homeo- and zinc finger domains. It has been reported that they play important roles in neural differentiation and promotion of apoptosis in several tissues of both humans and flies. In the Drosophila wing imaginal disc, Zfh2 is expressed in a dynamic pattern and previous results suggest that it is involved is proximal–distal patterning. In this report we go further in the analysis of the function of this gene in wing development, performing ectopic expression experiments and studying its effects in genes involved in wing development. Our results suggest that Zfh2 plays an important role controlling the expression of several wing genes and in the specification of those cellular properties that define the differences in cell proliferation between proximal and distal domains of the wing disc.     

A Malthusian curb on spatial structure in microorganism populations

That all organisms are born in the company of a parent but die alone is a fundamental biological asymmetry. It has been suggested that this provides a deep-rooted source of spatial pattern formation for microorganisms even at the scale of the population. Such a theory, however, neglects the strong influence in nature of the limited and spatially variable availability of food. The tendency, first recognized by Thomas Malthus in the 18th century, of a population to out-strip its food resources will eventually lead, through local starvation, to the suppression of a heterogeneity growing within a population. Using a generic model it is demonstrated that including local food limitation of breeding strongly dampens spatial structure otherwise resulting from birth and death. The extent of this damping is shown to be a function of the strength of the coupling between organisms and their food and of the total abundance of organic material. Moreover, this work provides an example of a density-dependent process acting to diminish spatial structure rather than to create it and highlights the rich variety of behaviour that is missed by continuum models which fail to represent such local dynamics.     

Model and analysis of chemotactic bacterial patterns in a liquid medium

 A variety of spatial patterns are formed chemotactically by the bacteria Escherichia coli and Salmonella typhimurium. We focus in this paper on patterns formed by E. coli and S. typhimurium in liquid medium experiments. The dynamics of the bacteria, nutrient and chemoattractant are modeled mathematically and give rise to a nonlinear partial differential equation system. We present a simple and intuitively revealing analysis of the patterns generated by our model. Patterns arise from disturbances to a spatially uniform solution state. A linear analysis gives rise to a second order ordinary differential equation for the amplitude of each mode present in the initial disturbance. An exact solution to this equation can be obtained, but a more intuitive understanding of the solutions can be obtained by considering the rate of growth of individual modes over small time intervals. Received: 10 March 1998 / Revised version: 7 June 1998     

A computational model of chemotaxis-based cell aggregation

We present a computational model that successfully captures the cell behaviors that play important roles in 2-D cell aggregation. A virtual cell in our model is designed as an independent, discrete unit with a set of parameters and actions. Each cell is defined by its location, size, rates of chemoattractant emission and response, age, life cycle stage, proliferation rate and number of attached cells. All cells are capable of emitting and sensing a chemoattractant chemical, moving, attaching to other cells, dividing, aging and dying. We validated and fine-tuned our in silico model by comparing simulated 24-h aggregation experiments with data derived from in vitro experiments using PC12 pheochromocytoma cells. Quantitative comparisons of the aggregate size distributions from the two experiments are produced using the Earth Mover's Distance (EMD) metric. We compared the two size distributions produced after 24 h of in vitro cell aggregation and the corresponding computer simulated process. Iteratively modifying the model's parameter values and measuring the difference between the in silico and in vitro results allow us to determine the optimal values that produce simulated aggregation outcomes closely matched to the PC12 experiments. Simulation results demonstrate the ability of the model to recreate large-scale aggregation behaviors seen in live cell experiments.     

Complex pattern formation of marine gradient bacteria explained by a simple computer model

We report on the formation of conspicuous patterns by the sulfide-oxidizing bacterium Thiovulum majus and a recently described vibrioid bacterium. These microaerophilic bacteria form mucus veils on top of sulfidic marine sediment exhibiting regular spaced bacterial patterns (honeycombs, interwoven bands, or inverse honeycombs). A simple qualitative computer model, based on chemotaxis towards oxygen and the ability of the bacteria to induce water advection when attached, can explain the formation of the observed patterns. Our study shows that complex bacterial patterns in nature can be explained in terms of chemotaxis and resource optimisation without involvement of cell-cell signalling or social behavior amongst bacteria.     

Generation of bilateral symmetry in Anthozoa: a model

Polyps of Anthozoa usually display bilateral symmetry with respect to their mouth opening, to their pharynx, and in particular to the arrangement of their mesenteries. Mesenteries, which are endodermal folds running from the apical to the basal end of the body, subdivide the gastric cavity into pouches. They form in a bilateral symmetric sequence. In this article I propose that early in polyp development the endoderm subdivides successively into three different types of compartments. A mesentery forms at the border between compartments. Two of the compartments are homologous to those of Scyphozoa. They form by mutual activation of cell states that locally exclude each other. The third compartment leads to siphonoglyph formation and is an evolutionary innovation of the Anthozoa. The mechanism that controls the number and spatial arrangement of the third type of compartment changes the radial symmetry into a bilateral one and occasionally into a different one. The dynamics of its formation indicate an activator-inhibitor mechanism. Computer models are provided that reproduce decision steps in the generation of the mesenteries.