Mathematical Model For Pattern Formation Research Articles

Enhanced Waddington landscape model with cell-cell communication can explain molecular mechanisms of self-organization.

The molecular mechanisms of self-organization that orchestrate embryonic cells to create astonishing patterns have been among major questions of developmental biology. It is recently shown that embryonic stem cells (ESCs), when cultured in particular micropatterns, can self-organize and mimic the early steps of pre-implantation embryogenesis. A systems-biology model to address this observation from a dynamical systems perspective is essential and can enhance understanding of the phenomenon. Here, we propose a multicellular mathematical model for pattern formation during in vitro gastrulation of human ESCs. This model enhances the basic principles of Waddington epigenetic landscape with cell-cell communication, in order to enable pattern and tissue formation. We have shown the sufficiency of a simple mechanism by using a minimal number of parameters in the model, in order to address a variety of experimental observations such as the formation of three germ layers and trophectoderm, responses to altered culture conditions and micropattern diameters and unexpected spotted forms of the germ layers under certain conditions. Moreover, we have tested different boundary conditions as well as various shapes, observing that the pattern is initiated from the boundary and gradually spreads towards the center. This model provides a basis for in-silico modeling of self-organization. https://github.com/HFooladi/Self_Organization. Supplementary data are available at Bioinformatics online.

Investigation of branching structure formation by solutions of a mathematical model of pattern formation in coral reefs

The Journal of the National Science Foundation of Sri Lanka publishes the results of research in all aspects of Science and Technology. The journal also has a website at http://www.nsf.gov.lk/. 2021 Impact Factor: 0.682The JNSF provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge.Cover :Leatherback (a), green (b - adult & f - hatchling), hawksbill (c) and olive ridley (d) turtles who nest in Sri Lankan beaches, green turtle crawl marks (e) and by-catch of sea turtles in fisheries (c & d)Photo credits: SJ Perera (a & c), WPN Perera (b & e), K Ekaratne (d & f)

A new mathematical model for pattern formation by cranial sutures

Cranial sutures are narrow mesenchymal tissues that connect skull bones to each other. Given that they serve as growth centers in the skull, these undifferentiated tissues play crucial roles in skull development. Cranial sutures are also of clinical importance, because the premature fusion of skull bones results in a pathological condition called craniosynostosis. In newborns, skull sutures are wide and straight; during adolescence, they become thinner and start winding to form an interdigitating pattern. From a functional aspect, as the degree of interdigitation becomes larger, the strength of the connection between bones increases. However, the mechanisms underlying the maintenance of mesenchymal narrow bands or formation of interdigitation remain poorly understood. In the present study, we presented a new mathematical model that can reproduce the suture width maintenance and interdigitation formation. We can predict the width of the mesenchyme bands and wavelengths of suture interdigitations from the model.

A computational analysis of bone formation in the cranial vault in the mouse.

Bones of the cranial vault are formed by the differentiation of mesenchymal cells into osteoblasts on a surface that surrounds the brain, eventually forming mineralized bone. Signaling pathways causative for cell differentiation include the actions of extracellular proteins driven by information from genes. We assume that the interaction of cells and extracellular molecules, which are associated with cell differentiation, can be modeled using Turing’s reaction–diffusion model, a mathematical model for pattern formation controlled by two interacting molecules (activator and inhibitor). In this study, we hypothesize that regions of high concentration of an activator develop into primary centers of ossification, the earliest sites of cranial vault bone. In addition to the Turing model, we use another diffusion equation to model a morphogen (potentially the same as the morphogen associated with formation of ossification centers) associated with bone growth. These mathematical models were solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data showing skull bone formation in mouse at different embryonic days in mice carrying disease causing mutations and their unaffected littermates. The results show that the relative locations of the five ossification centers that form in our model occur at the same position as those identified in experimental data. As bone grows from these ossification centers, sutures form between the bones.

A mathematical model for pattern formation of glioma cells outside the tumor spheroid core

Glioblastoma is the most common and the most aggressive type of brain cancer. The median survival time from the time of diagnosis is approximately one year. Invasion of glioma cells from the core tumor into the surrounding brain tissue is a major reason for treatment failure: these migrating cells are not eliminated in surgical resection and cause tumor recurrence. Variations are seen in number of invading cells, and in the extent and patterns of migration. Cells can migrate diffusely and can also be seen as clusters of cells distinct from the main tumor mass. This kind of clustering is also evident in vitro using 3D spheroid models of glioma invasion. This has been reported for U87 cells stably expressing the constitutively active EGFRVIII mutant receptor, often seen expressed in glioblastoma. In this case the cells migrate as clusters rather than as single cells migrating in a radial pattern seen in control wild type U87 cells. Several models have been suggested to explain the different modes of migration, but none of them, so far, has explored the important role of cell–cell adhesion. The present paper develops a mathematical model which includes the role of adhesion and provides an explanation for the various patterns of cell migration. It is shown that, depending on adhesion, haptotactic, and chemotactic parameters, the migration patterns exhibit a gradual shift from branching to dispersion, as has been reported experimentally.

Formation of molecular species of mitochondrial cardiolipin: 2. A mathematical model of pattern formation by phospholipid transacylation

Formation of the unique molecular species of mitochondrial cardiolipin requires tafazzin, a transacylase that exchanges acyl groups between phospholipid molecular species without strict specificity for acyl groups, head groups, or carbon positions. However, it is not known whether phospholipid transacylations can cause the accumulation of specific fatty acids in cardiolipin. Here, a model is shown in linear algebra representation, in which acyl specificity emerges from the transacylation equilibrium of multiple molecular species, provided that different species have different free energies. The model defines the conditions and energy terms, under which transacylations may generate the characteristic composition of mitochondrial cardiolipin. It is concluded that acyl-specific cardiolipin patterns could arise from phospholipid transacylations in the tafazzin domain, even if tafazzin itself does not have substrate specificity.

A Mathematical Model of Pattern Formation in the Vascular Cambium of Trees

The beautiful patterns apparent in wood grain have their origin in the alignment of fusiform initial cells in the vascular cambium of trees. We develop a mathematical model to describe the orientation of fusiform initial cells, and their interaction with the plant hormone indole-3-acetic acid (auxin). The model incorporates the following four assumptions: (1) auxin is actively transported parallel to the long axis of the initials, (2) auxin diffuses perpendicular to the long axis of the initials, (3) the initials tend to orient parallel to the flux of auxin through the cambium, and (4) adjacent initials tend to orient parallel to one another. Each assumption is justified on the basis of available evidence and cast in mathematical form. Our main result is a pair of nonlinear differential equations that describe the coupling between the distribution of auxin in the cambium and the orientation of fusiform initials. Numerical solutions to the equations show qualitative resemblance to the wood grain patterns observed at branch junctions, wounds and knots, and topological defects.

Analysis of concentration and temperature patterns on catalytic surfaces

A simple mathematical model for pattern formation on isothermal as well as nonisothermal catalytic surfaces is developed and analyzed. The model accounts for diffusion of the species, conduction of heat, convection from the fluid phase, and a bimolecular Langmuir–Hinshelwood type kinetic expression. The isothermal model is shown to exhibit stationary concentration patterns for typical sets of parameters. The nonisothermal model exhibits stationary temperature and concentration patterns only for near stoichiometric composition of the reactants (three equation model). The calculations show that these stationary patterns exist in regions near the ignition and extinction points and are most likely to form during ignition or extinction of the surface. It is also found that moving concentration and temperature patterns exist near the Hopf bifurcation point of the ignited homogeneous branch. The moving patterns predicted for realistic values of the transport and kinetic parameters are concentration patterns with almost constant temperature distribution on the surface. The typical size of the patterns and the period of oscillation are estimated in terms of the physicochemical parameters.

A mathematical model of pattern formation

This paper presents an explicit mathematical model describing pattern formation in monolayer epithelia. The approach is a generalization of the equations describing soap bubble configurations (Plateau, 1873; Thompson, 1917; Almgren & Taylor, 1976) that allows adjacent cells to adhere with differing intensities (Steinberg, 1962, 1978). The model is a system of simultaneous non-linear equations that considers cell-cell interactions in a two-dimensional sheet. The implementation involves using the equations of the model to predict explicitly the energy-minimizing configuration of a system of cells, based on the adhesivity of their membranes. The model can thus be used to explore the effects of varying adhesions on the dynamics of pattern formation. Following Chichilnisky (1985), such a descriptive system is introduced in this paper, and its predictive properties explored.

A mathematical model for pattern formation in biological systems

A system of reaction-diffusion equations with applications in biological problems is presented and studied theoretically and numerically. Solutions demonstrating the following types of behaviour have been obtained: (i) time constant, spatially inhomogeneous, (ii) time periodic, spatially homogeneous, and (iii) time periodic, travelling wave-like. Some applications in embryological problems are discussed.

A mathematical model of pattern formation by swimming microorganisms.

Bioconvection in suspensions of Tetrahymena pyriformis and Crypthecodinium cohnii is described and 2 new patterns, the toroid and the cat's-eye, which appear in shallow suspensions of C. cohnii, are reported. Except in very dense cultures, bioconvection does not arise unless the depth of the suspensions or the mean concentration exceed certain critical values, other things being equal. A mathematical model describing the hydrodynamics of suspension of negatively geotactic microorganisms is described which predicts the existence of critical depths and concentrations. The equations presented admit solutions describing the "polka-dot" patterns seen at low organism concentration in suspensions slightly deeper than the critical value. The discussion here is limited to the case of fairly dilute suspensions, but the basic approach can be applied also to richer cultures.