Pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients

Reaction-diffusion models for biological pattern formation have been studied extensively in a variety of embryonic and ecological contexts. However, despite experimental evidence pointing to the existence of spatial inhomogeneities in various biological systems, most models have only been considered in a spatially homogeneous environment. The authors consider a two-chemical reaction-diffusion mechanism in one space dimension in which one of the diffusion coefficients depends explicitly on the spatial variable. The model is analysed in the case of a step function diffusion coefficient and the insight gained for this special case is used to discuss pattern generation for smoothly varying diffusion coefficients. The results show that spatial inhomogeneity may be an important biological pattern regulator, and possible applications of the model to chondrogenesis in the vertebrate limb are suggested

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns

Travelling waves in wound healing

We illustrate the role of travelling waves in wound healing by considering three different cases. Firstly, we review a model for surface wound healing in the cornea and focus on the speed of healing as a function of the application of growth factors. Secondly, we present a model for scar tissue formation in deep wounds and focus on the role of key chemicals in determining the quality of healing. Thirdly, we propose a model for excessive healing disorders and investigate how abnormal healing may be controlled

Corneal epithelial wound healing

We propose a reaction-diffusion model of the mechanisms involved in the healing of corneal surface wounds. The model focuses on the stimulus for increased mitotic and migratory activity, specifically the role of epidermal growth factor. We determine an analytic approximation for the speed of travelling wave solutions of the model in terms of the parameters and verify the results numerically. By comparing the predicted speed with experimentally measured healing rates, we conclude that serum-derived factors can alone account for the overall features of the healing process, but that the supply of growth factors by the tear film, in the absence of serum-derived factors, is not sufficient to give the observed healing rate. Numerical solutions of the model equations also confirm the importance of both migration and mitosis for effective wound healing. By modifying the model, we obtain an analytic prediction for the healing rate of corneal surface wounds when epidermal growth factor is applied topically to the wound

The role of cell-cell adhesion in wound healing

We present a stochastic model which describes fronts of cells invading a wound. In the model cells can move, proliferate, and experience cell-cell adhesion. We find several qualitatively different regimes of front motion and analyze the transitions between them. Above a critical value of adhesion and for small proliferation large isolated clusters are formed ahead of the front. This is mapped onto the well-known ferromagnetic phase transition in the Ising model. For large adhesion, and larger proliferation the clusters become connected (at some fixed time). For adhesion below the critical value the results are similar to our previous work which neglected adhesion. The results are compared with experiments, and possible directions of future work are proposed.Comment: to appear in Journal of Statistical Physic

A mathematical model for collagen fibre formation during foetal and adult dermal wound healing

Adult dermal wounds, in contrast to foetal wounds, heal with the formation of scar tissue. A crucial factor in determining the nature of the healed tissue is the ratio of collagen 1 to collagen 3, which regulates the diameter of collagen fibres. We develop a mathematical model which focuses on the stimulus for collagen synthesis due to the secretion of the different isoforms of the regulatory chemical transforming growth factor $\beta$. Numerical simulations of the model lead to a value of this ratio consistent with that of healthy tissue for the foetus but corresponding to scarring in adult wound healing. We investigate the effect of topical application of TGF$\beta$ isoforms during healing and determine the key parameters which control the difference between adult and foetal repair

Enhanced three-factor security protocol for consumer USB mass storage devices

The Universal Serial Bus (USB) is an extremely popular interface standard for computer peripheral connections and is widely used in consumer Mass Storage Devices (MSDs). While current consumer USB MSDs provide relatively high transmission speed and are convenient to carry, the use of USB MSDs has been prohibited in many commercial and everyday environments primarily due to security concerns. Security protocols have been previously proposed and a recent approach for the USB MSDs is to utilize multi-factor authentication. This paper proposes significant enhancements to the three-factor control protocol that now makes it secure under many types of attacks including the password guessing attack, the denial-of-service attack, and the replay attack. The proposed solution is presented with a rigorous security analysis and practical computational cost analysis to demonstrate the usefulness of this new security protocol for consumer USB MSDs

Wound healing in the corneal epithelium: biological mechanisms and mathematical models

Corneal epithelium has a highly specialised wound-healing response. The biological aspects of this repair process are reviewed, and methods of modelling it mathematically are described. A model which focuses on the source of epidermal growth factor (EGF) within a healing wound is described. By considering mathematical representations of a number of possible source terms, it is shown that the EGF present in the tear film is insufficient to explain the observed rate of healing, and experimental approaches are suggested for distinguishing between other sources. Also, the simulation of exogenous addition of EGF using the model is described. An issue that has been the subject of considerable debate in the literature is the role of eyeball curvature. The mode is used to show that this curvature is not significant for either the speed or form of healing in the epithelium. In conclusion, a comparison is made between wound healing in the corneal epithelium with that in the epidermis of the skin. Possible directions for future modelling work are considered