Mathematical Modelling of Natural Phenomena
Preface The past decade has seen explosive growth in the use of mathematical and computational tools to address problems in the biosciences and medicine. The challenges are novel and unique: how do we cope with the vast amount of experimental data being generated, the huge number of components involved and their complex interactions? Increasingly, theoreticians are beginning to make real advances in this field and much of their success has stemmed from changes in the nature of their research.
... is now very clear that for interdisciplinary research to have a true impact, an intimate engagement with the science is requiredthe mathematics cannot be done in a vacuum. The articles in this special issue bear testament to that approach as, to varying degrees, they assess model behaviour critically in the light of experimental data. The paper by Othmer et al. presents a detailed review of pattern formation in three classical experimental systems, the fruitfly Drosophila, the bacterium E. coli and pigmentation patterning in fish. It presents a range of models that have been proposed in each case, compares and contrasts them in terms of their properties, and critiques them in the context of experimental data. The paper also addresses the effects of domain growth on patterning, a subject area that has received surprisingly little attention given its importance. It has been known for decades that reaction-diffusion models can produce arrays of repeating patterns similar to those observed in nature. Headon and Painter review this mainly in the context of feather germ patterning in the chick and discuss in detail how such models must now be validated and extended in light of the new biological findings that allow concrete definitions of the hypothetical chemical players proposed by these models. They define a number of challenges that must be met by both the experimental and theoretical communities if we are to gain greater insights into the underlying mechanisms that give rise to repeated patterns during morphogenesis. Cai et al. study homeostasis and control of cell population levels in the epidermis. Specifically, they investigate multi-scale regulation between extracellular secreted factors and intracellular models and delimit regions in parameter space where the model predicts homeostasis. The focus in their paper is on the proto-oncogene, c-Myc, an intracellular transcription factor which affects both proliferation and differentiation. Miura and Tanaka study how capillary networks form in human umbilical vein endothelial cells. This problem has attracted much attention and a number of modelling approaches based on different biological hypotheses have been proposed. To validate/invalidate such models it is important to determine biologically realistic parameter values within the models. This paper shows how, with the rapid recent advances in imaging techniques, it is now becoming possible to obtain