UBC Theses and Dissertations
Reaction-diffusion theory of localized structures with application to vertebrate organogenesis Holloway, David M.
Biological pattern formation is addressed within the framework of Turing's (1952) chemical theory. His (linear) reaction-diffusion equations for chemical concentration have spatially inhomogeneous solutions. These stationary chemical waves are a possible mechanism for the spatial patterning of cellular differentiation in embryogenesis. Chemical mechanisms which give Turing dynamics have nonlinear rate equations. Much of the dynamics can be predicted from linear analysis, but solution is generally only possible numerically. Numerical methods for reaction-diffusion equations, especially implicit techniques, are reviewed. This work is specifically concerned with the localization of pattern. Reaction-diffusion models tend to form repeated peaks. Under some circumstances, however, patterns of localized peaks surrounded by large areas of low concentration can be formed. The requirements for this are developed and explored through comparison of two models, the Brusselator and the Gierer-Meinhardt. It is found that both can give localized pattern if the ratio of the component diffusivities is high. This leads to partly-ordered pattern, linking Turing theory with the inhibitory field theory of Wigglesworth (1940). However, the Gierer- Meinhardt gives less ordered pattern in two dimensions and is more susceptible to gradients in precursor concentration than the Brusselator. The different dynamics in the two models, underlying the different behaviours, are discussed. These principles of pattern localization in reaction-diffusion models are applied to heart formation in the axolotl, Ambystoma mexicanum. A form of the Gierer-Meinhardt model is used (but derived from a chemical mechanism), due to its greater tendency to form localized pattern. Heart formation proceeds by induction of the mesoderm tissue by the underlying endoderm, followed by more localized cardiomyocyte differentiation in the mesoderm. This sequence is modelled by treating induction as a process which establishes the distribution of a precursor in the reaction-diffusion mechanism postulated to be responsible for cardiomyocyte differentiation. Normal development, and numerous transplant and explant experiments with wild-type and cardiac-lethal mutant (c) tissues are successfully modelled. Agreement with experiment is achieved in terms of spatial patterns and timing of events. Predictions for future experiments are given, as well as some suggestions on the probable chemical nature of the morphogens, with some relationship to signal transduction mechanisms.
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