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Mathematical models for biological pattern formation in two dimensions

University of British Columbia

The general problem of pattern formation in biological development is described. One possible type of general solution, reaction-diffusion theory, is outlined, together with the difficulties involved in definite experimental confirmation or rejection of it. The methods for linear analysis of reaction-diffusion equations are presented. An approximate analytical technique for solving nonlinear reaction-diffusion equations close to the onset of the pattern-forming instability is discussed. The technique, known as the adiabatic approximation, is applied to the problem of stripe formation in two spatial dimensions. This analytical result demonstrates the existence of a class of pattern-forming models which preferentially select striped patterns. This class of models is defined by the requirement that the reaction terms contain only odd powers of the departures from the homogeneous steady state. The nonlinear properties of reaction-diffusion models in two spatial dimensions have been extensively studied numerically using finite-difference methods. These computations support the results of the analysis and extend their validity to a wider range of parameter values. Additional nonlinear effects, for example elimination of pattern defects, orientation of pattern using monotonic gradients, and the stripe/spot transition, have been investigated computationally. Two specific instances of biological stripe formation are considered: the expression of segmentation genes in the early embryogenesis of Drosophila melanogaster, and the development of ocular dominance domains in the primary visual cortex of some higher vertebrates. Mechanisms by which the segmentation patterns could arise by reaction diffusion are discussed. Existing models for ocular dominance, which take the form of integro-differential equations, are analyzed using the adiabatic approximation. A fundamental connection between a symmetry in the visual system and stripe selection is established.

Thesis/Dissertation

application/pdf

2008-12-23

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http://hdl.handle.net/2429/3291

Physics

University of British Columbia

UBCV

DSpace