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Mathematical models for biological pattern formation in two dimensions Lyons, Michael J.
Abstract
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.
Item Metadata
Title |
Mathematical models for biological pattern formation in two dimensions
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1992
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Description |
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.
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Extent |
5214877 bytes
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Genre | |
Type | |
File Format |
application/pdf
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Language |
eng
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Date Available |
2008-12-23
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0085628
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
1992-05
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Media
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Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.