Nov 6, 2014

Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: Criteria for successful colonization

BioRxiv : the Preprint Server for Biology
Matthew J Simpson

Abstract

Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction-diffusion process on 0 < x < L ( t ), where L ( t ) is the length of the growing domain. Comparing our exact solutions with numerical approximations confirms the veracity of the method. Furthermore, our examples illustrate a delicate interplay between: (i) the rate at which the domain elongates, (ii) the diffusivity associated with the spreading density profile, (iii) the reaction rate, and (iv) the initial condition. Altering the balance between these four features leads to different outcomes in terms of whether an initial profile, located near x = 0, eventually overcomes the domain growth and colonizes the entire length of the domain by reaching the bound...Continue Reading

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Mentioned in this Paper

Antigens, Differentiation
Phosphoric diester hydrolase
Diffusion Weighted Imaging
Embryonic Development
Differential Diagnosis
Cell Proliferation
Diethylstilbestrol monophosphate
Analysis
Aminomethyltransferase
Embryonic Development Aspects

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