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Flux ratios for biological membranes and reciprocity theorems for linear operators

Published online by Cambridge University Press:  17 February 2009

A. J. Bracken ,
K. Holmåker ,
L. V. Maloney  and
L. Bass
A. J. Bracken
Affiliation:
Department of Mathematics, University of Queensland, Brisbane, Australia.
K. Holmåker
Affiliation:
Department of Mathematics, Chalmers University of Technology and University of Göteborg, Sweden.
L. V. Maloney
Affiliation:
School of Mathematics, University of New South Wales, Sydney, Australia.
L. Bass
Affiliation:
Department of Mathematics, University of Queensland, Brisbane, Australia.
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Abstract

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The constancy in time of the ratio of unidirectional tracer fluxes, passing in opposite directions through a membrane that has transport properties varying arbitrarily with the distance from a boundary face, has been established recently for successively more sophisticated mathematical models of tracer transport within the membrane. Such results are important in that, when constancy is not observed experimentally, inferences can be drawn about the dimensionality of distributions of transport properties of the membrane. The known theoretical results are shown here to follow from much more general theorems, valid for a wide class of models based on linear-operator equations, including elliptic and hyperbolic partial differential equations as well as the essentially parabolic equations of interest in membrane transport problems. These theorems have the general character of “reciprocity theorems” known for a long time in other areas, such as mechanics, acoustics and elasticity. The general results obtained here clarify the conditions on membrane properties under which constancy of a flux ratio can be expected. In addition, flux ratio theorems of a new type are proved to hold under suitable conditions, for the normal components of flux vectors at points on either side of a membrane, as distinct from previously established theorems for total fluxes through membrane faces. Possible new experiments are suggested by the analysis.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 3 , January 1989 , pp. 278 - 297
Copyright
Copyright © Australian Mathematical Society 1989

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