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A flux ratio theorem for multicomponent linear diffusion equations. (English) Zbl 1006.92014

Summary: H.H. Ussing [G. Giebisch et al. (eds.), Membrane Transport in Biology, 115-140 (1978)] considered the steady flux of a single chemical component diffusing through a membrane under the influence of chemical potentials and derived from his linear model an expression for the ratio of this flux and that of the complementary experiment in which the boundary conditions were interchanged. Here, an extension of Ussing’s flux ratio theorem is obtained for \(n\) chemically interacting components governed by a linear system of diffusion-migration equations that may also incorporate linear temporary trapping reactions. The determinants of the output flux matrices for complementary experiments are shown to satisfy an Ussing flux ratio formula for steady state conditions of the same form as for the well-known one-component case.

MSC:

92C30 Physiology (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
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References:

[1] Ussing, H. H., Interpretation of tracer fluxes, (Giebisch, G.; etal., Membrane Transport in Biology, Vol. 1 (1978), Springer: Springer New York), 115-140
[2] Bass, L.; Bracken, A. J., The flux ratio equation under nonstationary boundary conditions, Mathematical Biosciences, 66, 87-92 (1983) · Zbl 0522.92001
[3] Bass, L.; Bracken, A. J.; Hilden, J., Flux ratio theorems for nonstationary membrane transport with temporary capture of tracer, J. Theor. Biol., 118, 327-338 (1986)
[4] Bass, L.; McNabb, A., Flux ratio theorems for nonlinear membrane transport under nonstationary boundary conditions, J. Theor. Biol., 133, 185-191 (1988)
[5] McNabb, A.; Bass, L., Flux theorems for linear multicomponent diffusion, IMA Journal of Applied Mathematics, 44, 155-161 (1990) · Zbl 0716.73074
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