Transp1sol.Distrib2F

Model number: 0009

A two region one solute facilitated transporter kinetic model including binding steps and transmembrane flips for free and occupied transporters. Model accounts for single site binding without competition in the setting for axially-distributed capillary-tissue exchange.

Detailed Description

ERROR: A2:T is in proportion to NGrid.

Transp1sol.Distrib2F is an axially distributed, three state transporter model for one solute, input via Flow, with a membrane between two mixing chambers. The capillary plasma region, volume V_p, has flow F_p, first order consumption G_p, and axial diffusion (dispersion) D_p. Radial diffusion is assumed instantaneous (short radial distances).

The Interstitial Fluid Region, with volue V_ISF, is axially distributed. The gradients are dissipated by axial diffusion D_ISF. Consumption, G_ISF, is first order.

Model Verification: Total mass is conserved: substrate in solution is totalled as SubstrateV, and substrate bound to transporter as SubstrateM, for membrane bound. Total transporter conservation is forced through the equation for T2. The Michaelis-Menten reaction is at 50% of maximum at the Km, shown on the JSim PlotPage labled MM.

Assumptions: Compartmental assumptions apply to the solutions on either side of the membrane. These are: instantaneously stirred tank, no concentration gradients, no diffusion limitation for reactants. Also, it is assumed that reactions are first order with fixed rates, not fractal.

Relevant Equations

Partial Differential Equations

${ \frac {d}{dt}A1(t,x) = k_{\text{\small{off A1}}}T_{\text{\small{A1}}}S_{\text{\small{of V1}}} - k_{\text{\small{on A1}}}A1 T1 S_{\text{\small{of V1}}} - \frac {Flow L \frac {dA1}{dx}}{V1} - \frac {PS_{\text{\small{g}}}(A1 - A2)}{V1} + D_{\text{\small{A2}}} \frac {d^2 A1}{dx^2}$

${ \frac {d}{dt}A2(t,x) = k_{\text{\small{off A2}}}T_{\text{\small{A2}}}S_{\text{\small{of V2}}} - k_{\text{\small{on A2}}}A2 T2 S_{\text{\small{of V2}}} + \frac {PS_{\text{\small{g}}}(A1 - A2)}{V2} + D_{\text{\small{A2}}} \frac {d^2 A2}{dx^2}$

${ \frac {d}{dt}T1(t,x) = - k_{\text{\small{on A1}}}A1 T1 + k_{\text{\small{off A1}}}T_{\text{\small{A1}}} - k_{\text{\small{T12}}}T1 + k_{\text{\small{T21}}}T2$

${ \frac {d}{dt}T_{\text{\small{A1}}}(t,x) = k_{\text{\small{on A1}}}A1 T1 - k_{\text{\small{off A1}}}T_{\text{\small{A1}}} - k_{\text{\small{TA12}}}T_{\text{\small{A1}}} + k_{\text{\small{TA21}}}T_{\text{\small{A2}}}$

${ \frac {d}{dt}T_{\text{\small{A2}}}(t,x) = k_{\text{\small{on A2}}}A2 T2 - k_{\text{\small{off A2}}}T_{\text{\small{A2}}} + k_{\text{\small{TA12}}}T_{\text{\small{A1}}} - k_{\text{\small{TA21}}}T_{\text{\small{A2}}}$

${ T2 = T_{\text{\small{tot}}} - T_{\text{\small{A1}}} - T_{\text{\small{A2}}} - T1$

${ PS_{\text{\small{c}}} = \frac {PS_{\text{\small{cMAX}}}}{1 + \frac {A1}{K_{\text{\small{dA1}}}}}$

WARNING: An additional thermodynamic constraint is not included in the model. For a passive transporter, the transport rate constants should satisfy the following constraint:

${ \frac {k_{\text{\small{TA12}}} k_{\text{\small{T21}}} k_{\text{\small{on A1}}} k_{\text{\small{off A2}}}}{k_{\text{\small{TA21}}} k_{\text{\small{T12}}} k_{\text{\small{off A1}}} k_{\text{\small{on A2}}}} = 1 = {\text {test}} = \frac {k_{\text{\small{TA12}}} k_{\text{\small{T21}}} k_{\text{\small{dA2}}}}{k_{\text{\small{TA21}}} k_{\text{\small{T12}}} k_{\text{\small{dA1}}}}$

This constraint ensures that the model runs to equilibrium at steady-state. If these ratios deviate from 1, the model will run to a steady-state net concentration gradient. This would be the case if the transporter is coupled to an energy source, which is not explicitly modeled here.

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References

Klingenberg M. Membrane protein oligomeric structure and transport function. Nature 290: 449-454, 1981.

Stein WD. The Movement of Molecules across Cell Membranes. New York: Academic Press, 1967.

Stein WD. Transport and Diffusion across Cell Membranes. Orlando, Florida: Academic Press Inc., 1986.

Wilbrandt W and Rosenberg T. The concept of carrier transport and its corollaries in pharmacology. Pharmacol Rev 13: 109-183, 1961.

Schwartz LM, Bukowski TR, Ploger JD, and Bassingthwaighte JB. Endothelial adenosin transporter characterization in perfused guinea pig hearts. Am J Physiol Heart Circ Physiol 279: H1502-H1511, 2000.

Foster DM and Jacquez JA. An analysis of the adequacy of the asymmetric carrier model for sugar transport. Biochim Biophys Acta 436: 210-221, 1976.

Dawson CA, Linehan JH, Rickaby DA, and Roerig DL. Inﬂuence of plasma protein on the inhibitory effects of indocyanine green and bromcresol green on pulmonary prostaglandin E1 extraction. Br J Pharmac 81: 449-455, 1984.

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Key Terms

Two compartment, facilitated transporter, transmembrane, flow, two region, single transporter, one solute, no competition, axially distributed, radial diffusion, tutorial

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Copyright (C) 1999-2009 University of Washington. From the National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061. Academic use is unrestricted. Software may be copied so long as this copyright notice is included. This software was developed with support from NIH grant HL073598. Please cite this grant in any publication for which this software is used and send one reprint to the address given above.

Model development and archiving support at physiome.org provided by the following grants: NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.