Transp2sol.Distrib2F

Model number: 0012

An axially-distributed facilitating transporter for two competing solutes, including binding steps, with input via flow. Shows countertransport facilitation/inhibition Enzymatic conversion in V_ISF.

Detailed Description

Transp2sol.Distrib2F is an axially distributed, six state transporter model for two solutes in competition, 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. In the Interstitial Fluid Region, A is reacted to form B in an enzymatic reaction approximated by a Michaelis-Menten expression, without any accounting for binding of substrate or product to the enzyme. When the rates of conformational state change for transmembrane flipping of TA and TB are high compared to that of uncomplexed transporter T, the model behaves much like an obligatory countertransporter, exchanging B for A across the membrane.

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

${ \frac {d}{dt}A_{\text{\small{p}}}(t,x) = - \frac {F_{\text{\small{p}}}L}{hV_{\text{\small{p}}} \frac {dA_{\text{\small{p}}}}{dx}} + S_{\text{\small{of V1}}}(k_{\text{\small{off A1}}}T_{\text{\small{A1}}} - k_{\text{\small{on A1}}}A_{\text{\small{p}}}T1) - \frac {PS_{\text{\small{g}}}(A_{\text{\small{p}}}-A_{\text{\small{ISF}}})}{hV_{\text{\small{p}}}} - \frac {G_{\text{\small{pA}}}A_{\text{\small{p}}}}{hV_{\text{\small{p}}}} + D_{\text{\small{pA}}} \frac {d^2 A_{\text{\small{p}}}}{dx^2}$

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

${ \frac {d}{dt}B_{\text{\small{p}}}(t,x) = - \frac {F_{\text{\small{p}}}L}{hV_{\text{\small{p}}} \frac {dB_{\text{\small{p}}}}{dx}} + S_{\text{\small{of V1}}}(k_{\text{\small{off B1}}}T_{\text{\small{B1}}} - k_{\text{\small{on B1}}}B_{\text{\small{p}}}T1) - \frac {PS_{\text{\small{g}}}(B_{\text{\small{p}}}-B_{\text{\small{ISF}}})}{hV_{\text{\small{p}}}} - \frac {G_{\text{\small{pB}}}B_{\text{\small{p}}}}{hV_{\text{\small{p}}}} + D_{\text{\small{pB}}} \frac {d^2 B_{\text{\small{p}}}}{dx^2}$

${ \frac {d}{dt}B_{\text{\small{ISF}}}(t,x) = S_{\text{\small{of V2}}}(k_{\text{\small{off B2}}}T_{\text{\small{B2}}} - k_{\text{\small{on B2}}}B_{\text{\small{ISF}}}T2) + \frac {PS_{\text{\small{g}}}(B_{\text{\small{p}}}-B_{\text{\small{ISF}}})}{hV_{\text{\small{ISF}}}} - \frac {G_{\text{\small{ISFB}}}B_{\text{\small{ISF}}}}{hV_{\text{\small{ISF}}}} + D_{\text{\small{ISFB}}} \frac {d^2 B_{\text{\small{ISF}}}}{dx^2}$

${ \frac {d}{dt}T1(t,x) = -( k_{\text{\small{on A1}}} A_{\text{\small{p}}} + k_{\text{\small{on B1}}}B_{\text{\small{p}}})T1 + k_{\text{\small{off A1}}}T_{\text{\small{A1}}} + k_{\text{\small{off B1}}}T_{\text{\small{B1}}} - k_{\text{\small{T12}}}T1 + k_{\text{\small{T21}}}T2$

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

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

${ \frac {d}{dt}T_{\text{\small{B1}}}(t,x) = k_{\text{\small{on B1}}}B_{\text{\small{p}}}T1 - k_{\text{\small{off B1}}}T_{\text{\small{B1}}} - k_{\text{\small{TB12}}}T_{\text{\small{B1}}} + k_{\text{\small{TB21}}}T_{\text{\small{B2}}}$

${ \frac {d}{dt}T_{\text{\small{B2}}}(t,x) = k_{\text{\small{on B2}}}B_{\text{\small{ISF}}}T2 - k_{\text{\small{off B1}}}T_{\text{\small{B2}}} + k_{\text{\small{TB12}}}T_{\text{\small{B1}}} - k_{\text{\small{TB21}}}T_{\text{\small{B2}}}$

${ T2 = T_{\text{\small{tot}}} - T_{\text{\small{A1}}} - T_{\text{\small{A2}}} - T_{\text{\small{B1}}} - T_{\text{\small{B2}}} -T1$

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

Mass conservation check for the closed system of V_p and V_ISF

${ Substrate_{\text{\small{V}}} = V_{\text{\small{p}}}(A_{\text{\small{p}}} + B_{\text{\small{p}}}) + V_{\text{\small{ISF}}}(A_{\text{\small{ISF}}} + B_{\text{\small{ISF}}})$

${ Substrate_{\text{\small{M}}} = Surf(T_{\text{\small{A1}}} + T_{\text{\small{A2}}} + T_{\text{\small{B1}}} + T_{\text{\small{B2}}})$

${ Substrate_{\text{\small{Tot}}} = Substrate_{\text{\small{V}}} + Substrate_{\text{\small{M}}}$

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References

Sangren WC and Sheppard CW. A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment. Bull Math Biophys 15: 387-394, 1953 (This gives an analytic solution for the two-region model.)

Goresky CA, Ziegler WH, and Bach GG. Capillary exchange modeling: Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970. (This gives another derivation of the analytical form, and uses the model in both single and multicapillary models.

Bassingthwaighte JB. A concurrent flow model for extraction during transcapillary passage. Circ Res 35: 483-503, 1974. (This gives numerical solutions, which are faster than the analytic solutions, and imbeds the model in an organ with tissue volums conserved, and with arteries and veins. The original Lagrangian sliding fluid element model with diffusion.)

Guller B, Yipintsoi T, Orvis AL, and Bassingthwaighte JB. Myocardial sodium extraction at varied coronary flows in the dog: Estimation of capillary permeability by residue and outflow detection. Circ Res 37: 359-378, 1975. (Application to sodium exchange in the heart.)

Goresky CA. Hepatic membrane carrier transport processes: Their involvement in bilirubin uptake. In: Chemistry and Physiology of Bile Pigments. Washington, D.C.: Publishing House U.S. Government, 1977, p. 265-281.

Silverman M and Goresky CA. A unified kinetic hypothesis of carrier-mediated transport: Its applications. Biophys J 5: 487-509, 1965.

Related Models

Two solute, 2 region, facilitated transport with enzymatic conversion and flow.
Two solute, 2 region, axially distributed, facilitated transport with enzymatic conversion, and flow.
Gentex: A General tissue exchange model.

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Key Terms

transporter, Michaelis-Menten, capillary-tissue exchange, axial gradients, solute-solute competition, permeability surface area, BTEX, spatially distributed, convection, diffusion, reaction.

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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.