Two compartment master transporter model with choices:

Flow: yes or no,

Solutes: A only, A and B;

Conversion A to B: none, linear, Michaelis-Menten (MM);

Transporters: Passive, MM A1 1-sided, MM A1,A2 two 1-sided,MM A1,A2 one 2-sided, MM A,B 2-sided, and T1&T2 (facilitated).

Looping on parameters flow, solutes, convert, and transporter (all lower case) allows comparisons to be made between different model formulations.

## Figure

The choices producing this diagram include no flow, two solutes. a linear conversion of A to B in the second compartment, and T1T2 facilitated transport. Solute A is initialized to 1 mM in the first compartment (A1(0) = 1 mM). A is transported from V1 to V2 by a transporter that can bind to either A or B. The transporter can flip from side to side in the unbound form T, or in the bound forms, TA and TB. In V2, A is converted to B by a linear reaction process.

Note that B in compartment one (B1) is greater than B in compartment 2 (B2) where B is produced. B has been transported against a gradient, from low concentration to high concentration.

## Description

TransComp2 is a general model for transporter between two compartments. The model has five basic choice options: FLOW (1) YES: The model has a flow in compartment 1. (2) NO: The model has no flow. SOLUTES (1) A only: The model has only one solute, A. (2) A and B: The model has two solutes, A and B. CONVERT (1) None: There is no conversion of A2 into B2. (2) G2aTOb: If there are two solutes, the loss from A2 and gain to B2 is a linear process given by G2aTOb/V2*A2. (3) GmaxA2,KmA2: If there are two solutes, the loss from A2 and gain to B2 is a Michaelis-Menten process given by GmaxA2*A2/(V2*(1+A2/KmA2)). TRANSPORTER (1) PSa,PSb, Passive: The transport is by a passive process. A1 change = PSa/V1*(A2-A1). A2 change = PSa/V2*(A1-A2). B1 change = PSb/V1*(B2-B1). B2 change = PSb/V2*(B1-B2). (2) PSmaxA1, TKmA1 MM A1 1-SIDED: The transport of A1 and A2 is a Michaelis-Menten process governed by only the A1 concentration. A1 change = (PSmaxA1/V1)/(1 + A1/TKmA1)*(A2-A1). A2 change = (PSmaxA1/V2)/(1 + A1/TKmA1)*(A1-A2). (3) PSmaxA2, TKmA2 MM A1 A2 2 1-SIDED: The transport of A1 and A2 are Michaelis-Menten processes where each side has a 1-sided transporter for A1 and A2. A1 change = (PSmaxA2/V1)/(1 + A2/TKmA2)*(A2 )+ (PSmaxA1/V1)/(1 + A1/TKmA1)*( -A1). A2 change = (PSmaxA2/V2)/(1 + A2/TKmA2)*( -A2)+ (PSmaxA1/V2)/(1 + A1/TKmA1)*(A1 ). (4) PSmaxA, TKmA MM A1 A2 2-SIDED: The transport of A1 and A2 is a Michaelis-Menten process where the transporter is 2-sided: A1 change = (PSmaxA/V1)/(1 + (A1+A2)/TKmA)*(A2-A1). A2 change = (PSmaxA/V2)/(1 + (A1+A2)/TKmA)*(A1-A2). (5) PSmaxAB, TKmAB MM A&B 2-SIDED: The transport of A and B is governed by a single Michaelis-Menten process where the transporter is dependent on all four species: A1 change = (PSmaxAB/V1)/(1 + (A1+A2+B1+B2)/TKmAB)*(A2-A1). A2 change = (PSmaxAB/V2)/(1 + (A1+A2+B1+B2)/TKmAB)*(A1-A2). B1 change = (PSmaxAB/V1)/(1 + (A1+A2+B1+B2)/TKmAB)*(B2-B1). B2 change = (PSmaxAB/V2)/(1 + (A1+A2+B1+B2)/TKmAB)*(B1-B2). (6) T1&T2, facilitated: A free transporter T flips between side 1 and side 2. (T1<->T2) For solute = A only, the change in concentrations are A1 change = SoV1*(koffA1*TA1 - konA1*A1*T1). A2 change = SoV2*(koffA2*TA2 - konA2*A2*T2). T1 change = (koffA1*TA1-konA1*A1*T1) - kT12*T1 + kT21*T2; TA1 change= (konA1*A1*T1 - koffA1*TA1 - kTA12*TA1 + kTA21*TA2) ; TA2 change=(konA2*A2*T2 - koffA2*TA2 + kTA12*TA1 - kTA21*TA2) ; T2 change = Ttot - TA1 - TA2 - T1. For solute = A and B, The equations for A1, A2, TA1, and TA2 are unchanged. The equations for T1 and T2 are changed and equations for B1, B2, TB1, and TB2 are added: T1 change = (koffA1*TA1-konA1*A1*T1) - kT12*T1 + kT21*T2 +(koffB1*Tb1-konB1*B1*T1); T2 change = Ttot - TA1 - TA2 - TB1 -TB2 - T1. B1 change = SoV1*(koffB1*TB1 - konB1*B1*T1). B2 change = SoV2*(koffB2*TB2 - konB2*B2*T2). TB1 change= (konB1*B1*T1 - koffB1*TB1 - kTB12*TB1 + kTB21*TB2) ; TB2 change= (konB2*B2*T2 - koffB2*TB2 + kTB12*TB1 - kTB21*TB2) ; STAT_FlowYes: STATISTICS: FLOW must be YES for this calculation to be performed. (1) A only Cin = Ain, Cout = Aout. (2) B only Cin = Bin, Cout = Bout. (3) A and B Cin = Ain+Bin, Cout = Aout+Bout; The area, mean transit time and relative dispersion are calculated for Cin and Cout. In addition, the system transit time and relative dispersion are calculated. WARNING: An additional thermodynamic constraint is not included in the model. For a passive transporter, the transport rate constants should satisfy the following constraints: kTA12*kT21*konA1*koffA2 ------------------------ = 1 (1) see TestA kTA21*kT12*koffA1*konA2 kTB12*kT21*konB1*koffB2 ------------------------ = 1 (2) see TestB kTB21*kT12*koffB1*konB2 These constraints ensure that the model runs to equlibrium 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 a energy source, which is not explicitly modeled here.

## Equations

## Not displayed here.

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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.

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