/*
* Mathematical Models of Ionic Transport in the Distal Tubule
* of the Rat
*
* Model Status
*
* This CellML model is a description of Chang and Fujita's 2001
* mathematical model of an anion exchanger in the distal tubule
* of the rat: it is one component of an overall model of acid/base
* transport in a distal tubule.
*
* Model Structure
*
* ABSTRACT: The purpose of this study is to develop a numerical
* model that simulates acid-base transport in rat distal tubule.
* We have previously reported a model that deals with transport
* of Na(+), K(+), Cl(-), and water in this nephron segment (Chang
* H and Fujita T. Am J Physiol Renal Physiol 276: F931-F951, 1999).
* In this study, we extend our previous model by incorporating
* buffer systems, new cell types, and new transport mechanisms.
* Specifically, the model incorporates bicarbonate, ammonium,
* and phosphate buffer systems; has cell types corresponding to
* intercalated cells; and includes the Na/H exchanger, H-ATPase,
* and anion exchanger. Incorporation of buffer systems has required
* the following modifications of model equations: new model equations
* are introduced to represent chemical equilibria of buffer partners
* [e.g., pH = pK(a) + log(10) (NH(3)/NH(4))], and the formulation
* of mass conservation is extended to take into account interconversion
* of buffer partners. Furthermore, finite rates of H(2)CO(3)-CO(2)
* interconversion are taken into account in modeling the bicarbonate
* buffer system. Owing to this treatment, the model can simulate
* the development of disequilibrium pH in the distal tubular fluid.
* For each new transporter, a state diagram has been constructed
* to simulate its transport kinetics. With appropriate assignment
* of maximal transport rates for individual transporters, the
* model predictions are in agreement with free-flow micropuncture
* experiments in terms of HCO reabsorption rate in the normal
* state as well as under the high bicarbonate load. Although the
* model cannot simulate all of the microperfusion experiments,
* especially those that showed a flow-dependent increase in HCO
* reabsorption, the model is consistent with those microperfusion
* experiments that showed HCO reabsorption rates similar to those
* in the free-flow micropuncture experiments. We conclude that
* it is possible to develop a numerical model of the rat distal
* tubule that simulates acid-base transport, as well as basic
* solute and water transport, on the basis of tubular geometry,
* physical principles, and transporter kinetics. Such a model
* would provide a useful means of integrating detailed kinetic
* properties of transporters and predicting macroscopic transport
* characteristics of this nephron segment under physiological
* and pathophysiological settings.
*
* The original paper reference is cited below:
*
* A numerical model of acid-base transport in rat distal tubule,
* Hangil Chang and Toshiro Fujita, 2001, American Journal of Physiology,
* 281, F222-F243. PubMed ID: 11457714.
*
* reaction_diagram2
*
* [[Image file: chang_2001b.png]]
*
* State diagram of the anion exchanger. In this model, the anion
* transporter (E) has a single binding site to which Cl- and HCO3
* - competitively bind. Only the bound forms of the transporter
* are able to cross the membrane. (Symbols with the asterisk (*)
* represent conformations facing the cytosol, symbols without
* indicate conformations facing the extracellular environment.)
*/
import nsrunit;
unit conversion on;
// unit millimolar predefined
unit flux=1 meter^(-3)*second^(-1)*mole^1;
unit first_order_rate_constant=1 second^(-1);
unit second_order_rate_constant=1 meter^3*second^(-1)*mole^(-1);
math main {
//Warning: the following variables were set 'extern' or given
// an initial value of '0' because the model would otherwise be
// underdetermined: E, ECl, EHCO3, ECl_, EHCO3_, E_
realDomain time second;
time.min=0;
extern time.max;
extern time.delta;
real E(time) millimolar;
//Warning: Assuming zero initial condition; nothing provided in original CellML model.
when(time=time.min) E=0;
real k1 second_order_rate_constant;
k1=1.0E8;
real k2 first_order_rate_constant;
k2=7.87E6;
real k3 second_order_rate_constant;
k3=1.0E8;
real k4 first_order_rate_constant;
k4=8.28E6;
real ECl(time) millimolar;
//Warning: Assuming zero initial condition; nothing provided in original CellML model.
when(time=time.min) ECl=0;
real EHCO3(time) millimolar;
//Warning: Assuming zero initial condition; nothing provided in original CellML model.
when(time=time.min) EHCO3=0;
real HCO3 millimolar;
HCO3=1.0;
real Cl millimolar;
Cl=1.0;
real k9 first_order_rate_constant;
k9=5.14E5;
real k10 first_order_rate_constant;
k10=9.26E4;
real ECl_(time) millimolar;
//Warning: Assuming zero initial condition; nothing provided in original CellML model.
when(time=time.min) ECl_=0;
real k11 first_order_rate_constant;
k11=3.24E5;
real k12 first_order_rate_constant;
k12=5.83E4;
real EHCO3_(time) millimolar;
//Warning: Assuming zero initial condition; nothing provided in original CellML model.
when(time=time.min) EHCO3_=0;
real E_(time) millimolar;
//Warning: Assuming zero initial condition; nothing provided in original CellML model.
when(time=time.min) E_=0;
real k5 second_order_rate_constant;
k5=1.0E8;
real k6 first_order_rate_constant;
k6=7.87E6;
real k7 second_order_rate_constant;
k7=1.0E8;
real k8 first_order_rate_constant;
k8=8.28E6;
real HCO3_ millimolar;
HCO3_=1.0;
real Cl_ millimolar;
Cl_=1.0;
real J_Cl_influx(time) flux;
real Ki_Cl millimolar;
Ki_Cl=0.528;
real Ki_HCO3 millimolar;
Ki_HCO3=0.423;
//
//
E:time=(k2*ECl+k4*EHCO3-(k1*Cl*E+k3*HCO3*E));
//
ECl:time=(k1*Cl*E+k10*ECl_-(k2*ECl+k9*ECl));
//
EHCO3:time=(k3*HCO3*E+k12*EHCO3_-(k4*EHCO3+k11*EHCO3));
//
E_:time=(k6*ECl_+k8*EHCO3_-(k5*Cl_*E_+k7*HCO3_*E_));
//
ECl_:time=(k5*Cl_*E_+k9*ECl-(k6*ECl_+k10*ECl_));
//
EHCO3_:time=(k7*HCO3_*E_+k11*EHCO3-(k8*EHCO3_+k12*EHCO3_));
//
J_Cl_influx=((k9*ECl-k10*ECl_)*(1+Cl_/Ki_Cl+HCO3_/Ki_HCO3)^(-1));
//
}