Membrane transport models with fast and slow reactions: general analytical solution for a single relaxation

Membrane transport models are usually expressed on the basis of chemical kinetics. The states of a transporter are related by rate constants, and the time-dependent changes of these states are given by linear differential equations of first order. To calculate the time-dependent transport equation, it is necessary to solve a system of differential equations which does not have a general analytical solution if there are more than five states. Since transport measurements in a complex system rarely provide all the time constants because some of them are too rapid, it is more appropriate to obtain approximate analytical solutions, assuming that there are fast and slow reaction steps. The states of the fast steps are related by equilibrium constants, thus permitting the elimination of their differential equations and leaving only those for the slow steps. With a system having only two slow steps, a single differential equation is obtained and the state equations have a single relaxation. Initial conditions for the slow reactions are determined after the perturbation which redistribute the states related by fast reactions. Current and zero-trans uptake equations are calculated. Curve fitting programs can be used to implement the general procedure and obtain the model parameters.