NMR spin exchange kinetics at equilibrium in membrane transport and enzyme systems

A set of differential equations is formulated to describe the rapid exchange (time scale, approximately 0.01 to approximately 10 s) of a labelled solute across the membranes of cells in suspension. The labelling is achieved with nuclear magnetic resonance by exposure of the system to a high intensity radio-frequency pulse, and the excited nuclei relax to the equilibrium state with a short half life. An analytical expression for the decay of the magnetic resonance signal is presented; the solution involves the determination of eigenvalues, of an array of Laplace-Carson transformed differential equations, by use of the general solution of a quartic polynomial. Simulations of the behaviour of the exchange system using various conditions of cell number, rate constants and nuclear magnetic relaxation times are presented. The marked concentration dependence of the extent of reaction at a given time has not previously been reported for nuclear magnetic resonance exchange systems and is a feature anticipated from the known saturability of several membrane transport systems including glucose transport into human erythrocytes. The theory is readily generalized to other model systems by appropriate reinterpretation of the physical meaning of various parameters; the general form of the solution holds in many biological contexts other than membrane transport and includes equilibrium enzyme kinetics.