Kuchel P W, Chapman B E
J Theor Biol. 1983 Dec 21;105(4):569-89. doi: 10.1016/0022-5193(83)90220-5.
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.
建立了一组微分方程来描述标记溶质在悬浮细胞的细胞膜间的快速交换（时间尺度约为0.01秒至约10秒）。通过将系统暴露于高强度射频脉冲，利用核磁共振实现标记，被激发的原子核以短半衰期弛豫至平衡态。给出了磁共振信号衰减的解析表达式；该解涉及通过使用四次多项式的通解来确定拉普拉斯 - 卡森变换微分方程阵列的特征值。给出了使用不同细胞数量、速率常数和核磁共振弛豫时间条件对交换系统行为的模拟。给定时间反应程度对浓度的显著依赖性此前尚未在核磁共振交换系统中报道，这是从包括葡萄糖转运到人类红细胞在内的几种膜转运系统已知的饱和性所预期的一个特征。通过对各种参数的物理意义进行适当重新解释，该理论很容易推广到其他模型系统；该解的一般形式适用于除膜转运之外的许多生物学情境，包括平衡酶动力学。
