Jafri M S, Keizer J
Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, MD 21205, USA.
Bull Math Biol. 1997 Nov;59(6):1125-44. doi: 10.1007/BF02460104.
Oscillations in cytosolic Ca2+ concentrations in living cells are often a manifestation of propagating waves of Ca2+. Numerical simulations with a realistic model of inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ wave trains lead to wave speeds that increase linearly at long times when (a) IP3 levels are in the range for Ca2+ oscillations, (b) a gradient of phase is established by either an initial ramp or pulse of IP3, and (c) IP3 concentrations asymptotically become uniform. We explore this phenomenon with analytical and numerical methods using a simple two-variable reduction of the De Young-Keizer model of the IP3 receptor that includes the influence of Ca2+ buffers. For concentrations of IP3 in the oscillatory regime, numerical solution of the resulting reaction diffusion equations produces nonlinear wave trains that shows the same asymptotic growth of wave speed. Due to buffering, diffusion of Ca2+ is quite slow and, as previously noted, these waves occur without appreciable bulk movement of Ca2+. Thus, following Neu and Murray, we explore the behavior of these waves using an asymptotic expansion based on the small size of the buffered diffusion constant for Ca2+. We find that the gradient in phase of the wave obeys Burgers' equation asymptotically in time. This result is used to explain the linear increase of the wave speed observed in the simulations.
活细胞中胞质钙离子浓度的振荡通常是钙离子传播波的一种表现。用肌醇1,4,5 -三磷酸（IP3）诱导的钙离子波列的真实模型进行数值模拟，当（a）IP3水平处于钙离子振荡范围内，（b）通过IP3的初始斜坡或脉冲建立相位梯度，且（c）IP3浓度渐近地变得均匀时，会导致波速在长时间内线性增加。我们使用解析和数值方法，通过对包含钙离子缓冲剂影响的IP3受体的De Young - Keizer模型进行简单的双变量简化来探究这一现象。对于处于振荡状态的IP3浓度，所得反应扩散方程的数值解产生非线性波列，其显示出相同的波速渐近增长。由于缓冲作用，钙离子的扩散相当缓慢，并且如前所述，这些波在没有明显的钙离子大量移动的情况下发生。因此，遵循Neu和Murray的方法，我们基于钙离子缓冲扩散常数的小尺寸，使用渐近展开来探究这些波的行为。我们发现波的相位梯度在时间上渐近地服从伯格斯方程。这一结果用于解释模拟中观察到的波速的线性增加。
