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Agonist-induced calcium waves in oscillatory cells: A biological example of Burgers' equation

Bulletin of Mathematical Biology
Publication Date
DOI: 10.1016/s0092-8240(97)00073-6


Abstract Oscillations in cytosolic Ca 2+ concentrations in living cells are often a manifestation of propagating waves of Ca 2+. Numerical simulations with a realistic model of inositol 1, 4, 5-trisphosphate (IP 3)-induced Ca 2+ wave trains lead to wave speeds that increase linearly at long times when (a) IP 3 levels are in the range for Ca 2+ oscillations, (b) a gradient of phase is established by either an initial ramp or pulse of IP 3, and (c) IP 3 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 IP 3 receptor that includes the influence of Ca 2+ buffers. For concentrations of IP 3 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 Ca 2+ is quite slow and, as previously noted, these waves occur without appreciable bulk movement of Ca 2+. 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 Ca 2+. 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.

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