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

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.