AGONIST-INDUCED CALCIUM WAVES IN OSCILLATORY CELLS - A BIOLOGICAL EXAMPLE OF BURGERS-EQUATION

Oscillations in cytosolic Ca2+ concentrations in living cells are ofte n a manifestation of propagating waves of Ca2+. Numerical simulations with a realistic model of inositol 1,4,5-trisphosphate (IP3)-induced C a2+ wave trains lead to wave speeds that increase linearly at long tim es when (a) IP3 levels are in the range for Ca2+ oscillations, (b) a g radient of phase is established by either an initial ramp or pulse of IP3, and (c) IP3 concentrations asymptotically become uniform. We expl ore this phenomenon with analytical and numerical methods using a simp le two-variable reduction of the De Young-Keizer model of the IP3 rece ptor that includes the influence of Ca2+ buffers. For concentrations o f IP3 in the oscillatory regime, numerical solution of the resulting r eaction 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 with out 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+. W e find that the gradient in phase of the wave obeys Burgers' equation asymptotically in time. This result is used to explain the linear incr ease of the wave speed observed in the simulations. (C) 1997 Society f or Mathematical Biology.