Numerical Approach to Fast Reactions in Reaction-Diffusion Systems: Application to Buffered Calcium Waves in Bistable Models

A numerical approach to modeling a biochemical system that includes processes with significantly different time scales has been developed within the Virtual Cell environment (J. Schaff et al., 1997, Biophys. J.73, 1135). The key features of the algorithm are time splitting of slow and fast processes and pseudo-steady approximation based on stoichiometry analysis. We apply the method to study the effect of fast calcium buffering on the properties of self-sustaining calcium waves in living cells. Numerical results for one-dimensional traveling waves in one-variable bistable models are compared with theoretical predictions. The effect of a mobile buffer on calcium waves appears to strongly depend on buffer affinity and system excitability. In systems with low excitability, the buffer can stop the traveling wave and make it move in the opposite direction, which means physiologically that the wave becomes self-extinguishing. We then consider traveling waves in a more realistic two-variable model (the Li–Rinzel model). This system exhibits a new feature: in the mode of low excitability, under certain conditions, it undergoes bifurcation with the buffer concentration as a bifurcation parameter. As a consequence, for some buffer concentrations, there exist two stable traveling waves with very different velocities. Finally, to study how a fluorescent indicator, which acts as a mobile buffer, might affect the fertilization calcium waves in eggs, we run three-dimensional simulations within the Li–Rinzel model using realistic parameters, geometry, and initial conditions. The results indicate strong interaction of a fluorescent dye with initiating calcium spikes. As a result, a fluorescent dye added to visualize calcium dynamics in a cell causes a delay in wave formation and, at sufficient concentration, can prevent a wave.