A Ca²⁺ puff model based on integrodifferential equations

The calcium signalling system is important for many cellular processes within the human body. Signals are transmitted within the cell by releasing calcium (Ca $_{}^{2+}$ ) from the endoplasmic reticulum (ER) into the cytosol via clusters of Ca $_{}^{2+}$ channels. Mathematical models of Ca $_{}^{2+}$ release via inositol 1,4,5-trisphosphate receptors (IP₃R) are used to compute Ca $_{}^{2+}$ transients in regions that are difficult to measure directly. In particular, accounting for the data on Ca $_{}^{2+}$ puffs as stochastic Ca $_{}^{2+}$ release events in models remains challenging. Parameterising Markov models for representing the IP₃R with steady-state single channel data obtained at fixed combinations of the ligands Ca $_{}^{2+}$ and inositol-trisphosphate (IP₃) has previously been demonstrated to be insufficient. However, by extending an IP₃R model based on steady-state data with an integral term that incorporates the delayed response of the channel to varying Ca $_{}^{2+}$ concentrations we succeed in generating realistic Ca $_{}^{2+}$ puffs. By interpreting the integral term as a weighted average of Ca $_{}^{2+}$ concentrations that extend over a time interval of length  $\tau$ into the past we conclude that the IP₃R requires a certain amount of memory of past ligand concentrations.