Equations for InsP₃ Receptor-mediated [Ca²⁺]_i Oscillations Derived from a Detailed Kinetic Model: A Hodgkin-Huxley Like Formalism

Abstract

The nine-variable De Young-Keizer model (1992) for [Ca²⁺]_i oscillations mediated by InsP₃ receptor channels in endoplasmic reticulum (ER) membrane is analyzed and reduced to a two-variable system. The different time scales in the three basic channel gating processes, namely InsP₃ regulation, Ca²⁺ activation, and Ca²⁺ inactivation, are revealed and characterized. The method of multiple scales is used in solving the equations on a succession of faster time scales and reducing them to a 2D system. The reduced system, (V_cy/f_cy) dC/dt = -P_IP3Rm³_∞h³ (C - C_O) - P_L (C - C_O ) - J_pump(C); dh/dt = (h_∞ - h)/τ_h, is analogous in form to the Hodgkin-Huxley equations for plasma membrane electrical excitability. [Ca²⁺]_i dynamics in this model thus involve ER membrane-associated excitability. The reduced system has a bifurcation diagram almost identical to that of the original system and retains the most important dynamic features of the latter. The analysis also shows that the reduced system becomes simpler when the different gating processes are more independent from each other, i.e. when the rates for Ca²⁺ binding at the site associated with one gating process are independent of occupancy at the other two binding sites. Assuming further that binding of InsP₃ does not depend on Ca²⁺ occupancy at the inactivation site, we obtain a "minimal" form yet retain significant ability to reproduce experimental observations.