The effect of residual Ca2+ on the stochastic gating of Ca2+-regulated Ca2+ channel models

Single-channel models of intracellular Ca2+ channels such as the inositol 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca2+-dependent transitions are mediated by a constant background [Ca2+] as opposed to a dynamic [Ca2+] representing the formation and collapse of a localized Ca2+ domain. This assumption neglects the fact that Ca2+ released by open intracellular Ca2+ channels may influence subsequent gating through the processes of Ca2+-activation or -inactivation. We study the effect of such “residual Ca2+” from previous channel opening on the stochastic gating of minimal and realistic single-channel models coupled to a restricted cytoplasmic compartment. Using Monte Carlo simulation as well as analytical and numerical solution of a system of advection-reaction equations for the probability density of the domain [Ca2+] conditioned on the state of the channel, we determine how the steady-state open probability (popen) of single-channel models of Ca2+-regulated Ca2+ channels depends on the time constant for Ca2+ domain formation and collapse. As expected, popen for a minimal model including Ca2+ activation increases as the domain time constant becomes large compared to the open and closed dwell times of the channel, that is, on average the channel is activated by residual Ca2+ from previous openings. Interestingly, popen for a channel model that is inactivated by Ca2+ also increases as a function of the domain time constant when the maximum domain [Ca2+] is fixed, because slow formation of the Ca2+ domain attenuates Ca2+-mediated inactivation. Conversely, when the source amplitude of the channel is fixed, increasing the domain time constant leads to elevated domain [Ca2+] and decreased open probability. Consistent with these observations, a realistic De Young-Keizer-like IP3R model responds to residual Ca2+ with a steady-state open probability that is a monotonic function of the domain time constant, though minimal models that include both Ca2+-activation and -inactivation show more complex behavior. We show how the probability density approach described here can be generalized for arbitrarily complex channel models and for any value of the domain time constant. In addition, we present a comparatively simple numerical procedure for estimating popen for models of Ca2+-regulated Ca2+ channels in the limit of a very fast or very slow Ca2+ domain. When the ordinary differential equation for the [Ca2+] in a restricted cytoplasmic compartment is replaced by a partial differential equation for the buffered diffusion of intracellular Ca2+ in a homogeneous isotropic cytosol, we find the dependence of popen on the buffer time constant is qualitatively similar to the above-mentioned results.