Linearized buffered Ca2+ diffusion in microdomains and its implications for calculation of [Ca2+] at the mouth of a calcium channel

Abstract

Immobile and mobile calcium buffers shape the calcium signal close to a channel by reducing and localizing the transient calcium increase to physiological compartments. In this paper, we focus on the impact of mobile buffers in shaping steady-state calcium gradients in the vicinity of an open channel, i.e. within its "calcium microdomain." We present a linear approximation of the combined reaction-diffusion problem, which can be solved explicitly and accounts for an arbitrary number of calcium buffers, either endogenous or added exogenously. It is valid for small saturation levels of the present buffers and shows that within a few hundred nanometers from the channel, standing calcium gradients develop in hundreds of microseconds after channel opening. It is shown that every buffer can be assigned a uniquely defined length-constant as a measure of its capability to buffer calcium close to the channel. The length-constant clarifies intuitively the significance of buffer binding and unbinding kinetics for understanding local calcium signals. Hence, we examine the parameters shaping these steady-state gradients. The model can be used to check the expected influence of single channel calcium microdomains on physiological processes such as excitation-secretion coupling or excitation-contraction coupling and to explore the differential effect of kinetic buffer parameters on the shape of these microdomains.

Fig. 1.

Effect of multiple buffers and their contribution to the flux of calcium. In A, starting with 2 mm ATP (top trace), different buffers are added successively, and their range of buffering is visualized by plotting δ[Ca²⁺] · r (calculated using Eq. EAI.10) over r to eliminate the “ 1/r -dependence” of the calcium concentration deflections. In the absence of EGTA or BAPTA, the ATP kinetics is showing up between 10 and 50 nm from the channel, whereas the endogenous buffer shows up in the range of 50–200 nm. For larger distances, the equilibrium buffer properties according to Equation 12determine the concentration of calcium. Although EGTA and BAPTA have similar binding ratios and hence equilibrium buffering powers, EGTA is kinetically limited in buffering within 100 nm compared with BAPTA because of its on-rate, which is two orders of magnitude smaller than the on-rate of BAPTA.  In B, the contributions to calcium flux of the individual calcium-carrying species are plotted as a function of distance for the case of 2 mm ATP, 0.5 mm endogenous buffer, and 2 mm EGTA. The total flux corresponds to 3.1 × 10⁶ ions/sec. Depending on its length-constant, there is a well defined range within which each buffer is maximally exerting its kinetically limited buffering action. ATP, for instance, has a binding ratio of only 0.9. Nevertheless, it captures >40% of the calcium ions within 30–200 nm. Farther away from the source, the calcium is “handed over” to the endogenous buffer and then to EGTA. At large distances from the channel, 99.9% of the total flux is carried by EGTA. C plots the spatial changes of the fluxes (given in B), normalized to their peak values according to Equation17; for Ca²⁺, the absolute value of the flux changes is plotted. The curves peak at the corresponding buffer length-constants, demonstrating the intuitive notion that the length-constants are points at which maximal changes of fluxes are occurring. In this way, one gets a fingerprint of the present buffers, indicating their range of kinetic action.

Fig. 2.

Comparison of the steady-state rapid buffer approximation with the linearized solution to the steady-state buffered diffusion problem. A depicts [Ca²⁺] and B [CaBAPTA] as a function of distance in the presence of 1 mm BAPTA with 150 fA single-channel current. The rapid buffer approximation is calculated according to Equation 11 of Smith (1996). Because chemical equilibrium is assumed everywhere in Smith’s model, [Ca²⁺] is massively underestimated close to a channel where no equilibrium can be reached. The linearized approach explicitly accounts for kinetics and thus is void of this problem. The problem is also reflected in B, where 1.8 μm[Ca²⁺] saturates almost 90% of the buffer, whereas we get only 10 μm maximal increase in [CaBAPTA] (i.e., 1% of total BAPTA), close to the channel. The reason for this shortcoming is the steep calcium gradient produced by BAPTA, with which the rapid buffer assumption cannot cope, whereas the linear approach holds with only 1% saturation. As expected, for distances larger than the length-constant of BAPTA, the two curves converge.

Fig. 3.

Transient gradients as a fraction of steady-state gradients. Pseudocolor images of the ratio of the actual concentration over the steady-state concentration of the corresponding species for 2 mm EGTA. A, δ[Ca²⁺]/δ[Ca²⁺] (t = ∞). B, δ[CaEGTA]/δ[CaEGTA] (t = ∞). Superimposed on the images are the lines where a constant fraction (indicated by the numbers) of the steady-state concentrations are achieved.