Modeling the Apparent Diffusion Constant of Calcium Ions Emanating From a Channel: Implications for Calcium Wave Propagation.

Ionized calcium (Ca2+) is an intracellular messenger in nearly all eukaryotic organisms. Calcium waves have been observed in many cell types, including airway epithelia, eggs, hepatocytes, exocrine pancreas, cultured glial cells, and neuronal cells. These waves travel across or between cells without slowing or getting smaller, and so cannot be generated by pure passive diffusion. Rather, the waves must be amplified locally. This leads to the questions: what substance mediates the local amplification process, and over what distance does that substance act? Here I bring together previous measurements of calcium diffusion and wave propagation to show that if calcium itself mediates amplification, it acts locally at a distance of 0.2 pm or less. In most calcium wave-generating preparations, the source of Ca2+ is an internal store that is sensitive to the second messenger inositol 1,4,5triphosphate (IP,). Receptor activation leads to the activation of heterotrimeric G proteins in the plasma membrane, which in turn activate the enzyme phospholipase C to produce IP3. In this signaling cascade, both IP3 and Ca2+ can contribute to the production of a regenerative calcium wave. IP3 binds to receptors in the endoplasmic reticulum, which open to allow Ca*+ to pass into the cytoplasm. Conversely, micromolar levels of calcium augment the activity of phospholipase C and the IP3 receptor. Therefore, the speed of calcium waves might be rate-limited by the diffusion of IP3, Ca2+, or both, from one release site to neighboring ones, in the direction of propagation. Previously I showed that BAPTA and its analogs, all of which have fast on-rates (k, IO” Mm’s ‘), slow calcium wave propagation in cultured neuronal cells. In contrast, the “slow” buffer EGTA (k, lOh M-Is-‘) does not slow waves (1). Together, these experiments show that an exogenous buffer must bind calcium quickly if it is to impede Ca2+ wave propagation. This supports a mechanism for wave propagation in which the local diffusion of calcium is a rate-limiting step. This conclusion is important in interpreting estimates of the diffusion constant (D) of the rate-limiting factor from wave front parameters. D can be inferred from measurements of the length and speed of a traveling wave by using the relation D = Lv + @ (2-4; J. Sneyd and L. V. Kalachev, in prep.), where @ is a positive correction term that depends on the specific excitatory mechanism, so that D > Lv. In agonist-evoked calcium waves in N 1 E115 neuroblastoma cells, wave front analysis gives Lv = 140 to 700 pm2/s (1). These values are near the diffusion constant of IP3 (280 pm2s) or free calcium (200-600 pm2/s), but not near the D of buffered calcium in cytoplasm (lo-20 pm’/s). Therefore, the estimate of D from Lv is reconcilable with calcium as the rate-limiting factor only if calcium is free, not buffered, at the time it closes the feedback loop. The apparent slowing of calcium diffusion in cytoplasm can be explained by the presence of immobile calcium buffers that slow the movement of calcium ions by binding them (5,6). At first, when calcium ions emerge from an ion channel, they diffuse freely. Then they begin to come into equilibrium with endogenous buffers. The movement of the ions is then slowed if the buffer is less mobile than the free calcium ion. When the ions have come into equilibrium with buffers (at long distances or after long times), the apparent rate of diffusion is the weighted average D,,, = Dcat + Dmer (1 feq), where feq is the fraction of calcium ions which are free at binding equilibrium, Dca is for calcium diffusing freely in cytoplasm, and Dbuiier is for the calcium-buffer complex. A consequence of this equation is that D,,, must lie between Dc, and Dbuirer. At low calcium levels, Dapp has been measured to be about 10 Fm2/s (6,7). Therefore, D burrer is less than 10 lm2/s. Dbuffer can also be calculated directly for elevated levels of calcium from the published data of Allb&ton et al. (6) because they give D,,, as a function of both free calcium and the total amount of calcium added. Dbuffer calculated this way increases with increasing [Cali and is lowest at resting levels of calcium. These low rates of buffer diffusibility are consistent with the observation that in chromaffin cells, buffers do not wash out even after extended whole-cell patch clamp recording (8). Because these numbers are so much smaller than Dc,, the endogenous buffers can be regarded as immobile (Dbuffer = 0). The spatial distribution of calcium ions diffusing into a nonsaturating buffer can be calculated by regarding calcium ions emanating from a pore as starting in a free state and passing back and forth between free and bound states with binding and unbinding rate constants k, and km. D,,, , the apparent diffusion constant of calcium, can be calculated by considering the averaged behavior of the ions. The average fraction of ions that are free at a given time, f&,(t), is an exponential that relaxes toward binding equilibrium with time constant r = I/(k + [B],k-) (ref. 9; Fig. IA). Integrating f&t) over time will give the total average amount of time that calcium ions are free. Then, for the case of an immobile buffer,