Mean field strategies induce unrealistic non-linearities in calcium puffs

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

Using two dyes to observe the competition of Ca2+ trapping mechanisms and their effect on intracellular Ca2+ signals

The specificity and universality of intracellular [Formula: see text] signals rely on the variety of spatio-temporal patterns that the [Formula: see text] concentration can display. [Formula: see text] liberation through inositol 1,4,5-trisphosphate receptors ([Formula: see text]) is key for this variety. In this paper, we study how the competition between buffers of different kinetics affects [Formula: see text] signals that involve [Formula: see text] release through [Formula: see text]. The study also provides insight into the underlying spatial distribution of the channels that participate in the signals. Previous works on the effects of [Formula: see text] buffers have drawn conclusions 'indirectly' by observing the [Formula: see text]-bound dye distributions in the presence of varying concentrations of exogenous buffers and using simulations to interpret the results. In this paper, we make visible the invisible by observing the signals simultaneously with two dyes, [Formula: see text] and [Formula: see text], each of which plays the role of a slow or fast [Formula: see text] buffer, respectively. Our observations obtained for different concentrations of [Formula: see text] highlight the dual role that fast buffers exert on the dynamics, either reducing the intracluster channel coupling or preventing channel inhibition and allowing the occurrence of relatively long cycles of [Formula: see text] release. Our experiments also show that signals with relatively high [Formula: see text] release rates remain localized in the presence of large [Formula: see text] concentrations, while the mean speed of the elicited waves increases. We interpret this as a consequence of the more effective uncoupling between [Formula: see text] clusters as the slow dye concentration increases. Combining the analysis of the experiments with numerical simulations, we also conclude that [Formula: see text] release not only occurs within the close vicinity of the centers of the clearly identifiable release sites ([Formula: see text] clusters) but there are also functional [Formula: see text] in between them.

Frequency and relative prevalence of calcium blips and puffs in a model of small IP₃R clusters

In this work, we model the local calcium release from clusters with a few inositol 1,4,5-trisphosphate receptor (IP3R) channels, focusing on the stochastic process in which an open channel either triggers other channels to open (as a puff) or fails to cause any channel to open (as a blip). We show that there are linear relations for the interevent interval (including blips and puffs) and the first event latency against the inverse cluster size. However, nonlinearity is found for the interpuff interval and the first puff latency against the inverse cluster size. Furthermore, the simulations indicate that the blip fraction among all release events and the blip frequency are increasing with larger basal [Ca(2+)], with blips in turn giving a growing contribution to basal [Ca(2+)]. This result suggests that blips are not just lapses to trigger puffs, but they may also possess a biological function to contribute to the initiation of calcium waves by a preceding increase of basal [Ca(2+)] in cells that have small IP3R clusters.

Termination of calcium puffs and coupled closings of inositol trisphosphate receptor channels

Calcium puffs are localized Ca(2+) signals mediated by Ca(2+) release from the endoplasmic reticulum (ER) through clusters of inositol trisphosphate receptor (IP3R) channels. The recruitment of IP3R channels during puffs depends on Ca(2+)-induced Ca(2+) release, a regenerative process that must be terminated to maintain control of cell signaling and prevent Ca(2+) cytotoxicity. Here, we studied puff termination using total internal reflection microscopy to resolve the gating of individual IP3R channels during puffs in intact SH-SY5Y neuroblastoma cells. We find that the kinetics of IP3R channel closing differ from that expected for independent, stochastic gating, in that multiple channels tend to remain open together longer than predicted from their individual open lifetimes and then close in near-synchrony. This behavior cannot readily be explained by previously proposed termination mechanisms, including Ca(2+)-inhibition of IP3Rs and local depletion of Ca(2+) in the ER lumen. Instead, we postulate that the gating of closely adjacent IP3Rs is coupled, possibly via allosteric interactions, suggesting an important mechanism to ensure robust puff termination in addition to Ca(2+)-inactivation.

Buffer regulation of calcium puff sequences

Puffs are localized Ca(2 +) signals that arise in oocytes in response to inositol 1,4,5-trisphosphate (IP3). They are the result of the liberation of Ca(2 +) from the endoplasmic reticulum through the coordinated opening of IP3 receptor/channels clustered at a functional release site. The presence of buffers that trap Ca(2 +) provides a mechanism that enriches the spatio-temporal dynamics of cytosolic calcium. The expression of different types of buffers along the cell's life provides a tool with which Ca(2 +) signals and their responses can be modulated. In this paper we extend the stochastic model of a cluster of IP3R-Ca(2 +) channels introduced previously to elucidate the effect of buffers on sequences of puffs at the same release site. We obtain analytically the probability laws of the interpuff time and of the number of channels that participate of the puffs. Furthermore, we show that under typical experimental conditions the effect of buffers can be accounted for in terms of a simple inhibiting function. Hence, by exploring different inhibiting functions we are able to study the effect of a variety of buffers on the puff size and interpuff time distributions. We find the somewhat counter-intuitive result that the addition of a fast Ca(2 +) buffer can increase the average number of channels that participate of a puff.

A stochastic model of calcium puffs based on single-channel data

Calcium puffs are local transient Ca(2+) releases from internal Ca(2+) stores such as the endoplasmic reticulum or the sarcoplasmic reticulum. Such release occurs through a cluster of inositol 1,4,5-trisphosphate receptors (IP3Rs). Based on the IP3R model (which is determined by fitting to stationary single-channel data) and nonstationary single-channel data, we construct a new IP3R model that includes time-dependent rates of mode switches. A point-source model of Ca(2+) puffs is then constructed based on the new IP3R model and is solved by a hybrid Gillespie method with adaptive timing. Model results show that a relatively slow recovery of an IP3R from Ca(2+) inhibition is necessary to reproduce most of the experimental outcomes, especially the nonexponential interpuff interval distributions. The number of receptors in a cluster could be severely underestimated when the recovery is sufficiently slow. Furthermore, we find that, as the number of IP3Rs increases, the average duration of puffs initially increases but then becomes saturated, whereas the average decay time keeps increasing linearly. This gives rise to the observed asymmetric puff shape.

Intracellular calcium signals display an avalanche-like behavior over multiple lengthscales

Many natural phenomena display “self-organized criticality” (SOC), (Bak et al., 1987). This refers to spatially extended systems for which patterns of activity characterized by different lengthscales can occur with a probability density that follows a power law with pattern size. Differently from power laws at phase transitions, systems displaying SOC do not need the tuning of an external parameter. Here we analyze intracellular calcium (Ca²⁺) signals, a key component of the signaling toolkit of almost any cell type. Ca²⁺ signals can either be spatially restricted (local) or propagate throughout the cell (global). Different models have suggested that the transition from local to global signals is similar to that of directed percolation. Directed percolation has been associated, in turn, to the appearance of SOC. In this paper we discuss these issues within the framework of simple models of Ca²⁺ signal propagation. We also analyze the size distribution of local signals (“puffs”) observed in immature Xenopus Laevis oocytes. The puff amplitude distribution obtained from observed local signals is not Gaussian with a noticeable fraction of large size events. The experimental distribution of puff areas in the spatio-temporal record of the image has a long tail that is approximately log-normal. The distribution can also be fitted with a power law relationship albeit with a smaller goodness of fit. The power law behavior is encountered within a simple model that includes some coupling among individual signals for a wide range of parameter values. An analysis of the model shows that a global elevation of the Ca²⁺ concentration plays a major role in determining whether the puff size distribution is long-tailed or not. This suggests that Ca²⁺-clearing from the cytosol is key to determine whether IP₃-mediated Ca²⁺ signals can display a SOC-like behavior or not.