Intra-cluster percolation of calcium signals

Abstract

Calcium signals are involved in a large variety of physiological processes. Their versatility relies on the diversity of spatiotemporal behaviors that the calcium concentration can display. Calcium entry through inositol 1,4,5-trisphosphate (IP3) receptors (IP3R's) is a key component that participates in both local signals such as "puffs" and in global waves. IP3R's areusually organized in clusters on the membrane of the endoplasmic reticulum and their spatial distribution has important effects on the resulting signal. Recent high resolution observations of Ca2+ puffs offer a window to study intra-cluster organization. The experiments give the distribution of the number of IP3R's that open during each puff without much processing. Here we present a simple model with which we interpret the experimental distribution in terms of two stochastic processes: IP3 binding and unbinding and Ca2+-mediated inter-channel coupling. Depending on the parameters of the system, the distribution may be dominated by one or the other process. The transition between both extreme case sis similar to a percolation process. We show how, from an analysis of the experimental distribution, information can be obtained on the relative weight of the two processes. The largest distance over which Ca2+-mediated coupling acts and the density of IP3-bound IP3R's of the cluster can also be estimated. The approach allows us to infer properties of the interactions among the channels of the cluster from statistical information on their emergent collective behavior.

Figure 1. Distribution of puff sizes: transition between Ca-dominated to IP-binding dominated stochasticity.

Solid circles: distribution of puff sizes, , obtained with our model for , , and three values of : (A), (B) and (C). Histograms (in grey): corresponding distributions of available channels, for the same parameter values. All distributions were computed from 1000 realizations for each set of parameters.

Figure 2. Distribution of puff sizes: change of behavior with the radius of influence and comparison with observations.

We show the probabiliy, , of having a puff with open channels obtained with our model for , , and (solid circles), (open circles) and (triangles). Each curve corresponds to 500 realizations of the model. We observe a transition from a Ca-dominated to a IP-binding dominated stochasticity distribution as increases. Superimposed with bars: experimental data taken from Fig. 4D of .

Figure 3. Percolation transition: when all available channels open during a puff.

A: Probability that all available IPR's become open, , as a function of the dimensionless radius of influence, , for (circles), (squares) and (triangles). B: (circles), (squares) and (triangles) as functions of . The values of and for the case with are indicated in A with one and two asterisks, respectively.

Figure 4. Change of behavior with event size.

A: for data obtained with our model using , , and (solid circles). Complementary cumulative Poisson distributions, , for (inverted triangles), (triangles), (squares), (rhombes). B: Error of approximating by the various for (see text for definition) as a function of . Symbols are the same as in A. From this figure we choose as the one that provides the best fit to the tail of . The error in the case is larger than 0.02 in most cases and falls outside the region displayed in the figure. C: for the four values of that we tested. We see that .