Derivation of Ca2+ signals from puff properties reveals that pathway function is robust against cell variability but sensitive for control

Abstract

Ca(2+) is a universal second messenger in eukaryotic cells transmitting information through sequences of concentration spikes. A prominent mechanism to generate these spikes involves Ca(2+) release from the endoplasmic reticulum Ca(2+) store via inositol 1,4,5-trisphosphate (IP(3))-sensitive channels. Puffs are elemental events of IP(3)-induced Ca(2+) release through single clusters of channels. Intracellular Ca(2+) dynamics are a stochastic system, but a complete stochastic theory has not been developed yet. We formulate the theory in terms of interpuff interval and puff duration distributions because, unlike the properties of individual channels, they can be measured in vivo. Our theory reproduces the typical spectrum of Ca(2+) signals like puffs, spiking, and bursting in analytically treatable test cases as well as in more realistic simulations. We find conditions for spiking and calculate interspike interval (ISI) distributions. Signal form, average ISI and ISI distributions depend sensitively on the details of cluster properties and their spatial arrangement. In contrast to that, the relation between the average and the standard deviation of ISIs does not depend on cluster properties and cluster arrangement and is robust with respect to cell variability. It is controlled by the global feedback processes in the Ca(2+) signaling pathway (e.g., via IP(3)-3-kinase or endoplasmic reticulum depletion). That relation is essential for pathway function because it ensures frequency encoding despite the randomness of ISIs and determines the maximal spike train information content. Hence, we find a division of tasks between global feedbacks and local cluster properties that guarantees robustness of function while maintaining sensitivity of control of the average ISI.

Fig. 1.

Summary of the modeling concept and comparison with traditional concepts. The stochastic hierarchic model takes advantage of the structural and functional hierarchy formed by channels, channel clusters, and the cell. It subsumes the dynamics of the lower structural level into waiting time distributions on the next higher one. Channels cause the IPI distribution ψ_o and puff duration distributions ψ_c of clusters, and clusters generate the ISI and spike length (SL) distributions on cell level. The waiting time distributions on cluster level can be measured in vivo. That circumvents the problems arising from using parameter values from in vitro experiments for cell simulations, as deterministic rate equation models usually do. The involvement of many channels is required for the validity of rate equation models, such that average deterministic dynamics apply. Therefore, they are based on the assumption of continuous channel densities neglecting channel clustering. The additional assumption of fast Ca²⁺ diffusion entails neglecting spatial gradients and a mathematical description of cell behavior by ordinary differential equations. But this is in contradiction to the steep concentration gradients occurring during Ca²⁺ release. See SI Text for details.

Fig. 2.

Various patterns of Ca²⁺ signals can be inferred from properties of single clusters. (A) Configurations of open clusters (red) in the tetrahedral geometry. (B) Left: Waiting time distribution for the opening of a cluster in the rest state (0) and with N_o = 1, 2, or 3 clusters open. The probability to open early increases with N_o (number at lines). Because N_o determines [Ca²⁺], this corresponds to CICR. Right: Probability to open the second cluster at [IP₃] as indicated. It is very unlikely that the second cluster opens before the first one closes with small [IP₃]. (C) Stochastic simulations reveal different types of Ca²⁺ signals at various [IP₃]: only puffs (Upper Left), spiking (Upper Right), bursting (Lower Left), and overstimulation (Lower Right). The colors indicate [IP₃] as in B, p = 3.85 s^-1. (D) The average ISI T_av depends on many different parameters. Here, we show (Left) its dependence on the number of channels per cluster N_ch (black dots) and the cluster distance a (blue triangles), and (Right) its dependence on [IP₃] (black dots) and the channel closing rate γ (blue triangles). Parameter values not mentioned in this legend are given in Table 1.

Fig. 3.

Conditions for spiking. (A) Spiking occurs for values of the coupling and the channel closing rate γ between the red and black symbols. They show the long-T_av criterion for spiking C_l (red) and the short-T_av criterion C_s (black), respectively (see Conditions for Spiking and Bursting). Coupling values smaller than C_l entail essentially only local puffs, and coupling values larger than C_s cause the regime of overstimulation. The spike range increases with γ. We investigated both the four-cluster model (squares) and the eight-cluster model (circles). C_l values are similar for both models, whereas C_s values depend on the number of clusters N_cl. The spike range becomes smaller for a larger number of clusters involved in spike nucleation. Each pair of squares and circles indicates the critical () at the standard parameters (Table 1), whereas coupling was changed by varying cluster distance (values range from 0.5 μm to ∞). The triangles show the critical values for the four-cluster model obtained when we varied by changing [IP₃]. That leads to very similar results. (B) The N_clth root of the short-T_av criterion C_s is shown. It suggests that C_s depends essentially exponentially on N_cl, because the N_clth root of C_s is similar for the four-cluster model (squares) and the eight-cluster model (circles).

Fig. 4.

Characteristics of the relation between average (T_av) and standard deviation (σ) of ISIs. The slope of the σ–T_av relation with constant rate for the first puff λ₀ is always 1. (A) Modification of the σ–T_av relation by the delayed puff rate given by Eq. 1. For moderate values of T_av, the slope of the σ–T_av relation decreases with ξ (upper triangles, ξ = 0.1 s^-1; lower triangles, ξ = 10^-3 s^-1). The relations are identical for the four-cluster model (black), the eight-cluster model (red), and the eight-cluster model with randomly shifted vertex positions (pink). (B) The minimal ISI T_min increases with decreasing ξ. T_min sets the deterministic part of the ISI, which accounts for the regular oscillations often observed in experiments with high stimulation (4, 15). Here, we show that T_min naturally occurs in a stochastic model. The four-cluster model has longer T_mins than the eight-cluster model. T_min has been identified with the smallest observed T_av in simulations.