Calcium signals driven by single channel noise

Abstract

Usually, the occurrence of random cell behavior is appointed to small copy numbers of molecules involved in the stochastic process. Recently, we demonstrated for a variety of cell types that intracellular Ca2+ oscillations are sequences of random spikes despite the involvement of many molecules in spike generation. This randomness arises from the stochastic state transitions of individual Ca2+ release channels and does not average out due to the existence of steep concentration gradients. The system is hierarchical due to the structural levels channel--channel cluster--cell and a corresponding strength of coupling. Concentration gradients introduce microdomains which couple channels of a cluster strongly. But they couple clusters only weakly; too weak to establish deterministic behavior on cell level. Here, we present a multi-scale modelling concept for stochastic hierarchical systems. It simulates active molecules individually as Markov chains and their coupling by deterministic diffusion. Thus, we are able to follow the consequences of random single molecule state changes up to the signal on cell level. To demonstrate the potential of the method, we simulate a variety of experiments. Comparisons of simulated and experimental data of spontaneous oscillations in astrocytes emphasize the role of spatial concentration gradients in Ca2+ signalling. Analysis of extensive simulations indicates that frequency encoding described by the relation between average and standard deviation of interspike intervals is surprisingly robust. This robustness is a property of the random spiking mechanism and not a result of control.

Figure 1. IP₃R properties and clustering generate a hierarchical system.

A: form channel clusters (green dots) that are randomly scattered across the membrane of the ER and separated by 1 to 7 in the cell. B: Compared with inter-cluster distances, channels (orange) within a cluster are tightly packed in the ER membrane and are strongly coupled by (red). Channels within a cluster are lumped into one source term (green sphere) with radius , which depends on the number of open channels (see text). C: Single consist of four subunits the dynamics of which is described by the DeYoung-Keizer model. The 8 subunit states form a cube and subunit state transitions correspond to the edges. D: The dependent activation and inhibition of are key elements of induced release. Combined with the spatial clustering, the resulting hierarchical structure transforms fast fluctuating single channel dynamics (blips) first into locally amplified cluster signals (puffs) and then into cellular release spikes. (Local concentrations are determined 10 nm apart from the release site.)

Figure 2. Spatially resolved Ca²⁺ dynamics.

An initial puff induces release of adjacent clusters by diffusion and induced release leading to a global spike. The puff to spike transition is visualized by the iso-concentration surface of 2 during a spike. Time is indicated on the panels (see Video S1).

Figure 3. Stochasticity of Ca²⁺ oscillations.

A: An experimental example of oscillations in an astrocyte. The varying ISIs demonstrate the stochasticity of spiking. B,C: Simulations of the cellular dynamics of a cell with 47 clusters each having a random number of channels between 4 and 16 for different base level concentrations and the standard parameters given in Table 1. For a low base level of 30 nM spiking is rather slow and irregular (B). For an increased base level of 50 nM spiking becomes faster and more regular (C). D: The simulated − relation, where dots correspond to spike trains of single cells having different and concentration (see Figure 5 in Text S1), is in accordance with the experimentally observed one supporting the wave nucleation mechanism. E,F: The dependence of the average period on the concentration and the resting concentration obtained in simulations show that regular spiking is more likely if one concentration is high.

Figure 4. Spontaneous Ca²⁺ signals in individual astrocytes measured under identical conditions (upper row) and simulations of a cell with 32 clusters with different parameters (red line, middle row) exhibit good agreement in the cytosolic Ca²⁺ concentration.

The parameter changes between the simulations account for the variability of the cells in the experiment. The lumenal concentration is shown in blue (middle row). The channel dynamics (lower row) is shown as the number of open channels (black) and inhibited subunits (magenta). A: Fast and regular spiking occurs by array enhanced coherence resonance where the simulated cell spikes as soon as enough channels are in the excitable state again. Spikes occur before the cell reaches its resting state as can be seen from the time course of the fraction of inhibited subunits. This is caused in simulations by a high base level concentration nM and a concentration of 0.12 . B: Spontaneous oscillations exhibit often a more irregular spiking. This is achieved in simulation for the same cellular setup as in A by a base level concentration of nM, which is lower than the standard value of 50 nM (Table 1). That decreases the probabilities for an initial event and spikes compared to panel A. The cell reaches the resting state before some of the spikes. C: A bursting like behavior is observed for decreased SERCA activity () in simulations, since remains longer in the cytosol. D: For a even smaller SERCA activity of , signals obtained in simulations exhibit plateau responses with superimposed oscillations which are also found in experiments. Simulation parameters are given in Table 1 if not stated here.

Figure 5. Buffers render spiking more irregular by decreasing spatial coupling.

A: Astrocytes were measured several minutes for reference values (red) before loading with 20 nM BAPTA-AM during the break and restarting the measurement (blue). Fast and regular spiking is shifted to a slower and more irregular one. B: Simulation of a cell containing 32 clusters with two different EGTA concentrations shown in red and blue respectively exhibit an analogous behavior. C: An increase of 10 EGTA increases and for a population of simulated cells with different cell properties, very similar to experimental observations. D: increases with increasing EGTA (magenta) and BAPTA (black) concentration for a given cell. The value of the increase depends on the single channel current. Squares correspond to 0.12 pA and dots to 1.2 pA. E: Corresponding − dependence of simulations in panel D. BAPTA and EGTA lead to a similar − dependence for the smaller current (squares), whereas the increased current decreases the slope to 0.6. F: A single channel current of 0.12 pA leads to a population slope of 1 rather independent of spatial arrangement of clusters (gray), stimulation strength (light red) and pump strength (light blue) where the population slopes arise due to 10 different buffer concentrations ( simulations for each condition). For the larger current of 1.2 pA the slope decreases to 0.6 and is again relatively independent of other physiologic parameters. This may explain the experimentally observed cell specific slopes . Parameters used in simulations are given in Table 1 if not explicitly stated here.