Reliable encoding of stimulus intensities within random sequences of intracellular Ca2+ spikes

Abstract

Ca(2+) is a ubiquitous intracellular messenger that regulates diverse cellular activities. Extracellular stimuli often evoke sequences of intracellular Ca(2+) spikes, and spike frequency may encode stimulus intensity. However, the timing of spikes within a cell is random because each interspike interval has a large stochastic component. In human embryonic kidney (HEK) 293 cells and rat primary hepatocytes, we found that the average interspike interval also varied between individual cells. To evaluate how individual cells reliably encoded stimuli when Ca(2+) spikes exhibited such unpredictability, we combined Ca(2+) imaging of single cells with mathematical analyses of the Ca(2+) spikes evoked by receptors that stimulate formation of inositol 1,4,5-trisphosphate (IP3). This analysis revealed that signal-to-noise ratios were improved by slow recovery from feedback inhibition of Ca(2+) spiking operating at the whole-cell level and that they were robust against perturbations of the signaling pathway. Despite variability in the frequency of Ca(2+) spikes between cells, steps in stimulus intensity caused the stochastic period of the interspike interval to change by the same factor in all cells. These fold changes reliably encoded changes in stimulus intensity, and they resulted in an exponential dependence of average interspike interval on stimulation strength. We conclude that Ca(2+) spikes enable reliable signaling in a cell population despite randomness and cell-to-cell variability, because global feedback reduces noise, and changes in stimulus intensity are represented by fold changes in the stochastic period of the interspike interval.

Fig. 1. Ca²⁺ spikes are stochastic and vary between cells.

(A) Many extracellular signals activate GPCRs coupled to Gα_q proteins, which stimulate phospholipase Cβ (PLCβ) and the production of IP₃. Binding of IP₃ and Ca²⁺ to IP₃Rs triggers release of Ca²⁺ into the cytosol. The increase in [Ca²⁺]_i can then activate neighboring IP₃Rs to generate a Ca²⁺ wave. Repetitive initiation of Ca²⁺ waves generates sequences of Ca²⁺ spikes that vary in frequency. Information is encoded in the properties of these spike sequences and decoded by downstream effectors. PM, plasma membrane. (B) Ca²⁺ signals in HEK293 cells and hepatocytes. HEK293 cells were stimulated with carbachol (CCh, 30 μM), and hepatocytes with phenylephrine (1 μM) or vasopressin (10 nM). Top: [Ca²⁺]_i is shown for typical cells as fura-2 fluorescence ratios (F₃₄₀/F₃₈₀). Middle: Individual ISIs of the traces above. Bottom: For each cell, average ISI (T_av) and its standard deviation (σ) provide a single point on the T_av-σ relation. The ratio of axes scales is preserved in the three T_av-σ plots to allow direct comparison of their slopes. (C) The ISI comprises the spike duration and the refractory period (the sum of which is T_min), and the stochastic period (T_av-T_min). T_av and σ are linearly related with slope α. σ_min is the standard deviation of sequences with T_av = T_min. (D) Box plots of T_av from HEK 293cells stimulated with CCh (10 μM, n = 81; 30 μM, n = 135; or 50 μM, n = 50) or hepatocytes stimulated with phenylephrine (1 μM, n = 60) or vasopressin (10 nM, n = 77). Bold lines indicate medians, boxes show interquartile ranges, whiskers show minima and maxima. Results from hepatocytes and HEK293 cells stimulated with 30 μM CCh were taken from panel B.

Fig. 2. Robust signal-to-noise ratios of stochastic Ca²⁺ signals.

(A) Diagram of the pathway that stimulates Ca²⁺ spikes in HEK293 cells and the perturbations tested. CCh stimulates PLC producing IP₃ that releases Ca²⁺ from ER through IP₃Rs. U73122 inhibits PLC. PTH stimulates adenylyl cyclase (AC) and formation of cAMP, which increases IP₃R sensitivity. Cyclopiazonic acid (CPA) reversibly inhibits the sarco/endoplasmic reticulum Ca²⁺-ATPase (SERCA). (B) [Ca²⁺]_i in a HEK293 cell exposed to two sequential stimuli. In this representative experiment, PTH was added to cells in the continued presence of CCh. The presence of CCh (30 μM) and PTH (100 nM) is indicated by the black and red bar, respectively. (C) The change in Ca²⁺ spike frequency (1/T_av) in HEK293 cells following a perturbation of the signaling pathway. The experiments were performed similarly to the experiment shown in B, but with two successive challenges with CCh alone (30 μM) (control, 21 cells), or CCh alone followed by CCh with PTH (100 nM, 31 cells), U73122 (100 nM, 35 cells), or CPA (10 nM, 33 cells) (see fig. S3A for the responses recorded from each cell). Results (means ± SEM) show the change in Ca²⁺ spike frequency (1/T_av2-1/T_av1) between successive challenges as a percentage of the first frequency (1/T_av1). CPA had no effect on [Ca²⁺]_i in unstimulated cells (62 ± 16 nM and 58 ± 6 nM, mean ± SEM, before and 10 min after CPA addition; n = 48 cells). For all except the successive CCh stimuli (control), the change in spike frequency was significant (p < 0.05, Student’s t test). (D) T_av-σ relations for HEK 293 cells stimulated with CCh in the presence of PTH, U73122, or CPA at the concentrations described in C. (E) The slopes of the T_av-σ relation (α ± 95% confidence intervals) in hepatocytes or HEK 293 cells exposed to the conditions indicated (p < 0.05, F-test).

Fig. 3. Stimulation steps are encoded by fold changes in the average stochastic period of the ISI.

(A) The relationship between T_av1 and ΔT_av calculated from simulations of IP₃-evoked Ca²⁺ spikes based on the stochastic model of Thurley et al. (21, 22) (see Materials and Methods). (B) [Ca²⁺]_i in a single HEK293 cell subjected to a paired stimulation protocol. The cell was stimulated as shown with 30 μM CCh before its removal and replacement with 200 μM CCh. (C) The relationship between T_av1 and ΔT_av for HEK293 cells successively stimulated with the indicated CCh concentrations (μM) in the paired stimulation protocol. (see table S2 for spiking behavior of cells included in the analysis). (D) The relationship between T_av1 and ΔT_av for hepatocytes stimulated with 0.6 μM and then 1 μM phenylephrine. (E) Pearson’s correlation coefficients (ρ) and explained uncertainties (u_ex, Eq. 8) for T_av1-ΔT_av relations for HEK293 cells stimulated with the indicated steps in CCh concentration (μM), or hepatocytes stimulated with 0.6 μM and then 1 μM phenylephrine. (F) Comparisons of the average deviation of individual cell behavior from T_pop [CV(T_pop2)] and Eq. 2 [CV(β)], and the coefficient of variation of the integral ratio [CV(IR)] for the paired stimulation protocols. The code (a-g) applies to panels E and F. (G) The relationship between T_av1 and T_av2 calculated from individual HEK 293 cells stimulated first with 30 μM and then with 150 μM CCh (data from C). The dashed line shows the population average (T_pop2) of T_av2. The solid line shows Eq. 2 in the form T_av2 = (1-β)T_av1 + βT_min. (H) Fold changes (β ± 95% confidence intervals, Eq. 2) calculated from the slopes of T_av1 - ΔT_av relations for all steps in CCh concentration. Symbols are color-coded to indicate the initial CCh concentration (red 30 μM, blue 50 μM). The line shows the exponential relationship between the fold change (β) and Δ[CCh] (Eq. 6), with γ being the only fit parameter, γ = 7.84 ± 0.37 mM⁻¹ (mean ± 95% confidence interval).

Fig. 4. Fold changes determine a universal concentration-response relation for Ca²⁺ spikes evoked by stimulation of GPCRs.

(A) Population average (T_pop) of T_av for HEK293 cells at each CCh concentration (means ± SEM). Line drawn using the parameter value γ = 7.84 mM⁻¹ (from the fit to Eq. 6 in Fig. 3H), and T_min = 57 s (the average value of T_mins from the 6 paired-stimulation experiments shown in Fig. 3C), but with no additional curve-fitting. (B, C) The relationship between T_pop and ligand concentration for hepatocytes is exponential. Hepatocytes (32) were stimulated with phenylephrine (B) or vasopressin (C). (D) The relationship between T_pop and ligand concentration for insect salivary gland stimulated with 5-HT (33) is exponential. Lines in B-D are best fits in parameters T_min and γ to Eq. 7: For hepatocytes γ = 1.059 μM⁻¹, T_min = 61 s (phenylephrine) and γ = 0.279 μM⁻¹, T_min = 44 s (vasopressin); and for salivary gland γ = 0.319 nM⁻¹, T_min = 16 s.