Research ArticleCalcium signaling

Reliable Encoding of Stimulus Intensities Within Random Sequences of Intracellular Ca2+ Spikes

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Science Signaling  24 Jun 2014:
Vol. 7, Issue 331, pp. ra59
DOI: 10.1126/scisignal.2005237


Ca2+ is a ubiquitous intracellular messenger that regulates diverse cellular activities. Extracellular stimuli often evoke sequences of intracellular Ca2+ 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 Ca2+ spikes exhibited such unpredictability, we combined Ca2+ imaging of single cells with mathematical analyses of the Ca2+ 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 Ca2+ spiking operating at the whole-cell level and that they were robust against perturbations of the signaling pathway. Despite variability in the frequency of Ca2+ 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 Ca2+ 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.


Signaling from receptors in the plasma membrane requires a strong correlation between the extracellular stimulus and downstream intracellular events if information is not to be lost. The mechanisms by which changes in extracellular stimulus intensity are reliably converted into graded changes in cellular activity have not been fully resolved. It is unclear, for example, whether individual cells reliably transmit changes in the intensity of an extracellular stimulus to a graded change in cellular activity or whether the correlation between stimulus intensity and cellular response is realized largely by the average behavior of many cells (1). The complex spatiotemporal organization of the changes in cytosolic free Ca2+ concentration ([Ca2+]i) evoked by receptors that stimulate formation of inositol 1,4,5-trisphosphate (IP3) enables Ca2+ to regulate many cellular events (28). IP3 receptors (IP3Rs) are intracellular Ca2+ channels located on the endoplasmic reticulum (ER). Opening of IP3Rs is stimulated by IP3 and Ca2+ (9), allowing them to propagate Ca2+ signals regeneratively. As IP3 concentrations increase, openings of single IP3Rs lead to coordinated opening of clustered IP3Rs, and then to cytosolic Ca2+ waves (10). Repetitive initiation of these waves generates sequences of Ca2+ spikes, the frequency of which often increases with stimulus intensity (2, 11, 12). This repetitive spiking behavior is not limited to Ca2+ signaling because there are examples of other signaling pathways in which sustained stimuli evoke pulsatile intracellular signals or responses (13, 14). The location, amplitude, duration, and frequency of Ca2+ signals are likely to convey information (28, 12). However, most supporting evidence comes from biochemical analyses (6), experimentally imposed Ca2+ spikes (4, 6), or analysis of cell populations (4). Few examples directly demonstrate that individual cells can mount graded responses to changes in extracellular stimulus intensity by decoding Ca2+ spikes (5, 7, 8).

In various cells, extracellular stimuli evoke trains of Ca2+ spikes for which the interspike intervals (ISIs) are not predictable, because each ISI has a large random component (15). Thus, we investigated whether there are features of the relationship between stimulus and ISI that enabled reliable encoding of stimulus intensity in the spike frequency. We analyzed Ca2+ spikes by imaging single cells exposed to ligands that stimulate IP3-evoked Ca2+ signals. We then performed mathematical analysis to identify properties of the sequences of spikes that correlated with stimulus intensity, and tested those properties for robustness against variability between cells.


Ca2+ spike sequences exhibit temporal randomness and large cell-to-cell variability

We performed Ca2+ imaging of individual cells exposed to ligands that activate phospholipase C (PLC) through G protein (heterotrimeric guanine nucleotide–binding protein)–coupled receptors (GPCRs), thereby stimulating production of IP3 and Ca2+ release from the ER (Fig. 1A). We imaged human embryonic kidney (HEK) 293 cells exposed to carbachol (CCh), an agonist of muscarinic acetylcholine receptors, or rat primary hepatocytes exposed to either vasopressin or phenylephrine, an agonist of α1-adrenoceptors (Fig. 1B, upper). We measured the ISI for sequences of spikes occurring in individual cells (Fig. 1B, middle), and plotted the average ISI (Tav) and its SD (σ) for each condition (Fig. 1B, bottom) (abbreviations and symbols are listed in table S1).

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

(A) Many extracellular signals activate GPCRs coupled to Gαq proteins, which stimulate 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 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 (Tav) and its SD (σ) provide a single point on the Tav-σ relation. The ratio of axes scales is preserved in the three Tav-σ plots to allow direct comparison of their slopes. (C) ISI comprises the spike duration and the refractory period (the sum of which is Tmin), and the stochastic period (TavTmin). Tav and σ are linearly related with slope α. σmin is the SD of sequences with Tav = Tmin. (D) Box plots of Tav from HEK293 cells 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, and whiskers show minima and maxima. Results from hepatocytes and HEK293 cells stimulated with 30 μM CCh correspond to data shown in (B).

Repetitive Ca2+ spiking in stimulated HEK293 cells, hepatocytes, and many other cells (2, 11, 12, 16) creates a false impression that Ca2+ spikes are predictable (15, 17). All biological processes include some variability (18), but we found that in stimulated HEK293 cells and hepatocytes, a distinguishing feature of the Ca2+ spikes was the correlation between the variability of the ISI within each cell (σ) and Tav. This relationship between σ and Tav has been described previously for Ca2+ spikes in HEK293 cells, astrocytes, microglia, processed lipoaspirate (PLA) cells (15), and endothelial progenitor cells (17). The correlation is captured in the Tav-σ relation (Fig. 1B, bottom):σ=α(TavTmin)+σmin(1)where Tmin is the sum of the spike duration and refractory period, and σmin is the SD of ISI sequences with Tav = Tmin. The ISI has three components (Fig. 1C): spike duration, refractory period, and an interval before the next spike initiates that terminates stochastically (15). For Ca2+ spikes, the relationship between Tav and σ reveals the contribution of such a stochastic process to the ISI. This stochastic period shortens as the stimulus intensity increases until it becomes so brief that a spike initiates almost as soon as a cell emerges from the refractory period (15). With high stimulus intensities, spike sequences would have an average ISI equal to Tmin, and the ISIs would become almost uniform, with a small σ (σmin). Under all other conditions, the stochastic component makes a major contribution to the ISI.

We observed that there is also variability in Tav between cells treated with the same stimulus (Fig. 1D). Under various conditions, this variability between cells often exceeded Tav for the cell population, indicating that there is no consistent relationship between stimulus intensity and Tav that applies to all cells. These data indicated that there are at least two potential impediments to transferring information reliably from extracellular signals through repetitive Ca2+ spikes: individual cells differ in their sensitivities to stimuli, and within cells, Ca2+ spike sequences have a stochastic or random component.

The signal-to-noise ratio is increased by negative feedback and robust against cell-to-cell variability

The ISI comprises fixed components (Tmin: the spike duration and the refractory period) and a stochastic period (Fig. 1C, bottom). By analyzing the temporal randomness of the ISI, we can determine how the probability of a spike occurring changes with time after the refractory period. Therefore, we examined many sequences of stimulus-evoked Ca2+ spikes to determine the variability of the ISI. If the probability of a spike occurring stepped immediately to its final value, the occurrence of each spike would be unaffected by the timing of any preceding Ca2+ spike. The slope of the Tav-σ relation (α) would then be 1 (15), that is, the ISIs would be described by a Poisson distribution. Once the refractory period had passed, the timing of the next Ca2+ spike would then behave like the random decay of radioactive atoms. Under this condition, the signal-to-noise ratio (α−1), and thus the information content of the signal, would have their minimal values (19, 20). There is, however, an interval after the refractory period when the cell recovers from negative feedback inhibition toward the maximal probability of firing another Ca2+ spike (eq. S4, fig. S1, and text S1: Mathematical description of ISI distributions) (15, 17, 19). This lingering inhibition after the refractory period delays the time at which the information content of the signal becomes minimal. The time scale of recovery from feedback inhibition is, therefore, critical in determining the reliability of signaling: slow recovery from feedback inhibition improves reliability by increasing signal-to-noise ratios (reducing α) (eqs. S3 and S6, fig. S2, and text S1: Mathematical description of ISI distributions) (21, 22).

For hepatocytes and HEK293 cells, α is less than 1 (Fig. 1B), indicating the involvement of feedback inhibition with a recovery time after the refractory period that is at least as long as the ISI (23). Because the ISI can range from tens of seconds to several minutes, the recovery is too slow to result from known properties of clustered IP3Rs: the intervals between Ca2+ signals evoked by clusters of IP3Rs, often described as Ca2+ puffs, are no more than a few seconds (23). The feedback inhibition must, therefore, be “global,” that is, a consequence of the Ca2+ spike invading the entire cell, rather than the result of local Ca2+ signaling from clusters of IP3Rs. Such global mechanisms could include regulation of IP3 metabolism (12), changes in the sensitivity of IP3Rs to IP3 or Ca2+ (9), or store depletion. The value of α is different for HEK293 cells and hepatocytes, and different for hepatocytes stimulated with different stimuli (Fig. 1B). That different stimuli produced different values of α suggested that the slowly reversing feedback mechanism must regulate signaling events specifically associated with GPCRs.

Because a high signal-to-noise ratio is advantageous, we examined how the ratio was affected by perturbations of steps within the Ca2+ signaling pathway. Theoretical analyses predict that α should be unaffected by such perturbations unless they affect the global feedback mechanism (21, 24), the molecular targets of which have not yet been identified (12). Thus, changes in PLC activity, for example, would not be expected to affect α unless PLC contributed to the global feedback. Conversely, changing the Ca2+ sensitivity of a sensor mediating the feedback by, for example, altering the activity of IP3 3-kinase, a kinase that inactivates IP3 (25), or influencing coupling of the GPCR to G proteins and downstream signaling events would affect α. Because the theoretical predictions have not been investigated experimentally, we perturbed various steps in the GPCR-mediated Ca2+ signaling pathway to determine their effects on ISI. We then calculated α to determine the consequences of the perturbations on the signal-to-noise ratio.

In the HEK293 cells, CCh activates muscarinic acetylcholine receptors coupled to Gαq, which stimulate PLC and thus IP3-mediated Ca2+ release from the ER. In the same cells, a stably expressed receptor for human parathyroid hormone (PTH) couples to Gαs, which stimulates adenylyl cyclase to promote accumulation of adenosine 3′,5′-monophosphate (cAMP). cAMP directly increases the sensitivity of IP3Rs to IP3 (26). We analyzed the ISI of spikes triggered by CCh in the presence or absence of U73122 to inhibit PLC (27), PTH to increase IP3R sensitivity (26), or cyclopiazonic acid (CPA) to inhibit the sarcoplasmic/endoplasmic reticulum Ca2+–ATPase (SERCA) (28) that pumps Ca2+ into the ER (Fig. 2A). Ca2+ imaging analysis showed that these manipulations changed the frequency of CCh-evoked Ca2+ spikes: PTH increased their frequency (shortened Tav), whereas U73122 and CPA decreased Ca2+ spike frequency (increased Tav) (Fig. 2, B and C, and fig. S3A). Despite substantial changes in Tav, none of these perturbations changed α (Fig. 2, D and E). Varying the concentration of CCh also changed Tav without affecting α (fig. S3B). Thus, cells can produce Ca2+ spikes with a constant signal-to-noise ratio despite perturbations to the signaling pathway that alter the frequency of the spikes. We found that the signal-to-noise ratio is specific for each cell type (Figs. 1B and 2E) and that within hepatocytes, it varies with the stimulus (Fig. 2E). Thus, the downstream mechanisms that decode Ca2+ spikes receive signals in which the signal-to-noise ratio is defined, and it is robust to variations within the pathway that triggers the spikes.

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 bars, respectively. (C) Change in Ca2+ spike frequency (1/Tav) in HEK293 cells after 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/Tav2 − 1/Tav1) between successive challenges as a percentage of the first frequency (1/Tav1). 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) Tav-σ relations for HEK293 cells stimulated with CCh in the presence of PTH, U73122, or CPA at the concentrations described in (C). (E) Slopes of the Tav-σ relation (α ± 95% confidence intervals) in hepatocytes or HEK293 cells exposed to the conditions indicated (*P < 0.05, F-test).

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

We found that Ca2+ spikes in two different cell types responding to three different extracellular stimuli exhibited a stochastic component in the ISI (Figs. 1B and 2D), and Tav for cells exposed to identical stimuli varied widely between individual cells (Fig. 1D). With this amount of variability, how might Ca2+ spikes encode extracellular stimulus intensities?

In sensory perception, Weber’s law proposes that the ability to detect a small change in stimulus is a constant fraction of the initial stimulus (29). Therefore, we evaluated whether fold changes in the stochastic period of the ISI might encode stimulus intensities in a way that was insensitive to the variability between cells. Such fold changes in ISI would more reliably encode stimulus intensities than absolute values only if the fold changes were less variable than the absolute values. As a starting point, we define the fold change (β) as the change of the stochastic period relative to its initial value:(Tav1Tmin)(Tav2Tmin)Tav1Tmin=β(2)where Tav1 and Tav2 are average ISIs with different stimulus intensities, and Tmin is the same as in Eq. 1 (see Fig. 1C and table S1). Equation 2 implies a linear relationship between ΔTav = Tav1Tav2 and Tav1Tmin. The linear relationship is also predicted by an independently derived stochastic model, which is illustrated in Fig. 3A, and includes only basic properties of IP3R clusters (21) (see Materials and Methods). We refer to this relationship between the change of the stochastic period (ΔTav) relative to its initial value as the “encoding relation,” because as we show later, β reliably encodes changes in stimulus intensity:

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

(A) Relationship between Tav1 and ΔTav 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) Relationship between Tav1 and ΔTav 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) Relationship between Tav1 and ΔTav for hepatocytes stimulated with 0.6 μM and then 1 μM phenylephrine. (E) Pearson’s correlation coefficients (ρ) and explained uncertainties (uex, Eq. 8) for Tav1Tav 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 Tpop [CV(Tpop2)] and Eq. 2 [CV(β)], and the coefficient of variation of the integral ratio [CV(IR)] for the paired stimulation protocols. Codes a to g apply to (E) and (F). (G) Relationship between Tav1 and Tav2 calculated from individual HEK293 cells stimulated first with 30 μM and then with 150 μM CCh [data from (C)]. The dashed line shows the population average (Tpop2) of Tav2. The solid line shows Eq. 2 in the form Tav2 = (1 − β)Tav1 + βTmin. (H) Fold changes (β ± 95% confidence intervals, Eq. 2) calculated from the slopes of Tav1Tav 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).


We tested whether this relation (Eq. 3) applies to Ca2+ spiking using a paired stimulation protocol in which we stimulated HEK293 cells with one concentration of CCh and then switched the medium to one with a higher concentration (Fig. 3B). The amplitude of the Ca2+ spikes remained similar when the concentration of CCh was increased: for the largest step (30 to 200 μM CCh), the peak amplitude was 154 ± 21 nM (n = 23 cells, 292 spikes) for the first stimulation, and 155 ± 18 nM (632 spikes) for the second. Changes in stimulus intensity are not, therefore, encoded by spike amplitudes. Instead, Tav changed with stimulus intensity in accord with Eq. 3, and the stimulation step affected β, where β is the slope of the lines (Fig. 3C and table S2). The smallest step in CCh concentration (from 50 μM to 100 μM) caused only small changes in Tav (Fig. 3C right, red line). The practicable duration of experiments limits the reliability with which we can measure such small ΔTav, because the length of the spike sequences sets the precision with which we can determine Tav. Thus, the linearity of the relationship between ΔTav and Tav1Tmin was less clear for this small stimulation step than for larger steps. Nevertheless, even with such small changes in stimulus intensity, β increased with the size of the concentration step (Fig. 3C right, table S2), and the value of β obtained fits well in the context of the additional concentration-response data (Figs. 3H and 4A). Analysis of hepatocytes sequentially stimulated with 0.6 μM and then 1 μM phenylephrine showed a similar linear relationship between ΔTav and Tav1Tmin (Fig. 3D). We assessed the linearity of the relationship between ΔTav and Tav1Tmin using Pearson’s correlation coefficient (ρ) and analysis of explained uncertainty (uex) (see Materials and Methods) (30). A value of 1 for ρ or uex establishes a perfect linear correlation. These analyses (Fig. 3E) confirmed that the relation between Tav1Tmin and ΔTav is linear and that β increases with the stimulus step in agreement with our theoretical predictions (Fig. 3A).

We assessed whether stimulation steps were more reliably encoded by fold changes (β) in the average stochastic period of the ISI (TavTmin) or by absolute responses (Tav) by quantifying the variability of each (Fig. 3F), using the data shown in Fig. 3 (C and D). To illustrate the methods used, we describe this analysis for HEK293 cells stimulated first with 30 μM and then with 150 μM CCh as an example (Fig. 3G). We plotted Tav1 and Tav2 from each single cell. If absolute responses more reliably encoded the stimulation steps, all values of Tav2 should be similar to the population average (that is, Tav2 = Tpop2, dashed line in Fig. 3G). If fold changes more reliably encoded the stimulation steps, the data should obey Eq. 2, which we rewrite for the purpose of this analysis as Tav2 = (1 − β)Tav1 + βTmin (Fig. 3G, solid line). To determine whether stimulation steps were more reliably encoded by absolute responses or fold changes, we compared the root mean square distances of the data points from these lines. We divided the distance by Tpop2 to obtain the coefficient of variation (CV), which can then be compared across the different experiments (a to g) in Fig. 3F (see Materials and Methods, Eqs. 9 and 10). On the basis of this analysis, we found that the relative deviation of Tav2 from its population average Tpop2 [CV(Tpop2), Eq. 9] was consistently larger than its relative deviation from the encoding relation [CV(β), Eq. 10] (Fig. 3F). Thus, we concluded that β represents individual cell behavior better than does the population average (Tpop), and β encodes stimulation more reliably than does the average ISI.

Because spike amplitudes and durations were unaffected by stimulus intensity, the integral ratio (IR), that is, the ratio of the area beneath the Ca2+ spikes occurring during the stationary phases of responses to the first stimulus relative to that for the second stimulus, is given by Eq. 11 (see Materials and Methods). To determine whether absolute values or fold changes in the integrated Ca2+ signals more reliably encoded differences in stimulation intensity, we compared the CV(Tpop2) with the CV of the integral ratio CV(IR) for HEK293 cells and hepatocytes stimulated with paired steps in CCh or phenylephrine concentration (Fig. 3F). This comparison showed that fold changes of the integrated Ca2+ signal are less variable than is Tav2. Thus, this analysis indicated that fold changes in the average stochastic period of the ISI and fold changes in the integrated Ca2+ signal more reliably encode stimulus changes than does the average ISI (Tav).

Fold changes reliably encode stimulus intensity through an exponential relationship between the stimulus concentration and response

For HEK293 cells exposed to paired steps in CCh concentration, we observed that β depended only on the step size (Δ[CCh]) and not the initial CCh concentration. This is illustrated in Fig. 3H, where the relationship between Δ[CCh] and β was the same whether the first challenge was with 30 μM (red symbols) or 50 μM CCh (blue). We showed in eq. S7 (see text S2: Mathematical derivation of the concentration-response relation) that this observation and Eq. 2 result in the differential equation:d(TavTmin)d[CCh]=γ(TavTmin),  γ=βΔ[CCh]|Δ[CCh]=0(4)

The solution to Eq. 4 is the concentration-response relation, which predicts an exponential dependence of Tav on CCh concentration:Tav=eγ([CCh][CCh]ref)(Tav,refTmin)+Tmin(5)

Tav,ref is the value of the average ISI measured at a reference CCh concentration ([CCh]ref), and γ describes the sensitivity of the stochastic period of Tav to CCh. Cell-to-cell variability appears in Eq. 5 in the variability of Tav,ref, which captures differences between individual cells in the response of the average stochastic period to CCh. Because Eq. 5 applies to average ISIs, it does not conflict with the randomness of individual ISIs. Inserting Eq. 5 into Eq. 2 shows that these results entail an exponential dependence of β on stimulation step Δ[CCh] = [CCh] − [CCh]ref:β=1eγΔ[CCh](6)

Equation 6 describes the measured data well: The relationship between Δ[CCh] and the experimentally determined fold changes (β) fitted to the exponential function using the fit parameter γ confirmed the reliability with which β describes cell behavior (Fig. 3H).

Analysis of the effects of different CCh concentrations on the population average (Tpop) of Tav provided additional support for our suggestion that Eq. 5 appropriately describes the concentration-response relationship. If Eq. 5 correctly describes single-cell behavior, all cells contributing to the population average Tpop must obey the same exponential dependence. Consequently, Tpop is not the sum of exponentials; rather, it obeys a single exponential function:Tpop=eγ([CCh][CCh]ref)(Tpop,refTmin)+Tmin(7)

Therefore, we analyzed the dependence of Tpop derived from analysis of the Ca2+ spikes evoked by different concentrations of CCh in HEK293 cells (Fig. 4A) and found that the relationship was well described by the single exponential function (Eq. 7) with values for γ and Tmin obtained from Fig. 3H and Fig. 3C, respectively. This fit of the experimental data to Eq. 7 is inconsistent with β and γ varying substantially between individual cells.

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

(A) Population average (Tpop) of Tav 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 Tmin = 57 s (the average value of Tmins from the six paired-stimulation experiments shown in Fig. 3C), but with no additional curve fitting. (B and C) Relationship between Tpop and ligand concentration for hepatocytes is exponential. Hepatocytes (31) were stimulated with phenylephrine (B) or vasopressin (C). (D) Relationship between Tpop and ligand concentration for insect salivary gland stimulated with 5-HT (32) is exponential. Lines in (B) to (D) are best fits in parameters Tmin and γ to Eq. 7: for hepatocytes, γ = 1.059 μM−1, Tmin = 61 s (phenylephrine), and γ = 0.279 μM−1, Tmin = 44 s (vasopressin); and for salivary gland, γ = 0.319 nM−1, Tmin = 16 s.

Equation 7 follows from the encoding relation (Eq. 3) and the independence of β from the initial stimulus intensity (Fig. 3H, and see text S2: Mathematical derivation of the concentration-response relation). Conversely, if we had started with this exponential concentration-response relationship (Eq. 7) and found that it accurately described the responses of cell populations, it follows that Eqs. 2, 5, and 6 would apply, and that β would be independent of the initial stimulus intensity. To test if Eq. 7 accurately described experimentally determined concentration-response relations, we used our measurements of Tpop from CCh-stimulated HEK293 cells (Fig. 4A) and published data for hepatocytes (31) and insect salivary glands (32). Data from Rooney et al. (31) provided measurements of the frequency of the Ca2+ spikes evoked by different concentrations of phenylephrine or vasopressin in hepatocytes (Fig. 4, B and C). Data from Rapp and Berridge (32) with blowfly salivary glands provided measurements of the frequency of the Ca2+-regulated changes in transepithelial membrane potential evoked by different concentrations of 5-hydroxytryptamine (5-HT) (Fig. 4D). For the published data, we derived Tpop from the frequency of the spiking responses, and then used least-square nonlinear regression to fit the relationships between stimulus concentration and Tpop (Fig. 4, B to D), with both Tmin and γ as fitting parameters (the values are provided in the legend of Fig. 4). In each case, the effects of stimulus intensity on Tpop were well described by Eq. 7: The concentration-response relation was exponential, the response to maximal stimulation was the sum of spike duration and refractory period (Tmin, Fig. 1C), and stimulation controlled the average stochastic period of the ISI (TavTmin).


All biological systems are variable (18, 3335). Heterogeneity between cells can have beneficial effects, such as contributing to adaptability (36), increasing the range of sensitivities to extracellular stimuli (1), and providing robustness (37). However, heterogeneity may also distort relationships between stimulus and response. We found that the frequency of Ca2+ spikes in individual cells varied with stimulus intensity, but that no consistent relationship applied to all cells in the population (Fig. 1D). We identified two general features of receptor-regulated Ca2+ spiking that enable effective encoding of stimulus intensity.

First, Ca2+ spikes occurred with a reliable signal-to-noise ratio. Our observation that the slope (α) of the Tav-σ relation was constant between cells exposed to the same type of stimulus formed the basis for this conclusion (Fig. 2E and fig. S3B). Furthermore, α determines the quality of signal transmission (19): A small α enables more information to be transmitted by spike sequences. We showed that the value of α was unaffected by stimulus intensity (fig. S3B), as predicted earlier by mathematical modeling (21), nor was α affected by perturbations of the signaling pathway, like inhibition of PLC, sensitization of IP3Rs, or inhibition of Ca2+ uptake by the ER (Fig. 2), that do not affect recovery from global negative feedback. The value of α, and thus the signal-to-noise ratio, is predicted to depend on the time scale of recovery from global negative feedback (19, 21), which our results suggested is likely to operate close to GPCRs, because α was different for different extracellular stimuli (Fig. 1E).

Second, changes in extracellular stimulus intensity (ligand concentration) were encoded by fold changes in the average stochastic period of the ISI (TavTmin), rather than by Tav itself. The fold change (β) was similar for each cell, but it varied with stimulus step size, and β correlated better with stimulus intensity than did Tpop (Fig. 3F). Many effectors, including transcription factors, integrate Ca2+ signals (1, 3, 17, 38). Fold changes in the stochastic period of the ISI lead to integrated Ca2+ signals that also occur as approximate fold changes (IR in Fig. 3F). Thus, downstream effectors that respond to integrated Ca2+ signals will also benefit from the reliability of β (3, 4, 38). Our observation that stimulus intensities are encoded by fold changes in Ca2+ signals may provide the explanation for downstream cellular activities responding with fold changes (3, 4, 38).

Our findings resulted in a fundamental equation (Eq. 4), which can be applied to all signals that generate a fold change in response. The solution to this equation for Ca2+ signaling (Eqs. 5 and 7) defines an exponential dependence of Tpop on stimulus intensity, with individual cells having similar exponents (γ). Our analysis of three different cell types responding to four different ligands suggested that this relationship is generally applicable to GPCR-stimulated PLC signaling pathways (Fig. 4). Other signaling pathways also exhibit fold changes in responses after stimulation (14): epidermal growth factor–mediated regulation of extracellular signal–regulated kinase 2 (ERK2) has been reported to produce fold changes in the peak of nuclear ERK2 (39), Wnt signaling produces fold changes in the abundance of the transcriptional regulator β-catenin (40), and tumor necrosis factor stimulates fold changes in the nuclear abundance of the transcription factor nuclear factor κB (NF-κB) (41). We suggest, therefore, that measuring the fold changes according to Eq. 2 for nuclear ERK2, β-catenin, or nuclear NF-κB and solving the corresponding Eq. 4 would provide the concentration-response relations for these systems (eq. S8).

We determined that, for Ca2+ spiking, temporal randomness, cell-to-cell variability, and fold changes are unified by an exponential concentration-response relationship that applies to individual cells and the population average (Eqs. 5 and 7). The same features—temporal randomness (42), cell variability (39, 42), and fold changes (39)—have been reported for ERK2 activity, but without a mathematical analysis leading to the concentration-response relation. These analogies between very different signaling systems suggest that we have identified a signaling concept that extends beyond Ca2+ spiking.

We conclude that despite variability between cells and in the ISIs within cells, there are generic features of Ca2+ spikes that enable repetitive spikes to encode stimulus intensities. Slow recovery from global feedback provides spike sequences with reliable signal-to-noise ratios. Changes in stimulus intensity are encoded by fold changes (β) in the average stochastic period of the ISI, which generate exponential concentration-response relationships defined by the stimulus sensitivity (γ).



Cell culture media, fura-2 AM, and fluo-4 AM were from Invitrogen. Human PTH (residues 1 to 34) was from Bachem. Other chemicals, including Arg8-vasopressin, were from Sigma-Aldrich.

Single-cell imaging of [Ca2+]i in HEK293 cells

HEK293 cells stably transfected with human type 1 PTH receptor (26) were cultured in Dulbecco’s modified Eagle’s medium/Ham’s F-12 supplemented with l-glutamine (2 mM), fetal calf serum (10%), and G-418 (800 μg/ml). For imaging experiments, the cells were plated onto 22-mm round glass coverslips coated with 0.01% poly-l-lysine. Cells were loaded with fura-2 AM (2 μM) for 45 min at 20°C in Hepes-buffered saline (HBS), washed, and imaged after a further 45 min. HBS had the following composition: 135 mM NaCl, 5.9 mM KCl, 1.2 mM MgCl2, 1.5 mM CaCl2, 11.6 mM Hepes, 11.5 mM glucose, pH 7.3. Single-cell fluorescence measurements were performed at 20°C in HBS as previously described (26). Fura-2 fluorescence ratios were collected at 5-s intervals and calibrated to [Ca2+]i after correction for background fluorescence determined after addition of MnCl2 (1 mM) and ionomycin (1 μM) to quench fura-2 fluorescence.

Single-cell imaging of [Ca2+]i in hepatocytes

Hepatocytes were isolated by collagenase perfusion of livers from adult male Sprague-Dawley rats and used on the same day for Ca2+ imaging measurements (5, 31). Cells were plated on poly-l-lysine–coated coverslips for 30 min in HBS solution (HBSS). HBSS comprised 121 mM NaCl, 5 mM NaHCO3, 4.7 mM KCl, 1.2 mM KH2PO4, 2 mM CaCl2, 1.2 mM MgSO4, 10 mM glucose, 100 μM sulfobromophthalein, 0.25% (w/v) fatty acid–free bovine serum albumin, 25 mM Hepes, pH 7.4, at 37°C. Cells were loaded with fura-2 AM (8 μM) for 30 min in the presence of 0.02% (v/v) Pluronic F-127 and immediately transferred to an imaging chamber at 37°C. Pairs of fura-2 fluorescence images using excitation at 340 and 380 nm were collected at 3-s intervals with a cooled charge-coupled device camera and corrected for background fluorescence by release of cytosolic fura-2 using digitonin (25 μg/ml) before calculation of fluorescence ratios for individual cells.

Analysis of Ca2+ spikes

In some sequences of Ca2+ spikes, the stationary phase was preceded by an initial transient in the ISI (see Fig. 3B for an example). Only the stationary components of spike sequences were used for analyses of ISI. In some recordings, there was a linear trend through the entire ISI sequence (see top panels of Fig. 1B). Such a trend, which may reflect slow changes in cellular sensitivity, is not the result of a stochastic process, but it would exaggerate the randomness of ISIs (increase σ) if it were not removed. Therefore, when calculating σ, these trends were removed by subtracting a linear fit to the trend. All other analyses (for example, calculation of Tav) were performed without this correction.

After normalization using a running average, spikes were identified as an increase in F340/F380 fluorescence of at least 20% from the basal fluorescence ratio. Because long stationary sequences are required to determine σ reliably, only cells in which at least 12 spikes were recorded were used to determine Tav-σ relations. Within the text, Tav refers to the average ISI recorded from a single cell (SD, σ), and Tpop is the average of the Tavs from a population of identically stimulated cells (see table S1 for definitions and abbreviations). Where only averages of ISIs were analyzed, stationary trains with fewer Ca2+ spikes (>7) were accepted. Table S2 summarizes the properties of the data used.

Paired Student’s t tests were used to compare Tav values from cells exposed to sequential stimuli. F tests were used to compare the slopes of Tav-σ relations. Statistical tests used Prism (version 5, GraphPad Software); P < 0.05 was considered significant.

Calculation of the encoding relation

The building blocks of intracellular Ca2+ signals are Ca2+ puffs. These are brief, local increases in [Ca2+]i that result from the coordinated opening of a few clustered IP3Rs (43). Ca2+ spikes occur when the probability of initiating a Ca2+ puff increases, allowing regenerative propagation of a Ca2+ wave across the cell. In previous publications, we developed a stochastic model for IP3-evoked Ca2+ signals in terms of the distributions of interpuff intervals and puff durations (20, 21), which we use here to calculate the Tav values entering the encoding relations in Fig. 3A in the same way as described in (20, 21). The distributions of both the interpuff intervals and puff durations are generalized exponential functions (see eq. S1 with Tmin = 0 for this type of distribution). In the model, each cell has identical clusters of IP3Rs, the parameters that determine the distribution of puff durations are fixed, and the values of the parameters of the interpuff interval distributions specify individual cells. The dependence of the interpuff interval distribution parameters on IP3 concentration was calculated from the DeYoung-Keizer model (21). The parameter ν (see eq. S1) consists of a constant and an additive part that depends on IP3 concentration. Both have values typically in the range from 1.0 to ~4.0. We captured cell-to-cell variability by varying the constant part in a way that provided ν values from 1 to 3.85 at the first stimulation level. The range of values of Tav1 in Fig. 3A arises from this cell-to-cell variability. Stimulation steps were modeled by increasing the IP3 concentration from 0.5 μM by the steps specified in Fig. 3A. Average ISIs were calculated as the average first passage time for the transition from 0 to 4 open clusters. All other parameter values are given in (21).

Statistical analysis of the encoding relation

The encoding relation (Eq. 3) predicts a linear relationship between individual Tav1 and Tav2 values. The linear fit was assessed in two ways. Pearson’s correlation coefficient (ρ) measures how close data points are to a straight line, whereas the explained uncertainty (uex) evaluates how much of the variance in the experimental data can be explained by fitting to a model like Eq. 3. In a regression model, it is generally true that (yiy¯)2=(y^iy¯)2+(y^iy^i)2 with the mean y¯=1Ni=1Nyi and the predicted value y^i=axi+b. The explained uncertainty uex is defined as the fraction of the total variance that remains if the regression error is subtracted (31):uex=(yiy¯)2(yiy^i)2(yiy¯)2(8)

The explained uncertainty thereby reports the extent to which a model adequately describes the experimental variation. For ρ and uex, a value of 1 indicates a perfect linear correlation between values. We used both ρ and uex to assess the linearity of the Tav1Tav relations for HEK293 cells and hepatocytes exposed to steps in stimulus concentration using our paired stimulus paradigm.

Our analysis of whether absolute responses (Tav) or fold changes (β) provided the most reliable encoding of stimuli was performed for the fold change β = ΔTav/(Tav1Tmin) (Eq. 3). The analysis is equally valid for the fold change of stochastic periods (Tav2Tmin)/(Tav1Tmin), which is equal to 1 − β (see Eq. 2). We compared the coefficients of variation CV(Tpop2) and CV(β):CV(Tpop2)=1Ncell1j=1Ncell(Tav2(j)Tpop2)2Tpop2(9)

Tav2(j) is the average ISI for an individual cell (j) responding to a second stimulus. Tpop2 is the average Tav2 from a population of Ncell cells.CV(β)=1Ncell1j=1Ncell[(1β)Tav1(j)Tav2(j)+βTmin]222β+β2Tpop2(10)

Tav1(j) and Tav2(j) are the average ISIs for an individual cell (j) responding to a first and second stimulus, respectively. Tmin is defined in Eq. 1. β is determined by ΔTav/(Tav1Tmin) (see Eqs. 2 and 3). The expression in the numerator of Eq. 9 is the root mean square orthogonal distance of the data points from the line Tav2 = Tpop2, and the numerator in Eq. 10 is the root mean square orthogonal distance of the data points from the line Tav2 = (1 − β)Tav1 + βTmin.

In general, the integrated Ca2+ signal comprises the integral over the initial transient and over the stationary part of the spike sequence. Because transients vary between cells and with stimulation conditions, no general statement about their integral is possible. For paired stimuli, the ratio of the integrated stationary signal approximately equals the ratio of Ca2+ spike frequencies, because spike amplitudes were unaffected by stimulus intensity, and we did not detect any obvious change in spike duration. The IR of the stationary parts of the spike trains is thereforeIR=f2f1=Tav1Tav2=11β(1TminTav1)(11)

Mathematical descriptions of ISI distributions are provided in Supplementary Materials.


Text S1. Mathematical description of ISI distributions.

Text S2. Mathematical derivation of the concentration-response relation.

Fig. S1. ISI distributions for hepatocytes.

Fig. S2. Tav-σ relation.

Fig. S3. Responses of individual HEK293 cells to stimulation.

Table S1. Symbols and abbreviations.

Table S2. Selection of data for paired stimulation experiments.


Funding: Supported by grants from the Wellcome Trust (101844) and Biotechnology and Biological Sciences Research Council (BB/H009736) to C.W.T., DFG (GRK 1772) to G.M., EMBO ASTF (Application Short-Term Fellowship) 398-09 to K.T., and Infusino Endowment to A.P.T. Author contributions: M.F., C.W.T., and A.S. designed the study. K.T., S.C.T., V.L.P., and A.M. collected experimental data and analyzed the data with C.W.T. and A.P.T. Mathematical analyses were performed by K.T., G.M., and A.S. and were supervised by M.F. The paper was written by C.W.T. and M.F. with input from all other authors. Competing interests: The authors declare that they have no competing interests.
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