In order to study some of the dynamic properties of the chloride channels we have recorded calcium-activated chloride currents in Xenopus laevis oocytes, which have been evoked by serum under different external pH stimuli (pH = 0.5, pH = 0.7 and pH = 0.9). Thus, we had 21 time series in total, each one of them formed by 130,000 discrete data points. Figure 1 shows three representative experimental signals obtained by means of the patch-clamp technique, under three different pH conditions, Ringer’s solution at pH 5.0, 7.0 and 9.0 (acid, neutral and basic pH).

Figure 1: Calcium-activated chloride currents in Xenopus laevis oocyte. Three prototype experimental Cl− currents obtained from the same cell at different conditions: (a) pH 5.0 (n10), (b) pH 7.0 (n11), (c) pH 9.0 (n12). Each chloride time series has 130,000 points (sampling interval 2 milliseconds), which correspond to a period of time of 260,000 milliseconds duration. The vertical axis (Φ) corresponds to the measures of currents in nanoampers (nA). Full size image

To confirm that oscillations monitored in Xenopus oocytes by application of Fetal Bovine Serum corresponded with Ca2+-dependent Cl− currents, three different experiments were performed. First, oocytes generating oscillations were voltage-clamped at 4 different voltages (either −60, −40, −20 or at 0 mV). As it is illustrated in Fig. 2a, currents reversed near to −20 mV, in accordance with the reversal potential of Cl− in oocytes. Second, the reversal potential observed was shifted toward more positive potentials when the external Cl− concentration was reduced, this is shown in Fig. 2b. In this case, oocytes were held to either −30 mV (first column) or 0 mV (second column), while they were superfused with solutions containing 100%, 50% or 0% of Cl− (NaCl was substituted proportionally by Na2SO4 in Ringer solution and, osmolarity compensated adding sucrose). It is clear that reversal potential is close to −30 mV in 100% Cl−, while in 0% Cl− oscillations continued being in inward direction at 0 mV, indicating that reversal potential in this condition is more positive. An intermediate case occurs with 50% Cl− solution, where the shift in reversal potential by reducing external Cl− is predicted by the Nernst equation. And finally, it was demonstrated that Cl− currents were Ca2+-dependent. Intraoocyte injection of the calcium chelator ethylene glycol-bis(2-aminoethylether)N,N,N’,N’,-tetraacetic acid (EGTA) abolished completely oscillatory currents, according to Ca2+-dependent Cl− currents.

Figure 2: Ca2+-dependent Cl− current validation. (a) Xenopus oocyte held at either −60, −40, −20 or 0 mV. Reversal potential of oscillatory currents corresponded to a value close to −23 mV. (b) Oscillatory current reversal potential were dependent on external Cl− concentration, traces show currents in oocytes held at −30 mV or 0 mV in 3 different solutions containing 100%, 50% or 0% Cl−, reversal potential shifted toward more positive potentials as external Cl− concentration decreased. (c) Cytoplasmic injection of EGTA, a Ca2+ chelator, completely eliminated the oscillatory Cl− current. Full size image

First, to test for the presence of long-term correlations in the experimental chloride data we have used the root-mean square (rms) fluctuation F(l). For uncorrelated data, the exponent α for the relationship F(l) ~ lα is equal to 0.5; in contrast α > 0.5 indicates the presence of positive long-range correlations and α < 0.5 implies long-term anti-correlations. According to this method, we have divided the 130,000 data points of each time series in 6 non-overlapping windows with k = 5, performing the rms fluctuation method on every window for each of the 21 experimental chloride series and fitting F(l) within the range l = 1, …, l max (see Methods for more details). The values of l max were systematically increased in 100 points, which correspond to 1 second, and the reliability of the rms correlation exponent α was calculated by means of the R2 parameter, which measures the goodness fit (also called the coefficient of determination).

Second, in order to discern whether the experimental Cl− currents exhibit non-trivial correlations, we have fixed a threshold criterion of R2 ≥ 0.99. The obtained α values were calculated for every window on each time series, and the results ranged between 0.75 and 1, being 0.927 ± 0.048 (mean ± SD) the global mean of all the experimental chloride series. These non-trivial correlations encompassed between 1,500 and 6,500 evoked chloride values (mean of 3,809.5 ± 1,298.8), which correspond to periods of time ranging between 3 and 13 seconds (mean of 7.66 ± 2.6). Boundary times where achieved on the series n17 (experiment 6, pH = 7.0) and n2 (experiment 1, pH = 7.0) respectively. The mean rms correlation coefficients (α), as well as the number of evoked chloride values under the non-trivial correlation regimen (N), with their respective correlation times (T c ) for all the experimental series are given in Table 1. Figure 3 shows an example of rms fluctuation analysis applied to three calcium-activated chloride responses of the same oocyte (n1, n2 and n3 time series belonging to the experiment 1) for their T c times on a single window. In all three cases, the obtained α values were significantly different to 0.5, and for at least 10, 13 and 12 seconds respectively, the evoked chloride dynamics presented non-trivial long-term correlations. Alternatively, long term correlations were also observed by calculating the autocorrelation function from the time series (Supplementary Information).

Table 1 The first column shows the number of the experiment, each one corresponding to a single oocyte. Full size table

Figure 3: Root mean square fluctuation analysis applied to experiment 1 on a single window. Log-log plot of the rms fluctuation F versus l step. The red points depict the results of the original data for each value of l, while the black lines represent the regression lines. (a) α = 0.88 (n1), (b) α = 0.92 (n2) and (c) α = 0.83 (n3). Corresponding (respectively) R2 adjustment coefficients were 0.9915, 0.9921 and 0.9976. The high values of α and R2 indicate non-trivial long-term correlations for each chloride time series during 10, 13 and 12 seconds respectively. Full size image

Next, we have studied the long-range correlations for α ≥ 0.6. The analysis showed exponents ranging between 0.6008 and 0.9718, which respectively correspond to the time series n1 (pH = 5.0, l max = 2,200) and n17 (pH = 7.0, l max = 1,200). The global average was 0.774 ± 0.108. All the means of α values, R2 adjustments, and the l max are given in Table 2. It can be observed that the values of α decrease slowly as l max increases. This behavior is illustrated in Fig. 4a, where the average for the 21 time series, as a function of l max , are represented; all the corresponding values of the Fig. 4 are displayed on Table 3.

Table 2 The first column shows the number of the experiment, each one corresponding to a single oocyte. Full size table

Figure 4: Long-term correlations across different windows lengths. (a) Global average versus different values of l max (varying from 1 to 24 seconds). (b) as a function of l max (varying from 25 to 40 seconds). The error bars represent the standard deviation at each step. It can be observed that all Cl− time series change from positive to negative correlation near l max = 28 seconds. Full size image

Table 3 The first and third columns represent different values of l max , ranging from 1 to 40 seconds. Full size table

In addition, we have observed a critical transition around l max = 28 seconds, where the behavior of the Cl− currents changes from positive to negative correlations (Fig. 4b). It can be observed that as l max increases, all the α exponent values decreased, and for the maximum window length (l max = 40, corresponding to 20,000 time points), the α values were lower than 0.5 ( = −0.051 ± 0.283) indicating anti-correlations in all cases; concretely, α values ranged between −0.885 and 0.349, which belong to n2 (experiment1, pH = 7.0) and n7 time series (experiment 3, pH = 5.0) respectively.

Finally, we performed a rms fluctuation analysis without the separation of the data in shorter windows, thus considering all the points for each experimental time series, observing anti-correlations for all the cases ( = −0.01 ± 0.1).

Moreover, we have examined whether the chloride currents are described by a fractional Gaussian noise (fGn) or a fractional Brownian motion (fBm) by calculating the slope of the Power Spectral Density plot41. The signal exhibits power law scaling if the relationship between its Fourier spectrum and the frequency is approximated asymptotically by S(f) ≈ S(f 0 )/fβ, where S(f 0 ) and β are constant values. If −1 < β < 1 the signal corresponds to an fGn. In particular, when β = 0, the power spectrum is flat, as is the case for white noise in which the time series is composed of a sequence of independent random values. If 1 < β < 3 the signal corresponds to a fBm. The analysis of the Power Spectral Density plot revealed that the experimental series are characterized by a power-law scaling with β ranging within 1.507 and 2.991, which suggests that all the series are described by fBm (β values are given in Table 4).

Table 4 The first column shows the number of the experiment, each one corresponding to a single oocyte. Full size table

Additionally, an analysis of the classical descriptive statistics of the experimental data has been included in the Supplementary Information).

Next, we have checked whether the chloride time series show persistent or anti-persistent long-term memory by calculating the Hurst exponent. Although several tools exist for estimating the long-term memory from fBm time series, one of the most reliable methods is the bridge detrended Scaled Windowed Variance analysis (bdSWV) (see Methods for more details). After bdSWV analysis, the resulting Hurst exponents had a mean value of 0.191 ± 0.101, implying long-range memory and an anti-persistence effect in all the experimental data sets (Table 4). In addition, an ANOVA test revealed that Hurst exponent values were significantly different for time series corresponding to pH = 9.0 in comparison to pH = 7.0 (p-value = 10−5) and pH = 5.0 (p-value = 10−4), but no significant distinction was found between pH = 7.0 and pH = 5.0 (p-value = 0.42). Notice that the obtained values of H are very low, showing a high degree of anti-persistence (strong trend-reversing), so that an increasing trend in the experimental data values will tend markedly to be followed by a decreasing trend, or a decrease on average will be followed by a robustly increasing trend.

In order to estimate the significance of our results, we have performed a shuffling procedure that defines the null-hypothesis. If the original time series exhibits a memory structure (H ≠ 0.5), after the shuffling such structure will disappear, thus re-applying a new Hurst analysis on the shuffled data should provide values of H close to 0.5. According to this procedure, for each experimental time series (21 in total), we performed a thousand random permutations, which allowed building the null-hypothesis of no correlations. In total, we generated 21,000 random series from the original data belonging to the seven experiments with Xenopus laevis oocytes. After shuffling, the results show a mean Hurst exponent of 0.499 ± 0.01, indicating the absence of long-term memory i.e., the informational memory structures in all shuffled series was completely lost. Notice that after shuffling, the series became Gaussian white noise (fGn series with , and for this case the use of bdSWV is not justified. Instead, Dispersion Analysis is the most recommendable tool for this kind of series41,42 (for more details see Methods).

Figure 5a illustrates the regression lines of a bdSWV process applied to an example of experimental series giving H = 0.104 (experiment 5, n13, pH = 5.0), which indicates a strong anti-persistent memory. After randomly permuting all the 130,000 points contained in this time series n13, the Dispersion Analysis gave H = 0.492, which indicates a breakdown for the long-term memory (Fig. 5b). In Fig. 5c, we represent 100 Hurst exponent values corresponding to 100 shuffled series, obtained from shuffling the experimental data. It can be observed that, after shuffling, the long-term memory disappears completely in all the time series ( = 0.498 ± 0.01). For illustration purposes, Fig. 5c shows, rather than the 21,000 obtained values of Hurst exponent, only 100 of them. The informational memory structures in all shuffled series were completely broken-down, and therefore, the memory structure that characterizes the experimental data could not be found by chance. Finally, in order to calculate the values of Hurst exponent from short data periods, we used the Detrended Fluctuation Analysis (DFA), because the bdSWV is recommended for data sizes greater than 212, whilst for data sets with less than 28 points bdSWV has been shown to be unreliable43. The DFA analysis showed that for time periods ranging between 2 and 5 seconds all the experimental time series exhibit persistent behavior with H > 0.5 being the global mean of = 0.697 ± 0.11, which indicates that the properties of persistent memory dominate at short time intervals of the calcium-activated chloride currents in Xenopus laevis oocytes.