Non-stationary noise analysis
A brief discussion of the application of non-stationary noise to these data will be given here. For more complete discussions see previous publications (Sigworth, 1980; Heinemann and Conti, 1992; Silberberg and Magleby, 1993; Lingle, 2006). It is assumed that there are N identical receptors, which are simple two-state channels – either closed (zero conductance) or open to a single level (conductance g). If each receptor has probability P of being open [and so (1 −P) of being closed], then the predictions of the binomial distribution are that the mean current is
where i is the single channel current, i=gΔV and ΔV is the driving force on ion movement through the channel.
The variance of the current is
The variance is zero when P equals either 0 or 1, and reaches a maximum at P= 0.5. At very low levels of activation, when P << 1, the variance is the product of the single channel current (i) times the mean current (I) so the ratio of the variance to the mean provides an estimate of i. When P is increased (i.e. at higher concentrations of agonist) a plot of the variance against the mean is parabolic (Sigworth, 1980), as described by
In principle therefore it is possible to estimate the single channel current (and if ΔV is known the single channel conductance) and the number of channels present in the cell. However, there are several factors that can make the experimental results or the interpretation less than perfect. One is technical: is the bandwidth of the recording sufficient to capture all (or the majority) of the variance? Others are concerned with the assumption that binomial statistics can be used. Is it reasonable that this receptor is a two-state channel (1 open and 1 closed state)?
The technical issue has been addressed experimentally (see below) and is not a significant problem. The assumption of a two-state channel is clearly more problematic. First, it is known that the nicotinic α4β2 receptor shows more than one conductance level (Buisson et al., 1996; Kuryatov et al., 1997; Buisson and Bertrand, 2001; Curtis et al., 2002; Nelson et al., 2003), and that the channel kinetics are more complex than a simple two-state scheme (Buisson et al., 1996). At least one source of complexity lies in the fact that there are at least two forms of the receptor, depending on subunit stoichiometry (α43β22 and α42β23; Nelson et al., 2003; Moroni et al., 2006). Previous studies have not clearly defined the properties of the two stoichiometric forms. However, it is known that the α43β22 (‘low-sensitivity’) form has an EC50 (concentration producing half-maximal response) for activation by ACh of ∼100 µM whereas the α42β23 (‘high-sensitivity’) form has an EC50 of ∼1 µM (Nelson et al., 2003; Moroni et al., 2006).
The results from non-stationary noise analysis of macroscopic currents provide a weighted average of the functional parameters for the two forms and kinetic complexities can affect the shape of the variance–mean relationship. Accordingly, the data are a first step towards obtaining a more quantitative understanding of the functional properties of this receptor.
Responses to low concentrations of agonists
Typical responses to the application of 1 µM ACh are shown in Figure 1A. In this cell, 10 sequential applications of 1 µM ACh were made, at 30 s intervals. The mean currents were relatively stable, although there was a small amount of desensitization apparent in each response. These responses were fit with single exponentials declining to a constant and the variance from the fit lines calculated as described in the Methods. The resulting variance–mean relationship is shown in Figure 1B. The different responses gave quite similar median values for the variance and the mean. For ease of comparison with values reported in the literature, the ratio of the variance to the mean (the estimate of i) has been converted to an estimate for the mean single channel conductance (g) by dividing by the holding potential (−60 mV); it was assumed that under these ionic conditions the reversal potential is 0 mV. The median estimates for g for these 10 responses ranged from 19 to 23 pS (on average 21 ± 1 pS; mean ± SD). In three other cells to which repeated applications were made the medians were 23 ± 2 pS (8 applications), 31 ± 4 pS (8) and 29 ± 3 pS (8). Accordingly, the values obtained can be repeated for a given cell.
Figure 1. Responses of a cell to 1 µM ACh. (A) A cell was exposed to 10 applications of 1 µM ACh. Two responses are shown (the second in red, the ninth in blue). Superimposed on the data are fitted single exponential curves declining to a constant (yellow lines). (B) Plots of the variance against the (fit) mean current are shown for 10 responses for this cell. Data for the records shown in (A) are in larger symbols, coloured to match the traces in (A). The thin lines show slopes given by the median average single channel current for individual responses, while the thick line shows the slope given by the mean of the values for the 10 individual responses. Each point corresponds to the mean for 1628 samples over a ∼163 ms time period. Holding potential −60 mV.
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It is unlikely that the filtering (2 kHz low pass) reduced the estimate of the variance. One test was to acquire data at 20 kHz sampling, after analogue filtering at 10 kHz. The digitized records were then either analysed with no further filtering or filtered at 2 kHz low pass using the Gaussian filter function in pClamp. For four cells this was done at concentrations of 0.3 and 1 µM ACh. The ratio of the subtracted variances for the original (10 kHz) data to that for data refiltered at 2 kHz was 1.04 ± 0.13 (mean ± SD; 8 pairs compared), indicating that there was no systematic reduction in the variance with the normal filtering. In contrast, the baseline variance was much greater for the records filtered at 10 kHz, on average 13 ± 8 times the baseline variance filtered at 2 kHz. In the case of responses to low concentrations it is also possible to estimate a total variance by analysis of stationary noise. In three cells long (50 s) applications of 3 µM ACh were made, and the last half of the record (after the initial decline in current) was analysed. The difference spectra after subtraction of baseline spectra were fit by either the sum of 2 Lorentzians (2 cells) or 3 Lorentzians (1 cell). The half-power frequencies corresponded to time constants of about 2 ms, 10 to 20 ms and, when three components were fit, an additional component of about 70 ms. The variance in each component was calculated from the relationship σ2=S(0)fcπ/2, where S(0) is the power at zero frequency and fc is the half-power frequency. The total variance is the sum of the variance in each component. Because the fitted values were used, the filter did not affect these estimates. For the three cells, the spectral estimates of variance were 13, 52 and 23 pA2, while the median estimates from the non-stationary analysis were 13, 50 and 22 pA2 respectively. The finding that the spectral and median estimates were similar suggests that the bandwidth was appropriate. A full analysis of power spectra was not undertaken for these experiments.
In almost all cases data were obtained at a holding potential of −60 mV. In four cells responses to 1 µM ACh were recorded at multiple holding potentials (−60, −80, −100 and −120 mV). The slope of a line fit to the estimated i versus V gave estimates for the single channel conductance which were, on average, about 1.5 times the chord conductance calculated assuming a reversal potential of 0 mV; for the data at −60 mV the value was 1.5 ± 0.3. The reversal potential calculated from the fit was −26 ± 5 mV. This may result from the previously reported inward rectification of these receptor currents (Buisson et al., 1996; Sabey et al., 1999). The data suggest that the calculated values for g (the chord conductance at −60 mV assuming a reversal potential of 0 mV) were underestimated.
The variance and mean currents (obtained at −60 mV) were analysed for responses to 0.3 or 1 µM ACh for a total of 159 responses from 72 cells. The mean calculated value for g was 18 ± 6 pS (72 cells). The distribution of values for g is shown in Figure 2 and is well described by a Gaussian function, with no indication of multiple modes in the distribution.
Figure 2. Single channel conductance for receptors activated by low concentrations of ACh. The values for the average single channel conductance of responses elicited by 0.3 or 1 µM ACh are shown, with a superimposed Gaussian curve generated from the mean and SD of the data (18.2 ± 6 pS, n= 72).
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We also examined the responses to low concentrations of nicotine, cytisine and 5-I A-85380. Cytisine was chosen because it has been reported to be relatively selective for the low-sensitivity form of this receptor, α43β22 (Moroni et al., 2006; Mineur et al., 2009). 5-I A-85380 has been reported to have a much higher potency for the high-sensitivity, α42β23 form (Zwart et al., 2006). ACh and nicotine are expected to activate a mixture of receptor types, although when the response is a small fraction of the maximal response it might be expected that the high-sensitivity form would be preferentially activated.
The estimates for g were similar for ACh, nicotine (100 or 300 nM; 17 ± 5 pS; 22 cells) and 5-I A-85380 (10 or 100 nM; 17 ± 2 pS; 8 cells). The estimated g was significantly higher for cytisine (1 or 3 µM; 24 ± 4 pS; 10 cells; P= 0.012 by anova with Bonferroni correction). We also compared the agonists on the same cell, relative to ACh. In these paired comparisons, the relative estimated g for nicotine and 5-I A-85380 was close to 1 (1.1 ± 0.1 for nicotine; n= 5 comparisons; 1.0 ± 0.1 for 5-I A-85380; n= 5 comparisons). Again, the estimate for cytisine was significantly larger than that for ACh (1.3 ± 0.3; n= 10 comparisons, P= 0.007).
We examined the effects of two potentiating drugs, physostigmine (Sabey et al., 1999; Smulders et al., 2005) and 17β-oestradiol (Paradiso et al., 2001; Curtis et al., 2002). The mean current and the variance were obtained for responses to 0.3 µM ACh in the absence and presence of 15 µM physostigmine or 10 µM 17β-oestradiol. As shown in Figure 3 the potentiators increased the mean current relative to the control response of the cell, but did not increase the estimated g. Both these results indicate that the potentiators do not act by changing the average size of the currents through open channels, and provide an additional pharmacological test that the currents originate from nicotinic α4β2 receptors. Both physostigmine and 17β-oestradiol inhibited responses at higher concentrations; depending on the mechanism and kinetics of the inhibition mechanism this might increase or reduce the measured variance.
Figure 3. The values of the single channel conductance in the presence of a potentiator relative to that in the same cell, in the absence of potentiator. The potentiators increased the mean response (relative current) but not the single channel conductance (relative conductance). The responses were elicited by 0.3 µM ACh alone or with coapplication of 15 µM physostigmine or 10 µM 17β-oestradiol, and the peak response and single channel conductances were estimated. The regression coefficients for relative conductance on relative current were 0.29 for 17β-oestradiol and −0.28 for physostigmine. Neither coefficient differed from 0 (P > 0.05).
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Responses to high concentrations of agonists
Responses to low concentrations of agonists were analysed in terms of the ratio of the variance to the mean, and provided estimates of the single channel current.
We then examined the relationship between the variance and the mean as a function of increasing agonist concentration to determine the probability that the receptor is active at a maximal concentration of agonist. Figure 4A shows responses of a cell to applications of ACh, covering the range from 0.3 to 100 µM ACh. The mean was estimated for these responses by fitting an exponential declining to a constant value (see Methods), and the variance estimated from the deviations from the fit value. Figure 4B shows a plot of the variance against the fit, binned into 21 equal duration bins for each response. The variance initially increased as the response increased, but then declined at the largest responses. The line in Figure 4B shows a fit of a parabola to the data, providing estimates of −0.93 pA for the single channel current (15 pS) and 1051 for the number of agonist-activatable receptors on the cell.
Figure 4. The decrease in variance with responses to high concentrations of ACh. (A) The responses of a cell to six applications of ACh (2 s duration at 30 s intervals) are shown, with fitted single exponential curves declining to a constant superimposed. (B) A plot of the variance against the (fit) mean is shown, together with the fit parabola (parameter values of i=−0.93 pA and 1051 active receptors). The yellow diamonds show the mean and variance for brief segments of data at the peak of three responses to 300 µM ACh; these values were not used in fitting the parabola (see text). Each point (except those at 300 µM ACh) shows the mean of 543 samples over ∼54 ms. Holding potential −60 mV.
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In general, data for ACh concentrations less than 300 µM provided similarly shaped plots of the variance against the mean. However, the response to 1 mM ACh often did not agree well with parabolas fit to the data at lower concentrations. The variance often showed little change as the response declined during the application, and could lie below the parabola. The reason for this divergence at a high ACh concentration is not known, but could reflect the channel-blocking activity of ACh (Sine and Steinbach, 1984; Maconochie and Steinbach, 1995; Paradiso and Steinbach, 2003). However, not all cells showed this difference, so a full explanation must await further study. In any case, parabolas were only fit to ACh data obtained with 300 µM or lower concentrations.
Possible artefacts introduced by low-pass filtering might contribute more heavily to responses obtained with higher concentrations of ACh, because the relaxations would be expected to be more rapid. Accordingly, the set of data obtained with higher bandwidth (described earlier) was analysed for concentrations of ACh ranging from 0.3 µM to 300 µM. The baseline variance increased by 12-fold (±6-fold), but the additional variance during the response increased by only 1.1-fold (±0.1-fold) when the 10 kHz bandwidth was compared with that at 2 kHz. More importantly, the parameters for fit parabolas were not different (ratios of fit single channel current were 1.01 ± 0.01 and numbers of receptors were 1.01 ± 0.01). Accordingly it is unlikely that low-pass filtering significantly affected the shape of the variance–mean relationship.
The average values for the parameters of fit parabolas were 18 ± 5 pS and 698 ± 446 activated receptors (n= 55 cells). The estimated single channel conductance was quite comparable to the values obtained at low ACh concentrations (see above).
We applied long pulses of ACh (50 s) to some cells. When 100 µM was applied, a plot of the variance versus the mean showed a typical parabolic relationship, which overlapped the data obtained with a lower concentration of ACh (3 µM) (Figure 5).
Figure 5. Variance–mean relationship for long duration responses. Plots of the variance against the (fit) mean for response of a cell to one long application of 100 µM ACh and one long application of 3 µM ACh are shown. The line through the data shows the fit of a parabola to both data sets, giving values of i=−1.52 pA and 291 agonist-activatable receptors. The yellow diamonds show the mean and variance for brief segments at the peak of the responses; these data were not used in fitting the parabola. Similar data were obtained with applications to three other cells. Each point (except the peak responses) shows the mean of 2849 samples over ∼285 ms. Holding potential −60 mV.
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Data were also obtained using nicotine as an agonist. It is well known that nicotine is a channel blocker (Rush et al., 2002; Paradiso and Steinbach, 2003), and we noted that responses to 100 and 300 µM nicotine showed a slowing in the decay of current after it had been removed (data not shown). Accordingly, only the data obtained with concentrations of nicotine lower than 100 µM were analysed. A plot of the variance against the mean current was well described by an inverted parabola (Figure 6). The average values for the parameters of fit parabolas were 17 ± 5 pS and 628 ± 103 activated receptors (n= 18 cells). The estimates were similar to those made using ACh, and the estimated single channel conductance with low nicotine concentrations was quite comparable to the values obtained with ACh.
Figure 6. Variance–mean relationship for responses elicited by nicotine. (A) The responses of a cell to five applications of nicotine (Nic, 2 s duration at 30 s intervals), with fitted single exponential curves declining to a constant superimposed. (B) A plot of the variance against the (fit) mean, together with the fit parabola (parameter values of i=−1.0 pA and 232 receptors). The yellow diamonds show the mean and variance for brief segments of data at the peak of one response to 100 µM nicotine; this value was not used in fitting the parabola. Each point (except that at 100 µM nicotine) shows the mean of 527 samples over ∼53 ms. Holding potential −60 mV.
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In five cells we obtained data with both ACh and nicotine. The variance–mean plots were quite similar, as shown in Figure 7. For these cells, the ratio of the estimated mean single channel conductance for nicotine to that for ACh was 0.87 ± 0.04, which differed significantly from a ratio of 1 (P < 0.01, 2-tailed t-test). The ratio for numbers of agonist-activatable receptors was 1.2 ± 0.5, which did not differ from 1. Overall, the data for nicotine and ACh were quite comparable.
Figure 7. Variance–mean relationships for a single cell activated by ACh or nicotine (Nic). Plots of the variance against the (fit) mean are for the responses of a cell to the indicated concentrations of ACh (red) and nicotine (blue), together with the fit parabolas. For responses to ACh the parameter values were i=−1.23 pA and 390 active receptors, and for responses to nicotine i=−1.06 pA and 439 active receptors. For this cell ACh was applied before nicotine, and both were applied in a series of ascending concentrations. In other experiments the order was reversed, with no discernable effect. Each point shows the mean of 537 samples over ∼54 ms. Holding potential −60 mV.
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We made repeated series of applications to some cells, and fit the separate data sets with parabolas, to estimate the variability between sets. On average, the parameters for the parabolas were quite similar. The fit values for single channel conductance normalized to the value for the first data set for that cell were 1.1 ± 0.5 (31, 8) for ACh and 0.99 ± 0.09 (9, 4) for nicotine (mean ± SD, number of ratios, number of cells). For the number of receptors the ratios were 1.0 ± 0.4 (31, 8) and 0.9 ± 0.2 (9, 4). Although these mean values indicate that the parameter estimates were reproducible, there was quite a bit of variation with a minimal ratio of 0.3 (number of receptors using ACh) and a maximal ratio of 2.8 (channel conductance, again using ACh). Accordingly, a single data set from a single cell may show extreme values on fitting.
The maximal probability of being open was estimated from 2 ratios. Firstly, we used the largest value for the fit (mean) current used in fitting the parabola to the variance–mean data and normalized it to the predicted maximal current. This gave ratios of 0.8 ± 0.2 (for 100 µM ACh, 34 cells) and 0.8 ± 0.1 (10 µM nicotine, 12 cells). Both of these ratios differed significantly from 1. Alternatively, we used the largest amplitude peak response obtained from the concentration–response data and normalized it to the maximal current predicted by the parabola. This gave ratios of 1.1 ± 0.3 (for ACh, 47 cells) and 1.1 ± 0.4 (nicotine, 18 cells). Neither of these ratios differed significantly from 1.
We only used low concentrations of 5-I A-85380 and did not fit parabolas to the data. Data with cytisine were obtained over concentrations from 1 µM to 100 µM. Cytisine elicited a small maximal response (see next section and Moroni et al., 2006; Mineur et al., 2009), suggesting it is a partial agonist. Plots of the variance against the mean did not decline at the larger responses, as would be expected for a low efficacy agonist. A parabola could be fit to the data, but in all cases predicted a small number of activated receptors (102 ± 107, 12) (data not shown). The reason for this is not known, but could reflect some channel-blocking activity for cytisine.
In the course of this work we obtained concentration–response data for these agonists. The main objective was to obtain non-stationary noise, and so the data were not obtained at appropriate concentrations to fully describe the relationships. However, the data were fit with the Hill equation (Figure 8). Overall, the data for ACh were described by an EC50 of 53 ± 53 µM (34 cells) and a Hill coefficient of 0.8 ± 0.2 (median values of 38 µM; range 3–287 µM and 0.8; 0.5–1.4). For nicotine, the values were EC50 4 ± 2 µM (18 cells) and Hill coefficient 1.2 ± 0.6 (median values 3.6 µM; 0.3–9 µM and 1.0; 0.8–3.2). For cytisine, the mean EC50 was 6 ± 2 µM (22 cells) (median value 5 µM; 2–13) and the Hill coefficient was constrained to 1 for the fitting. As discussed before, nicotine was able to elicit the same maximal response as ACh. However, cytisine produced a fit maximal response of only 0.06 ± 0.03 times the response to high ACh concentrations (300 or 1000 µM). This value is lower than previous estimates (0.08 to 0.3; Moroni et al., 2006; Mineur et al., 2009).
Figure 8. Concentration–response relationships for the agonists employed. Plots of the normalized response against the applied agonist concentration are shown for ACh, nicotine, cytisine and 5-I A-85380 (5-iodo-3-(2(S)-azetidinylmethoxy) pyridine ). The responses to ACh and nicotine were normalized to the fit maximum value for that cell and the responses to cytisine and 5-I A-85380 were normalized to the response of the same cell to a high concentration of ACh (300 or 1000 µM). Then the normalized responses at each concentration were averaged to produce the data shown. The curves are those predicted by the Hill equation with the median parameter values given in Results. The data for 5-I A-85380 were not fit. Each point shows the mean ± SD for normalized responses from 4 to 34 cells.
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The EC50 for ACh suggests that most of the response reflects activation of the low-sensitivity type of receptor (Buisson and Bertrand, 2001; Nelson et al., 2003). To obtain a rough estimate of the prevalence of the two forms, data (from cells for which at least 6 concentrations were tested) were fit with the sum of two simple binding curves with EC50 values of 0.9 µM and 60 µM (Buisson and Bertrand, 2001). The mean fraction in the high-sensitivity response was 0.3 ± 0.3 (24 cells), with a median of 0.17 and range from 0.00 to 0.97. This suggests most cells express predominantly low-sensitivity receptors. In agreement with this, 100 nM 5-I A-85380 elicited a response of 0.07 ± 0.03 times the response to 300 µM ACh (6 cells). At this concentration 5-I A-85380 would be expected to activate most high-sensitivity receptors and few low-sensitivity receptors (Zwart et al., 2006).