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Figure 2.The effect of holding potential on peak current response and pEC50 values. Illustrative current-voltage (I/V) data from two different oocytes that responded with low (a,c,e) and high (b,d,f) maximal peak current responses. (a,b) Current-voltage (I/V) plots at different 5-HT concentrations (■ 10 µM; △ 6 µM; + 3 µM; ● 2 µM; ⬦ 1 µM; ▲ 0.3 µM; × 0.1 µM). (c, d) Peak current shown as a function of 5-HT concentration at differing holding potentials (−80 to + 40 mV in 10 mV steps). The fitted curves represent the Partial Model in which pEC50 and nH parameters are constrained to be the same across all holding potentials for each oocyte. The pEC50 is indicated by the dotted line. (e, f) The relationship between the fitted parameters, Imax, pEC50 and nH, and the holding potential. The values shown are from the Full Model in which the parameters can adopt different values for each holding potential. The figures illustrate the variance in the unconstrained estimates of pEC50 and nH. Likelihood Ratio Tests comparing the Full and Partial Models suggest that there is no difference in the pEC50 and nH values between different holding potentials (Low responder: P = 0.74; High responder: P = 0.93). The straight line gradients (mean ± SE) of the data in Panel (e) are: Imax, 0.031 ± 0.001 (P = 3 × 10-11); pEC50, −0.00051 ± 0.00025 (P = 0.063); nH, −0.0100 ± 0.0062 (P = 0.14), and in Panel (f) are: Imax, 0.17 ± 0.0004 (P = 5 × 10-24); pEC50, −0.00015 ± 0.00029 (P = 0.62); nH, −0.0013 ± 0.0045 (P = 0.78). Note the differing y axis scales in the panels.
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Figure 3. A representation and review of P2X2 data from Clyne et al. (2003). Imax and pEC50 from (a) native P2X2 receptors shown in Fig. 3B of Clyne et al.12 and (b) C-terminal 76 amino acid deleted P2X2 receptors shown in Fig. 4A of Clyne et al.12. Values were estimated by visual inspection and represented in a similar fashion to Fig. 1a in this report. (c) A simulation of the predicted response (fitted to the 3PL equation) resulting from two distinct populations of channels each with similar expression levels, Hill coefficients of 2.4, but with different EC50 values equal to 6.6 µM (red) and 37 µM (blue). The mixed population of channels has an intermediate EC50 of 15.6 µM and an apparently lower Hill coefficient of approximately 1.4. (d) Predicted change in apparent Hill coefficient (nH) and EC50 for a mixed population of the two channel subtypes defined in Panel (c), as the proportion of those subtypes changes from 0 to 100%. (e) A representation of Fig. 3B from Clyne et al.12 with the x axis shown on a log scale. The line is the hyperbolic model fitted by Clyne et al.12 indicating a high EC50 of 37 µM, a low EC50 of 6.6 µM and a mid-point Imax of 6.108 µA, as described in their manuscript. (f) An equivalent representation of Fig. 4A from Clyne et al.12. The line is the fitted hyperbolic model indicating a high EC50 of 45.1 µM, a low EC50 of 4.8 µM and a mid-point Imax of 0.637 µA. Panels (e,f) illustrate how the parameter estimates lie in relation to the data. Note that Clyne et al.12 constrained the lower EC50 values to be no smaller than the lowest measured EC50 and consequently the low EC50 limits (dashed lines) pass through the lowest value data points.
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Figure 4. Modelling the formation of dimers from individual channels, A. (a) The dimerisation model assumes mass action association and dissociation of A channels. [AM] is the concentration of A monomers, and [AD] the concentration of dimers formed from two A channels. The association constant (KA) is the ratio of the forward (k+1) and backward (k−1) rate constants and is determined by the concentrations of AD and AM at equilibrium (Eq. 1 in Figure). [ATOT] is the concentration of all A channels whether they form a monomer or dimer (Eq. 2). (b) The model predicts that, as [ATOT] increases, the probability that the channels form dimers also increases. Thus, when [ATOT] = the dissociation constant (Kd = 1/KA), the proportion of A channels forming either a monomer or a dimer is 0.5 (‘a’ on figure). When [ATOT] is high, channels are therefore more likely to form dimers. When [ATOT]·KA = 10 (‘b’), the proportion of A channels forming dimers is 0.80, whereas when [ATOT]·KA = 0.1 (‘c’), the proportion is 0.15. Models describing the probabilities of dimer (Eq. 3) and monomer (Eq. 4) formation clearly reveal that these are dependent on [ATOT].
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Figure 5.A representation and review of data from Taleb and Betz (1994). (a) Representation of a mixed population of glycine receptors with differing EC50 and nH values derived from Fig. 2B of Taleb & Betz15. The mixed population (thick, black line) is the sum of two distinct populations (Red: Imax = 7.2 µA, EC50 = 50 µM, nH = 4.1; Blue: Imax = 20.1 µA, EC50 = 289 µM, nH = 3.1). At maximal glycine concentrations, the low EC50 population conducts approximately 25% of the total current and the high EC50 population conducts approximately 75% of the total current. The EC50 values are sufficiently different that the mixed population has a biphasic pattern. A monophasic concentration-response model derived from such a scenario would resemble the dashed line and thus have an intermediate EC50 of 194 µM, and a lower nH of 1.7. (b) Predicted changes in overall EC50 and nH as the proportions of channels in low and high EC50 states vary. The dashed line indicates the scenario depicted in Panel (a).
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Figure 6. Effect of pooling concentration-response data on estimates of the Hill coefficient (nH). (a) Six simulated data sets are shown (- - - -) each with 11 data points (○) ranging from 0.1 to 30 µM, a normalised maximum of 1, and a Hill coefficient of 3.9. The pEC50 values differ and are: 6.20, 6.06, 5.92, 5.78, 5.64, 5.50. The mean ± SD of the six data points at each concentration are shown. A line of best fit (OLS) through the means is a 3PL model (——) with the maximum constrained to 1. The estimated pEC50 = 5.85 and nH = 2.26. It is evident that the estimated nH is biased down compared to the individual concentration-response data, and that there is substantial variance in the central part of the mean concentration-response curve. This pattern is expected when data are normalised and the pEC50s vary from one dataset to another. The extent of the bias is difficult to predict, but depends on the variance in the pEC50 values, the number of datasets, and the number and location of the data points. (b) Data from each of the 95 oocytes in our study were normalised to the response induced by 10 µM 5-HT. The mean ± SD for each 5-HT concentration was calculated from these normalised values. A 3PL model (——) was fitted (OLS) to the mean values with the maximum constrained to 1. The estimated pEC50 = 5.77 and nH = 2.69. The dashed line (- - - -) shows the 3PL model defined by estimates shown in Table 1 (pEC50 = 5.78; nH = 3.94).
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