Evidence for two distinct processes in the final stages of neurotransmitter release as detected by binomial analysis in calcium and strontium solutions

Abstract

The statistical parameters underlying acetylcholine (ACh) release were studied using Ca(2+) and Sr(2+) ions to promote ACh secretion. Experiments were performed at frog neuromuscular junctions using electrophysiological recording techniques. Increases in asynchronous ACh release, reflected as the frequency of occurrence of miniature end-plate potentials (MEPP(f)), were evoked by high potassium depolarization in either Ca(2+) or Sr(2+) solutions. Increases in MEPP(f) mediated by Ca(2+) were of very low probability and well-described by a Poisson distribution whilst similar MEPP(f) increases mediated by Sr(2+) were best described as a simple binomial distribution. From the binomial distribution in Sr(2+) solutions, values for the average probability of release (p) and the number of releasable ACh quanta (n) may be determined (whereby mean MEPP(f) = np). In Sr(2+) solutions, values of p were independent of both bin width and of the value of n, suggesting that both n and p were stationary. Calculations of p using the simple binomial distribution in Sr(2+) solutions gave theoretical values for the third moment of the mean which were indistinguishable from the experimental distribution. These results, in conjunction with Monte Carlo simulations of the data, suggest that spatial and temporal variance do not measurably affect the analysis. Synchronous ACh release evoked by nerve impulses (end-plate potentials, EPPs) follow a simple binomial distribution in both Ca(2+) and Sr(2+) solutions. Similar mean levels of synchronous ACh release (m, where m = np) were produced by lower values of p and higher values of n in Ca(2+) as compared to Sr(2+). The statistical analyses suggest the presence of two different Ca(2+)-dependent steps in the final stages of neurotransmitter release. The results are discussed in accordance with (i) statistical models for quantal neurotransmitter release, (ii) the role of Sr(2+) as a partial agonist for evoked ACh release, and (iii) the specific loci that may represent the sites of Ca(2+) and Sr(2+) sensitivity.

Figure 1. Poisson distribution of MEPPs in Ca²⁺ solutions and binomial distribution of MEPPs in Sr²⁺ for similar levels of asynchronous release in high K⁺ solutions (10 ms bin width)

A, ▪, MEPP frequency distributions recorded in 0.5 mm Ca²⁺, with 12 mm KCl (± 1 s.e.m.). B, ▪, MEPP frequency distributions recorded in 0.5 mm Sr²⁺, with 10 mm KCl, from the same end-plate (± 1 s.e.m.). The MEPP frequencies obtained for both treatments were statistically indistinguishable (see Table 1 and text). Poisson distributions obtained from the mean frequency of MEPPs (eqn (1)) produced excellent descriptions of the Ca²⁺ MEPP data (A, □). The data in A are the averaged distributions from 4552 MEPPs. In contrast, the Poisson distribution did not produce good fits of the MEPPs recorded in Sr²⁺, but was well described by the binomial distribution (□). The theoretical binomial distribution in Sr²⁺ (□) was predicted by eqn (6) using the variance and mean of the MEPP distribution. The calculated value for p was 0.049 and n = 4.31 (χ² test, P < 0.05). The data in B are the averaged distributions from 5914 MEPPs. As can be seen from the inset, the differences between the Poisson distribution and these data are not readily apparent. The degree of significance results from the very small errors in the data (error bars are barely discernible on the graphs). See text for a further description of the statistical tests used in these experiments. C and D, results of Monte Carlo simulations. In C, the Monte Carlo simulation (▪) was generated using Poisson's equation (eqn (1)) and the mean value of the MEPP_f determined experimentally in A. In D, the Monte Carlo simulation (▪) was obtained using the estimates of n and p determined experimentally from B to generate a binomial distribution. Values of the variance and mean (as determined from the events generated by the Monte Carlo simulations (▪)) were then used to generate theoretical distributions (□). The population of events obtained by the Monte Carlo simulation of the Poisson distribution using the mean MEPP_f determined from the simulated data in C was well described by the theoretical Poisson distribution (This was similar to the data in Ca²⁺ solutions). Monte Carlo simulations of the binomial distribution were made by using values of p = 0.049 and n = 4.31 as determined experimentally in Sr²⁺ solutions, see B above). The estimates for p and n obtained from the Monte Carlo simulations were p = 0.067 and n = 3.15.

Figure 2. Poisson distribution of MEPPs in Ca²⁺ solutions and binomial distribution of MEPPs in Sr²⁺ for similar levels of asynchronous release in high K⁺ solutions (100 ms bin width)

The results are similar to Fig. 1, only using a 100 ms bin width. A, MEPP frequency distributions recorded in 0.5 mm Ca²⁺, with 12 mm KCl (▪). B, MEPP frequency distributions recorded in 0.5 mm Sr²⁺, with 10 mm KCl, from the same end-plate (▪). The MEPP frequencies obtained for both treatments were similar. Poisson distributions obtained from the mean frequency of MEPPs (see Methods) produced good descriptions of the data in Ca²⁺ (□). The data in A are the averaged distributions from 11 291 MEPPs. In contrast, the Poisson distribution did not produce good fits of the MEPPs recorded in Sr²⁺ (inset) but was well described by the binomial distribution (□) predicted by the variance and mean of the MEPP distribution. The values obtained for B gave: p = 0.089 ± 0.004 and n = 37.9 (χ² test, P < 0.05). The data in B are the averaged distributions from 10 905 MEPPs. C and D, results of Monte Carlo simulations. In C, the Monte Carlo distribution (▪ ± s.e.m.) was obtained using the same mean frequency as the data recorded in A in conjunction with Poisson's equation (eqn (1)). In D, the Monte Carlo simulation was generated using the estimates of n and p from B to produce a binomial distribution (▪). Values of the variance and mean of the Monte Carlo distributions were then used to provide estimates of the Poisson and binomial distributions (□). The population of events obtained using Monte Carlo simulation of the Poisson distribution was well-described by the theoretical Poisson distribution (□). The Monte Carlo simulation of the binomial distribution ((D▪) using the estimates of p = 0.09 and n = 38 - see B above) was similar to the theoretical distribution generated using the variance and the mean of the Monte Carlo distribution (□). This simulation, using the variance and the mean of the Monte Carlo data, also yielded values of p = 0.095 and n = 37 (□).

Figure 3. ACh release evoked by action potentials follows a binomial distribution in Ca²⁺ (A) and Sr²⁺ (B) solutions

The concentrations of Ca²⁺ (0.3 mm) and Sr²⁺ (2 mm) were chosen to approximate equiactive concentrations of these ions for evoked released (m). The mean and variance of the EPP amplitudes (EPP and S²_EPP respectively) and the mean and variance of the MEPP amplitude (MEPP and σ²_MEPP, respectively) were used to in conjunction with eqn (12) to calculate p. The value of n was calculated by eqn (13). For calculation of error estimates for these parameters, see McLachlan (1978) and Methods. In Ca²⁺ (A), the values for m, n and p (± 1 s.e.m.) were respectively 6.7 ± 0.2, 57.3 ± 39 and 0.11 ± 0.08. In Sr²⁺ (B), values for m, n and p (± 1 s.e.m.) were respectively 4.9 ± 0.1, 17.9 ± 5.0 and 0.27 ± 0.06. The curved lines show expected frequency plot for the binomial distribution (dark line) and the Poisson distribution (grey line - see Methods). For the Ca²⁺ data (A), P > 0.05 for the χ² fit to the binomial distribution and P < 0.005 for the χ² fit to the Poisson distribution. For the Sr²⁺ data (B), P > 0.5 for the fit to the binomial distribution and P < 0.001 for the fit to the Poisson distribution. The statistical analyses thus demonstrate that there is no justification for the hypothesis that the data deviate from a binomial distribution. For all seven experiments (Table 5), the binomial distributions provided a better fit to these data than the Poisson distribution (χ² squared statistic). These data depict experiment 1 in Table 5.