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The Journal of Neuroscience, March 1, 2002, 22(5):1648-1667

Calcium Secretion Coupling at Calyx of Held Governed by Nonuniform Channel-Vesicle Topography

Christoph J. Meinrenken¹, J. Gerard G. Borst², and Bert Sakmann¹

¹ Max Planck Institute for Medical Research, 69120 Heidelberg, Germany, and ² Swammerdam Institute for Life Sciences, University of Amsterdam, 1098 SM Amsterdam, The Netherlands

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Phasic transmitter release at synapses in the mammalian CNS is regulated by local [Ca²⁺] transients, which control the fusion of readily releasable vesicles docked at active zones (AZs) in the presynaptic membrane. The time course and amplitude of these [Ca²⁺] transients critically determine the time course and amplitude of the release and thus the frequency and amplitude tuning of the synaptic connection. As yet, the spatiotemporal nature of the [Ca²⁺] transients and the number and location of release-controlling Ca²⁺ channels relative to the vesicles, the "topography" of the release sites, have remained elusive. We used a time-dependent model to simulate Ca²⁺ influx, three-dimensional buffered Ca²⁺ diffusion, and the binding of Ca²⁺ to the release sensor. The parameters of the model were constrained by recent anatomical and biophysical data of the calyx of Held. Comparing the predictions of the model with previously measured release probabilities under a variety of experimental conditions, we inferred which release site topography is likely to operate at the calyx: At each AZ one or a few clusters of Ca²⁺ channels control the release of the vesicles. The distance of a vesicle to the cluster(s) varies across the multiple release sites of a single calyx (ranging from 30 to 300 nm; average ~100 nm). Assuming this topography, vesicles in different locations are exposed to different [Ca²⁺] transients, with peak amplitudes ranging from 0.5 to 40 µM (half-width ~400 µsec) during an action potential. Consequently the vesicles have different release probabilities ranging from <0.01 to 1. We demonstrate how this spatially heterogeneous release probability creates functional advantages for synaptic transmission.

Key words: active zone; buffer; diffusion; glutamate; heterogeneity; synapse; transmitter release; vesicle; domain

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

During fast synaptic transmission the release of neurotransmitter from vesicles in presynaptic terminals is controlled by calcium ions (Ca²⁺) (Katz, 1969). Brief influx of Ca²⁺ through voltage-gated channels causes a transient rise in the intracellular concentration of Ca²⁺ ([Ca²⁺]). Within <1 msec this rise in [Ca²⁺] causes vesicles to fuse with the presynaptic membrane, releasing transmitter. The transient rise in [Ca²⁺] is less pronounced with increasing distance from the Ca²⁺ channels. Therefore, a vesicle (i.e., the Ca²⁺ sensor controlling its release) must be located sufficiently close to one or more Ca²⁺ channels. The exact distance between channels and vesicles critically determines the time course and amplitude of the [Ca²⁺] signal at the vesicles and thus determines the time course and amplitude of the release rate (Augustine and Neher, 1992). Therefore, understanding the functional and spatial organization of the release-controlling Ca²⁺ channels relative to vesicles (henceforth "topography of release sites") is crucial for a quantitative description of synaptic transmission itself (Augustine, 2001).

Because direct measurements of the local [Ca²⁺] transients are not (yet) available, our understanding of the local [Ca²⁺] dynamics during action potentials (APs) must rely on experimental data on release and on quantitative models (Neher, 1998a). A number of studies have investigated the significance of channel/vesicle location (Yamada and Zucker, 1992; Cooper et al., 1996; Gil et al., 2000), some of them quantifying release on the basis of measured Ca²⁺ sensitivity of the release-controlling Ca²⁺ sensor (Chow et al., 1994; Klingauf and Neher, 1997; Bennett et al., 2000). However, the considerable number of poorly known parameters in the models often has defeated attempts to derive with certainty the topography of release sites (Neher, 1998a).

We present a time-resolved model that simulates Ca²⁺ influx, three-dimensional buffered Ca²⁺ diffusion, and the binding of Ca²⁺ to the release sensor for the calyx of Held (henceforth, calyx), a giant terminal in the medial nucleus of the trapezoid body (MNTB) of mammalian brainstem (Forsythe et al., 1995). The model is based on recent anatomical and biophysical data, which constrain key parameters of the simulations. We infer the topography of release sites by comparing simulated transmitter release with previous experimental data (effects of added exogenous Ca²⁺ buffers BAPTA and EGTA as well as effects of altered Ca²⁺ channel gating). The topographic analysis indicates that the "readily releasable" pool of vesicles is heterogeneous with respect to its release probability. Heterogeneous release probability of vesicles has been observed at the calyx (Wu and Borst, 1999; Sakaba and Neher, 2001b). We show that, other than being an intrinsic property of the release apparatus, the heterogeneity may arise to a large extent from variability in the distances between vesicles and release-controlling Ca²⁺ channels at different release sites of a single calyx.

Our model reproduces the experimental data, including previously unexplained effects of exogenous Ca²⁺ buffers, only if the spatial nonuniformity is included explicitly in the calculations (discussed in Quastel et al., 1992). To demonstrate further the functional significance of the proposed nonuniformity, we investigated its effects on synaptic delay and on release during consecutive APs.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Inferring the topography

The model simulates the time course (0-5 msec for a single AP; room temperature) of Ca²⁺ influx, three-dimensional buffered Ca²⁺ diffusion, and phasic transmitter release for a calyx at the developmental stage postnatal days 8-10. Parameters in the simulations were constrained by electrophysiological and morphological measurements of the calyx (Table 1). The only remaining crucial but unknown parameters were the conductance of single Ca²⁺ channels and the channel-vesicle topography at release sites. We assumed a topography and then set the single Ca²⁺ channel conductance such that the predicted release probability for physiological conditions is the same as that observed in the experiments (see Results for values). We then simulated release under nonphysiological conditions: added exogenous Ca²⁺ buffers, lowered [Ca²⁺] of the extracellular solution, and reduced open probability of Ca²⁺ channels (using, for each topography, the same single channel conductance as that used for the physiological condition). The predicted effects of the nonphysiological conditions on release critically depend on the assumed topography. Testing different hypothetical topographies, we compared the results of the model with the experimental data (Table 2) and thus inferred whether a particular topography is likely to be present at the calyx or not.

View this table: [in this window] [in a new window]   Table 1.   Experimental calyx data used to constrain model parameters

View this table: [in this window] [in a new window]   Table 2.   Experimental calyx data used to infer topography of release sites

Numerical simulations

In the simulations the calyx volume was split up into subcompartments ("reaction volumes") of identical size around active zones (AZs; see Results for dimensions). For the periodic grid topography (see Results), boundary conditions (to adjoining compartments) for the buffered diffusion were periodic (side walls only). For all other simulations the boundaries were "closed," and the location of the channel cluster and the vesicles on the AZ, the radius of the circular AZ, as well as the stochastic Ca²⁺ currents through channels were different in each subcompartment (see Results). Dimensions of the compartments were chosen sufficiently large so that reflections of Ca²⁺ at the walls affect only volume average [Ca²⁺] but not local [Ca²⁺] near the vesicles (see below). The local [Ca²⁺] transients around individual AZs in the model were assumed to be independent. The model was implemented as Ansi C code, running on a Silicon Graphics Oregon 2000 computer (processor MIPS RP1200, 300 MHz). A single simulation (0-5 msec) took ~45 min to complete.

Hodgkin-Huxley model for Ca²⁺ channel gating and time course of I_Ca

To simulate Ca²⁺ influx through individual channels in response to APs, the model uses a two-gate Hodgkin-Huxley model with parameters that were fit for the calyx. The gates of the channels and the resulting currents are driven by AP waveforms (equations and parameters as in Borst and Sakmann, 1998). For some simulations with low channel open probability, a third gate was added (see Results). As the time course for Ca²⁺ entry at each channel location [i_Ca(t)], the simulations used either the "uniform i_Ca mode" or the "stochastic i_Ca mode." In the uniform i_Ca mode all channels are "open" and i_Ca(t) is the same for all channels, with a time course matching that of the whole-cell Ca²⁺ current (predicted by the Hodgkin-Huxley Model). In the stochastic i_Ca mode the model varies i_Ca(t) for each channel stochastically. In this mode the individual channel locations contribute different i_Ca(t), and some remain closed. Individual i_Ca(t) are simulated by using Monte Carlo-type pseudo-stochastic sampling to determine the open and closed times of the two gates [random number generator, ran(2) (Press et al., 1988); time step for i_Ca(t), 1 µsec]. Whenever a channel is open, the current is given according to the electrical driving force and the conductance. Single channel conductance was varied for different topographies (see above) but was the same for all channels. Note that values for single channel conductance given in Results refer to open channels, whereas values for channel current i_Ca [given as the peak amplitude of i_Ca(t), i_Ca, peak] refer to the average across all open or closed channels. To simulate experiments with reduced [Ca²⁺] of the extracellular solution, we reduced the channel conductance to match the reduction of whole-cell I_Ca observed in the experiments.

Buffered diffusion of Ca²⁺

At time 0, Ca²⁺ and buffers were at resting concentrations and at spatial equilibrium. Standard equations (3.20-3.23 in Smith, 2001) for diffusion and buffering were solved numerically (forward Euler finite difference). We assumed unrestricted diffusion of Ca²⁺ and buffers around AZs (i.e., barriers, particularly nondocked vesicles in the vicinity of AZs, were neglected). To validate the unrestricted diffusion assumption (Glavinovic and Rabie, 2001) for the calyx, we analyzed the three-dimensional reconstruction of 31 of the ~600 AZs (K. Sätzler, L. Söhl, J. Bollmann, J. Borst, M. Frotscher, B. Sakmann, and J. Lübke, unpublished data). In a dome-like control volume around each AZ (200 nm distance from the edge of the control volume to the nearest point on AZ), there were, on average, 62 vesicles (not docked, i.e., not readily releasable). These vesicles occupied ~6% of the dome-like control volume. Therefore, their effect as diffusion barriers is negligible.

Only for the simulation of consecutive APs (see Results), a linear extrusion mechanism was used, which reduces the [Ca²⁺] of each voxel separately by ([Ca²⁺]  [Ca²⁺]_rest)··t per time step, where [Ca²⁺]_rest = 50 nM (see below),  = 400 Hz is the pump rate, and t is the time step.

Spatial resolution was as follows: (5 nm)³ voxels for the first six layers on the membrane (0-30 nm), (10 nm)³ for 30-90 nm, and (20 nm)³ for the remainder. After 1.5 msec, when [Ca²⁺] gradients have dissipated, spatial resolution was decreased by collapsing neighboring voxels into (10 nm)³, (20 nm)³, and (40 nm)³. For those voxels (of the first layer) located "above" the presumed Ca²⁺ channels, [Ca²⁺] = i_Ca(t)·t/(V·F) per time step was added to [Ca²⁺] of the voxel (t, time step; V, voxel volume; F, Faraday's constant). Time step was 10 fsec to 0.68 nsec, such that the relative change in concentration of any substance in any voxel during any time step did not exceed ±1% (autoadaptive); dt_max = 0.15·(dx)²/220 µm²/sec, where dx = 5 nm for times up to 1.5 msec and 10 nm thereafter.

Unless otherwise indicated, the model solution contained (control condition): free Ca²⁺ at a starting concentration of [Ca²⁺]_rest = 50 nM (Helmchen et al., 1997) and diffusion coefficient D_Ca = 220 µm²/sec (Albritton et al., 1992); endogenous fixed buffer (termed EFB, unspecified identity, single Ca²⁺ binding site) with a binding ratio of 40 (Helmchen et al., 1997), total concentration [EFB]_total = 80 µM, affinity K_D = 2 µM (varied in sensitivity analyses between 200 nM and 200 µM; see below), and a forward Ca²⁺ binding rate, k_on = 5·10⁸ per Msec (Klingauf and Neher, 1997) (sensitivity analysis below); ATP with [ATP]_total = 0.58 mM, K_{D, Ca} = 200 µM, k_{on, Ca} = 5·10⁸ per Msec (Baylor and Hollingworth, 1998) (with k_{on, Ca} corrected for temperature; sensitivity analysis below), D_ATP = 220 µm²/sec. Kinetic parameters of ATP are for the binding of ATP to Ca²⁺ only (not Mg²⁺). The presence of 4 mM Mg-ATP in the pipette during the experiments (Borst et al., 1995) was accounted for by reducing the concentration of total ATP available for Ca²⁺ binding to 0.58 mM. The remaining ATP was assumed to stay bound to Mg²⁺ during the [Ca²⁺] transient (K_{D, Mg} = 100 µM) and thus unavailable for Ca²⁺ buffering (slow off-rate of Mg-ATP, k_{off, Mg} = 150-390/sec; Baylor and Hollingworth, 1998). For some simulations, mobile exogenous buffers were added at varying concentrations: BAPTA (k_on = 4·10⁸ per Msec, K_D = 220 nM, D_BAPTA = 220 µm²/sec; Naraghi and Neher, 1997) and EGTA. The binding kinetics of EGTA are strongly pH-dependent. We thus used two sets of parameters: "EGTA" (k_on = 10·10⁶ per Msec, K_D = 70 nM; Nägerl et al., 2000) or "EGTA-2" (k_on = 2.5·10⁶ per Msec, K_D = 180 nM; Naraghi and Neher, 1997). D_EGTA = D_EGTA-2 = 220 µm²/sec.

Because the buffered diffusion algorithm uses neither the steady-state assumption nor the rapid buffer or the linearized buffer approximation, it correctly simulates any local/global depletion of unbound buffers ("buffer saturation"; Naraghi and Neher, 1997). Because [Ca²⁺] in the simulations is generally low (micromolar range), mobile buffers deplete only marginally. For example, in the simulation with added 1 mM BAPTA (see Fig. 7D), at the time of peak Ca²⁺ influx and at the center of the channel cluster (where depletion is the strongest), the concentration of unbound BAPTA is still 90% of the volume average concentration (94% for unbound ATP). In contrast, unbound endogenous fixed buffer (EFB) is depleted locally because it binds Ca²⁺ without being replenished by diffusion. As a result, unbound EFB (at the cluster center and at the peak of the Ca²⁺ current) is depleted to 6% of the volume average concentration.

Net Ca²⁺ influx into calyx volume versus modeled subcompartments

In the reference topography simulating physiological conditions with Release Model A, the conductance was 14.52 pS per channel cluster, corresponding to an average I_Ca, peak = 0.66 pA per cluster. (The conductance for the simulations with the less Ca²⁺-sensitive Release Model B was 37.04 pS.) This corresponds to 0.26 fC or 12 µM unbuffered Ca²⁺ entering each subcompartment (volume, 0.110 µm³), thus increasing volume average [Ca²⁺] to 379 nM, as observed in experiments (see Table 1). The 600 subcompartments (for 600 AZs) contributed a total of 0.16 pC Ca²⁺ (per AP and assuming one cluster per AZ), which is 17% [42% in case of Release Model B] of the whole-cell value observed in experiments (see Table 1). The total modeled volume (600 subcompartments or ~17% of 400 µm³) corresponds to this ratio. Therefore, the increase in volume average [Ca²⁺] in the subcompartments was the same as that experimentally observed for the whole calyx. This approach indirectly accounts for Ca²⁺ that enters through channels located away from release sites, assuming that these channels do not (significantly) affect local [Ca²⁺] transients at release sites but only the volume average [Ca²⁺] (see Discussion). To comply with the experimental measurement of volume average [Ca²⁺], [Ca²⁺] was 12 µM in every simulation; whenever Ca²⁺ influx for the control condition was changed (for topographies other than the reference topography and for Release Model B; see Results), the volume of the subcompartments was adjusted accordingly. [The model does not include contributions to [Ca²⁺] from the release of Ca²⁺ from intracellular stores. At the calyx this contribution during a single AP is marginal at most (Helmchen et al., 1997).]

Release

We defined release probability P_r (as a percentage) as the fraction of all readily releasable vesicles that are released during a single AP (phasic release only; see Table 1 for size of readily releasable pool). "Release site" is defined as the functional entity of one readily releasable vesicle and the one or more Ca²⁺ channels controlling its release. At the calyx (~600 AZs) a single AZ contains, on average, more than one release site (Sätzler, Söhl, Bollmann, Borst, Frotscher, Sakmann, and Lübke, unpublished data). Although different release sites at the same AZ may be controlled by the same Ca²⁺ channels, the model assumes that release from individual sites is stochastically independent. Under this assumption more than one vesicle during a single AP may be released from a single AZ (Auger and Marty, 2000; Sun and Wu, 2001). However, a single release site can release at most one vesicle per AP (the model does not include recovery of the readily releasable pool). Because it is a relative measure, predicted P_r does not depend on the total number of vesicles in the readily releasable pool (= number of release sites) nor on the number of AZs or the number of release sites per AZ.

To quantify P_r in response to a transient increase in [Ca²⁺], the model uses, alternately, two kinetic schemes (Fig. 1): Release Model A (Bollmann et al., 2000) or Release Model B (Schneggenburger and Neher, 2000). Note that, because it is less Ca²⁺-sensitive, Release Model B predicts higher absolute [Ca²⁺] transients than Release Model A. However, relative spatial profiles of the transients are almost identical (marginal depletion of mobile buffers; see above). Therefore, the conclusions on the release site topography are valid for either release model. For clarity, figures generally show [Ca²⁺] transients and P_r of simulations with Release Model A. Analogous results with Release Model B are given in the text only. At time 0, the Ca²⁺ binding sites of the sensors were equilibrated with [Ca²⁺]_rest. The readily releasable pool was "full." For each assumed vesicle location (location at which the membrane of the vesicle is closest to presynaptic membrane), the time course of the local [Ca²⁺] transient predicted by the reaction-diffusion scheme (measured in the voxel ~10 nm above the presynaptic membrane) was translated into a release rate versus time. This was done for each vesicle individually, assuming that (1) release sites are independent and (2) binding of Ca²⁺ to the release-controlling Ca²⁺ sensor does not affect [Ca²⁺] (Yamada and Zucker, 1992). The differential equations of the model describing the relative state occupancies were solved numerically (forward Euler finite difference), using a variable time step such that, during any time step, the absolute change of the relative occupancy of any state was at most ±0.5%. The time integral (0-5 msec) of the release rate of an individual vesicle is P_r, vesicle. The (heterogeneous) release rates of individual vesicles were averaged into an average release rate versus time. The time integral of this rate (0-5 msec) is the predicted average release probability of all vesicles in the calyx (P_r, calyx). Different release site topographies predict different P_r, calyx because they correspond to different distributions of channel-to-vesicle distances for the readily releasable pool. The average release rate was converted to an EPSC, as described by Bollmann et al. (2000) (convolution of release rate with quantal EPSC).

View larger version (13K): [in this window] [in a new window]   Figure 1.   Intrinsic Ca2+ sensitivity of transmitter release. Shown is Pr for a single vesicle when exposed to a [Ca2+] transient with a time course equal to that of whole-cell ICa (full width at half-maximum, 383 µsec) and with a peak amplitude of [Ca2+]vesicle. The thin lines indicate release probability during APs according to Release Model A (Pr, vesicle = 25% at [Ca2+]vesicle = 8.8 µM; Bollmann et al., 2000) or Release Model B (Pr, vesicle = 10% at [Ca2+]vesicle = 35 µM; Schneggenburger and Neher, 2000).

Sensitivity of P_r to kinetics and concentrations of endogenous buffers

The model includes the mobile Ca²⁺ buffer ATP as well as one fixed buffer. The fixed buffer simulates the effect of one or more not further identified fixed or poorly mobile buffers. We varied concentration and binding kinetics of the endogenous buffers to estimate the sensitivity of predicted P_r to these parameters. Even drastic variations in the parameters of endogenous buffers (two to three orders of magnitude) do not change the predicted P_r, calyx to a degree that would compromise our results on release site topography. The reference simulation for each parameter variation is the simulation for the reference topography (see Results; control condition, P_r, calyx = 25%).

Endogenous fixed buffer. We varied [EFB] between 10 and 10,000% of the reference value (80 µM) while adjusting K_D, EFB such that [EFB]/K_D, EFB (approximate binding ratio) was kept unchanged at 40. This variation changes the predicted P_r, calyx between 41 and 18%, respectively. During an AP the EFB locally depletes/equilibrates with [Ca²⁺] (see above) and thus is rendered ineffective as a local sink for Ca²⁺. Therefore, EFB has only a small effect on the direct attenuation of the local [Ca²⁺] transients that control phasic release of transmitter. This shows that the predictions of the model of P_r are accurate even if the concentration of EFB in the presynaptic volume were not spatially uniform.

ATP. Two sets of parameter variations for ATP were investigated. (1) Keeping [ATP]/K_D, ATP constant, we changed both [ATP] and K_{D, ATP} between 10 and 1000% of their respective reference values (see above). This changes P_r, calyx between 37 and 22%, respectively. (2) Keeping [ATP]·k_on (buffer) product constant, we changed both [ATP] and (inversely) k_{on, ATP}, between 10 and 1000% of the reference values. This changes P_r, calyx between 44% (low [ATP], high k_{on, ATP}) and 12% (low k_on, _ATP, high [ATP]). Complete removal of ATP from the model calyx results in P_r, calyx = 48%. For simulations with the added exogenous buffers EGTA or BAPTA, the presence of ATP changes the predicted P_r, calyx even less because the effect of ATP is small compared with that of the exogenous buffer (see Fig. 4D).

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

In the first part of Results, we infer which channel-vesicle topography characterizes release sites at the calyx (developmental stage postnatal days 8-10). In the second part, we simulate the spatiotemporal pattern of AP-evoked [Ca²⁺] transients and of phasic transmitter release at the calyx. In the third part, we illustrate the functional significance of the proposed release site topography for synaptic transmission at this fast synapse.

Topography of release sites

Overview

Because the location of Ca²⁺ channels at the calyx is not known, there is a multitude of conceivable topographic arrangements of channels relative to readily releasable vesicles. We analyzed various topographies for their compatibility with measured phasic release probabilities (P_r) under different experimental conditions (Table 3): (1) reducing the [Ca²⁺] transients by dialyzing the calyx with exogenous Ca²⁺ buffers EGTA or BAPTA and (2) reducing the Ca²⁺ influx by altering the gating of the Ca²⁺ channels.

View this table: [in this window] [in a new window]   Table 3.   Summary of findings on topography

(1) The efficacy of Ca²⁺ buffers in reducing the [Ca²⁺] transient around a Ca²⁺ channel depends on the diffusion distance from the channel (Neher, 1986). For the calyx it was found previously that there is no channel-to-vesicle distance, for which the theoretical efficacy of BAPTA versus EGTA is consistent with experimental data (Naraghi and Neher, 1997). Although we confirm this result, we find further that the observed buffer efficacies can be explained by assuming that the channel-to-vesicle distance is different for different release sites of the same calyx.

(2) When the open probability of Ca²⁺ channels at the calyx is reduced, P_r is reduced in a supralinear manner. On the basis of this experimental finding, it was concluded previously that the majority of readily releasable vesicles at the calyx is controlled by more than one Ca²⁺ channel per vesicle (Borst and Sakmann, 1999a). We find further that the multiple Ca²⁺ channels controlling a vesicle, rather than being distributed evenly in the membrane, are likely to be organized in clusters of Ca²⁺ channels.

On the basis of these findings, we suggest that phasic transmitter release at the calyx is governed by the following nonuniform topography of release sites (henceforth "reference topography"). Readily releasable vesicles are controlled by clusters of Ca²⁺ channels, with one or a few clusters per AZ. For any one release site the distances between the vesicle and the individual channels of a cluster are similar. However, vesicles at different release sites are located at a broad range of distances from the channel cluster (average distance ~100 nm; coefficient of variation > 0.5). Details of the findings on topography are explained below as Properties I-III.

Property I: The distance between a vesicle and its release-controlling Ca²⁺ channel(s) varies across different release sites of the same calyx

For the calyx the distance between a vesicle and the Ca²⁺ channel(s) controlling its release is not known. Knowing the sensitivity of the Ca²⁺ sensor does not solve this problem, because the conductance of Ca²⁺ channels in the calyx is unknown. The [Ca²⁺] transient that drives vesicle release could be supplied by a channel at distance D and with conductance C or, alternatively, from a channel twice as far away but with approximately twice the conductance (see below). This scaling behavior prevents one from inferring at what distances from Ca²⁺ channels the vesicles are likely to be located. However, the scaling is broken when experimental data on the effects of added exogenous Ca²⁺ buffers EGTA or BAPTA are taken into account.

When a calyx is loaded with EGTA or BAPTA, P_r is reduced in a concentration-dependent manner. For example, 10 mM EGTA reduces P_r to ~0.45 of P_r in the native calyx. BAPTA (1 mM), which binds Ca²⁺ faster than EGTA, reduces P_r to ~0.35 of P_r in the native calyx (see Table 2). The reduction of P_r by the buffers is ascribed to the effect of the buffers on [Ca²⁺] transients. EGTA and BAPTA intercept part of the Ca²⁺ diffusing from the inner mouth of the channels to the vesicles, thereby reducing the peak amplitude of the [Ca²⁺] transients reaching the vesicles ([Ca²⁺]_vesicle). In combination with experimental data, diffusion calculations thus may be used to infer characteristic distances between release-controlling Ca²⁺ channels and vesicles. This has been addressed previously for the calyx (Naraghi and Neher, 1997). Here we confirm the earlier result while offering a different interpretation.

We begin by assuming that every releasable vesicle at an AZ in the calyx is located at some fixed distance from a single Ca²⁺ channel that controls its release (e.g., 80 nm) (Fig. 2A). The Ca²⁺ current through each channel gives rise to a [Ca²⁺] transient, with a peak amplitude decaying rapidly with increasing distance from the channel. Therefore, to yield similar [Ca²⁺]_vesicle (and thus P_r) for different vesicle distances, we increased the single channel conductance for increasing distance (Fig. 2B). Single channel conductance was chosen such that the predicted release probability for a vesicle (P_r, vesicle) was 25% under control conditions ("control" means that the model calyx contained only endogenous fixed buffer and ATP). For any given distance the simulations for added buffers used the same channel conductance as that used for the control condition. The time course of the single channel current [i_Ca(t); approximately Gaussian] was calculated with a Hodgkin-Huxley Model. In these simulations i_Ca(t) was the same for all channels (uniform i_Ca mode; see Materials and Methods). Depending on the conductance, peaks of i_Ca (per channel) varied from 0.0157 pA (for distance of 5 nm) to 0.936 (for distance of 125 nm). After the addition of exogenous buffers, the predicted [Ca²⁺]_vesicle is strongly dependent on distance (Fig. 2C). This is expected because, for a single Ca²⁺ channel domain (Neher, 1986), the reduction in [Ca²⁺]_vesicle relative to [Ca²⁺]_vesicle under control conditions is stronger the farther away from the channel the vesicle is located.

View larger version (28K): [in this window] [in a new window]   Figure 2.   Effect of channel-to-vesicle distance on release probability. A, Reaction volume used in the simulation; all numbers are given in nanometers (not drawn to scale). Height (400 nm) is equal to the thickness of the calyx, and the width corresponds approximately to the distance between neighboring AZs (see Table 1). Partial 5 and 20 nm grids indicate varying spatial resolution (see Materials and Methods). Single Ca2+ channel is located at the center in the first voxel layer on the membrane (same for all channel-to-vesicle distances). Readily releasable vesicle is located on the membrane at 80 nm from the channel (example only). B, Peak Ca2+ current per channel required to yield release probability (Pr, vesicle) of 25% for the control condition [control = endogenous fixed buffer (EFB) and ATP only] if all vesicles were located at the same distance from the Ca2+ channel that controls their release. C, Predicted peak [Ca2+] at the location of the vesicle for the same assumption as in B. Traces 1-4 represent four different buffer conditions. Trace 1, EFB and ATP (control condition). Trace 2, EFB, ATP, and 10 mM EGTA-2. Trace 3, EFB, ATP, and 10 mM EGTA. Trace 4, EFB, ATP, and 1 mM BAPTA. D, Predicted Pr, vesicle for the same assumption as in B. Traces 1-4 for the buffer conditions are the same as in C. The dashed box indicates the distance range in which Pr, vesicle in the presence of EGTA, but not BAPTA, is similar to the experiment. Vertical dashed line indicates average vesicle distance (118 nm) of the reference topography used in the second part of Results.

Figure 2D shows P_r, vesicle as a function of the distance from the Ca²⁺ channel. As expected, for distances of 10 nm or less the efficacy of EGTA in reducing release is marginal. By comparing the differential effect of added exogenous buffers EGTA and BAPTA at any one distance, Figure 2D confirms a previous result for the calyx (Naraghi and Neher, 1997). There is no distance at which the predicted P_r, vesicle is consistent with experimental results for both EGTA and BAPTA loading (assuming either parameter set, EGTA or EGTA-2). For any distance at which the predicted reduction of P_r, vesicle by 10 mM EGTA (Fig. 2D, trace 3) is similar to that observed in the experiments (25-40 nm) (Fig. 2D, dashed box), the predicted reduction of 1 mM BAPTA is much stronger than that observed in the experiments. At distances >125 nm the predicted reduction of P_r, vesicle of neither BAPTA nor EGTA is consistent with the experiments. (Fig. 2 shows simulations with Release Model A. Analogous simulations with Release Model B yielded the same result.)

In comparing the P_r, vesicle in Figure 2D with experiments (which measure the average release probability across all vesicles in the calyx) we assumed implicitly that all readily releasable vesicles in the calyx are located at the same distance from their release-controlling channel. Under this assumption the experimental observation could not be explained. We therefore propose that the distance of individual vesicles to their release-controlling Ca²⁺ channel(s) varies across different release sites. This explanation indeed is suggested by the experiments. Although some vesicles seem to be located sufficiently far from Ca²⁺ channels to be affected by the slowly binding buffer EGTA (10 mM), other vesicles in the same calyx may be located closer to channels so that even the fast binding buffer BAPTA (1 mM) reduces their release probability only moderately.

To test this assumption, we assumed a simple distribution of distances. Every AZ is a circular area with the same radius of 125 nm (see Table 1; for the simple distribution the variation of AZ sizes was neglected). Every AZ has only a single Ca²⁺ channel controlling the release. The channel is located at the center of the AZ. Vesicles are located at random anywhere on the AZs, with a distance ranging between 0 and 125 nm from the center, the average distance being 83 nm (Fig. 3A). The predicted average P_r, vesicle across all vesicles is 25.0% for the control condition, 15.4% when adding 10 mM EGTA, and 7.7% when adding 1 mM BAPTA (i_Ca, peak = 0.53 pA per channel, Release Model A). Even for this simple distribution the predicted average P_r, vesicle is similar to the P_r observed in experiments with added buffers (Fig. 3B). The quantitative mechanism that favors variable location of vesicles in reproducing the experimental data will be illustrated further below.

View larger version (20K): [in this window] [in a new window]   Figure 3.   Measured buffer efficacies are reproduced (within ±2 SEM) when assuming variable channel-to-vesicle distance. A, Reaction volume used in the simulation; all numbers are given in nanometers (not drawn to scale). Partial 5 and 20 nm grids indicate varying spatial resolution (see Materials and Methods). Single Ca2+ channel is located at the center in the first voxel layer on the membrane. Readily releasable vesicles (three examples shown) are located randomly anywhere on the AZ (all AZs are a circular area with a 125 nm radius centered on the Ca2+ channel). B, Comparison of model-predicted Pr with measured Pr. Filled columns, Predicted average Pr, vesicle for added exogenous buffer as ratio of control [control = endogenous fixed buffer (EFB) and ATP only]. Open columns, Measured data for calyx. Error bars indicate ± 1 SEM (see Table 2).

Property II: Release-relevant Ca²⁺ sources are separated by diffusion distances >200 nm

During APs at the calyx many Ca²⁺ channels open for every vesicle that is released (Borst and Sakmann, 1996). The experimental findings on the effects of added exogenous buffers on P_r may be used to infer a minimum distance between the Ca²⁺ channels that control phasic release. For these simulations the channels were arranged on a regular grid covering the entire presynaptic membrane (henceforth "periodic grid topography") (Fig. 4A). The separation of neighboring channels (grid constant d, 60 nm in Fig. 4A,B) was varied, adjusting the single channel conductance such that the predicted release probability for the control condition was 25%. Simulations used the uniform i_Ca mode.

View larger version (36K): [in this window] [in a new window]   Figure 4.   Effect of Ca2+ channel spacing on release probability. A, Reaction volume with the periodic grid topography used in the simulation; all numbers are given in nanometers (not drawn to scale). Partial 5 and 20 nm grids indicate varying spatial resolution. Dashed lines indicate repetition of volume with the use of periodic boundary conditions (see Materials and Methods). Ca2+ channels are located on a uniform grid with grid constant d (example shows d = 60 nm). Readily releasable vesicles are located randomly anywhere on the membrane. B, Spatial profiles of [Ca2+] (traces 1-3, snapshot at time of peak ICa) and of Pr, vesicle (traces 4-6, after 5 msec; right axis) generated by a d = 60 nm regular grid of channels under different buffer conditions; single Ca2+ channel current the same as in D. Channels are at 30 and 90 nm. Traces 1, 4, Endogenous fixed buffer (EFB) and ATP (control condition). Traces 2, 5, EFB, ATP, and 10 mM EGTA. Traces 3, 6, EFB, ATP, and 1 mM BAPTA. Squares at left axis indicate average [Ca2+] across the membrane (7.4, 4.3, and 1.9 µM for control, EGTA, and BAPTA, respectively). Triangles at right axis indicate average Pr, vesicle (= Pr, calyx) across the membrane (25, 2.4, and 0.13% for control, EGTA, and BAPTA, respectively). C, Peaks of [Ca2+] transients for different grid constants and buffer conditions (1, 4, and 7; 2, 5, and 8; 3, 6, and 9 as in B; single Ca2+ channel current same as in D). 1-3, Peak of transient at 10 nm above a channel. 4-6, Peak of average transient across the membrane. 7-9, Peak of transient at half-grid constant from two neighboring channels. D, Predicted average Pr, vesicle across the membrane (= Pr, calyx) as a function of grid constant d. Traces 1-6 show Pr, calyx for six different buffer conditions. Trace 1, EFB and 50 µM BAPTA. Trace 2, EFB and ATP (control condition). Trace 3, EFB, ATP, and 10 mM EGTA-2. Trace 4, EFB, ATP, and 10 mM EGTA. Trace 5, EFB and 1 mM BAPTA. Trace 6, EFB, ATP, and 1 mM BAPTA. Trace 7, Peak iCa per channel required to achieve Pr, calyx ~25% for the control condition (right axis).

Using d = 60 nm as an example, Figure 4B illustrates how the Ca²⁺ domains generated by individual channels in a regular grid combine into a net [Ca²⁺] transient (left axis), how this transient is modified in the presence of exogenous buffers, and how this affects the predicted release probability of vesicles at different hypothetical locations in the grid (right axis). For d = 60 nm, there is no location at which P_r, vesicle in the presence of 1 mM BAPTA is similar to that measured in the experiments (i.e., 0.35 of P_r, vesicle under control conditions). In the immediate vicinity of a channel the peak of the [Ca²⁺] transient in the presence of 1 mM BAPTA is approximately one-half of control. At this location, however, P_r, vesicle is only 0.05 of that under control conditions. For all other locations the release-reducing effect of 1 mM BAPTA relative to control is even stronger. This means that there can be no distribution of vesicles within the d = 60 nm grid that would be in agreement with the experimentally observed P_r (vesicles at fixed distance to the next channel, random locations, or any other distribution).

Extending the analysis, we varied d between 5 nm (uniform influx through the membrane) and 500 nm, thus varying the diffusion distances for Ca²⁺ between different Ca²⁺ channels. Keeping the total Ca²⁺ influx into the model calyx constant, we varied the single channel conductance as d². This leaves the peak of the average [Ca²⁺] transient (average across all locations in grid) nearly constant (8-9 µM for control, 4.5-4.9 µM for 10 mM EGTA, 1.9-2.2 µM for 1 mM BAPTA). However, the spatial profile of the [Ca²⁺] transients becomes increasingly nonuniform, leading to a steeper gradient between the spatial peaks of [Ca²⁺] in the direct vicinity of channels and the troughs of [Ca²⁺] between neighboring channels. For d = 480 nm, the peak of the predicted [Ca²⁺] transient ([Ca²⁺]_peak) in the direct vicinity of channels is 350 µM, the peak of the average transient is 9.0 µM, and the peak at distance d/2 from two neighboring channels is 4.1 µM (Fig. 4C).

Similar to the analysis for Property I, we begin by investigating whether there is any hypothetical location in the grid of channels (i.e., all vesicles located at the same distance from nearest channel) for which the predicted P_r, vesicle is in agreement with the experiments with added exogenous buffers. At d = 5 nm, [Ca²⁺]_peak in the presence of 1 mM BAPTA is 0.24 of that of control. P_r, vesicle is only 0.0024 of control, i.e., >100 times smaller than that observed in the experiments (implying a supralinearity of n = 4.2 when expressing P_r, vesicle  [Ca²⁺]ⁿ_peak). As seen in Figure 4, B and C, the reduction of [Ca²⁺]_peak in the presence of exogenous buffers relative to [Ca²⁺]_peak under control conditions is weaker the closer the location to any one channel and the larger the d. For example, if vesicles and channels were colocalized (in a grid of d = 200 nm), then [Ca²⁺]_peak in the presence of 1 mM BAPTA would be 0.75 of [Ca²⁺]_peak under control conditions, and P_r, vesicle in the presence of 1 mM BAPTA would be ~0.75⁴ = 0.32 of P_r, vesicle under control conditions (in agreement with the experiments). For the same assumption, however, predicted P_r, vesicle in the presence of 10 mM EGTA would be too high (0.7 of control). Summarizing, there is no location within a grid of Ca²⁺ channels (for any d) at which the predicted effects of BAPTA and EGTA are both in agreement with the experiments.

Similar to the simulations for Property I, the agreement between measured and modeled P_r improves when a distribution of different locations of vesicles within the grid is considered. As a simple distribution of vesicles we assumed that vesicles are located at random anywhere within the grid of channels (exact dimensions of AZs, for the moment, were neglected). Then the predicted release probability of the calyx (P_r, calyx) is given by the average P_r, vesicle across all locations in the grid. Given such a simple distribution, P_r, calyx in the control condition is nearly constant for all d. However, the predicted efficacy of added exogenous buffers in reducing P_r, calyx changes by more than two orders of magnitude (Fig. 4D). When compared with experimental data, the results for EGTA (both parameter sets) as well as for BAPTA show that d is likely to be at least 200 nm. Otherwise, the expected effect of either exogenous buffer would be far stronger than that observed in the experiments. To eliminate the uncertainty introduced by the role of ATP (for which the kinetic parameters and concentration in the native calyx are uncertain), we repeated the simulations without ATP and simulated the effect of BAPTA and endogenous fixed buffer alone (50 µM vs 1 mM BAPTA; see Materials and Methods for effect of endogenous fixed buffer). The results also imply d > 200 nm. (Fig. 4 shows simulations with Release Model A. Analogous simulations with Release Model B yielded similar results, suggesting d > 250 nm.)

In the simulations assuming a periodic grid of Ca²⁺ channels, we have, so far, used the uniform i_Ca mode (i.e., every channel of the grid opens during an AP). However, because only 10-20% of all Ca²⁺ channels may open during a single AP (Colecraft et al., 2001), the topographic pattern of open Ca²⁺ channels is different from the pattern of all, i.e., open or closed channels. In particular, the average diffusion distance of Ca²⁺ between open channels may be large (>200 nm), even if these open channels are part of a periodic channel grid with d < 200 nm.

To illustrate this, we might consider, for example, the following situation. Each vesicle is surrounded by a large uniform grid of Ca²⁺ channels (Yamada and Zucker, 1992). We used a grid of 10 × 10 channels, with a grid constant of 50 nm. The readily releasable vesicle was located in the middle of the channel field, in our example colocalized with one of the Ca²⁺ channels ("nonperiodic grid topography") (Fig. 5A). Time course and amplitude of the Ca²⁺ currents were varied across channels (stochastic i_Ca mode; see Materials and Methods). To simulate low open probability of Ca²⁺ channels during APs, we added a third gate to the two-gate Hodgkin-Huxley Model. The third gate, for which gating was independent of the membrane potential, either opened for the entire course of an AP (probability p_{open, max}) or remained closed. Thus, the peak open probability of a single Ca²⁺ channel during an AP was p_open = 69% · p_open, max, where 69% is the predicted peak open probability of the two-gate Hodgkin-Huxley Model for the AP waveform (see Materials and Methods) and p_{open, max} was varied between 100 and 15%. To yield P_r, calyx ~25% for the control condition, we set single channel conductance to 0.40 pS for p_open = 69% and scaled it as 1/p_open up to 2.67 pS for p_open = 10.4% (Release Model A). This left the average total Ca²⁺ influx per AP unchanged, independent of p_open (i_Ca, peak was 0.018 pA per channel, averaged across all open and closed channels).

View larger version (13K): [in this window] [in a new window]   Figure 5.   Effect of Ca2+ channel open probability on release probability. A, Reaction volume with the nonperiodic grid topography (100 channels on d = 50 nm grid) used in the simulation (height of reaction volume, 500 nm). Readily releasable vesicle is colocalized with the Ca2+ channel at the center of the grid. B, Predicted Pr, vesicle after adding 1 mM BAPTA as a ratio of Pr, vesicle under the control condition (control = endogenous fixed buffer and ATP only) as a function of the open probability of Ca2+ channels (popen) assumed in the three-gate channel model. For decreasing popen, single channel conductance was increased as 1/popen (see Property II for values). Results for ratios show mean ± SEM after 200-1000 Monte Carlo simulations for each data point (stochastic iCa mode). The ratio for popen = 100% is the one predicted by the uniform iCa mode (i.e., all channels open). Control condition: Pr, vesicle = 30% ± 1.5% for popen = 10.4%; Pr, vesicle = 35% ± 1.5% for popen = 69%; Pr, vesicle = 35.2% for popen = 100%.

Figure 5B shows predicted effects of added 1 mM BAPTA on P_r, calyx. The results demonstrate how the effective topography (open Ca²⁺ channels relative to vesicle) changes with decreasing p_open and how this affects whether the topography reproduces the experimental data. For p_open = 69%, the predicted release-suppressing effect of added 1 mM BAPTA is much stronger than that observed in the experiments (as expected from the result for d = 50 nm in Fig. 4D). P_r, calyx with added 1 mM BAPTA is 0.04 ± 0.005 of P_r, calyx under control conditions, i.e., approximately eight times lower than measured in the experiments (all values given as mean ± SEM after 200-800 Monte Carlo simulations). This is expected, because for high p_open the diffusional distances between open Ca²⁺ channels are too small. In contrast, for p_open = 10%, most open Ca²⁺ channels are separated by diffusion distances >200 nm. Thus the predicted effect of 1 mM BAPTA is consistent with experimental data (P_r, calyx reduced to 0.26 ± 0.06 of control). (As indicated in Table 3, the nonperiodic grid topography, although it is consistent with experiments with added BAPTA, is not consistent with experiments measuring the apparent Hill coefficient m; see Property III.)

For diffusion distances d of several hundred nanometers, required i_Ca, peak per channel is on the order of 1 pA (Fig. 4D, right axis). This current is ~5-10 times higher than the usual upper estimates for single Ca²⁺ channels (conductance ~2 pS at physiological conditions; Gollasch et al., 1992; Church and Stanley, 1996). For large d, therefore, we replaced each single channel with a cluster of 10 channels, each conducting only one-tenth of the original channel (channel-to-channel distance within cluster, ~15 nm). Predicted effects of added exogenous buffers remained almost unchanged (data not shown). The [Ca²⁺] transient provided by a single large channel is, for many quantitative arguments, indistinguishable from that of a cluster of channels. Henceforth, we will refer to either as a Ca²⁺ source.

In summary, we conclude that, for any hypothesized location of vesicles within a large field of channels or channel clusters, the average distance between neighboring (open) Ca²⁺ sources is likely to be >200 nm.

Buffer effects in single channel versus multiple channel topographies

Our result, at a first glance, may seem to contradict previous interpretations of experiments with added exogenous buffers. Previously, a substantial effect of the kinetically slow buffer EGTA in reducing P_r, particularly when compared with the effects of kinetically fast BAPTA, was used to infer relatively large diffusion distances for Ca²⁺ between release-controlling channels and vesicles (Borst and Sakmann, 1996). This is true when assuming that a vesicle is triggered by a single Ca²⁺ channel/source. The larger the distance from a single Ca²⁺ source, the stronger the predicted relative reduction of phasic release by mobile buffers (Neher, 1998b) (Fig. 2D). However, for the case of a grid of channels, the effect is more complicated. The larger the grid distance d, i.e., the larger the average diffusion distance between vesicles and their nearest channels, the less efficient the buffers are in reducing phasic release. This can be understood by approximating [Ca²⁺]_vesicle by adding up, for any one location in the grid, the single domains of all other channels (linearized steady-state approximation; Neher, 1998b). The smaller the d, the more spatially uniform are the [Ca²⁺] transients (along the membrane) and thus the more generated not only by a single nearby channel but by a large number of channels at different locations (Fig. 4C). In addition, because P_r, vesicle is a nonlinear function of [Ca²⁺]_vesicle, it is not sufficient to consider the effect of added exogenous buffers on [Ca²⁺]_vesicle alone. Instead, when considering distributions of vesicle distances, the effect of the buffers on the average release probability must be considered, too. (As seen in Fig. 4, the effect of the buffers on the average [Ca²⁺]_vesicle is independent of d, whereas the effect on average P_r, vesicle changes by more than two orders of magnitude.)

The above finding relates primarily to the probable location of Ca²⁺ sources with respect to other Ca²⁺ sources, not to the location of vesicles. As shown, inferring diffusion distances between channels and vesicles from exogenous buffer experiments depends on previous assumptions on how many Ca²⁺ channels contribute to the local [Ca²⁺] at the vesicle (single channel domain vs multiple channels domain). Therefore, we will address the question of how many Ca²⁺ channels control the release of a vesicle in a separate, independent argument.

Property III: The majority of vesicles is controlled by clusters of ~10 or more Ca²⁺ channels

Experimental modifications of the stochastic gating of Ca²⁺ channels and/or their partial blockage by toxins affect phasic transmitter release at the calyx (see Table 2). Such experiments have been used to conclude that the majority of vesicles at this synapse is controlled by more than one channel (Borst and Sakmann, 1999a; Wu et al., 1999). Extending these results, we estimate how many Ca²⁺ channels are likely to control the release of a single vesicle and how these channels are located.

Time-independent model without diffusion. We begin with a simple model (Yoshikami et al., 1989), which assumes the following: (1) The release probability (P_r, vesicle) of every vesicle is proportional to the nth power of [Ca²⁺]_vesicle, P_r, vesicle  [Ca²⁺]ⁿ_vesicle. [Ca²⁺]_vesicle is supplied by one or more channels controlling the release of the vesicle. (2) The total number of channels controlling each vesicle (N), open or closed, is the same at every release site. (3) During an AP a channel either opens (probability p_open) or remains closed (no multiple opening). (4) Each open channel contributes the same [Ca²⁺] to [Ca²⁺]_vesicle. Assuming that sites are independent, we get: P_r, calyx = (P_r, vesicle)_{average  k = 1}^Np(k)·kⁿ, where p(k) is the probability that, at any single site, k of the N channels open during an AP (binomial distribution). Because channel gating is stochastic, the number of open channels at each release site varies around the average number (p_open·N). This causes a variance of [Ca²⁺]_vesicle across sites. Therefore, only the average [Ca²⁺]_vesicle of all release sites is reduced proportionally to p. If n  1, the effect on release is nonlinear, i.e., the average release probability of all vesicles in the calyx (P_r, calyx) is not proportional to the average [Ca²⁺]_vesicle raised to the nth power. Instead_, P_r, calyx  p_open^m [the notation of m vs n follows the one in Wu et al. (1999), defined in Table 2]. Therefore, the apparent degree of supralinearity of release versus Ca²⁺ influx measured experimentally (m) may be different from n and depends on how the influx is varied (Yoshikami et al., 1989; Quastel et al., 1992). The discrepancy of m versus n is larger the higher the coefficient of variation (CV) of [Ca²⁺]_vesicle across release sites. CV, which depends on the number of Ca²⁺ channels per site, is largest for the case of a single channel, for N = 1, m = 1, independent of n (Yoshikami et al., 1989; Augustine et al., 1991). As the number of channels controlling each site increases, the CV of [Ca²⁺]_vesicle decreases and m converges to n.

N estimated with time-independent model. In presynaptic voltage-clamp recordings at the calyx, Borst and Sakmann (1999a) reduced p_open for Ca²⁺ channels by modifying the AP waveform. To simulate these experiments with the time-independent model, we calculated P_r, calyx for p_open = 69% (for the physiological AP waveform) (Fig. 6A) and for p_open = 42% (step-like AP waveform) and determined m according to P_r, calyx  p_open^m (both p_open given by a two-gate Hodgkin-Huxley Model fit to the calyx as in Borst and Sakmann, 1998). We assumed n = 3.3 (maximum possible m predicted by time-dependent diffusion model; see below). As expected, m predicted by the time-independent model was equal to 1 for N = 1 and converged to 3.3 for large N (Fig. 6B, solid line). On the basis of the experimental finding that m ~ n, the predictions for m suggest that phasic transmitter release for the majority of vesicles is likely to be controlled by at least 10 Ca²⁺ channels per vesicle. Although in the model m is a function of p_open (high vs low), the inferred minimum number of channels N is insensitive to the exact values for p_open. We also tested p_open = 75 versus 25% (yielding N > 11) and p_open = 20 versus 10% (yielding N > 14).

View larger version (21K): [in this window] [in a new window]   Figure 6.   Effect of the number of Ca2+ channels on release probability. A, ICa predicted by Hodgkin-Huxley Model for different AP waveforms. Trace 1, Physiological AP. Trace 2, Step AP to reduce peak channel open probability (popen). Trace 3, Average ICa (used in uniform iCa mode) for 12 channel cluster (physiological AP); half-width of ICa = 383 µsec; popen = 69%. Trace 4, Same as trace 3 but for step AP; popen = 42%. Trace 5, Example of stochastic ICa (used in stochastic iCa mode) for cluster of 12 channels (physiological AP). Trace 6, Same as trace 5 but for step AP. B, Effects of the number of channels per cluster on apparent Hill coefficient m (see Property III for details). Solid line, m predicted by time-independent model. Squares, m predicted by time-dependent model with the stochastic iCa mode (for all N the vesicle locations are as described for the reference topography). Error bars indicate ± SEM after 400 Monte Carlo simulations for each data point. Open circles, Apparent Hill coefficient predicted by time-dependent model with the uniform iCa mode. C, Number and position of Ca2+ channels in clusters of different N (1-100) used in the simulation. Grid indicates (5 nm)3 voxels on the membrane; the black circles indicate a voxel with a channel. The cluster with N = 100 channels is the same as the cluster with N = 1 channel except that the same voxel holds 100 channels instead of 1 (unrealistic channel density). Reaction volume, as well as location of readily releasable vesicles and of cluster centers (indicated by the black circle in N = 1 cluster) on AZs, is as in Figure 7A. D, Coefficient of variation of the total Ca2+ influx through a cluster [time integral of ICa(t), 0-5 msec] as a function of the number of channels per cluster (physiological AP waveform).

Time-dependent model with diffusion. The simplified model above neglects time-dependent buffered diffusion of Ca²⁺ as well as the exact time-dependent response of vesicles to transient [Ca²⁺]. To confirm the result on N, we implemented stochastic channel gating into the three-dimensional time-dependent model used to infer Properties I and II. Vesicles and clusters of Ca²⁺ channels were located randomly on AZs as described for the reference topography. Time course and amplitude of the Ca²⁺ currents were varied across channels (stochastic i_Ca mode; see Materials and Methods). Channel kinetics were driven either by the physiological AP waveform, resulting in p_open = 69%, or by the modified step-like waveform (p_open = 42%). Total conductance per channel cluster was 4.8 pS or 4.8 pS/N per channel. [4.8 pS is 0.33 of the conductance used to simulate release under control conditions. The lower conductance, which simulates lower [Ca²⁺] of the extracellular solution, was used so that n_model ~3 (average n observed in the experiments; see Table 2).]

N estimated with time-dependent model. When channels are gated by the step-like AP, the total influx into the calyx is 61% of that under the physiological AP. The average release probability P_r, calyx is reduced to 0.61^m, where the value of m is strongly dependent on N (Fig. 6B). We show results for seven cluster types with N varying between 1 and 100 channels per cluster (Fig. 6C). To confirm that the predicted change in m is attributable to the change in the CV (Fig. 6D) of the total Ca²⁺ influx per cluster and not attributable to different channel-to-vesicle distances inherent in clusters of varying N, we repeated the simulations for all seven clusters. This time we did not vary individual channel currents stochastically but used the uniform i_Ca mode. As expected, the predicted Hill coefficient is ~3.3 for all N (Fig. 6B, open circles).

The time-dependent model confirms the finding on N derived from the time-independent model. However, for N = 4-12, an additional effect is revealed. Because the Ca²⁺ channels in the cluster cannot all be in the same location in the membrane, the diffusion distance to the vesicle they control varies and thus the [Ca²⁺] that each open channel contributes to the combined [Ca²⁺]_vesicle varies as well. Therefore, variable diffusion distances of the channels controlling a particular vesicle raises the CV of [Ca²⁺]_vesicle (for the same N), thus increasing the discrepancy between the measured values of m versus n. To illustrate this further, we may consider again the nonperiodic grid topography simulated in Property II (Fig. 5A). At high p_open = 69%, the prediction for m (2.5 ± 0.1) is consistent with the experiments, because at high p_open each vesicle is controlled by a large number of Ca²⁺ channels at similar distances (m given as mean ± SEM after 200-800 Monte Carlo simulations). At low p_open = 10%, a vesicle still is controlled by ~10 Ca²⁺ channels (for the grid of 100 channels). However, in the grid these channels are not located at similar distances from the vesicle. Therefore, the predicted m for p_open = 10% is only 1.37 ± 0.14, far lower than that measured in the experiments.

In summary, we conclude that N ~10 or more Ca²⁺ channels control the phasic release of a single vesicle at the majority of release sites at the calyx and, further, that these channels are located at similar distances from the vesicle they control. For the N = 12 cluster in Figure 6B, the average vesicle, located at 118 nm from the cluster center, is located at 90-140 nm from an individual channel.

It should be noted that the above simulations are based on the assumption that the Hodgkin-Huxley Model is a sufficiently accurate description for gating and current of single Ca²⁺ channels. The model is probably not accurate for all Ca²⁺ channel subtypes. Just as variable distance between channels and vesicles increases the variance of [Ca²⁺]_vesicle (see above), different subtypes of Ca²⁺ channels with presumably different gating and/or conductances would produce the same effect, thus further increasing the discrepancy between m and n (for the same N). Similarly, if p_open during APs is lower than 69% assumed in the above simulations, more channels per cluster would be needed to achieve sufficiently low CV of the [Ca²⁺] reaching the vesicles. Hence our estimate N ~ 10 should be viewed as a lower limit.

Property synthesis: Phasic release is controlled by one or a few channel clusters per AZ, and vesicles are located at variable distance from the cluster(s)

Properties I-III describe three specific "requirements" on the topography of release sites at the calyx. Although there are many conceivable topographies that would exhibit one or two of these properties, very few topographies exhibit all three properties simultaneously. Further combining these requirements with recent anatomical data of the location and size of AZs in the presynaptic membrane of the calyx, one can infer a single, probable topography for the calyx.

Electron microscopic (EM) reconstruction of the calyx shows that AZs are separated by 200-800 nm from the next closest AZ (see Table 1). The average radius of an AZ is 125 nm. Therefore, >200 nm separation of Ca²⁺ sources (Property II) suggests that phasic release for the majority of AZs is controlled by a single source of Ca²⁺ per AZ, a source being a single Ca²⁺ channel or a group of Ca²⁺ channels.

As for the distances between Ca²⁺ channels and vesicles, the simulations of the effects of added exogenous buffers (Property I) and of reduced Ca²⁺ channel open probability (Property III) suggest seemingly contradicting properties. Simulations for exogenous buffer suggest nonuniformity (variable distances), whereas simulations for reduced p_open suggest uniformity (similar distances). However, these two properties are not mutually exclusive and may be treated as independent requirements on the topography. The nonuniformity relates to the distances that different releasable vesicles have to the Ca²⁺ source that controls the release of the vesicles. The uniformity relates to the distances that one vesicle has to the several individual channels of the Ca²⁺ source.

To satisfy both requirements in one topography, we suggest that Ca²⁺ channels at AZs appear in clusters, with 10 or more channels per cluster and with a maximum distance between any two channels in the cluster of ~50 nm. Ten Ca²⁺ channels on a circular area with a diameter of 50 nm correspond to one channel per 14 × 14 nm² membrane area. This is consistent with estimates of channel-to-channel distances for other synapses (Stanley, 1997). For any one vesicle located, for example, 100 nm away from the center of the cluster, the distances between the Ca²⁺ channels and this vesicle are "similar" (75-125 nm). As a second, independent property of the reference topography, we suggest that different releasable vesicles (at the same AZ or at other AZs) are located at different distances from the Ca²⁺ channel cluster. Note that the simulations to reproduce the measured effects of added exogenous buffers on P_r cannot predict the exact distribution of cluster-to-vesicle distances. However, the simulations indicate that the CV of the distribution must be ~0.5 or larger.

Spatiotemporal pattern of [Ca²⁺] transients and phasic transmitter release

We have derived the topographic properties likely to be found at release sites of the calyx. However specific, these properties do not define the topography in every detail. In particular, they cannot define the exact distances between readily releasable vesicles and their Ca²⁺ sources; different distributions of cluster-to-vesicle distances may result in similar net P_r, calyx and thus may reproduce the experimental data equally well. Below, we propose a possible specific topography, which is consistent with the above properties and the anatomic data. Using this topography in the model, we simulate the physiological [Ca²⁺] transients that control phasic transmitter release at AZs.