 |
Previous Article | Next Article 
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 |
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 |
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 |
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.
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 ICa
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
[iCa(t)], the simulations used either the "uniform iCa mode" or
the "stochastic iCa mode." In the
uniform iCa mode all channels are
"open" and iCa(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 iCa mode the model varies
iCa(t) for each channel
stochastically. In this mode the individual channel locations
contribute different iCa(t), and some remain
closed. Individual iCa(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
iCa(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 iCa [given
as the peak amplitude of
iCa(t),
iCa, 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 ICa 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+] = iCa(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);
dtmax = 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 DCa = 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 KD = 2 µM (varied in sensitivity analyses between 200 nM and 200 µM; see
below), and a forward Ca2+ binding rate,
kon = 5·108 per Msec (Klingauf and Neher,
1997 ) (sensitivity analysis below); ATP with
[ATP]total = 0.58 mM,
KD, Ca = 200 µM, kon, Ca = 5·108 per Msec (Baylor and Hollingworth,
1998 ) (with kon, Ca corrected for
temperature; sensitivity analysis below),
DATP = 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
(KD, Mg = 100 µM) and thus unavailable for
Ca2+ buffering (slow off-rate of Mg-ATP,
koff, Mg = 150-390/sec; Baylor and
Hollingworth, 1998 ). For some simulations, mobile exogenous buffers
were added at varying concentrations: BAPTA
(kon = 4·108 per Msec,
KD = 220 nM, DBAPTA = 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"
(kon = 10·106 per Msec,
KD = 70 nM;
Nägerl et al., 2000 ) or "EGTA-2"
(kon = 2.5·106 per Msec,
KD = 180 nM;
Naraghi and Neher, 1997 ). DEGTA = DEGTA-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 ICa, 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
Pr (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 Pr
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 Pr 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
Pr 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 Pr, 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 (Pr, calyx).
Different release site topographies predict different
Pr, 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 Pr 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
Pr to these parameters. Even drastic
variations in the parameters of endogenous buffers (two to three orders
of magnitude) do not change the predicted Pr, 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,
Pr, calyx = 25%).
Endogenous fixed buffer. We varied [EFB] between 10 and
10,000% of the reference value (80 µM) while adjusting
KD, EFB such that
[EFB]/KD, EFB (approximate binding ratio) was kept unchanged at 40. This variation changes the predicted
Pr, 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 Pr 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]/KD, ATP
constant, we changed both [ATP] and KD,
ATP between 10 and 1000% of their respective reference values (see above). This changes
Pr, calyx between 37 and 22%,
respectively. (2) Keeping [ATP]·kon
(buffer) product constant, we changed both [ATP] and (inversely)
kon, ATP, between 10 and 1000% of the
reference values. This changes
Pr, calyx between 44% (low [ATP],
high kon, ATP) and 12% (low
kon, ATP, high
[ATP]). Complete removal of ATP from the model calyx
results in Pr, calyx = 48%. For
simulations with the added exogenous buffers EGTA or BAPTA, the
presence of ATP changes the predicted
Pr, calyx even less because the
effect of ATP is small compared with that of the exogenous buffer (see
Fig. 4D).
 |
RESULTS |
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
(Pr) 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.
(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, Pr
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,
Pr is reduced in a
concentration-dependent manner. For example, 10 mM EGTA reduces Pr to ~0.45 of
Pr in the native calyx. BAPTA (1 mM), which binds Ca2+ faster than EGTA, reduces
Pr to ~0.35 of
Pr in the native calyx (see Table 2). The
reduction of Pr 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 Pr) 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
(Pr, 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 [iCa(t);
approximately Gaussian] was calculated with a Hodgkin-Huxley Model.
In these simulations iCa(t)
was the same for all channels (uniform
iCa mode; see Materials and Methods).
Depending on the conductance, peaks of
iCa (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
Pr, 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
Pr, 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 Pr, 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 Pr, 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 Pr, 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 Pr, 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
(iCa, peak = 0.53 pA per channel,
Release Model A). Even for this simple distribution the predicted
average Pr, vesicle is similar to the
Pr 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 Pr 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
iCa 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
Pr, vesicle in the presence of 1 mM BAPTA is similar to that measured in the
experiments (i.e., 0.35 of
Pr, 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, Pr, 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
Pr (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 Pr, 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. Pr, 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 Pr, vesicle [Ca2+]npeak).
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 Pr, vesicle
in the presence of 1 mM BAPTA would be
~0.754 = 0.32 of
Pr, vesicle under control conditions
(in agreement with the experiments). For the same assumption, however,
predicted Pr, 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 Pr 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
(Pr, calyx) is given by the average Pr, vesicle across all locations in
the grid. Given such a simple distribution,
Pr, calyx in the control condition is
nearly constant for all d. However, the predicted efficacy of added exogenous buffers in reducing
Pr, 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 iCa 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
iCa 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 popen, max) or remained
closed. Thus, the peak open probability of a single
Ca2+ channel during an AP was
popen = 69% · popen, 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
popen, max was varied between
100 and 15%. To yield Pr, calyx
~25% for the control condition, we set single channel conductance to 0.40 pS for popen = 69% and scaled it
as 1/popen up to 2.67 pS for
popen = 10.4% (Release Model A). This
left the average total Ca2+ influx per AP
unchanged, independent of popen
(iCa, 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
Pr, calyx. The results
demonstrate how the effective topography (open
Ca2+ channels relative to vesicle) changes
with decreasing popen and how this
affects whether the topography reproduces the experimental data. For
popen = 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).
Pr, calyx with added 1 mM BAPTA is 0.04 ± 0.005 of Pr, 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
popen the diffusional distances between open Ca2+ channels are too small.
In contrast, for popen = 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
(Pr, 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 iCa, 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 Pr, 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 Pr, 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
Pr, 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 (Pr, vesicle)
of every vesicle is proportional to the nth power of
[Ca2+]vesicle,
Pr, vesicle [Ca2+]nvesicle.
[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
popen) or remains closed (no multiple
opening). (4) Each open channel contributes the same
[Ca2+] to
[Ca2+]vesicle.
Assuming that sites are independent, we get:
Pr, calyx = (Pr, vesicle)average k = 1Np(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
(popen·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
(Pr, calyx) is not proportional to
the average
[Ca2+]vesicle
raised to the nth power. Instead,
Pr, calyx popenm [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 popen for
Ca2+ channels by modifying the AP
waveform. To simulate these experiments with the time-independent
model, we calculated Pr, calyx for
popen = 69% (for the physiological AP
waveform) (Fig. 6A) and
for popen = 42% (step-like AP
waveform) and determined m according to
Pr, calyx popenm (both
popen 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 popen (high vs low), the inferred
minimum number of channels N is insensitive to the exact
values for popen. We also tested
popen = 75 versus 25% (yielding
N > 11) and popen = 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
iCa mode; see Materials and Methods).
Channel kinetics were driven either by the physiological AP waveform,
resulting in popen = 69%, or by the
modified step-like waveform
(popen = 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 nmodel ~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 Pr, calyx is reduced to
0.61m, 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
iCa 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
popen = 69%, the prediction for
m (2.5 ± 0.1) is consistent with the experiments,
because at high popen 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
popen = 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 popen = 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
popen 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
popen 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 Pr 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
Pr, 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.
|