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Brevity of the Ca²⁺ Microdomain and Active Zone Geometry Prevent Ca²⁺-Sensor Saturation for Neurotransmitter Release

¹Departments of Physics, Simon Fraser University, Burnaby and ²Department of Biology, University of Victoria, Victoria, British Columbia, Canada

Submitted 10 March 2005; accepted in final form 4 May 2005

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
The brief time course of the calcium (Ca²⁺) channel opening combined with the molecular-level colocalization of Ca²⁺ channels and synaptic vesicles in presynaptic terminals predict sub-millisecond calcium concentration ([Ca²⁺]) transients of 100 µM in the immediate vicinity of the vesicle. This [Ca²⁺] is much higher than some of the recent estimates for the equilibrium dissociation constant of the Ca²⁺ sensor(s) that control neurotransmitter release, suggesting release should be close to saturation, yet it is well known that release is highly sensitive to changes in Ca²⁺ influx. We show that due to the brevity of the Ca²⁺ influx the binding kinetics of the Ca²⁺ sensor rather than its equilibrium affinity determine receptor occupancy. For physiologically relevant Ca²⁺ currents and forward Ca²⁺ binding rates, the effective affinity of the Ca²⁺ sensor can be several-fold lower than the equilibrium affinity. Using simple models, we show redundant copies of the binding sites increase effective affinity of the Ca²⁺ sensor for release. Our results predict that different levels of expression of Ca²⁺ binding sites could account for apparent differences in Ca²⁺ sensor affinities between synapses. Using Monte Carlo simulations of Ca²⁺ dynamics with nanometer resolution, we demonstrate that these kinetic constraints combined with vesicles acting as diffusion barriers can prevent saturation of the Ca²⁺-sensor(s) for neurotransmitter release. We further show the random positioning of the Ca²⁺-sensor molecules around the vesicle can result in the emergence of two distinct populations of the vesicles with low and high release probability. These considerations allow experimental evidence for the Ca²⁺ channel-vesicle colocalization to be reconciled with a high equilibrium affinity for the Ca²⁺ sensor of the release machinery.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
It is well established that Ca²⁺ influx into the presynaptic nerve terminal through voltage-gated Ca²⁺ channels triggers fusion of the synaptic vesicles and neurotransmitter release. Recent molecular biological studies indicate direct physical and functional interactions between some voltage-gated Ca²⁺ channels and vesicle-associated proteins (Spafford and Zamponi 2003). High-resolution electron microscopic studies at vertebrate neuromuscular junctions provide strong evidence for colocalization, on the order of 10–30 nm between the vesicle and intra-membrane particles (thought to be Ca²⁺ channels), consistent with the separations predicted by binding of channels to synaptic proteins (Harlow et al. 2001). Such extreme colocalization of Ca²⁺ channels with vesicles (and their associated release machinery) leads to high local [Ca²⁺] of order 100 µM at the vesicle (Simon and Llinas 1985). Also overlap of the Ca²⁺ microdomains from a few open channels can produce high [Ca²⁺] of over 100 µM at a region larger than a single-channel microdomain (Yamada and Zucker 1992). Using a low-affinity calcium-dependent photoprotein, high Ca²⁺ microdomains on the order of 200–300 µM close to the plasma membrane have been observed in the squid giant synapse (Llinas et al. 1992).

Neurotransmitter release appears to be triggered by the cooperative action of Ca²⁺ ions on three to five Ca²⁺ binding sites (Bollmann et al. 2000; Dodge and Rahamimoff 1967; Heidelberger et al. 1994; Schneggenburger and Neher 2000). The classical view is that at least one of the binding sites of the Ca²⁺ sensor has a low affinity with a dissociation constant (K_d) of 100 µM (Zucker 1993). Assuming [Ca²⁺] of 100 µM at the Ca²⁺-sensor, a low affinity is predicted from the fact that the Ca²⁺ sensor is not saturated under normal physiological conditions. Increasing [Ca²⁺] in the bath increases release and decreasing [Ca²⁺] (even by small amounts) decreases release substantially (Dodge and Rahamimoff 1967). A low equilibrium binding affinity is consistent with caged Ca²⁺ photolysis experiments performed using goldfish retinal bipolar neurons (Heidelberger et al. 1994). However, this classical view of a low-affinity binding site was challenged by similar experiments in which caged Ca²⁺ was used to stimulate release from the calyx of Held (Bollmann et al. 2000; Schneggenburger and Neher 2000). In this case, [Ca²⁺] dependence of release for this synapse was best fitted by a model with five Ca²⁺ binding sites with K_d of 10 µM and two Ca²⁺-independent steps (Bollmann et al. 2000). The nonsaturation of release by the high local [Ca²⁺] produced during an action potential is paradoxical if one assumes release is controlled by high-affinity Ca²⁺ binding sites, especially for synapses with highly colocalized channels and vesicles such as vertebrate neuromuscular junctions (Harlow et al. 2001). This problem can be avoided in principle in an active zone with noncolocalized channel vesicles because the [Ca²⁺] would be lower at the sensor if the vesicles and channels are further apart. Thus in the calyx of Held, a model that assumes a nonuniform channel-vesicle topography with average separations of 100 nm is successful in describing most aspects of release (Meinrenken et al. 2002). However, because of the steep dependence of release on [Ca²⁺], it becomes difficult to impossible to accommodate observed release probabilities if this separation is reduced by a few tens of nanometer.

Although the affinity of the Ca²⁺ sensors has been actively debated, their chemical kinetics has been given less attention. The on rate of the Ca²⁺-sensor molecule remains poorly known, although some recent studies suggest it is 10⁷-10⁸ M^–1s^–1 (Davis et al. 1999; Millet et al. 2002). Models of Ca²⁺-triggered release implemented in different studies have employed a large range of k_on values [for example, 10⁶ M^–1s^–1 (see Bertram et al. 1999), 10⁷ M^–1s^–1 (see Bennett et al. 2000) and 5 x 10⁸ M^–1s^–1 (see Tang et al. 2000)]. The binding kinetics will be crucial to determining the results of release models and release probability if the reaction with Ca²⁺ does not reach equilibrium. In the context of cytoplasmic Ca²⁺ buffering, it has been established that the binding kinetics rather than the affinity determines the effectiveness of a particular buffer for inhibiting Ca²⁺-triggered release (Adler et al. 1991).

Among the vesicle-associated proteins, synaptotagmin is the best Ca²⁺-sensor candidate for Ca²⁺-triggered release (Augustine 2001; Chapman 2002). Yet the role of different binding sites of synaptotagmin, their chemical affinity, kinetics, position relative to the vesicle and Ca²⁺ channel, and whether all the Ca²⁺-binding sites needed for release are on the same molecule or on different molecules continue to be debated (Bai and Chapman 2004; Koh and Bellen 2003). The cooperation of three to eight SNARE complexes may be needed for fusion (Han et al. 2004; Hua and Scheller 2001). There is a synaptotagmin associated with each SNARE complex, and evidence suggests their interaction plays a role in fusion (Bai et al. 2004). If we assume up to five binding sites on each synaptotagmin molecule and three to eight synaptotagmin molecules per vesicle, the potential number of binding sites could be well above the observed Ca²⁺ cooperativity. Consequently, the Ca²⁺ binding sites that need to bind to Ca²⁺ to trigger release could be distributed around the vesicle (Stewart et al. 2000). Release models with Ca²⁺ ions binding to a subset of a larger number (>5) of potential binding sites to cause release have not yet been considered.

In this study, we address the possibility of nonsaturation of release in a synapse with colocalized active zone topography and high affinity Ca²⁺ sensor(s) using a detailed assessment of the temporal and spatial aspects of neurotransmitter release. In the first part of this study, our analysis relies primarily on the fact that the Ca²⁺ sensor(s) are exposed to high [Ca²⁺] for only a fraction of a millisecond during and immediately after the channel opening (Llinas et al. 1982, 1995; Simon and Llinas 1985). Therefore chemical equilibrium may not be achieved, and the forward binding kinetics of the sensor, rather than the equilibrium affinity, becomes the critical variable determining the extent of Ca²⁺ binding. Next, we introduce two simple models that allow for the possibility of more than five binding sites for release consistent with biochemical evidence as discussed in the preceding text. We explore the effect of extra binding sites on release probability using simple rate equations. Then using a Monte Carlo simulation of Ca²⁺ dynamics with nanometer spatial resolution, we assess the effect of the vesicle as a barrier to free diffusion of Ca²⁺ ions on release. Finally using the Monte Carlo analysis we explore the effect of the position and number of potential binding sites around the docked vesicle on release.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Invasion of the nerve terminal by an action potential opens voltage-gated Ca²⁺ channels. Ca²⁺ ions enter and diffuse within the nerve terminal where they bind reversibly to fixed and mobile endogenous buffers. Remaining free Ca²⁺ ions can bind to the Ca²⁺-sensor sites on the release machinery to initiate fusion of synaptic vesicles. The release follows the onset of the brief influx of Ca²⁺ with a delay of a fraction of millisecond (Llinas et al. 1981; Sabatini and Regehr 1996).

For most of this study, following Bollmann et al. (2000), we used a general release model with five independent Ca²⁺ binding sites and two Ca²⁺-independent fusion steps

(1)

where X represents the Ca²⁺ sensors, which comprise five Ca²⁺ binding sites, XCa₅* is an intermediate state prior to fusion, and F represents a complete fusion event. k_on and k_off are the on rate and off rate of the reaction. The fusion rates, , , and were fixed to 30,000, 8,000, and 40,000 s^–1, respectively, as estimated in Calyx of Held (Bollmann et al. 2000). We call this release model 1.

The number of potential Ca²⁺ binding sites per docked vesicle could exceed the observed Ca²⁺ cooperativity of release because several synaptotagmin molecules, each with five potential binding sites, may comprise each docking complex. One of the simplest generalizations of release model 1 is to assume several (m > 5) identical binding sites per vesicle, with binding to any five sites allowing fusion to proceed. In this case the release model presented in Eq. 1 will be slightly modified

(2)

where the forward rates are now dependent on m, the number of available binding sites. Release model 1 is a special case of release model 2 for which the number of binding sites is equal 5 (m = 5).

Another possibility is to assume five clusters of binding sites each comprising k independent sites. Release proceeds only if at least one Ca²⁺ is bound to each cluster. In this model (release model 3), there are more than five binding sites (5k in total), but unlike model 2, more than five Ca²⁺ ions will likely bind before the conditions required to initiate fusion are achieved.

In the first part of the study (Figs. 1 and 2), we assume the [Ca²⁺] at the sensor is constant and that it persists for either 0.3 or 1 ms during the opening of the channel and briefly afterward. These are good approximations because [Ca²⁺] reaches its steady state in the immediate vicinity of the channel within microseconds of the opening of the channel and dissipates rapidly after closing (Llinas et al. 1995). The release probability predicted for a given set of rate constants can be calculated from the rate equations describing the reaction scheme in Eq. 1 (or Eq. 2). Figure 1 is produced using this procedure for 10,000 different choices of k_on [Ca²⁺] and K_d (=k_off/k_on). Figure 2 is produced by solving Eq. 1 for different values of [Ca²⁺] and k_on and Eq. 2 for different values of [Ca²⁺] and m.

View larger version (65K): [in this window] [in a new window]   FIG. 1. Release probability as a function of on rate and affinity of the Ca2+ sensor in response to a single channel opening that lasts 0.3 ms for release model 1. The [Ca2+] dependence of the rate of forward reaction in Eq. 1 makes it natural to plot release probability as a function of the Kd (dissociation constant of the sensor) divided by [Ca2+] against kon (on rate of the sensor) times [Ca2+]. So for a given [Ca2+], this graph specifies the minimum kon that is required for release probability to be dependent only on Kd. This condition is indicated by the regions where the constant release probability curves become parallel to the kon x [Ca2+] axis. Estimated kon and Kd values for Calyx of Held in the presence of 15 µM () or 200 µM () [Ca2+].

View larger version (40K): [in this window] [in a new window]   FIG. 2. Release probability as a function of [Ca+2] for a Ca2+ sensor with Kd = 10 µM. Release probability using release model 1 for different values of the on-rate with Ca2+ current durations of 0.3 ms in (A) and 1 ms in (B). In A and B, kon values are indicated on the graph beside each trace in M–1s–1. The intersection of the vertical bold line and the trace of kon = 1010 M–1s–1 defines the Peq for each panel. The intersection of the horizontal bold line and each trace indicates the effective affinity associated with each combination of on-rate and TCa. C: release probability using release model 2 with a Ca2+ current duration of 0.3 ms assuming the number of binding sites varies from 5 to 25 with kon = 108 M–1s–1.

The Monte Carlo simulation employed for the second part of this study is adopted from Shahrezaei and Delaney (2004). In the Monte Carlo simulation method, the motion of each individual Ca²⁺ or buffer molecule is followed as it diffuses inside the nerve terminal. This is not done at the level of actual Brownian motion but rather at a coarser level, using random walk theory. Ca²⁺ ions bind to the binding sites of the Ca²⁺ sensor with some probability if they get closer than a certain distance. Therefore in this part the forward reaction rate is diffusion limited.

Because of the stochastic nature of the Monte Carlo method, repeated trials are required to assess the average behavior of the system. The steady-state concentration profile in this study (Fig. 3B) is the result of averages of 1,000 1-ms-long Monte Carlo runs corresponding to 10⁸ independent sampling events.

View larger version (36K): [in this window] [in a new window]   FIG. 3. Effect of geometry of release site on release probability. A: schematic view of the arrangement of Ca2+ channel (red), vesicle (orange), the Ca2+-sensor(s) molecule synaptotagmin (green) and SNARE complex (light blue). Association of the channel and the SNAREs allows for possible functional interactions between them. B: steady-state [Ca2+] after opening of the Ca2+ channel and constant [Ca2+] contour profiles for a cross section through the channel at (0, 0), the center of the vesicle at (20 nm, 25 nm) and the docking site at (20 nm, 0). C: schematic view of possible Ca2+-sensor site positions a to e around the vesicle (top). Release probability for release model 1 for a single-channel opening for different choices for the position of 5 Ca2+ binding sites. Results for 5 binding sites clustered at a single point, a, b, c, d, or e (red bars) or some binding sites at point a with the rest at e (green bars) are shown. D: schematic view of a uniform distribution of 5 clusters of Ca2+-sensor sites at positions a to e around the vesicle (top). Red bars are release probabilities assuming k binding sites (k = 1 to 5) located at each position with binding of at least one Ca2+ at each cluster required for release (model 3). Green bar is release probability assuming 10 binding sites, equivalent to k = 2 but with binding to any 5 sites being sufficient to cause release (model 2).

View larger version (24K): [in this window] [in a new window]   FIG. 4. Two populations of vesicles. A: histogram of release probabilities for release model 1 assuming random choices for the position of the Ca2+-sensor molecule(s) for 2 cases of clustered and unclustered binding sites. B: histogram of release probabilities for release model 2 assuming random choices for the position of the Ca2+-sensor molecule(s)'s binding sites (unclustered) with 10 or 20 binding sites.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

Due to brevity of the Ca²⁺ influx, the binding kinetics of the Ca²⁺ sensor can be critical to determining release probability. To address the effect of binding kinetics on release, we use release model 1, which has five independent Ca²⁺ binding sites and two final Ca²⁺-independent fusion steps (Eq. 1). We examine the release probability in response to a single-channel opening for a wide range of k_on and K_d of the Ca²⁺ sensor. The release probability is dependent on the equilibrium affinity of the site (1/K_d) and independent of the on rate, if equilibrium between free Ca²⁺ and Ca²⁺ binding sites is achieved. Figure 1 shows that for a given [Ca²⁺], if k_on is greater than a critical value, equilibrium will be largely achieved. The regime, where binding between Ca²⁺ and the sensor is at equilibrium, is evident where the constant release probability contours are parallel to the k_on[Ca²⁺] axis (Fig. 1). Importantly, though, if the on rate is smaller than this critical value, the release probability will depend strongly on the on rate and will be almost independent of the K_d. This corresponds to the portion of Fig. 1 where the constant release probability contours are parallel to the K_d/[Ca²⁺] axis. The forward rate of reaction is proportional to both k_on and [Ca²⁺] (Eq. 1). Consequently a site that experiences lower [Ca²⁺] needs a higher k_on to achieve equilibrium. For example, for a set of sensors with K_d = 10 µM that transiently experience [Ca²⁺] = 200 µM, equilibrium conditions are achieved for k_on of 3 x 10⁸ M^–1s^–1. If the same set of sensors experience [Ca²⁺] = 15 µM, equilibrium would be achieved for k_on of 2 x 10⁹ M^–1s^–1.

The time constant for the exponential approach to equilibrium for the reaction between a single Ca²⁺ binding site and free Ca²⁺ is described by the following

(3)

Although the cooperativity between the binding sites for fusion makes Eq. 1 more complicated than a simple collection of five equal and independent binding sites, _eq provides a good approximation of the approach to equilibrium. A good rule of thumb is that the kinetic rates of the Ca²⁺-sensor are important if the duration of the current (T_Ca) is comparable to or smaller than the equilibrium time constant of the reaction (_eq).

To investigate the effect of the on rate of the Ca²⁺ sensor, we fixed the K_d at 10 µM and looked at the release probability as a function of the [Ca²⁺] at the sensor. We did this for two durations of Ca²⁺ current (T_Ca) 0.3 and 1 ms (Fig. 2, A and B) and five different values for the on rate between 10⁶ and 10¹⁰ M–¹s^–1. It should be noted that in reality, there is a limit to the on-rate because the arrival of new molecules at the binding site is limited by diffusion. This effect is absent from our analysis in this section because we are just solving the rate equations (Eq. 1). The diffusion-limited rate for Ca²⁺ reactions in the cytoplasm is thought to be 5 x 10⁸ to 10⁹ M^–1s^–1 and that limits how fast the binding site can function and also the effect of multiple binding sites. For k_on = 10¹⁰ M^–1s^–1 (which is well above the diffusion limited rate) the Ca²⁺ reaction is close to equilibrium because higher k_on values do not increase the release probability. So for Ca²⁺ durations of 0.3–1 ms, the release probability for k_on = 10¹¹ M^–1s^–1 or higher would liealmost on top of the line for k_on = 10¹⁰ M^–1s^–1. At lower on rates, release probability is reduced significantly. Although this effect is larger for the shorter T_Ca of 0.3 ms (Fig. 2A), it is still significant for T_Ca of 1 ms (Fig. 2B). Because there is more time for the Ca²⁺ reaction to fully equilibrate and there is more time for the final Ca²⁺-independent fusion step to proceed, release probabilities for the same equilibrium affinity and binding kinetics are higher for longer duration currents. Therefore for a given equilibrium affinity the release probability is invariant under the following scaling of the parameters by a constant (c): T_Ca T_Ca x c, k_on k_on/c, /c, /c, and /c. In the limit of large fusion rates (, , and ), this scaling freedom simplifies to the scaling of the duration of the Ca²⁺ influx and the on rate of the sensor. The estimates for the Ca²⁺-independent fusion rates under physiological conditions (Bollmann et al. 2000) are not fast enough to put us in the preceding mentioned simplified scaling regime with the result that Fig. 2B is not just a simple scaling transformation of Fig. 2A.

The on rate of the Ca²⁺ binding and the duration of the [Ca²⁺] transient combine to create an "effective affinity" that is lower than the true equilibrium affinity. To compare the results of our release model under equilibrium and nonequilibrium conditions, we define "P_eq" as the release probability at [Ca²⁺] = K_d for a fast Ca²⁺ sensor (that is whereequilibrium between Ca²⁺ and its binding sites is rapidly achieved). P_eq depends on the duration of the Ca²⁺ current transient. For example, for K_d = 10 µM and Ca²⁺ current duration of 0.3 ms, P_eq is 0.2, whereas for a current duration of 1 ms, it increases to 0.5 (Fig. 2, A and B). Using P_eq we define the effective dissociation constant of a Ca²⁺ sensor with a finite on rate as the [Ca²⁺] that produces a release probability equal to P_eq. By this definition, the effective dissociation constant of a Ca²⁺ sensor with K_d = 10 µM and k_on = 10⁸ M^–1s^–1 is 50 µM if T_Ca = 0.3 ms, which is fivefold larger than its equilibrium dissociation constant (Table 1).

Release probability is proportional to [Ca²⁺]ⁿ for small [Ca²⁺] and n, the power of this [Ca²⁺] dependence, reflects the Ca²⁺ cooperativity of release. Because in the models we used five Ca²⁺ ions are needed for release, the Ca²⁺ cooperativity (n, the limiting slope seen in Fig. 2 for low [Ca²⁺]) is exactly 5. This slope for low [Ca²⁺], where the relationship in a log-log plot is linear, is independent of the duration of the current, the number of potential binding sites (m) and whether the Ca²⁺ reaction is at equilibrium or not. This is consistent with the fact that measurements performed under conditions where the time scales for [Ca²⁺] experienced at the Ca²⁺ sensor are different obtain similar results for the Ca²⁺ cooperativity of release (Dodge and Rahamimoff 1967; Heidelberger et al. 1994). However, if the range of concentrations used is close to saturation, then longer durations of current could result in an apparently lower cooperativity. This phenomenon has been observed in the context of glutamate binding to N-methyl-D-aspartate receptors (Chen et al. 2001).

Geometry of active zone

The micro-geometry of the vesicle channel Ca²⁺ sensor(s) complex is critical for release because Ca²⁺ influx through the channel produces a very steep [Ca²⁺] gradient in the vicinity of the channel (Fogelson and Zucker 1985; Simon and Llinas 1985). To use the results of our kinetic analysis in the context of a spatially realistic model of neurotransmitter release, we employ a Monte Carlo method with nanometer resolution to simulate the Ca²⁺ microdomain of a Ca²⁺ channel in the presence of a colocalized docked vesicle (Shahrezaei and Delaney 2004). We assume the docked vesicle is in contact with the presynaptic membrane and 20 nm separated from the channel by virtue of its interaction with SNARE proteins. We assume the Ca²⁺ sensor(s) are situated at a radius of 10 nm around the contact point, which is a plausible model considering our current knowledge of the structure of the release machinery complex (Fig. 3A ). A mobile endogenous buffer (0.5 mM) with a rather high affinity (K_d = 2 µM), fast kinetics (k_on = 3 x 10⁸ M^–1s^–1) and slow diffusion (diffusion coefficient for free and bound buffer is 27.5 µm²s^–1) is included (Burrone et al. 2002). The [Ca²⁺] profile at the immediate vicinity of the channel and around the vesicle is not strongly dependent to the choice of the buffer for moderate concentrations of the buffer (Shahrezaei and Delaney 2004). This is the case as long as the buffer binding kinetics are Ca²⁺ diffusion limited. We assume a Ca²⁺ current of 0.3 pA through the channel that lasts for 0.3 ms (280 Ca²⁺ ions) (Stanley 1993). The [Ca²⁺] profile close to the channel reaches to >90% of its steady-state value within tens of microseconds. Because the vesicle diameter is comparable with the channel-vesicle distance, the synaptic vesicle acts as a diffusion barrier and modifies the shape of the steady-state Ca²⁺ microdomain in its vicinity (Fig. 3B). [Ca²⁺] on the channel side of the vesicle is higher than the corresponding [Ca²⁺] in the absence of the vesicle due to the reflection of Ca²⁺ by the vesicle. Conversely blocking of Ca²⁺ diffusion by the vesicle reduces [Ca²⁺] on the opposite side of the vesicle from the channel. Therefore the Ca²⁺ sensor is potentially exposed to markedly different [Ca²⁺] depending on its position around the vesicle, differing by as much as 13-fold (from 15 to 200 µM) (Shahrezaei and Delaney 2004).

To explore the effect of the position of the Ca²⁺ sensor on its saturation, we fixed the parameters of our release models to those measured in the calyx of Held (K_d = 10 µM, k_on = 3 x 10⁸ M^–1s^–1) (Bollmann et al. 2000). For release model 1, with all Ca²⁺ binding sites on the channel side of the vesicle, the release probability for a single-channel opening is 0.99 and release is highly saturated (the reaction in this case has reached to an equilibrium). Shielding only one of the binding sites from the channel by placing it behind the vesicle protects the release from saturation and reduces the release probability to 0.53 (Fig. 3C). With all the Ca²⁺ binding sites on the other side of the vesicle, the release probability drops to 0.02 even though the [Ca²⁺] level is still higher than the K_d (equilibrium is not achieved due to briefness of the Ca²⁺ transient and [Ca²⁺] is lower than the effective affinity). These results show that release can be extremely sensitive to the position of the Ca²⁺-sensor(s).

Uniform distribution of five Ca²⁺ binding sites around the vesicle (every 72° as illustrated in Fig. 3D, top) results in a release probability of 0.3. To test the effect of more than five binding sites, we employed models 2 and 3. When we consider five clusters of k (2–5) binding sites release probability increases by an amount depending on the number of extra binding sites (k) and type of release model (Fig. 3D). For model 3 where binding to at least one binding site of each cluster around the vesicle is needed, release probability is smaller than model 2 where binding to any five binding sites is sufficient to trigger release. Also for model 3, the effectiveness of adding more binding sites is reduced at the point where the diffusion-limited rate is approached. Thus increasing from 5 binding sites to 10 has a more significant effect on release than going from 10 to 20 (Fig. 3D).

An unexpected result is the emergence of two populations of vesicles with distinct release probabilities when one allows a variable position for the Ca²⁺-binding sites (Fig. 4 ). We consider two cases, first, all five binding sites clustered at a single random position and second, five binding sites at different random positions around the vesicle. The average release probability is slightly greater for the multi-site case (0.55) than the single clustered site case (0.49), but the variance is almost the same. Interestingly the histogram for both geometries is bimodal; most of the time the release probability for the vesicle is either very high or very low (Fig. 4A). Therefore if the position of the Ca²⁺-sensor sites around the vesicle is not constrained to the channel, then vesicles are roughly divided into two populations with high and low release probabilities. This result is quite robust against small variations of parameters and as long as the effective dissociation constant for the Ca²⁺ sensor is between the ranges of concentrations that occur around the vesicle then the histogram of release probabilities remains bimodal. This bimodal distribution can be understood from a simple geometrical perspective. The gradient of [Ca²⁺] with respect to position around the circumference of the circle defining potential binding sites is the smallest at positions close to and far from the channel (because at these positions the circumference is almost perpendicular to the channel-vesicle axis). Therefore there will be disproportionately more locations corresponding to large or small [Ca²⁺] because the Ca²⁺ sensors are assumed to be distributed randomly around the vesicle.

To see the effect of having more binding sites randomly situated around the vesicle on the population of vesicles with low and high release probability, we use model 2 with 10 and 20 binding sites (Fig. 4B). The population of vesicles with low release probability will decrease, but interestingly, there still exist two distinct populations of vesicles with low and high release probability. Also doubling the number of binding sites from 10 to 20 does not change the average release probability (0.7) and the shape of the histogram significantly. This is because the reaction approaches the diffusion-limited rate of arrival of Ca²⁺ ions to the potential sensor sites around the vesicle.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
In summary we suggest that it is possible to reconcile the colocalized active zone topography with a high-affinity Ca²⁺ sensor for transmitter release to produce release probabilities less than unity with a high sensitivity to changes in Ca²⁺ influx. This is a direct consequence of the fact that during an action potential the Ca²⁺ sensor is exposed to a brief Ca²⁺ transient, and in general the sensor does not reach chemical equilibrium with free Ca²⁺ ions. In addition to the kinetics of the reaction, the geometry of some active zones may help to prevent saturation of the sensor since the sensor may not be exposed to [Ca²⁺] levels as high as traditionally assumed due to the effect of vesicles acting as local diffusion barriers. This blocking effect would be particularly relevant in linear active zones like those of amphibian or mammalian neuromuscular junction.

The effective affinity of the Ca²⁺ sensor under physiological conditions could be different from the actual affinity of the sensor molecule, depending on the duration of the Ca²⁺ transient and number of binding sites involved in the release. For a brief, sub-millisecond Ca²⁺ transient, it is likely that the Ca²⁺ binding sites of the release machinery do not reach chemical equilibrium. Consequently, the on rate, not the equilibrium affinity, of the site may be the relevant quantity in determining effective affinity and as a result the release probability (Fig. 1). Efforts to characterize the Ca²⁺-sensor molecules controlling release should closely consider their Ca²⁺ binding kinetics because these are expected to play an essential role in determining the release properties under physiological conditions. Also, the number of binding sites involved and their relationship to release need to be examined to be able to construct a realistic model of Ca²⁺-triggered release.

Ca²⁺ uncaging experiments that employ [Ca²⁺] elevations longer than the action-potential-evoked Ca²⁺ influx reveal the true Ca²⁺ cooperativity of release and the functional equilibrium affinity of the Ca²⁺ sensor (Bollmann et al. 2000; Heidelberger et al. 1994; Schneggenburger and Neher 2000), but it is more difficult to obtain a reliable estimate of the Ca²⁺-sensor molecule on rate and its affinity from these experiments. k_on is estimated from the dependence of the delay to release onset on the [Ca²⁺] after the flash. Although the delay can be measured directly, [Ca²⁺] within a few milliseconds after the flash must be modeled using the binding kinetics of the indicator dyes, which have been reasonably well characterized. Attempts to derive the [Ca²⁺] experienced by a typical Ca²⁺ sensor during an action potential (Bollmann et al. 2000; Schneggenburger and Neher 2000) are limited by the quality of the estimate of the on rate because for a brief Ca²⁺ transient, the release probability depends directly on the product of the on rate and [Ca²⁺] (Fig. 1). Consequently, if the estimate of k_on was high by some factor, then the estimated [Ca²⁺] at the Ca²⁺ sensor would be underestimated by the same amount. Recently, using rapidly decaying [Ca²⁺] elevations generated by Ca²⁺ uncaging technique, the estimate of kinetic properties of the Ca²⁺ sensor has improved, and a lower on rate has been suggested (Bollman and Sakmann 2005). The new technique provides a more reliable method of studying kinetic properties of the Ca²⁺ sensor of release machinery.

The binding rates are the limiting factor in any reaction that has limited time to progress. A relevant biological example is the nonsaturation of N-methyl-D-aspartate (NMDA) receptors by action-potential-evoked glutamate release in hippocampal synapses that occurs despite the high affinity of the receptor for glutamate (McAllister and Stevens 2000). Consistent with the ideas presented here, Chen et al. (2001) demonstrated that NMDA receptors due to their slow kinetics have relatively lower potency under brief application of glutamate than under equilibrium conditions.

It appears that cooperation of a few SNARE complexes, along with their associated synaptotagmin molecules, is needed for Ca²⁺ triggered neurotransmitter release (Bai et al. 2004; Han et al. 2004). It is apparent that because each synaptotagmin molecule has five Ca²⁺ binding sites, the total number of binding sites potentially available for release is much more than the observed Ca²⁺ cooperativity of release. We introduced two simple models for release (models 2 and 3) that despite having extra binding sites produce the same Ca²⁺ cooperativity as model 1 with five binding sites. We showed that extra binding sites for the release model increase the effective on rate of the Ca²⁺ sensors, but still the functional on rate is constrained by the diffusion-limited rate for a Ca²⁺ binding site, which is generally accepted to be 5 x 10⁸-10⁹ M^–1s^–1. This result suggests the apparent high affinity observed in Calyx of Held compared with retinal bipolar cell could be due to the number of available binding sites rather than the affinity of the binding site itself.

It is possible that Ca²⁺ binding sites are distributed around the docking point of the vesicle on a few sensor molecules (Stewart et al. 2000). Using a Monte Carlo simulation with nanometer spatial resolution, we show that shielding even one of the binding sites from the Ca²⁺ source by placing it behind the vesicle can significantly reduce the probability of release. This is because the vesicle acts as barrier to free diffusion of Ca²⁺ and modifies the shape of the Ca²⁺ microdomain. If more than one channel is contributing to release, the microdomain of the closest channel would be the most affected and binding sites opposite the closest channel would be the hardest to occupy (Shahrezaei and Delaney 2004).

Our Monte Carlo simulation reveals a novel result that in the presence of a nonuniform [Ca²⁺] around the vesicle variation in the position of the sensor around the vesicle can produce two distinct populations of vesicles with high and low release probabilities. This heterogeneity of release probabilities does not require different types of Ca²⁺-sensor molecules or distinct populations of colocalized and noncolocalized vesicle-channel populations; it arises solely from the micro-geometry of the vesicle-channel-Ca²⁺ sensor(s) complex. The two distinct populations would exist even if the number of available binding sites changes, although the size of one population versus another could change. Conceivably, Ca²⁺-dependent interactions between the release machinery and the Ca²⁺ channels (Spafford and Zamponi 2003) could cause small rearrangements of the Ca²⁺ sensor(s) relative to the channel to increase the size of the high-release-probability vesicle population. This is a potential mechanism for activity-dependent enhancement of neurotransmitter release.

	GRANTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
This work was supported by National Sciences and Engingeering Research Council Canada Grant RGPIN 121698.

	ACKNOWLEDGMENTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
We thank T. Murphy and E. Stanley for valuable comments on the manuscript.

	FOOTNOTES

 
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Address for reprint requests and other correspondence: V. Shahrezaei, Dept. of Physics, Simon Fraser University., 8888 University Dr., Burnaby, B.C. V5A 1S6, Canada (E-mail: vshahrez{at}sfu.ca)

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