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Facilitation through Buffer Saturation: Constraints on Endogenous Buffering Properties

Victor Matveev ^* , Robert S. Zucker  and Arthur Sherman 

^* Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102; Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892; and Molecular and Cell Biology Department, Neurobiology Division, University of California, Berkeley, California 94720

Correspondence: Address reprint requests to Arthur Sherman, LBM, NIDDK, NIH, Bethesda, MD 20892-5621. Tel.: 301-496-4325; Fax: 301-402-0535; E-mail: asherman{at}nih.gov.

	ABSTRACT

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Synaptic facilitation (SF) is a ubiquitous form of short-term plasticity, regulating synaptic dynamics on fast timescales. Although SF is known to depend on the presynaptic accumulation of Ca²⁺, its precise mechanism is still under debate. Recently it has been shown that at certain central synapses SF results at least in part from the progressive saturation of an endogenous Ca²⁺ buffer (Blatow et al., 2003), as proposed by Klingauf and Neher (1997). Using computer simulations, we study the magnitude of SF that can be achieved by a buffer saturation mechanism (BSM), and explore its dependence on the endogenous buffering properties. We find that a high SF magnitude can be obtained either by a global saturation of a highly mobile buffer in the entire presynaptic terminal, or a local saturation of a completely immobilized buffer. A characteristic feature of BSM in both cases is that SF magnitude depends nonmonotonically on the buffer concentration. In agreement with results of Blatow et al. (2003), we find that SF grows with increasing distance from the Ca²⁺ channel cluster, and increases with increasing external Ca²⁺, [Ca²⁺]_ext, for small levels of [Ca²⁺]_ext. We compare our modeling results with the experimental properties of SF at the crayfish neuromuscular junction, and find that the saturation of an endogenous mobile buffer can explain the observed SF magnitude and its supralinear accumulation time course. However, we show that the BSM predicts slowing of the SF decay rate in the presence of exogenous Ca²⁺ buffers, contrary to experimental observations at the crayfish neuromuscular junction. Further modeling and data are required to resolve this aspect of the BSM.

	INTRODUCTION

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Short-term synaptic facilitation (SF) is the transient increase of synaptic response that can be elicited by a single action potential (AP) or a short train of APs, and decays on timescales from tens to hundreds of milliseconds. It is observed in a variety of systems, from invertebrate neuromuscular junctions to neocortical synapses (reviewed in Magleby, 1987; Fisher et al., 1997; Zucker, 1994 and 1999; Zucker and Regehr, 2002). The pioneering work of Katz and Miledi (1968) and Rahamimoff (1968) showed that SF depends on the extracellular Ca²⁺. However, there is still no agreement on the exact nature of the SF mechanism. Arguably, the simplest explanation is that SF results from the accumulation of free residual Ca²⁺ ([Ca²⁺]_res) in the presynaptic terminal. This possibility is supported by the extensive evidence that SF is sensitive to manipulations reducing the free intracellular Ca²⁺ (reviewed in Zucker and Regehr, 2002). However, early modeling work (Chad and Eckert, 1984; Simon and Llinás, 1985; Fogelson and Zucker, 1985; Roberts, 1994) demonstrated that the Ca²⁺ concentration in the vicinity of an open Ca²⁺ channel "domain" can reach very high values, from tens to hundreds of µM. This is much higher than the residual [Ca²⁺] increase caused by a single AP, believed to range from 10 nM to 1 µM (see reviews by Stanley, 1997; Neher, 1998b; Zucker, 1996 and 1999; and Zucker and Regehr, 2002). If the Ca²⁺-dependent release machinery is located in close proximity to a Ca²⁺ channel, as suggested by the evidence of molecular interactions between the proteins mediating exocytosis and the Ca²⁺ channel proteins (reviewed in Stanley, 1997; Sheng et al., 1998; Catterall, 1999; Fisher and Bourque, 2001; Jarvis and Zamponi, 2001), it is hard to argue that such a small increase in free Ca²⁺ would lead to significant SF. Even taking into account several experimental indications that the Ca²⁺ affinity of the secretory site may be in the 5–20-µM range rather than the 100-µM range (Delaney and Tank, 1994; Ravin et al., 1999; Bollmann et al., 2000; Schneggenburger and Neher, 2000; Ohnuma et al., 2001), it would not be sufficient to explain the high magnitude of SF observed in many systems. One way to resolve this problem is to assume the presence of a separate high-affinity Ca²⁺-sensitive site responsible for SF, distinct from the main secretory trigger, and located far enough from a Ca²⁺ channel, where the residual [Ca²⁺] is comparable to the peak [Ca²⁺] (Tang et al., 2000). We have recently shown that the properties of SF observed at the crayfish neuromuscular junction (NMJ) can be explained by such a model with two Ca²⁺-dependent release-controlling sites, given a sufficient separation between the two sites (150 nm), and under the additional assumptions of high tortuosity near the active zone and immobilization of Ca²⁺ buffers by the cytoskeleton (Matveev et al., 2002).

An alternative solution is the bound residual Ca²⁺ hypothesis, which postulates that SF results from the buildup of Ca²⁺ bound to the vesicle release sensors (Yamada and Zucker, 1992; Bertram et al., 1996; see also Bennett et al., 1997; Dittman et al., 2000). In fact, this possibility had been suggested in the pioneering studies of Katz and Miledi (1968) and Rahamimoff (1968). Within this framework, the decay time course of SF is determined by the kinetics of Ca²⁺ unbinding from the SF sensor. However, without including the effects of free Ca²⁺ in the model, this possibility is inconsistent with the above-mentioned evidence that SF is sensitive to manipulations of free intracellular Ca²⁺.

Recently, a third, qualitatively different model of SF was put forward. It was proposed (Klingauf and Neher, 1997; Neher 1998a,b) that the fast presynaptic Ca²⁺ transients elicited by successive depolarizing pulses might themselves increase, as a result of a gradual reduction in the ability of endogenous Ca²⁺ buffers to reduce such transients, due to their progressive saturation (binding) by the residual Ca²⁺ accumulating from pulse to pulse. In this case SF would result from an increase in the size of AP-induced [Ca²⁺] peaks, rather than an increase in free [Ca²⁺]_res, or an increase in the Ca²⁺-bound state of the release sensor (Table 1). An appealing aspect of the buffer saturation mechanism (BSM) is that it does not assume the presence of a facilitatory Ca²⁺ binding site distinct from the main secretory site, a condition required by the free residual Ca²⁺ model described above. Recently BSM has been shown to play a major role in SF at calbindin-containing neocortical and hippocampal synapses (Blatow et al., 2003), and was also inferred to contribute to SF at the calyx of Held synapse, albeit to a smaller degree (Felmy et al., 2003). The growth of Ca²⁺ transients resulting from buffer saturation has also been observed in Purkinje cells (Maeda et al., 1999) and dentate granule cell axons (Jackson and Redman, 2003). Although the buffer saturation hypothesis has been analyzed in depth by Neher (1998a) using approximate analytic methods (see also Bennett et al., 2000; Trommershäuser et al., 2003), there has yet been no rigorous systematic analysis of the conditions necessary to achieve significant SF within this framework. In this work we use computer simulations of the presynaptic spatiotemporal Ca²⁺ dynamics to study the properties of the BSM. The numerical approach allows us to consider the widest range of buffering conditions, whereas available analytic techniques rely on approximations that have limited applicability (Neher, 1998a; Pape et al., 1998; Smith et al., 1996, 2001). Our general findings are in agreement with the experimentally observed properties of SF resulting from buffer saturation, as described by Blatow et al. (2003).

	METHODS

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Geometry
For simplicity, we represent the presynaptic crayfish motor bouton by a sphere 3 µm in diameter (Fig. 1), which approximates the shape and size of real boutons (Cooper et al., 1995). The surface of the bouton is covered with active zones (AZs), with a density of 1 AZ/2.56 µm² (Cooper et al., 1995; Tang et al., 2000). It is assumed that all AZs are equivalent, and are distributed uniformly over the surface of the spherical bouton. This allows us to restrict our simulations to a conical region 1.8 µm in diameter at its base, surrounding a single AZ, yielding a cone base area close to 2.56 µm². We impose reflective boundary conditions for Ca²⁺ and buffer(s) on the sides of the cone, thereby assuming that the Ca²⁺ and buffer fluxes flowing into the cone from the neighboring AZ regions are balanced by the equal fluxes flowing out of the cone. This simplified geometry allows us to perform simulations in rotationally symmetric spherical (r, ) coordinates, instead of solving the problem in full three dimensions. A similar method has been used by Klingauf and Neher (1997). The reduction from three to two dimensions significantly reduces the computational intensity of the simulations, allowing us to study the behavior of facilitation as two parameters are varied at a time (Figs. 3, 5, 6, 8, 9, and 13). This conical geometry is a rough approximation because the surface of a sphere cannot be tiled with disks, but the inaccuracy introduced by this simplification does not affect the qualitative features of our results.

View larger version (21K): [in this window] [in a new window]   FIGURE 1  Model of a presynaptic crayfish motor bouton. The crayfish motor terminal is approximately represented as a sphere 3 µm in diameter (bottom), with active zones (AZ) distributed evenly over the spherical surface. Ca2+ diffusion and buffering are only simulated in the conical region of the sphere surrounding a single AZ. The ring of vesicles around the model AZ is shown for illustration only, and is not included in the simulation. Current density g() (see Eqs. 2 and 4) is assumed to be uniformly distributed over a disk surface area 160 nm in diameter (top). Asterisks mark the locations at which the Ca2+ concentration is sampled during the simulations. These sites represent locations of the putative Ca2+ "sensors" (possible vesicle locations). We record Ca2+ concentration 20 nm below the membrane surface, and at lateral distances of 20, 60, and 100 nm away from the edge of the Ca2+ current influx (corresponding sites are marked 1–3).

View larger version (31K): [in this window] [in a new window]   FIGURE 3  FCT as a function of buffer affinity and total buffer concentration. FCT is measured as the ratio of the fifth and the first Ca2+ transients, P5/P1, at a distance of 60 nm from the edge of the Ca2+ channel cluster (site 2 in Fig. 1); (A) koff = 0.4 ms–1 and (B) koff = 0.1 ms–1. The remaining parameter values are the same as in Fig. 2 (DB = 0.2 µm2 ms–1, ICa = 11.7 pA). FCT is constant along the contours separating the differently shaded bands, and its value is indicated on each of the contours. FCT grows with decreasing KD (increasing kon). Given a fixed value of KD (kon), maximal FCT is achieved at some finite value of Btotal, denoted (+) The parameter points corresponding to panels A–C in Fig. 4. Note the similarity between data in panels A and B, and the equal range of kon values in the two cases (right scale). The diagonal line marks the set of parameter points corresponding to 0 = Btotal/KD = 500. Increasing (decreasing) ICa is equivalent to sliding left (right) along this line (see text for details). For example, the value of FCT for Ca = 5.85 pA, half the value used in this plot, at the parameter point corresponding to the solid circle (Btotal = 350 µM, KD = 0.7 µM), is equal to FCT for the doubled values of ICa, Btotal, and KD (ICa = 11.7 pA, Btotal = 700 µM, KD = 1.4 µM), corresponding to the open circle, yielding FCT 1.8.

View larger version (35K): [in this window] [in a new window]   FIGURE 5  Dependence of five-pulse FCT on the values of kon and Btotal, for a fixed value of the buffering ratio. (A) 0 = 500; (B) 0 = 50. Parameter region corresponding to SF magnitude observed at the crayfish NMJ lies inside the area bounded by contour FCT = 2.0 (see text for details). Hyperbolic curves mark parameter points of constant koff. (A) The two curves correspond to the diagonal lines in Fig. 3, A and B. Increasing (decreasing) ICa is equivalent to sliding left (right) along these lines. Because the value of 0 is fixed, KD is determined by the value of Btotal. Other parameters are the same as in Figs. 2–4 (DB = 0.2 µm2 ms–1, ICa = 11.7 pA, d = 60 nm).

View larger version (37K): [in this window] [in a new window]   FIGURE 6  FCT increases with increasing distance from the Ca2+ channel cluster. (A) Five-pulse FCT as a function of Btotal and kon, at distances of (a) 20 nm, (b) 60 nm, and (c) 100 nm away from the edge of the channel cluster (locations marked 1–3 in Fig. 1). Data in panel Ab are the same as in Fig. 5 A. (B) [Ca2+] time course for parameter values corresponding to the points marked with an asterisk in A (Btotal = 700 µM, KD = 1.4 µM, kon = 0.5 µM–1 ms–1). Note the difference in scale along the concentration axis. Other parameter values are the same as in Figs. 2–5 (DB = 0.2 µm2 ms–1, ICa = 11.7 pA).

View larger version (22K): [in this window] [in a new window]   FIGURE 8  Dependence of FCT on buffer mobility. Shown is the five-pulse FCT as a function of the buffer diffusion coefficient, DB, and Btotal (KD), for 0 = 500. Panel B extends the data in panel A into the region of very high Btotal, and uses logarithmic scale to emphasize the region of very small DB. Note the two FCT peaks, one at large DB and small Btotal, and another peak at DB close to zero and very large Btotal. Parameter values are kon = 0.8 µM–1 ms–1, ICa = 11.7 pA, and d = 60 nm. The set of points along the top edge of all panels (DB = 0.2 µm2 ms–1) correspond to the horizontal line kon = 0.8 µM–1 ms–1 in Figs. 5 and 6 Ab).

View larger version (35K): [in this window] [in a new window]   FIGURE 9  FCT as a function of Btotal and kon in the fixed buffer case, for a constant value of the buffering capacity, 0 = 500. This is the same as Fig. 6, except that DB = 0. Note that much higher Btotal is required to achieve significant facilitation, and the slower decay of the Ca2+ transients, as compared to the mobile buffer case (cf. Fig. 6).

View larger version (25K): [in this window] [in a new window]   FIGURE 13  Dependence of FCT on the properties of a mobile buffer, in the presence of an immobile low-affinity buffer with fixed characteristics. Five-pulse FCT is shown as a function of kon and Btotal of the mobile buffer, for mobile buffering capacity of (A) 0 = 500 and (B) 0 = 50. The buffering capacity of the immobile low-affinity buffer is = 50, in both panels. The properties of the immobile buffer are fixed: Btotal = 750 µM, KD = 15 µM, and kon = 0.1 µM–1 ms–1. Other parameters are the same as in Figs. 2–4 (DB = 0.2 µm2 ms–1, ICa = 11.7 pA, d = 60 nm). Note that introducing an immobile low-affinity buffer reduces FCT, as compared to the single-buffer case (Fig. 5). The magnitude of this reduction is more dramatic when the capacities of the two buffers are comparable (cf. panel B and Fig. 5 B).

Equations describing buffered diffusion of Ca²⁺
For simplicity, in most of our simulations we assume the presence of a single dominant Ca²⁺ buffer, with the Ca²⁺ binding reaction described by

(1)

where k_on and k_off are, respectively, the binding and the unbinding rates of the Ca²⁺-buffer compound. This leads to the following reaction-diffusion equations for the Ca²⁺ concentration, [Ca²⁺], and the concentration of the free (unbound) buffer, [B]:

(2)

(3)

Here D_B and D_Ca are the diffusion coefficients in cytosol of the buffer and Ca²⁺, respectively. We choose D_Ca = 0.2 µm² ms^–1 (Allbritton et al., 1992). Following the standard convention, in Eqs. 2 and 3 we have assumed that the initial distribution of the buffer is spatially uniform, and that the diffusion coefficient of the buffer is not affected by the binding of a Ca²⁺ ion. Under these assumptions the sum of the bound and the unbound buffer concentrations is constant in space and time, and is equal to the total buffer concentration, B_total. This allows one to eliminate the equation for the evolution of the concentration of the bound buffer, [CaB] = B_total – [B]. Equations 2 and 3 are extended in a straightforward way when simulations include more than one buffer. In this case, each of the buffers evolves according to an equation identical to Eq. 3, and Eq. 2 is expanded to include the binding and the unbinding terms for each of the buffers present (e.g., see Smith et al., 2001).

The last term in Eq. 2 represents the Ca²⁺ influx, where F is Faraday's constant, I_Ca(t) is the (inward) calcium current per active zone, and (r – r_b) is the Dirac delta function, with r_b = 1.5 µm equal to the bouton radius (this describes a Ca²⁺ current flowing through the surface membrane). The surface density function g() determines the distribution of the Ca²⁺ current over the AZ surface; it is assumed to be uniform across disk surface area 160 nm in diameter (S_Ca), and is zero everywhere else, as illustrated in Fig. 1 (see justification above). The integral of g() over the S_Ca area should be equal to one, from which we obtain

(4)

where is the angle corresponding to the edge of the Ca²⁺ influx area: = 80 nm/ r_b = 0.0533 rad. I_Ca(t) is represented as a train of simple 1-ms-long constant current pulses. Unless otherwise noted, we set I_Ca magnitude to 11.7 pA. This value is a factor of 2 greater than the estimate of I_Ca = 5.85 pA obtained by measuring residual [Ca²⁺] per AP (Yamada and Zucker, 1992; Tang et al., 2000), and is within the range of I_Ca = 10–50 pA inferred from the measurements of [Ca²⁺] rise rate during sustained stimulation (Tank et al., 1995; both values are translated to a 1-ms-long current pulse). In Results, we discuss the behavior of the model for variable magnitude of I_Ca.

Reflective boundary conditions hold for both [Ca²⁺] and [B] at all boundaries, except for the boundary condition for [Ca²⁺] on the top surface of the cone, which assumes Ca²⁺ extrusion by surface pumps:

(5)

Here M is the maximal pump rate, and K_P is the pump dissociation constant. We use values of K_P = 0.2 µM (Dipolo and Beauge, 1983; Carafoli, 1987) and M = 0.01 µM µm ms^–1 (except for Fig. 12, A and B). At low [Ca²⁺], this yields a Ca²⁺ clearance time constant of (1 + ₀) V K_P/(M S) = 5 s, where V and S are the volume and the surface area of the bouton (Fig. 1), and ₀ = 500 is the resting state endogenous buffering capacity at the crayfish NMJ (Tank et al., 1995). This agrees with the experimental value of Ca²⁺ clearance rate obtained by Tank et al. (1995). In Results, we will discuss the possibility of higher Ca²⁺ extrusion rates. The resting buffering capacity ₀ (also known as the endogenous buffering ratio or the binding ratio) is an important characteristic of the buffering strength, defined as (Irving et al., 1990; Neher and Augustine, 1992)

(6)

View larger version (15K): [in this window] [in a new window]   FIGURE 12  Experimental and simulated effect of Fura-2 application on the decay time course of SF. Shown is the SF magnitude for the last pulse in a 100-Hz five-pulse train, for a variable time interval between the fourth and the fifth pulses, t4-5, for (A) mobile buffer simulation, (B) fixed buffer simulation, and (C) experimental data from Fig. 3 B of Tang et al. (2000). SF is computed as in Fig. 7. Parameter values in A are the same as in Fig. 7 A, except for the faster Ca2+ extrusion (see text), leading to a reduction of SF at t4-5 = 10 ms from 19-fold to 14-fold. Parameter values in B are the same as in Fig. 11. The biexponential fits in A are given by 9.4 exp(–t/57 ms) + 5.3 exp(–t/202 ms) (•), and 3.7 exp(–t/111 ms) + 3.3 exp(–t/416 ms) (). In B, fits are 41 exp(–t/7.9 ms) + 7.1 exp(–t/44 ms) + 0.29 exp(–t/6330 ms) (•), and 21.7 exp(–t/7.1 ms) + 1.16 ().

Numerical simulations
The reaction-diffusion Eqs. 2–33 were solved using the Calcium Calculator (CalC) software developed by one of us (V.M.). CalC uses an alternating-direction implicit finite-difference method, second order accurate in spatial and temporal resolution (Morton and Mayers, 1994). To preserve the accuracy of the method in the presence of the buffering term, the equations for [Ca²⁺] and [B] are solved on separate time grids, shifted with respect to each other by half a time step (Hines, 1984). CalC uses an adaptive time-step method; the spatial grid size is adjusted to limit the error tolerance to at most 1–2% (grid of 36 x 36 points in the mobile buffer case, and 70 x 70 in the fixed buffer case). CalC is freely available from http://web.njit.edu/matveev, and runs on Linux, SGI, and Windows/Intel platforms. To ensure reproducibility of this work, the commented simulation script files generating the data reported here will be made available on the above web site.

	RESULTS

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Facilitation of Ca²⁺ transients: an example
Fig. 2 simulates the buffer saturation mechanism of SF, showing the facilitation of Ca²⁺ transients (FCT) in response to a train of five equal Ca²⁺ current steps. This growth in simulated peak [Ca²⁺] transients is caused by the residual Ca²⁺ accumulating from pulse to pulse and gradually binding and depleting the free buffer, thereby reducing more and more the ability of the buffer to shunt the AP-triggered Ca²⁺ transients. Note that the free [Ca²⁺]_res is negligibly low (see the inset), because most of the residual Ca²⁺ is in fact bound to the buffer, due to a high buffering ratio in our example (₀ = B_total/K_D = 1000). In this and most of the following figures, concentration is shown at a distance of 60 nm from the edge of the Ca²⁺ influx region (site 2 in Fig. 1); dependence on distance will be analyzed later on (see Fig. 6).

View larger version (20K): [in this window] [in a new window]   FIGURE 2  Demonstration of the facilitation of Ca2+ transients caused by buffer saturation. Ca2+ (middle panel) and buffer (bottom panel) concentration time courses produced in response to a 5-pulse train of 1-ms-long Ca2+ current pulses (top panel), as measured at a distance of 60 nm from the edge of the Ca2+ channel cluster (site 2 in Fig. 1). The growth in the Ca2+ transient between the fifth and the first Ca2+ pulses, P, depends on the degree of buffer saturation achieved right before (B) and during (BP) the final pulse. See text for details. Parameters are Btotal = 500 µM, KD = 0.5 µM, kon = 0.8 µM–1 ms–1, DB = 0.2 µm2 ms–1, and ICa = 11.7 pA.

The conditions for high FCT
We will quantify FCT by the ratio of the amplitudes of the fifth and the first peaks in Ca²⁺ concentration:

(7)

To achieve significant FCT, one needs a large increase in the Ca²⁺ transient magnitude (P), and a comparatively small value of the first Ca²⁺ transient, P₁ (see Fig. 2). These two requirements lead to two different sets of constraints on model parameters:

1. The requirements for small P₁ are straightforward and include all conditions that increase the buffering efficiency: i), large B_total; ii), fast binding (large k_on); iii), high buffer mobility, D_B; iv), small I_Ca.
   
2. The conditions for large P are twofold: i), significant buffer saturation/depletion B should be achieved before the fifth pulse; as discussed above, this requires moderately high values of ₀ (₀ > 10), and high I_Ca; ii), for this reduced buffer availability to make a difference, the buffer should be close to saturation during the last pulse, as well: the free buffer remaining at the end of the last pulse, B_P, should be small. In other words, initially the buffer should be effective in limiting the rise in Ca²⁺, but at the end of the train, Ca²⁺ transients should be able to strongly saturate the buffer. This condition is easier to satisfy for low B_total and fast Ca²⁺-buffer binding (high k_on).
   

Notice that one requirement in the two groups coincides (large k_on), whereas the conditions on B_total and I_Ca are in contradiction. Therefore, one may expect the FCT to grow monotonically with increasing k_on, but its behavior with respect to B_total and I_Ca may be nonmonotonic. The results presented below confirm this conclusion.

FCT peaks at a finite value of B_total, and grows monotonically with increasing k_on
Fig. 3 shows the dependence of the five-pulse FCT, defined according to Eq. 7, as a function of K_D and B_total, the first two model parameters in Table 2. In this figure the magnitude of the unbinding rate k_off is kept fixed (k_off = 0.4 ms^–1 for panel A, k_off = 0.1 ms^–1 for panel B), so the value of k_on varies along with the value of K_D, according to the relation k_on = k_off/K_D. The similarity between FCT values in Fig. 3, A and B, demonstrates that FCT is only weakly dependent on k_off, and is predominantly determined by the values of k_on and B_total (note that the range of values of k_on, indicated along the right margin, are equivalent in the two panels). However, the lower the value of k_off, the lower should be K_D to achieve a given FCT magnitude.

View larger version (29K): [in this window] [in a new window]   FIGURE 4  FCT in response to five Ca2+-current steps, for three different values of the total buffer concentration (marked by + in Fig. 3 A). (A) Btotal = 200 µM, (B) Btotal = = 500 µM, and (C) Btotal = 1000 µM. Shown are [Ca2+] (upper panels), and the buffer concentration (bottom panels) time courses. Notice the difference in scale along the concentration axis. The size of the first Ca2+ peak is scaled to the same level for easy comparison. FCT achieved in B is higher than in either A or C. Buffer affinity is KD = 0.5 µM (kon = 0.8 µM–1 ms–1), corresponding to the bottom edge of Fig. 3 A. Other parameters are the same as in Figs. 2 and 3 A (koff = 0.4 ms–1, DB = 0.2 µm2 ms–1, ICa = 11.7 pA, d = 60 nm). Data in panel B are the same as in Fig. 2.

As the above analysis suggests, the optimal buffer concentration should depend on the number of pulses in the stimulation train. One should expect to increase with increasing train length, because more buffer is required to produce the same saturation effect for a greater total calcium influx, which is proportional to the number of pulses. Our simulations confirm this conclusion, and show a decrease in of 10% when the number of pulses is decreased from five to three (data not shown).

FCT for a fixed value of buffering ratio
Even though the values of most of the model parameters summarized in Table 2 are not readily available from experiment, the rest-state endogenous buffering capacity, ₀ (see Eq. 6), can be estimated experimentally using fluorescent dye techniques (Neher and Augustine, 1992). For zero Ca_rest, ₀ is given simply by the ratio B_total/K_D. We have chosen a value of 500, used in Tang et al. (2000) and Matveev et al. (2002), and close to the estimate of 600 inferred using the fluorescent Ca²⁺ sensitive dye experiments at the crayfish NMJ by Tank et al. (1995). In Fig. 3, the set of points corresponding to ₀ = 500 lie along the diagonal lines.

Constraining the value of ₀ allows us to eliminate one of the parameters from the pair {K_D, B_total}. That is, having restricted ourselves to a constant value of ₀ (the diagonal in Fig. 3), in Fig. 5 we can explore the dependence of FCT on an independent variation of k_on and B_total (the latter now also determines K_D, because B_total = ₀K_D). In the new parameter space of Fig. 5, the locus of points corresponding to any given value of k_off is a hyperbolic curve. As an example, Fig. 5 A shows the two curves of points that correspond to k_off = 0.4 ms^–1 and k_off = 0.1 ms^–1. These two hyperbolae are the images of the fixed-₀ diagonal lines in Fig. 3, A and B, respectively. Thus, Fig. 5 A compresses all the fixed-₀ information contained in a whole family of figures like Fig. 3, A and B. Moving vertically from one hyperbolic curve to the other in Fig. 5, we see that faster buffers would give greater FCT. Finally, the FCT peak in Fig. 5 A shows that the maximal magnitude of FCT is still achieved at a certain finite value of B_total (= 500 K_D), even under the fixed ₀ constraint.

Interestingly, we find that the dependence of FCT on k_on and B_total is qualitatively as well as quantitatively similar at different values of ₀. This follows from our earlier result that FCT is primarily determined by the values of B_total and k_on (Fig. 3). From the relationship ₀ = B_total/K_D = (B_total k_on)/k_off it is clear that increasing k_off, which leaves FCT almost unchanged (Fig. 3), is equivalent to a decrease in ₀. Therefore, lowering the value of ₀, say to ₀ = 50 (a 10-fold change), and simultaneously increasing k_off 10-fold (10-fold decrease in K_D), results only in a moderate reduction of the FCT magnitude (cf. Fig. 5, A and B).

Dependence on I_Ca is redundant
Apart from the value of ₀, the value of I_Ca per action potential per AZ can also be estimated using fluorescent Ca²⁺ indicator dye measurements. However, the accuracy of the Ca²⁺ influx measurements is not high, because it depends on the optical calibration of the fluorescent Ca²⁺-sensitive dye used in the measurements, on the estimates of its concentration, and on the independent estimate of the endogenous buffering capacity, ₀. Further, one may expect significant variability in I_Ca between different AZs, as well as between different NMJ boutons. Therefore, it is important to consider how the variation in the value of I_Ca would affect FCT.

As it turns out, the dependence of FCT on the magnitude of Ca²⁺ influx per AP is redundant, because knowing the dependence of FCT on K_D and B_total, for a given value of k_off (Figs. 3 and 5), gives full information about its behavior with respect to I_Ca. This is a consequence of the freedom in choosing the units of concentration. The FCT magnitude should not be affected if the three parameters with units of concentration, K_D, B_total, and I_Ca (current equals concentration times volume per unit time), are rescaled by the same factor. Thus, if the FCT magnitude is calculated for a given value of the Ca²⁺ current, I_Ca, then at any other value of the Ca²⁺ current, _Ca, it is given by

(8)

where a I_Ca/_Ca. It follows then that changing the value of the Ca²⁺ current is equivalent to sliding along the line of constant B_total/K_D ratio (Figs. 3 and 5).

We have to emphasize that the above redundancy condition only holds for the FCT, and not the facilitation of neurotransmitter release. To obtain an expression similar to Eq. 8 for SF proper, given a particular choice of a Ca²⁺-dependent exocytosis scheme, the affinity of the release mechanism to Ca²⁺ would have to be scaled along with I_Ca, K_D, and B_total (i.e., it would have to be included in the list of arguments that are multiplied by factor a on the right-hand side of Eq. 8). However, we assume a simple power-law relationship between release and [Ca²⁺], thus avoiding this complication (see "Relationship between FCT and facilitation of release"). A similar caveat concerns the case of nonzero resting free [Ca²⁺] value, Ca_rest: in general, Ca_rest would have to be included in the argument list on the left- and right-hand sides of Eq. 8. However, the procedure of subtracting out the Ca_rest value, described in the Appendix, preserves the scaling freedom expressed by Eq. 8 in the case of nonzero values of Ca_rest (the arguments in Eq. 8 are to be understood as the "primed" zero-Ca_rest values: see Appendix). Finally, we note that the Ca²⁺ pump dissociation constant K_P (Eq. 5) constitutes yet another concentration parameter, however it mostly affects the decay time course of SF (to be discussed below), and has only moderate influence on the FCT magnitude for short-duration stimulation trains considered here.

Relationship between FCT and facilitation of release
To compare our simulation results with the magnitude of SF available from experiment, we have to convert the FCT values into a measure characterizing facilitation of neurotransmitter release. We prefer not to constrain ourselves to a particular choice of a Ca²⁺-dependent release model, because doing so would require introduction of additional parameters into our problem. Instead, we will assume that the synaptic response is given by the fourth power of [Ca²⁺]. Then, SF is likewise given by the fourth power of FCT:

(9)

Note that this equation neglects the inevitable saturation of synaptic response, which would occur at sufficiently high levels of [Ca²⁺]. Further, the cooperativity value of 4 lies at the high end of experimental range obtained for the crayfish NMJ (Landò and Zucker, 1994; Wright et al., 1996; Ravin et al., 1999; Vyshedskiy and Lin, 2000). Therefore, all facilitation estimates in this paper are to be understood as upper bounds on physiologically attainable facilitation.

Based on experimental observations, we require a five-pulse SF magnitude of 18–20 (see Figs. 1 C and 2 C of Tang et al., 2000). This corresponds to FCT of P₅/P₁ 2.1 (because 2.1⁴ 19.4). Therefore, in the contour plots of Figs. 3 and 5, the parameter region yielding physiological magnitude of SF lies within the area bounded by the contour of FCT = 2.0.

FCT grows with increasing distance between Ca²⁺ channel cluster and release site
There is substantial evidence for a close association between the vesicle release machinery and the presynaptic Ca²⁺ channels, suggesting their mutual proximity at many synapses (for reviews, see Stanley, 1997; Sheng et al., 1998; Catterall, 1999; Fisher and Bourque, 2001; Jarvis and Zamponi, 2001). At the crayfish NMJ, morphological studies suggest that vesicles are clustered along the edge of the AZ (Cooper et al., 1996; Atwood et al., 1997; Atwood and Karunanithi, 2002). However, at certain synapses, like the calyx of Held, vesicles may be scattered at various distances from the Ca²⁺ channel cluster (Meinrenken et al., 2002; Sätzler et al., 2002). Further, recent experiments on the sensitivity of phasic response to an application of a fast exogenous buffer (1,2-bis(2-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid (BAPTA)) suggest that there is a significant separation between Ca²⁺ channels and secretion triggers at several strongly facilitating, but not at depressing, neocortical and hippocampal synapses (Rozov et al., 2001; Blatow et al., 2003). For any synapse type, it is possible that various obstacles to diffusion (such as vesicles themselves) and the tortuosity effects associated with the dense cytoskeletal and protein mesh within the AZ may significantly increase the effective diffusional distance between the Ca²⁺ channel and the release-triggering Ca²⁺ sensor. Therefore, we simulated the Ca²⁺ transients at various distances from the Ca²⁺ influx region. Fig. 6 A demonstrates that the magnitude of FCT is increased by an increase in the channel-sensor separation. This result is easy to understand, because the buffer is able to bind more Ca²⁺ when it is acting over a longer diffusional distance (see review by Neher, 1998b), and thereby is more effective in reducing the first response, P₁, leading to higher FCT. Sufficient SF is achieved at distances of at least 40–60 nm from the Ca²⁺ channel cluster.

Facilitation growth is supralinear
As discussed above, Figs. 5 and 6 A demonstrate that the physiological magnitude of five-pulse SF can be achieved within a certain range of values of k_on and B_total (K_D). However, we also need to establish whether BSM can reproduce the supralinear accumulation time course of SF observed at the crayfish NMJ (Figs. 1 C and 2 C of Tang et al., 2000; Fig. 2 C of Winslow et al., 1994). Traces in Fig. 6 B show that the simulated growth of FCT for a chosen parameter point (marked by an asterisk in Fig. 6 A) is indeed supralinear, at all distances from the Ca²⁺ channel cluster, and therefore facilitation of release (Eq. 9) grows supralinearly as well, in agreement with experiment. Fig. 7 A demonstrates the supralinear time course of SF corresponding to the [Ca²⁺] trace shown Fig. 6 Bb, yielding a physiological SF magnitude. In fact, we find that, for a mobile buffer, the parameter region yielding high facilitation coincides with the region of high supralinearity (data not shown). Therefore, BSM can successfully reproduce the SF magnitude and its accumulation time course observed experimentally at the crayfish NMJ.

View larger version (16K): [in this window] [in a new window]   FIGURE 7  Facilitation grows supralinearly in the mobile buffer case. Comparison of (A) simulation results with (B) experimental data from Fig. 3 C of Tang et al. (2000), showing SF for a five-pulse 100-Hz train, with and without Fura-2. In A, simulations are performed for parameter values corresponding to the point marked by an asterisk in Fig. 6 Ab (distance = 60 nm, Btotal = 700 µM, KD = 1.4 µM, kon = 0.5 µM–1 ms–1, DB = 0.2 µm2ms–1, ICa = 11.7 pA). Facilitation is normalized to zero for the first pulse: ([Ca2+]n/[Ca2+]1)4 – 1 (Eqs. 11 and 12). The [Ca2+] trace corresponding to the control simulation curve in A is shown in Fig. 6 Bb. The following parameter values are used for the Fura-2 in A: Btotal = 200 µM, KD = 360 nM, kon = 0.27 µM–1 ms–1, and DB = 0.118 µm2 ms–1 (Tang et al., 2000).

FCT at lower buffer mobility
Until now we have assumed that the Ca²⁺ buffer is highly mobile, with a diffusion coefficient comparable to that of ATP, BAPTA, or EGTA (Naraghi and Neher, 1997). There are several studies that agree with such high mobility of endogenous buffers, for instance the experiments at goldfish retinal bipolar cells (Burrone et al., 2002), Purkinje cells (Maeda et al., 1999), and frog saccular hair cells (Roberts, 1993). However, endogenous buffers were found to be immobile or poorly mobile in many other preparations, including the calyx of Held (Helmchen et al., 1997), neocortical, and hippocampal pyramidal cells (Helmchen et al., 1996; Ohana and Sakmann, 1998; Lee et al., 2000), axons of Aplysia neurons (Gabso et al., 1997), and bovine adrenal chromaffin cells (Neher and Augustine, 1992; Zhou and Neher, 1993; Xu et al., 1997). At the crayfish NMJ, the mobility of endogenous Ca²⁺ buffers is not known. To study the dependence of the BSM on buffer mobility, we repeated our simulations for variable values of B_total and the D_B, the only parameter in Table 2 yet to be varied, for fixed values of ₀ and k_on (₀ = 500, k_on = 0.8 µM^–1 ms^–1). Fig. 8 A shows that FCT declines dramatically as D_B is decreased, which agrees with the conditions for high FCT that were formulated in the section entitled "Facilitation of Ca²⁺ transients: an example." At the same time, there is an increase in the optimal buffer concentration required to achieve maximal FCT. This is because a buffer with lower mobility is saturated much more readily, so greater B_total is required to prevent oversaturation. However, we find that the transition to the fixed buffer case (D_B = 0) is nonmonotonic, and FCT starts increasing as D_B drops below 0.01 µm² ms^–1 (Fig. 8 B), suggesting the existence of a distinct facilitation regime at very low values of D_B.

FCT in the fixed buffer case
At low mobility and high B_total it becomes possible for the buffer to absorb most of the Ca²⁺ influx before it reaches the Ca²⁺ sensor, strongly decreasing the first few responses, and thereby leading to a high FCT magnitude. This regime is most pronounced when the buffer is completely immobile; as Fig. 8 B shows, FCT rapidly decreases even for small nonzero values of D_B. Fig. 9 A illustrates the behavior of FCT for D_B = 0, as a function of k_on and B_total (= 500 K_D), at different distances from the Ca²⁺ channel cluster (compare with the mobile buffer case, Fig. 6). FCT is seen to be relatively weakly dependent on k_on, for sufficiently high values of k_on. This is because the effective diffusional distance from the point of influx to the Ca²⁺ sensor is significantly increased in the presence of a fixed buffer, assuring that the Ca²⁺ ions will undergo multiple binding-unbinding steps for a wide range of k_on values. For this reason, FCT is sensitive to the buffer affinity, K_D, and not to k_on, as in the mobile buffer case. Therefore, FCT is also sensitive to the value of buffering capacity, ₀. We find that large ₀ is required to achieve high FCT in the fixed buffer case (data not shown).

The [Ca²⁺] traces shown in Fig. 9 B demonstrate that the free [Ca²⁺]_res makes a more significant contribution to Ca²⁺ peaks, as compared to the case of a mobile buffer (cf. Fig. 6 B). In fact, we find that the regions of high FCT and high free [Ca²⁺]_res significantly overlap (data not shown). When [Ca²⁺]_res is nonnegligible, Eq. 9 should be modified to include its contribution to facilitation:

(10)

where R₄ equals [Ca²⁺]_res achieved at the end of the fourth interpulse interval (see Fig. 9 Bb). The R₄/P₁ ratio should remain much smaller than one, because the delayed asynchronous release 10 ms after the fourth pulse is known to be much smaller than the peak synaptic response achieved during the first pulse. The contribution of [Ca²⁺]_res increases with increasing B_total (K_D), and with increasing distance from the Ca²⁺ channel cluster. In fact, for sufficiently high B_total values and at large distances, the fast Ca²⁺ transient becomes completely shunted by the buffer, and the peak in Ca²⁺ concentration is significantly delayed with respect to the end of the pulse (in Fig. 9 Bc, the delay is 1.5 ms). Under these conditions the BSM would predict a physiologically unrealistic latency between the end of the Ca²⁺ influx and the peak of neurotransmitter release. At a given distance, there is only a narrow range of B_total values where the magnitude of facilitation (Eq. 10) reaches experimentally observed levels, whereas the free [Ca²⁺]_res contribution remains small (R₄/P₁ 0.2). For a distance of 60 nm (Fig. 9 Ab), the corresponding range is 10–12 mM, approaching unphysiologically high concentrations.

Fig. 10 demonstrates the spatial distribution of the Ca²⁺ ions and the buffer at different times after the cessation of an AP, for a parameter point marked with a + in Fig. 9 A. Contrary to the mobile buffer case, it takes tens of milliseconds rather than a fraction of one millisecond for the concentrations to reequilibrate in space, so the free [Ca²⁺]_res elevation and the buffer saturation region are still significantly localized by the end of a 10-ms-long interpulse interval. This illustrates that it is a local rather than global buffer saturation that underlies FCT in the fixed buffer case.

View larger version (23K): [in this window] [in a new window]   FIGURE 10  Slow spatial reequilibration of Ca2+ and fixed buffer after an action potential. Panels on the left (right) demonstrate the Ca2+ (buffer) concentration profiles, respectively, at different times (indicated on the corresponding contours) after the cessation of a 1-ms-long Ca2+ current pulse. The profiles are shown along an axis situated 20 nm below the synaptic membrane, and running parallel to it (dot-dashed line in Fig. 2). The inset shows [Ca2+] profiles on a finer concentration scale, at later times. The short-dashed vertical line marks the edge of the Ca2+ influx region (distance of 80 nm), whereas the long-dashed lines correspond to the three sites marked by asterisks in Fig. 1 (distances of 100 nm, 140 nm, and 180 nm from the AZ center, or 20 nm, 60 nm, and 100 nm from the edge of the influx region). Parameter values correspond to a point marked by a + in Fig. 9 A. At the end of the 10-ms-long interpulse interval, there is still a significant nonuniform (localized) saturation of the fixed buffer, which is in equilibrium with the locally elevated residual free Ca2+ (inset).

View larger version (13K): [in this window] [in a new window]   FIGURE 11  Facilitation grows sublinearly in the fixed buffer case. Shown is the simulated SF in response to a five-pulse 100-Hz AP train, arising from the saturation of a fixed buffer, with and without Fura-2. Note the sublinear accumulation time course, contrary to the mobile buffer case and the experimental data (cf. Fig. 7). Parameter values correspond to a point marked by + in Fig. 9 Ab (Btotal = 11 mM, KD = 22 µM, kon = 0.2 µM–1 ms–1; d = 60 nm). Facilitation is normalized to zero for the first pulse. Fura-2 parameters are the same as in Fig. 7 A. The [Ca2+] trace corresponding to the control simulation curve is shown in Fig. 9 Bb. The residual free Ca2+ contributes 12% to the fifth [Ca2+] peak in the control simulation, and 7% in the Fura-2 simulation.