Neurotransmitter release: Facilitation and three-dimensional diffusion of intracellular calcium

Bulletin of Mathematical Biology Vol. 54, No. 5, pp. 875 894, 1992. Printed in Great Britain.

0092 8240/9255.00+0.00 Pergamon Press Ltd © 1992 Society for Mathematical Biology

NEUROTRANSMITTER RELEASE: FACILITATION AND THREE-DIMENSIONAL DIFFUSION OF INTRACELLULAR CALCIUM G. HOVAV, H. PARNAS and I. PARNAS The Department of Neurobiology and The Otto Loewi Center for Cellular and Molecular Neurobiology, The Hebrew University, Jerusalem, Israel In order to account for the time courses of both evoked release and facilitation, in the framework of the Ca 2+ hypothesis, Fogelson and Zucker (1985, Biophys. J. 48, 1003 1017) suggested treating diffusion of Ca 2 +, once it enters through the Ca 2÷ channels, as a three-dimensional process (three-dimensional diffusion model). This model is examined here as a refined version of the" Ca 2 +-theory'' for neurotransmitter release. The three-dimensional model was suggested to account for both the time course of release and that of facilitation. As such, it has been examined here as to its ability to predict the dependence of the amplitude and time course of facilitation under various experimental conditions. It is demonstrated that the three-dimensional diffusion model predicts the time course of facilitation to be insensitive to temperature. It also predicts the amplitude and time course of facilitation to be independent of extraceUular Ca 2+ concentration. Moreover, it predicts that inhibition of the [Na+]o~--~[Ca2+]~ exchange does not alter facilitation. These predictions are not upheld by the experimental results. Facilitation is prolonged upon reduction in temperature. The amplitude of facilitation declines and its duration is prolonged upon increase in extracellular Ca 2 + concentration. Finally, inhibition of the [Na+]o~--~[Ca2 +]i exchange prolongs facilitation but does not alter the time course of evoked release after an impulse.

Introduction. Evoked release of neurotransmitter is known to depend on the concentration of intracellular Ca 2÷ (Katz and Miledi, 1977; Llinas et al., 1981; Connor et al., 1986). This dependence on Ca 2+ led to the development of the Ca 2 +-hypothesis for neurotransmitter release. This widely held hypothesis asserts that Ca 2 ÷ is the only limiting factor in the process of release. Facilitation, the average increase in release induced by the second of a pair of pulses, was suggested to reflect the residual Ca 2 + remaining from the first pulse (Katz and Miledi, 1968). The residual Ca 2 ÷ hypothesis has been confirmed by a large body of experiments (Mallart and Martin, 1967; Rahamimoff, 1968; Magleby, 1973; Bittner and Sewell, 1976; Charlton and Bittner, 1978; Bittner and Schatz, 1981; H: Parnas et al., 1982; I. Parnas et al., 1982). Facilitation at room temperature has been shown to last between several tens and several hundreds of milliseconds. In contrast, in several synaptic systems, at room temperature, the entire process of release as measured by synaptic delay 875

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histograms lasts about 2 ms (Katz and Miledi, 1965; Barrett and Stevens, 1972; Datyner and Gage, 1980; Dudel, 1984; Parnas et al., 1989). At lower temperatures both the duration of release after a single impulse and the duration of facilitation are prolonged, but to a different extent; evoked release lasts approximately 10-15 ms (Dudel, 1984; Parnas et al., 1989), and facilitation can last as long as 1 s (H. Parnas et al., 1982; I. Parnas et al., 1982). If the time courses of evoked release and of facilitation are determined by the same factor, i.e. the concentration of intracellular Ca 2 + in the vicinity of the release sites, then it is difficult to reconcile their strikingly different time courses. This difficulty, among others, led to the emergence of an extended hypothesis for release of neurotransmitter, the CaZ+-voltage hypothesis, according to which evoked release is determined by two factors. One, as for the Ca 2 + hypothesis, is the level of intracellular Ca 2 +. The other factor becomes available upon depolarization, and together with Ca z + evokes release. It is this other factor that determines the time course of evoked release (Parnas and Parnas, 1986; and see review by H. Parnas et al., 1990). A different approach to resolve the difficulty mentioned above was undertaken by several investigators according to whom evoked release is terminated rapidly owing to the initial fast diffusion of C a 2 + away from the release sites. Facilitation, on the other hand, can last much longer, owing to the much slower diffusion at later times (Fogelson and Zucker, 1985; Zucker and Stockbridge, 1983; Stockbridge and Moore, 1984). Zucker and Stockbridge (1983) indeed showed that the C a 2 + hypothesis could account for the short time course of release and for twin pulse facilitation. However, results were not satisfactory when a train of pulses preceded the test pulse after which facilitation was measured. To address this problem, Fogelson and Zucker (1985) suggested that one must take into account the non-homogeneous distribution of the Ca 2 + channels and hence the three-dimensional details of how the Ca 2 + concentration is affected by diffusion. Many investigators feel that Fogelson and Zucker's 1985 paper showed that what can be termed the "three-dimensional (3D) diffusion" version of the Ca 2 + hypothesis provides a satisfactory explanation of the various phenomena. Others do not agree, and there is an ongoing debate concerning the relative merits of the C a 2 + and Ca 2 +-voltage hypotheses. In view of this, it is worth examining in considerable detail all the implications of the Fogelson and Zucker (1985) model. Fogelson and Zucker (1985) employed analytical procedures to ascertain the behavior of both release and facilitation, which narrowed the range of cases that could be studied. In contrast, in Parnas et al. (1989) and in the present paper we solve the three-dimensional model numerically. This makes it possible to extend the study to examine the ability of the three-dimensional

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diffusion model to account for key experimental results associated with evoked release and with facilitation. In Parnas et al. (1989) we contrasted the three-dimensional diffusion model with the experimental findings associated with evoked release. We found there to be a qualitative discrepancy between the predictions of the threedimensional model and the experimental results. Contrary to the experimental results, the three-diffusion model predicts that the time course of release is insensitive to temperature, but is affected by variations in either extra- or intracellular calcium concentrations. In the present paper we extended the study to account for experiments associated with facilitation, and we again found that the predictions of the three-dimensional model differ significantly from the experimental results. In particular, contrary to experiments, the three-dimensional model predicts the time course of facilitation to be independent of temperature and of the concentration of extracellular calcium. Additional discrepancies between the three-dimensional model predictions and the experimental results are presented.

Summary of Experimental Results. The experimental results presented in Figs 1-4 were chosen according to two criteria. Firstly, they address the main factor that determines facilitation. Second, they address the fundamental assumptions of the three-dimensional diffusion model. As mentioned above, residual Ca 2 + remaining from a preceding pulse is the main cause of facilitation. Facilitation must therefore be sensitive to conditions that alter the intracellular Ca 2 + concentration. Examples of such experiments are depicted in Figs 1-3. The experiment in Fig. 4 addresses the key assumption of the three-dimensional diffusion model, that diffusion is the dominant process in regulating intracellular Ca 2 + Figure 1 shows that short term facilitation, Fs (obtained at short intervals between the pulses), decreases as the extracellular calcium concentration, [-Ca2 +]o, is raised. This holds for both the crayfish (Fig. la) and frog (Fig. lb) neuromuscular junctions (Fig. la, from H. Parnas et al., 1982; Fig. lb, from Rahamimoff, 1968). Another aspect shown is the duration of facilitation (Tf), the time it takes for facilitation to decay from any initial level back to 1 (or close to 1). T f i n the crayfish was shown to increase as [Ca 2 +]o increases (Fig. lc, from H. Parnas et al., 1982). This trend is also evident in the frog (Fig. ld, from Rahamimoff, 1968), even though a full curve relating T f t o I-Ca2+]o was not established. Together, these e~perimental results were taken to indicate that the dominant Ca 2 + removal processes saturate at higher concentrations, and that release also saturates at higher intracellular Ca 2 + concentrations (Parnas and Segel, 1980; H. Parnas et al., 1982).

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Figure 1. Dependence of various aspects of facilitation on [Ca 2 +]o : (a) Short term facilitation, Fs (7 ms), at various [Ca z +]o in the crayfish neuromuscular junction. Redrawn from H. Parnas et al. (1982). (b) Same type of experiment as in (a) (Fs 5 ms) but in the neuromuscular junction of the frog. Redrawn from Rahamimoff (1968). (c) Dependence of the duration of facilitation, Tf, on [Ca2+]o in the neuromuscular junction of the crayfish. Redrawn from H. Parnas et al. (1982). (d) The time course of facilitation at two levels of [Ca2+]o; 0.2 mM (solid line) and 0.4 mM (dashed line). Redrawn from Rahamimoff (1968). The horizontal line in (a), (b) and (d) represents the base line level of facilitation, i.e. when facilitation reaches a level of 1. Facilitation in Figs 1-4 was measured as the ratio between the quantal content elicited by the second pulse to that elicited by the first. The quantal content was measured using conventional procedures. The dependence of facilitation on depolarization is depicted "in Fig. 2. Experiments in which equal twin pulses were employed at various depolarizations generated facilitation curves with a complicated pattern (Dudel, 1990a,b). Such a pattern does not permit a straightforward interpretation of the results and is therefore not suitable for the purpose of testing the model. A more convenient method is to use the protocol of the test pulse facilitation. Here release of a constant test pulse is measured after a variable first pulse (prepulse). Facilitation is defined in this case as the ratio of the release produced by the test pulse given after the first pulse to that of the test pulse alone (Dudel e t al., 1983). As the amplitude of the test pulse is constant, presumably the same amount of Ca 2 + enters during this pulse when it is given alone or after the first pulse. Therefore an increase in release of the test pulse when given after the first pulse reflects the residual Ca 2 + Figures 2a (from Dudel et al., 1983) and 2b (from Zucker and Lando, 1986) show that test pulse facilitation first increases as depolarization (first pulse amplitude) is raised, but then declines upon a further increase in first pulse depolarization. These results were interpreted to reflect the dependence of the

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influx of Ca 2 + induced by the first pulse on depolarization (Dudel et al., 1983; Zucker and Lando, 1986). As depolarization increases, the influx of C a 2 + is larger, due to the opening of more channels• However, as depolarization continues to increase, the driving force for Ca 2 + entry is reduced and less C a 2 + enters the terminal. In experiments such as those described in Fig. 2 the level of depolarization corresponding to a given pulse amplitude is not known. However, such experiments give the trend of behavior of facilitation as depolarization varies•

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T h e r e f o r e o n l y the relative m a g n i t u d e s o f d e p o l a r i z a t i o n , n o t the e x a c t values, are of i m p o r t a n c e . T h e b e h a v i o r of the d u r a t i o n of facilitation, Tf, r e s e m b l e s t h a t o f the s h o r t t e r m facilitation (Fig. 2a). T h e curves o f facilitation a n d o f T f e v e n p e a k at the

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same prepulse amplitude. These observations provide additional support for the residual Ca 2 ÷ hypothesis in general, and demonstrate that the duration of facilitation reflects the influx of Ca 2 ÷ during the first of the two pulses. The inserts in Fig. 2a,b show that the quantal content of the first pulse, the prepulse, continues to increase even at those depolarizations at which the test pulse facilitation declines. Another line of experiments (Fig. 3) addresses the Ca 2 ÷ removal processes. Lowering extracellular Na ÷ concentration to half its normal value greatly prolongs facilitation. This result corroborates earlier findings showing that inhibition of the [Na+]o~-+[Ca 2 +]i exchange prolongs facilitation (I. Parnas et al., 1982). However, while facilitation is greatly prolonged, the quantal content and the time course of evoked release are not altered at all by lowering [Na +]o (Fig. 3, insert, Fig. 3 is taken from Arechiga et al., in press). Finally, an important experiment for the validation of the three-dimensional diffusion model is to test the dependence of facilitation on temperature. The results of such experiments are depicted in Fig. 4 (from Zengal et al., 1980). It is seen that the time constant of decay of facilitation decreases as temperature is raised. Obviously, such a dependence casts doubt on the assumption that diffusion plays a major role in regulating the time course of facilitation. To summarize, the experiments presented in Figs 1-4 indicate that, as suggested by Katz and Miledi (1968), residual Ca 2 ÷ is the main mechanism that determines the time course of facilitation. Moreover, it seems that Ca 2 ÷ diffusion cannot be the dominant mechanism that determines the level of intracellular Ca 2÷ concentration available for evoked release and facilitation. The natural question, therefore, is why investigate whether a three-dimensional diffusion model can account for the experimental results presented in Figs 1-4. Obviously, threedimensional diffusion cannot overcome the inherent characteristics of a simplified one-dimensional diffusion, i.e. weak dependence on temperature and linear dependence on concentration. However, the authors of the three-dimensional model hoped that the combination of localized entry and accumulation of Ca 2 +, together with three-dimensional diffusion, would account for the time courses of both release induced by a single pulse and facilitation. We therefore examined in more detail the ability of the three-dimensional diffusion model to account for the experimental results presented in Figs 1-4. The Three-dimensional Diffusion Model. The three-dimensional diffusion model of Fogelson and Zucker (1985) has already been discussed by us (Parnas et al., 1989). For the sake of clarity, a brief description of its main aspects is provided below.' Figure 5 summarizes the main features of the model together with the equations and the standard values of the various parameters as employed by Fogelson and Zucker (1985). C a 2+ channels are situated in patches in the upper membrane (see Fig. 5b). The release sites are located

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Equation (5.1) describes conditions at the opposite membrane, - Y o . Equation (5.2) describes the conditions of the boundaries of the x and z axes. Finally, equation (5.3) describes the dependence of release, R, on C. The meaning and standard values of the various parameters, as in Fogelson and Zucker (1985) are given below. D, the coefficient of diffusion = 0.6 #m2//~s (Hodgkin and Keynes, 1957); P, the pump rate = 0.08 ms -1; /~, the buffering effect=500; n, the cooperativity in release=4; K, a constant in equation (5.3)=1. A detailed description of the parameters off(x, z) and g(t) can be found in Parnas et al. (1989). f(x, z) is the sum of delta-like functions that are situated on the channels (represented by a Gaussian function), and g(t) is a function acquiring a positive depolarization dependent value during the pulse and zero following the pulse. Three standard levels of depolarization were employed: low depolarization corresponding to - 2 5 mV, with a 9(0 during the pulse of 1 and 4 open channels; medium depolarization which corresponds to - 15 mV, with a g(t) of 0.8 and 36 open channels; and high depolarization to 0 mV, in which case 64 channels in the active zone are opened and 9(t) is 0.5. F o r Fig. 10, higher depolarizations were employed. In this case, too, 64 channels were open, and g dropped below 0.5.

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between the C a 2 + channels. The influx of Ca 2 + is determined by two main components. The first, the function f(x, z) (see upper equation in Fig. 5), describes the density of open channels at a given depolarization, and the degree of localization of the influx at the mouth of the channel. The second, the 9(t) function (see same equation), describes the magnitude of the influx (given the properties described above). Hence, g(t) depends on both depolarization and e x t r a c e l l u l a r C a 2+ concentration, and acquires the value zero when the channels are closed. The distribution of Ca 2+ inside the terminal is governed by the lower equation in Fig. 5. Notice that the buffers, fi, are considered in this equation but not in the upper equation. Other equations related to the model are listed in the legend of Fig. 5. Predictions of the Three-dimensional Diffusion Model Spatio-temporal behavior of intracellular Ca 2+. One of the major consequences of the localized entry of Ca 2 ÷ coupled with its three-dimensional diffusion (Fogelson and Zucker, 1985; Simon and Llinas, 1985) is the nonuniform spatial distribution of intracellular calcium. Accordingly, Ca 2÷ indicators such as Fura 2 might detect an elevated concentration of global intracellular calcium while the local concentration of Ca 2 ÷ near a release site might be below the level required for release (see Blaustein, 1988). Figure 6 shows that such a situation is unlikely. The three-dimensional model predicts that the highest Ca / ÷ concentration will always be below the cell membrane through which Ca 2 + entered (Fig. 6). Moreover, the three-dimensional model, with its parameters presumably chosen to best fit experimental results, predicts that the overall concentration of Ca 2 ÷ will decline sharply shortly after the end of the pulse. Other simulations (not shown) indicate that 1 ms after the end of the pulse, even though some Ca 2 + is still detected below the points of entry, no Ca 2 + is evident at any depth below the membrane. These results negate the proposition that Ca 2 + concentration is lower in the vicinity of the release sites (required to terminate release) in spite of the high levels of Ca 2 ÷ detected by the Ca 2 + indicators.

The dependence of facilitation and its duration on extracellular Ca 2+ concentration. As mentioned, the effect of variations in extracellular Ca 2 + concentration can be studied by modifying the value of 9(t) (see Fig. 5). Varying 9(t) does not change the time course of the C a 2 + current; it only alters its amplitude. Figure 7 depicts the effects of modifying 9(t) on the magnitude and the duration of facilitation, at two levels of depolarization. Facilitation in the simulations is defined as the ratio between the peak amplitude of the release, R, [-see equation (5.3) in caption of Fig. 5] resulting from the second pulse, R2, and the peak amplitude of R resulting from the first pulse, R 1 . Thus:

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Figure 7 shows that the three-dimensional diffusion model predicts both the amplitude of facilitation and its duration to be independent of the value of 9(t), that is of extracellular C a 2 + concentration. This result holds both for low (Fig. 7a) and high (Fig. 7b) depolarizations. This predicted independence of facilitation is in contrast to the experimental results depicted in Fig. 1.

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The inserts in Fig. 7 illustrate the temporal distribution of intracellular Ca 2 ÷, hereafter denoted by C, following the first and the second pulses. In the three-dimensional model, the rate of release, R, is proportional to C" [see equation (5.3) in the legends of Fig. 5]. Consequently, the inserts in fact also describe the time course of release. Comparing the time course of release to that of facilitation shows that while facilitation lasts longer than release, the difference between the two by no means agrees with what is found experimentally. For example, in Fig. 8a, at the high g (solid line), release of the first pulse lasts more than 5 ms and facilitation lasts slightly more than 10 ms. Similar time courses for evoked release and facilitation are predicted by the three-dimensional diffusion model under all conditions. In contrast, experiments show a difference of up to three orders of magnitude between the two (I. Parnas et al., 1982).

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The experimentally observed reduction in the amplitude of short term facilitation as extracellular Ca 2÷ increases was interpreted to reflect a saturative dependence of release on Ca 2 ÷ concentration (Rahamimoff, 1968; Parnas and Segel, 1980; H. Parnas e t al., 1982). In the three-dimensional diffusion model, the dependence of release, R, on Ca 2 ÷ concentration was not taken to be saturative [see equation (5.3) in the legend of Fig. 5]. It is therefore not surprising that it fails to account for the experimental results depicted in Fig. 1.

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The dependence of facilitation on the distance of the release site from a Ca 2 + channel. The variable in Fig. 8 is the distance of the release site from the C a 2 + channel. When the distance is 70 nm, a higher level of facilitation is obtained (solid line) than when the distance is 50 n m (dashed line). The difference between the two curves is reduced as the time interval between the pulses increases. The results seen in Fig. 8 illustrate a characteristic property of the threedimensional diffusion model. W h e n diffusion is the main process that regulates the temporal distribution of C, the higher C is the faster it will diffuse away. As C is higher at the release site located 50 nm away from the channel (compare dashed to solid line in the insert of Fig. 8), it diffuses away faster from that release site. Consequently, at an interval of 4 ms between the pulses, the residual Ca 2 + remaining is virtually the same at the two release sites despite an initially higher level at the closer release site. Dependence of release and test pulse facilitation on depolarization. Two aspects of release and facilitation and their dependence on depolarization were tested. The first is the finding of "hysteresis" by Llinas e t a l. (1981 ). Here, using the squid giant synapse and voltage clamp conditions, these researchers found that for the same Ca 2 + current, release was higher at higher depolarizations. Three explanations were advanced: (a) membrane depolarization affects release in addition to its known effect on Ca z + entry (Llinas et al., 1981; Dudel et al., 1983); (b) the finding is an artifact of the Voltage Clamp Technique (Augustine et al., 1985); and (c) the finding can be explained with Ca 2+ micro-domains and three-dimensional Ca 2+ diffusion (Simon and Llinas, 1983; Fogelson and Zucker, 1985). Accordingly, the Ca 2 + currents measured with voltage clamp techniques do not reflect the Ca 2 + concentration actually present below the release sites. At the higher depolarizations, even though less Ca 2 + enters per channel, more Ca 2 + is available for release because of overlap of Ca 2+ from the larger number of channels opened at the higher depolarization. Without entering into the debate concerning whether the "hysteresis" is an artifact of the voltage clamp technique, we tested whether the threedimensional diffusion model is capable, as hoped, to predict a higher release at a higher depolarization for the same Ca 2 + current. We found that the answer is negative. In the three-dimensional model, the Ca 2 + current, that is the Ca 2 + influx, is given by the influx per channel, governed by the parameter 9, times the number of open channels. The dependence of the Ca z + influx on membrane depolarization is ttepicted in Fig. 9 (see . . . . in insert). It can be seen that the concentration of Ca z + near a release site (solid line in insert) exactly matches the influx of C a 2 +. Release (dashed line in insert), as expected, is the same for equal Ca 2 + currents (influx).

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A

-Z5mV ~ tesl pulse

"~ 20

o

[ ..,' U".\I,

[ ,

oI

~ -25

/

,, -,.

I

~ -15

..

,

, 0 •

,-~ a

,Jo,.,

~ >0 "

i >>0 (mV)

.~ 20 -.~--

I0-

0

I I -25 -15 0 >0 >>0 Pre pulse depolarizofion ( m V )

Figure 9. Computer simulation of the dependence of facilitation on the depolarization of the first pulse. The protocol of the "experiment" is outlined at the left. The second, test, pulse is kept constant at - 25 mV. The first, prepulse, acquires variable levels of depolarization. The corresponding number of open channels and value of ,q(t) are as described in Fig. 5. At depolarizations higher than 0 mV ( > 0 mV and ~>0 mV), 64 channels are open, but the 9(t) values are 0.25 and 0.1, respectively. Insert: the dependence of intracellular Ca 2 + concentration, C (solid line), and release, R (dashed line), on first pulse amplitude.

A second point pertains to test pulse facilitation and its dependence on first pulse depolarization. Test pulse facilitation has a bell shaped dependence on first pulse amplitude (see Fig. 2). Attention must be drawn to the fact that as facilitation declines, release of the first pulse continues to increase (see insert in Fig. 2). Dudel et al. (1983), who first observed this phenomenon, explained it as being analogous to the "hysteresis" of the curve relating release to membrane depolarization. As test pulse facilitation reflects residual Ca 2 +, they concluded that lower facilitation reflects smaller Ca 2 + entry during the first pulse and that the higher release seen at the higher depolarization is the result of a direct membrane depolarization effect. Zucker and Lando (1986) claimed that the result is an artifact of the loose macropatch system. Again, without entering into this debate, which has already been discussed in several papers (see for example Parnas and Parnas, 1986), we tested whether the three-dimensional diffusion model can explain this experimental finding. As Fig. 9 shows, the answer is negative. The facilitation and evoked release show the same dependence on prepulse depolarization. In particular, both decline at precisely the same depolarization (compare curve and insert in Fig. 10). We conclude that the three-dimensional diffusion model cannot account for either the

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889

experimentally observed hysteresis (Llinas et al., 1981) or the dissociation in the dependence of release and test pulse facilitation on depolarization (Dudel et al., 1983).

(a)

(b) Model Assumed

Saturotive Release

Facilitation F at shortand longintervals.

Duration of facilitation Tf

Assumed

Non-Saturative Removal

F

L- -[ [Ca]Li

-(K~+[coIII)

Mode I

Tf

d [Call = _~[Ca] i

i

dt

[Cole

[ca] o Non-Saturative

Saturative

Release

Removal

L: X [Ca] Li

Tf

d[Co]~ =_ ~[Ca]~

i i I

dt

[Ca]o

Kc-[Ca]i

/ [CaJo

Figure 10. Predictions concerning the behavior of various aspects of facilitation when different model equations are assumed. (a) The effect of the model equation assumed for release, L, on the dependence of facilitation on [Ca2+]o. (b) The effect of the model equation assumed for removal of the entered Ca 2+ on the dependence of the duration of facilitation, Tf, on [Ca2+]o . Redrawn from H. Parnas et al. (1982).

The effect of the pump on the time course of release and of facilitation. The three-dimensional diffusion model includes a pump as the only other means, in addition to diffusion, of regulating the concentration of intracellular C a 2 + near the membrane (see upper equation in Fig. 5). However, with the standard parameters employed in the model by its authors, the pump contributes very little to C a 2 + regulation. Such low rates of pump activity were probably taken in order for diffusion to play the major role in determining the distribution of Ca 2 +. Indeed, simulations of the three-dimensional diffusion model show that reduction of the pu'mp rate below its standard value does not prolong the time course of evoked release or the magnitude and time course of facilitation (not shown). In contrast, Fig. 3 shows that slowing the extrusion of C a 2 + (in the experiments, by blocking the [Na+]o*--fl-Ca2+] i exchange) significantly

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prolonged facilitation and had no effect on the magnitude or the time course of release elicited by the first pulse. If the rate of the pump is increased tenfold (over its standard value), the pump contributes significantly to the temporal regulation of intracellular Ca 2 +. Under such conditions, in contrast to experiments, both the time course of evoked release and that of facilitation are shortened in parallel.

The dependence of facilitation on temperature. The three-dimensional diffusion model, employing its standard parameters, predicts a much too short time course of facilitation (see for example Figs 7 and 8). Even in the frog neuromuscular junction, where facilitation is relatively short, it lasts, at room temperature, approximately 100ms at the normal [CaZ+]o concentration (1.8 raM) (Rahamimoff, 1968; Mallart and Martin, 1967). In'the crayfish, facilitation lasts approximately 1 s under physiological conditions of [Ca2+]o = 13.5 mM and at around 16°C (H. Parnas et al., 1982). Lowering the temperature in the three-dimensional diffusion model does not result in prolongation of the too short facilitation, since under standard conditions it is mainly diffusion that regulates the Ca z + concentration near the release sites, and diffusion depends very weakly on temperature. For the same reason the three-dimensional diffusion model fails to account for the experimental results of Fig. 4, namely that the time constant of decay of facilitation decreases as temperature is raised. Discussion. The main experimental results associated with facilitation are summarized below.

(1) As the extracellular calcium concentration, [Ca2+]o, is raised, short term facilitation of identical pulses decreases, but the duration of facilitation increases (Fig: 1). (2) Test pulse facilitation shows a bell-shaped dependence on the level of depolarization of the first pulse. Both the short term test pulse facilitation and the duration of the test pulse facilitation show the same dependence on depolarization. In particular, the maximal amplitude of both is obtained at the same level of depolarization (Fig. 2a,b). On the other hand, release of the first pulse continues to increase even at those prepulse depolarizations which produced smaller facilitation. (3) Inhibition of the [Na +]o~-+[Ca2 +]i exchange significantly prolongs the duration of facilitation but does not alter the time course of evoked release (Fig. 3). (4) The decay constant of facilitation decreases as temperature is raised. These experimental results are of interest because they shed light on the

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OF Ca 2 ~

891

nature of the processes that underlie facilitation. In particular, the sensitivity of facilitation to the inhibition of the [Na+]o~--+[Ca2+]i exchange suggests that facilitation is indeed determined by the intracellular level of Ca 2 +. This level is presumably not primarily regulated by diffusion, as is evident from the strong sensitivity of the time course of facilitation to temperature. The three-dimensional model can partially account only for result No. 2. It correctly predicts a bell-shaped dependence of facilitation on depolarization, but in contrast to experiments release shows this same dependence on depolarization. In order to better understand the origin of the discrepancies between the experimental results and the predictions of the three-dimensional diffusion model, we present below predictions from a simplified version of the Ca 2 + theory (H. Parnas et al., 1982). Figure 10 depicts the expected behavior of facilitation under different conditions. The dependence of release, L, is given as a saturative or nonsaturative function of Ca 2+ (Fig. 10a). Similarly, the functions describing removal of Ca 2 ÷ are given as saturative and non-saturative (Fig. 10b). If the rate of removal of Ca 2 ÷ depends linearly on intracellular Ca 2 ÷ concentration (Fig. 10b), then the duration of facilitation, Tf, in contrast to experiments (Fig. lc), should be independent of [Ca2+]o (Fig. 10b, upper part). This prediction holds both for a conventional linear equation for removal (Fig. 10b, upper part), as well as for the more elaborate three-dimensional diffusion model (Fig. 7). On the other hand, when a saturative function is assigned for removal, the duration of facilitation, similarly to experiments, increases with [Ca2+]o (Fig. 10b, lower part). Moreover, if release is a non-saturative function of Ca 2 +, then the short term facilitation is independent of [Ca 2 +]o, both for uniform distribution of internal Ca 2 ÷ (Fig. 10a, lower part), and for the localized high concentration of Ca 2÷, as postulated in the threedimensional diffusion model (Fig. 7). In contrast, facilitation declines as [Ca2+]o is raised (as in the experiments, Fig. la,b) if release is a saturative function of Ca 2 +. We thus see that none of the discrepancies existing between experiments and some of the predictions from a simplified Ca 2÷ theory (Fig. 10a, lower part, and b, upper part) were solved by employing the more complicated three-dimensional diffusion model. In conclusion, the defect of the three-dimensional diffusion model must be ascribed to two of its key assumptions: (1) Ca 2 ÷ is the only factor determining release; (2) diffusion is the main process that regulates the spatio-temporal distribution of [Ga 2 +]i which is related to release. It must be stressed that the failure of the three-dimensional diffusion model to account for the experimentally observed behavior of facilitation does not diminish the validity of the residual Ca 2 ÷ theory for facilitation (Katz and Miledi, 1968). A large body of evidence has been accumulated to demonstrate that residual free or complexed

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Ca 2 + is indeed the dominant factor in regulating facilitation (see Introduction and review by H. Parnas et al., 1990; Parnas and Segel, 1989). Recently, a few other discrepancies between experimental results and predictions obtained from the three-dimensional diffusion model have been pointed out (Delaney et al., 1989). However that paper, as well as earlier ones (for example, Fogelson and Zucker, 1985), failed to pinpoint the fundamental cause of the observed discrepancy. The purpose of this analysis was not to question whether Ca 2 + diffuses away once it enters through the channels. It is the relative weight of the different C a 2 + removal mechanisms which is questioned. We conclude that.diffusion of C a 2 + cannot be the major mechanism for C a 2 + removal that determines the time course of facilitation. Other mechanisms such as the [Na+]o+--~[Ca2+]i exchange are involved in determining the duration of facilitation. Furthermore, none of the Ca 2 + removal mechanisms, and certainly not diffusion, can account for both the rapid termination of evoked release and its insensitivity to treatments that greatly alter the intracellular Ca 2+ concentration, and concomitantly to the much longer duration of facilitation. Some of the discrepancies between the three-dimensional diffusion model and the experimental results can be corrected by slightly modifying the threedimensional diffusion model without altering its main assumptions. For example, if R, the release equation, is modified to become a saturative function o f C a 2 +, then facilitation would decline a s [ C a 2 + ] o is raised (H. Parnas et al., 1982). However, the rest of the inconsistencies between predictions and results cannot be overcome unless the principal assumptions of the model, namely that diffusion is the major factor governing the spatio-temporal distribution of i n t r a c e l l u l a r C a 2 + and release depends solely on Ca 2 +, are abandoned. In this and in a previous paper (H. Parnas et al., 1989), we showed difficulties of the C a 2+ hypothesis for neurotransmitter release in accounting for experimental results associated with evoked release and facilitation. We further showed that refinements such as consideration of three-dimensional diffusion and localized points of entry of C a 2+ (Fogelson and Zucker, 1985) did not solve any of the problems encountered with a simpler version of the C a 2 + hypothesis. It is possible that implementation of other, as yet untried, ideas may overcome the existing difficulties. However, to date, only the alternative Ca 2 +-voltage hypothesis for neurotransmitter release has been able to predict a wide range of existing experimental results associated with both evoked release and facilitation (see review by H. Parnas et al., 1990). Hanna Parnas would like to thank the National Institute of Health for their support of this research. Itzchak Parnas is the Greenfield Professor of Neurobiology, supported by the Goldie Anna Trust Fund. We are also grateful to Frances Bogot for preparing the manuscript for publication.

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LITERATURE Arechiga, H., A. Cannone, H. Parnas and I. Parnas. Blockage of synaptic release by brief hyperpolarizing pulses• J. Physiol. (London), in press. Augustine, G. J., M. P. Charlton and S. J. Smith. 1985. Calcium entry and transmitter release at voltage clamp nerve terminals of squid. J. Physiol. (London) 36"/, 163-181. Barrett, E. F. and C. F. Stevens. 1972. The kinetics of transmitter release at the frog neuromuscular junction. J. Physiol. 227, 691 708. Bittner, G. D. and V. L. Sewell. 1976. Facilitation of crayfish neuromuscular j unctions. J. comp. Physiol. 109, 287-308. Bittner, G. D. and R. A. Schatz. 1981. An examination of the residual calcium theory for facilitation of transmitter release. Brain Res. 210, 431-436. Blaustein, M. P. 1988. Calcium transport and buffering in neurons. TINS 11,438M43. Connor, J. A., R. Kretz and E. Shapiro. 1986. Calcium levels measured in a presynaptic neurone of Aplysia under conditions that modulate transmitter release. J. Physiol. (London) 375, 625-642. Charlton, M. P. and G. D. Bittner. 1978. Facilitation of transmitter release at squid synapses. J. 9en. Physiol. 72, 471-486. Datyner, N. B. and P. W. Gage. 1980. Phasic secretion of acetylcholine at a mammalian neuromuscular junction. J. Physiol. (London) 303, 299-314. Delaney, K. R., R. S. Zucker and D. W. Tank. 1989. Calcium in motor nerve terminals associated with post tetanic@otentiation. J. Neurosci. 9(10), 3558-3567. Dudel, J. 1984. Control of quantal transmitter release at frog's motor nerve terminals. I. Modulation by de- or hyperpolarizing pulses. Pflfigers Arch. 402, 235-243. Dudel, J. 1990a. Calcium and depolarization dependence of twin-pulse facilitation of synaptic release at nerve terminals of crayfish and frog muscle. Pfliigers Arch., in press. Dudel, J. 1990b. Twin pulse facilitation in dependence on pulse duration and calcium concentration at motor nerve terminals of crayfish and frogs. Pfliigers Arch., in press. Dudel, J., I. Parnas and H. Parnas. 1983. Neurotransmitter release and its facilitation in crayfish muscle. VI. Release determined by both intracellular calcium concentration and depolarization of the nerve terminal. Pfliigers Arch. 399, 1-10. Fogelson, A. L. and R. S. Zucker. 1985. Presynaptic calcium diffusion from various arrays of single channels. Biophys. J. 48, 1003 1017. Hodgkin, A. L. and R. D. Keynes. 1957. Movement of labelled calcium in squid giant axons. J. Physiol. (London) 138, 153-281. Katz, B. and R. Miledi. 1965. The measurements of synaptic delay and the time course of acetylcholine release at the neuromuscular junction. Proc. R. Soc. Lond. B161, 483-495. Katz, B. and R. Miledi. 1968. The role of calcium in neuromuscular facilitation. J. Physiol. (London) 195, 481-492. Katz, B. and R. Miledi. 1977. Suppression of transmitter release at the neuromuscular junction. Proc. R. Soc. Lond. B196, 465-469. Llinas, R., I. Z. Steinberg and K. Walton. 1981. Relationship between presynaptic calcium current and post synaptie potential in squid giant synapse. Biophys. J. 33, 323-352. Lustig, C., H. Parnas and L. A. Segel. 1989. Neurotransmitter release: Development of a theory for total release based on kinetics. J. theor. Biol. 136, 151-170. Magleby, K. L. 1973. The effect of repetitive stimulation on facilitation of transmitter release at the frog neuromuscular junction. J. Physiol. (London) 234, 327-352. Mallart, A. and A. R. Martin. 1967. An analysis of facilitation of transmitter release at the neuromuscular junction of the frog. J. Physiol. (London) 193, 679-694. Parnas, H. and L. Segel. 1980. A theoretical explanation for some effects of calcium on the facilitation of neurotransmitter release. J. theor. Biol. 84, 3-29. Parnas, H., J. Dudel and I. Parnas. 1982. Neurotransmitter release and its facilitation in crayfish. I. Saturation kinetics of release and of entry and removal of calcium. Pfliigers Arch. 393, 1-14.

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Parnas, H. and L. A. Segel. 1989. Facilitation as a tool to study the entry of calcium and the mechanism of neurotransmitter release. Prog. Neurobiol. 32, 1-9. Parnas, H., G. Hovav and I. Parnas. 1989. The effect of Ca 2+ diffusion on the time course of neurotransmitter release. Biophys. J. 55, 859-874. Parnas, H., I. Parnas and L. A. Segel. 1990. On the contribution of mathematical models to the understanding of neurotransmitter release. Int. Rev. Neurobiol. 32, 1-50. Parnas, I., H. Parnas and J. Dudel. 1982. Neurotransmitter release and its facilitation in crayfish. II. Duration of facilitation and removal processes of calcium from the terminal. Pfliigers Arch. 393, 232-236. Parnas, I. and H. Parnas. 1986. Calcium is essential but insufficient for neurotransmitter release: The calcium-voltage hypothesis. J. Physiol. (Paris) 81,289-305. Rahamimoff, R. 1968. A dual effect of calcium ions on neuromuscular facilitation. J. Physiol. (London) 195, 471-480. Simon, S. M. and R. R. Llinas. 1985. Compartmentalization of the submembrane calcium activity during calcium influx and its significance in transmitter release. Biophys. J. 48, 485498. Stockbridge, N. and J. W. Moore. 1984. Dynamics of intracellular calcium and its possible relationship to phasic transmitter release and facilitation at the frog neuromuscular junction. J. Neurosci. 4, 803-811. Zengel, J. E., K. L. Magleby, J. P. Horn, D. A. McAfee and P. J. Yarowsky. 1980. Facilitation, augmentation and potentiation of synaptic transmission at the superior cervical ganglion of the rabbit. J. 9en Physiol. 76, 213-231. Zucker, R. S. and L. Lando. 1986. Mechanisms of transmitter release: voltage hypothesis and calcium hypothesis. Science 231,574-579. Zucker, R. S. and N. Stockbridge. 1983. Presynaptic calcium diffusion and the time courses of transmitter release and synaptic facilitation at the squid giant synapse. J. Neurosci. 3, 1263 1269.

R e c e i v e d 27 F e b r u a r y 1990 R e v i s e d 30 A p r i l 1991