O?WJi22’92

SS.OQ+ 0.00

PergamonPressLrd t‘ 1991, IRRO

THE RELATTON BETWEEN TRANSMITTER RELEASE AND Ca”” ENTRY AT THE MOUSE, MOTOR NERVE TERMINAL: ROLE OF STOCHASTIC FACTORS CAUSING HETEROGENEITY D.M.J.QuAsTEL,Y.-Y.CJUA~* Medicine. 2176 Health

Mall, Vancouver,

The Canada

of British V6T

Abstract-The

relation quanta1 transmitter and pr~s~n~~ptic +,%a’+ entry the mouse junction was making use the finding in the of Ba” of nerve end-plate potential stimuli or brief nerve terminal depolarizations elicit “tails” of raised miniature frequency (,&,) that reflect entry of Ba” per pulse. and hence effectiveness of pulses in opening Ca’+.‘Ba” channels; at the same time these pulses elicit end-plate potentials. With nerve stimulation in the presence of Ba?) and Ca’+ and modulation of release by raised Mg’+ or b~kallamycin. slopes of log quantal content (m) vs log apparent Ba? + entry per pulse were close to 4, which is the same as the Hill coefficient for Ba’ ’ cooperativity derived from other data. With depolarizing pulses of varied intensity, however. similar plots gave slopes close to 2, with Ba?+ alone or in a mixture of Ca’+ and Ba’+. Thus, the relation between transmitter release and Ca”” (or Ba’+ ) entry apparently depends upon how entry is varied: varying the numbers of channels opened is not the same as varying ion entry per channel. A mathematical model was developed to examine the consequences of heterogeneity of local Ca” (or Ba’+ ) between release sites, arising because of stochastic variation of number and time course of Ca’+ channels opened per site: the experimental results were consistent with this model. It was therefore concluded that release is normally governed by intracellular Ca2 + closeto points of Ca” entry through channels; stochastic factors give rise to more release than if Ca’ ~ were homogeneously distributed. tf Ca’ f channels are uniformly close to release sites the average number of channels opened per site per action potential may he as low as 4.

It is generally agreed that depolarization-release coupling in nerve terminals depends critically on enlry of Ca’+ Into the pfcsynaptic terminal via voltage-sensitive Ca’ ’ channels’5~“’ and some sort of cooperativity ofCa’+ in the release process is implicit in the finding that end-plate potentials (EPPs) at the neuromuscular junction are steeply graded with Ca’ +.‘).i4ft has been unclear, however, whether the apparent cooperativity reAects only the intrac~llu~~?r mechanism by which CaZ+ induces release.“~” To resolve this question we have used an indirect measure of Ca” channel opening and ion entry, namely, the “Ra’ “-tail” of high miniature end-plate potential (MEPP) frequency that follows trains of nerve terminal action potentials or nerve terminal depolarizations in the presence of Ba?‘.‘” This evidently represents Ba’+ a&c~~mu~ation within the terminal”” and the intensity of a ‘“tail” therefore reflects the number of channels that have been opened per nerve terminal action potential or nerve terminal

*Prcscnt address: De~rtment of Ph~~acol~~~y~ Sun Yatsen University of Medical Sciences, Zhongshan Road II, Guctngzhou, The People’s Republic of China. *Present address: Department of Physiotogy. Australian National University. Canberra, Australia. Ahh~c~iu!io~~: BAPTA, l,2-bis(p-aminophenoxylethanl:)M..ni.!ni’,R”-tetra-acetic acid; EPP, end-plate potential: MEPP. miniature end-plate potential.

depolarization, if this number remains Constant during a train.7” Previously. this method was employed to obtain the relation between fprolonged) presynaptic depolarization and channel opening in the presence of Ba’^ and absence of Ca”. Together with the observed relation between depolarization and relcasc in the presence of Ca’ ‘. this yielded a slope of 2 for the graph of log release in Ca’” vs log fhanncl opening.“’ in contrast to a Hill coefficient (tr) for Ha’ ’ of 4 or 5, derived from the relation between “*tail” intensity and number of nerve terminal action potcntials or nerve terminal depolarizations in a train.“’ In the present experiments we have used brief nerve terminal depoiarizations. producing “direct*’ EPPs, and nerve terminal action potentials in the presence of both Ca’ * and Ba’ ‘, or Ba’” atonc. to obtain simultaneous measures of release (quanta1 content of EPPs) and ion channel opening (from the Ba’ ’ tail). The data confirm that transmitter release indeed grades approximately with the second power of Ca’_ channel opening (and ion entry), but this applies equally to Ba’+ and Ca’+-mediated release. and only when each are varied by altering the rate of channel opening or number of channels opened per pulse. When ion entry per channel is varied by added Mg’ ’ or bekanamycin release grades with the fourth power of ion entry. The value of 4 coincides with estimates

D. M. J.

658

QCASTEL cd d

of the Hill coefficient expressing the relation between exocytosis rate and internal Ca2+ in rat melanotrophs. 38 At the squid giant synapse, where Ca’.’ entry can be measured concurrently with measurement of transmitter release manifest in excitatory postsynaptic potentials,2~5~17~‘* data suggest the same n and also that the relation between release and Ca” entry depends to some extent upon how entry is varied.2 We find that this behaviour is predicted by a model based simply on a power relation between release probability and local intracellular Ca*+ and the concept of “Ca-domains”,4 when one takes into account stochastic factors that produce heterogeneity between release sites of local Ca’+ concentration after voltage-gated channels open to admit Ca2+. EXPERIMENTAL

PROCEDURES

All experiments were performed on hemidiaphragms removed from ether-killed mice, mounted on Sylgard and superperfused as previously described7,r0 at room temperature (25528°C). Standard bathing solution contained 150mM Nat, 5mM K+, 24mM HCO;, 125mM Cl-. 1 mM H,PO, , 11mM glucose, and was bubbled with 95% 0,-S% CO,, to which was added CaZ+, Mgz+, Bar+ etc. as required. In some experiments Cl- was replaced by NO;, with no obvious difference in the results except to make MEPPs larger.6,” Intracellular recording from muscle fibres was conventional, using KCI-filled microelectrodes; a Mingograf ink writer provided permanent records from which EPPs were measured and MEPPs counted. Recordings were also made using a microcomputer, which permitted verification that measurements of EPPs and counts of quanta immediately following stimulation pulses were within the time frame (about 1 ms) in which a quanta1 event was defined as an EPP. For direct depolarization of nerve terminals we used large (-20 pm) tipped micropipettes filled with 4 M NaCl and agar, placed in proximity to the nerve terminal by monitoring the increase in MEPP frequency (f,) and decrease in MEPP height produced by depolarizing current.’ With this method, any non-uniformity of nerve terminal polarization is visible in non-uniformity of MEPP height modulation by applied currents;’ only uniformly polarized junctions were selected for study. Intensity, duration, and repetition rate of direct polarizations in the presence of Ba*+ were chosen to avoid generation off, more than about 400/s, at which there was difficulty in counting MEPPs, to avoid “explosions” of extremely high f,, probably indicative of regenerative responses in the nerve terminal, and to keep well below threshold for eliciting contractures in the muscle fibre. In these experiments solutions contained tetrodotoxin and 4-aminopyridine to suppress voltage-sensitive Na+ and K’ conductances. Suction electrodes were employed for nerve stimulation. In all cases MEPP frequency (&) has been expressed as MEPP/s. Calculation of Ba2+ entry per pulse

Apparent Ba*+ entry into nerve terminals was calculated from the “Ba*+-tail”, i.e. the raisedf, following a train of nerve or nerve terminal stimuli in the presence of Ba*+; justification for this procedure is found elsewhere.30 In general, when Ba2+ entry was varied by varying intensity and/or duration of depolarizing pulses, an effort was made to alter the number of pulses in trains so as to produce tails of approximately the same magnitude (usually about 100 MEPPs in the period 0.2-2.5 s after the train) so that apparent Ba2+ entry per pulse was derived primarily on the

basis of the relative number of pulses required 1~1product the same tail However, since for any series of identical pulses the l/4 power of average& over a defined period in the tail grew linearly with the number of pulses,” essentiaily identical values were obtained using the increment of /‘g 4 (=f,‘!” -f$“) per pulse as a measure of apparent Ba’ ’ entry, wherex is the meanf, over the period 02 2.5 s after the train and ,f, is mean spontaneous fi (from 20 s control periods before each train), both expressed in units of i: I. The latter measure was generally employed so as not to exclude data from trains producing relatively small or large tails. Calculation of quanta1 content of end-plate potentiuO

A minor problem encountered with respect to measurement of EPPs was the increase in /k that occurs during a train of nerve stimuli or nerve terminal depolarizations in the presence of Ba2+, which may reach a level where it is difficult to distinguish between quanta1 units of the EPP and “spontaneous” MEPPs. To avoid this, in all experiments EPPs were measured only for those pulses in a train before f, rose to where there could be any confusion of MEPPs with quanta1 units of EPPs. A more difficult problem was how to deal with the rise in quanta1 content of EPPs (m) that occurs concurrently with rise in f, during a train of nerve terminal action potentials or nerve terminal depolarizations, presumably as a result of Ba2+ entry. Here we obtained “corrected” values for initial quanta1 content, based on the following. In other experiments,‘3.28it was observed that in the presence of both Ca” and Ba*+ or in Ba 2+ alone nerve stimulation causes increase in MEPP frequency (f,) and quanta1 content (m) in accord with a constant value for (1000m)‘;4 -SL”,” and during a train fz” grows linearly with the number of pulses; ” this agrees with a “residual-Ba2+” mode1 in which release rate is a-power function of the concentration(s) of Ca’+, or. in this case, Ca*+ and Ba2+, at critical sites within the nerve terminal.“~2’.3DThus, one can calculate (using a microcomputer) the total number of quanta expected over any portion of a stimulation train, given any (guessed) initial m and observed values off, before and after the train. By trial and error one finds the value for initial rn that produces the observed number of quanta. In practice it was found that correction of tn values by this procedure made little difference to the final results; slopes of log m vs log per pulse increment infi4 were not significantly affected. Neither did correcting quanta1 content in the same way but using n = 5, instead of 4, or using 2OOOminstead of 1000m. have any discernible effect on these slopes. When EPPs were fairly large (no failures), amplitude either rose little or tended to fall during a pulse train, presumably because of depletion of available transmitter.‘” Therefore, measurement of only the first few EPPs in a train sufficed for an estimate of quanta1 content; mean EPP height was corrected for non-linear summation by the formula of Martin’9~26and divided by mean MEPP height to give quanta1 content. Mathematical modelling

To calculate the hypothetical release expected from a nerve terminal when Ca2+, entering via channels, is not uniformly distributed among release sites, one needs not only an equation for the relation between release rate and local [Ca2+] but also a distribution curve, P(c) vs c, where P(c) is the probability that a site has a given concentration of Ca2+, c. In each variant of the model this distribution curve was defined by three variables: (1) p. the mean number of channels opened per release site; (2) c,, the resting level of Ca2+, present at sites without any increment of Ca*+ produced by channel opening; and (3) C,, the mean increment in Ca’+ produced by each channel opened. However, the distribution curve also depends upon whether the increment in [Ca2+] per site per channel is uniform (variant I),

Transmitter

release and CaZ+ entry

exponentially distributed (variant II) or varies also because channels are randomly located with respect to release sites (variant III)-these assumptions introduce degrees of heterogeneity of local Cal+ between release sites that are defined by the model rather than by arbitrary parameters. For the relation between release and [Ca’+] (c) we used:

r =

: = kc”

(la)

(’ = cp + (‘”

(lb)

1- exp( -z6[),

(lc)

where : is the instantaneous release rate but r is the release expected from one site in time 6r (arbitrarily taken as 0.5 ms). In these equations, c, the total Ca’+ at a site, has two components, c,,, resting Ca’+ per site (assumed uniform and constant), which is assumed to be reflected in “resting” /,,, (./,) and cp, the Ca” added when channels open. The Hill coefficient, U, k and br are constants; the attribution of,f,, to c’r)leads to calculated results that are independent of the chosen value of k. Equation (Ic) serves to limit release to a maximum of 1 quantum per site in time 6t. From the above equations the probability of release by a site in time 6z can he calculated given n. a value for.fb (usually chosen as 0.1 :s or I IS) and the ratio of cp to co, Summing I over all possible values of c. with weighting according to its probability distribution curve, produces the expected release per site; we convert this to quanta1 content (m) by assuming (arbitrarily) that there are 1000 release sites per nerve terminal (cf. Ref. IO). A numerical example may help to clarify this procedure. Suppose that n = 4. .C, is 1:s (5 x lO~‘isite/0.5 ms). that a presynaptic depolarization opens an average of one channel per site, and each channel contributes to local Ca’+ an increment (C,) equal to IO co. Then, if one channel opened at every site, the instantaneous release rate at each would be increased by a factor of II’ to 0.00732/sitei0.5 ms. This gives [equation (Ic)] a probability of release in 0.5ms of I ~ exp( -0.00732) = 0.00729 and nr = 7.29. However. with stochastic channel opening some sites will be associated with no open channels. some with one. some with two. some with three, etc. In the simplest case (I-uniform Ca” increment per channel) the instantaneous release rates [z in equation (la)] at sites with zero. one, two. three etc. channels are the spontaneous release rate multiplied by I, I I3 , 21’. 31”. etc. By equation (Ic) these translate respectively into release probabilities of 5 x IO- ‘, 0.00729, 0.0926, 0.370. 0.757 (for four channels), 0.966 (for five channels) and 1.0 (for six or more channels). The overall release probability is obtained by adding the products of these numbers and the corresponding probabilities of occurrence of zero. one. two, three. etc. channels. which, for a Poisson distribution of numbers of open channels, are exp(-AL). p exp(--p), /~‘cxp( -/1):2, /L’ exp(-p):‘3 etc., with p = I. The resulting sum is 0.0576 (m = 57.6); about 66% of the release comes from sites where three or more channels havne opened. although this occurs at only 8% of sites. For a binomial distribution of channel opening with N = 4 and an average of one channel per site the probabilities of zero, one, two, three and four channels are respectively. 4’. 4pq’, 6p’q’. 4p’y and p4, where p = 0.25 and y = 0.75. The summation gives ,?I = 42.9. In our calculations we run through a range of possible values of mean Ca’+ contribution per channel to a release site (C‘,) and a range of possible values of p, mean number of channels per site. For model variant I the calculations are straightforward (see above) since cp takes discrete values (0. C,. 2C,, 3C, etc.). By equations (1a-c), these give values I,. each of which is the expected (average) release from a site with i open channels. Overall release by the nerve terminal for any 11 is then given by the sum of p,r, (where p, is the probability that i channels will be open). multiplied by 1000 (the assumed number of release sites). For a Poisson distribution of numbers of channels opening per site pii = rxp( -/r ). p, = ~rp,~.p2 = i~p,:2. p, = ppz:3 etc.. while for

h59

a binomial distribution p,‘s are defined by the binomial coefficients and p (=p/N, where N is the number of available channels). With variants II and III, however. the contribution of Ca’+ to a site by a single channel varies. For these it is necessary to construct for each number i of open channels (one, two, three etc.), a distribution function P,(.\-) vs x, where .Y extends over a wide range (up to 100 was sufficient) and the mean of the distribution is i. For any given C,. every value .X has a corresponding (‘p ( =.YC,). giving [by equations (la-c)] a corresponding release probability I(\.). Summing P,(s). r(s) over all possible values of .Y gives I,. In model variant II we assume that for each channel the contribution of Ca’+ to a nearby release site is exponentially distributed. because of an exponential distribution of channel open times. while in variant III channel contribution also varies because channels are at varymg distances from release sites. For both of these the procedure is first to construct the distribution curve P,(s) for Ca’+ distribution at sites where only one channel has opened; the probability distribution P,(s) for sites with any number i channels is then obtained as the convolution of P,_ ,(.Y) with P,(s). For II (exponential distribution). log(P,(\-)) falls linearly with I; the slope of the decline can have any convenient value. Successive convolutions. to obtain Pz(.v). Pl(_\-). etc. are most easily performed by integration and normalization to give unit area. For III, to obtain P,(.u) one starts with a function P( I-). where r is the mean contribution of a channel relative to the maximum possible (for channels at zero distance from the site). The actual Ca’+ contribution of a channel to a site will be a complex function of time and distance (tl). but from a hemispherical diffusion model we obtain that the tnne integral of [Ca’+] is almost proportional to cxp( -0). after an instantaneous injection at distance tl. Then. the fraction of channels (assumed proportional to the surface area) that will give Ca” :site between any chosen hmits is proportional to & ~ d$ where d, and (I? are defined by the limits. For example. limits 0.1 I and 0.12 of maximum give tl, = -log(O.l I) and dz = -log(O.l2). To avoid the inlinite value when one limit is 0, one must ignore the region below a cut-off point that is an arbitrary fraction of the maximum. The resulting curve of P(J,) vs J’ is normalized by dividing the sum of the values P(J,). Each value J’ is now considered as the mean of an exponential distribution curve. Q, (1 ). with area umty (see above) and one obtains a curve P,(V) vs .Y by summing P(),).Q, (_Y). From this. successive convolutions give the distribution curves for sites with various numbers of open channels. In the illustrated results for variant III the chosen cut-oh fraction was 5%. giving a mean c‘, 17%; of that for a channel at the release site. Choosing other values for the minimum cut-off has a verv simple effect once the cut-oh’is below IO?‘,, or so of the maxjmum. The lower the cut-off the lower is the average C, as a fraction of the maximum and the more channels one counts (although many are ineffective) as producing a given nr. When 11 is weighted hy this fraction and (, is weighted inversely by this fraction. plots of calculated quanta1 content (m) vs weighted 11 and weighted C, become independent of the cut-off. Model variant III is inexact in that the formulae do not give the heterogeneity arising from a random distribution of Ca” channels. However. experimenting with other relations between Ca” contribution per channel and rl showed the overall results to be very insensitive to the relation chosen---the major factor is the coefficient of variation of the distribution curve P,(x). In practice the above calculations for II and III depended upon substituting discontinuous for theoretical continuous distributions: the accuracy was checked by verifying that increasing the number of points in each array corresponding to theoretical distribution curves made no apprcciahle difference to the calculated results.

660

D. M. .I. OUASTEL

In calculating the modification by “residual ion” ol magnitude and time course of transmitter release (Fig. 10). no attempt was made to deal with the evolution in time of the variance of ion distribution between sites, since this will depend upon factors that are not known--diffusion rate. internal ion binding, life-time of channels. It would also be a difficult computational problem. Instead. it was simply assumed that C, evolves in proportion to e -“’- e h’, with rate constants 2/ms and 20/ms, stochastic factors being constant in time. However, release at any one site was limited to 1 quantum over the whole time period. All calculations were done on a microcomputer with programs written in “C“.

RESULTS Experimental results “Direct” end-plate potentials-varied nerve terminal depolarizations. In these experiments nerve terminals were subjected to trains of brief nerve terminal depolarizations of varied amplitude and/or duration in the presence of Ba2+ alone or a mixture of Ba2+ and Ca2+. These evoke “direct” EPPs during the train, mainly attributable to Ca2+ entry when Ca2+ is present, and a rise in MEPP frequency (f,), which continues as a “tail” after the train, attributable to Ba2+.30 Each train provides a value for the quanta1 content of the EPPs (m) and of increment offi per pulse. On the assumption that the “Ba2+ tail” represents Ba’+ accumulation in the nerve terminal, with equal entry per pulse, 3o the latter measure is proto portional to Ba’+ entry per pulse and proportiona the number of Ca2+/Ba2+ channels opened per pulse. Two examples of graphs of log m vs log increment of f!,(“ per pulse, in the presence of Ca2+ and Ba’+, are shown in Fig. 1; in A some of the determinations were

h

0.2

I

increment

o.k---T-n

of fm’14

per 100

pulses

Fig. 1. Log-log plots of quanta1 content (m) vs increment of fi’ per pulse, for direct EPPs elicited by brief nerve terminal depolarizations in the presence of 1 mM Ca’+ and 2 mM Ba2+. Each point represents the value of m early in a train of nerve terminal depolarizations and “increment of f!! ner 100 pulses” is the value of ( ff’4 - fY4) multiplied by 100 and divided by the number of p&es in”the train, where f; is the mean& at 0.2-2.5 s after the train andf, isf, before trains. At the junction on the left, m was modulated mainly by varying pulse height, with some pulses of 0.3 ms and others at 0.4 ms. At the other junction pulse height was kept constant but duration varied from 0.5 to 0.9ms. In both examples the concentration of 4-aminopyridine was 1 mM and the concentration of tetrodotoxin was lO_‘M. The regression lines have slopes of 2.25 and 1.7 respectively. Expts 8657 and 85221d. .

.

.

_

rl N/

made at a pulse duration of 0.3 ms and others ;II 0.4 ms, the major modulation being of pulse height. while in B the pulse height was held constant but duration varied over the range 0.5SO.9 ms. Because of an effective nerve terminal time constant of about 1 ms,” the latter method of varying the pulse probably amounts to a change in extent rather than duration of depolarization. It may be noted that any slow drift of the position of the polarizing electrode relative to the end-plate, between the trains of stimuli. will merely change the polarizing voltage per applied current,’ and not jeopardize determination of the relation between m and Ca” channel opening. In both these examples the same result arises; EPP quanta1 content (m) modulates with inferred Ca’ +channel opening (as manifest in relative Ba” entry per pulse) with an apparent Hill coefficient (n) of about 2, over a range of m of nearly two orders of magnitude. As listed in Table 1. these results were typical of those from nine junctions where this determination was carried out. A value for II close to 2 coincides with that reported’” on the basis of a somewhat different method, in which channel opening per nerve terminal depolarization (long pulses) was determined in Ba2+ without Ca” and transmitter release was measured as f, during prolonged nerve terminal depolarizations in Ca’ L without Ba’ ‘. However, this result was misinterpreted as indicating that the Hill coefficient for Ca’+ was less than for Ba’ ’ That Ca*+ and Ba *+ do not differ in this way is illustrated in Fig. 2, which is for a similar experiment done in the absence of Ca”. where the EPPs are presumably produced mainly by Ba” rather than by a small amount of contaminating Ca” in the solution?’ It is notable that quanta1 content (m) is much lower than when Ca*+ is also present (Fig. I), at any given value of per pulse increment in ,ft,‘. Nevertheless, the slope of the log-log plots is again close to 2, with the same result holding for the six junctions where this experiment was performed (Table 1). Thus, EPPs generated (presumably) by Ba2+ show the same apparent Hill coefficient as EPPs generated by Ca”, although for Ba” the “true” cooperativity for release, as evidenced by the Ba2+-tails, has n =4 or 5.3” Indirect end-plate potentials-nerve stimulation. Evidently, in the above experiments ion entry per depolarizing pulse was varied by varying the number of voltage-gated channels opened. We therefore sought a method of varying Ca’ +/Ba’ ’ entry in another way, the important proviso being that the entry of both ions should be altered to the same extent, so that a graph as in Figs 1 and 2 should continue to be a graph of log m vs log Cal+ entry per pulse (except for a parallel shift on the abscissa). For this we used three agents known to reduce transmitter release reversibly, apparently by competing with Ca2+ for entry, namely raised Mg*+,‘,” bekanamycin24.40 and Cd’+ .‘2,35j9 In principle, if two agents (say B and C) compete for the same binding

Transmitter Table

release and Ca”

661

entry

1. Slopes of log m vs log “Ba’+ entry per pulse” for “direct“ and “indirect” end-plate potentials Indirect Direct 1 mM Ca2+. 2mM Ba’+

EPPs 0 Ca’+, 1mM Ba’+

I .70 2.25 1.58 1.66 2.16 I .36 1.64 1.90 1.87 1.79 0.28 0.10

Mean S.D. S.E.M.

EPPs

1mM Ba’+, 0.1 mM Ca’+ Bekanamycin

Mg”

1.61 2.78 2.07 I .44 2.79 I .83

4.06 3.28 3.96 3.95 3.69 4.37

3.57 3.89 3.76 3.71 3.69 4.34 3.93

2.09 0.58 0.24

3.88 0.37 0.15

3.84 0.25 0.10

Each value represents the slope of a plot of log m vs log (increment off‘!,, per pulse), obtained at a single end-plate from measurement of EPPs and Ba2+-tails produced by trains of stimuli. Direct EPPs were elicited by focal nerve terminal depolarizations of I ms or less in duration, and varied by varying the magnitude and/or duration of the current applied. Indirect EPPs were elicited by nerve stimulation and varied by adding Mg’+ (to 3, 5, 7 mM) or bekanamycin (10, 20, 40 I’M). [Mg”] was generally 1 mM. dissociation constants K,, and Kc. in the presence of a third competitor (say D) then the ratio of the liganded forms [CR]/[BR] is given by the formula: site R. with

[BR]/[R,] = h/( I + h + c + d); [CR]/[R,] = c/( 1 + h + c + d); and, therefore. [BR]/[CR] = h/c where h = [B]/K,, c = [Cl/K,, d = [D]/K,, and [R,] is the total concentration of receptor R. Thus, adding the third competitor D will reduce [BR] and [CR] to the same extent. Figure 3 shows results from a single junction where EPPs and Ba” tails were produced by trains of nerve stimuli in the control solution (0.1 mill Ca’+/l mM Ba’+/l mM Mg’+, pH 6.8) and with added Mg’+,

Cd’+, or bekanamycin, plotted in the same way as in Figs 1 and 2. Notably, points for bekanamycin and Mg’+ fall on the same line in the log-log plot, while with Cd’+ points fall on a line with a lesser slope. As listed in Table I. with raised Mg’ + and bekanamycin, the slope of log m vs log Ca”/Ba’+ entry was consistently close to 4 (3.88 + 0.15 and 3.84 _+0. IO respectively). With Cd’+ the mean slope at six junctions was 2.22 k 0.17 (S.E.M.). The simplest interpretation of this result is that with bekanamycin and ME’+, which have relatively low affinity for the (hypothetical) site at which they compete with Ca” or Ba”. there is in effect reduction of ion entry per channel (although the channel may in fact fluctuate between blocked and unblocked states), so that the true cooperativity of Ca’+ and Ba” at the internal release site appears, Why this might not occur with Cd’+ is considered in the Discussion.

0.1

:,,d,

2

10

incremlaont in fll

I4 ',er 100

20

pulses

Fig. 2. Similar to Fig. 1, with 1 mM Ba’+ and no Cal+. Bathing solutions contained 0.5 mM tetraethylammonium in addition to tetrodotoxin and 4-aminopyridine. At both junctions EPPs and “Ba ‘+-tails” were modulated by varying pulse intensity. The regression lines have slopes of 2.8 and 1.8 respectively. Expts 8661 lA,B.

,

.

1 increase in f, 'I4

2

5 / 100 stimuli

Fig. 3. Plots similar to Fig. I but for nerve stimulation and m modulated by addition of Cd’+ (triangles: 0.5 and 1/tM). Mg’+ (filled circles: 3, 5, 7 mM), or bekanamycin (squares: 10, 20, 40 PM). Note that the slope is less for Cd? ’ than for Mg2+ and bekanamycin. All data from the same junction. Expt. 851152.

662

I>.

2mMBo

1mMBa

M. .I

QIIASTLL

0.5mMBo

1000 fm

: 100:

-

1

r’ 0

0 0

:

0.9

f

0

. .

0.1 0

-5 Polarking

-10 current

-15

-20

(pA)

Fig. 4. Modulation off, at an end-plate during depolarizing pulses, in the presence of 1mM CaZf (open symbols: no BaZ+) and in the presence of 0.5, I, and 2 mM BaZ+ (filled symbols: right to left). Series with Ba’+ were done in ascending order of concentration, and a control series in Ca2+ was done before and after each series in BaZ+ (right to left). In similar experiments no reduction in f, in 0.5 mM Ba*+ by depolarizing pulses was observed..The change in position of the control curves (1 mM Cazf) probably represents slight drift of the polarizing electrode relative to the end-plate; this will also cause some lateral displacement of the curves in Ba’+. Note consistently steeper slope of log& vs applied current in the presence of Ba*+, relative to that with Ca’+.

Increase of miniature end-plate potential ,frequency by prolonged nerve terminal depolarizations in the presence of Ba2+ or Ca2+. Figure 4 shows a typical example of a result obtained using prolonged nerve terminal depolarizations; fmwas counted during each depolarizing pulse once an equilibrium frequency had been attained. At this junction it was possible to obtain series at various levels of Ba2+, with controls the in 1 mM Ca2+ between each series. Although experiment is flawed by the non-return to control values after each test with Ba*+, presumably because of slight movement of the polarizing electrode, and perhaps because excessive stimulation in Ba2+ tends to cause irreversible changes,30 it is clear that the maximum slope of log .f, vs polarizing current was consistently about twice as steep in the presence of Ba*+ as it was in Ca2+. Such a large change in slope cannot be explained by an alteration by Ba2 + of the effective input resistance of the nerve terminal,’ due to blockade of Ca-sensitive K+ channels.4’ Thus, unless the presence of Ca*+ or Ba*+ itself alters the relation of channel opening to presynaptic depolarization, the slope of log.f, vs log channel opening must be about twice as steep in the presence of Ba*+ as it is in the presence of Ca* + Since this slope is about 4 in the presence of Ba2+, with respect to accumulated Ba2’-” it follows that the slope is about 2 for Cal+, at an external concentration of 1 mM. Calculations and computations In this section we examine the consequences of disposing with the mathematical convenience that comes from the simplifying assumption that release sites are homogeneous with respect to changes in local Ca2+ when Ca*+-channels are opened, which

VI N/

has been a feature of previous models” ” ‘X cunccrning the relation of release to Ca’ ’ entry. Instead. UC consider a model in which the heterogeneity of C’;I,among release sites arises from stochastic factors. variation in the numbers of channels opened in proximity to each release site and variation m the life-time of channels, and from spatial distribution 01‘ channels with respect to release sites. This can be done by computing the theoretical distribution ot Ca*+ between release sites corresponding to v,arious sources of heterogeneity and calculating the sum of release expected from different sites. This approach might explain the low slope of log m vs log ion entry when the number of channels opening is varied (“channel-slope”) as can be seen from two artificially simple cases. First, imagine a reductio-ad absurdam where each release site can be supplied with Ca*+ by only one channel, that may or may not be opened by a depolarizing pulse. Then. although the release probability at each site is graded with the nth power of the amount of Ca2’ that enters, overall release is linear with the number of channels that open per pulse. Somewhat more realistically. suppose that there are a large number of channels associated with each site and the number of channels opened by a pulse is a Poisson variable with mean 11. Local Ca2+ takes values of 0, Cc, 2C, etc., with a mean of ,L&,, C, being the [Ca”] contributed by one channel. Then, if the release rate at each site is simply proportional to [Ca”14. the overall rate is proportional to the fourth moment of a Poisson distribution: r = kC& + 711’ + 611~+ klJ). The “Caslope”, dlog r/dlog C, is always 4, while the “channelslope” dlog ridlog p varies between I and 4. with a value of 2 at p = 0.384. Evidently, the latter “model” is highly oversimplified in many respects: (a) the actual number of Ca channels at a release site must be limited and the numbers of channels opening must be binomially rather than Poisson distributed; (b) other sources of Ca2+ heterogeneity are ignored; (c) release rate at any one release site must have a maximum; and (d) no account is taken of “resting” internal Ca’ + Computations were therefore made using a more complex model. Because some of the results were by no means obvious intuitively, they will be presented in some detail below. In general. calculations were done with three stages of discarding the assumption of homogeneity: (I) numbers of channels/site vary stochastically but each channel always contributes the same Ca’ ’ to its site; (II) numbers of channels/site vary stochastically and the contribution of Ca’+/channel per site also varies with channel lifetime, which is a priori exponentially distributed. (III) contribution of Ca2+/channel/site also varies because channels are randomly distributed in the membrane.

663

Transmitter release and Ca’+ entry For comparison, computations were also carried out for the case where all release sites have the same increment of Ca2 +. This presumably becomes realistic when (a) internal Ca2+ (or BaZ+ etc.) has equilibrated or (b) numbers of channels providing CaZ+ to reiease sites are so large that the coefficient of variation between sites is negligible. In order, variants I-III of the model represent increasing variation of Ca’* between release sites. For equivalent numbers of channels~site (p) and mean Ca’+/sitejchannel (C,), the variance for II and III (with “cut-off” as defined in Experimental Procedures) are respectively 2.0 and 3.6 times that for I, for which the variance is pCf for a Poisson distribution. In each case mean total Ca2+ per release site and its variance grow linearly with /L and therefore the coefficient of variation declines with p; at sufficiently high ,Urelease is the same as if Cal+ were homogeneously distributed. A source of variance that we have not taken into account is that introduced by the evolution in time of the numbers of channels that are open and closed; when we modelled release produced by a Ca” transient fitting the sum of two exponentials (as for Fig. lo), it showed overall release varying with p and C, much the same as for “square pulses”, but this model assumed constant heterogeneity of Ca2+ between sites during phasic release. In a preliminary version of these models,?’ spontaneous~~ (fo) and variance of Cal’ contribution per channel (II and III) were not taken into account. These factors make a substantial difference to the computed results and attributing ,fOto an arbitrary level of 100 nM “resting” Ca’+ allows scaling nominal values of Ca2+ at release sites in plausible units. P~ei2~rnena to he .~~rnu~a~ed. To begin, we consider that all normal quanta1 transmitter release is governed by the same laws whether mediated by Ca2+ or Ba? + (or Sr’+). which differ only quantitatively. Then, a model should account for each of the following phenomena. (A) A linear relation between the nth root of release rate (_&,f and (hypothetical) accumulated intracellular ion. with n equal to 4 or 5 (Ba2+,30 Sr” ?). (8) A slope of log m vs log ion entry of 4 when ion entry per channel is varied and channels are opened by presynaptic action potentials (Fig. 3, Table 1). CC) A slope of log m vs log ion entry close to 2 when the number of channels per pulse is varied by varying the intensity of direct depolarizing pulses (Figs 1 and 2, Table 1). (D) Growth of “phasic” release (in the EPP) with.f;, when the latter is increased by accumulated Ba’+ or Sr-“. in close accord with r(t)’ 4 -_fk” being a function of time which is unaltered as ,f, increases (Ba “.l’.ZK Sr’+ ‘). This relation is equivalent to: r(t) = k(c(t) + c,)4 where r(f) is an ion transient introduced by each pulse and kcf is ,f, resulting from the sum of

baseline and accumulated ions (cI). This equation also describes observation (A). Functional relation between release and local CaL” .

In view of phenomena (A), (B) and (D) we simply put the instantaneous release probability (2) at each release site proportional to the n th power of local Ca2+ concentration, where n = 4 or 5, i.e. I = kc”, where k is a proportionality factor. This equation may appear oversimplified but its use may be justified on two grounds: (i) it minimizes parameters. and (ii) realistic models lead to the same equation (see Appendix). It should be noted that this attributes a spontaneous “resting” f,, ,foq to a baseline level of active ion, as is implicit in phenomenon (A) above. We also assume that any one release site has (by definition) only one quantum; the probability that this quantum is released in time 6r is r = 1 - exp(-zdt). This gives equations (1) in Experimental Procedures, repeated below, when c is divided into two components, “resting” Ca’+ , co. and added Ca:+ per pulse, c,,: z=kc”:

c’=c,+c,;

r=I-exp(--6~).

It is notable that the only variable in the above equations is c,/c,, given an assumed ,f,i and St (0.5 ms), 1000 release sites at a terminal, and n = 4 or 5. From these equations some relationships may be calculated analytically. In particular, the “Caslope” dlog vjdlog cp is less than n-it is less than dlog z/dlog cp = nc,/(c, + co) and cp/(cp + r;)) is always appreciably less than 1 when n = 4 or 5, for transmitter release in any realistic range. For example, at m = 10 the mean release rate (in 0.5 ms) is 200,000 times a spontaneous rate (,f;,) of (say) 0.1/s; c;;/c; = 5 x 10-h. With fr =4 this gives cp/(c, + c,,) = 0.95, limiting the slope to 3.X. With Jo = l/s, the slope at m = 10 is about 3.7. At higher c,/co, dlog =/dlog rp is higher but dlog ridlog z (always less than I) becomes limiting. With stochastic factors taken into account (see below) the maximum slopes of Iogm vs log mean Caichanneljsite were somewhat less and an.f; = l/s with n = 4 is untenable (“Ca-slope” always ~3.6). In modelling we have therefore always used ,f;= 0.I in conjunction with R = 4. This is less than usually observed (about 0.3-i/s) but it is possible that a significant component of “spontaneous” release arises from Ca” channels opening spontaneously, rather than from “resting” internal Ca’ ’ This extra release is implicitly included in cp in the above equations. Heferogeneit_v qf Ca’+ per release site. In the hope of fitting phenomenon (C) we assume that at any release site Ca’” is contributed probab~Iisti~~lly (see above). For stochastic variation in the number of channels (which is of primary importance) we choose either a fixed number of channels available and a binomial distribution of zeros, ones, twos etc. or a Poisson distribution. In the event, using binomial rather than Poisson distributions was found to have

664

D. M. J.

O~JAST~L et ml

little effect on calculated responses except to impose a low maximum on release at low Ca’+ entry per channel. More complicated model variants. Calculations were also carried out with sub-variants of I-III in which more complexity was introduced; none of these had a substantial effect on the calculated results. In particular, there was little effect of heterogeneous composition of the nerve terminal in terms of either channel density per site or effectiveness of Ca’+ at sites. Thus, heterogeneity of the kind described by Robitaille and Tremblay33 at the frog junction will not significantly add to the effect of stochastic factors. Model III implicitly includes the idea that Ca* + may spill over from one Ca domain to others.42 By way of confirmation, calculations were made assuming that at all release sites local Ca” is also augmented by a constant amount proportional to mean entry. This also had a negligible effect. Another factor that (surprisingly) turned out to have scarcely any visible effect on graphs of release vs Cc was regular modulation of Ca” entry per channel with number of channels opened by a pulse, which is to be expected if some ion entry occurs while the nerve terminal is still depolarized. Units ofhypothetical internal Cal + Merely in order to express Ca*+ at release sites in terms of plausible units we assume (arbitrarily) that the resting Ca” concentration in the nerve terminal, giving rise to the assumed ,fo (0.1 or l/s), is 100 nM. Computed results. Graphs illustrating some of the results are shown in Figs 5-l I. Figure 5 shows a typical family of curves of calculated quanta1 content, m, vs p (mean number of open channels/site), in this case with model variant II where n = 4. The solid lines running from left to right are for a Poisson distribution of p and mean Ca’+,channel/site (C,) of 800, 400. 200, 100, 50, and 25 nM. Notably. at high

3 2 1 E 0

0

CT -1 A2 -2 -3 -4

-6

-5

-4

-3

-2

-1

0

1

2

loglo (channels/site)

Fig. 5. Theoretical curves of log m vs log channels per site @) with model variant II (exponentially distributed Ca’+/channel) with n = 4 andf, = 0.1/s calculated for mean Ca*+/channel/site (C,) = 800, 400, 200, 100, 50 and 25 nM (from left to right), assuming that f0 represents 100 nM Ca”. Lines represent values calculated using a Poisson distribution of numbers of channels/site; points assume a binomial distribution with N = 5. Dashed lines (on right) represent calculated values of m for a homogeneous distribution of Ca’+ between release sites.

C, and low p, m can be very much higher than if (‘awere equally distributed (dashed lines). This arises because most release is from sites with much larger than average Ca2+. The points, calculated for a binomial distribution with N = 5 available channels/site, fall essentially on or just below values for a Poisson distribution. Lower values of N (not shown) reduce the m attainable in each curve and exaggerate the difference between the binomial and Poisson but whatever the value of N, when it was at all possible to obtain m = 1 at a channel slope = 2, 11 and C, at this point were close to those with a Poisson distribution In Figs 7-11 all plots pertain to a Poisson distribution of numbers of open channels, since using binomial distributions made little difference except to limit maximum release. Figure 6A-D shows graphs of m vs mean added Ca*+/site (PC,), illustrating that predicted release is far from a constant function of this quantity; release “efficiency” grows as /* is reduced and the heterogeneity of Ca’ + per site is increased. Similarly, comparing model variant III with 11, both with n = 4 (Fig. 6B and Fig. 6A, respectively) curves are shifted upwards (and to the left) by the increased heterogeneity introduced in model III. Figure 6C and D (calculated with n = 5 and ,f, = l/s) shows the same tendency. At very high C,, m can be less than if Ca’+ were equally distributed between sites-- this occurs because for some release sites there arc no open channels. In Fig. 7 are shown the combinations of Cc that give m = I, for I (circles), II (filled circles) and III (squares), all calculated with a Poisson distribution of numbers of open channels and n = 4. In each case. at high enough C,, m = 1 (=O.OOl quantarelease site) is obtained at =O.OOl channels/site. In most of the range of C, the more varied the distribution of Ca’+jsite the fewer the number of channels per site (p) needed to produce m = 1. In Fig. 8A and B are shown values of dlog nr/dlog curves on the left) and of dlog C, (“Ca-slope”, m/dlog p (“channel-slope”, curves to the right), plotted vs p, for combinations of p and CL that produce m = 1. The former slope should be about 4 to fit phenomenon (B) above and the latter slope should be about 2 to fit phenomenon (C). The graphs are with n = 4 (Fig. SA). and with n = 5 and,/; = I :‘s (Fig. SB). With n = 5 and fi =0.1/s (not shown) the values of “Ca-slope” were consistently about 10% higher than those shown in Fig. 8B. With n = 4 and J(i = 0.1, the maximum Ca-slope is just under 3.8, suggesting that true n may be 5. or that true “spontaneous” release, due to c* is less than 0.1/s. Although these results show that it is indeed possible to obtain values of “channel-slope” close to 2 at the same time as a “Ca-slope” of about 3.8 or somewhat more (with n = 5). for each model this occurs only at a particular C, and particular LL. Moreover, plots of log m vs log p are not linear: a

Transmitter

665

release and Ca’+ entry 100

A 3-

lO[

0 .e$

2-

>

lE o0 G-1 . 9 -2. -3

1.

: 6 '0

0.1 0.01 0.001

-

L 10

-4. -5" -4

-3

8 -2

”

-1

0

log,,_,(mean added

1

j 2

" 3

4

. . . ..I 100

.

1 5

Cc/site -nM)

. .

..I

,....I

,.-

1000

10000

Co/channel/site

- nM

100000

Fig. 7. Plot of combinations of n (open channels per release site) and C, (Ca' +/channel/site) that produce m = I. for homogeneous Ca’+/site (dotted line) and various degrees of heterogeneity: I (circles), II (filled circles) and III (squares). Calculated with n = 4 and 1,)= 0. I ‘s.

B 32lE o0 g-1 _o -2 -3

-

-4 -51 -4

b -3

”

-2

’

-1

loglO(meon

b 3

”

0 added

1

2

'1 4

5

Co/site -nM)

L

typical example is shown in Fig. 9. Thus. phenomenon (C) cannot be exactly reproduced with any of these models, but it is arguable that non-linearity of the magnitude indicated in Fig. 9 would not be easy to detect experimentally. Values of 1~and PC, (mean added Ca’ ’ per site) at which channel-slope (dlog m:dlog ~1) = 2 at m = I (corresponding roughly to the experimental data) are listed in Table 2. In every case. of course. most of the release comes from sites with higher than

A

3-

vs Co/channel

4r

(ry

2l-

E cl

.. /

o-

z-1.

I

-2

-

-3

-

-41 -4

“S

no. of

channels

”

-3

-2

8

-1

loglo (mean

“‘I 0 added

8

1

2

k/site

3

4

1

0’

5

-nM)

“““‘I

0.01

“I’

0.1

’

s 1

” .‘....,.t

10

““‘..I

100

site at m=l

channels per release B

n

u

5r

3l

-41 -4

”

-3

-2

b

-1

loglO(mean

L

0

added

a

1

&/site

c

2

c

3

‘1

4

vs

Co/channel

0.01 5

channels

0.1

1

10

100

per release site at m=l

-nM)

Fig. 6. (A) Same as Fig. 5 but with m plotted vs mean total Ca:+ per release site (=/cc,). Dashed line is for homogeneous Ca’+ distribution among release sites, where m depends only on PC,. (B) Same as A but with IllLexponentially and spatially distributed Ca’+/channel. (C) Same as A but with n = 5 and & = l/s. (D) Same as C but with III.

Fig. 8. Calculated logglog slopes (at m = I) of vr vs Ca” entry per channel (C,: curves on left) and vs number of channels (n: curves on right) plotted vs mean number 01 channels;‘release site at m = I (which varies with C, ), for model variants I. II and III (left to right). (A) Calculated with n = 4 and f;, = 0.1 :s. (B) Calculated with II = 5 and /,, = I s.

666

D. M. J.

f

QUASTEL

0?

r

10 m 1

1 Channels

many ineffective distant channels are count& In hut the mean increase in [Ca’+] at release sites is much the same, 2800 nM. With n = 5 the number of channels is about three (II) and the increase in Ca’+ at release sites turns out to be about I100 nM (if& = 0. I :‘s) or 700 nM (if,/;, is I/s) for all model variants. Increase of ‘phasic” transmitter release bJ, residual ion. The graphs in Fig. 10 pertain to phenomenon (D), namely, the effect of repeated nerve stimulation

0.1

0.1

et al

10 per release site

Fig. 9. Example of log-log plot of calculated m vs p, mean number of channels per release site (variant III, n = 4, fo = 0.1/s, C, = 135 nM). Note deviation from linearity: the line is drawn with a slope of 2. average local Ca2+ (&‘, + 100 nM). For III, each quoted value of C, is 17% of the mean concentration produced by those channels at zero distance from the release site. With a homogeneous distribution of Ca2+ between release sites, a channel-slope of 2 at m = 1 cannot be obtained at all; m = 1 is produced by values of PC, of 1089 nM for n = 4/f, = 0. I/s and 625 nM for n = S/f0 = l/s. The range of C, over which dlog mldlog p was between 1.5 and 2.5 (at m = 1) was in all cases between two-fold (for III) and 3.3-fold (for I). Thus, the experimental finding of channel-slopes close to 2 (Table 1) suggests that ion concentrations (chosen to produce easily measurable EPPs and Ba2+-tails) were rather fortuitously selected to be in the correct range to give a channel-slope close to 2. To extrapolate from the numbers in Table 2 to a normal EPP, which in the mouse has an m value of about 100 (representing a release rate about 2 x IO6 times anSo of 0.1/s) in 2 mM Ca2+ (double that used in these experiments), we assume C, twice as large as in 1 mM Ca2+ (experimental condition). The number of channels normally opened by a nerve impulse turns out in each case to be about 2.8 times that listed in Table 2 and m is about 5 times that which would occur if incoming Ca2+ were evenly distributed between release sites. With channels close to release sites (II) and n = 4, the presynaptic action potential opens an average of about four channels/release site and mean local increase in [CaZ+] at release sites is normally about 2500 nM (iffO reflects 100 nM Ca2+). With III the number of channels depends upon how

in the presence of Ba2+ or Sr2+, which causes increase in f,, presumably reflecting accumulation of the ion in the presynaptic cytoplasm, to augment quanta1 contents (phasic release) in close accord with the relation r(t)“4 = l~“~c(t) +f fl. where r(t) is release rate as a function of time and k’*4~(t) is a rapid transient function of time, presumably reflecting influx of divalent ion through channels opened by a stimulus, which is unaltered as f, is increased.‘,‘3“x Briefly, in terms of the present model, calculated results show that this formula is not exact (at 77 = 4) unless fb = 0 and p is so high that ion distribution between sites is nearly uniform; at low p (and high C,) residual ion can contribute only negligibly to the A 11 10

9

6

-a

s7 ~6

5

0’

0

1

2

3

2

3

latency (ms)

0 11 10

-+ 6

‘2 Y

7 6

0’

Table 2. Calculated values of p and PC, giving m = 1 and “channel-slope” = 2.0 I

n

fo

P

4 5 5

0.1 0.1 1.0

0.43 0.25 0.29

II ~__ .-

62) 406 153 99

p 1.48 1.10 1.19

___~ _(2, 451 196 123

III

p

(%i,

3.7 3.2 3.3

500 234 142

All quoted values of &, (mean added Ca*+ per release site) assume that resting MEPP frequency (,fO) reflects 1OOnM Ca*+.

0

1 Iatsncy (ms)

Fig. 10. Calculated modulation of “phasic” release by previously accumulated Ca*+ (or Ba2+ or Sr*+) which has raised baselinef,. The graph is of the l/4 power of r(t) vs time. In the calculation it is assumed that local intracellular Ca*+ starts to rise at 0.5 ms after a stimulus and the-increase over baseline level is proportional to e-#’ -e-*’ with a = 2/ms and b = 20/ms. The lines are repetitions of the line formed by points for release when there has been no accumulated Ca*+ (lowest points), raised to fit baseline_&, (A) III, n = 4, p = 10, C, = 51 nM. (B) II, n = 5, p = 3.2, C, = 51 nM.

fast phase of r(f), However, the discrepancy between the predictions of the present model and the above

dwing #2~~~~~i~ed dqmiarization. In Fig. 4. data were shown with respect to the ~~a~#ur of ,f,:, in response to prolonged nerve terminal depotarizations in the presence of Ba”’ or Ca”‘; slopes of Jog!“,, vs applied current were steeper in Ba2+. This is obviously to be expected from the present modd if release in Ba’+ reflects accumulated ion, while release in CL%? i is dominated by transient Ca’+ at release sites close to channels. However, it is not obvious that the fatter should he the case for a prolonged depoiarizafion causing repeated ck*nneI opening. Tf. for example, incoming Cazi were diluted JO-fold (or bound t.o the same extent) after entry, and removed with a time constant of 250 ms (as with ST* “f an opening rate of Io(f chan~eJs~reJease site/s would cause bulk internal Ca’+ to increase by an amount 2.5 times the average transiently contributed by one channel to a release site. This rough calculation is, however, m~s~~adjng. The piots in Fig. 1f were mm puted (III, with tl = 4 and ,/;, = 0.I) assuming that steady depolarization is equivalent to very rapid repeated stimulation at very low p and that steadyfi, is fimited to ftXN,is. At Cc = 800 nM (on the f&f, predicted .f, cafcuJated for w 1: 10 dilution, a I :40 dilution and no accumulation at all are virtually identical over most of the range of channels& That is.,fiR is mainly due to Iocat rather than a~~~~uJa~~o~ Cal ’ . At the other extreme, Cl = 50 nM (right) with 1: IO dilution .fA almost entirely reflects accumulated CaZ ’ . Similar families of curves were found with all

0

1

2 3 4 loglo (channels/s)

5

6

-7

Fig. 1L~~~~~r~~~~~#n af ~~~~~~~~C~~ zccumutatedC$ + $0 /, causedby protonged depolarkation. Caknfations wet-c with 111 (n = 4._/, = 0. I/s),

assuming a t : 10 dilution of Ca’+ after entry (filled circles), I:40 dilution (empty circles) and iafinitc dilution (no accumulated Ca’+: dashed lines), k.x ~~~~s~t~~c~~~~e~ (C,) = XC@. 260 and 50 nM. 3% di%rence between the dashed Iines and the points represents rhc contribution made by accumulated channel opening (for the whole

ion at various

rates of”

terminal). Similar curves were seen with model variants I and II.

variants of the model (I-III), and with n = 4 or S, and altering the diiution or time constant of removal merefg changes the range of CV for the ~r~~sj~~~~ from release dominated by Iocal Ca2+ (tow stopc) or accumulated Ca*+ (high slope). Experimentally. such shifts in the slope of log.f, vs prcsynaptic depolarization, with altered external Ca’+, were seen by Cooke cl al.’ Simulated responses (not shown) to a pralongcd “square-pulse” increase in rate of channel opening have two phases corrgs~~nd~ng to the eo~~r~bu~jon of transient and bufk Ca”+ to overall release; the relative magnitude of the fast phase grows with C,. Perhaps surprisingly, it turns out that in the off-response the relative magnibxie of the fast phase (in a plot of,fi?r f 1 vs time) is always smaller than in the on-response, i.e. the relative contribution of bulk Ca” to release grows as depolarization is maintained. Behaviour correspondjng to these predictions has been seen experintentatJy with Si+?

In the present experiments we obtained graphs relating quantaJ transmitter reJe?tse t0 inflUX of Cd” or BaZ+ into the nerve terminal using as an indirect measure of the Jatter the “Ba” -taif”‘” produced by trams of nerve impulses or “direct” nerve terminal depolariztktions. This tail of raisedJ, reflects accumulated Ba”+ “’ and therefore reflects the number of channels opened per p&e when external Ba? + and Ca” are held constant, bur ion eniry per channel when the number of channels per pulse is constant and a blocking agent impedes ion entry, Since the goal of the experiment was to obtain graphs of log release rate {qnantat content) ws log ion entry, it is of no consequence that the method cannot provide an absoXute rather than rclativc measure of ion entry. Moreover. the indirectness of the measure of inn entry has one ad~~a~tage. Any ~hy~othetj~~~ ion entry that does not supply ions to the neighbourhood of release sites will not be “seen” by the method, The most striking result is that the apparent Hiif coctTicient for cooperativity. the slope of Jog PS vs fog ion entry’pulse. depends upon how entry is varied. having a value close to 4 when entry is inhibited with Mg” or bekanam~cjn. but much tess, close to 2, when the number of channels opened per pulse is modulated by varying the intensity of nerve terminal depolarization and the number of Ca channels opened. Some of the data of Augustine and thariton’ for the squid giant synapse show the same tendency. In principle, the true Will coclhcient should appear when the ion accumulates to produce a steady concentration within the terminal. as presumably occurs during the &a’+-rail’” and afso when ion infius per channel is varied. On this basis the coefficient has a true value of 4 or 5 for both ions (Table I). The low slope with varied puJse intensity {“channel-slope”)

implies that reducing the number of channels opened by a pulse causes less drop in release than expected if release depends only upon total divalent ion entry and the consequent average intracellular concentration. This is in the opposite direction to that expected from the “calcium-voltage hypothesis” of Parnas et al.,z0.z3in which a reduced intensity of depolarization should have a more than expected effect to reduce release. It is also opposite to that predicted by the model of Zucker and Fogelson,4’ who calculated the theoretical effect of overlapping “Ca-domains”. It is also not explicable in terms of varied Ca entry per channel with varied pulse intensity (see Results). Our conclusion that “channel-slope” is low with “direct” EPPs depends upon the assumptions that ion entry per pulse is at least roughly constant in a train and that Ba’+ and Ca2-l enter through the same channels; evidence for this has been provided previously.“,3” In other experiments, we have found that apparent Ba’+ entry per pulse is independent of train length over a wide range of intensities and durations of presynaptic polarization. In order to explain the low “channel-slope” (about 2) found with “direct” EPPs on the basis of progressively falling pulse effectiveness, however, it would be necessary to suppose that entry/pulse declined in trains more with small than with large depolarizations. However, in other experiments with Cd’+ there was evidence of a use-dependence in Cd * + blockade; this could explain, at least in part, the low slope of log m vs log Ba” entry seen with Cd’+. Alternatively, Cd’ ’ which acts with relatively high potency, might in effect reduce the number of channels opened by a brief nerve terminal depolarization. That the true Hill coefficient for Ca’+ (or BaZ’ ) cooperativity might not appear when the number of channels per pulse is varied arises directly from the concept of “Ca-domains” of Chad and Eckert4 once one considers the possibility that each release site may be supplied with Cal+ by only a few channels. With only one channel per site, total release would be linear with the number of channels opened regardless of the relation of release probability to local Cal+. The computations in the Results deal primarily with working out the theoretical consequences if the number of channels that open at each site is not large, giving rise to heterogeneity of local Cal’ between sites. These show effects that are not at all small. Transmitter release for EPPs is largely governed by local Ca2+, contributed by channels near release sites, rather than overall internal Ca2+. This is implied by the finding that loading nerve terminals with BAPTA (a Ca *+ chelator) can block facilitation at frog junctions without blocking the EPP.16 The importance of local Ca2+ is also a feature of many models (e.g. Ref. 42) but seems not to be generally accepted (e.g. Ref. 21). Our computations indicate that a low “channelslope” of log m vs log ion entry is explicable solely in terms of the EPP being generated by local rather than

bulk Ca”, with local Ca“ varying because of stochastic factors. If Ca’+ channels are all located in close proximity to release sites,“s.32 variant II of the model (variance in local Ca” due to stochastic opening and closing of channels close to release sites) must approximate reality more closely than model III (variance also increased because channels are randomly placed). To the extent that channels with unusually long durations contribute less than in proportion to thcii duration, or (hypothetically) long channel lives are prevented by local high Ca’+, the effect will be to reduce the variance of Ca2 + between sites (towards I). On the other hand, overall variance should be increased by variance related to the rather small numbers of Ca atoms (a priori Poisson distributed) that may enter through each open channel-- -a channel with a conductance of 0.20 pS at 1 mM c‘a’ . at a driving potential of 100mV would inject only six Ca” ions (10 ~” moles) in 0. I ms and this number will be apriori Poisson distributed. A channel lifetime of the order of 0.3 ms or less is indicated by a rate constant of decline of the fourth root of “phasic” release of about 0.3 ms3 This source of variance was not taken into account in the calculations+ it would tend to move curves derived from I and II towards II and III. From the computed results, it appears that if the low observed “channel-slope” is to be attributed to “stochastic heterogeneity” of local Ca’+, the average number of channels opened per release site by an action potential is only about three or four. This implies a rather high “efficiency” of the release process. For example, if average injection of c’a” through a channel (at 1 mM) is six atoms or 10 -2’ moles (see above), 300 nM Ca’+,site/channel (see Results) represents dilution into a volume of about 3 x IO- ” 1 (corresponding to a hemisphere of 0.25 pm radius), which is not particularly small. The simultaneous opening of 10 such channels at a site would provide 3000 nM Ca’* (and virtually guarantee release) by the presence of only 60 atoms of Ca’ ’ in this volume. This suggests that rather few (perhaps only one) Ca,X moieties (see Appendix) at a site are required to induce release of its quantum. Recently, results of experiments with calcium buffers at the squid giant synapse have been interpreted as suggesting that release of transmitter is normally activated by internal Ca’+ at a concentration of hundreds of PM or more.’ This estimate is likely to be too high if the calculations did not take into account heterogeneity of local Ca’+ between release sites, and if equations appropriate to equilibrium kinetics were employed for the nonequilibrium situation close to the site of Ca’+ entry via a channel. However, Ca2+ imaging in the squid terminal has shown local Ca2+ in microdomains in the range of 10m4 M. lRaIt is possible that transmitter release at the squid synapse requires much more Cal ’ than at the mouse terminal, or that, in the squid

669

Transmitter release and Ca’+ entry synapse, Ca” entry is normally far in excess of that required for release of most quanta. Alternatively, if local Ca’ + concentrations during release in the mouse nerve terminal are really so high, our calculations err in attributing spontaneous release (fO) to “resting” intracellular Cal+ (at a nominal 100 nM). If this is the case, the main effect on our calculations is to remove the calibration of internal Ca’+ concentrations involved in release. The major result remains, however, the heterogeneity of Ca’+ between release sites which will make release higher than it would be if Ca” were uniformly distributed, and can lead to a slope of log release vs log Ca entry which is less when Ca” entry is varied by varying numbers of channels opened than when entry is varied by varying Ca’+ entry per channel. It should be emphasized that the model presented in the Results section contains only one arbitrary equation-the power relation between release probability and Ca?’ concentration [equation (la)] and only one arbitrary parameter introduced to fit experimental results, the Hill coefficient n = 4 or 5 (which is needed to fit a wide variety of data); the added heterogeneity with each variant of the model arises only from stochastic and geometric considerations and is parameter-free. Thus, the observed fit of phenomena (A)-(D) (see Results) to the model. in any of the its variants, is remarkably good considering the simplicity of the equations. It is also of interest that the model makes predictions that were not expected and are experimentally testable. In particular it predicts that if some active ion accumulates in the cytoplasm during prolonged nerve terminal depolarization there will be an alteration in the slope of log.f;,, vs log (channels/s) over a moderate range of external Ca’+. with curves at the extremes of low or high Ca entry per channel becoming those characteristic of release dominated by accumulated ion or by local ion entry (Fig. I I). Previously unexplained data (Fig. I IA in Ref. 6) show just such a shift in the

slopes of log fm vs applied nerve terminal depolarizations in the range 0.2552 mM external Ca” The model also predicts asymmetric fast and slow components of response to a step change in number of channels open, when an ion accumulates appreciably in the cytoplasm. closely resembling those seen experimentally with step depolarizations in the presence of Sr’ +,j It should be noted, however, that there exist other experimental observations, especially regarding facilitation and potentiation, that cannot be fitted to the model unless one postulates change in the parameter k in equation (la), which causes “multiplicain both spontaneous and cvokcd tive” change release.” CONCLUSION

In summary, the experimental results are in good accord with the predictions of a model in which the probability of transmitter release at a release site is proportional at all times to the fourth or fifth power of the local concentration of Ca” (or a substitute) but this power function fails to appear in the relation between overall release and ion entry when one varies entry by varying the number of channels opened by depolarization. This arises from stochastic factors leading to heterogeneity of local Ca’ ’ between release sites, because release is normally caused mainly by local Ca”. contributed by entry through small numbers of channels close to a release site, rather than by increase in bulk Ca” concentration in the nerve terminal. This also applies to relatively low-level elevated MEPP frequency produced by nerve terminal depolarization in the presence of Ca’+, which results from intermittently high concentrations of Ca’+ close to its entry sites, rather than a build-up of intracellular concentration, as occurs with Ba’ ’ Ach,lo~~ledR~m~nr.v-This work was supported by grants from the Muscular Dystrophy Association 01‘Canada and the Medical Research Council of Canada.

REFERENCES I. .Adler E. M., Augustine

2. 3. 4. 5. 6. 7. 8. 9. IO.

G. J., Duffy S. N. and Charlton M. P. (1991) Alien intracellular calcium chelators attenuate neurotransmitter release at the squid giant synapse. J. Neurosc~i. 11, 149661507. Augustine G. J. and Charlton M. P. (1986) Calcium dependence of presynaptic calcium current and post-synaptic response at the squid giant synapse. J. Ph~~siol. 381, 619-640. Bain A. 1. and Quastel D. M. J. (1992) Quanta1 transmitter release mediated by strontium at the mouse motor nerve terminal. J. Physiol. 450, 63-87. Chad J. E. and Eckert R. (1984) Calcium domains associated with individual channels can account for anomalous voltage relations of Ca-dependent responses. Biophy.v. J. 45, 993-999. Charlton M. P.. Smith S. J. and Zucker R. S. (1982) Role of presynaptic calcium ions and channels in presynaptic facilitation and depression at the squid giant synapse. J. Ph,v.viol. 323, 1733193. Cooke J. D., Okamoto K. and Quastel D. M. J. (1973) The role of calcium in depolarization secretion coupling at the motor nerve terminal. .I. Ph_rsiol. 228, 459-497. Cooke J. D. and Quastel D. M. J. (1973) Transmitter release by mammalian motor nerve termtnals in response to focal polarization. J. Physiol. 228, 3777405. Del Castillo J. and Katz B. (1954) The effect of magnesium on the activity of motor nerve endings. J. Phy.siol. 124, 553-559. Dodge F. A. and Rahamimoff R. (1967) Cooperative action of calcium ions in transmitter release at the neuromuscular junction. J. Physiol. 193, 419-432. Elmqvist D. and Quastel D. M. J. (1965) A quantitative study of end-plate potentials in isolated human muscle. J Physiol. 178, 505 -529.

I I, Fogelson

A. L. and Zucker R. S. (1985) Presynaptic calcium ditfusion from various ‘irrays ot smplc ~tiannci.\ Implications for transmitter release and synaptic facilitation. Bi@ys. J. 48, 1003~~1017. 12. Forshaw P. J. (1977) The inhibitory effect of cadmium on neuromuscular transmission in the rat. Eur. ./. P/rrrrnrrr(,. 42, 371l377. 13. Guan Y.-Y., Quastel D. M. J. and Saint D. A. (1988) Single Ca’+ entry and transmitter release systems at the neuromuscular synapse. Synupse 2, 558 -564. 14. Jenkinson D. H. (1957) The nature of the antagonism between Ca and Mg ions at the neuromuscular junction. J. Physiol. 138, 438-444. 15. Katz B. (1969) The Reieuse qf Neurul Transmitter Suhstunces. Charles C. Thomas, Springfield, IL. 16. Kijima H. and Tanabe N. (1988) Calcium independent increase of transmitter release at frog end-plate by trinitrobenzene sulphonic acid. J. Physiol. 403, 1355149. 17. Llinas R. and Nicholson C. (1975) Calcium role in depolarizationsecretion coupling: an aequorin study in squid giant synapse. Proc. natn. Acad. Sri. U.S.A. 72, 187 190. calcium currents in squid giant synapse. Biophyv. J. 33, 18. Llinas. R., Steinberg 1. 2. and Walton K. (1981) Presynaptic 2W 322. 18a. Llinas R., Sugimori M. and Silver R. B. (1991) Imaging preterminal calcium microdomains in the squid giant synapse. Biol. Bull. 181, 316-317. 19. Martin A. R. (1955) A further study of the statistical composition of the end-plate potential. J. Physiol. 130, 114 -122. 20. Parnas H., Dude1 J. and Parnas 1. (1986) Neurotransmitter release and its facilitation in crayfish. VII. Another voltage dependent process beside Ca entry controls the time course of phasic release. Pfiiiger ‘s Arch. ges. Phy.siol. 406, 12 I- 130. 21. Parnas H., Parnas 1. and Segel L. A. (1986) A new method for determining cooperativity in transmitter release. .I. rheor. Bin/. 119, 481-499. 22. Parnas H. and Segel L. A. (1981) A theoretical study of calcium entry in nerve terminals, with application to neurotransmitter release. J. theor. Biol. 91, 1255169. hypothesis for neurotransmitter release. Biol. Chcm. 29, 85-93. 23. Parnas I. and Parnas H. (1988) The “Ca-voltage” 24. Pittinger C. B. and Adamson R. (1972) Antibiotic blockade of neuromuscular function. A. Rev. Phormac. 12, 169.-184. 25. Pumplin D. W., Reese T. S. and Llinas R. (1981) Are the presynaptic membrane particles the calcium channels? Proc. natn. Acud. Sci. U.S.A. 78, 7210--7213. of end-plate potentials and currents for non-linear summation. Crm. J. Physiol. 26. Quastel D. M. J. (1978) Correction Pharmac. 51, 702-709. to blockade of nerve-released transmitter at synapses. Proc. R. 21. Quastel D. M. J. (1988) Receptor kinetics pertaining Sot. Lond. B 233, 461. 475. in transmitter release. 28. Quastel D. M. J., Bain A. I., Guan Y.-Y. and Saint D. A. (1989) Ionic cooperativity In Neuromuscular Junction (eds Selling L. C., Libelius R. and Thesleff S.). Elsevier, Amsterdam. in calcium entry and calcium action and its implications 29. Quastel D. M. J. and Saint D. A. (1986) Calcium co-operativity with regard to facilitation at the mouse motor nerve terminal. In Symposium on Calcium, Neuronal Function and Transmitter Release (eds Rahamimoff R. and Katz B.), pp. 141-158. Martinus Nijhoff, Boston, MA. release at mouse motor nerve terminals mediated by temporary 30. Quastel D. M. J. and Saint D. A. (1988) Transmitter accumulation of intracellular barium. J. Physiol. 406, 55--73. 31. Quastel D. M. J., Sastry B. R. and Steeves J. D. (1981) Focal excitation of motor nerve terminals. Abstract.\, Eighth Int. Gong. Pharmac., p. 646. release sites 32. Robitaille R., Adler E. M. and Charlton M. P. (1990) Strategic location of calcium channels at transmitter of frog neuromuscular synapses. Neuron 5, 773.-779. release along the length 33. Robitaille R. and Tremblay J. P. (1989) Frequency and amplitude gradients of spontaneous of the frog neuromuscular-junction. Synupsc-3, 29ll307. of motor nerve terminals. Pf@er’.r Arch. 34. Saint D. A.. McLarnon J. G. and Ouastel D. M. J. (1987) Anion permeability ges. Physiol. 409, 258 -264. . M. and Urakawa N. (1982) Mechanism of cadmium-induced blockade of 35. Satoh E., Asai F., Itoh K., Nishimura neuromuscular transmission. Eur. J. Pharmuc. 77, 251-257. the asynchronous release of acetylcholine quanta by motor 36. Sihnsky E. M. (1978) On the role of barium in supporting nerve impulses. J. Physiol. 274, 157 171. of calcium-dependent acetylcholine secretion. Phurmw. Rev. 37, 37. Silinsky E. M. (1985) The biophysical pharmacology 81 -132. in single melanotrophs A. and Almers W. (1990) Cytosolic Ca’+, exocytosis, and endocytosis 38. Thomas P., Suprenant of the rat pituitary. Neuron 5, 723-733. blocking action of cadmium and manganese in isolated frog striated muscles. Eur. J. 39. Toda N. (1976) Neuromuscular Phurmac. 40, 67-75. T., Molgo J. and Lemeignan M. (1981) Bekanamycin blockade of transmitter release at the motor nerve 40. Uchiyama terminal. Eur. J. Phurmac. 72, 271-280. 41. Woll K. H. (1982) The effect of internal barium on the K current of the node of Ranvier. @‘iger’s Arch. ge.7. PhJ,.Gol. 393, 318321. between transmitter release and presynaptic calcium influx when 42. Zucker R. S. and Fogelson A. L. (1986) Relationship calcium enters through discrete channels. Proc. natn. Arud. Sci. U.S.A. 83, 3022-3036. (Accepted 8 June 1992)

APPENDIX: KINETICS OF Ca*+ ACTION TO INDUCE TRANSMITTER RELEASE The simple power relation [equation (la)] between z and C can be regarded as a simplification of equations derived for more realistic models.9.*3

(i) Ca*+ combines with a receptor proportional to [CaXy: c =co+cp;

y = [CaX] = [X,]c/(K

X and release + c);

rate is

: = (kr)

where [X,] is the total concentration of receptor X, K is the dissociation constant of CaX, and k is a constant. However

Transmitter

release

plausible, this model can be shown to be untenable for the following reasons. Differentiation gives dlog z/dlog c,, (which is always less than “Ca-slope”) equal to n(l +c/K)-‘.c,/c, and the term (1 +c/K)-’ becomes severely limiting unless K % c,,. For example, even at K = 2OOc, (giving I = IO”,f, at c,, = 200~~) with n = 4 the maximum slope obtainable is 3.4. A further consideration also rules out this model for the situation where K 9 c. If X is a macromolecule in the region between a vesicle and the presynaptic membrane, its numbers at any one release site will be limited and at c < K the numbers of [CaX] at any one site will be a stochastic variable with a low mean. Then ,I is no longer simply proportional to (k~)“, but must be obtained by summing p,z,, where p, is the probability of i CaX complexes at a site and z, is proportional to i”. With a mean of 100 X molecules available per release site and Poisson distributed numbers of CaX, we calculate maximum dlog:idlog cp of 3.3 using II = 4, and about 3.6 for n = 5. (ii) A scheme with a single receptor binding n Ca atoms with equal aflinity at each stage:

The derivative with respect to cp is the same as above but the difficulty associated with stochastic distribution of numbers of active X does not arise. Thus, this model can apply if c + K, i.e. the affinity of the receptor for Cal+ is so low that there is effectively no saturation in the physiological

671

and Ca’+ entry

range. In this case equation (la) provides an adequate approximation. (iii) A scheme in which X binds n Ca atoms. in which Ca,X is much more stable than CaX, Ca2X. etc.: c = c0 + c,:

J = [X]c”/(K” + F’);

z = k>

Differentiation gives dlog z/dlog cI, = n( I + c”:K”) ‘cF/c. The second term now has much less influence until (’ approaches K, and it turns out that for any value of K that can give a maximum M of 500 or more (from 1000 sites) in conjunction with “Ca-slopes” close to n at lower m, values of m (below 100 or so) are indistinguishable from those calculated using equation (I a). This scheme gives a different equation for J if there are more X molecules than Ca’+ ions in the vicinity of a release site;” if K is chosen too low the model is untenable because dlog tn idlog cp becomes low, while with high values of K the results are indistinguishable from those produced with equation (la). It may be noted that k in equation (la) includes both k and K in the above schemes. Yet another model that can yield a slope of log M vs log cp close to 4 is one where release grows exponentially6 with [CaX]. With this model, however, in order to fit observations closely, it was necessary to choose K rather exactly. and this value must be the same for Ba” 1Ca?’ and Sr’+. which seems a priori unlikely. This model was therefore not pursued further.