mouths before diffusional equilibration can occur at the

Transcription

1 PRSYNAPTI ALIUM DIFFUSION FROM VARIOUS ARRAYS OF SINGL HANNLS Impliations for Transmitter Release and Synapti Failitation AARON L. FOGLSON AND ROBRT S. ZUKR Departments ofmathematis and Physiology-Anatomy, University ofalifornia, Berkeley, alifornia 9472 ABSTRAT A one-dimensional model of presynapti alium diffusion away from the membrane, with ytoplasmi binding, extrusion by a surfae pump, and influx during ation potentials, an aount for the rapid deay of phasi transmitter release and the slower deay of synapti failitation following one spike, as well as the very slow deline in total free alium observed experimentally. However, simulations using this model, and alternative versions in whih alium uptake into organelles and saturable binding are inluded, fail to preserve phasi transmitter release to spikes in a long tetanus. A three-dimensional diffusion model was developed, in whih alium enters through disrete membrane hannels and ats to release transmitter within 5 nm of entry points. Analyti solutions of the equations of this model, in whih alium hannels were distributed in ative zone pathes based on ultrastrutural observations, were suessful in prediting synapti failitation, phasi release to tetani spikes, and the aumulation of total free alium. The effets of varying alium buffering, pump rate, and hannel number and distribution were explored. Versions appropriate to squid giant synapses and frog neuromusular juntions were simulated. Limitations of key assumptions, partiularly rapid nonsaturable binding, are disussed. INTRODUTION The brief duration of transmitter release and the longer failitation of release by subsequent ation potentials have been attributed to the diffusion of alium away from the presynapti plasma membrane after influx during a spike (Zuker, 1982; Zuker and Stokbridge, 1983; Stokbridge and Moore, 1984; Zuker, 1985a). Suh a alium diffusion model has been extended to aount for both synapti failitation and the aumulation of total presynapti alium as measured spetrophotometrially during and after a tetanus (Zuker, 1984, 1985b). However, this model inorretly predits a great prolongation of phasi transmitter release evoked by late spikes in a tetanus due to the aumulation of a residual "ative," or submembrane, alium that reahes too high a level ompared to the peak ative alium in a single spike. In this model, alium enters uniformly through the surfae membrane and diffuses radially in one dimension. This formulation fails to reognize that alium atually enters through disrete hannels, leading to pronouned nonuniformities in the submembrane alium onentration at the end of a spike (Simon et al., 1984; had and kert, 1984). Diffusion away from hannel mouths is slowed to -.1 /um2/ms by the rapid binding of >95% of alium in ytoplasm (Brinley, 1978). This allows alium to diffuse only -5 nm in the 2,us between the peak of alium influx in a spike and the beginning of transmitter release (Llinas, 1977). Release must our near hannel BIOPHYS. J. Biophysial Soiety * /85/12/13/15 Volume 48 Deember mouths before diffusional equilibration an our at the surfae. The peak ative alium onentration must therefore be substantially higher than indiated by the one-dimensional diffusion model. A more realisti model might predit a more rapid deay of ative alium and phasi release, even in late spikes in a tetanus, while leaving the time ourses of residual alium, failitation, and total presynapti alium unaffeted. To test these ideas, we have developed a three-dimensional alium diffusion model, in whih alium enters through disrete hannels and diffuses internally in three dimensions from hannel mouths to nearby release sites. MTHODS One-Dimensional Model To simulate presynapti alium movements at squid giant synapses, we solved the diffusion equation in ylindrial oordinates. It was assumed that alium enters uniformly through the whole surfae membrane of the terminal and diffuses only in a radial diretion (i.e., longitudinal and irumferential gradients were assumed to be negligible). Diffusion was slowed from its value in aqueous solution (D -.6 um2/ms) by rapid binding to uniformly distributed fixed binding sites, with a ratio of bound to free alium (fl) of 4 (Brinley, 1978). A boundary ondition at the surfae inluded terms for the influx of alium during ation potentials and at rest, and its removal by a first-order pump. The pump rate (P) was set to.8,m/ms to remove alium from ytoplasm at a rate similar to that observed experimentally at squid giant synapses (harlton et al., 1982). Resting alium influx was set to 4 fm/m2-s, resulting in a resting free alium level of 2 nm (DiPolo et al., 1976). alium influx during ation potentials was represented by a 1-ms square pulse of 1 $1. 13

2 nm/m2-s, orresponding to an inward alium urrent of -21 na for 1 ms in a terminal 7,m long and 5,m in diameter (Llinas et al., 1982). Stokbridge and Moore (1984) have shown that this approximation to the time ourse of alium urrent yields results similar to those obtained using a more aurate representation of the alium urrent. The diffusion equation was evaluated numerially by finite-differene methods. A submembrane shell thikness of 1 nm was used, whih gave results within 1% of those obtained using 1-nm shells. Details of this model have been published previously (Zuker and Stokbridge, 1983). The previous publiation ontained an error in q. 4: the terms for pumping (P) and influx (Ji.) must be divided by (1 +,B) to aount for the effet of ytoplasmi alium binding on extrusion and the fration of entering alium that remains free. In the earlier paper, impliit or bakwards differene methods (Moore et al., 1975) were used to solve the finite differene equations with relatively large time and distane steps. In the present simulation, the finite differene equations were solved expliitly, to more easily aommodate nonlinear (saturating) binding and pumping. Three-Dimensional Model In this model, alium enters the presynapti terminal through an array of disrete hannels. As in the one-dimensional model, alium is rapidly bound to immobile, nonsaturable ytoplasmi sites and alium is extruded at the terminal's surfae. We represented the terminal as a ylinder with retangular ross-setion, with the hannels grouped in thousands of disrete ative zone pathes on the terminal's synapti fae (Fig. 1 A and B). We onsidered the movement of alium in a rod-like element extending through the terminal and with one end ontaining a path of hannels (Fig. 1 ). The sides of this element are the boundaries between adjaent elements of the terminal. By symmetry, there is no net diffusion of alium aross these boundaries. Let (x, y, z, t) be the onentration hange above resting level of free alium at a point in the element. Then the equations of our model are d D At (1+) ~~~~~(1) O D - + P=f(x, z) g(t), aty = yo (2) -D +P=,aty=-yo (3) dy ' =, at x = ±xo (4) Ox a -=, at z = ±zo (5), zz (x,y,z, ) = (6) R(x, z, t) = k (x, yo, z, t)n (7) Here, D, P, and 11 represent the same parameters as in the onedimensional model. The funtion g(t) desribes the time ourse of alium influx through the hannels and is usually taken to be a train of square pulses. The funtionf (x, z) desribes the distribution of hannels on the synapti fae, y - yo, of the element. Sine the hannels are muh smaller than the path,f (x, z) is effetively a sum of delta-like funtions entered on the hannels. The fator 1/(1 +,1) appears in q. 1 beause only the free alium an diffuse. qs. 2 and 3 reflet the balane of alium fluxes at the surfae membrane, and qs. 4 and 5 express the absene of diffusion aross the sides of the element. The funtion R in q. 7 is the rate of transmitter release, and therefore the amplitude of the postsynapti response; we assume it is proportional to a presribed power of the submembrane free alium onentration at seleted points near F Jm... 71eg FrJi FIGUR 1 Unit elements of presynapti terminal in whih diffusion equation was solved in three-dimensional model. (A) Retilinear ylinder representing squid terminal and showing oordinate system. (B) Slie of terminal, with 26 ative zones in a row in the synapti fae. () lement of terminal, ontaining one ative zone. (D) lement of terminal for simulations in whih alium hannels are dispersed uniformly throughout synapti fae. () lement of terminal for simulations in whih alium hannels are onentrated in a entral strip of the synapti fae. (F) Retilinear ylinder representing frog motor neuron terminal. (G) lement of terminal ontaining one ative zone. In - and G, the dots represent alium hannels, and release sites are 5 nm from hannel mouths. the alium hannels (ative alium). qs. 1 to 6 may be solved for (x, y, z, t) by separation of variables and superposition, using Duhamel's Priniple to go from the solution for a time-independent alium influx, g(t) - 1, to that for the time-dependent ase (Berg and MGregor, 1966). The most immediate form of the solution obtained in this way is (x,y, z, t) = 1 where 4 g(r) SI(y, t - T) S2(X, Z, t - T)dT, (8) S( t os (wiyo) os (wily) o' t = (YoD[ e[-dw.?(t-r)l/(1+p) [ 2 + -j sin (W21Yo) sin (wv2y) +e[-b(yo [](W ([2 4 + P2] +Dp) - e(d~1t)/1~ (9) 14 BIOPHYSIAL JOURNAL VOLUM

3 S2(X, Z, t - T) = : wl, and w2, 1 = 1, 2, 3,.. and ank OS n- k- / os (-) e[-d2k(tt-)]/(1+p), (1) XO n2 k2,ynk2 _+2_ ( 1) o Xo are, respetively, the positive solutions of Do1l p_ = ot (wllyo) and -Dw2j = tan (wo21yo), (12) f(x, z) = 7 a,,k os S(k (13) i- k- ZO XO The series S2(x, y, t, r) onverges very slowly, espeially for small (t - r), beause the funtion f(x, z), whose Fourier oeffiients (a,,*) appear in this series, is very loalized. This series may be resummed using the Poisson Summation Formula (ourant and Hilbert, 1953) to obtain a muh more rapidly onverging series suitable for numerial alulations. The result of this resummation is the solution (x,y,z, t) = 1 where 1 f g(r) SI(Y, t - T) S3(X, Z, t - r)dt, (14) S3(X,Z t-r) = fdx',/ dz'f(x', z') e-(z_z)2+(x-x')21/4(t-v) 47ra(t - r). e-xiw-"(x+x')/xo+xx'/xoi/o,(t-,r) + e-xj[#2-#(x#-x)1xol/ao(t-r)} V--o + ezl[uu(z)/zo1/4tqt)l] (15) and or D/(l + s) is the effetive diffusion oeffiient. It is the possibility of arrying out this resummation for both the x and z series that led us to use a retangular ross-setion for our domain. In the analogue of q. 8 for a irular ylinder, the r and series do not deouple, so only the z series may be resummed. In using q. 14, we expliitly assume that the hannels are points, so that f(x, z) is a sum of delta funtions. Thus, S3(x, z, t, r) beomes a sum, over the hannels (xj, zj) in our path, of the expression within the large square brakets in q. 15 (x', z') is replaed by (x,, Zj). For numerial alulation using the formula, the time integral is evaluated using Gaussian Quadrature (Atkinson, 1978), and eah series is trunated at a term beyond whih preliminary alulations indiated that the remainder would be muh smaller than the numerial integration error. In both one- and three-dimensional models, failitation (F) of a response (R2) is measured as its frational inrease ompared with an unfailitated response (RI), F - R2/RI - 1. RSULTS One-dimensional Model Tetani Failitation. Fig. 2 A shows preditions of failitation of synapti responses to suessive spikes in a train of 1 ation potentials at 2 Hz. In performing these simulations, transmitter release was assumed to depend instantaneously on the square of the free submembrane alium onentration (harlton et al., 1982; Zuker, 1982; Zuker and Stokbridge, 1983). The deay of failitation that would be seen by responses to single test ation potentials at various intervals after the tetanus is also shown as the falling phase of the upper urve in Fig. 2 A. This may be ompared to the deay of failitation following a single ation potential, also shown in Fig. 2 A. The deay of failitation following the tetanus and a single impulse are shown on a faster time sale in Fig. 2 B. Several properties of failitation are evident from this simulation: (a) Failitation delines with fast and slow omponents. When plotted on a 5-ms time sale, the deay of failitation following a spike (Fig. 2, urve 5) an be desribed roughly as the sum of two exponentially deaying omponents, one with a time onstant of -5 ms and a magnitude of about.8 (Fig. 2, line 7), the seond with a time onstant of -1 ms and magnitude of about.25 (Fig. 2, line 6). These are similar to the experimentally measured properties of failitation at squid giant synapses (harlton and Bittner, 1978). (b) Following a tetanus, failitation (Fig. 2, urve 1) delines with similar phases (Fig. 2, lines 3 and 4) plus a third slower phase (Fig. 2, line 2), having a time onstant of several seonds and a magnitude of about 1.5. () Failitation to suessive spikes rises rapidly at first, and then ontinues to grow slowly. The growth of failitation has omponents similar to those of its deay. (d) The faster omponents of failitation rise to a level of about 3 at the end of a tetanus. The latter three properties are diffiult to measure at the squid giant synapse, due to the dominane of depression in suh tetani (Kusano and Landau, 1975), but they are similar to the properties of tetani failitation reported at neuromusular juntions (Mallart and Martin, 1967; Magleby, 1973; Magleby and Zengel, 1976; Magleby and Zengel, 1982). Thus the one-dimensional alium diffusion model is reasonably suessful in prediting the harateristis of tetani failitation and its posttetani deay. Presynapti Average Free alium. The predited rise of the total number of free alium ions inside the terminal during a tetanus, and its subsequent deline, are plotted in Fig. 2. This may be ompared to the hange in average free ytoplasmi alium measured spetrophotometrially with arsenazo III (harlton et al., 1982). The simulated tetani total alium rises with a slightly delining slope to a peak of 1,MM, similar to the - experimental order-of-magnitude measurement of -3,uM for similar tetani. Posttetanially, total alium delines FOGLSON AND ZUKR Presynapti alium Diffusionfrom Single hannels 15

4 U. Iā IL ' a 2 U a a 2 4 I1 a3 a FIGUR 2 Simulations of synapti transmission at squid giant synapse, using one-dimensional model and assuming two alium ions ooperate to release a quantum of transmitter. In all panels, tetani simulations are plotted as solid lines and single spike simulations as dashed lines. (A) Growth of failitation to suessive spikes in a tetanus (2 Hz, 5 s) and deay of failitation measured at various intervals after the end of the train (at time ) or after a single spike. Failitation is the frational inrease in transmitter released by a tetani or posttetani spike, ompared to release by a single spike. (B) Deay of posttetani (urve 1) and postspike (urve 5) failitation on a faster time sale. The deays have been deomposed into a series of exponentials (posttetani: lines 2-4; postspike: lines 6 and 7) by peeling suessive omponents, slowest first. () hange in average presynapti free alium onentration during a tetanus and one spike, normalized to the same peak. Left ordinate refers to tetani response, right ordinate to single spike response. (D) Deay of square of ative alium (phasi transmitter release) following last spike in a tetenus (solid urve) and a single spike (dashed urve). with a half-time of -7 s, similar to experimental results. The simulated total alium deays somewhat more slowly than that following a single spike, beause in a tetanus alium has diffused further from the membrane and so takes longer to be extruded at the membrane than does alium that has entered during a single ation potential. Unfortunately, the measurement of the deay of total alium after one ation potential is too rude to determine 16 whether this aspet of the simulation is observed experimentally. Time ourse of Transmitter Release. Fig. 2 D illustrates the deline in simulated transmitter release for one spike and following the last spike in the tetanus. At the end of one ation potential, ative alium reahes 1.53,uM. Phasi release evoked by one ation potential drops to BIOPHYSIAL JOURNAL VOLUM

5 1% of its peak in 4 ms. As expeted, this is somewhat faster than the postsynapti urrent delines (harlton et al., 1982), a result of the kinetis of losing of the postsynapti hannels. However, the tetani simulation predits that even 1 ms after the last spike, transmitter release is ourring at nearly the same level as at the peak for the first spike. This is beause residual alium after a tetanus deays very slowly beause of the redued spatial alium gradient after prolonged alium influx, and so residual alium remains a large fration of the peak alium reahed in a single spike. Residual alium is 1.35,gM at 1 ms after the last spike,.76,m at 1 s, and.41,um at 5 s after the end of the tetanus. This aspet of the model is independent of the relation assumed between transmitter release and ative alium. It predits a ontinual release of transmitter for -5 ms after the tetanus at a higher level than that at the peak of the first spike. Suh behavior has never been observed at any hemial synapse, and is the ruial point of failure of the one-dimensional model. alium Uptake. Various alternative formulations of the one-dimensional model have been devised and used to simulate tetani failitation, time ourse of transmitter release, and total free presynapti alium. For example, alium ould be removed not at the surfae, but everywhere throughout ytoplasm by a first-order uptake mehanism, suh as is reported for mitohondria and endoplasmi retiulum. Mitohondria take up alium too slowly to aount for the observed deline of presynapti alium (Brinley et al., 1978), but uptake into endoplasmi retiulum ours with a time onstant of seonds (Blaustein et al., 1978), similar to that observed at squid synapses. Suh an uptake system aounts as well as does a surfae pump for the tetani arsenazo measurements. With internal uptake, unlike surfae extrusion, presynapti alium delines at the same rate following one spike or a tetanus. As mentioned above, the experimental data are insuffiient to distinguish these alternatives. Tetani failitation and the transmitter release following one or tetani spikes behave similarly for both forms of alium removal: failitation is still well desribed, while posttetani transmitter release persists too long. Saturable Buffers. A seond variant of the onedimensional model that we onsidered was to allow the ytoplasmi buffer to saturate. Versions with 5,M of a low laffinity buffer (KD = 2,MM, f. Alema et al., 1973), or with 4,M of a high affinity buffer (KD = 1 nm, see Baker and Shlaepfer, 1978), or with both buffers present simultaneously, have been simulated. In these simulations, the diffusion equation and orresponding differene equations were modified as desribed by Barish and Thompson (1983). We found that neither type of saturable buffer substantially affeted the time ourse of tetani failitation. The peak submembrane alium onentrations reahed in ation potentials were muh higher when saturable buffers were present, espeially in the ase of a high-affinity, low-apaity buffer, and the phasi deay of submembrane alium ourred about twie as fast. Residual alium, on the other hand, was little affeted. Failitation, whih depends on the ratio of residual to peak alium, was therefore redued. The most interesting result of the simulations using saturable buffers appeared in the total alium preditions. A high-affinity, low-apaity buffer aused the total alium signal to display a sharp drop to 5% in 2 ms, and a further rapid deline to 25% in 2 ms, due to the diffusion of unbound alium past saturated submembrane buffer to unsaturated buffer where it was bound (see onnor and Nikolakopoulou, 1982). The kinetis of arsenazo's response to alium might be too slow (see Disussion) to detet the initial 2-ms spike in the total alium signal, but it should report most of the subsequent large drop in free alium in the next 2 ms. Sine no suh deline in the arsenazo signal is seen (harlton et al., 1982), the existene of a high-affinity, low-apaity buffer may be questioned. Brinley (1978) gives other reasons to doubt the presene of suh a highly saturable buffer. Three-dimensional Model The hief failure of all versions of the one-dimensional model is that residual ative alium after a tetanus remains higher than the peak ative alium reahed in the first spike for almost 1 ms after the end of the tetanus. We expeted that near points of alium entry through disrete hannels, where transmitter release must our (see Introdution), the peak alium reahed after eah ation potential must be muh higher than predited by a model of alium entering uniformly aross the surfae. To test this idea, we developed the three-dimensional model of alium entering through disrete hannels. We assume that the hannels open nearly simultaneously during the rising phase of the ation potential, and remain open for exatly 1 ms before losing, and do not reopen in an ation potential. Sine the average hannel lifetime is 1 Ims, similar to the duration of total alium urrent (Llina's et al., 1982; Lux and Brown, 1984), these seem reasonable approximations. Disposition of alium hannels. We must deide where to put the alium hannels and transmitter release sites in our model. Pumplin et al. (1981) found that ontat between presynapti and postsynapti ells at the squid giant synapse ours at ative zones, whih are roughly irular pathes.65,um2 in area. The total area of synapti ontat ranged from 3,7 to 13,,gm2. ah of the approximately 1, ative zones ontained about 1, randomly sattered intramembranous partiles, whih are thought to orrespond to alium hannels. The likely number of alium hannels to open in an ation potential is far fewer than the total of I7 partiles, FOGLSON AND ZUKR Presynapti alium Diffusionfrom Single hannels 17

6 however. The alium urrent in an ation potential is -24 na for 1 ms (Llinas et al., 1982). If the single hannel urrent during the ation potential (at - mv) is -.4 pa (Lux and Brown, 1984), then only 6, alium hannels, or 6% of the number of partiles thought to represent alium hannels, open in a spike. Fig. 1 A- show how this distribution of alium hannels in ative zones was represented in most of our simulations. Ative zones onsisted of square pathes,.8,um on a side, and ontaining 64 hannels eah in a square array, with 18 nm between alium hannels opened during an ation potential. The 9,464 ative zones were arranged in a square pattern of 26 by 364 zones over the fae of the presynapti terminal (5,um x 7,um) in ontat with the postsynapti axon. ah ative zone was in the enter of a square region of membrane 1.93,um on a side, whih was one fae of an element of the terminal in whih the diffusion equation was solved. The element was a retilinear rod, 1.93,tm on a side and 5,um long, extending through the ytoplasm to the opposite surfae of the terminal. alium entered through the 64 open hannels as point soures in the synapti fae, and was removed from the front and bak surfae of eah element. By symmetry, there would be no net diffusion of alium aross the sides of the elements to neighboring idential elements. If it is also assumed that there is no diffusion or removal of alium aross the sides of the elements on the edge of the terminal, i.e., aross the sides and ends of the terminal, then all elements will behave identially. Otherwise, edge zones would behave somewhat differently. Sine only 8% of the ative zones are on the edge, they will ontribute little to synapti transmission. Loation of Ative alium liiting Release. Synapti vesiles are also lustered at ative zones (Pumplin and Reese, 1978). As explained in the Introdution, the short synapti delay suggests that transmitter release ours within -5 nm of open alium hannels. We imagine that transmitter release ours at speifi vesile attahment sites, whih, like alium hannels, are integral membrane proteins. These proteins, -5 nm in radius, must be separated by at least 1 nm. If vesile fusion and exoytosis are to our without displaing alium hannels, then vesiles with a 3 nm radius would have to fuse at least 35 nm from a alium hannel. We therefore imagine that the relevant ative alium onentration that ats to release transmitter is the submembrane free alium onentration -5 nm from alium hannel mouths opened during ation potentials. This is similar to the spaing between sites of vesile fusion and intramembranous partiles thought to represent alium hannels at frog neuromusular juntions (Heuser et al., 1979). Predition ofative alium in a Tetanus. Fig. 3 A shows the submembrane alium at the enter of an ative zone, equidistant from the four entermost open 18 alium hannels. After one spike, ative alium drops from a peak of 3,M to 22 zm in 1 ms. Fig. 3 B shows the peak ative alium onentration at the end of eah spike and the alium onentration just before eah spike in a 5-s tetanus of 1 ation potentials at 2 Hz. The posttetani deay of residual ative alium is also shown. The values of pump rate (P), binding ratio (,B), and total alium influx in an ation potential are the same as in the one-dimensional simulation. It is evident that the residual alium after a tetanus, or before the last spike in a tetanus, is now muh less than the peak alium in the first spike. The reason is that the peak ative alium after one spike is nearly 2 times as big as that predited if alium enters uniformly aross the entire presynapti membrane. Ative alium drops very rapidly after eah spike, diffusing in three dimensions away from hannel mouths, until submembrane equilibrium is established. Then residual alium drops muh as it did in the one-dimensional model, due to radial diffusion into the interior of the terminal. It is now a muh smaller fration of peak ative alium in a spike than it was in the one-dimensional simulation. hoie of alium Power Governing Release. To onvert ative alium to transmitter release or predit synapti failitation, we need the relation between alium and release. It is lear that with residual alium now a smaller fration of peak alium in a spike, there will be less failitation than in the one-dimensional simulation, or than is observed experimentally. To ompensate for this redution in relative residual alium, a higher stoihiometry is needed for the reation of alium with release sites to release transmitter. Our simulations with the onedimensional model assumed a stoihiometry of 2, based on measurements of the relationship between transmitter release and alium urrent in voltage-lamp experiments (harlton et al., 1982). More reent measures suggest powers of 3 or more (Smith et al., 1985). The dependene of transmitter release on external alium also shows a power law dependene with exponents of from 2 to greater than 4 in different speimens (Katz and Miledi, 197; Lester, 197). Parnas et al. (1982) and Barton et al. (1983) have shown that this measurement is likely to underestimate the true stoihiometry of the reation. Preliminary alulations (Zuker and Fogelson, 1986) indiate that the release vs. alium-urrent relation also underestimates the alium-release stoihiometry. We have hosen a value of n = 5 in our simulations with the three-dimensional model, whih produes a failitation similar to that observed experimentally. Simulated Failitation, Time ourse of Release, and Total alium. Fig. 3 and D show the tetani growth, posttetani deay, and postspike deay of failitation on two time sales. Failitation behaves very muh as in the one-dimensional simulation (Fig. 2 A and B) and as reported experimentally (harlton and Bittner, 1978). The BIOPHYSIAL JOURNAL VOLUM

7 Tine (ms) a a IL o o "I3 ạ _ 1 1 I. B TIme (s).2 o o -._a o D F~~~~~~~~~~ FIGUR 3 Simulations of synapti transmission at squid giant synapse using three-dimensional model. A alium ooperativity of 5 is assumed in Figs Transmitter release sites and alium hannels are loated in ative zone pathes of presynapti membrane. (A) Deay of ative alium following one spike. (B) Upper urve onnets peaks of ative alium in suessive spikes in a tetanus (2 Hz, 5 s) and subsequent deay of ative alium at end of tetanus (at time ). Lower urve onnets troughs of ative alium just before eah suessive spike. () Growth and deay of tetani (solid urve) and deay of postspike (dashed urve) failitation on a slow time sale. (D) Deay of posttetani (solid urve) and postspike (dashed urve) failitation on faster time sale and their exponential omponents (numbers as in Fig. 2 B). () Deay of fifth power of ative alium (phasi transmitter release) following last (solid urve) and first (dashed urve) tetani spikes. Ordinate normalized to peak of first spike (equal to 1). In ', the two urves are normalized to the same peak, to show differenes in time ourse. (F) Average presynapti alium onentration during and after a tetanus. early deay of failitation (Fig. 3, urves 1 and 5) has rapid (5.7 ms, Fig. 3, lines 4 and 7) and slow (66 ms, Fig. 3, lines 3 and 6) omponents following either a tetanus or one spike, with a ombined magnitude of 3 measured at an interval of 2 ms after 1 spike. There is an even briefer omponent of intense failitation, orresponding to that measured experimentally by prolonging the depolarization of terminals (Katz and Miledi, 1968). Although not expliitly illustrated in Fig. 3 D, this omponent is evident from the fat that the initial amplitudes of the two illustrated omponents of failitation (Fig. 3, lines 6 and 7) sum to less than the initial magnitude of failitation (Fig. 3, urve 5). A slow omponent of failitation (Fig. 3, line 2), with a deay onstant of 7 s and magnitude.32, appears during and after a tetanus. Thus, tetani failitation aumulates in a tetanus in the three-dimensional model muh as it does in the onedimensional model. It is diffiult to measure failitation in isolation at squid giant synapses to long tetani. harlton and Bittner (1978) report that failitation rises within five spikes at 1 Hz to a plateau level of We have also simulated brief trains at 1 Hz, and our fifth spike exhibited a failitation of.84. Fig. 3 shows the predited time ourse of ative alium raised to the fifth power for the first and last spike in the tetanus. learly, there is now little differene between these two spikes. If the hange in ative alium is the rate limiting step in transmitter release, this should resemble the time ourse of release and be faster than the deay of postsynapti urrent. In fat, the time ourse of release is similar to this at frog neuromusular juntions (Barrett and Stevens, 1972), and postsynapti urrent at squid giant synapses is slower than the predited release funtion (harlton et al., 1982). We have predited the arsenazo signal, indiating total free alium, by alulating the total alium influx minus total alium efflux aross the surfae at eah time, then integrating this net flux over time to get the total alium in the ell, and dividing by the binding ratio and the volume to get the average free alium onentration. The threedimensional model (Fig. 3 F) behaves muh like the one- FOGLSON AND ZUKR Presynapti alium Diffusionfrom Single hannels 19

8 dimensional model (Fig. 2 ), and the simulations resemble experimental results (harlton et al., 1982). Submembrane alium at Different Loations. Fig. 3 deals with the submembrane alium at the enter of an ative zone, 76 nm from eah of four open alium hannels. We have also looked at the submembrane alium at various distanes from a hannel loated near the enter of an ative zone. Between 4 and 76 nm from suh a alium hannel, the form and magnitude of the alium transient are similar (differing by <3%) for both single spikes and a tetanus. This is beause, by the end of a 1 ms influx, alium has already diffused away from hannel mouths, and the submembrane alium is nearly uniform in the enter of an ative zone. We have also omputed the submembrane alium near hannels at various positions in an ative zone. For release sites within three rows of the enter of an ative zone (eight-by-eight rows of hannels), the peak alium reahed in both the first and last spike in a tetanus, and the rate of deay of residual alium, are all within 5% of the values at the enter of the ative zone. Thus, 5% of the ativated hannels evoke release in a way nearly idential to those at the enter (Fig. 3). Outside of this region, alium onentrations are suffiiently lower that they are not well represented by Fig. 3 A and B. Between the outer two rows of hannels, the peak alium onentration for either the first or the last spike in a tetanus reahes 75-9% of the value for the enter of the path (less near a orner than at the middle of a row). It follows from the nonlinear dependene of release on ative alium that these and more peripheral release sites (beyond the outer row of hannels) ontribute little to synapti transmission. ffet of hannel lustering. Why are alium hannels and transmitter release sites lustered into ative zone pathes? Would transmission be the same if the release mahinery were distributed uniformly over the presynapti surfae that is in ontat with the postsynapti membrane? We approahed these questions by onsidering the synapse's behavior for eah of three onfigurations of synapti mahinery: In the first, 6, alium hannels are distributed uniformly in the synapti fae of the presynapti terminal. The diffusion element of the terminal is now a slab, with nonsynapti and synapti faes of 5 gm x 242 nm and a length of 5,m, ontaining 26 hannels in a single row (Fig. 1 D). In the seond onfiguration, the 6, hannels are onentrated into a strip running longitudinally along the middle of the synapti fae and having an area of 7,,m2, the same as the total area of ative zones used in Fig. 3. This time, the diffusion element is a slab with dimensions of 5 Am x 18 nm on the synapti fae and a length of 5,m, whih ontains a entral 1 Mm long row of 92 hannels (Fig. 1 ). The third onfiguration has the hannels lustered in ative zones as desribed above (Fig. 1 B). 11 In Fig. 4 the behavior of these three distributions of synapti mahinery is ompared. ah row illustrates ative alium and synapti failitation during a tetanus, and time ourse of release for the first and last spikes in the tetanus. In Fig. 4 A, the solid lines show simulations in whih hannels are dispersed uniformly throughout the synapti fae 242 nm apart in a square array. We illustrate simulations at 5 nm from a hannel mouth. Similar results were obtained between 5 and 5 nm from the nearest hannel, so that the preise loation of vesile fusion within the harateristi diffusion distane from hannel mouths is not ritial. The dashed lines show simulations at a point midway between (54 nm from eah of) two hannels in a row for the ase of hannels onentrated in a single synapti strip with the same density as in ative zones. Again, very similar results were observed at the edge of an element, 76 nm from eah of four hannels, or as lose as 5 nm to a hannel mouth. Similar results were also obtained near all but the outer two hannels in the 92 lines of hannels in the synapti strip. Finally, the solid lines in Fig. 4 B illustrate simulations 76 nm from eah of four hannels when hannels are lustered in ative zones. As mentioned previously, similar results were obtained at points between 4 and 76 nm from hannels not at the edge of an ative zone. Thus the results are truly representative of the behavior of submembrane alium at likely points of vesile fusion near the majority of alium hannels for all three hannel arrays. In all these simulations, f8 = 4 and P =.2,um/ms. Several differenes are evident among the three arrays: Peak ative alium in a spike rises from 8 MM for hannels distributed uniformly in the synapti fae to just over 3,MM for hannels onentrated in a single strip or in dispersed ative zones. Ative alium deays fastest from uniformly distributed and widely dispersed hannels. vidently, onentrating hannels inreases the peak ative alium and slows the initial rapid deay as a result of the inreased overlap of alium diffusing from neighboring hannels. With uniformly dispersed hannels, a point whih is 5 nm from one hannel an be no nearer than 92 nm to a seond hannel, and the next nearest hannels would be 247 nm away. Here, ative alium and transmitter release are dominated by one alium hannel. When hannels are onentrated into a strip or ative zones, it is impossible for a point to be >76 nm from a hannel, and then the point is equidistant from four. This leads to higher peaks and slower deays. Ative alium aumulates with several time onstants during a tetanus and deays afterwards with similar time onstants. This leads to similar multiomponent failitations for all hannel arrays. onentrating the hannels into a strip speeds the time onstants and redues the magnitude of failitation. This is beause, at late times, the strip ats on a larger sale like a line soure from whih alium diffuses in two dimensions, while the widely dispersed hannels at as a plane soure from whih BIOPHYSIAL JOURNAL VOLUM

9 A '3 a 4. 2 I..2 U 'a ) Bo i2 ' o 2.2 'aa I- I- I I 4,'.... I, f,-- T Time (a) ) f T - II I I I I #r-s -i '-- - ^ { Time (81 la al 1.1 1I I.1k % 4t 1-. A : "", U 1 - s Time (a) I^ 4 S. a - a - o - a 41 -'a 2 1l o.. II...,. 4.,. 1. I FIGUR 4 ffet of hannel lustering and pump rate on synapti transmission. Left olumn shows ation potential peaks and pre-spike troughs of ative alium in a tetanus, and subsequent posttetani deay. Middle olumn shows tetani growth and posttetani deay of failitation. Right olumn shows phasi transmitter release following last spike (upper urve) and first spike (lower urve) in tetanus, normalized to peak of first spike. (A) Simulations in whih alium hannels and release sites are dispersed throughout synapti fae (solid lines) or onentrated into a entral strip (dashed lines). (B) Simulations in whih alium hannels and release sites are lustered in ative zones with the same density as in the strip (solid lines), and simulations in whih release mahinery is uniformly dispersed, but with a pump rate tenfold higher than in A (dashed lines). diffusion is one-dimensional. The most interesting result, however, is that whether hannels are uniformly dispersed or onentrated in a strip, posttetani ative alium drops muh too slowly to be onsistent with the deay of transmitter release. This is the point of failure of the onedimensional model. However, when hannels are loated in separated ative zones, ative alium does dissipate rapidly enough after a tetanus. This is beause eah ative zone ats at late times like a disrete disk soure, from whih alium diffuses in three dimensions after eah spike even on the larger sale. Hene, transmitter release deays almost as rapidly after the last spike in the tetanus as after the first spike, and tetani failitation is redued to levels more onsistent with experimental observation. ffets of Buffer Ratio and Pump Rate. Another way to redue residual alium to levels onsistent with a rapid deay of posttetani transmitter release is to inrease the pump rate (P). A 1-fold enhanement of pump rate (to.2,um/ms) in the widely dispersed hannel array leads to fast omponents of failitation and time ourse of transmitter release (dashed lines in Fig. 4 B) similar to those for the ative zone onfiguration of lustered hannels. This inrease in pump rate had little effet on the deay of ative alium following a single spike, but the slowest phase of failitation (augmentation) was faster and less prominent. We have also explored the effet of varying the ratio of bound to free alium (f3). Fig. 5 presents simulations using the ative zone onfiguration of alium hannels, but with,8 set to 16 (solid lines) or 1 (dashed lines). In all ases, ative alium is measured midway between the entral four hannels in an ative zone, and P =.2,um/ms. Several effets are evident: Dereasing the proportion of alium that is bound inreases the peak ative alium. The inrease is less than proportional to the hange in $, beause reduing,b not only inreases the proportion of entering alium that remains free, but also speeds diffusion of alium away from hannel mouths, speeding the deay of transmitter release (if alium ativity is rate limiting). With,B as high as 16, diffusion is suffiiently slowed that the peak in ative alium 76 nm from four FOGLSON AND ZUKR Presynapti alium Diffusionfrom Single hannels 111

10 D ~4-.2~~~~~~~~~~~~~~~~~~~~ o 2 al So Time (a) -4-2 FIGUR 5 ffet of ytoplasmi alium binding ratio on synapti transmission. Ative alium in a tetanus, tetani failitation, and phasi release to first and last spikes are shown on left, middle, and right as in Fig. 4. The ratio of bound to free alium is 16 (solid lines) or 1 (dashed lines). alium hannels were lustered in ative zones. hannels ours a full milliseond after the end of the alium influx. Thus, unlike hanges in pump rate, hanges in : have a large effet on the rapid early deay of ative alium following a spike. The effet of reduing on the slow omponent of alium aumulation in a tetanus, and its posttetani deay, is more omplex. Reduing a is equivalent to inreasing the pump rate in so far as extrusion is onerned. This is beause the pump must ompete with the alium buffer for free alium. Reduing A thus speeds removal of residual alium, like inreasing P, and results in less slow failitation or augmentation. However, reduing A is less effetive than inreasing P, beause reduing also speeds diffusion. alium then moves more rapidly away from the surfae to where it is less subjet to pumping ation. Simulations of Frog Motor Neuron Terminals. Sine it is not possible to observe synapti failitation without depression at the squid giant synapse in response to long tetani, we have adapted the three-dimensional diffusion model to frog neuromusular juntions, where failitation has been more thoroughly haraterized. For this purpose, we hose as an element for alium diffusion one ative zone, 1,um wide, in a long motor neuron terminal with a square ross setion of 1.5,um on a side (Fig. 1 Fand G). ah ative zone ontains four rows of intramembranous partiles thought to represent alium hannels (Heuser et al., 1979). These run perpendiular to the axis of the terminal in the synapti fae, and are loated 25 and 35 nm, respetively, on either side of the enterline of the ative zone. Synapti vesiles release their ontents -5 nm lateral to eah outer row of partiles, so we onentrate on submembrane alium at this distane from an outer row. Sine there are no measures of alium binding, extrusion, uptake, or influx at motor nerve terminals, the hoie of values for these parameters is somewhat arbitrary. We used values of (1) and P (.3,um/ms) within the 112 range of reported values for squid axons. Deiding the number of alium hannels opened by an ation potential is more diffiult. There are about 1 intramembraneous partiles in eah transverse row. If a similar proportion of these opens during ation potentials as in the squid synapse, we would expet to have about six open hannels per row, or 25 per ative zone. We have tried simulations with from 1 to 8 hannels per ative zone. In Fig. 6 A and B, we present simulations of ative alium after an ation potential and during and after a tetanus. In this ase there were three hannels in the inner rows, and two in the outer rows, staggered and spaed 6 nm apart in eah row (Fig. 1 G). Using a fourth-power relation between ative alium and transmitter release (Dodge and Rahamimoff, 1967), we predit the time ourse of transmitter release for the first and last spikes in the tetanus (Fig. 6 ), tetani failitation and its subsequent deay (Fig. 6 D), and the early postspike and posttetani deay of failitation. (Fig. 6 ). Several results are remarkable: Ative alium rises to a peak of 7.8 AM in one spike, and 11.6,uM at the end of a 2-Hz, 5-s tetanus. Transmitter release evoked by the last spike drops to 25% of its peak at the end of the first spike in 1 Ims, so phasi release is preserved with only modest prolongation in a tetanus. Failitation deays after a spike (Fig. 6, urve 4) and a tetanus (Fig. 6, urve 1) with fast (Fig. 6, lines 4 and 7) and slow (Fig. 6, lines 3 and 6) omponents of 1 and 266 ms, with magnitudes of.55 and.8 measured 3 ms after one spike. These failitation properties are similar to those observed at frog neuromusular juntions (Mallart and Martin, 1967; Magleby, 1973), exept that measured failitation deays initially somewhat more slowly than the simulations suggest (time onstant of ms). As in the squid simulations, there is an early, brief, and intense phase of failitation (not shown as a peeled omponent) orresponding to experimental observation (Katz and Miledi, 1968). During and after a tetanus, a very slow omponent of failitation appears (Fig. 6, line 2), with a time onstant of -2.8 s and BIOPHYSIAL JOURNAL VOLUM

11 17Ir..*.. I I I % a o - - L)._ ) : 1l i B- N U) ) - L- O- as : o I I I I I JI I I I I n o IL U FIGUR 6 Simulations of synapti transmission at frog neuromusular juntion, using three-dimensional model. A alium ooperativity of 4 is assumed. (A) Deay of ative alium following one spike. (B) Peaks and troughs of ative alium in a tetanus and posttetani deline of ative alium. () Phasi transmitter release following last (solid urve) and first (dashed urve) tetani spikes, normalized to peak of first spike. (D) Tetani and posttetani (solid urve) and postspike (dashed urve) failitation on a slow time sale, and their exponential omponents. () Posttetani and postspike failitation on a fast time sale (numbers as in Fig. 2 B). magnitude of 2.5. This may be ompared to augmentation we measure submembrane alium in the line 5 nm at frog neuromusular juntions, whih has a time onstant lateral to the outer row of hannels. Points opposite a of 4-9 s and magnitude of 1-3 following a similar tetanus hannel in the outer row (5 nm away) or opposite a (Magleby and Zengel, 1976). hannel in the inner row (6 nm away) yield virtually We are unaware of measurements of the time ourse of idential results. In both ases, ative alium is dominated phasi release evoked by spikes late in a tetanus. Datyner by influx from one hannel, but influened by diffusion and Gage (198) reorded virtually idential time ourses from adjaent hannels and from the nearest hannel on of release for the first through third spikes at 65 Hz. They the other side of the ative zone. A point on the line of saw only a slight (1%) prolongation of the foot of release, vesiles midway between hannels in the inner and outer at one-tenth the peak. In our simulation of release to three rows (-9 nm from eah of two hannels) reahes a peak spikes at this frequeny (not illustrated) we also observed alium level 12% lower than opposite a hannel, and so is about a 1% prolongation of the foot at one-tenth the 65% as likely to ause transmitter release. It makes little peak. differene near whih of the open hannels in the ative We have heked the dependene of the results on where zone alium is measured. FOGLSON AND ZUKR Presynapti alium Diffusion from Single hannels 113

12 Inreasing the number of open hannels to between 5 and 2 in eah row (2 and 8 per ative zone) auses the ative zone to behave more like a line soure at late times. Augmentation and the slow omponents of failitation then deay more rapidly. Due to the signifiant overlap of diffusion from adjaent hannels, early release time ourse after a tetanus is slowed, dropping to only half the peak of the first spike in 2 ms. 3 and P an be adjusted to ompensate for any one of these diffiulties in simulations with higher hannel densities, but only at the expense of worsening other aspets of the fit to observation. Thus, simulations with 1 open hannels per ative zone more losely resemble experimental data. DISUSSION Aomplishments of the Simulations Our study is hardly the first to treat transmitter release and synapti failitation as onsequenes of phasi and residual alium. The experimental evidene that phasi release depends on alium influx (Katz, 1969) and that failitation and augmentation depend on alium aumulation (Katz and Miledi, 1968; Rahamimoff et al., 198; harlton et al., 1982) is already quite strong. The interpretation of this evidene is not altered by the present simulations. Others have also argued that phasi release is kinetially driven by hanges in ative alium (e.g., Llinas et al., 1981) and that failitation and augmentation an be explained by residual ative alium and a high aliumrelease stoihiometry (e.g., Parnas et al., 1982; Magleby and Zengel, 1982; Zuker and Lara-strella, 1983). What is unique about alium diffusion models is that they attempt to predit the magnitude and time ourse of phasi and residual alium from a onsideration of physial and hemial proesses known to regulate alium in neurons. A simple one-dimensional model (Zuker and Stokbridge, 1983; Stokbridge and Moore, 1984) predits aurately the radial distribution of residual alium after repetitive ativity. However, it fails to predit the magnitude and time ourse of ative alium at transmitter release sites near alium hannel mouths right after ation potentials. A more realisti three-dimensional model, in whih alium enters through disrete hannels arranged in patterns suggested by ultrastrutural observations, is more suessful in prediting some harateristis of synapti transmission. In partiular, the three-dimensional model predits that posttetani residual alium remains a fration of the peak ative alium in a single spike. It yields reasonable levels of tetani failitation in frog and squid with a alium-release stoihiometry of 4 or 5, onsistent with experimental estimates. Suh a degree of alium ooperativity generates too muh tetani failitation in the one-dimensional model. One surprising result is that with a realisti hoie of alium extrusion rate, alium hannels had to be lustered into separated ative zones to keep residual alium from aumulating too 114 heavily near alium hannels. This may be why the synapti mahinery is lustered in this fashion. Another differene between one- and three-dimensional models is that the latter predits peak ative alium levels of >3,uM in late tetani spikes in the squid, while the former predits ative alium levels of only 3,uM. This has important impliations for the sensitivity of the release mahinery to alium onentration. It might appear that an inrease of ative alium of nearly 1, times from its resting level of -5 nm might redue the driving fore on alium suffiiently to retard the alium influx in late tetani spikes. However, appliation of onstant field theory to alium hannels indiates that the alium urrent will be redued by only.3% by this elevation of alium near hannel mouths. Our simulations are not very stiff, in that fourfold hanges in parameters suh as buffer binding ratio, alium pump rate, hannel number or density, or distane of release sites from hannel mouths affet quantitatively but not qualitatively the time ourses and magnitudes of alium aumulation, phasi transmission, and failitation. With parameter values hosen to be onsistent as muh as possible with diret experimental measurements, the simulations resemble remarkably losely the dynami properties of synapti transmission at highly failitating synapses, suh as frog neuromusular juntions. Limitations of the Model The fit to experiment is not perfet, however. For example, the slowest omponent of failitation (augmentation) deayed with a time onstant of 2.8 s, ompared to a typially observed time onstant of 7 s (Magleby and Zengel, 1976). Reduing the pump rate, or inreasing the binding ratio, inreases this time onstant in our simulations, but then the predited magnitude of augmentation is too high. The early omponent of failitation in frog, measured beginning -5 ms after an ation potential (the typial absolute refratory period), was -12 ms in our simulations, ompared with an average of 35 ms observed experimentally. Finally, the tail of residual alium shortly after a tetanus, although suffiient to aount for failitation, predits a, large inrease in frequeny of spontaneously released quanta of transmitter. This predited inrease in miniature postsynapti potential frequeny is observed at rayfish neuromusular juntions (Zuker and Lara-strella, 1983), but not at those of frog (Zengel and Magleby, 1981). Several other qualities of synapti transmission are not explained by our model: (a) The time ourse of transmitter release has a high Qlo (harlton and Atwood, 1979). At least at low temperature, it is learly not a diffusion-limited proess. Perhaps the dynamis of exoytosis determine the time ourse of transmitter release. We make no attempt to represent these dynamis in our model, nor to aount for the form of the time ourse of phasi release and its synapti delay. (b) We do not treat effets of hanging BIOPHYSIAL JOURNAL VOLUM

13 external alium, magnesium, and temperature on phasi release and failitation. Suh effets ould be due to influenes on various proesses (exoytoti mahinery, alium buffering, alium extrusion, transmitter mobilization, et.). Little information is available to hoose between the alternatives, so their inorporation into our model is premature. () The evidene for a speifi role of presynapti membrane potential in regulating transmitter release (Llinas et al., 1981; Dudel et al., 1983) is treated ritially in two separate publiations (Zuker and Lando, 1986; Zuker and Fogelson, 1986). No suh role is inluded in the present model. These imperfetions are proof that the model is not a omplete and totally aurate representation of the proesses ontrolling transmitter release. Many simplifying assumptions are ertainly inorret: uniform, nonsaturating, and instantly equilibrating alium buffering, linear nonsaturable extrusion only at front and bak surfaes, no internal uptake of alium, and a simple nonsaturating power-law relation between alium and transmitter release. The model inludes no provision to aount for synapti depression, suh as a limited, but refillable, releasable transmitter store. And there was no attempt to aount for tetani and posttetani potentiation, whih appear to depend on the presynapti entry and aumulation of sodium ions (Rahamimoff et al., 198). Possible effets of internal membrane surfae harge on submembrane alium onentration were also ignored. Thus, the present three-dimensional model must be regarded as only one step toward understanding some of the physial proesses that determine some of the properties of transmitter release, and, in partiular, an improvement over the onedimensional model used heretofore. One harateristi of our model is that the same alium hannels are assumed to open in suessive impulses. Sine hannel ativation is a stohasti proess, and there appear to be many more submembrane partiles or available alium hannels than open hannels, this is an unrealisti assumption. This turns out not to matter, however, sine by the time of a subsequent ation potential the submembrane residual alium has fully equilibrated throughout an ative zone, so it makes no differene where in the ative zone the next hannels to open are loated. One serious limitation to our analyti solution of the diffusion equation in three dimensions is that it relies on linear solution methods. Thus, we annot inorporate the effets of saturable extrusion and buffering, as we ould in the one-dimensional model solved by numerial approximation methods. The inability to treat saturable and noninstantaneous buffers is partiularly disappointing, beause even a low-affinity, high-apaity buffer is likely to be saturated near alium hannel mouths, where the loal inrease in total (bound plus free) alium is -1 mm. It is worth onsidering the probable onsequenes of suh loal buffer saturation: (a) If alium buffers saturate near alium hannels, the loal peak free alium onentration will be even higher, whereas residual alium will be unaffeted (as shown by our simulations with the one-dimensional model). onsequently, failitation will be redued. This effet may be ompensated by adjustment of the buffer ratio (p3) or the pump rate (P). (b) A saturated buffer does not retard diffusion, so initial diffusion of alium away from hannel mouths will be even faster than in our simulations, until loal alium drops to levels below the buffer apaity. This makes the initial deay of ative alium, and phasi transmitter release, even faster. () In our simulations, raising a slowed failitation but prolonged posttetani time ourse of release. If alium hannels are surrounded by a partiularly high buffer onentration (loal high p3), whih is saturable, the rapid deay of alium in a spike (above saturation levels) will be preserved, while the early omponent of failitation will be slowed (due to higher,b). This may be the reason why the deay of failitation was too fast at early times in our simulations of frog neuromusular juntion. (d) Finally, a saturable buffer has impliations for free alium measurements using arsenazo. The diffusion of alium from loally saturated to unsaturated buffer regions away from hannel mouths will be aompanied by a rapid drop in total free presynapti alium (see onnor and Nikolakopoulou, 1982). However, it is unlikely that arsenazo will report this alium spike aurately. The relaxation time onstant for the reation of arsenazo with alium has been variously estimated as ms (Sarpa et al., 1978; Ogawa et al., 198; Dorogi et al., 1983). Unfortunately, none of these measurements were made under onditions of arsenazo onentration, ioni strength, magnesium onentration, or alium buffering that prevail in ytoplasm, so the assoiation and dissoiation rates for aliumarsenazo omplexes may be substantially different under physiologial onditions. Nevertheless, on present evidene one would expet arsenazo to respond to a true step in total free alium (ase of nonsaturable buffer) with an absorbane hange that rises to a plateau in perhaps 1 ms. On the other hand, a spike of free alium near hannel mouths, arising from loal buffer saturation, and lasting - only 1 ms, would be like an impulse so far as arsenazo is onerned, and would result in a step inrease in absorbane deaying in - 1 ms. If there is a alium onentration spike superimposed on a step, suh as would our with a saturable buffer, the signal's rising phase in response to the step might anel the falling phase responding to the impulse, resulting in something like the plateau response observed (harlton et al., 1982). In other words, the situation most onsistent with arsenazo's reation kinetis and the observed response is a spike in free alium suh as would our with loal alium buffer saturation in the immediate region of alium hannel mouths. These onsiderations emphasize the primitive nature of both the present simulations and our ability to measure hanges in presynapti alium affeting transmitter release. FOGLSON AND ZUKR Presynapti alium Diffusionfrom Single hannels 115

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