Determining Ca²⁺-sensor binding time and its variability in evoked neurotransmitter release

Abstract

The speed and reliability of neuronal reactions are important factors for proper functioning of the nervous system. To understand how organisms use protein molecules to carry out very fast biological actions, we quantified single-molecule reaction time and its variability in synaptic transmission. From the synaptic delay of crayfish neuromuscular synapses the time for a few Ca²⁺ ions to bind with their sensors in evoked neurotransmitter release was estimated. In standard crayfish saline at room temperature, the average Ca²⁺ binding time was 0.12 ms for the first evoked quanta. At elevated extracellular Ca²⁺ concentrations this binding time reached a limit due to saturation of Ca²⁺ influx. Analysis of the synaptic delay variance at various Ca²⁺ concentrations revealed that the variability of the Ca²⁺-sensor binding time is the major source of the temporal variability of synaptic transmission, and that the Ca²⁺-independent molecular reactions after Ca²⁺ influx were less stochastic. The results provide insights into how organisms maximize reaction speed and reliability.

High-speed biological functions such as neurotransmitter exocytosis depend on fast dynamics of molecular interactions. To understand how these interactions affect the speed and reliability of synaptic transmission, we studied the reaction time of Ca²⁺-dependent and Ca²⁺-independent molecular reactions in the process of synaptic transmission, and to what extent the uncertainty of neurotransmitter release is due to Ca²⁺-sensor binding and due to other protein reactions

Proteins required for evoked neurotransmitter release include at least synaptotagmin and SNARE proteins (SNARE: SNAP receptors) (Schiavo et al. 1992; Blasi et al. 1993; Littleton et al. 1993). Synaptotagmin acts as the major Ca²⁺ sensor in Ca²⁺ triggering of synaptic vesicle fusion (reviewed by Chapman, 2002), and the SNARE complex brings closer the membranes to be fused (Sollner et al. 1993; Lin & Scheller, 1997). In addition, several other proteins such as Unc proteins and complexin were also found to be necessary for neurotransmitter release (Sollner, 2003). Although the chemical features of these proteins have been well studied, due to their complex arrangement at the neurotransmitter release site, the in vivo reaction probabilities of the proteins are poorly understood.

In the first step of synaptic transmission, whether a primed synaptic vesicle can be released depends on the interaction of only a few Ca²⁺ ions with their sensors. The probability of Ca²⁺-sensor reaction is particularly important for neuronal functions because the high Ca²⁺ concentration at the release site lasts for only a few milliseconds. This reaction probability may well set the speed limit of interneuronal communications and even the signal processing capacity of neural networks. Since it has not been possible to measure the early change in the free Ca²⁺ concentration at the release site ([Ca²⁺]_RS) following an action potential, Ca²⁺-sensor binding rate was estimated by simulating synaptic responses induced with prolonged elevation in [Ca²⁺]_RS (Bollmann et al. 2000; Schneggenburger & Neher, 2000; Millar et al. 2005). We realized that although it is impossible to measure the reaction time of a few molecules in most biological and chemical reaction systems, synaptic transmission is an event where the total reaction time of a few molecules in the process can be accurately determined owing to the magnification of the reaction by thousands of neurotransmitter molecules released from one synaptic vesicle. This feature of synaptic transmission allowed us to quantify the reaction time of a few molecules in physiological synaptic transmission.

Traditionally, synaptic transmission is studied from the magnitude of synaptic response. The measured total response amplitude is a nonlinear sum of quantal response amplitude, and the quantal amplitude is proportional to the product of the probabilities of all molecular reactions in synaptic transmission. Estimations of individual reaction probabilities from the total synaptic response amplitude often carry huge errors. Synaptic delay, on the contrary, is the linear sum of the Ca²⁺-dependent and Ca²⁺-independent reaction times; and linear summation allows more accurate estimation of individual reaction times and their variances. By analysing the variability of Ca²⁺-dependent and Ca²⁺-independent reaction times in synaptic transmission, we found that the uncertainty of synaptic transmission is mostly due to the Ca²⁺-dependent molecular reactions. This indicates that most of the Ca²⁺-independent reactions in synaptic transmission such as lipid bilayer fusion and other possible protein interactions including SNARE protein binding occur with a relatively high probability.

Methods

Neuromuscular preparation

Crayfish (Procambarus clarkii) of two inches were purchased from Atchafalaya Biological Supply Co. (Raceland, LA, USA), housed in fresh water at room temperature for up to 3 months and fed a diet of fish food pellets. For each experiment, one of the first pair of walking legs was quickly detached from a crayfish, partially by autotomy. Care was taken to ensure that animal discomfort was kept to a minimum. The neuromuscular preparations of leg muscle extensor were dissected from the meropodite segment in crayfish salines with different Ca²⁺ concentrations as required for electrophysiological recordings. See Hoyle & Wiersma (1958) for anatomy of the walking leg. The frozen animal tissue was destroyed by a qualified biological hazards control company.

The normal crayfish saline contained (mm): 205 NaCl, 5.4 KCl, 13.5 CaCl₂, 2.7 MgCl₂, 10 Hepes (N-[2-hydroxyethyl]piperazine-N′-[2-ethanesulphonic acid]) and 10 glucose, pH 7.39–7.41. When salines with different Ca²⁺ concentrations were used, equinormal NaCl was added or reduced to maintain the salines/ osmolarity, and Mg²⁺ concentration was kept constant. For salines containing Sr²⁺, glucose concentration was adjusted to achieve equal osmolarity for the salines.

Electrophysiology

To induce synaptic activities, the crayfish leg nerve was stimulated through a platinum wire placed close to the proximal end of the nerve. Single square pulses of 0.3 or 0.5 ms were applied at 0.1 Hz. Both the axon action potentials and muscle fibre excitatory postsynaptic potentials (EPSPs) were recorded with intracellular sharp microelectrodes filled with 3 m KCl (resistance: ∼3 MΩ) as described before (Hua & Charlton, 1999).

Single nerve terminal spikes and postsynaptic responses were recorded extracellularly with a method described by Millar et al. (2002). To locate the phasic nerve terminals, the preparation was incubated with 2 μm 4-(4-(diethylamino)styryl)-N-methylpyridinium iodide (4-Di-2-Asp; Molecular Probes, Eugene, OR, USA) in a saline for 4 min in order to visualize nerve terminals under a fluorescence microscope (modified from Olympus BX40). Micropipette electrodes (filled with a saline, resistance: ∼1.0 MΩ) were placed under a 40× water immersion objective (W.D. 3.3 mm) on single phasic terminals at an angle of about 30 deg for extracellular focal recording. The tip of the micropipettes had been ground at 30 deg using a micropipette beveller (BV-10, Sutter Instrument Co., Novato, CA, USA) to obtain an opening of 5–15 μm across, and the opening was heat polished to reduce possible damage to the preparation.

Standard preamplifiers (model 1600 of A-M Systems, Carlsborg, WA, USA) were used for both intracellular and extracellular recordings. To increase the time resolution, the low-pass filter of the amplifiers was set at 50 kHz. Tomahocq computer software (Thomas A. Goldhorpe, University of Toronto, Canada), combined with an analog/digital interface (LM-12, Dagan, Minneapolis, MN, USA), was used for data acquisition. The interface allowed a sampling rate of 100 kHz.

All the experiments were conducted at room temperature (23–25°C). To avoid incomplete change of Ca²⁺ concentration at the nerve terminals due to pipette seal, each preparation was kept only at one Ca²⁺ concentration for extracellular recordings.

Data analysis

The minimum synaptic delay was measured as the time period from the negative peak of the presynaptic spikes to the start time of the excitatory postsynaptic currents (EPSCs). Since individual nerve spikes were often too small to be measured accurately, 10 traces were averaged for the measurement of spike peaks. This average was allowed because the variation of the nerve terminal spike timing was very small compared to the synaptic delay variation. The start time of individual EPSCs and EPSP maximum derivative were determined using routines written in MATLAB (The Mathworks, Natick, MA, USA). The recordings after the stimulus were first smoothed with a median filter (temporal resolution: 0.1 ms). For recordings of EPSCs, a section of the data that was continuously below two standard deviations (∼0.1 mV) of the baseline fluctuation was extracted. The time corresponding to the beginning of the extracted data section was defined as the EPSC starting time.

Curve fitting was conducted using the nonlinear regression function of SigmaStat 3.0 (Systat Software Inc., San Jose, CA, USA). For curve fitting using the maximum derivative of EPSPs, 30 EPSPs were recorded from each muscle fibre, and the mean of the maximum derivative of all the EPSPs at the same condition was used for curve fitting.

Results

Estimating Ca²⁺ concentration at the release site

To estimate the reaction time of Ca²⁺ with its sensor in synaptic transmission, we need to know the concentration of free Ca²⁺ ions at the release site, [Ca²⁺]_RS. Since it is impossible to directly measure the submembrane Ca²⁺ concentration for a few milliseconds immediately after an action potential, we estimated [Ca²⁺]_RS as a function of the extracellular Ca²⁺ concentration ([Ca²⁺]_o):

(1)

where [Ca²⁺]_RS-max is the maximum [Ca²⁺]_RS which is reached when [Ca²⁺]_o is infinite and calcium influx is fully saturated, and K_D-Ca is the [Ca²⁺]_o at which [Ca²⁺]_RS is half-maximal. This equation is based on the equilibrium binding reaction between Ca²⁺ ions and Ca²⁺ channels. The K_D-Ca for Ca²⁺ influx has been estimated for several types of neurons (Akaike et al. 1978; Carbone & Lux, 1987). Assuming that neurotransmitter release is triggered by the binding of n Ca²⁺ ions with their sensors, synaptic responses at low Ca²⁺ influx are proportional to ([Ca²⁺]_RS)ⁿ. The synaptic response, R, at a given [Ca²⁺]_o can be expressed as:

(2)

where β is a constant; β([Ca²⁺]_RS-max)ⁿ is the maximum synaptic response (R_max) which is achieved when [Ca²⁺]_o is infinite. A similar version of eqn (2) was introduced by Dudel (1981). This equation provides a means to estimate the value of K_D-Ca from the synaptic responses at various [Ca²⁺]_o. Later we used K_D-Ca to estimate Ca²⁺-sensor binding time (see the next section of Results).

We first tested the synaptic responses of the extensor muscle fibres at various [Ca²⁺]_o. Each muscle fibre of the extensor is innervated by two excitatory axons: the phasic and tonic axons (both shown in Fig. 1A). At low-frequency stimulation, the neurotransmitter release probability of the phasic terminals is much higher than that of the tonic terminals (Msghina et al. 1998); therefore the EPSPs induced at 0.1 Hz were caused by glutamate released from the phasic terminals (see Hua et al. 2007 for further verifications). To minimize the error in K_D-Ca estimation due to nonlinear summation of the postsynaptic responses and the saturation of Ca²⁺ sensor and neurotransmitter receptors, the maximum derivative of the EPSPs was measured instead of EPSP amplitude to quantify the synaptic responses. At [Ca²⁺]_o ranging from 1.7 mm to 20.3 mm, the EPSP maximum derivative increased with increasing [Ca²⁺]_o (Fig. 1D). Intracellular recordings showed no significant effect of changes in [Ca²⁺]_o on action potentials of the phasic axon. Examples of the phasic axon action potentials and muscle fibre EPSPs at 1.7 mm and 20.3 mm[Ca²⁺]_o were shown in Fig. 1B and C, respectively (5 experiments for [Ca²⁺]_o of 1.7, 6.8 (not shown) and 20.3 mm each). After recording synaptic responses at eight [Ca²⁺]_o values, the value of K_D-Ca was estimated by fitting eqn (2) to the mean EPSP maximum derivative at each [Ca²⁺]_o. Figure 1D shows the mean synaptic responses and the best-fit curve with a K_D-Ca of 5.8 mm (s.e.m. of estimation: 2.4 mm, with 2.7 mm Mg²⁺). This value is within the range of Ca²⁺ dissociation constants for neuronal Ca²⁺ channels (2–10 mm at various [Mg²⁺]: Carbone & Lux, 1987; Akaike et al. 1989; Aibara et al. 1992; Church & Stanley, 1996; Schneggenburger et al. 1999; Foster et al. 2002).

Figure 1. Intracellular recordings of the activities of crayfish motor axons and muscle fibres.

A, fluorescence image of the phasic and tonic nerve terminals on an extensor muscle fibre (the image is shown in colour in the online version of this paper). B, action potentials of the phasic axon and EPSPs at 1.7 and 13.5 mm[Ca²⁺]_o. C, action potentials of the phasic axon and EPSPs at 20.3 mm and 13.5 mm[Ca²⁺]_o. Recordings in B and C were from different preparations. Recordings at 13.5 mm[Ca²⁺]_o were taken from the same axon and muscle fibre after the recordings at 1.7 mm or 20.3 mm[Ca²⁺]_o. D, the synaptic responses at various [Ca²⁺]_o. The mean maximum derivative of EPSPs (max derivative) was normalized to that in normal crayfish saline. For calibration the recordings in normal saline were also taken after each preparation was thoroughly washed in normal crayfish saline. Data were from five preparations for each [Ca²⁺]_o. Error bars show the s.d. The line is the best fit of eqn (2) to the mean maximum derivative.

Equation (2) requires that the Ca²⁺ sensors and postsynaptic neurotransmitter receptors are not saturated at the maximum derivative of EPSPs. These saturations have been observed at some mammalian neuronal synapses (Foster et al. 2002). The synaptic response curve in Fig. 1D shows that the synaptic responses saturated at a higher level of [Ca²⁺]_o. This saturation has three possible components: the saturations of Ca²⁺ influx, Ca²⁺ sensor and postsynaptic neurotransmitter receptors. In addition, saturation of intracellular Ca²⁺ buffer may add an opposite effect on the synaptic response. If the Ca²⁺ sensors or postsynaptic neurotransmitter receptors were saturated when the synapses generated their maximum derivative of EPSPs, the estimated K_D-Ca value would not reflect the half-maximal [Ca²⁺]_RS correctly. We explored the significance of the saturation of Ca²⁺ sensors and neurotransmitter receptors by reducing Ca²⁺ influx. Several divalent cations are known to compete with Ca²⁺ for interaction with Ca²⁺ channels (reviewed by Hagiwara & Byerly, 1981). Among these ions, Sr²⁺ causes little synaptic response in our preparation. Therefore we used Sr²⁺ to reduce Ca²⁺ influx induced by single action potentials. The [Ca²⁺]_RS in the presence of Ca²⁺ and Sr²⁺ can be calculated as:

(3)

where K_D-Sr is the dissociation constant of Sr²⁺ for Ca²⁺ channels. From eqns (2) and (3), the synaptic response in the presence of Sr²⁺, denoted as R_Sr, is expressed as:

(4)

where R_max is the maximum synaptic response. If the saturation of the synaptic response shown in Fig. 1D was solely due to the saturation of Ca²⁺ influx, reducing Ca²⁺ influx by adding Sr²⁺ should not change the value of estimatedK_D-Ca. But if the saturation of the synaptic response was largely due to the saturation of Ca²⁺ sensors or postsynaptic receptors, reducing Ca²⁺ influx by adding Sr²⁺ would reduce the saturation of the sensors or receptors, thus leading to a larger K_D-Ca as estimated with eqn (4). We then tested K_D-Ca in the presence of Sr²⁺.

Replacing Ca²⁺ with 10 mm Sr²⁺ blocked synaptic responses (example shown in Fig. 2A). In four experiments, the EPSP amplitude with 10 mm Sr²⁺ and without Ca²⁺ was 0.39 ± 0.07 mV, which is very close to the amplitude in Ca²⁺-free saline without Sr²⁺ (0.38 ± 0.11 mV, mean ±s.d.). This observation confirmed that Sr²⁺ barely causes any synaptic response following low-frequency nerve stimulation. We next examined the effect of Sr²⁺ on Ca²⁺-dependent synaptic transmission by measuring synaptic responses with and without [Sr²⁺]_o. This test was repeated at different [Ca²⁺]_o, for each of which the ratio of [Sr²⁺]_o/[Ca²⁺]_o was fixed at 3/13.5. The synaptic responses measured as EPSP maximum derivative were inhibited most (by 26%) at 20.3 mm[Ca²⁺]_o. The inhibition of synaptic responses by Sr²⁺ is shown in Fig. 2B and plotted in Fig. 2C. From eqns (2) and (4), this inhibition can be predicted as:

(5)

Equations (4) and (5) share parameters of K_D-Ca, K_D-Sr and n. By multiple-function curve fitting of the two equations to the mean synaptic responses plotted in Fig. 2C and D (with K_D-Ca, K_D-Sr and n shared for the two equations), K_D-Ca was estimated to be 6.0 mm (s.e.m. of estimation: 3.2 mm). This value is very close to the K_D-Ca estimated without Sr²⁺ (5.8 mm), indicating little saturation of Ca²⁺ sensor and/or postsynaptic receptors at the maximum derivative of EPSPs; therefore the K_D-Ca reflects the feature of Ca²⁺ influx.

Figure 2. Synaptic responses in the presence of Sr²⁺.

A, synaptic responses in saline of 10 mm Ca²⁺ and saline of 10 mm Sr²⁺. The recordings were from the same muscle fibre. B, Sr²⁺ inhibits synaptic transmission. The recordings were from the same muscle fibre. In A and B, the axon was stimulated at 0.1 Hz and each trace is an average of 5 recordings. C, inhibition of synaptic response by Sr²⁺ at various [Ca²⁺]_o. The ratio of the EPSP maximum derivative in the presence of Sr²⁺ (R_Sr) to that in salines without Sr²⁺ (R) was plotted to [Ca²⁺]_o. The line is the best fit of eqn (5). D, synaptic responses at various [Ca²⁺]_o and [Sr²⁺]_o. The EPSP maximum derivatives were normalized to that in saline of 13.5 mm Ca²⁺ (without Sr²⁺). The line is the best fit of eqn (4). In C and D, concentrations of [Sr²⁺]_o are 3/13.5 times [Ca²⁺]_o for each saline; data are means of 5–7 experiments for each [Ca²⁺]_o and errors are s.d.

Determining mean time for Ca²⁺-sensor binding

With the estimated K_D-Ca value, we next estimated the Ca²⁺-dependent molecular reaction time in synaptic transmission from the minimum synaptic delays at various [Ca²⁺]_o. Synaptic activities were recorded extracellularly by placing pipette electrodes on single phasic nerve terminals. Following each stimulus, a presynaptic current and an excitatory postsynaptic current (EPSC) were detected as voltage signals (shown in Figs 3A and 4A). When the axon was stimulated at 0.1 Hz, the minimum synaptic delay, which was the delay of the first quantum, was stable for the period of experiments as long as [Ca²⁺]_o was not changed. In normal crayfish saline, the minimum synaptic delay of 15 single terminals from 10 preparations was 0.61 ± 0.13 ms (mean ±s.d.), which was comparable to the synaptic delays for other types of synapses (Katz & Miledi, 1965; Llinás et al. 1981; Sabatini & Regehr, 1996; Waldeck et al. 2000).

Figure 3. The synaptic delay is Ca²⁺ dependent.

A, the nerve terminal spikes and postsynaptic responses at [Ca²⁺]_o of 13.5 mm and 3.4 mm as labelled. The arrows point to the negative peak of the nerve terminal spikes and the onset of the postsynaptic responses. The large noise of the recordings was due to 50 kHz low-pass filter used for recording. B, mean synaptic delays at various [Ca²⁺]_o. Data were collected from 10–15 single nerve terminals for each [Ca²⁺]_o. Errors are s.e.m. The curve is the best fit of eqn (9) to the mean synaptic delays.

Figure 4. The temporal distribution of the first quantum delay.

A, EPSCs induced with 7 stimuli at 0.1 Hz from the same location on a muscle fibre. The stimuli were applied at 0.1 ms and recordings were taken at 1.0 ms. B, distribution of the first quantum starting time. All the 44 EPSCs induced with 50 stimuli (at 0.1 Hz) were included. Examples of EPSCs were shown in A. C, the compound EPSC of 44 EPSCs in B. The arrow points to the nerve terminal spike. The time scale is the same for B and C for comparison.

Synaptic responses were then induced in eight salines differing in [Ca²⁺]_o, and synaptic delays were measured. The presynaptic signal and EPSCs at two Ca²⁺ concentrations are shown in Fig. 3A. The minimum synaptic delay at eight [Ca²⁺]_o values (plotted in Fig. 3B) revealed that the synaptic delay was [Ca²⁺]_o dependent, with longer delay at lower Ca²⁺ concentrations, and that the delay approached a steady level at high [Ca²⁺]_o. These features allowed us to analyse Ca²⁺-dependent and Ca²⁺-independent components of the minimum synaptic delay.

The synaptic delay is the total time for the following events in series: Ca²⁺ channel opening, Ca²⁺ influx and Ca²⁺-sensor binding, the possible SNARE protein interaction and other molecular reactions not well-defined yet, lipid membrane fusion, neurotransmitter release and diffusion, and postsynaptic response generation. The synaptic delay and its variance were mainly attributed to presynaptic events, since neurotransmitter diffusion across the synaptic cleft and postsynaptic events take little time (Katz & Miledi, 1965). Among the presynaptic events, the rates of Ca²⁺ influx and Ca²⁺-sensor binding are [Ca²⁺]_o dependent. Since [Ca²⁺]_o is much higher than [Ca²⁺]_RS, the Ca²⁺-channel interaction takes much less time than Ca²⁺-sensor interaction; therefore the Ca²⁺-dependent portion of the synaptic delay is approximately the Ca²⁺ sensor binding time. For a first-order reaction with respect to Ca²⁺ binding site, the probability of the time length from Ca²⁺ influx to Ca²⁺-sensor binding is exponentially distributed with the mean time length, t, and this time length is the inverse of the product of the binding rate constant and Ca²⁺ concentration (Colquhoun, 1971b). Thus the mean time required for the binding of Ca²⁺ to Ca²⁺ sensor is expressed by the equation:

(6)

where k is the binding rate constant. If the quantal release requires multiple Ca²⁺-sensor binding reactions that do not overlap in time and if no unbinding occurs before the subsequent event in vesicle fusion, the total Ca²⁺-sensor binding time can be calculated as:

(7)

where d denotes the minimum synaptic delay, c is the time for all the Ca²⁺-independent reactions in synaptic transmission, and N is the number of the binding sites required for one vesicle fusion. The unbinding rate of the Ca²⁺ sensor was not included in eqn (7) because the unbinding rate for the putative Ca²⁺ sensor in neurotransmitter release was estimated to be low (Bollmann et al. 2000) relative to our estimated Ca²⁺-sensor binding time, and only the vesicles that experienced no Ca²⁺ unbinding have a high probability to be released first after each nerve terminal spike. The minimum synaptic delay we measured was the delay of these vesicles.

Combining eqns (1) and (7) we have the equation for synaptic delay:

(8)

where γ≡ 1/N+ 1/(N− 1) + 1/(N− 2) +…+ 1/(N−N+ 1).

Let e₁=c+γ/(k[Ca²⁺]_RS-max), and e₂=γK_D-Ca/(k[Ca²⁺]_RS-max), eqn (8) can be simplified into:

(9)

Equation (9) was then fitted to the mean minimum synaptic delay at each [Ca²⁺]_o (Fig. 3B); the best fits of e₁ and e₂ are 0.58 ms (s.e.m. 0.01 ms) and 0.47 ms mm (s.e.m. 0.03 ms mm), respectively. From e₁, e₂ and K_D-Ca= 5.8 mm, the Ca²⁺-independent portion of the minimum synaptic delay, c, was calculated to be 0.50 ms (c =e₁−e₂/K_D-Ca). The estimate of c carries errors of e₁, e₂ and K_D-Ca. From these errors, we estimated the s.e.m. of c to be less than 0.03 ms.

From eqn (9), e₁ is the minimum synaptic delay when Ca²⁺ current is saturated at infinite [Ca²⁺]_o, and is the lower limit of the synaptic delay at a particular temperature. The value of e₂ is a constant for [Ca²⁺]_o-dependent change in synaptic delay. With these two parameters, the minimum synaptic delay can be calculated if one knows [Ca²⁺]_o (d = 0.58 ms + 0.47 ms mM/[Ca²⁺]_o). From eqns (8) and (9), the ratio of e₂ over K_D-Ca (=γ/(k[Ca²⁺]_RS-max)) denotes another constant (0.08 ms, s.e.m.: 0.03 ms), which is the Ca²⁺-sensor binding time at [Ca²⁺]_RS-max (see eqn (6) for the binding time). Thus the synaptic delay can also be calculated as 0.50 ms + 0.08 ms × (1 +K_D-Ca/[Ca²⁺]_o).

Once we know the Ca²⁺-independet reaction time, how long it takes for a single Ca²⁺ ion to bind to a binding site can be calculated if we know how many Ca²⁺ ions are required for one quantum and whether they bind with the sensors simultaneously or in series. Since Dodge & Rahamimoff (1967) proposed that ∼4 Ca²⁺ ions were required for each quantum, experimental findings suggested that 1–5 Ca²⁺ ions are required to release 1 vesicle of neurotransmitter (Llinás et al. 1976; Charlton et al. 1982; Zucker & Stockbridge, 1983; Bollmann et al. 2000; Schneggenburger & Neher, 2000). In normal crayfish saline, if only 1 Ca²⁺ is required for a quantum, the binding time calculated from eqn (8) is 0.12 ms. If, on the other hand, 5 Ca²⁺ ions are required to bind with their identical sensors in sequence, the individual Ca²⁺-sensor binding time is 0.05 ms. If each binding increases the subsequent binding affinity, or the binding times overlap, each binding should take 0.05–0.12 ms.

Using minimum synaptic delay measurements, we also examined the delay distribution of the first quantal event after nerve spikes. At all the Ca²⁺ concentrations tested, single action potentials often caused multiple quanta from single phasic nerve terminals, but most of the quanta were relatively synchronized (see Fig. 4A). The delay distribution of EPSCs induced with 50 stimuli at 0.1 Hz (6 failures; [Ca²⁺]_o: 16.9 mm; examples shown in Fig. 4A) is shown in Fig. 4B, and these EPSCs were summed to construct a compound EPSC (Fig. 4C) illustrating an EPSC of a muscle fibre with 50 phasic terminals. The distribution of the minimum synaptic delay showed that the single terminal EPSCs started during the period from the onset to the negative peak of the compound EPSC. This distribution was common at various [Ca²⁺]_o. This means that the first quanta we analysed occurred up to the negative peak of the compound EPSC. These first quanta should result from the exocytosis of the vesicles that are relatively homogeneous in their priming status and their access to Ca²⁺ source. In our estimation of the Ca²⁺-independent component of synaptic delay, we assumed that [Ca²⁺]_RS is constant during Ca²⁺ channel opening and is very low before channel opening and after channel closing. This assumption is based on findings of simulation studies that in the vicinity of Ca²⁺ channels, [Ca²⁺]_RS reaches a steady level in a few microseconds after channel opening and reduces quickly when the channel closes (Simon & Llinás, 1985; Shahrezaei & Delaney, 2004). Consistent with this notion, the frequency of the first quanta increased promptly after a short period following the nerve spike and most of the first quanta occurred during a period of 400 μs in Fig. 4.

Determining synaptic delay variance

The synaptic delay of the phasic synapses had a considerable variance. This can be seen from Fig. 5A, which shows EPSCs induced with five consecutive stimuli of 0.1 Hz in normal crayfish saline. The variance was even larger at lower [Ca²⁺]_o. Studies on frog neuromuscular synapses suggested that the variance of synaptic delay reflects a presynaptic probabilistic process (Barrett & Stevens, 1972). We wanted to know to what extent the variance was associated with Ca²⁺-sensor binding. Since the minimum synaptic delay was caused by two types of non-overlapping processes: the Ca²⁺-dependent and Ca²⁺-independent reactions, the variance of the minimum synaptic delay should be the sum of the variances of the two processes. From eqn (8) the variance of the minimum synaptic delay (see Colquhoun, 1971a) is:

(10)

in which var(c) and var(γ/(k[Ca²⁺]_RS-max)) × (1 +K_D-Ca/[Ca²⁺]_o)² are the variances of [Ca²⁺]_o-independent and [Ca²⁺]_o-dependent portions of the synaptic delay, respectively. In eqn (10), K_D-Ca is a constant and [Ca²⁺]_o does not change for a particular saline. Because Ca²⁺-sensor binding carries considerable uncertainty, k is the major source of the variance. Whether γ changes from trial to trial is not known. Therefore γ is treated as a possible source of variance. The variance of the minimum synaptic delay (var(d)) was calculated for each [Ca²⁺]_o and plotted in Fig. 5B. By fitting eqn (10) to the means of the variances, the variance of the Ca²⁺-independent portion of the synaptic delay was estimated to be 0.011 ms² (s.e.m. of estimation: 0.002 ms²). Figure 5B shows the curve fitting to the experimental data.

Figure 5. The variance of the synaptic delay.

A, five recordings from a phasic nerve terminal and the postsynaptic fibre in normal saline were superimposed and aligned at the stimulation artifact. The synaptic delay has a large variance (with one failure). The nerve spikes prior to the synaptic responses showed little variance. B, the variance of the synaptic delay is Ca²⁺ dependent. The variance of the minimum synaptic delay from 10–14 muscle fibres at each Ca²⁺ concentration is plotted. The curve is the best fit of eqn (10) to the variance.

The variance of the Ca²⁺-sensor binding time now can be calculated form eqn (10). In normal crayfish saline, the calculated total variance of minimum synaptic delay was 0.021 ms², meaning that 0.010 ms², or 48% of the delay variance, was from Ca²⁺-sensor binding. At 1.7 mm[Ca²⁺]_o, the variance of the Ca²⁺ sensor binding time was 89% of the total delay variance. The large variance of Ca²⁺-sensor binding time is consistent with the probabilistic feature of Ca²⁺ binding with its sensor. If the Ca²⁺-sensor binding is close to a diffusion-controlled reaction as in vitro (Millet et al. 2002), a major source of this variance should be the random walk of Ca²⁺ ions.

Discussion

Although it has been shown that synaptic delay is Ca²⁺-influx dependent for mammalian neuronal synapses (Sabatini & Regehr, 1996; Bollmann et al. 2000; Schneggenburger & Neher, 2000), the synaptic delays of some neuromuscular junctions of both mammals and crustaceans are known to be Ca²⁺ independent (Datyner & Gage, 1980; Parnas et al. 1989; Dudel et al. 1991). In this study, we found that the delay is Ca²⁺ dependent at crayfish phasic neuromuscular junctions. This discrepancy may be attributed to the difference in the synapses and the range of [Ca²⁺]_o tested. The crayfish phasic neuromuscular synapses have a very high release probability which allowed us to measure the synaptic delay over a wide range of [Ca²⁺]_o.

In this study, we estimated Ca²⁺-sensor binding time from the synaptic delay of the first quanta in synaptic transmission. The synaptic vesicles that gave rise to these first quanta should be relatively uniform in their molecular arrangement for fusion. The analysis and conclusions of this study should apply only to these vesicles, which are probably already docked (or very close) to Ca²⁺ channels at the moment of Ca²⁺ channel opening. This docking is made possible by syntaxin, a protein integrated to the nerve terminal membrane. Syntaxin interacts with both Ca²⁺ channels (Bezprozvanny et al. 1995) and the vesicle associated membrane protein (VAMP, synaptobrevin). Due to this docking, the minimum distance between a vesicle and a Ca²⁺ channel has been estimated to be less than 20 nm (Shahrezaei & Delaney, 2004). Equation (8) cannot be used for vesicles with various distances from Ca²⁺ channels: the reaction time of Ca²⁺-sensor binding for these vesicles highly depends on the distances between the vesicles and Ca²⁺ channels, and the synaptic delay for these vesicles would more likely be affected by Ca²⁺ unbinding rate.

The challenge in estimating molecular reaction times and variances is to reduce unknown factors in the process so that the estimates are relatively accurate. We strived to reduce the number of free parameters in curve fitting. In the equation for Ca²⁺-sensor binding time we have only two unknown parameters (see eqn (9)). This equation provided a unique set of estimates with reasonable estimation errors. Other factors related to estimation error are discussed as follows. Synaptic vesicle fusion is believed to be caused by multiple Ca²⁺ bindings, but it is not known whether all the binding sites bind with Ca²⁺ with the same probability. Our method does not assume a fixed binding probability for all the binding sites, and the cooperativity of Ca²⁺-sensor binding does not affect the accuracy of the estimations of the total Ca²⁺-binding time. With our approach, the value of K_D-Ca for [Ca²⁺]_o–[Ca²⁺]_RS conversion was estimated from synaptic responses, and therefore the Ca²⁺ influx that was not related to neurotransmitter release was not included. The major error source for our estimation resides in the estimation of K_D-Ca value. With three free parameters in eqn (2), the estimation of K_D-Ca has a relatively large error. Another possible source of estimation error is the assumption that [Ca²⁺]_RS is constant during Ca²⁺-sensor reaction. If more than one Ca²⁺ channel contributes to the fusion of the same vesicle, [Ca²⁺]_RS cannot be a constant value for Ca²⁺ sensor binding. However, this assumption affects [Ca²⁺]_RS estimations at all Ca²⁺ concentrations to a similar extent, and therefore would not affect the main conclusions of this study. Lastly, the assumption that no unbinding occurs for the first quanta may affect the accuracy of the estimation.

The length and variance of the Ca²⁺-independent reaction time in synaptic transmission reflect the overall reaction probability of several proteins and lipid molecules involved in synaptic transmission. A simulation model by Schneggenburger & Neher (2000) provided a lower limit estimate of the rate constant for the Ca²⁺-independent events in vesicle fusion (6000 s⁻¹, mammalian neurons). With our method, these events take 0.50 ± 0.03 ms in normal saline. While this 0.5 ms accounts for 81% of the total synaptic delay for the first quanta in normal crayfish saline, the variance of these reactions is only 52% of the total delay variance. The low variability of the Ca²⁺-independent reaction time is important for reliability of synaptic transmission. Indeed, neurons cannot afford a large number of stochastic processes in synaptic transmission. The release machinery and the structure of lipid bilayer are believed to be very similar between crustaceans and mammals. For both, the Ca²⁺-independent portion of the synaptic delay is composed of the time for Ca²⁺ channels to open, the time for Ca²⁺ and molecular machinery to cause vesicle fusion with the nerve terminal membrane, the subsequent neurotransmitter release and the brief time for generation of the postsynaptic response. Among these events, Ca²⁺ channel opening is known to be a probabilistic process; the variability of the timing of channel opening should contribute substantially to the synaptic delay variance. This leaves very little variance for other protein reactions including SNARE complex formation. Therefore, the stochastic feature of the vesicular release cannot be explained with SNARE protein interactions. Furthermore, the best way to achieve a low variability of reaction time is to keep the proteins that interact following Ca²⁺-sensor binding very close but separated from each other before Ca²⁺ influx. This idea has been expressed in a model of partial SNARE complex (Hua & Charlton, 1999) combined with a negative switch in vesicle fusion (Hua et al. 2007). The model allows very short synaptic delay and low variability of Ca²⁺-independent protein reactions in synaptic transmission.