We start from the observation that published data on neurotransmitter release for different synapses and experimental setups (Barrett and Stevens, 1972; Kyoung et al., 2011; Diao et al., 2012; Heidelberger et al., 1994; Schneggenburger and Neher, 2000; Lou et al., 2005; Bollmann et al., 2000; Miki et al., 2018; Duncan et al., 2010; Kochubey et al., 2009; Wölfel et al., 2007; Sun et al., 2007; Voets, 2000; Sakaba, 2008; Yang and Gillis, 2004; Beutner et al., 2001; Fukaya et al., 2021) can all be encompassed by a unifying kinetic scheme:

Scheme 1

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In this kinetic scheme, synaptic vesicle fusion proceeds through two parallel reaction pathways. Both pathways contain fast steps of rate constants ${\displaystyle \{k_{fast,i}\}}$. One of the pathways contains an additional, slow, step of rate constant ${\displaystyle k_{slow}\ll \{k_{fast,i}\}}$. The pathways originate in the ‘fast’ and ‘slow’ vesicle pools of sizes $n_{tot1}$ and $n_{tot2}$, respectively. The interpretations of the fast and slow steps as well as the individual states in this unifying kinetic scheme for different experimental setups are summarized in Figure 1 and detailed below.

In the context of in vivo experiments (Heidelberger et al., 1994; Schneggenburger and Neher, 2000; Lou et al., 2005; Bollmann et al., 2000; Miki et al., 2018; Duncan et al., 2010; Kochubey et al., 2009; Wölfel et al., 2007; Sun et al., 2007; Voets, 2000; Sakaba, 2008; Yang and Gillis, 2004; Beutner et al., 2001; Fukaya et al., 2021), Scheme 1 concretizes into the kinetic scheme in Figure 1B and D. The fast pool represents the readily releasable pool (RRP) comprised of vesicles that are docked on the presynaptic terminal (state $D$) and fuse readily upon $C{a}^{2+}$ influx (Kaeser and Regehr, 2017). The slow pool represents the reserve pool (state $R$), which supplies vesicles to the RRP ($R\to D$) with slow rate k₂. Fusion of an RRP vesicle ($\mathrm{\dots }\to F$) requires $N$ independent SNARE assemblies tethering the vesicle at the cell membrane to concurrently undergo a conformational transition. This transition is $C{a}^{2+}$-dependent and involves a single rate-limiting step (Hui et al., 2005) of rate constant $k_1([C{a}^{2+}])$. Note that $N$ is defined broadly as the critical number of independent SNARE assemblies per docked vesicle. Each of the $N$ independent assemblies may consist of a single SNARE or may represent a ‘super-assembly’ of multiple SNAREs that undergo the conformational transition cooperatively (Acuna et al., 2014; Wang et al., 2014; Grushin et al., 2019; Tagliatti et al., 2020; Zhu et al., 2021).

In the context of in vitro experiments (Kyoung et al., 2011; Diao et al., 2012), Scheme 1 becomes the kinetic scheme in Figure 1C and E. All vesicles are initially docked (states D₁ and D₂) but adopt different morphologies (Figure 1C) and, consequently, fuse through different pathways (Gipson et al., 2017). Vesicles in a point contact with the membrane (state D₁) fuse rapidly upon $C{a}^{2+}$-triggered SNARE conformational transition, mimicking RRP vesicles in vivo. Vesicles in an extended contact (state D₂) become trapped in a hemifusion diaphragm intermediate (state $H$), escape from which ($H\to F$) constitutes the slow step k₂.

In all these experiments, the delay due to steps $I_N\to F$ is negligible compared to both fast and slow steps k₁ and k₂. Note that a scheme with $N$ independent and concurrent steps of rates k₁ (Figure 1B and C) is equivalent to a scheme with $N$ sequential steps of rates $N{k}_1,(N-1)k_1,\mathrm{\dots },k_1$ (Figure 1D and E).

Despite the differences in the details of the fusion process in vivo and in vitro described above, the mathematical equivalence of the corresponding kinetic schemes enables their treatment through a unifying theory. We will assume that the calcium influx is triggered by an action potential that arrives at the presynaptic terminal at $t=0$. The microsecond timescales (much faster than neurotransmitter release) of the opening of voltage-gated $C{a}^{2+}$ channels and diffusion of $C{a}^{2+}$ ions across the active zone justify treating the $[C{a}^{2+}]$ rising as instantaneous. Since the typical width of $[C{a}^{2+}]$ profile is $\sim 1-10ms$ (Bean, 2007) while most vesicles fuse within $t\sim 100\mu s$ (Katz and Miledi, 1965), $[C{a}^{2+}]$ can be treated as approximately constant during the fusion process. The theory is thus applicable both for step-like and for spike-like $[C{a}^{2+}]$ profiles, as well as for responses to long sequences of spikes of the duration shorter than the timescale $k_2^{-1}$ of RRP replenishment. With the above assumptions, the theory is developed in detail in Appendix 1. Below, we present analytic expressions derived from the theory for the key outputs of the experiments that probe synaptic transmission at the single-synapse level in vivo and in vitro. These expressions relate experimentally measurable characteristics of synaptic transmission to the molecular parameters of synaptic release machinery, thereby enabling the extraction of these parameters through a fit to experimental data.

An informative characteristic of synaptic transmission is the average release rate. Defined as the average (over an ensemble of repeated stimuli) rate of change in the number of fused vesicles, this quantity is usually reported in experiments on the kinetics of neurotransmitter release (Schneggenburger and Neher, 2000; Bollmann et al., 2000; Kyoung et al., 2011; Diao et al., 2012). The rate equations for the kinetic scheme in Scheme 1 yield the exact solution for the average release rate:

where $p_{1,2}(t)$ are the probability distributions for the fusion time in the fast and slow pathways, $N$ is the necessary number of independent SNARE assemblies, and $n_{tot1}$ and $n_{tot2}$ are the sizes of the fast and slow pools, respectively. We use the standard notation for binomial coefficient $\left(\genfrac{}{}{0pt}{}{N}{m}\right)\equiv \frac{N!}{m!(N-m)!}$.

In practice, the average release rate is obtained from the average cumulative release $⟨n(t)⟩$, which is defined as the average number of vesicles fused by time $t$ and can be measured directly through electrophysiological recording on the postsynaptic neuron (Schneggenburger and Neher, 2000; Lou et al., 2005; Bollmann et al., 2000; Wölfel et al., 2007; Kochubey et al., 2009; Duncan et al., 2010; Miki et al., 2018) or through fluorescence imaging in synthetic single-vesicle systems (Kyoung et al., 2011; Diao et al., 2012). Integrating Equation 1 yields the exact solution for average cumulative release:

where ${\displaystyle F_{1,2}(t)={\int }_0^t{p}_{1,2}(t)dt}$ are cumulative distributions for the fusion time in the fast and slow pathways and are given by Equations 15; 18. In vivo, $F_1(t=T)={(1-e^{-k_{1}T})}^N$ is the fusion probability for an RRP vesicle after an action potential of duration $T$ (Neher, 2015; Miki et al., 2018; Malagon et al., 2016). We also derived the full probability distribution of cumulative release by time $t$ (Appendix 1), which, although at present is challenging to measure experimentally, contains more information than the average values in Equations 1; 2.

Experiments indicate a separation of timescales, $k_2\ll k_1$ (Neher, 2010; Kaeser and Regehr, 2014), which yields useful asymptotic behaviors for AP-evoked neurotransmitter release. At short times, $t\ll 1/k_1,1/k_2$, the release rate in Equation 1 is $\frac{d⟨n(t)⟩}{dt}\sim t^{N-1}$, which can be readily fit to data to extract the number $N$ of independent SNARE assemblies necessary for fusion. At intermediate times, $1/k_1\ll t\ll 1/k_2$, cumulative release in Equation 2 becomes $⟨n(t)⟩\approx n_{tot1}+n_{tot2}{k}_{2}t$, which can be used to determine the RRP size, $n_{tot1}$, by extrapolation (Neher, 2015). At long times, $t\sim 1/k_2\gg 1/k_1$, cumulative release is $⟨n(t)⟩\approx n_{tot1}+n_{tot2}(1-e^{-k_{2}t})$. As expected, the cumulative release on the intermediate and long timescales is independent of the number $N$ of SNARE assemblies and conformational rate k₁ of an assembly as all the fast steps have been completed.

A measure of sensitivity of a synapse to ${\displaystyle [C{a}^{2+}]}$ is the peak release rate (Schneggenburger and Neher, 2000; Lou et al., 2005; Bollmann et al., 2000). The time at which the peak is reached is found from Equation 1 using ${\displaystyle k_2/k_1\ll 1:t_{max}\approx k_1^{-1}\left[\ln N+(n_{tot2}/n_{tot1})\left((N-1)/N^3\right)(k_2/k_1)\right]}$ . The peak release rate is then

Now we must establish an explicit form for the calcium-dependence of the rate constant of SNARE conformational transition $k_1([C{a}^{2+}])$ in Equations 1–3. We utilize the formalism of reaction kinetics (Kramers, 1940) generalized to the presence of a bias field (Dudko et al., 2006). The formalism treats a conformational transition as thermal escape over a free energy barrier along a reaction coordinate. In the present context, the role of the reaction coordinate is fulfilled by the average number $n_{Ca}$ of $C{a}^{2+}$ ions bound to a SNARE assembly at a given $[C{a}^{2+}]$, assuming that this average follows the dynamics of the conformational degree of freedom of the SNARE assembly. The generic shape of the free energy profile with a barrier that separates the two conformational states of a SNARE assembly is captured by a cubic polynomial (Appendix 1—figure 1). The effect of calcium on the free energy profile is incorporated in analogous manner to the $pH$-dependence of Gibbs free energy of a protein, taking into account the contributions both from the electrostatic energy and from entropy (Schaefer et al., 1997; Zhang and Dudko, 2015; Mostafavi et al., 2017). As shown in Appendix 1, the rate constant of the conformational transition of the SNARE assembly is then

Here, k₀ is the rate constant and $\mathrm{\Delta }{G}^{‡}$ is the activation barrier for SNARE conformational transition, and $n_{Ca}^{‡}$ is the number of $C{a}^{2+}$ ions bound to a SNARE assembly at the transition state, with all three parameters corresponding to a reference calcium concentration ${[C{a}^{2+}]}_0$. Equation 4 provides a quantitative explanation for the remarkable temporal precision of neurotransmitter release. Indeed, the argument of $\exp (\mathrm{\dots })$ is the change in the barrier height at a given $[C{a}^{2+}]$ relative to the reference state. The logarithm of calcium concentration, $\text{ln}[C{a}^{2+}]$, is the external force that lowers the barrier (concentrations appear logarithmically because the relevant ‘force’ on the molecule comes from the chemical potential, and this helps us to understand how changes in concentration by many orders of magnitude have sensible, graded effects). Equation 4 shows that the rate k₁ is exponentially sensitive to this external force, and so are the release rate (Equation 1) and its peak (Equation 3) that are both proportional to k₁. This exponentially strong sensitivity of the release rate to the force that drives the release explains, quantitatively, the precisely timed character of synaptic release: synaptic fusion machinery turns on rapidly upon $C{a}^{2+}$ influx during the action potential and terminates rapidly upon $C{a}^{2+}$ depletion (Sudhof, 2011).

Equations 1 and 4 reveal that the number of independent SNARE assemblies ${\displaystyle N=2}$ per vesicle provides the optimal balance between stability and temporal precision of release dynamics (Sinha et al., 2011). Indeed, at $N=1$, the release is hypersensitive to sub-millisecond $[C{a}^{2+}]$ fluctuations caused by stochastic opening of $C{a}^{2+}$ channels (note the high release rate on the sub-millisecond timescale at $N=1$ in Appendix 1—figure 2B). On the other hand, at $N>2$, the peak of release following an action potential is delayed. The optimality of $N=2$ is further supported by the least squares fit of the experimental data (Kochubey et al., 2009) to Equation 1 with different values of ${\displaystyle N}$: ${\displaystyle N=2}$ results in the smallest fitting errors for all calcium concentrations used in the experiment (Appendix 3—table 1). However, the theory also reveals that incorporating additional independent SNARE assemblies beyond $N=2$ may be advantageous for the synapses that require robustness against slower $[C{a}^{2+}]$ fluctuations, beyond the sub-millisecond timescale. Indeed, the presynaptic calcium channels are diverse in their intrinsic properties and their interactions with regulatory proteins, and, as the result, generate $[C{a}^{2+}]$ fluctuations on a wide range of timescales, $0.5ms-20ms$ (Perez-Reyes, 2003; Dolphin and Lee, 2020). The shift of the peak release to longer timescales that accompanies an increase in $N$, as seen in Appendix 1—figure 2B, allows the synapses to ‘avoid’ correspondingly longer-timescale fluctuations in $[C{a}^{2+}]$. This point is illustrated further in Appendix 1—figure 2C: in synapses with the larger values of $N$, the RRP vesicle release (Equation 2) remains low over longer timescales, thereby providing robustness against slower $[C{a}^{2+}]$ fluctuations.

In the presence of cooperative interactions among SNAREs that form super-assemblies, k₁ in Equation 4 represents the effective transition rate of a super-assembly. Appendix 1—figure 2D illustrates how cooperativity between SNAREs results in a steeper increase of the rate k₁ with increasing $[C{a}^{2+}]$, and hence in a faster vesicle release. Specifically, every additional SNARE in the super-assembly is estimated to increase the release rate by a factor of $\sim 100$ (Appendix 1—figure 2D), a result consistent with the previous work (Manca et al., 2019) that utilized a different approach.

Now that we have closed-form expressions for the key characteristics of the neurotransmitter release dynamics in hand, we can establish a universal relation for the sensitivity $r$ of a synapse to the strength $c$ of the trigger. Nondimensionalization of Equations 3 and 4 gives:

where $c\equiv \frac{2n_{Ca}^{‡}{k}_{B}T}{3\mathrm{\Delta }{G}^{‡}}\ln \frac{[C{a}^{2+}]}{{[C{a}^{2+}]}_0}$ and $r\equiv {\left({\frac{a}{{(1-c)}^{1/2}}\frac{d⟨n(t)⟩}{dt}|}_{t_{max}}\right)}^{\frac{k_{B}T}{\mathrm{\Delta }{G}^{‡}}}$ are the dimensionless calcium concentration and peak release rate, and $a\equiv {\left(1+\frac{1}{N-1}\right)}^{N-1}/(n_{tot1}{k}_0)$. If the scaling law in Equation 5 indeed captures universal principles of synaptic transmission, data from different synapses should collapse onto the curve given by Equation 5. This prediction is tested in the section ‘Application of the theory to experimental data’ below.

A postsynaptic response to the action potential events is measured by the peak value of the postsynaptic current (PSC). Using the well-established conductance-based model (Destexhe et al., 1994), the average of the peak PSC can be shown to be proportional to the total number of released neurotransmitters (Katz and Miledi, 1965):

where $T$ is the duration of the action potential ($\sim 1ms$) and $\gamma$ depends only on the properties of the postsynaptic neuron. As our focus is on the AP-evoked neurotransmitter release in synaptic transmission, $\gamma$ can be regarded as a constant and postsynaptic receptor saturation can be neglected, so that $⟨n(T)⟩$ and ${\overline{I}}_{PSC}$ can be used interchangeably. Note that the presynaptic factors affect the postsynaptic response through $⟨n(t)⟩$ as described by Equation 2, and include $C{a}^{2+}$-sensitivity of different $C{a}^{2+}$ sensors in SNAREs (captured through $N$, ${\displaystyle k_0}$, $n_{Ca}^{‡}$ and $\mathrm{\Delta }{G}^{‡}$) and the sizes of both vesicle pools ($n_{tot1}$ and $n_{tot2}$). Equations 2, 4 and 6 relate the presynaptic action potential to the postsynaptic current response and thus complete our framework for synaptic transmission. Detailed derivations of Equations 1–6 are given in Appendix 1.

To validate the developed analytic theory, we first compare its predictions to data generated through numerical simulations of the kinetic scheme in Scheme 1. A simple least squares fit reliably recovers input parameters of the simulations (Appendix 2—figure 1 and Appendix 2—figure 2). Next, we test the robustness of the theory by comparing it to modified simulations, in which deviations from the assumptions underlying (Equations 1–4) are introduced. The modified simulations incorporate (i) the finite-capacity effect of RRP and (ii) heterogeneity of $[C{a}^{2+}]$ among different release sites. For deviations within physiological range, the analytic expressions still reliably recover the input parameters (Appendix 2—figure 3). Details of the simulations are given in Appendix 2.

The availability of analytic expressions for measurable quantities enables direct application of the theory to experimental data. A fit of the peak release rate vs. $[C{a}^{2+}]$ with Equations 3 and 4 was performed for a range of synapses to extract a set of parameters $\{\mathrm{\Delta }{G}^{‡}$, $n_{Ca}^{‡}$, $k_0\}$ for each synapse. These parameters were then used to rescale the peak release rate and calcium concentration to get the dimensionless quantities $r$ and $c$ that appear in Equation 5. We utilized the experimental data from in vivo measurements on (i) the calyx of Held, a large synapse (diameter $\sim 20\mu m$) in the auditory central nervous system, at different developmental stages (Schneggenburger and Neher, 2000; Lou et al., 2005; Bollmann et al., 2000; Sun et al., 2007) (ii) parallel fiber - molecular layer interneuron (PF-MLI), a small synapse ($\sim 1\mu m$) in the cerebellum (Miki et al., 2018) (iii) the photoreceptor synapse (Duncan et al., 2010) (iv) the inner hair cell (Beutner et al., 2001) (v) hippocampal mossy fibre (Fukaya et al., 2021) (vi) the cerebellar basket cell (Sakaba, 2008) (vii) the retina bipolar cell (Heidelberger et al., 1994) (viii) the chromaffin cell (Voets, 2000) and (ix) insulin-secreting cell (Yang and Gillis, 2004), as well as (x) two in vitro measurements (Kyoung et al., 2011; Diao et al., 2012). Figure 2 demonstrates that the data from all these synapses collapse on a single curve given by Equation 5, consistent with the prediction of the theory. Even though these synapses have been known to have a huge variation in their release rates (up to 10 orders of magnitude) due to the different underlying calcium sensors (Cohen and Atlas, 2004; Kerr et al., 2008; Johnson et al., 2010; Kochubey et al., 2016) and different couplings between the SNAREs and their regulatory proteins or calcium channels (Kasai et al., 2012; Vyleta and Jonas, 2014; Stanley, 2016), our theory reveals that all these rates can be brought into a compact, universal form (Equation 5). The universal collapse is an indication that synaptic transmission in different synapses is governed by common physical principles and that these principles are captured by the present theory. Variability across synapses on the molecular level is captured through the distinct sets $\{\mathrm{\Delta }{G}^{‡},n_{Ca}^{‡},k_0\}$ for each synapse. Notably, the generality of Equation 5 spans beyond the context of synaptic transmission: the same scaling has appeared in another, seemingly unrelated, instance of biological membrane fusion – infection of a cell by an enveloped virus (Zhang and Dudko, 2015).

Figure 2

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While a single SNARE can maximally bind $n_{Ca}{}_{max}=4-5C{a}^{2+}$ ions (Radhakrishnan et al., 2009; Brunger et al., 2018), the fit of some of the experimental data on the calyx of Held analyzed in Figure 2 produces the transition state values of $n_{Ca}^{‡}>5$ (Appendix 3—table 2). This result indicates that each SNARE assembly in these synapses is in fact a super-assembly containing two or more cooperative SNAREs. We further note that, since the number of calcium ions bound to a SNARE at the transition state is generally less than the maximum occupancy for the SNARE, $n_{Ca}^{‡}<n_{Ca}{}_{max}$, the synapses with the values of $n_{Ca}^{‡}$ less than but close to five are likely to contain SNARE super-assemblies as well. Interestingly, if we assume that these synapses have the optimal number $N=2$ of the super-assemblies, and note that the typical rate $k_1\approx 4m{s}^{-1}$ at $[C{a}^{2+}]=10\mu M$ would require $\sim 3$ SNAREs per super-assembly (see Appendix 1—figure 2D), then the theory estimates that each docked vesicle contains 2 superassemblies × 3 SNAREs/superassembly $=6$ SNAREs total. This estimate is consistent with the sixfold symmetric structure recently found using cryoelectron tomography analysis in cultured hippocampal neurons (Radhakrishnan et al., 2021).

The utility of the theory as a tool for extracting microscopic parameters of synaptic fusion machinery is further illustrated in Figure 3A–E. A fit of in vivo data for cumulative release at different levels of $[C{a}^{2+}]$ (Wölfel et al., 2007) with Equation 2 extracts the rate of conformational transition of the SNARE assembly, $k_1([C{a}^{2+}])$ (Figure 3A). A fit of the rate with Equation 4 extracts activation barrier and rate at reference concentration ${[C{a}^{2+}]}_0$ (Figure 3B) of the SNARE assembly. Fits of in vitro data (Kyoung et al., 2011; Diao et al., 2012) with Equations 1 and 2 are shown in Figure 3C,D. In Figure 3C, the content mixing occurrence, defined in Kyoung et al., 2011 as the average release rate normalized by the total number of vesicles, $\frac{d⟨n(t)⟩}{dt}/(n_{tot1}+n_{tot2})$, is fitted with Equation 1. In Figure 3D, the rapid burst magnitude, defined in Diao et al., 2012 as the ratio of the numbers of vesicles fused within the first $1s$ and within $50s$ after calcium trigger, $⟨n(t=1s)⟩/⟨n(t=50s)⟩$, is fitted with Equation 2. Figure 3E demonstrates that Equation 4 yields a significantly better agreement with the experimental data on the frog neuromuscular junction (Barrett and Stevens, 1972) than the empirical fourth-power model (Barrett and Stevens, 1972; Jackman and Regehr, 2017) that was originally used to describe these data. In contrast to the fourth-power model, Equation 4 accounts for the saturation effect in the dose-response curve of a SNARE assembly at high-calcium concentrations (see, e.g. the nonlinearity in the rate as a function of calcium concentration on the double logarithmic plots in Figure 2 and Figure 3B).

Figure 3

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The parameter values extracted from the fits in Figure 2 and Figure 3 as well as the least-square fitting algorithm for extracting these parameter values are provided in Appendix 3.

Our theory casts the synaptic fusion scenarios observed in different experimental setups into a unifying kinetic scheme. Each step in this scheme has its mechanistic origin in the context of a given experimental setup. In the context of in vivo experiments, distinct vesicle pool dynamics are taken into account (Alabi and Tsien, 2012; Yang and Calakos, 2013; Kaeser and Regehr, 2017) to quantitatively explain the different timescales observed in the vesicle release dynamics (Kaeser and Regehr, 2014; Neher, 2017; Rozov et al., 2019): vesicles from the readily releasable pool (RRP) fuse readily once the critical number of SNARE complexes undergo conformational transitions upon $C{a}^{2+}$ influx (fast step), while the reserve pool supplies vesicles to the RRP (slow step). In the context of in vitro experiments, different timescales in vesicle release dynamics are due to the observed distinct states of docked vesicles (Diao et al., 2012; Kweon et al., 2017; Gipson et al., 2017): the vesicles that are in a point contact with the membrane fuse readily upon $C{a}^{2+}$-triggered SNARE conformational transition (fast step), while the vesicles that are in an extended contact become trapped in a hemifusion diaphragm state prior to fusing with the membrane (slow step). Although the presence of these distinct docked states in vivo is still under debate (Neher and Brose, 2018; Brunger et al., 2018), the realization that both of the fusion scenarios can in fact be mapped onto the same kinetic scheme allowed us to capture these scenarios through a unifying analytical theory. The fact that each fusion step in the kinetic scheme has a concrete mechanistic interpretation makes the theory directly predictive in both in vitro and in vivo experiments.

The calculated measurable quantities include: (i) cumulative release, which quantifies the number of vesicles fused during a given time interval following the action potential, (ii) temporal profile of the release rate, which measures the rate of change in the number of fused vesicles, (iii) peak release rate, which is a measure of sensitivity of a synapse to the trigger, and (iv) the calcium-dependent rate of SNARE conformational change. A least-squares fit of data with these expressions yields the activation energy barrier and rate constant for SNARE conformational change at any calcium concentration of interest, the critical number of SNARE assemblies necessary for fusion, and the sizes of the readily releasable and reserve vesicle pools.

Since the pioneering efforts to quantitatively describe synaptic transmission (Katz and Miledi, 1965; Dodge and Rahamimoff, 1967), multiple models have been developed, such as the “five-site” model and its variants (Klingauf and Neher, 1997; Schneggenburger and Neher, 2000; Bollmann et al., 2000; Sakaba, 2008; Kochubey et al., 2009; Voets, 2000; Beutner et al., 2001) and the dual $C{a}^{2+}$ sensor models (Sun et al., 2007; Pan and Zucker, 2009). These models provided valuable insights into the action-potential-triggered neurotransmitter release in the particular synapses for which they have been developed. However, the existing models have at least two fundamental limitations. First, the system-specific nature of these models limits their applicability beyond specific systems, so that the description of synapses with different calcium-response properties requires the use of different models. In contrast, the present theory is applicable to a wide variety of synaptic types, despite the differences in their fusion pathways, different calcium sensors that they implement (Wolfes and Dean, 2020) and different couplings between their regulatory proteins (Kasai et al., 2012; Gramlich and Klyachko, 2019). Indeed, recent experiments have suggested that the calcium-response properties of synapses are much more diverse than had been thought previously (Özçete and Moser, 2021; Gómez-Casati and Goutman, 2021; Schröder et al., 2021). Second, the existing models did not produce analytic expressions for the key observables that emerge from the experiments, which limits the predictive value of these models, their utility in extracting information from the experiments, and their ability to reveal the organizational principles of synaptic transmission. In contrast, the present theory yields analytic expressions for the key measurable characteristics of synaptic transmission, which can be used as the tools for extracting the essential molecular parameters of synaptic release machinery through a direct fit to experimental data. Thus, the predictive power of the present theory in describing synaptic transmission in vastly different synapses through a unifying framework is complemented by the utility of the theory as a tool for extracting the molecular parameters that uniquely identify each synapse. The theory links the underlying molecular diversity of synapses to the distinct phenomenological responses observed in experiments, and thus constitutes a constructive step toward a yet more complete description of synaptic transmission (Stevens, 2000).

The theory presented here has several limitations. (i) Our treatment of the vesicle replenishment rate k₂ as a constant is justified by its weak sensitivity to the intracellular calcium concentration compared to that of k₁, as found in recent experiments (Wölfel et al., 2007; Kobbersmed et al., 2020; Kusick et al., 2020). However, in the response to a tetanic stimulus, where the asynchronous component of the release becomes dominant, the calcium-dependence of k₂ may no longer be negligible. Explicitly taking this dependence into account in the theory will allow the extraction of the parameters for post-tetanic potentiation. (ii) The theory describes synaptic transmission at the level of a single synapse. The theory was motivated by the experimental setups that are capable of probing synaptic transmission at the single-synapse level and is applicable both to giant synapses with many active zones in sensory systems (Borst and Soria van Hoeve, 2012) and to small synapses with few active zones in the brain (Harris and Weinberg, 2012; Figure 2). However, a postsynaptic neuron usually receives inputs from many synaptic connections, and the cellular response is an integration of these inputs. The analytic expressions presented above can be directly applied to integrated multiple synaptic inputs in the cases where the molecular features of the presynaptic and postsynaptic sides are similar across the synapses, for example when the synapses originate from the same axon and connect to nearby dendritic regions of a postsynaptic neuron (Branco and Staras, 2009). The theory can be extended to account for the effects of heterogeneous presynaptic inputs by applying the derived expressions to each synapse separately with an individual set of microscopic parameters for each synapse. (iii) We treated the postsynaptic response as a linear function of neurotransmitter release (Equation 6). Such a treatment is sufficient to explain the experimental data on neurotransmitter release (Figure 2 and Figure 3) and the paired-pulse ratio in short-term plasticity (Figure 4) through a single, unifying framework. The theory can be extended to account for the nonlinearity of postsyanptic response by replacing Equation 6 with a relevant nonlinear function. Such an extension will enable the elucidation of the details of active dendritic integration of heterogeneous synaptic inputs.

The rate-limiting step in the initiation of fusion of the synaptic vesicles that are docked on the presynaptic membrane is the conformational transition of the critical number of SNARE assemblies tethering the vesicles to the membrane (Kaeser and Regehr, 2014). We derived the calcium-dependence of the SNARE conformational rate from the classical reaction-rate theory (Kramers, 1940) which we generalized to include an external trigger – calcium influx. The resulting analytic expression reveals that the SNARE conformational rate, and hence both the vesicle release rate and the peak of the release rate, are all exponentially sensitive to the force that drives the release – the logarithm of calcium concentration (the logarithmic scale arises naturally due to the several-orders-of-magnitude changes in $[C{a}^{2+}]$ following an action potential). This result provides a quantitative explanation for the remarkable synchrony of synaptic vesicle fusion: since the rising of calcium concentration after an action potential occurs on a microsecond timescale and is thus essentially instantaneous on the timescale of synaptic release, the exponential sensitivity of the release rate to this nearly instantaneous trigger ensures an ultra-rapid initiation of vesicle fusion upon calcium influx. Likewise, the exponential sensitivity of the release rate to the trigger ensures that the fusion process terminates rapidly upon calcium depletion (Brunger et al., 2018).

Unlike the conventional model that postulates $k_{SNARE}\sim {[C{a}^{2+}]}^4$, our expression in Equation 4 naturally accounts for the saturation effect at intermediate-to-high calcium concentrations (Figure 2 and Figure 3B), which is the typical regime for the AP-evoked neurotransmitter release. In the limit of $\ln \left(\frac{[C{a}^{2+}]}{{[C{a}^{2+}]}_0}\right)\ll 1$, the asymptotic expansion of Equation 4 recovers the power-law $k_1([C{a}^{2+}])\sim {\left(\frac{[C{a}^{2+}]}{{[C{a}^{2+}]}_0}\right)}^{n_{Ca}^{‡}}$, indicating that a power-law description is only valid for the initial rise of the release rate in response to calcium. Moreover, the power exponent $n_{Ca}^{‡}$ is not a universal number (e.g., 4) but rather it depends on the details of the molecular constitutes of the SNARE complexes in a given synapse, such as different calcium sensors from synaptotagmin family (Wolfes and Dean, 2020) and different couplings between the regulatory proteins (Kasai et al., 2012; Stanley, 2016) (Appendix 3—table 2).

The theory further reveals how the kinetics of vesicle fusion are affected by the critical number of SNARE assemblies per vesicle. Given the lack of general consensus (Südhof, 2013; Sinha et al., 2011; van den Bogaart and Jahn, 2011; Brunger et al., 2018), the theory makes no assumptions about the specific number of SNAREs necessary for fusion, and the number itself can serve as a free parameter when sufficient data is available for a robust fit. Interestingly, however, the theory suggests that $N=2$ independent SNARE assemblies per vesicle provide the optimal balance between stability and precision of release dynamics. Indeed, on the one hand, in the presence of a single SNARE, the high values and an exponentially-steep temporal dependence of the release rate make the rate highly sensitive to sub-millisecond calcium fluctuations, and thus a very fine tuning of the calcium concentration would be necessary to prevent instability of the fusion process. On the other hand, the values of $N$ greater than two lead to longer delays in the peak of the release rate following an action potential, thus reducing the temporal precision of vesicle release. Furthermore, a least-squares fit of the release rate from the experiment (Kochubey et al., 2009) with the theory at different values of $N$ reveals that $N=2$ indeed results in the smallest fitting errors for all calcium concentrations. The generality of this result can be determined as more data on the release dynamics for different synapses becomes available. The theory further suggests that incorporating additional SNARE assemblies beyond $N=2$ may be advantageous for the synapses that require robustness against slow $[C{a}^{2+}]$ fluctuations (Mohrmann et al., 2010).

The theory can account for cooperativity between SNAREs and can help identify the presence of SNARE super-assemblies (Radhakrishnan et al., 2021). Mathematically, this is due to the formal definition of the parameter $N$ as the number of independent reaction steps needed for fusion. Each such step may represent a conformational transition of a single SNARE (in the absence of cooperativity) or of a multi-SNARE super-assembly (i.e. an assembly of cooperative SNAREs). The calcium-dependent release rate $k_1([C{a}^{2+}])$ in Equation 4 should be regarded as the transition rate for each independent SNARE unit: if individual SNAREs act independently, k₁ is the transition rate of a single SNARE and $N$ is the number of SNAREs per vesicle; alternatively, if multiple SNAREs undergo conformational change cooperatively, k₁ is the effective transition rate of a super-assembly and $N$ is the number of the super-assemblies per vesicle. The theory allows one to detect the presence of super-assemblies through the values of $n_{Ca}^{‡}$ extracted from the fit: if $n_{Ca}^{‡}$ is larger than the number of $C{a}^{2+}$ binding sites for a single SNARE ($n_{Ca}{}_{max}=5$), it is an indication that a super-assembly of more than one SNARE is present. Applying this criterion produced evidence for the presence of such super-assemblies in several experimental data sets analyzed in this study. More detailed measurements will be needed to get a more direct estimate of the number of SNAREs in each super-assembly. One approach is to perform single-molecule measurements of the kinetics of a single SNARE under different calcium concentrations, fit the resulting rate $k_1([C{a}^{2+}])$ with Equation 4 to extract the value of $n_{Ca}^{‡}$ for the single SNARE, and to compare this value with the value of $n_{Ca}^{‡}$ extracted from a fit with Equation 4 of in vivo data to get an estimate for the number of SNAREs in each super-assembly. The theory suggests that synapses may have more than 2 SNAREs while still having the optimal value of $N=2$: the SNAREs in these synapses may form $N=2$ super-assemblies, each comprising more than one SNARE.

The fact that, in all chemical synapses, the delay time from the action potential triggering to vesicle fusion is determined by the conformational transition of preassembled SNARE complexes, and that the conformational transition itself occurs through a single rate-limited step, suggests possible universality in synaptic transmission across different synapses despite their structural and kinetic diversity. Our theory made this intuition precise through a non-dimensionalized scaling relationship between the peak release rate and calcium concentration (Equation 5), which is predicted to hold for all synapses irrespective of their variability on the molecular level. In statistical physics, the significance of universality is that it indicates that the observed phenomenon (here, synaptic transmission) realized in different systems is governed by common physical principles that transcend the details of particular systems.

The universal relation was tested using published experimental data on a variety of synapses, including in vivo measurements on the calyx of Held studied at different developmental stages, parallel fiber-molecular layer interneuron, the photoreceptor synapse, the inner hair cell, the hippocampal mossy fiber, the cerebellar basket cell, the retina bipolar cell, the chromaffin cell, and the insulin-secreting cell, as well as a reconstituted system. Despite more than an order of magnitude difference in the size of these synapses, ten orders of magnitude variation in the dynamic range of synaptic preparations, and a range of calcium concentrations spanning more than three orders of magnitude, the data for the sensitivity of the synapses to the trigger collapsed onto a universal curve, as predicted by the theory. The collapse serves as an evidence that the established scaling of the normalized peak release $r$ with calcium concentration $c$, $r=\exp \left[1-{(1-c)}^{3/2}\right],$ is indeed universal across different synapses. At the same time, the unique properties of specific synapses are captured by the theory through the distinct sets of parameters of their molecular machinery: the critical number of SNAREs, their kinetic and energetic characteristics, and the sizes of the vesicle pools. The practical value of the theory as a tool for extracting microscopic parameters of synapses was further illustrated by fitting in vivo and in vitro data for cumulative release and for the average release rate at different calcium concentrations. Compared to previous work based on phenomenological formulas (Kochubey et al., 2011), the mechanistic nature of the present theory allows it to be further tested by independently measuring the microscopic parameters of synaptic fusion machinery $\{\mathrm{\Delta }{G}^{‡},n_{Ca}^{‡},k_0\}$ through single-molecule experiments (Gao et al., 2012; Oelkers et al., 2016) and the postsynaptic response through electrophysiological recording experiments.

We applied the theory to establish quantitative connections between the molecular constituents of synapses and synaptic function. Previous quantitative analyses of the experimental data on short-term plasticity were based either on the empirical fourth-power model (Magleby, 1973) or on custom models that are only applicable to specific calcium sensors (Klingauf and Neher, 1997; Pan and Zucker, 2009). The present theory provides analytic expressions for the paired-pulse ratio (Equations 7 and 8) that can be directly compared with the existing experimental data on a variety of synapses (Müller et al., 2007; Jackman et al., 2016; Turecek and Regehr, 2018). As an illustration of the functional implications of the theory, we tested two prevalent hypotheses for the mechanism of synaptic facilitation: syt7-mediated facilitation and buffer saturation. Our results support the facilitation sensor (syt7) as the dominant mechanism for short-term facilitation over most of the interstimulus timescales in the Schaffer collateral, perforant path, corticothalamic, cerebellar granule cell, and retinal ribbon synapses, in agreement with (Jackman et al., 2016; Turecek and Regehr, 2018) but contrary to an earlier study that has suggested other mechanisms for facilitation in the retinal ribbon synapse (Luo et al., 2015). The theory also identified the regimes where the proposed mechanisms fail to account for the observed facilitation. In particular, the syt7-mediated facilitation cannot explain data at $>500ms$ for cerebellar granule cell and corticothalamic cell synapses, plausibly due to a dominant effect of buffer saturation in this regime (Kawaguchi and Sakaba, 2017; Rebola et al., 2019). Likewise, the failure of the syt7 mechanism to explain facilitation in Schaffer collateral and perforant path synapses at $<10ms$ suggests a significant contribution of the calcium current facilitation in this regime (Nanou et al., 2016). We limited the discussion of the short-term plasticity to the two mechanisms of synaptic facilitation and to the data on the paired-pulse ratio as illustrative examples, but other mechanisms can be explored in an analogous manner. For example, spike-broadening effects (Cho et al., 2020) and calcium-dependent vesicle recycling (Marks and McMahon, 1998) can be incorporated into the theory by introducing variations in $T$ and k₂, respectively.

The theory enabled a quantitative description of how short-term facilitation, depression, or coexistence of multiple forms of plasticity in a given synapse emerge from the interplay between the molecular-scale factors such as the timescales of RRP recovery and buffer dissociation as well as the sensitivity of $C{a}^{2+}$-sensors. In contrast to phenomenological models of short-term plasticity (Tsodyks and Markram, 1997; Fuhrmann et al., 2002; Rosenbaum et al., 2012), the mechanistic nature of the present theory reveals the connection between temporal filtering of synaptic transmission and calcium-sensitivity of synaptic fusion machinery, and shows how diverse short-term facilitation/depression modes emerge from the diversity of the molecular constituents.

While one intuitively expects that there must be a tradeoff between the maximum transmission rate and fidelity of a synapse, our theory turns this intuition into a quantitative relation (Equation 10). The trade-off relation shows how transmission failure can be controlled by changing the microscopic properties of the vesicle pool and SNARE complexes. The relation further shows that the probability of synaptic failure decreases exponentially with increasing the synapse size, which makes large synapses significantly more reliable than small synapses in transmitting signals. Furthermore, the established condition for the maximal synaptic efficacy (Equation 12) reveals that, for large synapses, the parameter range of near-optimal performance is broad, indicating that no fine tuning is needed for these synapses to maintain near-optimal transmission (Figure 4I). This finding may also be relevant to small synapses: although a small size of their individual RRPs makes them less reliable in transmitting signals individually, trans-synaptic interactions that couple many nearby small synapses may result in a large ‘effective’ RRP (Bailey et al., 2015) and thus enable small synapses to collectively maintain near-optimal transmission without fine-tuning. Altogether, the results of the theory provide a quantitative basis for the notion that the molecular-level properties of synapses are not merely details but are crucial determinants of the computational and information-processing synaptic functions (Südhof, 2013). Limitations of the theory and possible routes to generalize it to other settings are also discussed.

Other biological processes, including infection by enveloped viruses, fertilization, skeletal muscle formation, carcinogenesis, intracellular trafficking, and secretion, have features that are very similar to those in synaptic transmission, despite the bewildering number and structural diversity of the molecular constituents involved (Harrison, 2017). These processes occur through membrane fusion that (i) requires overcoming high energy barriers, (ii) is controlled by proteins that undergo a conformational transition once exposed to a trigger, (iii) is facilitated by the energy released during this transition, which reduces the fusion timescale by orders of magnitude. The theory presented here can be generalized to encompass these processes while engaging with the diversity of specific systems. The mapping from molecular mechanisms to cellular function, provided by the present theory, is a step toward a more complete framework that would bridge mechanisms with function at the multicellular scale (e.g. neuronal circuits and tissues) and further at the scale of an organism.