Physical determinants of vesicle mobility and supply at a central synapse

We investigated vesicle mobility within cerebellar MFTs using FRAP (Figure 1) at near-physiological temperature (35°C) in acute slices obtained from young-adult (P22-33) VGLUT1^Venus knock-in mice (Herzog et al., 2011). In the cerebellar input layer, VGLUT1 is expressed in MFTs arising from several pre-cerebellar nuclei (Gebre et al., 2012; Hisano et al., 2002). Whole-cell patch-clamp recordings from postsynaptic granule cells (GCs) confirmed EPSC amplitudes, kinetics and short-term plasticity of cerebellar MFT-GC synapses in VGLUT1^Venus mice were similar compared to wild-type mice (Figure 1—figure supplement 1). Moreover, previous work showed the expression and subcellular distribution of VGLUT1^Venus is indistinguishable from that of native VGLUT1 in wild-type mice (Herzog et al., 2011). The advantages of studying vesicle mobility using VGLUT1^Venus knock-in mice rather than lipophilic dyes such as FM1-43 include the ability to use acute slices rather than primary cultures, and the capacity to monitor fluorescence of both mobile and immobile vesicles.

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MFTs expressing VGLUT1^Venus were visualized with conventional fluorescence microscopy (Figure 1A). To assay vesicle mobility, we positioned a diffraction-limited laser focal spot within a MFT and recorded fluorescence using the confocal spot detection method. The intensity of a brief bleaching pulse was set to produce a modest reduction in fluorescence (~35%) to ensure the bleached volume did not extend significantly beyond the core of the illumination point-spread function of our microscope (iPSF; FWHM xy = 0.30 µm, z = 1.32 µm; e⁻² volume = 0.31 µm³). Fluorescence was monitored before and after the bleaching pulse using brief low-intensity probe pulses that produced little cumulative bleaching (Figure 1B, red circles). Since the iPSF was considerably smaller than the MFTs (Figure 1A, blue spot), which are typically 7 × 10 µm, we made multiple FRAP recordings from several locations within the same MFT (Figure 1B). While the individual FRAP measurements were variable, fluorescence almost always exhibited a strong recovery within 10 s (grey lines) indicating unbleached and bleached vesicles were free to move in and out of the confocal volume. The mean fluorescence recovery was determined for each MFT by averaging the individual FRAP measurements (black circles). To determine whether fluorescence recovery varied between MFTs, we calculated the fluorescence at two times, 1 s (f_1s) and 5 s (f_5s), during the recovery. Distributions of f_1s and f_5s across 62 MFTs were unimodal with mean 35 ± 2 and 63 ± 2%, respectively (Figure 1C). We therefore calculated a population mean FRAP curve by performing a weighted mean across all 62 MFTs (Figure 1D). The time of half recovery (t_1/2) of the average FRAP curve was 0.8 s and 77% of the bleached fluorescence recovered within 10 s. These results indicate the majority of vesicles within MFTs are highly mobile.

To test whether our FRAP measurements reflected the movement of vesicles within MFTs, we performed several manipulations known to slow or speed vesicle mobility. Reducing the temperature from 35° to 21°C slowed the fluorescence recovery (Figure 2A; f_1s = 35 ± 2 vs. 25 ± 3%, respectively, p<0.01; n = 62 vs. 36 MFTs). Moreover, the Q₁₀ of the t_1/2 was 1.5, which is closer to that expected for passive diffusion (Q₁₀ ~ 1.3) than for active transport in axons (Q₁₀ ~ 3; Forman et al., 1977). Disruption of the actin cytoskeleton with 10 μM cytochalasin-D and 10 μM latrunculin-B (n = 60 MFTs) sped the fluorescence recovery compared to control conditions (f_1s = 45 ± 2 vs. 35 ± 2%, respectively, p<0.001). In contrast, 5 μM jasplakinolide (n = 44 MFTs), a peptide that stabilizes actin filaments in vitro (Bubb et al., 2000), slowed fluorescence recovery compared to control conditions (f_1s = 27 ± 2 vs. 35 ± 2%, p<0.01). These results suggest actin filaments within MFTs (Hirokawa and Yorifuji, 1989) have a modest effect on vesicle mobility.

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Average FRAP curves for single MFTs for various conditions.

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Average percent fluorescence recovered at 1 s and 5 s after bleaching.

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Parameters file for best-match finite-difference FRAP simulation.

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Application of okadaic acid (OA; n = 42 MFTs), a nonspecific phosphatase inhibitor, dramatically sped the fluorescence recovery (Figure 2B; f_1s = 64 ± 5 vs. 35 ± 2%, p<0.001, unequal-variance t-test), consistent with other studies (see Table 1, legend; Jordan et al., 2005; Shtrahman et al., 2005; Gaffield et al., 2006). Moreover, OA decreased the immobile fraction from 25% to 10–15%, indicating the immobile fraction can be attributed to immobile vesicles rather than large immobile organelles.

To examine the potential role of synapsin in MFTs (Hirokawa and Yorifuji, 1989) we applied the Cdk5 inhibitor roscovitine (n = 45 MFTs), which has been shown to increase vesicle mobility in hippocampal synapses from wild-type but not synapsin-knockout mice (Orenbuch et al., 2012). We found 50 µM roscovitine had no detectable effect on the rate of fluorescence recovery (Figure 2B; f_1s = 38 ± 3 vs. 35 ± 2%, p=0.3) suggesting that, as for ribbon synapses (Mandell et al., 1990), there is little synapsin-based vesicle clustering at the centre of MFTs.

Finally, we repeated our FRAP experiments using paraformaldehyde-fixed slices (n = 17 MFTs), where subcellular components, including vesicles, were immobilized by cross-linking, and found fluorescence recovery was nearly absent (Figure 2B). However, there was a small but consistent increase in fluorescence after 2 s due to slow tissue drift (Figure 2—figure supplement 1A–C). A simple linear correction for the drift allowed a more accurate estimate of the control FRAP curve (Figure 2—figure supplement 1D). Together, these results confirm our FRAP measurements reflect the diffusion of vesicles within MFTs, suggest vesicle mobility is reduced by the presence of a network of actin filaments in the cytoplasm, and indicate synapsin has little impact on vesicle mobility within the interior of MFTs.

Vesicle mobility is often quantified by calculating the ‘effective’ diffusion coefficient, also known as the long-time self-diffusion coefficient (D_long), which reflects the mobility of a diffusant on long time scales (van Blaaderen et al., 1992). We quantified D_long of vesicles in MFTs by fitting our average control FRAP curve to an analytical solution of the diffusion equation (Axelrod et al., 1976). This gave D_long = 0.018 ± 0.005 µm²/s for the raw control FRAP curve and 0.025 ± 0.003 µm²/s after correcting for tissue drift (Figure 2C, red line; Figure 2—figure supplement 1D; Table 1). In contrast to the diffusion model, which matched our data closely, the fit of a model of directed flow (Axelrod et al., 1976) was poor (Figure 2C, yellow line). Moreover, the fit of the weighted sum of the diffusion and flow models converged on diffusion (98%), suggesting diffusion underlies vesicle mobility in MFTs. Because some of the assumptions underlying the analytical diffusion model only approximated our experimental conditions, we also used 3D finite-difference reaction-diffusion simulations that explicitly modelled these conditions (Materials and methods). The FRAP simulation that best matched the drift-corrected data had a D_long similar to that obtained with the analytical solution (p=0.84, F-test; Figure 2C, blue line). A comparison of our best estimate for D_long in MFTs (0.025 µm²/s) with that estimated at other synapses (Figure 2D) indicates vesicle mobility is higher in MFTs than at other conventional central synapses and the neuromuscular junction (NMJ), but comparable to that measured at ribbon synapses.

Densely packed organelles within presynaptic terminals are expected to lead to crowding effects (Wilhelm et al., 2014) that could affect vesicle mobility (Gaffield et al., 2006). To quantify vesicle and mitochondrial densities within the central region of MFTs, where our FRAP measurements were predominantly made, we performed serial-section EM (Figure 3A). Quantitative analysis of 3D volumes (0.2–0.4 µm³, n = 3) revealed a vesicle density of 3930 ± 262 per µm³ (or 118 ± 8 per µm² in 2D) in regions not occupied by mitochondria, consistent with previous measurements from MFTs (Palay and Chan-Palay, 1974). To convert this density into a volume fraction, we estimated the mean volume occupied by a vesicle by performing high-resolution 3D electron tomography on MFTs. Vesicle diameters exhibited an approximately normal distribution with a mean of 41.1 ± 0.2 nm (256 vesicles around 3 AZs; Figure 3—figure supplement 1), again consistent with previous measurements (Palay and Chan-Palay, 1974). Taking into account vesicle membrane proteins (Takamori et al., 2006), and the cubic relationship between vesicle radius and volume, indicates the mean volume occupied by a vesicle is equivalent to a sphere with 44 nm diameter. Using this diameter, we computed a vesicle volume fraction of 0.17 ± 0.01 of the non-mitochondrial volume (Figure 3B). The average volume fraction occupied by mitochondria was 0.28 ± 0.04. Hence, this analysis shows the central region of the MFT is a highly crowded environment with 40% of the volume occupied by vesicles and mitochondria.

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Density of vesicles and mitochondria.

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To investigate the impact of organelle crowding on vesicle mobility in MFTs, we modelled vesicle diffusion in 3D using a hard-sphere Monte Carlo algorithm (Cichocki and Hinsen, 1990) that explicitly simulated the movement of individual vesicles and their collisions (Figure 3C). In these simulations, vesicle movement on each time step was determined by the short-time vesicle diffusion coefficient (D_short) which defines the rate of diffusion before collisions occur. Vesicles were not allowed to overlap with themselves or mitochondria. Since simulations mimicked conditions of live tissue, we corrected our measured vesicle diameter for the 11% tissue shrinkage in fixed tissue (Korogod et al., 2015) giving an equivalent diameter of 49 nm. The spatial extent of the bleaching reaction was set by the iPSF, and the average fluorescence during the probe pulses was computed by spatially weighting the fluorescence according to the measured confocal PSF (cPSF) of our microscope (Figure 3C, right; Figure 3—figure supplement 2). Approximately 300 vesicles were located within the cPSF (Figure 3C, left). When the timing and intensity of the bleaching reaction were set to match the bleaching and probe pulses used in our FRAP experiments, FRAP curves from the Monte Carlo simulations (Figure 3D) exhibited remarkably similar behaviour to our experimental FRAP curves (Figure 1B).

Only two parameters were varied in these experimentally constrained simulations: D_short and the fraction of immobile vesicles. To find the values of these two parameters that produced a FRAP curve that best matched our experimental FRAP curve, we used a parameter search and chi-square (${\displaystyle {\chi }^2}$) criterion (Figure 4A,B). The log of the ${\displaystyle {\chi }^2}$ calculated from the simulated and experimental FRAP curves had a minimum at D_short = 0.060 µm²/s and a 25% immobile fraction (68.3% confidence intervals 0.055–0.070 µm²/s and 24–25%, respectively). To better understand how vesicle collisions reduce vesicle mobility, we examined how vesicle diffusion, computed from the mean square displacement (MSD) of the mobile vesicles, changed as a function of time (D(t) = MSD/6t; Figure 4C). At short times, before any vesicle collisions, D(t) ≈ D_short (0.060 µm²/s). At longer times, however, as vesicles collided with themselves and mitochondria, D(t) fell to a steady-state value of 0.025 µm²/s. Hence, over the 10 s period of our FRAP experiment, D(t) reached a similar D_long estimated from our analytical and finite-difference approaches (Figure 2C). Indeed, comparison of the FRAP curve computed from our best-match Monte Carlo simulation with that computed from our best-match finite-difference simulation showed a close match (Figure 4D). These results indicate steric interactions introduce a pronounced time dependence to vesicle diffusion and over long timescales vesicle collisions reduce vesicle mobility by a factor of 2.4 (i.e. D_long/D_short = 0.42).

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Parameters file for best-match Monte-Carlo FRAP simulation.

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To understand how steric interactions affect vesicle mobility under different conditions, we explored the effect of different vesicle volume fractions, immobile vesicle fractions and mitochondria volume fractions on vesicle mobility. In the absence of immobile vesicles and mitochondria, increasing the vesicle volume fraction from 0 to 0.5 reduced vesicle mobility (measured as D_long/D_short) in a near-linear manner (Figure 5A, red circles) as previously reported (black squares; Cichocki and Hinsen, 1990). Adding immobile vesicles further reduced the diffusion coefficient of the mobile vesicles (Gaffield et al., 2006) and as the fraction of immobile vesicles increased, this effect became increasingly nonlinear. Finally, adding mitochondria reduced vesicle mobility to a similar extent across all vesicle densities (blue open circles). For conditions within MFTs, the reduction in vesicle mobility due to collisions between mobile vesicles was 30%, while that due to collisions with immobile vesicles and mitochondria was 13% and 32%, respectively. These results show vesicle collisions are a major determinant of vesicle mobility at a central synaptic terminal.

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Hydrodynamic interactions have long been known to be a major determinant of the self-diffusion of nanoscale beads in crowded colloidal suspensions (van Blaaderen et al., 1992; Segrè et al., 1995). Unlike steric interactions, hydrodynamic interactions occur on a very fast timescale, so their effect on vesicle diffusion can essentially be considered instantaneous. To estimate the effect of hydrodynamic interactions on vesicle diffusion in MFTs, we used an analytical approach that is accurate across a wide range of volume fractions (Tokuyama and Oppenheim, 1994). We quantified the effect of hydrodynamic interactions by calculating the ratio of D_short to the ‘free’ unhindered diffusion coefficient of a single vesicle in cytoplasm (D_cyto). As shown in Figure 5B, D_short/D_cyto shows an approximately linear relationship with vesicle volume fraction (red line). However, introduction of randomly dispersed immobile vesicles at 25% produced a nonlinear dependence between D_short/D_cyto and the total vesicle volume fraction (blue line; Freed and Muthukumar, 1978; Michailidou et al., 2009). For the vesicle volume fraction found at the centre of MFTs, D_short/D_cyto = 0.47 (dashed blue line) suggesting hydrodynamic interactions reduce vesicle mobility by two-fold within MFTs.

As hydrodynamic interactions act on a microsecond timescale, the vesicle diffusion coefficient D(t) is expected to decrease from D_cyto to D_short almost instantaneously (Figure 5C). On the other hand, vesicle collisions take longer to occur, so their effect on D(t) is expected to occur on a millisecond to second timescale. At these longer timescales, both hydrodynamic and steric interactions are therefore present. The different timescales of hydrodynamic and steric interactions suggest they can be treated as independent processes, allowing their combined effect to be calculated via a simple multiplication: D_long/D_cyto = D_short/D_cyto × D_long/D_short (Figure 5D; red circles and black line; Medina-Noyola, 1988; van Blaaderen et al., 1992). For the conditions at the centre of MFTs, the combined effect of hydrodynamic interactions and vesicle collisions resulted in an 80% reduction in vesicle mobility (D_long/D_cyto = 0.19; dashed blue line). This suggests the diffusion coefficient of dilute vesicles in cytoplasm, in the absence of crowding effects, is 0.127 µm²/s, which is only a factor of 3.5 lower than that of dilute 50 nm beads in the cytoplasm of 3T3 cells at 37°C (0.45 µm²/s; Luby-Phelps et al., 1987). A likely explanation of the lower value of D_cyto for vesicles than beads is that vesicle protein interactions with the cytomatrix slow vesicle mobility.

To understand how hydrodynamic and steric interactions could influence vesicle supply at AZs, we quantified the morphological properties of 14 AZs using 3D reconstructions from high-resolution serial-section EM (Figure 6A,B). The AZ area, closely opposed to the postsynaptic density, varied across synaptic contacts (0.009–0.039 µm²) with a mean of 0.017 ± 0.002 µm² (n = 14). This is smaller than most other central synapses which can have 10-fold larger areas (Xu-Friedman and Regehr, 2004; Harris and Weinberg, 2012; Holderith et al., 2012). When expressed as the diameter of a circular disk, our AZ measurements correspond to a range of 108–222 nm, with mean 145 ± 8 nm.

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AZ area and vesicle densities from 3D AZ reconstructions.

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To determine the spatial extent of vesicles near the AZ, we designated those regions in the 3D reconstructions that contained vesicles as freely diffusible space, and all other regions as non-diffusible space (Materials and methods). We refer to the region of diffusible space extending from each AZ face as the vesicle ‘cloud’. Vesicle density within the vesicle cloud was calculated either by counting the number of vesicle centre points falling within its boundaries, or by computing the volume fraction occupied by the vesicles (Figure 6C, bottom; red circles and blue line respectively). The vesicle density reached 5652 ± 377 per µm³ at 30 nm from the AZ. This corresponds to a volume fraction of 0.25, assuming a vesicle diameter of 44 nm in fixed tissue, which is 50% higher than the volume fraction at the center of MFTs (0.17). Vesicle density declined with distance from the AZ, converging to the density estimated at the centre of the MFT for distances >100 nm. These results suggest an accumulation of vesicles extends ~2 vesicle diameters from the AZ. The total number of vesicles increased monotonically with distance from the AZ, from 2 ± 1 within 22 nm, to 48 ± 6 within 100 nm and 252 ± 34 within 300 nm (Figure 6C, top). These results show that, while there are few docked vesicles at the AZ, hundreds of vesicles reside nearby.

The similarity in vesicle density within the cloud and in the MFT interior (Figure 6C, bottom; red circles and dashed horizontal line, respectively), plus the irregular shape of the vesicle clouds, suggests diffusional barriers imposed by intracellular organelles and/or the plasma membrane are present in the AZ region. To examine this, we quantified the spatial dependence of the vesicle density assuming all of the space surrounding the AZ is diffusible (Figure 6C, bottom; black circles). This approach gave a similar estimate of the vesicle density close to the AZ, but beyond 150 nm from the AZ the density rapidly declined, reaching 15% of the peak value by 400 nm from the AZ. This suggests that, beyond 150 nm, intracellular organelles, cytoskeletal barriers and/or curvature of the plasma membrane create a significant amount of non-diffusible space (Figure 3A). Indeed, quantification of the mean distance from the AZ to the nearest mitochondrial membrane revealed mitochondria are located within 270 ± 20 nm (n = 29) of the AZ. These results suggest that the diffusible space available to vesicles is restricted near the AZ of MFTs and that vesicle density only increases above that found in the MFT interior within 100 nm of the AZ release face.

To examine the effectiveness of diffusion in supplying vesicles to AZs, we modelled vesicle diffusion in the vicinity of the 14 reconstructed AZs using our measured 3D geometries (Figure 7A). We assumed D_cyto was identical to that measured in the centre of the MFT and regions outside the vesicle cloud consisted of non-diffusible space since these regions contained no vesicles in the EM reconstructions. The most distal regions of the vesicle cloud were coupled to a large reserve of vesicles to mimic replenishment from the centre of the MFT. Hydrodynamic interactions were simulated using analytical expressions that accounted for local variations in vesicle density and the effect of the plasma membrane, which reduced vesicle mobility as vesicles approached the AZ (Figure 7—figure supplement 1). In our initial simulations, vesicles that collided with the AZ were instantaneously ‘released’ without any delay. Under these conditions the vesicle release rate equals the rate at which vesicles are supplied to the AZ via diffusion, which we here simply call the vesicle supply rate.

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Vesicle supply rates and pool sizes computed from Monte Carlo AZ simulations.

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Parameters file for one Monte-Carlo AZ simulation of EM series #3.

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At the earliest simulation times (0–2 ms) the vesicle supply rate varied between 25–4000 s⁻¹, with a mean of ~1000 s⁻¹ (Figure 7B–D). The average vesicle supply rate declined to ~200 s⁻¹ by 10 ms when ~5 vesicles had been released (Figure 7F). These values are well above the experimental vesicle reloading rates of 40–80 s⁻¹ per functional release site estimated at MFT-GC synapses during brief bursts of high-frequency stimulation (Saviane and Silver, 2006; Hallermann et al., 2010) and obtained during continuous depolarization of the presynaptic terminal (Ritzau-Jost et al., 2014). Even when vesicles were accumulated close to the AZ by simulating the effects of vesicle-vesicle binding due to fine protein connectors (Figure 7—figure supplement 2; Siksou et al., 2007; Fernández-Busnadiego et al., 2013), which reduced vesicle mobility by five-fold close to the AZ (Figure 7—figure supplement 1), the maximal vesicle supply rates remained higher than the experimental vesicle reloading rates (Figure 7D). These results suggest that at early times vesicle diffusion does not limit vesicle reloading at the AZ.

At intermediate times (~100 ms), when ~15 vesicles had been released, vesicle supply rates to the AZ approached the experimental vesicle reloading rates (Figure 7C,E,F), raising the possibility vesicle reloading could be limited by diffusion during sustained release. To investigate this, we focussed on later times (50–100 s) when the cumulative number of vesicles supplied to the AZ increased linearly with time, indicating the supply rate was constant (Figure 7G, blue line). The slope of the linear portion of the cumulative supply functions gave a mean vesicle supply rate of 7 ± 1 s⁻¹ (Figure 7G,H) which is remarkably similar to the vesicle release rates per release site measured at MFTs (8 s⁻¹; Saviane and Silver, 2006) and vestibular nerve synapses (7 s⁻¹; McElvain et al., 2015) during long trains of stimuli. Moreover, the back-extrapolation method for estimating the size of the pool of vesicles that can be depleted during the train (i.e. RRP+RP) gave an average of 237 ± 40 vesicles (Figure 7I), similar to values obtained for EPSC trains at MFTs (300; Saviane and Silver, 2006) and vestibular nerve synapses (200; McElvain et al., 2015). The close agreement between our experimentally constrained models of vesicle diffusion at AZs and experimental measurements suggests vesicular release is diffusion-limited during sustained high-frequency signalling once the RRP and RP are depleted. Examination of vesicle density close to the AZ during sustained release revealed that it falls dramatically, reaching a steady state after ~20 s (Figure 7—figure supplement 1, control conditions). However, vesicle mobility increased substantially during vesicle depletion due to reduced hydrodynamic and steric interactions, resulting in a shallow concentration gradient and a constant vesicle supply rate at late times.

To better understand the role of physical factors in vesicle supply to MFT AZs, we first removed hydrodynamic interactions from our simulations. This increased the vesicle supply rate at early, intermediate and late times (Figure 7C–E,H), indicating fast hydrodynamic interactions between vesicles play a key role in slowing vesicle supply to the AZ. Interestingly, removing steric interactions (by shrinking vesicles to points) had little additional effect to removing hydrodynamic interactions at early and intermediate times (Figure 7C–E). However, at late times, removing vesicle collisions increased vesicle supply to the AZ by a factor of 2 (Figure 7C,G,H) consistent with their effect on vesicle mobility over long time scales (Figure 5D). These results show the physical interactions arising from vesicle crowding limit the maximal vesicle supply rate to the AZ.

Next, we examined how the restricted diffusible space in the vicinity of the AZ affected vesicle supply. To do this we ‘opened up’ the 14 AZ geometries by removing the non-diffusible space surrounding each AZ and filled it with vesicles at the density measured at the centre of MFTs, thereby combining the RP and reserve pools (Figure 7—figure supplement 2). For open geometries, the average vesicle supply rate at early times was the same as that for the control geometries (Figure 7C). However, after ~1 s, the average vesicle supply rate for the open geometries levelled off to 55 s⁻¹, resulting in a seven-fold higher supply rate than the control (Figure 7G,H). Back extrapolation produced a pool size estimate of only 16 ± 3 vesicles (Figure 7I). These results suggest the presence of diffusion boundaries arising from membrane invaginations, intracellular organelles and possibly the actin cytomatrix (Sankaranarayanan et al., 2003; Guillet et al., 2016) limit diffusion-mediated vesicle supply to the AZ. Moreover, by determining the limiting vesicle supply rate during sustained release and thus the size of the pool of vesicles that can be depleted before release and supply rates equalize (excluding those replenished during that time), the geometry of the diffusible space close to the AZ determines the size of the functionally defined RP.

To examine how vesicle docking and priming (molecular and positional) and release probability affect our conclusions, we simulated 100 Hz MFT spike trains (Figure 8) assuming 1 or 2 release sites per AZ and a release probability of 0.5 (Saviane and Silver, 2006; Hallermann et al., 2010). To examine the limiting case, we assumed the total time for docking and priming (τ_d+p) was equal to the measured RP→RRP vesicle reloading time at these AZs (16.7 ms; Figure 8—figure supplement 1). As expected, the initial release rate was lower than that for conditions of instantaneous release and, after the RRP was released, the vesicle release rate became limited by τ_d+p and the release probability (Figure 8A–C). However, at later times during the train (>1 s), the release rate converged with that for instantaneous release, indicating that it had become limited by the diffusion-mediated vesicle supply (Figure 8A). Back extrapolation analysis of cumulative release at late times during the 100 Hz train gave estimates for the limiting vesicle release rate and RP size that were similar to that obtained with instantaneous release, irrespective of whether 1 or 2 release sites were present (Figure 8D,F,G). These more realistic simulations of release during 100 Hz trains therefore support our finding that vesicle release becomes diffusion-limited during sustained high-frequency signalling. Furthermore, back extrapolation analysis of the open-geometry condition revealed that the release rate rapidly becomes limited to ~30 s⁻¹, resulting in a RP pool of only 2 ± 1 vesicles (i.e. close to the RRP; Figure 8E–G). While the exact values of the functional pool estimated from back extrapolation of the open-geometry condition may be prone to some error, because of its small size and the fact that the replenishment rate is higher than the release rate (Neher 2015), the more than 100-fold reduction in the RP estimate compared to that for measured vesicle clouds shows that the diffusible space around the AZ is a major determinant of the RP pool size.

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Our results show that crowding within presynaptic terminals gives rise to two types of physical interaction that slow vesicle mobility. Fast hydrodynamic interactions arising from fluid displacement influence vesicle mobility on both short (microsecond) and long (millisecond to second) timescales. In contrast, effects of vesicle collisions (Gaffield et al., 2006) are only felt on long time scales. The combined effects of these interactions result in a vesicle diffusion coefficient that evolves with a characteristic time course that depends on the vesicle density (Figure 5C). Given the strength of hydrodynamic interactions in crowded environments, it is surprising their effects have only recently been investigated in biological systems in the context of protein diffusion (Doster and Longeville, 2007; Ando and Skolnick, 2010). Our results extend these studies by showing hydrodynamic interactions are likely to be a major determinant of vesicle diffusion in synaptic terminals. Indeed, taking account of hydrodynamic and steric interactions arising from vesicle crowding explains much of the discrepancy between D_long of vesicles and 50 nm beads in cytoplasm (Luby-Phelps et al., 1987). Hence, our results identify the main physical determinants of vesicle mobility in a central axon terminal and highlight the need to consider hydrodynamic interactions within crowded intracellular environments.

By quantifying how hydrodynamic and steric interactions vary with vesicle volume fraction and the fraction of immobile vesicles, it is possible to predict the vesicle mobility near the AZ and across different types of synapses (Table 2). At MFTs our quantitative model of vesicle mobility predicts that vesicle diffusion slows down from 0.025 μm²/s within the MFT interior to 0.012 μm²/s in the vicinity of the AZ. Thus, physical interactions slow vesicle mobility even in the absence of binding to tethers and connectors (which could slow diffusion further to 0.002 μm²/s). For ribbon-type bipolar cells, which have a vesicle density comparable to MFTs, our model of vesicle mobility predicts D_long = 0.014 μm²/s, which matches well to the value measured in goldfish (0.015 μm²/s; Holt et al., 2004), but cannot account for the exceptionally high mobility found in lizard (Rea et al., 2004). Nevertheless, our framework does predict the low vesicle mobility at the NMJ (predicted 0.006 μm²/s vs. measured 0.005 μm²/s; Gaffield and Betz, 2007), where vesicle density is more than double that within the MFT (0.33 vs. 0.17 volume fraction). Measurements of vesicle mobility in small central synapses are more numerous but highly variable. Our modelling predicts that D_long is low within the middle of the vesicle cluster in small boutons (0.002 μm²/s). This is similar to some FRAP-based estimates (Shtrahman et al., 2005) and recent single-vesicle tracking measurements (0.003 μm²/s; Lee et al. 2012). Moreover, our predicted D_long falls within the wide range of estimates of vesicle mobility using fluorescence correlation spectroscopy (Figure 2D, pink symbols; Yeung et al., 2007; Jordan et al., 2005), but these estimates depend on the diffusion/binding model applied (Jordan et al., 2005; Shtrahman et al., 2005; Yeung et al., 2007). Synapsin-based immobilization of vesicles appears prevalent in small boutons (Orenbuch et al., 2012), consistent with reports that most vesicles are immobile (Shtrahman et al., 2005; Lee et al., 2012). Moreover, there is a rapid vesicle exchange with the axon (Staras et al., 2010; Herzog et al., 2011) where mobility is high (Lee et al., 2012). Our simulations suggest that the low mobility of vesicles in small central synapses is largely due to strong steric and hydrodynamic interactions arising from the high vesicle density. Moreover, although the presence of tethers/connectors will restrict vesicle mobility further, their effects will only be strong if they have long lifetimes (i.e. slow unbinding). While the properties of vesicle mobility within small synapses is still uncertain, our results show that to understand the variations in vesicle mobility across synapses, it is necessary to account for the physical interactions arising from organelle crowding.

Our 3D EM reconstructions revealed the AZs of MFTs are small and surrounded by a cloud of vesicles that is highly variable in shape and size. Our analysis shows that on average there are only 2 vesicles at the AZ face, suggesting few vesicles are docked and release ready at each AZ. This potentially explains why multi-vesicular release (Wadiche and Jahr, 2001) is not a dominant form of release at MFTs (Sargent et al., 2005). Away from the AZ face, the vesicle density falls to a level comparable to the density measured at the centre of the MFT. The accumulation of vesicles at the AZ face has a similar spatial extent as the AZ cytomatix protein Bassoon (~100 nm), which speeds vesicle reloading at MFTs (Hallermann et al., 2010) possibly via vesicle clustering (Mukherjee et al., 2010). Our experimentally constrained simulations suggest diffusion can supply vesicles to the AZ at the onset of high-frequency signalling (<100 ms) faster than the measured vesicle reloading rates of 40–80 s⁻¹ (Figure 7C), even in the presence of protein filament connectors and tethers (Siksou et al., 2007; Fernández-Busnadiego et al., 2013). This suggests factors other than diffusion, such as docking and priming, limit vesicle reloading at the onset of sustained release. Ca²⁺-dependent mechanisms that speed docking and priming (Neher and Sakaba, 2008) would therefore be most effective at accelerating vesicle reloading at the early stages of high-frequency signalling, but increasing vesicle mobility or reducing the immobile fraction would be effective later, once vesicle depletion occurs.

Previous studies of vesicular release at cerebellar MFTs (Saviane and Silver, 2006; Hallermann et al., 2010; Sargent et al., 2005) and vestibular nerve synapses (McElvain et al., 2015) suggest that each AZ has an RRP of 1–2 vesicles and a RP of ~200–300 vesicles (Figure 8—figure supplement 1). Remarkably, 3D AZ reconstructions predict the size of the RRP and our simulations predict the RP from the spatial positions of the vesicles, the diffusible space surrounding the AZ and our model of vesicle mobility. Furthermore, once the RP is depleted, our simulations predict a vesicle supply rate of 7 s⁻¹, which matches the release rates of 7–8 s⁻¹ recorded at these synapses (Saviane and Silver, 2006; McElvain et al., 2015). Because glutamate refilling is 100-fold slower than these rates (Hori and Takahashi, 2012), rapid local endocytosis (Watanabe et al., 2013) is unlikely to be responsible for these limiting rates. Our results therefore suggest that, during sustained high-frequency release, vesicle supply from the reserve pool to the AZ is limited by vesicle diffusion from the interior of the MFT. Thus, diffusion rather than docking and priming potentially limits the rate at which continuous sensory variables, such as head velocity (Arenz et al., 2008) and joint angle (van Kan et al., 1993), are transmitted. Moreover, the fact that increasing diffusional access to the AZ effectively eliminates the functionally defined RP indicates the shape and extent of the vesicle cloud associated with an AZ is a major determinant of the RP, because it sets the rate of supply from the reserve pool. This structural property could therefore contribute to the heterogeneous functional properties of MFTs (Sargent et al., 2005) originating from different precerebellar nuclei (Chabrol et al., 2015). Thus, our results provide the structural basis for a functionally defined vesicle pool and show diffusion ultimately limits vesicle supply during sustained high-frequency signalling at a central synapse.

The generation and general characterization of the VGLUT1^Venus (Slc17A7^ct(venus)Nbr) knock-in mouse line was published previously (Herzog et al., 2011); all experiments using this line were performed with littermates derived from crossing homozygous VGLUT1^v/v mice (F2 SV129/ola x C57BL/6 genetic background). All animal experiments were conducted in strict accordance with the United Kingdom Home Office Animals Scientific Procedures Act of 1986, and approved by the UCL ethics review board. All mice were anaesthetized with ketamine or isoflurane during surgical procedures.

Two C57Bl6 mice (P28 and P30) were anaesthetized with ketamine (35 mg) and transcardially perfused with 0.9% saline, then with 2% paraformaldehyde and 1% glutaraldehyde in 0.1 M Na-acetate buffer (pH = 6) for 2 min, then with 2% paraformaldehyde and 1% glutaraldehyde in 0.1 M Na-borate buffer (pH = 8) for one hour. Four VGLUT1^Venus mice (all P28) were anaesthetized with ketamine (35 mg) and transcardially perfused with 0.9% saline, then with 4% paraformaldehyde and 0.1% or 0.05% glutaraldehyde in 0.1 M Na-phosphate buffer for 25 min. After perfusion, brains were dissected and 60 µm sections were cut from the cerebellar vermis. Sections from VGLUT1^Venus mice were immunoreacted for either VGLUT1 or GFP using anti-VGLUT1 (1:500 dilution; Synaptic Systems Cat# 135 302, RRID:AB_887877; Goettingen, Germany) or anti-GFP (1:1000 dilution; Millipore Cat# AB3080P, RRID:AB_91338; Billerica, Massachusetts) primary antibodies, respectively. Sections were then washed, incubated with biotinylated or 0.8 nm gold-coupled (Aurion, Wageningen, The Netherlands) secondary antibodies. Reactions were visualized with either silver enhancement (Aurion SE-EM kit) or a DAB reaction. Following reactions, sections from the C57Bl6 mice were washed in 0.1 M PB then treated with 1% OsO₄ and 1% uranyl acetate before dehydration and embedding in Epoxy resin. Small blocks from the sections were re-embedded and 30 nm serial ultrathin sections were cut for 3D reconstruction (Holderith et al., 2012). Images were taken with a Jeol JEM1011 electron microscope equipped with a bottom-mounted CCD camera (Cantega; Olympus Soft Imaging Solutions, Münster, Germany). 3D reconstructions and measurements were performed using Synapse Web Reconstruct (RRID:SCR_002716; http://synapses.clm.utexas.edu/tools/reconstruct/reconstruct.stm). MFTs from VGLUT1^Venus mice were qualitatively analysed at low and high magnification and compared to those obtained from C57Bl6 mice. All GFP- and VGLUT1-immunopositive MFTs were large, contained clusters of mitochondria and had vesicle densities similar to those found in C57Bl6 mice. The cloud of vesicles surrounding the AZs were also apparent with variable size. The proximity of mitochondria to AZs was also highly variable, similar to that found in C57Bl6 mice (data not shown).

Serial sections (200 nm) from C57Bl6 mice were cut and collected onto copper slot grids. Fiducial markers were introduced at both sides of the grids (Imig et al., 2014). Single-axis tilt series were acquired in FEI Tecnai G2 Spirit BioTWIN transmission EM operating at 120 kV and equipped with an Eagle 4K HS digital camera (FEI, Eindhoven, The Netherlands). Tomographic volumes were reconstructed using IMOD (Kremer et al., 1996; RRID:SCR_003297) and exported as z-stacks for analysis. Vesicle diameters were measured using Reconstruct, defined as the distance between the outer parts of the membrane bilayers at the plane where the diameter was largest.

Vesicle count was computed as a function of distance from the AZ face for 14 3D reconstructions. To do this, the distance from a vesicle’s centre to the nearest point of the AZ face was measured for each vesicle and a histogram of the distances constructed using 22 nm bins. In order to compute the vesicle density as a function of distance to the AZ, we limited the diffusible space to the outermost extremity of the vesicle ‘cloud’. To do this, we divided the space surrounding the AZ into 44 nm voxels and defined voxels as diffusible only if they contained any part of a vesicle (Figure 7A). The vesicle density was computed within the cloud as a function of distance from the AZ by sorting the diffusible voxels as a function of their distance to the AZ and counting the number of vesicle centre points that fell within each bin and dividing by the sum of the voxel volumes. The vesicle density was also calculated by subdividing the 44 nm voxels into 4.4 nm voxels and computing the volume of voxels that fell within the vesicles and dividing by the total volume of the diffusible voxels. To calculate the vesicle density when assuming all space surrounding the AZ is diffusible, we divided the vesicle count as a function of distance from the AZ (22 nm bins, as described above) by the space surrounding the AZ in 22 nm thick bands.

Parasagittal slices of the cerebellar vermis were prepared (Nielsen et al., 2004) from VGLUT1^Venus knock-in mice (P22–33, n = 19). FRAP in the cerebellar slices was performed on a custom spot confocal system (Prairie Technologies, Middleton, Wisconsin; DiGregorio et al., 2007) at 35°C unless stated otherwise. Data was acquired and analyzed using NeuroMatic (RRID:SCR_004186; http://www.neuromatic.thinkrandom.com) that runs within the IGOR Pro environment (RRID:SCR_000325; WaveMetrics, Portland, Oregon). Laser light (488 nm) was focussed to a diffraction-limited spot with 100x objective lens (1.0 NA, Olympus). The bleaching pulse was 28.5 µW (after the objective) of 0.5 ms duration and probe pulses were 0.04 µW of 2 ms duration. The ACSF contained (in mM) 125 NaCl, 2.5 KCl, 2 CaCl₂, 1 MgCl₂, 1.25 NaH₂PO₄, 26 NaHCO₃ and 25 glucose. Okadaic acid (2 μM), cytochalasin-D (10 μM) plus latrunculin-B (10 μM), jasplakinolide (2 or 5 μM) and roscovitine (50 µM) were added to the ACSF where specified. FRAP recordings were discarded if they had an unstable baseline fluorescence or large jumps in fluorescence. MFTs were discarded if they had less than 3 FRAP recordings.

To test for potential phototoxicity, we repeated our measurements using half the laser power during the bleaching pulse (16 MFTs, 156 total locations, 1 mouse), which produced ~13% bleaching from baseline fluorescence, rather than ~35% under control conditions, and computed f_1s and f_5s values. However, no significant differences were found for half-power conditions compared to control conditions for f_1s (40 ± 5 vs. 35 ± 2%, respectively, p=0.4, unequal-variance t-test) and f_5s values (59 ± 7 vs. 63 ± 2%, p=0.6, unequal-variance t-test), consistent with results of finite-difference FRAP simulations for full- and half-power bleaching (not shown). We also tested for time-dependent changes in the FRAP recordings within single MFTs by comparing f_1s and f_5s distributions computed from the first 4–10 and last 4–10 recordings taken from a single MFT. Restricting the analysis to those MFTs with at least 8 recordings (n = 47, 6 mice), we found no difference between f_1s values of the first and last recordings (38 ± 2 and 37 ± 2%, respectively, p=0.8, paired t-test) and f_5s values of the first and last recordings (63 ± 3 and 63 ± 3%, p=0.8, paired t-test) indicating the laser was not inducing time-dependent changes in vesicle mobility within single MFTs.

To test for photoactivation of bleached Venus (McAnaney et al., 2005), we compared FRAP recordings with 21 probe pulses after the large bleaching pulse (n = 82; Figure 1B, top) to recordings with 11 probe pulses (n = 79; recorded within the same MFTs from a P45 and P51 mouse) and found no significant difference (f_1s = 33 ± 5 vs. 41 ± 4%, respectively, p=0.17, unequal-variance t-test). Since photoactivation of bleached Venus should result in a faster fluorescence recovery with a larger number of probe pulses, these results demonstrate the small brief probe pulses used in our FRAP experiments did not cause photoactivation of bleached Venus.

For each population of FRAP recordings from multiple MFTs, a final weighted average (X_i) and variance (σ_i²) for a given probe pulse i was computed across MFTs (e.g. Figure 1D; black circles) using the following equations:

where j is the MFT index number, n_j the number of recordings for a given MFT, and x_ji the average of these n_j recordings.

A theoretical fluorescence recovery curve for pure diffusion can be described as follows (Axelrod et al., 1976):

where f₀ is the fluorescence before bleaching, K is the bleaching parameter that determines the level of fluorescence immediately after bleaching (f_b = (1−exp(−K))/K), and τ_D is the characteristic diffusion time defined as τ_D = ω²/4D_long. Parameter ω is the half-width of the illumination beam at e⁻² of the peak height which we estimated to be 0.23 µm from our iPSF. We computed the F_K summation for n = 0–19, which was sufficient to approximate the infinite series. To determine D_long, Equation (2) was fitted to our normalized control data by letting parameters K and D_long vary while fixing f₀ = 1 and ω = 0.23 µm. To allow a variable steady-state fluorescence (f_inf) during the fit, F_K was transformed as follows: f_K = f_b + (f_inf − f_b)(F_K − f_b)/(1 − f_b). Results of the fit were: D_long = 0.018 ± 0.005 µm²/s, K = 0.94 ± 0.05, f_inf = 0.917 ± 0.011. After drift correction, results of the fit were: D_long = 0.025 ± 0.003 µm²/s, K = 0.97 ± 0.03, f_inf = 0.883 ± 0.004.

One caveat of using Equation (2) to estimate D_long is that its accuracy relies on the following assumptions about our experimental paradigm: (1) the bleaching pulse is brief compared to the rate of diffusion, (2) bleaching is a simple irreversible first-order reaction, (3) the bleaching volume is small compared to the total volume of the synaptic terminal (4) iPSF and cPSF have the same Gaussian intensity profile in the xy direction and are infinite in the z direction (i.e. both are the same Gaussian beam), (5) fluorescence detection causes no additional bleaching during the fluorescence recovery phase, (6) there is a homogeneous concentration of diffusant and (7) diffusion is isotropic. While assumptions (1), (2) and (3) are reasonable for our experimental paradigm, assumptions (4–7) may not be strictly correct. To test assumptions (4) and (5), we used a 3D finite-difference reaction-diffusion simulation approach (see below) that explicitly modelled the bleaching and probe pulses and the spatial properties of the iPSF and cPSF of our microscope (Figure 3—figure supplement 2). Comparison of the fit to our drift-corrected control data (where D_long = 0.025 µm²/s) to a finite-difference simulation with D_long = 0.025 µm²/s showed a close agreement, with only a slightly slower rate of recovery for the finite-difference simulation (not shown; t_1/2 = 0.58 vs. 0.66 s). Only a small increase in D_long to 0.028 µm²/s of the finite-difference simulation was necessary to produce matching FRAP curves (Figure 2C). Similarly, to test assumptions (4–7), we used a 3D Monte Carlo simulation approach (see below) that included long cylindrical mitochondria, which form non-diffusible regions and are therefore likely to introduce anisotropic diffusion. Nevertheless, we found a close agreement to the finite-difference simulation that simulates a homogeneous concentration with isotropic diffusion (Figure 4D). These results suggest that the assumptions required to apply Equation (2) are reasonable for our experimental paradigm and it should therefore produce a reasonable estimate of D_long.

FRAP experiments were simulated in 3D space using an extended version of D3D, an in-house finite-difference reaction-diffusion simulator (Nielsen et al., 2004; DiGregorio et al., 2007; Nakamura et al., 2015). The voxel size was 50 nm and the time step (dt) was set by a stability restriction parameter (Crank, 1975) which was tested to be sufficiently small. The expression for iPSF was the same as that derived for the measured emission PSF of our microscope (Figure 3—figure supplement 2A,B), but using a light wavelength of 488 nm instead of 515 nm. The expression of cPSF (a Gaussian function with FWHM_xy = 255 nm and FWHM_z = 916 nm) was derived from fluorescence measured from 110 nm beads (Figure 3—figure supplement 2C). Both iPSF and cPSF were positioned so their peaks were aligned at the centre of the simulation geometry (Figure 3C). Values of iPSF and cPSF were computed at the centre of each voxel, and these values were used for bleaching and fluorescence detection. The e⁻² volume of iPSF (0.31 µm³) was computed by summing the volume of voxels with an iPSF value > e⁻². The e⁻² volume of the cPSF was 0.15 µm³, which is small compared to the volume of the MFTs (30–60 µm³; Jakab and Hámori, 1988; Kim et al., 2013). To take advantage of xy symmetry and reduce simulation time, we simulated one-quarter of the entire space.

Brownian motion of vesicles, including steric interactions, was simulated using a 3D Monte Carlo algorithm for non-overlapping hard spheres (Cichocki and Hinsen, 1990). At the start of the simulation, mobile vesicles were given the same D_short, the mean vesicle step size (dr) was set to a small fraction (0.5–5.0 nm) of the vesicle diameter (49 nm), and dt was computed via Einstein’s relation in 3D space:

If a displaced vesicle resulted in overlap with another vesicle, or a non-diffusible voxel, the vesicle stayed at the same location; otherwise the vesicle was moved to the new location. Vesicles were not allowed to overlap with the simulation borders, except when computing D(t), in which case periodic boundary conditions were used.

For FRAP simulations, mitochondria were simulated as cylindrical regions of non-diffusible voxels with 0.28 µm diameter and 2.25 µm length, randomly placed throughout the cubic simulation space in clusters of 2–3 (Figure 3C). Vesicles were given an initial fluorescence (f) of unitary and this was then scaled by the following reaction during the bleaching and probe pulses:

where t is time, dt is the time step and k is the bleaching rate that is scaled by a normalized 3D spatial weighting function defined by iPSF described above. Average fluorescence of all vesicles was computed according to a normalized 3D spatial weighting defined by cPSF described above. The vesicle step size dr was set to 2 nm which was small enough to avoid discretization error (Figure 4—figure supplement 1A). We compared simulated FRAP curves for cubic geometries in the range 1.5–3.0 µm and found a 2 µm cube produced negligible border effects for simulations with D_short < 0.080 µm²/s and was therefore sufficiently large to simulate the centre of a large MFT (Figure 4—figure supplement 1B).

For FRAP simulations with added drift (Figure 2—figure supplement 1B), mitochondria were simulated as non-diffusible volumes specified by x, y and z coordinates, rather than non-diffusible voxels as shown in Figure 3C, so that drift could be applied to the mitochondria’s x, y and z coordinates. If a vesicle moved outside the geometry it was returned to the opposite side of the geometry (i.e. periodic boundary conditions) with a fluorescence value of 1.0 to simulate unbleached vesicles moving into the simulation space.

One potential caveat to our quantification of vesicle mobility is the possibility our measured FRAP represents the movement of small clumps of vesicles rather than individual vesicles. However, this should not unduly affect our results since the vesicle volume fraction, rather than the vesicle size, primarily determines the hydrodynamic and steric contributions to D_long (Medina-Noyola, 1988). Moreover, inspection of EMs of the interior of the MFT suggests vesicles are dispersed. Interestingly, vesicles exhibit a negative potential, which generates an electrostatic repulsive force, which could explain why vesicles do not tend to aggregate in clumps (Ohsawa et al., 1981).

For AZ simulations, we used the 3D reconstructions from our EM data. The diffusible space was determined by the outer surface of the vesicle cloud as described above. In one of the 14 AZ reconstructions, a low vesicle density resulted in space close to the AZ being ‘filled in’ by our algorithm. As the presence of non-diffusible space so close to the AZ seemed unlikely, we expanded the diffusible space surrounding this AZ by two vesicle diameters (the dimensions of the high-vesicle-density zone; Figure 6C). To remove the effects of fixation shrinkage in our EM data, the geometries, including AZs, were scaled up by 11% (Korogod et al., 2015). After scaling, vesicle diameters and voxel widths were 49 nm.

Because we used a uniform vesicle diameter of 49 nm, rather than a distribution, small overlaps between vesicles occurred. To alleviate this problem we used an algorithm (Lubachevsky and Stillinger, 1990) that first shrank vesicles until there were no overlaps and then allowed them to randomly move, slowly expanding until they reached a diameter of 49 nm. To maintain high vesicle densities close to the AZ (Figure 6C) vesicles <200 nm from the AZ were not allowed to move more than 24.5 nm from their original location; all other vesicles were not allowed to move more than 49 nm. We verified the average vesicle density near the AZ was similar before and after removing vesicle overlaps (Figure 7—figure supplement 1A). To create a vesicle reserve surrounding the reconstruction geometries, the geometries were expanded and populated with vesicles at a 17% volume fraction of which 25% were immobile (Figure 7A). Final simulations had ~13,000 mobile vesicles. To create ‘open’ geometries, non-diffusible voxels surrounding the AZ were converted to diffusible space and populated with vesicles as for the reserve. The final simulations had ~17,000 mobile vesicles. For all AZ simulations, dr = 5 nm, which was sufficiently small to avoid significant discretization error (Figure 7—figure supplement 3A,B). Simulations were repeated 20 times for each AZ and averaged.

To investigate how filament connectors might influence vesicle properties near the AZ, we performed simulations whereby vesicles <150 nm from the AZ rapidly formed connectors with neighbouring vesicles (average 1.5 connectors per vesicle; Siksou et al., 2007; Fernández-Busnadiego et al., 2013) if they were <10 nm of one another. Once connected, vesicles were not allowed to diffuse more than 10 nm from each other, thereby simulating flexible protein filaments (Graydon et al., 2014). A connector on rate of 10,000 s^-1 was used and each bound connector had a lifetime that was randomly sampled from an exponential distribution with mean equal to the inverse of the connector unbinding rate (1000 or 10 s^-1). Filament tethers to the AZ (Fernández-Busnadiego et al., 2013) were simulated in a similar fashion. In this case, vesicles became ‘tethered’ to the AZ if they were <8 nm from it. The same on and off rates of the connectors were used and no more than 2 vesicles could be tethered to the AZ at one time.

Because the vesicle density increases close to the AZ (Figure 6C) the effects of vesicle-vesicle hydrodynamic interactions are expected to be larger near the AZ. Hence, for those vesicles within the vesicle cloud, we used the local vesicle volume fraction surrounding each vesicle to compute the vesicle’s D_short and dr on every time step of the AZ simulations. The local vesicle density was computed within a distance of 4 vesicle radii from the vesicle’s centre (Urbina-Villalba et al., 2003). D_short was computed from the ratio D_short/D_cyto (Figure 5B) assuming our estimate of D_cyto = 0.127 µm²/s and 0.04 immobile vesicle volume fraction.

To compute the vesicle supply rates in Figure 7, release events were counted within the following windows: 0-10 ms in 2 ms bins, 10-100 ms in 10 ms bins, 100-1000 ms in 100 ms bins, 1-10 s in 1 s bins, 10-100 s in 10 s bins. Release counts were converted to rates by dividing the bin count by the bin duration and the number of simulation repetitions (20). The cumulative vesicle counts were computed within the same windows. The vesicle release rates and cumulative vesicle counts in Figure 8 were computed in the same manner, except for using a single 10 ms bin within the first window as this corresponds to the stimulus interval of the 100 Hz train. Final release rates and counts were displayed as continuous functions by drawing lines between the midpoint of each consecutive bin.

Effects of hydrodynamic interactions on vesicle mobility from vesicle-vesicle interactions were determined with analytical expressions for D_short/D_cyto as a function of the vesicle volume fraction (Φ). For conditions of all mobile vesicles, D_short/D_cyto (here denoted as Γ_m) was computed using the analytical expression of Tokuyama and Oppenheim (1994) (Figure 5B; red line):

where D_cyto is equivalent to D₀ of Tokuyama and Oppenheim and Φ_m is the mobile vesicle volume fraction. For a mixture of mobile and immobile vesicles, D_short/D_cyto (here denoted as Γ_mix) was computed using the self-consistent equation of Freed and Muthukumar (1978) up to the squared term:

where ζ is the drag coefficient (related to Stokes-Einstein equation D = k_BT/ζ, k_B being the Boltzmann's constant and T absolute temperature), η is viscosity, r is the vesicle radius and c_im is the density of immobile vesicles (count per µm³) which was converted to the immobile vesicle volume fraction (Φ_im). However, Equation (6) describes conditions for small Φ_m. For conditions with large Φ_m, hydrodynamic interactions from the immobile vesicles are expected to be less due to ‘screening’ effects from the mobile vesicles. To account for this, we applied the analytical model of Michailidou et al. (2009) to compute Γ_mix in the presence of a large Φ_m (Figure 5B; blue solid line) denoted as ${\displaystyle {\mathrm{\Gamma }}_{\text{mix}}^{\mathrm{\prime }}}$:

where Γ_ff is the far-field-only D_short, i.e. the short-time self-diffusion coefficient in the absence of near-field hydrodynamic interactions, derived from previously published computer simulations (Banchio and Brady, 2003). Note that when Γ_ff ≈ Γ_m (e.g. with a low Φ_m) then Equation (7) reduces to ${\displaystyle {\mathrm{\Gamma }}_{\text{mix}}^{\mathrm{\prime }}\approx {\mathrm{\Gamma }}_{\text{m}}{\mathrm{\Gamma }}_{\text{mix}}}$.

The combined long-time effects of steric and hydrodynamic interactions, expressed as D_long/D_cyto, was computed for conditions of all mobile vesicles using the analytical expression of Tokuyama and Oppenheim (1994):

where Φ_m is the mobile vesicle volume fraction and H(Φ_m) is defined in Equation (5). This equation was used only for comparison purposes in Figure 5D.

For AZ simulations, the effects of hydrodynamic interactions from a membrane wall (denoted as β) were computed via analytical expressions for diffusion toward a wall (β_⊥; Brenner, 1961) and parallel to a wall (β_||; Goldman et al., 1967):

where △_z is the shortest distance between the wall (i.e. AZ) and edge of the vesicle and △_z+r is the distance between the wall (i.e. AZ) and the centre of the vesicle (i.e. △_z+r = △_z + r). To account for diminished hydrodynamic interactions from the wall due to a high Φ_m, we again used the analytical model of Michailidou et al. (2009) to apply a correction factor to β, denoted as β':

Values for β' were then used to appropriately scale the vesicle steps in the x, y and z directions computed during each time step.

Data are presented as mean ± standard error of the mean (SEM) and fit parameters as ± standard deviation (STDV). Experimental and simulation results were compared with a chi-square criterion and experimental means were compared using the Student’s t-test (unpaired two-tailed equal-variance unless stated otherwise) where p<0.05 was considered significant. Model comparisons were computed via an F-test. No statistical method was used to predetermine sample sizes.

Java code to reproduce the finite-difference and Monte Carlo reaction-diffusion simulations is available at https://github.com/SilverLabUCL/D3D_eLife.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Nils Brose for supplying the VGLUT1^Venus knock-in mice, Stefan Hallermann for providing electrophysiological data on wild-type mice, Noemi Holderith for performing electron tomography and Diccon Coyle for providing technical assistance. We would also like to thank J Ashmore, D Attwell, T Fernández-Alfonso, F Lanore, B Marin, A Valera and T Younts for their comments on the manuscript.

Animal experimentation: Animal experiments were conducted in strict accordance with the United Kingdom Home Office Animals Scientific Procedures Act of 1986, and approved by the UCL ethics review board. All mice were anaesthetized with ketamine or isoflurane during surgical procedures.

1. Received: February 9, 2016
2. Accepted: August 14, 2016
3. Accepted Manuscript published: August 19, 2016 (version 1)
4. Accepted Manuscript updated: August 23, 2016 (version 2)
5. Version of Record published: September 15, 2016 (version 3)

© 2016, Rothman et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.