Sequential and parallel vesicle pool models

In order to explain the complex Ca²⁺-dependent properties of regulated exocytosis, we investigated a mathematical model with sequentially arranged pools (Sequential Pool Model, SPM), separated by Ca²⁺-dependent steps [11] (Fig. 1A). Our model assumes that some vesicles reside in a Non-Releasable Pool (NRP) from which they can undergo reversible priming to a Readily-Releasable Pool (RRP). From there, vesicles can irreversibly converge to the fused state (F). Vesicles enter the NRP by recruitment from a much larger Depot Pool (Fig. 1A). This recruitment is Ca²⁺-dependent, driven by a Michaelis-Menten Ca²⁺-dependent forward process (k₁) with a K_d of 2.3 µM, as in previous models of the adrenal chromaffin cell [16], [32]. At rest, the supply of vesicles to the RRP from the NRP can be slow, because the demand is low. However, during repetitive stimulation, RRP vesicles may soon be depleted unless Ca²⁺ speeds up their re-supply, as observed experimentally [9], [32]. Here, we investigated the possibility that the process that enhances priming (i.e. transition from NRP to RRP) is Ca²⁺-dependent catalysis. A catalyst lowers the activation energy barrier of a reaction (Fig. 1A), thereby simultaneously increasing both forward (k₂) and backward (k₋₂) rates (see Materials and Methods). Thus, unlike the selective increase of the forward rate, a catalyst speeds the transition without affecting the proportion of NRP- to RRP-vesicles in equilibrium. This complies with data in the adrenal chromaffin cells, as shown below. The Ca²⁺ trigger for fusion is modeled as a sequential sensor binding 3 Ca²⁺ ions, followed by a Ca²⁺-independent, but very fast, conversion to the Fused state [16] (rate constant k₄, not shown in Fig. 1A; see Fig. 2A). In thermodynamic terms, the Ca²⁺ trigger affects the energy barrier for fusion. This barrier is high in the absence and low in the presence of Ca²⁺, resulting in strongly Ca²⁺-dependent fusion rates (Fig. 1A).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 1. Two different models of secretion. A. Sequential Pool Model (SPM).

The Depot, Non-Releasable Pool (NRP), Readily-Releasable Pool (RRP) and Fused (F) states are represented by local minima of the free energy. Reactions between these states require the crossing of transition states that represent local maxima of the energy landscape. A Michaelis-Menten Ca²⁺-dependent step fills the NRP from the Depot Pool (recruitment, brown background). A Ca²⁺-dependent catalyst decreases the energy of the transition state for priming (transition to the RRP state) in a Ca²⁺-dependent manner, thus increasing the rates of both forward and backward reactions (priming, green background). A downstream clamp arrests release by imposing a high-energy transition state for fusion, which can be removed by Ca²⁺ binding to the associated sensor, thereby enabling fusion (fusion, blue background). The details of the cooperative Ca²⁺ sensor for fusion is given in Fig. 2A. B. Parallel Pool Model (PPM). In this model, the same Michaelis-Menten Ca²⁺-dependent step fills the SRP from the Depot Pool, as in the SPM (recruitment, brown background). Both the Slowly-Releasable Pool (SRP) and the Readily-Releasable Pool (RRP) are releasable, with different cooperative Ca²⁺ sensors (fusion, dark and light blue background, shown in detail in Fig. 2B). The transition between SRP and RRP is Ca²⁺-independent (after [16] ).

https://doi.org/10.1371/journal.pcbi.1003362.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 2. Both the SPM and the PPM predict a biphasic burst of release.

A. Top panel: Full Sequential Pool Model (SPM), including details of the cooperative Ca²⁺ sensor. Abbreviation: R = RRP. Lower panels: The model was solved in the steady-state using parameters in Table 1, and simulated over time with a step increase in [Ca²⁺]_i from 0.5 to 25 µM at 0.5 s (grey lines). The development of the pools (bottom panel) shows fast depletion of all RRP-states (blue), concomitant with the fastest phase of burst release (middle panel). The NRP (red) is depleted slower and only partially, giving rise to the slower phase of burst release. During the subsequent sustained component the recruitment of vesicles from the Depot, followed by maturation through the NRP and RRP states, support ongoing release. The dotted blue line in the middle panel is a fit of a sum of two exponential functions plus a straight line to the model simulation. B. Top panel: Full Parallel Pool Model (PPM), including details of the fast and slow cooperative Ca²⁺ sensors. Abbreviations: R = RRP; S = SRP. Lower panels: The model was solved in the steady-state using parameters in Table 2, and simulated over time with a step increase in [Ca²⁺]_i from 0.5 to 25 µM at 0.5 s. The development of the pools (bottom panel) shows fast depletion of all RRP-states (blue), concomitant with the fastest phase of burst release (middle panel). The SRP is depleted slower, giving rise to the slower phase of burst release. During the subsequent sustained component recruitment of vesicles from the Depot to the SRP is followed by fusion through the slow Ca²⁺ sensor, without maturation to the RRP state (middle panel: F_s is fusion through the slow pathway, F_r is fusion through the fast pathway). The dotted blue line in the middle panel is a fit of a sum of two exponential functions plus a straight line to the model simulation.

https://doi.org/10.1371/journal.pcbi.1003362.g002

We modeled release from chromaffin cells using the SPM to investigate whether previously obtained experimental data could be explained by the sequential Ca²⁺-dependent actions we suggest. For comparison, we simulated the Parallel Pool Model (PPM), where this had not already been done in the literature [16], [32] (Fig. 1B). In the PPM, parallel fusion of two releasable pools, the Slowly-Releasable Pool (SRP) and the (fast) Readily-Releasable Pool (RRP), is assumed. The Ca²⁺ sensors for RRP and SRP fusion are separate sequential and cooperative sensors each binding three Ca²⁺ ions (see Materials and Methods). Like in the SPM, supply of vesicles from the larger Depot Pool is a Ca²⁺-dependent Michaelis-Menten process [16]. The parameters of this step as well as the ones for the Ca²⁺-binding and fusion from the RRP were identical in the two models. Each model was solved in the steady-state at the beginning of each simulation, and then allowed to evolve by numerical integration, driven by the Ca²⁺ signal characteristic for each stimulation protocol. The cumulative release was calculated, from which release rates and pool sizes were determined by fitting a sum of exponential functions. Parameters were taken or estimated from published data, as described in Materials and Methods, and are listed in Table 1 (SPM) and Table 2 (PPM), respectively.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. SPM model parameters for chromaffin cells.

https://doi.org/10.1371/journal.pcbi.1003362.t001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. PPM model parameters for chromaffin cells.

https://doi.org/10.1371/journal.pcbi.1003362.t002

Ca²⁺ uncaging experiments and biphasic release

Pioneering work [6], [16], [32] using Ca²⁺ uncaging and cellular capacitance recordings resulted in the identification of fast and slow release components in adrenal chromaffin cells. After parameter estimation, we simulated both models (SPM, PPM) using an abrupt Ca²⁺-step from 0.5 µM to 25 µM (Fig. 2). Both models displayed biphasic capacitance responses, in agreement with published data. The fastest phase is referred to as the ‘fast burst’ of release and has a rate of ∼50 s⁻¹ under these conditions. The subsequent slow burst of release is approximately 10-fold slower. Following these two components, release persists in the so-called sustained phase, which is nearly linear. In the ‘classical’ PPM, the fast burst component is caused by the fusion of vesicles from the RRP, whereas the slow burst component owes itself to the fusion of SRP-vesicles (Fig. 2B). Finally, when both RRP and SRP have been depleted, newly recruited vesicles (from the Depot Pool) fuse continuously as long as the intracellular Ca²⁺ concentration ([Ca²⁺]_i) remains high, giving rise to the sustained component of release. A particularity of the PPM is that fusion during the sustained phase is almost entirely through the SRP, since the SRP-to-RRP conversion is relatively slow and Ca²⁺-independent (Fig. 2B, middle panel: F_s gives the fusion through the slow pathway, F_r gives the fusion through the fast (rapid) pathway) [33].

In the sequential model (SPM), the fast burst component corresponds mainly to the rapid depletion of RRP-vesicles, whereas the slow burst component corresponds mainly to NRP-vesicles that mature to the RRP-state before fusing (Fig. 2A). This maturation step is slower than fusion from the RRP and it is Ca²⁺-dependent, driven by the Ca²⁺-dependent catalyst, giving rise to the slow burst phase. It should be noted that the identification of the fast component with the RRP and the slow component with the NRP is only approximate, because the size and kinetics of the two phases depend on the entire system (Materials and Methods). Therefore, to identify fast and slow burst components we fitted simulated traces with a sum of exponentials in order to compare these values to the ones obtained from experiments in the same way. Finally, when both the NRP and RRP are empty, sustained release is driven by the upstream Ca²⁺-dependent reaction from the Depot Pool (k₁, Fig. 1). Other than in the PPM, these vesicles also fuse through the RRP, which is the only releasable state in the SPM.

Since both models could reproduce biphasic burst release and the sustained component, we next compared our new model with data that had previously been fitted with the PPM [16].

Vesicle pools and fusion kinetics

Systematic variation of the Ca²⁺ signal driving the release model allowed the determination of amplitudes and rate constants of fast and slow burst components as a function of pre- and post-flash Ca²⁺ concentrations (data points in Fig. 3A, 3B, and 3C, from [16]). Experimentally it was found that the time constants depended on the Ca²⁺ levels reached after Ca²⁺ uncaging (Fig. 3A), whereas the number of vesicles released in each phase was relatively independent of post-flash [Ca²⁺]_i (data points in Fig. 3B, from [32]). Our model accounts well for the kinetic data (model simulations are shown as lines in Fig. 3; for examples of simulated capacitance traces see insert in Fig. 3B). In further agreement with the SPM, both slow and fast burst components were augmented with moderate increases of resting (pre-flash) [Ca²⁺]_i below 0.7 µM, owing to the Ca²⁺-sensitive supply rate (k₁ in Fig. 1) from the Depot (Fig. 3C, model simulation are lines, data are points), while at higher pre-flash [Ca²⁺]_i the size of both components decreased, due to the partial depletion of the underlying pools (NRP, RRP) through release before the uncaging event. However, the relative proportion of fast and slow components remained fairly constant (Fig. 3C). The invariance in fast-to-slow amplitude when changing the Ca²⁺ concentration originally served as an argument for a direct slow fusion pathway in the PPM, since it seemed inconsistent with a Ca²⁺-dependent interconversion between fast and slow vesicles. However, we show here that when modeling Ca²⁺-dependent priming as a catalytic process, the invariance is maintained in a sequential model (lines in Figs. 3C and 3B), except for very low post-flash Ca²⁺ concentrations (Fig. 3B), where the distinction between fast and slow burst components becomes experimentally challenging.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. The SPM fits experimental data from chromaffin cells.

A. The fast (blue) and slow (red) fusion rates are sensitive to the Ca²⁺ levels reached after Ca²⁺-uncaging (data points from [16], and model predictions of the SPM in solid lines). B. In contrast, the sum of the fast and slow burst component ( = total burst, black, left hand side), or the fraction of fast- to total burst release (blue, right hand side) are fairly insensitive to post-flash Ca²⁺ (data points, from [16], and model predictions of the SPM in solid lines). The inset shows examples of model simulations (top panel, Ca²⁺ signal; bottom panel, the calculated capacitance responses). C. The amplitudes of the fast (blue) and slow (red) burst components are a bell-shaped function of the resting Ca²⁺ levels (data points from [16] and model predictions in solid lines). The increase at low [Ca²⁺]_i is due to the Ca²⁺-dependent recruitment from the Depot Pool, whereas the decrease at higher [Ca²⁺]_i is due to partial pool depletion. D. Bottom panel: following selective depletion of the fast component, its recovery is slow and at the cost of the slow component, in agreement with experiments (data points from [32] and model predictions of the SPM in solid lines). The top panel illustrates our simulation paradigm: a stimulation that selectively depleted the R-state (the major determinant of the fast component) was followed by a second stimulation that depleted the total releasable pool at varying intervals (Δt). (See Table 1 for parameter values).

https://doi.org/10.1371/journal.pcbi.1003362.g003

Another stimulation paradigm probed the recovery of the fast burst component in chromaffin cells [32]. The stimulation paradigm combined depolarizations and Ca²⁺ uncaging to allow a selective depletion of the fast component without affecting the slow component. It was found that the recovery of the fast component occurred at the loss of the slow component (data points in Fig. 3D, from [32]). This behavior is also an inherent feature of our sequential model: the refilling of fast (RRP) vesicles after selective depletion occurred at the cost of NRP vesicles, which are the major source of the slow burst (lines in Fig. 3D).

In the above, we simulated a model where only a single Ca²⁺ bound to the priming catalyst (cooperativity one). We also constructed a version of the model with a cooperativity of two (Materials and Methods). Fitting this model to the data also resulted in a satisfactory fit (Fig. S1, Table S1). The fit to the fast and slow burst components as a function of preflash [Ca²⁺]_i was a little better (Fig. S1C), but the fit to the fast burst fraction as a function of postflash [Ca²⁺]_i was a little worse (Fig. S1B). Thus, the data do not allow a clear conclusion as to whether the catalyst has a cooperativity of one or two. Therefore, we continued exploring the simplest model with cooperativity one.

We conclude that the now classical data of Thomas Voets on fast and slow release phases and fusion kinetics in mouse chromaffin cells can all be satisfactorily fit by our sequential model (SPM), as well – as previously shown [16] – by the PPM.

Recovery of the RRP after strong stimulation

One fundamental difference between the PPM and the SPM is the recovery behavior of the RRP after its depletion. In the PPM, the refilling of the RRP from the SRP is Ca²⁺-independent. This Ca²⁺-independence was concluded from the observation of parallel enhancement of the fast and slow burst by increasing steady state [Ca²⁺]_i in the sub-micromolar range before stimulation [16] (Fig. 3C). However, as shown above, such a behavior is also inherent to the SPM, as long as the Ca²⁺-dependent acceleration of priming is catalytic. Therefore, the two models can be distinguished by probing the recovery of the RRP under conditions of elevated Ca²⁺: according to the SPM, its recovery should be sped up, whereas the PPM predicts Ca²⁺-independent - and thus slower - recovery.

Recently, we used a dual-uncaging protocol to investigate the recovery of the RRP after Ca²⁺-uncaging had emptied both the fast and the slow burst components [34]. Applying a second Ca²⁺ uncaging flash at variable inter-stimulus intervals (ISI) while measuring Ca²⁺-relaxation (Fig. 4A, top panel) allowed estimating the RRP recovery (points in Fig. 4D). Note the difference in approach to the previous selective depletion of the RRP under conditions where Ca²⁺ presumably relaxed to baseline much faster (Fig. 3D). We found that already at 8 s the fast component had recovered to 18.1±4.2% of the initial value, and at 22 s recovery was at 35.3±4.9% (symbols in Fig. 4D, note that all lines in Fig. 4 represent simulations, not data). This behavior is inconsistent with the PPM (broken line in Fig. 4D), which featured a full ∼10 s delay, before the first recovery of the fast component was visible, owing to the lack of Ca²⁺-accelerated refilling of the RRP. This is also clearly appreciated from the simulations of the capacitance increases (Fig. 4A, B), which lack a fast component in the PPM at these time points. Looking at the evolution of the pools in the PPM (Fig. 4C) it is clear that the RRP does not start to recover appreciably within the first 10 s. In contrast, the SPM in fact predicted the faster recovery of the fast component without the need to adjust parameters (i.e. the same parameters were used here as in Fig. 2 and 3) (Fig. 4D). On the simulation of single traces, the fast burst is clearly seen at short interstimulus intervals (Fig. 4A, B). In spite of the quicker recovery of the fast component in the SMP, its recovery still lagged behind the recovery of the slow component (Fig. 4B and Fig. S2). The faster recovery of the slow pool is a fundamental feature of neurotransmission also found in the Calyx of Held synapse [35].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. The SPM explains recovery of the fast component after strong stimulation.

A. Top panel: double Ca²⁺ uncaging protocol (points, data; lines, Ca²⁺ signal used to drive the models). Lower panels: simulated capacitance traces for the two models (PPM and SPM, Figure 1) during the double uncaging protocol with Inter-Stimulus Interval (ISI) 8 s. B. Capacitance changes elicited by the 2^nd uncaging event after different ISIs and simulated by both models (left panel, PPM; right panel, SPM). C. Development of the pools over time in the two models (PPM and SPM), following the first uncaging event. For the PPM, the SRP and RRP vesicles not bound to Ca²⁺ are shown. For the SPM, the RRP not bound to Ca²⁺ is shown. It is noticeable that recovery of the RRP is delayed in the PPM compared to the SPM. D. Recovery of the fast burst component (identified by fitting of exponentials) in the experiment (points: experimental data from [34]), and as predicted by the two models (PPM: dotted line; SPM: full line). Only the SPM predicts the fast recovery of the fast component seen experimentally.

https://doi.org/10.1371/journal.pcbi.1003362.g004

The faster recovery in the SPM is a potentially physiologically important property, which allows secretory cells to regain potency for fast release quickly after stimulation. Clearly, the SPM is more consistent with this type of experiment. The recovery ‘pause’ in the PPM due to the Ca²⁺-independent SRP-to-RRP conversion is a fundamental feature of that model and it would therefore not be possible to account for this behavior while still accounting for the data in Fig. 3 by mere adjustment of the model's parameters. We therefore sought to modify the PPM in such a way that this important property could be reproduced by a model with parallel fusion pathways by speeding the recovery of the RRP in a Ca²⁺-dependent manner. As described above, this cannot be achieved by an increase in the SRP-to-RRP forward conversion rate alone, as this would change steady-state pool sizes (inconsistent with Fig. 3). Therefore we made use of our idea of a Ca²⁺-dependent catalyst, which is used in the SPM, and incorporated catalysis in the SRP-to-RRP inter-conversion, so that once again forward and reverse rates were Ca²⁺-dependent (Fig. S3A). Indeed, such a model (PPM+cat) was also able to account for the general Ca²⁺-dependence of release (Fig. S3A–C, Table S2) while the catalysis of the SRP-RRP inter-conversion sped the recovery of the fast component at elevated Ca²⁺ to a closer agreement with the experimental data (Fig. S3D).

In conclusion, the classical PPM failed in reproducing the faster recovery of the fast component after strong stimulation, while the SPM or the modified PPM+cat, which both feature Ca²⁺-dependent catalysis in RRP replenishment, matched the data more closely. In our opinion this still argues for the SPM rather than the PPM+cat, because the SPM is simpler, has fewer parameters and does not require a second fusion pathway, while it can account for all experiments investigated.

Perturbation of the last fusion step affects release components and fusion rates

Thus far, the SPM is in line with experimental data describing vesicle pools, fusion kinetics and RRP recovery. The recovery after strong stimulation particularly favors the SPM over the PPM. Another difference between the models is in the predicted effect of a change in the fast release rate. In the PPM, release rates and amplitudes are independent. Therefore, a change in any one of these parameters is expected to leave all others (almost) unaffected. In contrast, in the SPM model, the amplitudes and release rates are all functions of the elementary rate constants k₂, k₋₂ and k₃ (see Materials and Methods). Therefore, the time constants and amplitudes of fast and slow burst release are interdependent (due to their individual dependence on k₂, k₋₂ and k₃) and a change in k₃ is expected to affect both fast and slow release. This is demonstrated in Fig. 5A where we have calculated the time constants of fast and slow burst release while varying the final fusion rate k₃ (lines are model simulations). Moreover, the relative contribution of fast and slow release depends on the time constant of fast release (Fig. 5B), reducing k₃ will therefore also change the ratio of fast to slow release.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. The SPM predicts the observed inter-dependence of release rates and amplitudes following regional SNARE mutation.

(A and B) Evaluation of the SPM model in comparison to experimental data obtained after C-terminal SNARE mutation. A. According to the SPM model, the time constants of both fast (blue line) and slow burst release (red line) depend on the fusion rate k₃. The data points (from [25]) show the measured time constants of fast (blue edges) and slow (red edges) release in wildtype (control, black squares) and synaptobrevin-2 mutants (Layer +8 mutants, purple triangles) bearing a mutation in layer +8 (residue 84: L to A, D, N or G). B. According to the SPM model, release changes from mainly fast (blue) to mainly slow (red) with increased values of the fast time constant. This is in line with experimental data from C-terminal SNARE mutants (purple triangles), which show an increase in the fast time constant of release as well as an increase in slow (red edges) and a decrease in fast (blue edges) burst release. This behavior cannot be accounted for by a single parameter change in the Parallel Pool Model (PPM, simulation, dashed lines). C. Comparison of the SPM and the PPM models. Ci. Top panel: the SPM model fits well to average experimental data [25] and accounts for differences between control (wildtype synaptobrevin-2, black) and mutant (an L84A mutant, purple) by a change of a single parameter (bottom panel). Cii. Top panel: the PPM fits well to average experimental data [25], but the effect of L84A mutation requires the adjustment of multiple parameters (white, unevenly dashed line and bottom panel). Model predictions in panel A and B were calculated with k₃ as independent variable, the relationship between τ_fast and k₃ given in Methods and with values of k₂ and k₋₂ determined by a global fit to the data in panels A and B (k₂ = 5.26 s⁻¹ and k₋₂ = 3.80 s⁻¹). Error bars represent SEM.

https://doi.org/10.1371/journal.pcbi.1003362.g005

It is commonly assumed that one of the last events before vesicle fusion is the assembly of the final interaction ‘layer’ (layer +8) in the SNARE complex. In the SPM this molecular event is primarily reflected by the rate constant k₃ (although it might affect k₂ as well). Mutagenesis studies of layer +8 in synaptobrevin-2 previously performed by us [25] indeed resulted in the identification of an increase in the fast time constant (data points in Fig. 5A). In the PPM one might expect that the release rates of both fast and slow vesicles are decreased if both fusion reactions required the C-terminal interaction of synaptobrevin-2. However, changes in the proportion of fast to slow burst release are not expected. In contrast to this we found that mutation of layer +8 resulted in a shift of the relative contribution of fast and slow release; as fast release became slower, the amplitude of the fast component decreased while the amplitude of the slow component increased ([25]; data points in Fig. 5B). With the SPM we can explain the connection between these effects in the framework of a simple model (solid lines in Fig. 5): C-terminal SNARE destabilization reduces k₃ (illustrated by the purple arrow in Fig. 5Ci), resulting in an increased time constant for fast release (τ_fast, Fig. 5A) and a simultaneous change in fast and slow contribution to release (Fig. 5B). The PPM cannot account for the simultaneous change of release rates and amplitudes by a single parameter change, because pool sizes and release rates are independent. Therefore, changes in the fast release rate are predicted to leave the relative contribution of fast and slow release unchanged (dashed lines in Fig. 5B are simulations of the PPM), which is not easily reconciled with the experiment. The PPM can still mathematically describe the data, but for this it is necessary to assume that several rates are changed by the single point mutation, including one that changes the relative sizes of the RRP and SRP (purple arrows in Fig. 5Cii). Hence, while both models fit the data, the SPM delivers a straightforward explanation for the observed relationship of fast release rates and amplitudes in general and relates the main effect of C-terminal SNARE mutation to a single rate constant (k₃) in particular.

Thus, a sequential model featuring an upstream Ca²⁺-dependent catalyst with a downstream Ca²⁺ sensor predicts the effects of SNARE mutations.

Deletion of synaptotagmin-1: Unclamping release, or destabilizing the RRP?

One argument for parallel pool or parallel sensor models has been the observation that after deletion of the fast Ca²⁺ sensor in various systems slow release components persist. This is also the case in mouse adrenal chromaffin cells, where deletion of syt-1 leads to ablation of the fast burst component, whereas the slow burst component and the sustained component both remain [17], [31], [36]. Clearly, a Parallel Pool Model, where the RRP would utilize syt-1 and the SRP a separate Ca²⁺ sensor (even though no such molecule has been found yet) is an easy and reasonable way to account for this finding. How can one account for it in a sequential model?

Recent in vitro and in vivo experiments have emphasized the function of syts and/or complexins as clamps on release [26]–[30]. Therefore, one idea is that these proteins – probably working in concert – are able to arrest SNARE complex assembly. Ca²⁺-binding to syt would release the clamp and allow the SNARE complex to complete its assembly, leading to rapid membrane fusion. In our model, the clamp would generate the final energy barrier for fusion (Fig. 1A), which could be removed by the binding of three Ca²⁺ ions to syt-1 (Fig. 2A). In this scenario one can model the syt-1 null data by removing the final fusion barrier, making RRP vesicles fuse with the maximal rate (k₄) (Fig. 6A, top panel). This fast – and now Ca²⁺-independent – fusion from the RRP leads to chronic pool depletion. As a result of the missing clamp, the NRP vesicles in effect become releasable, with the transition from NRP to RRP being rate limiting for fusion. This transition is sped up by the Ca²⁺-dependent catalyst (Fig. 1), resulting in Ca²⁺-dependent, but much slower, release rates. We simulated our model under these conditions driven by a Ca²⁺ uncaging event (Fig. 6A, bottom panels). Indeed, our model predicted a missing fast component of release, and the persistence of both slow and sustained release components [17], [31], [36]. The slow component was driven by the sequential transition of vesicles from the NRP to the RRP to the F-state, following an increase in [Ca²⁺]_i. The slow component of release was somewhat smaller in the syt-1 null than in the WT case, because of partial depletion of the NRP at rest (comp. Fig. 6A and Fig. 2A). Because of the lack of the last energy barrier, ongoing release at rest depleted the RRP (and the NRP partially) and led to an increase in the spontaneous release rate from 1.7 fF/s to 6.9 fF/s.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 6. Destabilizing or spontaneously fusing the RRP in the SPM qualitatively mimics the syt-1 knockout phenotype.

A. In the first version of the model simulating syt-1 null data, we assumed that the last barrier to fusion is gone (top panel, grey dotted line shows the energy landscape in the WT case), and the RRP vesicles fuse independently of Ca²⁺ with the maximal rate constant, k₄ (middle panel, compare to Fig. 2A). The SPM was solved in the steady-state, and simulated over time with a step increase in [Ca²⁺]_i from 0.5 to 25 µM at 0.5 s. The capacitance increase displays a slow and a sustained component, but no fast component (bottom panels, compare to Fig. 2A), in qualitative agreement with experimental data (see text). Bottom panel: the temporal development of pools shows that the RRP-state was depleted even before the Ca²⁺ step. The NRP was also partially depleted, leading to a smaller slow component than in the wildtype case (compare to Fig. 2A). B. In the second version of the model simulating syt-1 null data, we assumed that the RRP would be destabilized and thus we increased the k₋₂ by a factor of 10 (top panel, grey dotted line shows the energy landscape in the WT case). Simulating a step increase in [Ca²⁺]_i from 0.5 to 25 µM revealed a large slow burst of release, whereas the fast burst is small and hardly noticeable. The NRP size is even a bit larger than in the WT case (bottom panel, compare to Fig. 2A). Note that, in both versions of the syt-1 null, it is the Ca²⁺-dependent catalyst between the NRP and the RRP which drives secretion and determines its Ca²⁺ sensitivity.

https://doi.org/10.1371/journal.pcbi.1003362.g006

Another way of accounting for the selective loss of the fast burst component in the SPM model is to assume that syt-1 stabilizes the RRP state, by lowering its free energy. This is in line with previous data showing that syt-1 overexpression increases the RRP/SRP ratio [31]. We simulated this by increasing k₋₂ by a factor of 10 (Fig. 6B). Thermodynamically, this corresponds to increasing the energy state of the RRP, plus the subsequent energy barrier. Simulating this situation resulted in only a very small fast burst, followed by a slow burst and a sustained component (Fig. 6B). Under these circumstances, the NRP was a bit larger than in the WT case (Fig. 2A), leading to a slightly larger slow burst. In this version of the model, the properties of the Ca²⁺ triggering step remained unchanged. This might appear unrealistic for a simulation of the syt-1 null, because syt-1 is usually supposed to provide the Ca²⁺ binding sites. However, since most vesicles are present in the NRP, and the NRP-to-RRP transition is rate limiting for the majority of release, the properties of the Ca²⁺ sensor can in fact be changed simultaneously without noticeably changing the kinetics of overall release. In other words, when the majority of vesicles are present in the NRP, the model is pretty insensitive to the details of the RRP-to-F conversion.

We conclude that the SPM offers at least two possibilities for qualitatively accounting for the selective loss of the fast burst component, as observed in the syt-1 null, without assuming parallel release sensors or vesicle pools: either by assuming that the RRP is emptied because of spontaneous fusion, or by assuming that the RRP is destabilized, so that most vesicles reside in the NRP instead.

A Ca²⁺-dependent catalyst for priming

The Parallel Pool Model (PPM) assumed that Ca²⁺-dependent priming (refilling of the SRP) indirectly led to refilling of the RRP through a Ca²⁺-independent process (Fig. 1B). A Ca²⁺-independent SRP-to-RRP inter-conversion was assumed, because this could account for the parallel increase in SRP and RRP size with increasing Ca²⁺ concentrations at rest (Fig. 3C). If merely the forward rate were Ca²⁺-dependent, then the RRP-to-SRP ratio would change with [Ca²⁺]_i, which was inconsistent with the experiment. However, the slow, Ca²⁺-independent RRP refilling reaction made it impossible to account for the slow burst of release by fusion through the RRP state alone. Instead, it was assumed that the SRP-vesicles could fuse directly through a separate pathway, which in turn necessitated a separate Ca²⁺ sensor for this pool. This arrangement seemed to be confirmed when deletion of syt-1 led to a specific deletion of the fast burst of release [17].

We solved the problem of the invariant slow-to-fast release ratio in a different way: by assuming that the NRP-to-RRP conversion is driven by a Ca²⁺-dependent catalyst. This in effect makes both the forward and the reverse reactions Ca²⁺-dependent. Moreover, because the catalyst acts on the transition state (Fig. 1A and Materials and Methods), it follows immediately that the Ca²⁺-dependence of forward and reverse rates are identical, ensuring the invariance of the NRP/RRP ratio. The catalytic process then speeds up refilling of the RRP to the degree that fusing the upstream NRP vesicles after transit through the RRP accounts for the slow burst of release in uncaging experiments, while this transition is slow at rest. This removes the need to assume a separate release pathway for those vesicles. Thus, in our model the ‘slow’ vesicles are no longer releasable, which is why we renamed this state Non-Releasable (NRP). Moreover, the Ca²⁺-dependent NRP-to-RRP conversion speeds up RRP refilling, which makes it possible to explain the faster recovery kinetics upon strong stimulation (Fig. 4) that could not be explained in the PPM. Thus, modeling the Ca²⁺-dependence of priming as a catalytic process explains a range of phenomena. Ca²⁺-dependent refilling of the RRP is a physiologically important mechanism, which ensures that the same signal that causes release from the RRP also speeds up refilling. We also managed to modify the PPM by the incorporation of a Ca²⁺-dependent, catalytic acceleration of RRP refilling (PPM+cat, Fig. S3), which could account for the data in a parallel model with additional parameters. This emphasizes the advantage of modeling RRP replenishment as a catalytic process independent of whether release commences in parallel or not. Moreover, catalysis accelerates priming under high-use conditions while strictly preventing ‘overfilling’ of the RRP, which is characteristic of models with non-catalytic Ca²⁺-dependent priming [12].

Since vesicle priming coincides with SNARE-complex formation [25], [37], the molecular counterpart of the catalyst should be sought amongst Ca²⁺-dependent proteins which have been shown to facilitate SNARE-complex assembly; likely candidates are Munc18, Munc13 and CAPS proteins [23], [38]. Indeed, it was recently suggested that Munc13-1 can reduce the energy barrier for SNARE-complex formation [24], and stimulates both opening and closing of syntaxin [22] implying catalytic action. Furthermore, Munc13/CAPS are Ca²⁺-dependent proteins known to affect vesicle priming [38]–[42]. Reinterpreting these proteins as catalysts for vesicle priming means that they have to be reclassified as enzymes, which might appear provocative or unusual. But it is not hard to see how proteins interacting with the SNAREs might provide an alternative pathway for assembly (and disassembly), by providing a surface stabilizing an intermediate conformation, and such mechanisms are frequently discussed in the literature. Opening and closing of syntaxin might well be stimulated by the action of a catalyst, which stabilizes an intermediate configuration. Another catalyzed step could be the initial formation of the ternary SNARE-complex, which involves the interaction of a very short, transiently formed alpha-helix in synaptobrevin with the SNAP-25:syntaxin dimer [43], [44]. Stabilization of the alpha-helix is a candidate mechanism for the catalysis of vesicle priming.

It is not necessary for the validity of our model that a protein can be found that only acts catalytically without inducing pool size effects. Instead, the catalyst may be a molecule on the plasma membrane, the association to which is necessary for priming, but which is sped up by Ca²⁺. But – unlike our simple assumption of sufficient abundance – its number may be limited, thus limiting the number of RRP vesicles. This might play a role in those cell types (e.g. neurons) where the RRP size appears to be limited by a fixed number of ‘release sites’.

Implications for release in the syt-1 null

We introduced a specific interpretation of the release sensor, as a clamp that can be lifted by Ca²⁺ (Fig. 1). This idea aligns with recent findings and ideas that the fusion trigger consists of syt-1 and complexin in combination [27], [45], [46], even though those proteins might have additional upstream functions as well [36], [47]. Complexin and syt-1 appear to arrest SNARE-complex assembly, effectively setting up an energy barrier for fusion, which is removed when Ca²⁺ binds to syt-1. In order to explain the syt-1 null phenotype we made two separate assumptions. In the first, we assumed that the fusion barrier was removed. This caused depletion of the RRP and effectively converted the NRP into a releasable pool, because the downstream barrier was removed. Fusion in this ‘barrier-less’ model resulted in a ‘slow burst’ followed by a normal sustained component, which qualitatively matches data from syt-1 null chromaffin cells. This follows immediately from the model itself, since fusion from the NRP is the source of the slow burst in the wildtype case. In the other simulation of the syt-1 null, we assumed that syt-1 was responsible for stabilizing vesicles in the RRP, and in its absence the RRP would be nearly empty. As a result, the slow burst of release dominates after Ca²⁺ uncaging. The two models for the syt-1 null are distinguishable based on the size of the slow burst, which is larger in the second model, and the frequency of spontaneous release, which is increased in the first and decreased in the second case. The larger slow burst in the second model fits somewhat better with published data from the syt1 null [17], [31], [36], however it should be noted that syt-1 most likely has several functions in secretion [47], which could cause secondary changes in release amplitude. Therefore, we conclude that both models qualitatively account for the syt-1 null data.

In both versions of our syt-1 null model, the ‘slow Ca²⁺ sensor’ driving release in the syt-1 null is the priming catalyst, which is arranged upstream of RRP and moonlights as a Ca²⁺ sensor for release when the NRP-to-RRP transition becomes rate limiting, as it has previously been suggested [48]. This introduces a specific interpretation of the syt-1 null. In most cases, the slow release component has been assumed to either originate from a separate releasable pool (the SRP in the case of the chromaffin cell), or from an alternative sensor, which only gains access to the release machinery in the absence of syt-1. Our interpretation would explain why the ‘slow Ca²⁺ sensor’ for release – though heavily searched for – thus far could not be identified: because it is also important for fast release and the two sensors are not additive. Investigations of syt-7, which has slower Ca²⁺ (un-)binding kinetics than syt-1 [49] indeed showed that syt-7 also affected fast release in chromaffin cells [50] while its deletion in central neurons was apparently without effect [51]. Thus syt-7 does not seem to fit the description of a parallel sensor in the chromaffin cell, but the situation might be different in some neurons, including the zebrafish neuromuscular junction [52]. A recent study implicated Doc2α in asynchronous release in central neurons, but it cannot be the slow Ca²⁺ sensor itself, because asynchronous release was still Ca²⁺-dependent when Doc2α was mutated to be Ca²⁺-insensitive [53]. Thus, a testable prediction of our model is that the elusive secondary sensor should be found among Ca²⁺-dependent proteins driving priming. The alternative hypothesis of another syt-like molecule, taking the place of syt-1 or syt-2 in their absence, is also possible, but additional Ca²⁺-dependent priming reactions need to be assumed to account for the enhanced recovery of the RRP during elevated Ca²⁺ (Fig. 4D, Fig. S3), which in our model are both direct consequences of the priming catalyst.

Conclusion: Parallel or sequential models?

Our work documents that a sequential model can describe salient wildtype and mutant phenotypes in adrenal chromaffin cells. By this, we do not mean to imply that parallel models are inherently wrong, or even unlikely, and future experiments might revive the need for such models. However, at a point in time where it seems that molecular manipulations and experiments more often than not lead to the suggestion of separate vesicle pools or sensors, we think it is important to point out that small changes in the assumptions behind sequential models might lead to similar phenotypes. Specifically, we have shown that the assumptions of catalytic action of the priming machinery, and stabilization of the RRP by syt-1, remove the necessity of assuming two fusogenic vesicle pools in the adrenal chromaffin cell. It is of note that recent experiments in cultured hippocampal neurons and in the Calyx of Held synapse led to the conclusion that the vesicles fusing during the slow phase of release are recruited to the fastest releasable pool [54], [55], in agreement with our model.

When considering neurons, other factors will contribute to kinetic diversity of release. Chief amongst these is the variable distance between RRP vesicles and Ca²⁺ channels, which might lead to kinetically distinct release phases [56]. The experiments we analyzed here were mostly performed using a spatially homogeneous Ca²⁺ signal (uncaging), which makes it easy to derive pool behavior. Future efforts will be needed to understand whether a sequential model when combined with spatial heterogeneity has explanatory power in the world of neurons.

Kinetic release model for adrenal chromaffin cells

To model the release from chromaffin cells we set up the reaction rate equations, which consist of a set of ordinary differential equations describing the temporal change in the population of each state (see below). We followed the suggestion by Thomas Voets [16] to simplify the scheme by assuming that the size of the Depot is so large, that it does not effectively decrease by the release (infinite Depot). We used the same parameters determined by Thomas Voets to describe the Ca²⁺ sensitivity of vesicle supply from the Depot. We also used the parameters of the release sensor found for the RRP vesicles (three site binding model [16]) to describe the release from the RRP state in our model. The full secretion model is shown in Fig. 2A and the values of all parameters can be found in Table 1. The steady state values were solved analytically under the assumption of constant NRP, RRP, RRPCa, RRPCa₂ and RRPCa₃ (and Depot) sizes at rest, where RRPCa_n represents RRP vesicles with n Ca²⁺ bound.

We suggest that the transition between the NRP and RRP states is catalyzed by a Ca²⁺-dependent catalyst. Therefore, the rate constants k₂ and k₋₂ are functions of the Ca²⁺ concentration. A catalyst acts by increasing the rate of a reaction through the lowering of the activation energy barrier (Fig. 1). Yet a catalyst does not change the equilibrium of a reaction as it accelerates both the forward and the reverse rates simultaneously. Both reactions also occur without catalysis, but with low rates (k₂₀ and k₋₂₀). We suggest that the overall rate constants k₂ and k₋₂ are functions of the un-catalyzed (Ca²⁺-independent) rate constants k₂₀ and k₋₂₀ and functions of the faster, catalyzed (Ca²⁺-dependent) reaction rates k_2cat and k_−2cat. The relative contribution of the catalyzed rate in turn depends on the probability (g(Ca²⁺)) of the catalyst to have bound the required number of Ca²⁺ ions in order to become activated. For simplicity, we assumed that the supply of catalyst is not limited (this could for instance mean that each vesicle bears a catalyst). The overall rate constants k₂ and k₋₂ take the following form:(1.1)(1.2)We assume instantaneous binding of Ca²⁺ to the catalyst, so that the catalyst is in equilibrium with Ca²⁺ at all times. In general, a catalyst might bind n Ca²⁺ ions in order to be activated:The overall dissociation constant is given by:(1.3)For an individual Ca²⁺-binding step:(1.4)The fraction of activated (bearing the correct number n of Ca²⁺ ions) to total catalyst (g) is(1.5)Experimentally, the relative concentration of the active catalyst is difficult to access, whereas the Ca²⁺-levels are well defined. g(Ca²⁺) can be rearranged:(1.6)(1.7)(1.8)For simplicity, it will be assumed that n Ca²⁺ can bind independently and with identical dissociation constants:(1.9)Then g(Ca²⁺) can be further simplified:(1.10)(1.11)In most of our work we assumed the simplest case of a Ca²⁺-cooperativity of one (n = 1). Then equation (1.11) takes the simplified form:(1.12)Since a catalyst does not interfere with the free energy of RRP and NRP (same in the presence and absence of Ca²⁺), the following relationship holds true:(1.13)which implies that(1.14)Thus, it is sufficient to have information about the relative sizes of the NRP and RRP at equilibrium (depend mostly on the k₂₀ to k₋₂₀ ratio), the uncatalyzed re-supply rate of vesicles (k₂₀) and the asymptotic rate of NRP-to-RRP conversion at high Ca²⁺-levels to calculate all four rates.

Estimation of parameters

We estimated the K_D and the k_2cat values of the catalyst from the previous study by Thomas Voets (data points in Fig. 3A [16]). The k_2cat corresponds to the asymptotic rate of the slow component at high Ca²⁺ concentrations. As can be seen from the data points in Fig. 3A, this corresponds roughly to a rate constant of around 20 s⁻¹. According to equation (1.12), the K_D for a cooperativity of n = 1 corresponds to the Ca²⁺ concentration, where g(Ca²⁺) = 0.5 and the slow component has reached 50% of its asymptotic rate (a value of k₂₀+0.5*k_2cat, where we can assume that k₂₀ is negligible). As can be seen from the data points in Fig. 3A, this corresponds to a value of roughly 100 µM. Now, k₂₀ was calculated by using equation (1.1) and investing the observation that at resting Ca²⁺ concentrations (500 nM), the rate k₂ is equal to 0.12 s⁻¹ [32]. Finally, k₋₂₀ and k_−2cat were calculated using equations (1.2) and (1.14) by investing the observations that k₋₂ at resting Ca²⁺ concentration is 0.1 s⁻¹ and at steady state the k₂/k₋₂ ratio is 1.2/1 [32]. All parameters can be found in Table 1.

Numerical integration of differential equations

The kinetic equations for the model presented in Fig. 2A take the following form:(2.1)(2.2)(2.3)(2.4)(2.5)(2.6)where k_1Max = 55 fF/s is a constant because of the assumption of constant Depot size [16]. The system of differential equations was integrated numerically using a fifth order Runge Kutta method with Cash-Karp coefficients and adaptive step size. For this purpose, we used a custom made macro in IGOR Pro (version 6.22A, WaveMetrics Inc.). The procedure of Runge Kutta was adapted from Numerical Recipes [57].

Kinetic analysis

The sizes and time constants of the fast and slow release components were determined by fitting a sum of exponentials to 5 s of simulated cumulative release. The sustained release that is typically observed after the burst phase of secretion was approximated by a line. The fit function used had the following form:(3.1)A₀ was fixed to the baseline value immediately before the onset of the Ca²⁺ step and t₀ was set to the inflection point of the cumulative release after stimulus onset. All other parameters were free. A₁ and τ₁ correspond to the amplitude and time constant of the fast, and A₂ and τ₂ to the size and time constant of the slow burst component, respectively. A₃ is the slope of the sustained component and t is time. The rate constants displayed in Fig. 3A are the inverse of the time constants.

Refilling after depletion of the fast component in chromaffin cells

The temporal change of the two kinetic components during recovery of the fast component of release was characterized and compared to the experimental values presented in [32]. In the experiments, Voets and colleagues used a paradigm of voltage depolarizations that selectively depleted the fast component. Subsequently, the sizes of both the fast and the slow component were probed by a flash experiment (which releases both components) at varying inter-stimulus intervals [32]. Unfortunately, the precise Ca²⁺ signal at the site of the vesicle in response to the depolarization is not known. In order to test whether our model in principle would allow for a selective depletion of the fast- without major depletion of the slow component, we resorted to a simplified stimulus and probed the effect of a Ca²⁺ step from 0.5 µM to 25 µM on the NR and R states. In order to mimic the experimental conditions of the previous study we were looking for a point in time after onset of the stimulus, where the majority of fast vesicles were depleted, whereas the majority of the slow component's amplitude was still left intact, which was satisfied at the condition shown in the top panel of Figure 3D. Ca²⁺ was then stepped back to resting (0.5 µM) levels and the system was allowed to re-equilibrate for various inter-stimulus intervals before a second step to 25 µM Ca²⁺ was applied. Then, the sizes and time constants of the two components were again obtained by the fitting of exponentials (see above).

Model comparison

For the comparison between the PPM and the SPM, the data presented in [25] were evaluated. We made use of the fact that that pre- and post-flash Ca²⁺ levels were similar in all conditions, allowing us to solve the model's differential equations under the assumption of constant k₂, k₋₂ and k₃ (Fig. 2A). Therefore these values are merely functions of the genetic alteration. Since we are only interested in the kinetics of burst release, we further disregarded refilling, arriving at the simple model:The kinetic equations can be written in matrix form:(4.1)Only the fused vesicles (F) contribute to the cell's increase in membrane capacitance (C_M) during secretion.

To arrive at the initial conditions we assumed a fixed amount of vesicles in the burst phase (the sum of [NRP] and [RRP]), V_tot. We further assumed that the NRP and RRP states are in equilibrium and that the fusion rate is essentially zero (k₃ = 0) before stimulation, such that:(4.2)(4.3)(4.4)After solving the differential equations analytically one can express the cellular capacitance change (ΔC_M, which is equal to F(t) when calculating in capacitance units) in terms of the kinetic rate constants k₂, k₋₂ and k₃ (same rate constants as depicted in Fig. 1):(4.5)with(4.6)Accordingly, the size of the fast and slow release components and their time constants are:(4.7)(4.8)(4.9)(4.10)In Figure 5C, the equation (4.5) was used to fit the total burst (1^st second) of release from chromaffin cells expressing wildtype and mutant synaptobrevin-2, using a custom made macro in Igor Pro 6.22A (Wave Metrics). The onset of the capacitance increase was determined manually.

The data of the measured time constants and release amplitudes following mutation of synaptobrevin-2 shown in Figure 5 were taken from Table S1 in [25]. The lines in Figure 5 represent model predictions with constant k₂ and k₋₂ while varying k₃. The values of k₂ and k₋₂ were determined by a global fit of the model to the data sets shown in Figure 5. The behavior of the Parallel Pool Model was probed by simulating responses to Ca²⁺ uncaging (from 0.5 µM to 25 µM) for 5 s while varying the fusion rate (k_RRP = 1/τ_RRP) in the following model:(4.11)The cumulative release was subsequently fit using the above routine. The starting values were chosen to be [RRP]₀ = 61.3 fF and [SRP]₀ = 52.3 fF.