Abstract
Synaptic vesicle fusion is a crucial step in the neurotransmission process. Neurotransmitter-filled vesicles are pre-docked at the synapse by the mediation of ribbon structures and SNARE proteins at the ribbon synapses. An electrical impulse triggers the fusion process of pre-docked vesicles, leading to the formation of a fusion pore and subsequently resulting in the release of neurotransmitter into the synaptic cleft. In this study, a continuum model of lipid membrane along with lubrication theory is used to determine the traverse time of the synaptic vesicle under the influence of hydrodynamic forces. We find that the traverse time is strongly dependent on how fast the driving force decays or grows with closure of the gap between the vesicle and the plasma membrane. If the correct behaviour is chosen, the traverse time obtained is of the order of a few hundred milliseconds and lies within the experimentally obtained value of approximately 250 ms (Zenisek D, Steyer JA, Almers W. 2000 Nature 406, 849–854 (doi:10.1038/35022500)). We hypothesize that there are two different force behaviours, which complies with the experimental findings of pre-fusion docking of synaptic vesicles at the ribbon synapses. The common theme in the proposed force models is that the driving force has to very rapidly increase or decrease with the amount of clamping.
Keywords: synaptic vesicle fusion, hydrodynamics, vesicle docking, lubrication theory, SNARE proteins
1. Introduction
For a continuous neurotransmission process, neurotransmitter-filled synaptic vesicles are resupplied and primed for fusion at the ribbon synapse. The vesicles are approximately 40–50 nm in diameter and are highly fusogenic [1]. These vesicles are docked at the synapse by ribbon structures [2,3] and proteins of the SNARE (soluble N-ethylmaleimide-sensitive factor attachment protein receptor) family [4–8]. Specifically, there are three SNARE family proteins involved in the docking process: synaptobrevin 2 (Syb), syntaxin 1 (Syx) and SNAP-25 [9]. Syb is attached to the synaptic vesicle at one end, and is referred to as ‘v-SNARE’ [10]. Attached to the plasma membrane are the ‘t-SNAREs’: Syx and SNAP-25 [11,12]. During the pre-fusion docking process, the ribbon structure drives the vesicle to the synapse so that v-SNARE and the t-SNAREs are within striking distance of each other, allowing clamping to initiate [13].
Using fluorescence microscopy, Zenisek et al. [13] showed that the time to dock a vesicle is about 250 ms. From this, they concluded that it takes about 250 ms for the pre-fusion docking machinery to close the gap between the membranes from approximately 20 nm [13] to approximately 2 nm [14]. The separation of approximately 2 nm is often considered to be the distance below which electrostatic and hydration pressures dominate [14–17].
As the membranes get closer than approximately 2 nm, Ca2+ can play an important role in leading the system towards fusion [18–20]. At this stage, the vesicle is referred to as being fusion ready, and it waits for an electrical impulse to initiate the fusion process [13,21]. During the pre-fusion docking process, the ribbon structure drives the vesicle to the synapse so that v-SNARE and the t-SNAREs are within striking distance of each other, allowing clamping to initiate [13]. Neuronal ‘porosomes’ have also been proposed to be present at the clamping site of these excretory cells [22–25]. The interaction distances between the porosomes and the synaptic vesicles are believed to be less than 2 nm, and this interaction is proposed to be crucial for vesicle fusion.
This paper focuses on the dynamics of docking beyond this separation (greater than 2 nm), where the dominant impeding force for the docking comes from hydrodynamics. As the vesicle is pulled towards the plasma membrane, water has to be squeezed out. For this to occur, an outwards pressure gradient is required. This pressure deforms the membranes and affects the traverse time. We emphasize that our model does not apply for separations of less than 2 nm because electrostatic and hydration pressures dominate at this length scale and local structures such as neuronal porosomes can significantly affect the fusion process.
In this paper, we model this phase of vesicle docking where hydrodynamics forces are dominant. Our results show that the traverse time is very sensitive to the time history of the driving force. To agree with the time measured by Zenisek et al. [13], the driving force should either decay or rise very rapidly with the gap closure. The length scale over which this rapid force change takes place is of the order of a few nanometres.
2. Material and methods
2.1. Model: geometry, fluid flow and membrane mechanics
The synaptic vesicle is assumed to be a rigid sphere of radius R. The entire compliance of the system comes from the plasma membrane of the neuron. The justification for this is that cryo-electron micrsocopy images of synaptic vesicles docked near the plasma membrane show very little deformation [26,27] in the vesicles. This could be due to a high osmotic pressure across the synaptic vesicle membrane [28], pertaining to the high concentration of neurotransmitter inside it. In addition, the vesicle is expected to be stiffer because it is much smaller than the plasma membrane. Therefore, our model lumps all the deformation to the much larger plasma membrane, which, before deformation, is modelled as a flat circular disc with radius l. It is held at the edge 
by a tension force, which is the pretension, 
, of the neuron. Varying this pretension will allow us to change the compliance of the system. The plasma membrane is modelled as a lipid bilayer using a continuum theory developed by Jenkins [29,30] and Steigmann and co-workers [31,32]. The continuum theory has been used in the literature for mechanistic modelling of the lipid membrane-associated processes in red blood cell shape analysis [29], receptor-mediated endocytosis [33], micropipette aspiration of and curvature sorting of proteins [34–37], two-component lipid membrane systems [38,39], vesicle adhesion [40–43] and synaptic vesicle fusion [14,44,45].
The fluid layer between the vesicle and the plasma membrane is assumed to be sufficiently thin so we can use Reynold's equation in elasto-hydrodynamic lubrication theory to model the flow [46–51]. This theory is well suited for the present problem for the following reasons.
(1) The flow between the synaptic vesicle and the plasma membrane is in the low Reynold's number regime, hence it can be assumed to be laminar.
(2) The film thickness is relatively small compared with the size of the synaptic vesicle.
As in Zenisek et al. [13], we assume that the driving force to overcome hydrodynamics is provided by the ribbon structure and SNARE complex. Because the vesicle is rigid, the driving force can be represented by a point force F acting on the south pole of the vesicle, as shown in figure 1a,b. The magnitude of this force depends on the number of SNARE complexes, n, acting on the vesicle, which can vary from 
to 11 [52–56]. This membrane model is coupled with a hydrodynamic solver based on lubrication theory. This allows us to determine the flow as well as the traverse time of the vesicle given the time history of F.
Figure 1.
(a) Synaptic vesicle docking mediated by the ribbon structure and SNARE proteins; (b) the driving force is represented by a single force acting on the south pole of the vesicle; and (c) a deformed plasma membrane under the hydrodynamic and driving force. Horizontal arrows indicate the direction of fluid flow. (Online version in colour.)
2.2. Lubrication theory
The problem is axisymmetric. For axisymmetric flow, Reynold's equation is [57]
 |
2.1 |
where r is the radial distance from the symmetry axis (z-axis) of the vesicle (figure 1c), 
is the gradient operator in cylindrical coordinates, 
is the thickness of the fluid film, 
is the fluid pressure and 
is the dynamic viscosity of water. The film thickness 
is related to the deflection of the membrane 
and the shape of the sphere by
 |
2.2 |
where 
is the vertical separation between the lowermost point on the sphere and the plasma membrane at 
, where the membrane deflection 
is zero, as shown in figure 1c. We assume l to be much greater than the region where pressure is significant. The second term on the right-hand side of equation (2.2) approximates the local shape of the sphere by a paraboloid.
2.3. Elastic deformation of the plasma membrane: calculation of w
The deformation of the membrane is coupled to the flow via the fluid pressure, 
, which acts normal to the surface. The governing equations of the lipid membrane in the full form can be used to obtain the deformation in the plasma membrane for a general pressure distribution. For the present work, the governing equations have been linearized under the small deflection assumption (see the electronic supplementary material for details). The linearized equation is found to be
 |
2.3 |
where 
is the bending rigidity of the lipid membrane. Equation (2.3) is solved analytically (see the electronic supplementary material for details) and the deflection 
is found to be
 |
2.4 |
where 
and 
are the modified Bessel functions of the first and second kind, respectively.
2.4. Numerical solution
Details of the numerical methods are given in the electronic supplementary material. Briefly, the rigid sphere (vesicle) is moved towards the plasma membrane at a prescribed rate, 
. For a given 
and 
, equation (2.1) is solved iteratively to determine the pressure distribution, 
, at a given time. The force F is calculated using the force balance equation
 |
2.5 |
3. Results
Much insight can be gained by considering the special case of an undeformable plasma membrane. This case provides a lower bound for the traverse time. Also, because the solution is exact, we can study analytically how the traverse time depends on the variation of clamping force with time. Details of the solution are provided in the electronic supplementary material; here we state the key results.
3.1. Undeformable plasma membrane limit
, 
, denotes the time history of the force, where n is the number of SNARE complexes and 
is the force exerted by one SNARE complex. Let us first assume that the force acting on the vesicle is a constant independent of time, that is,
 |
3.1 |
where 17 pN is the peak force exerted by one SNARE complex [58]. The traverse time 
is found to be (see the electronic supplementary material for details)
 |
3.2 |
where 
and 
are the separations between the vesicle and the plasma membrane at 
and 
, respectively. The presence of the logarithmic function indicates that the traverse time is insensitive to the initial separation. Figure 2 shows the dependence of the traverse time on the number of SNARE complexes. For 
, it is about 200 ns, which is six orders of magnitude smaller than the experimental value of 250 ms [13].
Figure 2.
Traverse time of the vesicle for a constant force. (Online version in colour.)
Experiments suggested that the SNARE clamping force is not a constant, but varies with distance between the clamps [58,59]. This motivates us to use a clamping force that varies with distance between the sphere and the membrane; for the present case, this distance is 
. We assume that
 |
3.3 |
where m governs the rate of decay of the force (larger m implies faster decay) and 
, the separation between the synaptic vesicle pool and the plasma membrane [13] is used as a scaling parameter for the separation and 
. Equation (3.3) states that the clamping force is maximum when the SNARE complex zipping starts and reduces to a very small value towards the end of docking. The traverse time for this particular force history is (see the electronic supplementary material for details)
 |
3.4 |
Figure 3 plots 
versus the number of SNARE complexes n. It shows that the traverse time is very sensitive to the rate of decay of the SNARE's clamping force. To agree with the experimental result of Zenisek et al. [13], the force has to decay rapidly with separation, with m between 5.5 and 6.5.
Figure 3.
Traverse time of the vesicle versus the number of SNAREs using equation (3.4). m governs the rate of decay of the SNARE force with distance. (Online version in colour.)
3.2. Deformable membrane
The result in the previous section shows that the decay of the clamping force is the dominant factor controlling the traverse time. For example, increasing the number of SNARE complexes or the clamping force will not change the traverse time by several orders of magnitude. However, one may still argue that membrane deformation can also increase the traverse time; here we study this possibility by solving equation (2.1) in conjunction with equation (2.4). All calculations are performed with 
Pa s (kinematic viscosity of water) and R = 20 nm. We fixed the bending stiffness of the membrane to be 
[60]. This means that the compliance of the system depends on the pretension 
. We vary the decay constant, m, the number of SNARE complexes n and 
, which controls the compliance of the system.
Figure 4 plots the logarithm of the traverse time 
versus the decay exponent m, for 
and for three different plasma membrane tension values, 
. Consistent with the undeformable membrane case, the traverse time is extremely sensitive to the decay constant m. This result is in agreement with the result in our previous section, that is, the force to dock the vesicle must be derived from the posterior part of the SNARE complex. In particular, varying the complaince of the system by changing 
has negligible impact on the traverse time compared with that of m.
Figure 4.
Traverse time with varying decay exponent, for 
. (Online version in colour.)
Figure 5 shows the effect of varying the number of SNARE complexes, n, on the traverse time, with 
and three different values of 
. The traverse time is a straight line in a log–log plot, indicating that the traverse time is inversely proportional to n, which is consistent with equation (3.4) (undeformable membrane). Again, variation in membrane pretension 
has little effect on traverse time, indicating that the compliance of the system plays a secondary role in controlling the traverse time. Clearly, the traverse time is much more sensitive to the decay constant than the number of SNARE complexes and is of the same order of a few milliseconds.
Figure 5.
Traverse time with a varying number of SNAREs. (Online version in colour.)
Shi et al. [61] have shown that one SNARE complex is sufficient for fusion and three SNARE complexes are needed to keep the nascent fusion open. Therefore, it is interesting to note that figure 3 shows that the traverse time decreases approximately by a factor of 2 as the number of SNARE complexes n increases from one to three; after that increasing n does not significantly change the traverse time. This behaviour is also consistent with figure 5, which shows that the docking time is proportional to 
.
The effect of pretension 
is shown in figure 6. As one increases the pretension, the traverse time very slowly approaches the undeformable membrane limit given by equation (3.4). Our numerical result shows that the traverse time is approximately constant for 
. This is about three orders of magnitude smaller than the rupture strength (
; [62,63]) of lipid membranes. This result again supports the fact that compliance or pretension is not an important factor compared with the decay behaviour in determining the traverse time of the synaptic vesicle.
Figure 6.
Traverse time with varying number of pretension, for number of SNAREs (a) 
(b) 
and (c) 
. (Online version in colour.)
As expected the traverse time of the synaptic vesicle is always higher for deformable membranes. Our numerical result shows that, when the synaptic vesicle is docked approximately 2 nm from the plasma membrane, the deformation in the plasma membrane was negligible. This result is in agreement with the cryo-electron microscopy images of the synapse, which shows negligible deformation in the plasma membrane, with the synaptic vesicle docked and ready for fusion [26,27].
3.2.1. Different force model
From the comparison of the rigid and compliant models of the plasma membrane, a very clear conclusion can be drawn that the traverse time is extremely sensitive to how the driving force varies with distance between clamps. The elasticity of the membrane and the magnitude of the force have relatively small effects on the traverse time. The force behaviour described in the previous analysis is in agreement with the force behaviour obtained from SNARE un-zipping experiment [58,59]. However, it is possible that the SNARE complex zipping behaviour can be very different and the ribbon structure might drive the vesicle for a part of its traversal. In the light of this, we propose to study another force model. As we have demonstrated that the compliance of the membrane has little effect on the traverse time, all results in this section are obtained using the undeformable plasma membrane limit where an exact solution can be found.
The force behaviour is assumed to have the form
 |
3.5 |
This force model is based on the idea that, as the SNARE complex zips, the force increases. The clamping force reaches a maximum value 
when the synaptic vesicle is within 
proximity of the plasma membrane, after which strong repulsive forces (electrostatic and hydration forces) act to resist docking (not modelled by equation (3.5)). The parameter m in equation (3.5) controls the rate of increase of the clamping force.
Figure 7a plots the traverse time versus the number of SNARE complexes for three different values of m. It should be noted that a change in the value of m impacts the traverse time much severely than changing the value of SNARE complexes. In order to reach a traverse time of 250 ms, we find 
. The variation of force versus h0 is shown for 
in the insert of figure 7a, indicating that the SNARE force changes in a matter of few nanometres. For most of the traverse, the vesicle is driven towards the plasma membrane by the ribbon structure, and for the last stretch SNARE proteins take over the task.
Figure 7.
Traverse time of synaptic vesicles versus (a) the number of SNARE complexes when 
and (b) 
when the number of SNARE complexes, 
. (Online version in colour.)
Figure 7b shows the effect of varying the peak force which can be applied by the SNARE complex on the traverse time. The peak force was varied in the range as reported in the literature [58,59]. The impact is not yet as big as varying the exponent of force behaviour.
4. Conclusion
Using elasto-hydrodynamic lubrication theory, we determine the docking time of a synaptic vesicle against the plasma membrane of a neuron. This docking is driven by the force exerted by the SNARE complexes and is resisted by hydrodynamics.
The main conclusions are as follows.
(1) The decay of the clamping force of the SNARE complex is the most important factor in determining the traverse time of the synaptic vesicle.
(2) The effect of membrane pretension and the number of SNARE complexes is negligible in comparison with the impact of the decay in the clamping force.
(3) The rapid clamping force decay indicates the possibility that the posterior segment of the SNARE complex is the force-generating machinery.
(4) The clamping force decays down to very small values by the end of the docking process, when the vesicle is fusion ready. This result suggests that other force-regulating machineries are needed to bring the synaptic vesicle to the plasma membrane for exocytosis. Complexin (Cpx) [64–68] and synaptotagmin (Syt) [69] are believed to be a force-regulating agent that initiates the further clamping of the SNARE complex on the arrival of the electrical impulse. Recall that our model is only valid for separations of more than 2 nm. Beyond this separation, other interactions such as hydration pressure, electrostatics, Ca2+ ions and porosomes will play a major role in the fusion process.
Supplementary Material
Acknowledgements
We acknowledge Dr Patricia Bassereau at Institut Curie, Paris, France, for her valuable discussions, which formed the basis of the present work. We thank Anand Jagota and Maria Bykhovskaia for discussions and inputs.
Data accessibility
The theoretical framework and computational scheme supporting this study are provided as the electronic supplementary material accompanying this paper.
Authors' contributions
C.Y.H. and P.S. participated in formulating the problem, generating the solution strategy, interpreting the results and drafting the manuscript. P.S. developed the computational framework and generated the results.
Competing interests
We declare we have no competing interests.
Funding
C.Y.H. and P.S. acknowledge support by the National Institutes of Health under award no. R01MH099557.
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Supplementary Materials
Data Availability Statement
The theoretical framework and computational scheme supporting this study are provided as the electronic supplementary material accompanying this paper.