In this study, we have considered how the strength of adhesion between a vesicle and a plasma membrane would affect bending of the vesicle and the tension at the vesicle–membrane contact region. We have obtained a non-trivial result that strong adhesion would produce modifications in the vesicle membrane curvature resulting in compressive forces within the vesicle–membrane contact region, and this, in turn, may favour a metastable state with a transiently opened pore, that is, a ‘kiss-and-run’ fusion mode. As such, this is the first study, to our knowledge, that develops a mechanical model for the ‘kiss-and-run’ fusion mechanism.
The effects of local modifications in membrane curvature on the fusion process have been extensively studied (see Zimmerberg & Kozlov [60
] for a review). In particular, it was demonstrated that Ca2+
binding to the Ca2+
fusion sensor synaptotagmin induces high positive curvature in target membranes [61
], which could trigger membrane rupture and pore opening. However, the question of possible deformations of the vesicle owing to the formation of the vesicle–membrane contact region has not been explored. We show here that even very subtle deformations may produce compressions in the contact region which may affect the character of the fusion.
Our study explored the relationship between the dimensionless adhesion, α, and tensions in the vesicle–membrane contact region. In agreement with a previous study [52
], we find if adhesion energy is less than a critical value, (α < 1), the SNARE complex will dock the vesicle onto the membrane with negligible change of shape. If adhesion exceeds this critical value (α > 1), we predict the formation of a distinct contact region with compressive in-plane forces in the membrane; the stronger the adhesion the stronger the compression. It is a non-trivial result that the nature of the in-plane force in the contact region is compressive, and thus it will not by itself favour pore opening or expansion. This result agrees with an experimental study [62
], which used spectroscopy methods to demonstrate that tension decreases during fusion and eventually may become negative (compressive).
Furthermore, we predict that there exists a second critical value, such that if adhesion energy exceeds it, a partially fused state will be pinned and stabilized. We have obtained a family of equilibrium solutions for fixed rp/R (), which predicts the character of the fusion process upon opening a pore. For a typical size of a synaptic vesicle (b, dotted line), if α < ~2, an opened pore will be unstable, since there is no stable solution available to trap a partially opened pore. Thus, the kiss-and-run mode will not be possible. Let us consider now a stronger adhesion (α > ~2). For any fixed α, we have a family of solutions for different values of rp/R. If rp/R is sufficiently large (top left region), the pore is unstable and the fusion proceeds in a full collapse mode. However, if rp/R lies in the top right region, we find that the total potential energy decreases as rp decreases. That is, the pore will tend to fluctuate in a biased way towards reducing its size. However, it may be prevented from immediately resealing by short-range hydration repulsion (not included in our model). Thus, if a pore opens for α > ~2, we expect that it will fluctuate and eventually reseal, and may in the interim release the vesicle's contents. Larger pores may also reach the unstable boundary during these fluctuations thus being transferred to the region of collapse. The larger the value of α, the higher the probability that the pore will reseal instead of reaching the unstable boundary. Thus, increasing molecular adhesion would favour the kiss-and-run release mode.
Thus, our model suggests that the balance between kiss-and-run and full collapse fusion can be regulated by adhesion strength. This finding has important implications for understanding the fusion process. It is generally believed that stronger adhesion forces would monotonically favour fusion, and thus the fusion clamp mechanism corresponds to reducing adhesion, for example, by means of partial SNARE unzipping. Thus, models have been developed that explain the clamping of fusion by complexin via its interfering with full zippering of the SNARE complex [63
]. Our results demonstrate that, in fact, fusion can be clamped by increasing the adhesion, since for sufficiently large adhesion the compressive force can prevent the pore expansion.
Our model predicts that the adhesion forces driving the full collapse fusion most efficiently will correspond to the values of α somewhere in the range between 1 and 2. It is thus of interest to estimate α for a typical synaptic vesicle. This estimate can be made based on the expression α
, where the work of adhesion Wad
is the adhesive energy per unit area of the adhesive region, c
is the membrane bending stiffness, and R
is the vesicle size. The latter estimates are the easiest to obtain, with bending stiffness being in the range 10–20 kB
], and synaptic vesicle radius being in the range 15–25 nm [64
]. Estimating a value for Wad
, however, is far from trivial. Several studies employed atomic force microscopy (AFM) to evaluate the forces exerted by SNARE proteins on the membrane–vesicle complex [68
]. The estimate for the SNARE zippering energy obtained by AFM is 35 kB
] or possibly larger, given that an additional energy barrier has to be overcome as the distance between lipid bilayers decreases [73
]. This energy serves to overcome a substantial repulsion energy of membrane bilayers which was estimated to be of the order of 40 kB
]. Thus, although the adhesive and repulsive energies themselves are quite large, the regulation of fusion itself is expressed by the parameters Wad
and α and occurs over a delicately balanced small range of energies (just a few times higher than the thermal noise). A reasonable estimate of this energy was provided in Abdulreda and co-workers [69
], showing that the SNARE complex would reduce the barrier for lipid merging by 1.3 kB
T. Similarly, it is hard to obtain an accurate estimate of the membrane–vesicle contact region. Electron tomography studies [76
] sometimes show a contact of approximately 10 nm. However, this can only be taken as an upper limit for the contact region, since the spatial resolution is not sufficient to evaluate this accurately, and merging of membrane and vesicle electron density does not necessarily mean that lipids are in real contact. A more accurate estimate was obtained employing a combination of AFM and electron microscopy [79
], and it gave a linear diameter of a porosome of 4–5 nm. Thus, accepting 5 nm diameter for the adhesive region, bending stiffness of 15 kB
T, the synaptic vesicle radius of 20 nm, and Wad
of 1.3 kB
T, we obtain the estimate of α = 1.8, that is within the suggested range.
It is important to note that since α is proportional to WadR2
, where R
is the radius of the vesicle, our model predicts that fusion of larger vesicles is more likely to involve the transient metastable pore-opened state, given that the same work of adhesion is applied. This agrees with a demonstrated transient fusion state at neurosecretory adrenal cells [32
] with larger vesicles (approximately 200 nm diameter), as opposed to fusion of smaller synaptic vesicles (30–50 nm diameter) which mostly involves full collapse, with kiss-and-run mode being detectable only under a very narrow range of experimental conditions. Of course, one can argue that it is harder to detect kiss-and-run events at synapses. However, analysis of shapes and sizes of quantal events at adrenal chromaffin cells demonstrated that events with a pre-spike foot, those involving a transient metastable pore opening prior to full collapse, are larger than events without a foot [80
]. This observation agrees with the prediction of our model that fusion of larger vesicles is more likely to involve a metastable transient state with a partially opened pore.
It has to be noted that we do not model explicitly the individual SNARE complexes. Instead, our model uses a ‘smeared’ or distributed representation of adhesion forces applied by the SNARE complexes. The advantage of this ‘smeared’-specific energy representation of the adhesive action of SNARE is that it results in a tractable axisymmetric problem formulation and clear predictions in terms of a single dimensionless parameter. This simplification is also made by necessity; although the general formulation of our model can accommodate discrete forces, how many SNARE complexes per vesicle–membrane complex mediate fusion is still a matter of debate. Recent studies of model systems suggest that two or three SNARE complexes may be sufficient to trigger fusion [81
], and that fusion can even be activated by a single SNARE complex [84
]. However, it is still unclear whether this occurs in vivo
. Importantly, it was demonstrated that the fusion process can be dramatically accelerated when a larger number of SNARE complexes (5–10) hold the membrane–vesicle complex together [86
]. Thus, although fusion can be potentially triggered by only a few SNARE complexes, it is likely that action potential release in vivo
is mediated by a larger number of SNARE complexes per vesicle–membrane complex, that is, in the order of 5–10.
Our model represents adhesion as forces uniformly distributed over the membrane–vesicle contact region. This is a reasonable approach if we assume that a number of SNARE complexes (5–10) are distributed over the perimeter of the contact region. However, if the fusion is mediated by only 1–3 SNARE complexes, discrete adhesion points should be introduced. In this case, we expect that the two main conclusions of our model; that forces in the contact region are compressive and that large adhesion can stabilize an opened pore, will survive qualitatively though they may change quantitatively.
Since the adhesion strength will ultimately depend on the number of zippered SNARE complexes, we can hypothesize a scenario where a vesicle would be initially docked by a small number of SNARE complexes, and at that point it can either fuse spontaneously or form extra SNARE complexes strengthening adhesion forces, possibly with a participation of Ca2+-unbound/partially bound synaptotagmin and complexin, and thus clamp the fusion. A subsequent action potential would then trigger ‘unclamping’ and stimulate pore opening. Our model provides quantitative support for such a scenario, suggesting that increasing adhesion energy promotes fusion clamping.
Finally, we would like to discuss whether the negative (compressive) force in the contact region predicted by our model is likely to significantly perturb the membrane curvature and thus promote the stalk formation and the fusion process. More specifically, we ask whether the compression is sufficient to buckle the contact region. According to Timoshenko & Woinowsky-Krieger [88
], the critical buckling force (per unit length) for a circular plate is:
is the bending stiffness of the plate, a
is the radius of the plate and k
is a numerical factor and is 3.83 for the clamped boundary condition and 2.16 for the simply supported boundary condition. In our case, the D
should be interpreted as c
is the contact radius rc
, i.e. the normalized critical compressive force is:
As the adhesion energy increases, rc
increases and thus Ncr
decreases. Since the compressive force at the contact edge increases with adhesion, buckling is more likely to occur as the adhesion increases. Using k
= 3.83 (clamped condition), for α
= 5, ncr
= 18.5 and
; for α
= 17, ncr
= 11.0 and
. Using k
= 2.16 (simply supported condition), for α
= 5, ncr
= 5.9 and
; for α
= 7, ncr
= 4.9 and
. That is, even for the weaker simply supported condition, the critical value of adhesion required to buckle the contact region is greater than 5. This quantity is significantly larger than the value required for a transition from full collapse to a ‘kiss-and-run’ mode (approximately two for synaptic vesicles, as discussed above). Our model would therefore predict that in-plane compression in the contact region is insufficient to cause buckling for the smaller vesicles that undergo full collapse, but may play this role for larger neurosecretory vesicles.