Mechanics of post-fusion exocytotic vesicle

Abstract

Exocytosis is an important cellular process controlled by metabolic signaling. It involves vesicle fusion to the plasma membrane, followed by the opening of a fusion pore, and the subsequent release of the vesicular lumen content into the extracellular space. While most modeling efforts focus on the events leading to membrane fusion, how the vesicular membrane remodels after fusing to plasma membrane remains unclear. This latter event dictates the nature and the efficiency of exocytotic vesicular secretions, and is thus critical for exocytotic function. We provide a generic membrane mechanical model to systematically study the fate of post-fusion vesicles. We show that while membrane stiffness favors full-collapse vesicle fusion into the plasma membrane, the intravesicular pressure swells the vesicle and causes the fusion pore to shrink. Dimensions of the vesicle and its associated fusion pore further modulate this mechanical antagonism. We systematically define the mechanical conditions that account for the full spectrum of the observed vesicular secretion modes. Our model therefore can serve as a unified theoretical framework that sheds light on the elaborate control mechanism of exocytosis.

Introduction

An emerging feature of cancer is a derailed membrane trafficking process (1, 2). Mounting evidence has established a link between exocytosis and invasive tumor growth (1, 3). Exocytosis secretes cellular content into extracellular space, which is a fundamental process of normal cellular metabolism and communication between cells (e.g., at neuronal synapse) (4). Uncontrolled exocytosis in cancer cells results in excessive secretion that relays crucial information for fostering growth, migration, matrix degradation, and tumor development (1). It is thus of great interest to understand how exocytosis works normally and why it goes wrong. Addressing these questions may thus provide a window of understanding cancer and tumor development. In this regard, we can learn from the established choreography of exocytosis in neurons, chromaffin cells, and other cell lines.

Exocytosis involves sequential episodes of a secretory vesicle fusion with the plasma membrane, followed by opening of a fusion pore, and the subsequent releasing of the lumen content (4). While seminal theoretical efforts have been dedicated to the events that lead to membrane fusion (5), the fate of post-fusion vesicle is not comprehensively addressed. Experiments, on the other hand, define the fusion pore as a critical bifurcating point that can evolve in two distinct ways. In the so-called “full–collapse fusion” pathway, a fusion pore expands, which collapses the secretory vesicle into the plasma membrane and causes complete release of the luminal content (6). Alternatively, a fusion pore can shrink, triggering vesicle fission. This phenomenon is termed “kiss-and-run”; it is believed to be a mechanism for efficient membrane recycling at synapses without de novo regeneration (6). Full-collapse fusion and kiss-and-run coexist as two modes of vesicular secretions in many systems (4, 6). Under certain conditions, cells can switch to one mode at the expense of the other as an adaptive response to cope with signaling needs (7–9). Thus, the fate of the post-fusion vesicle dictates the nature of cellular secretion and the function of exocytosis.

A multitude of proteins, lipids, and second messengers (e.g., Ca²⁺) govern the different fates of post-fusion vesicles and hence the inter-conversions between them (8, 9). These key components typically interact in a complicated manner, making it difficult to pinpoint the exact roles of the individual players, precluding mechanistic insights. Instead of focusing on the molecular details, one way to better understand this highly entangled situation is to take a top-down approach. Whatever the molecular activities of these players are, they must ultimately manifest as mechanical action upon the membrane, as fusion pore dynamics is ultimately a mechanical process. Mechanics of the post-fusion vesicle, however, has not yet been theoretically investigated in a systematic fashion.

In this work we take on a modeling effort with the purpose of delineating the mechanical conditions that support full-collapse fusion and kiss-and-run. Our results suggest a tripartite relationship between the vesicle size, the membrane stiffness, and the osmotic pressure in dictating different fates of a post-fusion vesicle. Increasing membrane stiffness promotes full-collapse fusion, while intravesicular pressure that swells the post-fusion vesicle drives vesicle fission. Additionally, vesicle size modulates the relative contributions of membrane stiffness and intravesicular pressure: smaller vesicles favor full-collapse while larger ones tend to pinch off from the plasma membrane. Our findings identify the constraints for the mechanical control of post-fusion vesicles, from which functions of individual exocytotic players can be inferred. Our work thus sheds some light on the control mechanism of exocytosis, and its possible role in tumor development.

Model development

The model starts with a vesicle that has already fused to the cell membrane with a small opening pore (Figure 1A). The vesicle in consideration here is typically on the order of ten to a few hundred nanometers in diameter, whereas the model approximates the outer cell membrane by a large membrane vesicle of 20 microns in diameter. We study the subsequent shape changes of the post-fusion vesicle under the influence of membrane mechanics, including surface tension, bending modulus, and intravesicular pressure (Figure 1B). The model adopts the Helfrich-like formulation (10), and depicts the membrane energy J as Eq. (1):

$$J(M)=\frac{\kappa }{2}{\oint }_M{H}^{2}dS+{\oint }_M\sigma dS+{\int }_M{p}_{0}dV+{\int }_{\mathit{\text{Intravesicle}}}pdV$$ Eq. (1)

where H is the mean curvature; the membrane bending modulus κ scales the energy penalty for membrane deformation, σ is the surface tension, p₀ is the osmotic pressure exerted on the entire membrane, and p is the intravesicular pressure that only locally applies to the vesicle.

Figure 1.

Model schematics. (A) Overview of model setup. We define the osmotic pressure as positive when it points from the vesicle lumen toward the cytoplasm, and as negative when it points from the cytoplasm toward the vesicle lumen. Schematics are not to scale. (B) Zoom-in view of post-fusion vesicle in the model. Left: Computational meshwork that represents the surface of the membrane in simulation. The mesh was generated via COMSOL Multiphysics software. The minimum element size was set to 0.5 nm to ensure curvature resolution around the pore area. Right: Representation of post-fusion vesicle in cylindrical coordinates for analytic purposes.

The model approximates the membrane together with the post-fusion vesicle as a closed elastic sheet with homogeneous mechanical properties. In terms of fusion process, this continuum description of membrane is valid as long as the fusion pore is larger than the thickness of the lipid monolayer, as demonstrated by recent MD simulations (11). Given the small size of the vesicle (~ 100 nm) and relatively fast diffusion of lipids (~ 10 μm²/sec), any spatial heterogeneity or mechanical strain of the membrane could be rapidly smoothed out or relaxed by the lipid diffusion with a sub-millisecond timescale. In contrast, the timescale for post-fusion vesicle dynamics is typically ~ 10s – 100s millisecond or more (e.g., as revealed in (12)), so lipid hydrodynamic flow can be ignored. Such timescale separation further lends support for the model treatment that simplifies the membrane as a homogeneous elastic sheet. On a similar token, the model treats the intravesicular pressure as a localized effect. This spatially non-uniform pressure could cause mechanical strain within the plane of membrane. Given rapid lipid diffusions and the large cell size (typically ~ 20 μm), a spatially heterogeneous mechanical strain of the cell membrane could be relaxed by lipid diffusion within 10s seconds. In comparison, with the vesicle of 10s –100s nm in diameter as in the model, the duration of typical membrane fusion process is ~ sub-second. Due to this timescale separation, we approximate the effects of the intravesicular pressure to be localized to the vesicle.

The dynamic evolution of the membrane shape is characterized by the corresponding functional derivative of the membrane energy (Eq. (2)), which minimizes the total energy of the system (13). This formalism assumes that the system is near equilibrium without the involvement of active energy (e.g., ATP consumption) so that near-equilibrium statistical mechanics applies.

$$\xi V=\kappa \left({\Delta }_{M}H+H{\left|{\nabla }_M\overrightarrow{n}\right|}^2-\frac{1}{2}{H}^3\right)+\sigma H+p_0+{\left.p\right|}_{\text{intravesicle}}$$ Eq. (2)

where V is the velocity of the membrane shape change in the normal direction of the membrane surface; $\overrightarrow{n}$ is the unit normal vector that points toward the cytosol of the local membrane surface M. $\xi =2\times {10}^{9}Pa\cdot s/m$ is the effective viscous drag coefficient of the membrane shape change (14). p only applies to the post-fusion vesicle. To obtain the membrane shape changes, we integrate the dynamic equation over time, starting from the initial condition of the post-fusion vesicle configuration like that in Figure 1C. The initial fusion pore adopts an hourglass-like shape that smoothly interpolates between the tangents of the vesicular bulb and the larger membrane vesicle. The initial configuration is represented by a triangulated mesh (Figure 1B).

Here, we consider a scenario in which the total surface area of plasma membrane and the enclosed volume are conserved. While these conservations are global constraints, numerical calculation is done on discrete local lattices. To this end, we adopted the established computational scheme to calculate the membrane shape evolution (15–17). This semi-implicit finite element scheme treats the membrane tension σ and the global osmotic pressure p as Lagrange multipliers, and ensures the conservations of the total membrane surface area and the enclosed volume during the time course of simulation, respectively (Figure S1). We thus interpret the sequence of mesh displacements computed at each time step from this scheme as the evolution of the post-fusion exocytotic system.

An important note is that modulation of these global constraints is equivalent to changes in global membrane tension and osmotic pressure. We show that such modulations do not significantly impact the evolution of the post-fusion vesicular membrane in our system (Figure S1). This is because the size of the post-fusion vesicle is very small relative to the whole system (10s – 100s nm vs. 10s μm, Figures 1A and 1B). Additionally, at the post-fusion vesicle, the relative contribution to energy from membrane tension and bending modulus depends on lengthscale. A good estimate of this lengthscale is ~ (bending modulus/membrane tension)^1/2. Given that membrane bending modulus in cell is 100’s k_BT (e.g., (18, 19)) and membrane tension is measured to be ~ 10⁻⁵ – 10⁻⁴ N/m (e.g., (18–21)), this results in a lengthscale ~ several μm, beyond which membrane tension is the dominant factor for controlling membrane shape, below which membrane bending is the determinant. As most of post-fusion vesicle is in the range of 100s nm or less, membrane bending modulus is thus a more relevant factor in dictating the membrane shape change of the vesicle.

Due to the above considerations, we will focus more on the effects of membrane bending modulus, intravesicular pressure, and membrane geometry on the fate of post-fusion vesicles. Before detailing the results, let us elaborate on the physical basis of variations in these control parameters and their connections with reality.

First, the membrane stiffness reflects the effective bending modulus of a cortex in cells, which largely consists of a lipid bilayer in tight connection with an actomyosin meshwork. In this picture the effective bending modulus mainly stems from the rigidity of the actomyosin meshwork, which is in the range of 100’s k_BT (18, 19). In contrast, the membrane bending modulus of a lipid bilayer is typically ~ 10’s k_BT (22). Variations in the membrane stiffness therefore largely correspond to modulations of the actomyosin meshwork (18, 19).

Second, we assume that there is an intravesicular pressure that locally applies on the post-fusion vesicle. This model assumption is motivated by the following experimental observations. As a secretory vesicle typically contains a high concentration of signaling molecules, the osmotic pressure could point from within the lumen toward the cytosol (positive pressure) (23), or adapt to be isotonic with cytoplasm (zero pressure) (24). Reducing such pressure could correspond to the tensional pressure built-up mediated by actomyosin contraction (25, 26). Conversely, increasing this intravesicular pressure could reflect the modulation from the external osmolarity controlled in experiments (27–29); alternatively, the intravesicular matrix swelling can contribute to this pressure built-up in the cases like dense-core vesicle fusion (23, 30–35). In dense-core vesicles, neuropeptids are packaged into a dense core matrix, and free neurotransmitter molecules are stored in the space between the dense core matrix and the vesicular membrane. When free neurotransmitter molecules release through the initial fusion pore, the neuropeptide molecules associated with the core begin to be exchanged with hydrated extracellular ions (33), inducing swelling of the dense core matrix without a change in vesicle membrane surface area (30, 35). This resulting intravesicular pressure could further push the vesicular membrane toward the cytoplasm. Regardless its origin, the intravesicular pressure only applies to the vesicular membrane of the small vesicle in the model (Figure 1A). Additionally, due to the transient nature of post-fusion vesicles, it is difficult to directly measure the intravesicular pressure in current experiments. But, as a rough estimate of the intravesicular pressure, we borrow the experiments showing that actomyosin cortex can generate ~ 0.1 – 1 kPa stress to drive cell rounding-up in mitosis (e.g.,(36, 37)). We therefore use these values as the typical range of the intravesicular pressure in the model.

Last, the secretory vesicle size could vary greatly among different cell types from tens of nanometers to microns. And the amperometry measurements of single vesicle fusion events reveal a trickling of vesicle content release before the full discharge (12, 38–40). This trickling phenomenon is believed to coincide with the initial fusion pore opening (12). The duration and amount of luminal content release of the trickling appear to be quite heterogeneous among different model systems, which lend the experimental supports for variations of the initial pore size in the model.

With the above physical underpinnings of the model parameters, we now set out to explore their individual effect on post-fusion vesicles.

Results

We now study how the membrane stiffness, the intravesicular pressure, and the geometry influence the dynamics of the post-fusion vesicle. We stress from the outset that the predicted fates of post-fusion vesicle by the current model only reflects the preference driven by intrinsic membrane mechanics, whereas it will additionally be regulated by the actions of membrane fusion proteins in vivo. The value of our membrane-centric model is to delineate a mechanical role played by some of the membrane fusion proteins in addition to their conventional role in promoting membrane fusion events.

The model can readily recapitulate distinct fate of post-fusion vesicles. The series of snapshots in Figure 2 illustrate the typical cases of post-fusion vesicle fates, and provide a sense of the dynamic membrane shape changes in these processes. These findings point to an intricate relationship between the vesicle dimension and initial pore size in dictating the evolution of the post-fusion vesicle.

Figure 2.

Representative cases of post-fusion vesicle shape changes from model simulations. (A) Fission of post-fusion vesicle. Vesicle size is 80 nm; initial pore size is 10 nm; membrane bending modulus is 20 k_BT; and intravesicular pressure is 1.5 kPa. In the model simulation, when the fusion pore evolves down to 1 nm in diameter, we deem this state sets the stage for spontaneous membrane fission between the two opposing membrane leaflets. Note that the last snapshot in this plot is a schematic, instead of a real simulation result. (B) Smaller dimensions confer faster full-collapse fusion. Vesicle size is 40 nm; initial pore size is 5 nm; membrane bending modulus is 400 k_BT; and intravesicular pressure is −1.5 kPa. (C) Full-collapse vesicle fusion. Vesicle size is 80 nm; initial pore size is 10 nm; membrane bending modulus is 400 k_BT; and intravesicular pressure is −1.5 kPa. (D) Balance between membrane bending and intravesicular pressure confers a static fusion pore. Vesicle size is 120 nm; initial pore size is 25 nm; membrane bending modulus is 50 k_BT; and intravesicular pressure is 0 kPa. For (A)-(D), the scale bar is 10 nm.

On a more systematic ground, let us first investigate the dependence of post-fusion vesicle fate on its initial pore size and vesicle size, with a constant membrane stiffness at 200 k_BT and a constant positive intravesicular pressure (~ 1 kPa). Figure 3A shows that for a given vesicle size, full-collapse fusion requires a sufficiently large initial pore opening. Otherwise the inherent membrane mechanics tend to close up the fusion pore, leading to the pinching off of the vesicle from the plasma membrane. Conversely, with a fixed size of the initial fusion pore, increasing vesicle size is predicted to favor the fusion pore shrinkage and, hence, vesicle fission rather than full-collapse fusion.

Figure 3.

Phase diagram study of post-fusion vesicle fate. (A) Initial pore size vs. vesicle size. The intravesicular pressure is fixed at 1 kPa, and the bending modulus is fixed at 200 k_BT. (B) Membrane bending modulus vs. intravesicular pressure. The initial pore size is fixed at 10 nm, and the vesicle diameter is kept at 100 nm. (C) Intravesicular pressure decrease confers full-collapse fusion. Here the phase diagram is similar to that in (A), except that the distinct contour lines correspond to the respective phase boundaries between full-collapse fusion and vesicle fission at different intravesicular pressures, while the membrane bending modulus is fixed as 200 k_BT. The full-collapse fusion regime is above the corresponding phase boundary, whereas the vesicle fission is below it. (D) Vesicle size modulates the antagonism between membrane bending and intravesicular pressure. Here, the phase diagram is similar to that in (B), except that the distinct contour lines correspond to the respective phase boundaries between full-collapse fusion and vesicle fission at different vesicle sizes, while the initial pore size is fixed at 10 nm. The full-collapse fusion regime is below the corresponding phase boundary, whereas the vesicle fission is above it. Note that for (A) and (C), the “Geometrically incompatible regime” refers to the scenarios that the fusion pore is wider than the diameter of the vesicular bulb, which by definition the post-fusion vesicle has already fully collapsed into the plasma membrane.

Since the neck region of a post-fusion vesicle is highly deformed, widening the fusion pore is expected to relax the stress and, hence, the subsequent full-collapse fusion is easy to understand. However, our simulation also delineates the conditions for spontaneous fission, which is a bit counter-intuitive. We would like to further analytically dissect the physical reason behind it. Since this prediction is insensitive to the constraints on the total membrane surface area and the enclosed volume (Figure S1), it solely stems from the membrane bending energy that scales ~ (mean curvature)². Because the post-fusion vesicular membrane under consideration is axi-symmetric, we will now describe the membrane shape of the post-fusion vesicle in cylindrical coordinates for analytic purposes (see the right panel in Figure 1B). Here, the mean curvature, H, is the sum of the curvature in the radial direction $\left(\text{sin}\psi \left(s,t\right)/r\left(s,t\right)\right)$ and that in the longitudinal direction $\left({\psi }^{\prime }\left(s,t\right)\right)$, where s is the arc length along the membrane, t is the time. $r\left(s,t\right)$ is the radius of the membrane shape at the arch length, s, and time, t. $\psi \left(s,t\right)$ is the corresponding angle between the horizontal direction and the tangent direction so that $\psi \left(\text{s}\right)\in \left(0,\pi \right)$, and ${\psi }^{\prime }\left(s,t\right)$ is the derivative of the angle by the arc length. For an initial membrane configuration, its shape evolution is driven by the steepest descend of the bending energy $J=\frac{\kappa }{2}\oint H^{2}dS=\int ds\frac{\kappa }{2}{\left(\frac{\text{sin}\psi (s,t)}{r(s,t)}+{\psi }^{\prime }(s,t)\right)}^{2}2\pi r(s,t)$. The dynamic equation pertaining to the radius is:

$$\begin{gathered} \xi \frac{\partial r(s,t)}{\partial t}=-\frac{\delta J}{\delta r(s,t)} \\ =\pi \kappa \left(\frac{\text{sin}\psi (s,t)}{r(s,t)}+{\psi }^{\prime }(s,t)\right)\left(\frac{\text{sin}\psi (s,t)}{r(s,t)}-{\psi }^{\prime }(s,t)\right) \end{gathered}$$ Eq. [3]

where $\xi$ is the viscous drag coefficient for membrane shape change and hence is positive. From this formula, we can deduce how the radii change initially (increases or decreases), akin to linear stability analysis. For a membrane tubule where $\psi \left(s\right)=\pi /2$, the longitudinal curvature is zero $\left({\psi }^{\prime }\left(s,t\right)=0\right)$; so the mean curvature is just the radial curvature $\left(1/r\left(s,t\right)\right)$. Therefore, the membrane tubule tends to expand its diameter. However, for a toroid-shaped membrane like post-fusion vesicular membranes, the radial curvature is always positive $\left(\text{sin}\psi \left(s,t\right)/r\left(s,t\right)\ge 0\right)$, while the longitudinal curvature is negative at the neck region $\left({\psi }^{\prime }\left(s,t\right)<0\right)$. In fact, for the post-fusion vesicular membranes that undergo spontaneous fission, they all start out with a very high aspect ratio – a large bud and a narrow neck. In these cases, the absolute value of the longitudinal curvature is larger than the radial curvature at the neck region; consequently, the mean curvature is initially negative, i.e., $\left(\text{sin}\psi \left(s,t\right)/r\left(s,t\right)+{\psi }^{\prime }\left(s,t\right)\right)<0$, whereas $\left(\text{sin}\psi \left(s,t\right)/r\left(s,t\right)-{\psi }^{\prime }\left(s,t\right)\right)>0$. Together, the sign of the right hand side of the Eq. [3] is negative. That is, when the local mean curvature is negative, the neck radius, $r(s,t)$, initially tends to decrease to minimize the bending energy. For the ensuing full dynamics, our simulation result shows that during the spontaneous fission event, the local mean curvature at the neck keeps on evolving toward zero from a highly negative value (Figure S2A), and the total energy continues to decrease (Figure S2B). Therefore, spontaneous fission is intrinsic to the geometry of a post-fusion vesicle that has a negative mean curvature at the neck. This prediction is consistent with other theory papers, in which minimizing Helfrich bending energy has similarly indicated this kind of spontaneous fission (e.g., (17)). A further note is that such inherent tendency is certainly modulated by other mechanical factors, e.g., the bending moduli and intravesicular pressure etc, although these factors within their respective physical ranges may not impact the fate of spontaneous fission in certain phase diagram regimes (see later in Figure 3B).

We note that during vesicle fusion, the vesicle and plasma membranes undergo series of topological change that eventually result in a fusion pore with the Ω-shape post-fusion vesicle connecting to the plasma membrane. During this membrane remodeling processes, the fusion pore start to open from a small diameter in nm-range, comparable to the size of a lipid or protein that re-arranges its position/conformation leading to the formation of the fusion pore. Indeed, experiments can detect nascent fusion pore size as small as in nm-range (38, 41–44). When the diameter of a nascent fusion pore is smaller than the width of lipid monolayers, i.e., 2–3 nm, previous computational works show that continuum description of membrane mechanics becomes inaccurate (8). In this limit, our model is invalid and we have to leave accurate descriptions of the fate of post-fusion vesicle to our future investigation. On the other hand, if the nascent fusion pore is larger than 2–3 nm, then our model results should hold up. In this latter scenario, our model predicts that with such a small fusion pore, a post-fusion vesicle larger than 20 nm in diameter will undergo fission. In reality, most exocytotic vesicles are larger than 20 nm in diameter and have nascent fusion pore size in nm-range. And yet, many of these fused vesicles routinely undergo full collapse and are completely incorporated to the plasma membrane. These experimental observations do not necessarily contradict our model, because our predicted mode – either full-collapse or kiss-and-run – only reflects the intrinsic mechanical preference by the membrane itself. In vivo, however, post-fusion vesicle is not a bare membrane; instead, it associates with many key membrane fusion proteins. It is known that some of these proteins not only drives membrane fusion but also prevents the fusion pore from resealing. For instance, whereas one SNARE per vesicle could trigger membrane fusion, it takes at least three SNAREs per vesicle to keep the fusion pore open that will otherwise reseal (45). Our model prediction is thus consistent with this experimental observation. That is, while membrane itself prefers to close a very small fusion pore, it is the proteineous structures that hold the neck and prevent the fusion pore from collapsing. In this context, as the fusion pore opening is caused by the action of an elaborate ensemble of membrane fusion machinery (4), our results in Figure 3A also suggest that in order to achieve full-collapse fusion, the fusion machinery needs to adaptively open up a fusion pore in accordance to the secretory vesicle size. Conversely, when the secretory vesicle is artificially enlarged, the same operation of the membrane fusion machinery is predicted to shift the balance toward kiss-and-run rather than full-collapse fusion (Figure 3A), which is in line with experimental observations (34). Moreover, our model suggests that the smaller the vesicular dimension is, the larger is the driving force from membrane bending, hence the rapid dilation of the fusion pore (compare Figures 2B with 2C). This is consistent with the experimental observations that the lifetime of the initial fusion pore inversely correlates with the vesicle size (12, 46). Furthermore, the model predicts that near the phase boundary in Figure 2A, where membrane bending force and intravesicular pressure are approximately in balance, the fusion pore could persist for an extended period of time (Figure 2D). This predicted state could account for the so-called “kiss-and-hold” state of post-fusion vesicles observed in osmolarity manipulation experiments (27), or the stable Ω-shape post-fusion vesicle observed in (47).

With the size effects revealed in Figure 3A, we next study in detail how the fate of the post-fusion vesicle depends on membrane stiffness and pressure, while keeping the dimensions of the initial vesicle and the initial pore fixed. In Figure 3B, all these simulations start with the initial vesicle diameter of 100 nm, and the initial fusion pore diameter of 10 nm. Our results in Figure 3B indicates that for any intravesicular pressure, there is a threshold membrane stiffness, only above which full-collapse fusion can be realized. Conversely, with constant membrane stiffness, increasing this pressure promotes vesicle fission. This phase diagram study provides some perspectives on the mechanical condition of exocytosis. During exocytosis, the intravesicular pressure originating from the vesicular lumen could be positive and thus tend to swell the vesicle. This tightening effect in general supports membrane fusion (48), as the fusion is inhibited by a relaxation of membrane (48). But once the fusion pore opens, the same pressure, which confers membrane fusion, now drives vesicle fission according to our model (Figure 3B). This necessitates a stiffer membrane for the full-collapse fusion (Figure 3B), providing an explanation for the critical roles of actomyosin contraction and the buildup of a cortical actin meshwork in collapsing the post-fusion vesicle to fully integrate into the plasma membrane (25). In this context, our model suggests that a cell could adaptively remodel its cortex to overcome the osmotic pressure and fully collapse the fused vesicles in exocytosis. In the future it would be interesting to explore the underlying mechanistic pathway of such an adaptation.

The model also emphasizes the importance of intravesicular pressure in controlling the fates of post-fusion vesicles (Figure 3B). First, the releasing of the vesicular content after the fusion pore opening could reduce the pressure. If this pressure decrease is sufficiently fast, then the model predicts that the post-fusion vesicle can fully collapse into the plasma membrane even without increasing the membrane stiffness (Figure 3C). Second, in some cases the intravesicular pressure could increase upon fusion pore opening. For instance, when dense-core vesicles form a fusion pore, the matrix-associated molecules exchange with hydrated extracellular ions, which causes the matrix to swell (33). This results in an additional intravesicular pressure that pushes the vesicular membrane toward the cytoplasm, driving vesicle fission. The dynamics of this matrix swelling could therefore determine the duration of the fusion pore and the fate of the post-fusion vesicle (25). Third, the pressure in our model could be also modulated by the salt concentration in the extracellular space. A hypotonic condition (lower salt concentration) will reduce the effective osmotic pressure impinging upon the post-fusion vesicle, whereas a hypertonic condition (higher salt concentration) will increase the osmotic pressure. According to our model (Figure 3B), hypotonic and hypertonic solutions will thus promote or inhibit full-collapse fusion, respectively. This model implication could explain the observed osmolarity effects on vesicular secretion (27–29). Last, the pressure effect certainly depends on the size of vesicle. Our calculation shows that with a much smaller vesicle (e.g., 60 nm in diameter), the threshold membrane stiffness only marginally hinges on the pressure (Figure 3D). In this limit the membrane stiffness becomes the most essential control parameter for post-fusion vesicle fate. Given that the effective membrane stiffness in vivo is typically ~ 100s k_BT due to the presence of cortex (19), our result suggests that smaller vesicles tend to undergo full-collapse fusion on their own, larger ones require additional coordination between compressing pressure and membrane bending modulus (Figure 3D). This finding highlights differential mechanical requirements of full-collapse fusion for vesicles of different dimensions, consistent with the essential role of actomyosin contraction in collapsing large fused vesicles, e.g. in the regulated exocytosis in the acinar cells of the salivary glands (25). Conversely, the model suggests that for a given set of membrane mechanics (e.g., fixed membrane stiffness and intravesicular pressure), enlargement of the vesicle could promote fission of the fused vesicle and reduce vesicular secretion, whereas reducing the vesicle size could promote full-collapse vesicle fusion and full release of the vesicle content. This insight is consistent with the observation of the inverse correlation between synaptic activity and vesicle sizes (49, 50).

Discussions

To put this work in perspective, we provide a generic mechanical model that describes the post-fusion vesicle based on a complete set of mechanical parameters. In contrast, earlier models, while insightful, only incorporate a subset of these characters. These earlier endeavors either only focus on toroid fusion pore without describing the full vesicle and plasma membrane, or depict a relatively complete geometry of post-fusion vesicle but without investigating the effect of vesicular geometry (51–53). Our integrated approach allows us to go beyond the previous ones, and has the following merits: it (1) yields generic phase diagrams of post-fusion vesicle fates that recapitulate the full range of observed vesicular secretion modes, (2) defines distinct mechanical requirements for membrane fusion machinery for different modes of vesicle fusion, and (3) serves a unified framework to understand exocytosis in different systems that have different vesicular dimensions and mechanics.

Certainly, the fate of post-fusion vesicle, either kiss-and-run, full-collapse fusion, or remaining a stable Ω-shape, critically hinges on the specific vesicular geometry and the mechanical conditions, which could vary from system to system. However, on the qualitative ground, the predicted impact of the tripartite interplay between membrane stiffness, intravesicular pressure, and vesicular geometry on the fate of post-fusion vesicle, as illustrated by Figure 3, is expected to hold up. The value of this model thus is to provide a vintage point from a mechanical perspective so that one could map out the inner workings of vesicle fusion in vivo, which is obviously a more complicated mechanochemical process. For instance, our phase diagram study suggests that cell can fine-tune the vesicle size and the membrane mechanics to achieve kiss-and-run (Figure 3), which renders a controlled and limited exocytotic secretion. In this context, the excessive and uncontrolled exocytotic secretion in cancer cells may stem from a mismatch between the vesicle geometry and mechanics, which may instead favors full-fusion collapse. We suggest this possibility that calls for more elaborated experiments in the future. This realization also indicates that one could put the derailed exocytosis back under control by externally manipulating the membrane mechanics and the intravesicular pressure. While the malfunction of exocytosis is just one facet of the entire picture of cancer, our model suggests that regulation of cell mechanics and exocytotic vesicle geometry could play an important role in this paradigm.

From an energy standpoint, our model treats the evolution of post-fusion vesicular membrane shape changes as a downhill process. However, both membrane fission and fusion in vivo require energy inputs to overcome the energy barrier of the topological changes. As the model starts with a post-fusion vesicle configuration, it may not be in subject to the energy barrier issue for its subsequent evolution. The question now is: Can this model faithfully describe the membrane fission, i.e., kiss-and-run event? The answer is yes, based on the following considerations. For in vivo membrane fission, e.g., in mammalian endocytosis, dynamin GTPase-driven membrane constriction is necessary, but not sufficient for several reasons. First, dynamin level drops off during or even before the final membrane fission event (e.g., (54)). Second, as such GTP-driven membrane fission is nonleaky (55), this process must pass through a hemi-fission stage, at which the two apposing membrane leaflets touch each other and the lumen in-between is squeezed out (56). And yet, dynamin can at most restrict a membrane tubule down to a diameter, at which the apposing membrane leaflets are still ~ 4 – 10 nm apart (e.g., (57–59)). The current understanding is that dynamin and/or along with other proteins constricts the membrane tubule to a size, from which the membrane spontaneously evolves to hemi-fission and subsequently undergoes fission (56). This view is in line with the theoretical study suggesting that a highly constricted membrane neck (down to a lumen of 3 nm in diameter, but prior to hemi-fission) has roughly the same elastic energy as that of the hemi-fission configuration (60). As such, thermal fluctuation can readily drive the highly constricted membrane pass the hemi-fission state and subsequently, pinch off the membrane. That is, thermal fluctuation is sufficient for the final step of membrane fission, without additional energy input. Based on this understanding, we assume in the paper that the membrane fission will ensue when the apposing membrane leaflets are less than 1 nm apart. More importantly, the model in Figure 3B already imposes a highly constricted membrane shape of post-fusion vesicle as the initial configuration. This certainly needs energy input in reality, and is equivalent to the GTP-driven process. It is just that the subsequent shape evolution does not need the additional energy. In this regard, Figure 3B shows that this super-constricted post-fusion vesicle intrinsically prefers to undergo fission in low-membrane bending modulus regime. And this post-fusion vesicle will undergo full-collapse fusion when the membrane becomes very rigid. This result is consistent with experiments that increasing membrane rigidity hinders membrane fission (61), whereas reduced membrane rigidity facilitates the pinching off (62). In contrast, when the initial membrane shape is not highly constricted (i.e., the pore size approaches to the diameter of vesicular bulb), Figure 3D shows that the membrane could undergo full-collapse fusion even with a soft membrane, which is exemplified in Figure S3. This comparison further underlines the importance of interplay between membrane mechanics and vesicular membrane geometry in dictating the fate of post-fusion vesicles.

Pulling back into a wider view, our model provides a starting point to reconcile some of the conflicting results in the exocytosis field. For instance, some experiments show that increasing calcium concentration shifts the mode of exocytosis from full-collapse fusion to kiss-and-run (9), and at least one demonstrates the opposite trend (8). Despite the possible differences in the intrinsic biology, such as different cell types, the different fates of post-fusion vesicles are ultimately driven by mechanical processes, and here our model offers an interesting standpoint. In (9), the extracellular calcium concentration increase (~ 100 mM) favors the kiss-and-run. While the osmolarity is kept roughly constant in the experiment, the presence of calcium ion could affect the mechanical properties of the system in several aspects. First, calcium could impact the charge density on membrane, which could be softened or stiffened, depending on the calcium concentration and the lipid composition of the membrane (e.g., (63, 64)). Second, changing calcium concentration could alter the ionic conditions and also perhaps the local pH. Exchanging ions and changing pH via the fusion pore opening could lead to swelling of the intravesicular matrix (e.g., (33)), which could contribute to the outward pressure. According to our model, this intravesicular pressure built-up favors re-sealing of the fusion pore and pinching off the vesicle (Figure 3B). As multiple factors could play out in this context, our model calls for a more systematic evaluation of the extracellular Ca²⁺–mediated mechanical contributions to exocytotic processes. On the intracellular side, the rise of intracellular free calcium concentration promotes full-collapse fusion in (8). While this small calcium concentration (~ 10s – 100s μM range) per se does not contribute to notable change in osmotic pressure or membrane rigidity, it is known to drive actin coating around the post-fusion vesicle and the associated actomyosin contraction (65). This contractile force could exert a sufficient negative pressure that compresses the vesicle and thus drives the full-collapse fusion (Figure 3B). The above elaboration presents a plausible explanation for the seemingly conflicting experimental results, and punctuates the important role of detailed mechanics in orchestrating diverse cellular responses such as those in exocytosis.

As a starting point, the current model is inevitably simplistic. For instance, it describes membrane elasticity of post-fusion vesicles, which is in reality a composite structure mixing both proteins and lipids, e.g., at the fusion pore. While our finding, on a coarse-grained level, could be used to annotate the mechanical role of the key proteins in this paradigm, it is still an open question of how to faithfully model the detailed dynamics of such composite structure. Moreover, the secretory granules with different synaptotagmin isoforms in the same cell have different fusion modes (66). As different synaptotagmins possess distinct biochemical properties, e.g., calcium-binding affinities (67), it remains to see exactly how these biochemical characters influence the mechanics of post-fusion vesicles and how our model can explain this more realistic and yet more heterogeneous cellular context. Further, this model so far only investigates how mechanics and geometry of membrane influence the fate of post-fusion vesicles, rather than the entire process of exocytosis. To better appreciate the complexity and regulation of exocytosis, it is of our future project to model the mechanochemical process of exocytosis, similar to our work in endocytosis (68). Ultimately exocytosis and endocytosis are coupled in time and space (69, 70). It holds an even greater interest to quantitatively understand how the two opposing membrane trafficking processes coordinate with each other spatio-temporally. This will be a significant modeling undertaking that will require organically integrating the essential biochemical pathways with their mechanical actions. In this regard, our current mechanical model of post-fusion vesicles sets the stage for future model developments.