The shape of cellular organelles, such as the Golgi complex, results from the generation and stabilization of the curvature of their membranes (Shibata et al., 2009). According to the elastic model of membrane bending (Helfrich, 1973; Campelo et al., 2014), energy is required to induce local changes of the membrane curvature from its preferred or spontaneous curvature. This energy is generally supplied by specialized lipids and/or membrane proteins, usually referred to as membrane curvature generators (Zimmerberg and Kozlov, 2006; McMahon and Gallop, 2005; Kozlov et al., 2014). Moreover, the amount of bending energy associated with local curvature deviations is proportional to the local bending rigidity of the membrane (Helfrich, 1973). As a consequence, both local variations in the amounts of curvature generators present on the membrane and in the bending rigidity of the membrane can influence the morphology of Golgi cisternae. Taking into account these considerations, our aim here is to establish a physical model for the Golgi membrane morphology, with a special focus on understanding the mechanisms by which SM metabolism controls the overall shape of the Golgi cisternae. Similar models based on the Helfrich bending energy have been widely used in the past to describe the shapes of lipid vesicles (Seifert, 1997). In such models, transitions from flat cisterna-like vesicles to curled vesicles, named stomatocytes, were promoted upon a reduction in the volume-to-area ratio. We previously proposed that reduction in the volume-to-area ratio of the Golgi cisternae could in fact be responsible for the observed flat-to-curled cisternae transition upon short-chain ceramide treatment (Duran et al., 2012). However, in our subsequent studies, we observed no obvious increase in the overall volume-to-area ratio of the Golgi cisternae during such a morphological transition (van Galen et al., 2014). Moreover, Golgi cisternae have an extremely low volume-to-surface ratio, which does not account for the reported flat configurations according to the aforementioned models (Seifert et al., 1991; Miao et al., 1991; Seifert, 1997). Altogether, this prompted us to propose an alternative model that takes into account two contributions that could potentially influence a role in SM-regulated shaping of the Golgi cisternae: (i) the presence of small, rigid, and highly dynamic membrane nanodomains enriched in sphingolipids and cholesterol; and (ii) the SM-dependent recruitment to or release from the Golgi membranes of budding factors and other membrane curvature generators essential for the formation of transport carriers. We first qualitatively describe how each of these two contributions can influence the shape of a Golgi cisterna.

Nanodomain partitioning-mediated mechanism

Experimental evidence has suggested the existence of nanoscopic SM-enriched liquid-ordered domains at the Golgi membranes, although direct visualization has remained challenging (Gkantiragas et al., 2001; Klemm et al., 2009; Duran et al., 2012; Bankaitis et al., 2012; Deng et al., 2016). Based on in vitro data, such liquid-ordered membrane domains are expected to have a higher bending rigidity than the surrounding liquid-disordered membrane and are therefore less prone to accommodate membrane curvature (Roux et al., 2005; Heinrich et al., 2010; Dimova, 2014). In essence, the presence of large amounts of such rigid nanodomains at the highly curved rim of a flat Golgi cisterna is associated with a large bending energy penalty. There are two ways to reduce this bending energy. The first one is by partitioning these rigid nanodomains away from the rim to the flatter regions of the Golgi cisterna (Figure 1A). However, such inhomogeneous curvature-driven nanodomain redistribution is entropically unfavorable. Therefore the balance between the bending energy and the entropic free energy dictates the optimal distribution of nanodomains along the cisterna membrane (Figure 1A). The second possibility is to decrease the surface area of the highly curved rim, hence reducing the overall bending stress of the rim, by globally changing the shape of the Golgi cisterna from a flat to a curled configuration, while maintaining the total surface area of the cisterna (Figure 1A). This morphological transition is associated with a reduction of the bending energy of the rim, but also with an increase in the bending energy of the central region of the cisternae (Knorr et al., 2012; Helfrich, 1974). Again, the balance between these two opposite contributions to the overall bending energy determines the optimal cisternae shape (Figure 1A). Obviously, these two means of decreasing the cisternae free energy are not mutually exclusive. Instead, a combination of both lateral nanodomain partitioning and a change in Golgi cisternal shape might possibly result from a decrease in the amounts of SM-enriched nanodomains present at the Golgi membranes (Figure 1A).

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In order to quantify the relative effect in Golgi shaping of the two mechanisms proposed above, we developed a mathematical formulation of the free energy of a Golgi cisterna as a function of its shape and nanodomain distribution on the membrane. The major assumption of our model is that the time scale of mechanical equilibration of the cisterna shape is much smaller than that of the changes in lipid and protein composition through membrane fluxes. Non-equilibrium shapes should be considered only if the composition would change faster than the shape relaxed to the new equilibrium state (Sens and Rao, 2013). We can estimate the mechanical relaxation time as a combination of the characteristic viscosity, bending rigidity and length scale of the cisterna, ${\tau }_{mech}=\eta R^3/\kappa ~ 1 ms$ (Allain et al., 2004). On the other hand, the rates of the composition changes based on the fluxes through the Golgi cisternae have been theoretically inferred from experimental data (Dmitrieff et al., 2013), resulting in a characteristic compositional relaxation time by means of membrane fluxes, ${\tau }_{flux} ~ 100 s$. Since ${\tau }_{mech}\ll {\tau }_{flux}$, the cisternae shape is assumed to be mechanically equilibrated for every instant composition. Since the composition is in a steady state, the shape is in mechanical equilibrium corresponding to this steady state composition. Hence, the equilibrium configuration of a cisterna is assumed to correspond to a free energy minimum.

We consider the cisterna membrane to have a shape of a sheet bound by a rim (Figure 1C). The sheet part is represented by two parallel membranes with inter-membrane distance, $2h$, much smaller than the sheet lateral dimension, $r_{flat}$. The maintenance of the narrow luminal space of Golgi cisternae can result from protein arrays bridging the two parallel membranes of a cisterna, which have been visualized by cryo-electron tomography (Engel et al., 2015). Alternatively, membrane adhesion between adjacent cisternae has been shown to be required to keep the narrow luminal space in HeLa cells (Lee et al., 2014). The rim shape is modeled by an open toroid of a cross-sectional radius, $r_{rim}$, merging the sheet boundary (Figure 1C). The distance between the bridging and/or stacking protein scaffolds forming the flat part of a cisterna and the cisterna edge sets the cross-sectional radius of the cisterna rim. The sheet part of the cisterna can curve into spherical segments of variable radii, $R$, which is accompanied by the corresponding changes of the rim perimeter, $L=2\pi r_{gap}$, the latter measured as the length of the toroidal axis. We will use the shape parameter $r_{gap}$ as a means to quantitate the degree of cisternal curling.

The free energy responsible for the cisterna shape includes the elastic bending energy of the membrane sheet and rim and also the entropic cost of a non-homogeneous partitioning of rigid nanodomains between the sheet and the rim. The bending energy is computed based on the Helfrich model (Helfrich, 1973), in which the membrane elastic properties are characterized by the bending modulus, $\kappa$, and the spontaneous curvature, $J_s$, the latter describing the intrinsic tendency of the membrane to curve (see the Materials and methods for a complete description of the model). We assume that the membrane spontaneous curvature is generated by specific proteins or proteins complexes (such as the components of the budding machinery) bound to or inserted in the outer membrane monolayer. These proteins occupy a fraction ${\varphi }_{budding}$ of the outer monolayer area, and are characterized by an effective individual spontaneous curvature, ${\zeta }_{budding}$, which has typical values in the range ${\zeta }_{budding}\approx 0.5-0.75 n{m}^{-1}$ (Campelo et al., 2008). The membrane spontaneous curvature is given by ${\displaystyle J_s=\frac{1}{2}{\varphi }_{budding}{\zeta }_{budding}}$ (the factor ½ accounting for the resistance of the internal monolayer to curving of the external one) and can vary along the membrane in accord with variation of the area fraction, ${\varphi }_{budding}$. (Campelo et al., 2008).

The membrane nanodomains are considered to occupy a fraction ${\displaystyle \mathrm{\Phi }}$ of the overall membrane area. The nanodomains are assumed to have a vanishing spontaneous curvature and a bending rigidity greatly exceeding that of the surrounding membrane (Roux et al., 2005; Heinrich et al., 2010). The domains can freely partition between the rim and the sheet parts of the system. The detailed presentation of the system free energy is given in the Materials and methods section. In essence, the relative contribution between the free energy of the rim and that of the rest of the cisterna mainly governs the transitions between flat and curled shapes, in analogy to other membrane systems (Helfrich, 1974; Lipowsky, 1992; Knorr et al., 2012).

We determined the equilibrium shape of a Golgi cisterna by minimizing the free energy for a given set of geometric and elastic parameters (see Table 1) upon specific assumptions. First, we consider the budding machinery to localize exclusively at the rims of the Golgi cisternae, so that the membrane of the sheet part the cisterna has a vanishing spontaneous curvature $J_{s,mid}=0$ while the rim membrane is characterized by $J_{s,rim}=J_s$ (see Materials and methods). Second, we assumed that the area fraction of the curvature generators in the rim, ${\varphi }_{budding}$, ranges between 0% and 10%, so that the variation range of the membrane spontaneous curvature in the rim is $0\le J_s\le 0.033 n{m}^{-1}$. And third, the overall membrane area fraction covered by the nanodomains, ${\displaystyle \mathrm{\Phi }}$, varies over a wide range of values, ${\displaystyle 0\le \mathrm{\Phi }\le 0.4}$.

Our model predicts that, depending on the values of the two parameters, $J_s$ and ${\displaystyle \mathrm{\Phi }}$, the minimal energy state of the system can correspond to a flat cisterna or a highly curled cisterna. In an alternative situation, referred below as the bistability state and considered in more detail in the next section, both the flat and curled shapes correspond to local energy minima. The parameter ranges corresponding to the three possible states of the system are summarized in a shape diagram (Figure 2A). Qualitatively, the model predicts that increasing the spontaneous curvature of the rim membrane by augmenting the amount of the curvature generators in the rim favors a shape transition from the curled to the flat cisterna configuration. In other words, large numbers of the budding factors at a Golgi cisterna favor flat rather than curled cisternae (see Figure 1B). Moreover, according to our computations, transitions between curled and flat cisternae are almost insensitive to variations of the nanodomain area fraction, ${\displaystyle \mathrm{\Phi }}$ (Figure 2A). The reason being that the free energy required to partition large amounts of rigid nanodomains away from the curved rim cannot be counterbalanced by the relaxation of the bending energy (see Appendix).

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Additionally, we considered the situation where the budding machinery is homogeneously distributed along the whole Golgi cisterna, $J_{s,rim}=J_{s,mid}=J_s$, and compared the model predictions with the previous case of the curvature generators concentrated only at the rim (Figure 2A). Since the flat-to-curled cisterna transition is associated with an increase in the surface area of the central part of the cisterna, the presence of curvature generators therein leads to an increase in the total bending energy of this area, thereby opposing cisterna curling. Indeed, our results show no overall qualitative difference from the results in Figure 2A, but only a general shift of the shape transitions between the flat and curled cisterna configurations towards the lower values of the membrane spontaneous curvature, as well as a reduction of the parameter space occupied by the transition area (Figure 2—figure supplement 1). In the Appendix, we present an analytical estimation of the extent of this shape transition shift.

In summary, the model predicts that the presence of membrane curvature generators at the Golgi cisterna is necessary to stabilize a flat morphology and a partial release of such curvature generators destabilizes the flat shape in favor of a curled cisterna shape. In the next sections we expand, describe and analyze these results in more detail.

The bistability region of the shape diagram (Figure 2A, left, shaded region) encompasses a set of values of the spontaneous curvature of the cisterna rim, $J_s$, and of the nanodomain area fraction, ${\displaystyle \mathrm{\Phi }}$, for which the free energy has two local minima. Each of these two local minima corresponds to a locally stable cisterna configuration. This is illustrated in Figure 2B, which represents the total free energy of a Golgi cisterna as a function of the shape parameter, $r_{gap}$, for two sets of the parameter values within the bistability region (see Figure 2A). The two local minima of the energy correspond, respectively, to a highly curled and a flat cisterna (Figure 2B, left panel). Moreover, an energy barrier separates the two locally stable shapes. Hence, at any transition from a curled to a flat cisterna shape the system needs to overcome an energy barrier ${\displaystyle \mathrm{\Delta }{F}_{curl-flat}=F_{max}-F_{curl}}$; whereas transition from the flat to the curled morphology requires crossing an energy barrier ${\displaystyle \mathrm{\Delta }{F}_{flat-curl}=F_{max}-F_{flat}}$ (see Figure 2B). We computed the values of these energy barriers for both flat-to-curled and curled-to-flat transitions, which are shown in Figure 2A (right panel). Our results predict a relatively broad range of parameters within the bistability region of the shape diagram where the shape transition can occur by crossing relatively low energy barriers, comparable to the typical few $k_{B}T$ energies of thermal fluctuations (see the color-coded plot in Figure 2A, right panel).

Our model predicts that the Golgi cisterna shape transition must be driven, most effectively, by variations in the amount of curvature generators present on the membranes of the Golgi rims (Figure 2A). To explore this mechanism in more depth, we considered a situation where the rim spontaneous curvature, $J_s$, is allowed to vary within a range, $0\le J_s\le 0.033 n{m}^{-1}$, whereas the membrane area fraction covered by nanodomains, ${\displaystyle \mathrm{\Phi }}$, is taken to be constant and equal to 0.2. Then, for each value of $J_s$, we compute (see Materials and methods) (i) which of the two possible cisterna shapes corresponds to the free energy minimum, that is, the equilibrium state; (ii) the energy value in the equilibrium state determined with respect to the flat cisterna configuration, and (iii) for the bistability range, the energy barrier between the two minimal energy states. The results, shown in Figure 3A, indicate that for large values of the spontaneous curvature,${\displaystyle 0.024n{m}^{-1}<J_s\le 0.033n{m}^{-1}}$, the only possible stable cisterna configuration is the flat one (solid blue line, Figure 3A). An intermediate range of spontaneous curvatures, ${\displaystyle 0.016n{m}^{-1}<J_s<0.024n{m}^{-1}}$, corresponds to the bistability state where both curled and flat Golgi cisternae (orange and blue lines, respectively, Figure 3A) are locally stable shapes separated by a free energy barrier. Finally, for lower values of the spontaneous curvatures, ${\displaystyle 0\le J_s<0.016n{m}^{-1}}$, the only stable shape is that of a curled Golgi cisterna (solid orange line, Figure 3A).

Figure 3

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Next we focused on the shape bistability region, ${\displaystyle 0.016n{m}^{-1}<J_s<0.024n{m}^{-1}}$, and computed the energy at the peak of the energy barrier between the two shapes, $F_{max}$ (Figure 3A, black solid line). This allowed us to compute the energy barrier required to be overcome for transition from a curled to a flat cisterna, ${\displaystyle \mathrm{\Delta }{F}_{curl-flat}}$; and the energy barrier of the inverse transition from the flat to the curled, ${\displaystyle \mathrm{\Delta }{F}_{flat-curl}}$ (Figure 3B). In addition, we computed the relative redistribution of nanodomains between the rim to the sheet part of the cisterna for both flat and curled cisternae shapes. These results show that there is a preferential partitioning of nanodomains from the highly curved rim region to the flatter sheet region of the cisterna, and that this partitioning is enhanced upon a reduction in the spontaneous curvature of the Golgi cisterna (Figure 3C). Notably, these results further predict the non-homogeneous distribution of nanodomains along the Golgi membranes to be more pronounced in curled as compared to flat cisternae (Figure 3C).

A hallmark of bistable systems is exhibition of hysteresis in the transition between the two states (Bhalla and Iyengar, 1999; Kholodenko, 2006; Katira et al., 2016). This means that the system retains some kind of memory of its dynamic evolution. As a result, for the same parameter values two different outputs can be expected depending on how the system dynamically evolved to that situation. In our case, for the same value of the membrane spontaneous curvature in the cisterna rim, $J_s$, two different cisterna configurations can form depending on how $J_s$ changed in time before reaching the final value.

To demonstrate this behavior, we considered a reversible trajectory of the system in the parameter space beginning from a situation where the flat cisterna morphology is the only stable shape (point one in the hysteresis diagram shown in Figure 3D) to a situation where only curled cisternae represent the locally stable shapes (point five in the hysteresis diagram shown in Figure 3D). For the sake of simplicity, we considered a trajectory where the nanodomain area fraction is kept constant and equal to ${\displaystyle \mathrm{\Phi }=0.2}$. Upon a gradual reduction of the value of $J_s$, the flat morphology ceases to be the global minimum of the free energy (point two in Figure 3D). However, a large energy barrier kinetically traps the system in the flat configuration, preventing it from acquiring its preferred curled morphology (Figure 3B). Further reduction of $J_s$ diminishes the energy barrier until the transition from a flat to a curled cisterna starts to be kinetically feasible (point 3a in Figure 3D) and even further to a point where the flat configuration does not correspond to a local energy minimum so that the curled state is the only equilibrium state of the system (point 3b in Figure 3D). The transition of the Golgi cisterna from the flat to the curled configuration is stochastically triggered at some point of the trajectory between the two values of the spontaneous curvature indicating the boundaries of the bistability region (Figure 3D). Once the system is out of the bistability region, further reduction of the membrane spontaneous curvature only subtly changes the shape of the curled cisternae in a continuous and smooth manner (until point five in Figure 3D). In the inverse process where the value of membrane spontaneous curvature, $J_s$, changes back to the initial value (point one in Figure 3D), the cisterna shape will remain in a curled configuration until $J_s$ reaches large enough values within the bistability region for which the curled-to-flat shape transition becomes kinetically feasible and the Golgi cisternae abruptly flattens (somewhere between points 6a and 6b in Figure 3D).

In summary, the results shown in Figure 3D indicate that, in certain conditions, upon recovery of the amount of the curvature generators present at the Golgi membranes, the shape of the Golgi cisternae can be kinetically trapped in a curled morphology, different from the initial flat shape.

Next, we examined in more detail how a reduction in the amount of nanodomains at the Golgi membranes contributes to the Golgi cisterna shaping. Our model predicted that a reduction in the membrane area fraction covered by the nanodomains, ${\displaystyle \mathrm{\Phi }}$, does not affect the flat-to-curled Golgi cisterna transition (see Figure 2A). Nevertheless, within the bistability region, a reduction in ${\displaystyle \mathrm{\Phi }}$ reduces the energy barriers for both flat-to-curl and curl-to-flat transitions (Figure 2A, right panel). To quantify the extent of this effect, we considered two different values of the spontaneous curvature of the rim, $J_s$, within the bistability region in Figure 2A and computed the dependence of the transition energy barrier on ${\displaystyle \mathrm{\Phi }}$. The results, presented in Figure 2—figure supplement 2, show that both the energy barrier required to flatten a curled cisterna, ${\displaystyle \mathrm{\Delta }{F}_{curl-flat}}$, and the energy barrier required to curl a flat cisterna, ${\displaystyle \mathrm{\Delta }{F}_{flat-curl}}$, increase with the amount of rigid nanodomains on the Golgi membranes.

One of the parameters used to compute the shape diagram in Figure 2A is the size of the SM-enriched rigid lipid nanodomains, $R_d$ (see Table 1 and Materials and methods). In order to quantify the sensitivity of our results to the value of this parameter, we computed the Golgi cisterna shape diagram for two extreme values of the nanodomain size, $R_d=20 nm$and $R_d=2 nm$, respectively. The results, shown in Figure 4, indicate that the effect of the nanodomain size in controlling Golgi cisterna shape is relatively minor, and it mainly plays a role in controlling Golgi cisterna morphology by increasing the sensitivity of the shape transition to the nanodomain area fraction (compare Figure 2A with Figure 4A and B). Moreover, we also computed the shape diagram for a cisterna of four times larger surface area (two fold larger cisterna radius, $r_{flat}=1000 nm$). The results of the model for this situation, shown in Figure 4—figure supplement 1, indicate that the effect of a larger cisterna surface area on the shape diagram is in shifting the transition curves to higher values of the spontaneous curvature of the cisterna rim. However, changing the surface area of the Golgi cisterna does not increase the sensitivity of flat-to-curled transitions to the area fraction covered by nanodomains, ${\displaystyle \mathrm{\Phi }}$ (Figure 4—figure supplement 1).

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Addition of short-chain ceramide to HeLa cells leads to a local change in lipid homeostasis at the Golgi membranes (Duran et al., 2012). Besides the reduction in the levels of long-chain SM and the concomitant increase in short-chain SM levels, this treatment also causes an increase in the amounts of diacylglycerol (DAG) at the Golgi membranes by 30% (from 1.4 to 1.8 mol%) (Duran et al., 2012). DAG is a lipid characterized by an effective molecular spontaneous curvature having a large negative value, ${\zeta }_{DAG}\simeq -1 n{m}^{-1}$ (Szule et al., 2002; Leikin et al., 1996), and exhibiting a very fast flip-flop rate (Ganong and Bell, 1984; Bai and Pagano, 1997). The latter allows DAG molecules to homogeneously redistribute between the two membrane monolayers unless an active mechanism imposing DAG inter-monolayer asymmetry exists.

Could it be that an increase in the levels of DAG provides an alternative mechanism for the observed morphological changes of the Golgi complex? One possibility is that DAG partitions non-homogeneously within each monolayer of the Golgi cisterna membrane in such a way that the DAG distribution in the cytosolic monolayer has an opposite character to that in the luminal monolayer. Specifically, the top part of the cytosolic monolayer is enriched in DAG at the expense of the bottom part of this monolayer, whereas the bottom part of the luminal monolayer gets enriched in DAG at the expense of its top part. This DAG partitioning, which keeps the overall inter-monolayer symmetry unchanged (Figure 5A), might help stabilize the curvature of the sheet part of the cisterna, hence, reducing the overall bending energy of the curled state (Figure 5A). However, such a non-homogeneous DAG distribution along the membrane monolayers is entropically unfavorable and its extent, as well as its effect on the cisterna shape, has to be determined by minimization of the total free energy of the system accounting for both the bending elastic energy and the entropic contributions (see Materials and methods for details). In order to quantitatively evaluate these effects, we assumed that the total amount of DAG is symmetrically distributed between the luminal and cytosolic monolayers of the Golgi membrane. We numerically computed the cisterna configurations corresponding to a minimum of the total free energy (see Materials and methods, Equation 21) for a wide range of both the DAG mole fraction, ${\varphi }_{DAG}$, within the system, $0\le {\varphi }_{DAG}\le 0.05$, and of the spontaneous curvature of the cisterna rim membrane, ${\displaystyle 0\le J_s\le 0.033n{m}^{-1}}$. The results, presented in Figure 5B, show that there is a very weak dependency of the preferred cisterna shape on DAG levels. Moreover, our theoretical model predicts that, for the experimentally observed increase of DAG levels from 1.4% to 1.8% (Duran et al., 2012), would not lead to a morphological transition of a flat cisterna to the curled form (Figure 5B). Finally, we computed the relative distribution of DAG amongst the different monolayers of the Golgi cisterna. These results show that the DAG distribution is almost homogeneous for all the cisterna monolayers (Figure 5C), due to the relatively large entropy cost of inhomogeneous partitioning of such small molecules, in agreement with previous studies (Derganc, 2007; Sorre et al., 2009). Taken together, these results suggest that the increase in the levels of DAG is not the driving force for the observed curling of the Golgi cisternae.

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One of the main predictions of our model is that the flat-to-curled cisterna transition results from a reduction in the amounts of curvature generators present at the Golgi membrane (Figure 2A). We decided to experimentally test this prediction by investigating whether short-chain ceramide treatment, which specifically disrupts SM homeostasis at the Golgi membranes and is known to drive formation of curled Golgi cisternae (van Galen et al., 2014), induces the release of peripheral membrane proteins implicated in generation of membrane curvature in the course of transport carrier formation at the Golgi membranes. It has been recently found that the metabolic generation of SM at the Golgi membranes results, via a DAG-triggered signaling pathway, in the local reduction of PI(4)P levels causing the release of PI(4)P-binding proteins, such as the clathrin-adaptor protein γ-adaptin, without affecting the localization of COPI components (Capasso et al., 2017). Hence, we investigated whether short-chain ceramide treatment induces the release of endogenous clathrin heavy chain (clathrin-HC), one of the components of clathrin-coated vesicles, the subunits of which polymerize into a cage-like triskelion structure involved in bending the underlying membrane (Kirchhausen, 2000). To this end, HeLa cells were treated with short-chain ceramide (D-cer-C6, 20 µM) for different times, after which the cells were fixed and the localization of endogenous clathrin-HC and the TGN protein p230 was monitored by immunofluorescence microscopy. We observed that, already after 30 min of treatment with short-chain ceramide, the pool of clathrin-HC initially present at the Golgi membranes was mostly released (Figure 6A,B). These results indicate that, indeed, the formation of curled cisternae upon the treatment with short-chain ceramide proceeds in parallel with a decrease in the amounts of curvature inducers at the Golgi membranes.

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We next assessed the relative timing of the release of the clathrin coats from the Golgi membranes with respect to the flat-to-curled Golgi cisternae transition induced by short-chain ceramide treatment. For this purpose, we analyzed the ultrastructural morphology of the Golgi complex by immuno-electron microscopy after different times of short-chain ceramide treatment. Our results show that Golgi cisternae curling starts already after 30 min of D-cer-C6 treatment, and that after 4 hr of treatment, virtually no flat stacks are observed (Figure 6C,D). In agreement with our previous analysis (van Galen et al., 2014), cisternae curling occurs towards the trans-Golgi cisternae/TGN, as monitored by the localization of specific late Golgi markers (Figure 6C). We extracted from these images the radius of flat cisternae, $r_{flat}=510\pm 21 nm$ (average ± SEM; N = 22) and the radius of curling in the 4 hr D-cer-C6 treated Golgi cisternae, $R=270\pm 26 nm$ (average ± SEM; N = 14), which are in agreement with the condition of area conservation (Equation A2). Taken together, these results indicate that short-chain ceramide treatment leads to the release of clathrin coats thus reducing the spontaneous curvature of the Golgi membranes, which, as our model predicts, leads to the curling of the Golgi cisternae.

Our model predicts the existence of a bistability region in the cisterna shape diagram, a relatively large range of parameters where both flat and curled Golgi cisternae correspond to locally stable shapes (Figure 2A). As we showed, the transitions between the flat and curled configurations within the bistability region are expected to have a hysteretic character (Figure 3D). To experimentally test whether Golgi cisterna shape transition induced by short-chain ceramide treatment exhibits hysteresis, we performed short-chain ceramide washout experiments to monitor the timing of recovery of both the Golgi morphology and the amounts of clathrin present at the Golgi membranes as the cells return to steady conditions.

We first investigated how the dynamics of recovery of the Golgi cisternae shape during short-chain ceramide washout correlates with the recruitment of clathrin coats to the Golgi membranes. To this aim, we pre-treated HeLa cells with D-cer-C6 for 30 min, after which the cells were extensively washed and incubated with normal growth medium for different times. Then the cells were fixed and the intracellular localization of clathrin-HC was monitored by immunofluorescence microscopy. The results of this experiment show that a 2 hr short-chain ceramide washout is sufficient to recover similar levels of clathrin-HC at the Golgi membranes as to those found in untreated cells (Figure 6A,B). Recovery of normal clathrin-HC levels at the Golgi membranes after a 4 hr D-cer-C6 treatment is slower, and occurs in about 6 hr (Figure 6—figure supplement 1).

Next, we performed analogous washout experiments to monitor the timing of Golgi cisternae shape recovery. We added D-cer-C6 to HeLa cells for 30 min, after which the cells were extensively washed and incubated in complete medium without D-cer-C6 for different times. The cells were then fixed and the ultrastructure of the Golgi cisternae was visualized by immuno-electron microscopy (Figure 6C). Our results show that the curled-to-flat Golgi cisterna transition during short-chain ceramide washout occurs at a much slower kinetics as the recovery of the clathrin coats to the Golgi membranes (Figure 6B,D). Indeed, even 16 hr after the short-chain ceramide washout –a condition where the Golgi membranes already recovered their stationary pools of clathrin coats– both flat and curled Golgi cisternae can still be observed (Figure 6C,D).

To confirm these observations, we used an alternative approach to quantitate the dynamics of the curled-to-flat Golgi shape transition promoted during short-chain ceramide washout. We took advantage of the fact that the curling of the Golgi cisternae induced by short-chain ceramide is accompanied by a lateral segregation of different Golgi-resident proteins, such as TGN46 and p230, and that the level of this segregation can be quantitatively assessed by immunofluorescence microscopy (van Galen et al., 2014). Although the observed protein segregation correlates with changes in Golgi membrane morphology, its driving mechanisms still remain unknown (van Galen et al., 2014). To monitor the dynamics of protein segregation during Golgi shape recovery after short-chain ceramide treatment, HeLa cells were treated with D-cer-C6 for 30 min or 4 hr, after which the cells were extensively washed, and incubated in normal medium without short-chain ceramide for different times before being fixed. Then, the intracellular localization of the two trans-Golgi membrane proteins p230 and TGN46 was monitored by immunofluorescence microscopy and the relative colocalization of the two proteins was quantitated by means of the Pearson's correlation coefficient. Our results confirm that cells pre-treated with short-chain ceramide for 30 min required about 12 hr to recover the initial levels of p230 and TGN46 colocalization, whereas a longer 4 hr pre-treatment with D-cer-C6 required about 24 hr for a complete recovery (Figure 6E).

Taken together, these results indicate that, after short-chain ceramide washout, the recovery of the levels of clathrin-HC at the Golgi membranes (which we suggest parallels the recovery of the initial values of the membrane spontaneous curvature) occurs much faster than the recovery of the flat cisternae morphology and of protein colocalization. This is indicative of a hysteretic behavior of the transition from flat-to-curled cisternae and reverse, as our model predicts.

The Golgi complex in mammalian cells and in most eukaryotes consists of multiple stacks of flattened disc-like cisternae (Klumperman, 2011). Our previously (van Galen et al., 2014) and presently reported ultrastructural data shows that normal Golgi morphology is altered in cells where SM metabolism had been disrupted. Under those conditions, the original flat-like cisternae curl into a concentric stacked onion-like structure. Based on those data, we describe the morphology of a single flat Golgi cisterna as explained in the main text (Figure 1C). In the Appendix we describe the effect of having multiple stacked cisternae. The overall Golgi curling is geometrically characterized by the radius of curvature of the Golgi cisterna, $R$, which tends to infinity for a completely flat Golgi cisterna (Figure 1C). Alternatively, the degree of Golgi curling can also be described by the distance between the center of the cisterna rim and the axis of symmetry, $r_{gap}$ (Figure 1C). Hence, $r_{gap}=r_{flat}$ for a completely flat cisterna, and $r_{gap}\to r_{rim}$ for highly curled cisternae. Based on the ultrastructural data, we assume that the total surface area of a Golgi cisterna membrane, $A$, the luminal thickness of the cisterna, $2h$, and the radius of the rim, $r_{rim}$, do not change as a result of cisternae deformation (van Galen et al., 2014). In an unconstrained system, the natural tendency of the system to minimize the bending energy is by adjusting the radius of the rim to match the spontaneous curvature. In our model, the radius of the rim cross-section is set by the distance between a protein scaffold forming the flat part of a cisterna and the cisterna edge. For example, if such scaffold would extend beyond the cisterna, e.g. if a cisterna would be formed by flattening of a big liposome between two infinite flat rigid plates, the edge cross-sectional radius would be simply equal to the cisterna half-thickness, independently of the spontaneous curvature of the rim. The fact that the rim is somewhat swollen as compared to the cisterna thickness reflects the distance to which the scaffold edge approached the cisterna edge. In this model, the membrane spontaneous curvature could influence the detailed shape of the edge cross-section profile resulting in its deviation from the circular shape. This would be the case for the shapes where $r_{rim}$ is comparable to $r_{gap}$. Taking into account this effect would result in corrections of the rim energy but on a semi-quantitative level of description we neglect these corrections. The molecular identity of such protein scaffold could be the spacer proteins maintaining the luminal thickness and/or the stacking factors keeping the subsequent cisternae stacked. Interestingly, it has been experimentally shown that knockdown of the stacking/tethering factors GRASP55/65 or Golgin45/GM130 in HeLa cells leads to the cisternae unstacking, and swelling of the lumen and rims of the Golgi cisternae (Lee et al., 2014). Altogether this means that the total area, $A=A\left(r_{gap}\right)$, which is the sum of the area of the central part of the cisterna, $A_{mid}\left(r_{gap}\right)$, and the area of the rim, $A_{rim}\left(r_{gap}\right)$, is the same regardless of the cisternae morphology, that is, for all values $r_{rim}<r_{gap}\le r_{flat}$. On the one hand, the total surface area of the rim region, $A_{rim}\left(r_{gap}\right)$, can be obtained by using the area element,

where $\left\{\phi ,\theta \right\}$ are toroidal coordinates (see Figure 1—figure supplement 1). Then, the surface area of the rim is written as

where ${\displaystyle {\phi }_0=arcsin\left(1-2{r_{gap}}^2/{r_{flat}}^2\right)}$ and $\alpha =arcsin(h/r_{rim})$ (see Figure 1—figure supplement 1). On the other hand, we can express the area of the central part of the cisterna as

Thus, for the flat morphology, where $r_{gap}=r_{flat}$, we can write

which sets the constrained value of the total surface area of a Golgi cisterna. Finally, by using Equations (2), (3), and (4) we can mutually relate the two parameters describing cisternae curling, $R$ and $r_{gap}$, as

In the Appendix we derive simpler version of these equations by imposing a few approximations. This approximate theory will allow us to have analytical estimations of the main numerical results of this article.

Our model considers that the total free energy of a Golgi cisterna, $F$, has two contributions: the first one is the free energy of lipid nanodomain partitioning, $F_{part}$, and the second one is the membrane bending energy, $F_{bend}$.

The free energy of lipid nanodomain partitioning is an entropic term associated with a possible non-homogeneous distribution of liquid-ordered nanodomains along the membrane. This free energy term is given by

where $T$ is the absolute temperature and $S_{part}$ is the translational entropy associated with the lateral distribution of nanodomains. For the sake of simplicity, we model these domains as small, uniformly sized circular membrane patches of radius $R_d$. The total cisternae membrane area fraction covered by such nanodomains is given by ${\displaystyle \mathrm{\Phi }=\pi R_d^2{N}_d/A}$, where $N_d$ is the number of nanodomains. We use the area fraction covered by nanodomains, ${\displaystyle \mathrm{\Phi }}$, as a free parameter in our model. We also have to take into consideration the dynamics of the SM-enriched nanodomains, which can be continuously formed, reabsorbed and also diffuse along the membrane. However, these processes occur at much faster time scales (of the order of nanoseconds) (Eggeling et al., 2009) than the typical time scale of global shape remodeling of the Golgi membranes (of the order of seconds) (Bankaitis et al., 2012). Hence, for the purpose of finding how SM levels control the shape of the Golgi membranes, we can disregard domain dynamics and consider that the rigid nanodomains optimally and instantly redistribute along the membrane during cisternae deformation. Taken together these premises, we can write down an expression for the entropy of nanodomain partitioning, using a mean-field approach, as (Boucrot et al., 2012; Kozlov and Helfrich, 1992; Shemesh et al., 2003; Andelman et al., 1994; Derganc, 2007; Bozic et al., 2006)

where $k_B=1.38\times {10}^{-23}J\bullet K^{-1}$ is the Boltzmann constant, $a_{dom}=\pi R_d{}^2$ is the surface area of a single domain, and $\varphi \left(\overrightarrow{x}\right)$ is the local area fraction of SM nanodomains on the membrane, the area average of which is the total area fraction covered by nanodomains, ${\displaystyle \mathrm{\Phi }=\frac{1}{A}\int \varphi \left(\overrightarrow{x}\right)dA}$. We consider, for the sake of simplicity, that differential partitioning can only occur between the central region of the cisterna and the cisterna rim, where the membrane curvatures are considerably different from each other. The average nanodomain area fractions in each of these two regions are given by ${\displaystyle {\mathrm{\Phi }}_{mid}=\frac{1}{A_{mid}}\int \varphi \left(\overrightarrow{x}\right)dA}$, and ${\displaystyle {\mathrm{\Phi }}_{rim}=\frac{1}{A_{rim}}\int \varphi \left(\overrightarrow{x}\right)dA}$, where the integrals are performed over the central and rim areas of the cisterna, respectively. Conservation of the total area fraction covered by nanodomains leads to the following relationship

where $A_{rim}$ and $A_{mid}$ are, respectively, the surface area of the rim and central regions of the cisterna (Equations (2) and (3), respectively). Altogether, we can write Equation (6) as

where ${\displaystyle {\mathrm{\Phi }}_{mid}}$ is given by Equation (8).

The second contribution to the total membrane free energy comes from the energy of membrane bending, given by the Helfrich Hamiltonian (Helfrich, 1973). The membrane bending energy per unit area is given by

where $\kappa$ and $\overline{\kappa }$ are the bending modulus and the modulus of Gaussian curvature, respectively; $J$ and $K$ are the total and Gaussian curvatures of the membrane, respectively; and $J_s$ is the spontaneous curvature of the membrane. We assume that a priori there is no spatial correlation between the distribution of rigid nanodomains and of curvature generators. Indeed, SMS-mediated synthesis of SM is restricted to the luminal leaflet of the trans-Golgi membranes (Huitema et al., 2004), whereas recruitment of curvature generators occurs on the cytosolic side of the membrane. Although transbilayer lipid coupling between phosphatidylserine and long acyl chain lipids has been shown to occur at the level of the plasma membrane (Raghupathy et al., 2015), and a specific SM species has been shown to bind transmembrane cargo receptor at the Golgi membranes (Contreras et al., 2012), it is not clear how SM-rich nanodomains in the inner leaflet of the trans-Golgi membranes are coupled to the budding effector recruitment at the opposed side. The total bending energy of the cisterna is the integral of the free energy density, Equation (10), along the area of the cisterna,

For a laterally inhomogeneous membrane formed by a set of rigid nanodomains, the elastic moduli vary from regions of high membrane order to regions of low membrane order. We can locally describe the bending modulus of such a membrane as

where $\kappa \left(\overrightarrow{x}\right)$ is the local bending modulus, which has the meaning of an average over soft and rigid membrane domains (Kozlov and Helfrich, 1992; Markin, 1981). Moreover, ${\kappa }_{ld}$ and ${\kappa }_{lo}$ are the bending rigidities of a purely liquid-disordered membrane and of a purely liquid-ordered membrane, respectively. Based on different experimental studies (Roux et al., 2005; Heinrich et al., 2010), we take the values of these rigidities as ${\kappa }_{ld}=20 k_{B}T$ and ${\kappa }_{lo}=80 k_{B}T$. Similarly, ${\overline{\kappa }}_{ld}$ and ${\overline{\kappa }}_{lo}$ represent the moduli of Gaussian curvature of purely liquid-disordered and liquid-ordered membranes, respectively. Theoretical considerations, as well as indirect experimental evidence estimate the value of the modulus of Gaussian curvature to be $\overline{\kappa }={\alpha }_{\overline{\kappa }}\kappa$, where ${\alpha }_{\overline{\kappa }}$ is a proportionality factor that ranges between −0.2 and −0.83 (Templer et al., 1998; Siegel and Kozlov, 2004). It should be noted that the Gauss-Bonnet theorem cannot be applied to the Gaussian curvature term of the free energy Equation (11) since the modulus of Gaussian curvature is not constant along the membrane area (Allain et al., 2004). This term is therefore not a topological invariant for laterally inhomogeneous membranes, and therefore needs to be explicitly taken into account. To compute the bending energy in the geometry illustrated in Figure 1C, we separately consider the contributions to the elastic free energy of the rim region and of the central region of the cisterna, $F_{bend}=F_{bend}^{mid}+F_{bend}^{rim}$. These two regions are characterized by having an approximately constant total curvature. Hence, as mentioned above, we assume that there is a differential partitioning of SM-rich liquid-ordered nanodomains between the central and the rim regions of the cisternae. The details of the derivation of the expression for the free energy of bending of the cisternae central part, $F_{bend}^{mid}$, and of the cisternae rim, $F_{bend}^{rim}$, are found in the next section.

Finally, the total membrane free energy is given by

where the individual contributions to the total free energy are given by Equations (17), (20), and (6). In the model we consider that the spontaneous curvature of the membrane could take different values at the rim and central regions, $J_{s, rim}$ and $J_{s,mid}$, respectively. The total membrane free energy Equation (13) depends on a set of geometric parameters describing the cisterna morphology, $\left\{r_{gap},r_{flat},r_{rim},h\right\}$; a set of nanodomain-related parameters ${\displaystyle \left\{R_d,\mathrm{\Phi },{\mathrm{\Phi }}_{rim}\right\}}$; and a set of parameters describing the elastic properties of the membrane, $\left\{{\kappa }_{ld},{\kappa }_{lo},{\alpha }_{\overline{\kappa }},J_{s,rim},J_{s,mid}\right\}$. A thermodynamic treatment of the curvature effectors could in principle be incorporated into the model. However, in order to reduce the amount of free variables in the model, we distinguished two extreme situations: (i) the membrane bending proteins, contributors to the membrane spontaneous curvature, are only localized at the rims of the Golgi cisternae, implying that $J_{s,mid}=0$ and $J_{s,rim}=J_s$, which could be explained by the fact that membrane recruitment of some of these proteins is highly sensitive to membrane curvature (Antonny, 2011); and (ii) the budding machinery is homogeneously distributed along the whole Golgi membrane, $J_{s,mid}=J_{s,rim}=J_s$. As we showed, the results are qualitatively similar when considering the presence of curvature generators in the central part of the Golgi cisternae, but the shape transition quantitatively shifts. The reason for such a shift comes from the fact that, since the total surface area is conserved, curling of a cisterna leads to an increase in the surface area of the central part of the cisterna, concomitant with a decrease in the surface area of the rim. Hence, increasing $J_{s,mid}$ leads to an increase in the bending energy of this region, thereby penalizing cisterna curling and eventually shifting the flat-to-curled cisterna transition towards smaller values of the spontaneous curvature. In the Appendix we present an analytical estimation of this shift, under certain approximations. In summary, of all the above-mentioned parameters, there are only two free parameters that can change as a result of membrane deformation. The first describes the level of cisternae membrane curling, $r_{gap}$, and the second is associated to the level of nanodomain partitioning between high- and low-curvature membrane regions, ${\displaystyle {\mathrm{\Phi }}_{rim}}$. Therefore, the optimal gap aperture radius and partitioning of the nanodomains between the rim and the middle part of the cisterna, ${\displaystyle \left\{r_{gap}^{\ast },{\mathrm{\Phi }}_{rim}^{\ast }\right\}}$, correspond to the global minimum of the total free energy in the entire parameter space,

The values of the rest of the parameters are fixed and determined from other studies or vary within a range of possible values (see Table 1).

The total and Gaussian curvatures along the surface of the central part of the Golgi cisterna are given by $J_{mid}=\pm 2/R$, and $K=1/R^2$, respectively, where the plus and minus signs in the total curvature value correspond to the bottom and top membrane patches of the central part of the cisterna. Since $h\ll r_{flat}$, the area of these bottom and top membrane surfaces are, to a first approximation, equal, and therefore we can write

where ${\kappa }_{mid}$ and ${\overline{\kappa }}_{mid}$ represent, respectively, the bending rigidity and the modulus of Gaussian curvature at the central part of the cisterna; and $J_{s,mid}$ is the spontaneous curvature in the central part of the cisterna. Based on Equation (12), we can write $1/{\kappa }_{mid}={\text{Φ}}_{mid}/{\kappa }_{lo}+\left(1-{\text{Φ}}_{mid}\right)/{\kappa }_{ld}$ and ${\overline{\kappa }}_{mid}={\alpha }_{\overline{\kappa }}{\kappa }_{mid}$. Using these expressions together with Equations (4), and (8), we can rewrite the bending free energy of the middle region Equation (15) as

In toroidal coordinates $\left\{\phi ,\theta \right\}$ (see Figure 1—figure supplement 1), the total and Gaussian curvatures at the rim surface are given, respectively, by

Similarly, the total bending free energy in the rim area is given by

where ${\kappa }_{rim}$ and ${\overline{\kappa }}_{rim}$ are the bending rigidity and the modulus of Gaussian curvature at the cisternae rim area, respectively; and $J_{s,rim}$ is the spontaneous curvature at the rim. This morphology is similar to the one corresponding for fusion pores (Chizmadzhev et al., 1995; Kozlov et al., 1989). Using Equation (12), we can write ${\displaystyle 1/{\kappa }_{rim}={\mathrm{\Phi }}_{rim}/{\kappa }_{lo}+\left(1-{\mathrm{\Phi }}_{rim}\right)/{\kappa }_{ld}}$ and ${\displaystyle {\overline{\kappa }}_{rim}={\alpha }_{\overline{\kappa }}{\kappa }_{rim}}$. The limits of integration in Equation (19) when the area element is expressed in toroidal coordinates (Equation (1)) are the same as those in Equation (2). This last integral cannot be analytically solved, so we will compute it numerically. In the Appendix, an analytical approximation is obtained under certain simplifying assumptions.