Sphingomyelin metabolism controls the shape and function of the Golgi cisternae

1 Sphingomyelin metabolism controls the shape and function of the Golgi cisternae 1 2 Felix Campelo1, Josse van Galen2,3, Gabriele Turacchio4, Seetharaman Parashuraman4, Michael M. 3 Kozlov5, Maria F. Garcia-Parajo1,6 and Vivek Malhotra2,3,6 4 5 1 ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 6 Castelldefels (Barcelona), Spain. 7 2 Centre for Genomic Regulation (CRG), The Barcelona Institute of Science and Technology, 8 Barcelona, Spain. 9 3 Universitat Pompeu Fabra (UPF), Barcelona, Spain. 10 4 Institute of Protein Biochemistry, National Research Council of Italy, Naples, Italy 11 5 Department of Physiology and Pharmacology, Sackler Faculty of Medicine, Tel Aviv University, Tel 12 Aviv, Israel 13 6 Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain. 14 15 Corresponding authors: 16 Vivek Malhotra 17 Tel: +34-93 316 0235 18 Fax: +34-93 396 9983 19 E-mail: [email protected] 20 21 Felix Campelo 22 Tel: +34-93 554 2225 23 Fax: +34-93 553 4000 24 E-mail: [email protected] 25 2 The flat Golgi cisterna is a highly conserved feature of eukaryotic cells, but how is this 26 morphology achieved and is it related to its function in cargo sorting and export? A physical 27 model of cisterna morphology led us to propose that sphingomyelin (SM) metabolism at the 28 trans-Golgi membranes in mammalian cells essentially controls the structural features of a Golgi 29 cisterna by regulating its association to curvature-generating proteins. An experimental test of 30 this hypothesis revealed that affecting SM homeostasis converted flat cisternae into highly curled 31 membranes with a concomitant dissociation of membrane curvature-generating proteins. These 32 data lend support to our hypothesis that SM metabolism controls the structural organization of a 33 Golgi cisterna. Together with our previously presented role of SM in controlling the location of 34 proteins involved in glycosylation and vesicle formation, our data reveal the significance of SM 35 metabolism in the structural organization and function of Golgi cisternae. 36 3 INTRODUCTION 37 The Golgi complex plays a central role in protein processing, sorting and transport (Emr et al., 2009). 38 In higher eukaryotes the Golgi complex consists of multiple stacks of polarized flattened cisternae 39 (Klumperman, 2011). Cisternae polarization allows for the directional transport and sequential 40 processing of newly synthesized proteins arriving at the cis-face of the Golgi complex from the 41 endoplasmic reticulum (Glick and Luini, 2011; Stanley, 2011). At the trans-Golgi network (TGN), 42 fully processed proteins are sorted and exported to other compartments of the secretory pathway or for 43 secretion (Guo et al., 2014). Remarkably, despite the large influx and efflux of membrane-bound 44 transport carriers, the overall morphology of the Golgi complex remains essentially unaltered. How is 45 the shape of the Golgi cisternae maintained and how does it relate to the function of this organelle? 46 Golgi cisternae are characterized by having a relatively large area-to-volume ratio to accommodate the 47 large numbers of incoming and outgoing transport carriers and also to efficiently regulate the 48 enzymatic reactions occurring at the Golgi membranes (Klumperman, 2011). Moreover, a Golgi 49 cisterna consists of two geometrically distinct regions with very different membrane curvatures: the 50 central cisterna part, which is almost flat with the seldom presence of fenestrations or pores; and the 51 highly bent rims of the cisterna. How the different functions of the Golgi membranes (namely, protein 52 processing and transport) are organized between these two regions is not yet fully understood. 53 We previously reported that disruption of sphingomyelin (SM) organization specifically at the 54 Golgi membranes –by SM synthase-mediated formation of short-chain SM at the trans-Golgi 55 membranes– leads to inhibition of transport carrier formation (Duran et al., 2012) and also to defects in 56 transmembrane protein glycosylation and localization (van Galen et al., 2014). Interestingly, we 57 showed that these effects occur concomitantly with an overall reduction in the lateral lipid order of the 58 Golgi membranes (Duran et al., 2012) as well as with striking alterations in the morphology of Golgi 59 cisternae, which abandon their typical flat morphology and become highly curled (van Galen et al., 60 2014). Based on our findings we suggested that short-chain SM might not be able to generate liquid-61 ordered nanodomains at the Golgi membranes (Duran et al., 2012). However, it is still unclear whether 62 there is any causal relation between the ability of SM to control lateral Golgi membrane organization 63 and the observed changes in the morphology of the Golgi cisternae. 64 Motivated by these experimental observations we decided to investigate the physical 65 mechanisms by which SM metabolism controls Golgi cisternae morphology, with a general aim at 66 4 understanding whether the shape and the function of the Golgi complex are two sides of the same coin 67 and how they relate to each other. Curling of a flat Golgi cisterna has, from a physical point of view, 68 severe consequences. A flat cisterna has a large surface area at its rim with a very high membrane 69 curvature thereby bearing large elastic stresses (Shibata et al., 2009). Hence, cisterna curling is 70 accompanied by a change in the distribution of the membrane elastic stresses. The quantitative analysis 71 of the extent of these stresses and how they can be sustained within the overall cisterna morphology 72 requires a physical description of the Golgi cisterna free energy. Here we present a biophysical model 73 that describes the free energy of a Golgi cisterna to elucidate the mechanisms by which SM 74 homeostasis mechanically regulates the shape of the Golgi complex and therefore its function. In the 75 following, we describe the model, the results derived from it and the experimental validation of the 76 model's predictions follow. 77 5 RESULTS 78 79 Theoretical results 80 Qualitative description of the proposed mechanisms of SM-regulated Golgi morphology 81 The shape of cellular organelles, such as the Golgi complex, results from the generation and 82 stabilization of the curvature of their membranes (Shibata et al., 2009). According to the elastic model 83 of membrane bending (Helfrich, 1973; Campelo et al., 2014), energy is required to induce local 84 changes of the membrane curvature from its preferred or spontaneous curvature. This energy is 85 generally supplied by specialized lipids and/or membrane proteins, usually referred to as membrane 86 curvature generators (Zimmerberg and Kozlov, 2006; McMahon and Gallop, 2005; Kozlov et al., 87 2014). Moreover, the amount of bending energy associated with local curvature deviations is 88 proportional to the local bending rigidity of the membrane (Helfrich, 1973). As a consequence, both 89 local variations in the amounts of curvature generators present on the membrane and in the bending 90 rigidity of the membrane can influence the morphology of Golgi cisternae. Taking into account these 91 considerations, our aim here is to establish a physical model for the Golgi membrane morphology, with 92 a special focus on understanding the mechanisms by which SM metabolism controls the overall shape 93 of the Golgi cisternae. Similar models based on the Helfrich bending energy have been widely used in 94 the past to describe the shapes of lipid vesicles (Seifert, 1997). In such models, transitions from flat 95 cisterna-like vesicles to curled vesicles, named stomatocytes, were promoted upon a reduction in the 96 volume-to-area ratio. We previously proposed that reduction in the volume-to-area ratio of the Golgi 97 cisternae could in fact be responsible for the observed flat-to-curled cisternae transition upon short-98 chain ceramide treatment (Duran et al., 2012). However, in our subsequent studies, we observed no 99 obvious increase in the overall volume-to-area ratio of the Golgi cisternae during such a morphological 100 transition (van Galen et al., 2014). Moreover, Golgi cisternae have an extremely low volume-to-surface 101 ratio, which does not account for the reported flat configurations according to the aforementioned 102 models (Seifert et al., 1991; Miao et al., 1991; Seifert, 1997). Altogether, this prompted us to propose 103 an alternative model that takes into account two contributions that could potentially influence a role in 104 SM-regulated shaping of the Golgi cisternae: (i) the presence of small, rigid, and highly dynamic 105 membrane nanodomains enriched in sphingolipids and cholesterol; and (ii) the SM-dependent 106 recruitment to or release from the Golgi membranes of budding factors and other membrane curvature 107 6 generators essential for the formation of transport carriers. We first qualitatively describe how each of 108 these two contributions can influence the shape of a Golgi cisterna. 109 110 (i) Nanodomain partitioning-mediated mechanism. Experimental evidence has suggested the existence 111 of nanoscopic SM-enriched liquid-ordered domains at the Golgi membranes, although direct 112 visualization has remained challenging (Gkantiragas et al., 2001; Klemm et al., 2009; Duran et al., 113 2012; Bankaitis et al., 2012; Deng et al., 2016). Based on in vitro data, such liquid-ordered membrane 114 domains are expected to have a higher bending rigidity than the surrounding liquid-disordered 115 membrane and are therefore less prone to accommodate membrane curvature (Roux et al., 2005; 116 Heinrich et al., 2010; Dimova, 2014). In essence, the presence of large amounts of such rigid 117 nanodomains at the highly curved rim of a flat Golgi cisterna is associated with a large bending energy 118 penalty. There are two ways to reduce this bending energy. The first one is by partitioning these rigid 119 nanodomains away from the rim to the flatter regions of the Golgi cisterna (Figure 1A). However, such 120 inhomogeneous curvature-driven nanodomain redistribution is entropically unfavorable. Therefore the 121 balance between the bending energy and the entropic free energy dictates the optimal distribution of 122 nanodomains along the cisterna membrane (Figure 1A). The second possibility is to decrease the 123 surface area of the highly curved rim, hence reducing the overall bending stress of the rim, by globally 124 changing the shape of the Golgi cisterna from a flat to a curled configuration, while maintaining the 125 total surface area of the cisterna (Figure 1A). This morphological transition is associated with a 126 reduction of the bending energy of the rim, but also with an increase in the bending energy of the 127 central region of the cisternae (Knorr et al., 2012; Helfrich, 1974). Again, the balance between these 128 two opposite contributions to the overall bending energy determines the optimal cisternae shape 129 (Figure 1A). Obviously, these two means of decreasing the cisternae free energy are not mutually 130 exclusive. Instead, a combination of both lateral nanodomain partitioning and a change in Golgi 131 cisternal shape might possibly result from a decrease in the amounts of SM-enriched nanodomains 132 present at the Golgi membranes (Figure 1A). 133 134 (ii) SM-dependent release of budding effectors. The budding machinery, such as the COPI and clathrin 135 coats, is responsible for generation of large membrane curvatures that are required to form transport 136 carriers at the Golgi membranes (Kirchhausen, 2000; Campelo and Malhotra, 2012). It is reported that 137 7 budding events are more frequent at the rims of the Golgi cisternae, where the membrane is already 138 curved thus facilitating this process (Rothman, 2010; Lavieu et al., 2013; Popoff et al., 2011; Engel et 139 al., 2015). In addition to their role in transport carrier formation, recruitment of components of the 140 budding machinery to the cisterna rims helps stabilizing these highly bent membranes (Figure 1B). 141 Hence, a reduction in the amounts of membrane curvature generators present at the cisterna rim leads 142 to an increase of the bending energy penalty of the rim, which could in turn be relaxed by a flat-to-143 curled shape transition (Figure 1B). 144 145 Prediction of flat and curled cisterna configurations 146 In order to quantify the relative effect in Golgi shaping of the two mechanisms proposed above, we 147 developed a mathematical formulation of the free energy of a Golgi cisterna as a function of its shape 148 and nanodomain distribution on the membrane. The major assumption of our model is that the time 149 scale of mechanical equilibration of the cisterna shape is much smaller than that of the changes in lipid 150 and protein composition through membrane fluxes. Non-equilibrium shapes should be considered only 151 if the composition would change faster than the shape relaxed to the new equilibrium state (Sens and 152 Rao, 2013). We can estimate the mechanical relaxation time as a combination of the characteristic 153 viscosity, bending rigidity and length scale of the cisterna, =/ ~ 1 ms (Allain et al., 2004). 154 On the other hand, the rates of the composition changes based on the fluxes through the Golgi cisternae 155 have been theoretically inferred from experimental data (Dmitrieff et al., 2013), resulting in a 156 characteristic compositional relaxation time by means of membrane fluxes,  ~ 100 s. Since 157 ≪, the cisternae shape is assumed to be mechanically equilibrated for every instant 158 composition. Since the composition is in a steady state, the shape is in mechanical equilibrium 159 corresponding to this steady state composition. Hence, the equilibrium configuration of a cisterna is 160 assumed to correspond to a free energy minimum. 161 We consider the cisterna membrane to have a shape of a sheet bound by a rim (Fig. 1C). The 162 sheet part is represented by two parallel membranes with inter-membrane distance, 2ℎ, much smaller 163 than the sheet lateral dimension, . The maintenance of the narrow luminal space of Golgi cisternae 164 can result from protein arrays bridging the two parallel membranes of a cisterna, which have been 165 visualized by cryo-electron tomography (Engel et al., 2015). Alternatively, membrane adhesion 166 between adjacent cisternae has been shown to be required to keep the narrow luminal space in HeLa 167 8 cells (Lee et al., 2014). The rim shape is modeled by an open toroid of a cross-sectional radius, , 168 merging the sheet boundary (Fig. 1C). The distance between the bridging and/or stacking protein 169 scaffolds forming the flat part of a cisterna and the cisterna edge sets the cross-sectional radius of the 170 cisterna rim. The sheet part of the cisterna can curve into spherical segments of variable radii, , which 171 is accompanied by the corresponding changes of the rim perimeter, =2, the latter measured as 172 the length of the toroidal axis. We will use the shape parameter  as a means to quantitate the degree 173 of cisternal curling. 174 The free energy responsible for the cisterna shape includes the elastic bending energy of the 175 membrane sheet and rim and also the entropic cost of a non-homogeneous partitioning of rigid 176 nanodomains between the sheet and the rim. The bending energy is computed based on the Helfrich 177 model (Helfrich, 1973), in which the membrane elastic properties are characterized by the bending 178 modulus, , and the spontaneous curvature, , the latter describing the intrinsic tendency of the 179 membrane to curve (see the Materials and Methods for a complete description of the model). We 180 assume that the membrane spontaneous curvature is generated by specific proteins or proteins 181 complexes (such as the components of the budding machinery) bound to or inserted in the outer 182 membrane monolayer. These proteins occupy a fraction  of the outer monolayer area, and are 183 characterized by an effective individual spontaneous curvature, , which has typical values in 184 the range ≈0.5 − 0.75  (Campelo et al., 2008). The membrane spontaneous curvature is 185 given by =  (the factor ½ accounting for the resistance of the internal monolayer 186 to curving of the external one) and can vary along the membrane in accord with variation of the area 187 fraction,  (Campelo et al., 2008). 188 The membrane nanodomains are considered to occupy a fraction Φ of the overall membrane 189 area. The nanodomains are assumed to have a vanishing spontaneous curvature and a bending rigidity 190 greatly exceeding that of the surrounding membrane (Roux et al., 2005; Heinrich et al., 2010). The 191 domains can freely partition between the rim and the sheet parts of the system. The detailed 192 presentation of the system free energy is given in the Materials and Methods section. In essence, the 193 relative contribution between the free energy of the rim and that of the rest of the cisterna mainly 194 governs the transitions between flat and curled shapes, in analogy to other membrane systems (Helfrich, 195 1974; Lipowsky, 1992; Knorr et al., 2012). 196 9 We determined the equilibrium shape of a Golgi cisterna by minimizing the free energy for a 197 given set of geometric and elastic parameters (see Table 1) upon specific assumptions. First, we 198 consider the budding machinery to localize exclusively at the rims of the Golgi cisternae, so that the 199 membrane of the sheet part the cisterna has a vanishing spontaneous curvature ,=0 while the rim 200 membrane is characterized by ,= (see Materials and Methods). Second, we assumed that the 201 area fraction of the curvature generators in the rim, , ranges between 0 and 10%, so that the 202 variation range of the membrane spontaneous curvature in the rim is 0≤≤0.033 . And third, 203 the overall membrane area fraction covered by the nanodomains, Φ, varies over a wide range of values, 204 0≤Φ≤0.4. 205 Our model predicts that, depending on the values of the two parameters,  and Φ , the 206 minimal energy state of the system can correspond to a flat cisterna or a highly curled cisterna. In an 207 alternative situation, referred below as the bistability state and considered in more detail in the next 208 section, both the flat and curled shapes correspond to local energy minima. The parameter ranges 209 corresponding to the three possible states of the system are summarized in a shape diagram (Figure 2A). 210 Qualitatively, the model predicts that increasing the spontaneous curvature of the rim membrane by 211 augmenting the amount of the curvature generators in the rim favors a shape transition from the curled 212 to the flat cisterna configuration. In other words, large numbers of the budding factors at a Golgi 213 cisterna favor flat rather than curled cisternae (see Figure 1B). Moreover, according to our 214 computations, transitions between curled and flat cisternae are almost insensitive to variations of the 215 nanodomain area fraction, Φ (Figure 2A). The reason being that the free energy required to partition 216 large amounts of rigid nanodomains away from the curved rim cannot be counterbalanced by the 217 relaxation of the bending energy (see Appendix). 218 Additionally, we considered the situation where the budding machinery is homogeneously 219 distributed along the whole Golgi cisternae, ,=,=, and compared the model predictions 220 with the previous case of the curvature generators concentrated only at the rim (Figure 2A). Since the 221 flat-to-curled cisterna transition is associated with an increase in the surface area of the central part of 222 the cisterna, the presence of curvature generators therein leads to an increase in the total bending 223 energy of this area, thereby opposing cisterna curling. Indeed, our results show no overall qualitative 224 difference from the results in Figure 2A, but only a general shift of the shape transitions between the 225 flat and curled cisterna configurations towards the lower values of the membrane spontaneous 226 16 that Golgi cisternae curling starts already after 30 min of D-cer-C6 treatment, and that after 4 h of 405 treatment, virtually no flat stacks are observed (Figure 6C,D). In agreement with our previous analysis 406 (van Galen et al., 2014), cisternae curling occurs towards the trans-Golgi cisternae/TGN, as monitored 407 by the localization of specific late Golgi markers (Figure 6C). We extracted from these images the 408 radius of flat cisternae, =510±21  (average ± SEM; N=22) and the radius of curling in the 4h 409 D-cer-C6 treated Golgi cisternae, =270±26  (average ± SEM; N=14), which are in agreement 410 with the condition of area conservation (Eq. (A2)). Taken together, these results indicate that short-411 chain ceramide treatment leads to the release of clathrin coats thus reducing the spontaneous curvature 412 of the Golgi membranes, which, as our model predicts, leads to the curling of the Golgi cisternae. 413 414 Experimental evidence for hysteresis of Golgi cisternae morphology during recovery from short-415 chain ceramide treatment 416 Our model predicts the existence of a bistability region in the cisterna shape diagram, a relatively large 417 range of parameters where both flat and curled Golgi cisternae correspond to locally stable shapes 418 (Figure 2A). As we showed, the transitions between the flat and curled configurations within the 419 bistability region are expected to have a hysteretic character (Figure 3D). To experimentally test 420 whether Golgi cisterna shape transition induced by short-chain ceramide treatment exhibits hysteresis, 421 we performed short-chain ceramide washout experiments to monitor the timing of recovery of both the 422 Golgi morphology and the amounts of clathrin present at the Golgi membranes as the cells return to 423 steady conditions. 424 We first investigated how the dynamics of recovery of the Golgi cisternae shape during short-425 chain ceramide washout correlates with the recruitment of clathrin coats to the Golgi membranes. To 426 this aim, we pre-treated HeLa cells with D-cer-C6 for 30 min, after which the cells were extensively 427 washed and incubated with normal growth medium for different times. Then the cells were fixed and 428 the intracellular localization of clathrin-HC was monitored by immunofluorescence microscopy. The 429 results of this experiment show that a 2h short-chain ceramide washout is sufficient to recover similar 430 levels of clathrin-HC at the Golgi membranes as to those found in untreated cells (Figure 6A,B). 431 Recovery of normal clathrin-HC levels at the Golgi membranes after a 4h D-cer-C6 treatment is slower, 432 and occurs in about 6h (Figure 6-figure supplement 1). 433 17 Next, we performed analogous washout experiments to monitor the timing of Golgi cisternae 434 shape recovery. We added D-cer-C6 to HeLa cells for 30 min, after which the cells were extensively 435 washed and incubated in complete medium without D-cer-C6 for different times. The cells were then 436 fixed and the ultrastructure of the Golgi cisternae was visualized by immuno-electron microscopy 437 (Figure 6C). Our results show that the curled-to-flat Golgi cisterna transition during short-chain 438 ceramide washout occurs at a much slower kinetics as the recovery of the clathrin coats to the Golgi 439 membranes (Figure 6B,D). Indeed, even 16h after the short-chain ceramide washout –a condition 440 where the Golgi membranes already recovered their stationary pools of clathrin coats– both flat and 441 curled Golgi cisternae can still be observed (Figure 6C,D). 442 To confirm these observations, we used an alternative approach to quantitate the dynamics of 443 the curled-to-flat Golgi shape transition promoted during short-chain ceramide washout. We took 444 advantage of the fact that the curling of the Golgi cisternae induced by short-chain ceramide is 445 accompanied by a lateral segregation of different Golgi-resident proteins, such as TGN46 and p230, 446 and that the level of this segregation can be quantitatively assessed by immunofluorescence 447 microscopy (van Galen et al., 2014). Although the observed protein segregation correlates with 448 changes in Golgi membrane morphology, its driving mechanisms still remain unknown (van Galen et 449 al., 2014). To monitor the dynamics of protein segregation during Golgi shape recovery after short-450 chain ceramide treatment, HeLa cells were treated with D-cer-C6 for 30 min or 4 h, after which the 451 cells were extensively washed, and incubated in normal medium without short-chain ceramide for 452 different times before being fixed. Then, the intracellular localization of the two trans-Golgi membrane 453 proteins p230 and TGN46 was monitored by immunofluorescence microscopy and the relative 454 colocalization of the two proteins was quantitated by means of the Pearson's correlation coefficient. 455 Our results confirm that cells pre-treated with short-chain ceramide for 30 min required about 12 h to 456 recover the initial levels of p230 and TGN46 colocalization, whereas a longer 4 h pre-treatment with 457 D-cer-C6 required about 24 h for a complete recovery (Figure 6E). 458 Taken together, these results indicate that, after short-chain ceramide washout, the recovery of 459 the levels of clathrin-HC at the Golgi membranes (which we suggest parallels the recovery of the initial 460 values of the membrane spontaneous curvature) occurs much faster than the recovery of the flat 461 cisternae morphology and of protein colocalization. This is indicative of a hysteretic behavior of the 462 transition from flat-to-curled cisternae and reverse, as our model predicts. 463 18 DISCUSSION 464 465 The architecture of the Golgi complex in higher eukaryotes has been the subject of extensive research 466 using numerous experimental approaches, including electron and immunofluorescence microscopy 467 techniques (see (Klumperman, 2011) for a review). Generally, the Golgi complex in mammalian cells 468 consists of a set of 4-8 flat-like cisternae stacked to each other (Emr et al., 2009). Each of these 469 cisternae has a relatively flat central part and a highly curved rim. What is the connection between the 470 shape of Golgi cisternae and the functions of this organelle? It has been suggested that the large 471 surface-to-volume ratio of the Golgi cisternae helps accommodating the continuous influx and efflux of 472 transport carriers to and from these membranes (Griffiths et al., 1989; Glick and Nakano, 2009). 473 Moreover, two of the principal functions of the Golgi complex –protein glycosylation and transport– 474 need to be spatially and timely organized to ensure their efficiency. Thus, it has been proposed that 475 processing events localize preferentially at the central flat part of the Golgi cisternae, whereas transport 476 carrier formation occurs at the rims of the Golgi membranes (Rothman, 2010; Popoff et al., 2011; van 477 Galen et al., 2014; Engel et al., 2015). 478 We previously showed the importance of SM homeostasis in protein organization and 479 function at the Golgi membranes (Duran et al., 2012; van Galen et al., 2014). In particular, we revealed 480 that selective disruption of SM organization at the Golgi membranes leads to (i) an overall reduction in 481 the lateral order of the Golgi membranes (Duran et al., 2012), (ii) a strong inhibition of transport carrier 482 formation at the Golgi complex (Duran et al., 2012), and (iii) a defect in the formation of functional 483 enzymatic domains caused by the physical segregation between Golgi resident enzymes and their 484 substrates (van Galen et al., 2014). Intriguingly, these effects parallel an abrupt change in the 485 morphology of the Golgi complex, which turns from a stack of flat cisternae into an onion-like stack of 486 highly curled cisternae (van Galen et al., 2014). In the present study we aimed at resolving the 487 mechanism by which SM metabolism controls the morphology of the Golgi cisternae. Our approach 488 consisted in the elaboration of a theoretical biophysical model of Golgi cisternae morphology that 489 describes the membrane free energy including the contributions arising from the elastic energy of 490 membrane bending and the entropic contribution of lateral partitioning of SM-rich nanodomains. Our 491 model explains the existence of two distinct families of Golgi cisternae shapes, flat and highly curled 492 cisternae. Moreover, our model predicts the existence of a flat-to-curled shape transition triggered by a 493 19 reduction in the amounts of membrane curvature generators at the Golgi membranes. We 494 experimentally tested this theoretical prediction and observed that clathrin, a protein involved in the 495 assembly of the large membrane curvatures required for the formation of clathrin-coated vesicles at the 496 Golgi membranes, was less present at the Golgi membranes as a result of SM homeostasis alteration. 497 Another key prediction of our model is that the flat-to-curled Golgi cisterna transition is 498 practically insensitive to changes in the amounts of SM-rich rigid nanodomains (Figure 2A). These 499 results indicate that such rigid membrane domains are not sufficient to stabilize the flat cisterna 500 configuration in the absence of membrane curvature generating proteins stabilizing the highly curved 501 rim (see orange region in the shape diagram, Figure 2A). However, our model predicts a non-502 homogeneous partitioning of such lipid nanodomains along the Golgi membrane (Figure 3C). 503 Specifically, rigid nanodomains tend to be concentrated at the central, flatter part of the cisterna rather 504 than at the rim. The extent of the nanodomain redistribution from the rim to the central part of the 505 cisterna is gradually magnified upon the decrease in the amounts of curvature generating proteins at the 506 rim for both flat and curled cisterna configurations (Figure 3C). We hypothesize that such 507 redistribution could be a causal link to the previously reported defects in protein glycosylation in cells 508 where SM homeostasis had been altered (van Galen et al., 2014). According to this conjecture, lateral 509 segregation of Golgi-resident enzymes from their substrates would follow from the lateral 510 repartitioning of lipid nanodomains along the surface of the Golgi cisterna. We propose that during 511 short-chain ceramide washout, three processes with different dynamics occur simultaneously (see 512 Figure 7): (i) recovery of normal levels of membrane curvature generators, which, for a 30' treatment, 513 takes about 2h (Figure 6B); (ii) recovery of the amounts of nanodomains, which we propose is in direct 514 correlation to the recovery of protein colocalization at the Golgi membranes, takes about 12 hours 515 (Figure 6E); and (iii) recovery of the flat Golgi morphology, which takes about 16 h (Figure 6D). 516 Testing this hypothesis requires further experimental work, the results of which will advance our 517 understanding of the role that lipid homeostasis and membrane lateral organization play in regulating 518 the functions of the Golgi complex. 519 520 DAG acts as a signaling effector rather than as a molecular shaper to control Golgi membrane 521 morphology 522 20 DAG is a key regulator involved in transport carrier biogenesis at the Golgi membranes (Baron and 523 Malhotra, 2002; Bard and Malhotra, 2006; Fernández-Ulibarri et al., 2007; Malhotra and Campelo, 524 2011). On the one hand, DAG is a conical lipid, which has the ability on its own to generate negative 525 (positive) membrane curvature if asymmetrically enriched in the cytosolic (luminal) leaflet of a 526 membrane (Carrasco et al., 2007; Leikin et al., 1996; Szule et al., 2002). On the other hand, protein 527 kinase D (PKD), a protein that controls the fission of TGN-to-cell surface transport carriers (Malhotra 528 and Campelo, 2011; Campelo and Malhotra, 2012), is directly recruited to the TGN by binding to DAG 529 (Maeda et al., 2001; Liljedahl et al., 2001; Baron and Malhotra, 2002). It has been reported that low 530 levels of DAG leads to defects in protein export from the TGN (Baron and Malhotra, 2002) and to 531 abnormal Golgi morphology (Litvak et al., 2005). In contrast, local increase in DAG levels leads to the 532 activation of PKD (Malhotra and Campelo, 2011). Active PKD phosphorylates a number of substrates 533 at the TGN membranes, including the lipid kinase PI4KIIIβ (Hausser et al., 2005) and the lipid 534 transport proteins CERT (Fugmann et al., 2007) and OSBP (Nhek et al., 2010), thus regulating the 535 local lipid homeostasis. The theoretical results presented here (Figure 5) indicate that changes in the 536 DAG levels do not directly promote a flat-to-curled Golgi cisternae transition by redistributing along 537 the membrane and changing its curvature. Importantly, it has been recently shown that a local burst in 538 DAG levels at the Golgi membranes caused by increased sphingolipid metabolic flow leads to a peak 539 of activation of PKD, which in turn, through a downstream signaling cascade, results in the 540 consumption of PI(4)P and the consequent release of PI(4)P-binding proteins (Capasso et al., 2016). 541 Phosphoinositides are mostly localized away from SM-rich rigid nanodomains (Arumugam and 542 Bassereau, 2015), thus suggesting that this signaling event occurring at the cytosolic side of the 543 membrane is spatially uncoupled from SM-rich rigid nanodomains in the lumenal leaflet. Based on 544 these results and on our results showing that clathrin coats are released from the Golgi membranes 545 prior to short-chain ceramide-mediated flat-to-curled Golgi cisterna transition, we propose that SM 546 metabolism, through the by-product DAG, indirectly controls Golgi morphology by means of a 547 downstream PKD-dependent signaling cascade rather than by playing a direct mechanical part in 548 membrane bending. 549 550 Membrane curvature generators dynamically stabilize the flat shape of Golgi cisternae 551 21 One of the main results of our model is that the release of membrane curvature-generating proteins 552 leads to the destabilization of the flat Golgi cisterna configuration, triggering a morphological 553 transition towards a curled configuration. Being the central hub of the secretory pathway, the Golgi 554 complex recruits a number of different curvature-generating proteins to efficiently sustain transport 555 carrier formation (Cruz-Garcia et al., 2013; Bonazzi et al., 2005; Campelo and Malhotra, 2012). Our 556 results suggest that the role of such proteins is twofold. First, they induce membrane curvature to 557 accommodate the secretory cargoes into nascent budding carriers prior to their fission from the Golgi 558 membranes. Second, this dynamic series of budding and scission events serves to stabilize the highly 559 bent rims of the Golgi cisternae. As such, we propose that the shape and function of Golgi membranes 560 are maintained by membrane curvature generators via a positive feedback loop where the highly bent 561 Golgi rims provide optimal nucleation sites for the budding of transport carriers. At the same time, the 562 machinery involved in this process maintains and stabilizes a functionally optimal Golgi cisternae 563 morphology. Does the presence of clathrin coats on the Golgi membranes represents the main driving 564 force for flat cisternae stabilization? Or are there other curvature-inducing proteins involved? Altered 565 SM metabolic flow at the Golgi membranes did not affect the localization of COPI components 566 (Capasso et al., 2016) to those membranes. However, other curvature generators might be released in 567 addition to clathrin due to the defects in SM metabolism. It has been recently reported that knockdown 568 of the two Golgi-localized PI(4) kinases in Atg5 knockout cells induces curling of the Golgi cisternae 569 (Yamaguchi et al., 2016). Moreover, it has been shown that components of the COPI machinery are 570 released from the Golgi membranes in HeLa cells incubated at 15ºC, a situation that parallels curling of 571 the Golgi cisternae towards the cis side of the stack (Martinez-Alonso et al., 2005). Altogether, we 572 propose that maintenance of the flat Golgi cisternae morphology requires a combined effort of different 573 classes of curvature generating proteins and that the release of a subset of these proteins can lead to the 574 destabilization of the flat configuration. We suggest that the breaking of the stack symmetry upon 575 cisternae curling is driven by the initial release of rim stabilizers from a few cisternae (trans-cisternae 576 in our experiments, cis-cisternae in (Martinez-Alonso et al., 2005)), which is then followed by the other 577 cisternae. From a theoretical perspective, the effect of including multiple cisternae with different levels 578 of budding effectors is analyzed and discussed in the Appendix. In brief, our results indicate that Golgi 579 curling must parallel some level of release of curvature generators from all Golgi cisternae. Further 580 experiments are needed to test this. 581 22 Our model predicts the existence of shape bistability within a certain range of values of the 582 membrane spontaneous curvature and of the membrane area fraction covered by rigid nanodomains 583 (Figure 2A). Within this region of the parameter space, both flat and curled cisterna configurations 584 correspond to locally stable shapes (Figure 2B). This means that the system can be kinetically trapped 585 in a metastable configuration, which corresponds to a local but not global minimum of the cisterna free 586 energy before it relaxes to the globally stable configuration. Such transition from a metastable to a 587 stable configuration needs to overcome an energy barrier. If the value of the energy barrier is relatively 588 small, transition to the stable configuration can be overcome by thermal fluctuations of the Golgi 589 cisterna shape. Such thermally-triggered transitions follow Arrhenius kinetics, according to which an 590 average transition time, τ, can be estimated as τ=/, where  is a characteristic time scale of 591 Golgi cisterna fluctuations and Δ is the height of the energy barrier (Hänggi et al., 1990; Morlot et al., 592 2012). As mentioned above, we can estimate ≈1  from hydrodynamic arguments as =/ 593 (Allain et al., 2004), where  is the cytosol viscosity,  is a typical length scale of a Golgi cisterna, and 594  is the bending rigidity of the membrane. We can qualitatively compare this to our experimental 595 results on the curled-to-flat Golgi cisterna transition during short-chain ceramide washout (Figure 6). 596 Those results indicated that 14 hours after the recovery of normal clathrin levels at the Golgi 597 membranes, about 50% of Golgi stacks were composed of flat cisternae (Figure 6D). According to the 598 aforementioned kinetics, such shape transition time corresponds to an energy barrier for the curled-to-599 flat transition of Δ≈18 . If we compare this estimation for the curled-to-flat transition to the 600 numerical results of our model (Figure 2A), we find that the spontaneous curvature at the Golgi 601 cisterna rims in the fully recovered state should be of the order of ≈0.0225 . Following the 602 relationship between the membrane spontaneous curvature and the area fraction covered by the 603 curvature generators (Campelo et al., 2008), the predicted cisterna spontaneous curvature corresponds 604 to a membrane area fraction covered by curvature generators of 5−10%, which is a physiologically 605 reasonable estimation. Moreover, we showed that the time of recovery of Golgi protein localization 606 after short-chain ceramide treatment depends on the duration of the treatment (Figure 6E). These 607 different recovery times, we suggest, can be explained by the fact that the longer the treatment with 608 short-chain ceramide, the lower the levels of long-chain SM at the Golgi membranes. Hence, a longer 609 time would be required for the Golgi membranes to recover their normal levels of SM-rich 610 nanodomains and, according to our hypothesis, of Golgi protein recovery. 611 23 Likewise, our model predicts that the Golgi cisterna shape transition is, in thermodynamic 612 terms, a first-order transition because the transition is discontinuous in the shape parameter (in our case, 613 the distance between the center of the cisterna rim and the axis of symmetry, ) (Figure 3D). This 614 indicates that once the transition from a flat cisterna (=500 ) to a highly curled cisterna 615 (40 <<100 ) is triggered, the transition is abrupt because no cisternae of intermediate 616 curling correspond to a locally stable configuration (Figure 3D). Although it is hard to extract 617 quantitative information of the curled cisterna gap opening size, , from the ultrathin sections 618 (Figure 6C) to compare with the theoretical predictions, our ultrastructural analysis of the Golgi 619 morphology qualitatively showed that Golgi curling indeed occurs in an abrupt manner (Figure 6C). 620 In summary, we have presented a theoretical biophysical model of Golgi cisterna morphology, 621 which describes the existence of stable flat and curled Golgi cisternae for different values of the 622 membrane spontaneous curvature. We experimentally validated some of the model's predictions. In 623 particular, our model helps explaining the mechanisms by which a reversible flat-to-curled Golgi 624 cisternae transition is induced upon disruption of SM homeostasis by short-chain ceramide treatment. 625 Flat Golgi cisternae in untreated HeLa cells have stationary levels of different curvature-inducing 626 proteins, such as components of the clathrin-coated vesicle machinery (Figure 7, top left cartoon). 627 Moreover, a certain amount of small, dynamic, SM-enriched rigid nanodomains might be present in the 628 membrane, and slightly enriched in the central flat part of the cisterna (Figure 7, top left cartoon). 629 Treatment of cells with D-cer-C6 has a twofold effect on the Golgi membrane properties: it causes the 630 release of clathrin from the membranes (Figure 6A,B) and decreases the lateral order of the Golgi 631 membranes (Duran et al., 2012) (Figure 7, top right cartoon). The results of our model show that the 632 decrease in membrane spontaneous curvature (through the release of curvature generating proteins such 633 as clathrin) but not a reduction in the number of rigid nanodomains alters the cisterna free energy 634 profile to a situation where the flat cisterna configuration is unstable and hence a fast flat-to-curled 635 cisterna transition occurs (Figure 7, right). Short-chain ceramide washout leads to the recruitment of 636 clathrin back to the Golgi membranes (Figure 6A,B) and, we suggest, also leads to the recovery of the 637 initial levels of SM-enriched rigid nanodomains (Figure 7, bottom left). Under these conditions the 638 system free energy profile presents shape bistability, so the Golgi cisternae are kinetically trapped in 639 the curled configuration. Hence, the curled-to-flat cisterna transition is slow because it requires the 640 energy barrier to be overcome by thermal fluctuations (Figure 7, left). 641 24 Overall, the model presented in here together with some of its experimental validation 642 underscore the crucial role of SM metabolism in regulating the structural morphology and function of 643 the Golgi cisternae. We foresee that future experimental work along these lines will strengthen our 644 predictions and will help to understand better the different factors governing the shape and function of 645 the Golgi complex. 646 25 MATERIALS AND METHODS 647 648 Reagents and antibodies 649 N-hexanoyl-D-erythro-sphingosine (D-cer-C6) was obtained from Matreya and dissolved in pure 650 ethanol (Merck) to a 10 mM stock solution. Sheep anti–human TGN46 was obtained from AbD 651 Serotec (Bio-Rad / AbD Serotec Cat# AHP500, RRID:AB_324049). Mouse anti-p230 was obtained 652 from BD (BD Biosciences Cat# 611280, RRID:AB_398808). Goat anti-Clathrin-HC antibody was 653 from Santa Cruz (Santa Cruz Biotechnology Cat# sc-6579, RRID:AB_2083170). Alexa Fluor–labeled 654 secondary antibodies were obtained from Invitrogen. 655 656 Cell culture 657 HeLa cells, obtained from ATCC, were cultured in DMEM (Lonza) containing 10% FCS. HeLa cells 658 stably expressing the plasmid encoding the first 100 amino acids of rat mannosidase-II in the pEGFP-659 N1 vector (HeLa-MannII-GFP cells) were described previously (Sutterlin et al., 2005; van Galen et al., 660 2014). All cell lines were periodically checked for mycoplasma contamination. 661 662 Immunofluorescence microscopy 663 For clathrin-HC immunostaining, samples were fixed and permeabilized in methanol for 6 min at -664 20ºC. For p230 immunostaining, samples were fixed with 4% formaldehyde in PBS for 20 min and 665 permeabilized with 0.2% Triton X-100 in PBS for 30 min. Fixed cells were then blocked in 2% BSA in 666 PBS for 30 min before antibody staining. Cells were then sequentially incubated for 1h at room 667 temperature first with primary and then with secondary antibodies diluted in blocking buffer. Samples 668 were analyzed with a confocal system (TCS SP5 II CW STED; Leica) in confocal mode using a 100x, 669 1.4 NA objective and HyD detectors (Leica). Alexa Fluor 488–, 568-, 594-conjugated secondary 670 antibodies were used. Images were acquired using the Leica software and converted to TIFF files using 671 ImageJ (version 1.43; National Institutes of Health). Two-channel colocalization analysis was 672 performed using ImageJ, and the Pearson’s correlation coefficient was calculated using the Manders’ 673 coefficients plugin developed at the Wright Cell Imaging Facility (Toronto, Ontario, Canada). 674 675 Immunoelectron microscopy 676 32 where the individual contributions to the total free energy are given by Eqs. (17), (20), and (6). In the 851 model we consider that the spontaneous curvature of the membrane could take different values at the 852 rim and central regions, , and ,, respectively. The total membrane free energy Eq. (13) 853 depends on a set of geometric parameters describing the cisterna morphology, ,,,ℎ; a set 854 of nanodomain-related parameters 󰇝,Φ,Φ󰇞; and a set of parameters describing the elastic 855 properties of the membrane, ,, ,,,,. A thermodynamic treatment of the curvature 856 effectors could in principle be incorporated into the model. However, in order to reduce the amount of 857 free variables in the model, we distinguished two extreme situations: (i) the membrane bending 858 proteins, contributors to the membrane spontaneous curvature, are only localized at the rims of the 859 Golgi cisternae, implying that ,=0 and ,=, which could be explained by the fact that 860 membrane recruitment of some of these proteins is highly sensitive to membrane curvature (Antonny, 861 2011); and (ii) the budding machinery is homogeneously distributed along the whole Golgi membrane, 862 ,=,=. As we showed, the results are qualitatively similar when considering the presence 863 of curvature generators in the central part of the Golgi cisternae, but the shape transition quantitatively 864 shifts. The reason for such a shift comes from the fact that, since the total surface area is conserved, 865 curling of a cisterna leads to an increase in the surface area of the central part of the cisterna, 866 concomitant with a decrease in the surface area of the rim. Hence, increasing , leads to an increase 867 in the bending energy of this region, thereby penalizing cisterna curling and eventually shifting the flat-868 to-curled cisterna transition towards smaller values of the spontaneous curvature. In the Appendix we 869 present an analytical estimation of this shift, under certain approximations. In summary, of all the 870 above-mentioned parameters, there are only two free parameters that can change as a result of 871 membrane deformation. The first describes the level of cisternae membrane curling, , and the 872 second is associated to the level of nanodomain partitioning between high- and low-curvature 873 membrane regions, Φ. Therefore, the optimal gap aperture radius and partitioning of the 874 nanodomains between the rim and the middle part of the cisterna,  ∗,Φ ∗, correspond to the 875 global minimum of the total free energy in the entire parameter space, 876 877  ∗,Φ ∗=min∀,,Φ. (14) 878 879 33 The values of the rest of the parameters are fixed and determined from other studies or vary within a 880 range of possible values (see Table 1). 881 882 Mathematical expression of the bending free energy of a membrane cisterna 883 The derivation of total bending energy of a Golgi cisterna is detailed here, taking separately the 884 contributions from the central part of the cisterna and of the cisterna rim. In the Appendix we also 885 present simplified analytical expressions for the free energy of the cisterna, obtained under certain 886 approximations. 887 888 (i) Central region 889 The total and Gaussian curvatures along the surface of the central part of the Golgi cisterna are given 890 by =±2/, and =1/, respectively, where the plus and minus signs in the total curvature 891 value correspond to the bottom and top membrane patches of the central part of the cisterna. Since 892 ℎ≪, the area of these bottom and top membrane surfaces are, to a first approximation, equal, and 893 therefore we can write 894 895   = 󰇡 +,󰇢+  , (15) 896 897 where  and  represent, respectively, the bending rigidity and the modulus of Gaussian 898 curvature at the central part of the cisterna; and , is the spontaneous curvature in the central part of 899 the cisterna. Based on Eq. (12), we can write 1/=Φ/+(1−Φ)/ and =900  . Using these expressions together with Eqs (4), and (8), we can rewrite the bending free energy 901 of the middle region Eq. (15) as 902 903   =. /( ) /()()/(). (16) 904 905 (2) Rim region 906 In toroidal coordinates 󰇝,󰇞 (see Figure 1-figure supplement 1), the total and Gaussian curvatures at 907 the rim surface are given, respectively, by 908 34 909 = , (17) 910 911 = . (18) 912 913 Similarly, the total bending free energy in the rim area is given by 914 915   =∬󰇣 −,+󰇤, (19) 916 917 where  and  are the bending rigidity and the modulus of Gaussian curvature at the cisternae 918 rim area, respectively; and , is the spontaneous curvature at the rim. This morphology is similar to 919 the one corresponding for fusion pores (Chizmadzhev et al., 1995; Kozlov et al., 1989). Using Eq. (12), 920 we can write 1/=Φ/+(1−Φ)/ and = . The limits of integration in 921 Eq. (19) when the area element is expressed in toroidal coordinates (Eq. (1)) are the same as those in 922 Eq. (2). This last integral cannot be analytically solved, so we will compute it numerically. In the 923 Appendix, an analytical approximation is obtained under certain simplifying assumptions. 924 925 Strategy of computations 926 To obtain the local stable shapes of Golgi cisternae as a function of the spontaneous curvature and of 927 the total area fraction of nanodomains, our strategy is to compute the membrane free energy Eq. (13) as 928 a function of the gap aperture, , and the nanodomain area fraction at the cisternae rim, Φ. Then, 929 for all values of the gap aperture, <≤, we found the optimal distribution of 930 nanodomains at the cisternae rim, Φ ∗, by minimization of the free energy with respect to this 931 parameter, 932 933 ,Φ ∗()=min∈[,],Φ(), (20) 934 935 35 Equation (20) represents the partitioning-optimized free energy as a function of the gap aperture length 936 (that is, as a function of the cisterna morphology). Depending on the parameter values, the free energy 937 of the cisterna Eq. (20) has one or two local minima, which correspond to curled or flat cisternae 938 morphologies. Moreover, to compute the energy barriers, we used that 939 =max∈[,],Φ ∗(), =(,Φ ∗()), and =940 ,Φ ∗(). 941 942 Free energy including DAG redistribution along the membrane 943 The free energy of a Golgi cisterna, including the bending energy term taking into account a different 944 distribution of DAG molecules between the top, bottom and rim regions of both the luminal and 945 cytosolic monolayers (see Figure 5A), as well as the associated entropic free energy penalty of a non-946 homogeneous distribution of DAG molecules reads as, 947 948 =949  ∑− , −, −,∈󰇝,,󰇞+950  ∑∑ , , +1−, 1−, ∈󰇝,,󰇞 ∈󰇝,󰇞, (21) 951 952 where , are the spontaneous curvatures of the top, bottom, and rim bilayers (index i), ≈953 0.6  is the area per DAG molecule (Shemesh et al., 2003), and ,  is the DAG area fraction in 954 the cytosolic or luminal monolayers (index j) of the top, bottom, and rim bilayers (index i). 955 Considering that the total amount of DAG is symmetrically distributed between the luminal and 956 cytosolic monolayers of the Golgi membrane, these quantities are related to the total membrane DAG 957 area fraction, , as 958 959 ∑, ∈󰇝,,󰇞=∑, ∈󰇝,,󰇞=. (22) 960 961 36 Acknowledgements 962 We thank members of the Garcia-Parajo lab and Giovanni D'Angelo for valuable discussions. FC and 963 MGP acknowledge support by the Spanish Ministry of Economy and Competitiveness (“Severo Ochoa” 964 Programme for Centres of Excellence in R&D (SEV-2015-240522) and FIS2014-56107-R), BFU2015-965 73288-JIN, AEI/FEDER;UE, Fundacion Privada Cellex, HFSP (GA RGP0027/2012), EC FP7-NANO-966 VISTA (GA 288263) & LaserLab 4 Europe (GA 654148). MMK is supported by the Israel Science 967 Foundation. VM acknowledges support from the Spanish Ministry of Economy and Competitiveness 968 through the Programme ‘Centro de Excelencia Severo Ochoa 2013-298 2017’ (SEV-2012-0208); and 969 support from the CERCA Programme / Generalitat de Catalunya. VM is an Institució Catalana de 970 Recerca i Estudis Avançats (ICREA) professor at the Center for Genomic Regulation and the work in 971 his laboratory is funded by grants from MINECO’s Plan Nacional (BFU2013-44188-P), Consolider 972 (CSD2009-00016), and European Research Council (268692). The project has received research 973 funding from the European Union. This paper reflects only the author’s views. The Union is not liable 974 for any use that may be made of the information contained therein. 975 976 37 References 977 978 Abramowitz, M., and I. Stegun. 1964. Handbook of mathematical functions: with formulas, graphs, 979 and mathematical tables. 980 Allain, J.M., C. Storm, A. Roux, M. Ben Amar, and J.F. Joanny. 2004. Fission of a multiphase 981 membrane tube. Phys Rev Lett. 93:158104. 982 Andelman, D., M. Kozlov, and W. Helfrich. 1994. Phase transitions between vesicles and micelles 983 driven by competing curvatures. Europhys. Lett. 984 Antonny, B. 2011. 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Traffic. 6:32–44. 1126 48 Φ, for two different values of the spontaneous curvature, =0.0175  (solid lines), and =1297 0.0225  (dashed lines). 1298 1299 Figure 4 - figure supplement 1. Effect of the cisterna surface area on the Golgi cisterna shape 1300 diagram. Shape diagrams for different values of the area fraction covered by nanodomains, Φ, and of 1301 the membrane spontaneous curvature, , for two different values of the flat cisterna radius, =1302 500  (solid lines), and =1000  (dashed lines). 1303 1304 Figure 6 - figure supplement 1. Relative localization of Clathrin-HC at the Golgi area in D-cer-1305 C6 treated cells. HeLa cells were treated with 20 µM D-cer-C6 for 0h or 4h, after which the cells 1306 were extensively washed and incubate in normal medium for the indicated times. Then the cells were 1307 fixed and the localization of Clathrin-HC and the TGN marker protein p230 was monitored by 1308 immunofluorescence microscopy and the results quantitated. The bars show the intensity of Clathrin-1309 HC in the Golgi area relative to the intensity in the rest of the cytoplasm, for at least 15 cells from 3 1310 different experiments. Bars represent average values and error bars are the S.E.M. 1311 49 Table 1. Model parameters. 1312 Parameter Value(s)  500 nm; 1000 nm  30 nm ℎ 15 nm  5 nm; 2 nm; 20 nm Φ 0 - 0.4  20 kBT  80 kBT   -0.83   0 - 0.033 nm-1 1313 1314 50 APPENDIX 1315 1316 In this Appendix, we derive and discuss analytical approximations of the Golgi morphology model, and 1317 we compare them to the numerical results presented in the text. 1318 1319 Approximate analytic expression for the total bending energy of a cisterna 1320 Our first aim is to obtain an analytically treatable, approximate expression for the total bending energy 1321 of a cisterna. For this purpose, we start by deriving a simplified expression of the total surface area of 1322 the cisterna rim, Eq. (2). Assuming that ≫ℎ, we get 1323 1324 =4(−). (A1) 1325 1326 This approximation is most accurate for the case of a flat configuration, where =. Using Eq. 1327 (A1) together with Eq. (3), we can obtain a simplified version of Eq. (5), which describes the 1328 relationship between the radius of Golgi curling, , and the length of the cisterna opening, , 1329 1330 = () ()≈ / . (A2) 1331 1332 Next, we derive approximate expressions for the bending energy of the central and rim regions of a 1333 Golgi cisterna, respectively. We consider that the role of the Gaussian curvature is minor, hence we 1334 take =0. First, by plugging Eq. (A2) and Eq. (3) in Eq. (15), we get 1335 1336   =,+161−/ . (A3) 1337 1338 The following step is to generate an approximate expression for the non-analytical bending energy of 1339 the rim region, Eq. (19). We assume again a vanishing modulus of the Gaussian curvature, =0. For 1340 slightly curled cisternae, we could assume that ≫, which would lead to an expression for the 1341 total curvature of the rim. Eq. (17), =1/. However, this expression does not hold for highly 1342 51 curled cisternae, where ~ (see Figure 1 - figure supplement 1). In such situation, we can 1343 approximate the expression for the total curvature of the rim as 1344 1345 = −  󰆓, (A4) 1346 1347 where ′=−. Since this expression introduces only a relatively small error of the order of 1348 /≪1 for almost flat cisternae, we decided to use Eq. (A4) as a relatively good approximation 1349 for the whole range of values of . Hence, using Eqs. (A1), (A4), we can write down an analytical 1350 approximation of Eq. (19) as 1351 1352   =2(−)󰆒󰆒 , 󰆒  , (A5) 1353 1354 which, for relatively flat cisternae, ≫, can be further simplified as 1355 1356  ≫=2(−),  =2 , (A6) 1357 1358 where we defined 1359 1360 =(−),  , (A7) 1361 1362 as an effective line tension of the rim, in analogy to the formal description of the lipid bilayer edge 1363 energy arising during the formation of small vesicles upon sonication (Helfrich, 1974) and of budding 1364 membrane domains (Lipowsky, 1992). Finally, the total bending energy of the cisterna is the sum of 1365 the contributions from the central and rim regions, =  + . 1366 1367 Analytical expression for the cisterna shape transitions 1368 The free energy of nanodomain partitioning, , has a logarithmic dependence on the nanodomain 1369 area fractions, Φ and Φ, thus posing a challenge for an analytic treatment (see Eq. (9)). 1370 52 Nevertheless, the degree of nanodomain redistribution only appears in the expression for the total 1371 bending energy of a cisterna through the bending rigidities,  and , given by Eq. (12). Hence, 1372 to get an analytical approximation of the numerically-computed shape diagram boundaries (see Figure 1373 2A), we consider the two extreme situations of nanodomain partitioning, for which we will compute 1374 the total bending energy: The first situation corresponds to no nanodomain partitioning, that is to 1375 homogeneous distribution of nanodomains along the entire cisterna. From a physical point of view, this 1376 situation corresponds to when the total free energy of the cisterna is dominated by the entropic 1377 contribution of nanodomain partitioning. The second case corresponds to the complete partitioning of 1378 the nanodomains to the central part of the cisterna. From a physical point of view, this corresponds to 1379 the bending energy dominating the total free energy of the cisterna. We later discuss about the relative 1380 importance of the entropic part of the total free energy of a cisterna as compared to the bending energy. 1381 In both cases, the total free energy of a cisterna reads as 1382 1383 =  +  , (A8) 1384 1385 where   and   are given by Eqs. (A3) and (A5), respectively. If we further assume that the 1386 spontaneous curvature of the central region of the cisterna is zero, we can write Eq. (A8) as 1387 1388 =16 1− +2(−)󰆒󰆒 , 󰆒  . (A9) 1389 1390 Let us first estimate the critical values of the rim spontaneous curvature that are associated 1391 with a loss of stability of flat cisternae, ,∗. Mathematically, this corresponds to the situation where 1392 the free energy Eq. (A9) has a local maximum at =, 1393 1394  , , ,∗=0, (A10) 1395 1396 the solution of which leads to the expression for the spontaneous curvature of the rim, 1397 1398 53 ,∗ ≃1−4   , (A11) 1399 1400 where we assumed that ≫. Interestingly, from this expression we can compute the critical 1401 effective line tension, Eq. (A7), as ∗=,∗=8 /, where =2 is an effective 1402 total bending rigidity of the central part of the cisterna, given that it is made of two parallel bilayers. 1403 This rough estimation matches with previously calculated analytical values for critical line tensions 1404 required to form spherical vesicles by sonication (Helfrich, 1974) or budded membrane domains 1405 (Lipowsky, 1992). 1406 We next estimate the critical value of the rim spontaneous curvature associated with a loss of 1407 stability of a highly curled cisterna, ,∗∗. Basically, this condition corresponds to a situation where 1408 the total free energy Eq. (A9) ceases to have a local minimum for a highly curled cisterna configuration. 1409 The local extremes of the free energy function are given by the solutions of the equation  =0, 1410 which can be approximated as 1411 1412 −16    + 1−,−(−)=0, (A12) 1413 1414 which is a cubic equation in , with coefficients =−16    , = 1−,, 1415 =0, and =−(−). Loss of a local minimum for a highly curled configuration, 1416 mathematically, corresponds to Eq. (A12) having only one real solution (and two complex conjugate 1417 solutions) instead of three real solutions. The transition in the number of real solution of a cubic 1418 equation corresponds to when the discriminant of the cubic equation, Δ=−4−27, is equal 1419 to zero (see, for instance, (Abramowitz and Stegun, 1964)). This condition, after some algebra, leads to 1420 1421 ,∗∗ =1−2√3   /. (A13) 1422 1423 We can now estimate the values of the transition spontaneous curvatures, Eqs. (A11) and (A13), which 1424 correspond to the aforementioned extreme situations of complete partitioning and homogeneous 1425 distribution of nanodomains. 1426 54 Homogeneous nanodomain distribution: This situation corresponds to the condition Φ=Φ=Φ, 1427 and therefore ==. Under these circumstances, it is straightforward to see that the 1428 transition zones do not depend on the area fraction covered by nanodomains, Φ, or on the value of the 1429 bending rigidity of the membrane, , which is the only energy scale of the system. Hence, Eqs. (A11) 1430 and (A13) can be, respectively, expressed as 1431 1432 ,∗, =1−4  , (A14) 1433 1434 and 1435 1436 ,∗∗, =1−2√3  /. (A15) 1437 1438 Complete nanodomain partitioning: As explained above, this situation corresponds to the condition 1439 Φ=0, and Φ≃Φ. According to Eq. (12), the bending rigidities at the rim and central part of 1440 the cisterna are expressed as = and =/1−Φ(1−/). Now, we can rewrite 1441 Eqs. (A11) and (A13) as 1442 1443 1444 ,∗, =1−4   (/) , (A16) 1445 1446 and 1447 1448 ,∗∗, =1−2√3   (/) /. (A17) 1449 1450 In Appendix 1-figure 1, we plot these analytical approximations of the transition zones of the shape 1451 diagram, Eqs. (A14)-(A17), and compare them to the numerical computations shown in Figure 2A. 1452 These results show that our analytical approximation is good enough to qualitatively estimate the shape 1453 55 diagram of flat-to-curled cisterna transitions, as well as the shape bistability. However, these results 1454 differ quantitatively from the numerical solutions by about 20% (see Appendix 1 - figure 1). 1455 1456 Analytical description of the relative contribution of the entropic partitioning free energy to the 1457 total free energy of the system 1458 Our numerical results showed that the contribution of the nanodomain area fraction to the Golgi 1459 cisterna shape diagram is relatively minor (Fig. 2A). Moreover, increasing the size of the nanodomains 1460 increased the sensitivity of the shape transitions to the amount of nanodomains (Fig. 4). To gain some 1461 more insight into the role of the nanodomain area fraction on the Golgi cisterna shape transition, we 1462 aimed at understanding the relative contribution to the overall cisterna free energy of the entropic free 1463 energy of nanodomain partitioning and the bending energy. For the sake of simplicity, we focus on the 1464 flat cisterna configuration, corresponding to =. In this case, and assuming that ≪, 1465 we can express the bending energy of the cisterna, Eq. (A9), as 1466 1467 =2(−) 1−,, (A18) 1468 1469 where, according to Eq. (12), 1470 1471 = ()= ()(), (A19) 1472 1473 where ξ=(Φ−Φ)/Φ is the nanodomain partitioning coefficient, which ranges between ξ=0 for 1474 a non-partitioned homogeneous nanodomain distribution to ξ=1 for fully partitioning of nanodomains 1475 away from the cisterna rim. Noteworthy, the bending energy Eq. (A18) is a monotonically decreasing 1476 function for 0<ξ<1, as long as >. Also assuming that ≪, we can express the 1477 partitioning free energy, Eq. (9), as 1478 1479 =  [ΦΦ+(1−Φ)(1−Φ)]+2(−) [ΦΦ+1480 (1−Φ)(1−Φ)], (A20) 1481 1482 56 where Φ=Φ(1−ξ). In this case, the partitioning free energy, Eq. (A20) is a monitonically 1483 increasing function for 0<ξ<1, as long as Φ<1/2. Given this opposite monotonicity between the 1484 bending energy, Eq. (A18) and the partitioning free energy, Eq. (A20), the relative contribution of 1485 these two energies will dictate whether the partitioning entropy is the dominant part of the free energy, 1486 and therefore the nanodomains will tend to be homogeneously distributed along the cisterna; or the 1487 bending energy dominates and an extensive partitioning is to be expected. To have a rough analytical 1488 estimation of this relative contribution, we look for the condition where the growth slopes of the two 1489 energies cancel each other out when there is no nanodomain partitioning, that is 1490 1491   󰇻=0, (A21) 1492 1493 which, after some algebra, leads to the expression of a critical nanodomain size, ∗, 1494 1495 ∗=  √(/) (/)/() ,, (A22) 1496 1497 below which entropy dominates and low nanodomain partitioning is expected; and above which 1498 bending energy dominates and nanodomains are mostly distributed away from the cisterna rim to 1499 minimize the overall free energy. In Appendix 1-figure 2, we plot the value of the critical nanodomain 1500 size as a function of the spontaneous curvature of the rim and/or the area fraction covered by 1501 nanodomains. These results indicate that, indeed, the free energy penalty of nanodomain redistribution 1502 dominates over the bending energy for small values of the nanadomain size, and therefore large 1503 partitioning away from the rim is entropically prevented. 1504 1505 Analytical estimation of the role of the distribution of membrane curvature generators in the 1506 cisterna shape diagram 1507 In Figure 2 - figure supplement 1, we numerically showed that the presence of curvature generators on 1508 the central part of the Golgi cisterna does not qualitatively change the shape diagram but shifts the flat-1509 to-curled transitions to lower values of the spontaneous curvature. To better understand this change and 1510 have a rough analytical estimation of the extent of the shift, we proceeded to compare the total bending 1511 57 energy (assuming that there are no rigid nanodomains) of a flat cisterna to a simplified situation of a 1512 completely curled cisterna without a rim (that is, corresponding to two concentric spheres, separated by 1513 a distance 2ℎ and with a mean radius ). In the flat configuration, the surface area of the rim is 1514  =4(−), while the surface area of the central part is  =2 , and the 1515 total surface area of the cisterna is = + . In the completely curled configuration, there is 1516 no rim, so the total surface area is = =8 . Surface area conservation leads to =1517   1+2(−)/. The bending energies for the rim and central parts of the flat cisterna 1518 are, respectively, 1519 1520  =  󰇡 −,󰇢  , (A23) 1521  =  ,  . (A24) 1522 1523 The bending energy of the completed curled cisterna made of two concentric spheres with 1524 approximately opposite total curvature ±1/ is 1525 1526  =  󰇡 −,󰇢  . (A25) 1527 1528 The condition where the two shapes have the same total bending energy can be mathematically 1529 expressed as  + = , characterized by a rim spontaneous curvature , () . Plugging all 1530 these expression together, we obtain the following expression, 1531 1532 1−, () −,= ()  . (A26) 1533 1534 When the spontaneous curvature of the central part is zero, ,=0 , we obtain 1535 1536 , () (,=0)=  −2 () . (A27) 1537 1538