A minimal self-organisation model of the Golgi apparatus

The design principles dictating the spatio-temporal organisation of eukaryotic cells, and in particular the mechanisms controlling the self-organisation and dynamics of membrane-bound organelles such as the Golgi apparatus, remain elusive. Although this organelle was discovered 120 years ago, such basic questions as whether vesicular transport through the Golgi occurs in an anterograde (from entry to exit) or retrograde fashion are still strongly debated. Here, we address these issues by studying a quantitative model of organelle dynamics that includes: de-novo compartment generation, inter-compartment vesicular exchange, and biochemical conversion of membrane components. We show that anterograde or retrograde vesicular transports are asymptotic behaviors of a much richer dynamical system. Indeed, the structure and composition of cellular compartments and the directionality of vesicular exchange are intimately linked. They are emergent properties that can be tuned by varying the relative rates of vesicle budding, fusion and biochemical conversion.

The highly dynamical nature and compact structure of a stacked Golgi makes it difficult to 39 determine how cargo-proteins are transported in an anterograde fashion from the cis to trans-side 40 of the stack and how this transport is coupled to processing. Several models have been proposed 41 to explain the transport dynamics of Golgi cargo (See sketch in Fig.1 A). They mostly belong to  is a local mechanism of identity conversion distinct from the maturation of entire compart-146 ments, which is also affected by the dynamics of budding and fusion. To limit the number of 147 parameters, the biochemical conversion rate m is the same for the two reactions (cis→medial 148 and medial→trans) and is independent of the concentration. Introducing cooperativity in the 149 conversion process is expected to increase the robustness of the self-organisation process 150 (Vagne and Sens, 2018a). 151 In its simplest form, our model contains only four parameters: the rates of injection, fusion, 154 budding, and biochemical conversion ( , f , b , m ). By normalizing time with the fusion rate, 155 we are left with three parameters: = ∕ f , b = b ∕ f and m = m ∕ f . The dynamics of the 156 system is entirely governed by these stochastic transition rates, and can be simulated exactly using 157 a Gillespie algorithm (Gillespie, 1977), described in App.1. 158

159
Mean-field description of the system. 160 The complexity of the system prohibits rigorous analytical calculation. Nevertheless, analytical 161 results can be obtained for a number of interesting quantities in certain asymptotic regimes where 162 simplifying assumptions can be made. We present below some of these derivations. 163 De-novo formation and steady-state composition in the well-sorted limit 164 The composition of the system at steady-state is difficult to compute due to the fact that the exit of 165 components To estimate the typical size of compartments, we assume that for each species, a single large 178 compartment of size interacts through budding and fusion with a number v of vesicles of the 179 same identity (so that = + v ). The compartments size for each species then satisfies: To limit the number of parameters, we fix the average system's size = cis + medial + trans to a set value = 300 in the main text, which is suitable for Golgi ministacks whose total area is of the order of 1 m 2 (the area of a mammalian Golgi ribbon is much larger) (Yelinek et al., 2009), corresponding to about a few hundreds of vesicles of diameter ∼ 10 − 50nm. This is obtained by adjusting the influx to the variation of the other parameters according to: We show in App.1 Fig.1 that the overall structure of the phase diagrams for the compartments size 181 and composition is robust upon variation of the average system's size.

182
The de-novo formation of the system can be approached with Eq.3. Starting with an empty 183 system with an influx (= ) of cis vesicles, the system grows stochastically to reach the steady-184 state size after a time of order 1∕ f . The evolution of the size of the system with time obtained 185 from numerical simulations, shown in App.4, Fig.1 190 Although Eq.6 is only valid in the well-sorted limit, it permits a satisfactory control of the average 191 system size over the entire range of parameters, with only a 10% variation for small budding rates 192 (App.5, Fig.1A). At steady-state, the system size exhibits stochastic fluctuations around the mean 193 value that depend on the parameters (App.5, Fig.1A). The large fluctuation (about 30%) for low 194 budding rate b stem from the fact that compartments are large and transient, as explained below. 195 Compartment size and composition in the maturation-dominated regime. 196 For low values of the budding rate b , compartments do not shed vesicles and evolve in time 197 independently of one another in a way dictated by a balance between fusion-mediated growth 198 and biochemical maturation. Their size and composition can be approximatively calculated by 199 assuming that compartments grow in size by fusion from a constant pool of vesicles containing the 200 same number v of vesicle for all three identities. Calling cis ( ), medial ( ) and trans ( ) the amount of 201 components of the three identities in a given compartment of total size = cis + medial + trans , and 202 neglecting vesicle budding, the mean-field equations satisfied by these quantities are: Starting with a vesicle of cis identity for = 0: cis (0) = 1, medial (0) = trans (0) = 0, the compartment size evolves linearly with time: ( ) = 1 + v , and the composition of each species satisfies: The fraction of each species in the compartment is thus independent on v , and reads: These results show that in this regime, a given compartment smoothly evolves from a mostly cis 205 to a mostly trans identity over a typical time 1∕ m . The maximum amount of medial identity is 206 obtain for m = 1 and is medial = 1∕ ≃ 0.37. Therefore, the mechanism does not lead to pure medial 207 compartments. These analytical results are confirmed by the full numerical solution of the system 208 in the low budding rate regime shown in App.5, p.34 and discussed in detail below.

209
Steady-state organisation 210 The steady-state organisation and dynamics of the system is described in terms of the average 211 size and purity of compartments (introduced here and detailed in the App.2). The stationary size 212 distribution of compartments is a decreasing function of the compartment size, which typically 213 shows a power law decay at small size, with an exponential cut-off at large size ( Fig.2A) beyond which it is unlikely to find a compartment (Vagne et al., 2015). 219 The typical compartment size varies with parameters as shown in Fig.2B. Increasing the ratio of For all panels, ER = TGN = 1. See also App.5, p.34 for further characterizations of the steady-state organisation, App.7, p.39 for the role of composition of the exit compartment, and App.8, p.41 for results with alternative budding and fusion kinetics. trans) from a perfectly mixed composition. = 0, 1∕2, 1 correspond respectively to compartments 229 that contain the same amount of the three identities, the same amount of two identities, and a single 230 identity (App.2, page p.25). The purity of the system is the average purity of each compartment 231 weighted by its size, ignoring vesicles. The dependency of the average purity with the parameters 232 is shown in Fig.2C. As for the size of the compartments, it is non-monotonic in the biochemical 233 conversion rate, and regions of lowest purity are found for intermediate biochemical conversion  disruption of molecular gradients in the system, which we interpret, in the limits of our system, 369 as a lower purity due to slower sorting kinetics following an impairment of the budding dynamics. 370 Importantly, the correlation between the size and purity of the compartments predicted by our 371 model is in agreement with the observation that decreasing the budding rate by altering the activity 372 of COPI (a budding protein) leads to larger and less sorted compartments (Papanikou et al., 2015). 373 In yeast again, over-expression of Ypt1 (a Rab protein) increases the transition rate from early to 374 transitional Golgi, and increases the co-localization of early and late Golgi markers (Kim et al., 2016). 375 We interpret this as a decrease of the purity of Golgi cisternae upon increasing the biochemical 376 conversion rate by unbalancing the ratio of budding to conversion rates. This suggests that the   Here is the nomenclature used in the main text and Appendix sections. 520   Budding flux of vesicles decorated by the identity , from a compartment   The nomenclature we use in this section is presented in the Tables 1, 2 and 3 in p.15. We propose a coarse-grained, discrete model of the Golgi Apparatus. The smallest element is a vesicle, which defines the unit size of the system. Other compartments are assemblies of fused vesicles; their size is an integer corresponding to an equivalent number of vesicles that have fused to build this compartment. Each unit of membrane is of either cis, medial or trans-identity. A vesicle has a unique identity, and a compartment of size is a assembly of discrete membrane patches of different identities. We let the system self-organizes between two boundaries, the ER and the TGN, that are two biological boundaries of the Golgi (Presley et al., 1997; Klemm et al., 2009). This theoretical Golgi is self-organized as we do not impose any topological or functional constraints, such as the number of compartments or the directionality of vesicular trafficking. The evolution of the system is only dictated by three mechanisms that are budding, fusion and biochemical conversion (short description in the main text, complete description bellow). In order to keep the model as simple as possible, we neglect all mechanical and spatial dependencies. This means that we do not take into account motion of compartments in space, or mechanical properties of the membranes. Any two compartments have the possibility to fuse together, with a rate that depends on their relative composition (see below), but not on their size. An alternative model where compartments cannot fuse with one another is presented in App.8 on p.42. Biochemical conversion 706 In the Golgi, it is known that cargo (Kelly, 1985) and cisternae (Day et al., 2013) undergo biochemical modifications over time. Changes in membrane identity are driven by maturation cascades of proteins such Rab GTPases (Suda et al., 2013). These biochemical conversions can be directly observed using live microscopy, and have been best characterized in the yeast Golgi (Matsuura-Tokita et al., 2006). Biochemical conversion of membrane identity is a complex and possibly multi-step process involving different kinds of enzymes. As we are mostly interested in the interplay between biochemical conversion (which mixes different identities within a single compartment) and sorting mechanisms, we coarse-grain the biochemical conversion events into simple, one step, stochastic processes: cis→medial and medial→trans. To simplify the model, we choose the same rate m for both conversions.
Each patch of membrane (in a vesicle or a compartment, and whatever the surrounding patches are) has the same transition rate.  (Takamori et al., 2006). This involves at least two steps: the budding of a vesicle from a donor compartment (described in the next section), and its fusion with a receiving compartment. The fusion process itself requires close proximity between the receiving compartment and the vesicle, followed by the pairing of fusion actors resulting in membrane fusion. In the current model, the rate of encounter of two compartments is constant, and equal to f , while the actual fusion event depends on the biochemical composition of the fusing compartments. In vivo, some of the key proteins involved in the fusion process, such as the SNAREs or tethering proteins are known to closely interact with membrane markers like Rab proteins (Cai et al., 2007). This is thought to accelerate fusion events between compartments of similar biochemical identities and decrease it between compartments of different identities, a process often called homotypic fusion (Marra et al., 2007). To take this into account, fusion is modulated by the probability that both compartments exhibit the same identity at the contact site, corresponding to Eq.1 in the main text. The fusion rate is maximum ( f ) for two identical compartments, and vanishes between two compartments with no common identity. The exchange of vesicles between compartments of different identities (Glick and Luini, 2011), or the fusion of compartments of different identities (Marsh et al., 2004), are sometimes regarded as heterotypic fusion. Such processes are allowed within our homotypic fusion model, provided the different compartments share at least some patches of similar identities. Budding is the formation of a new vesicles from a large compartment. We assume that each budding vesicle is composed of a single membrane identity, which is consistent with the high specificity of the in vivo budding machinery (Bonifacino and Glick, 2004). In our model, the flux of vesicles budding from a compartment depends on the size and composition of the compartment. Compartments smaller than 2 cannot bud a vesicle. Compartments of size 2 can split into two vesicles. For larger compartments, we consider a general budding flux for vesicles of identity (for = cis, medial and trans) of the form: One expects the budding flux to follow a Michaelis-Menten kinetics; linear with the number of patches of a given identity ( ( ) = ) at small concentration, and saturating to a constant at high concentration (Vagne and Sens, 2018b). This scenario is consistent with the fact that budding proteins interact very dynamically with the membrane, attaching and detaching multiple times before budding a vesicle (Hirschberg et al., 1998). Previous theoretical works suggest that vesicular sorting of different membrane species is more efficient in the saturated regime (Dmitrieff and Sens, 2011). Since one of the main features of the Golgi is to segregate different biochemical species into different compartments, we assume a very high affinity between membrane patches and their budding partners, setting ( i ) = 1 if i > 0 and ( i = 0) = 0 (corresponding to Eq.2 of the main text). The budding flux for a given identity thus depends on the total size of the compartment rather than on the amount of that particular identity. This leads to a total budding flux b × × id for a compartment carrying id different identities. The results of simulations with this linear budding scheme are discussed in App.8 on p.41. In cells, the Golgi is placed between two intra-cellular structures that are the ER and the TGN (Presley et al., 1997; Klemm et al., 2009). Fluxes of material leaving the Golgi thus include retrograde fluxes toward the ER, carrying immature components such as cis and medial-Golgi enzymes or recycling ER enzymes, and anterograde fluxes toward the TGN of mature components, such as processed cargo that properly underwent all their post-translational maturation steps (Boncompain and Perez, 2013). The exiting fluxes are accounted for by allowing the different compartments in the system to fuse with the boundaries. All compartments, from vesicles to the largest ones, can fuse homotypically with the ER or the TGN to exit the system. Thus, these boundaries are modelled as stable compartments, containing a fraction ER of cis components for the ER and TGN of trans components for the TGN. These fraction are simply written when they are assumed to be the same. This allows immature (cis) compartments to undergo retrograde exit, and mature (trans) components to undergo anterograde exit. In the main text we focus on the case = 1. Lower values of reduce fusion with the boundaries and increase the residence time of components in the system. This increases the average size of compartments, and increases fluctuations in the system, as large compartments exiting the system induce large fluctuations in the instantaneous size and composition of the system. The impact of is studied in some details in App.7 p.39. The Golgi receives material from different compartments like the endosomal network, lyzosomes, etc... (Boncompain and Perez, 2013). As we are primarily interested in characterizing the relationship between the structure and dynamics of the Golgi, and in relating these to the rates of individual biochemical conversion and transport processes, we focus here on the secretory role of the Golgi, and only account for the incoming flux of immature components coming from the ER. We define a rate of injection of cis-vesicles in the system. As the parameters of the system are varied, the injection rate is varied as well in order to keep the total system's size to an almost constant value. This constraint is only approximately enforced using Eq.6, which is only strictly valid when all compartments are pure. The impact of the system's size on the structure of the Golgi is shown on Fig.1A of the current appendix. Increasing the total size increases the number of compartments and hence the total fusion flux between compartments. Consequently, compartments are larger in larger systems, and tend to be (slightly) less pure, as fusion increases mixing. In the main text, we restrict ourselves to a system size of = 300. One should remember that Eq. 6, used to predict the system's size , assumes that compartments are pure (perfectly sorted). Fig.1B -current appendix, shows that this assumption fails to predict for systems where the budding rate is low compared to the fusion rate (highly interacting compartments). In this regime, compartments have a great probability to fuse together, creating hybrid cis/medial/trans-structures. This makes medial-patches sensitive to the interaction with the ER and the TGN, and creates an exit flux for these patches that is not observed in a pure regime. Such systems exhibit a larger exit flux and thus a lower . Note this is only true for biochemical conversion rates lower than the fusion rate, as the system is saturated with trans-patches for high m (see Eq.4). steps decreases with increasing reaction rates. Since we are interested by steady-state quantities, we average over a fixed number of steps rather than a given physical time.
As shown in Appendix 4, the steady-state of the system is reached after a physical time of order 1∕ f . Depending on the parameters, the steady-state is reached after between 10 3 and 10 6 simulation steps. To characterise the steady-state, we disregards the transient regime and data are recorded after 10 6 steps. The full simulation typically last at least 10 7 steps. This arbitrary amount is a good compromise between computation time and the need to accumulate sufficient amount of data to obtain enough statistics on all the measured quantities. In practice, the time needed to reach steady-state can be shortened by starting the simulation from a vesicular system with the predicted amount of cis, medial and transspecies (with Eq.4). Because simulation steps have different durations, one should be careful when computing time averages. Two different approaches can be used. Either we weight each configuration by the duration between two consecutive time-steps, or we re-sample data to get a fixed time-step between observations, Δ . We checked that both procedures give the same results, and adopted the second approach for the analysis. As the steady-state of the system is reached after a physical time of order 1∕ f , we take Δ = 1∕ f . Computation of the purity 866 The purity of a compartment is defined such that its value is 0 for a perfectly mixed compartment containing the same amount of cis, medial, and trans-species, and it is equal to 1 for a pure compartment containing a single species. With the fraction of the species in the compartment, the purity is defined as: On a triangular composition space where each corner corresponds to a pure compartment, is the distance from the center of the triangle, see Fig.1 -current appendix. When we show snapshots of the system, each compartment is represented as a sphere whose area is proportional to the compartment size (defined as an equivalent number of vesicles). The composition of a compartment is represented as a color following the color code shown in Fig.1 -current appendix. The purity of the system is a global average (over time and over all compartments) of the purity of individual compartments, weighted by their sizes and ignoring vesicles (that are always pure as they are composed of one unique identity). Each compartment is represented as a sphere with an area equal to the area of a vesicle (the smallest elements in this snapshot) times the number of vesicles that fused together to create the compartment. The color of the compartment reflects its composition, according to the color code on the triangular composition phase space: 100% cis at the top, medial at the bottom right, trans at the bottom left. The purity of each of these compartments is its distance from a perfectly mixed compartment at the center of the triangle (middle arrow). This distance is normalized so that it equals 1 for a perfectly pure compartment (bottom arrow). Among other mechanisms, this theoretical organelle self-organizes by budding and fusion of components. In that sense, it is a scission-aggregation structure which should follow the laws dictating the behavior of this class of systems. One of these laws is the fact that the size distribution for small compartments should follow a power-law. Because of the scission, compartments cannot grow indefinitely and the power-law ends by an exponential cutoff (Turner et al., 2005; Vagne et al., 2015). Thus, the size distribution of compartments of size should, on a first approximation, follow this general formulation: where is the cut-off size and is an exponent that has been calculated to be = 3∕2 in a similar system (Turner et al., 2005). There are multiple ways to characterize the average size of the distribution, using ratios of moments + 1 over of the distribution: It turns out that for ≫ 1, In order to have ⟨ ⟩ ∼ , we need to choose the exponent such that > − 1 = 1∕2 in the present case. In the main text, we adopt = 1 and define the characteristic size of the distribution as: As shown in Fig.2A of the main text, the calculated average is in good agreement with simulations' data. To discriminate between the two models of Golgi's dynamics that we can find in the literature, we need to quantify whether the vesicular transport is anterograde or retrograde. Indeed, the "Vesicular transport" model assumes that cargo is transported sequentially from cis to medial to trans-compartments while resident enzymes remain in place, meaning that the vesicular flux is anterograde. On the other hand, "Cisternal maturation" models assume that cargo remain inside cisternae, and resident enzymes are recycled, thus requiring a retrograde vesicular flux. One way to measure this flux in our simulations is to follow passive cargo molecules and quantify whether they move to more or less mature compartments, as they get carried by vesicles. To do so, we record all events that affect cargo molecules. Every time a compartment buds a cargo, we store the composition of this compartment and compare it to the one in which the vesicle later fuses. Both compositions are a vector ⃗ , with three components that are the fraction ( equals cis, medial and trans) of the donor and acceptor compartments. Defining ⃗ the composition of the donor compartment, and ⃗ the composition of the acceptor, we can compute the enrichment ⃗ as: The sum of both ⃗ components equals 1 (as they are fractions of each identities), and the sum of ⃗ components equals ⃗ 0, which simply means one cannot gain in fraction of any identity without loosing the same amount of the others. We can now compute ⟨ ⃗ ⟩ to calculate the mean enrichment in cis, medial and trans-species, between a budding event and the next fusion event. However, and because of back fusion events (fusion into the same compartment that previously budded the vesicle), ⟨ ⃗ ⟩ components can be close to 0. This is particularly true for pure, sorted systems. In that case, a budded vesicle has the same identity as its donor compartment, and thus has a great probability to fuse back with the same compartment or one of very similar composition. That is why, non-treated data do not allow to discriminate well between an anterograde and a retrograde regime when the purity (and thus b ) is high (Fig.1A -App.6, p.37). As we are primarily interested in the sign of these vectors' components, we normalize ⃗ using the 1 norm ( ∑ | | = 1). The vesicular flux discussed in the previous section can be directly displayed on the triangle of composition we introduced in Fig.1 -current appendix. Instead of computing ⟨ ⃗ ⟩ for all ⃗ , we can bin data with respect to the donor compartment composition (typically triangular bins of size Δ = 0.1) and calculate the average enrichment for each bin of donor compartments.
To get rid of back fusion effects in this quantification, we remove the vectors ⃗ for which all components are smaller (in absolute values) than the binning mesh-grid. For each binned composition we can now compute the mean enrichment vector, and plot this vector on the 2D triangular composition space. To emphasize the dominant fluxes in this vector field, the opacity of vectors is set proportionally to the flux of transported cargo per unit time (normalized by the total amount of cargo-proteins in the system). This can be seen on Fig.4 main text. We want to build a simple model to explain how the purity of the system is affected by the parameters. This is challenging as all events (budding, fusion and biochemical conversion) influence the purity of a compartment. We seek to determine the purity transition, namely the values of b and m for which a pure system becomes impure. We make the following approximations, based on the assumption that the system is almost pure: (i) trans-compartments are pure as they cannot be contaminated by biochemical conversion. (ii) cis and medial-compartments share the same purity. (iii) The total amount of the different species cis , medial and trans follows the steady-state repartition of cis, medial and trans-species in a pure limit, given by Eq.4. Most of the components of the system are within compartments. Weighting these with the fraction of the different species in the system (from Eq.4) gives To compute the average contamination ⟨ ⟩, we consider the different events that affect the purity of a cis-compartment of size contaminated by a fraction of medial-patches, surrounded by a number v of cis-vesicles, and a number ′ v of medial-vesicles. These events are sketched in Fig.1 -current  The rate ( ) (normalized by the fusion rate) for each mechanism, and the extent to which it affects , are: We can now compute the temporal variation of :̇ = ∑ ( ) ( ). We concentrate on the limit of large compartments: ≫ 1. For simplicity, we also assume that ≫ ( v − ′ v ), so that the net contribution of vesicle fusion to the purity variation, which scales like ( ′ v − v )∕ , is negligible. Neglecting stochastic fluctuations, the mean-field evolution of the fraction of contaminating species is:̇ Below, we discuss separately the case where = 1 (fast fusion with the ER boundarydiscussed in the main text), and the case where = 0 (no exit through the ER -discussed in App.7, p.39). In what follows, we propose a simplified treatment of this process, focusing on the case = 0 for simplicity, and neglecting the contribution of vesicle fusion as discussed above. For a compartment of size contaminated by medial patches, we identify 2 types of mechanisms (depending on their amplitude). First, a smooth drift in the contamination due to biochemical conversion (with a rate m * ( − medial )) and budding (with a rate b ). Second, a jump in the contamination due to fusion with the neighbor compartment (rate medial ( − medial )∕ 2 ). We compute below the mean contamination, assuming that the cis-compartment spends a time conta undergoing small fluctuations ≃ 0 close to the pure state, before fusing with a medial-compartment of composition 1 − . The resulting compartment with ≃ 1∕2 then spends a time deconta undergoing a decontamination process, which depends on the balance between budding and biochemical conversion.
We first calculate the average value of the contamination around = 0 due to biochemical conversion and budding, disregarding inter-compartment fusion. The probability ℙ( medial ) of finding medial in the cisterna satisfies: In the limit ≫ medial , this becomes: The steady-state solution is and the average contamination due to budding and biochemical conversion is: These fluctuations in composition allow fusion between compartments, at a rate (1 − ) ≃ (for ≪ 1). The average waiting time for an inter-compartment fusion event is thus After inter-compartment fusion, the compartment is at = 1∕2, which is an unstable fixed point according to Eq.17. Any small fluctuation in composition leads to a smooth decontamination that satisfieṡ = ( m − b )(1 − 2 ), disregarding inter-compartment fusion. The decontamination time deconta and the average contamination during the process ⟨ ⟩ deconta , can be estimated by integratinġ , from = 1∕2 − 1∕ to = 0: , ⟨ ⟩ deconta = 2 + log( ∕2) − 1 2 log( ∕2) We can now calculate ⟨ ⟩ as a temporal average of : When log( ∕2) is close to 1 (typically true for the range of that is interesting here, ∼ 100), ⟨ ⟩ can be simplified to: Injecting this result into Eq.16 gives the approximate purity boundary for ER = 0: Despite numerous simplifications, the model gives good predictions on the purity, both for ER = TGN = 1 (Fig.2 -main text) and for ER = 0, TGN = 1 (Fig.1 -App.7, p.39). Depending on the normalized biochemical conversion rate m , we can discriminate three regions in the purity phase diagram:  Fig.1 -current appendix. The organelle forms de-novo, and reaches the steady-state after a time of order a few 1∕ f . The steady-state is reached under any parameter values, but the fluctuations around the steady-state depend on the parameter, as discussed in Appendix 5. The analytic calculation presented in Eqs.3 (red line in Fig.1 above) is very accurate for high budding rate, but gives a 10% error for low budding rate. Transit kinetics of passive cargo. 1117 The cargo used as marker of vesicular transport directionality can be used to quantify the transit kinetics of a passive cargo across the self-organised organelle. Fig.2 -current appendix shows the amount of cargo present in the system as a function of time during a typical "pulse-chase" experiment, in which a given number of cargo molecules is allowed to exit the E.R. at = 0, by joining the vesicles constituting the influx . The amount of cargo in the system initially increases, then decreases in a close-to-exponential fashion. The export kinetics resemble the one observed by iFRAP (Fig.2 i of Patterson et al. (2008)) or using the RUSH technics (Fig.3 e of Boncompain et al. (2012)). The fluctuations of the system around its steady-state are characterized by computing the temporal standard deviation of the total system's size and purity, shown in Fig.1A-B -current appendix. Fluctuations decrease as the budding rate b increases, owing to the fact that compartments are on average smaller for larger budding rate, so that the removal of a compartment by fusion with the boundaries has a smaller impact on the state of the system. We note in Fig.1A -current appendix, that the total size of the system depends on the budding rate b , despite the fact that the influx is varied according to Eq.6 to limit variations of the system's size. This is due to the fact that Eq.6 is valid only in the limit of pure compartments (large b ). For smaller values of the budding rate, medial-compartments contaminated by cis and trans-species and may thus exit the system by fusing with the boundaries, which decreases the total size for low b . For the regime we are interested in, namely m ∼ ∼ 1, this phenomenon is practically negligible, but its impact on the average size is more severe for lower m (App.1, p.23). We also characterize the variation of the size of compartments and their abundance as a function of their composition. In the main text (Fig.4), variations of the total size (number of membrane patches) and the number of compartments is shown as a function of their composition, in the composition triangular space defined in App.2, p.25. This is obtained by binning the composition space in such a way that the characterization is both relatively precise and statistically significant (binning with a mesh of 1∕30). The size and number of compartments are averaged over the simulation time (once the steady-state is reached) for each composition bin. These results are reproduced in Fig.1C -current appendix, where the average size of compartment, computed as the ratio of average size over average number of compartments, is also shown. For low values of the budding rate b , both the size and the number of compartments follow the theoretical line obtained from the "cisternal maturation" limit (Eq.9). Compartments also grow linearly in time, in agreement with the linear prediction of Eq.7. Their sizes are shown in Fig.1D - cis-vesicles are continuously being injected in the system. On the other hand, medial and trans-vesicles fuse with cis/medial and cis/trans-compartments, as pure medial-compartments are absent, and pure trans-compartments are few and transitory. In this limit, the net enrichment is completely dominated by the retrograde transport of cis-vesicles, and is positive in cis-identity and negative in trans-identity. For intermediate budding rate ( b ∼ 0.1 → 1), and as b increases, components can be efficiently sorted as they mature, since the budding and biochemical conversion rates are of the same order. Medial compartments are rather unstable, as they can fuse with both cis-rich and trans-rich compartments. Their reformation involves medial vesicles leaving cis-rich and trans-rich compartments to fuse with each other or small medial-rich compartments. Fig.1C -current appendix, shows that this is the main contribution to the overall vesicular flux, thereby defining the centripetal net vesicular flux shown in Fig.4 -main text. Note that they primarily exit trans-compartments for b ∼ 0.1, and cis-compartments for b ∼ 1, denoting a transition from retrograde to anterograde vesicular fluxes. For high budding rate ( b ≫ 1), compartments are very pure, and most budding events lead to vesicle back fusion from the donor compartment (Fig.1A -current appendix). A new phenomenon can be observed, which is the anterograde transport of cis and medial-vesicles fusing with medial and trans-compartments, respectively. This is due to vesicle biochemical conversion after budding, as discussed below.  1254 The biochemical conversion of vesicles after their budding from an immature compartment has been presented as a mechanism to prevent back fusion with the donor compartment and to promote anterograde vesicular transport (see Discussion -main text). This mechanism is naturally included in our model, as vesicles have the same biochemical conversion rate than any membrane patch that belong to bigger compartments. One can notice on Fig.1B -current appendix, that there is a major increase of the anterograde cis-to-medial and medial-to-trans vesicular fluxes between b = 10 and b = 10 2 . This is due to the biochemical conversion of the vesicles after their budding. The budding vesicle has the same identity than the donor compartment, but undergoes biochemical conversion before undergoing back fusion and fuses with a more mature compartment.  Fig.1C -current appendix, shows the net vesicular flux (for the same simulation than in the main text) ignoring the events where vesicles undergo biochemical conversion after their budding. The net vesicular flux vanishes in the vector field for high budding rates when these events are removed. We interpret this result considering dimers and trimers are dominant when b is large. Disregarding vesicle biochemical conversion, such compartments emit as many immature vesicles fusing with immature compartments as mature vesicles fusing with mature compartments, yielding a vanishing net vesicular flux. However, this is not true for the net enrichment which still displays an anterograde signature of the vesicular transport. Indeed, the explanation of the vesicular transport considering the minority identity (see Discussion section in the main text) is still relevant in this regime; compartments larger than 3 vesicles can generate an anterograde transport by budding patches of membrane that just underwent biochemical conversion. Consequently, the net enrichment displays the characteristics of an anterograde vesicular flux even in the absence of vesicle biochemical conversion. system. This increases their size, as they have more time to aggregate, and increases the fluctuations in the system, as larger compartments have a stronger impact each time they exit or fuse together.

Role of vesicles biochemical conversion
The impact of on the fraction of cis, medial and trans-species in the system is rather straightforward if compartments are pure (Eq.4). In this case, the exit flux can be exactly computed, and Eq.4 shows that the different species are in equal amount at steady-state when m = . If m ≫ , the membrane patches have time to undergo biochemical conversion before exiting the system, which is then dominated by trans-species. The purity of the compartment is high, but this is a rather uninteresting limit as the system is dominated by a single species. If m ≪ , compartments are recycled out of the system before transmembrane patches appear, and the system is dominated by cis and medial-species, which are of equal amount at steady state if the system is perfectly sorted. The same conclusion can be reached in the slow budding regime when compartments are not pure. Let's consider a cis-rich compartment of size contaminated by a fraction medial of medial-species, and let's disregard, for simplicity, budding and fusion with other compartments. The number of medial-patches increases by biochemical conversion with a flux m (1− medial ), and decreases by compartment fusion with the boundary, which removes all medial-patches with an average flux (1 − medial ) × medial . At steady-state, one thus expects medial ≃ m ∕ , suggesting that the lowest purity will be observed for m ≃ . The results of simulations, shown in Fig.1A-B -current appendix, confirm this prediction. The transition of purity occurs when sorting mechanisms become more efficient than mixing mechanisms. As a first approximation mixing occurs by biochemical conversion and sorting by budding, yielding the prediction that the purity transition occurs when b ≃ m . Additional mixing mechanisms include the fusion of two slightly impure compartments of different compositions. Such slow events become relevant if compartments remain in the system for a long time, i.e. if fusion with the boundaries is slow (small values of ). In this case, the transition of purity occurs for b > m . Phase diagram of the purity of the system for a system where = 10 −2 can be found in Fig.1B -current appendix. As expected the diagram is centered around m = , with a transition of purity between b ∼ 10 −2 and b ∼ 10 −1 (and thus b > m ). As it tunes the residing time of compartments in the system, also impact the average size of compartments. Decreasing the value of increases the residence time and leads to larger compartments, as predicted by Eq.4. Note however that for small budding rates, when compartments are mixed and their exit through the boundary is difficult to estimate, the size distribution deviates from the single-component ideal distribution. The Golgi being a highly polarized organelle, it could be argued that the parameters controlling the fusion with the cis face (ER) and the trans face (TGN), ER and TGN could be different. An obvious way to reduce the retrograde exit of material from the ER is to reduce ER . To investigate the role of this asymmetry, we use an extreme regime of ER = 0 and TGN = 1. This prevents the recycling of impure compartments by fusion with the ER and broadens the low purity region of the parameter space, as shown in Fig.1C - proportional to the size of the compartment for linear budding, and is thus smaller than for saturated budding, where it is proportional to the size times the number of species. Saturated budding thus promotes systems that are at the same time well sorted, and with large compartments.
Even though the purity transition is shifted toward higher values of b the link between purity and vesicular flux is qualitatively conserved (Fig.1C -current appendix). For low purity, the system is retrograde with an enrichment in cis-species and a depletion of more mature species, and for high purity it is anterograde. However the retrograde flux is less marked than with a saturated budding rate (no clear depletion in trans-species). Indeed, efficient sorting relies on the capacity to export the minority component out of a compartment. Within the saturated budding scheme, the rate of export is proportional to the size of the compartment and does not depend on the number of minority components to export. Within the linear budding scheme however, it is proportional to the number of minority components. In the low b regime, trans-rich compartments are large. They emit a large vesicular flux of immature components within the saturated budding scheme, but this flux is much smaller within the linear budding scheme. This explains the qualitative difference between the enrichment curves in the low b regime for the saturated budding scheme (large depletion in trans identity - Fig.3 -main text) and the linear budding scheme (almost no change in trans identity - Fig.1C -current appendix). Non-fusing compartments 1385 Fusion between compartments is an important ingredient of the model discussed in the main text. It could be argued that the fusion of large cisternae with one another could be a much slower process than fusion involving much smaller transport vesicles. We have performed simulations where fusion between compartments is prohibited if both compartments are larger than a vesicle. Note that in the model, such compartments are still able to fuse with the boundaries, so that the composition of the system can still be predicted by Eq. 4 The results of this model are shown in Fig.1D-F -current appendix. The size distribution of non-fusing compartment shows a broad peak, instead of the power-law seen in the case where compartments can fuse. As compartments can only grow by vesicle fusion and not by compartments fusion, their size is smaller than in the previous case. The compartments size is results from a balance between the rate at which compartments are recycled (fuse with one boundary) and the rate at which they grow by vesicle fusion. Consequently, the size distribution only weakly depends on b , as compartments have very little time to fuse with vesicles before exiting the system (Fig.1-D  As expected, and because we remove the possible interactions between compartments, the centripetal flux we observe between the previously described anterograde and retrograde is abolished. This is particularly visible if we plot the (normalized) mean enrichment in cis, medial and trans-species seen by cargo as they are transported via vesicular fluxes ( Fig.1-F -current appendix). As before, the system goes from a purely retrograde flux, characterized by a depletion in trans-species and an enrichment in cis-species, to an anterograde flux, characterized by an enrichment in trans-species and a depletion in cis-species. However, the transitional centripetal regime we previously described, characterized by an enrichment in medial-species, vanishes. This suggests this particular regime is indeed resulting from the possible fusion between compartments, as it destabilizes medial-compartments that have a great probability to fuse with slightly impure cis or trans-compartments. Thus the transport is anterograde as soon as the purity transition occurs (Fig.1E -current appendix).