A minimal self-organisation model of the Golgi apparatus

Abstract

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.

Introduction

The Golgi apparatus is an intracellular organelle at the crossroad of the secretory, lysosomal and endocytic pathways (Heald and Cohen-Fix, 2014). One of its most documented functions is the sorting and processing of many proteins synthesized by eukaryotic cells (Lippincott-Schwartz et al., 2000). Proteins translated in the endoplasmic reticulum (ER) are addressed to the Golgi, where they undergo post-translational maturation and sorting, before being exported to their final destination. The Golgi itself is composed of distinct sub-compartments, called cisternae, of different biochemical identities (Stanley, 2011). From the ER, cargo-proteins successively reside in cisternae of the cis, medial and trans-identities, after which they exit the Golgi via the trans-Golgi network (TGN). In some organisms such as the yeast Saccharomyces cerevisiae cisternae are dispersed throughout the cell (Suda and Nakano, 2012) and each cisterna undergoes maturation from a cis to a trans-identity (Losev et al., 2006; Matsuura-Tokita et al., 2006). In most other eukaryotes, cisternae are stacked in a polarized fashion, with cargo-proteins entering via the cis-face and exiting via the trans-face (Boncompain and Perez, 2013). This polarity is robustly conserved over time, despite cisternae constantly exchanging vesicles with each other, the ER and the TGN (Lippincott-Schwartz et al., 2000). Pharmacological treatments that alter the structure and stacking of Golgi compartments in mammalian cells affect the processing of certain proteins, and in particular glycosylation (Hu et al., 2005; Shen et al., 2007; Xiang et al., 2013; Bekier et al., 2017).

The highly dynamical nature and compact structure of a stacked Golgi makes it difficult to determine how cargo-proteins are transported in an anterograde fashion from the cis to trans-side of the stack and how this transport is coupled to processing. Several models have been proposed to explain the transport dynamics of Golgi cargo (See sketch in Figure 1A). They mostly belong to two classes: one involving stable compartments exchanging components through fusion-based mechanisms and one involving transient compartments undergoing en-block maturation and in which fusion mechanisms are dispensable for cargo transport. Historically (Malhotra and Mayor, 2006), the 'Vesicular transport’ model postulated that cisternae have fixed identities and cargo progresses from one cisterna to the next via anterograde vesicular transport, while the 'Cisternal Maturation’ model postulated that cisternae themselves undergo maturation from the cis to the trans-identity and physically move through the stack, while Golgi resident proteins remain in the stack by moving toward the cis-face by retrograde vesicular transport. Several extensions of these two models have been proposed, including cisternal maturation with tubular connections (Trucco et al., 2004; Rizzo et al., 2013), rapid partitioning within Golgi cisternae between processing and exporting sub-compartments (Patterson et al., 2008), or the cisternal progenitor model (Pfeffer, 2010), in which stable cisternae exchange material by the fusion and fission of sub-compartments defined by their composition of Rab GTPases, which evolve over time through exchange with the cytosol. The strengths and weaknesses of these different models are nicely reviewed in Glick and Luini, 2011. Variations of these models, such as the ‘rim progression’ model also exist (Lavieu et al., 2013). It is noteworthy that these models do not provide a quantitative analysis of the generation and maintenance of Golgi compartments, nor do they attempt to relate the Golgi structure (number and size of compartments) and transport dynamics. Therefore, much remains unknown regarding the mechanisms that dictate the organisation and dynamics of the Golgi.

Although there is a large diversity in Golgi structures and dynamics among different species, the physiological function of the organelle as a sorting and processing hub is common to all species, suggesting that important mechanisms controlling Golgi dynamics are conserved. Works in evolutionary biology and biophysics have attempted to describe these mechanisms (Klute et al., 2011; Sens and Rao, 2013). Different classes of mathematical models have been proposed, from models of vesicle budding and fusion based on rate equations (Binder et al., 2009; Ispolatov and Müsch, 2013; Mukherji and O'Shea, 2014) - some specifically including space (Sachdeva et al., 2016), to continuous, discrete and stochastic models of protein sorting in the Golgi (Gong et al., 2010; Vagne and Sens, 2018b) and computer simulations based on membrane mechanics (Tachikawa and Mochizuki, 2017). To account for the ability of the Golgi to reassemble after mitosis (Wei and Seemann, 2017), many of these studies have sought to describe the Golgi as a self-organized system (Binder et al., 2009; Kühnle et al., 2010). However, the manner in which the kinetics of the Golgi two main functions (namely the transport and biochemical conversion of its components) interplay with one another to yield a robust steady-state has so far received much less attention (Sachdeva et al., 2016; Vagne and Sens, 2018b).

Most existing models discuss transport through a pre-existing compartmentalised Golgi structure. The main goal of our model is to explain how this compartmentalised structure and the flux within it can spontaneously emerge through self-organisation. We propose here a model in which both Golgi self-organisation and vesicular transport are solely directed by (local) molecular interactions resulting in composition-dependent budding and fusion. In particular, we neglect the spatial structure of the Golgi, and assume that any two components of the system can interact with one another (see the Discussion section for the relevance of this approximation). Thus, the directionality of vesicular transport is an emergent property of the self-organisation process that co-evolves with the size and number of Golgi sub-compartments, rather than being fixed by arbitrary rules. This approach brings new conceptual insights that highlight the link between the large-scale steady-state organisation of the Golgi, the directionality of vesicular transport, and the kinetics of individual processes at the molecular scale. We show that a wide variety of organelle phenotypes can be observed upon varying the rates of the local processes. In particular, our model uncovers a correlation between the size and composition (purity) of Golgi sub-compartments. Besides, it emphasizes that composition-dependent exchange between compartments yields complex vesicular flux patterns that are neither purely anterograde (as in the ‘vesicular transport’ model) nor purely retrograde (as in the ‘cisternal maturation’ model).

Model

The model, shown in Figure 1 and fully described in the Appendix 1 Detailed model is a coarse-grained representation of the Golgi Apparatus. The parameters used in the model are summarised in Appendix 1—table 1, and the quantities used to charaterise a compartment and the steady-state of the system are summarised in Appendix 1—table 2 and Appendix 1—table 3, respectively. The system is discretised at the scale of small vesicles, which define the unit size of the system. Vesicles can fuse together or with existing compartments to form larger compartments and can bud from compartments. Compartments consist of a number of membrane patches (of size unity) with distinct identities. The biochemical identity of a membrane patch is defined in a very broad sense by its composition in lipids and proteins that influence its interaction with other patches, such as Rab GTPases (Grosshans et al., 2006) or fusion proteins such as SNAREs (Cai et al., 2007). Consistently with the three main biochemical identities reported for Golgi membranes (Day et al., 2013), vesicles and membrane patches are of either cis, medial or trans-identity, and can undergo irreversible biochemical conversion from the cis to the trans-identity. The system is fed by a constant influx $J$ of cis vesicles coming from the ER, and vesicles and compartments can fuse homotypically with the ER (itself of cis identity) and the TGN (of trans identity).

Figure 1.

Model of golgi structure and transport.

(A) Top: classical representation of the structure of the Golgi, showing some of the important protein actors. The three main membrane identities are shown in different colors (cis: blue, medial: orange, trans: green): ER = Endoplasmic Reticulum, ERGIC = ER Golgi intermediate Compartment, TGN = Trans Golgi-Network, PM = Plasma Membrane. Bottom: Sketches of the four main models of Golgi transport (see text). (B) Our quantitative model of Golgi self-organisation. The left boundary is the ER, composed of a cis-membrane identity, and the right boundary is the TGN, composed of a trans-membrane identity. Golgi compartments self-organise via three stochastic mechanisms: Fusion: (1) All compartments can aggregate using homotypic fusion mechanisms: the fusion rate is higher between compartment of similar identities. (2) Each compartment can exit the system by fusing homotypically with the boundaries. Budding: (3) Each compartment larger than a vesicle can create a vesicle by losing a patch of membrane. Biochemical conversion: (4) Each patch of membrane undergoes a conversion from a cis to a trans-identity. New cis-vesicles (0) bud from the ER at a constant rate. In the sketch, the boundaries also contain neutral (gray) membrane species that dilute their identity (impact of this dilution in Appendix 7).

Compartments are defined by their size $n$ (a number of patches) and their composition, the fractions ${\varphi }_i$ (with $i=cis,medial,trans$) of patches of the three different identities composing it. These quantities are dynamically controlled by three, composition dependent, microscopic mechanisms – budding, fusion and biochemical conversion – as described below.

Fusion: Homotypic fusion – the higher probability of fusion between compartments of similar identities – is a feature of cellular organelles in general (Antonin et al., 2000) and the Golgi apparatus in particular (Pfeffer, 2010; Bhave et al., 2014). This is controlled here by a fusion rate between compartments proportional to the probability that they present the same identity at the contact site. The total fusion rate for two compartments $(a)$ and $(b)$ with composition ${\varphi }_i^{(a)}$ and ${\varphi }_i^{(b)}$ for each identity $i$ (with $i$ equals cis, medial and trans) is then:

(1)

$K_{\mathrm{f}}\times \sum _i{\varphi }_i^{(a)}{\varphi }_i^{(b)}$where $K_{\mathrm{f}}$ is the fusion rate between two identical compartments. Fusion with the boundaries follow the same rules, with the ER containing a fraction ${\alpha }_{\mathrm{ER}}$ of cis patches and the TGN a fraction ${\alpha }_{\mathrm{TGN}}$ of trans patches. Numerical results described in the main paper are for ${\alpha }_{\mathrm{ER}}={\alpha }_{\mathrm{TGN}}=1$, and the role of the composition of the boundaries is discussed in the Appendix 7.

Budding: Budding is the emission of a vesicle from a larger compartment. Composition-dependent vesicular budding is an important sorting mechanism in cellular traffic (Bonifacino and Glick, 2004). Owing to the high specificity of the budding machinery, each vesicle is assumed to be composed of a single membrane identity. The budding flux for any components $i$ (equals cis, medial or trans) present in a compartment is assumed not to depend on the number $n_i=n{\varphi }_i$ of patches of identity $i$, but on the total size of the compartment:

(2)

$J_{\mathrm{b},i}=K_{\mathrm{b}}\times n.$

This budding kinetics corresponds to the saturated regime of a Michaelis-Menten reaction kinetics, and is appropriate to the case where budding actors (e.g. coat proteins) bind non-specifically to the membrane of the compartment and find their budding partners (a patch of a particular identity) by diffusion (Vagne and Sens, 2018b). An alternative budding scheme where the budding flux is linear with the number of patches of a given identity ($J_{\mathrm{b},i}=K_{\mathrm{b}}{n}_i$) is discussed in Appendix 8.

Biochemical conversion: Each membrane patch is assumed to undergo stochastic biochemical conversion from a cis to a medial to a trans-identity. This local mechanism of identity conversion is motivated by the Rab cascade, during which the membrane identity evolves over time through molecular exchange with the cytosol, as was observed in endosomes (Rink et al., 2005) and in the Golgi of Yeast (Losev et al., 2006; Matsuura-Tokita et al., 2006). The Rab conversion mechanism has not been demonstrated in mammalian Golgi, but is thought to be universally important for the specificity of membrane trafficking (Grosshans et al., 2006), and many of the proteins involved in the Yeast Rab cascade are conserved in mammalian cells (Klute et al., 2011). Rab conversion is at the basis of the cisternal progenitor model (Pfeffer, 2010), and it is thus of fundamental importance to understand its potential implication on Golgi structure and dynamics with the use of a quantitative model. We stress that our model of biochemical conversion is not limited to Rab GTPases, but can also correspond to the modification of lipid components by enzymes in the Golgi (McDermott and Mousley, 2016). This is a local mechanism of identity conversion distinct from the maturation of entire compartments, which is also affected by the dynamics of budding and fusion. To limit the number of parameters, the biochemical conversion rate $K_{\mathrm{m}}$ is the same for the two reactions (cis→medial and medial→trans) and is independent of the concentration. Introducing cooperativity in the conversion process is expected to increase the robustness of the self-organisation process (Vagne and Sens, 2018a).

In its simplest form, our model contains only four parameters: the rates of injection, fusion, budding, and biochemical conversion ($J,K_{\mathrm{f}},K_{\mathrm{b}},K_{\mathrm{m}}$). By normalizing time with the fusion rate, we are left with three parameters: $j=J/K_{\mathrm{f}}$, $k_{\mathrm{b}}=K_{\mathrm{b}}/K_{\mathrm{f}}$ and $k_{\mathrm{m}}=K_{\mathrm{m}}/K_{\mathrm{f}}$. The dynamics of the system is entirely governed by these stochastic transition rates, and can be simulated exactly using a Gillespie algorithm (Gillespie, 1977), described in Appendix 1. 

Codes used for the simulation can be found in Source code 1.

Results

Mean-field description of the system

The complexity of the system prohibits rigorous analytical calculation. Nevertheless, analytical results can be obtained for a number of interesting quantities in certain asymptotic regimes where simplifying assumptions can be made. We present below some of these derivations.

De-novo formation and steady-state composition in the well-sorted limit

The composition of the system at steady-state is difficult to compute due to the fact that the exit of components from the system, through fusion with the boundaries, depends on the composition of the exiting compartments, which cannot be derived analytically in the general case. This calculation becomes straightforward if the system is well sorted and all compartments are pure. As explained below, this corresponds to the limit of high budding rate: $k_{\mathrm{b}}\gg k_{\mathrm{m}}$. In the well-sorted limit, only cis components exit through the cis boundary and only trans components exit through the trans boundary, and mean-field equations can be derived for the total amount [Formula/expression omitted. See PDF], [Formula/expression omitted. See PDF] and [Formula/expression omitted. See PDF] of each species:

(3)

${\displaystyle \frac{\mathrm{\partial }{N}_{\mathit{\text{cis}}}}{\mathrm{\partial }t}=j-N_{\mathit{\text{cis}}}({\alpha }_{\text{ER}}+k_{\mathrm{m}}),\frac{\mathrm{\partial }{N}_{\mathit{\text{medial}}}}{\mathrm{\partial }t}=k_{\mathrm{m}}(N_{\mathit{\text{cis}}}-N_{\mathit{\text{medial}}}),\frac{\mathrm{\partial }{N}_{\mathit{\text{trans}}}}{\mathrm{\partial }t}=k_{\mathrm{m}}{N}_{\mathit{\text{medial}}}-{\alpha }_{\text{TGN}}{N}_{\mathit{\text{trans}}}}$

With $j$ the injection rate of new cis-vesicles in the system and $k_{\mathrm{m}}$ the biochemical conversion rates, both normalized by the fusion rate, and ${\alpha }_{\text{ER}}$ (${\alpha }_{\text{TGN}}$) is the fraction of cis (trans) species promoting homotypic fusions with Golgi components in the ER (TGN). At steady-state, the total amounts of cis, medial and trans-species are fixed by the balance between influx, exchange, biochemical conversion and exit:

(4)

${\displaystyle N_{\mathit{\text{cis}}}=N_{\mathit{\text{medial}}}=\frac{j}{{\alpha }_{\text{ER}}+k_{\mathrm{m}}},N_{\mathit{\text{trans}}}=\frac{k_{\mathrm{m}}}{{\alpha }_{\text{TGN}}}\frac{j}{{\alpha }_{\text{ER}}+k_{\mathrm{m}}}}$

To estimate the typical size of compartments, we assume that for each species, a single large compartment of size $n$ interacts through budding and fusion with a number $n_{\mathrm{v}}$ of vesicles of the same identity (so that $N=n+n_{\mathrm{v}}$). The compartments size for each species then satisfies:

(5)

${\displaystyle \frac{\mathrm{\partial }n}{\mathrm{\partial }t}=n_{\mathrm{v}}-k_{\mathrm{b}}n\underset{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}}{\overset{}{\to }}n=\frac{N}{1+k_{\mathrm{b}}}}$

To limit the number of parameters, we fix the average system’s size [Formula/expression omitted. See PDF] to a set value $N=300$ in the main text, which is suitable for Golgi ministacks whose total area is of the order of 1 μm² (the area of a mammalian Golgi ribbon is much larger) (Yelinek et al., 2009), corresponding to about a few hundreds of vesicles of diameter ~ 10-50 nm. This is obtained by adjusting the influx to the variation of the other parameters according to:

(6)

${\displaystyle {\displaystyle j=\frac{N({\alpha }_{\text{ER}}+k_{\mathrm{m}}){\alpha }_{\text{TGN}}}{2{\alpha }_{\text{TGN}}+k_{\mathrm{m}}}}}$

We show in Appendix 1—figure 1 that the overall structure of the phase diagrams for the compartments size and composition is robust upon variation of the average system’s size.

The de-novo formation of the system can be approached with Equation 3. Starting with an empty system with an influx $J(=K_{f}j)$ of cis vesicles, the system grows stochastically to reach the steady-state size after a time of order $1/K_{\mathrm{f}}$. The evolution of the size of the system with time obtained from numerical simulations, shown in Appendix 4—figure 1 for different sets of parameters, agrees with Equation 3. At steady-state, the influx of vesicles is balanced by an outflux of material. The exit kinetics can be quantified by looking at the fate of a pulse of cargo released from the ER at a given time. Appendix 4—figure 2 shows that the exit kinetics is exponential, in agreement with results of FRAP (Patterson et al., 2008) or pulse-chase (Boncompain et al., 2012) experiments.

Although Equation 6 is only valid in the well-sorted limit, it permits a satisfactory control of the average system size over the entire range of parameters, with only a 10% variation for small budding rates (Appendix 5, Appendix 5—figure 1). At steady-state, the system size exhibits stochastic fluctuations around the mean value that depend on the parameters (Appendix 5, Appendix 5—figure 1). The large fluctuation (about 30%) for low budding rate $k_{\mathrm{b}}$ stem from the fact that compartments are large and transient, as explained below.

Compartment size and composition in the maturation-dominated regime

For low values of the budding rate $k_{\mathrm{b}}$, compartments do not shed vesicles and evolve in time independently of one another in a way dictated by a balance between fusion-mediated growth and biochemical maturation. Their size and composition can be approximatively calculated by assuming that compartments grow in size by fusion from a constant pool of vesicles containing the same number $n_{\mathrm{v}}$ of vesicle for all three identities. Calling [Formula/expression omitted. See PDF], [Formula/expression omitted. See PDF] and [Formula/expression omitted. See PDF] the amount of components of the three identities in a given compartment of total size [Formula/expression omitted. See PDF], and neglecting vesicle budding, the mean-field equations satisfied by these quantities are:

[Formula omitted. See PDF]

Starting with a vesicle of cis identity for ${\displaystyle t=0:n_{\mathit{\text{cis}}}(0)=1}$, [Formula/expression omitted. See PDF], the compartment size evolves linearly with time: $n(t)=1+n_{\mathrm{v}}t$, and the composition of each species satisfies:

(8)

${\displaystyle \begin{array}{ll}{\displaystyle n_{\mathit{\text{cis}}}(t)}& ={\displaystyle (1+n_{\mathrm{v}}t)e^{-k_{\mathrm{m}}t}}\\ {\displaystyle n_{\mathit{\text{medial}}}(t)}& ={\displaystyle (1+n_{\mathrm{v}}t)k_{\mathrm{m}}t{e}^{-k_{\mathrm{m}}t}}\\ {\displaystyle n_{\mathit{\text{trans}}}(t)}& ={\displaystyle (1+n_{\mathrm{v}}t)\left(1-(1+k_{\mathrm{m}}t)e^{-k_{\mathrm{m}}t}\right)}\end{array}}$

The fraction of each species in the compartment is thus independent on $n_{\mathrm{v}}$, and reads:

(9)

${\displaystyle {\varphi }_{\mathit{\text{cis}}}(t)=e^{-k_{\mathrm{m}}t},{\varphi }_{\mathit{\text{medial}}}(t)=k_{\mathrm{m}}t{e}^{-k_{\mathrm{m}}t},{\varphi }_{\mathit{\text{trans}}}(t)=\left(1-(1+k_{\mathrm{m}}t)e^{-k_{\mathrm{m}}t}\right).}$

These results show that in this regime, a given compartment smoothly evolves from a mostly cis to a mostly trans identity over a typical time $1/K_{\mathrm{m}}$. The maximum amount of medial identity is obtain for $k_{\mathrm{m}}t=1$ and is ${\varphi }_{\mathrm{medial}}=1/e\simeq 0.37$. Therefore, the mechanism does not lead to pure medial compartments. These analytical results are confirmed by the full numerical solution of the system in the low budding rate regime shown in Appendix 5 and discussed in detail below.

Steady-state organisation

The steady-state organisation and dynamics of the system is described in terms of the average size and purity of compartments (introduced here and detailed in the Appendix 2). The stationary size distribution of compartments is a decreasing function of the compartment size, which typically shows a power law decay at small size, with an exponential cut-off at large size (Figure 2A). This is expected for a non-equilibrium steady-state controlled by scission and aggregation (Turner et al., 2005) and means that there exist many small compartments and very few large ones. The typical compartment’s size is defined as the ratio of the second over the first moments of the size distribution. This corresponds to a size close to half the size of the exponential cut-off (see Appendix 2) beyond which it is unlikely to find a compartment (Vagne et al., 2015).

Figure 2.

Steady-state of the self-organized model of Golgi apparatus.

(A) Size distribution of compartments for a biochemical conversion rate $k_{\mathrm{m}}=1$ and different values of the budding rate $k_{\mathrm{b}}$. The red bar shows the characteristic compartment size. (B) Phase diagram of the size of the compartments as the function of $k_{\mathrm{b}}$ and $k_{\mathrm{m}}$. Black lines: theoretical prediction (Equations 4,6) (C) Phase diagram of the purity of the system as a function of $k_{\mathrm{b}}$ and $k_{\mathrm{m}}$. Dashed line is $k_{\mathrm{m}}=k_{\mathrm{b}}$, black lines are a theoretical prediction (see Appendix 3). (D) System snapshots for $k_{\mathrm{m}}=1$ showing the mixed regime (square - $k_{\mathrm{b}}={10}^{-2}$), the sorted regime (triangle - $k_{\mathrm{b}}=1$), and the vesicular regime (circle - $k_{\mathrm{b}}={10}^2$) - see text. (E) Average compartment purity as a function of their average size, obtained by varying the budding rate $k_{\mathrm{b}}$ for $k_{\mathrm{m}}=1$ (black dots: simulation results – gray line: guide for the eyes). For all panels, ${\alpha }_{\mathrm{ER}}={\alpha }_{\mathrm{TGN}}=1$. See also Appendix 5 for further characterizations of the steady-state organisation, Appendix 7 for the role of composition of the exit compartment, and Appendix 8 for results with alternative budding and fusion kinetics.

The typical compartment size varies with parameters as shown in Figure 2B. Increasing the ratio of budding to fusion rate $k_{\mathrm{b}}$ decreases the compartment size (Turner et al., 2005). The size depends on the biochemical conversion rate $k_{\mathrm{m}}$ in a non-monotonic manner: for a given budding rate, the size is smallest for intermediate values of the biochemical conversion rate ($k_{\mathrm{m}}\simeq 1$). The conversion rate controls the composition of the compartment and hence their fusion probability. This dependency is well explained by the simple analytical model discussed above (Equations 3,4) that approximates the system by three pure compartments interacting with a pool of vesicles (solid lines in Figure 2B).

The purity of a given compartment is defined as the (normalized) Euclidean distance of the composition of the compartment (the fractions ${\varphi }_i$ in the different i-identities, $i=$ cis, medial and trans) from a perfectly mixed composition. ${\displaystyle P=0,1/2,1}$ correspond respectively to compartments that contain the same amount of the three identities, the same amount of two identities, and a single identity (Appendix 2). The purity of the system is the average purity of each compartment weighted by its size, ignoring vesicles. The dependency of the average purity with the parameters is shown in Figure 2C. As for the size of the compartments, it is non-monotonic in the biochemical conversion rate, and regions of lowest purity are found for intermediate biochemical conversion rates $k_{\mathrm{m}}\simeq 1$, when all three identities are in equal amount. The purity increases with the budding rate $k_{\mathrm{b}}$ in a sigmoidal fashion (see also Figure 3A). This can be understood by analyzing the processes involved in mixing and sorting of different identities. In a first approximation, mixing occurs by biochemical conversion and sorting occurs by budding of contaminating species, suggesting a transitions from low to high purity when the budding rate reaches the biochemical conversion rate ($k_{\mathrm{b}}\simeq k_{\mathrm{m}}$). A ‘purity transition’ is indeed observed in an intermediate range of biochemical conversion rates (${\displaystyle 0.1<k_{\mathrm{m}}<10}$, see Figure 2C). It is most pronounced for $k_{\mathrm{m}}\simeq 1$. Beyond this range, the system is dominated by one identity and compartments are always pure. The purity variation can be qualitatively reproduced by an analytical model (Appendix 3) that accounts for the competition between biochemical conversion and budding, but also includes fusion between compartments and with the exits (solid lines in Figure 2C).

Figure 3.

Directionality of vesicular transport, displayed varying the budding rate $k_{\mathrm{b}}$ for a biochemical conversion rate $k_{\mathrm{m}}=1$ (equal amount of all species in the system).

(A) System purity as a function of the budding rate – a cut through the purity phase diagram shown Figure 2C. (B) Normalized enrichment in cis (blue), medial (orange) and trans (green) identities between the acceptor and donor compartments during vesicular exchange (solid lines are guides for the eyes). See Appendix 6 for non-normalized fluxes. (C) Sketches showing the direction of the dominant vesicular fluxes – consistent with data of A and B and data from Appendix 6—figure 1 – omitting less contributive transports. Before the purity transition (low values of $k_{\mathrm{b}}$ - purity ~ 0.5) the vesicular flux is retrograde. After the transition (high values of $k_{\mathrm{b}}$ - purity ~ 1) the vesicular flux is anterograde. Around the crossover ($k_{\mathrm{b}}\sim 0.1$), the vesicular flux is centripetal and oriented toward medial-compartments. The centripetal flux disappears if inter-compartments fusion is prohibited, see Appendix 8.

In summary, three types of organisation can be found, mostly controlled by the ratio of budding over fusion rate $k_{\mathrm{b}}$: a mixed regime at low $k_{\mathrm{b}}$, where compartments are large and contain a mixture of identities, a vesicular regime at high $k_{\mathrm{b}}$, where compartments are made of only a few vesicles and are very pure, and a sorted regime for intermediate values of $k_{\mathrm{b}}$, where compartments are both rather large, and rather pure. Snapshots of these three steady-states are shown in Figure 2D. The relationship between average size and average purity of compartments is shown in Figure 2E. The existence of a well-defined intermediate regime with large sorted compartments is promoted by our assumption of ‘saturated budding’ where the budding flux of any identity only depends on the compartment size (Equation 2). If the budding flux is linear with the number of patches of a given identity, the purity transition occurs for larger values of the budding rate $k_{\mathrm{b}}$, where compartments are smaller (see Appendix 8).

Vesicular transport

Vesicular exchange between compartments is quantified by following the dynamics over time of passive cargo injected from the ER. Each cargo in a compartment of size $n$ has a probability $1/n$ to join a vesicle budding from this compartment. When this vesicle fuses with another compartment or a boundary, the composition difference $\mathrm{\Delta }{\varphi }_i$ between the receiving and emitting compartment (a number between −1 and 1) is recorded for the three i-identities ($i=$cis, medial, trans). Averaged over all budding/fusion events, this defines the enrichment vector $\overrightarrow{E}$, whose three components $E_i$ are normalized for readability so that ${\sum }_i|E_i|=1$ (see Appendix 6—figure 1 for non-normalized enrichment). Vesicular flux is predominantly anterograde if ${\displaystyle E_{\mathit{\text{cis}}}<0}$ and ${\displaystyle E_{\mathit{\text{trans}}}>0}$, while it is predominantly retrograde if ${\displaystyle E_{\mathit{\text{cis}}}>0}$ and ${\displaystyle E_{\mathit{\text{trans}}}<0}$.

We focus on systems containing comparable amounts of each species at steady-state ($k_{\mathrm{m}}=1$ - see Equation 6). The enrichment in cis, medial and trans-identities are shown in Figure 3B as a function of the budding rate $k_{\mathrm{b}}$. The most striking result is the high correlation between purity of the compartment and the directionality of vesicular transport. For low values of the budding rate ($k_{\mathrm{b}}\ll 1$), compartments are well mixed (purity $\lesssim 1/2$) and the vesicular flux is retrograde, with a gain in cis-identities and a loss in trans-identities. For high values of the budding rate ($k_{\mathrm{b}}\gg 1$), compartments are pure (purity $\simeq 1$) and the vesicular flux is anterograde, with a gain in trans-identities and a loss in cis-identities. The archetypal behaviors most often discussed in the literature are thus, within the limits of our model, asymptotic regimes for extreme values of the ratio of budding to fusion rates. Remarkably, the cross over between these two asymptotic behaviors is rather broad ($k_{\mathrm{b}}\sim 0.1-1$) and displays a more complex vesicular transport dynamics, mostly oriented toward medial compartments.

In our model organelle, vesicle budding and fusion are biased solely by local compositions, which is subjected to irreversible biochemical conversions. The interplay between these microscopic processes gives rise to an irreversible flux of vesicles across the system. To better understand the directionality of vesicular exchanges, the net vesicular flux leaving compartments with a given composition is represented as a vector field in the compartments composition space in Figure 4. This is shown as a triangle plots in which each point corresponds to a fraction of cis, medial and trans components in a compartment, and the three vertices of the triangle correspond to pure compartments (see Appendix 2). The variation of the vesicular flux with the budding rate $k_{\mathrm{b}}$ (Figure 4) is consistent with that of the composition enrichment in the different identities (Figure 3B). As $k_{\mathrm{b}}$ increases, the flux evolves from being mostly retrograde at low $k_{\mathrm{b}}$ (from trans-rich toward less mature compartments) to being anterograde at high $k_{\mathrm{b}}$ (from cis to medial, and medial to trans-compartments). In between, the vesicular flux is centripetal toward mixed compartments, leading to a net enrichment in medial-identity. The relationship between structure and transport is explained below, and a dissection of this fluxes with respect to the composition of the vesicles is shown in Appendix 6. The net flux leaving a particular region of the triangular phase space is proportional to the total system’s size (number of compartments times their size) with this composition, while the flux arriving at a particular region depends on the number of compartments with that composition. Both quantities are shown on Figure 4.

Figure 4.

Relationship between the structure of the system and the vesicular fluxes.

Distribution of the total size of the system (truncated scale, maximum value in brackets) and number of compartments, and the vesicular flux between them, shown in a triangular composition space. Each point represents a given composition of compartments and the triangle vertices are compartments of pure cis, medial and trans identity. Arrows show the vesicular flux. The base of each arrow is the composition of the donor compartment and the tip is the average composition of receiving compartments (ignoring back fusion). The opacity of the arrows is proportional to the flux of transported cargo going through this path per unit time ($1/K_{\mathrm{f}}$), normalized by the total number of cargo-proteins. The dashed line on the top-left triangle is the theoretical prediction given by Equation 9. See also Appendix 6 for further characterizations of the vesicular transport, and Appendix 8 to discuss the impact of other budding and fusion implementations on the vesicular fluxes.

At low budding rate ($k_{\mathrm{b}}\ll 1$ - top row in Figure 4), the system is in the mixed regime, where compartments are large (hundreds of vesicles in size) but mixed (purity $\lesssim 1/2$). The system is dominated by compartment maturation, and the compartment distribution is spread around a path going from purely cis to purely trans-compartments, consistent with a dynamics where each compartment maturates independently from the other and given by Equation 9. - see top left panel of Figure 4. The majority of the transport vesicles are emitted by trans-rich compartments, which concentrate most of the components of the system, leading to a retrograde vesicular flux. Note that although vesicular flux is clearly retrograde in the maturation-dominated regime, as seen from the increase in cis identity and the decrease in trans identity between emitting and receiving compartment (Figure 3B), there is no net flux of vesicles from medial-rich to cis compartments in this limit (Figure 4). This differs from the classical ‘cisternal progression and maturation’ mechanism sketched in Figure 1A, and is due to the fact that the compartments that contain the highest fraction of medial identity in this regime are well mixed (${\varphi }_{cis}\simeq {\varphi }_{medial}\simeq {\varphi }_{trans}\simeq 1/3$). The vesicular flux leaving such compartments is equally split into cis, medial and trans vesicles, which fuse homotypically with compartments enriched in their identity, yielding a vanishing net flux averaged over the three identities. There is nevertheless a strong retrograde flux of cis vesicles toward cis compartments, as shown in Appendix 6—figure 1. In the classical ‘cisternal maturation’ mechanism, retrograde transport is needed to recycle cis-Golgi components from more mature compartments. Within the present model, this can be achieved by targeting such components to $cis$ vesicles budding from more mature compartments.

At high budding rate ($k_{\mathrm{b}}\gg 1$ - bottom row in Figure 4), the system is in the vesicular regime, where compartments are fairly pure, but rather small, with a size equivalent to a few vesicles. Compartments are exclusively distributed along the cis/medial and medial/trans-axes of composition, and accumulate at the vertices of the triangle. Membrane patches that just underwent conversion bud quickly from the donor compartment and fuse with pure compartments with the appropriate medial or trans-identity, explaining the clear anterograde vesicular transport. This feature is further reinforced under very high budding rate, where the vesicular flux is dominated by budding vesicles undergoing biochemical conversion before fusion, which prohibits their back fusion with the donor compartment.

In the intermediate regime (middle row in Figure 4), the distribution of mixed compartments is more homogeneous along the line describing the maturation of individual compartments - both in terms of size and number of compartments. Cis-rich and trans-rich compartments emit comparable amount of transport vesicles, while large medial-rich compartments are absent. This leads to a centripetal vesicular flux with an enrichment in medial identity (see Figure 3B). The lack of large medial-rich compartment is due to the fact that they can be contaminated by either cis or trans-species and fuse with cis-rich or trans-rich compartments. If inter-compartment fusion is prohibited, the enrichment in medial-identity in the intermediate regime is abolished, with a direct transition from anterograde to retrograde vesicular transport upon increasing $k_{\mathrm{b}}$ (see Appendix 8 and the Discussion section).

A graphical sketch consistent with our numerical observations for the typical size and composition of different compartments and for the dominant vesicular fluxes between them in the different regimes is shown in Figure 3C.

Discussion

The genesis and maintenance of complex membrane-bound organelles such as the Golgi apparatus rely on self-organisation principles that are yet to be fully understood. Here we present a minimal model in which steady-state Golgi-like structures spontaneously emerge from the interplay between three basic mechanisms: the biochemical conversion of membrane components, the composition-dependent vesicular budding, and inter-compartment fusion. As our model does not include space, we do not refer to the spatial structure of a stacked Golgi, but rather to the self-organisation of Golgi components into compartments of distinct identities undergoing vesicular exchange. We show that directional vesicular exchanges between compartments of different identities spontaneously emerge from local biochemical interactions, despite the absence of spatial information. This is in line with recent observations that the Golgi functionality is preserved under major perturbation of its spatial structure resulting from land-locking Golgi cisternae to mitochondria (Dunlop et al., 2017). Indeed, cargo transport still proceed, and is merely slowed down by a factor of two (from 20 min to 40 min) in land-locked Golgi despite the absence of spatial proximity between cisternae. This suggests that spatial information is less crucial to the Golgi function than biochemical information. Our main result is that the steady-state organisation of the organelle, in terms of size and composition of its compartments, is intimately linked to the directionality of vesicular exchange between compartments through the mechanism of homotypic fusion. Both aspects are controlled by a kinetic competition between vesicular exchange and biochemical conversion.

The results presented in Figure 2 can be used to explain a number of experimental observations. The predicted role of the budding and fusion rates is in agreement with the phenotype observed upon deletion of Arf1 (a protein involved in vesicle budding), which decreases the number of compartments and increases their size, particularly that of trans-compartments which seem to aggregate into one major cisterna (Bhave et al., 2014). A comparable phenotype has been observed upon mutation of NSF (a fusion protein), which produces extremely large, but transient, trans-compartments (Tanabashi et al., 2018), thereby increasing the stochasticity of the system. This is consistent with an increase of the fusion rate according to our model (see Appendix 5 for a quantification of the fluctuations in our model). Modifying the expression of VPS74 (yeast homolog of GOLPH3) leads to an altered Golgi organisation, comparable to the ARF1 deletion phenotype (Iyer et al., 2018). Both $\mathrm{\Delta }$arf1 and $\mathrm{\Delta }$vps74 present an enlargement of the Golgi cisternae and a disruption of molecular gradients in the system, which we interpret, in the limits of our system, as a lower purity due to slower sorting kinetics following an impairment of the budding dynamics. Importantly, the correlation between the size and purity of the compartments predicted by our model is in agreement with the observation that decreasing the budding rate by altering the activity of COPI (a budding protein) leads to larger and less sorted compartments (Papanikou et al., 2015). In yeast again, over-expression of Ypt1 (a Rab protein) increases the transition rate from early to transitional Golgi, and increases the co-localization of early and late Golgi markers (Kim et al., 2016). We interpret this as a decrease of the purity of Golgi cisternae upon increasing the biochemical conversion rate by unbalancing the ratio of budding to conversion rates. This suggests that the wild-type Yeast Golgi is closed to the line $k_{\mathrm{m}}=k_{\mathrm{b}}$ shown in Figure 2.

One crucial model prediction is that the size and purity of Golgi compartments should be affected in a correlated fashion by physiological perturbations. This could be tested experimentally by further exploring the phase diagrams of Figure 2B–C by simultaneously varying the ratios of biochemical conversion over fusion rate $k_{\mathrm{m}}$ and of budding over fusion rate $k_{\mathrm{b}}$. Compartment purity can be altered without changing their size by acting on $k_{\mathrm{m}}$, while it should be correlated with a change of size by acting on $k_{\mathrm{b}}$. We thus predict that the purity decrease concomitant with the size increase observed in Papanikou et al., 2015 upon impairing COP-I activity (decreasing $k_{\mathrm{b}}$) could be reversed without a change of cisterna size upon decreasing the biochemical conversion rate $k_{\mathrm{m}}$, for example by impairing Ypt1 activity (as done in Kim et al., 2016). Along the same line, we predict that the decrease of purity observed by Kim et al., 2016 upon Ypt1 over-expression (which increases the early to transitional conversion rate) should not be associated to change of cisternae size (at steady-state) if altering Ypt1 does not modify the budding rate $k_{\mathrm{b}}$. We can predict further that the loss of purity phenotype could be reversed upon increasing the budding rate by over-expressing COP-I, but that this would be accompanied by an decrease of cisternae size. This experimental proposal is represented graphically on the size and purity phase diagrams in Appendix 9—figure 1, with somewhat arbitrary arrows, to give a feel for the way our model may be used to design experimental strategies.

The directionality of vesicular transport is intimately linked to the steady-state organisation of the organelle; vesicular transport is anterograde when compartments are pure, and it is retrograde when they are mixed. This can be intuitively understood with the notion of ‘contaminating species’. If a compartment enriched in a particular molecular identity emits a vesicle of that identity, the vesicle will likely fuse back with the emitting compartment by homotypic fusion, yielding no vesicular exchange. One the other hand, an emitted vesicle that contains a minority (contaminating) identity will homotypically fuse with another compartment. If budding is faster than biochemical conversion, the contaminating species is the one that just underwent conversion, and its budding and fusion with more mature compartment leads to an anterograde transport. In such systems, compartments are rather small and pure. If biochemical conversion is faster than budding, the contaminating species is the less mature one, leading to a retrograde transport. Such systems are rather mixed with large compartments. Thus, the directionality of vesicular exchange is an emergent property intimately linked to the purity of the compartments. The medial-rich compartments are special in that regard: for intermediate values of the purity ($k_{\mathrm{b}}\sim k_{\mathrm{m}}$), they may be contaminated both by yet to be matured cis-species and already matured trans-species and can fuse both with cis-rich and trans-rich compartments. If inter-compartment fusion is allowed, large medial-rich compartments are relatively scarce, and emit few vesicles. On the other hand, medial-vesicles emitted by cis/medial and medial/trans-compartments may fuse together to form (small) medial-rich compartments. For intermediate values of the budding rates, this vesicular flux dominates vesicular exchange and leads to a centripetal vesicular flux towards medial-compartments (details in Appendix 6). If inter-compartment fusion is prohibited, which for instance corresponds to a situation where Golgi cisternae are immobilized (Dunlop et al., 2017), medial-rich compartments are present at steady-state, the centripetal vesicular flux is less intense, and the purity transition is accompanied with a direct transition from retrograde to anterograde vesicular flux (Appendix 8). Note that vesicular transport of cargo through the Golgi is not bound to follow the net vesicular flux between compartments discussed above. Indeed, the net flux is the average of the flux of vesicles of the three identities. A given cargo follows this flux - on average - if it does not interact preferentially with membrane of particular identity. On the other hand, a cargo that is preferentially packaged into vesicles of trans identity (for example) will be transported toward trans-rich compartment even if the net vesicular flux is mostly retrograde.

Regardless of the details of the model, we find that systems showing a well defined polarity with well sorted cisternae exhibit anterograde vesicular fluxes, whereas systems with mixed compartments exhibit retrograde fluxes. The former dynamics is expected when biochemical conversion is the slowest kinetic process and compartments are long-lived, while the latter is expected when vesicular exchange is slow and the system is composed of transient compartments undergoing individual maturation. One can relate this prediction to the difference in organisation and dynamics between the Golgi of S. cerevisiae and the more organized Golgi of higher organisms such as vertebrates. Maturation of Golgi cisternae has been directly observed in S. cerevisiae (Losev et al., 2006), with colocalization of different identity markers within single cisternae (low purity) during maturation, whereas vesicular transport phenotypes have been indirectly observed (Dunlop et al., 2017) or inferred through modeling (Dmitrieff et al., 2013) in mammalian cells. Consistent with our predictions, Golgi dynamics is one to two orders of magnitude faster in S. cerevisiae Golgi cisternae (as measured by the typical maturation rate of cisternae $\sim 1/\min .$ [Losev et al., 2006; Matsuura-Tokita et al., 2006]) than in mammalian cells (as measured by the typical Golgi exit rate of cargo $\sim 1/20\min .$ to $\sim 1/40\min .$ Bonfanti et al., 1998; Hirschberg et al., 1998; Patterson et al., 2008; Boncompain et al., 2012]). At this point, one may speculate a link between structure and function through kinetics. A well sorted and polarized Golgi is presumably required to accurately process complex cargo. The glycosylation of secreted cargo is key to the interaction of a vertebrate cell with the immune system of the organism it belongs to Ryan and Cobb, 2012, and glycans appear to be more diverse in higher eukaryotes - which also possess a highly organized Golgi - than in unicellular eukaryotes like yeast (Wang et al., 2017). The Golgi organisation in S. cerevisiae could thus be the result of an adaptation that has favored fast transport over robustness of processing, leading to a less organized Golgi characterized by cisternal maturation and retrograde vesicular transport. Remarkably, the slowing down of Golgi transport when S. cerevisiae is starved in a glucose-free environment coincides with the Golgi becoming more polarized (Levi et al., 2010), strengthening the proposal derived from our model of a strong connection between transport kinetics and steady-state Golgi structure.

The present model is designed to account for crucial dynamical processes at play in the Golgi with a limited number of parameters. It is sufficient to yield robust self-organised structures which can reproduce a number of experimental observations, but cannot be expected to account for the full richness of Golgi dynamics, especially under severe perturbation. The model includes a single kind of exchange mechanism based on vesicles. Exchange could also proceed via the pinching and fusion of larger cisterna fragments (Pfeffer, 2010; Dmitrieff et al., 2013), or via tubular connection between compartments (Trucco et al., 2004), although the relevance of the latter to Golgi transport is still debated (Glick and Luini, 2011). Exchange mechanisms that do not establish stable connections able to relax concentration gradient between compartments by diffusion could be included as extension of our vesicle exchange mechanism (at the expense of introducing additional parameters to characterise their composition and size-dependent nucleation, scission and fusion rates). Stable Golgi tubulation without tubule detachment is observed under brefeldin A treatment, which prevent the membrane association of COPI coat. This eventually leads to the disappearance of the Golgi by quick resorption into the ER whenever a tubular connection is established between them, in a way suggesting a purely physical (tension-driven) mechanism (Sciaky et al., 1997). According to our model, preventing COPI vesicle budding should lead to larger cisternae, as discussed above. The brefeldin A phenotype can be reconciled with our framework by invoking the fact that larger cisternae should have a higher surface to volume ratio and hence a lower membrane tension, decreasing the force required to nucleate membrane tubules and increasing the probability of direct tubular connection with the ER. Such tension effects are clearly beyond the scope of the present model, but this phenotype could be qualitatively reproduced by invoking an increase of the ER fusion parameter ${\alpha }_{\mathrm{ER}}$ with the size of the compartment due to membrane mechanics considerations.

The model does not account for the possibility of lateral partitioning between processing and exporting domains within compartments which could play an important role in regulating cisterna composition and cargo processing (Patterson et al., 2008). A detailed kinetic model of organisation based on this concept and supported by extensive quantitative live cell imaging experiments is proposed in Patterson et al., 2008 to account for the distinct composition of the different Golgi cisterna and the kinetics of cargo transport. This rapid partitioning model (RPM) is very different in scope with the model we propose here, since it focuses on the transport of lipids and cargo molecules in a pre-established Golgi apparatus with a prescribed structure, while the present model describes de-novo Golgi formation and maintenance, through self-organised fusion and scission mechanisms between dynamic compartments. The two models share important similarities, such as the fact that no directionality is assumed a priori for intra-cisternal vesicular transport, and that export is allowed from every cisterna, which is crucial to reproduce the exponential kinetics observed in experiments. The RPM is much more precise in terms of the microscopic description of the biochemical processes, and consequently involves many parameters. Combining our simple model of Golgi self-organisation with the detailed description given by the RPM for the kinetics of lipid, enzymes and cargo within the Golgi is the logical next step to push forward our understanding of Golgi self-organisation.

In summary, we have analyzed a model of self-organisation and transport in cellular organelles based on a limited number of kinetic steps allowing for the generation and biochemical conversion of compartments, and vesicular exchange between them. Although we kept the complexity of individual steps to a minimum, our model gives rise to a rich diversity of phenotypes depending on the parameter values, and reproduces the effect of a number of specific protein mutations. We identify the concomitance of a structural transition (from large and mixed to small and pure sub-compartments) and a dynamical transition (from retrograde to anterograde vesicular exchange). Within our model, anterograde vesicular transport is accompanied by many spurious events of vesicle back-fusion. For very high budding rates, compartments are very pure and anterograde transport is dominated by vesicles undergoing biochemical conversion after budding from a compartment. In our model, such vesicle biochemical conversion events, which have been described in the secretory pathway as a way to direct vesicular traffic and to prevent vesicles back-fusion (Lord et al., 2011), only occur when compartments are very small (high budding rate). Adding composition-dependent feedback involving specific molecular actors to tune budding and fusion rates - for instance to reduce or prevent the budding of the majority species - will displace the purity transition to lower budding rates and thus extend the regime of anterograde transport to larger sub-compartment steady-state sizes. Cisternal stabilizers like GRASP or vesicles careers like Golgins, known to already be present in the ancestor of eukaryotes (Barlow et al., 2018), are obvious candidates. Although more complex models, and in particular the inclusion of spatial dependencies, are surely relevant to dynamics of the organelle, the fundamental relationship between kinetics, structure and transport highlighted by our model is a universal feature of the interplay between biochemical conversion and vesicular exchange in cellular organelles.