General features of the model

We assume that

1. The Golgi consists of a stack of cisternae, which move in an anterograde direction or “mature”, carrying with them their SNAREs, enzymes (such as glycosyltransferases), and proteins that are being processed. The latter will not be considered here. Once every time units, a new cisterna is added to the cis end of the stack, while the most mature cisterna dissolves and disappears from the trans end of the stack. The new cisterna is formed by coalescence of ER-derived vesicles and contains fixed concentrations of SNAREs and enzymes.
2. Along with cisternal progression, vesicles containing SNAREs and Golgi enzymes continuously bud from each cisterna. We assume that the vesicles provide local transport and can only fuse with the neighboring cis (less mature) and trans (more mature) cisternae, and with the progenitor. Indeed, in the stacked mammalian Golgi, coil-coiled vesicular tethering factors which span the distance between adjacent cisterna are thought to grab vesicles even prior to their release from the donor cisterna and prevent them from reaching more distant cisternae [18], [19]. We will later relax this restriction and consider transport in a non-stacked Golgi, as it exists, for example, in the yeast Saccharomyces cerevisiae.
3. SNAREs and Golgi enzymes are uploaded into a vesicle via competitive binding to a fixed number of vesicular sites. We assume that the vesicular transport results primarily in the movement of cargo without any significant change in the volume and budding surface area of the cisternae. This is supported by observations that the size of all cisternae is similar [20] and our estimates that taking into account the vesicular transport of membrane itself would not significantly alter the results.

Functioning of the model hinges on two general principles: Establishment and maintenance of a directed retrograde vesicular flux and sorting of the vesicular cargo via competition for binding sites.

Establishment of a cistrans SNARE gradient that mediates retrograde vesicular flow

To reveal the universality of the proposed self-establishing mechanism of vesicular traffic directionality we first consider the simplest possible setup, a single cognate SNARE pair and vesicle type. We assume that the rate of vesicular fusion is proportional to the product of the concentrations of the SNAREs present in vesicles and cisternae, respectively. The precise nature of SNARE molecules does not have to be specified here. We can even consider the SNAREs as mere proxy for fusion-specifying factors. The probability for a vesicle to fuse with a given cisterna depends solely on the cisternal concentration of compatible SNAREs, and cisternae with higher SNARE concentration have a higher probability to absorb vesicles. A retrograde vesicular flux thus requires a cistrans gradient in cisternal SNAREs.

We propose that key to a robust cistrans SNARE gradient is the observation that all systems, living and otherwise, function with a loss. As Golgi cisternae mature they inevitably lose active SNARE molecules. Such a decay of active SNAREs breaks the symmetry between the otherwise identical cisternae in a systematic way: The older trans cisternae contain less SNAREs than the younger cis cisternae. The SNARE loss can occur by escape of SNARE-carrying vesicles that fuse with the ER thus recycling their content. However, some of the cisternal SNARE decay is likely due to irreversible loss that requires some new SNARE synthesis to replenish the system.

The “seed” SNARE gradient generated in this manner sets a preference for vesicles to fuse with cis rather than trans cisternae, thus initiating the directed vesicular transport. As SNAREs are transported retrograde, their cistrans gradient is further enhanced. When the vesicular flux becomes balanced by the anterograde transport of SNAREs due to cisternal maturation, the system comes to a steady state. Indeed, we show both numerically and analytically that the seed gradient, created by the temporal decay of SNAREs, is self-enhancing (Figs. 2a and 5, which is presented in Methods, and Eqs. (11, 12, 13)). Importantly, while vesicular transport significantly increases the seed gradient produced by SNARE loss, without that loss vesicular transport by itself cannot produce or maintain any gradient, (see Eq. (13) and subsequent illustrations in Methods). This is in accordance with the results of [14] that the single SNARE pair/single coat minimal system cannot spontaneously break the initial symmetry of compartments. The constant progression of cisternae is equally important for maintaing the steady state SNARE gradient and directional vesicular flux. Without the progression, the seed SNARE gradient would have been equilibrated via vesicular transport.

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Figure 2. Self-generated concentrations of SNAREs and enzymes.

Panel A: Steady state concentration of cisternal SNARE vs the number of cisterna for: both the loss and the vesicular transport mechanisms are enacted (solid line), only the loss mechanism operates (dashed line), only vesicular transport occurs (dotted line). All concentrations are sampled immediately before the cisternal shift event, when the number of each cisterna is incremented by one. The definitions of parameters are given in Methods. Here and in all following plots it is assumed that and , i.e. all concentrations are expressed in the units of initial concentrations and the time is expressed in units of the cisternal maturation period. Solid line: and , dashed line: and , dotted line: and , for all curves . Panel B: Distribution of Golgi enzymes: cis (solid line), medial (dashed line) and trans (dotted line) established as a result of competition for incorporation into vesicles. Vesicular flux is controlled by the gradient of cisternal SNAREs shown by the solid line in the left panel, vesicles from the first cisterna can exit the Golgi and fuse with the ER. The parameters for the enzyme transport are , , , and .

https://doi.org/10.1371/journal.pcbi.1003125.g002

We note that at steady state the vesicular flux does not depend on the concentration of SNAREs in the vesicles: Lower concentrations of vesicular SNAREs are compensated by a higher steady state number of vesicles. Naturally, a vesicle should contain a minimum number of vesicular SNARE molecules to ensure any fusion at all.

The calculation of the steady state SNARE gradient and vesicular flow are presented in the Methods section.

Establishment of Golgi enzyme peaks in cis, medial and trans cisternae via SNARE-mediated retrograde vesicular traffic

Next, we investigated how retrograde vesicular flow, created by the cisternal SNARE gradient, maintains the inhomogeneous steady state distribution of Golgi enzymes during cisternal maturation. To this end, we further developed the principle proposed by Glick et al. that attributes the different cisternal enzyme profiles to the competition of enzymes for the binding sites in vesicles [10]. For simplicity, we assume three categories of Golgi enzymes with peaks in cis, medial and trans cisternae, and with strong, intermediate and weak affinities for vesicular binding sites, respectively (Schematically depicted in Fig. 1B). Unlike in earlier models [10] and [21], the fraction of binding sites occupied by each type of enzyme is determined by mass action equilibrium. Also, in contrast to [10] and [21] where a number of ad hoc assumptions about vesicular flow were used, we “couple” the enzyme-carrying capacity to the self-established vesicular flow described above. Hence, while each vesicle competitively uploads enzymes according to their dissociation constants, its fusion probability is determined by the cisternal SNARE gradient shown by the black curve in Fig. 3A. To study the competition mechanism in its simplest form, we assume here that the SNARE distribution is unperturbed by enzyme uploading.

We find that the distribution of enzymes radically depends on whether vesicles originating from the first cisterna can exit the Golgi and fuse with its cis neighbor, the ER. If we permit ER recycling, the desired cis-medial- trans 3-mode steady state localization can be reproduced (Fig. 3B). Enzymes that have the highest affinity for the vesicular coat concentrate in the cis Golgi. A substantial fraction of these enzymes is loaded from the first cisterna into ER-bound vesicles and leaves the Golgi. In more central compartments, where cis enzymes are depleted, medial Golgi enzymes outcompete the weaker-binding trans enzymes for space in the vesicles. As a result, those enzymes advance with the maturing cisterna until the mid-Golgi where their concentration peaks, and then become effectively loaded into retrograde vesicles. Finally, the weak-binding enzymes can only incorporate into vesicles when all stronger-binding competitors are depleted. Their concentration peaks in the penultimate cisterna. The ultimate cisterna, equivalent to the disintegrating cisterna or trans Golgi network (TGN), exhibits a somewhat lower enzyme concentration as it does not receive any incoming retrograde vesicular traffic.

On the other hand, if none of the enzyme-carrying vesicles can escape to the ER, the cisternal distribution of all enzymes converges onto a single peak form (Fig. 6, presented in the Methods section). In this case the overall steady state abundance of enzymes increases with their affinity for vesicular binding: The stronger-binding enzymes are more efficiently retrieved to younger cisternae and thus better avoid being flushed out with the disintegrating trans cisterna than the weaker-binding enzymes. As a consequence of their higher concentration, the stronger-binding enzymes do not get sufficiently removed from younger cisternae to achieve their cis-Golgi peak and at the same time they do not give their weaker competitors any chance to enter the retrograde transport vesicles in the later cisternae. Hence, all enzymes peak at the trans face of the Golgi. Thus, recycling of enzymes to the ER is necessary for establishing the cis-medial- trans enzyme segregations. At the same time, we observe that the steady state SNARE distribution and resulting intra-Golgi vesicular flux is only weakly affected by the presence or absence of ER-recycling. This is because both scenarios feature an inherent loss mechanism, which breaks the intra Golgi conservation of SNAREs.

We also observe that, as discussed in [21], the competition-based enzyme segregation is rather sensitive to the variation of model parameters. Thus, it is possible that mechanisms have evolved to make the cisternal enzyme distribution more robust. One such mechanism, the change in enzyme affinity for vesicular binding sites with cisternal age, has been studied in [21] and could easily be incorporated into the a more detailed versions of our model.

The quantitative details of the calculation of the steady state enzyme concentrations, including Fig. 6, which illustrates enzyme distribution in the absence of ER-recycling, are presented in Methods.

The two Golgi SNARE pairs can function with a single vesicle type to establish their own gradients and the observed Golgi enzyme peaks in cis, medial and trans

We now apply the general mechanisms of fusion asymmetry and competitive vesicle binding to explain the specific SNARE and enzyme distributions as they are actually observed in the mammalian Golgi. The important adjustment to our basic model is that the Golgi apparatus features not one, but two cognate SNARE pairs. The first pair, which we label , consists of the monomer SNARE rBet1 with its trimer SNARE partner Membrin/ERS24/Syntaxin5. The second pair, labeled , consists of the monomer SNARE GS15, compatible with the trimer SNARE complex of Gos28, Ykt6 and Syntaxin5. There is solid experimental evidence for both pairs to be incorporated in COPI vesicles [17], [22] and to participate in vesicular traffic of Golgi resident proteins [23], [24], [25].

To reproduce three Golgi enzyme peaks in concurrence with the experimentally observed distributions of the and SNARE pairs we introduce an additional specification at this point, namely that the monomeric SNAREs rBet1 and GS15 mediate fusion only when present on the vesicle, and the trimeric-SNARE complexes only when present in the cisternae. In the following paragraph we provide a justification for the functional allocation of SNAREs as vesicular and cisternal.

In the Golgi, only the SNARE proteins actually have a cistrans distribution [17] such as shown in Fig. 2A. The SNARE Gos28 also decreases from cis to trans [26]; however, its cognate monomeric SNARE partner GS15 accumulates in the trans-most cisternae instead [17], and the GS15 yeast homologue Sft1p is also enriched in the late Golgi [16], [23], (see Fig. 1C). If GS15 and the Gos28-Ykt6-Synt5 complex could both function as fusiogenic SNAREs in the cisternae, our model of vesicular flux would imply that Golgi enzymes known to depend on this SNARE pair for vesicular traffic undergo anterograde rather than retrograde transport. The anterograde vesicular enzyme transport does little to improve enzyme segregation as the cisternal maturation already moves enzymes in trans direction. More importantly, the anterograde vesicular transport makes the enzyme recycling impossible. Our allocation agree with in vivo observations: monomeric SNAREs act indeed most often as vesicle- or v-SNAREs and the trimeric SNAREs generally function at the target membrane (and are therefore typically referred to as t-SNAREs), [27]. But we also have a mechanistic explanation for why trimeric Golgi SNAREs function in the cisternae rather than the vesicles: When we consider the monomeric and trimeric SNAREs of a cognate SNARE pair separately, the SNARE that is most abundant in the vesicle determines which of the cisternal SNAREs the vesicle engages with. If the monomeric SNARE is more abundant in a vesicle than the trimeric SNARE, it will specify that the vesicle fuses with the cisterna which has the highest amount of cognate trimeric SNAREs, regardless of its monomeric SNARE concentration. Thus, when monomeric and trimeric SNARE partners differ significantly in their affinity for vesicles, the one with higher affinity becomes the v-SNARE, leaving the other to function in the cisternae. This is the case especially for the SNARE pair as Syntaxin 5, the limiting partner in both and trimeric SNARE complexes, is at least 4 times less abundant than the monomer GS15 in COPI vesicles (See Fig. 7B in [17]). Syntaxin-5's apparent poor affinity for Golgi vesicles explains its observed homogenous distribution in Golgi cisternae. However, the other constituents in and trimers are more efficiently transported by vesicles, thus maintaining the cisternal SNARE gradient. It follows from these observations that the two Syntaxin 5 containing trimeric Golgi SNAREs function as t-SNAREs.

In addition to the two Golgi SNAREs, we consider a third v-SNARE, which mediates the fusion of Golgi-derived vesicles with the ER. It is ERS24, which thus has a dual function as part of a t-SNARE complex in intra Golgi transport and as v-SNARE in Golgi-to-ER transport. The corresponding ER t-SNARE does not leave the ER and is therefore not considered here [15].

Apart from the SNARE specifications, we implemented a similar set of minimal assumptions as for the single SNARE scenario:

1. The rate of vesicular fusion with Golgi cisternae is determined by both and SNARE pairs and is proportional to the sum of the products of the cognate v-SNARE and t-SNARE concentrations.
2. All SNAREs and Golgi enzymes compete for the same binding sites in the vesicles. This is in agreement with the findings for ER-derived COPII vesicles, the only instance where cargo-competition for coat binding has been elucidated to date [28].
   However, our model reproduces the enzyme segregation as well in the case when the enzymes compete only with each other and not with SNAREs for vesicular binding sites, as shown in Fig. 2.
3. Vesicles fuse locally, i.e. with the cis and trans neighbors of the progenitor cisterna and the progenitor cisterna itself. In addition to fusing with Golgi cisternae, vesicles also fuse with the ER with a rate that is controlled by the product of the concentrations of the vesicular ER v-SNARE and a fixed concentration of ER t-SNAREs. Due to the expansive volume of the ER that brings it in proximity to the entire Golgi apparatus, we assume that all vesicles can fuse with the ER independent of their originating cisterna.
4. The age-dependent decrease (loss) of the cisternal concentration of the t-SNARE, which is essential for triggering the retrograde directionality of the traffic, occurs due to transport of vesicles to the ER. Therefore, no additional decay term is introduced for it. A small age-dependent decay term is introduced for the t-SNARE (see Eq. (19)).

We found a good qualitative agreement between our results and the experimentally observed concentration profiles. With the proper choice of dissociation constants (Table 1), t-SNARE decay rate, and vesicular transport intensity, the model functions in the following way: The strong coat-binding affinities of and ER SNAREs effectively package them into vesicles that bud from the younger cis cisternae. These vesicles have a high probability to fuse with the ER due to a substantial concentration of the ER SNAREs. These vesicles also recycle a good fraction of strong-binding cis enzymes, and a part of medial enzymes to the ER. The recycling of t-SNAREs to the ER seeds a cisternal gradient, which is responsible for the mostly retrograde direction of vesicular transport in the early cisternae. The recycling of t-SNAREs to the ER is poor, yet when coupled with age-dependent decay, the t-SNAREs extends the cistrans t-SNARE gradient to the trans Golgi. In more mature cisternae where the and ER SNAREs and cis-enzymes are depleted, the vesicles incorporate the weaker-binding molecules, such as medial and, to a lesser extent, trans enzymes and SNAREs. These vesicles have a much lower probability to reach the ER and transport their cargo mostly to younger Golgi cisternae. Finally, the trans-most cisternae bud vesicles that contain predominantly trans enzymes and v-SNAREs. These cargoes are transported mostly retrograde, but hardly reach the ER. The total fraction of each protein that is retained in the Golgi (as compared to that recycled to the ER) can be appreciated by its concentration in the trans-most cisterna in Fig. 3B. Since the figures represent the situation before the last cisternal maturation step and removal of the last cisternae, the protein concentration that remains in the Golgi is equal to the initial concentration (set equal to one for all molecules), minus the loss to the ER.

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Table 1. Dissociation constants for binding to vesicular sites that yield the plots depicted in Fig. 3.

https://doi.org/10.1371/journal.pcbi.1003125.t001

Our prediction that cis-enzymes, and ER SNAREs recycle through the ER at a higher level than SNAREs and trans enzymes is indeed born out by numerous experimental observations in yeast and mammalian cells. Cis but not trans Golgi markers accumulated in the ER upon an acute ER-exit block [23], [29] or in the ER-derived intermediate compartment (ERGIC) after a temperature-induced exit block from this compartment [30], [24].

Based on the SNARE dissociation constants that yielded the experimentally observed protein gradients (Table 1) we further predict that monomeric ERS24, which functions as ER-v-SNARE, has the highest affinity of all SNAREs for the COPI coat, followed by the v- and t-SNAREs, (rBet1p and the proteins Syntaxin5 and Membrin, which together with ERS24 make up the t-SNARE complex). Indeed, ERS24 is much higher concentrated in COPI vesicles than any of the other v-SNAREs (Fig. 8B in [17]). Syntaxin 5 is translated as a long and short version in mammalian cells [31]. The longer form features a known ER-retrieval signal and we predict that it is this form that predominantly functions in the t-SNARE complex and is more efficiently incorporated into COPI vesicles then the short form that likely functions mostly as t-SNARE, which has a higher dissociation constant than the t-SNARE.

So far we assumed that vesicles only fuse with the immediate neighbors of their progenitor cisternae. A stacked Golgi, however, is not a requirement for Golgi asymmetry and cisternal maturation, which are also observed in S. cerevisiae where individual Golgi cisternae are scattered throughout the cytoplasm [32], [33], [34], [35]. Removing the local fusion restriction, and allowing vesicles to fuse with any cisterna and the ER depending on their SNARE concentrations, we achieve only poor enzyme segregation with all enzyme maxima shifted towards younger cisternae, (Fig. 4).

We suggest therefore, that a realistic description of fusion probability in S. cerevisiae must include a factor that considers fusion preferences related to cisternal age although it might be less stringent than the nearest neighbor limitation of a Golgi stack. Golgi scattering occurs when novel cisternae emerge from multiple, short-lived transitional ER (tER) sites rather than from a single, stable tER [36]. If individual tER sites release multiple cisternae in short succession before ceasing their activity, the diffusion limits imposed by the ER-network could maintain sister cisternae that are close in age in proximity to each other, thus ad hoc generating a series of maturing Golgi cisternae that remain separate from those generated in parallel by other tER sites. Evidence from a recent study by Nakano et al in S. cerevisiae supports this prediction: When due to altered ER-morphology the motility of Golgi elements away from the ER-exit site(s) is impeded, cis and trans Golgi elements could be seen in close proximity to each other and to ER-exit sites [37]. A position-age correlation is also apparent from the more coarse-grain viewpoint: Consider the emission of Golgi elements from multiple scattered ER exit sites and their subsequent one-dimensional diffusion in the cytoplasmic half-space away from ER membrane. The average distance from the ER membrane of a Golgi element at time t after emission scales as . Thus, the older Golgi elements are on average further away from the ER than the younger ones. Real-time imaging maps of the spatial relationship between yeast Golgi cisternae that exchange cargo should provide the experimental framework to make our model applicable to Golgi systems with scattered compartments where we expect the enzyme distribution to be somewhere in between the examples shown in Fig. 3 and Fig. 4.

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Figure 3. Self-generated steady state distributions of alpha and beta SNAREs and enzymes as it is observed in the Golgi apparatus.

Panel A: t-SNARE (solid line) and v-SNARE (circles) coinciding with t-SNARE, and t-SNARE (dashed line) and v-SNARE (dotted line) vs. the cisternal number . Panel B: cis (solid line), medial (dashed line) and trans (dotted line) Golgi enzymes normalized by their maximum value vs number of cisterna . The parameters are: Decay rates are zero for all substances except for t-SNARE for which , the vesicular transport coefficient , and dissociation constants for vesicular binding are for ER v-SNARE, for cis t- and v-SNAREs, for trans t-SNARE, for trans v-SNARE, for cis enzymes, for median enzymes, and for trans enzymes. Initial concentrations of all substances in the first cisterna are , and the concentration of t-SNARE in ER is 0.7.

https://doi.org/10.1371/journal.pcbi.1003125.g003

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Figure 4. Steady state concentrations of three classes of enzymes in the unrestricted fusion scenario.

Golgi enzyme concentrations are normalized by their maximum value. The parameters are: Decay rates for the cis and trans t-SNARE are , the vesicular transport coefficient , and dissociation constants for vesicular binding are: for the ER v-SNARE, for the cis t- and v-SNAREs, for trans t-SNARE, for trans v-SNARE, for cis enzymes, for medial enzymes, and for trans enzymes. Initial concentrations of all proteins in the first cisterna are , and the concentration of the t-SNARE in the ER is 0.7.

https://doi.org/10.1371/journal.pcbi.1003125.g004

Establishment of a cistrans SNARE gradient that mediates retrograde vesicular flow

Here we do not specify the nature of t- and v- SNAREs, simply calling fusiogenic molecules present in a vesicle and cisterna v-SNAREs and t-SNAREs. The chemical distinction between t- and v-SNAREs will be stated later. However, to keep the same notations throughout the paper, we use the specific and notations already here. Small letters denote the vesicular concentrations of a molecule with the subscript referring to the parental cisterna. So and are concentrations of v- and t-SNAREs, and is the concentration of the th Golgi enzyme in a vesicle that emerged from the th cisterna. Capital letters , , and denote the concentrations of these substances in cisterna number . We number the compartments in the cis to trans direction, so the youngest cisterna has number one.

The number of vesicles that bud from the th compartment per unit time, , is assumed to be proportional to the area of the compartment ,(1)where is the budding rate constant which depends on the concentration and activity of coat proteins.

A vesicle emitted from the th cisterna fuses with the th cisterna with a probability proportional to the product of the concentrations of the SNARE in the vesicle and the SNARE in the cisterna . The number of vesicles that fuse with the cisterna per unit time is(2)with being the fusion rate constant. The assumption of local transport restricts a vesicle emitted by the th cisterna to fuse with the th, th, and th cisternae. The trans-most cisterna does not receive any retrograde vesicular input. The time evolution of the population of vesicles emitted by the th compartment is described by the rate equation which includes both the budding and fusion terms.(3)At steady state, the concentration of vesicles emitted by the th compartment becomes(4)Hence, an increment in SNARE concentration in the st cisterna due to the vesicular flux from the th cisternae is(5)A dimensionless factor describes how the cargo is “diluted” when a vesicle fuses with its target cisterna and is equal to the ratio of vesicle to compartment surface areas. Assuming mass-action equilibrium between the vesicular binding sites and its cargo (t-SNARE) and that budding of a single vesicle does not significantly alter the cisternal concentration of t-SNARE, the amount of t-SNARE in a vesicle isHere is the concentration of cargo binding sites in a vesicle and is the dissociation constant for binding between cargo and such sites. Eq. (5) indicates that the steady state flux of vesicles does not depend on the v-SNARE concentration and is only determined by the budding rate and t-SNARE distribution. In the following we assume that the volume and the budding area of the compartments remains constant, . Relaxing this assumption does not substantially change the results.

The rate equation that describes the evolution of the t- SNARE concentration in the th compartment reads(6)The first term describes the loss of the t-SNARE with the per molecule rate . To keep it general, the loss term collects all mechanisms of t-SNARE decay approximately described by first-order kinetics, such as degradation, loss of mis-targeted vesicles, etc. Thus, here the lost vesicles are not accounted for in Eqs. (3, 4), but are only included in the first term in Eq. (6). The second line describes the outgoing vesicular transport from the th cisterna to its th and th neighbors, and the last two lines represents the incoming flux from the same neighbors to the th cisterna.

To complete the description of t-SNARE distribution, the vesicular transport equation (6) has to be complemented by the definition of cisternal dynamics: Every time units the running number of each cisterna is incremented by one, so that the th cisterna becomes st. A new first cisterna with a given initial concentration of t-SNARE is added to the cis end of the stack, while the trans-most cisterna is removed.

We measure cisternal concentrations of SNAREs and other molecules in the units of their initial concentrations in the first cisterna and the natural unit of time is the period of cisternal maturation . This is equivalent to setting these quantities equal to one. Then the t-SNARE distribution is described by three parameters: decay rate , the vesicular transport coefficient , and the dissociation constant .

This cisternal maturation scenario together with Eq. (6) are implemented numerically as a simple Euler update. For reasonable values of parameters the system quickly converges to a steady regime: In each cisterna concentrations of t- and v-SNAREs undergo periodic evolution with the period . Plots of the cisternal distributions of the t-SNARE are presented in Fig. 2A in the Results.

Analytic solution for the asymptotic steady state cisternal concentrations of SNAREs

Consider a hypothetical system where the number of cisternae is non-biologically large. For older cisternae, the concentrations of SNAREs are small, , so the uptake of a SNARE into a vesicle is proportional to the concentration of the SNARE in the progenitor cisterna, . In the asymptotic regime, i.e., sufficiently far from the first and the last cisterna, we seek a solution of Eq. (6) in the form(7)After substitution into (6) it yields(8)where(9)We look for the periodic solution in a sense that each time units, after the addition of a new cisterna and dissolution of the most mature cisterna, the system returns to the same state. So the th cisterna at the time must be identical to the cisterna at the time ,(10)This yields the following equation for ,(11)which is solved numerically.

We observe that in the asymptotic regime, the steady state t-SNARE concentration decays exponentially with the number of cisterna,(12)with the coefficient being the solution of Eq. (11). Simulations confirm our theoretical prediction given by (7, 11), see Fig. 5.

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Figure 5. Distribution of SNARE for large number of cisternae.

The steady state gradient has the exponentially decaying form, where depends on two dimensionless groups of parameters, which caracterizes the decay of the SNARE and which characterizes the vesicular transport of SNARE: and (solid line) with the best fit given by , and (dashed line) with the best fit given by . Theoretical values of determined from (11) are indistinguishable from the values obtained as the best fit for the simulations. The dotted line corresponds to the case of zero loss, and , and illustrates the absence of a concentration gradient. To reveal the exponential decay of the SNARE concentration, we purposefully consider a non-biologically high number of compartments and analyze the SNARE concentration away from both the cis and trans ends of the stack, where the boundary effects can play a role.

https://doi.org/10.1371/journal.pcbi.1003125.g005

The necessity of the loss term for establishing the gradient by breaking the initial symmetry between the cisternae is clearly revealed by the following analytic argument: For a small loss rate (), the expansion of the steady state exponent reads(13)Hence for independent of the intensity of the vesicular transport characterized by . Indeed, without breaking the initial similarity between cisternae, a vesicle would fuse with any of its three target compartments with the same probability, so that vesicular flux into a compartment would be equal to the vesicular flux out of this compartment. In other words, the vesicular transport can only enhance the initial difference in concentrations between cisternae, created by some other mechanism, rather than create this difference de novo.

Establishment of Golgi enzyme peaks in cis, medial and trans cisternae via SNARE-mediated retrograde vesicular traffic

The transport of Golgi enzymes with cisternal concentrations , where labels an enzyme class, is described by an equation analogous to Eq. (6). The difference is that instead of a single vesicular cargo type (t-SNARE), we now consider three classes of competitors for vesicular seats. Thus, for each , replaces the in the left-hand side and replaces in the right-hand side of Eq. (6),(14)Here (only in this subsection) we assume that the enzyme transport does not affect the vesicular flow, which is established by the autonomously evolving t-SNARE distribution, described by (6). The concentration of enzyme uploaded to a vesicle is determined by the mass action equilibrium(15)Here are vesicle-th enzyme dissociation constants, the last equation states that the total number of the vesicular binding sites is equal to the number of free sites plus the number of sites occupied by enzymes of all three classes. Solving (15), one finds ,(16)which are subsequently substituted into Eq. (14), As with SNAREs, each newly formed (first) cisterna is assumed to be loaded with Golgi enzymes with given concentrations, We assume that there is no temporal decay of enzymes, so is put equal to zero in the transport equations.

When the retrograde vesicular transport is counterbalanced by the anterograde cisternal progression, the enzyme distribution reaches its steady state. The nature of the steady state depends on the boundary conditions imposed on the cis side of the Golgi stack: An “open” boundary condition is implemented as a zeroth cisterna (ER) with a fixed concentration of t-SNAREs which can fuse vesicles (see Fig. 2B), while under the “closed” boundary conditions vesicles do not escape the Golgi, (see Fig. 6).

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Figure 6. Enzyme segregation depends on the open boundary condition.

Steady state distribution of the same enzymes as in Fig. 2 if the “Closed boundary conditions” on the cis end of the stack are assumed: No vesicles can exit the Golgi. The parameters are the same as in Fig. 2.

https://doi.org/10.1371/journal.pcbi.1003125.g006

The two Golgi SNARE pairs can function with a single vesicle type to establish their own gradients and the observed Golgi enzyme peaks in cis, medial and trans

Putting together the two mechanisms considered above, we introduce a realistic model of Golgi transport. It describes the evolution of 8 distinct types of molecules: and sets of t- and v-SNAREs controlling intra-Golgi fusion, a v-SNARE for fusion with the ER, and cis, medial, and trans types of enzymes. For brevity of equations, we use the universal notations and for cisternal and vesicular concentrations of each of the eight molecules, . At the same time, in the fusion rate terms we retain the specific notations for t- and v-SNAREs with superscripts “”, “” and “ER” denoting the affiliation of particular SNAREs. The evolution of the cisternal concentration , of each type of molecule is described by the rate equation similar to (14) with two important distinctions. First, the rate of fusion of a vesicle with a cisterna, previously given by (2), is now proportional to the sum of the products of the concentrations of cis and trans SNAREs [14].(17)The increment in the vesicular cargo concentration in the th cisterna due to the vesicular flux from the th cisternae is (compare to (5)),(18)Here is the vesicular concentration of molecule defined by mass-action equilibrium (16) between vesicular binding sites and all eight competing molecules. The last term in the denominator corresponds to the fusion of vesicles with the ER, which is the second distinction of the considered mechanism from the model case analyzed above. The ER t-SNARE concentrations is considered to remain constant and vesicles originating from any cisterna can fuse with the ER. Assembling together all gain and loss mechanisms for the cisternal concentration of , we write the complete system of kinetic equations that describe the vesicular transport.(19)

The escape of a fraction of vesicles from the Golgi to the ER provides one part of a loss mechanism necessary for seeding the gradient of t-SNAREs. Yet we do not exclude the possibility of other mechanisms of t-SNARE decay, so the remains present in the rate equation. In the simulations, we set for the t-SNARE equal to a small value, while the decay coefficients for all other substances are put equal to zero.

Vesicular transport without nearest neighbor fusion restriction

To model the vesicular transport in yeast, we used an equation similar to Eq. (19) where the restriction of nearest neighbor fusion was relaxed,(20)(21)A typical steady state distribution of enzymes produced by the unrestricted vesicular fusion is shown in Fig. 4.