To assess and rationalize mechanisms of TANGO1 assembly into rings at ERES and their contribution to procollagen export, we formulate general hypotheses built on accumulated experimental data that will serve as the foundation for our two modeling approaches.

Hypothesis 1: TANGO1 family of proteins and COPII coat proteins are described as two distinct membrane-bound species. Given the complexity of the system and the lack of quantitative biochemical data regarding the interactions amongst the different TANGO1 and COPII components, we assume for our modeling purposes that the TANGO1 family of proteins (TANGO1, cTAGE5 and TANGO1-Short) can be effectively described as a single species, to which we will simply refer as TANGO1. Similarly, for simplicity, we describe all the subcomponents of the COPII inner and outer coats as a single species. For our purpose, a more detailed description would only add complexity and free parameters without providing relevant biophysical insights.

Hypothesis 2: COPII subunits polymerize on the ER membrane forming a coat of specific curvature. COPII coat assembly at the ERES is relatively well characterized (Peotter et al., 2019). Polymerization of COPII subunits into spherical lattices induces curvature on the underlying membrane, typically resulting in spherical carriers of 60–90 nm (Faini et al., 2013; Aridor, 2018; Peotter et al., 2019). However, for bulky cargoes such as procollagens, complementary mechanisms are required to prevent the premature completion of the small carriers and to enable the packaging of large molecules.

Hypothesis 3: TANGO1 molecules have a propensity to assemble into rings as a result of protein-protein interactions between TANGO1-family proteins. This hypothesis is based on the observations that: (i) TANGO1 is seen in ring-like filamentous assemblies of specific size by STED nanoscopy (Raote et al., 2017; Roux, 2018); (ii) there is a direct 1:1 binding between TANGO1 and cTAGE5 CC2 domains (Saito et al., 2011); (iii) TANGO1-Short and cTAGE5 can form oligomers and oligomeric complexes together with Sec12 and TANGO1 (Maeda et al., 2016); (iv) TANGO1 and TANGO1-Short can directly homo-dimerize by their CC1 domains (Raote et al., 2018); and (v) super-resolution live lattice SIM imaging of TANGO1 in D. melanogaster larval salivary gland shows filament growth in ring formation (Reynolds et al., 2019). A physical consequence of this hypothesis is that deviations from the preferred TANGO1 ring curvature are associated to an elastic energetic cost: the ring is subject to internal strains and stresses and therefore resists bending away from its preferred curvature (Figure 1B,C).

Hypothesis 4: TANGO1 has a binding affinity to the peripheral COPII coat subunits, helping stabilize the COPII domain boundary by effectively reducing its line energy. This hypothesis is supported by the observations that proteins of the TANGO1 family bind to the COPII components Sec23, Sec16, and Sec12 (Saito et al., 2009; Ma and Goldberg, 2016; Hutchings et al., 2018; Raote et al., 2018). In physical terms, this hypothesis is equivalent to state that TANGO1 acts as a linactant by filling unsatisfied binding sites of the COPII coat edges and therefore effectively lowering its line tension (Figure 1B; Trabelsi et al., 2008, Saleem et al., 2015; Glick, 2017). Combined with hypothesis 1, another consequence of the TANGO1-COPII affinity is the mechanical coupling between the COPII coat edge and TANGO1.

 Qualitatively, our generic model for TANGO1 ring assembly can be described as a competition between two different driving forces: the resistance to bending of TANGO1 filamentous assemblies (hypothesis 3) and the binding affinity of TANGO1 proteins for peripheral COPII subunits (hypothesis 4). In the case of low TANGO1 bending resistance and/or high TANGO1-COPII binding affinity, TANGO1 will easily assemble around COPII patches, forming a ring. In the following, we refer to this process as capping of the COPII patch by TANGO1, the analogue to ‘wetting’ in soft-matter physics (Figure 1B,C). As a result of capping, the linactant effect of TANGO1 on COPII-coated ERES reduces the line energy, thus limiting further growth of the COPII lattice and the size of the TANGO1 rings (Figure 1B). In contrast, in the case of high TANGO1 rigidity and/or low TANGO1-COPII affinity (for instance, in cells expressing mutants of TANGO1 with reduced or abrogated interaction to COPII proteins), capping of COPII coats by TANGO1 rings will be energetically unfavorable, preventing any linactant effect (Figure 1B).

Here, we summarize our dynamic model of transport intermediate formation at ERES. The detailed, mathematical description of the model is presented in Appendix 1. The model extends the work of Tozzi et al., 2019 to include two membrane-bound species representing TANGO1 and COPII, respectively.

The ER membrane is represented as a continuous elastic surface on which both protein species can diffuse, interact, and induce membrane curvature. To account for these coupled chemical-mechanical processes, we follow Onsager’s variational formalism of dissipative dynamics (Arroyo and Desimone, 2009; Arroyo et al., 2018). The first step is to define the free energy of the system, which in our case is the sum of the following mechanical and chemical-related energetic contributions:

1. Membrane bending energy: the energetic cost to bend the membrane away from a preferred spontaneous curvature, known as Helfrich energy (Helfrich, 1973). Here, we assume that the spontaneous curvature is proportional to the COPII surface density, so that when the surface is saturated with COPII species (fully polymerized lattice), the membrane tends to adopt a given total curvature of $2/R_c$ (where $R_c$ is the radius of curvature of the polymerized COPII coat).

2. TANGO1 ring elastic energy: the energetic cost to bend a TANGO1 filament away from a preferred ring curvature.

3. Entropic free energy: the energetic cost for COPII and TANGO1 to mix or phase separate, here represented by a Flory-Huggins type of mixing free energy.

4. Species self-interaction energy: the energetic gain for COPII and TANGO1 to self-recruit or self-repel.

5. Interfacial energy: the energetic cost for COPII and TANGO1 domains to have an interface, physically similar to a line energy.

6. TANGO1-COPII affinity energy.

7. TANGO1 affinity for COPII domain interfaces (linactant effect), effectively reducing COPII line energy.

Next, we identify the dissipative mechanisms such as those associated with protein diffusion on the membrane surface, and with membrane in-plane shear stress. In addition, the system exchanges energy in the form of external supply of species (fixed chemical potential at the boundaries) and mechanical power. Finally, membrane inextensibility is ensured with a Lagrange multiplier field that has the physical interpretation of the membrane tension.

With these definitions (see Appendix 1), we obtain the rate of change of the system at each time point by numerically minimizing the rate of free energy release, energy dissipation, and energy exchange by the system. To facilitate the computational scheme, we formulate the problem in axisymmetric conditions. Note that this is not a limitation of the model’s physics itself but only of its numerical implementation.

The generality of Model A allows us to explore in a thermodynamically consistent manner a large range of possible dynamics of the system, including protein phase-separation, self-organization, spontaneous bud growth, and accessible stable states. However, the cost of this generality is a relative complexity that makes the physical understanding of the system more challenging. For this reason, in the next section, we describe a simplified but analytically tractable model.

To develop a theoretical model of transport intermediate formation at an ERES, we extend and adapt the approach from Saleem et al., 2015 on clathrin-coated vesicle formation to include the contributions of TANGO1 proteins in modulating COPII-dependent carrier formation. The detailed mathematical description is given in Appendix 2.

The main simplifying assumptions of this model are that (i) the system is at mechanical and thermodynamic equilibrium; (ii) the COPII coat is assumed to be much more rigid that the lipid bilayer, and therefore imposes a fixed radius of curvature leading to formation of spherical membranes of fixed radius $R_c$; and (iii) the buds are supposed axisymmetric. The second assumption allows us to limit the family of possible shapes to partial spheres connected to a flat surface, and greatly simplifies the theoretical treatment of the system. As shown below, this assumption is supported by the type of geometries obtained from Model A, which does not assume any a priori geometry.

We consider a continuous lipid membrane with a bending rigidity (${\kappa }_b$), under a certain lateral membrane tension ($\sigma$). COPII assembly into a coat is driven by a binding chemical potential (${{\mu }_c}^0$) and is opposed by the membrane bending energy. We consider that the major contribution to the binding chemical potential comes from coat-coat polymerization and not from coat-membrane binding (Saleem et al., 2015), so the binding energy ${{\mu }_c}^0$ represents the polymerization energy due to COPII-COPII binding, and will be referred to as the COPII polymerization energy in what follows. COPII units can therefore freely exchange with the rest of the membrane, which acts as the reservoir of COPII units at a fixed chemical potential. The edge of the COPII lattice has a line tension (${\lambda }_0$) due to free COPII polymerization sites. TANGO1 protein interactions are assumed to favor the formation of a linear filament of preferred curvature ($c_0$) and bending rigidity (${\kappa }_T$). Due to its affinity to COPII coat peripheral subunits, TANGO1 filaments can satisfy a fraction ($\omega$) of the edge length, effectively reducing the COPII coat line tension by $\Delta \lambda$. Finally, we also account for the mechanical work of an outward-directed force ($N$) which favors transport intermediate elongation.

The main advantage of this model is that the assumption on the carrier geometry allows us to analytically express the dependency of given quantities of interest on the model parameters, facilitating the physical interpretation of the system. For instance, writing the above mechanisms in terms of the free energy of the system, and taking a bare flat membrane as the reference state, the total free energy change of the system accounting for TANGO1 and COPII can be written as (see Appendix 2):

where $A_m$ is the membrane surface area, $A_c$ is the membrane surface area coated by COPII, $A_p$ is the surface area of the carrier projection onto the flat membrane, $\rho$ is the radius of the neck of the COPII coat, and $z_{max}$ is the height of the carrier (see Appendix 2—figure 1). For practical reasons, we define the dimensionless shape parameter $\eta =z_{max}/\left(2\mathrm{}{R}_c\right)$, so that $\eta =1$ for a single sphere of COPII spontaneous curvature. Equation 1 is plotted in Figure 1—figure supplement 1 for capped ($\omega =1$) and uncapped ($\omega =0$) carriers, highlighting the influence of COPII binding chemical potential on the energetically accessible shapes of the system.

We first make use of the generality of the computational dynamic model (Model A) to study the formation of TANGO1 assemblies at ERES. We start by investigating the ability of COPII complexes alone to generate spherical carriers. This corresponds to the control case where TANGO1 is treated as an inert species that only contributes to the entropic energy (no chemical nor mechanical interaction with COPII, the membrane, nor with itself). As shown in Figure 2A, after an initial nucleation stage (see Appendix 1), the COPII coat grows by recruiting more COPII species as it bends the membrane into a bud (Figure 2A(i–iii), Figure 2—video 1). As a metric to follow the bud growth with time, we track the height of the bud ($z_{max}$) as a function of time. As seen in Figure 2A, for inert TANGO1, the shape parameter – which we have earlier defined as $\eta =z_{max}/\left(2\mathrm{}{R}_c\right)$ – increases continuously until $\eta \simeq 1$ (Figure 2(ii)), before an abrupt event where $\eta$ jumps to 1.4 and then drops back to 1. This event corresponds to the neck closure (Appendix 1, Figure 2A(iii)), characterized by a snap-through instability (Hassinger et al., 2017).

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In contrast, functional TANGO1 drastically modifies the dynamics and outcome of bud formation (see Figure 2B and Figure 2—video 2). First, the nucleation stage is shortened, suggesting that TANGO1 can facilitate the nucleation of the COPII coat in the presence of a small force perturbation at the membrane. Second, instead of growing to the point where the neck closes, we find that the shape parameter reaches a maximum around $\eta \simeq 0.9$, before slightly decreasing to a stable height around $\eta \simeq 0.8$. As seen from the corresponding snapshots (Figure 2B(i–iii)), this dynamic evolution is accompanied by the self-assembly of a high-density TANGO1 ring-like structure around the COPII coat that stabilizes the bud into a shallow, incomplete transport intermediate. These simulation results are in agreement with experimental observations of TANGO1 rings encircling COPII components in procollagen-rich membrane patches (Raote et al., 2017; Raote et al., 2018).

To test if the formation of a stable incomplete transporter is compatible with the hypothesis that TANGO1 facilitates the biogenesis of large transport intermediate for procollagen export, we applied transient membrane tension reductions to shallow buds stabilized by TANGO1 rings. Such scenario is motivated by our previous experimental data showing that (i) ERGIC53-containing vesicles are recruited to the ERES by TANGO1 via tethering complexes (Nogueira et al., 2014; Santos et al., 2015; Raote et al., 2018), presumably enabling their fusion to the ER membrane and (ii) cells depleted of the tethering factors recruited by TANGO1 to ERES showed a reduction of ~70–80% in collagen secretion (Raote et al., 2018).

To mimic the presence of procollagen molecules inside the growing carrier preventing the total closure of the neck, we arbitrarily set a minimum neck radius threshold of 7.5 nm (see Appendix 1 for details). Starting from a stable shallow bud of shape parameter, $\eta \simeq 0.8$, at a large membrane tension of $\sigma =0.006k_{B}T/{nm}^2$ (Figure 3D(i)), we apply an ad hoc transient tension reduction as follows. First, the tension is reduced to $\sigma =0.003k_{B}T/{nm}^2$, leading to bud re-growth. Second, when the bud height reaches $\eta =1.5$ (just after the beginning of neck closure), the membrane tension is gradually set back to its original value of $\sigma =0.006k_{B}T/{nm}^2$ over a 10 ms time ramp (Figure 3A–C). We find that the carrier reaches a new equilibrium state at $\eta \simeq 2.8$, corresponding to a 'key-hole' shape of about 195 nm height (Figure 3D(ii)). Next, continuing from this new equilibrium state, we repeat the transient tension reduction protocol, this time initiating the ramp to recover high tension for $\eta =3.5$, just after a new neck is formed. This time, the bud continues growing in a pearled shape despite the recovery of high membrane tension (Figure 3D(iii,iv), and Figure 3—video 1). Interestingly, the '2-pearl' shape corresponds to a height of about 300 nm, a typical length of trimerized procollagen I molecules (McCaughey and Stephens, 2019).

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Altogether, our computational results support the scenario by which TANGO1 self-organizes around COPII coats and stabilizes shallow buds at physiological membrane tensions. This mechanism is enabled by TANGO1 affinity to COPII, TANGO1 filament bending rigidity, and the modulation of COPII coat line energy. The stability of shallow buds might facilitate procollagen recruitment and packaging by TANGO1 as well as the recruitment of ERGIC membranes to the ERES. We suggest that procollagen located at the neck of the growing carrier acts as a means to sterically prevent carrier scission. Transient membrane tension reduction, possibly mediated by fusing ERGIC membranes to the ER, allows the buds to grow from shallow to elongated pearled transport intermediates of sizes compatible with the encapsulation of procollagen molecules.

Although the generality of Model A allows us to explore ranges of possible dynamic behaviors of the TANGO1-COPII system, its computational nature and inherent complexity makes its complete and rigorous analysis prohibitively challenging. To overcome this difficulty and gain a deeper physical insight, we developed an analytically tractable model that relies on two key simplifying assumptions: (i) given the shapes obtained with Model A (Figures 2 and 3), we constrain our analysis to carrier geometries that can be approximated by a stack of spheres connected to a flat membrane patch; and (ii) we focus on the local equilibrium states of the system. The details of this equilibrium analytic model (Model B) are summarized in the Model Development section and detailed in Appendix 2.

We start by examining the effect of membrane tension on the accessible equilibrium configurations of the system as a function of the two degrees of freedom of Model B: the transport intermediate shape characterized by the shape parameter ($\eta$), and the capping fraction ($\omega$). This geometric simplification allows to write the free energy change per unit area (Equation 1) as (see Appendix 2):

where $\tilde{\mu }={{\mu }_c}^0-{2\kappa }_b{/R_c}^2+N/\left(2\pi R_c\right)$ is the effective chemical potential, which depends on the polymerization energy of the coat on the membrane, ${{\mu }_c}^0$, the bending rigidity of the membrane, ${\kappa }_b$, and the applied pulling/pushing force, $N$; $\tilde{\lambda }\left(\omega \right)=\left({\lambda }_0-\omega \Delta \lambda \right)/R_c+4\omega {{\tilde{\kappa }}_T\left(c_0{R}_c\right)}^2$, is the effective line tension of the coat; ${\tilde{\kappa }}_T={\kappa }_T/8{R_c}^3$ is the renormalized bending rigidity of the TANGO1 filament; and $n$ is the number of fully-formed buds (see Appendix 2).

We plot in Figure 4A the free energy per unit area, ${\Delta f}_c$ (Equation 2), as a function of the shape parameter $\eta$, at two membrane tensions corresponding to experimentally measured value of the ER membrane, ${\sigma }_{ER}=0.003k_{B}T/{nm}^2$, and of Golgi membranes, ${\sigma }_{GC}=0.0012k_{B}T/{nm}^2$, respectively (Upadhyaya and Sheetz, 2004). In each case, the free energy per unit area for complete capping (TANGO1 rings forming around COPII patches, $\omega =1$) and no capping (no TANGO1 rings around COPII patches, $\omega =0$) are shown. The free energy per unit area of the transport intermediate, ${\Delta f}_c$, presents multiple local minima separated by energy barriers, indicating that global and locally stable shapes can coexist for a given set of parameters. At 'high' (ER) membrane tension, the global minimum of the free energy corresponds to a capped shallow bud ($\eta \simeq 0.35$), while locally stable shapes are also found for complete spherical buds connected to a capped shallow bud ($\eta \simeq 1.5$). In contrast, at 'low' (Golgi) membrane tension, the global free energy minimum is shifted to the capped complete spherical bud shape (pearled tube), while the incomplete bud state is now a local minimum. It should be noted that due to the geometric restriction of Model B, the energy barrier separating locally stable shapes tends to infinity for fully capped coats. However, our results indicate that the uncapping of the coat reduces the energy barrier, suggesting that a capping-uncapping transition might be necessary for a bud to grow. Alternatively, growth of a pearled structure from a shallow bud could also occur through unduloid-like shapes with a fully capped neck, which are not considered by model B, but are indeed observed with model A (see e.g. Figure 3D). The parameters controlling the capping-uncapping transition are studied in a later section.

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To better visualize the effect of membrane tension on the shift in globally and locally stable shapes, we plot the optimal shape parameters, ${\eta }^{*}$, (Figure 4B) and energy barriers between stable incomplete buds (${\displaystyle {\eta }^{\ast }<1}$) and large transport intermediates (${\displaystyle {\eta }^{\ast }>1}$) (Figure 4C) as a function the membrane tension. The effect of a transient decrease in membrane tension from ${\sigma }_{ER}$ to ${\sigma }_{GC}$ on a capped stable shallow bud is represented by the green arrows in Figure 4B. Upon tension reduction, the initially globally-stable shallow bud becomes a local but not global minimum (Figure 4A,B). Concomitantly, the free energy barrier to transition from an incomplete bud to a large transport intermediate, ${∆f}_{\mathrm{1,2}}$, is reduced, and the reciprocal free energy barrier to transition back from a large to shallow bud, ${∆f}_{\mathrm{2,1}}$, is increased (Figure 4C). At low membrane tension, the growth of the metastable bud can be triggered by a small perturbation – such as thermal fluctuations or a mechanical perturbation in the system – and might be facilitated by transient uncapping (Figure 4A). By assuming Arrhenius kinetics, we can estimate an average transition time as ${\tau }_{\mathrm{1,2}}=t_0{e}^{{∆F}_{\mathrm{1,2}}/k_{B}T}$, where $t_0~1ms$ is a characteristic time scale of membrane shape dynamics (Campelo et al., 2017); and ${∆F}_{\mathrm{1,2}}\approx {∆f}_{\mathrm{1,2}}\pi {R_c}^2$ is the overall free energy barrier of the shape transition. For the conditions detailed in Figure 4C, we obtain ${∆F}_{\mathrm{1,2}}\left({\sigma }_{ER}\right)\approx 33k_{B}T$ and ${∆F}_{\mathrm{1,2}}\left({\sigma }_{GC}\right)\approx 8k_{B}T$, and hence the transition time is decreases drastically upon tension reduction, from ${\tau }_{\mathrm{1,2}}\left({\sigma }_{ER}\right)~{10}^{9}min$ to ${\tau }_{\mathrm{1,2}}\left({\sigma }_{GC}\right)~0.1min$. Note that our equilibrium model can only predict the free energy barriers but not the actual dynamics of the transitions between metastable and stable states. Then, as the membrane tension recovers the 'high' ER value, the free energy barrier to transition back to a shallow bud remains relatively large (${∆f}_{\mathrm{2,1}}\left({\sigma }_{ER}\right)>{∆f}_{\mathrm{1,2}}\left({\sigma }_{GC}\right)$), preventing the carrier to recover its shallow, globally stable shape and hence possibly being kinetically trapped in a metastable configuration. Similar estimations as above, give that ${∆F}_{\mathrm{2,1}}\left({\sigma }_{ER}\right)\approx 25k_{B}T$, and ${\tau }_{\mathrm{2,1}}\left({\sigma }_{ER}\right)~{10}^{6}min$. By increasing these energy barriers, TANGO1 capping can contribute to preventing large transport intermediates from shrinking back to shallow shapes during subsequent cycles of membrane tension reduction. Overall, the tension-mediated growth of large transport intermediates is effectively a ratchet-like mechanism, where the difference between the energy barriers guarantees an almost unidirectional evolution of the system leading to the transport intermediate growth.

The previous results were computed for a typical set of parameters of the TANGO1-COPII system (see Appendix 2—table 1). Next, we studied how robust was the stabilization of an incomplete carrier by a TANGO1 ring with respect to the COPII polymerization energy. To do so, we computed the local minima of the overall energy per unit area (Equation 2) for a range of values of the COPII polymerization energy, ${{\mu }_c}^0$, and the membrane tension, $\sigma$. The results are shown in Figure 5A in the form of a shape diagram for the optimal shape parameter, ${\eta }^{*}$. We find that at low COPII polymerization energy, the coat fails to bend the membrane into a bud, resulting in a flat membrane (${\eta }^{*}=0$). At high COPII polymerization energy, however, the system adopts a fully budded structure. Interestingly, we identify a range of intermediate COPII binding energies, (${\displaystyle 0.0285k_{B}T/n{m}^2<{\mu }_c^0<0.0315k_{B}T/n{m}^2}$) at which a stable shallow bud (${\eta }^{*}<1$) can be brought to a large pearled shape (${\eta }^{*}>1$) by membrane tension reduction. Although this range of parameters might seem relatively narrow, similar narrow ranges have been shown to regulate clathrin-coated vesicle formation (Saleem et al., 2015; Hassinger et al., 2017).

Figure 5

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As a comparison, we made use of the dynamic computational model (Model A) to compute the final carrier states and shape parameters for various values of the membrane tension, $\sigma$, and the COPII self-interaction coefficient, ${\mathrm{\chi }}_c$. The resulting shape diagrams are shown in Figure 5B for both inert and functional TANGO1. Note that the COPII self-interaction parameter ${\mathrm{\chi }}_c$ in Model A is conceptually equivalent to ${{\mu }_c}^0$ in the equilibrium Model B. In agreement with the results obtained with Model B (Figure 5A), we find that at low COPII self-interaction, the COPII coat either does not nucleate during the initial perturbation stage, or does nucleate but cannot provide enough bending energy to stabilize the membrane curvature against the membrane tension, thus resulting in a flat membrane (${\eta }^{*}=0$). At high COPII self-interaction, the nucleated coat successfully works against membrane tension to generate membrane curvature, generating a closed spherical bud. At intermediate COPII self-interaction, we find a transition region where the deformed membrane stabilizes as an open, shallow bud. Interestingly, the presence of functional TANGO1 dramatically widens the parameter space at which stable shallow buds are obtained. Importantly, the phase boundary at the transition from shallow to closed bud now spans a larger range of membrane tension and COPII self-interaction values (Figure 5B,C).

Overall, despite the different assumptions underlying the computational dynamic and equilibrium models, their qualitative agreement highlights the relevance of their common underlying physical mechanisms to TANGO1-mediated assembly of procollagen-containing transport intermediates.

As seen in Figures 4 and 5, the capping-uncapping transition of COPII coat by TANGO1 plays an important role in determining the transition between equilibrium-budded states. Therefore, we set out to examine the influence of TANGO1 model parameters on its capping ability.

We define ${\eta }^{tr}$ as the shape parameters at which the different capping-uncapping transitions occur, which corresponds to the solutions to the equation ${\Delta f}_c\left({\eta }^{tr},\omega =0\right)={\Delta f}_c\left({\eta }^{tr},\omega =1\right)$. Similarly, it is also informative to define the neck opening radius at which the capping-uncapping transition occurs, ${\rho }^{tr}$, which can be analogously found as the stationary points of the free energy (Equation 1) with respect to the capping fraction ($\partial {\Delta f}_c/\partial \omega =0$). From these equivalent definitions we obtain

for $c_0>-1/\xi$. Here $\xi =\sqrt{{\kappa }_T/\left(2\Delta \lambda \right)}\approx 22nm$ is a TANGO1-related length-scale (see parameter values in Appendix 2—table 1). These analytical results highlight how capping of COPII components by TANGO1 is promoted by large values of the TANGO1-COPII interaction, $\Delta \lambda$, and prevented by large TANGO1 filament bending rigidities, ${\kappa }_T$ (Figure 6—figure supplement 1A). Additionally, if the bud opening radius $\rho$ is equal to the radius of curvature imposed by the spherical polymerization of COPII, $R_c$, we can define a critical value of the TANGO1 linactant strength below which there is no capping ${\Delta \lambda }^{tr}=\frac{{\kappa }_T}{2}{\left(\frac{1}{R_c}-c_0\right)}^2$. This is illustrated in Figure 6A where the value of the critical COPII boundary radius ${\rho }^{tr}$ (Equation 4) is plotted as a function of TANGO1 filament bending rigidity (${\kappa }_T$) and linactant strength ($\Delta \lambda$). For $\rho >{\rho }^{tr}$, TANGO1 caps the COPII domains, whereas for $\rho <{\rho }^{tr}$, no capping nor ring formation occurs.

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We then computed the optimal shape parameter and wetting fraction for a large range of values of TANGO1 bending rigidity, ${\kappa }_T$, linactant strength, $\Delta \lambda$, and spontaneous curvature, $c_0$. These are presented in the form of shape diagrams in Figure 6B,C and Figure 6—figure supplement 2A. We find that, although none of these properties control the transition between shallow buds ($\eta <1/2$) and large intermediates ($\eta >1$), they are key in determining the capping-uncapping transition. Remarkably, we find once again that TANGO1 capping significantly expands the regions where stable shallow buds are energetically favored. In addition, the presence of a TANGO1 ring capping COPII subunits induced the formation of shapes with larger neck radii (Figure 6B,C, and Figure 6—figure supplement 1B). These results provide further support to our hypothesis that by capping COPII coats, TANGO1 stabilizes the formation of open buds preventing therefore the formation of fully budded spherical carriers. Such a stable wide neck would facilitate procollagen packing into the nascent intermediate.

Finally, we examined how the application of a force directed toward the cytosolic side of the bud can facilitate the growth of a TANGO1-capped transport intermediate. The existence of such a force, although speculative in the context of procollagen export, could have several possible origins. For instance, molecular motors linking the membrane to the cytoskeleton could potentially produce such pulling forces. Another hypothetical source of force is linked to the folding of triple-helical procollagen in the ER lumen. TANGO1 recruits procollagen by binding its chaperone HSP47 (Ishikawa et al., 2016), which preferentially interacts with triple-helical procollagen (Tasab et al., 2000). Procollagen folds into a rigid triple-helix from the C terminus to the N terminus in a zipper-like manner (Engel and Prockop, 1991). Therefore, HSP47, and correspondingly TANGO1, binding sites are generated on the procollagen as it folds. We thus hypothesize that the interface between TANGO1/HSP47 and procollagen could act as a zipper helping in the elongation of triple helical rigid procollagen. The C termini of procollagens are often associated with the ER membrane (Beck et al., 1996). It is therefore possible that a ring of TANGO1 would constrain the procollagens in a bundle with the folded C termini oriented toward the growing tip of the bud. As procollagens fold, newly generated HSP47/TANGO1-binding sites will be more distant to the growing tip of the bud. The net result of this process would be procollagen applying a force directed toward the cytosol (and exerting an opposite reaction force on the TANGO1 ring). The isotropic TANGO1 ring structure would prevent the generation of any net torque and therefore of any tilt of the procollagen molecules.

Following the same protocol as above, we computed the optimal shape parameter as a function of the intensity of an outward point force for various COPII polymerization energies and membrane tensions. As shown in Figure 7A,B, we found that having a point force shifts the capping-uncapping and transport intermediate growth transitions toward lower COPII binding energies and higher membrane tensions. Interestingly, we can obtain an analytical estimate of the transition between incomplete shallow buds and large transport intermediates. To that end, we consider the situation where the free energy of an incomplete bud of shape parameter $\eta$, is equal to the free energy of a larger intermediate with an extra pearl, given by the shape parameter $\eta +1$. Equation 2 then gives

which is independent of the structural details of the TANGO1 ring, as we have seen before (Figure 6B,C). The solution to this parametric equation is represented by the plane in light blue shown in Figure 7C. For completeness, in Figure 7C we also show the numerically found transition surfaces between flat membranes and shallow buds (dark blue surface, Figure 7C) and between large capped intermediates and full uncapped intermediates (orange surface, Figure 7C). Additionally, we rely on Equation 5 to define the critical force $N^{*}=2\pi R_c\left(\sigma -{{\mu }_c}^0+2{\kappa }_b{/R_c}^2\right)$ at which the transition to a large transport intermediate is triggered. For the experimental values reported in Appendix 2—table 1and ${{\mu }_c}^0=0.03k_{B}T/{nm}^2$, we obtain $N^{*}~0.34\frac{k_{B}T}{nm}=1.4pN$, which, for comparison, is in the same order of magnitude of the force generated by molecular actin polymerization (Footer et al., 2007), and an order of magnitude smaller than typical force required to extract a membrane tethers (Derényi et al., 2002). Taken together, our results indicate that the formation of large transport intermediates can be greatly facilitated in the presence of TANGO1 by forces of physiological ranges oriented toward the cytosol. This mechanism could be complementary to the membrane tension regulation mechanism described above.

Figure 7

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Here we delineated a feasible biophysical mechanism of how TANGO1 contributes to the formation of procollagen-containing transport intermediates at the ER. In particular, we presented two complementary models to study this process. The first model (Model A) is a dynamic model that does not impose any particular geometry for the intermediate shape. We next complemented this dynamic model with a more simplified but analytically tractable equilibrium model (Model B), which assumes that COPII polymerizes into lattices of spherical geometry. Physically, these models can be understood in terms of a competition between different driving forces. Each of these forces can either prevent or promote the elongation of procollagen-containing transport intermediates. The results obtained with these two complementary models of large transport intermediate formation reinforce the notion that TANGO1 rings serve to modulate the formation of COPII carriers for the export of bulky cargoes.

We previously showed that TANGO1 forms rings at ERES surrounding COPII components (Raote et al., 2017). We also revealed interactions that are required for TANGO1 ring formation, which are also important to control TANGO1-mediated procollagen export from the ER (Raote et al., 2018). However, it still remained unclear how TANGO1 rings organize and coordinate the budding machinery for efficient procollagen-export. Here, we propose that TANGO1 rings form at ERES by assembling as a filamentous structure made of different components of the TANGO1 family of proteins (Figure 1B,C). Evidence for the existence of linear assemblies of transmembrane proteins has indeed been reported in the context of transmembrane actin-associated (TAN) lines that couple outer nuclear membrane components to actin cables (Luxton et al., 2010). The components of the TANGO1 filament have an affinity for COPII subunits (Saito et al., 2009), leading to a capping of the periphery of a COPII lattice (Glick, 2017; Raote et al., 2018). This effectively reduces the line energy of the COPII coat and therefore we propose that, by capping COPII lattices, TANGO1 filaments can act as linactants (Figure 1B,C). In the context of HIV gp41-mediated membrane fusion, linactant compounds, such as vitamin E, lower the interfacial line tension between different membrane domains to inhibit HIV fusion (Yang et al., 2016).

When the association of TANGO1 with COPII subunits was abrogated –either by expressing TANGO1-∆PRD or by silencing the expression of SEC23A–, TANGO1 formed either smaller rings or long linear filamentous structures or planar clusters (Raote et al., 2018). In cells expressing TANGO1-∆PRD, the interaction between one of the filament components, TANGO1, and the COPII subunits is abolished. However, a TANGO1 filament can still be formed because this mutant still interacts with other TANGO1 or cTAGE5 proteins (Raote et al., 2018). In this situation, the filament proteins cTAGE5 (Saito et al., 2011; Saito et al., 2014) and TANGO1-Short (Maeda et al., 2016) can still bind the COPII component Sec23A. Because the affinity to bind to the peripheral COPII subunits is reduced, the filaments should be less line-active, and therefore less able to cap COPII components. Our theoretical results presented here predict that decreasing the TANGO1 filament linactant strength, $∆\lambda$, prevents capping of the peripheral COPII components and therefore results in a lower probability of having open carriers (Figure 6), which can lead to the observed defects in terms of ring structure and procollagen export.

Our results show that the formation of TANGO1 rings helps stabilize the COPII bud neck (Figures 2 and 5). This, we suggest, could allow for an efficient recruitment of procollagen molecules by TANGO1 to the exit site. It is not clear how long it takes for a procollagen molecule to fully fold into an export-competent triple helix after its translation into the ER, and it is possible that TANGO1 can act as a sensor of procollagen folding to couple it with the export machinery. However, in experiments where type I procollagen triple-helical stabilization and ER export were synchronized, fully folded procollagen required ~15 min to reach the early Golgi compartment (McCaughey et al., 2019). Since the half-lives of COPII components on the ER membrane are of the order of a few seconds (Forster et al., 2006), it seems likely that there is a mechanism in place to stabilize open nascent carriers, which helps to fully pack complex cargoes. Our theoretical results (Figure 2) support the proposed mechanism by which TANGO1 rings arrest the growth of standard COPII carriers and stabilize open shallow buds for the efficient packaging of procollagens (Figure 8A,B).

Figure 8

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Once a TANGO1 ring forms and stabilizes a shallow bud, the results of our model indicate that carrier expansion can proceed via at least two different, non-exclusive scenarios (Figure 8C,D): (i) local reduction of the membrane tension (Figures 3–5) and (ii) appearance of a directed force applied at the growing carrier and pointing toward the cytosol (Figure 7). Additionally, it is also possible that carrier growth occurs by an increase in the polymerization ability of COPII coats (Figure 5). It is possible that TANGO1 can directly or indirectly control each of these mechanisms (Ma and Goldberg, 2016; Raote et al., 2018). Notably, the TANGO1 ring properties, such as the linactant strength or the filament bending rigidity, are not drivers for the incomplete bud to large transport intermediate transition (Figure 6) but rather, they enhance the regions of the parameter space where carrier shapes with wide-open necks are stabilized, as opposed to fully formed vesicles or unassembled coats (Figure 5A,B).

The results of our models indicate that a transient reduction of the ER membrane tension can induce a transition from a stable shallow bud encircled by a TANGO1 ring to a large transport intermediate, that is not reabsorbed upon recovery of the tension (Figures 3 and 4). It is worth noting that, despite the different modeling and implementation choices taken in these two approaches, the values of the membrane tension at which the predictions agree only differ by a factor two. This difference could certainly be further reduced by undergoing a thorough analysis of the computational model's parameter space. However, given the computational cost of such study and the uncertainty of the experimental estimates for the ER and Golgi membrane tensions, seeking such a quantitative match between two qualitative models would only bring very limited insights. Finally, whether and how such tension regulation occurs at the level of the ERES still remains to be fully resolved. Nevertheless, we suggest a possible way by which TANGO1 could act as the membrane tension regulator at the ERES. We propose that the fusion of ERGIC53-containing membranes tethered by the TANGO1 TEER domain (Raote et al., 2018) would be the trigger for carrier growth. In particular, the ER-specific SNARE protein Syntaxin18 and the SNARE regulator SLY1, both of which are involved in membrane fusion reactions at the ER, are also required for procollagen export in a TANGO1-dependent manner (Nogueira et al., 2014). Fusion of ERGIC membranes to the sites of procollagen export would lead to a local and transient reduction of the membrane tension (Sens and Turner, 2006). In this scenario, TANGO1 would act as a regulator of membrane tension homeostasis to control procollagen export at the ERES (Figure 8C,E). We have recently proposed that ERGIC membranes fuse directly or adjacently to the growing transport intermediate to allow for membrane addition and tension release; and showed that compartment mixing can be arrested by the TANGO1 ring serving as a diffusion barrier (Raote et al., 2020). In our models, we have not included the effects of a partial diffusion barrier at the base of the growing carrier. Although these effects could lead to changes in the dynamics of carrier growth as discussed below, we do not expect any major qualitative changes in our proposed mechanisms of carrier elongation.

From the free energy profile shown in Figure 4A, we also noticed that the system needs to overcome an energy barrier to reach the globally stable state. Hence, depending on the dynamics of the system, this can be kinetically trapped into a locally stable deep bud (yellow shape in Figure 4A, right panel). Although our analytical equilibrium model cannot account for the dynamics of such transitions, we can give some estimates. First, mechanical equilibration of the membrane shape (${\tau }_{mech}$) can be theoretically estimated to be of the order of milliseconds (see e.g. Campelo et al., 2017; Sens and Rao, 2013), which is also in accordance to our own results from model A, where shape changes are found within few milliseconds (Figure 3). Second, the diffusive behavior of membrane tension has been measured at the plasma membrane of HeLa cells, with a diffusion coefficient of $D_{\sigma }=0.024{\mu m}^2/s$ (Shi et al., 2018), which gives a characteristic diffusion time of the order of ${\tau }_{\sigma }~1s$ (using characteristic length scales of ${\displaystyle \sim 100-500nm}$). Since ${\tau }_{\sigma }\gg {\tau }_{mech}$, membrane shape equilibrates much faster than the recovery of the membrane tension. Moreover, our recent findings that TANGO1 can induce a diffusion barrier at the base of the growing bud (Raote et al., 2020), suggest that the dynamics of tension equilibration can be even slower.

In parallel, we also hypothesize a situation where TANGO1 rings help pushing procollagen molecules into the growing carrier and couple this pushing force to procollagen folding, through the chaperone HSP47 (Figure 8D,E). Because HSP47 chaperone assists in folding (and hence in rigidifying) procollagen, it is tempting to speculate that the trimerization of procollagen occurs concomitantly to the its export and that the physical interaction between TANGO1 rings and procollagen/HSP47 could serve as a means to couple procollagen folding to force production. Although the existence of this pushing force is largely speculative, it could, according to our model, promote formation of a large intermediate and hence TANGO1 could act as a sensor of procollagen folding to couple it with the export machinery.

Finally, the results of our dynamic model show that the shape of the elongated carrier is a pearled tubule (Figure 3D and Figure 3—video 1). The fission of the upstream pearls is prevented in our model by a steric hindrance of the procollagen molecules packed inside the tubule (Figure 8E). Since TANGO1 is the receptor molecule that binds procollagen and packs it to facilitate its export (Saito et al., 2009), TANGO1 is indirectly required to prevent the fission of upstream necks. The results of our dynamic model indicate that during the elongation of the structure (Figure 3D and Figure 3—video 1). We previously proposed that the large transport intermediate is converted into a tunnel connecting ER to the ERGIC/early Golgi cisternae (Raote and Malhotra, 2019). If so, an appealing possibility is that, at this stage, the role of a TANGO1 ring would be to prevent lipid mixing between these transiently connected organelles by acting as a diffusion barrier (Raote et al., 2020). Finally, after procollagen delivery to the acceptor compartment, fission of the neck between the first two pearls of the tubule would not be sterically prevented anymore and therefore the tunnel could be disconnected. Future work will be required to help elucidate the precise control of the timing between these proposed fusion and fission events.

What controls organelle size in the context of intracellular trafficking? There has been a lot of work on what set the size of organisms, the size of tissues in an organism, and the size of cells in a tissue (Guertin and Sabatini, 2006; Marshall et al., 2012). However, there has been less work toward elucidating what sets the size of organelles relative to the cell. Extensive cargo transfer while trafficking bulky cargoes such as collagens leads to large amounts of membrane being transferred from one organelle to another. To maintain organelle homeostasis, loss of membrane from a compartment has to be concomitantly compensated by membrane acquisition from the biosynthetic pathway or by trafficking from other organelles; the arrival and departure of membrane at each compartment has to be efficiently balanced. How is this homeostatic balance controlled? Changes in membrane tension have been described to affect rates of exocytosis and endocytosis at the plasma membrane (Apodaca, 2002; Kosmalska et al., 2015; Wu et al., 2017) and of growth of membrane buds and tubes in general (Derényi et al., 2002; Cuvelier et al., 2005). Interestingly, a theoretical model has also established a crucial role for membrane tension in modulating the budding of clathrin-coated vesicles (Hassinger et al., 2017). Furthermore, it has been recently proposed that Atlastin-mediated maintenance of ER membrane tension is required for the efficient mobility of cargo proteins (Niu et al., 2019); and that the formation of intra-lumenal vesicles in endosomes is also regulated by membrane tension (Mercier et al., 2020). However, control of endomembrane trafficking by membrane tension still remains challenging to study experimentally and hence remains incompletely understood. We suggest that inter-organelle hubs, such as a TANGO1-scaffolded ERES-ERGIC, control the local tension homeostasis at specific membrane sub-domains and regulate the membrane flux between these organelles.

We have proposed and analyzed a theory by which TANGO1 assembles into a ring at the ERES to facilitate cargo export. The results of the theoretical biophysical models presented are compatible with our current understanding of the biology of this complex process. Importantly, our analyses are meant to help design experiments to further test our hypotheses. We speculate that a TANGO1 ring functions as a hub to collect and concentrate procollagens, to stabilize open COPII lattices, and to recruit and fuse membranes thus alleviating the ERES tension while promoting the export of procollagens from the ER. Although many of these individual events are based on experimental evidence, we lack an understanding of how collagens are collected and then percolated into the growing intermediate, whether the binding of TANGO1 to the rims of growing COPII coats acts as a linactant, the involvement of tension, the mechanism of tension sensing, and finally how these events are coordinated to cause procollagen export. It is important to be able to ask these questions because they highlight the gaps in our understanding of the process of cargo export at the ER. We propose a set of experimental procedures to test our hypotheses. The new experimental data will undoubtedly improve our understanding of how cells engage to export cargoes based on their size, volume, and the overall cellular needs.

In summary, we proposed a theoretical mechanical model that explains how TANGO1 molecules form functional rings at ERES, and how these TANGO1 rings assemble the machinery required to form a large transport intermediate commensurate with the size of procollagens. We envision that our hypotheses and the predictions of our model will guide new lines of experimental research to unravel mechanisms of COPII coats organization for the export of complex cargoes out of the ER.

In this Appendix, we derive a dynamic model of a lipid bilayer whose spontaneous curvature is dictated by the diffusion and interaction of two membrane-bound species, and specialize it to COPII and TANGO1.

The proposed model extends the work of Tozzi et al., 2019 to account for a second membrane-bound species and apply it to TANGO1-COPII complex assembly. We therefore focus our exposé on this novel aspect, and direct readers interested in the detailed underlying theory to Tozzi et al., 2019; Arroyo et al., 2018; Rahimi and Arroyo, 2012; Torres-Sánchez et al., 2019.

Our modeling approach is based on Onsager’s variational formalism of dissipative dynamics (Arroyo et al., 2018; Rahimi and Arroyo, 2012). The fundamental principle consists in describing the time evolution of the system through a minimization process of energy released, energy dissipated, and energy exchanged by the system. In other words, if $\dot{ℱ}$ is the rate of change of free energy, $𝒟$ is the total dissipation potential, and ${𝒫}_{\text{ext}}$ is the power input, the rate of change of the system can be obtained by minimizing at each time point the Rayleighian functional

In this section, we define the energetic contributions to each of these quantities.

We consider a lipid bilayer as a material surface parametrized by $\mathbf{𝐫}({\theta }^{\alpha },t)$, where (${\theta }^1,{\theta }^2$) are the Lagrangian surface coordinates, and t is the time. The state variable associated with the mechanical energy of the system is $\mathbf{𝐫}$, while the state variables associated with the chemical energy of the systems are the local area fractions of COPII and TANGO1, ${\varphi }_c$ and ${\varphi }_t$, respectively. Note that these latter are bounded and should satisfy ${\varphi }_c>0$, ${\varphi }_t>0$, and $0\le {\varphi }_c+{\varphi }_t\le 1$.

The bending energy of the membrane is described by the classical Helfrich model with spontaneous curvature (Helfrich, 1973; Campelo et al., 2014; Chabanon et al., 2017)

where $\kappa$ is the bending modulus, J is the total curvature (twice the mean curvature, H) of the membrane, and the integration is performed over the entire membrane patch, $\mathrm{\Gamma }$. Here we consider a local spontaneous curvature $C_c{\varphi }_c$ resulting from the presence of COPII complexes. We assume a linear dependence of the spontaneous curvature on COPII local coverage, with $C_c=2/R_c$ being the maximum spontaneous curvature induced by a full coverage of COPII (${\varphi }_c=1$) with preferred radius of curvature $R_c$.

Appendix 1—figure 1

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The total curvature J is a function of the state variable $\mathbf{𝐫}$. Briefly, from standard differential geometry (Deserno, 2015; do Carmo, 2016) we have that the tangent vectors at each point of the membrane are ${\mathbf{𝐠}}_{\alpha }=\partial \mathbf{𝐫}/\partial {\theta }^{\alpha }$. They define the natural basis of the tangent space, from which the covariant components of the metric tensor are obtained $g_{\alpha \beta }={\mathbf{𝐠}}_{\alpha }\cdot {\mathbf{𝐠}}_{\beta }$. Additionally, the unit normal to the surface is $\mathbf{𝐧}=({\mathbf{𝐠}}_1\times {\mathbf{𝐠}}_2)/\sqrt{g}$, where $g=det(g_{\alpha \beta })$. From these definitions, one gets the components of the second fundamental form $k_{\alpha \beta }=\mathbf{𝐧}\cdot \partial {\mathbf{𝐠}}_{\alpha }/\partial {\theta }^{\alpha }$, whose invariants are the total curvature $J=\text{tr}\mathbf{𝐤}=k_{\alpha \beta }{g}^{\alpha \beta }$, and the Gaussian curvature $K=det\mathbf{𝐤}=k_{\alpha \beta }{g}^{\beta \gamma }$. Here $g^{\beta \gamma }$ are the components of the inverse of the metric tensor, obtained from $g_{\alpha \beta }{g}^{\beta \gamma }={\delta }_{\alpha }^{\gamma }$. Note that for simplicity, we have neglected the contribution of the Gaussian curvature to the bending energy in Equation (A2).

Based on our experimental observations (Raote et al., 2018), TANGO1 proteins are assumed to favor a ring-like conformation with a specific radius of curvature ${\rho }_t$ and a preferred angle with the plane of the ring ${\theta }_t$ (see Appendix 1—figure 1). As detailed later, we will restrict the model to axisymmetric shapes. Therefore for clarity, here we directly write an axisymmetric expression of the functional for the TANGO1 ring stiffness as

where ${\kappa }_{\rho }$ and ${\kappa }_{\theta }$ are, respectively, the stiffness coefficients associated with the ring curvature $1/\rho$ and angle with the membrane $\theta$.

We consider two distinct membrane-bound species representing COPII and TANGO1 proteins. They are described by continuous surface fractions ${\varphi }_c$ and ${\varphi }_t$, respectively. We consider the entropic mixing energy of the two proteins to be represented by a Flory–Huggins type energy such as

where a_p is the molecular area of the proteins, assumed for simplicity to be identical for both proteins. The self-interaction and line energy of COPII proteins are

where ${\chi }_c$ is negative for attractive interactions. The parameter ${\mathrm{\Lambda }}_c$ ensures a length-scale associated with the interface of the COPII domain: the spatial gradient of ${\varphi }_c$ is smoother for large values of ${\mathrm{\Lambda }}_c$.

Similarly, we write the self-interaction and line energy of TANGO1 proteins as

Finally, the interactions between COPII and TANGO1 proteins are

where the first term represents the affinity between the two proteins and the second term represents the affinity of TANGO1 for the COPII coat boundary. Combining the second term of Equation (A7) with the second term of Equation (A5), one can see that TANGO1 modulates the interfacial energy of COPII with an effective parameter ${\mathrm{\Lambda }}_c+{\mathrm{\Lambda }}_{ct}{\varphi }_t$.

The total free energy is the sum of the protein and membrane contributions:

Dissipation of mechanical energy occurs through in-plane shear stress of the lipid bilayer as it dynamically deforms under the action of a Lagrangian velocity of the membrane $\mathbf{𝐕}=\partial \mathbf{𝐫}/\partial t=\mathbf{𝐯}+v_n\mathbf{𝐧}$, where $\mathbf{𝐯}$ and v_n are its tangential and normal components respectively. This membrane ‘flow’ results in the time evolution metric tensor, whose time derivative in Lagrangian setting is the rate-of-deformation tensor of the surface (Torres-Sánchez et al., 2019)

The first term, involving the membrane tangential velocity $\mathbf{𝐯}$ represents the contribution of the tangential flow to the membrane deformation. The last term, involving the normal velocity v_n, accounts for the shape change of the membrane. The rate of change of local area is $\text{tr}(\mathbf{𝐝})=\nabla \cdot \mathbf{𝐯}-v_{n}J$, which is zero for an inextensible membrane. We consider the lipid bilayer to be in a fluid phase that can be approximated by an interfacial viscous Newtonian fluid (Arroyo and Desimone, 2009). Neglecting intermonolayer friction, and assuming membrane inextensibility, the dissipation potential by in-plane shear stress takes the form

where ${\eta }_m$ is the in-plane viscosity of the lipid bilayer.

Dissipation of chemical energy occurs by protein diffusion along the membrane surface, described by the species diffusive velocities relative to the Lagrangian coordinates, w_c and w_t for COPII and TANGO1, respectively. Assuming for simplicity that the two proteins have the same molecular drag coefficient $\xi$ and surface area a_p, the chemical dissipative potential of the system is

Given that the typical length scale of our system is well below the Saffman-Delbrück length scale (∼1–10 µm), we can safely neglect dissipation arising from the friction between the membrane and the cytosol [Arroyo and Desimone, 2009]. The total dissipation potential of the system is $𝒟={𝒟}_{\text{mech}}+{𝒟}_{\text{chem}}$.

The free energy per unit area, Equation (1) depends on a number of structural, biochemical, and mechanical parameters, which we can split in three groups: (i) membrane-associated parameters, (ii) coat-associated parameters, and (iii) TANGO1-associated parameters (Appendix 2—table 1). As for the membrane-associated parameters, we have the lateral tension, $\sigma$, and the bending rigidity, ${\kappa }_b$.

Regarding membrane-associated parameters, we use the experimentally measured values of the standard membrane tension of the ER, ${\sigma }_{ER}=0.003k_{B}T/{nm}^2$ (Upadhyaya and Sheetz, 2004); and of the membrane bending rigidity, ${\kappa }_b=20k_{B}T$ (Niggemann et al., 1995; Appendix 2—table 1).

Regarding coat-associated parameters, we use the size of the standard spherical COPII vesicle, $R_c=37.5nm$ (Miller and Schekman, 2013). The line tension and the binding free energy of the polymerizing COPII coat, ${\lambda }_0$ and ${{\mu }_c}^0$, respectively, have not been, to the best of our knowledge, experimentally measured. Nevertheless, for clathrin coats, which lead to the formation of vesicles of a size comparable to the standard COPII vesicles, these values have been recently measured, yielding a value of ${\lambda }_{clathrin}=0.05pN$ for the line tension and of ${{\mu }_{clathrin}}^0=0.024\pm 0.012k_{B}T/{nm}^2$ for the binding free energy (Saleem et al., 2015). We use these values as starting estimations for COPII coats, which we will then vary within a certain range (Appendix 2—table 1). It is also informative to estimate the binding energy of COPII polymerization per molecule. To obtain this energy in units of $k_{B}T$ for the polymerization of a single COPII unit, we use the characteristic size of the Sec13-31 heterotetramers, $~30nm$ (Stagg et al., 2008; Zanetti et al., 2013), which gives a characteristic area of a COPII unit, $a_{COPII}~{\left(30nm\right)}^2$. This gives that ${{\mu }_c}^0{a}_{COPII}~22k_{B}T$, very similar to the estimated lateral interaction energy per triskelion of the clathrin coat (Saleem et al., 2015).

Finally, regarding the TANGO1-associated parameters, which are associated to different protein-protein interactions, we have the bending rigidity of the TANGO1 filament, ${\kappa }_T$; the preferred curvature of the filament, $c_0$; and the TANGO1-COPII binding energy/linactant strength of TANGO1, $\mathrm{\Delta }\lambda$. The elastic parameters of the TANGO1 filament, ${\kappa }_T$ and $c_0$, depend on the chemistry of the bonds between the different proteins within a TANGO1 filament. As we lack experimental data on the value of these parameters, we consider them within a wide range of reasonable values. Typical values of the bending rigidity of intracellular filaments formed by protein-protein interactions, such as intermediate filaments, are of the order of ${\kappa }_{IF}=2000pN\cdot {nm}^2$ (Fletcher and Mullins, 2010), which we consider as an upper limit for the rigidity of a TANGO1 filament. In addition, by taking ${\kappa }_T=0$, we can exploit our model to study the case where TANGO1 proteins do not form a cohesive filament by attractive lateral protein-protein interactions, but individual proteins can still bind COPII subunits and hence act as monomeric linactants. For our analytical analysis, we will start by taking a zero spontaneous curvature of the TANGO1 filament, $c_0=0$, and later study it within a range given by twice the radius of experimentally measured TANGO1 rings, -${\displaystyle 0.02n{m}^{-1}<c_0<0.02n{m}^{-1}}$. For the value of $\mathrm{\Delta }\lambda$, we can make an upper limit estimate, by considering that the TANGO1-COPII binding energy should be lower than the corresponding binding energy between polymerizing COPII components, that is, $\mathrm{\Delta }\lambda l_1<{{\mu }_c}^0{l}_1{l}_2$, where $l_1\approx 16nm$ and $l_2\approx 10nm$ are the lateral dimensions of the inner COPII coat components Sec23/24 (Matsuoka et al., 2001). Hence, our estimation gives that ${\displaystyle \mathrm{\Delta }\lambda <0.24k_{B}T/nm,}$ and therefore we use as the initial value for our analysis half of the upper limit value, $\mathrm{\Delta }\lambda =0.12k_{B}T/nm$, which is 10-fold larger than the bare line tension of the coat (Appendix 2—table 1).