Golgi apparatus self-organizes into the characteristic shape via postmitotic reassembly dynamics

Significance

To date, there is no experimental method for examination of detailed morphological dynamics of a cellular organelle, because of the small size. The Golgi apparatus, a membrane-bounded organelle, decomposes into small fragments during cell division, and the fragments reassemble to form the characteristic Golgi shape in daughter cells, which is also unobservable. To understand the reassembly process, we developed a three-dimensional morphological dynamics simulator based on a coarse-grained physical membrane model. Our simulation reproduced the fine Golgi shape and revealed self-organizing properties of the reassembly process. Moreover, it visualizes the whole reassembly dynamics in detail. The proposed model will bring various insights into the Golgi architecture, which can be examined both experimentally and by extended simulations.

Abstract

The Golgi apparatus is a membrane-bounded organelle with the characteristic shape of a series of stacked flat cisternae. During mitosis in mammalian cells, the Golgi apparatus is once fragmented into small vesicles and then reassembled to form the characteristic shape again in each daughter cell. The mechanism and details of the reassembly process remain elusive. Here, by the physical simulation of a coarse-grained membrane model, we reconstructed the three-dimensional morphological dynamics of the Golgi reassembly process. Considering the stability of the interphase Golgi shape, we introduce two hypothetical mechanisms—the Golgi rim stabilizer protein and curvature-dependent restriction on membrane fusion—into the general biomembrane model. We show that the characteristic Golgi shape is spontaneously organized from the assembly of vesicles by proper tuning of the two additional mechanisms, i.e., the Golgi reassembly process is modeled as self-organization. We also demonstrate that the fine Golgi shape forms via a balance of three reaction speeds: vesicle aggregation, membrane fusion, and shape relaxation. Moreover, the membrane fusion activity decreases thickness and the number of stacked cisternae of the emerging shapes.

The Golgi apparatus is a membrane-bounded cellular organelle acting as a center of the membrane trafficking system in a eukaryotic cell. Passing through the Golgi apparatus, cargo proteins and lipids transported from the endoplasmic reticulum are processed, modified, sorted, and sent to their destinations. The Golgi apparatus basically consists of a series of stacked cisternae—flat membrane sacs of discoid shape—whereas trans-most cisternae normally show tubular network structure (trans-Golgi network). This basic morphology is highly conserved among eukaryotic species, with a few exceptions (1, 2). At the same time, the progression of membrane traffic through the Golgi apparatus is a highly dynamic membrane-reorganizing process that involves formation of new cisternae on the cis face, elongation of membrane tubules for connection of adjacent cisternae, vesicle budding and fusion for retrograde transport of Golgi-resident molecules, and fragmentation of the trans-Golgi network (2). Moreover, the Golgi apparatus in mammalian cells fragments into vesicles at the start of mitosis, and these vesicles reassemble to form Golgi in each daughter cell at the end of mitosis (3–5). Although chemical and molecular characteristics of the Golgi apparatus are being intensively studied at present, the mechanism underlying the morphology and the above behavior remains unknown. The Golgi apparatus is too small to allow examination of its morphological dynamics by live imaging, thus making elucidation of these mechanisms difficult.

A typical morphological anomaly of the Golgi apparatus is fragmentation, which is observed in Golgi-associated protein deletion mutants, stress-exposed cells, and diseases (5–10). Via the fragmentation, Golgi components completely decompose and lose their morphological characteristics such as the structural anisotropy and long-range coherence. In contrast, there are few reports of moderately decomposed states (11). Therefore, this all-or-none response is a fundamental characteristic of the Golgi architecture. Mitotic fragmentation and reassembly can also be regarded as the transition between the two characteristic states.

What does the all-or-none response mean? The Golgi morphology may be resistant to various perturbations until a destructive influence appears, followed by a breakdown (nonresponse). If this is true, then conservation of Golgi morphology among eukaryotes may also be a consequence of the above robustness. Total loss of the Golgi apparatus in several organisms can be regarded as nonresponses (12). Considering the reverse process of Golgi breakdown, formation of Golgi morphology from the endomembrane system can also be an abrupt transition. The fine stacked Golgi structure requires sophisticated arrangement of various proteins and membrane faces, and the abrupt emergence means that these ordered processes occur simultaneously. Thus, the abrupt emergence, if it takes place, is characterized by the self-organization of proteins and lipid membranes. The robustness in biological systems is often a result of self-organization (13).

There are several theoretical studies on Golgi structures and functions on the premise of its self-organization (14–17). Nonetheless, the self-organizing formation process itself is based on the dynamics of endomembrane morphology and has never been examined. The aim of this study was to identify the dynamic behaviors of the Golgi apparatus by physical modeling and to determine whether the process is mediated by self-organization. In particular, we focused on the Golgi reassembly process for the following reasons: (i) The Golgi reassembly process involves building of the whole Golgi structure de novo, and (ii) an in vitro assay of Golgi reassembly from Golgi-derived vesicles has been developed successfully (18, 19). The second reason implies that the Golgi reassembly is described by physicochemical processes, and that background cellular structures are not necessary. Thus, reproducing the reassembly process by physical modeling is a promising approach. The Golgi reassembly process in vivo has two phases: formation of initial Golgi ministacks and later Golgi ribbon formation (5). The former process is believed to be independent from the microtubule cytoskeleton and to be reconstituted in vitro (20). The Golgi ministacks possess common features of the Golgi apparatus among eukaryotes. Thus, our computational study is aimed at reproducing the process of formation of Golgi ministacks.

Model

The Coarse-Grained Membrane Model.

We performed physical simulations of dynamic behaviors of endomembrane morphology using a coarse-grained membrane model. We used the dynamical triangulation membrane (DTM) model, which describes the shape of a lipid membrane by means of a triangular polygon with minimal (15 nm) and maximal (25.5 nm) bond lengths (21) (Fig. 1A). The membrane has excluded volume interaction with the thickness 9 nm. Although the thickness of a typical lipid bilayer membrane is ∼5 nm, we took membrane-integrated proteins into account and increased the thickness (22). We also introduced a restriction for the membrane area in the range $170N_f\pm 90\sqrt{N_f}{\text{nm}}^2$, where $N_f$ is the number of triangles for the membrane.

Fig. 1.

The DTM model. (A) A schematic of a lipid membrane and DTM model. The green sphere indicates a membrane-bending protein. (B) Four processes of deformation in Monte Carlo steps. (C–E) Examples of typical equilibrium shapes at $\mathrm{\Delta }\text{P}=7.0\cdot {10}^{-4}\hspace{0.17em}\text{atm}$ and different values of ${\epsilon }^{adsp}$: (C) 5.0 k_BT, (D) 1.0 k_BT, and (E) 0.0 k_BT. Protein-attaching vertices are surrounded by green color. (See SI Text.)

We also introduced proteins that attach to the vertices of the DTM. In particular, we considered proteins preferentially attaching to a bent membrane (membrane-bending proteins). The attachment of these proteins locally destabilizes the flat membrane and induces a bent shape, thus modifying the morphology of the membrane. These proteins are expected to stabilize the rim structures of a discoid membrane such as Golgi cisternae. The distributions of proteins dissolved in the cytoplasm were not modeled; instead, the concentration in the cytoplasm was set arbitrarily. Thus, proteins appear at a membrane vertex with a probability proportional to the concentration, and disappear in the course of membrane dynamics. We did not specify the excluded volume of the proteins.

We considered four sources of energy for the DTM: Helfrich free energy for membrane bending elasticity ($E^{bend}$), osmotic pressure energy ($E^{OP}$), protein adsorption energy ($E^{adsp}$), and membrane adhesion energy ($E^{adh}$),

$$E^{total}=E^{bend}+E^{OP}+E^{adsp}+E^{adh}.$$ [1]

The Helfrich free energy is given as $E^{bend}=2\kappa {\displaystyle \int }{\left\{C\left(A\right)-C_0\right\}}^{2}dA$, where $\kappa$ is bending modulus, $C\left(A\right)$ is the mean curvature at each point on the membrane, and $C_0$ is spontaneous curvature. The integration is performed across the membrane. The Helfrich free energy on the DTM is obtained according to Gompper and Kroll (21) (SI Text). The bending modulus was specified: $\kappa =20\hspace{0.17em}{k}_{B}T$ (23), where $k_B$ is the Boltzmann coefficient, and $T$ is the absolute temperature. Without the membrane-bending protein, the flat membrane is stable $\left(C_0=0\right)$. Spontaneous curvature properties of protein-attached vertices are modified locally to $C_0=0.04\left[1/\text{nm}\right]$, which is similar to the curvature of Golgi cisternae rims (24, 25).

Osmotic pressure energy is an energy caused by the osmotic pressure difference between inside and outside of Golgi cisternae. Although the precise value of osmotic pressure in the Golgi lumen has not been reported, Golgi membranes are known to have several ion transport channels (26), and a luminal pH change causes osmotic swelling of Golgi cisternae (27). Osmotic pressure is expected to be an efficient force to decrease the Golgi lumen volume and to form flat cisternae. The energy is given as $E^{OP}=\mathrm{\Delta }P\cdot V$, where $\mathrm{\Delta }P$ is the osmotic pressure difference, and $V$ is the volume enclosed by the membrane (see SI Text for calculations behind the DTM model).

We supposed that membrane-bending proteins are soluble in the cytoplasm and attach to the membrane with a certain potential energy. Thus, we introduced protein adsorption energy $E^{adsp}={\epsilon }^{adsp}\cdot N_p$, where ${\epsilon }^{adsp}$ is the adsorption energy per protein, and $N_p$ is the number of proteins attaching to the membrane. The ${\epsilon }^{adsp}$ is composed of two energy sources: (i) the physical energy of the interaction between the protein and membrane and (ii) the chemical potential of the protein in the cytoplasm. Thus, although the physical interaction energy is supposed to be negative, ${\epsilon }^{adsp}$ can be positive, depending on the concentration in the cytoplasm.

We introduced membrane adhesion energy among vertices, which represents both cisternae stacking and vesicle tethering, $E^{adh}={\displaystyle {\sum }_{i>j}^{N_v}-{\epsilon }^{adh}}\cdot H\left(L_{adh}-\left|{\overrightarrow{r}}_i-{\overrightarrow{r}}_j\right|\right)H\left(-0.5-{\overrightarrow{n}}_i\cdot {\overrightarrow{n}}_j\right)$, where $N_v$ is the total number of vertices, ${\epsilon }^{adh}$ is the energy of microadhesion between two vertices; ${\overrightarrow{r}}_i$ and ${\overrightarrow{n}}_i$ are the position vector and the normal vector of vertex $i$, respectively; $H$ s are Heaviside functions; and $L_{adh}=22.5\hspace{0.17em}\text{nm}$ is the range of the adhesion force. The first Heaviside function evaluates the distance between vertices, and the second Heaviside function determines whether the two vertices face each other.

We used the Monte Carlo method (Metropolis method) to describe the membrane deformation dynamics. We consider four processes of deformation: (i) a vertex shift, (ii) bond flip, (iii) protein adsorption/desorption, and (iv) membrane fusion (Fig. 1B). The combination of processes i and ii describes basic deformation processes of a fluid membrane with DTM (21). Process iii changes the distribution of proteins on the membrane including the total number of attached proteins. Process iv is the fusion of two facing membranes accompanying elimination of two vertices and bond reconnection (Fig. 1B, Fig. S1, and SI Text). Because the membrane fusion process is a nonequilibrium phenomenon activated by ATP (5), we did not take into account the energetic barrier for the process iv trials. One Monte Carlo step corresponds to $N_v$ steps of process i trials for randomly chosen vertices, $0.3N_v$ steps of process ii trials for randomly chosen bonds, and $0.3N_v$ steps of process iii and iv trials for randomly chosen vertices.

Fig. S1.

Schematic figure of the fusion process (same as shown in Fig. 1B) with indices of vertices.

Conditions for Stable Cisternae.

Before starting the Golgi reassembly simulations, we considered physical conditions in our membrane model for stabilization of Golgi-ministack–like (Golgi-like) structures. First, we analyzed conditions for a single cisterna formation. A cisterna of Golgi ministacks represents a thin discoid structure with rumen thickness ∼20 nm (22), regulated rim thickness ∼50 nm (24), and variable width up to about 500 nm (28). There are two remarks from the energetic viewpoint. Thinness indicates the considerable force that compresses the lumen volume, and the cisterna rim with regulated thickness independent of the width requires some molecules to stabilize the curved rim membrane with regulated curvature (29, 30). Thus, we introduced an osmotic pressure difference (to compress the lumen volume) and membrane-bending proteins, which attach to the membrane and increase the local spontaneous curvature (to stabilize the rims).

Let us first consider the scenario of high adsorption energy, where the number of attached proteins is too small to modify the membrane morphology. Applying osmotic pressure to the spherical membrane to some extent results in stomatocyte structure with a drastic decrease in lumen volume (31). This transition was observed in our simulations (Fig. 1C). Decreasing the adsorption energy, we obtained a discoid membrane with proteins densely attached at the rim (Fig. 1D). A further decrease in the adsorption energy increased the number of attached proteins, and a tubular structure membrane emerged (Fig. 1E). Thus, a single-cisterna-like discoid membrane was formed under large osmotic pressure and in a certain range of the adsorption energy (Fig. S2 and SI Text).

Fig. S2.

A flat single-cisterna-like discoid structure formation. (A and B) The dependence of ${\psi }_{last}$ on adsorption energy ${\epsilon }^{adsp}$ at (A) $\mathrm{\Delta }P=5.0\cdot {10}^{-4}\hspace{0.17em}\text{atm}$ and (B) $\mathrm{\Delta }P=7.0\cdot {10}^{-4}\hspace{0.17em}\text{atm}$ with the spherical initial condition, and (C and D) those with the discoid initial condition. (E) The phase diagram of parallel alignment on $\left({\epsilon }^{adsp},\mathrm{\Delta }P\right)$ space with a spherical initial condition. Shapes at ${\psi }_{last}>0.3$ were determined to have flat structures, and the number of flat structures was counted in five simulations for each parameter set. The number is indicated by a color (green to yellow) and the size of a filled circle; the corresponding relation is shown above the figure.

A Golgi-like structure is obtained by piling up of several discoid membranes. Obviously, the membrane adhesion is necessary to avoid scattering of the membranes. In addition, the membrane fusion should have a restriction for the following reason. In our model, two facing membranes satisfying the geometric condition spontaneously fuse with each other via a nonequilibrium process. If we apply this process to a Golgi-like structure, a number of fusion processes will occur among neighboring flat faces of membranes, and a Golgi-like structure will disappear. Thus, we concluded that a restriction for the fusion process is necessary.

The restriction should suppress the fusion among flat faces of membranes; meanwhile, it should not suppress the fusion between a cisterna rim and newly arriving vesicles. Given the difference in physical properties among these membranes, we introduced the curvature-dependent fusion restriction model; local membranes with the curvature greater than threshold $C_{th}\left[{\left(15\hspace{0.17em}\text{nm}\right)}^{-1}\right]$ can fuse. Suppose that the fusion restriction originated from the biased distribution of fusion-mediating proteins on the membrane. Our model implies that the distribution of these proteins depends on the membrane curvature.

The Setup for Vesicle Assembly Simulations.

To mimic the reassembly process of a Golgi ministack in a mammalian cell, we modeled a small cubic space with a periodic boundary (length = 900 nm) and introduced vesicles one by one at interval $\tau$, which is inversely related to the vesicle density. Vesicles were placed close to the preexisting membrane, so that the vesicles and membrane adhered to each other. The direction of the new vesicle position from the center of mass of the preexisting membrane was random. A vesicle was composed of 42 vertices and 80 triangles with diameter ∼60 nm (32). Seventy-four vesicles were added in the course of the simulations. After the last vesicle was added, $3\cdot {10}^5$ Monte Carlo steps were calculated for shape relaxation.

The set of control parameters to be tested is as follows: $\mathrm{\Delta }P$, osmotic pressure; ${\epsilon }^{adsp}$, protein adsorption energy; ${\epsilon }^{adh}$, membrane adhesion energy; $\tau$, vesicle addition interval; and $C_{th}$, curvature threshold for fusion restriction.

We hypothesized a Golgi-like structure as a stack of parallel membrane disks, and introduced the order parameter of the parallel alignment $\psi \left(t\right)$,

$$\psi \left(t\right)=\frac{4}{N_v\left(N_v-1\right)}{\displaystyle \sum _{i>j}^{N_v}}\left(\left|{\overrightarrow{n}}_i\cdot {\overrightarrow{n}}_j\right|-0.5\right).$$ [2]

If a membrane or an assembly of membranes is statistically isotropic, then $\psi \left(t\right)\approx 0$. On the other hand, if it is completely parallel, then $\psi \left(t\right)=1$. In each simulation, the maximum value of $\psi \left(t\right)$ after the last vesicle addition is referred to as ${\psi }_{max}$, whereas $\psi \left(t\right)$ at the end of simulation is referred to as ${\psi }_{last}$.

Results

Fine Golgi-Like Structures Under Fusion Restriction.

After some preliminary simulations (SI Text), we chose four physical parameters $\left(\mathrm{\Delta }P,{\epsilon }^{adsp},{\epsilon }^{adh},\tau \right)=\left(\left\{6.0\cdot {10}^{-4},7.0\cdot {10}^{-4}\right\},\left\{3.0,4.0\right\},1.0,{10}^4\right)$ and tested various curvature thresholds C_th = 0.4 to 1.0 or without the fusion restriction. Six calculations were performed for each combination of parameters. We found that ${\psi }_{max}$ depended on $C_{th}$ and took higher values (∼0.6) at C_th = 0.6 to 0.85 (Fig. 2A and Fig. S3). Several examples of time series of $\psi \left(t\right)$ are shown in Fig. 3B, and the corresponding membrane assembly structures are shown in Fig. 2 D–M. Simulations ending up with high ${\psi }_{last}$ values clearly yielded piles of flattened membrane sac structures (Fig. 2 G and H and Movies S1 and S2), i.e., fine Golgi-like structures were spontaneously generated. In such a case, a small discoid membrane formed at an early stage (Fig. 2D), and subsequently added vesicles fused to enlarge the disk or formed other layers using the former structure as scaffolding (Fig. 2 D–G and Movie S1). Without the fusion restriction, the membrane assembly finally formed spongy structures (Fig. 2J). In a very early phase, it had discoid structure with pores (Fig. 2I). Nevertheless, additional vesicles did not generate other layers but always fused to the former membrane. As a result, a large interconnected aggregate of membranes was formed (Movie S3). This situation indicates that the fusion restriction is necessary to keep the layers apart. On the other hand, a strong fusion restriction ($C_{th}=1.0$) kept many small membrane fragments at the end of simulation (Fig. 2K and Movie S4). The strong fusion restriction did not permit enough fusion events.

Fig. 2.

The curvature threshold for membrane fusion and membrane assembly structures. Six simulations were performed in each combination of parameters. (A) Dependence of maximum parallel alignment on curvature threshold. Results on four combinations of parameters are overlapping without distinction (Fig. S3 shows separate plots); “w/o” indicates without fusion restriction. The box-and-whisker plot indicates the minimum, the first quantile, the median, the third quantile, and the maximum values. (B) Several examples of time series of $\psi \left(t\right)$. (C) A scatter plot of ${\psi }_{max}$ and ${\psi }_{last}$ from all simulations. Colors indicate curvature threshold corresponding to A. (D–M) Examples of formed membrane shapes sampled from B. (D–G) Time series of the process of formation of a fine Golgi-like structure, $C_{th}=0.8$. Width of the largest cisternae in G is 453 nm. (H) The fine Golgi-like structure, $C_{th}=0.8$, with width 399 nm. (I) Early formed discoid structure and (J) final porous structure, without fusion restriction. (K) Final structure with a high threshold, $C_{th}=1.0$. (L) A temporally formed fine structure, with width 520 nm, and (M) its collapsed structure, $C_{th}=0.75$.

Fig. S3.

Dependence of maximal parallel alignment ${\psi }_{max}$ on the curvature threshold $C_{th}$, for vesicle assembly simulations. The cases where $\left({\epsilon }^{adh},\tau \right)=\left(1.0,{10}^4\right)$ and for parameter sets $\left(\mathrm{\Delta }P,{\epsilon }^{adsp}\right)=\left(7.0\cdot {10}^{-4},3.0\right),\left(7.0\cdot {10}^{-4},4.0\right),\left(6.0\cdot {10}^{-4},3.0\right),\hspace{0.17em}\text{and}\hspace{0.17em}\left(6.0\cdot {10}^{-4},4.0\right)$ are shown in A, B, C, and D, respectively. Six simulations were performed for each combination of parameters. The box-and-whisker plot indicates the minimum, the first quantile, the median, the third quantile, and the maximum values. Fig. 2A displays the overlap of these four graphs. (E) Examples of emergent membrane structures at the end of simulations. (Top) Four fine Golgi-like structures, and (Middle and Bottom) eight disrupted structures.

Fig. 3.

Characterization of fine structures. Further analysis of 50 samples in Fig. 2 at ${\psi }_{max}>0.5$. (A) Schematic representation of the anisotropic axis and size measurements used for thickness calculation. (B) Evolution of anisotropic axis; 50 time series are shown by color lines, and the first quantile, the median, and the third quantile for them are shown with the black line with error bars. (Inset) A scatter plot of $\psi \left(t\right)$ and $\lambda \left(t\right)$ at t = 20. (C) Thickness ($\mathrm{\Theta }$) and curvature threshold ($C_{th}$). (D and E) Samples of structures at t = 20. (F) The smallest and (G) largest $\mathrm{\Theta }$ structures, with width 602 and 413 nm, respectively.

Some simulations showed that $\psi \left(t\right)$ increased once and decreased after a while (Fig. 2B). The causal deformations are shown in Fig. 2 L and M (Movie S5). Although a well-pronounced flattened structure formed in midcourse of the simulation, the size of one membrane was larger than that of the others. Because the large membrane had no scaffold to maintain the flat structure, the thermal fluctuation made it fold, and $\psi \left(t\right)$ decreased. The relations between ${\psi }_{max}$ and ${\psi }_{last}$ for all simulations in Fig. 2A are shown in Fig. 2C, which indicates that simulations with ${\psi }_{max}>0.5$ kept the parallel alignment structures to their ends (${\psi }_{last}>0.4$).

Description of Processes of Fine-Structure Formation.

Given that simulations with ${\psi }_{max}>0.5$ yielded stable Golgi-like structures (50 simulations in Fig. 2A), we further examined their formation processes. Golgi-like structures have structural anisotropy characterized by an axis perpendicular to membrane disks, which we call the anisotropic axis. To describe formation of the anisotropic axis, we introduced axis movement index $\lambda \left(t\right)$,

$$\lambda \left(t\right)=\left|\overrightarrow{a}\left(t-1\right)\times \overrightarrow{a}\left(t\right)\right|,$$ [3]

where $\overrightarrow{a}\left(t\right)$ is the unit vector spanning the anisotropic axis at each time point, and the unit of time is ${10}^4$ Monte Carlo steps (Fig. 3A and SI Text); $0\le \lambda \left(t\right)\le 1$ and lower $\lambda \left(t\right)$ mean a smaller change of the axis direction. Fig. 3B shows time series of $\lambda \left(t\right)$ for these 50 simulations. After short disturbing phases, the axis movements decreased and had low values in most cases. The disturbing phase corresponds to continuing emergence and collapse of the structural anisotropy of initial small membranes under the influence of thermal fluctuations. Subsequent slow movement indicates that the axis was established stably and only slow diffusion of axis direction was observed. Fig. 3B, Inset shows the snapshots $\lambda \left(t\right)$ and $\psi \left(t\right)$ at $t=20$, which indicates that the anisotropic axis was already established, whereas the parallel alignment was still not well pronounced, in most simulations. Thus, we concluded that axis anisotropy is established at an early stage. Fig. 3 D and E depicts examples of membrane shapes at t = 20.

The rate of successful formation of fine Golgi-like structures seems to change little in the range C_th = 0.65 to 0.85 (Fig. 2A). Here, we examined other morphological dependences on $C_{th}$ in this range. We introduced another order parameter dealing with thickness,

$$\mathrm{\Theta }=\sqrt{\frac{{{\displaystyle \sum }}_i^{N_v}{\left\{\left({\overrightarrow{r}}_i-\overrightarrow{R}\right)\cdot \overrightarrow{a}\right\}}^2}{{{\displaystyle \sum }}_i^{N_v}\left[{\left|{\overrightarrow{r}}_i-\overrightarrow{R}\right|}^2-{\left\{\left({\overrightarrow{r}}_i-\overrightarrow{R}\right)\cdot \overrightarrow{a}\right\}}^2\right]}},$$ [4]

where $\overrightarrow{R}$ is the position vector of the center of mass. It was calculated for the shape taking ${\psi }_{max}$ in each simulation. The numerator is the SD of vertices projected on the anisotropic axis ($\langle l\rangle$ in Fig. 3A), and the denominator is the SD of vertices on the plane perpendicular to the anisotropic axis ($\langle w\rangle$ in Fig. 3A). Thus, if many small disks are piling up, Θ becomes large, whereas, if few large disks are piling up, Θ becomes small. We found that Θ depends on $C_{th}$ among stable structures (Fig. 3C). The structure with the smallest Θ consisted of only three large disks and some membrane fragments (Fig. 3F), whereas the structure with the largest Θ consisted of six small disks (Fig. 3G).

Dependence of the Structure on Physical Parameters.

Dependence of ${\psi }_{\text{max}}$ on osmotic pressure $\mathrm{\Delta }P$ and adsorption energy ${\epsilon }^{adsp}$ are examined below (Fig. 4 A and B), with other parameters $\left({\epsilon }^{adsp},{\epsilon }^{adh},\tau ,C_{th}\right)=\left(4.0,1.0,{10}^4,\left\{0.75,0.80,0.85\right\}\right)$ and $\left(\mathrm{\Delta }P,{\epsilon }^{adh},\tau ,C_{th}\right)=\left(7.0\cdot {10}^{-4},1.0,{10}^4,\left\{0.75,0.80,0.85\right\}\right)$, respectively. The rate of fine structures increased with $\mathrm{\Delta }P$ in the range $1.0–5.0\times {10}^{-4}\hspace{0.17em}\text{atm}$, and lumen volume also decreased in this range (Fig. 4C). The ${\epsilon }^{adsp}$ values control the number of attaching proteins (Fig. 4D), and fine structures formed in the medium range: ${\epsilon }^{adsp}$ = 2 to 5 k_BT. Smaller ${\epsilon }^{adsp}$ resulted in a tubular structure (Fig. 4G and Movie S6), whereas larger ${\epsilon }^{adsp}$ destabilized the rim structures of disks, thus forming nested membrane spheres (Fig. 4H and Movie S7). $\mathrm{\Delta }P$ and ${\epsilon }^{adsp}$ are control parameters for a single cisterna shape (Fig. S2), and shape deviations similar to those mentioned above are also observed there. Therefore, we concluded that $\mathrm{\Delta }P$ and ${\epsilon }^{adsp}$ are tuned mainly for the flat single-cisterna formation.

Fig. 4.

Dependence of structure on physical parameters. (A−F) Cases of $C_{th}=0.75,\hspace{0.17em}0.80,\hspace{0.17em}\text{and}\hspace{0.17em}0.85$ are overlapping, with colors corresponding to Fig. 2A. Five simulations were performed for each parameter set. (A) Dependence of ${\psi }_{max}$ and (C) luminal volume on $\mathrm{\Delta }P$. (B) Dependence of ${\psi }_{max}$ and (D) the number of attached proteins on ${\epsilon }^{adsp}$. (E) Dependence of ${\psi }_{max}$ on $\tau$ at $\left(\mathrm{\Delta }P,{\epsilon }^{adsp},\hspace{0.17em}\text{and}\hspace{0.17em}{\epsilon }^{adh}\right)=\left(7.0\cdot {10}^{-4},\hspace{0.17em}4.0,\hspace{0.17em}\text{and}\hspace{0.17em}1.0\right)$. (F) Dependence of ${\psi }_{max}$ on ${\epsilon }^{adh}$ at $\left(\mathrm{\Delta }P,{\epsilon }^{adsp},\hspace{0.17em}\text{and}\hspace{0.17em}\tau \right)=\left(7.0\cdot {10}^{-4},\hspace{0.17em}4.0,\hspace{0.17em}\text{and}\hspace{0.17em}{10}^4\right)$. (G) Structure at ${\epsilon }^{adsp}=1.0\hspace{0.17em}{k}_{B}T$. (H) Structure at ${\epsilon }^{adsp}=6.0\hspace{0.17em}{k}_{B}T$. (I) Structure at $\tau =100$. (J and K) Structures at ${\epsilon }^{adh}=30.0\hspace{0.17em}{k}_{B}T$. The box-and-whisker plot indicates the minimum, the first quantile, the median, the third quantile, and the maximum values.

In our simulations, vesicle addition interval $\tau$ is inversely related to vesicle density of the hypothetical physical conditions. We found that the reassembly structure depended on the interval, and a short interval (dense vesicle conditions) did not yield well-pronounced alignment structures (Fig. 4E). Under these conditions, vesicles were piled up before enough membrane shape relaxation took place. As a result, multiple core structures were formed in several spots, and the entire system failed to establish a single anisotropic axis (Fig. 4I and Movie S8).

An increase in adhesion energy was also found to destroy fine layered structures (Fig. 4F). With greater adhesion energy, a cisterna tended to grow along with other cisternae. Thus, when there was a difference in size between two neighboring cisternae, a larger one wrapped around the other one to increase the adhesion area, and nested structures were formed (Fig. 4 J and K and Movies S9 and S10).

Thickness Control Mechanism.

Here we further explore how $C_{th}$ controls the thickness of emerging shapes $\mathrm{\Theta }$ depicted in Fig. 3C. Higher $C_{th}$ means stronger restriction for fusion and decreases the frequency of fusion events. Does the frequency of fusion events directly affect $\mathrm{\Theta }$? $\mathrm{\Theta }$ and the number of piled membrane disks correlate, and a larger number of disks means a lesser frequency of fusion events. The frequency of fusion events is controlled directly by modifying the Monte Carlo step. We redefined the number of membrane fusion trials (process iv) in one Monte Carlo step as $0.3N_v\alpha$, where $\alpha$ is the fusion activity, and we examined the dependence of membrane structures on $\alpha$. We chose other parameters as in Fig. 2. Thus, $\alpha =1.0$ corresponds to Fig. 2. We confirmed that the number of fusion events increased with $\alpha$ (Fig. S4A).

Fig. S4.

(A) The number of fusion events depending on fusion activity $\mathrm{\alpha }$, with physical parameters $\left(\mathrm{\Delta }P,{\epsilon }^{adsp},{\epsilon }^{adh},\tau \right)=\left(\left\{6.0\cdot {10}^{-4},7.0\cdot {10}^{-4}\right\},\left\{3.0,4.0\right\},1.0,{10}^4\right)$ and without fusion restriction. Three simulations were performed for each combination of parameters. (B and C) Dependence of ${\psi }_{max}$ on fusion activity (B) at $C_{th}=0.6$ and (C) without fusion restriction, with physical parameters $\left(\mathrm{\Delta }P,{\epsilon }^{adsp},{\epsilon }^{adh},\tau \right)=\left(\left\{6.0\cdot {10}^{-4},7.0\cdot {10}^{-4}\right\},\left\{3.0,4.0\right\},1.0,{10}^4\right)$. Three and five simulations were performed for each combination of parameters for B and C, respectively. The box-and-whisker plot indicates the minimum, the first quantile, the median, the third quantile, and the maximum values.

First, unexpectedly, the decrease in $\alpha$ increased the rate of fine structures at $C_{th}=0.4$ in the range α = 0.01 to 1.0 (Fig. 5A). This increase in the rate was not observed in the cases of $C_{th}=0.6$ (Fig. S4B). Without the fusion restriction, a decrease in $\alpha$ never resulted in flat structures (Fig. S4C). Thus, the decrease in $\alpha$ compensated the insufficient fusion restriction with a smaller threshold. Then, we calculated $\mathrm{\Theta }$ for successfully formed flat structures (${\psi }_{max}>0.5$) at $C_{th}=0.6$ (29 simulations), and found that $\mathrm{\Theta }$ inversely correlated with a logarithm of $\alpha$ (Fig. 5B). Thus, we can conclude that the decrease in fusion activity increased $\mathrm{\Theta }$.

Fig. 5.

Dependence of structure on fusion activity $\alpha$. Five simulations were performed for each parameter set. Colors correspond to Fig. 2A. (A) Dependence of ${\psi }_{max}$ on $\alpha$ at $C_{th}=0.4$. The box-and-whisker plot indicates the minimum, the first quantile, the median, the third quantile, and the maximum values. (B) Dependence of thickness on $\alpha$ at $C_{th}=0.6$.

Discussion

We performed a vesicle assembly simulation using a coarse-grained physical membrane model, aiming to reproduce Golgi reassembly in the postmitotic phase. We successfully reconstructed the self-organization process of formation of Golgi-like structures. On the basis of the preliminary assumption of stability of an interphase Golgi structure, we introduced rim stabilizer proteins and the assumption of the curvature-dependent fusion restriction. These two stabilization factors were sufficient as additional assumptions to generate the self-organization process (Fig. 2). The self-organization was resistant to changes in physical parameters (Fig. 4 A, B, and E). These data imply that the physical stability of a static Golgi structure also ensures the stable Golgi reassembly process.

Although our simulation setup implied isotropic aggregation of vesicles, a Golgi-like structure is characterized by anisotropy in morphology. The emergence of anisotropy is a necessary step for the formation process. Our simulations revealed that anisotropic axes were established in small membranes at early stages (Fig. 3B). They acted as scaffolds for building of the whole Golgi-like structures.

We did not characterize all of the properties of collapse processes, because some time scales of structural decay events seemed to be much longer than our simulations. In general, longer simulation promotes fusion events and causes the shape to deviate from a well-pronounced layered structure in our setup. In vivo, the balance of fusion and fission is expected to maintain the stable shape, and a subsequent process such as Golgi ribbon formation will take place. Accordingly, we did not analyze the slower collapse events any further. Within our simulation time scales, finer structures above a certain level (${\psi }_{max}\ge 0.5$) seemed to attain stable states (Fig. 2C). Our simulations produced various abnormal Golgi morphologies, which, in vivo, may also be maintained through the balance of fusion and fission processes (33) or may be fragmented.

If the vesicle addition frequency was high (Fig. 4I) or the frequency of fusion events was low because of the strong fusion restriction (Fig. 2K) or low fusion activity, a number of vesicles aggregated before clear-cut flat core structures formed. This aggregation blocked formation of a fine core structure in some cases and yielded multiple core structures in other cases. This finding indicates that the balance of three process speeds—the vesicle aggregation speed, membrane fusion speed, and shape relaxation speed—is necessary for stable formation of Golgi-like structures.

Misfire structures, in which one membrane wrapped another and formed a spherical shape, were frequently observed under various conditions (Figs. 2M and 4 H and J). Nested spheres of a membrane were also observed (Fig. 4K); similar structures were reported in a cell-free reassembly experiment (18). They are similar to the stomatocyte, which has a stable shape of a homogeneous membrane with high osmotic pressure (Fig. 1C) (31). Various driving forces were present that promote formation of stomatocyte-like structures, depending on the conditions. Small adsorption energy resulted in fewer membrane-bending molecules (Fig. 4D); this situation became similar to the homogeneous membrane. Large adhesion energy made the membrane grow along with another membrane and caused them to wrap each other (Fig. 4 J and K). Occasionally, one cisterna grew more than others (Fig. 2M). The larger cisternae were destabilized by thermal fluctuations and wrapped other membranes.

Stronger membrane adhesion blocked flat growth of cisternae and prevented formation of fine Golgi-like strictures (Fig. 4 J and K). Under the successful formation conditions, tethering energy between single vertices was comparable to the thermal fluctuation $\left({\epsilon }^{adh}=1k_{B}T\right)$. Weaker adhesion (${\epsilon }^{adh}<1k_{B}T$) does not anchor vesicles against the fluctuation. The adhesion strength should be within a fine range where membrane deformation is small but the anchoring works.

Membrane-bending proteins were introduced for stabilizing cisterna rims. Although there are no detailed reports about rim stabilizer proteins, a number of candidates can be listed. One candidate is coatomer protein complex (COP I) (34). It assembles on the Golgi membrane and deforms it to form vesicles. In geometric representation, the vesicle formation itself is the process causing the substantial curvature of the membrane. Thus, it is natural for COP I to behave as a rim stabilizer. GTPase Arf1, which mediates COP I coatomer assembly, also induces membrane curvature (35). Another candidate is phospholipase A2, which associates with the cytoplasmic side of the Golgi membrane and generates positive curvature directly or by producing lysophosphatidylcholine (36).

Furthermore, we do not have clear evidence of biased distributions of fusion-mediating proteins such as SNAREs. Nonetheless, the distribution of occurrences of the fusion events is doubtlessly biased, and the fusion between flat faces of cisternae is not known. Thus, there must be a certain molecular mechanism to generate this bias. Some membrane-tethering proteins associated with Golgi (GMAP210, giantin, golgin-84, and CASP) are known to localize at cisterna rims (5). GMAP210 contains an Amphipathic Lipid Packing Sensor motif that interacts with a curved membrane. Rab proteins, binding partners of tethering proteins, are also reported to bind a curved membrane (37). These tethering proteins may promote fusion of a membrane among curved membranes.

There is no consensus on how lumen volume is compressed and a flat cisterna is formed. There are two candidates of forces: osmotic pressure and membrane adhesion. Osmotic pressure is a logical choice of a force to compress lumen volume. Golgi cisternae are known to have several ion transport channels (26), and a luminal pH change causes osmotic swelling of Golgi cisternae (27) although the precise pH value is difficult to determine. Our model stipulates osmotic pressure as the dominant force for lumen compression. Membrane adhesion is a force increasing areas of two mutually adhering membranes. A pile of flattened membranes has lower adhesion energy than does a row of round membranes. Thus, membrane adhesion basically acts as the flattening force. Lee et al. (11) reported that a knockdown of adhesion proteins caused a swelling of Golgi cisternae. In contrast, our simulations indicated that stronger adhesion affecting the membrane morphology prevents the formation of fine Golgi-like strictures (Fig. 4F). A knockdown of adhesion proteins also changed the efficiency of membrane trafficking, which may have changed the physical properties of Golgi, as indicated in ref. 11.

To perform a more realistic simulation, for example, one can readily introduce variations of adhesion and fusion processes on the basis of molecular differences among cis-, medial, and trans-membranes in our simulations. Identification of rim stabilizer proteins or the molecular mechanism of fusion restriction can also be incorporated. These model expansion phenomena will prolong the shape relaxation time; consequently, another improvement in the simulation algorithm will be needed.

Energy Calculations in the DTM Model

The Helfrich free energy of the DTM model was obtained according to Gompper and Kroll (21)

$$E^{bend}=\frac{\kappa }{2}{\displaystyle \sum _i}{\sigma }_i{\left[\frac{1}{{\sigma }_i}{\displaystyle \sum _{j\left(i\right)}}\frac{{\sigma }_{ij}}{l_{ij}}\left({\overrightarrow{r}}_i-{\overrightarrow{r}}_j\right)-C_0{\overrightarrow{n}}_i\right]}^2,$$ [S1]

where $i$ is the vertex index, $j\left(i\right)$ is the vertex index that is connected with vertex $i$, ${\overrightarrow{r}}_i$ is the position vector, ${\overrightarrow{n}}_i$ is the normal vector, $l_{ij}$ is the distance between vertices $i$ and $j$, ${\sigma }_{ij}$ is the distance of the corresponding edge in the dual lattice, and ${\sigma }_i$ is the area enclosed by the dual lattice.

Lumen volume calculation for DTM model is as follows:

$$V=\frac{1}{6}{\displaystyle \sum _{\left(i,j,k\right)}^{N_f}}\det \left({\overrightarrow{r}}_i,{\overrightarrow{r}}_j,{\overrightarrow{r}}_k\right),$$ [S2]

where $\left(i,j,k\right)$ is a set of vertices constituting a triangle, $N_f$ is the number of triangles, and $\det \left({\overrightarrow{r}}_i,{\overrightarrow{r}}_j,{\overrightarrow{r}}_k\right)$ is the determinant of the matrix consisting of these position vectors.

In our model, the unit of membrane adhesion energy ${\epsilon }^{adsp}$ is defined for a pair of vertices. The mean number of connections for a vertex was six, and the mean area of a triangle mesh was ∼ $170\hspace{0.17em}{\text{nm}}^2$ in our simulations. Thus, the membrane adhesion energy per area was $0.002\cdot {\epsilon }^{adsp}\left[k_{B}T/{\text{nm}}^2\right]$.

Details of Deformation Processes

We considered four processes of deformation: (i) a vertex shift, (ii) bond flip, (iii) protein adsorption or desorption, and (iv) membrane fusion (Fig. 1B). In process i, vertex is shifted randomly according to the spherical uniform distribution with the radius of 1.875 nm. If a vertex with a bond connection less than 2 or greater than 12 emerges in process ii or iv, it is rejected. In process iii, if a chosen vertex contains a protein or has no protein, protein desorption or adsorption is tested, respectively.

Because the fusion process iv is the largest deformation process including change in membrane topology, and the results strongly depend on it, here we explain the detailed procedure of this process (Fig. S1). First, a vertex ($v_a$) is randomly chosen. If the vertex ($v_{\alpha }$), which is the closest to the chosen vertex and not connected to it, is not closer than the maximal bond length (25.5 nm), the trial of the fusion process is rejected. If not rejected, a pair of vertices ($v_b,v_{\beta }$), which is connected to $v_a$ and $v_{\alpha }$, respectively, and is the closest among possible combinations, is selected. A third pair of vertices ($v_c,v_{\gamma }$), which forms triangles with $\left(v_a,v_b,v_c\right)$ and $\left(v_{\alpha },v_{\beta },v_{\gamma }\right)$, respectively, and is closer than the other candidates ($v_c\text{'},v_{\gamma }\text{'}$), is also selected. It is checked that all pairs ($v_a,v_{\alpha }$), ($v_b,v_{\beta }$), and ($v_c,v_{\gamma }$) have no common connecting vertex. Then two new vertices ($v_x,v_y$) are placed at the midpoints of ($v_a,v_{\alpha }$) and ($v_b,v_{\beta }$), respectively, and the reconnections of bonds are practiced as shown in Fig. S1. Then, it is checked that all bond lengths are between the minimal and maximal bond length, and that no bonds and triangles violate the volume exclusion.

There can be various fusion procedures. Another procedure was proposed by Gompper and Kroll (38), in which the number of triangles increases at the fusion process. On the other hand, the process proposed here conserves the number of triangles so that the membrane area does not change. Changes in other details can modify membrane dynamics locally. For example, the fusion efficiency change may be altered. However, we think these changes do not affect the results of global morphology, because a large change in fusion efficiency had little effect on morphology, as shown in Fig. 5 and Fig. S4.

Single-Cisterna Simulations

We performed Monte Carlo simulations for a single connected membrane with 1,002 vertices, in which three energies ($E^{bend}$, $E^{OP}$, and $E^{adsp}$) were taken into account and three deformation processes—(i) a vertex shift, (ii) bond flip, and (iii) protein adsorption or desorption—were examined. Shapes after $1.5\cdot {10}^6$ Monte Carlo steps were evaluated. We prepared two initial conditions: spherical membrane without attachment of membrane-bending proteins (spherical initial condition) and a single-cisternae-like discoid membrane shown in Fig. 1D (discoid initial condition).

For Fig. S2, we chose parameter ranges $\mathrm{\Delta }P=\sim 1.0-7.0\cdot {10}^{-4}\hspace{0.17em}\left[\text{atm}\right]$ and ${\epsilon }^{adsp}=\sim 0.0-6.0\hspace{0.17em}\left[k_{B}T\right]$, performed five simulations for each parameter set, and calculated the order parameter of parallel alignment at the ends of simulations ${\psi }_{last}$ [Eq. 2]. Fig. S2 A and B shows the dependence of ${\psi }_{last}$ on adsorption energy ${\epsilon }^{adsp}$ at $\mathrm{\Delta }P=5.0\cdot {10}^{-4}\hspace{0.17em}\left[\text{atm}\right]$ and $\mathrm{\Delta }P=7.0\cdot {10}^{-4}\hspace{0.17em}\left[\text{atm}\right]$, respectively, with the spherical initial condition. Fig. S2 C and D shows those with the discoid initial condition. High ${\psi }_{last}$ indicates flat structures like a single-cisternae-like discoid membrane, low ${\psi }_{last}$ at smaller ${\epsilon }^{adsp}$ indicates tubular structures, and low ${\psi }_{last}$ at larger ${\epsilon }^{adsp}$ indicates stomatocyte structures. Fig. S2E shows the phase diagram of parallel alignment on $\left({\epsilon }^{adsp},\mathrm{\Delta }P\right)$ space with the spherical initial condition. We determined that shapes at ${\psi }_{last}>0.3$ are flat and counted how many simulation runs yielded flat structures in each parameter set. The number is represented by the size and color of a filled circle. These results indicate that flat discoid structures are observed in wide ranges of parameters. Moreover, comparing Fig. S2A to Fig. S2C and comparing Fig. S2B to Fig. S2D, we found that in the ${\epsilon }^{adsp}$ ranges, the tubular, flat and stomatocyte structures emerged, depending on initial conditions. Thus, these structures are metastable states in wide ranges of parameters.

Membrane Structure Dependence on Fusion Restriction

Before we examined membrane structure dependence on the fusion restriction (Fig. 3), we roughly surveyed the five-dimensional parameter space (ΔP, ${\epsilon }^{adsp}$, ${\epsilon }^{adh}$, τ, and C_th) and estimated values for four parameters (ΔP, ${\epsilon }^{adsp}$, ${\epsilon }^{adh}$, and τ) to give fine Golgi-like structures frequently. Two parameters, ${\epsilon }^{adh}$ and $\tau$, were determined to be $1.0$ and ${10}^4$. For the two other parameters, $\mathrm{\Delta }P$ and ${\epsilon }^{adsp}$, we estimated that structure formation was robust in the ranges $\mathrm{\Delta }P=\sim 6.0\cdot {10}^{-4}$ to $\sim 7.0\cdot {10}^{-4}$ and ${\epsilon }^{adsp}$ = ∼3.0 to ∼4.0, thus both boundary values were adopted for calculations. Fig. 2A shows the results obtained from these four combinations of parameters (ΔP, ${\epsilon }^{adsp}$) = (6.0⋅10⁻⁴, 3.0), (6.0⋅10⁻⁴, 4.0), (7.0⋅10⁻⁴, 3.0), and (7.0⋅10⁻⁴, 4.0) together, and separate versions of it are shown in Fig. S3 A−D. These separate graphs indicate the structure robustness in these parameter ranges; fine Golgi-like structures are frequently formed in similar regions of C_th = ∼0.7 to ∼0.85, although disrupted structures are also occasionally formed. Fig. S3E shows examples of emergent membrane structures at the end of simulations; four fine Golgi-like structures (Top) and eight disrupted structures (Middle and Bottom) are shown. The examples also indicate that similar fine and disrupted structures emerged from different parameter values in these ranges.

Calculation of the Anisotropic Axis

If membranes are parallelly aligned, the set of normal vectors for vertices is distributed widely in the direction perpendicular to the aligned membranes. Thus, we calculated the gyration tensor for the set of normal vectors for vertices, with components

$$g_{mn}={\displaystyle \sum _i^{N_v}}{n}_i^m{n}_i^n,$$ [S3]

where the normal vector of vertex $i$ is given as ${\overrightarrow{n}}_i=\left(n_i^1,n_i^2,n_i^3\right)$. The eigenvector corresponding to the maximal eigenvalue of the tensor indicates the direction of normal vector expansion, i.e., the direction of the anisotropic axis. Accordingly, we calculated the eigenvector by the Jacobi algorithm, and the normalized vector is referred to as $\overrightarrow{a}\left(t\right)$.