Golgi apparatus self-organizes into the characteristicshape via postmitotic reassembly dynamicsMasashi Tachikawaa,b,1 and Atsushi Mochizukia,b,c,d

aTheoretical Biology Laboratory, RIKEN, Wako 351-0198, Japan; bInterdisciplinary Theoretical Science Research Group, RIKEN, Wako 351-0198, Japan;cCore Research for Evolutionary Science and Technology, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan; and dInterdisciplinaryTheoretical and Mathematical Science Program, RIKEN, Wako 351-0198, Japan

Edited by Jennifer Lippincott-Schwartz, Howard Hughes Medical Institute, Ashburn, VA, and approved April 12, 2017 (received for review November 30, 2016)

The Golgi apparatus is a membrane-bounded organelle with thecharacteristic shape of a series of stacked flat cisternae. Duringmitosis in mammalian cells, the Golgi apparatus is once fragmentedinto small vesicles and then reassembled to form the characteristicshape again in each daughter cell. The mechanism and details of thereassembly process remain elusive. Here, by the physical simulationof a coarse-grained membrane model, we reconstructed the three-dimensional morphological dynamics of the Golgi reassembly pro-cess. Considering the stability of the interphase Golgi shape, weintroduce two hypothetical mechanisms—the Golgi rim stabilizerprotein and curvature-dependent restriction on membrane fusion—into the general biomembrane model. We show that the character-istic Golgi shape is spontaneously organized from the assembly ofvesicles by proper tuning of the two additional mechanisms, i.e., theGolgi reassembly process is modeled as self-organization. We alsodemonstrate that the fine Golgi shape forms via a balance of threereaction speeds: vesicle aggregation, membrane fusion, and shaperelaxation. Moreover, the membrane fusion activity decreases thick-ness and the number of stacked cisternae of the emerging shapes.

Golgi apparatus | self-organization | physical biology modeling |computer simulation

The Golgi apparatus is a membrane-bounded cellular organelleacting as a center of the membrane trafficking system in a

eukaryotic cell. Passing through the Golgi apparatus, cargo proteinsand lipids transported from the endoplasmic reticulum are pro-cessed, modified, sorted, and sent to their destinations. The Golgiapparatus basically consists of a series of stacked cisternae—flatmembrane sacs of discoid shape—whereas trans-most cisternaenormally show tubular network structure (trans-Golgi network).This basic morphology is highly conserved among eukaryotic spe-cies, with a few exceptions (1, 2). At the same time, the progressionof membrane traffic through the Golgi apparatus is a highly dy-namic membrane-reorganizing process that involves formation ofnew cisternae on the cis face, elongation of membrane tubules forconnection of adjacent cisternae, vesicle budding and fusion forretrograde transport of Golgi-resident molecules, and fragmenta-tion of the trans-Golgi network (2). Moreover, the Golgi apparatusin mammalian cells fragments into vesicles at the start of mitosis,and these vesicles reassemble to form Golgi in each daughter cell atthe end of mitosis (3–5). Although chemical and molecular char-acteristics of the Golgi apparatus are being intensively studied atpresent, the mechanism underlying the morphology and the abovebehavior remains unknown. The Golgi apparatus is too small toallow 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 de-letion mutants, stress-exposed cells, and diseases (5–10). Via thefragmentation, Golgi components completely decompose and losetheir morphological characteristics such as the structural anisot-ropy and long-range coherence. In contrast, there are few reportsof moderately decomposed states (11). Therefore, this all-or-noneresponse is a fundamental characteristic of the Golgi architecture.

Mitotic fragmentation and reassembly can also be regarded as thetransition between the two characteristic states.What does the all-or-none response mean? The Golgi mor-

phology may be resistant to various perturbations until a destructiveinfluence appears, followed by a breakdown (nonresponse). If thisis true, then conservation of Golgi morphology among eukaryotesmay also be a consequence of the above robustness. Total loss ofthe Golgi apparatus in several organisms can be regarded as non-responses (12). Considering the reverse process of Golgi break-down, formation of Golgi morphology from the endomembranesystem can also be an abrupt transition. The fine stacked Golgistructure requires sophisticated arrangement of various proteinsand membrane faces, and the abrupt emergence means that theseordered processes occur simultaneously. Thus, the abrupt emer-gence, if it takes place, is characterized by the self-organization ofproteins and lipid membranes. The robustness in biological systemsis 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). None-theless, the self-organizing formation process itself is based on thedynamics of endomembrane morphology and has never been ex-amined. The aim of this study was to identify the dynamic be-haviors of the Golgi apparatus by physical modeling and todetermine whether the process is mediated by self-organization. Inparticular, we focused on the Golgi reassembly process for thefollowing reasons: (i) The Golgi reassembly process involvesbuilding of the whole Golgi structure de novo, and (ii) an in vitroassay of Golgi reassembly from Golgi-derived vesicles has beendeveloped successfully (18, 19). The second reason implies that

Significance

To date, there is no experimental method for examination ofdetailed morphological dynamics of a cellular organelle, becauseof the small size. The Golgi apparatus, a membrane-bounded or-ganelle, decomposes into small fragments during cell division, andthe fragments reassemble to form the characteristic Golgi shapein daughter cells, which is also unobservable. To understand thereassembly process, we developed a three-dimensional morpho-logical dynamics simulator based on a coarse-grained physicalmembrane model. Our simulation reproduced the fine Golgi shapeand 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 ar-chitecture, which can be examined both experimentally and byextended simulations.

Author contributions: M.T. and A.M. designed research; M.T. performed research;M.T. analyzed data; and M.T. and A.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: mtach@riken.jp.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1619264114/-/DCSupplemental.

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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 apromising approach. The Golgi reassembly process in vivo has twophases: formation of initial Golgi ministacks and later Golgi ribbonformation (5). The former process is believed to be independentfrom the microtubule cytoskeleton and to be reconstituted in vitro(20). The Golgi ministacks possess common features of the Golgiapparatus among eukaryotes. Thus, our computational study isaimed at reproducing the process of formation of Golgi ministacks.

ModelThe Coarse-Grained Membrane Model. We performed physical simulations ofdynamic behaviors of endomembrane morphology using a coarse-grainedmembrane model. We used the dynamical triangulation membrane (DTM)model, which describes the shape of a lipidmembrane bymeans of a triangularpolygon 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 tookmembrane-integrated proteins into account and increased the thickness (22).We also introduced a restriction for the membrane area in the range170Nf ± 90

ffiffiffiffiffiffiNf

pnm2, where Nf is the number of triangles for the membrane.

We also introduced proteins that attach to the vertices of the DTM. In par-ticular, we considered proteins preferentially attaching to a bent membrane(membrane-bending proteins). The attachment of these proteins locally desta-bilizes the flat membrane and induces a bent shape, thus modifying the mor-phology of the membrane. These proteins are expected to stabilize the rimstructures of a discoid membrane such as Golgi cisternae. The distributions ofproteins dissolved in the cytoplasmwerenotmodeled; instead, the concentrationin the cytoplasmwas set arbitrarily. Thus, proteins appear at a membrane vertexwith a probability proportional to the concentration, and disappear in the courseofmembrane dynamics.We did not specify the excluded volume of the proteins.

We considered four sources of energy for the DTM: Helfrich free energy formembrane bending elasticity (Ebend), osmotic pressure energy (EOP), proteinadsorption energy (Eadsp), and membrane adhesion energy (Eadh),

Etotal = Ebend + EOP + Eadsp + Eadh. [1]

The Helfrich free energy is given as Ebend = 2κR fCðAÞ−C0g2dA, where κ is

bending modulus, CðAÞ is the mean curvature at each point on the membrane,and C0 is spontaneous curvature. The integration is performed across the mem-brane. The Helfrich free energy on the DTM is obtained according to Gompperand Kroll (21) (SI Text). The bending modulus was specified: κ= 20  kBT (23), wherekB is the Boltzmann coefficient, and T is the absolute temperature. Without themembrane-bending protein, the flat membrane is stable ðC0 = 0Þ. Spontaneouscurvature properties of protein-attached vertices are modified locally toC0 = 0.04½1=nm�, which is similar to the curvature of Golgi cisternae rims (24, 25).

Osmotic pressure energy is an energy caused by the osmotic pressuredifference between inside and outside of Golgi cisternae. Although theprecise value of osmotic pressure in the Golgi lumen has not been reported,

Golgi membranes are known to have several ion transport channels (26), anda luminal pH change causes osmotic swelling of Golgi cisternae (27). Osmoticpressure is expected to be an efficient force to decrease the Golgi lumenvolume and to form flat cisternae. The energy is given as EOP =ΔP ·V, where ΔPis the osmotic pressure difference, and V is the volume enclosed by the mem-brane (see SI Text for calculations behind the DTM model).

We supposed that membrane-bending proteins are soluble in the cyto-plasm and attach to the membrane with a certain potential energy. Thus, weintroduced protein adsorption energy Eadsp = «adsp ·Np, where «adsp is theadsorption energy per protein, and Np is the number of proteins attachingto the membrane. The «adsp is composed of two energy sources: (i) thephysical energy of the interaction between the protein and membrane and(ii) the chemical potential of the protein in the cytoplasm. Thus, althoughthe physical interaction energy is supposed to be negative, «adsp can bepositive, depending on the concentration in the cytoplasm.

We introduced membrane adhesion energy among vertices,which represents both cisternae stacking and vesicle tethering,Eadh =

PNvi>j−«

adh ·HðLadh −��~ri −~rj

��ÞHð−0.5−~ni ·~njÞ, where Nv is the totalnumber of vertices, «adh is the energy of microadhesion between two ver-tices; ~ri and ~ni are the position vector and the normal vector of vertex i,respectively; H s are Heaviside functions; and Ladh = 22.5  nm is the range ofthe adhesion force. The first Heaviside function evaluates the distancebetween vertices, and the second Heaviside function determines whetherthe two vertices face each other.

We used the Monte Carlo method (Metropolis method) to describe themembrane deformation dynamics. We consider four processes of de-formation: (i) a vertex shift, (ii) bond flip, (iii) protein adsorption/desorption,and (iv) membrane fusion (Fig. 1B). The combination of processes i and iidescribes basic deformation processes of a fluid membrane with DTM (21).Process iii changes the distribution of proteins on the membrane includingthe total number of attached proteins. Process iv is the fusion of two facingmembranes accompanying elimination of two vertices and bond reconnec-tion (Fig. 1B, Fig. S1, and SI Text). Because the membrane fusion process is anonequilibrium phenomenon activated by ATP (5), we did not take intoaccount the energetic barrier for the process iv trials. One Monte Carlo stepcorresponds to Nv steps of process i trials for randomly chosen vertices, 0.3Nv

steps of process ii trials for randomly chosen bonds, and 0.3Nv steps ofprocess iii and iv trials for randomly chosen vertices.

Conditions for Stable Cisternae. Before starting the Golgi reassembly simula-tions, we considered physical conditions in our membrane model for stabili-zation of Golgi-ministack–like (Golgi-like) structures. First, we analyzedconditions for a single cisterna formation. A cisterna of Golgi ministacks rep-resents a thin discoid structure with rumen thickness ∼20 nm (22), regulatedrim thickness ∼50 nm (24), and variable width up to about 500 nm (28). Thereare two remarks from the energetic viewpoint. Thinness indicates the con-siderable force that compresses the lumen volume, and the cisterna rim withregulated thickness independent of the width requires some molecules tostabilize the curved rim membrane with regulated curvature (29, 30). Thus, weintroduced an osmotic pressure difference (to compress the lumen volume)and membrane-bending proteins, which attach to the membrane and increasethe local spontaneous curvature (to stabilize the rims).

Let us first consider the scenario of high adsorption energy, where thenumber of attached proteins is too small to modify the membrane mor-phology. Applying osmotic pressure to the spherical membrane to someextent results in stomatocyte structure with a drastic decrease in lumenvolume (31). This transition was observed in our simulations (Fig. 1C). De-creasing the adsorption energy, we obtained a discoid membrane withproteins densely attached at the rim (Fig. 1D). A further decrease in theadsorption energy increased the number of attached proteins, and a tubularstructure membrane emerged (Fig. 1E). Thus, a single-cisterna-like discoidmembrane was formed under large osmotic pressure and in a certain rangeof the adsorption energy (Fig. S2 and SI Text).

A Golgi-like structure is obtained by piling up of several discoid membranes.Obviously, the membrane adhesion is necessary to avoid scattering of themembranes. In addition, the membrane fusion should have a restriction for thefollowing reason. In ourmodel, two facingmembranes satisfying the geometriccondition spontaneously fuse with each other via a nonequilibrium process. Ifwe apply this process to a Golgi-like structure, a number of fusion processes willoccur among neighboring flat faces of membranes, and a Golgi-like structurewill disappear. Thus, we concluded that a restriction for the fusion processis necessary.

The restriction should suppress the fusion among flat faces of membranes;meanwhile, it should not suppress the fusion between a cisterna rim andnewly arriving vesicles. Given the difference in physical properties among

Fig. 1. The DTM model. (A) A schematic of a lipid membrane and DTMmodel. The green sphere indicates a membrane-bending protein. (B) Fourprocesses of deformation in Monte Carlo steps. (C–E) Examples of typicalequilibrium shapes at ΔP= 7.0 ·10−4   atm and different values of «adsp: (C) 5.0kBT, (D) 1.0 kBT, and (E) 0.0 kBT. Protein-attaching vertices are surrounded bygreen color. (See SI Text.)

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these membranes, we introduced the curvature-dependent fusion restrictionmodel; local membranes with the curvature greater than thresholdCth½ð15  nmÞ−1� can fuse. Suppose that the fusion restriction originated fromthe biased distribution of fusion-mediating proteins on the membrane. Ourmodel implies that the distribution of these proteins depends on themembrane curvature.

The Setup for Vesicle Assembly Simulations. Tomimic the reassembly process ofa Golgi ministack in a mammalian cell, we modeled a small cubic space with aperiodic boundary (length = 900 nm) and introduced vesicles one by one atinterval τ, which is inversely related to the vesicle density. Vesicles were placedclose to the preexisting membrane, so that the vesicles and membrane ad-hered to each other. The direction of the new vesicle position from the centerof mass of the preexisting membrane was random. A vesicle was composed of42 vertices and 80 triangles with diameter ∼60 nm (32). Seventy-four vesicleswere added in the course of the simulations. After the last vesicle was added,3 ·105 Monte Carlo steps were calculated for shape relaxation.

The set of control parameters to be tested is as follows: ΔP, osmoticpressure; «adsp, protein adsorption energy; «adh, membrane adhesion energy;τ, vesicle addition interval; and Cth, curvature threshold for fusion restriction.

We hypothesized a Golgi-like structure as a stack of parallel membranedisks, and introduced the order parameter of the parallel alignment ψðtÞ,

ψðtÞ= 4NvðNv − 1Þ

XNv

i>j

���~ni ·~nj

��− 0.5�. [2]

If a membrane or an assembly of membranes is statistically isotropic, thenψðtÞ≈ 0. On the other hand, if it is completely parallel, then ψðtÞ= 1. In eachsimulation, the maximum value of ψðtÞ after the last vesicle addition is re-ferred to as ψmax, whereas ψðtÞ at the end of simulation is referred to as ψ last.

ResultsFine Golgi-Like Structures Under Fusion Restriction. After some pre-liminary simulations (SI Text), we chose four physical parametersðΔP, «adsp, «adh, τÞ= ðf6.0 · 10−4, 7.0 · 10−4g, f3.0, 4.0g, 1.0, 104Þ andtested various curvature thresholds Cth = 0.4 to 1.0 or without thefusion restriction. Six calculations were performed for each combina-tion of parameters. We found that ψmax depended on Cth and tookhigher values (∼0.6) at Cth = 0.6 to 0.85 (Fig. 2A and Fig. S3). Severalexamples of time series of ψðtÞ are shown in Fig. 3B, and the corre-sponding membrane assembly structures are shown in Fig. 2 D–M.Simulations ending up with high ψ last values clearly yielded piles offlattened membrane sac structures (Fig. 2G andH andMovies S1 andS2), i.e., fine Golgi-like structures were spontaneously generated. Insuch a case, a small discoid membrane formed at an early stage (Fig.

2D), and subsequently added vesicles fused to enlarge the disk orformed other layers using the former structure as scaffolding (Fig. 2D–G and Movie S1). Without the fusion restriction, the membrane as-sembly finally formed spongy structures (Fig. 2J). In a very early phase,it had discoid structure with pores (Fig. 2I). Nevertheless, additionalvesicles did not generate other layers but always fused to the formermembrane. As a result, a large interconnected aggregate of mem-branes was formed (Movie S3). This situation indicates that the fusionrestriction is necessary to keep the layers apart. On the other hand, astrong fusion restriction (Cth = 1.0) kept many small membrane frag-ments at the end of simulation (Fig. 2K and Movie S4). The strongfusion restriction did not permit enough fusion events.Some simulations showed that ψðtÞ increased once and de-

creased after a while (Fig. 2B). The causal deformations are shownin Fig. 2 L and M (Movie S5). Although a well-pronounced flat-tened structure formed in midcourse of the simulation, the size ofone membrane was larger than that of the others. Because thelarge membrane had no scaffold to maintain the flat structure, thethermal fluctuation made it fold, and ψðtÞ decreased. The relationsbetween ψmax and ψ last for all simulations in Fig. 2A are shown inFig. 2C, which indicates that simulations with ψmax > 0.5 kept theparallel alignment structures to their ends (ψ last > 0.4).

Description of Processes of Fine-Structure Formation. Given thatsimulations with ψmax > 0.5 yielded stable Golgi-like structures(50 simulations in Fig. 2A), we further examined their formationprocesses. Golgi-like structures have structural anisotropy char-acterized by an axis perpendicular to membrane disks, which wecall the anisotropic axis. To describe formation of the anisotropicaxis, we introduced axis movement index λðtÞ,

λðtÞ= j~aðt− 1Þ×~aðtÞj, [3]

where~aðtÞ is the unit vector spanning the anisotropic axis at eachtime point, and the unit of time is 104 Monte Carlo steps (Fig. 3Aand SI Text); 0≤ λðtÞ≤ 1 and lower λðtÞ mean a smaller change ofthe axis direction. Fig. 3B shows time series of λðtÞ for these50 simulations. After short disturbing phases, the axis movementsdecreased and had low values in most cases. The disturbing phasecorresponds to continuing emergence and collapse of the struc-tural anisotropy of initial small membranes under the influence ofthermal fluctuations. Subsequent slow movement indicates thatthe axis was established stably and only slow diffusion of axis

Fig. 2. The curvature threshold for membrane fusion and membrane assembly structures. Six simulations were performed in each combination of param-eters. (A) Dependence of maximum parallel alignment on curvature threshold. Results on four combinations of parameters are overlapping without dis-tinction (Fig. S3 shows separate plots); “w/o” indicates without fusion restriction. The box-and-whisker plot indicates the minimum, the first quantile, themedian, the third quantile, and the maximum values. (B) Several examples of time series of ψðtÞ. (C) A scatter plot of ψmax and ψ last from all simulations. Colorsindicate curvature threshold corresponding to A. (D–M) Examples of formed membrane shapes sampled from B. (D–G) Time series of the process of formationof a fine Golgi-like structure, Cth = 0.8. Width of the largest cisternae in G is 453 nm. (H) The fine Golgi-like structure, Cth = 0.8, with width 399 nm. (I) Earlyformed discoid structure and (J) final porous structure, without fusion restriction. (K) Final structure with a high threshold, Cth = 1.0. (L) A temporally formedfine structure, with width 520 nm, and (M) its collapsed structure, Cth =0.75.

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direction was observed. Fig. 3B, Inset shows the snapshots λðtÞ andψðtÞ at t= 20, which indicates that the anisotropic axis was alreadyestablished, whereas the parallel alignment was still not well pro-nounced, in most simulations. Thus, we concluded that axis an-isotropy is established at an early stage. Fig. 3 D and E depictsexamples of membrane shapes at t = 20.The rate of successful formation of fine Golgi-like structures

seems to change little in the range Cth = 0.65 to 0.85 (Fig. 2A). Here,we examined other morphological dependences on Cth in this range.We introduced another order parameter dealing with thickness,

Θ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNvi

n�~ri −~R

�·~a

o2

PNvi

h��~ri −~R��2 − n�

~ri −~R�·~a

o2i

vuuut , [4]

where ~R is the position vector of the center of mass. It was calcu-lated for the shape taking ψmax in each simulation. The numeratoris the SD of vertices projected on the anisotropic axis (hli in Fig.3A), and the denominator is the SD of vertices on the planeperpendicular to the anisotropic axis (hwi in Fig. 3A). Thus, ifmany small disks are piling up, Θ becomes large, whereas, if fewlarge disks are piling up, Θ becomes small. We found that Θdepends on Cth among stable structures (Fig. 3C). The structurewith the smallest Θ consisted of only three large disks and somemembrane fragments (Fig. 3F), whereas the structure with thelargest Θ consisted of six small disks (Fig. 3G).

Dependence of the Structure on Physical Parameters. Depen-dence of ψmax on osmotic pressure ΔP and adsorption energy«adsp are examined below (Fig. 4 A and B), with other pa-rameters ð«adsp, «adh, τ,CthÞ= ð4.0, 1.0, 104, f0.75, 0.80, 0.85gÞ andðΔP, «adh, τ,CthÞ= ð7.0 · 10−4, 1.0, 104, f0.75, 0.80, 0.85gÞ, respec-tively. The rate of fine structures increased with ΔP in the range1.0–5.0× 10−4   atm, and lumen volume also decreased in this range(Fig. 4C). The «adsp values control the number of attaching proteins(Fig. 4D), and fine structures formed in the medium range: «adsp =2 to 5 kBT. Smaller «adsp resulted in a tubular structure (Fig. 4G andMovie S6), whereas larger «adsp destabilized the rim structures ofdisks, thus forming nested membrane spheres (Fig. 4H and MovieS7). ΔP and «adsp are control parameters for a single cisterna shape(Fig. S2), and shape deviations similar to those mentioned above arealso observed there. Therefore, we concluded that ΔP and «adsp aretuned mainly for the flat single-cisterna formation.

In our simulations, vesicle addition interval τ is inversely relatedto vesicle density of the hypothetical physical conditions. We foundthat the reassembly structure depended on the interval, and a shortinterval (dense vesicle conditions) did not yield well-pronouncedalignment structures (Fig. 4E). Under these conditions, vesicleswere piled up before enough membrane shape relaxation tookplace. As a result, multiple core structures were formed in severalspots, and the entire system failed to establish a single anisotropicaxis (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 cis-terna tended to grow along with other cisternae. Thus, when therewas a difference in size between two neighboring cisternae, a largerone wrapped around the other one to increase the adhesion area,and nested structures were formed (Fig. 4 J and K and Movies S9and S10).

Thickness Control Mechanism. Here we further explore how Cthcontrols the thickness of emerging shapes Θ depicted in Fig. 3C.Higher Cth means stronger restriction for fusion and decreasesthe frequency of fusion events. Does the frequency of fusionevents directly affect Θ? Θ and the number of piled membranedisks correlate, and a larger number of disks means a lesserfrequency of fusion events. The frequency of fusion events iscontrolled directly by modifying the Monte Carlo step. Weredefined the number of membrane fusion trials (process iv)in one Monte Carlo step as 0.3Nvα, where α is the fusion ac-tivity, and we examined the dependence of membrane struc-tures on α. We chose other parameters as in Fig. 2. Thus, α= 1.0

Fig. 4. Dependence of structure on physical parameters. (A−F) Cases ofCth = 0.75,   0.80,   and  0.85 are overlapping, with colors corresponding to Fig.2A. Five simulations were performed for each parameter set. (A) De-pendence of ψmax and (C) luminal volume on ΔP. (B) Dependence of ψmax

and (D) the number of attached proteins on «adsp. (E) Dependence of ψmax onτ at ðΔP, «adsp,   and  «adhÞ= ð7.0 · 10−4,   4.0,   and  1.0Þ. (F) Dependence of ψmax

on «adh at ðΔP, «adsp,   and  τÞ= ð7.0 · 10−4,   4.0,   and  104Þ. (G) Structure at«adsp = 1.0  kBT. (H) Structure at «adsp = 6.0  kBT. (I) Structure at τ= 100. (J and K)Structures at «adh = 30.0  kBT. The box-and-whisker plot indicates the minimum,the first quantile, the median, the third quantile, and the maximum values.

Fig. 3. Characterization of fine structures. Further analysis of 50 samples inFig. 2 at ψmax > 0.5. (A) Schematic representation of the anisotropic axis andsize measurements used for thickness calculation. (B) Evolution of anisotropicaxis; 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 ψðtÞ and λðtÞ at t = 20. (C) Thickness (Θ) and curvaturethreshold (Cth). (D and E) Samples of structures at t = 20. (F) The smallest and(G) largest Θ structures, with width 602 and 413 nm, respectively.

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corresponds to Fig. 2. We confirmed that the number of fusionevents increased with α (Fig. S4A).First, unexpectedly, the decrease in α increased the rate of fine

structures at Cth = 0.4 in the range α = 0.01 to 1.0 (Fig. 5A). Thisincrease in the rate was not observed in the cases of Cth = 0.6 (Fig.S4B). Without the fusion restriction, a decrease in α never resultedin flat structures (Fig. S4C). Thus, the decrease in α compen-sated the insufficient fusion restriction with a smaller threshold.Then, we calculated Θ for successfully formed flat structures(ψmax > 0.5) at Cth = 0.6 (29 simulations), and found that Θ in-versely correlated with a logarithm of α (Fig. 5B). Thus, we canconclude that the decrease in fusion activity increased Θ.

DiscussionWe performed a vesicle assembly simulation using a coarse-grainedphysical membrane model, aiming to reproduce Golgi reassemblyin the postmitotic phase. We successfully reconstructed the self-organization process of formation of Golgi-like structures. On thebasis of the preliminary assumption of stability of an interphaseGolgi structure, we introduced rim stabilizer proteins and theassumption of the curvature-dependent fusion restriction. Thesetwo stabilization factors were sufficient as additional assumptionsto generate the self-organization process (Fig. 2). The self-organization was resistant to changes in physical parameters (Fig. 4A, B, and E). These data imply that the physical stability of a staticGolgi 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 inmorphology. The emergence of anisotropy is a necessary step for theformation process. Our simulations revealed that anisotropic axeswere established in small membranes at early stages (Fig. 3B). Theyacted as scaffolds for building of the whole Golgi-like structures.We did not characterize all of the properties of collapse pro-

cesses, because some time scales of structural decay events seemedto be much longer than our simulations. In general, longersimulation promotes fusion events and causes the shape todeviate from a well-pronounced layered structure in our setup.In vivo, the balance of fusion and fission is expected to maintainthe stable shape, and a subsequent process such as Golgi ribbonformation will take place. Accordingly, we did not analyze theslower collapse events any further. Within our simulation timescales, finer structures above a certain level (ψmax ≥ 0.5) seemedto attain stable states (Fig. 2C). Our simulations producedvarious abnormal Golgi morphologies, which, in vivo, may alsobe maintained through the balance of fusion and fission pro-cesses (33) or may be fragmented.If the vesicle addition frequency was high (Fig. 4I) or the fre-

quency of fusion events was low because of the strong fusion re-striction (Fig. 2K) or low fusion activity, a number of vesicles

aggregated before clear-cut flat core structures formed. Thisaggregation blocked formation of a fine core structure in somecases and yielded multiple core structures in other cases. Thisfinding indicates that the balance of three process speeds—thevesicle aggregation speed, membrane fusion speed, and shaperelaxation 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 undervarious conditions (Figs. 2M and 4 H and J). Nested spheres ofa membrane were also observed (Fig. 4K); similar structureswere reported in a cell-free reassembly experiment (18). Theyare similar to the stomatocyte, which has a stable shape of ahomogeneous membrane with high osmotic pressure (Fig. 1C)(31). Various driving forces were present that promote for-mation of stomatocyte-like structures, depending on the con-ditions. Small adsorption energy resulted in fewer membrane-bending molecules (Fig. 4D); this situation became similar tothe homogeneous membrane. Large adhesion energy made themembrane grow along with another membrane and causedthem to wrap each other (Fig. 4 J and K). Occasionally, onecisterna grew more than others (Fig. 2M). The larger cisternaewere destabilized by thermal fluctuations and wrapped othermembranes.Stronger membrane adhesion blocked flat growth of cisternae

and prevented formation of fine Golgi-like strictures (Fig. 4 J andK). Under the successful formation conditions, tethering energybetween single vertices was comparable to the thermal fluctuationð«adh = 1kBTÞ. Weaker adhesion («adh < 1kBT) does not anchorvesicles against the fluctuation. The adhesion strength should bewithin a fine range where membrane deformation is small but theanchoring works.Membrane-bending proteins were introduced for stabilizing

cisterna rims. Although there are no detailed reports about rimstabilizer proteins, a number of candidates can be listed. Onecandidate is coatomer protein complex (COP I) (34). It assembleson the Golgi membrane and deforms it to form vesicles. In geo-metric representation, the vesicle formation itself is the processcausing the substantial curvature of the membrane. Thus, it isnatural for COP I to behave as a rim stabilizer. GTPase Arf1,which mediates COP I coatomer assembly, also induces mem-brane curvature (35). Another candidate is phospholipase A2,which associates with the cytoplasmic side of the Golgi membraneand generates positive curvature directly or by producing lyso-phosphatidylcholine (36).Furthermore, we do not have clear evidence of biased distri-

butions of fusion-mediating proteins such as SNAREs. Nonethe-less, the distribution of occurrences of the fusion events isdoubtlessly biased, and the fusion between flat faces of cisternae isnot known. Thus, there must be a certain molecular mechanism togenerate this bias. Some membrane-tethering proteins associ-ated with Golgi (GMAP210, giantin, golgin-84, and CASP) areknown to localize at cisterna rims (5). GMAP210 contains anAmphipathic Lipid Packing Sensor motif that interacts with acurved membrane. Rab proteins, binding partners of tetheringproteins, are also reported to bind a curved membrane (37). Thesetethering proteins may promote fusion of a membrane amongcurved membranes.There is no consensus on how lumen volume is compressed

and a flat cisterna is formed. There are two candidates offorces: osmotic pressure and membrane adhesion. Osmoticpressure is a logical choice of a force to compress lumen vol-ume. Golgi cisternae are known to have several ion transportchannels (26), and a luminal pH change causes osmotic swellingof Golgi cisternae (27) although the precise pH value is difficultto determine. Our model stipulates osmotic pressure as thedominant force for lumen compression. Membrane adhe-sion is a force increasing areas of two mutually adhering

Fig. 5. Dependence of structure on fusion activity α. Five simulations wereperformed for each parameter set. Colors correspond to Fig. 2A. (A) Dependenceof ψmax on α at Cth = 0.4. The box-and-whisker plot indicates the minimum, thefirst quantile, the median, the third quantile, and the maximum values. (B) De-pendence of thickness on α at Cth = 0.6.

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membranes. A pile of flattened membranes has lower adhesionenergy than does a row of round membranes. Thus, membraneadhesion basically acts as the flattening force. Lee et al. (11)reported that a knockdown of adhesion proteins caused a swellingof Golgi cisternae. In contrast, our simulations indicated thatstronger adhesion affecting the membrane morphology preventsthe formation of fine Golgi-like strictures (Fig. 4F). A knockdownof adhesion proteins also changed the efficiency of membranetrafficking, which may have changed the physical properties ofGolgi, 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 stabilizerproteins or the molecular mechanism of fusion restriction can alsobe incorporated. These model expansion phenomena will prolongthe shape relaxation time; consequently, another improvement inthe simulation algorithm will be needed.

ACKNOWLEDGMENTS. We thank S. Suetsugu, T. Miura, N. Mizushima,N. Ishihara, P. Sens, M. S. Turner, M. Rao, and our lab members for helpfuldiscussions. We are grateful to T. Kohyama for technical advice on theDTM model. This work was supported by Grant-in-Aid 16H01454 fromJapan Society for the Promotion of Science (to M.T.).

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