Formation and Stability of Lipid Membrane Nanotubes Amir Houshang Bahrami† and Gerhard Hummer*,†,‡ †
Department of Theoretical Biophysics, Max Planck Institute of Biophysics, Max-von-Laue Str. 3, 60438 Frankfurt am Main, Germany Institute for Biophysics, Goethe University Frankfurt, 60438 Frankfurt am Main, Germany
‡
S Supporting Information *
ABSTRACT: Lipid membrane nanotubes are abundant in living cells, even though tubules are energetically less stable than sheetlike structures. According to membrane elastic theory, the tubular endoplasmic reticulum (ER), with its high area-to-volume ratio, appears to be particularly unstable. We explore how tubular membrane structures can nevertheless be induced and why they persist. In Monte Carlo simulations of a fluid−elastic membrane model subject to thermal fluctuations and without constraints on symmetry, we find that a steady increase in the area-to-volume ratio readily induces tubular structures. In simulations mimicking the ER wrapped around the cell nucleus, tubules emerge naturally as the membrane area increases. Once formed, a high energy barrier separates tubules from the thermodynamically favored sheet-like membrane structures. Remarkably, this barrier persists even at large area-to-volume ratios, protecting tubules against shape transformations despite enormous driving forces toward sheet-like structures. Molecular dynamics simulations of a molecular membrane model confirm the metastability of tubular structures. Volume reduction by osmotic regulation and membrane area growth by lipid production and by fusion of small vesicles emerge as powerful factors in the induction and stabilization of tubular membrane structures. KEYWORDS: lipid nanotube, nanotube stability, internal volume, endoplasmic reticulum, Monte Carlo simulations, MARTINI molecular dynamics
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ubular membrane structures are abundant in living cells1 yet challenging to form synthetically.2−5 This raises the question how biology creates and maintains membrane nanotubes. Biological membrane tubules form prominently in the endoplasmic reticulum (ER), Golgi, and certain mitochondria6−9 and are used for intercellular transport and communication.10−12 Recent super-resolution microscopy of the peripheral ER even revealed seemingly sheet-like structures as, actually, densely packed tubular arrays.13 However, their high area-to-volume ratio places these tubular structures deep within a regime where they are inherently unstable against transitions to sheet-like structures.14−16 To learn how membrane tubules can nevertheless be induced, and why they persist, we perform extensive Monte Carlo (MC) simulations of a fluid−elastic membrane model,17 complemented by molecular dynamics simulations of a molecular membrane model. The generation and stabilization of membrane nanotubes, in particular, of the ER, have attracted much attention.18−20 Membrane-shaping reticulon (Rtn) and DP1/Yop1p proteins have been found capable of generating and maintaining ER tubules.19−25 Deletion of Rtns and DP1/Yop1p in budding yeast indeed eliminated tubular structures from mother cells; however, bloated ER tubes still formed in daughter cells.26 Whether these proteins generate, sense, or stabilize membrane curvature in eukaryotes is thus actively investigated.26,27 © 2017 American Chemical Society
Membrane nanotubes can also be induced by molecular motors attached to the cytoskeleton.28,29 Although the cytoskeleton does not appear to play a significant role in stabilizing the ER tubes,21,30 it is associated with tube generation and the highly dynamic reorganization of the ER tubular network in animal cells.31 ER tubes have been found to retract or transition to cisternae upon microtubule depolymerization despite the presence of Rtns.32−34 Physical factors also play a key role in membrane tubulation. Tight packing of the excess membrane generated by lipid synthesis at the smooth ER,26,35 the regulation of the internal volume of the continuous ER by ion pumps and by restricting water flow,18,22 volume control of two-phase vesicles by osmotic deflation,36 area control,37 and membrane spontaneous curvature4,38−41 have all been associated with membrane tube formation. Coarse-grained simulations highlight the role of bound proteins42−44 and nanoparticles.45−47 Osmotically induced cell volume changes have been shown to inhibit ERto-Golgi transport in mammalian cells, where the extensive tubules observed under hypo-osmotic conditions disappeared under hyper-osmotic condition.48 Sheet-to-tubule transformaReceived: August 4, 2017 Accepted: September 5, 2017 Published: September 5, 2017 9558
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Figure 1. Stable vesicle shapes as a function of reduced volume, including sheet-like axisymmetric stomatocyte vesicles with reduced volumes (a) v = 0.3 and (b) v = 0.5; axisymmetric oblate discocytes with reduced volumes (c) v = 0.595 and (d) v = 0.647; and tubule-like axisymmetric prolate vesicles with reduced volumes (e) v = 0.7, (f) v = 0.8, and (g) v = 0.9. (h) Rescaled elastic energy E/8πκ of three shape branches as a function of reduced volume. The elastic energies represent the minimum value obtained in 10 SA simulations.
intermediate non-axisymmetric shapes, which in turn transforms to a stomatocyte for 0 < v ≲ 0.592 via a continuous transition through intermediate axisymmetric shapes.14,15 The prolate ellipsoid is an elongated tubular vesicle, and the oblate ellipsoid resembles a sheet-like flat vesicle with two opposing bilayers. Hereafter, we simply refer to these two locally stable conformations as the tube and the sheet. Figure 1 shows stable prolate, oblate, and stomatocyte vesicles from simulated annealing (SA) simulations corresponding to reduced volumes 0.2 ⩽ v ⩽ 0.9. Vertical lines indicate the two transitions between the three branches at reduced volumes v ≈ 0.652 and 0.592. Also shown is the elastic energy E of the three branches as a function of the reduced volume, rescaled by the energy 8πκ of a spherical vesicle. To explore the (meta)stability and shape fluctuations of these vesicle shapes, we performed MC simulations at ambient temperature in an NVT ensemble with a constant number of vertices and an effectively constant vesicle volume, for a membrane bending rigidity of κ/kBT = 30 (where kB is Boltzmann’s constant and T is the absolute temperature). We controlled the reduced volume by applying a harmonic restraint potential (Supplementary Text). In an alternative formulation,16 vesicle volume is controlled by osmotic pressure, which amounts to a Legendre transformation from an NVT-like ensemble in our case to an NΠT ensemble. In both cases, the vesicle area is preserved to account for the negligible role of the surface tension in shape fluctuations and membrane deformations compared to the bending rigidity.16 Starting from an initially spherical vesicle with v = 1, we gradually decreased the reduced volume to v = 0.12. As expected, the vesicle transformed into a prolate tube at reduced volumes 0.652 < v < 1, where the tube is the globally stable shape. Remarkably, upon further volume reduction, the tubular shape is preserved, even in the regime v < 0.652, where the sheet and stomatocyte are the stable shapes. A typical sequence of vesicle conformations is shown in Figure 2 and Supplementary Movie SM1. The lowest reduced volume of v = 0.12 is far below both the relative volumes of the transitions from prolate to oblate (v ≈ 0.652) and from oblate to stomatocyte (v ≈ 0.592), which implies that significant energy barriers separate these distinct shapes. As the reduced volume dropped, the undulating shapes of the membrane tubules persisted. Shape fluctuations of long tubular membranes occur on two distinct time scales. Whereas local membrane deformations depend on vertex diffusion and proceed on a rather fast time scale, tube undulations proceed on much slower time scales. A related separation of time scales occurs in long polymers, where the local dynamics at the monomer scale and the global dynamics of the polymer shape
tions of the ER have been observed during cell division, and linked with volume constraints forcing the ER into denser configurations.49 Vesicle fusion also increases the area-tovolume ratio50 and may thus be a factor in the formation of ER tubes.27,51 On a larger scale, the morphogenesis of endothelial tubes appears to be driven by intra- and intercellular vacuole fusion.52,53 Very recent in vivo experiments revealed a crucial role of vesicle fusion in maintaining the tubular ER structures.54 Membrane-shaping proteins likely act in concert with these physical factors to create and maintain tubular membrane structures. Here, we characterize the energetics underlying membrane nanotube formation and stabilization. For our MC simulations of membrane remodeling, we use a particle-based fluid−elastic membrane model46,55 as a discretized representation of the Helfrich continuum model17 without imposed axial symmetry. We focus first on tube-to-sheet transitions, which are known to break the axial symmetry14 and to encounter an energy barrier.16,56 Our high-resolution particle-based model allows us to probe the stability of very thin tubes relevant for the ER and synthetic membrane nanotubes. With molecular dynamics simulations of a molecular membrane model, we further assess the metastability of tubular shapes. We then focus on membrane−elastic factors in tube generation. We explore the induction of tubules by area growth for membranes containing spherical inclusions and for membranes inside a container, mimicking the ER wrapped around the nucleus and the inner membrane of ciliate mitochondria, respectively. The overall aim is to connect tubule induction and stabilization to the rich energetics of membrane shape transformations, with a particular emphasis on explaining the abundance of long and narrow tubes in living cells despite their inherent instability.
RESULTS AND DISCUSSION We quantified the energetics of membrane shape transformations with the Helfrich fluid−elastic membrane model.17 For axisymmetric shapes, complex “phase diagrams” of membrane shapes have been compiled with analytical models, as reviewed in refs 15 and 57. Triangulated membrane models make it possible to characterize the energetics also of highly asymmetric vesicle shapes.16,55,58 In our calculations, we used a dynamically triangulated membrane model46,55 to represent arbitrary membrane shapes. Tube Stability. Upon reduction of the reduced volume v of an initially spherical vesicle with v = 1, the vesicle undergoes two consecutive phase transitions between three stable axisymmetric shapes.14−16 The stable prolate ellipsoidal vesicle for 0.652 ≲ v < 1 transforms into the biconcave oblate ellipsoid for 0.592 ≲ v ≲ 0.652 via a discontinuous transition through 9559
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Figure 3. Oblate and prolate branches of vesicle shapes at reduced volume v = 0.55 for different area differences Δa. The energetic crossover between the two branches occurs at Δa ≈ 1.33. A high energy barrier H with paddle conformations (point A) separates tubular structures (point B) from sheet-like structures (point C). The elastic energies represent the minimum value obtained in 10 SA simulations. The inset provides a zoom-in on the prolate branch, indicating the variations in SA minimum energy values by error bars.
Figure 2. Tubule generation as a result of area-to-volume increase. The vesicle shapes shown are snapshots taken from MC simulations at ambient temperature. Upon increasing the area, an initially spherical vesicle with a reduced volume v = 1 (top left) gradually transforms into a tube. For reduced volumes v ⩾ 0.652, the tube remains energetically stable. Upon further reduction, v < 0.652, metastable tubes become trapped in a local energy minimum. At the smallest volumes, v = 0.3 to 0.12, long and narrow nanotubes form. Structures are scaled to fit the image.
8πκ of both branches at a typical reduced volume v = 0.55. The high resolution of the triangulated vesicle and the slow cooling during the SA simulations ensured small variations in the final energies (see inset in Figure 3 for a zoomed-in image). The transition from the prolate branch, which has lower energy for large area differences Δa ≳ 1.33, to the oblate branch always occurs to the left of the barrier top on the prolate branch (i.e., at smaller Δa; see point A in Figure 3). Figure 4 shows the elastic energies E/8πκ of the prolate branch for different reduced volumes as a function of Δa. At high reduced volumes, v ≳ 0.65, tubes are stable. For v ≲ 0.65, tubes are only metastable, separated by an energy barrier H from stable sheets for 0.592 ≲ v ≲ 0.652 and from stable stomatocytes for v < 0.592. The energy barrier H between tube and sheet does not vanish in the limit of small reduced volumes, even as the overall energetic driving force toward sheet-like states grows. The energy barrier remains at about ≈35 kBT as the reduced volume is lowered to v = 0.2 (insets in Figure 4ab; for κ = 20 kBT). These high energy barriers are unlikely to be crossed by thermal fluctuations on a seconds time scale. However, we noticed in our SA calculations that transitions can initiate also away from the ends of the tube, with only marginally higher energy barriers (see below). The effective barrier may thus be entropically reduced, roughly by kBT ln(L/ 2R), which becomes relevant only for very long tubes. Tubular membrane structures are metastable also in the presence of thermal fluctuations. To account for possible entropic effects, we calculated the free energy of the vesicles as a function of the area difference. The corresponding potentials of mean force (PMF) for reduced volumes of v = 0.2 and v = 0.3 are shown in Figures 7 and S1, respectively. In both cases, we obtained near-perfect agreement between the PMF and the elastic energies from SA simulations in the region of the tubular minimum and the barrier to sheet-like structures. These findings are consistent with our MC simulations of vesicles at finite temperature (Figure 2). Tubular vesicles, once formed, are thus metastable over the whole range of reduced volumes v < 1.
can differ widely. Indeed, in molecular dynamics simulations of a coarse-grained, solvent-embedded tubular vesicle, as described in detail below, we similarly observed relatively rapid relaxation of the local membrane structure and slow relaxation of the overall shape. Nonetheless, to ensure the fluidity of the membrane and vertex diffusion in our triangulated membrane model, we performed long simulations at ambient temperature starting from the tube with v = 0.208 in Figure 2 and continued with constant reduced volume and the same bending rigidity, κ/kBT = 30. The simulations confirm membrane fluidity, as reflected in vertex diffusion, and the slow dynamics of tubule undulations compared to local membrane shape fluctuations. Three pairs of vertices initially connected by mesh edges are followed in Supplementary Movie SM2. Their pair distance dynamics confirms the fluidity due to edge flipping, and their movement around the tube is consistent with a largely diffusive motion of the vertices inside the membrane. As shown in Supplementary Movie SM2, the undulation dynamics of the tube is very slow. During the long MC simulation with 2.7 × 107 steps, the initial W shape gradually relaxed to a V shape. Full sampling of the spectrum of the tubular undulations would require even longer simulations. This slow undulation dynamics explains why the undulating shapes of tubular structures were preserved in the simulations of Figure 2 and Supplementary Movie SM1 with about 107 MC steps. We also repeated the simulation in Supplementary Movie SM2 with a smaller bending rigidity, κ/kBT = 20, as shown in Supplementary Movie SM3. The overall shape of the tubular structure was only marginally influenced by the change in bending rigidity. The reason is that both local membrane deformations and tubular undulation are similarly affected by the bending rigidity. SA simulations at fixed reduced volume v resulted in two branches of vesicle shapes with different area differences Δa, corresponding to oblate (sheet-like) and prolate (tube-like) conformations. Figure 3 shows the rescaled elastic energy E/ 9560
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Figure 4. Rescaled elastic energies E/8πκ versus area difference Δa for (a) high reduced volumes 0.5 ⩽ v ⩽ 0.7 and (b) small reduced volumes 0.2 ⩽ v ⩽ 0.45 on the prolate branch. Tubular vesicles are separated from sheet-like structures by an energy barrier H > 0 for all reduced volumes. The rescaled energy barrier H/8πκ and its values for κ = 20 kBT are shown as insets. Each point on the energy curves was computed as the minimum of 10 SA simulations. For reference, (b) includes the elastic energy of a perfectly cylindrical tube (red line; Supplementary Text). The lower inset in (b) shows the dependence of the energy barrier H/8πκ on the ratio v/(L/r) of reduced volume v, length L, and radius r of the tubules.
The cause for the high energy barriers becomes apparent in the structures of the tube and the sheet, and of intermediates along the transition pathway (Figure 3 and Figure 5a−c). For a
Figure 5. Transition-state structures. (a) Side, (b) top, and (c) long-axis views of the paddle conformation of a vesicle with volume v = 0.55 corresponding to point A in Figure 3 at the top of the energy barrier. Tubule and sheet-like ends combine to a paddle structure. (d) Cross section of the sheet segment, with negative curvature in much of its area (e.g., in point D). (e) Cross section of positively curved tube segment (e.g., in point F).
Figure 6. Tubular vesicles corresponding to the metastable tube conformation (right) and vesicles at the top of the energy barrier (left) for different reduced volumes 0.2 ⩽ v ⩽ 0.6. The tubes resemble cylinders with uniform circular cross sections along the tube axis. Crossing the barrier requires an energetically unfavorable transition through intermediate paddle, spoon, or branched tube shapes at the top of the respective barriers for different reduced volumes.
typical reduced volume v = 0.55, the vesicle transforms from a cylindrical tube at area difference Δa ≈ 1.5 to a paddle shape at the top of the barrier (Δa ≈ 1.4). The paddle resembles a cylindrical tube at one end connected to a sheet-like structure at the other end (see Figure 5a−c). The tube and sheet conformations are distinguished by the distribution of the membrane principal curvatures, being positive at the tubular end (see point F in the cross section of Figure 5e) yet negative at the sheet end with its concave spoon shape (see point D in the cross section of Figure 5d). In the transition from tube to paddle conformations at v = 0.55, asymmetric and negatively curved sheet-like structures emerge at one end that are connected to more symmetric, positively curved tubular structures at the other. At smaller reduced volumes, between v = 0.4 and 0.3, the character of the transition state changes from a paddle to a spoon shape and between v = 0.3 and 0.2 from a spoon to an end-branched tube. By contrast, the metastable nanotubes maintain near-uniform circular cross sections for different reduced volumes (Figure 6 right). Slight deviations from uniformity at larger reduced volumes, v ⩾ 0.45,
actually diminish at smaller volumes, v < 0.45. The energetically costly deformations of the metastable tubular structures (Figure 6 right) into the spoon-like or branched segments at the transition states (Figure 6 left) give rise to the high barriers stabilizing tubular vesicles with near-uniform cross sections. Figure 7 for v = 0.2 (and Figure S1 for v = 0.3) shows shapes beyond the metastable tubular structure (Δa > 3.69), and below the transition state toward the sheet (Δa < 3.64). At high area differences, we find (Figure 7h) pearled, (g) bulged, and (f) cylindrical tubes. The number of pearls, n = 23, observed in Figure 7h is close to the value n = 25 expected from the reduced volume of n spheres, v = 1/√n = 0.2. Below the barrier top (e), the junction develops into a spoon (Figure 7b−d). In our MC simulations, upon relaxation of the restraints on Δa from high to lower values, initially pearled tubes readily transformed into uniform cylindrical tubes (Figure 7f−h). This 9561
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hemispherical tube ends. Importantly, this tubular structure remained stable during the entire 5.4 μs long simulation. Tube Generation. Membrane elasticity also defines the energetics of membrane tubule generation. As a possible driver of ER tubulation, the addition of lipids to the ER expands its area under constraints on volume and shape. Volume is controlled by osmoregulation and restricted water flow.18,22 As a key architectural element, the ER is wrapped around the cell nucleus as part of the nuclear envelope. We thus mimicked the growth of the ER in a minimal model of tubulation accounting for these constraints. In our dynamic MC membrane shape simulations, we gradually increased the membrane area while preventing the membrane from shrinking into the nucleus and maintaining the enclosed volume. We found that initial membrane protrusions away from the “nucleus” developed first into prolate tubes that then grew into long and narrow nanotubes with near-uniform circular cross sections. Figure 9 and Supplementary Movie SM4 show the
Figure 7. Tube-to-sheet transition resolved by area difference. (a) Elastic energy (blue points) and free energy of vesicle shapes as a function of the area difference Δa at a small reduced volume v = 0.2. The transition from (f) the metastable tube proceeds through (e) a branched tube at the top of the barrier toward the sheet vesicle. For small area differences, below the barrier, the three-way junction deforms into a spoon (b−d). For large area differences, unstable pearled vesicles form (g), with narrowing necks at increasing Δa (h).
finding indicates that nanotubes are also metastable upon fusion with small vesicles at their tips, the latter creating a pearllike structure that then relaxes to a tube. Molecular Dynamics Simulations. To confirm the metastability of tubular structures predicted by membrane elastic theory, we performed molecular dynamics simulations59 of a molecularly detailed description of lipid membranes in aqueous medium. An initial structure with a reduced volume of v ≈ 0.45 was obtained by fusing five spherical vesicles of equal diameter (Supplementary Text). In simulations using the coarse-grained MARTINI model,60 the resulting pearled-tube structure (Figure 8a) relaxed toward a tubular structure (Figure 8b). This relaxed structure closely resembles the corresponding elastic model in Figure 6 (right panel, v = 0.45), with a diameter-to-length ratio of ≈0.1 and slight expansions of the
Figure 9. Membrane tubulation in MC simulations of a vesicle wrapped around a spherical nucleus. As the area-to-volume ratio of the vesicle is gradually increased, constrained by the nucleus within (red; top left and right snapshots only), spontaneously formed protrusions grow into tubules.
vesicle evolution for increasing membrane area. In the specific example, five tubules formed spontaneously. Results for other simulations with different numbers of protrusions and for different rates of volume reduction are shown in Figure S2. The same argument about the preserved tubular shapes in Figure 2 also applies to the tubular structures in Figure 9 and Supplementary Movie SM4. Undulations of the tubular shape occur on much slower time scales compared to the membrane local fluctuations. The growth of tubular invaginations depends on kinetic parameters, the membrane bending rigidity, and the reduced volume around the nucleus. We performed several kinetic simulations with different volume reduction rates and constant confinement shrinkage rate at three different bending rigidities, κ/kBT = 10, 20, and 30. Our results show that the number of protrusions increases as the membrane rigidity is reduced and as volume reduction accelerates. These findings highlight the kinetic nature61 of invaginating protrusions (see Figure S3). For comparison, we also studied membrane tubulation for a vesicle contained inside a rigid container (Figure 10) mimicking
Figure 8. Coarse-grained molecular dynamics simulation of a POPC vesicle with reduced volume v ≈ 0.45 formed by fusing five spherical vesicles (a) just after fusion and (b) after 5.4 μs. The relaxed shape in (b) agrees well with the v = 0.45 structure in Figure 6 (right) obtained for the elastic membrane model. The tubular vesicle contains 10240 lipids. The 3.7 × 106 water particles are not shown. 9562
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CONCLUSIONS We used membrane elastic theory and simulations to characterize the energetics of lipid membrane tubulation, focusing on the formation and stabilization of the tubular ER. We showed that tubules can be induced quite readily by increasing the area of a membrane wrapped around a spherical inclusion modeling the cell nucleus. Control of the enclosed volume, however, is an important constraint. An increase in the area-to-volume ratio can result from lipid insertion, which is an ER function, and from volume-conserving fusion with vesicles. Continuous vesicle fusion mediated by the protein Sey1p has recently been found to play a significant role in establishing and preserving a tubular network.54 Once formed, we found that tubules are remarkably stable, even in the limit of long and narrow tubes. This result may seem surprising because in this limit the Helfrich membrane elastic energy strongly favors a shape transition from tubular to sheet-like structures.14,15 The reason for this high degree of metastability of tubules is that the transition to sheets occurs through non-axisymmetric structures14,15,56 of relatively high energy. Indeed, the barrier for this transition persists for long and narrow tubes. In macroautophagy, a key cellular process, special proteins of unusual shape appear to facilitate the tube-to-sheet transition in yeast phagophore formation (A.H. Bahrami, M.G. Lin, X. Ren, J.H. Hurley, G. Hummer, unpublished). A key event defining tubulation is the creation of initial protrusions that then develop into tubules. In cells, this energetically costly event is likely aided and controlled by proteins stabilizing negative curvature27 and/or by couplings to the cytoskeleton.32 We expect that also the successive evolution of the nascent bud into long narrow tubes (Figure 9 and Supplementary Movie SM4) will be greatly facilitated by membrane-associated proteins such as reticulons with a high affinity to narrow cylindrical membrane shapes. The cellular machineries driving membrane remodeling and stabilizing tubular structures can thus take advantage of the relative ease with which tubules can be induced by membrane area growth. The metastability of tubules with uniform circular cross sections then ensures that tubule-associated proteins such as Rtns and Yop1p can spread uniformly on the membrane and should thus be able to stabilize nanotubes against shape transformations even at relatively low area densities. Our results also suggest a possible way of inducing tubular structures in synthetic membranes such as polymersomes.3 The ability to control their flexibility, permeability, and chemical functionality makes synthetic tubules technologically attractive, including for drug delivery67,68 and as nanoreactors.69 According to our simulations, a gradual increase in the areato-volume ratio offers a promising route to growing tubular structures (see Supplementary Movies SM1 and SM4), for example, by osmotic deflation, by addition of membrane material, or by vesicle fusion. Once formed, they should be relatively stable as long as the area-to-volume ratio is controlled. Finally, looking forward on the methodological side, scaled up to even larger systems our flexible MC simulation approach should make it possible to study complex 3D motions also of more complex membrane structures. Of particular interest are the highly dynamic networks of interconnected membrane tubes in the ER.13,54
Figure 10. Tubulation in a vesicle confined within a sphere. Shown is the evolution of an initially spherical bud into a tubular vesicle upon volume reduction inside a fixed spherical confinement with Rc = 0.94R0, shown as red solid curve in the top left snapshot. The initial spherical bud transforms into a tubular bud as its volume is reduced. In practice, this is achieved by increasing the vesicle volume V at fixed confinement, which forces the bud volume to shrink. The reduced volume of the vesicle is linearly increased from v = 0.804 to v = 0.85 in 2 × 106 MC steps. The spherical bud transforms into a metastable tubular bud, where minimum energy states are the sheet and stomatocytes at intermediate and small reduced volumes, respectively.
tubule growth, e.g., in ciliate mitochondria.9 An initially spherical bud transforms into an interior tube upon volume reduction as its area-to-volume ratio grows. The resulting structures resemble nanotubes obtained by osmotic deflation.4,62 In our calculations, we studied tubulation in tension-free membranes. By contrast, membrane tension was found to be central to in vitro tubulation in phase separated vesicles.4,39,40 Tension also determines the tube size and the retraction force in active pulling of tubes from aspirated vesicles.63−65 However, in cellular environments, membrane tension is unlikely to play a significant role because membranes can readily shrink or expand.66 We also did not consider the effects of membraneassociated proteins that can shape membranes at sufficient coverage4,20 in the absence of aggregation.50 Similarly, we did not include membrane spontaneous curvature, which is associated with necklace structures4,39,40 not seen in the ER. Instead, we concentrated on free membranes without spontaneous curvature to establish the energetic background in which membrane-shaping proteins act. Finally, we do not have to consider specific tubule diameters. The reason is that the energetics in the Helfrich fluid−elastic membrane model17 is effectively scale invariant, allowing us to express results in terms of reduced areas, volumes, and energies. The membrane area-to-volume ratio thus emerges as a key factor both in the formation and stabilization of membrane tubes.
METHODS Building on the Helfrich fluid−elastic model of membrane bending energetics,17 we used a dynamically triangulated membrane model.46,55 9563
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ACS Nano Similar models have been used before to study asymmetric membrane shapes.16,55,58 We identified locally stable structures with SA simulations,70 characterized thermal fluctuations about them with MC simulations, and determined transition pathways and associated energetic barriers with restrained SA simulations. As order parameters, we used (1) the reduced volume v = 6(π)√πV/A3/2, where V is the enclosed volume and A the membrane area; and (2) the area difference Δa = ∮ dAM /2 πA defined as the integral of the mean curvature M = (c1 + c2)/2 over the vesicle surface, where c1 and c2 are the local principal curvatures.14,56,57 For a spherical vesicle of radius R, we have v = Δa = 1 and c1 = c2 = 1/R. An increase in the area-tovolume ratio corresponds to a decrease in the reduced volume v. Supplementary Text gives further information on the models and simulation methods, including the molecular dynamics simulations using the coarse-grained MARTINI60 model.
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ASSOCIATED CONTENT S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b05542. Supplementary Figures S1, S2, and S3 and supplementary text (PDF) Movie SM1 (AVI) Movie SM2 (AVI) Movie SM3 (AVI) Movie SM4 (AVI)
AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. ORCID
Gerhard Hummer: 0000-0001-7768-746X Notes
The authors declare no competing financial interest.
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