Shape transitions of high-genus fluid vesicles
aa r X i v : . [ c ond - m a t . s o f t ] D ec Shape transitions of high-genus fluid vesicles
Hiroshi Noguchi ∗ Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
The morphologies of genus-2 to -8 fluid vesicles are studied by using dynamically triangulatedmembrane simulations with area-difference elasticity. It is revealed that the alignments of themembrane pores alter the vesicle shapes and the types of shape transitions for the genus g ≥
3. At ahigh reduced volume, a stomatocyte with a circular alignment of g +1 pores continuously transformsinto a discocyte with a line of g pores with increasing intrinsic area difference. In contrast, at a lowvolume, a stomatocyte transforms into a ( g + 1)-hedral shape and subsequently exhibits a discretephase transition to a discocyte. I. INTRODUCTION
Cell organelles have a variety of morphologies. In someorganelles, lipidic necks or pores connect biomembranessuch that they hold a nonzero genus. For example, anuclear envelope consists of two bilayer membranes con-nected by many lipidic pores, which is supported by pro-tein complexes. The outer nuclear membrane is also con-nected by the endoplasmic reticulum, which consists oftubular networks and flat membranes. The fusion andfission of the membrane tubes change their genus. Itis important to understand the topology dependence oftheir morphologies.Since lipid bilayer membranes are basic componentsof biomembranes, lipid vesicles in a fluid phase are con-sidered a simple model of cells and organelles. The mor-phologies of genus-0 vesicles have been intensively studiedexperimentally and theoretically [1–9]. A red-blood-cellshape (discocyte) as well as prolate and stomatocyte, canbe reproduced by minimization of the bending energywith area and volume constraints. Other shapes suchas a pear and branched tubes are obtained by the ad-dition of spontaneous curvature or area-difference elas-ticity (ADE) [3–5]. In particular, the ADE model canreproduce experimentally observed liposome shapes verywell [7]. In contrast to the genus-0 vesicles, vesicles witha nonzero genus have been much less explored.Vesicle shapes with the genuses g = 1 and g = 2 werestudied in the 1990s [10–17]. For g = 1, a conformationaldegeneracy was found in the ground state of the bend-ing energy, where the vesicles can transform their shapesif the vesicle volume is allowed to freely vary [11]. For g ≥
2, a conformational degeneracy is obtained even witha fixed volume [14, 15, 17]. Phase diagrams of genus-1 and genus-2 vesicles were constructed by J¨ulicher etal. [13–15] for symmetric shapes. Recently, we found thatnonaxisymmetric shapes such as elliptic and handled dis-cocytes also exist in equilibrium for genus-1 vesicles [18].Vesicles with g ≫ ∗ [email protected] the only previous study of the shape transition of vesi-cles with g ≥
3. Thus, vesicle shapes for g ≥ ≤ g ≤ II. SIMULATION MODEL AND METHOD
The morphologies of fluid vesicles are simulated bya dynamically triangulated surface method [18, 20, 21].Since the details of the potentials are described inRef. [18] and the general features of the triangulatedmembrane can be found in Ref. [20], the membrane modelis briefly described here. A vesicle consists of 4000 ver-tices with a hard-core excluded volume of diameter σ .The maximum bond length is σ = 1 . σ . The volume V and surface area A are maintained by harmonic poten-tials U V = (1 / k V ( V − V ) and U A = (1 / k A ( A − A ) with k V = 4 k B T and k A = 8 k B T , where k B T is thethermal energy. The deviations in the reduced volume V ∗ = V / (4 π/ R A3 from the target values are less than0 . R A = p A/ π . A Metropolis Monte Carlo(MC) method is used for vertex motion and reconnectionof the bonds (bond flip).The bending energy of a single-component fluid vesicleis given by [1, 2] U cv = Z κ C + C ) dA, (1)where C and C are the principal curvatures at eachpoint in the membrane. The coefficient κ is the bendingrigidity. The spontaneous curvature and Gaussian bend-ing energy are not taken into account since the sponta-neous curvature vanishes for a homogeneous bilayer mem-brane and the integral over the Gaussian curvature C C is invariant for a fixed topology.In the ADE model, the ADE energy U ADE is added asfollows [3–5]: U ADE = πk ade Ah (∆ A − ∆ A ) . (2)The areas of the outer and inner monolayers of a bi-layer vesicle differ by ∆ A = h R ( C + C ) dA , where h is the distance between the two monolayers. Thearea differences are normalized by a spherical vesicleas ∆ a = ∆ A/ πhR A and ∆ a = ∆ A / πhR A to dis-play our results. The spherical vesicle with ∆ a = 0has ∆ a = 1 and U ADE = 8 π k ade . The mean cur-vature at each vertex is discretized using dual lattices[18, 20, 22, 23].In the present simulations, we use κ = 20 k B T and k ∗ ade = k ade /κ = 1. These are typical values for phospho-lipids [5, 7]. Most of the simulations are performed withthe bending and ADE potentials under the volume andarea constraints. In some of the simulations, k ∗ ade = 0and k V = 0 are employed in order to simulate the vesi-cles without the ADE energy and volume constraints,respectively. In the long time limit, the area differenceis relaxed to ∆ a = ∆ a , although it does not occur ona typical experimental time scale. The canonical MCsimulations of the ADE model are performed from dif-ferent initial conformations for various values of V ∗ and∆ a . To obtain the thermal equilibrium states, a replicaexchange MC (REMC) method [24, 25] with 8 to 24 repli-cas is employed for the genus-5 vesicles. Different replicashave different values of ∆ a or V and neighboring repli-cas exchange them by the Metropolis method. III. GENUS-5 VESICLES
We intensively investigate vesicles with the genus g = 5(Figs. 1–4). We categorize the vesicle shapes as stoma-tocyte and discocyte. A spherical invagination with anarrow neck into the inside of a spherical vesicle is atypical stomatocyte. Here, we also include a nonspheri-cal invagination [see Fig. 1(a)] and an invagination withwide necks [see Fig. 1(d)] into the stomatocyte. On thediscocyte, five pores can be aligned in circular, straight,or branched lines [see Figs. 1(e), 3(b), and 4(c)].When the volume constraint and the ADE potentialare removed ( k V = 0 and k ∗ ade = 0), the vesicle shape isdetermined by the bending energy with the topologicalconstraint. As mentioned in the Introduction, the lowestbending-energy states of the vesicles with g ≥ g = 5, the lowest energy states are a circular-cage stoma-tocyte, where g +1 pores are aligned in a circular line andthe pore size largely fluctuates [see Fig. 1(a)]. The vesi-cle shapes are distributed around V ∗ ≃ .
63 and ∆ a ≃ κ = 20 k B T ) so that theobtained shapes contain thermally excited states aroundthe lowest energy states.Stomatocytes are stable shapes with the volume con-straint for k ∗ ade = 0. A valley in the free-energy land-scape in the V ∗ –∆ a space is extended from the lowestenergy state to low reduced volumes along the solid (red) FIG. 1: Snapshots of genus-5 vesicles obtained by the simu-lation (a) without the volume constraint at k ∗ ade = 0, (b),(c)with the volume constraint at k ∗ ade = 0, and (d), (e) with thevolume constraint at k ∗ ade = 1 and ∆ a = 1 .
45. (a) Circular-cage stomatocyte at V ∗ = 0 .
63 and ∆ a = 0 .
98. (b) Stomato-cyte at V ∗ = 0 .
54 and ∆ a = 0 .
8. (c) Spherical stomatocyteat V ∗ = 0 . a = 0 .
74. (d) Cube at V ∗ = 0 . a = 1 .
41. (e) Discocyte at V ∗ = 0 . a = 1 .
32. Thetop and front views are shown. In the front view, the vesiclesare cut along a plane shown as a (red) straight line on the leftside in the top view. Their front halves are removed and thecross sections are indicated by the thick (red) lines. line in Fig. 2. This line is calculated using the REMCmethod for V ∗ with k V = 0 . V ∗ , it is found that the vesicletransforms from the circular-cage shape to a sphericalstomatocyte [see Fig. 1(c)], where g + 1 pores are dis-tributed on the vesicle surface. This transformation oc-curs as a continuous change. In the transient region at0 . . V ∗ . .
55, the vesicle has intermediate shapes, inwhich the positions of one or two pores often deviate froma plane [see Fig. 1(b)]. This shape transformation in thestomatocytes is characteristic for vesicles with g ≥ g ≤
2. Thus, the arrangement of thepores appears as a new factor for the high-genus vesicles.Next, we describe the stomatocyte–discocyte transi-tion as a function of ∆ a using the ADE model. It drasti-cally changes above or below the critical reduced volume V ∗ ≃ .
54 of the change in the pore arrangement on thestomatocyte. Figure 3 shows the free energy and shapesof the vesicles at V ∗ = 0 . V * ∆ a, ∆ a FIG. 2: Phase diagram of genus-5 vesicles. The (red) solidline represents the mean area difference h ∆ a i obtained fromthe simulation with the volume constraint at k ∗ ade = 0. Thecontour lines of the probability distribution P ( V ∗ , ∆ a ) =0 . , . , .
001 are obtained by the simulation withoutthe volume constraint at k ∗ ade = 0. The circles and squareswith the dashed lines represent the phase boundaries of ∆ a for stable and metastable discocytes, respectively. The errorbars are shown at several data points for h ∆ a i and at all datapoints for the phase boundaries. ulation. The lowest free-energy state is the circular-cagestomatocyte at ∆ a = 0 . F = 0, in this study.With increasing ∆ a , one of the pores in the cage stoma-tocyte gradually opens [see Fig. 3(a)], and subsequently,a discocyte with a straight line of g pores is formed [seeFig. 3(b)].The vesicle shapes are quantified by a shape param-eter, asphericity α sp , and the area difference ∆ a . Theasphericity α sp is defined as [26] α sp = ( λ − λ ) + ( λ − λ ) + ( λ − λ ) λ + λ + λ ) , (3)where λ ≤ λ ≤ λ are the eigenvalues of the gyrationtensor of the vesicle. The asphericity is the degree ofdeviation from a spherical shape: α sp = 0 for spheres, α sp = 1 for thin rods, and α sp = 0 .
25 for thin disks[18, 23]. The spherical stomatocyte and discocyte have α sp ≃ .
2, respectively. The circular-cage stomato-cyte has α sp ≃ .
05. As the stomatocyte transforms intothe discocyte, α sp and ∆ a changes abruptly. The tran-sition point (∆ a = 1 . ± . α sp curves in Fig. 3(d). When thediscocyte vesicle transforms into the open stomatocyte,the mirror symmetry breaks, and the slope of F (∆ a )changes. This is a second-order type of transition, but -0.20 1 1.5 < ∆ a > - ∆ a ∆ a (e) < α s p > (d) diskcage010 F / κ (c) FIG. 3: Dependence of the vesicle shapes on ∆ a for genus-5vesicles at V ∗ = 0 . k ∗ ade = 1. (a),(b) Snapshots at ∆ a =(a) 1 .
16 and (b) 1 .
45. (c) Free-energy profile F . (d) Meanasphericity h α sp i . (e) Mean area difference h ∆ a i compared tothe intrinsic area difference ∆ a . The error bars are shownat several data points. it is rounded by the thermal fluctuations. This charac-teristic of the transition is the same as the stomatocyte–discocyte transition at g = 0 and 1.The stomatocyte–discocyte transformation becomes adiscrete transition below the critical reduced volume V ∗ ≃ .
54. We obtained the coexistence of the stom-atocyte and discocyte using canonical MC simulations inthe right region of the dashed (gray) line with squaresin Fig. 2. The final shapes are determined by hystere-sis from the initial conformations. For example, cubicand discocyte vesicles are obtained at V ∗ = 0 . a = 1 .
45 [see Figs. 1(d) and (e)]. However, the canon-ical simulation cannot determine which shape is morestable. Both shapes can be maintained without trans-formation into the other shape in this region. Whenwe use the REMC simulation for ∆ a , the exchange be- -0.2-0.10 1 1.5 < ∆ a > - ∆ a ∆ a (f) < α s p > (e) disksph pyramid010 F / κ (d) FIG. 4: Dependence of the vesicle shapes on ∆ a for genus-5vesicles at V ∗ = 0 . k ∗ ade = 1. (a) Snapshot of a pentag-onal pyramid at ∆ a = 1 .
3. (b) Snapshot of a concave pen-tagonal pyramid at ∆ a = 1 .
6. (c) Snapshot of a discocyteat ∆ a = 1 .
6. (d) Free-energy profile F . The (blue) solid linerepresents F of the spherical and pyramidal stomatocytes; the(black) dashed line represents F of the discocyte. (e) Meanasphericity h α sp i . (f) Mean area difference h ∆ a i compared tothe intrinsic area difference ∆ a . The solid lines in (e) and (f)represent the data in equilibrium and the dashed lines repre-sent the data averaged for either pyramid or discocyte. Theerror bars are shown at several data points. tween the stomatocyte and the discocyte does not occurin equilibrium owing to a large free-energy barrier. Toovercome such an energy barrier, one more order param-eter is often added and the REMC or alternative gen-eralized ensemble method is performed in a partially orfully two-dimensional parameter space. Previously, weemployed the asphericity as an additional order param- eter for a constant radius of gyration in order to over-come the energy barrier between the two free-energyvalleys of the discocyte and prolate of a genus-0 vesi-cle [23]. For the current transition, however, we did notsucceed with similar strategies. Therefore, we take a de-tour via the stomatocyte–discocyte continuous transitionat V ∗ = 0 .
6. This type of detour has been used in free-energy calculations [27, 28]. For liquid–gas phase transi-tions the barrier can be avoided via supercritical fluids.For membranes, an external order field was used to in-vestigate the formation energy of a fusion intermediate[27].Here, we use three REMC simulations to estimatethe free-energy difference ∆ F v5 between the stomato-cyte at ∆ a = 0 .
75 and the discocyte at ∆ a = 1 . V ∗ = 0 .
5. First, the free-energy difference betweenthe stomatocytes at V ∗ = 0 . . F st /k B T = 15 . ± . F v6 /k B T = 192 . ± . a = 0 . k ade = 0) andthe discocyte at ∆ a = 1 .
55 for V ∗ = 0 . F dis /k B T =62 . ± . V ∗ = 0 . . a = 1 .
55 is calculated from the REMC simula-tion for V ∗ with k V = 0 . a = 1 .
55. Hence,the free-energy difference at V ∗ = 0 . F v5 = ∆ F v6 + ∆ F dis − ∆ F st = 239 . k B T ± . k B T . Wesimulate the stomatocyte and discocyte at V ∗ = 0 . − ∆ F/k B T ) as shown in Fig. 4(d). This cal-culation clarifies that the stomatocyte–discocyte trans-formation is a discrete transition at ∆ a = 1 . ± . V ∗ = 0 .
5. [see the solid (red) lines in Figs. 4(e) and(f)].Next, we investigate the vesicle shapes in detail for V ∗ = 0 .
5. With an increase in ∆ a , the sphericalstomatocyte transforms into a pentagonal pyramid [seeFig. 4(a)]. With a further increase, the side faces of thepyramid become concave, and the vesicle shape is rep-resented by a circular toroid connected to a sphere viafive narrow necks [see Fig. 4(b)]. This concave neckedshape prevents the opening of the bottom pore into adiscocyte. On the discocytes, pores are aligned in cir-cular, straight, or branched lines [see Figs. 1(e), 3(b),and 4(c)]. At V ∗ = 0 .
5, these three alignments coexist.As V ∗ decreases and increases, the circular and straightpore alignments appear more frequently, and the circularand straight alignments only exist for V ∗ = 0 . . FIG. 5: Snapshots of the genus-5 vesicles at V ∗ = 0 . k ∗ ade = 1. (a),(b) Budded stomatocytes at ∆ a = (a) 2 and(b) 2 .
2. The bird’s eye and front views are shown. (c) Buddeddiscocyte at ∆ a = 2 . ∆ a g (c) FIG. 6: Genus g dependence of the vesicle shapes at V ∗ = 0 . k ∗ ade = 1. (a) Snapshots of genus-3 vesicles at ∆ a = 1 . a = 1 .
5. (c) Upper( (cid:3) ) and lower ( ◦ ) boundaries of the coexistence regions as afunction of the genus g . The error bars are shown at all datapoints. transformation, the pore alignment does not change [seeFigs. 1(a) and 3(a),(b)]. In contrast, the transformationfrom the cubic to discoidal shapes at V ∗ = 0 . V ∗ = 0 . a &
2, outward buddings occur for both stoma-tocytes and discocytes. Since the buds are connected bymultiple membrane necks, the bud shapes are not alwaysspherical. For the stomatocytes, the circular-toroidal and spherical compartments are divided and the toroidalcompartment elongates at ∆ a = 2 [see Fig. 5(a)]. Withincreasing ∆ a , budding occurs in narrow regions of thetoroid [see Fig. 5(b)]. Discocytes become divided alongthe pore alignments [see Fig. 5(c)]. With further in-creasing ∆ a , more buds are formed. These buddedshapes coexist in a wide range of ∆ a because of thefree-energy barrier for the formation and removal of thenarrow necks. IV. DEPENDENCE ON GENUS
The discrete transitions from stomatocytes to disco-cytes are obtained for g ≥ g = 5 are general for g ≥
3. A g -fold pyramid-shaped vesicle coexists with adiscocyte [see the triangular and concave octagonal pyra-mids in Figs. 6(a) and (b)]. For g = 3, stomatocytestransform into discocytes at ∆ a & . V ∗ = 0 .
5. For g ≥
4, the coexistence region has no upper ∆ a limit. At∆ a &
2, the vesicles exhibit budding, as for g = 5. V. SUMMARY
We revealed that the stomatocyte–discocyte transfor-mation is a discrete shape transition at low reduced vol-umes V ∗ for 3 ≤ g ≤
8, while it is a continuous trans-formation at high V ∗ . This discreteness is caused by thealignment of g + 1 pores in the stomatocytes. At high V ∗ , the pores are aligned in a circular line, while thepores are distributed on the entire surface at low V ∗ . Asthe intrinsic area difference ∆ a increases, this spheri-cal stomatocyte transforms into polyhedral shapes. Thetransformations of these vesicles from polyhedral to dis-coidal shapes are the first-order transition.We found a continuous transformation from circular-cage to spherical stomatocytes. However, for vesicleswith g ≫
1, this transformation may be a discrete tran-sition since the symmetry of the pores is changed. Thephase behavior at g ≫ g + 1 faces. The vesicles canonly form a triangular pyramid for g = 3 and a cubeand pentagonal pyramid for g = 5. Thus, the possiblepolyhedra are limited and the shape can be controlledby V ∗ and ∆ a . Recently, the assembly and packing ofpolyhedral objects have received growing attentions [38].Cubic and other polyhedral vesicles can be interestingbuilding blocks, since their shapes are deformable andcan be controlled by the osmotic pressure.A mitochondrion consists of two bilayer membranes.The inner membrane has a much larger surface areathan the outer one and forms many invaginations calledcristae. As a model of such a confinement of an outervesicle, the morphology of a genus-0 vesicle under spher-ical confinement has been recently studied [8, 9]. Theconfinement induces various shapes such as double andquadruple stomatocytes, a slit vesicle, and vesicles oftwo or three compartments. Very recently, Bouzar et al.reported that the confinement transforms axisymmetric toroids into asymmetric shapes for genus-1 vesicles [39].For higher-genus vesicles, the confinement should simi-larly stabilize the stomatocytes with respect to the dis-cocytes since the stomatocytes are more compact. Acknowledgments
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