Detachment of fluid membrane from substrate and vesiculation
aa r X i v : . [ c ond - m a t . s o f t ] S e p Detachment of fluid membrane from substrate and vesiculations
Hiroshi Noguchi a,b ∗ a Institute for Solid State Physics, University of Tokyo,Kashiwa, Chiba 277-8581, Japan. b Institut Lumi`ere Mati`ere,UMR5306 Universit´e Lyon 1-CNRS, Universit´e de Lyon 69622 Villeurbanne, France. (Dated: September 24, 2019)The detachment dynamics of a fluid membrane with an isotropic spontaneous curvature from aflat substrate are studied by using meshless membrane simulations. The membrane is detachedfrom an open edge leading to vesicle formation. With strong adhesion, the competition betweenthe bending and adhesion energies determines the minimum value of the spontaneous curvaturefor the detachment. In contrast, with weak adhesion, a detachment occurs at smaller spontaneouscurvatures due to the membrane thermal undulation. When parts of the membrane are pinned on thesubstrate, the detachment becomes slower and a remained membrane patch forms straight or concavemembrane edges. The edge undulation induces vesiculation of long strips and disk-shaped patches.Therefore, membrane rolling is obtained only for membrane strips shorter than the wavelength fordeformation into unduloid. This suggests that the rolling observed for Ca -dependent membrane-binding proteins, annexins A3, A4, A5, and A13, results from by the anisotropic spontaneouscurvature induced by the proteins. I. INTRODUCTION
Lipid membranes supported on a solid substrate areconsidered as a model system for biological membranesand are extensively used to study immune reaction andprotein functions as well as membrane properties [1–5].Membranes are placed on a solid or polymer layer, anda wide range of the surface-specific analytical techniquescan be applied.The adhesion of vesicles onto a substrate is a typicalmethod for producing supported membranes. For theestablishment of this method, the adhesion process hasbeen investigated intensively by experiments [6–9] andcoarse-grained molecular simulations [10–12]. A vesicleadheres to the substrate, and the resultant high surfacetension induces membrane rupture, leading to membranespreading on the substrate. Moreover, the adhesion ofbicelles can result in the formation of supported mem-branes [13, 14].The opposite process, namely detachment, has beenthe subject of few studies. However, Boye et al. veryrecently reported that the annexin proteins can detachlipid membranes from a substrate [15, 16]. Various typesof detachment dynamics were observed. The annexins areCa -dependent membrane-binding proteins and havefunctions in endo/exocytosis and membrane repair [17–19]. The annexins A3, A4, A5, and A13 induce mem-brane rolling from open edges. In particular, thick rollsare grown with A4 and A5, while thin branched rollsare formed with A3 and A13. The annexins A1 and A2induce membrane blebbing and folding. The bleb ex-hibits a spherical shape and remains still connected tothe membrane patch. Moreover, the relation betweenthe observed rolling and the membrane repair has beendiscussed in Ref. 15. ∗ [email protected] Herein, the detachment dynamics of membranepatches from a flat substrate is studied by using mesh-less membrane simulations. Since the annexins inducethe rolling and blebbing, they almost certainly bendsthe bound membrane. The folding indicates that theannexins A1 and A2 also bind two membranes. Fur-thermore, the protein–protein interaction is also impor-tant. When the trimer formation of the annexins A4is inhibited, the membrane rolling does not occur [15].Annexins A5 forms a two-dimensional ordered array onthe membrane [18, 20], so that it may effectively solid-ify the membrane. Thus, the annexins definitely playvarious roles in the membrane detachment. However, inthis study, the isotropic spontaneous curvature is consid-ered as the minimum role, since the membrane bendingis required for the detachment. Certain proteins, such asthe Bin/Amphiphysin/Rvs (BAR) superfamily proteins,are known to bend along the domain; that is, induceanisotropic spontaneous curvature [21–24]. However, toour knowledge, experimental evidence of the annexinsbending the membrane anisotropically has not yet beenreported. Herein, the different types of detachment dy-namics occurring on a membrane with the isotropic spon-taneous curvatures are studied by means of simulation.The isotropic spontaneous curvature can also be inducedby the polymer anchoring and colloid adhesion [25–27].Moreover, the pinning effects on the membrane detach-ment are investigated. In experiments, parts of the mem-brane patch are often pinned in the original position. Fi-nally, the additional requirements for obtaining differenttypes of dynamics are discussed.The simulation model and method are described inSec. II. Various types of membrane models have been de-veloped for membrane simulations [28–30]. In this study,a spin type of the meshless membrane models is used,in which membrane particles self-assemble into a mem-brane [31]. This model was previously applied to mem-branes with an isotropic spontaneous curvature [32, 33]or anisotropic spontaneous curvature [34–37], as well astheir mixture [38]. One can efficiently simulate mem-brane deformation with membrane fusion and fission ina wide range of membrane elastic parameters. The simu-lation results of the membrane detachment without andwith pinning are described in Secs. III and IV, respec-tively. Finally, Sec. V presents a summary and discussionof this study.
II. SIMULATION MODEL AND METHOD
A fluid membrane is represented by a self-assembledone-layer sheet of N particles. The position and orien-tational vectors of the i -th particle are r i and u i , re-spectively. Since the details of the meshless membranemodel are described at length in Ref. 31, it is only brieflypresented here.The membrane particles interact with each other viathe potential U = U rep + U att + U bend + U tilt . The potential U rep is an excluded volume interaction with a diameter σ for all particle pairs. The solvent is implicitly accountedfor by an effective attractive potential as follows: U att k B T = ε mb X i ln[1 + exp {− ρ i − ρ ∗ ) } ] − C, (1)where ρ i = P j = i f cut ( r i,j ), C is a constant, k B T is thethermal energy, and ρ ∗ is the characteristic density with ρ ∗ = 6. f cut ( r ) is a C ∞ cutoff function and r i,j = | r i,j | with r i,j = r i − r j : f cut ( r ) = (cid:26) exp { A (1 + r/r cut ) n − ) } ( r < r cut )0 ( r ≥ r cut ) (2)where n = 12, A = ln(2) { ( r cut /r att ) n − } , r att = 1 . σ ,and r cut = 2 . σ . For low densities, U att is a pairwise at-tractive potential, while the attraction is smoothly trun-cated at ρ i & ρ ∗ . The bending and tilt potentials aregiven by U bend k B T = k bend X i 2. The bending rigid-ity κ is linearly dependent on k bend and k tilt but indepen-dent of ε mb for κ/k B T & 20. On the other hand, the line tension Γ of the membrane edge is linearly dependenton ε mb but independent of k bend and k tilt . Therefore,these two quantities are controlled individually. Here,we fix the ratio as k bend = k tilt = k : κ/k B T = 16, 34,and 52 for k = 10, 20, and 30 at ε mb = 4, respectively.The edge tension Γ σ/k B T = 3 . 89, 5 . 1, 6 . 2, and 7 . ε mb = 4, 5, 6, and 7 at k = 20, respectively. Experimen-tally, the edge tension is estimated from the membranepore formation: Γ = 4–40pN. [39–41] When the particlediameter is considered as membrane thickness, σ ≃ k B T /σ ≃ ε mb = 4 isused in this study. The ratio of the Gaussian modulus¯ κ to κ is constant [33]: ¯ κ/κ = − . ± . 1. The area a per particle in the tensionless membranes is slightlydependent on C as follows: a /σ = A + B ( C σ ) with { A, B } = { . , . } , { . , . } , and { . , . } for k = 10,20, and 30 at ε mb = 4, respectively.The solid substrate is set to z = 0 and interacts withthe membrane particles via the Lennard-Jones potential: U ad = X ε ad h(cid:16) σz i (cid:17) − (cid:16) σz i (cid:17) i , (5)which has an energy minimum at z = 2 / σ with a depthof ε ad . The membrane is initially equilibrated at C = 0.Further, C is transformed to a target value at t = 0.Membrane disks are simulated for N = 6400 and 25 600with radii of R disk /σ = 55 and 110, respectively, at C = 0, k = 20, and ε mb = 4. The limit of a largemembrane disk, namely a membrane strip, is also sim-ulated, in which the membrane is connected to itself bythe periodic boundary condition on the x axis. The striplength is L x = 160 σ with N = 25 600 unless otherwisespecified.To study the pinning effects, N pin membrane particlesare fixed in the initial position at k = 20, ε mb = 6, and N = 25 600. The detachment dynamics are simulated at N pin = 25, 50, 100, and 200, that is, the density of thepinned particle, φ pin = N pin /N = 0 . . . . ε mb = 4 for ε ad /k B T = 1, thisattraction strength is selected.Molecular dynamics with a Langevin thermostat is em-ployed [31, 42]. The numerical errors of the phase bound-aries and time evolution are estimated from 3 and 20 in-dependent runs, respectively. In the following sections,the results are displayed with the diameter of membraneparticles σ as the length unit and τ = σ /D as thetime unit, where D is the diffusion coefficient of isolatedmembrane particles. III. DETACHMENT OF FLUID MEMBRANEWITHOUT PINNING First, the detachment of a homogeneous fluid mem-brane with a spontaneous curvature is considered. The n c l t/ τ 1. (a) L x /σ = 160 and N = 25 600. (b) L x /σ = 40 and N = 12 800. The side viewis also presented at t/τ = 300 000. length increases temporally [see the middle-left snapshotin Fig. 1(a)]. This differs from the cup-to-vesicle transi-tion, in which the edge length decreases monotonically.The time evolution of the clusters is shown in Fig. 1(b).When membrane particles are closer than r att , it is con-sidered that they belong to the same cluster. Since sev-eral vesicles are simultaneously formed along the mem-brane edge [see the snapshot at t/τ = 2000 in Fig. 1(a)],a stepwise decrease in the size of the maximum cluster(membrane disk) appears. On average, the maximumcluster size decreases as h n max i / ≃ N / − bt (6)as shown in the inset of Fig. 1(b). This is due to theedge length decrease as L edge ≃ √ n max . The size ap-proximately decreases as dn max /dt = − b √ n max so thatEq. (6) is obtained. A similar decrease was obtainedduring membrane lysis, when the membrane dissolutionoccurs only from the membrane edge [43].These vesiculation dynamics are not qualitatively mod-ified for long membrane strips and small membrane disksof N = 6400. The membrane strip is simulated to investi-gate the straight edge as the limit of the large membranedisk. At L x /σ ≥ 80, the bent edges begin to undulateand vesicles are formed at C σ = 0 . L x /σ = 40 [see Fig. 2(b)and Movie S3 provided in ESI]. At L x /σ = 160, themembrane edge undulates into two or three bumps [seethe snapshot at t/τ = 4000 in Fig. 2(a)], and typicallyeach bump grows into one vesicle. This edge undulationis suppressed at L x /σ = 40. Although various condi-tions are examined, this rolling is obtained only for theshort strips. Thus, the straight free edges are always un-stabilized for the longer strips. This roll unstabilizationis caused by the formation of an unduloid-shaped mem-brane, which periodically undulates along the rotationalaxis and exhibits a constant mean curvature [44, 45]. The ρ σ (y - y ed )/ σ (c) 160 4012 z / σ (b) L x / σ =40 80160 FIG. 3. Membrane edge of the undetached membrane stripsat C σ = 0 . κ/k B T = 34, and ε ad /k B T = 0 . 5. (a) Snapshotsfrom bird’s eye and side views at L x /σ = 160. (b) Membraneheight z ( y ) and (c) density ρ ( y ) at L x /σ = 40, 80, and 160.The error bars are shown for several data points at L x /σ = 80. cylinder with a radius of 1 /C can be continuously trans-formed into unduloids by maintaining the mean curva-ture, and the transformation begins at a wavelength of l und = 2 π/C . Hence, the unduloid formation is sup-pressed for strips that are sufficiently shorter than l und .This threshold length agrees with the simulation results,since l und /σ ≃ 60 at C σ = 0 . 1. A similar instability ofa tubular lipid vesicle was observed in polymer anchor-ing [46].At a small C , the membrane remains on the substrate.The edges are slightly separated from the substrate andfluctuate along the edge ( x axis) as shown in Fig. 3(a).To quantitatively evaluate the edge shape, the membraneheight profile z ( y ) perpendicular to the edge is calculatedand presented in Figs. 3(b) and 4(a). The edge position y ed is defined as at the position in which the density ρ is half of that of the middle membrane region. Since theedge y position fluctuates along the x axis, the changesin the density and height become more gradual for longerstrips [see Figs. 3(b) and (c)]. As C increases or ε ad de-creases, the edge exhibits greater bending [see Fig. 4(a)]. ρ σ (y - y ed )/ σ (b) z / σ (a) C σ = 0.1 0.090.080.06 FIG. 4. Membrane height z ( y ) of the undetached membranestrips at κ/k B T = 34 and L x /σ = 40. The solid lines indicatethe data at C σ = 0 . 06, 0 . 08, 0 . 09, and 0 . ε ad /k B T =0 . 5. The dashed lines indicate the data at C σ = 0 . 06, 0 . . ε ad /k B T = 1. The errorbars are indicated for several data points at C σ = 0 . 08 and ε ad /k B T = 0 . It is noted that the height profile at y & y ed can besignificantly modified by the selection of the averagingmethods. At y & y ed , the slopes of the height profilesdecrease as indicated in Fig. 4(a), because the average istaken along the y axis. When the average is taken forthe z axis as y ( z ), the profile bends upwards. A similaraxis dependence is obtained for the profile of cup-shapedmembrane patches [33].The boundary of the membrane detachment is shownin Figs. 5, 6, and 7. Above or below the boundary curves,the membranes are detached or remained on the sub-strate, respectively. As expected, a higher curvature C is required for the detachment from a stronger adhesion(a greater ε ad ). As pointed out in Ref. 16, it is attributedto the competition between the bending energy κC / w ad = ε ad /a per unit membranearea. To examine this relation, the boundary is nor-malized as κC a /ε ad in Fig. 5(b). When the adhesionis stronger than the thermal fluctuations ( ε ad & k B T ),the plots exhibit approximately constant values, so thatthe detachment boundary can be determined by the ra-tio of the bending and adhesion energies. However, forweaker adhesions, the detachment occurs at a smaller C ,where the thermal fluctuations are not negligible. As thebending rigidity κ increases, the detachment curvaturedecreases at ε ad /k B T = 1 [see Fig. 6(a)]. However, inter-estingly, it increases for κ/k B T . 30 at ε ad /k B T = 0 . 01 0 1 2 κ C a / ε ad ε ad /k B T κ /k B T = 163452(b) strip C σ κ /k B T = 16 κ /k B T = 34N = 25600N = 6400strip(a) FIG. 5. Dynamic phase boundary of the membrane detach-ment. Above the curves, membranes are detached from thesubstrate and form vesicles. (a) The solid and dashed linesrepresent the data at κ/k B T = 16, and 34, respectively. Thered ( ◦ , × ), green ( △ , ▽ ), and blue ( (cid:3) , ⋄ ) colors indicate thedata for the membrane strips and disks at N = 25 600 and6400, respectively. (b) The detachment curvature is normal-ized as κC a /ε ad for the membrane strips at κ/k B T = 16( (cid:3) ), 34 ( △ ), and 52 ( ◦ ). the weak-adhesion condition, the membrane is detachedwhen the thermal undulation overcomes the adhesion.The detachment curvature is independent of the edgetension, Γ, for the membrane strips, because the edgelength does not vary during the initial detachment pro-cess. For a small disk with N = 6400, the detachmentcurvature increases slightly with increasing Γ as shown asa solid line in Fig. 7(b), where the edge length decreasesduring the detachment. Due to this effect, smaller mem-brane patches exhibit lower detachment curvatures [seeFigs. 5(a) and 6].It should be noted that the membrane fission into vesi-cles is slightly suppressed by an increase in Γ as indi-cated by dashed lines in Fig. 7(b). Above the bound-ary curves, the multiple vesicles are formed as shown inFig. 7(a). Between the detachment and fission curves,the membrane patch closes into a single vesicle. Thevesicle size resulted in the fission increases with increas-ing Γ. The mean vesicle size is h n ves i = 530, 690, and860 for Γ σ/k B T = 3 . 89, 5 . 1, and 6 . 2, respectively, underthe conditions of Fig. 1. The high edge tension acceler-ates the membrane closure and suppresses an increase inthe edge line length for the fission. This is opposite tothe case of a cup-to-vesicle transition, in which the highedge tension reduces the membrane sizes for the vesicleclosure [33].In the case that the substrate is removed at t = 0, the C σ κ /k B T (b) ε ad /k B T = 0.225600N = 6400strip C σ (a) ε ad /k B T = 125600N = 6400strip FIG. 6. Bending rigidity κ dependence of the detachmentboundary at (a) ε ad /k B T = 1 and (b) ε ad /k B T = 0 . 2. Fromtop to bottom: membrane strips and disks at N = 25 600 and6400. C σ Γσ /k B Tfis, stripfis, N = 25600detach, N = 6400(b) FIG. 7. Edge line tension, Γ, dependence of the phase bound-ary of the membrane detachment and fission at κ/k B T = 34.(a) Sequential snapshots of membrane fission at t/τ = 14 000,17 4000, and 22 000 from a membrane disk for C σ = 0 . ε ad /k B T = 0 . 2, and N = 25 600. (b) The dashed lineswith ( ◦ ) and ( (cid:3) ) represent the fission boundary for the mem-brane strips at ε ad /k B T = 0 . ε ad /k B T = 0 . N = 25 600, respectively. One or multiplevesicles are formed below or above these curves, respectively.The solid line with ( ⋄ ) represents the detachment boundaryof the membrane disks at ε ad /k B T = 0 . N = 6400. FIG. 8. Detachment dynamics of a pinned membrane at φ pin = 0 . C σ = 0 . κ/k B T = 34, ε ad /k B T = 1, ε mb /k B T = 6, and N = 25 600. (a) Sequential snapshotsat t/τ = 0, 5000, 10 000, 29 600, 31 000, and 60 000. Thepinned particles are displayed as blue spheres larger than themobile particles for clarity. (b) Time evolution of cluster sizes n cl . formation of one and multiple vesicles similarly occurs atlow and high spontaneous curvatures, respectively. Ex-perimentally, vesicles can be produced by the hydrationof dry lipid films [47, 48]. If one side of the membraneis fabricated to induce a spontaneous curvature by meanof polymer anchoring and so on, the fission of a detachedmembrane patch can lead to the formation of small vesi-cles. IV. DETACHMENT OF PINNED MEMBRANE Next, we describe the pinning effects on the detach-ment. Figures 8 and 9 show the detachment dynamics ofthe pinned membrane patches at ε ad /k B T = 1. When themembrane edge approaches a pinned particle, it locallysuppresses the detachment [see Fig. 8 and Movie S4 pro-vided in ESI]. As the membrane edges become trapped bythe pinned particles, the patch adopts a polygonal shapeand the pinned particles are located at the vertices. Oc-casionally, vesicles are still formed under fluctuations ofthe excess membrane area [see Fig. 8]. The mean vesiclesize becomes slightly smaller than the unpinned mem- < n m a x > t/ τ φ pin =(e) FIG. 9. Dependence of the detachment dynamics on pindensity φ pin at C σ = 0 . κ/k B T = 34, ε ad /k B T = 1, ε mb /k B T = 6, and N = 25 600. (a)–(d) Snapshots at t/τ = 60 000 for (a) φ pin = 0 . φ pin = 0 . φ pin = 0 . φ pin = 0 . h n max i for φ pin = 0,0 . . . . branes.As the pin density, φ pin , increases, larger membranepatches remain on the substrate [see Fig. 9]. At thelow densities of φ pin = 0 . 001 and 0 . φ pin = 0 . φ pin ≥ . ε ad is investi-gated as shown in Fig. 10. As ε ad decreases, the detach-ment becomes faster and the whole membrane can bedetached for the low density φ pin = 0 . ε ad . In-terestingly, the membrane patch can form concave edgesas well as straight edges [see Fig. 10(a) and (b)]. Similarconcave edges were observed in the experiments [15, 16].The edges are bent and more membrane region is de-tached from the substrate. The detachment boundaryis C bd0 σ = 0 . 15 at ε ad /k B T = 1. Hence, the simula-tion condition, C σ = 0 . 16, is close to the boundary and < n m a x > t/ τ φ pin = , ε ad /k B T =0.0078 10.0078, 0.50.002, 10.0078, 0.20, 1 (c) FIG. 10. Dependence of the detachment dynamics on adhe-sion strength ε ad at C σ = 0 . κ/k B T = 34, ε mb /k B T = 6,and N = 25 600. (a),(b) Snapshots for (a) ε ad /k B T = 0 . ε ad /k B T = 0 . φ pin = 0 . t/τ = 60 000.(c) Time evolution of the average of the maximum cluster size h n max i . The dotted, dashed-dotted, solid lines represent thedata for φ pin = 0, 0 . . ε ad /k B T = 1, 0 . 5, and 0 . becomes more distant with decreasing ε ad . In such far-from-equilibrium conditions, the membrane can be de-tached further even from the concave edge. V. SUMMARY AND DISCUSSION The detachment dynamics of membrane patches froma flat substrate were simulated. As the spontaneous cur-vature C increases, the membrane edge bends upwardsmore significantly. When C is higher than the thresh-old value, the membrane is detached from the edge andforms vesicles. The threshold curvature is C ∼ p w ad /κ for the strong-adhesion conditions but decreases for theweak-adhesion conditions. For small membrane patches,this curvature slightly decreases with an increase in theedge line tension. The pinning of the membranes ontothe substrate locally suppresses the membrane detach-ment and straight or concave membrane edges are formedbetween the pinned points.Boyes’ experiments [15, 16] showed several types of de-tachment dynamics. The membrane blebbing induced bythe annexins A1 and A2 is similar to the initial process ofthe vesicle formation. However, the membrane pinching-off into a vesicle is prevented. It is likely that the bindingof two neighboring membranes stabilizes the neck of thebleb by means of the annexins. Thus, we consider that the annexins A1 and A2 engender isotropic spontaneouscurvature of the membrane and also cause the binding oftwo membranes. They observed the membrane rolling forfour types of the annexins (A3, A4, A5, and A13). Weobtained the rolling only for strips shorter than the wave-length of the unduloid deformation, 2 π/C . Hence, weconclude that the isotropic spontaneous curvature can-not induce the membrane rolling, and these annexinspresumably induce the anisotropic spontaneous curva-ture. Moreover, membrane solidification is also a pos-sible cause of the rolling for the annexins A5, since theseform an ordered assembly on the membrane. Solid mem-branes cannot form vesicles. It is noted that an orderedsolid membrane can exhibit anisotropic spontaneous cur-vature but it is not orthogonal, ⁀ e.g., 0, 2 π/ 3, 4 π/ ACKNOWLEDGMENTS We thank Olivier Pierre-Louis (Univ. Lyon 1) for stim-ulating discussion and acknowledge the visiting professor-ship program of University of Lyon 1. This work was sup-ported by JSPS KAKENHI Grant Number JP17K05607. [1] M. Tanaka and E. Sackmann, Nature , 656 (2005).[2] E. T. Castellana and P. S. Cremer, Surf. Sci. Rep. ,429 (2006).[3] A. S. Achalkumar, R. J. Bushby, and S. D. Evans,Soft Matter , 6036 (2010).[4] A. Alessandrini and P. 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