ARTICLE pubs.acs.org/Langmuir
Surface Tension Driven Instabilities in Single-Component Saturated Fatty Acid Membrane Tubes E. Hatta* Nanoelectronics Laboratory, Graduate School of Information Science and Technology, Hokkaido University, Sapporo, 060-0814 ABSTRACT: Shape instabilities in single-component, saturated fatty acid membrane tubes have been investigated using phase contrast microscopy. These tubes were created in the course of a Langmuir monolayer collapse transition. Two types of shape instabilities were observed: (i) the excitation of a bending mode of a single tube, and (ii) topological changes of an assembly of tubes. The development of tube bending was accompanied by a shape transition from extended amphiphilic globules to confined ones that were transported in the tube. The evolution of bending instability has been analyzed as a balance among the bending energy, the surface tension energy of the tube, and the hydrodynamic dissipation energy by the surrounding fluid. Topological changes of an assembly of tubes were initiated by the formation of a membrane passage connecting two opposing tubes and followed by tube fusion and breaking. These changes were interpreted as a result of surface tension-gradient driven molecules transport on the tube surface.
’ INTRODUCTION Membranes in cylindrical geometry (e.g., tubes) often exhibit more drastic shape instabilities compared to spherical closed vesicles. One of such an example is the so-called “pearling instability” and has been found in some living cells1,2 and artificial membrane tubes.3,4 Pearling is an axisymmetric peristaltic modulation (e.g., a periodic change in diameter) of the cylindrical tube, characterized by a finite-amplitude and well-defined wavelength. A striking aspect of the instability that differentiates it from other dynamic phenomena in membranes is the nonlocal nature of the instability on the localized perturbation of the tube. Much attention has been paid to the dynamics of pearling instability from both theoretical5,6 and experimental7,8 viewpoints. Besides the pearling mode with the axial symmetry, another interesting, bending mode with the axial symmetry breaking is also possible from the linear stability analysis of the tube excitations.9 Although some theoretical papers9,10 reported on the structure of this mode, the dynamics of this excitation have still been less explored, compared to those of the pearling instability. Cylindrical membrane tubes serve as a fundamental unit in living matter for the transport of liquids, molecules, and cells from one site to another.11 Some tubes modify the transport materials while others act as passive pipes. In this respect, it is also important to examine how materials in the tube can be deformed and transported under such instability. The above excitation is a dynamic event of a single membrane tube. When two neighboring, fluctuating membrane tubes collide locally with each other, the formation of an interesting topological object, e.g., a membrane passage (the merger of two previously distant membrane topologies into one surface) is possible.12,13 The passages between membrane surfaces can be considered as topological defects or topological fluctuations, depending on their energy. If the passage is thermally activated, it is a topological r 2011 American Chemical Society
fluctuation; otherwise, it is a topological defect. There were some reports on the structure of the membrane passage previously.1416 The energetics for the opening of such a passage (very short junction) was also considered theoretically for the two membranes with different tensions.17 It was predicted that the exchange of molecules between the opposing membranes makes the lowering of the free energy of a high-tension membrane possible when the net free energy reduction is in the exponential regime by the surface tension driven exchange of molecules between a tense and a fluctuating membrane.17 This passage formation mechanism is unique in that it provides long-distance action far from the interaction zones such as passages, different from any direct interactions such as depletion or van der Waals ones in the contact region.18 In this mechanism, one interesting question is its dynamic nature, or how the flow of amphiphilic molecules between the two membranes along the passage affects the stability of original membranes and created passage themselves. Monomolecular films at the airwater interface (Langmuir monolayers) explore the third dimension from the water surface when they are compressed laterally beyond a limiting value of surface pressure (collapse transition).19,20 Upon monolayer collapse, soft collapsed structures such as buckling,21 twisted ribbons,22 straight23 and giant folds,24 and vesicles25 have been observed on micrometer length scales, while characteristic fracture evolutions have been found as a result of monolayer breaking on the molecular length scale.26,27 Depending on the subphase pH and the ions added into the subphase, fatty acid monolayer collapse leads to long-range cracking or surface roughening at some pH region (e.g., pH = 6.07.8).27 Received: April 5, 2011 Revised: July 18, 2011 Published: July 19, 2011 10400
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Langmuir In this paper, we report two types of surface tension driven instabilities of tubes formed in the collapse process of a singlecomponent, saturated fatty acid monolayer: (i) the development of bending mode of a tube induced by the mechanical adhesion to the neighboring tube and (ii) topological changes of an assembly of membrane tubes through the fusion and breaking process of tubes. In the former instability, the characteristic time required for the growth of a bending mode is given from a simple linear stability argument and compared with the result of video microscopy observation. The latter topological changes of tubes are discussed from the viewpoint of the Marangoni flow instability of the amphiphilic molecules on the membrane tube. It is stressed that the surface tension (gradient) plays a dominant role for these two shape instabilities in simple fatty acid membrane tubes.
’ EXPERIMENTAL SECTION Materials and Methods. Monolayers of octadecanoic acid (stearic acid, C18, 99% pure, Sigma Chemicals) dissolved in 0.5 mmol chloroform (97% pure, Kanto Chemicals) were spread onto a liquid subphase (Millipore Mill-Q system filtered water, 18.0 MΩ cm). All experiments were performed at 20.0 ( 0.2 °C. The pH values of the subphase from 6.0 to 8.0 were adjusted with NaHCO3. For comparison, the isotherm measurement and the PCM observation were made using the subphase including divalent Cd ions (1 mmol) and at pH 8.0. These materials were used without further purification. The tubes were monitored using phase contrast microscopy (PCM), (NIKON, OPTIPHOT-2) equipped with a CCD camera (Hamamatsu, C240077H) followed by an image processor (Hamamatsu, DVS-3000). The incident light was transmitted through the bottom of a homemade glass trough placed on the microscope stage. The motion of tubes was recorded at a frequency of 30 frames/s while constantly monitoring the πA isotherms with a Wilhelmy plate balance. The width of the trough used was 10 cm, and the monolayers were compressed at a fixed low rate of 13 Å2/(molecule 3 min) to diminish the kinetic effects of lateral compression. We can say that the formation of tubes is a metastable process, since the tube formation proceeds through metastable states away from equilibrium. The PCM images are projections through the monolayer. When the height of the fatty acid tube lies within the depth of the field of the objective, it is possible to view the whole tube. However, different parts of the tube often extend into the subphase, and in this case, not all of it is in focus at the same time. This causes the lowering of the image contrast at some places of the tube. By focusing into the subphase, it is possible to bring successive parts of the tube in focus, demonstrating that the tube extends into the subphase rather than into the air.
’ RESULTS Membrane tubes were created in many collapse regions and independent of the barrier compression direction at the intermediate stage of monolayer collapse only at the highest subphase pH value (pH = 8.0) under investigation. The formation of tubes was observed in the tilted, condensed (L2 and Ov) phases from the isotherm measured (Figure 1).28 These tubes were nearly straight, typically hundreds of micrometers in length, with diameters of 310 μm. Most of the tubes were likely to be attached to the monolayer. Some of the tubes were observed to detach from the monolayer and to drift freely within the subphase. Tubes generally exhibited strong thermal undulations, indicating fluidity of the tubes. Many tubes often exhibited a remarkable bending deformation with a well-defined amplitude
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Figure 1. Surface pressuremolecular area isotherms of stearic acid monolayers in the presence of (a) Na ions (2.5 mmol NaHCO3, pH 8.0) and (b) Cd ions (1 mmol CdCl2, pH 8.0) at 20 °C. The arrows indicate the surface pressures for which PCM images of stearic acid monolayers in the presence of (c) Na ions and (d) Cd ions at the final stage of monolayer collapse are shown. Blue and green dotted lines show the limiting areas for the monolayers. The region where the formation of fatty acid tubes was observed is marked in red.
Figure 2. Phase contrast microscopy image of a fatty acid membrane tube showing a bending instability. The scale bar length is 10 μm.
and wavelength (Figure 2). This instability was very likely to be induced by the sudden surface tension developed by the mutual adhesion between two opposing membrane tubes (region encircled by a rectangular section in Figure 3a). The flow of amphiphilic molecular globules was often observed in the tube under the instability. The detailed tracking of bending evolution showed a transition from extended globules in a relaxed tube (Figure 4a) to confined globules in a tensed tube (Figure 4e) as the bending instability developed. It was observed that membrane passages connecting two opposing tubes normally move along the tube surface, changing in number, position, and width (Figure 5). Some interesting topological changes of an assembly of membrane tubes were observed, initiated by the formation of a passage (Figure 6). The upper tube in which two large globules (indicated by two white arrows, Figure 6a) were trapped exhibited a strong undulation between them locally, approaching the lower tense tube. A membrane passage connecting the two opposing tubes formed immediately after colliding each other within the resolution of one video frame (1/30 s) (Figure 6b, dotted rectangular section). The upper left globule (indicated by white arrow in Figure 6c) gives some of its amphiphilic molecules to the lower tense tube and it restored some fluctuations. After this process, the suction of molecules from the lower tube leads to the formation of a globule (Figure 6d and e), while during this process the lower tense tube enhanced its fluctuations. The globule grew in size and moved toward the upper right globule 10401
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Figure 5. Phase contrast microscopy images of evolution of passages connecting two opposing membrane tubes. The scale bar length is 10 μm.
Figure 3. Phase contrast microscopy images to demonstrate the evolution of a bending instability driven by mutual adhesion between two opposing membrane tubes. Time of frames: (a) 0 s, (b) 0.03 s, (c) 0.06 s, and (d) 0.1 s. Contrast of the lower tube becomes lower due to the extension into the water subphase. The insets show the enlargements of the bending region. The both scale bar length is 50 μm.
Figure 6. Phase contrast microscopy images to show topological changes of membrane tubes initiated by the creation of a passage. Time of frames: (a) 0 s, (b) 1.5 s, (c) 1.8 s, (d) 1.92 s, (e) 2.49 s, (f) 2.76 s, (g) 3.99 s, and (h) 4.59 s. The scale bar length is 50 μm.
(Figure 6e and g, the breaking positions are shown by scissors). Another example of topological change of membrane tubes is shown in Figure 7. In this case, thin tubes (instead of a passage) were formed in pairs instantly within one video frame (1/30 s), connecting two opposing tubes (Figure 7a and b). After the upper end of the right thin tube slides along the original upper tube (Figure 7c), the two thin tubes grew in thickness and fused into one tube on the lower tube finally (Figure 7d).
’ DISCUSSION 1. Formation of Single-Component Saturated Fatty Acid Membrane Tube. The mechanisms by which cylindrical surfac-
Figure 4. Phase contrast microscopy images to demonstrate the growth (ae) and relaxation (f, g) of a bending instability. Each frame was extracted from the original PCM image. Time of frames: (a) 0 s, (b) 0.03 s, (c) 0.06 s, (d) 0.09 s, (e) 0.12 s, (f) 0.15 s, and (g) 0.18 s. The scale bar length is 50 μm.
(Figure 6f). Finally, the merging of two globules into one occurred (Figure 6g). During the sequence of globule formation, transportation, and fusion, the breaking of tubes was observed
tant bilayer tubes can be made of amphiphilic molecules are the subject of considerable interest. Most investigated amphiphilic tubes possess a chiral center, and chiral-symmetry breaking may play an important role in the formation of these tubes.2931 In nonchiral molecules, there have been very limited reports on tube formation.32 The detailed mechanism of the tube formation still remains unclear. The formation of tubes upon monolayer collapse at higher pH (∼8.0) in this study indicates that the degree of dissociation or the ionization state of fatty acid plays a role in the formation of the tubular structure. The variation of the dissociation of fatty acids at the airwater interface was investigated as a function of the subphase pH and for several cations.28,33 It was shown that monovalent Na ions show full dissociation at pH value of about 10 and the ionized fraction, i.e., monolayer dissociation χ is about 0.2 at pH value of 8. This value is much lower than that (χ ≈ 1.0) in divalent ions (e.g., Cd2+, Ca2+, and Ba2+) at its corresponding pH value. It is known that, in aqueous solutions of soaps, i.e., alkali salts of long chain fatty acids, the headgroups are partially ionized between pH values of 7.0 and 10.0, and the two kinds of headgroups (COOH and COO) form a hydrogen-bonded complex known acid soap.34 The bending modulus of the monolayer can be varied by the local 10402
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Figure 7. Phase contrast microscopy images to show the creation of a new tube mediated by the formation of a pair of thin tubes. Time of frames: (a) 0 s, (b) 0.03 s, (c) 0.27 s, and (d) 1.05 s. The scale bar length is 50 μm.
modulations in the ratio of constituents (e.g., carboxylate and carboxylic acid) in the monolayer, and this may lead to the formation of aggregates.35,36 Lower values of the bending modulus will favor soft, curved shapes, e.g., vesicles or tubes over flat folds. Kmetko et al. presented the systematic grazing incidence diffraction (GID) studies of the effect of pH on a fatty acid monolayer.37 They showed that the dilute solutions of both mono- and divalent ion change the structure of the monolayer, but in different ways. Dissolving divalent ions effectively leads to the lowering of the surface pressure at which tilted to untilted transition occurs, that is, compressing the monolayer into a tightly packed, untilted structure, while increasing pH in the subphase by adding a monovalent ion causes the monolayer to become disordered, in a similar way that increasing temperature would. They proposed that the increase in disorder is due to the formation of an acid soap. The presence of defects caused by the disorder would also facilitate buckling and folding that finally leads to the formation of tubes. We investigated the influence of divalent ions (Co2+, Cd2+, Mn2+, Ba2+) in the subphase on monolayer collapse previously (ref 27, second paper). In this case, the creation of any soft collapsed structures such as tubes was not observed; only long-range cracking and surface roughening were observed. The addition of monovalent ions into the subphase (and raising the pH value) thus seems to be a key factor for the spontaneous formation of soft tubes upon monolayer collapse from the tilted LC phases. Although it is certainly important to study tubular structures on a molecular basis to relate the chemical structure of the molecules to the membrane curvature of the tubes, we do not go further into the details of the tubular structure observed under this study and we focus on some features of the tubular dynamics. Here, in relation to the formation of tubular structures from the monolayer in the low surface pressure region, let us consider monolayer collapse observed in this study from the shapes of isotherms. The appearance of a “spike” or a ”plateau” at the maximum surface pressure, πc, attained during isotherm measurements has been considered to show the onset of the collapse process, by which the monolayer transforms to a bulk phase. The limit of the monolayer stability has been characterized by the magnitude of πc. The formation of our fatty acid tubes was, however, initiated by monolayer collapse from the tilted condensed phases much below the πc. In our isotherm measurements, the presence of monovalent (Na+) and divalent (Cd2+) ions at the same pH value (pH 8.0) caused the similar plateau surface pressure regions at the final stage of monolayer collapse (Figure 1a and b). The corresponding PCM images (Figure 1c and d) also show similar collapse patterns that extend across the trough, characteristic of the monolayer with a plateau region in the isotherm.27 The isotherms, however, show very different behavior below the πc. In the presence of divalent ions, the tilteduntilted transition along the isotherm occurs at a low surface pressure and the isotherm is featureless (Figure 1b), while the isotherm shows an intermediate slope region corresponding to the tilted liquidcondensed phases in the presence of monovalent Na ions.
The tube formation and the bending instability were observed in this region. The limiting area (0.194 nm2) of stearic acid monolayer in the Na+ case is smaller than that (0.205 nm2) in the Cd2+ case, although adding divalent ions into the subphase causes the monolayer to become a more tightly packed structure. This suggests that the monolayer loses many molecules through the formation of tubes by monolayer collapse in the low surface pressure region already before the monolayer reaches the πc. Some previous studies also revealed the growth of three-dimensional (3D) collapsed structures below the πc (e.g., at the steep part of the isotherm).3840 The role of the monolayer phase in the collapse mechanism has still been less clear. 2. Evolution of the Bending Mode in a Membrane Tube. Judging from visible thermal undulations of membrane tubes, a membrane can be described as a fluid, and we can use the Helfrich energy to describe the tube behavior observed in this study. To evaluate the growth rate of the bending mode initiated by applying a rapid pull on a tube through the sudden adhesion to the neighboring one, we try to obtain a scaling relation within a linear stability analysis. Let us consider the situation that a membrane tube of undeformed radius R0 and length l . R0 suspended in water undergoes shape instability by the competition between the bending energy and the surface tension energy. The bending free energy associated with deformations of the membrane tube with finite curvatures is given by the following Helfrich energy, Fb ( ) Z k 1 1 2 k Fb ¼ dS þ þ 2 R1 R2 R1 R2 where R1 and R2 are local radii of curvature of the membrane and k, k are the bending rigidity and the Gaussian bending rigidity, respectively.6 The second term proportional to k in the above equation is topologically invariant according to the Gauss-Bonnet theorem, therefore playing no role in the following discussion. A membrane tube has a finite surface tension by geometrical constraints and by stretching through the adhesion to the neighboring tube. The surface tension energy of the tube is given by Z dSσ Fσ ¼ We consider small deformations of a tube that undergoes bending instability from its cylindrical shape (Figure 8). The linear stability analysis allows us to restrict our attention to the perturbations of the Fourier series representation uk, m expðikz þ imjÞ uk, m ðj, zÞ ¼
∑
k, m
Since the cylindrical tube length L is much longer than the radius r, we neglect the contribution of the ends to the energy and impose periodic boundary condition along the tube axis. We can write the tube radius r(j, z) as rðj, zÞ ¼ R̅ þ uk, m ðj, zÞ ¼ R̅ þ
∑ uk, m expðikz þ imjÞ
k, m
ð1Þ 10403
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instability. This is a good approximation in the case where the applied tension σ greatly exceeds the threshold value for bending instability. Here, we consider the difference in surface area of the membrane tube with an axially symmetry breaking, bending mode (m = 1) and an axially symmetric, pearling mode (m = 0) under their instabilities to clarify the characteristics between these two modes. The surface area S under shape instabilities is written to the second order as !1=2 Z L Z 2π uj 2 dj dzr 1 þ 2 þ uz 2 S¼ r 0 0 ¼ S0 þ
Figure 8. Schematic (not to scale) of a bending tube with wavelength λ = 2π/k and average radius R. For comparison, the pearling state of a tube is shown in the inset.
where u , R, R is the average radius, k = 2πn/L, and m and n are * = uk,m integers. From the realization of r(j, z), the condition uk,m is required, where * signifies the complex conjugate. We require that the tube volume does not change under deformation due to the incompressibility of the water and the small permeability of the water through the bilayer surface. This means that the average radius R of the new tube becomes different from the original radius R0. We thus obtain the following simple relation between R and R0 to second order in u from the volume conservation Z L rðj, zÞ2 dz πR0 2 L ¼ π
R̅ ¼ R0 1
∑
∑
where S0 = 2πR0L is the surface area of the undeformed tube. In the soft, long-wavelength limit (k f 0), a bending mode (m = 1) leads to the excess area (ΔS = S S0 > 0) in the tube. A pearling mode (m = 0), in contrast, causes the area reduction (ΔS < 0) in the tube. This means that each deformation mode creates a negative or a positive surface tension on the tube, respectively. The destabilizing effect of surface tension in bending instability thus arises from the geometrical fact that soft and long wavelength deformations on a tube can increase the surface area while keeping its enclosed volume, thus satisfying the tendency of a negative surface tension to maximize area. To obtain the growth rate of bending mode, we write the time dependent perturbation radius as uk, m ðj, z, tÞ ¼ eωk, m t uk, m ðj, zÞ In the surface tension dominated regime, the time derivative of the energy gained by the tube in bending deformation is written as ∂ ð2Þ F ¼ πLσk2 R0 2ωk, 1 e2ωk, 1 t juk, 1 j2 ð3Þ ∂t σðk, m ¼ 1Þ On the other hand, considering the hydrodynamic viscous dissipation due to the surrounding fluid as the source of the energy loss of the tube ∂ Fηðk, m ¼ 1Þ ¼ L 3 fη 3 v ¼ L4πηωk, 1 2 e2ωk, 1 t juk, 1 j2 ∂t
0
!
i πL h 2 m þ ðkR0 Þ2 1 R0 k, m
ð4Þ
where η is the viscosity of water. Equating 3 to 4 gives the following growth rate
1 juk, m j2 2R0 2 k, m
In cylindrical coordinates (r, j, z), the differential area element is dS = dj dzr(1 + r2z + r2j/r2)1/2, and after some calculations, we obtain the excess free energy ΔF(2) over the unperturbed tube to order |uk,m|2 ΔF ð2Þ ¼ Fb ð2Þ þ Fσ ð2Þ πL k n 4 ¼ juk, m j2 2m þ 2ðkR0 Þ4 þ 4m2 ðkR0 Þ2 R0 k, m 2R0 2 n oi o ðkR0 Þ2 þ 3 5m2 þ σ m2 þ ðkR0 Þ2 1
∑
ð2Þ In the above equation, the bending energy terms stabilize the tube while the surface tension energy terms destabilize the tube. We neglect the bending (k) terms in (2) in the below argument and focus on the effect of the surface tension energy on the shape
ωk, 1 ¼
k 2 R0 σ 2η
The characteristic time required for the growth of bending mode of a tube is thus written as τk, 1 ∼
1 2η ¼ 2 ωk, 1 k R0 σ
ð5Þ
Putting ηwater ∼ 103 Ns/m2, R0 ∼ 3 μm, σ ∼ 107 106 J/m2,41,42 and λ(= 2π/k) ∼ (67.5) μm in eq 5, we obtain τk,1 ∼ (102 101) s as the growth time in bending evolution, being reasonably consistent with the characteristic experimental time (τexp ∼ 102 s). We make here some comments on the values of the tension used for estimating τk,1. The adhesion energy is generally balanced by the surface tension and the curvature energies of the membrane. In the tension-dominated regime, the adhesion energy is considered to be almost equal to the 10404
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Langmuir surface tension energy. We used the values in the mild tension regime (rather than those extracted from membranes under strong lateral tension) as those of the tension used in the calculation, since we observed an intermittent adhesion/unbinding process in the contact region of the tube in which bending instability was observed. We next compare this characteristic time with the time of development of the tension along the tube. The development of the tension along the membrane of a tube is written as6 dσ K R̅ d2 σ ¼ dτ 4η dz2 K being the compression (stretching) modulus of the membrane. From the above scaling, we see that at a distance L along the tube of radius R the time τσ required for the tension development along the tube is τσ ∼
4ηL2 K R̅
Putting R ∼ 3 μm, η = 103 Ns/m2, L ∼ 100 μm, and K ∼ 101 J/m2,43 we obtain τσ ∼ 104 s, which is much smaller than the characteristic time τk,1 of bending evolution. We assumed that the surface tension is constant and uniform along the tube in our calculation of the growth rate of bending excitation, since the propagation of tension is very fast from the above calculation. This cannot, however, possibly hold since globules move along the tube toward the direction of the adhesion region as bending instability develops. Since the motion of liquid within the tube is viscously coupled to that of molecules of the tube surface, the globular flow indicates the existence of shear force acting on the membrane by surface tension gradient (Marangoni force).44 The effect of surface tension gradient driven molecules flowing on the tube surface on the instability is left for further study. 3. Topological Instabilities of an Assembly of Tubes through the Fusion and Breaking. Two topological changes of membrane configuration, fusion and fission (breaking), are ubiquitous in many cellular processes. Significant driving forces would be required for changes in membrane configuration to occur since some shape change of membranes proceeds against the bending rigidity of the bilayer that constitutes a tube. For protein-coated membranes, some proteins may generate the driving force for the sequential stages in both processes.45,46 The mechanism and dynamics of fusion and fission in proteinfree membranes as in our study has still been less well characterized.47 Here, we consider the role of the surface tension gradient as a driving force for global topological changes of tubes. The membrane passage observed here is normally not a static object but a dynamic one, showing strong fluctuations in its position, shape, and size. In a previous study,16 it was shown that the calculated spectrum of eigenvalues of the passage contains three modes of low energy, two of which are translational modes and one of which is the breathing mode (in which their size changes). The calculation was made in the absence of amphiphilic molecular transport between two membranes. The allowance of the flow of molecules between the opposing membranes through the passage could make the motion and fluctuation of passage more dynamic. Here, we consider how the properties of membrane tube itself can drive its own fusion and breaking, changing the topology of membrane tubes. From video tracking, it is important to note that topological changes of membrane tubes are accompanied by the directional flow of the molecular
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globules in the tube (Figure 6) and tube thickening (Figure 7). This indicates that the surface tension gradient induced surfactant flow (Marangoni flow) on the tube strongly influences the topological changes of tubes. A theoretical description for the Marangoni flow in a simple system consisting of two vesicles connected by a nanotube was developed previously.44 When a tension gradient is created by the deformation of one vesicle, it was concluded that the Marangoni flow is the dominant mechanism over Laplace flow for transport in nanotubes as long as the length of the tube is much larger than its radius. The opposite case, e.g., the membrane flow through a short junction (or a passage) connecting two opposing membranes with different tensions, was also studied.48 In these papers, however, their shape instabilities that might be induced by surfactant flow on them were not taken into account. The present study demonstrates that the dynamics between two tubes is more remarkable when a significant material flow is present between them. The molecular exchange between two opposing membranes is likely to occur between a frustrated tense (higher tension, lower-density) membrane and a fluctuating (lower tension, higher-density) membrane.17 In this context, two globules (molecular reservoirs) trapped in the tube may work as a tension regulator of the membrane, since the tube undulation is enhanced locally between them (Figure 6b). The strong local undulation of the upper membrane tube is likely to be induced by the lowering of tube tension due to the molecules supplied from the two globules, assisted by thermal kicks from the surrounding fluid molecules, resulting in the formation of a passage connecting the two tubes. In Figure 7, a pair of thin tubes grown from the original tube connect with its opposing tube, but do not lead to the simple fusion between them, leading to the formation of a new long tube. This also seems to occur by the Marangoni transport of amphiphilic molecules as is clear from the thickening of grown tubes. In relation to the tube breaking event observed in Figure 6, recent studies have shown that it can be mediated by lipid phase separation in protein free mixed membranes49,50 and by dynamin in protein coated membranes.51 In the former physical mechanism, membrane tube breaking can be induced to release line tension energy between membrane domains that are composed of different lipids. Protein (dynamin) requires the energy to cross the energy barrier for breaking through GTP hydrolysis in the latter mechanochemical mechanism. The two breaking mechanisms above do not play a role for our system, however, since our tubes are composed of a single component and do not contain any proteins. For a single surfactant tube, we cannot expect the occurrence of tube breaking normally, since the relevant elastic and tension energies of the tube would be at most on the order of 105 106 N/m, which are far below the rupture tensile stress (∼103 N/m) of the bilayer. We thus must seek another mechanism describing how the properties of membrane itself lead to the tube breaking. We guess that the tension gradient on the tube surface is a good candidate for driving the tube breaking process. It may be important here to consider the relation between the membrane geometry (e.g., a cylindrical tube or a flat membrane) and tension under an applied external force to overcome the barrier for breaking.52 Compare the mechanical response of a membrane between the pulling of a tube along the tube axis and that of a tether perpendicular to a flat membrane. Prior to tube breaking, we can see the directional flow of a large globule merge with the upper right globule (Figure 6eg). The existence of such a directional flow of the globule must reflect the existence of a gradient of surface tension on the membrane. 10405
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Langmuir When the molecules flow from the low tension region to high tension region and the separation increases between molecules, attraction forces between molecules become insufficient to hold molecules together and the monolayer must break finally. In the case of a flat membrane, on the other hand, the pulling of a bud (the neck connecting the flat membrane and the curved one) from a flat membrane promotes breaking, while membrane tension tends to flatten the membrane due to its in-plane shear modulus, hindering the membrane breaking. The tension gradient along the membrane in cylindrical geometry may thus facilitate the tube breaking. From the above considerations, the Marangoni flow of molecules on the tube surface seems to be an effective mechanism for topological changes of tubes through fusion and fission in this system.
’ CONCLUSIONS We have found that membrane tubes are formed in the collapse process of a single-component saturated fatty acid monolayer in the tilted condensed phases much below the maximum surface pressure, πc, and in the presence of monovalent ions (Na+) in the subphase. It was observed that the tube shows bending instability with a small-amplitude and a welldefined wavelength when it approaches and adheres to the neighboring tube. The linear stability analysis of the instability mode taking into account the bending curvature energy and the surface tension energy of the membrane and the hydrodynamic viscous dissipation of the surrounding fluid describes bending evolution successfully. The globules of amphiphilic molecules flowing in the tube exhibit a transition from extended to confined ones as the bending mode develops. Tubes often show dramatic topological changes through tube fusion and breaking, accompanied by the internal globule transport. It is important to note that our membrane tubes are protein free and contain a single-component saturated fatty acid only. The observed membrane dynamics must thus be ascribed to physical properties of the membrane tube alone. This study demonstrates that membrane tension can be a global parameter to describe various shape transitions of simple fatty acid membrane tubes. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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