Coupling between line tension and domain contact angle in heterogeneous membranes

The compositional differences between domains in phase-separated membranes are associated with differences in bilayer thickness and moduli. The resulting packing deformation at the phase boundary gives rise to a line tension, the one dimensional equivalent of surface tension. In this paper we calculate the line tension between a large membrane domain and a continuous phase as a function of the thickness mismatch and the contact angle between the phases. We find that the packing-induced line tension is sensitive to the contact angle, reaching a minimum at a specific value. The difference in the line tension between a flat domain (that is within the plane of the continuous phase) and a domain at the optimal contact angle may be of order 40%. This could explain why previous calculations of the thickness mismatch based line tension tend to yield values that are higher than those measured experimentally.     

Effect of line tension on the lateral organization of lipid membranes

The principles of organization and functioning of cellular membranes are currently not well understood. The raft hypothesis suggests the existence of domains or rafts in cell membranes, which behave as protein and lipid platforms. They have a functional role in important cellular processes, like protein sorting or cell signaling, among others. Theoretical work suggests that the interfacial energy at the domain edge, also known as line tension, is a key parameter determining the distribution of domain sizes, but there is little evidence of how line tension affects membrane organization. We have investigated the effects of the line tension on the formation and stability of liquid ordered domains in model lipid bilayers with raft-like composition by means of time-lapse confocal microscopy coupled to atomic force microscopy. We varied the hydrophobic mismatch between the two phases, and consequently the line tension, by modifying the thickness of the disordered phase with phosphatidylcholines of different acyl chain length. The temperature of domain formation, the dynamics of domain growth, and the distribution of domain sizes depend strongly on the thickness difference between the domains and the surrounding membrane, which is related to line tension. When considering line tension calculated from a theoretical model, our results revealed a linear increase of the temperature of domain formation and domain growth rate with line tension. Domain budding was also shown to depend on height mismatch. Our experiments contribute significantly to our knowledge of the physical-chemical parameters that control membrane organization. Importantly, the general trends observed can be extended to cellular membranes.     

Line tension and stability of domains in cell-adhesion zones mediated by long and short receptor-ligand complexes

Submicron scale domains of membrane-anchored receptors play an important role in cell signaling. Central questions concern the stability of these microdomains, and the mechanisms leading to the domain formation. In immune-cell adhesion zones, microdomains of short receptor-ligand complexes form next to domains of significantly longer receptor-ligand complexes. The length mismatch between the receptor-ligand complexes leads to membrane deformations and has been suggested as a possible cause of the domain formation. The domain formation is a nucleation and growth process that depends on the line tension and free energy of the domains. Using a combination of analytical calculations and Monte Carlo simulations, we derive here general expressions for the line tension between domains of long and short receptor-ligand complexes and for the adhesion free energy of the domains. We argue that the length mismatch of receptor-ligand complexes alone is sufficient to drive the domain formation, and obtain submicron-scale minimum sizes for stable domains that are consistent with the domain sizes observed during immune-cell adhesion.     

The impact of line tension on the contact angle of nanodroplets

The line tension for a Lennard–Jones (LJ) fluid on a (9, 3) solid of varying strength was calculated using Monte Carlo simulations. A new perturbation method was used to determine the interfacial tension between liquid–vapour, solid–liquid and solid–vapour phases for this system to determine the Young's equation contact angle. Cylindrical and spherical nanodroplets were simulated for comparison. The contact angles from the cylindrical drops and Young's equation agree very well over the range of surface strengths and cylindrical drop sizes, except on a very weak surface. Tolman length effects were not observable for cylindrical drops. This shows that quite small systems can reproduce macroscopic contact angles. For spherical droplets, a deviation between the contact angle of spherical droplets and Young's equation was evident, but decreased with increasing interaction strengths to be negligible for contact angles less than 90°. Linear fitting of the contact angle data for varying droplet sizes showed no clear effect by line tension on contact angle. All calculated line tension values have a magnitude less than 4 × 10^? 12 J/m with both negative and positive signs. The best estimate of line tension for this system of LJ droplets was 1 × 10^? 13 J/m, which is smaller than the reported estimations in the literature, and is too small to be conclusively positive or negative in value.     

Vapour-to-liquid nucleation in cone pores

In this work, we perform a systematic study on the vapour-to-liquid nucleation in a cone pore by using comparatively the classical nucleation theory (CNT) and the constrained lattice density functional theory (constrained LDFT). Three different nucleation scenarios are observed depending on the contact angle θ and apex angle α: the spontaneous phase transition, nucleation with a critical nucleus in Cassie state and nucleation with a critical nucleus in Wenzel state. We also found that both the diagram for nucleation mechanisms and the reduced nucleation barriers with respect to the homogeneous nucleation given by the CNT are at least qualitatively consistent with those from the constrained LDFT. For an increasingly small critical nucleus, the difference between nucleation behaviours from two methods becomes significant due to the macroscopic approximations embedded in CNT, which fails to describe the curvature dependence of surface tension, the line tension effect and the space confinement effect.     

Line tension and structure of raft boundary calculated from bending,tilt, and lateral compression/stretching

Bilayer thickness in membrane domains enriched with sphingomielin and cholesterol (known as “rafts”) is bigger than thickness of neighboring membrane. Monolayers need to deform to compensate the thicknesses difference in the vicinity of the raft boundary. Line tension of the boundary of rafts associated with elastic deformations originating from the compensation of the thickness mismatch is calculated in the frame-work of the elasticity theory. In the calculations deformations of splay, tilt and lateral stretching/compression are considered. It is assumed that raft consists of two monolayer domains situated in the different membrane monolayers; it is also assumed that the boundaries of domains can shift in the lateral direction with respect to relative to each other. Dependence of the boundary energy of raft on the value of the relative shift of the boundaries is calculated. It is shown that the boundary energy is minimal when shift is equal to 4.5 nm. Dependence of the optimal shift on the mismatch of the monolayer thicknesses of raft and surrounding membrane as well as membrane shape in the vicinity of boundary are calculated. The calculated values of line tension are in a good agreement with available experimental data. Taking into account deformation of stretching/compression increases the accuracy of calculations by 30%; this exceeds the uncertainty of the line tension measurements by modern techniques.     

Lipid-glass adhesion in giga-sealed patch-clamped membranes.

Adhesion between patch-clamped lipid membranes and glass micropipettes is measured by high contrast video imaging of the mechanical response to the application of suction pressure across the patch. The free patch of membrane reversibly alters both its contact angle and radius of curvature on pressure changes. The assumption that an adhesive force between the membrane and the pipette can sustain normal tension up to a maximum Ta at the edge of the free patch accounts for the observed mechanical responses. When the normal component of the pressure-induced membrane tension exceeds Ta membrane at the contact point between the free patch and the lipid-glass interface is pulled away from the pipette wall, resulting in a decreased radius of curvature for the patch and an increased contact angle. Measurements of the membrane radius of curvature as a function of the suction pressure and pipette radius determine line adhesion tensions Ta which range from 0.5 to 4.0 dyn/cm. Similar behavior of patch-clamped cell membranes implies similar adhesion mechanics.     

Phase separation rates of aqueous two-phase systems: correlation with system properties

The kinetics of phase separation in aqueous two-phase systems have been investigated as a function of the physical properties of the system. Two distinct situations for the settling velocities were found, one in which the light, organic-rich (PEG) phase is continuous and the other in which the heavier, salt-rich (phosphate) phase is continuous. The settling rate of a particular system is a crucial parameter for equipment design, and it was studied as a function of measured viscosity and density of each of the phases as well as the interfacial tension between the phases. Interfacial tension increases with increasing tie line length. A correlation that describes the rate of phase separation was investigated. This correlation, which is a function of the system parameters mentioned above, described the behavior of the system successfully. Different values of the parameters in the correlation were fitted for bottom-phase-continuous and top-phase-continuous systems. These parameters showed that density and viscosity play a role in the rate of separation in both top continuous- and bottom continuous-phase regions but are more dominant in the continuous top-phase region. The composition of the two-phase system was characterized by the tie line length. The rate of separation increased with increasing tie line length in both cases but at a faster rate when the bottom (less viscous) phase was the continuous phase. These results show that working in a continuous bottom-phase region is advantageous to ensure fast separation.     

Stabilization of bilayer structure of raft due to elastic deformations of membrane

Line tension of the boundary of specific domains (rafts) rich in sphingomyelin was calculated. The line tension was calculated based on macroscopic theory of elasticity under assumption that the bilayer in raft is thicker than in the surrounding membrane. The calculations took into account the possibility of lateral shift of the domain boundaries located in different monolayers of the membrane. The line tension was associated with the energy of elastic deformations appearing in the vicinity of the boundary in order to compensate for the difference in the thickness of the monolayers. Spatial distribution of deformations and the line tension was calculated by minimization of elastic free energy of the system. Dependence of the line tension on the distance between the domains boundaries located in different monolayers was obtained. It was shown that the line tension is minimal if the distance is about 4 nm. Thus, membrane deformations stabilize the bilayer structure of rafts observed experimentally. The calculated value of line tension is about 0.6 pN for the difference between the monolayer thickness of raft and surrounding membrane of about 0.5 nm, which is in agreement with the experimental data available.     

Flicker spectroscopy of thermal lipid bilayer domain boundary fluctuations

This contribution describes measurements of lipid bilayer domain line tension based on two-dimensional thermal undulations of membranes with liquid ordered/liquid disordered phase coexistence and near-critical composition at room temperature. Lateral inhomogeneity of lipid and protein composition is currently a subject of avid research aimed at determining both fundamental properties and biological relevance of membrane domains. Line tension at fluid lipid bilayer membrane domain boundaries controls the kinetics of domain growth and therefore regulates the size of compositional heterogeneities. High line tension promotes membrane domain budding and fission. Line tension could therefore be an important control parameter regulating functional aspects of biological membranes. Here the established method of fluid domain flicker spectroscopy is applied to examine thermal domain wall fluctuations of phase-separated bilayer membranes. We find a Gaussian probability distribution for the first few excited mode amplitudes, which permits an analysis by means of appropriately specialized capillary wave theory. Time autocorrelation functions are found to decay exponentially, and relaxation times are fitted by means of a hydrodynamic theory relating line tensions and excited mode relaxation kinetics. Line tensions below 1 pN are obtained, with these two approaches yielding similar results. We examine experimental artifacts that perturb the Fourier spectrum of domain traces and discuss ways to identify the number of modes that yield reliable line tension information.     

Entropic part of the boundary energy in a lipid membrane

Two-phase lipid membrane is modeled with lipids of different bending rigidity of hydrophobic tails: domains consist of “rigid” lipid liquid condensed phase and are surrounded by the “flexible” lipid liquid expanded phase. Within the framework of the earlier proposed model of flexible strings, entropic contribution not including mismatch energy is calculated analytically. “Entropic” line tension is found to be weakly dependent on the domain radius. According to the model, it is shown that merely “entropic mismatch” is not enough for the domain formation. In the paper it is assumed that lipids at the boundary, on the average, have larger area than the ones in the volume. This leads to an increase of energy of boundary lipids. Cases of lipids with nearly the same bending rigidities and with strongly different ones are considered. Free energy is calculated using Taylor expansion on the difference of area of lipids at the domain’s boundary and in the volume. Based on the calculated boundary energy domain stability in the finite system is estimated.     

Line tension and interaction energies of membrane rafts calculated from lipid splay and tilt

Membrane domains known as rafts are rich in cholesterol and sphingolipids, and are thought to be thicker than the surrounding membrane. If so, monolayers should elastically deform so as to avoid exposure of hydrophobic surfaces to water at the raft boundary. We calculated the energy of splay and tilt deformations necessary to avoid such hydrophobic exposure. The derived value of energy per unit length, the line tension gamma, depends on the elastic moduli of the raft and the surrounding membrane; it increases quadratically with the initial difference in thickness between the raft and surround; and it is reduced by differences, either positive or negative, in spontaneous curvature between the two. For zero spontaneous curvature, gamma is approximately 1 pN for a monolayer height mismatch of approximately 0.3 nm, in agreement with experimental measurement. Our model reveals conditions that could prevent rafts from forming, and a mechanism that can cause rafts to remain small. Prevention of raft formation is based on our finding that the calculated line tension is negative if the difference in spontaneous curvature for a raft and the surround is sufficiently large: rafts cannot form if gamma < 0 unless molecular interactions (ignored in the model) are strong enough to make the total line tension positive. Control of size is based on our finding that the height profile from raft to surround does not decrease monotonically, but rather exhibits a damped, oscillatory behavior. As an important consequence, the calculated energy of interaction between rafts also oscillates as it decreases with distance of separation, creating energy barriers between closely apposed rafts. The height of the primary barrier is a complex function of the spontaneous curvatures of the raft and the surround. This barrier can kinetically stabilize the rafts against merger. Our physical theory thus quantifies conditions that allow rafts to form, and further, defines the parameters that control raft merger.     

Collective Cell Migration in Embryogenesis Follows the Laws of Wetting

Collective cell migration is a fundamental process during embryogenesis and its initial occurrence, called epiboly, is an excellent in vivo model to study the physical processes involved in collective cell movements that are key to understanding organ formation, cancer invasion, and wound healing. In zebrafish, epiboly starts with a cluster of cells at one pole of the spherical embryo. These cells are actively spreading in a continuous movement toward its other pole until they fully cover the yolk. Inspired by the physics of wetting, we determine the contact angle between the cells and the yolk during epiboly. By choosing a wetting approach, the relevant scale for this investigation is the tissue level, which is in contrast to other recent work. Similar to the case of a liquid drop on a surface, one observes three interfaces that carry mechanical tension. Assuming that interfacial force balance holds during the quasi-static spreading process, we employ the physics of wetting to predict the temporal change of the contact angle. Although the experimental values vary dramatically, the model allows us to rescale all measured contact-angle dynamics onto a single master curve explaining the collective cell movement. Thus, we describe the fundamental and complex developmental mechanism at the onset of embryogenesis by only three main parameters: the offset tension strength, α, that gives the strength of interfacial tension compared to other force-generating mechanisms; the tension ratio, δ, between the different interfaces; and the rate of tension variation, λ, which determines the timescale of the whole process.     

Effect of normalization and phase angle calculations on continuous relative phase

The purpose of this investigation was to determine if phase plot normalization and phase angle definitions would have an affect on continuous relative phase calculations. A subject ran on a treadmill while sagittal plane kinematic data were collected with a high-speed (180 Hz) camera. Segmental angular displacements and velocities were used to create phase plots, and examine the coordination between the leg and thigh. Continuous relative phase was calculated with a combination of two different amplitude normalization techniques, and two different phase angle definitions. Differences between the techniques were noted with a root mean square (RMS) calculation. RMS values indicated that there were differences in the configuration of the non-normalized and normalized continuous relative phase curves. Graphically and numerically, it was noted that normalization tended to modify the continuous relative phase curve configuration. Differences in continuous relative phase curves were due to a loss in the aspect ratio of the phase plot during normalization. Normalization tended to neglect the nonlinear forces acting on the system since it did not maintain the aspect ratio of the phase plot. Normalization is not necessary because the arc tangent function accounts for differences in amplitudes between the segments. RMS values indicated that there were profound differences in the continuous relative phase curve when the phase angle was normalized and a phase angle was calculated relative to the right horizontal axis.     

Thermodynamics of mechanosensitivity

Mechanosensitivity of ion channels is conventionally interpreted as being driven by a change of their in-plane cross-sectional area A(msc). This, however, does not include any factors relating to membrane stiffness, thickness, spontaneous curvature or changes in channel shape, length or stiffness. Because the open probability of a channel is sensitive to all these factors, we constructed a general thermodynamic formalism. These equations provide the basis for the analysis of the behaviour of mechanosensitive channels in lipids of different geometric and chemical properties such as the hydrophobic mismatch at the boundary between the protein and lipid or the effects of changes in the bilayer intrinsic curvature caused by the adsorption of amphipaths. This model predicts that the midpoint gamma(1/2) and the slope(1/2) of the gating curve are generally not independent. Using this relationship, we have predicted the line tension at the channel/lipid border of MscL as approximately 10 pN, and found it to be much less than the line tension of aqueous pores in pure lipid membranes. The MscL channel appears quite well matched to its lipid environment. Using gramicidin as a model system, we have explained its observed conversion from stretch-activated to stretch-inactivated gating as a function of bilayer thickness and composition.     

Interaction of a peptide model of a hydrophobic transmembrane alpha-helical segment of a membrane protein with phosphatidylcholine bilayers: differential scanning calorimetric and FTIR spectroscopic studies.

High-sensitivity differential scanning calorimetry (DSC) and Fourier transform infrared (FTIR) spectroscopy were used to study the interaction of a synthetic model hydrophobic peptide, Lys2-Gly-Leu24-Lys2-Ala-amide, and members of the homologous series of n-saturated diacylphosphatidylcholines. In the low range of peptide mole fractions, the DSC thermograms exhibited by the lipid/peptide mixtures are resolvable into two components. One of these components is fairly narrow, highly cooperative, and exhibits properties which are similar to but not identical with those of the pure lipid. In addition, the fractional contribution of this component to the total enthalpy change, the peak transition temperature, and cooperativity decrease with an increase in peptide concentration, more or less independently of acyl chain length. The other component is very broad and predominates in the high range of peptide concentration. These two components have been assigned to the chain-melting phase transitions of populations of bulk lipid and peptide-associated lipid, respectively. Moreover, when the mean hydrophobic thickness of the PC bilayer is less than the peptide hydrophobic length, the peptide-associated lipid melts at higher temperatures than does the bulk lipid and vice versa. In addition, the chain-melting enthalpy of the broad endotherm does not decrease to zero even at high peptide concentrations, suggesting that this peptide reduces but do not abolish the cooperative gel/liquid-crystalline phase transition of the lipids with which it is in contact. Our DSC results indicate that the width of the phase transition observed at high peptide concentration is inversely but discontinuously related to hydrocarbon chain length and that gel phase immiscibility occurs when the hydrophobic thickness of the bilayer greatly exceeds the hydrophobic length of the peptide. The FTIR spectroscopic data indicate that the peptide forms a very stable alpha-helix under all of our experimental conditions but that small distortions of its alpha-helical conformation are induced in response to any mismatch between peptide hydrophobic length and bilayer hydrophobic thickness. These results also indicate that the peptide alters the conformational disposition of the acyl chains in contact with it and that the resultant conformational changes in the lipid hydrocarbon chains tend to minimize the extent of mismatch of peptide hydrophobic length and bilayer hydrophobic thickness.     

Efficient tuning of potential parameters for liquid–solid interactions

Spherical and cylindrical water droplets on silicon surface are studied to tune the silicon–oxygen interaction. We use molecular dynamics simulations to estimate the contact angle of two different shaped droplets. We found that the cylindrical droplets are independent of the line tension as their three phases curvature is equal zero. Additionally, we compare an analytical model, taking into account or not the Tolman length and we show that for spherical small size droplets, this length is important to be included, in contrast to cylindrical droplets in which the influence of the Tolman length is negligible. We demonstrate that the usual convenient way to exclude linear tension in the general case can give wrong results. Here, we consider cylindrical droplets, since their contact angle does not depend on the droplet size in the range of few to 10ths of nanometres. The droplets are stabilised due to the periodic boundary conditions. This allows us to propose a new parameterisation for nanoscale droplets, which is independent the size of the droplets or its shape, minimising at the same time the calculation procedure. With the proposed methodology, we can extract the epsilon parameter of the interaction potential between a liquid and a solid from the nanoscaled molecular simulation with only as input the macrosized experimental wetting angle for a given temperature.     

Detailed mechanics of membrane-membrane adhesion and separation. I. Continuum of molecular cross-bridges

The mechanics of membrane-membrane adhesion are developed for the approximation that the molecular cross-bridging forces are continuously distributed as a normal stress (force per unit area). The significance of the analysis is that the finite range of the cross-bridging forces and the microscopic contact angle are not assumed negligible. Since the cross-bridging and adhesion forces are finite range interactions, there are two membrane regions: a free zone where the membranes are not subject to attractive forces; and an adherent zone where the membranes are held together by attractive stresses. The membrane is treated as an elastic continuum. The approach is to analyze the mechanics for each zone separately and then to require continuity of the solutions at the interface between the zones. Final solution yields the membrane contour and stresses proximal to and within the contact zone as well as the microscopic contact angle at the edge of the contact zone. It is demonstrated that the classical Young equation is consistent with this model. The results show that the microscopic contact angle becomes appreciable when the strength of adhesion is large or the length of the cross-bridge is large; however, the microscopic contact angle approaches zero as the membrane elastic stiffness increases. The solution predicts the width of the contact zone over which molecular bonds are stretched. It is this boundary region where increased biochemical activity is expected. In the classical model presented here, the level of tension necessary to oppose spreading of the contact is equal to the minimal level of tension required to separate the adherent membranes. This behavior is in contrast with that derived for the case of discrete molecular cross-bridges where the possibility of different levels of tension associated with adhesion and separation is introduced. The discrete cross-bridge case is the subject of a companion paper.     

Gramicidin channel kinetics under tension.

We have measured the effect of tension on dimerization kinetics of the channel-forming peptide gramicidin A. By aspirating large unilamellar vesicles into a micropipette electrode, we are able to simultaneously monitor membrane tension and electrical activity. We find that the dimer formation rate increases by a factor of 5 as tension ranges from 0 to 4 dyn/cm. The dimer lifetime also increases with tension. This behavior is well described by a phenomenological model of membrane elasticity in which tension modulates the mismatch in thickness between the gramicidin dimer and membrane.     

Bordered pit structure and vessel wall surface properties. Implications for embolism repair

The idea that embolized xylem vessels can be refilled while adjacent vessels remain under tension is difficult to accept if the cavitated vessels remain hydraulically connected to vessels under tension. A mechanism by which embolized conduits could be hydraulically isolated from adjacent conduits requires the existence of a non-zero contact angle and a flared opening into the bordered pit chamber such that a convex air-water interface forms at the entrance into the pit chamber. We measured the contact angle and pit chamber geometry for six species. The contact angle measured in the vessel lumen ranged between 42 degrees to 55 degrees, whereas the opening into the pit chamber ranged between 144 degrees and 157 degrees. If the surface properties within the pit chamber are similar to those in the lumen, a convex meniscus will form at the flared opening into the pit chamber. The maximum pressure difference between water in the lumen and gas in the pit chamber that could be stabilized by this interface was calculated to be within the range of 0.07 to 0.30 MPa.