Determination of bilayer membrane bending stiffness by tether formation from giant, thin-walled vesicles.

The curvature elastic modulus (bending stiffness) of stearoyloleoyl phosphatidylcholine (SOPC) bilayer membrane is determined from membrane tether formation experiments. R. E. Waugh and R. M. Hochmuth 1987. Biophys. J. 52:391-400) have shown that the radius of a bilayer cylinder (tether) is inversely related to the force supported along its axis. The coefficient that relates the axial force on the tether to the tether radius is the membrane bending stiffness. Thus, the bending stiffness can be calculated directly from measurements of the tether radius as a function of force. Giant (10-50-microns diam) thin-walled vesicles were aspirated into a micropipette and a tether was pulled out of the surface by gravitational forces on small glass beads that had adhered to the vesicle surface. Because the vesicle keeps constant surface area and volume, formation of the tether requires displacement of material from the projection of the vesicle in the pipette. Tethers can be made to grow longer or shorter or to maintain equilibrium by adjusting the aspiration pressure in the micropipette at constant tether force. The ratio of the change in the length of the tether to the change in the projection length is proportional to the ratio of the pipette radius to the tether radius. Thus, knowing the density and diameter of the glass beads and measuring the displacement of the projection as a function of tether length, independent determinations of the force on the tether and the tether radius were obtained. The bending stiffness for an SOPC bilayer obtained from these data is approximately 2.0 x 10(-12) dyn cm, for tether radii in the range of 20-100 nm.(ABSTRACT TRUNCATED AT 250 WORDS)     

Characterizing Cell Adhesion by Using Micropipette Aspiration

We have developed a technique to directly quantify cell-substrate adhesion force using micropipette aspiration. The micropipette is positioned perpendicular to the surface of an adherent cell and a constant-rate aspiration pressure is applied. Since the micropipette diameter and the aspiration pressure are our control parameters, we have direct knowledge of the aspiration force, whereas the cell behavior is monitored either in brightfield or interference reflection microscopy. This setup thus allows us to explore a range of geometric parameters, such as projected cell area, adhesion area, or pipette size, as well as dynamical parameters such as the loading rate. We find that cell detachment is a well-defined event occurring at a critical aspiration pressure, and that the detachment force scales with the cell adhesion area (for a given micropipette diameter and loading rate), which defines a critical stress. Taking into account the cell adhesion area, intrinsic parameters of the adhesion bonds, and the loading rate, a minimal model provides an expression for the critical stress that helps rationalize our experimental results.     

The human erythrocyte membrane skeleton may be an ionic gel. III. Micropipette aspiration of unswollen erythrocytes

We have carried out a theoretical analysis of micropipette aspiration of unswollen erythrocytes using the protein-gel-lipid-bilayer membrane model and taking into account that the modulus of area compression of the membrane skeleton may depend on the environmental conditions. Our analysis shows that the aspiration pressure needed to obtain a certain membrane projection length is strongly dependent on the ratio between the membrane skeleton modulus of area compression and the elastic shear modulus. Our analysis therefore predicts that micropipette aspiration of unswollen erythrocytes may be a sensitive method for detection of changes in this ratio. The analysis thus also shows that micropipette aspiration of unswollen erythrocytes can not be used to determine the membrane shear modulus unless something is known about the membrane skeleton modulus of area compression.     

The behavior of human neutrophils during flow through capillary pores

The passage times of individual human neutrophils through single capillary-sized pores in polycarbonate membranes were measured with the resistive pulse technique, and results were compared to those obtained from the micropipette aspiration of entire cells. Pore transit measurement serves as a useful means to screen populations of cells, and allows for protocols that measure time dependent changes to the population. Neutrophils exhibited a highly linear pressure/flow rate relationship at aspiration pressures from 200 Pa to 1,500 Pa in both the pore and pipette systems. Cellular viscosity, as determined by the method of Hochmuth and Needham, was 89.0 Pa.s for the pore systems and 134.9 Pa.s for the pipette systems. These results are in general agreement with recent values of neutrophil viscosity published in the literature. Extrapolation of the observed linear flow response revealed an apparent minimum pressure for whole cell aspiration significantly above the threshold pressure predicted by Evans' liquid drop model. However, whole cell aspiration was achieved in both the pore and pipette systems at pressures below this extrapolated minimum, although the calculated cellular viscosity was greatly increased. The implications of these two regimes of cell deformation is unclear. This behavior could be explained by shear thinning of the material in the cell body. However the origin of this phenomenon may be in the cortical region of the cell, which exhibits an elastic tension that may be deformation rate dependent.     

An axisymmetric boundary integral model for incompressible linear viscoelasticity: application to the micropipette aspiration contact problem

The micropipette aspiration test has been used extensively in recent years as a means of quantifying cellular mechanics and molecular interactions at the microscopic scale. However, previous studies have generally modeled the cell as an infinite half-space in order to develop an analytical solution for a viscoelastic solid cell. In this study, an axisymmetric boundary integral formulation of the governing equations of incompressible linear viscoelasticity is presented and used to simulate the micropipette aspiration contact problem. The cell is idealized as a homogeneous and isotropic continuum with constitutive equation given by three-parameter (E, tau 1, tau 2) standard linear viscoelasticity. The formulation is used to develop a computational model via a "correspondence principle" in which the solution is written as the sum of a homogeneous (elastic) part and a nonhomogeneous part, which depends only on past values of the solution. Via a time-marching scheme, the solution of the viscoelastic problem is obtained by employing an elastic boundary element method with modified boundary conditions. The accuracy and convergence of the time-marching scheme are verified using an analytical solution. An incremental reformulation of the scheme is presented to facilitate the simulation of micropipette aspiration, a nonlinear contact problem. In contrast to the halfspace model (Sato et al., 1990), this computational model accounts for nonlinearities in the cell response that result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region, and curvature of the edges of the micropipette (inner radius a, edge curvature radius epsilon). Using 60 quadratic boundary elements, a micropipette aspiration creep test with ramp time t* = 0.1 s and ramp pressure p*/E = 0.8 is simulated for the cases a/R = 0.3, 0.4, 0.5 using mean parameter values for primary chondrocytes. Comparisons to the half-space model indicate that the computational model predicts an aspiration length that is less stiff during the initial ramp response (t = 0-1 s) but more stiff at equilibrium (t = 200 s). Overall, the ramp and equilibrium predictions of aspiration length by the computational model are fairly insensitive to aspect ratio a/R but can differ from the half-space model by up to 20 percent. This computational approach may be readily extended to account for more complex geometries or inhomogeneities in cellular properties.     

Passive mechanical behavior of human neutrophils: power-law fluid.

The mechanical behavior of the neutrophil plays an important role in both the microcirculation and the immune system. Several laboratories in the past have developed mechanical models to describe different aspects of neutrophil deformability. In this study, the passive mechanical properties of normal human neutrophils have been further characterized. The cellular mechanical properties were assessed by single cell micropipette aspiration at fixed aspiration pressures. A numerical simulation was developed to interpret the experiments in terms of cell mechanical properties based on the Newtonian liquid drop model (Yeung and Evans, Biophys. J., 56: 139-149, 1989). The cytoplasmic viscosity was determined as a function of the ratio of the initial cell size to the pipette radius, the cortical tension, aspiration pressure, and the whole cell aspiration time. The cortical tension of passive neutrophils was measured to be about 2.7 x 10(-5) N/m. The apparent viscosity of neutrophil cytoplasm was found to depend on aspiration pressure, and ranged from approximately 500 Pa.s at an aspiration pressure of 98 Pa (1.0 cm H2O) to approximately 50 Pa.s at 882 Pa (9.0 cm H2O) when tested with a 4.0-micron pipette. These data provide the first documentation that the neutrophil cytoplasm exhibits non-Newtonian behavior. To further characterize the non-Newtonian behavior of human neutrophils, a mean shear rate gamma m was estimated based on the numerical simulation. The apparent cytoplasmic viscosity appears to decrease as the mean shear rate increases. The dependence of cytoplasmic viscosity on the mean shear rate can be approximated as a power-law relationship described by mu = mu c(gamma m/gamma c)-b, where mu is the cytoplasmic viscosity, gamma m is the mean shear rate, mu c is the characteristic viscosity at characteristic shear rate gamma c, and b is a material coefficient. When gamma c was set to 1 s-1, the material coefficients for passive neutrophils were determined to be mu c = 130 +/- 23 Pa.s and b = 0.52 +/- 0.09 for normal neutrophils. The power-law approximation has a remarkable ability to reconcile discrepancies among published values of the cytoplasmic viscosity measured using different techniques, even though these values differ by nearly two orders of magnitude. Thus, the power-law fluid model is a promising candidate for describing the passive mechanical behavior of human neutrophils in large deformation. It can also account for some discrepancies between cellular behavior in single-cell micromechanical experiments and predictions based on the assumption that the cytoplasm is a simple Newtonian fluid.     

Role of lamellar membrane structure in tether formation from bilayer vesicles.

A theoretical analysis is presented of the formation of membrane tethers from micropipette-aspirated phospholipid vesicles. In particular, it is taken into account that the phospholipid membrane is composed of two layers which are in contact but unconnected. The elastic energy of the bilayer is taken to be the sum of contributions from area expansivity, relative expansivity of the two monolayers, and bending. The vesicle is aspirated into a pipette and a constant point force is applied at the opposite side in the direction away from the pipette. The shape of the vesicle in approximated as a cylindrical projection into the pipette with a hemispherical cap, a spherical section, and a cylindrical tether with a hemispherical cap. The dimensions of the different regions of the vesicle are obtained by minimizing its elastic energy subject to the condition that the volume of the vesicle is fixed. The range of values for the parameters of the system is determined at which the existence of a tether is possible. Stability analysis is performed showing which of these configurations are stable. The importance of the relative expansion and compression of the constituent monolayers is established by recognizing that local bending energy by itself does not stabilize the vesicle geometry, and that in the limit as the relative expansivity modulus becomes infinitely large, a tether cannot be formed. Predictions are made for the functional relationships among experimentally observable quantities. In a companion report, the results of this analysis are applied to experimental measurements of tether formation, and used to calculate values for the membrane material coefficients.     

Bending elastic modulus of red blood cell membrane derived from buckling instability in micropipet aspiration tests.

Observation of cell membrane buckling and cell folding in micropipette aspiration experiments was used to evaluate the bending rigidity of the red blood cell membrane. The suction pressure required to buckle the membrane surface initially was found to be about one-half to two-thirds of the pressure that caused the cell to fold and move up the pipet. A simple analytical model for buckling of a membrane disk supported at inner and outer radii correlates well with the observed buckling pressures vs. pipet radii. The buckling pressure is predicted to increase in inverse proportion to the cube of the pipet radius; also, the buckling pressure depends inversely on the radial distance to the toroidal rim of the cell, normalized by the pipet radius. As such, the pressure required to buckle the membrane with 1 X 10(-4) cm diam pipet would be about four times greater than with a 2 X 10(-4) cm pipet. This is the behavior observed experimentally. Based on analysis of the observed buckling data, the membrane bending or curvature elastic modulus is calculated to be 1.8 X 10(-12) dyn-cm.     

Adhesivity and rigidity of erythrocyte membrane in relation to wheat germ agglutinin binding

Binding of the plant lectin wheat germ agglutinin (WGA) to erythrocyte membranes causes membrane rigidification. One of our objectives has been to directly measure the effects of WGA binding on membrane rigidity and to relate rigidification to the kinetics and levels of WGA binding. Our other objective has been to measure the strength of adhesion and mechanics of cell separation for erythrocytes bound together by WGA. The erythrocyte membrane rigidity was measured on single cells by micropipette aspiration. The slope of the suction pressure-length data for entry into the pipette provided the measure of the membrane extensional modulus. Data were collected for cells equilibrated with WGA solutions in the range of concentrations of 0.01- 10 micrograms/ml. Erythrocyte-erythrocyte adherence properties were studied by micropipette separation of two-cell aggregates. First, a "test" cell was selected from a WGA solution by aspiration into a small micropipette, then transferred to a separate chamber that contained erythrocytes in WGA-free buffer. Here, a second cell was aspirated with another pipette and maneuvered into close proximity of the test cell surface, and adhesive contact was produced. The flaccid cell was separated from the test cell surface in steps at which the force of attachment was derived from the pipette suction pressure and cell geometry. In addition, we measured the time-dependent binding and release of fluorescently labeled WGA to single erythrocytes with a laser microfluorometry system. The results showed that the stiffening of the erythrocyte membrane and binding of fluorescently labeled WGA to the membrane surface followed the same concentration and time dependencies. The threshold concentration for membrane stiffening was at approximately 0.1 microgram/ml where the time course to reach equilibrium was close to 1 h. The maximal stiffening (almost 30-fold over the normal membrane elastic modulus) occurred in concentrations greater than 2 micrograms/ml where the time to reach equilibrium took less than 1 min. The WGA binding also altered the normal elastic membrane behavior into an inelastic, plastic-like response which indicated that mechanical extension of the membrane caused an increase in cross-linking within the surface plane. Similar to the stiffening effect, we observed that the membrane adhesivity of cells equilibrated with WGA solutions greatly increased with concentration greater than 0.1 microgram/ml.(ABSTRACT TRUNCATED AT 400 WORDS)     

Time-dependent recovery of passive neutrophils after large deformation.

Experiments are performed in which a passive human neutrophil is deformed into an elongated "sausage" shape by aspirating it into a small glass pipette. When expelled from the pipette the neutrophil recovers its natural spherical shape in approximately 1 minute. This recovery process is analyzed according to a Newtonian, liquid-drop model in which a variational method is used to simultaneously solve the hydrodynamic equations for low Reynolds-number flow and the equations for membrane equilibrium with a constant membrane tension. The theoretical model gives a good fit to the experimental data for a ratio of membrane cortical tension to cytoplasmic viscosity of approximately 1.7 x 10(-5) cm/s (0.17 micron/s). However, when the cell is held in the pipette for only a short time period of 5 s or less, and then expelled, the cell undergoes an initial, rapid elastic rebound suggesting that the cell behaves in this instance as a Maxwell viscoelastic liquid rather than a Newtonian liquid with constant cortical tension.     

Electro-mechanical permeabilization of lipid vesicles. Role of membrane tension and compressibility.

A simple micropipet technique was used to determine the critical electric field strength for membrane breakdown as a function of the applied membrane tension for three different reconstituted membranes: stearoyloleoylphosphatidylcholine (SOPC), red blood cell (RBC) lipid extract, and SOPC cholesterol (CHOL), 1:1. For these membranes the elastic area expansivity modulus increases from approximately 200 to 600 dyn/cm, and the tension at lysis increases from 5.7 to 13.2 dyn/cm, i.e., the membranes become more cohesive with increasing cholesterol content. The critical membrane voltage, Vc, required for breakdown was also found to increase with increasing cholesterol from 1.1 to 1.8 V at zero membrane tension. We have modeled the behavior in terms of the bilayer expansivity. Membrane area can be increased by either tensile or electrocompressive stresses. Both can store elastic energy in the membrane and eventually cause breakdown at a critical area dilation or critical energy. The model predicts a relation between tension and voltage at breakdown and this relation is verified experimentally for the three reconstituted membrane systems studied here.     

Mechanical equilibrium of thick, hollow, liquid membrane cylinders.

The mechanical equilibrium of bilayer membrane cylinders is analyzed. The analysis is motivated by the observation that mechanically formed membrane strands (tethers) can support significant axial loads and that the tether radius varies inversely with the axial force. Previously, thin shell theory has been used to analyze the tether formation process, but this approach is inadequate for describing and predicting the equilibrium state of the tether itself. In the present work the membrane is modeled as two adjacent, thick, anisotropic liquid shells. The analysis predicts an inverse relationship between axial force and tether radius, which is consistent with experimental observation. The area expansivity modulus and bending stiffness of the tether membrane are calculated using previously measured values of tether radii. These calculated values are consistent with values of membrane properties measured previously. Application of the analysis to precise measurements of the relationship between tether radius and axial force will provide a novel method for determining the mechanical properties of biomembrane.     

Micropipette aspiration method for characterizing biological materials with surface energy

Many soft biological tissues possess a considerable surface stress, which plays a significant role in their biophysical functions, but most previous methods for characterizing their mechanical properties have neglected the effects of surface stress. In this work, we investigate the micropipette aspiration method to measure the mechanical properties of soft tissues and cells with surface effects. The neo-Hookean constitutive model is adopted to describe the hyperelasticity of the measured biological material, and the surface effect is taken into account by the finite element method. It is found that when the pipette radius or aspiration length is comparable to the elastocapillary length, surface energy may distinctly alter the aspiration response. Generally, both the aspiration length and the bulk normal stress decrease with increasing surface energy, and thus neglecting the surface energy would lead to an overestimation of elastic modulus. Through dimensional analysis and numerical simulations, we provide an explicit relation between the imposed pressure and the aspiration length. This method can be applied to determine the mechanical properties of soft biological tissues and organs, e.g., livers, tumors and embryos.     

A power-law rheology-based finite element model for single cell deformation

Physical forces can elicit complex time- and space-dependent deformations in living cells. These deformations at the subcellular level are difficult to measure but can be estimated using computational approaches such as finite element (FE) simulation. Existing FE models predominantly treat cells as spring-dashpot viscoelastic materials, while broad experimental data are now lending support to the power-law rheology (PLR) model. Here, we developed a large deformation FE model that incorporated PLR and experimentally verified this model by performing micropipette aspiration on fibroblasts under various mechanical loadings. With a single set of rheological properties, this model recapitulated the diverse micropipette aspiration data obtained using three protocols and with a range of micropipette sizes. More intriguingly, our analysis revealed that decreased pipette size leads to increased pressure gradient, potentially explaining our previous counterintuitive finding that decreased pipette size leads to increased incidence of cell blebbing and injury. Taken together, our work leads to more accurate rheological interpretation of micropipette aspiration experiments than previous models and suggests pressure gradient as a potential determinant of cell injury.     

Thermoelasticity of red blood cell membrane.

The elastic properties of the human red blood cell membrane have been measured as functions of temperature. The area compressibility modulus and the elastic shear modulus, which together characterize the surface elastic behavior of the membrane, have been measured over the temperature range of 2-50 degrees C with micropipette aspiration of flaccid and osmotically swollen red cells. In addition, the fractional increase in membrane surface area from 2-50 degrees C has been measured to give a value for the thermal area expansivity. The value of the elastic shear modulus at 25 degrees C was measured to be 6.6 X 10(-3) dyne/cm. The change in the elastic shear modulus with temperature was -6 X 10(-5) dyne/cm degrees C. Fractional forces were shown to be only on the order of 10-15%. The area compressibility modulus at 25 degrees C was measured to be 450 dyne/cm. The change in the area compressibility modulus with temperature was -6 dyne/cm degrees C. The thermal area expansivity for red cell membrane was measured to be 1.2 X 10(-3)/degrees C. With this data and thermoelastic relations the heat of expansion is determined to be 110-200 ergs/cm2; the heat of extension is 2 X 10(-2) ergs/cm2 for unit extension of the red cell membrane. The heat of expansion is of the order anticipated for a lipid bilayer idealized as twice the behavior of a monolayer at an oil-water interface. The observation that the heat of extension is positive demonstrates that the entropy of the material increases with extension, and that the dominant mechanism of elastic energy storage is energetic. Assuming that the red cell membrane shear rigidity is associated with "spectrin," unit extension of the membrane increases the configurational entropy of spectrin by 500 cal/mol.     

Estimation of viscous dissipation inside an erythrocyte during aspirational entry into a micropipette.

Viscous dissipation inside the erythrocyte during its aspirational entry into a micropipette is analyzed. The motion of the intracellular fluid is approximated by a flow into the micropipette orifice from a half space (the portion of the erythrocyte outside the micropipette). The stream function and intracellular pressure (p) in the half space are obtained as a function of radial and axial positions near the orifice. Solution of the boundary value problem for a uniform stream entering a circular hole gives p = 2 eta HQ/pi R3p, where eta H is the intracellular viscosity, Q is the total discharge, and Rp is the pipette radius. The results indicate that the moving erythrocyte membrane helps to drive the intracellular fluid into the orifice. For normal erythrocytes, p is only approximately 0.5% of the total aspiration pressure (delta P). The contribution of p to delta P, however, may become significant when there is a large increase in eta H due to a markedly elevated intracellular hemoglobin concentration or an alteration of the physical state of hemoglobin.     

Role of the membrane cortex in neutrophil deformation in small pipets.

The simplest model for a neutrophil in its "passive" state views the cell as consisting of a liquid-like cytoplasmic region surrounded by a membrane. The cell surface is in a state of isotropic contraction, which causes the cell to assume a spherical shape. This contraction is characterized by the cortical tension. The cortical tension shows a weak area dilation dependence, and it determines the elastic properties of the cell for small curvature deformations. At high curvature deformations in small pipets (with internal radii less than 1 micron), the measured critical suction pressure for cell flow into the pipet is larger than its estimate from the law of Laplace. A model is proposed where the region consisting of the cytoplasm membrane and the underlying cortex (having a finite thickness) is introduced at the cell surface. The mechanical properties of this region are characterized by the apparent cortical tension (defined as a free contraction energy per unit area) and the apparent bending modulus (introduced as a bending free energy per unit area) of its middle plane. The model predicts that for small curvature deformations (in pipets having radii larger than 1.2 microns) the role of the cortical thickness and the resistance for bending of the membrane-cortex complex is negligible. For high curvature deformations, they lead to elevated suction pressures above the values predicted from the law of Laplace. The existence of elevated suction pressures for pipets with radii from 1 micron down to 0.24 micron is found experimentally. The measured excess suction pressures cannot be explained only by the modified law of Laplace (for a cortex with finite thickness and negligible bending resistance), because it predicts unacceptable high cortical thicknesses (from 0.3 to 0.7 micron). It is concluded that the membrane-cortex complex has an apparent bending modulus from 1 x 10(-18) to 2 x 10(-18) J for a cortex with a thickness from 0.1 micron down to values much smaller than the radius of the smallest pipet (0.24 micron) used in this study.     

Optical evaluation of red blood cell geometry using micropipette aspiration.

Although red blood cell (RBC) geometry has been extensively studied by micropipette aspiration, the small size of RBC and pipettes vs. the optical resolution of light microscopy suggests the need to consider potential errors. The present study addressed such difficulties and investigated four specific problems: (1) use of exact equations to calculate RBC membrane area and volume; (2) calibration of the pipette internal diameter (PID); (3) correction for a noncylindrical pipette barrel; (4) diffraction distortion of the RBC image. The observed PID represents a cylinder lens enlargement that can be theoretically derived from the glass/buffer refractive index ratio (1.49/1.33 = 1.12). This enlargement was experimentally confirmed by: (1) studying pipettes bent to allow measurement through the barrel (horizontal) and at the orifice (vertical), with a resulting diameter ratio of 1.12 +/- 0.01; (2) and by replacing the surrounding buffer with immersion oil and hence abolishing the lens phenomenon (ratio = 1.12 +/- 0.02). In addition, use of aspirated oil droplets demonstrated a 3.2 +/- 0.2% error when the PID is focused at a sharp, maximum diameter. The average pipette cone angle was 1.49 +/- 0.09 degrees and varied considerably with pipette pulling procedures; calculated tongue geometry inside the pipette was affected by the noncylindrical pipette barrel. The RBC diffraction error, demonstrated by touching two aspirated cells held by opposing pipettes, was 0.091 +/- 0.002 microns. The PID, cone angle, and diffraction artifacts significantly (p < 0.001) affected calculated RBC geometry (average errors up to 5.4% for area and 9.6% for volume). Two new methods to calculate, rather than directly measure, the PID from images of a single RBC, during either osmotic or pressure manipulation, were evaluated; the osmotic method closely predicted the PID, whereas the pressure method markedly underestimated the PID. Our results thus confirm the need to consider the above-mentioned phenomena when determining RBC geometric parameters via micropipette aspiration.     

Viscoelastic properties of transformed cells: role in tumor cell progression and metastasis formation

The micropipette aspiration technique was used to investigate the deformation properties of a panel of nontransformed and transformed rat fibroblasts derived from the same normal cell line. In this method, a step negative pressure is applied to the cell via a micropipette and the aspiration distance into the pipette as a function of time is determined using video techniques. A standard solid viscoelastic model was then used to analyze the viscoelastic properties of the cell. From these results, it is concluded that a direct correlation exists between an increase in deformability and progression of the transformed phenotype from a nontumorigenic cell line into a tumorigenic, metastatic cell line.