Extensional flow of erythrocyte membrane from cell body to elastic tether. I. Analysis

This is the first of two papers on an analytical and experimental study of the flow of the erythrocyte membrane. In the experiment to be discussed in detail in the second paper, preswollen human erythrocytes are sphered by aspirating a portion of the cell membrane into a small micropipette; and long, thin, membrane filaments or "tethers" are steadily withdrawn from the cell at a point diametrically opposite to the point of aspiration. The aspirated portion of the membrane furnished a "reservoir" of material that replaces the membrane as it flows as a liquid from the nearly spherical cell body to the cylindrical tether. In this paper we show that an application of the principle of conservation of mass permits the tether radius (approximately 200 A or less) to be measured with the light microscope as the tether is formed and extended at a constant rate. A static analysis of the axisymmetric cell deformation and tether formation process reveals that the tether radius is uniquely determined by the isotropic tension in the membrane and the elastic constitutive (material) behavior of the tether itself. A dynamic analysis of the extensional flow process reveals that the tether radius must decrease as the velocity of the tether is increased and that the decrease depends on both the viscosity of the membrane and the elasticity of the tether. The analysis also shows that these two factors (membrane viscosity and tether elasticity) are readily decomposed and determined separately when flow experiments are performed at different isotropic tensions.

Extensional flow of erythrocyte membrane from cell body to elastic tether. II. Experiment

This is the second of two papers on an analytical and experimental study of the flow of erythrocyte membrane. In the experiments discussed here, preswollen human erythrocytes are sphered by aspirating a portion of the cell membrane into a small micropipette; and long, thin, membrane filaments or tethers are steadily withdrawn from the cell at a point diametrically opposite to the point of aspiration. The aspirated portion of the membrane furnishes a reservoir of material that replaces the membrane as it flows as a liquid from the nearly spherical cell body to the cylindrical tether. The application of the principle of conservation of mass permits the tether radius Rt to be measured with the light microscope as the tether is formed and extended at a constant rate. The tether behaves as an elastic solid such that the tether radius decreases as the force or axial tension acting on the tether is increased. For the range of values for Rt is these experiments (100 A less than or equal to Rt less than or equal to 200 A), the slope of the tether-force, tether-radius line is -1.32 dyn/cm. The surface viscosity of the membrane as it flows from cell body to tether is 3 x 10(-3) dyn.s/cm. This viscosity is essentially constant for characteristic rates of deformation between 10 and 200 s-1.

Analysis of adhesion of large vesicles to surfaces

An experimental procedure that can be used to measure the interfacial free energy density for the adhesion of membranes of large vesicles to other surfaces is outlined and analyzed. The approach can be used for both large phospholipid bilayer vesicles and red blood cells when the membrane force resultants are dominated by isotropic tension. The large vesicle or red cell is aspirated by a micropipet with sufficient suction pressure to form a spherical segment outside the pipet. The vesicle is then brought into close proximity of the surface to be tested and, the suction pressure reduced to permit adhesion, and the new equilibrium configuration is established. The mechanical analysis of the equilibrium shape provides the interfacial free energy density for the surface affinity. With this approach, the measurable range of membrane surface affinity is 10(-4)-3 erg/cm2 for large phospholipid bilayer vesicles and 10(-2)-10 erg/cm2 for red blood cells.

Calibration of beam deflection produced by cellular forces in the 10(-9)--10(-6) gram range

An experimental procedure and method of analysis are presented for calibration of a thin-beam force transducer. The beam transducer can be produced and calibrated with a minimum coefficient of 10 ng (10(-5) dyne) force per micron (10(-4) cm) deflection, i.e., kB approximately 0.1 dyne/cm. Since beam deflections on the order of 0.1 micron can be detected, forces of a few nanograms can be resolved. Such forces are common in mechanical experiments on microscopic bodies, e.g., biological cells, artificial membrane capsules, droplets, etc.

Minimum energy analysis of membrane deformation applied to pipet aspiration and surface adhesion of red blood cells

An experimental procedure is demonstrated which can be used to determine the interfacial free energy density for red cell membrane adhesion and membrane elastic properties. The experiment involves micropipet aspiration of a flaccid red blood cell and manipulation of the cell proximal to a surface where adhesion occurs. A minimum free energy method is developed to model the equilibrium contour of unsupported membrane regions and to evaluate the partial derivatives of the total free energy, which correspond to the micropipet suction force and the interfacial free energy density of adhesion. It is shown that the bending elasticity of the red cell membrane does not contribute significantly to the pressure required to aspirate a flaccid red cell. Based on experimental evidence, the upper bound for the bending or curvature elastic modulus of the red cell membranes is 10-12 ergs (dyn-cm). Analysis of the adhesion experiment shows that interfacial free energy densities for red cell adhesion can be measured from a lower limit of 10-4 ergs/cm2 to an upper limit established by the membrane tension for lysis of 5-10 ergs/cm2.

Temperature dependence of the viscoelastic recovery of red cell membrane

The time-dependent recovery of an elongated red cell is studied as a function of temperature. Before release, the elongated cell is in static equilibrium where external forces are balanced by surface elastic force resultants. Upon release, the cell recovers its initial shape with a time-dependent exponential behavior characteristic of a viscoelastic solid material undergoing large ("finite") deformation. The recovery process is characterized by a time constant, tc, that decreases from approximately 0.27 s at 6 degrees C to 0.06 s at 37 degrees C. From this measurement of the time constant and an independent measurement of the shear modulus of surface elasticity for red cell membrane, the value for the membrane surface viscosity as a function of temperature can be calculated.

Red cell extensional recovery and the determination of membrane viscosity

A theory of membrane viscoelasticity developed by Evans and Hochmuth in 1976 is used to analyze the time-dependent recovery of an elongated cell. Before release, the elongated cell is the static equilibrium where external forces are balanced by membrane elastic force resultants. Upon release, the cell recovers its initial shape with a time-dependent exponential behavior characteristic of the viscoelastic solid model. It is shown that the model describes the time-dependent recovery process very well for a time constant in the range of 0.1-0.13 s. The time constant is the ratio membrane surface viscosity eta:membrane surface elasticity mu. Measurements for the shear modulus mu of 0.006 dyne/cm give a value for the surface viscosity of red cell membrane as a viscoelastic solid material of eta = mu tc = (6-8) X 10(-4) poise . cm.

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.

Membrane mechanical properties of ATP-depleted human erythrocytes

Although the relations between the metabolic state and the mechanical properties of human red blood cells (RBC) continue to be of current interest, literature reports in this area are not in agreement. The present investigation was designed to determine several intrinsic mechanical properties of human RBC membranes before and after metabolic depletion via incubation at 37 degrees C for 24 hr. Using micropipette and flow channel techniques, three properties were measured: (1) mu, surface shear modulus of elasticity; (2) K, elastic area compressibility modulus; (3) etap, shear viscosity in the plastic domain. Our results indicate no significant differences in these parameters between fresh and ATP-depleted human RBC membranes. These present data are thus in disagreement with other literature reports indicating large changes in membrane mechanical properties consequent to metabolic depletion. A brief discussion of the possible reasons for this disagreement is presented.

Correlation between handedness and birth order: compilation of five studies

Bakan has suggested that left-handedness is the result of left hemishperic pyramidal motor dysfunction following perinatal hypoxia. To a degree support for the validity of this hypothesis rests on Bakan's (1971, 1977a) findings that left-handed college students were more likely the progeny of birth orders designated as "high-risk" than right-handed students. Attempts by others to replicate Bakan's data have been unsuccessful. To achieve a more powerful test of this relationship than has been provided by any single study, the data from the five studies which have considered it were pooled and tested. The resulting correlation between birth order and handedness was near zero.

Osmotic correction to elastic area compressibility measurements on red cell membrane

In a recent article (Biophys. J. 16:585, 1976), we reported measurements of the elastic area compressibility modulus or red cell membranes using micropipette aspiration on osmotically preswollen red cells. Subsequently, we have analyzed the effects of osmotic and hydrostatic pressure driving forces across the cell membrane in conjuction with the mass conservation equation; we find that the change in cell volume due to the reversible movement of water out of the cell can produce one-third of the movement of the cell projection in the pipette tip. Since the actual volume changes is too small to measure directly (about 1% of the total cell volume), we have used an indirect experimental method to provide critical evaluation of the analysis of cell volume change versus applied pressure; this is based on the model that the change in cell volume is inversely proportional to the cellular osmotic strength. We have increased the cellular cation concentration with a drug, nystatin, and measured the elastic area compressibility modulus corrected for osmotic volume changes as a function of cellular osmotic strength. We find that the corrected elastic are compressibility modulus is independent of cellular osmotic strength, which supports the model and calculated correction for the osmotic effect. The elastic area compressibility modulus is 450 dyn/cm at 25 degrees C instead of 300 dyn/cm, determined previously.

Elastic area compressibility modulus of red cell membrane

Micropipette measurements of isotropic tension vs. area expansion in pre-swollen single human red cells gave a value of 288 +/- 50 SD dyn/cm for the elastic, area compressibility modulus of the total membrane at 25 degrees C. This elastic constant, characterizing the resistance to area expansion or compression, is about 4 X 10(4) times greater than the elastic modulus for shear rigidity; therefore, in situations where deformation of the membrane does not require large isotropic tensions (e.g., in passage through normal capillaries), the membrane can be treated by a simple constitutive relation for a two-dimensionally, incompressible material (i.e. fixed area). The tension was found to be linear and reversible for the range of area changes observed (within the experimental system resolution of 10%). The maximum fractional area expansion required to produce lysis was uniformly distributed between 2 and 4% with 3% average and 0.7% SD. By heating the cells to 50 degrees C, it appears that the structural matrix (responsible for the shear rigidity and most of the strength in isotropic tension) is disrupted and primarily the lipid bilayer resists lysis. Therefore, the relative contributions of the structural matrix and lipid bilayer to the elastic, area compressibility could be estimated. The maximum isotropic tension at 25 degrees C is 10-12 dyn/cm and at 50 degrees C is between 3 and 4 dyn/cm. From this data, the respective compressibilities are estimated at 193 dyn/cm and 95 dyn/cm for structural network and bilayer. The latter value correlates well with data on in vitro, monolayer surface pressure versus area curves at oil-water interfaces.

Constitutive relation for red cell membrane. Correction

The intention of this note is to correct a subtle and somewhat esoteric error that the author discovered in his previous publications on membrane elastic behavior. The consitutive relation between membrane force resultants and large, elastic deformations of a membrane surface involves a strain tensor, characterizing the finite deformations. The original strain tensor that appeared in the equations was the Lagrangian strain tensor; however, the proper strain representation (also Lagrangian in nature because it is "measured" relative to the undeformed material state) is transformed by rotations of coordinates in the deformed material state (whereas the Lagrangian strain tensor is transformed by rotations of coordinates in the undeformed state). The principal membrane tensions are unchanged by this correction; the material elastic constants remain the same; and therefore, the material behavior in shear and isotropic tension is the same. However, the tensor, constitutive relation can be properly applied to coordinate systems other than the principal axis system.

Membrane viscoplastic flow

In this paper, a theory of viscoplasticity formulated by Prager and Hohenemser is developed for a two-dimensional membrane surface and applied to the analysis of the flow of "microtethers" pulled from red blood cells attached to glass substrates. The viscoplastic flow involves two intrinsic material constants: yield shear and surface viscosity. The intrinsic viscosity for plastic flow of membrane is calculated to be 1 X 10(-2) dyn-s/cm from microtether flow experiments, three orders of magnitude greater than surface viscosities of lipid membrane components. The fluid dissipation is dominated by the flow of a structural matrix which has exceeded its yield shear. The yield shear is the maximum shear resultant that the membrane can sustain before it begins to deform irreversibly. The yield shear is found to be in the range 2-8 X 10(-2) dyn/cm, two or three orders of magnitude smaller than the isotropic tension required to lyse red cells.