Elasticity of the human red blood cell skeleton

Biorheology 40 (2003) 247 251 247 IOS Press Elasticity of the human red blood cell skeleton G. Lenormand, S. Hénon, A. Richert, J. Siméon and F. Gallet Laboratoire de Biorhéologie et d Hydrodynamique Physico-Chimique, ESA 7057, CNRS, Universités Paris 6 and Paris 7, case courrier 7056, 2, place Jussieu, F-75251 Paris cedex 05, France Tel.: +33 1 44 27 77 82; Fax: +33 1 44 27 43 35; E-mail: henon@ccr.jussieu.fr. Abstract. We have measured by optical tweezers micromanipulations the area expansion and the shear moduli of spectrin skeletons freshly extracted from human red blood cells, in different controlled salinity conditions. At medium osmolarity (150 mosm/kg), we measure K C = 9.7 ± 3.4 µn/m, µ C = 5.7 ± 2.3 µn/m, K C/µ C = 2.1 ± 0.7. When decreasing the osmolarity, both K C and µ C decrease, while K C/µ C is nearly constant and equal to about 2. This result is consistent with the predictions made when modeling the spectrin skeleton by a two-dimensional triangular lattice of springs. From the measured elastic moduli we estimate the persistence length of a spectrin filament: ξ 2.5 nm at 150 mosm/kg. 1. Introduction The human red blood cell (RBC) undergoes strong stresses and deformations in the blood flow. Its elastic properties are due to its membrane, which has a great resistance to expansion attributed to the lipid bilayer, and a smaller resistance to shear attributed to a flexible two-dimensional protein skeleton, bound to the inner face of the lipid bilayer. This skeleton is made of spectrin tetramers, of length L 200 nm, forming a roughly triangular network [1]. The tetramers are linked together at the lattice sites by complex junctions, primarily composed of actin filaments and protein 4.1. The skeleton is attached to the lipid bilayer via transmembrane proteins, glycophorin C and band 3. Both the area expansion modulus K and shear modulus µ of the RBC membrane have been measured by various techniques [4,7,8], in particular by micropipette experiments, but the expansion and shear moduli of the isolated skeleton had never been measured before our work. We have used optical tweezers to measure the elastic moduli of free spectrin skeletons, freshly extracted from the RBC membrane, in different conditions of osmolarity. We compare our results to our previous measurement, with the same technique, of the shear modulus of the entire membrane [7], and to theoretical and numerical studies which have shown that for a triangular lattice of springs of stiffness k, constrained to in-plane deformations and in the small deformation regime, the shear modulus µ C is equal to k 3/4, and the area expansion modulus K C to k 3/2 = 2µ C [2,5,6,9]. Furthermore, from the measured moduli we infer the stiffness and the persistence length of a single spectrin tetramer. 2. Materials and methods 2.1. Optical tweezers Optical tweezers consist of a laser beam, focused by a large aperture microscope objective. Micrometric dielectric objects are trapped at the focusing point. Small latex or silica beads can be used as handles 0006-355X/03/$8.00 2003 IOS Press. All rights reserved

248 G. Lenormand et al. / Elasticity of the human red blood cell skeleton Fig. 1. Scheme of the experimental procedure (not to scale). A RBC with three silica beads bound to its periphery is seized by three optical tweezers. The skeleton is deformed by varying the distances between the traps. to apply pre-calibrated forces to biological material. Our setup has been described elsewhere in great details [7]. The restoring force F exerted on a trapped bead is pre-calibrated by applying a known viscous drag force to the bead. 2.2. Extraction of the skeleton in hypotonic buffers Fresh blood is obtained by fingertip needle prick. Red blood cells are washed three times in PBS. Silica microbeads (2.1 µm in diameter) are then added to the suspension. They stick spontaneously and irreversibly to the RBC membrane when incubated for one hour at 4 C. A flow chamber placed under the microscope allows the injection of the RBC suspension and of the different solutions used during the experiments. Under microscope, a RBC with three beads attached to its periphery is selected and the beads are trapped in three optical tweezers (see Fig. 1). Then hypotonic buffer (25 mosm/kg, or 100 mosm/kg, or 150 mosm/kg) is slowly injected to lower the osmolarity. A detergent solution (Triton- X100 in buffer 1 3 in volume) is then injected until bilayer dissolution. The detergent is finally rinsed out with buffer for about 10 min. After dissolution of the bilayer, the skeleton is invisible in bright field, but it remains attached to the beads. All the experiments are carried out at room temperature. 2.3. Measurements The skeleton is deformed by varying the distances between the traps. Each skeleton is submitted to a set of deformations, first increasing the distance between the traps, then decreasing it. For all the measurements, the forces exerted on the trapped beads vary from 1 to 8 pn. Since the skeleton is invisible, its deformation is interpolated from the beads positions. From the measured forces and deformations, we calculate the stress and strain, assuming that the deformed region is a triangle ABC (see Fig. 1) of a continuous, homogeneous, isotropic, elastic, bidimensional medium, and that the stress and strain are homogeneous over ABC, so that it keeps a triangular shape when deformed. By plotting the expansion

G. Lenormand et al. / Elasticity of the human red blood cell skeleton 249 (a) (b) Fig. 2. Stress plotted versus strain for a typical deformation of a freshly extracted skeleton in low osmolarity buffer (150 mosm/kg). (a) Expansion component of the stress versus area dilation S/S 0; (b) pure shear component of the stress versus shear strain. A linear behavior, in agreement with linear elasticity model, is obtained in both cases. The respective slopes give the area expansion modulus K C and the shear modulus µ C. The relative area increase is always smaller than 15%, and the shear deformation than 30%. stress (respectively shear stress) as a function of the expansion strain (respectively shear strain), we get K C (respectively µ C ). The actual stress is probably not homogeneous, especially in the regions where the skeleton sticks to the beads. This is a source of uncertainty in the determination of the elastic moduli. 3. Results All the measurements are performed within 15 min after extraction of the skeleton. Figure 2(a) and (b) show a typical measurement held at 150 mosm/kg. Linear relations are measured between stresses and deformations. Figure 3(a) and (b) show the results of all the measurements performed at 150 mosm/kg. These stack histograms clearly show two maxima. We explain this result by the fact that, depending on the positions of the beads and on the kind of deformation, we pull on either one or two sheets of the skeleton (see Fig. 4), thus measuring either once or twice the elastic moduli of a single sheet. Thus we fit the results by the sum of two Gaussian functions, the center of the second function being set to twice that of the first one. For a single sheet the measured values are K C = 9.7 ± 3.4 µn/m and µ C = 5.7±2.3 µn/m. Figure 3(c) shows the ratio K C /µ C, K C and µ C being measured for the same deformation of the same skeleton, and its fit by a single Gaussian function, which gives K C /µ C = 2.1 ± 0.7. In the same way, at 100 and 25 mosm/kg, the stack histograms of K C and µ C show two maxima. Both K C and µ C decrease when the osmolarity is decreased. On the contrary K C /µ C is nearly constant, and equal to 2. 4. Discussion Our measurements of the RBC skeleton elastic moduli have a relatively large dispersion. Nevertheless they show clear Gaussian distributions, and allow to obtain consistent measurements. The dispersion can

250 G. Lenormand et al. / Elasticity of the human red blood cell skeleton (a) (b) (c) Fig. 3. Stack histograms of the elastic moduli K C (a) and µ C (b) obtained in 150 mosm/kg buffer on 11 different skeletons submitted to about 35 different deformations. The histograms present two maxima, corresponding to the deformation of either a single sheet or two sheets of the skeleton (see Fig. 4). The values of the two maxima are in the ratio 2 : 1. The best fit with a double Gaussian is also represented. (c) Stack histogram of the ratio K C/µ C, K C and µ C being measured from the same deformation of a skeleton. The best Gaussian fit is shown. Fig. 4. Depending on the location of the beads, we pull on either one or two sheets of the skeleton, and measure either once or twice the elasticity of one sheet. be attributed to several effects: first, modifications of the structure of the spectrin and of the density of defects in the spectrin network with aging of the RBC can account for a natural dispersion [3]; secondly, the extent of the binding region between skeleton and beads can vary from one experiment to another, making the assumption of homogeneous strain and stress more or less valid. The measured values of the elastic moduli are, as expected, in the same order of magnitude as the shear modulus of the entire RBC membrane, µ = 2.5 ± 0.4 µn/m, as measured previously with the same technique [7]. An exact

G. Lenormand et al. / Elasticity of the human red blood cell skeleton 251 comparison between µ and µ C is difficult because they were not measured in the same conditions. First, measurements of K C and µ C in isotonic buffer are difficult, because at high ionic strength the electrostatic repulsions between the proteins are screened and the skeleton collapses [10]. Secondly, in the entire membrane, the skeleton is constrained by the lipid bilayer, and this has an influence on the value of the shear modulus. As the measured value of K C /µ C is close to 2, the theoretical value for a triangular lattice of springs, we use the formula K C = k 3/2 to estimate the stiffness k of a single spectrin filament. Then, assuming that the elastic behavior of spectrin tetramers is purely entropic, k can be related to the persistence length ξ of a spectrin filament, and to its contour length L, through the Worm Like Chain model [11] (strictly speaking applying to a free polymer not included into a network): k (3k B T )/(2ξL). Taking L 200 nm, ξ varies from 6nmat25mOsm/kgto 2.5 nm at 150 mosm/kg. This value is comparable to the estimate made from measurements on a free skeleton: ξ 6 nm [12]. The decrease of ξ when increasing the salinity is consistent with the fact that, when salt is added, the electrostatic repulsions inside the filament are more and more screened, and the filament becomes more and more flexible. 5. Conclusion The area expansion and shear moduli of isolated RBC skeletons have been measured for the first time, in controlled experimental conditions. They are of a few µn/m, and their ratio is in the order of 2, as expected for a bidimensional triangular lattice of springs. Furthermore from our measurements at the scale of an entire cell, we are able to infer an estimate of the persistence length of a single spectrin filament, which is equal to a few nm and decreases when the salinity increases. References [1] V. Bennett and D.M. Gilligan, The spectrin-based membrane skeleton and micron-scale organization of the plasma membrane, Annu. Rev. Cell. Biol. 9 (1993), 27 66. [2] D.H. Boal, Computer simulation of a model network for the erythrocyte cytoskeleton, Biophys. J. 67 (1994), 521 529. [3] D. Corsi, M. Paiardini, R. Crinelli, A. Bucchini and M. Magnani, Alteration of α-spectrin ubiquitination due to age dependent changes in the erythrocyte membrane, Eur. J. Biochem. 261 (1999), 775 783. [4] E.A. Evans, New membrane concept applied to the analysis of fluid shear- and micropipette-deformed red blood cells, Biophys. J. 13 (1973), 941 954. [5] J.C. Hansen, R. Skalak, S. Chien and A. Roger, An elastic network based on the structure of the red blood cell membrane skeleton, Biophys. J. 13 (1996), 146 166. [6] J.C. Hansen, R. Skalak, S. Chien and A. Roger, Influence of the network topology on the elasticity of the red blood cell membrane skeleton, Biophys. J. 72 (1997), 2369 2381. [7] S. Hénon, G. Lenormand, A. Richert and F. Gallet, A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers, Biophys. J. 76 (1999), 1145 1151. [8] R.M. Hochmuth and R.E. Waugh, Erythrocyte membrane elasticity and viscosity, Ann. Rev. Physiol. 49 (1987), 209 219. [9] Y. Kantor and D.R. Nelson, Phase transition in flexible polymeric surfaces, Physical Review A 36 (1987), 4020 4032. [10] G. Lenormand, S. Hénon, A. Richert, J. Siméon and F. Gallet, Direct measurement of the area and shear moduli of the human red blood cell membrane skeleton, Biophys. J. 81 (2001), 43 56. [11] J. Marko and E. Siggia, Stretching DNA, Macromolecules 28 (1995), 8759 8770. [12] K. Svoboda, C.F. Schmidt, D. Branton and S.M. Block, Conformation and elasticity of the isolated red blood cell membrane skeleton, Biophys. J. 63 (1992), 784 793.