Transactions on Biomedicine and Health vol 3, 1996 WIT Press, ISSN

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2 110 Simulation Modelling in Bioengineering molecules, such as laterally mobile membrane integral proteins and different glycolipids, see Gennis [1]. bilayer ^skeleton Figure 1: Schematic presentation of the red cell membrane crosssection (adapted from Cohen [3] ). In normal conditions the entire bilayer is underlaid with the skeleton. However, it was found that RBC vesicles, induced by detergents, are depleted in major components of the membrane skeleton, i.e. spectrin (proteins band 1 and band 2) (Figure 2) and actin (protein band 5), and enriched in membrane integral protein band 3, e.g. Hagerstrand & Isomaa [2]. Also, it was found that RBC vesicles have a lower cholesterol/phospholipid ratio and a higher phosphatidylserine/phospholipid ratio compared to the parent cell (Table 1). The aim of the present work is to elucidate a possible physical mechanisms leading to the formation of the membrane skeleton depleted RBC vesicles. Table 1. Phospholipid composition (given in percent) and cholesterol/phospholipid ratio (given as molar ratio) in parent RBC membrane and in RBC daughter vesicle membrane (adapted from Hagerstrand & Isomaa [2] ). phospolipid phospatidylethanolamine phospatidylserine phospatidylcholine sphingomyelin cholesterol/phospolipid pare nt + + cell ± 3 26 ± vesicle ± ± ±

3 Simulation Modelling in Bioengineering 111 a Figure 2: Protein profile of parent RBC membrane (a) and RBC daughter vesicle membrane (b) following sodium dodecyl sulphate-polyacrylamide gel electrophoresis. The numbers 1 and 2 denote the proteins band 1 and band 2 (spectrin) (adapted from Hagerstrand & Isomaa [2] ). 2 The role of membrane shear elasticity in partial skeleton detachment Recently, a possible physical interpretation of the partial detachment of the membrane skeleton in the budding region of the cell membrane and the consequent depletion of the membrane skeleton in RBC vesicles was given by Iglic [4]. Namely, it was shown that some cell shape changes can induce partial detachment of the skeleton. The partial detachment of the skeleton from the bilayer is energetically favourable if the consequent decrease of the skeleton expansion energy is larger than the corresponding increase of the bilayer-skeleton binding energy. However, in our previous work, i.e. Iglic [4], shear deformations of the skeleton were not taken into account. Therefore in the present analysis the shear energy of the RBC membrane skeleton is calculated during the budding process. The membrane bilayer greatly resists changes in surface area while the skeleton may dilate or condense relative to the constant density bilayer envelope, only the total area of the skeleton is conserved, e.g. Fischer [5], Mohandas & Evans [6]. Namely, the very recent experiments have shown that the membrane skeleton is locally compressible which leads to a new constitutive model for the membrane skeleton behaviour, e.g. Mohandas & Evans [6]. For

4 112 Simulation Modelling in Bioengineering the sake of simplicity, in this work the RBC membrane is considered within the model of immobilized boundaries, e.g. Markin & Kozlov [7]. Within this model the membrane skeleton cannot redistribute relative to the cytoplasmic surface of the bilayer, which prevents due to lateral incompressibility of bilayer any local area changes both of bilayer as well as of the skeleton, e.g. Lerche [8]. Therefore the shear energy of the skeleton is written using an approximative expression, e.g. Evans and Skalak [9] : jj, is the membrane area shear modulus, A,m is extension ratio along the meridional direction, where the membrane area element da is chosen sufficiently small so that we may consider it approximately flat. The local principal axes are chosen so that the shear resultants along the local area element edges are zero. The vesicle shape during the budding process is described by a geometrical model with two parameters. These parameters are the (1) Figure 3: The parameters characterizing the geometrical model of the vesicle shape. Curvilinear distance along the meridian (s) from the pole of the vesicle to the surface outside of the spiculum is defined in the area of vesicle sphere as ds = Rd d8 while in the area around the vesicle ds = dr%. Shaded area is the partial area of the vesicle (Ac) characterized by the angle 8.

5 Simulation Modelling in Bioengineering 113 radius of the vesicle Rd and the angle 80 determining the width of the vesicle neck (Figure 3). The shear energy of the vesicle Ws is calculated according to the method of Evans & Skalak [9]. While calculating the energy Ws the flat membrane is considered as an initial reference state, where it is assumed that in the reference state Ws = 0. The undeformed reference coordinate system is characterized by the radius TO originating from the symmetry axis of the vesicle and the polar angle 9 in the initially flat membrane. The deformed coordinate system is defined by the curvilinear distance along the meridian (s) which begins at the pole of the vesicle and eventually becomes the radial coordinate (rx) in the outer membrane surface. Because of the symmetry, the azimuthal angle is the same as the initial polar angle. The principal extension ratio along the meridian (km) is then given by Evans and Skalak [9] 1m = ds/dro = TQ/TX, (2) where it is taken into account that the area of the membrane (skeleton) is locally conserved, i.e. rodrody = rxdsdy. The total area of the membrane segment must be conserved too : fo o fx TO dro - J o rx ds. (3) The shear energy of the RBC vesicle Ws is expressed as the sum of two contributions, W, = Esp + Epi, (4) where Esp is the shear energy of the vesicle sphere, while Epi is the shear energy of the plane around the vesicle (see Figure 3). In order to calculate the energies Esp and Epi the corresponding values of the pricipal extension ratio Km in different points on the vesicle sphere and in its sorrounding area are calculated by using the equations (2) and (3) (see Figure 3) : vesicle sphere : Xm(&) = / = 1 + (1 + cos 8)^>in^8, (5) plane : 1^) = Avesfaix +1, rx > ^dsinso, (6) where the area of the vesicle sphere Ayes at given 80 Figure 3) is (see

6 114 Simulation Modelling in Bioengineering Ayes = 71/8 (^80 + (1 - COS&of) (7) Knowing the values of l/% the energies Esp and Epi can be then determined using the equation (1), where we integrate over the appropriate area of the vesicle and the area around the vesicle: E%, = JLI.T i (1%, + ^) ?!^ sins (/», (8) ; (^ + X^) ^ ^. (9) While we calculated the energy Epi we extended the integration area to infinity. In this way the energies Esp and Epi can be expressed as follows, - 5/8 + cosbo ), (10) A^ In (( 1 + Ay^^&m^»o)^). (11) Figure 4 shows the relative shear energy of the RBC vesicle Ws as a function of the angle 80 (Figure 5) determining the width of the vesicle neck. In addition, Figure 4 also shows the dependencies of the energies Esp and Epi normalized relative to the energy \ji4nr^ on the angle BQ. It can be seen in Figure 4 that the the shear energy of the RBC vesicle strongly increases with decresing width of the neck between the daughter vesicle and the parent cell. On the basis of presented results it can be concluded that the partial detachment of the skeleton from the bilayer in the budding region of the RBC membrane is energetically favourable since in this way the accumulated shear deformations in the vesicle neck are relaxed. 3 Discussion In the present work the study of the budding process was for the sake of clarity confined to the case in which only one daughter vesicle is created on the flat cell surface. In addition, we have for the sake of simplicity separated the phenomenon of the partial skeleton detachment from the mechanism of shape changes. For the real RBC membrane any change of the state of the skeleton also

7 Simulation Modelling in Bioengineering 115 B 10 o Figure 4: Relative shear energy of the vesicle VJs/^4nRd (A) and the relative energies Esp/p4%7% (B) and Ep\/[i4nRd (C) as functions of the angle Go (see Figures 3 and 5). affects the cell shape. The analysis of these effects was described recently, e.g. Iglic [10]. It was suggested recently that the redistribution of the membrane lipids and band 3 during the budding process are secondary to the partial detachment of skeleton network from the membrane, e.g. Hagerstrand & Isomaa [2]. The mechanism of this phenomena is not clear, however, the possibility of the direct binding between the skeleton and phospholipid molecules in intact RBC exists, e.g. Dressier [11], Maksymiw [12], Sikorski [13], Michalak [14], Bialkowska [15]. It was also shown by Svetina [16] that the lack of

8 116 Simulation Modelling in Bioengineering =143' Figure 5: Schematic presentation of the model vesicle shape for different values of the angle 80 (see Figure 3). skeleton-bilayer interaction may influence the lateral distribution of band 3 protein. A second possible cause of the different lipid and band 3 protein composition of RBC vesicles compared to the parent cell may be due to different curvatures of the parent cell and of the vesicles. Namely, lipid molecules and integral proteins have in general a different shape and a different net negative charge. Hence, the energy of interaction of these molecules with their membrane surrounding is different in the parent cell compared to the vesicles, e.g. Svetina [16], Kralj-Iglic [17], Bergelson & Barsukov [18], and as a consequence the area density of these molecules is different in the parent cell compared to the vesicles. 4 Conclusion The main outcome of the presented analysis is that the partial detachment of the skeleton in the budding region of the cell membrane is additionally favoured due to the accumulated shear deformations in the region of the neck between the daughter vesicle and the parent cell since they are relaxed in the detached state of the skeleton.

9 Simulation Modelling in Bioengineering 117 Acknowledgements We are grateful to Dr. Waugh R. and Dr. Kralj-Iglic V. for fruitful! discussions regarding the presented problem. References 1. Gennis, R.B. Biomembranes, Springer Verlag, Berlin and New York, Hagerstrand, H. & Isomaa B. Lipid and protein composition of exovesicles released from human erythrocyte following treatment with amphiphiles, Biochimica et Biophysica Ada, 1994, 1190, Cohen, C.M. The molecular organization of the red cell membrane skeleton, Seminars in Hematology, 1983, 20, Iglic, A., Svetina, S. & Zeks, B. Depletion of membrane skeleton in red blood cell vesicles, Biophysical Journal, 1995, 69, Fischer, T.M. Is the surface area of the red cell membrane skeleton locally conserved, Biophysical Journal, 1992, 61, Mohandas N. & Evans E. Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects. Annual Review of Biophysics and Biomolecular Sfrucfure, 1994, 23, Markin V.S. & Kozlov M.M. Mechanical properties of the red cell membrane skeleton : analysis of axisymmetric deformations, Journal of Theoretical Biology, 1988, 133, Lerche D., Kozlov M.M. & Meier W. Time-dependent elastic extensional RBC deformation by micropipette aspiration : redistribution of the spectrin network, European Biophysics Journal, 1991, 19, Evans, E. & Skalak, R. Mechanics and Thermodynamics of Biomembranes, CRC Press, Boca Raton, Iglic, A., Svetina, S. & Zeks, B. A role of membrane skeleton in discontinuous red blood cell shape transformations. Cellular & Molecular Biology Letters, 1996, 1, Dressier, V., Haest, C.W.M., Plasa, G., Deuticke, B. & Erusalimsky, J.D. Stabilizing factors of phospholipid asymmetry in erythrocyte membrane, Biochimica et Biophysica Ada, 1984, 775,

10 118 Simulation Modelling in Bioengineering 12. Maksymiw, S., Sui, S., Gaub, E. & Sackmann, E. Electrostatic coupling of spectrin dimers to phosphatidylserine containing lipid lamellae, abcaea%:sfr% 1987, 26, Sikorski, A.F., Michalak, K., Bobrowska, M. & Kozubek A. Interaction of spectrin with some amphipatic compounds, #ud/a m#;wca, 1987, 121, Michalak, K., Bobrowska, M. & Sikorski, A.F. Interaction of bovine erythrocyte spectrin with aminophospholipid liposomes, General Physiology and Biophysics, 1993, 12, Bialkowska, K., Zembron, A. & Sikorski A.F. Ankyrin inhibits binding of erythrocyte spectrin to phospholipid vesicles, Biochimica et Biophysica Acta, 1994, 1191, Svetina, S., Iglic, A., Kralj-Iglic, V. & Zeks, B. Cytoskeleton and red cell shape, Cellular & Molecular Biology Letters, 1996, 1, Kralj-Iglic, V., Svetina, S. & Zeks, B. Shapes of bilayer vesicles with membrane embedded molecules, European Biophysics Journal, 1996 (in print) 18. Bergelson, L.D. & Barsukov, L.I. Topological asymmetry of phospholipids in membranes, Science, 1977, 197,