2 A mechanism determining the stability of echinocytes

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1 Membrane shear elasticity and stability of spiculated red cells A. Iglic Faculty of Electrical and Computer Engineering, University of Ljublijana, Trzaska 25, Ljublijana, Slovenia 8291 Abstract In this work the stability of spiculated red blood cells, called also echinocytes, is studied. We assume that the stable echinocyte shape corresponds to the minimum of its membrane elastic energy. It is shown that by neglecting the membrane shear elasticity the calculated stable echinocyte shapes always have only one spicule. However by taking into account also the membrane shear elastic energy the calculated stable echinocyte shapes have many spicula in agreement with experimental observations. 1 Introduction The normal shape of red blood cells (RBC) is the biconcave discoid shape. Under various external conditions these normal RBCs (discocytes) may be transformed into various other shapes such as cup shapes or echinocyte shapes [1]. Echinocytes are spherical cells with spicula uniformly distributed over the surface (Fig.l). The aim of the presented work is to analyze a possible physical mechanism determining the stability of echinocytes. 2 A mechanism determining the stability of echinocytes The RBC membrane has extremely low permeability for water molecules. Therefore the RBC volume (V) is practically constant during the RBC shape transformation. Because of that and since RBC has

2 44 Computer Simulations in Biomedicine discocyte echinocyte Figure 1: Schematic presentation of discocyte and echinocyte red blood cell shapes. no internal structure, the RBC shape is the consequence of the membrane properties solely. The RBC membrane is essentially composed of two parts, the bilayer and the continuous network of proteins, also called the membrane skeleton (Fig.2). The bilayer which is composed of two layers of lipid molecules contains also some other molecules such as glycolipids and different membrane integral proteins. Both lipid layers are in close contact due to hydrophobic effect but are unconnected, so that they are free to slide over each other and may therefore respond differently to various external perturbations [1]. The skeleton is attached to the bilayer integral proteins at the inner side of the bilayer. Figure 2: Schematic presentation of the red cell membrane cross-section (adapted from Cohen [2])

3 Computer Simulations in Biomedicine 45 Because the area expansivity modulus of the skeleton is in normal conditions around four order of magnitudes smaller than the area expansivity modulus of the bilayer [3], the influence of the skeleton on the areas of the neutral surfaces of both lipid layers and on the area of the neutral surface of the bilayer, defined here as the mean membrane area (A), is neglected. It was shown that the stability of many different RBC shapes as well as the transformations beetwen these shapes can be well explained within the bilayer couple model [1,4,5]. In the view of this model the difference between the outer and the inner membrane lipid layer areas (AA) is taken to be an important parameter which influences the RBC shape, while the stable RBC shape at given A is determined by minimization of the membrane bending energy. For example, the effect of different amphiphilic molecules, which are bound to the inner (or outer) lipid layer and thus decrease (or increase) AA, on the RBC shape changes can be qualitatively well explained within the bilayer couple model [1]. It is expected that stable echinocyte shapes exist in the range of high AA. However, the existence of stable echinocyte RBC shapes can not be explained by minimization of the membrane bending energy at higer values of AA [6]. Here we propose that the generalization of the theoretical determination of the equilibrium RBC shapes should take into account also the membrane shear energy. Therefore in this work we shall determine the stable RBC shapes by minimizing the membrane elastic energy (W) consisting of both bending (Wb) and shear (Ws) contributions The bending energy can be expressed as [7] W = - B f (c + c f da b 2 * ^ 1 2^ ' ' where B is the bending elastic modulus of the RBC membrane, Ci and C2 are the two principal curvatures defined so that they are positive for a sphere and da is the infinitesimal membrane area element. The shear energy is [7] -* /I 1 \ 1 I,4 A \ / where p, is the membrane area shear modulus, while Ai and A.2 are the principal extension ratios of the membrane area element, which are chosen sufficiently small that we may consider them approximately flat. The local principal axes are chosen so that the shear resultants

4 46 Computer Simulations in Biomedicine along the local area element edges are zero [7]. The shear energy of the RBC membrane is contributed solely by the skeleton since the lipid bilayer has the properties of the two-dimensional liquid [7]. 3 Model The echinocyte shape can be approximately described by a geometrical model with five parameters. Those parameters are radius of the large sphere R, the number of axisymmetrical spicula distributed on the large sphere n, the length of the spiculum cylinder L, radius r and angle cp (Fig.3). In this model the bending energy Wb can be after some calculation expressed as [6]: Wb = 8%B + njcbl (r + R)sincp - r (4) 2n7iB(r + r((r + Rrsmi cp - arctg ((r +R)sincp + r) tg(% - ((r + Figure 3: The parameters characterizing the geometrical model of echinocyte shape [6]. While calculating the shear elastic energy Ws we shall assume that the area of the membrane is locally conserved, which means that Xl%,2 = 1 [7]. As a consequence Xi = A,2*. In the Mowing analysis we use the the definition Ai = Km- and A,2 = Km For the sake of simplicity we approximately calculate the shear elastic energy Ws as the sum of contributions of n single spicula:

5 Computer Simulations in Biomedicine 47 Ws = n [ J [ ( ) - 1 ] da ] where we integrate over the area of the single spicule and its sorroundings. While calculating the shear energy of the single spicule the infinite flat membrane is considered as an initial reference state having the shear energy equal to zero. For a given shape of the spicule, characterized by the parameters r, (p, L and R, the values of the extension ratio 1m in different points on the spicule and on the sorrounding area are calculated according to the method described in detail by Evans and Skalak [7]. In this way the energy Ws can be expressed as follows, Ws = n ( Ecp + Ecy + Eba + Epi ) (6) where Ecp is the shear energy of the cup of the spicule, Ecy is the shear energy of the spiculum cylinder, Eba is the shear energy of the spiculum base, while Epi is the shear energy of the plane around the spicule. The obtained expressions are: In2 - Ecy = n? Vp), (8) a Epi = \i [ a - n (r + p) ] In n ( r + p r (9) where p = (R + r) sincp - r and a = 2%p + 27ipL + + 7i^(r + p)r - 27ir^. The shear energy of the spiculum base (Eba) can be obtained only numerically by calculating the integral Eba = 2-2 %(% + ^ ) - 1 (2%[p + r(l - cosco)]r) dco, 2 m m where 4 Results 2 m ~ 7i[p 2%(r + p) rco - cosco)] As we mentioned before the aim of this work is to determine the stable echinocyte RBC shapes by minimization of the membrane energy.

6 48 Computer Simulations in Biomedicine The minimization procedure is carried out at given difference between the areas of the neutral surfaces of the two layers of the bilayer (AAo), given membrane area (Ao) and given cell volume (Vo): V(R,n,cp,L,R) = Vo, A(r,n,cp,L,R) = Ao, AA(r,n,(p,L,R) = AAo. (12) The expressions for V, A and AA [6] are not shown in this paper. At given Vo, Ao and AAo the parameters cp, L and R can be determined numerically as functions of r and n by solving the Equations (12). By minimizing the elastic energy of model echinocyte W with respect to parameter r, the elastic energy of the echinocyte as a function of number of spicula W(n) can be calculated (Fig.4a). It can be seen in Figure 4a that for p/b = 0 the stable echinocyte shape has only one spicule (nmin = 1) which is not in accordance with experimental observations [6]. However, for i/b # 0, the calculated echinocyte shape corresponding to the minimal elastic energy Wmin has more than one spicule (nmin >1) in agreement with the observed echinocyte shapes. Figure 4b shows the dependence of the number of spicula (nmin) corresponding to the minimal membrane cell elastic Wmin/8TlB W(n)/8TiB Figure 4: a: the calculated total echinocyte membrane elastic energy Was a function of the number of spicula n: (1) i/b = 0, nmin = 1, (2) M/B = ^ m-2, nmin = 25. b: the calculated number of echinocyte spicula nmin and the corresponding minimal energy Wmin (see Fig. 4a) as functions of the ratio p/b.

7 Computer Simulations in Biomedicine 49 energy (Wmin) as a function of the ratio i/b. The measured value of i is around " N/m [8], while the measured value of B is around "^ Nm [7], which gives us the value around 35-10^ m"^ for the ratio i/b. The chosen values of Vo = 90 p,m and Ao = 138 \im? in Fig.4 are the normal values for the RBCs, while the chosen value of AAo = 1.0 (urn is in the range of those AAo where other RBC can not exist due to geometrical restrictions [4,5]. 5 Conclusions The bilayer couple model and the requirement of the minimal RBC membrane bending energy can explain the stability of numerous observed RBC shapes [4,5,9]. However, these studies of RBC shapes [4,5,9] were limited to the range of area difference between both membrane lipid layers (AA) where spiculated RBC cells, i.e. echinocytes can not exist due to geometrical restrictions. Namely, echinocyte RBC shapes have due to topology of their surface much higher AA than other RBC shapes. Therefore in this work the analysis of the RBC shape stability was extended into the range of higher AA where the echinocytes can exist. It was shown that within the limitation of the presented geometrical model of echinocyte shape the requirement of the minimum bending energy can not explain the stability of echinocyte RBC shapes. However, by the inclusion of the membrane shear energy in the minimization procedure the existence of stable echinocyte shapes can be explained. References 1. Sheetz, M.P. & Singer, SJ. Biological membranes as bilayer couples. A mechanism of drug-erythrocyte interactions, Proceedings of the National Academy of Sciences USA, 1974, 72, Cohen, C.M. The molecular organization of the red cell membrane skeleton, Seminars in Hematology, 1983, 20, 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 Structure, 1994, 23, Svetina, S. & Zeks, B. Membrane bending energy and shape determination of phospholipid vesicles and red blood cells, European Biophysics Journal, 1989, 17,

8 50 Computer Simulations in Biomedicine 5. Svetina, S., Iglic, A. & Zeks, B. On the role of the elastic properties of closed lamellar membranes in membrane fusion, Annals of the New York Academy of Sciences, 1994, 710, Iglic, A., Zeks, B. & Svetina, S. A model for echinocyte red cell shape, presented on the 16th Yugoslav Symposium on Biophysics, Kranjska gora, Evans, E. & Skalak, R. Mechanics and Thermodynamics of Biomembranes, CRC Press, Boca Raton, Waugh, R. & Evans, E.A. Thermoelasticity of red blood cell membrane, Biophysical Journal, 1979, 26, Kralj-Iglic, V., Svetina, S. & Zeks, B. The existence of nonaxisymmetrical bilayer vesicle shapes predicted by the bilayer couple model, European Biophysics Journal, 1993, 22,