MACROSCOPIC FEATURES. d) Biological SystemsSOME THEORETICAL SHAPES OF RED BLOOD CELLS

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MACROSCOPIC FEATURES. d) Biological SystemsSOME THEORETICAL SHAPES OF RED

BLOOD CELLS

W. Helfrich, H. Deuling

To cite this version:

W. Helfrich, H. Deuling. MACROSCOPIC FEATURES. d) Biological SystemsSOME THEORETI-

CAL SHAPES OF RED BLOOD CELLS. Journal de Physique Colloques, 1975, 36 (C1), pp.C1-327-

C1-329. �10.1051/jphyscol:1975155�. �jpa-00216234�

MA CROSCOPIE FEA TURES. d ) Biological Systems

Classification Physics Abstracts

9.720 - 1.342

SOME THEORETICAL SHAPES OF RED BLOOD CELLS

W. HELFRICH and H. J. DEULING

Institut fur Theoretische Physik, Freie UniversiGt, D 1 Berlin 33, W. Germany

RBsumB. - On p r k n t e , pour les globules rouges du sang, quelques formes theoriques qui repro- duisent la transition bien connue entre le disque biconcave normalement observe et la sphkre. On suppose que la forme est contr8lk par I'Blasticitt de courbure de la membrane et le volume intkrieur a celleci. Pour que des formes aplaties plutBt qu'allongkes soient stables, la membrane doit avoir une courbure spontank opposk A celle de la sphkre.

Abstract. - Some theoretical shapes of red blood cells are presented, reproducing the well-known transition between the normally observed biconcave disk and the sphere. The curvature elasticity of the bounding membrane and the enclosed volume are assumed to control the shape. The membrane must have a spontaneous curvature opposite to the curvature of the sphere if flattened forms rather than elongated ones are to be stable.

Red blood cells, also called erythrocytes, possess a well-known biconcave-discoid shape under normal physiological conditions [I]. I t can be recognized with the aid of optical microscopes and, today, be studied in high resolution by means of scanning electron microscopy. This peculiar shape has been a long- standing puzzle which, to our knowledge, has hitherto not been completely solved. The purpose of the present note is to briefly review some previous expla- nations, indicate their deficiencies, give the ingredients of a satisfactory theory, and show some calculated shapes.

The rotational symmetry and smooth contour of red blood cells point to a control of the shape by the elasticity of the bounding membrane. The thickness of the semipermeable membrane (ca. 100 A) is much smaller than the dimensions of the cell (equatorial diameter ca. 8 p). An elastic interpretation is supported by the fact that via osmotic changes of the enclosed volume a discocytecan be transformed into a spherocyte, i. e. a sphere, and vice versa. Various attempts have been made to explain the transition in terms of models treating the membrane as a thin elastic rubber wall [2].

The relaxed state of the shell was taken to be either spherical or biconcave-discoid, but in both cases the theories were not really satisfying. In particular, they contradicted the observation that the membrane area remains practically constant in the transition.

In view of these difficulties Canham [3] proposed curvature elasticity as the shape-controlling factor, considering the membrane to be a two-dimensionally fluid bilayer. He took the local elastic energy density to be proportional to ( 1 1 ~ ~ ) ~ + (1/R2)2 where R, and R, are the radii of the two principal curvatures.

Varying a single parameter, he determined for a

restricted class of shapes (Cassini ovals) that of lowest curvature-elastic energy as a function of cell volume.

The membrane area was kept constant and the sphere represented the state of maximum volume and mini- mum elastic energy. (The biconcave-discoid shape o living red blood cells, which become spheres in the absence of metabolism, can be explained by osmosis due to active ion transport.) Comparing the computed shapes to those seen in the spherocyte-discocyte tran- sition, Canham found good agreement which, however, will be seen to be deceptive.

During our work we learned that some years ago Nehring and Saupe 141, being unaware of Canham's article, considered a much wider class of shapes, with rotational and equatorial reflection symmetry. They took the elastic energy density to be proportional to (l/R, + 1 / ~ , ) ~ and, again, calculated those shapes which a t given volume and fixed membrane area have the lowest elastic energy. For each volume they found both an oblate and a prolate shape, the former becoming biconcave-discoid and the latter dumbbell- like if the volume was sufficiently reduced. However, of the two shapes the elongated one always had the lower energy, thus being the stable solution, which is just the opposite of what one is looking for.

A complete formula for the curvature elastic energy of two-dimensionally fluid membranes must allow for a spontaneous curvature, i. e. an inherent tendency of the membrane to curve. As in a recent article [5] on the presumed elasticity of,lipid bilayers we may write for the elastic energy per unit area

where kc and kc are elastic moduli and c, = 1/R, and c2 = 1/R2 the principal curvatures. The spontaneous

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975155

C1-328 W. HELFRICH AND H. J. DEULING curvature co can be nonzero whenever the membrane

or its environment is asymmetric with respect tc the center plane of the membrane. The integral of c, c2 over the membrane surface is independent of the shape and size of the cell, so the second term in (1) is without effect and drops out of calculations of the shape. It may be noted that for the same reason Canham's expression for the energy density, though not correct, should lead to the same shapes as Nehring and Saupe's and as eq. (1) with c, = 0.

FIG. 1. - Cross sections through cell shapes calculated for

co

ro

- ^3. Only one quadrant is shown. There is rotational symmetry around the z axis and reflection symmetry at the x axis.

Volume and curvature-elastic energy are given in terms of the values for the sphere, the excess outside pressure in units of the

critical pressure. The surface area is kept constant.

Eq. (1) can be used to derive an Euler-Lagrange equation for the shape of rotationally symmetric cells. One may take the distance x from the axis of rotation as the independent variable and the tipping angle $ of the contour as the dependent variable (see Fig. I). The principal curvatures

dI) sin $

C,

= cos *

_dx ^,

^{= -} _x

are along the meridians (m) and parallels (p) of the surface. They and c, are counted positive if they are like the curvatures of the sphere. The integrand of the variational problem is

I:

^]

^{+ a p n 2 tan + .}

= -- k,(c, + cp - c,) + ^{I. --} _{cos I)}

The Lagrange multiplier 2 which turns out to represent a tensile stress results from the auxiliary condition of constant area. Ap is the excess outside pressure acting on the cell. It may also be viewed as a Lagrange multi- plier due to the auxiliary condition of constant

volume. The Euler-Lagrange equation was deduced to be [5]

+ ^{dpnx2 -} kc cos $ d(cm dx

2 nx cos t,b = 0 . ₍₄₎ It should be noted that the cell shape is independent of kc if the volume is prescribed instead of the pressure difference.

This second-order differential equation for $(x) may be separated into two of first order for c,(x) and cp(x), which gives a novel and more tractable form

Contours of the kind shown in figure 1 are obtained by a further integration,

Eq. (5) were solved numerically with the Runge- Kutta method in search of cell shapes symmetric with respect to the equatorial plane. Obviously, the sphe- rical shape is a solution for all values of Ap. In a plot of cell eccentricity s2 vs. excess outside pressure Ap a second branch of solutions crosses the Ap axis at the critical pressure [5] Ap, = (2 kc/r:) (6 - c, r,), where r, is the radius of the sphere. If the volume is taken as the independent variable (and not reduced too much) there are always two solutions, one a flattened and the other an elongated body. The total free energy E,,, of the membrane and the aqueous media obeys, along any branch of solutions,

where V is the volume and E the elastic energy of the cell. Consequently, for a given volume the solution at smaller Ap is the one of lower elastic energy and thus stable. (This holds a t least in a certain vicinity of Ap,.) For co=O the curve s,(Ap) has a falling slope which implies that the elongated shape is preferred in agreement with Nehring'and Saupe's result. However, the desired stability of the flattened forms can be achieved byj making the spontaneous curvature suffi- ciently negative. The critical spontaneous curvature coc is characterized by a vertical slope of s2(Ap) at s, ⁼ 0.

An analytical calculation based on detailed balancing of torque densities yielded c,, r , = - 39/23 and conforms with the computer results.

The shapes shown in figure 1 were obtained for

SOME THEORETICAL SHAPES OF RED BLOOD CELLS C 1-329 c, r , = - 3. At this value of ^c, one can be reasonably

sure that the oblate shapes are stable even if they differ strongly from the sphere. The figure shows a flattening of the sphere and a subsequent formation of biconcave disks as the volume is progressively reduced. The comparison with the photographs of the spherocyte-discocyte transition shown by Canham is very good. The two sides of the discocyte eventually touch each other at a certain volume, but the corres- ponding curve has been omitted in figure 1 because of multiple crossings with neighboring forms. A sequence of shapes was also computed for c, r,= - 6.

Here the volume increases again and the two concave halves move apart before touching each other as Ap surpasses a certain value. The details of the calcula- tions will be given in a full article [6].

From our results we draw the surprising conclusion

that nature provides the red cell membrane with a nega- tive spontaneous curvature 171. Its value is probably not too far from c, ⁼ - 3/ro which underlies figure 1.

The elastic energies given in the inset of the figure as well as simple estimates suggest that the curvature- elastic energy required to convert a disk into an elon- gated body is very small, amounting to a fraction of the elastic energy of the sphere, Eo = 8 7ckc(l - co r,/2)', estimated [5] to be of the order of 10 eV. The easy deformability of non-spherical shapes contrasts with the stiffness of the sphere which at fixed volume and surface area cannot be deformed. It seems to be very important in enabling red blood cells to pass through thin capillaries and narrow openings. One may specu- late that disk shapes as compared to elongated shapes have the extra advantage of not being trapped by holes unless they are forced into them by blood flow.

References

BESSIS, M., Living Blood Cells and their Ultrastructure (Springer-Verlag, Berlin) 1973.

See, e. g., FUNG, Y. C. and TONG, P., Biophys. 5.8 (1968) 175;

DANIELSON, D. A., J. Biomechanics 4 (1971) 61 1 ; BULL, B. in Red Cell Shape, ed. by M . Bessis, R. I. Weed,

and P. Leblond (Springer-Verlag. New York) 1973.

[3] CAKHAM, P. B., J. Theoret. Biol. 26 (1970) 61.

[4] NEHRING, J. and SAUPE, A., Comment at the Diskussions-

tagung der Bunsen-Gesellschaft, Konigstein im Tau- nus, March 20-22, 1974.

[S] HELFRICH, W., Z . Nuturforsch. 28c (1973) 693.

[6] DEULING, H. J. and HELFRICH, W., in preparation.

[7] The problem that part of

may be a time-dependent func-

tional of the deformation (for strong deformations of

the sphere) is discussed by HELFRICH, W., Z. Natur-

forsch. 29c (1974) 510.