Elastic theory of biomembranes

Thin Solid Films 393 Ž2001. 19᎐23

Elastic theory of biomembranes Ou-Yang Zhong-canU Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, PR China

Abstract This paper reports on some progress of vesicle shape study in the Helfrich curvature elasticity theory of fluid membranes which was recently extended to the complex structures of smectic liquid crystals. A general differential equation of surface is presented, which is the Euler᎐Lagrange equation for the variation problem ␦E⌽ dAs 0. Here ⌽ is any function of the principal curvatures, i.e. the generalized Helfrich curvature free energy. The application of the surface equations to dynamics of vesicles or microemulsion droplets is also discussed. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: Complex fluid; Soap bubble; Biomembrane; Liquid crystal; Minimal surface; Surface of constant mean curvature

1. Introduction Shape problems stem from real interface, membrane and lamellar systems in nature. Since the fundamental research by J. Plateau Ž1803., the shapes of soap films have been described by a variation problem ␦EdAs 0, or surfaces with Hs0

Ž1.

where A and H denote the area and the mean curvature for the soap surface. The beautiful experiments of Plateau in the middle of 19th century initialed a first ‘golden age’ in the study of minimal surface. In the investigations concerning the rise of a liquid in a capillary tube, T. Young Ž1805. and P.S. Laplace Ž1806. studied another variation problem ␦w ⌬ PHdVq ␭EdAx s 0, leading to a surface of constant mean curvature, H s ⌬ Pr2␭

Ž2.

Here, ⌬ P is the pressure difference between the U

Tel.: q86-10-62554467; fax: q86-10-62562587. E-mail address: [email protected] ŽO. Zhong-can..

airrliquid interface and ␭ the surface tension, and V is the volume of liquid phase. In fact, Eq. Ž2. can also serve to describe the shape of a closed soap bubble where V is the volume of the bubble. For a long time, the only known examples of such a surface were spherical and cylindrical surfaces. Great progress was made in 1841 by Delaunay, who found a beautiful way of constructing a surface of revolution with constant H: rolling a conic section on a straight line in a plane, and then rotating the trace of a focus about the line. After that, the study of surface with a constant mean curvature is very difficult as in the investigation in minimal surface and only in 1956 Alexandrov proved that the only closed embedded surfaces of constant mean curvature are spheres. All physically closed vesicles must be embedded, therefore, the shape of vesicle consisting of isotropic fluid membrane is known exactly, the sphere. In the 19th century S.D. Poisson Ž1812. designed the term of EH 2 dA to characterize elastic energy of thin shell. Its variation leads to a new surface equation ⵜ2Hq2 HŽ H 2 yK . s0

Ž3.

where ⵜ 2 is the Laplace᎐Beltrami operator, and K is the Gaussian curvature. As revealed by J.C.C. Nitsche, this form of the equation was derived by W. Schadow Ž1922. and extensively restudied by T.J. Willmore Ž1982.

0040-6090r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 0 8 4 - 7

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and many others. With the widening of the fields of study in soft matter, such as in biomembranes ŽBM. or fluid membrane ŽFM. Ž1973, Helfrich spontaneous curvature model., and in smectic-A liquid crystal ŽSA LC., Poisson’s theory of curvature elasticity has been greatly generalized and renewed, recently. In the following, some progress in the shape problem associated with the generalized Helfrich model in FM and SA LC are briefly reported. 2. Shape of blood cell and FM theory of Helfrich What is the FM theory of Helfrich? This may trace back to a long-standing problem in physiology: why the red blood cells ŽRBCs. in human bodies are always in a rotationally symmetric and biconcave shape ŽFig. 1.? To explain the configuration there were a series of approaches offered by biomechanical scientists. For example, Fung and Tong Ž1968. suggested that the thickness of cell BM varies from region to region in order to regulate the biconcave shape, but this model contradicts the observation under electron microscopes that the thickness is uniform. Lopez et al. Ž1968. proposed that the difference in electric charge distribution over the RBC surface could be a decisive factor in the shape formation. However, the measurement of Greer and Baker Ž1970. shows a uniform distribution of the charge. The shortcoming of these models basically comes from the fact that none of them recognized the material state of FM being in LC. It is W. Helfrich Ž1973., w1x who derived from curvature elastic theory in LC a form of free energy for FM as 2

F s Ž kr2. E Ž 2 H q c0 . dAq ⌬ PHdVq ␭EdA

Ž4.

In Eq. Ž4. the key step up for the Poisson’s potential is introduce of the new parameter c 0 , the Helfrich spontaneous curvature, which describes the asymmetry of the outer and inner monolayers of the FM and so of its environment. The variation of Eq. Ž4. in the case of rotational symmetry leads to a shape equation for axially symmetric vesicle as shown in Deuling and Hel-

Fig. 1. The biconcave discoidal shape of human red blood cell at rest.

Fig. 2. One quadrant of the cross-section of the contour of the analytical solution described by Eq. Ž6..

frich w2,3x. With a numerical method for solving the equation andror with a best fitting directly to the energy form wEq. Ž4.x it was surely shown that a biconcave disc shape strikingly like that of RBC can exist in the case of c 0 - 0. Using a general method of variation of surface in differential geometry one can derive from Eq. Ž4. the following general shape equation which is not restricted by rotational symmetry: w4,5x ⌬ Py 2␭ H q k Ž 2 H q c 0 .Ž 2 H 2 y 2 Ky c 0 H . q 2 kⵜ 2 H s 0

Ž5.

For the rigorous mathematicians, the numerical result is not relevant to the real proof. Does Eq. Ž5. really can lead to a solution of RBC shape? Since the presentation of Eq. Ž4. by Helfrich for 20 years, the first exact solution like RBC shape was found under the condition of ⌬ Ps ␭ s 0 as w6x z Ž ␳ . s z Ž0. q

␳

H0 tan␺ Ž ␳⬘. d␳⬘,

Ž6.

sin␺ Ž ␳ . s c 0 ␳ln Ž ␳r␳ B . where z Ž␳ . is the contour of the cross-section of the RBC along the rotationally symmetric axis and ␺ Ž␳ . the angle tangent to the contour at axis distance ␳ ŽFig. 2.. It can be proved easily by elementary differential calculus that Eq. Ž6. does show a biconcave shape if c 0 - 0. Unfortunately, the solution is singular at ␳ s 0 although the surface of the solution is completely smooth and integrable in energy. In other words, the rigorous proof on the existence of RBC shape within the Helfrich spontaneous curvature model is still a open question in mathematics. However, it is certain that the Helfrich spontaneous model does describe the right model of the FM shape. In 1990, the Zhong-can w7x predicted the torus solutions of Eq. Ž5., in which the radii of two generating circles r and R must satisfy rrRs 1r '2 or 0. This

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Fig. 3. The torus vesicles observed by Mutz and Bensimon in 1990, in which the radii of two generating circles satisfy '2 .

striking prediction has been confirmed experimentally by three groups working in physics, biochemistry and chemistry, respectively ŽFig. 3. w8᎐10x. Very recently, new analytic solutions, namely beyond Delaunay surfaces, were predicted w11x and are believed to be sought for experiment. Besides cylinders and spheres, the mentioned solutions are the only three types of analytic solutions for Eq. Ž5., so far, known in literature. Very recently, a periodic cylindrical surface for FM was discovered by Zhang and Zhong-can w12x. It was noted that the configuration of egg lecithin in excess water observed by Harbich and Helfrich w13x in 1984 is a feature of the periodic solution.

where ⌿ is any function of H. Nitsche also put forward the shape equation of ␦ F s 0 as ⵜ 2 ⌿H q 2 Ž 2 H 2 y K . ⌿H y 4 H ⌿ s 0

Ž8.

Here ⌿H s ⭸⌿r⭸H. For the special case of ⌿ s Ž kr2.Ž2Hq c 0 . 2 the differential equation reduces to Eq. Ž5. with ⌬ Ps ␭ s 0. However, the generalized equation remains not able to involve all the shape problem in soft matter. In the shape problem of liquid crystals one will need more generalized form of shape equation. 4. Equilibrium structures of smectic-A LC

3. Surface equation generalized by Nitsche The recent analytic work w4,5,7x in the shape problem of bilayer membranes has attracted great interest from mathematicians and the Helfrich FM theory has been regarded as the renewal of the Poisson’s theory of curvature elasticity w14x. In J.C.C. Nitsche’s encyclopedic book on minimal surface Žnew edition in 1989., w15x the Helfrich energy potential Eq. Ž4. was generalized to the following form F s E w ⌿ Ž H . y ␥K x dA

Ž7.

The shape on the equilibrium structures of smectic-A LC Ž SA LC. is also a long-lasting problem. Since the discovery by Friedel and Gradjean in 1910, the focal conic domain in SA has been known as Dupin cyclides, where the SA layers preserve the interlayer spacing. However, in 1934 Bragg w16x questioned why the cyclides are preferred to other geometrical structures under the same conditions. To answer the question, we need to solve the generalized Helfrich variation problem ␦ F s ␦E⌽ Ž D, H, K . dAs 0, where ⌽ is any function, the integral is taken over the inner surface A, and D is the thickness of the SA nucleus. The

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Euler᎐Lagrange equations for the variation include a surface-integral equation w17x which corresponds to ␦ Fr␦ D s 0 as well as a new surface-differential equation derived by surface variation w18x

ž 2 H y Kq 12 ⵜ / ⌽ 2

2

Hq

Ž 2 HKq ⵜ 2 . ⌽ K y 2 H ⌽ s 0 Ž9.

where ⌽ H s ⭸⌽r⭸H, ⌽ K s ⭸⌽r⭸K, ⵜ 2 remains the Laplace᎐Beltrami operator as Ž1r g .⭸i Ž g i j g ⭸j ., but ⵜ 2 is new one defined as Ž1r g .⭸i Ž KLi j g ⭸j .. Here g i j and L i j are associated with the first and second fundamental forms of the surface, respectively. This equation is believed to be the most general form for the variation problem of Poisson-like potential. Specify for the detailed form of ⌽ in SA , it was shown w18x that when the thickness of SA nucleus, D, achieves 2 ␥rg 0 in growth, where yg 0 and ␥ are the difference of Gibbs energy densities between SA and the isotropic phase and the interface tension, respectively, then Eq. Ž9. can reduce to Eq. Ž3.. From the Willmore conjecture Ž1982., the torus of Rrrs '2 as mentioned in Section 2 and its conformal transformations are the solutions of Eq. Ž3. with absolute minimum of curvature energy. Thus, the observed focal conic domains ŽFCDs. can grow from them as seeds. The conformal invariability of the seed surface solution ensures that the space can be fully filled by the FCDs.

'

'

'

'

5. Dynamics of FM vesicles and microemulsion droplets The obtained surface equation for FM wEq. Ž5.x, has been extended as a generalized Young᎐Laplace equation for describing the equilibrium configuration of a surfactant-loaded oilrwater interface. In 1992, Ljunggren and Eriksson have published a lengthy paper in Langmuir to study the problem with great details w19x. In 1993, the same Eq. Ž5. together with Navier᎐Stokes equation was also applied to investigate the dynamics of a three-component lamellar systems Ž i.e. oilrwaterrsurfactant . and regarded as an Extended Theory of Lamellar Hydrodynamics w20x. The dynamics for such systems ŽFM vesicles, red blood cells and microemulsion droplets. are closely related to the physiological rheology, clearance industry, and extraction of petroleum. Since the surface tension of the FM or surfactant-loaded interfaces is near zero in comparing with that of water, the systems can form large interfacial areas separating oil and water and their deformation is thereby governed mainly by their elastic bending energy wEq. Ž4.x. This is why the surface equation of FM wEq. Ž5.x, and its generalized form including viscous

effect of fluids in both sides of the interfaces have an important role in the study of the dynamics of the systems. In the literature there have been several controversial dynamical treatments of FM vesicles and microemulsion droplets. Schneider, Jenkins and Webb as well as many others w21᎐23x assumed so-called ‘stick’ boundary conditions to describe the balance of forces in the interfaces in which the total area and volume are treated as conserved, both for FM vesicles and microemulsion droplets. In contrast, Van der Linden et al. w24x insisted that only the area should be kept constant for FM vesicles, whereas only the volume kept constant for microemulsion droplets. To solve this difficulty, in 1993 Komura and Seki w25x noted that the general shape equation wEq. Ž5.x for FM is a satisfying boundary condition of balance of forces at the interfaces regardless of whether area or volume is constant and can be applied both to FM vesicles and microemulsion droplets having been called the generalized Laplace’s formula obtained by Zhong-can and Helfrich. They conclude that this choice is much simple and straightforward as compared to the previous treatments w21᎐24x. In order to construct the dynamical boundary condition that must be satisfied at the interface between two viscous fluids in motion, the surface equation wEq. Ž5.x is then generalized as ⌬ Py 2␭ H q k Ž 2 H q c 0 .Ž 2 H 2 y 2 Ky c 0 H . q2 kⵜ 2 H n s ⌬␴ = n

Ž 10 .

where n is the normal vector of the interface and ⌬␴ is the difference of the viscous stress tensors between the interface as ⌬␴ s ␴out y ␴in . Both ␴out and ␴in for incompressible fluids are defined by the relations with both flow fields as w25x Ž ␴out,in . ␣ ␤ s ␩out,in Ž ⵜvout,in . ␣ ␤ q Ž ⵜvout,in . ␤ ␣

Ž 11 .

where ␩out,in are the viscous constants of the fluids in both sides of the interface, respectively, vout,in are flow fields of the surrounding fluids which have to satisfy in addition the Navier᎐Stokes equation w26x. Obviously, for generalized differential equation of surface in complex FM, Eq. Ž9. the same extension as in Eq. Ž10. can be applied to the dynamical study in the most complex systems. 6. Summary The shape problem in FM and LC does offer a wide field in both physics and mathematics w27x. Although a series of successful results in theory compared with experiment have been obtained, the research in the

O. Zhong-can r Thin Solid Films 393 (2001) 19᎐23

field is still in its infancy. More progress in both theory and experiment is to be expected. Acknowledgements The project is supported by the National Natural Science Foundation of China. I also thank my Japanese colleagues Professors Y. Yamazaki, M. Iwamoto, A. Sugimura, H. Naito and S. Komura for their collaboration in both fields of LC and biomembranes. References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x

W. Helfrich, Z. Naturforsch. C 28 Ž1973. 693. H.J. Deuling, W. Helfrich, J. Phys. ŽParis. 37 Ž1976. 1335. H.J. Deuling, W. Helfrich, Biophys. J. 16 Ž1976. 861. O.-Y. Zhong-can, W. Helfrich, Phys. Rev. Lett. 59 Ž1987. 2486. O.-Y. Zhong-can, W. Helfrich, Phys. Rev. A 39 Ž1989. 5280. H. Naito, M. Okuda, O.-Y. Zhong-can, Phys. Rev. E48 Ž1993. 2304. O.-Y. Zhong-can, Phys. Rev. A 41 Ž1990. 4517. M. Mutz, D. Bensimon, Phys. Rev. A 43 Ž1991. 4525. A.S. Rudolph, B.R. Ratna, B. Kan, Nature 352 Ž1991. 52. Z. Lin et al., Langmuir 10 Ž1994. 1008.

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