Lecture III. Curvature Elasticity of Fluid Lipid Membranes. Helfrich s Approach

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1 Lecture III. Curvature Elasticity of Fluid Lipid Membranes. Helfrich s Approach Marina Voinova Department of Applied Physics, Chalmers University of Technology and Göteborg University, SE , Göteborg, Sweden In this lecture we discuss how to find a shape of closed lipid bilayer membrane (a vesicle)- an excellent model for studying mechanical properties of living cell membranes, in particular the red blood cells. In the lecture we will based on the so-called Helfrich theory of curvature elasticity (for the first time, published in W.Helfrich, Z.Naturforsch. C28 (1973) p.693), the most successful physical theory for description of equilibrium membranes shape. PACS numbers: CURVATURE ELASTICITY OF LIPID BILAYER MEMBRANES. The original derivation of elastic free energy of mechanically deformed membrane was based on the derivation of curvature elastic energy of liquid crystals. Wolfgang Helfrich deduced the expression for the elastic energy of curvature per unit area of the membrane as g c = (1/2)k(c 1 + c 2 c 0 ) 2 + kc 1 c 2, (1) where c 1, c 2 are two principal curvatures of the surface of the membrane as discussed in Lecture 1, and the constant c 0 is called the spontaneous curvature of the membrane surface. The total bending energy of the membrane F is often referred to as the total bending energy of the membrane, is given by F = g c da. (2) The constant c 0 can be attributed to the mean curvature of the membrane with asymmetric chemical composition of layers in bilayer or the environment and is closely related to the spontaneous splay of the liquid crystals.

2 2 The constant k is the bending rigidity and k is the elastic modulus of the Gaussian curvature K = c 1 c 2. By comparison with the curvature elasticity of liquid crystals, both k and k are found to be of order of the product of the elastic constants of lipid bilayer and the thickness of the membrane ( J). Equation (2) (together with (1)) is called the Helrich free energy of lipid membranes and is generally recognized as the basic quantity in dealing with the mechanical behavior of biomembranes in the liquid crystal phase. Later on, Helfrich gave a rigorous way to prove eqns.(1,2). In the lecture we introduce these eqns without proof. Bending deformations can be described in terms of curvature of the membrane surface. As we have seen in previous lectures, the curved properties of a surface at any point are described by two principal curvatures c 1 and c 2. One can show, that up to quadratic terms of c 1 and c 2 the bending energy expression is a complex combination of all the linear and the quadratic invariants. One can prove that in linear form only (c 1 + c 2 ) 2, c 1 c 2, c c 2 2, and (c 1 c 2 ) 2 terms appear. Taking the sum of the linear term c 1 + c 2 and the first two quadratic terms (c 1 + c 2 ) 2 and c 1 c 2, one can construct an energy density expression just like eqn. (1). A complete derivation of Helfrich s energy (1) can be done by using the rigorous 2D differential invariants. In differential geometry language, eqn.(1) gives F c = g c da = [(1/2)k(2H c 0 ) 2 + kk]da. (3) GENERAL SHAPE EQUATION. In equilibrium state the elastic free energy of the membrane must be at its minimum, i.e. the equilibrium energy of the system must be less than that in a slightly deformed state. For elastically deformed membrane Helfrich proposed that the equilibrium shape of a closed bilayer membrane (a vesicle) or the shape of a red blood cell membrane be given by minimum of the functional F Helfrich = F c + P dv + λ da = k/2 (2H c 0 ) 2 da + P dv + λ da, (4)

3 3 where F c is the curvature-elastic energy and k is the bending rigidity of the vesicle membrane. The surface element da and the volume element dv are given by the following expressions: da = g 1/2 dudv, (5) dv = 1 3 g1/2 ( Y n )dudv, (6) where Y is a point on a given surface expressed in terms of two independent parameters u and v, Y = Y (u, v), n is the unit normal of the surface, and g is g EG F 2 where E, F and G are coefficients of the first fundamental form I = ds 2 = Edu 2 + 2F dudv + Gdv 2. H is the mean curvature of the membrane surface H = (c 1 + c 2 )/2, and c 0 is the spontaneous curvature of the membrane. The value P may be considered as the pressure difference between the outside and the inside of the membrane or as the osmotic pressure, and λ may be considered as the tensile stress acting on the membrane or as the surface tension. Since water can penetrate through lipid bilayer membrane but salt can not, the difference in the salt concentration outside- and inside the vesicle C salt will produce the osmotic pressure difference P α C salt. The elastic membrane shell will increase in volume by taking water to compensate the osmotic stress and the surface tension (swelling process at hypotonic osmotic pressure), or vice versa, it will decrease it volume (shrinking process) at hypertonic osmotic pressure, respectively. These volume changes led to the visible shape transformations or the osmotic reaction of the cell membrane. In order to find the shape equation of the vesicle it is necessary to calculate the first variation δf Helfrich of the free energy F Helfrich. The variation of the free elastic energy in Helfrich s form gives us P 2λH + k(2h + c 0 )(2H 2 c 0 H 2K) + k (2H + c 0 ) = 0, (7) where is the Laplacian. This is just the general shape equation of the vesicle membrane.

4 4 SOLUTIONS OF GENERAL SHAPE EQUATION. The general shape equation (7) is a nonlinear differential equation of high order. Its two simple solutions are the sphere and the circular cylinder. For the surfaces of revolution, Deuling and Helfrich provided a catalog of numerically calculated membranes shapes found by minimization of the Helfrich s free energy. Below we attach the selected examples of predicted lipid vesicles and red blood cell shapes presented in ref.[2]. The general analysis on the equation is still lacking, however several particular solutions have been found. Among them, the exact solutions are the so-called Clifford torus and circular biconcave discoid showing that the Helfrich s theory can explain the existing shapes of red blood cells and also predict new shapes of membranes. EXERCISES. Exercise I. Calculate the Helfrich curvature elastic energy for the spherical membrane (in the relaxed tensile-free state i.e. for the λ = 0, P = 0). Exercise II. Calculate the Helfrich free energy for the cylinder membrane of radius r (in the relaxed tensile-free state, i.e. for λ = 0, P = 0). REFERENCES. 1.W.Helfrich, Z.Naturforsch. C28 (1973) p H.J.Deuling and W.Helfrich, The curvature elasticity of fluid membranes: a catalogue of vesicle shapes. J.Phys. (Paris), 37 (1976) C. E. Weatherburn, Differential Geometry, vol. XX, in Pure and Applied Mathematics, eds. R. Courant, L. Bers, J. J. Stoker M. P. de Carmo, Differential Geometry of Curves and Surfaces. Prentice-Hall, Ou-Yang Zhong-Can, Liu Ji-Xing, Xie Yu-Zhang. Geometric methods in the elastic theory of membranes in liquid crystal phase. Advanced Series on Theoretical Physical Science, v.2, World Scientific, S.Hyde,S.Andersson, K.Larson, Z.Blum, T.Landh, S.Lidin and B.W.Ninham. The Language of Shape. Elsevier, 1997.

5 5 FIG. 1: A biconcave discoidal shape of red blood cells (discocyte, upper picture) and a deformed discocyte through work of osmotic pressure ; the cell looks as a cup-shaped one (bottom picture)

6 FIG. 2: 6

7 FIG. 3: 7

8 FIG. 4: 8

9 FIG. 5: 9