Physics of Cellular Materials: Biomembranes

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1 Physics of Cellular Materials: Biomembranes Tom Chou 1 1 Dept. of Biomathematics, UCL, Los ngeles, C (Dated: December 6, 2002) Here I will review the mathematics and statistical physics associated with surfaces that form the boundaries of cells and internal organelles. Mechanical models such as Canham-Helfrich energy, bilayer couple (BC), and the area difference elasticity (DE) models will be presented. The concept of entropic tension will also be discussed. I. MEMBRNES Membranes define the boundaries of cells and internal organelles such as mitochondria, ER, Golgi bodies, chloroplasts, etc. They are comprised of amphiphillic molecules, those with parts that favor associations with water, and parts that repel water. Lipid bilayers are essentially two-dimensional incompressible fluids with a molecular thickness that imparts bending elasticity. Figure 1: schematic of the membrane surfaces in a typical cell. closer view of the plasma membrane shows a large number of proteins and structures associated with the lipid bilayer. molecular dynamics view of a small patch of lipid bilayer reveals the molecular structure. Thermodynamics Before presenting models of membrane bending, we consider the concept of surface tension of a membrane by looking at its thermodynamics. The origins of tension for material membranes like lipid bilayers is physically different from the familiar notion used to study e.g. the liquid-vapor interface. t an air-water interface, surface tension arises because of the difference between the energies of a water-water interaction compared to a water-air molecule interaction. t an air-water interface, the water molecules are coordinated with only about half the number of nearest neighbor water molecules. Bulk water has a lower energy than the waters at the interface. Thus, an interfacial energy arises due to the number of water-air interactions. Now consider lipid membranes. Lipid bilayers are inextensible so the number of lipid head-water interactions is relatively fixed unless the lipid molecules exchange with lipid molecules in the bulk. Therefore, we can define independent conjugate tensions for both material area, and the projected area p. Various free energies, or thermodynamic potentials, can be derived and used depending on the experimental measurement. Consider the patch of membrane shown in Fig. 2(b). It has a fairly fixed material area since membranes are nearly inextensible. Therefore, to a very good approximation, will be proportional to the total number of lipids in the patch

2 2 > (a) p Figure 2: (a) Molecular surface tension. (b) Fluctuations of a fixed material surface. (b) of membrane. This patch is fixed on a frame of area p, the projected area of the membrane patch. Both areas may be relevant. When one is considering mechanical pulling experiments on a thermally fluctuating membranes, the pulling force is the conjugate variable to p, while if one is concerned with say adsorption of protein on a membrane surface, the relevant area is. n inextensible material area can fluctuate only if the lipid constituents are exchanging with lipid molecules in the bulk solution. Let us call the thermodynamic variables conjugate to and p, γ and τ, respectively. The mechanical tension is τ, while the material surface tension is γ, much like an air-water surface tension. Therefore, we can distinguish four thermodynamic ensembles: I. (, p )-ensemble, isolated and framed (both and p fixed) II. (, τ)-ensemble, isolated, unframed membrane III. (γ, p )-ensemble, open, framed membrane IV. (γ, τ)-ensemble, open, unframed membrane. Isolated membranes cannot exchange lipid constituents with lipid molecules in the surroundings. Whether or not the bilayer effectively exchanges lipids depends on the time scale for expulsion and reincorporation of lipid molecules. If this is slow compared to the relevant experimental time scale, then one can assume is fixed. Experimentally, one typically has open, framed systems such as a long experiment in which a lipid is stretched across an orifice. The ensemble relevant for a biological vesicle in which lipids exchange slowly is the isolated, unframed system. In the (γ, p ) ensemble, the energy is H(γ,, p ) = H b + γ, (1) and depends on both the precise values of γ and. Here H b is the elastic energy of the membrane which we will take as the bending energy since we are assuming inextensibility. Nevertheless, in the fixed γ ensemble, the total material area fluctuates as lipids are expelled and readsorbed. Thus the partition function is Z(γ, p ) = C e H/k BT, (2) where the sum is performed over all possible configurations with fixed p, but any. The free energy in this case is then and the membrane mechanical tension is G o (γ, p ) = k B T ln Z(γ, p ), (3)

3 3 τ = G o (γ, p ) lim. (4) p p In the case of an isolated, unframed membrane, (, τ)-ensemble, we can find the relevant thermodynamic potential by applying Legendre transforms stepwise, and transform the (γ, p )-ensemble first to the (, p )-ensemble, then from the (, p )-ensemble to the (, τ)-ensemble. The first transform is effected by F (, p ) = G o (γ, p ) γ. (5) The value of γ that fixes the specific value of is found implicitly by solving F/ γ = 0, or = ( G/ γ) p. The next Legendre transformation is where τ is found from G i / p, or G i (, τ) = F (, p ) τ p, (6) τ = F p. (7) What is the meaning of tension τ for an isolated, unframed membrane patch? The mean projected area is can be found from F/ p = 0. If F (, p ) has a minimum for 0 < p / < 1, then τ = 0. lthough the membrane is shrunk due to thermal fluctuations, it is nonetheless a flat, extended object and has τ = 0 since Only when p = 0 can τ 0. F p = 0. (8) p= p B Energetics Now consider the bending part of the energy H b. The bilayer material as a wide aspect ratio, that is, its thickness is about 4-5nm and much smaller than the lateral size of the surfaces of interest. Therefore, we can treat the bending deformations of the membrane with ideas from plate or thin shell theory. Figure 3: Typical constituents of lipid bilayers standard model for this is one first written by Canham called the Helfrich free energy density: H b [S] = κ(s) 2 (C(S) C 0(S)) 2 + κ G (S)K(S), (9)

4 which is a local energy density due to local bilayer bending at surface position S. In (9), C(S) C 0 (S) is the mean curvature (measured from the midplane of the lipid) at S, K(S) is the local Gaussian curvature. The are the only two independent scalar deformations of an infinitely thin interface (they are the trace and determinant of the curvature tensor defining the second fundamental form). The parameters κ(s) and κ G (S) are microscopic parameters with units of energy. If there is no change in topology of a surface the κ G (S)K(S) is just a constant from the Gauss-Bonnet Theorem. Thus, it suffices to consider only the mean curvature term. The bending rigidity κ(s) can depend on surface position and composition (larger for higher cholesterol content) and is typically k B T or lipid bilayers. This is huge and in many situations it suffices to consider the low temperature regime, where thermal fluctuations are negligible. Thus, simple energy minimization does a good job in predicting vesicle shapes for example. The total bending energy of a surface is 4 H b = H b [S]dS = κ(s) 2 (C(S) C 0(S)) 2. (10) The bending rigidity is a function of the molecular details of the lipid bilayer. simple model for κ considers the bilayer as a continuum elastic plate. For a homogeneous medium, the bending rigidity is proportional to the cube of the plate thickness, κ d 3. However, lipid bilayers can have neutral surfaces (surfaces where bending does not result in stretching or compression) that are located anywhere from the outer edges (near the head groups) to the midplane. When the lateral size of the headgroups is large compared to that of the acyl chains, the area per lipid molecule is controlled by the headgroups and bending stresses are transduced along the two outer surfaces of the bilayer. In this extreme limit, κ d 2. The two leaftlets usually slide against each other so it has been necessary to consider the membrane as a two-layer system in order to qualitatively predict experimentally observed vesicle shapes. C Bilayer Couple Model The bilayer couple model treats each leaflet as independent plates that can slide over each other and that follow the Canham-Helfrich energy (12). However, for a closed vesicle for example, the two leaflets will have different numbers of lipid molecules and hence a different areas. nd since inextensibility demands that each leaftlet have the same area, there is an additional constraint on the area difference expressed in terms of the mean curvature d (C(S) C 0 (S))dS, (11) Exchange of lipids between the two leaflets has been ignored. occurring on time scales of 10min to days. This process has been measured and is very slow, D rea Difference Elasticity Model This is a generalization of the BC model to slight extensibility. Here, instead of hard fixing the area difference between the leaflets, we allow it a harmonic variation. lthough the elastic stretching energy is huge, and leaflet areas change slightly, the relative change is appreciable when compared to the area difference = + (± denote the outer and inner leaflets). The DE energy is H b = H b [S]dS = κ(s) 2 (C(S) C 0(S)) 2 + k π 2 d 2 ( 0) 2. (12) Note that k = 0 corresponds to the original spontaneous curvature model, while k limit yields the BC model. Other constraints can be straightforwardly applied by Legendre transforms. For example, if the interior volume of an impermeable vesicle and its midplane area are fixed, the thermodynamic potential is where S labels each surface configuration. Ω[S] = H b [S] + γ[s] + P V [S], (13)

5 5 E Membrane Proteins and Hydrophobic Mismatch Thus far, we have not considered local deformations in bilayer thickness. This has been shown to be important when considering how embedded or adsorbed proteins distribute themselves. Embedded proteins have different parts that like to be in contact with either the lipid tails or the polarizable, hydrogen bonding headgroups. Therefore, a protein can pinch a lipid bilayer much like the buttons on a mattress. In fact this is sometime called the mattress model. u+ u d~ 4 5nm Figure 4: Scematic depicting local membrane protein-induced lipid bilayer deformations. Using analogies with liquid crystals, an elastic energy can be written for small deformations u ±. H el = H b + ds B 2d (u + u ) 2 + ds κ(s) d [ ( 2 ) 2 ( u ) 2 ] u + γ 2 ds ( u u 2). (14) The deformation about a protein can be computed by imposing the appropriate boundary condition at the proteinmembrane contact region, and minimizing H el with respect to u ±. REFERENCES [1] P. B. Canham, J. Theor. Biol., 26, 61, (1970). [2] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch, 28C, , (1973). [3] E. Evans and R. Skalak, Mechanics and thermodynamics of of biomembranes (CRC Press, 1980). [4] E. Evans and W. Rawicz, Entropy driven tension and bending elasticity in condensed fluid membranes, Phys. Rev. Lett. 64, (1990). Seminal paper on measurements of entropic tension. [5] H. W. Huang, Deformation free energy of bilayer membrane and its effect on gramicidin channel lifetime, Biophys. J., 50, , (1986). [6] T.. Harroun, W. T. Heller, T. M. Weiss, L. Yang, and H. W. Huang, Theoretical analysis of hydrophobic matching and membrane-mediated interactions in lipid bilayers containing gramicidin, Biophys. J., 76, , (1999). [7] L. Peliti, in Fluctuating Geometries in Statistical Mechanics and Field Theory, F. David, P. Ginsparg, J. Zinn- Justin, (Eds.) Les Houches, Session LXII, 1994 (msterdam: Elsevier, 1996) cond-mat/ Good review article. [8] U. Seifert, Conformal Transformations of Vesicle Shapes, J. Phys. : Math. Gen. 24, L573-L578 (1991). [9] L. Miao, U. Seifert, M. Wortis, and H. G. Dbereiner, Budding Transitions of Fluid-Bilayer Vesicles: The Effect of rea-difference Elasticity, Phys. Rev. E 49, (1994). Detailed treatment of DE. First explicit application of DE. Good overview of all membrane models. [10] U. Seifert, The concept of effective tension for fluctuating vesicles, Z.Physik B 97, (1995). Very technical. [11] T. Chou, K. S. Kim, and G. Oster, Statistical thermodynamics of membrane bending-mediated protein-protein attractions, Biophys. J., 80, , (2001). Protein-induced bending of membranes. [12] U. Seifert, Configurations of fluid membranes and vesicles, dv. Phys. 46, (1997). Very good review article. [13] Erich Sackmann has also done many experiments on flicker analysis (power spectrum) of membranes.