Flexoelectricity of Model and Living Membranes

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Biochimica et Biophysica Acta 1561 (2001) 1^25 www.bba-direct.com Review Flexoelectricity of model and living membranes Alexander G. Petrov * Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee, 1784 So¢a, Bulgaria Received 25 April 2001; received in revised form 14 June 2001; accepted 20 June 2001 Abstract The theory and experiments on model and biomembrane flexoelectricity are reviewed. Biological implications of flexoelectricity are underlined. Molecular machinery and molecular electronics applications are pointed out. ß 2001 Else- vier Science B.V. All rights reserved. Keywords: Bilayer lipid membrane; Biomembrane; Curvature; Flexoelectricity; Mechanosensitivity; Electromotility; Membrane machine; Molecular electronics 1. Introduction Furthermore, the recognition of some speci¢c, mechanical pathways for energy transformation in bio- In 1943 Erwin Schro«dinger postulated that a living membranes is needed. cell should function like a mechanism [1]. He was Biomembranes constitute the basic building units then concerned about ‘solid’ parts of the cells, i.e. of the majority of cells and cellular organelles. It DNA molecules. appears that most membranes are built up according The present review aims at an extension of Schro«- to the general principles of lyotropic liquid crystal dinger’s postulate to the liquid crystal parts of the structures [2]. The widely accepted ‘£uid lipid^glob- cell, the biomembranes. In the ¢rst place this implies ular protein mosaic model’ [3^5] claims that lipids the recognition of the existence of an appropriate are organized in a bilayer in which the proteins are mechanical degree of freedom in a biomembrane. immersed. Model membranes containing lipids only, Abbreviations: AFM, atomic force microscopy; BLM, black lipid membrane; DPhPC, diphytanoylphosphatidylcholine; EYPC, egg yolk phosphatidylcholine; GMO, glyceryl monooleate; LC, liquid crystal; PC, phosphatidylcholine (lecithin); PE, phosphatidylethanol- amine; PS, phosphatidylserine; pp, peak-to-peak; rms, root mean square; UA, uranyl acetate; A, area per molecule; c0, spontaneous curvature; c1;2, principal curvatures; d, membrane thickness; E, electric ¢eld; e, proton charge; f, £exoelectric coe⁄cient; H, ¢eld coupling parameter; If , £exoelectric current; Ig, ¢rst harmonic current amplitude; I2g, second harmonic current amplitude ; K, curvature elastic modulus of a bilayer; kB, Boltzmann constant; N, number of molecules per unit area; PS, electric polarization per unit area; p, hydrostatic pressure di¡erence; T, absolute temperature; t, curvature elastic torque; U, voltage di¡erence; Uf , £exoelectric voltage; vV, surface potential, interfacial potential di¡erence; L, degree of dissociation of polar head; NN, neutral monolayer surface distance to the bilayer’s mid-surface; NH, head’s surface distance to the neutral monolayer’s surface; OL, dielectric constant of lipid core of bilayer; Ow, dielectric constant of water; O0, absolute dielectric permittivity of free space; R, viscosity; VD, Debye length; W, electric dipole moment; X, frequency; c, surface charge density; ca, surface tension; M, dielectric susceptibility; id, dipole surface potential; i0, charge surface potential; g, angular frequency * Fax: +359-2-975-36-32. E-mail address: [email protected] (A.G. Petrov). 0005-2736 / 01 / $ ^ see front matter ß 2001 Elsevier Science B.V. All rights reserved. PII: S0304-4157(01)00007-7 BBAMEM 85535 17-4-02 2 A.G. Petrov / Biochimica et Biophysica Acta 1561 (2001) 1^25 or lipids with just one type of protein, etc., are also well known. A membrane has a number of mechanical degrees of freedom: area stretching, thickness compression, shear deformation, chain tilting, and, notably, curvature deformation. The last one, curvature, is just a liquid crystal degree of freedom since membrane curvature is equivalent to a splay of lipid chains (under the condition that chains remain parallel to the local normal in each point of the curved bilayer). Fig. 1. Flexoelectric (curvature-induced) polarization PS and The ¢rst striking impression when looking at an sign convention about £exocoe⁄cient f: for the case shown f electron micrograph of a cell is that cell membranes will be positive. R1 and R2 are principal radii of membrane cur- very often tend to be strongly curved. Unlike tensile vature (from [7], with permission from the publisher). or compressive strain, it is very easy to sustain curvature strains (torques) in a membrane because of the smallness of its curvature elastic modulus K [6] the Helmholtz equation. Its curvature-dependent (typically a few kBT units). Examples include the part reads: highly convoluted cristae of inner mitochondrial v U ¼ P =O ¼ðf =O Þðc þ c Þð2Þ membranes in energized mitochondria, the edges of S 0 0 1 2 retinal rod outer segments and discs, the tylakoid By measuring simultaneously this potential di¡erence membranes of chloroplasts, the brush borders of in- and the curvature, one can determine a £exoelectric testinal epithelial cells, the spiculae tips in echyno- coe⁄cient of a given membrane. Like the piezoelec- cytes, the microvilli between contacting cells, the tric e¡ect in solid crystals, the £exoelectric one can stereocilia in the hearing organ, etc. In many cases also be manifested as a direct e¡ect (Eq. 1) and a this curvature is dynamic, e.g. there is good evidence converse e¡ect, featuring electric ¢eld-induced curva- that the peripheral protein network of a red blood ture [7]: cell membrane is able to produce a membrane curva- c þ c ¼ðf =KÞE ð3Þ ture at its contraction. 1 2 where E is the transmembrane electric ¢eld (strictly speaking, an averaged ¢eld according to Eq. 10 be- 2. Membrane curvature and £exoelectricity low) and K is the curvature elastic modulus. Eq. 3 is valid for a tension-free membrane (identically zero Flexoelectricity is a mechanoelectric phenomenon lateral tension), which is the case in an osmotically known from liquid crystal physics. In the case of a balanced cell. membrane £exoelectricity stands for curvature-in- Following [9] phenomenological equations of duced membrane polarization [10,7]: membrane £exoelectricity can be written in terms of ¢rst derivatives of a thermodynamic potential of PS ¼ f ðc1 þ c2Þð1Þ a £exoelectric body with curvature deformation and where PS is the electric polarization per unit area in electric ¢eld as independent variables: C/m, c1 and c2 are the two principal radii of mem- D H 31 E C brane curvature in m and f is the area £exoelectric t ¼ ¼ Kcþ3f E ð4aÞ D c coe⁄cient in C (coulombs), typically a few units of þ electron charge (Fig. 1). This e¡ect is manifested in D H P ¼ E ¼ f Dc þ M E ð4bÞ liquid crystalline membrane structures where an s D E þ s overall curvature is related to an orientational deformation of the splay type of membrane molecules where HE (cþ, E) is a thermodynamic potential (lipids, proteins) (cf. [8]). Across a polarized mem- called electric enthalpy, cþ = c1+c2 is total membrane brane a potential di¡erence develops according to curvature, Ms is area dielectric susceptibility ( = Md), BBAMEM 85535 17-4-02 A.G. Petrov / Biochimica et Biophysica Acta 1561 (2001) 1^25 3 D 2H f D ¼ E is direct flexocoefficient; that is, by an integral of the curvature derivative of D cþD E polarization distribution in a £at membrane at zero electric ¢eld, over membrane thickness. 2 D HE fC ¼ is converse flexocoefficient: ð5Þ 3.2. Converse £exocoe⁄cient D ED cþ Applying an electric ¢eld across a £at membrane, However, both £exocoe⁄cients must be equal be- its electric enthalpy term is given by the product of cause second derivatives are independent of the se- its preexisting polarization and the applied ¢eld: quence of di¡erentiation (Maxwell relation). There- Z fore, Eqs. 2 and 3 are written in terms of one and the HE ¼ 3 Pðz; cþÞEdz same £exocoe⁄cient f. Eq. 3 follows from Eq. 4a in the case of vanishing curvature elastic torque (t = 0), From the ¢rst phenomenological equation one gets while Eq. 1 follows from Eq. 4b when E =0. then for the torque (note that cþ =0): Z Z Exploring the molecular mechanisms of £exoelec- D D P tricity is a central task of the liquid crystal approach t ¼ 3 Pðz; cþÞEdz¼ 3E j0 dz ð8Þ D c D c in membranology [11,12,7]. The £exoelectric coe⁄- þ þ cient can be represented as an integral over the de- i.e. converse £exocoe⁄cient is: rivative of P (z), the distribution of the normal com- Z z D Pðz; c Þ ponent of polarization across the membrane (see f C ¼ þ j dz ð9Þ D c 0 below). Model distributions including electric mo- þ nopoles, dipoles and quadrupoles of lipids and pro- This proves the validity of the Maxwell relation for a teins have been considered [7], revealing respective membrane, and demonstrates that both direct and contributions to f. converse £exocoe⁄cients depend on the polarization distribution across a £at membrane and at zero electric ¢eld (there is no applied ¢eld at the direct £ex- 3. Flexoelectricity of membranes and polarization oe¡ect calculation). distribution Moreover, Eq. 8 provides a de¢nition for the aver- age electric ¢eld to be used in Eqs. 3, 4a and 4b: the A basic object in the theory is the distribution of averaging uses the curvature derivative of the polar- polarization along membrane normal, P(z,cþ). ization as a weighting function: If it is known, then £exocoe⁄cients can be ex- R D Pðz; c Þ pressed as integrals of its curvature derivative, as EðzÞ þ j dz D c 0 follows. E ¼ þ ð10Þ R D Pðz; c Þ þ dz D j0 3.1. Direct £exocoe⁄cient cþ Area polarization is expressed as: Z Z 4.

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