Elastic moduli and neutral surface for strongly curved monolayers. Analysis of experimental results

In this paper we apply our results concemmg a new view on a mechanical descnption of strongly curved amphiphllic interfaces to expenmental data available in the literature (Rand et al, 1990) Our theory predicts that, for cyhndncal interfaces, one has to take into account, three elastic moduh corresponding to area extension, bending and mtxed deformations We determine the values of the three elastic moduh for the mtemal dividing surface of strongly curved DOPE monolayer m the Hit-Phase Analysing for this purpose the expenmental we recognise lateral compressibility and bending m the same time As a result we recover the moduh that are closer to work done on bilayer than had been determined m precedent papers considenng only bending energy. Using this set of parameters we find the position of the neutral surface and the values of elastic moduh corresponding to the neutral surface We find the quantitative dependence of all the elastic moduh on the pos1tlon of the d1vlding surface and compare them with those previously obtained In«oduction. Recently several expenments [1-4] have been performed on interfaces with low surface tension and large curvature. Gruner [I], Rand [2] and coworkers performed a stimulating investigation of the mechanical properties of the strongly curved lipid monolayers, namely of the inverted hexagonal phase (Ha of pure lipid DOPE and of mixture between DOPE and DOPC In one senes of experiments only DOPE and water have been used. In a second senes a DOPE/DOPC mixture has been used as surfactant; however, to obtain an Hii-Phase, tetradecane has been added If water is in excess, the Hn-Phase has the equihbnum geometncal charactenstic, namely the lattice dimension measured by means of X-ray diffraction A simple calculation taking into account the known quantity of lipid and water in the system allows to determine the internal radius of lipid cylinder corresponding to the boundary between water volume and lipid monolayer. The radii of spontaneously formed lipid cylinders are small and show a typical value in the range of 2-3 nm depending on the (*) Permanent address AN Frulnkln Instituie of Electrochen~istry, USSR Academy of Science, 31Lemnsky Prosp Moscow, l17071, USSR rum JOURNAL DE PHYSIQUE II bt 9 compositions This means that the radius of curvature is comparable to the monolayer thickness of about approximately 1.5 nm The expenments by Rand and coworkers [2] have been performed in the following way An osmotic pressure difference between the mtemal volume of cylinder and the surrounding water solution was applied. The cylinder was deformed by this pressure difference. The deformation included the bending the change of mtemal radius of the monolayerand the extension-compression the change of area per lipid molecule at the intemal surface of the monolayer. The values of both deformations were calculated directly from the values measured expenmentally. The results of these expenments give a rare possibility to determine the elastic moduh of the strongly curved lipid monolayer. However the authors of expenmental investigations [2] analysed the data only with respect to a change of the curvature of the monolayer as the response to the action of applied pressure difference For this purpose they descnbed the monolayer in terms of the d1vlding surface situated inside the nonpolar region of the monolayer and keeping approximately its area constant dunng deformation. In this way any possible effects due to an area extension-compression are not considered. At the same time the expenmental data concemmg area extension-compression of the intemal surface of the monolayer give important additional information about the mechanical properties of the system. A more complete description of the mechanics of cylindrical monolayer was made by us in the precedent paper [5] showing that the following important points have to be taken into account m the analysis of the mentioned expenments First of all the choice of the dividing surface plays an essential role m the mechanical descnption of an interface. In general any arbitrary dividing surface is characterized by three elastic moduli corresponding to area extension, bending and mixed deformations. The simplest descnption of the monolayer by means of just two elastic modub (of area extension and of bending) is only possible for the neutral dividing surface, which by our definition is that of zero modulus of mixed deformation. However, the position of the neutral surface is a priori unknown and in the general case does not coincide with the position of the dividing surface of constant area. Secondly, the values of all elastic moduh strongly depend on the position of the dividing surface. This dependence is especially important for the bending modulus. For strongly curved monolayer the value of bending modulus corresponding to the intemal dividing surface is considerably smaller than that of the outer dividing surface In the present paper we will revisit the expenmental results concerning the mechanical deformation of lipid cylinders m Hjrphase of DOPE performed by Rand and coworkers [2]. For this purpose we will apply the complete descnption developed by us [5] First, the deformation of the intemal d1vldmg surface of the monolayer will by analysed. From the expenmental results we will determine three elastic moduh of the mtemal d1vldmg surface corresponding to area extension, bending and mixed deformations. These results allow us to determine the position of the neutral surface and m a next step to calculate the elastic moduli of the neutral surface The last part of the analysis is the quantitative calculation of the dependence of the set of elastic modub on the position of the dividing surface. All calculations will be performed without the need of a microscopic model of the structure of the lipid monolayer Main equations. THE SURFACE TENSION AND THE BENDING MOMENT Let US Consider a Cylindncal monolayer (Fig I). Let us assume a pressure difference between the outer medium and the internal volume of the cylinder, p = p~~ipm. For the descnption of the mechanical bt 9 ELASTICITY OF CURVED MONOLAYER ANALYSIS OF EXPERIMENT 1087