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Mechanics of inter-monolayer coupling in fluid surfactant bilayers Yeung, Anthony Kwok-Cheung 1994

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Mechanics of Inter-Monolayer Couplingin Fluid Surfactant BilayersbyAnthony K.C. YeungB.A.Sc., University of British Columbia, 1983M.A.Sc., University of British Columbia, 1987A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Physics)We accept this thesis as conformin to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1994©Anthony K.C. Yeung, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)___________________________Department of_________________________The University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)AbstractFluid surfactant bilayers are composed of two mono-molecular sheets that are weakly heldtogether by normal stresses. Except for viscous resistance, the monolayers are free to sliderelative to one another — giving rise to “hidden” degrees of freedom within the compositestructure. This thesis examines such an added level of complexity to the study of bilayer membranes. A continuum theory for monolayer-coupling is first developed, whichforms the theoretical basis for the work reported here and for future analyses on stratified fluid membranes. The phenomenological coefficient of dynamic (viscous) couplingbetween monolayers is then measured experimentally by a novel method called nanotetherextrusion. This technique, which is developed by myself for this present investigation, enables measurements of inter-monolayer viscous stresses in unsupported bilayers. Further,a useful spin-off from such a technique is the determination of bilayer bending rigidities.Finally, as an application of the present continuum model, the static and dynamic featuresof a bilayer vesicle’s Brownian shape undulations are predicted with the membrane conceptualized as a stratified composite structure. Throughout this thesis, it is shown thatthe effects of bilayer stratification (monolayer-coupling) are very important. For example,the conformational dynamics of bilayer membranes on mesoscopic length scales (roughlybetween 1 to 100 nanometers) are governed by the inter-monolayer viscous forces; in suchsituations, dissipation due the surrounding hydrodynamics plays only a secondary role.11ContentsAbstract iiTable of Contents iiiList of Tables viiList of Figures viiiAcknowledgements x1 Introduction 11.1 Overview of the Thesis 31.2 Structure of Amphiphilic Molecules 51.2.1 Components of phospholipids 51.2.2 The four major phospholipids and cholesterol 61.2.3 Motional freedom of hydrocarbon chains 71.3 Structure of Bilayers 81.3.1 Hydrophobic effect and condensed interfacial structures 81.3.2 The gel and fluid phases of bilayers 101.3.3 Disorder inside the fluid bilayer 111.3.4 Interdigitation 131.3.5 Effects of cholesterol 141.3.6 Molecular motions and time scales 152 Mechanics of Monolayer Coupling 172.1 Equations of Mechanical Equilibrium 202.1.1 Stresses, strains and axisymmetry 201112.1.2 Virtual work and the equilibrium equations2.1.3 Equilibrium equations for a fluid membrane2.1.4 Dividing one fluid membrane into two2.2 Free Energy of Deformation2.2.1 Elastic model for liquid crystalline membranes2.2.2 Pure stretch2.2.3 Pure bending and local curvature energy2.2.4 Coupled layers and global curvature energy2.2.5 Euler-Lagrange equations and mechanical equilibrium2.3 Interlayer Viscosity2.3.1 Phenomenological model and balance of tangential forces2.3.2 Kinematics2.3.3 Dynamic equation for a3 Measurement of Interlayer Drag3.1 General Description of Nanotether Extrusion3.2 Summary of Equations and Experimental Strategy .3.2.1 Two important equations3.2.2 Experiment 1: Static measurement3.2.3 Experiment 2: Dynamic measurement3.3 Methods3.3.1 Vesicles, solutions and beads3.3.2 Micropipette setup3.3.3 Experimental procedures3.4 Data and Discussion4 Thermal Undulations of Bilayer Vesicles4.1 Mean Square Amplitudes: Statics4.2 Correlation Function: Dynamics4.2.1 Equilibrium equations for the bilayer4.2.2 Interlayer drag and bilayer relaxation function .4.2.3 Hydrodynamics424345454647484850535558606263646522252729293031333538383941iv4.2.4 Solution. 654.3 Discussion 665 Summary 69Bibliography 72APPENDICESA From Virtual Work to Equilibrium Equations 77B Analysis of Tether Extrusion 80B.1 Dynamics of n in Vesicle-Tether Junction 80B.1 .1 Péclet number and order-of-magnitude arguments 80B.1.2 Integrating the low Péclet number equation 82B.1.3 Kinematics 83B.2 Balance of Forces 83B.2.1 Vesicle region 84B.2.2 Vesicle-tether junction 84B.2.3 Entering tether region 84B.3 Dynamics of on Sphere 85B.3.1 Simplifying dynamic equation for c 86B.3.2 Boundary value problem and general solution 86B.3.3 Constant rate of extrusion 88B.3.4 Constant tether force 88B.4 Other Dissipative Effects 89C Force Transducer: Analysis and Calibration 91C.1 Force Balance 91C.2 Numerical Solution 93C.3 Results and Calibration 95D Vesicle Undulations: Equilibrium Calculations 96vE Vesicle Undulations: Deterministic DynamicsE.1 Bilayer ShellE.1.1 GeometryE.1.2 KinematicsE.1.3 ForcesE.1.4 Modal expansionE.2 HydrodynamicsE.3 Matching Boundary ConditionsE.3.1 Laplace transformE.3.2 Matching kinematic boundary conditions: No-slipE.3.3 Matching stress boundary conditions: Force balance100• • . . 100100• . . . 102103104105• • . . 106• . . 106• . . . 107• . . 108F Relaxation Function for the Bending Moment 110viList of TablesI Summary of Results from Nanotether Experiment 57viiList of Figures1.1 Molecular structure of a phospholipid molecule 1131.2 Molecular structure of a cholesterol molecule 1141.3 Torsional potential between two CR2 groups 1151.4 Molecular shapes and condensed structures 1161.5 Molecular order profiles 1172.1 Axisymmetric geometry 1182.2 Stresses on a curved shell segment 1192.3 Defining the origins in the thickness direction 1202.4 Simple model for interlayer drag 1213.1 The nanotether experiment 1223.2 Experimental setup 1233.3 Photograph of nanotether experiment 1243.4 Tether pulling sequence 1253.5 Determination of tether radius 1263.6 Dynamic tether force vs. extrusion rate 127B.1 Magnified view of the vesicle-tether junction 128B.2 Axial force balance on nanotether 129C.1 Mechanical analysis on force transducer 130viiiC.2 Calibration of force transducer 131C.3 Experimental and theoretical transducer characteristics 132ixAcknowledgementsI would like to thank Evan Evans, my research supervisor, for introducing me to thisstimulating research problem and for his guidance and support.I also extend my sincere gratitude to all the people in the lab for their technical and, moreimportantly, emotional support throughout this ordeal:To Wieslawa Rawicz, who has given me the help when I most needed it; to Andrew Leung,for being the nicest person and for the liquid nitrogen; to David Knowles, for providingencouragement, comic relief and lamb chops; to Susan Tha, without whose assurance Iwould have seriously doubted the completion of this thesis; to Hans-Günter Döbereiner,for keeping me company in the lab during off-hours; to Ken Ritchie, who provides candies,cookies and comic relief of a more innocent nature and the list goes onI would also like to thank the members of my supervisory committee: Professors BoyeAhlborn, Ed Auld and Myer Bloom. They have been genuine scholars and gentlemen.xChapter 1IntroductionThe lipid bilayer is the fundamental component of all biological cell membranes. It is acondensed material that forms spontaneously as surfactant molecules of the right shapes(discussed later in this chapter) are introduced into an aqueous environment. Its structure consists of two mono-molecular surfactant layers that are weakly held together by vander Waals attraction — resulting in a molecularly thin composite(‘—i4 nm) whose surfacecan extend to macroscopic dimensions (exceeding 10 im). Iii addition to its biologicalrelevance, the unique properties of the bilayer have stimulated much interest in the community of condensed matter physics [Leibler, 1989; Bloom et al., 1991; Lipowsky, 1991].Being a two dimensional condensed material, the bilayer exhibits a rich set of phasesand phase transitions [Silver, 1985; Gennis, 1989]. In three dimensions, the 2-D fluid hasproperties of a liquid crystal — which has led to the “liquid crystal approach” to bilayerenergetics [Helfrich, 1973]. The fluid membrane is also unique for its extraordinary flexibility: Due to its “thinness”, the bilayer is a macroscopic body whose conformations aresusceptible to molecular (thermal) agitations. For a flaccid membrane, thermally-drivensurface undulations occur over a wide spectrum of wavelengths, with the macroscopicones clearly visible under the light microscope. Such surface undulations tend to increasethe configurational entropy of the system, giving rise to the notions of “entropy-drivenforces” and “fluctuation-induced repulsions” when the undulations are either suppressedby a lateral tension or confined in a narrow gap [Helfrich, 1978; Lipowsky and Leibler,1986; Evans and Rawicz, 1990]. In addition, equilibrium statistical and dynamic analyses of surface undulations have been reported [Schneider et al., 1984; Milner and Safran,1987; Faucon et al., 1989]. More recently, considerable work has been devoted to studying1shape transformations of fluid bilayers and constructing the corresponding conformational“phase diagrams” [Miao et al., 1991; Seifert et al., 1991; Käs et al., 1991].Despite the wide scope and sophistication of the above-mentioned works, almost all ofthem are based on the model in which the bilayer is considered a unit structure; that is,the two monolayers are assumed to be rigidly collected. This is in fact not true. Heldtogether by non-covalent forces, the monolayers are only prevented from separating inthe transverse direction. Laterally, they are free to slide relative to one another; suchmotions are resisted only by dynamic (viscous) forces as the two “hydrocarbon brushes”(inner parts of the monolayers) undergo relative motions. The effect of stratification wasrecognized many years ago with introduction of the non-local curvature energy for bilayers that form closed surfaces [Evans, 1974]. As will be shown in chapter 2, the non-localenergy is a subtle contribution that complements the widely adopted Helfrich Hamiltohan [Heifrich, 1973]; the two forms of curvature energy are of the same magnitude andtherefore must be treated on equal footing. Through careful observation of shape transformation sequences in fluid bilayer vesicles, it is becoming evident that the Hamiltonianbased on the unit-membrane assumption cannot by itself account for all of the conformational characteristics. It is then suggested that inclusion of the non-local term, whichreflects monolayer coupling, is necessary for explaining the experimental findings [Seifertet al., 1992]. While elastic aspects of coupling give rise to the non-local curvature energy, dynamic (viscous) coupling between monolayers can be important in determiningthe time-dependent features of shape changes. As will be shown, interlayer drag is thedissipative mechanism that dominates over hydrodynamics on mesoscopic length scales(below 0.1 tim). Thus, coupling of monolayers in the bilayer membrane introduces a newlevel of complexity to the physics of condensed fluid interfaces. To address this issue, bothstatic and dynamic features of monolayer coupling will be examined in this thesis.This thesis is a report on our investigation into the continuum mechanics of monolayercoupling within a fluid bilayer. Such an approach neglects individual properties of theconstituent molecules. Instead, the physical system is described by continuous fieldssuch as density, velocity and stress; these variables can be considered local averages ofmany microscopic systems (see discussion at beginning of chapter 2). The continuum ormacroscopic description complements other microscopic approaches: Based only on a few2constitutive parameters such as the elastic moduli, a continuum model can be used topredict collective behaviours of the physical system over very large length scales and verylong times (relative to molecular scales). An example of such applications is the correctprediction of entropy-driven forces in a fluctuating fluid membrane [Evans and Rawicz,1990]. That the membrane is fluid can be depicted as the vanishing of the in-plane shearelastic modulus. However, the continuum approach offers no insight into why the shearmodulus should vanish. Important characteristics associated with the membrane’s fluidstate such as the “melting” of hydrocarbon chains and the sudden increase in lipid mobilityare molecular features. Such microscopic information can be obtained, for example, frommolecular dynamics simulations or from experiments such as NMR spectroscopy andX-ray diffraction. It would be hopeless, however, to model Brownian undulations overlong distances using molecular dynamics calculations. In short, both microscopic andcontinuum approaches are essential as they complement each other in obtaining a morecomplete understanding of the physical phenomenon in question; this thesis concentrateson the latter in dealing with the mechanism of monolayer coupling.1.1 Overview of the ThesisIn chapter 2, the continuum mechanical theory for fluid bilayers is developed starting fromfirst principles. In particular, both the static and dynamic aspects of monolayer couplingare examined. It is generally true that mechanical theories for elastic bodies can be formulated either from an energetic approach or from the direct balance of forces; we showthe equivalence of the two formalisms in the present application to surfactant interfaces.A continuum description necessarily requires constitutive relations. It is shown in chapter2 that all the usual elastic moduli (area compressibility, bending rigidity and Gaussian“bending rigidity”) are derivable from a simple Taylor expansion, to first order, of theisotropic stress within the bilayer. In addition, constitutive parameters describing staticand dynamic coupling are introduced; they are the non-local bending rigidity Ic and theinterfacial drag coefficient b. Our continuum model is considered “complete” if both theseparameters are known. kc, is well characterized based on previous area compressibilitymeasurements [Evans and Needham, 1987]. On the other hand, the interfacial drag coefficient b is not known accurately — neither in magnitude nor in the way it changes with3other physical parameters such as temperature and chain structure. This leads logicallyto the next chapter.Chapter 3 deals with the experimental work that constitutes a major portion of thisthesis. It describes our measurement of interlayer viscous forces as a function of shearrates, from which the drag coefficient b is deduced. Bilayer test samples come in theform of vesicles (closed bags) with diameters typically around 20 ,um. Experiments onsuch vesicles are done with micromechanical techniques; i.e., the techniques of manipulating test samples using small suction pipettes. For this particular application, the use ofmicromanipulation involves pulling cylindrical bilayer tubes (diameters 50 nm) out ofspherical vesicle surfaces; for convenience, this will be called the “nanotether experiment”[Hochmuth and Evans, 1982; Bo and Waugh, 1989]. It is shown that interlayer drag cansimply be related to the tension force in the tether, with the latter on the order of microdynes. Thus, to get at the interlayer viscous forces, one needs an ultra-sensitive forcetransducer that is capable of measuring tether tensions accurately. A convenient way ofusing micromechanical techniques to construct such a force transducer is described. Further, a useful spin-off from the nanotether experiment is a method of measuring bendingrigidities of molecularly thin surfactant layers. The interlayer drag coefficient is monitored as a function of three parameters: temperature, difference in length between thetwo hydrocarbon chains of the surfactant molecules, and the incorporation of cholesterolinto the bilayer. These features will be further discussed in this introductory chapter.With the mechanical formalism of monolayer coupling developed in chapter 2 andthe interlayer drag coefficient b measured in chapter 3, we have obtained a continuumdescription of the stratified fluid bilayer. As an application of this mechanical model, therole of monolayer coupling in the thermal undulations of flaccid bilayer vesicles will beexamined. Chapter 4 presents such a theoretical analysis. The calculations are dividedinto two parts; they are (a) equilibrium statistics (spectrum of mean square amplitudes)based on the equipartition theorem, and (b) more complicated deterministic dynamicsthat require knowledge of the surrounding low Reynold’s number flow field. Similarcalculations have been published previously, but with monolayer coupling effects left out[Schneider et al., 1984; Milner and Safran, 1987]. Here, it is shown that the coupling1Microdyne (10 piconewton): Weight of 109gm; it is the typical force of a van der Waals “bond”.4effect can play a significant role in vesicle undulations. Finally, a general discussion andsummary of this work is given in chapter 5.Before going into the continuum mechanical formalism, the remainder of this chapterwill be spent on an introduction to the molecular structure of bilayer membranes. Thediscussion is very superficial and does not contain any new results. However, it doesprovide the non-biochemist reader (and the writer) a better understanding of the physicalsystem under study.1.2 Structure of Amphiphilic MoleculesSurfactants are more generally known as amphiphilic molecules or amphiphiles. Suchmolecules are characterized by their distinct hydrophobic (non-polar) and hydrophilic(polar) intramolecular regions. The two types of amphiphiles we discuss here are phospholipids and cholesterol, which are the most abundant amphiphiles found in biologicalcell membranes. Because of the large number of possible variations in phospholipid structure, we will start by looking at its components.1.2.1 Components of phospholipidsThe molecular structure of a phospholipid can be represented by a simple sketch as shownin figure 1.1: Every phospholipid is composed of a polar head group that is connected toa stiff backbone structure at the mid-section. The backbone is in turn linked to two nonpolar hydrocarbon chains that have, in most cases, 12 to 24 carbon atoms. A typicalphospholipid molecule has a length of 2 to 3 nm (with chains extended) and molecularweight of order 1000 Daltons.Hydrocarbon chains, also known as fatty acids, are series of CH2 groups linked byC—C single bonds. Occasionally, two carbon atoms are connected by C=C double bondsthat are almost always in the cis configuration [Stryer, 1981]. One end of the chainis terminated by an unreactive CH3 group while the other contains a carboxyl groupthat is capable of forming covalent bonding (ester or amide) with other compounds. Byconvention, carbon atoms are numbered beginning with the one adjacent to the carboxylgroup. Whereas carbon atoms connected by single bonds are free to rotate about thebond axis, such freedom is not allowed in C=C bonds. As a result, double bonds create5rigid kinks in a hydrocarbon chain that is otherwise much more flexible; these doublebonds have very strong influence on a bilayer’s physical properties. Fatty acids with onlysingle bonds are said to be saturated (with hydrogen). Generally, a fatty acid is identifiedby its length, the number of double bonds, and the locations of the double bonds. Somecommon saturated fatty acids are myristic acid (14 carbons), palmitic acid (16 carbons)and stearic acid (18 carbons). Another fatty acid we will encounter is oleic acid; it is a“mono-unsaturated” chain (one double bond) of 18 carbons with the C=C bond situatedat the ninth segment [Gennis, 1989].The backbone structure is most often a glycerol group. It can be thought of as a“three-hole socket” with each connection representing a covalent bond (ester). Thus, aphospholipid is formed by “plugging” two non-polar hydrocarbon chains and one polarhead group into this socket. Another backbone structure is the sphingosine; it is a “twohole socket” that comes with an attached hydrocarbon chain of length 12. Here, a two-chain phospholipid can be formed by plugging in a polar head group and one more fattyacid chain. The connection made with the fatty acid is an amide linkage. Chemicalstructures of these compounds can be found in many biochemistry textbooks (for example,see Stryer [1981]). Since these details are not essential to our discussion, they will not bereproduced here.The three most common polar head groups are: phosphocholine (PC), phosphoethanolamine (PE) and phosphoserine (PS). They are composed of a phosphate group (hencethe term “phospholipid”) that is linked to another small hydrophilic compound — whichcan be choline, ethanolamine, or serine. The linkage is through a covalent bond known asphosphate ester. Of the three head group structures, only PS has a net negative charge;the other two are electrically neutral. Again, chemical structures of these compounds canbe found elsewhere and will not be given here.1.2.2 The four major phospholipids and cholesterolThree of the four major phospholipids are formed from the glycerol backbone: The middleattachment site of glycerol is connected to a fatty acid which is labelled the “sn-2” chain.The other two connections are made with another fatty acid (the “sn-i” chain) and apolar head group. These three types of phospholipids are classified according to the head6group structure; they are the PC, PE and PS lipids. In addition, composition of thetwo fatty acids are specified in ascending “sn order” (sn-i followed by sn-2). Thus, a PCphospholipid with a stearic acid in the sn-i position and an oleic acid in sn-2 position iscalled an “SOPC” (1 -stearoyl-2-oleoyl-sn-glycero-3-phosphocholine) molecule.The fourth major class of phospholipids is sphingomyelin. Its backbone structure,the sphingosine, contains a 12-carbon fatty acid and has two connecting sites that arelinked to a PC head group and a fatty acid chain. This second fatty acid is usually muchlonger than the 12-carbon chain (typically 20 carbons). Variation in structure of thesphingomyelin is due only to the composition of this fatty acid.The other type of amphiphilic molecules we introduce here is cholesterol. As shownin figure 1.2, the major portion of the molecule is a rigid planar ring structure. Attachedto one end of this ring structure is a short but flexible hydrocarbon tail. A small polar(OH) group is connected to the other end of the ring; this is the only hydrophilic regionin the cholesterol molecule. Overall length of cholesterol is slightly less than 2 nm withthe short hydrocarbon tail extended. Unlike phospholipids, there is no variation in thestructure of cholesterol.1.2.3 Motional freedom of hydrocarbon chainsSince a major portion of the cholesterol molecule is occupied by the rigid ring structure,“intra-cholesterol” motions are not too interesting. Similarly, polar head groups of phospholipids are known to have restricted internal motions [Seelig and Seelig, 1980]. By farthe most exotic dynamics within phospholipids are the continual conformational changesin the hydrocarbon chains. Such motions are possible because azimuthal rotations aboutthe C—C bonds are allowed; for a free chain having 12 or more C—C bonds, this can resultin a large number of possible chain conformations. However, as illustrated in figure 1.3(a),not every angular position about the C—C bond is equally comfortable for the CR2 group:0 represents the trans (t) configuration as illustrated in figure 1.3(b); this energyminimum corresponds to the configuration in which all four carbon atoms are coplanar.Rotations of approximately + 120° from the trans state lead to the other two energy minima, which are called the gauche configurations (g+ and gj. The gauche states havepotentials of approximately one kT (T = 298K) higher than that of the trans state. In7order to get to these states, the CR2 group needs to overcome a barrier of -‘ 6 kT. Thus,at any instant, rotational angles corresponding to the trans or gauche states are stronglyfavoured over the neighbouring positions. On the other hand, a barrier of 6 kT is notenough to prevent rapid interconversions amongst the energy minima; typical rates of thisso-called trans-gauche isomerization is 1O’°Hz [Flory, 1969].Looking now at the entire chain, it is noted that a straight chain is one that has allits C—C bonds in the trans configuration (the all-trans state); this is the lowest energyconfiguration for the chain. A gauche C—C bond introduces a bend to the chain much likethe effect of a cis double bond, and combinations such as g+tg_ or gtg+ create kinks.Thus, we see that the free hydrocarbon chain can assume a large number of conformations,each with typical lifetime of 10’°sec.1.3 Structure of BilayersAfter a brief discussion on individual molecular structures, we turn now to the cooperativeeffects as large numbers of surfactants are present in aqueous environments. As we willsee, the unique properties of water is vital to the formation of surfactant aggregates.Different experiments have unanimously pointed to the “lively” nature of the fluid bilayerinterior; we will discuss the time scales and extents of such internal motions. The specialrole of cholesterol within bilayers is also considered.1.3.1 Hydrophobic effect and condensed interfacial structuresWhat makes water different from most other liquids is the dominance of hydrogen bondinteraction amongst its molecules [Israelachvili, 1985]. Contrary to the more common vander Waals interaction, hydrogen bonding is highly directional. The tetrahedral structure(4 nearest neighbours) of ice is a consequence of this directionality. Even in the liquidstate, a water molecule continues its attempts to maximize the number of hydrogen bondings with its neighbours; it is this tendency that is believed to give rise to the hydrophobiceffect.Consider the effect of having a foil-polar solute (such as an alkane) in water: Becausewater molecules adjacent to this solute cannot interact with it electrostatically, they mustseek to establish hydrogen bondings elsewhere. This causes a reorientation of the adja8cent and nearby water molecules in a way that allows them to make the most hydrogenbonds with each other. However, this organization of water molecules around the soluteis entropically very costly; the overall effect is a highly unfavourable free energy of solubilization, which is commonly called the hydrophobic effect. The hydrophobic effect causesnon-polar solutes to aggregate so as to minimize their contact with water. The forcesthat hold such condensed regions together are the familiar van der Waals forces, and ithas been argued that the notion of “hydrophobic forces” is a misconception [Hilderbrand,1979].Aggregation of surfactants in aqueous solutions is due to the hydrophobic/hydrophiliccharacter of the molecules. The resulting assemblies exhibit a large variety of forms, butthey all share the common feature of using the polar head groups to shield the hydrophobicchains from water. A few such arrangements are shown in figure 1.4. Much work hasbeen done on the thermodynamics of aggregation in attempts to predict the formationof these assemblies [Wennerström and Lindman, 1979]. However, a good intuition forself-aggregation can be obtained from just considering the molecular shapes and theirpacking properties that arise from density constraints. Here, the term “molecular shape”refers to the average space occupied by the molecule. In figure 1.4, the required molecularshapes for forming the various structures are shown. Such arguments based on molecularshapes have been further quantified by introduction of a “critical packing parameter”[Israelachvili et al., 1976].It is observed that lipids with PC head groups invariably form bilayers. This canbe explained by the fact that such head groups are somewhat larger than its adjacentglycerol backbone. Thus, with the two hydrocarbon chains slightly spread out and thelipid rotating about its long axis, the overall occupied space is indeed cylindrical (see figure1.4). Similarly, the reason that unsaturated PE lipids self-assemble as inverted micellesis because PEs form considerably smaller head groups. This feature, together with chainunsaturation (chains with C=C kinks tend to flop around more and thus occupy morespace), constitute a cone-shaped lipid molecule. Finally, lysolecithins, the one-legged PClipids, are unable to form bilayers but aggregate quite readily into micelles. It is easy tosee that, consistent with the scenario in figure 1.4, such lipids do have shapes of invertedcones.9In this thesis, we will oniy be concerned with fluid bilayer structures. More specifically,experiments are done on bilayers whose lipids contain PC head groups (PC lipids andsphingomyelins).1.3.2 The gel and fluid phases of bilayersThe bilayer is a water-mediated condensed material with complex intermolecular forces.In the hydrophobic interior, forces between hydrocarbon chains are fairly well understood;they are the van der Waals attraction and steric repulsion due to chain extension. Anattractive interaction exists at the hydrocarbon-water interface (where the hydrocarbonchains are attached to the glycerol backbone) due to the hydrophobic effect; this givesrise to an effective interfacial tension. Interactions in the headgroup region (i.e., from theglycerol backbone upward), on the other hand, are not as easily characterized. Being ahydrophilic environment, this region can soak up water and lead to hydrogen bond interactions. In addition, complicated electrostatic forces arise because the head groups aredipolar, possibly charged, and likely to have free or adsorbed ions from the solution. Thereare also steric repulsion interactions among head groups. The combined effect of theseforces is clearly one of cohesion. Depending on temperature and other thermodynamicparameters, molecules within a bilayer are ordered by intermolecular forces in distinctlydifferent manners; such phenomena are, of course, phase behaviours.As an example, let us consider a bilayer composed of PC lipids. At low temperatures,the bilayer is in a solid or gel state in which all hydrocarbon chains approach the all-transconfiguration. The chains are oriented parallel to each other and are usually tilted withrespect to the bilayer normal. As the temperature rises to the chain melting temperatureT, van der Waals attraction and interfacial tension can no longer hold the hydrocarbonchains in parallel bundles; the chains undergo an abrupt transition to become “floppy” andspread out. This results in a shortening of projected chain lengths (along the molecularaxes) and an increase in area occupied per molecule (by about 15 to 20 %); an averageof 4 to 5 gauche conformations are estimated to be present in each chain. The collectivetilt of hydrocarbon chains is also removed and the molecules are oriented, on the average,normal to the bilayer surface. Moreover, lipid molecules in such a state are free to diffuse— both in translation and rotation — within the monolayer planes. Physical states at10temperatures above T are said to be fluid or liquid crystalline. Chain melting is duemainly to the interplay between thermal randomization and cohesive organization withinthe hydrophobic interior. Not surprisingly, T increases with fatty acid chain lengths:For PC bilayers with saturated chains of 14, 16 and 18 carbons, chain melting occurs at24°C, 41°C and 55°C respectively [Mabrey and Sturtevant, 1976]. Lipids with even onedouble bond in one of the hydrocarbon chains have much lower T. Presumably, havingrigid knots in the otherwise flexible chains makes it very awkward to pack the chains intoparallel bundles (the gel state); the system is therefore much easier to disrupt. Typicalchain melting temperatures for unsaturated lipids can be below 0°C.For saturated PC lipids, there exists a “pre-transition” temperature T — roughly 10°Cbelow T — at which the bilayer goes through a less pronounced internal reorganization.This transition has a much lower enthalpy change compared to the chain melting transition; for this reason, chain melting is often called the “main” transition. Microscopically,bilayers at temperatures between T and T have periodic surface ripples with the hydrocarbon chains remaining in the frozen (all-trans) state. Such an effect is believed toresult from head group interactions; however, the phenomenon is not completely understood. Bilayers composed of unsaturated lipids do not undergo pre-transitions [Evans andNeedham, 1987].Since we are only interested in the fluid bilayer, effects that occur below the main transition temperature will be of no concern. All we need to make certain is that experimentsare done at temperatures well above T.1.3.3 Disorder inside the fluid bilayerThe bilayer’s fluid state results when intermolecular cohesive forces are overwhelmed bythermal agitations. Hence, we expect a certain degree of randomness within such a fluidstructure. As it turns out, experimental measurement of “disorder” reveals a very intuitivepicture: Molecular motions are progressively more disordered towards the centre of thebilayer.An important technique of measuring disorder in a fluid bilayer is nuclear magneticresonance (NMR). In particular, the method of deuterium NMR, which has been instrumental in providing detailed information on a bilayer’s internal structure and dynamics,11involves selectively replacing protons in a lipid molecule by 211 atoms. The most commontarget groups are the CR2 segments along the hydrocarbon chains. With this practicallynon-perturbing probe in place, the deuterium quadrupole splitting is detected which inturn can be directly related to the order parameter Smoi of the deuterated group.2 Theorder parameter is defined such that it is unity for a frozen all-trans chain and zero forcompletely isotropic motions; it is a direct measure of the amount of orientational disorderat local positions. Figure 1.5 shows typical order profiles of several lipid membranes atthe same reduced temperature of 0.061 [Seelig and Browning, 1978]. Here, the carbonatoms are numbered from the glycerol backbone; i.e., larger numbers correspond to Catoms closer to the bilayer centre. The important feature is that order in the first 6 to8 carbon atoms are roughly equal, while the last atoms show rapid increase in motionaldisorder as they approach the bilayer centre.Another powerful method that provides structural information on the fluid bilayer’sinterior is neutron diffraction. Accelerated neutrons (energies 0.1 eV, wavelengths ofabout 1 A) are scattered from bilayer samples to give diffraction patterns that reflect thebilayer’s internal structures. Because neutrons are scattered by atomic nuclei, transformation of the diffraction pattern amounts to a neutron scattering density profile acrossthe bilayer. More interestingly, because protons and deuterons have very different scattering properties, lipids that are selectively deuterated at different locations (as used in211 NMR studies) have neutron density profiles that contain easily identifiable deuteratedsites. Thus, the positions and positional fluctuations (locations and widths of the 211peaks) of different segments along the hydrocarbon chain can be obtained. The positional fluctuations have been shown to correlate well with the order parameter measuredby 2NMR techniques: For fluid DPPC (PC’s with two palmitic chains) bilayers, positional fluctuations of the CH2 segments are shown to increase by more than a factor oftwo towards the end of the chains [Zaccai et al., 1979].In addition to experimental studies, numerical simulations have also confirmed theNMR order profile. The first of such work is the mean field calculation by Marelja [1974]that gives excellent modelling of the order parameter as functions of chain position andtemperature. More recent molecular dynamics calculations have also produced similar2For a discussion on the technique, see Bloom et al.{1991]12order profiles across the bilayer [Pastor et al., 1988; Heller et al., 1993].1.3.4 InterdigitationIt is estimated that in a typical fluid bilayer, every hydrocarbon chain has, on the average,4 to 5 gauche configurations and the projected chain length is approximately 75% of thefully stretched value (the all-trans configuration) [Zaccai et al., 1979]. Because of thehigh degree of motional freedom near the bilayer centre, it is likely that hydrocarbonchains can extend beyond their monolayer region into the other half of the bilayer. Thisinterdigitation effect may be very important to inter-monolayer dynamic coupling — aneffect that will be studied experimentally in chapter 3. In the following, we will look atsome evidences that point to such an occurence.Neutron scattering studies on fluid DPPC (two 16-carbon chains) bilayers have successfully resolved the average positions of all CR2 chain segments [Zaccai et al., 1979].Positional fluctuations of the segments are also given for partially hydrated lipids up tothe twelfth CR2 segment. At the C-4 and C-12 segments, positional fluctuations are 1.5A and 3.4 A respectively (C-12 is closer to the bilayer midplane). Such values measurethe extent of thermal motions in the direction of the bilayer normal; they are expected toincrease: (a) towards the centre of the bilayer, and (b) on approach to water saturation inthe head group region (the case we are interested in). Unfortunately, no such data underthese conditions are given. However, it is safe to assume that the terminal CR3 groupscan have average positional fluctuations of at least 5 A. For a hydrocarbon interior thatis roughly 30 A in thickness, this can lead to very significant interdigitations.More recently, a new method of combining neutron and X-ray scattering data to obtainstructural information is introduced [Wiener and White, 1992]. Here, it is much clearerthat the terminal methyl groups are able to penetrate deep into the other monolayer —again, up to a depth of 5 A. CR2 groups close to the end are also shown to be capable ofpenetrating into the opposite side.Molecular dynamics simulations have further confirmed the existence of chain interpenetration. Recent simulations results of Helter et al.[1993] agree well with the abovementioned data by Wiener and White [1992]. In addition, another study by de Loof etal.[1991] shows molecular motions that are consistent with our picture of interdigitation.13It also provides valuable insight into the dynamics of such motions.1.3.5 Effects of cholesterolSo far, only single-component bilayers have been considered. We now discuss briefly theeffects of adding cholesterol into a fluid bilayer membrane. Although cholesterol cannotform bilayers on its own, it incorporates very readily into a lipid bilayer structure. Likethe phospholipids, cholesterol molecules orient themselves normal to the bilayer surface.The small polar OH group is believed to be located at the same level as the glycerolbackbone, while the rigid ring structure extends down to the level of the eighth or ninthCH2 group.Whereas single-component bilayers are characterized by relatively sharp gel-to-fluidphase transitions, such features begin to disappear upon incorporation of cholesterol intothe structure. With increasing cholesterol concentrations, “sharp features” correspondingto the main transition (for example, the peaks in calorimetric scanning plots) becomebroader and the associated transition enthalpies decrease. For cholesterol concentrationsabove 12.5 mol %, most PC bilayers behave as fluids even at temperatures well belowchain melting [Evans and Needham, 1987]. Such a phenomenon can be attributed tothe loss of positional order in the gel state due to lattice defects. With the presence ofimpurities (cholesterol), hydrocarbon chains in the gel state are forced to adopt moreirregular conformations; partial chain disorder persists even at temperatures below T.The other equally dramatic effect of cholesterol is the strengthening of bilayers: Introduction of cholesterol greatly increases a fluid bilayer’s area elastic modulus and rupturestrength; at the same time, water permeability across the structure is diminished [Evansand Needham, 1987]. All these indications suggest a disordered system influenced bystronger cohesive forces. It appears that, whereas cholesterol has randomizing effects inthe gel state, it actually helps to increase order in the hydrocarbon chains when the bilayer is fluid. This is supported by the fact that order parameters measured by 2H NMRare generally higher in the presence of cholesterol [Bloom et al., 1991]. The rigid ringstructure of cholesterol probably acts as conformational constraints on the hydrocarbonchains, thus decreasing the number of gauche configurations and overall random motions.It is likely that cholesterol can also enhance chain interdigitation in fluid bilayers.14In particular, because cholesterol orders hydrocarbon chains by decreasing their averagenumber of gauche configurations, the chains may lengthen and penetrate deeper into theopposite monolayer. Such an effect will be more pronounced for lipids with different chainlengths due to packing requirements.1.3.6 Molecular motions and time scalesEven a semi-detailed discussion on molecular motions and the various methods of detectionwould be out of the scope of this introductory section. Here, we only wish to point out thevarious types of lipid motions and give typical time scales for each. Relevance to inter-monolayer coupling is also discussed. For a review of the experimental methods employedin this broad field, the reader is referred to the comprehensive book on biomembranes byGennis [1989].Molecular motions in fluid bilayers can roughly be classified into three catagories:Lateral, rotational and conformational; conformational motions are primarily due to transgauche isomerizations in the hydrocarbon chains. In addition, there are two other formsof lipid motion that are at the extremes of characteristic time constants: They are (a)vibrational modes of the CH2 groups with characteristic times r “-‘ 10’4sec, and (b)exchange of lipids between monolayers — also known as lipid “flip-flop” — which has r onthe order of hours or even days. It should be noted that cholesterol flip-flop is usuallythought to be much faster than lipid flip-flop because of the small size of the polar groups[Alberts et al., 1989].Research into lateral and rotational motions of lipids is usually summarized in theform of diffusion constants. These constants are denoted Dtran and Drot which describelateral translation and rotation; they have units of cm2/sec and sec1 respectively. Therelevant relations are(r2) = 4Dtran t(02> = 2Drottwhere (r2) and (02) are the mean squares of displacement and angular rotation. It is upto the experimentalists to measure Dtra and Dr0t while the theorists attempt to derivethem on the basis of different models. For phospholipids in a fluid bilayer at “-‘ 30°C,15Dtran has values ranging from 10—8 to iOcm2/sec. This means a lipid will diffuse to itsnearest neighbour’s position in less than a microsecond (r 1 nm). Typical values of Drotare lO8sec’ or higher. Reciprocal of this number can be interpreted as the characteristictime of lipid rotation; thus, a lipid takes less than lO8sec to spin about its long axis.The rates of trans-gauche isomerizations vary with the position along the chain. Characteristic times of such motions vary from 10sec for C—C bonds near the glycerolbackbone to 101sec for bonds close to the terminal methyl group [Brown et al., 1979;Venable et al., 1993]. We are interested in the overall interdigitating frequency of thechains. Such a rate is likely to be slower than trans-gauche isomerization (by one or twoorders of magnitude) because chain extension is a cooperative effect that involves all C-Cbonds. As shown by molecular simulations, characteristic time for interdigitation doesappear to be around lO9sec [de Loof et al., 1991].For our experiment, which will be described in detail in chapter 3, the monolayers aremade to slide past each other at relative velocities of 103cm/sec or less. Given thatthe chains are spaced roughly a few A’s apart in the bilayer plane, the time to coverthis “lattice distance” is lO5sec. During this time, the hydrocarbon chains will havepenetrated the opposing monolayers at least iO times! This is effectively the rate ofmolecular collision that gives rise to macroscopic viscous effects.16Chapter 2Mechanics of Monolayer CouplingTo study the conformational characteristics of lipid bilayers, an appropriate mechanicaltheory is needed. Except for the recent work by Seifert and Langer [1993], all theoretical analyses on the mechanics of lipid bilayers have been developed from the energeticapproach that follow the original formalism of Heifrich [1973]. However, as discussed inchapter 1, such a description is only valid for elastic unit membranes; i.e., where the twomonolayers are assumed to be rigidly coupled (chemically bonded, for example). In morerealistic situations, the weakly bound leaflets in a bilayer can undergo relative sliding andthus lead to internal dissipation. Such a non-conservative system must be analyzed fromthe force-balance approach. Out of such a necessity, the mechanics of stratified mono-layers is developed here from the balance of forces. The work in this chapter establishesthe theoretical basis for analyzing surfactant membranes that are internally dissipativebecause of inter-monolayer viscous drag.A continuum mechanical description of the fluid bilayer is presented in this chapter.Such an approach neglects individual properties of the molecules. Instead, the bilayer ischaracterized by continuous fields such as density, displacement and stress in the macroscopic and mesoscopic 2 regimes. These fields represent averages over many microscopicsystems; the validity of continuum theories is founded on the criterion that these fieldshave negligible fluctuations. At first sight, treating the bilayer as a continuum seemsquestionable: It is well known that fractional deviations from thermodynamic averagesare of order N*2, where N is the number of microscopic systems. For a bilayer whose1Length scales larger than wavelength of visible light; i.e., 100 rim.2lntermediate regime between molecular and macroscopic length scales; roughly between 1 and 100nm.17surface density is typically several million molecules per um2, the continuum assumptionbreaks down in the mesoscopic regime (at length scales of lOnm). In the thicknessdimension, the situation is much worse — there are only two molecules! However, it mustbe recognized that the above-mentioned processes are spatial averages. In consideringtime averages, one needs to compare intrinsic molecular time scales to the experimental“sampling time”. Owing to thermal effects, lipid molecules within each monolayer areconstantly in rapid motions. For example, typical time for a lipid molecule to diffuse a“lattice distance” (‘-..‘ 1 nm) within the monolayer plane is 10 sec, while that needed forthe lipid to spin about its long axis is less than 10—8 sec. Thus, if the time scale of anexperiment is longer than, say, 10 sec, all one will see is a “smeared out” picture of themolecular motions. With large sampling times, continuum description for the monolayerscan be valid down to length scales of 1 nm [Bloom et al., 1991]. Along the thicknessdirection, however, the bilayer consists of two mono-molecular layers that are weakly heldtogether by van der Waals forces. Because exchange of lipid molecules between monolayers (lipid “flip-flop”) necessarily requires the polar head groups to traverse the bilayer’shydrophobic interior, such motions are strongly hindered. Typical times for such motionscan be up to hours or even days. Within usual experimental time scales (10isec for NMR.spectroscopy and 1 sec for mechanical experiments), the two monolayers will appear asdistinct structures. In such situations, the appropriate continuum picture for the bilayeris that of two shell-like layers being held together by a pressure normal to the membranesurface. The two layers can slide laterally relative to one another, and the rate of suchmotions is limited by interlayer viscous forces.Although the lipid bilayer is possibly the thinnest membranous material, it still haselastic resistance to bending due to its finite thickness. In many situations where the fluidbilayer is flaccid and therefore cannot support any in-plane forces, higher order bendingeffects become dominant factors in determining mechanical equilibrium of the condensedstructure. Early mechanical analyses on the bilayer were done almost thirty years ago.Conventional shell theories were put forth in an attempt to explain the bicoilcave shapeof red blood cells [Fung, 1966; Fung and Tong, 1968]; however, the predicted behaviourswere at odds with experimental findings. It was Canham [1970] who introduced thefirst successful energetic description of the lipid bilayer. He pointed out that, in view18of the molecular arrangement, it was incorrect to regard the lipid bilayer as “a slab ofisotropic material”. Canham’s proposal was put on firm theoretical grounding by Heifrich[1973]. Adopting formalisms in the physics of liquid crystals, Heifrich expressed the bilayerbending energy in terms of invariants of the curvature tensor up to second order. Thisis the energy required to splay or “fan out” lipid molecules as the membrane is bent.In rather loose terms, it has the functional form (c2), where c is local curvature ofthe bilayer and the symbol ( ) denotes averaging over the surface. Soon after, Evans[1974] introduced a more subtle but equally important curvature energy contribution tobilayers forming closed vesicles. This term accounts for the small difference in surfacearea between monolayers as the vesicle changes its shape; it has the form (c) 2• Becausethis energy depends on the overall conformation of the bilayer vesicle, it is called theglobal curvature energy. In contrast, the Hamiltonian that Heifrich proposed is known asthe local curvature energy. As will be shown, these two contributions to the total bendingenergy are of the same magnitude and therefore must be considered simultaneously. Twodecades after its developments, the above picture remains the accepted elastic theory forlipid bilayers.In the following, the surfactant bilayer is analyzed using the theory of thin shells. Ourstarting point is to evaluate the virtual work expended in an arbitrary displacement of sucha structure. The principle of virtual work is “midway” between the energetic approachand the force-balance approach to mechanics; as such, it forms a connection betweenthe two formalisms: On the one hand, once the statement of virtual work is written,it can be expanded to give the equations of mechanical equilibrium. The advantage ofsuch a derivation is that no restriction has yet been put on the nature of the forces; theequilibrium equations apply equally to elastic or dissipative systems. Pursuing in theother direction, when only conservative forces are involved, the statement of virtual workcan be integrated to give the free energy of deformation.In section 2.1, ordinary equilibrium equations for thin shells are derived using themethod of virtual work. The equations will then be specialized to liquid crystalline materials. This is the point of departure from conventional shell theories at which the bilayerwill no longer be treated as “a slab of isotropic material”. These equations of equilibriumwill be used in later chapters when dealing with bilayers that are intrinsically dissipative.19In section 2.2, the energetics of bilayer deformation are derived from a simple molecularmodel with two parameters (more precisely, parametric functions) [Heifrich, 1981; deGennes, 1990]. We will show how the different forms of curvature energy can be obtainedfrom this linear approach, and how elastic moduli are expressed in terms of the modelparameters. Moreover, Euler-Lagrange equations of the general curvature Hamiltonian[Evans, 1980] are shown to be equivalent to the equations of mechanical equilibrium.Finally, in section 2.3, the dynamics of inter-monolayer coupling is examined. Tangential motions between mono-molecular fluid layers represent hydrodynamics below thecontinuum limit. In such a situation, the conventional no-slip condition must necessarilybreak down — velocity is no longer a continuous function in space. The simple approachto this problem is to postulate an interlayer shear stress that is proportional to the relative velocity between the leaflets. It is shown that, for the lipid bilayer, interlayer dragmanifests itself as a diffusion of the differential strain field on the membrane surface; the“smoothing out” of perturbations in surface density is an over-damped process whoserate is driven by membrane elasticity and limited by interlayer viscosity. A generalizeddynamic equation will be derived, thus forming the theoretical framework for calculationspresented in chapters 3 and 4. The experimental part of this thesis involves measurementof the dynamic coupling between fluid monolayers.2.1 Equations of Mechanical EquilibriumIn this section, the principle of virtual work is used to derive equations of mechanicalequilibrium for thin shells. The equations will then be specialized to shell materials thatare of liquid crystalline nature. Equilibrium equations for stratified liquid crystallinestructures will also be examined. Since we are interested in the lipid bilayer, only double-layered sheets are discussed.2.1.1 Stresses, strains and axisymmetryTo simplify the analysis, only axisymmetric geometries are considered throughout thisthesis. Although this is somewhat of a restriction, most physically important situationsare represented. The coordinates of an axisymmetric surface are shown in figure 2.1: isthe axis of symmetry about which a meridian is rotated to generate the shell surface, while20the z-axis is oriented normal to this surface with the positive sense outward. In general,bending a shell results in compression on one face and dilation on the other. Somewhereinside the shell, there must be a surface on which there is no straining as the shell is bent;this will be referred to as the neutral surface. We define the neutral surface to be thesurface z = 0. s and q are respectively the curvilinear distance along the meridian andthe azimuthal angle; radial distance from the C-axis to the meridian is labelled r. If allthe forces and displacements are rotationally symmetric about the C-axis, as will be thecase here, the s and coordinates are also in the surface principal directions.We begin by looking at deformations in a thin shell. The idea of thinness should bemade more precise here: Let c be a typical curvature of the shell surface and H the shellthickness; the shell is thin when the quantity e = Hc is much less than unity. In describingdeformations of a thin shell, the basic assumptions adopted here are:1. As the shell is bent, material points originally along the surface normal remainaligned along the new normal direction.2. All “stretchings” within the shell are oriented parallel to the neutral surface.These are the so-called Kirchoff hypotheses; the assumptions become better as the shellgets thinner. Our approach here is to express the strain at any location within the shell interms of stretching and bending of the neutral surface. Let (Am, .Aqs) be the extension ratiosof the neutral surface in the s and q directions, and (Cm, c) be the curvatures likewise[Evans and Skalak, 1980]. Since these quantities are defined on the neutral surface z = 0,they are only functiolls of (s, ) but not of z. Next, let Am(, q, z) and A(s, q5, z) be theextension ratios in the s and directions at a distance z away from the neutral surface.These quantities are equal to Am and ), for a flat shell. When the curvatures are non-zero,it is easy to show that, consistent with Kirchoff’s hypotheses,Am Am(l+ZCm) (2.1)A = A(l+zc)Equations (2.1) describe a deformation field consistent with the Kirchoff conditions; it isvalid only for a unit structure. Because the bilayer is composed of two stratified continua,the extension ratios are, in general, discontinuous at the midplane (see section 2.3.2).21Turning now to the forces, we note that the internal stress (force per unit area) withinthe bilayer is a second order tensor [Landau and Lifshitz, 1986]; it is denoted by o’jj, wherethe subscripts (i,j) represent directions along the m, 5 or z coordinates. Because ofaxisymmetry,= 0zq5 = 0For the deformation field given by equations (2.1), the stress components zm and cr,do not contribute to any mechanical work. Consequently, they will not appear in thefollowing developments. These stress components are, however, not negligible as theyplay a crucial role in the direct (force balance) approach to shell mechanics (see Flügge[1973] for example). So far as the evaluation of mechanical work, the relevant stresscomponents are mm and o; they will be written as m and u from here on.2.1.2 Virtual work and the equilibrium equationsHaving considered stresses and deformations separately, we can now calculate the virtualwork 6w required to displace a thin shell. We will then equate 6w to the work doneon the body by external forces, from which the equations of mechanical equilibrium arederived. Before proceeding, it should be noted that the principle of virtual work is a meansof obtaining mechanical equilibrium conditions for a unit structure; it is not applicableto two or more unconnected bodies. The power of the method lies in the fact thatall internal forces within a mechanical system cancel each other and can therefore beignored. Here, the bilayer is considered the “unit structure” whose internal components(the monolayers) are, owing to viscous effects, always in dynamic equilibrium. It is inthis spirit that the “unit structure” is displaced virtually according to equations (2.1).Since this is all imaginary, we can consider the extension ratios varying continuously evenacross the bilayer midplane.Virtual work is, in general, force x virtual displacement. For a thin continuum satisfying Kirchoff’s conditions, 6w takes a seemingly more complicated but yet equivalentform [Evans and Skalak, 1980]:6w = fJdA0 fdz (m A 6Am + uAm 6A) (2.2)where is an elemental surface in the undeformed configuration. The notation 6( )22denotes variation with respect to deformations on the midplane, that is, with respect to.Am, A, cm and c. As such, variations of the extension ratios within the shell are, fromequation (2.1),6Am = 6,Xm (1 + z cm) + Z )‘m 6cm (2.3)6A, = 6A(1+zc) + zk6c,Using equations (2.1), (2.3) and the identity dA = ‘‘m A dA0, the expression for 6wbecomes6w = JJdA Jdz m [(i + z + z2g) + z (1 + z c) 6cm] (2.4)+ a [(1+z3+z2g) + z(l+zcm) 6c]where and g are respectively the mean and gaussian curvatures; they are defined asc—cm+cc ; g—c,c, (2.5)The next step is to integrate across the thickness dimension z. Once the z dependencies are “integrated out”, the shell is reduced to a two-dimensional surface coincidingwith the shell’s neutral surface. The internal stresses are replaced with effective stressresultants (tensions) and moment resultants. Thus, the dimensionality of the problem hasbeen reduced at the expense of introducing more surface fields. Let us first define thestress resultants and moment resultants as follows: Imagine exposing an edge of the shelland integrating the internal stress over the transverse cross sectional area to obtain theforce. The stress resultant is, by definition, the integrated force per unit length of theneutral surface. Since the force is dependent on the orientation of the exposed edge, thestress resultant is a surface tensorial quantity; we will denote its principal values by Tmand r, with the subscript indicating the appropriate direction. Likewise, the momentresultant is the integrated bending moment about the neutral surface, per unit length ofthe neutral surface; the principal values are denoted by Mm and M. To write this exactly,a differential force in the meridional direction isdfm= JUrn dx(z)dzas shown in figure 2.2. The azimuthal length dx1, is dependent on z because of thecurvature in the q direction; from simple geometry, dx(z) = dx1,(O) (1 + z c). According23to the above definition of stress resultants, Tm is precisely the ratio of dfm to dx(O).Similarly, Ts is the ratio of df to dxm(O):Tm =Ju(1+zc)dz ; T = Ja(1 +ZCm)dZ (2.6)Following an entirely similar argument, the moment resultants areMmJmz(1+Zc)dZ ; M=JUZ(l+ZCm)dZ (2.7)With the above expressions for stress resultants and moment resultants, equation (2.4)can be written as6w = JfdA [(Tm + mMm) + (T + cM) + Mm6Cm + M 6c] (2.8)From this equation, we see that, as expected, the moment resultants are conjugate tothe virtual changes in curvatures. However, terms conjugate to the virtual changes inextension ratios are a sort of “generalized tensions” having the form of actual tensionplus (curvature x moment resultant). This “complication” is a consequence of reducinga three dimensional body to a two dimensional surface.To arrive at the equilibrium equations, we will equate 6w to the work done by externalforces, which, in general, may be due to a normal traction p and a tangential tractionPt; Pn and p are defined as positive when acting in the positive z and s directions.Work done on the shell by the external forces is denoted by wext; a virtual change of thisquantity is6Wext = JJdA (pn 6x + Pt 6x) (2.9)where 6x and 6Xt are virtual displacements in the normal and tangential directionsrespectively. By writing 6w (equation (2.8)) in terms these same virtual displacementsand equating to SWet, the conditions for mechanical equilibrium are obtained:id d dr7n = TmCm + TçtC — —(rMm)—M-- (2.10)ld TdT 1 d M,drPt = TTm) — —— + Cmrds rds rds rd,sDetails of the calculations are given in appendix A. The first of equations (2.10) representsforce balance in the normal direction; in the absence of bending moments, it is just the24familiar “law” of Laplace. The second equation is equilibrium condition in the tangentialdirection.The stress and moment resultants can be decomposed into isotropic and shear components as follows:(Tm + r)1(2.11)AI (Mm + M) ; M3 (Mm — M) (2.12)With these definitions, equivalent forms of the equilibrium equations are1 d / dM’\ 1 d 1dM5 2M3 dr”\pn = + T (Cm — c) ———r— j — —— r + — I (2.13)r ds ds j r ds ds r ds jd’ dT5 2T3 dr dII 1dM5 2M3 dr—Pt =d.s ds rds ds \ds r dsEquations (2.10) and (2.13) can alternatively be derived from the direct balance of forcesand moments on a differential shell segment [Evans and Skalak, 1980]. There is no restriction on the origins of the forces involved; the equations are only statements of mechanicalequilibrium and therefore can be applied to either elastic or dissipative materials. Theonly requirements for the equations to be valid are the Kirchoff hypotheses, which, forshells as thin as the bilayers, are well justified.2.1.3 Equilibrium equations for a fluid membraneIt is necessary to have a precise mechanical characterization of the bilayer’s liquid crystalline state. For a two-dimensional surface having no resistance to bending, being fluidimplies rotational symmetry of the stress resultant about the surface normal; as such,Tm r, and therefore r5 0. However, the situation is slightly more complex inthe case of liquid crystalline shells. We will postulate that in its fluid state, a bilayerhas internal stresses that are rotationally symmetric about the surface normal. This isbased on the following observations: At temperatures above the phase transition, lipidmolecules are known to diffuse rapidly; in particular, the rotational frequency of a lipidabout its long axis is of order i0 Hz. For observation times larger than the reciprocalof this frequency, all intermolecular forces will average out to give rotationally symmetricpotentials about the molecular axes. Also, X-ray diffraction and neutron scattering stud25ies on fluid membranes have ruled out any collective tilt of molecular axes with respectto the bilayer normal.Expressed as an equation, the liquid crystalline bilayer has internal stresses such thatUm(Z) = u1s(z) u(z) (2.14)Deviations from this relation can be attributed to dynamic effects. In particular, thedeviatoric stress isDv— qs —Dxwhere is a bulk viscosity characterizing the interior of the bilayer and Dv/Dx is thein-plane shear rate. Typically, is that of an oil, which is - 1 dyn.sec/cm2 The shearrate, under the rather “extreme” condition of varying the velocity by 100 gum/sec over alength of 10 rim, is 10 sec’ . The resulting deviatoric stress is of order iO dyn/cm2 . Onthe other hand, the magnitude of cx is obtained by dividing the interfacial tension by shellthickness; using typical values of 10 dyn / cm for interfacial tension and 1 nm for thickness,cx is of order 108 dyn/cm2; it is four orders of magnitude larger than its deviatoric part.Equation (2.14) is therefore justified.The condition of rotationally symmetric stress distribution (about the surface normal)has interesting consequences on the equations of mechanical equilibrium. From equation(2.7), it is seen by neglecting the second moment of cx, the two moment resultants becomeequal:Mm = M M where M fdzzu(z) (2.15)It also follows that the shear resultant for a fluid shell is small but non-vanishing. Usingequations (2.6), (2.11) and (2.14),= M (cmc) (2.16)Substituting these expressions back into equations (2.13), the equilibrium conditions specific to a liquid crystalline shell arepn — M(Cm_C)2 — V2M (2.17)d lfdM de—Pt = —+--——M—ds 2 ds ds26where 2 is the surface Laplacian. For axisymmetric geometries, it has the form 2 =1 (ri). It is important to note that these equations remain valid for elastic as wellas viscous forces. The only requirement is that the internal stresses are rotationallysymmetric about the surface normal.2.1.4 Dividing one fluid membrane into twoThe development in this subsection will appear to be no more than a mathematicalexercise. We will divide the fluid shell into two regions — the upper half and the lower half.Stress and moment resultants within each layer are evaluated and equilibrium equationsare written in terms of these quantities. Motivation for this is, of course, that the bilayeris composed of two separate surfactant layers. Physical significance of the results will bediscussed in section 2.2.5.Up to now, integrations over the bilayer thickness are done with respect to z, with zranging from —H/2 to H/2 and z = 0 corresponding to the neutral plane. Here, we willuse the variables z and z to specify positions along the surface normal in the upper andlower layers, respectively. Origins of these coordinates are chosen at a distance h/2 awayfrom the bilayer midplane as shown in figure 2.3; h is a length smaller than the bilayerthickness H, but is otherwise arbitrary. Thus, we have—H/2 < z < H/2 (bilayer) (2.18)z = z — h/2 ; —h/2 < z < (H—h)/2 (upper half)Z1 z + h/2 ; — (H — h)/2 < zi < h/2 (lower half)Integrations with respect to the above variables are understood to range over the givenvalues; limits of integration will therefore not be specified.We will now calculate the bending moment resultant. Neglecting second moment ofstress, we have (equation (2.15))M = fdzzoThis can be written equivalently as3We will show in section 2.2.4 that it is convenient to let the surfaces z = h/2 and Z = _h/2 coincidewith the upper and lower neutral planes.27M = fdz (z + h/2) + fdzz (zi — h/2) a= Jdzu z a + Jdz1 + (fdz a — Jdz )Defining M+ and M_ to be the moment resultants of the upper and lower halves, thatis,M+ = fdzu z a ; M_ = fdzjand letting M be the coupling moment given byM = (fdzu — Jdz1) (2.19)we haveM M + M + M (2.20)Thus, the total moment is the sum of individual moments plus a coupling term. A similarexpression can be written for the isotropic stress resultant . From the definitions ofstress resultants (equation (2.6)) and using (2.14), the general expression for isotropictension is= Jdza (i + ëz) (2.21)Following the same procedure of dividing the integral into upper and lower parts, theisotropic tension can be written as= + + - + (2.22)where and are the isotropic tensions in the upper and lower layers; they are= fclzua (i + ; _ = fdza (i +The equilibrium equations can now be written in terms of these quantities. Substituting (2.22) and (2.20) into equations (2.17), we get= (+ + ) e — (M + M_) (Cm — c)2 — V2 (M + M) (2.23)+2CmM± — V2M±Pt (+ + -) + + M_) — (M + M_)dM+cds28Compared to equations (2.17), we see that the equilibrium conditions are modified byterms containing the coupling moment M±.2.2 Free Energy of DeformationEquation (2.4) gives the work done in the virtual displacement of a thin shell. If theforces within the shell are elastic, Sw represents the reversible work of deformation; atconstant temperature, this is just the variation of the Helmholtz free energy F. Thus,reversible virtual work can be integrated to obtain the free energy of deformation. Such aderivation of the free energy must be done with caution because we are no longer dealingwith virtual displacements. Discontinuities of the extension ratios at the bilayer midplanemust be accounted for where necessary.2.2.1 Elastic model for liquid crystalline membranesLet us assume the shell is fluid and therefore equation (2.14) applies. In such a case,equation (2.4) for the virtual work simplifies toSw = JJdA fdz a(z) ( + z ( ë + SeJ ‘\1+flJ+ z2 ( g + 6g (2.24)\1+an Jwhere o is the area dilation of the neutral plane defined asan = — 1 (2.25)We now need an elastic model for o(z). The area dilation a(z), which is a rotationalinvariant about the surface normal, is chosen as the measure of strain. Because areas ata distance z from the neutral surface are dilated by the factor (1 + zC +z2g), a(z) isa(z) = (1+a)(1+ze+z2g)— 1 (2.26)Note that a(O) = an as expected. For a linear elastic theory, the isotropic stress u(z)is expanded to first power in a(z). Here, the expansion is done about the planar andunstretched configuration (where a, e and g are all zero) which, in general, does not29correspond to the stationary state of the free energy. The phenomenological relation is[Heifrich, 1981; de Gennes, 1990](z) u0(z) + ic(z) a(z) (2.27)whereo0(z) and ic(z) are model parameters with the same dimensions as o (force per unitarea). The zero order term u0(z) is included because the reference configuration is notnecessarily stress free. However, we stipulate that the stress resultant in the unstretchedstate is zero:Jcro(z) dz = 0 (2.28)This condition suggests an equilibrium between interfacial effects that tend to draw thelipids together, and the close range steric effects that push the molecules apart. In general,these forces are not co-planar and may lead to higher non-zero moments ofu0(z) [Marelja,1974; Israelachvili et al., 1980]. From the above considerations, we see that a0 must changesign along z.i(z) is a variable compressibility along the thickness dimension. It is the coefficient in aharmonic expansion of the free energy density and is therefore positive definite. Physically,it represents the resistance to changes in area per molecule at different sections along thelipid. Let us now look at the cases of pure stretch and pure bending separately.2.2.2 Pure stretchHere, all curvatures are zero (a = a) and the monolayers are assumed to undergoidentical deformations. The bilayer can therefore be treated as a unit structure. Since weare dealing with elastic forces, the isothermal virtual work 6w becomes 6F3, where F8 isthe Helmholtz free energy associated with stretching. Equation (2.24) simplifies to6F3 = fJdA Jdz u(z) (2.29)Substituting (2.27) into (2.29) and using the condition of zero initial tension (equation(2.28)), we get= fJdAofdz a6a = JJdA0 Ka6awhere we have used dA = (1 + a) dA0. K is the area elastic modulus defined as thezeroth moment of ic(z):K = fi(z)dz (2.30)30Since ic(z) is a positive definite quantity, K > 0. Integrating the above expression for5F8, we arrive at the final result:F8 = ffdA0 Kc2 (2.31)For pure stretch, the isotorpic tension is simply= fudz = Ka (2.32)2.2.3 Pure bending and local curvature energyIn this subsection, curvature energy is derived with the bilayer treated as a unit structure.This must be considered the limiting case as the viscous drag between monolayers becomesinfinite. Similar derivation of the monolayer curvature energy can be done by replacingthe variable z with either z or z1 (see relations (2.18)).Pure bending is the situation where no lateral force is applied. The shell is bentwithout stretching the neutral surface. In such a case, a, = 0 and from equation (2.26),a(z) = zë-l- z2g. It follows that(z) = u + (zë + z2g) (2.33)Without dilation of the neutral surface, equation (2.24) simplifies to6F = JJdA Jdz a(z) ( Se + z2 Sg) (2.34)where F represents the Helmholtz free energy associated with bending. We substitute(2.33) into (2.34), and, in order to derive a linear elastic model, expand terms up tosecond order in (z x curvature):= JfdA fdz (z cr0 Sc + z2 Sc + z2 a0 Sg)= fJdA [Sc (Jzao(z)dz) + ke6c + kgSg]The constants k and kg are bending rigidities associated with the mean and gaussiancurvatures; they are given by the second moments of i(z) anda0(z):k = fz2ic(z)dz ; kg = fz2ao(z)dz (2.35)31Since i(z) is a positive definite quantity, k > 0. Comparing the above expression to(2.30), it is noted that k ‘s-’ Kh2. On the other hand, the sign and magnitude of k9 areunclear. Strictly speaking, kg cannot be considered an elastic modulus; together with thespontaneous curvature, they are the first and second moments of the initial stress o-0(z)(see equation (2.37) below). After integrating, the curvature free energy becomes= JJdA (ke2 + kgg + e.Jzao(z)dz) (2.36)We have mentioned that the planar configuration is not necessarily the stress free state forthe bilayer. To account for this, an initial stress u0(z) was introduced. We now minimizeF to obtain the spontaneous mean curvature c0. By differentiating equation (2.36) withrespect to the mean curvature and setting it to zero, we getkc0 = — fzu0(z) dz (2.37)For o-0(z) symmetric about the neutral surface, the spontaneous curvature vanishes. Interms of c0, the curvature energy (equation (2.36)) is= JfdA [k (2 — 2ëco) + kgg] (2.38)which is just the form introduced by Helfrich [1973]. This is called the local curvatureenergy.Let us also derive an expression for the neutral plane. The condition for pure bendingis that the lateral tension should vanish. For a cylindrically bent shell, the lateral tensionisr = Judz f(u0 + Kze) dzRecalling the assumption of zero initial tension (equation (2.28)), the neutral plane iscalculated from the relationfzk(z)dz = 0 (2.39)A constitutive relation for the bending moment can also be obtained as follows:M = JcTzdz = f(o + iczë) zdzUsing equations (2.37) and (2.35), the bending moment isM = 1c (e — c0) (2.40)322.2.4 Coupled layers and global curvature energyThe developments so far have assumed the fluid shell to be a unit structure. We now lookat the added effects of loosely holding two such layers together to form a closed surface(the bilayer vesicle). The only kinematic constraint is that the layers do not separatein the normal direction; relative tangential motion is permissible. By allowing moleculesto diffuse within the monolayer plane, all “irregularities” in surface density (grad a) willsmooth out until every lipid molecule occupies, on the average, the same area. Assumingthere is no exchange of particles across monolayers and given enough time, each monolayerwill have its distinct uniformly distributed surface density. Mismatch between top andbottom surface densities will lead to a coupling moment; this can be seen most easilyfrom the analogy of heating a bimetallic strip. A more rigorous derivation is as follows:General expression for the coupling moment is (equation (2.19))M = (Judz — fudzi)where c is given by equation (2.27). We will let a and cv_ be area dilations on thesurfaces zL, = 0 and zj = 0 respectively (see section 2.1.4 and figure 2.3). Area dilation inthe upper layer is thereforea(z) = a+ + ëzSubstituting this into (2.27), the integral of a in the upper layer isJ a dz = J cr dz + a+ J , dz, + f z ,The first term on the right hand side is identically zero because of relation (2.28) and theassumption that øc, is symmetric about the bilayer midplane. We will also require thelast term to be zero. In doing so, z, = 0 is chosen to be the neutral surface of the uppermonolayer (see equation (2.39)). This gives a definite expression for h in terms of thevariable compressibility i(z). We are therefore left withJ dz Km a ; Km = f K dz (2.41)Km is the area modulus for the monolayer. For tc(z) symmetric about the bilayer midplane, Km = 1<72 (see equation (2.30)). A similar expression can be written for thebottom layer:Jcrdz1 = Kma_ (2.42)33Thus, constitutive relation for the coupling moment isM= Krnh(2.43)where a is the differential area dilation from top to bottom:— (2.44)Equation (2.43) is a local relation which expresses proportionality between M and a;it remains valid even in dynamic situations where the quantities are time dependent.However, in the remaining of this subsection, we will only be concerned with the staticsituation where both M± and a± are uniform functions.We now evaluate the virtual work associated with the uniformly distributed or “global”coupling moment. Assuming pure bending (Sc 0) and neglecting second moments ofu(z), virtual change in free energy is, from equations (2.24) and (2.15),SF = JJMS3dAThe bending moment M is, according to (2.20),M=M+M+MIn evaluating the free energy, we will not be concerned with contributions from bendingmoments of individual layers (M+ and M_); each monolayer is a unit structure with localcurvature energy given by equation (2.38) in the previous section. The global curvatureenergy is due to the coupling moment:SFglobal= Jf M± S dAUsing expression (2.43) for M± and noting that this is uniform over the surface, we haveKmh 11 -SFglobal= 2 jj Sc dATo evaluate the above integral, we recognize that, to first order in (h x curvature), areadifference between the surfaces z = 0 and z1 = 0 (neutral planes of upper and lowermonolayers) iszA = fJhedA34Allowing also for an initial area difference zA0, the differential area dilation iszA-zA0= A= (ffhdA — A0) /A0 (2.45)For pure bending (6 = 0), variation of the above quantity is6a = JJ6CdASubstituting into the above expression for 6FglobaI , we getKmAo KmA0 2SFglobal= 2 ±Fgioai=Finally, the expression for c from equation (2.45) is used. The resulting global curvatureenergy is:Fglobal = )c A0 ( fJe dA — r)2(2.46)where- Kmh2 p LAOlc — , io— I. A.47UJlok is the non-local bending rigidity; it is of the same order of magnitude as k. Equation(2.46) is the global curvature energy introduced by Evans [1974].2.2.5 Euler-Lagrange equations and mechanical equilibriumBecause stretching a bilayer is energetically much more costly than bending it, conformations of a bilayer vesicle are, in most cases, entirely governed by the curvature energy.The most general form of curvature elastic energy is [Evans, 1980; Seifert et al., 1992]Fcurv = F1001 + FglobalWe will assume the bilayer has no spontaneous curvature; the local and global energiesare thereforeFiocai (2kcm) JJ e2 dAFglobal k A0 (* ff dA — r)2Here, kern is the bending rigidity of a monolayer. Since there are two monolayers, kern ismultiplied by 2 (Note that k 2kcm; see equation (2.35)). The curvature energy Fcurv35is dependent on the bilayer’s configuration; i.e., Frv = fj, c) where (.A, c; i = m,are the extension ratios and curvatures of the bilayer midplane. For the vesicle to retainits area and volume, we minimize instead the functionG(,c;7÷,7_,p) Fcurv(i,c) + ff(7++_)dA — fffpdV (2.48)7+ and “y_ are Lagrange mutipliers that ensure two dimensional incompressibility of theupper and lower layers, while p is for volume conservation. ‘ Minimization of the abovefunctional with respect to (A, c) leads to Euler-Lagrange equations that must agree withthe equations of mechanical equilibrium.Let us first minimize Fiocai. Proper use of Lagrange multipliers requires the minimization to be done as if there was no constraint. Thus, writing dA = (1 + a,) dA0, thevariation of Fiocai is6Fiocai = (2kcm) Jf (2e + e2 dAIn terms of virtual displacements 6Xt and 6x in the tangential and normal directions,= 6x — V2 (6x) — (c + c) 6x (2.49)= div(6x) + 6x (2.50)1 + Qfwhere2 ldfdf\ . ldV f = — — r_) ; div f = — — (rf)rds ds rdsUsing the relationsJJ (f V2g) dA = Jf (g V2 f) dA ; JJ (f divg) dA = ff (- g dAit is easy to show that6Fiocai = — JJdA [ (2kcm e) (Cm c)2 + V2 (2kcm )] sxn (2.51)Similarly, first variation of the non-local curvature energy is, from equation (2.46),6Fglobal = M 6 Jf dA/ 6Q \= M±ff(Se+e dA1+cv.j4Note that this gives rise to an apparent contradiction: The two monolayers are required to be incompressible, and yet they must be stretched/compressed slightly to create the a± field. This inconsistencycan be resolved by recognizing that while area conservation is a global approximation, local variations inc± represent higher order perturbations.36whereM= Krnh(ffhdA — LA0) /A0Using the above relations for 6ë and 6o/(1 + an), the first variation of Fglobaj 5SFgio&ai = JJdA (2CmCM± SXn) (2.52)Minimizing the remaining terms in equation (2.48) (with respect to ) and c) is nowstraightforward. The results areJJ(7+ + ) dA = JJdA (+ + 7-)= fJdA [—(7+ + 7-) + (+ + )• 6x] (2.53)—8 JJJp dV = JfdA (—p. 8x) (2.54)First variation of the constrained functional G (equation (2.48)) is obtained by summingequations (2.51) to (2.54). The resulting Euler-Lagrange equations areP = (7+ + 7)3 — (2kcm)(m — c)2 — V32 (2kcmCj + 2CmCM± (2.55)o =We now consider the equations of mechanical equilibrium (equations (2.23)). In thestatic situation, because the coupling moment M± is uniform, all terms involving dM±/dsdrop out. Also, without spontaneous curvature, the bending moment in each monolayeris equal to kcm ë; thus,M + M_ = 2kcmGiven these conditions, equations (2.23) reduce topn = (+ + ) — (2 kcm e) (Cm — c)2 — V2 (2 kcm ) + 2Cm c M—Pt =For p,-, = p and Pt = 0, the equations of mechanical equilibrium are equivalent to theabove Euler-Lagrange equations. It is interesting to note that the Lagrange multipliersfor constant area are just the isotropic tensions.372.3 Interlayer ViscosityIn the last section, equations of mechanical equilibrium are shown to reduce to the Euler-Lagrange equations by neglecting all terms containing grad M±. Such a situation can beconsidered the limiting case as t —* 00; that is, given enough time, lipid molecules withineach monolayer will diffuse collectively until the macroscopic density field is uniformlydistributed. This is a time-dependent process whose rate is driven by the two-dimensionalcompressibility modulus Km and limited by inter-monolayer viscosity. In what follows,we will present a phenomenological model for interlayer drag and show that relaxation oflocal perturbations in c is a diffusive process. A generalized dynamic equation is derivedthat embodies interlayer dynamics within the bilayer membrane.2.3.1 Phenomenological model and balance of tangential forcesFigure 2.4 depicts the situation in which relative motion between monolayers gives riseto an interlayer shear stress o. The simplest model for such a phenomenon is to assumeproportionality between o and the relative velocity:a3 = bv± ; v± v — v_ (2.56)b is the interfacial drag coefficient with dimensions dyn.sec/cm3. The major aim of thisthesis is to develop the technique for measuring this parameter.Let us examine the consequence of such a model on the dynamics of a±. We beginwith the condition for tangential force balance — the second of equations (2.17). Thisequation can be written equivalently asdi 1 dM— Pt = — — —eM) + e— (2.57)ds \. 2 j dswhere, according to (2.21) and (2.15),— eM = fa(z) dz (2.58)Equations of form given by (2.57) can be written for each monolayer. With relative motion, shear stresses experienced by the monolayers are equal and opposite at the midplane.However, the effective shear tractions at the top and bottom neutral surfaces are scaled38by area as follows:+ — 08 — — 0_sPt— 1 + ha/2, Pt— 1 — he/2Using (2.56) and (2.58), tangential force balance (equation (2.57)) on the two monolayersareb v± (1 + e) [ (J a dz) + e dM+] (upper monolayer) (2.59)— b v = (1 — e) [ (f a dz) + dM_] (lower monolayer) (2.60)We now evaluate the difference between the above equations. Since both M+ and M_have the form kcm + constant, their gradients are equal. Using (2.41) and (2.42) for theintegrals, the difference between (2.59) and (2.60) is, to first order in he,2bv = Km [(1 + e) — E.(i—Neglecting terms of order he in comparison to unity, the final expression isv = Dgrada (2.61)where D is the mechanically driven diffusivity defined asD2b(2.62)Note the subtle difference in use of the subscript “±“ for a and v:a Izh/2 — a Iz=—h/2 (2.63)V± = V IzrzO+ — VWe will look at the kinematic relation between these quantities in the next section.2.3.2 KinematicsIn equations (2.1), extension ratios (Am, A) at any location in the shell are given interms of the extension ratios (sm, )) at the midplane. Here, because the monolayers areuncoupled, we allow for discontinuities in the extension ratios at the midplane z = 0:At z = 0 : = ; =Atz=0:39According to equation (2.50),m div(6x) + E6z (2.64)=m+ div(6x) + ëxnBecause normal displacements of the monolayers are equal, the same Sx is used in theabove equations.Extension ratios away from the midplane (z 0) can be obtained from equation (2.1).On the upper and lower neutral surfaces z = ±h/2, we defineAtz=li/2: A —Am— mAtz=—h/2: AmAUsing equation (2.1),—— 11 h)(i + (2.65)m m+m ,= (i — c) ; A = (i - c)To first order in (h x curvature), changes in extension ratios at the upper and lower neutralsurfaces are— +6A —+ (2.66)mm m2tm’A-A- 2Now, from relation (2.63), c is associated with differences between quantities at theupper and lower neutral surfaces. Using (2.66) from above, we have___m I m= 6A 6A +5) h1++ = -p-- + (2.67)+ m m_ __A; 6A —+h=1+c- —The last step is to subtract the second of equations (2.67) from the first and substitute inequations (2.64). At the same time, infinitesimal changes in position are converted intotime derivatives; i.e., we replace 6 by d/dt. The resulting relation isda dë= divv + (2.68)402.3.3 Dynamic equation for cWe are now ready to write down the dynamic equation for a±. Taking the divergence ofequation (2.61) and using (2.68), we get±DV2+hIn addition, we note that transfer of material between monolayers (lipid flip-flop) canalso be a mechanism for relaxing the a± field. To first order, we assume such a ratebe proportional to the magnitude of a± (a measure of interlayer “frustration”). A term—ci,, a± is therefore added to the right hand side of the above equation, where c, is apositive constant with units sec1; the reciprocal of c, is the characteristic flip-flop time.With the time derivative d/dt separated into local and convective parts, the aboveequation is written as8c+ v5 = DV2 + h ( + v5 — (2.69)This dynamic equation for c is derived by Evans et al. [1992]. v is the velocity of thebilayer midplane; it is the average of v+ and v_. From the above equation, we see thatchanges in the mean curvature ë can be viewed as “sources” for the . field: Differentialdilation is created by the local rate of change in curvature and/or by the convection ofmaterial along a curvature gradient. On the other hand, the diffusive term D V2a± is a“sink” for smoothing out irregularities in the a distribution. The last term due to lipidflip-flop is also a sink for c; it is the only non-vanishing dynamic effect in the case wherea± is evenly distributed. Because this is usually a very slow mechanism, relaxation byflip-flop is neglected in most situations.Since the coupling moment M± is proportional to a± by equation (2.43), we can alsowrite the following dynamic equation for M:+ vsa = DV2M + k ( + vs) — (2.70)k is the non-local bending rigidity defined in (2.47).41Chapter 3Measurement of Interlayer DragThis chapter discusses the technique I have developed for measuring inter-monolayer viscous forces within a fluid bilayer; the goal is to determine the dynamic drag coefficient bintroduced in section 2.3.1. This method represents, for the first time, measurements ofinterlayer stresses in unsupported fluid bilayers. The procedures are now established andcan be applied to bilayers of many different compositions.Our method of creating interlayer drag is as follows: Because the monolayers areonly weakly held together, they will, as the bilayer passes through regions of curvaturevariations, undergo relative sliding which in turn gives rise to viscous effects. Sharpcurvature changes in the bilayer can be created using a technique known as nanotetherextrusion [Hochmuth and Evans, 1982; Bo and Waugh, 1989]. For measuring interlayerdrag, a transducer sensitive to forces on the microdyne level (weight of109gm) is needed.A transducer is constructed using the very material we experiment with; this so-calledfluid membrane force transducer will be described in a later section. Furthermore, auseful “by-product” from our nanotether technique is a novel way of measuring bendingrigidities of bilayers; this will also be explained in detail.Theoretical analysis of the tether experiment is based on the formalism developed inchapter 2. We have chosen to append these calculations to the end (appendix B) andquote only the important results in this chapter. Such a treatment is much clearer inillustrating predicted behaviours and leads logically to our experimental strategy.In the Results section, representative plots of experimental data are shown. Also,measured values of b and the bending rigidity are summarized for the following situations:• SOPC bilayers at temperatures 15°C, 25°C and 35°C;42• MOPC bilayers at room temperature; and• SOPC, MOPC and BSM (bovine brain sphingomyelin) bilayers — all containing 50molar % cholesterol at room temperature.3.1 General Description of Nanotether ExtrusionGeneral features of the nanotether experiment are illustrated here. The purpose is tointroduce the relevant variables and establish basic relations between them; details ofexperimental procedures will be postponed to a later section. Throughout this thesis, cgsunits will be used.Bilayer samples come in the form of vesicles (closed bags) of various sizes. We chooseto work with vesicles that are uni-lamellar and with excess area; i.e., those that haveonly one bilayer (as opposed to stacked bilayers) and which are flaccid and non-spherical.Because of the hydrophobic effect, solubility of the surfactants in aqueous solutions isextremely low (10’4mol 1_i or lower). For the duration of our experiment(r-1 sec) —and indeed for much longer times — the bilayer can be considered a closed system wherethe number of constituent particles is fixed. We will further assume that each vesicle hasfixed surface area and volume; justifications for these last assumptions are as follows:The bilayer exhibits elastic behaviour when stretched isotropically; the compressibilityrelation is, from section 2.2.2,r = Kar and K are respectively the isotropic tension and the area elastic modulus — both withunits dyri/cm; a is the fractional change in area. Typical values of K are of orderlO2dyn/cm [Evans and Needham, 1987]. For the tether experiment, the applied tensionis roughly 0.1 dyn/crn with deviations of less than 1% (lO3dyn/cm). From the aboveexpression, the resulting variation in area is estimated to be less than 1 part in iO;surface area of the bilayer vesicle can therefore be treated as constant. With regard tovolume conservation, it is noted that the bilayer is a semi-permeable membrane that allowseasy passage of water across its surface. Larger particles such as glucose, on the otherhand, are effectively blocked (compared to water, glucose molecules take iO times longerto pass through). Our vesicles are suspended in a 0.2 M glucose solution with an equal43concentration of solutes trapped inside. Any net displacement of water into or out of avesicle will amount to a difference in solute concentration across the bilayer membrane;the resulting osmotic stress can easily be as high as lO5dyn/cm2 (0.1 atm). Becausethe pressures involved in our experiment (.-.-‘ lO2dyn/crn)are orders of magnitude lowerthan such osmotic stresses, no appreciable amount of water can be forced through themembrane and, consequently, the volume remains unchanged. Thus, vesicle volume ismaintained for the most unlikely reason — the membrane’s permeability to the water.A tether experiment is schematically shown in figure 3.1. First, a bilayer vesicleis aspirated into the micropipette under a suction pressure The pipette size istypically 6 to 8 m in diameter and LFVS is a few hundred dyn/cm2 (100 dyn/cm21 mm of water). A vesicle whose diameter is larger than the pipette calibre is chosen.Under the constraints of fixed area and volume, the vesicle will be partially aspirated,with a spherical segment of radius R remaining outside. R is normally 10 1um (c.f.bilayer thickness h 4nm). r is the uniform isotropic tension in the bilayer; it isgiven byLFves RvTves = (3.1)2 (1 - R;/R)This expression is derived in appendix C as the special case of a more general formula.Since the values R and R can be measured quite accurately (to within a few %) andcan be controlled to within 1 %, Tves is an adjustable parameter in our experiment.The next step is tether pulling. A latex bead, whose surface can stick to the bilayer,is brought into contact with the aspirated vesicle surface. After a firm adhesion is established, the bead is pulled back along the axis of symmetry as shown. Attached to thebead is a tubular piece of bilayer material (tether) whose radius rt is typically 30 rim(3 x 106cm). Except local to the two end regions, whose extents are of order ri, thetether radius is uniform. 1 The small junction connecting the spherical vesicle segment tothe nanotether is the region of most interest: Over the extent of this region, the bilayercurvature increases by more than 100 fold; i.e., from ë = 2/Rn to ë 1/ri. Since inter-layer sliding is created by gradients in curvature, this is the region where all dissipativestresses are concentrated; away from the junction, the curvature is effectively uniform.1n principle, there may also be surface ripples on the tether. Because the tether is under tension,rippling is not likely to be a large effect.44The process of tether extrusion is characterized by the tether length L and the rateof extrusion V = dL/dt. Both of these are controllable parameters. The tension forcein the tether is denoted by ft (units of dynes). Because of the surface area and volumeconstraints, an increase in tether length L must be compensated by a decrease in thepipette projection length L. The geometric relation between these two quantities leadsto an expression for rt [Hochmuth and Evans, 1982]rt = —R (i— R’/R) (3.2)Since all quantities on the RHS are measurable, the tether radius can be calculated.3.2 Summary of Equations and Experimental StrategyTheoretical analysis of the tether experiment is based on the formalism developed inchapter 2. In particular, the general dynamic equation (equation (2.69)) is to be solved onthe entire vesicle surface. The most important region is within the vesicle-tether junctionwhere most of the interlayer viscous stresses are concentrated. As shown in appendix B,transport of the c field within this region can be characterized by the dimensionlessPéclet number P 2 defined as(3.3)where D = Km/2b is the mechanically driven diffusivity (equation (2.62)). The analysisin appendix B depends critically on the assumption that the Péclet number be smallcompared to unity. This condition will be verified at the end.3.2.1 Two important equationsBased on the inequality F << 1, two equations that form the basis of our experimentationare derived. We will quote the results here and leave the details to appendix B. The firstequation is one that expresses the tether radius in terms of membrane tension:rt = (kcm/Tves )1/2 (3.4)2The use of Péclet number is an idea borrowed from the field of transport phenomena [Landau andLifshitz 1982]. It is useful in the analysis of diffusive processes much like the Reynolds number is usefulin hydrodynamics. It should be noted that, although equation (2.69) has the form of a diffusion equation,strictly speaking, we are not dealing with thermally driven diffusive processes.45Referring to section 2.2.5, kern is the bending rigidity of a monolayer. The importantfeature of equation (3.4) is that the tether radius is independent of the rate of extrusionVt; this is a direct consequence of the small Péclet number assumption. Conversely, adependence of rt on V will mean P is not “small enough”, which in turn leads to thecollapse of our entire analysis.The other important equation relates the tether force to the kinematics. It has theform-ft = 2kcm— kF0 + bh21n() V + (3.5)27r rt rt 2Rwhere k,, I’ and Ii are defined in section 2.2.4. Again, this equation is only valid forP << 1. We discuss each of these terms separately.Using equation (3.4), the first term can be written as 2 kcrn/rt = 2 (kcmrves)”. Thisterm is due directly to the vesicle tension, which in turn is controlled by the pipettesuction pressure LFVeS. With regard to the second term, recall that F is a measure ofthe initial area difference between the monolayers. This pre-loaded stress shows up in thetether force and is actually quite significant. The sum of these first two terms on the RHSof (3.5), multiplied by 2r, represents the minimum force required to form a tether. Thisis the force corresponding to the situation where V = 0 and L is negligibly small; itsmagnitude is in the neighbourhood of microdynes.The last term is a non-local bending contribution to the tether force. For typicalparameters of kc, r’-’ 10’erg and R ‘—‘ 103cm, the tether force is a microdyne forL 300tm. For our experiment, L normally goes up to 50 m, giving rise to a tetherforce of about 0.1 to 0.2 udyn. This is in fact what we see experimentally.The third term is the one of most interest to us. It predicts a linear relationshipbetween the dynamic tether force and the rate of extrusion 4. The proportionalityconstant contains the parameter b that we wish to determine. Since this dynamic part offt only shows up when there is motion, it can easily be separated from the other threecontributions in equation (3.5).3.2.2 Experiment 1: Static measurementWe must first verify that the Péclet number is small. This is done by checking the validityof equation (3.4) for the range of extrusion speeds that we will use for the next experimemt46(measurement of interlayer drag). In particular, tethers are pulled as shown in figure 3.1at different speeds. For each pull, the values rj and Tves are determined and put intoequation (3.4) to evaluate kcm. A constant value of kcm for the chosen range of extrusionspeeds would justify the assumption of small Péclet number. The additional reward forthis, of course, is an accurate measurement of kcm. The tether radius rt is measuredaccording to the geometric relation (3.2) and the vesicle tension Tves is controlled by thesuction pressure according to (3.1).3.2.3 Experiment 2: Dynamic measurementThe next step is actual measurement of the interlayer drag coefficient b. Let us first definef° and fd as the initial-static and dynamic parts of the tether force:cm —f0 2r r — kcfoj (3.6)fd 27rbh2lnQ-3) V (3.7)The total tether force is therefore, according to (3.5),ft = f0 + fd + (3.8)We will postpone description of the force transducer to the next section and accept fornow that f can be measured with sufficient accuracy. As such, the contributions f° andfd can be determined separately: f0 is the tether force for V = 0 and L very short (lessthan 10 pm), and fd is the portion of the tether force that is associated with motion.With a knowledge of these quantities, we can deduce the following:• The initial area difference can be estimated from fo — 4kcm/Tt.• From the slope of fd vs. V, the parameter b can be calculated. Because thebilayer “thickness” h (more accurately, the separation between neutral surfaces ofthe monolayers) is not known precisely, the experimental result is presented in termsof b h2.473.3 Methods3.3.1 Vesicles, solutions and beadsBilayer VesiclesPreparation of vesicles has been a well established technique in Prof. E. Evans’ laboratory [Evans and Needham, 1987]. Phospholipids are purchased from Avanti Polar Lipids(Alabaster, AL) in dry powder form. The three types of lipids used here are: SOPC(18:0, 18:1) , MOPC (14:0, 18:1) and bovine brain sphingomyelin (BSM). One fatty acidin BSM is a 12-carbon chain while composition of the other fatty acid varies even withinthe same sample; the average length of this other chain is, according to the supplier,about 19 carbons. Cholesterol is purchased from Sigma Chemicals (St. Louis, MO); italso comes in dry powder form. Another type of phospholipids we incorporate into thebilayer are the chemically labelled PE lipids with two 16-carbon chains. A chemical groupcalled dinitrophenyl (DNP), which can be recognized by specific antibodies, is attachedto the PE head group. We put a small amount (0.5 mol %) of these labelled lipids intoall our bilayers. At the other end, anti-DNP proteins that bind specifically to the DNPgroups are attached to the bead surface to facilitate adhesion between the vesicles andthe beads. The DNP-labelled PE lipids are obtained from Molecular Probes, Inc (Eugene, Oregon). All surfactants are stored in solution form in known concentrations usingchiorofom/methanol (2:1) as a solvent. Since the molecular weights of the surfactants arealways provided, mixtures of specific molar ratios (e.g., SOPC/cholesterol 1:1) can easilybe prepared.To make vesicles, about 40 t1 of lipid solution (in chloroform/methanol) is spreadevenly on a clean teflon disc. The disc is then placed in a vacuum for at least 5 hoursto allow complete evaporation of the organic solvent, leaving only dry lipids on the disc.It has been found desirable to “pre-hydrate” the dried lipids before introducing themto aqueous environments. This is done by passing a stream of saturated water vapor,carried by an inert gas such as argon, over the disc surface for about 10 mm. After this,an aqueous solution containing the desired solutes is gently poured onto the disc (whichis already in a glass beaker). The entire assembly is stored at 35°C overnight. During this3The notation is (n1 m1, n2 : m2), where n1 and n2 are respectively the number of carbon atoms inthe sn-i and sn-2 chains; m1 and m2 are the number of double bonds in the respective chains.48time, bilayer vesicles will form spontaneously at the pre-determined solute concentration.SolutionsTwo different types of solutions are used in our experiment. The first type is used tohydrate the dry lipids (as described above); naturally, this will also be the solution thatoccupies the vesicle interior. Another type of solution is used for suspending the alreadyformed vesicles. Because only water can pass through the bilayer, solutions on the twosides of the bilayer are distinct. The small difference in refractive index between the twomedia greatly enhances the visibility of the vesicle. In this experiment, we choose to havea 0.2 M sucrose solution inside the vesicles. On the outside, a solution consisting primarilyof glucose is used for suspension; the osmolarity (a measure of osmotic activity) of thissolution is adjusted to be slightly higher — by about 5% — than the interior solution. Thiscauses the vesicles to deflate, thus giving us the desired excess area. Another advantageto such a combination is that the interior sucrose solution, being more dense than theoutside medium, causes the vesicles to sink to the bottom of our working chamber, thusmaking the vesicles more accessible. In addition to glucose, it is essential to includesmall amounts of non-adsorbing polymer (polyethylene glycol, or PEG) and salt (NaC1)in the suspending solution. These solutes help to create non-specific attraction betweenthe bilayer and the bead surface thus allowing antibody-antigen (DNP) reactions to takeplace [Evans, 1989].The interior sucrose solution (0.2M) is made by dissolving 6.64 gm of sucrose (MW342.30) into 100 ml of solution. The resulting osmolarity is 208 mOsm. The suspendingsolution contains 2.747 gm of glucose (MW 180.16) and 0.149 gm of NaC1 (MW 58.44)per 100 ml of solution; this is a 0.152 M glucose and 0.025 M NaC1 solution. In addition,0.5 % (by weight) of PEG (MW 19700; Scientific Polymer Products; Ontario, NY) isdissolved into the solution. The final solution has an osmolarity of 215 mOsm.Bead PreparationLatex beads are purchased from IDC (Interfacial Dynamics Corp., Portland, OR). Theparticular type we buy have chemically active aldehyde groups present on the bead surface. Bead diameters are 3.2 im and 5.4 ,um. Anti-DNP molecules are purchased fromMolecular Probes (Eugene, Oregon); it comes in solution form at 2 mg/ml. To attach theantibodies to the bead surface, latex beads are put into a dilute antibody solution (4049tg/ml) that is buffered at pH 7.4. The concentration of beads can vary, but is usuallyat ‘—0.l % solid. The mixture is left on a rocker for at least 2 hours. It is believed thatthe antibodies are covalently bound to the aldehyde groups during this incubation time.Beads that have not gone through such procedures do not stick to the bilayer vesicle.3.3.2 Micropipette setupA picture of the micropipette setup is shown in figure 3.2. This is the microscope stationat which all experiments discussed in this thesis are performed. Different componentsinvolved in such an experiment are discussed below:Making pipettesMicropipettes are small cylindrical glass tubes. They have two important functions in ourapplication: First, they are used as “micro-arms” to manipulate test samples on micronlength scales. In addition, through precise control of the suction pressure, micropipettescan be used to exert well defined forces on the samples. Such pipettes are made in thelaboratory as follows: Glass tubings with outside diameters of 1 mm and inner diametersof 0.7mm are purchased from Kimble Glass Inc. (Toledo, Ohio). By using a pipette puller,the glass tube is simultaneously heated and pulled axially to create very long thin tapersthat go down to sizes of less than 0.1 m. The tube is then cut transversely at properlocations to give the desired inner diameters. For our experiment, the pipettes are cutwith inner diameters of roughly 6 to 8 m. Because of the long taper, pipette diameterscan be considered uniform (i.e. cylindrical) over axial lengths of 10 ,um. Glass tubingsare pulled with a vertical pipette puller, manufactured by Kopf (Tujunga, CA), modelnumber: 700C. The heat and pulling force are adjustable in such a device. Followingthis, the pipettes are filled by boiling them in a desired aqueous solution; this is usually asolution similar to that we use for suspending the vesicles. When not in use, micropipettesare always stored in an aqueous environment. Refrigeration is also recommended to slowdown bacterial growth.Manipulating pipettesA micro-manipulation set includes a chuck for inserting the glass pipette, a receiver onwhich the pipette-chuck assembly is mounted, and a position controller (joystick) that islinked to the receiver through flexible hollow tubings. As such, the joystick is mechan50ically uncoupled from the receiver. Inside the receiver are three sets of bellows whoseextensions/contractions control the xyz translations of the pipette; the bellows are inturn controlled pneumatically by the “remote” joystick via the hollow tubings. Usingsuch a system, pipette motions up to 0.1 mm in all three directions can be made. Suchmotions are extremely smooth and free of hysteresis; fine manoeuvrings down to 0.1 mcan be made easily. Because the bellows are driven by air pressure, there is no need tofill the system with special fluids. There are usually two micromanipulating systems ateach microscope station — each controlling a pipette that inserts into either side of anopen chamber containing the vesicles. Such a system is ideal for “manual” work. Forour purposes, because one pipette has to be moved at well controlled speeds (for tetherextrusion), the micromanipulating set is replaced by an inchworm motor. This is a piezoelectric stepper motor that allows very accurate control of position and velocity (L and14 in equation (3.5)) in the direction along the pipette axis; digital readouts of thesevalues are also provided. The motor acquires the pre-set speed from resting position witheffectively zero rise time. The range of motion for such a motor can be from 0.1 1um to 2.5cm. Speeds are controllable from 0.0041um/s (according to the manufacturer) to as highas 2000 nm/s. Our speed range is typically 10 to 300 pm/s. The micromanipulationsystems we use are purchased from Microlnstruments (St. Louis, MO). The Inchwormmotor is bought from Burleigh Instruments Inc. (Fishers, NY), model number 1W 710.Pressure controlThe pipette suction pressure is controlled by a home-built manometer set that consists oftwo water reservoirs. Pressure is created from simple hydrostatics; i.e., from the differencein elevation between the two reservoirs. Thus, a difference in elavation of 1 cm creates ahydrostatic pressure of 980 dyn/cm2. Reservoir elevation is controlled by a micrometer(the measuring apparatus) to the nearest 0.005 nun, which corresponds to pressure variations of 0.5 dyn/cm2. To acquire zero reference pressure, we observe the movement of asmall particle deliberately placed inside a micropipette (note that the pipette is alreadyplaced in a chamber filled with water). Elevation of the entire manometer setup (including both reservoirs) is adjusted until the small particle is stationary. Such a method isaccurate to 0.5 dyn/cm2; i.e., from this reference position, a difference in reservoir elevation of 0.005 mm will cause the particle to move. The pressure is monitored by a pressure51transducer connected in series within the manometer. It is purchased from Validyne Engineering Corporation (Northridge, CA). The particular model we use is DP 103, fittedwith #10 diaphragm. Such an assembly has a linear range up to 880 dyn/crn2 with anaccuracy of 0.5%.Microscope, optics and image recordingThe microscope used here is an Inverted Leitz Diavert Microscope (Wild Leitz, Germany).It is illuminated by a 200 watt mercury arc lamp made by Oriel (Stamford, CT). Theultraviolet light from this intense source is first filtered out through a glass piece. Theremaining light is put through a band-pass filter centered at 546.1 nm with band width of10 nm (supplier: Corion; Holliston, MA). The objective is a 40x lens that uses Hoffmanmodulation contrast microscopy (Modulation Optics Inc.; East Hills, NY). Such a systemconverts the otherwise undetected phase information into amplitude differences. For ourvesicles, because there is a slight difference in refractive index between the interior sucrose solution and the suspending glucose/salt solution, visibility is greatly enhanced byusing such an optical setup. The experiment, as viewed from a 25x eyepiece, is recordedthrough a high contrast b/w camera (Dage-MTI Inc.; Michigan, IN) onto a SONY 3/4inch video cassette recorder (model VO 5800). During playback, such a recorder is capable of making single frame forward and reverse search, thus facilitating detailed analysisof the experiment. Dimensions on the monitor are measured with a position analysermanufactured by Vista Electronics (LaMesa, CA), model number 305. This device produces digital readouts of distances according to the separation of parallel lines (verticalor horizontal) which are movable on the monitor; such a device is sometimes called the“video caliper”.Force transducerThe transducer for measuring tether forces is definitely home-made. It is composed ofan aspirated vesicle attached to a bead — much like the nanotether assembly — but withthe bead-vesicle contact made macroscopically large to prevent tethering. The situation isshown in figure C.1(a,b). Because the radii of curvature involved are much larger than thebilayer thickness (no tethers), we can make the membrane approximation in analysing themechanics of this problem; i.e., all terms involving the bending moment M in equations52(2.17) can be dropped. It can be shown that the applied force ft scales asf rSwhere r is the membrane tension. For ft on the order of lO6dyn and r 0.Oldyn/cm(an easily attained level), the deflection is roughly a 1um. This is sufficiently large formeasurement. Since we are close to the optical resolution, the error in length measurements, as described above, is roughly ±0.1tm. This means the estimated forces willhave errors of about 10%.The above relation is an equality if we included a factor of order unity on the right handside. In appendix C, we will show how this factor can be calculated numerically. Alsopresented in the appendix is the experimental verification of the analysis.3.3.3 Experimental proceduresThe general experimental strategy is discussed in section 3.2. We will give here details ofthe actual procedures. In general, it is desirable to carry out micropipette experiments freeof unwanted influences from other vesicles. For this reason, all our experiments involvetwo open chambers that are separated by a small air gap. One chamber contains thesupply of vesicles while the other is an empty chamber (i.e., with only aqueous solution)in which the actual experiment is performed. Because we cannot simply take vesiclesacross the air gap, they are transferred to the empty chamber by first inserting them ina large pipette. This large pipette, when taken across the air gap, retains the solutionwithin and therefore the vesicles are protected. The chamber we use has temperaturecontrol features: It is designed so that water from a heat bath can circulate around it(without coming into contact, of course), thus bringing the ambient temperature of thechamber to the desired level. Vesicles in the empty chamber are set up as shown in figure3.3. In this picture, the nanotether between the bead and the opposing vesicle cannot beseen because of its mesoscopic dimension (diameter ‘-‘ 40 rim). This picture is taken at amagnification lower than our “operating value” in order to show all the features.Static measurementA discussion on this experiment is outlined in section 3.2.2. Here, we want to measurethe tether radius at different speeds of extrusion. According to equation (3.2), rt can be4The accuracy can be improved significantly by more sophisticated image processing techniques.53obtained by knowing the relation between the tether length L and the pipette projectionlength L. This is done by aspirating a vesicle into the stationary pipette (i.e., the pipettemanipulated by pneumatic controls) while holding a bead in the moving pipette (the oneconnected to the inchworm motor). The tension in the vesicle is set to roughly 0.1 dyn/cmby adjusting the pressure according to equation (3.1); exact value of the suction pressureis needed for the analysis. In this experiment, the transducer vesicle, as shown in figure3.3, is not needed.A tether with a length of 5 ,um is first pulled from the vesicle; this is easily doneby touching the “sticky” bead to the vesicle surface and pull back. From this position,the tether is pulled back at different speeds while the camera is focussed on the bilayerprojection inside the pipette. The shortening of L is very obvious; it decreases by about amicron for an increase in L of 50 to 100 microns. Although not visible on the monitor, thetether length can be obtained from a digital readout on the inchworm control panel; thevalue of 14 can be obtained similary. Values needed for this experiment are the suctionpressure (adjustable parameter), the pipette diameter (measured off the monitor), theextrusion speeds (adjustable parameter), and of course, the dependence of L on L.The range of speeds chosen here depends on the type of lipids involved: For all lipidsother than sphingomyelin (SOPC and MOPC — with and without cholesterol), speeds ofextrusion are taken up to 300 rim/s. Because sphingomyelin/cholesterol (sm/chol) hasa much higher interlayer drag, extrusion speeds are kept lower to avoid breaking of thetether; the speeds for pulling sm/chol tethers are kept below 80 sum/s. These speedranges are carried over to the following dynamic measurements.Dynamic measurementA discussion on this experiment is outlined in section 3.2.3. The experimental setupis as shown in figure 3.3. We want to measure the deflection of the transducer (andhence the tether force) for different rates of extrusion 14. Here, the transducer is aspirated into the stationary pipette while the test vesicle is held by the inchworm-controlledpipette. As before, the tension in the test vesicle is set at 0.1 dyn/crn. Tension in thetransducer, however, is set to a lower value: typically 0.03 dyn/cm. Pressure readingsof both manometers are recorded. During experiment, the camera is focussed on thetransducer bead while the test vesicle is pulled back. As shown in section B.4, the effects54of “conventional hydrodynamics” (due to the bilayer’s surface viscosity and the externalhydrodynamic drag) are not of importance. In particular, contribution of the bilayer’ssurface viscosity to the measured tether force is typically a few %; the dynamic tetherforce is due almost entirely to interlayer drag.As in the case for static measurements, a short tether of length ‘•‘-‘ 5 um is first pulledout. The force in this short tether is f0 as defined in section 3.2.3. Such a force is firstmeasured. From this position, the test vesicle is pulled back at different speeds whiledeflection of the transducer is recorded. A typical plot of such a process is shown infigure 3.4. Notice that during the pulling, there is a slight increase in the tether force,as predicted by the term proportional to L in equation (3.8). Because this is almost atthe resolution of our detection, such an effect cannot be quantified accurately. However,the dynamic force fd, as defined in section 3.2.3, is obtained easily from the recoil ofthe transducer once the inchworm motor is stopped. This value is recorded and will berelated to V in the analysis. The ranges of extrusion speeds V is the same as those usedin the static experiment.3.4 Data and DiscussionTypical plots of L vs. are shown in figure 3.5. Using equation (3.2), the tether radiusVt can be determined from the slopes of such graphs. From the linearity of these plots, itis concluded that the tether radius is uniform throughout the extrusion process. However,the possibility of small surface ripples on the nanotether cannot be ruled out. In such acase, this present method will register the average radius of such geometries.In all cases, the tether radius is independent of the extrusion rate to within 10 %. Thefundamental assumption of P << 1 is therefore justified (see section 3.2.1). For each typeof lipid (i.e., each row in table I), 5 to 7 vesicles are tested. Each vesicle is pulled at fourdifferent rates to check the dependence (or independence) of the tether radius Vj on theextrusion rate 4. With the measured values of rt, bending rigidities are calculated usingequation (3.4). Average values of the bilayer bending rigidity, quoted here as 2kcm, aresummarized in table I.55For the dynamic experiment, typical plots of fd vs. V are shown in figure 3.6. It isremarkable that the resulting velocity dependence is so linear. This supports the validity ofthe simple phenomenological model introduced in section 2.3.1. From the slopes of theseplots, the interlayer drag coefficient b (given here as b h2) can be calculated (equation(3.7)). Again, the results are summarized in table I, where the interlayer drag coefficientis given as bh2. As expected, cholesterol acts to increase interlayer drag. This is probablydue to cholesterol’s tendency to “straighten out” hydrocarbon chains, thus enhancingtheir interpenetrating ability into the opposite monolayer (sections 1.3.4 and 1.3.5). It isalso seen that cholesterol has larger effects on the interlayer drag of MOPC (an increase of75% in bh2) than it has on SOPC (an increase of 36% in bh2). The difference betweenthese two lipids lies in their chain length mismatch: An MOPC lipid consists of a 14-carbon chain and an 18-carbon chain, while SOPC has two 18-carbon chains. The factthat cholesterol has different effects on these two lipids can be rationalized by arguingthat stiffened chains of unequal lengths are more likely to interpenetrate due to packing(i.e., density) constraints.A surprising result shown in table I is the independence of the drag coefficient b h2 ontemperature for SOPC bilayers. We have not investigated temperature effects on othertypes of lipids. It is noted that the interior of the bilayer is basically a hydrocarbon oil.Such oils usually have viscosities that are very sensitive to temperature changes; typically,a two-fold decrease in viscosity results as the temperature increases from 15°C to 35° C.It is therefore difficult to rationalize our null result. This point will be further discussedin chapter 5.Assuming h is 4 nm, the interlayer drag coefficient for SOPC is 10 dyn.s/crn3.Valuesof b for other lipids can be calculated similarly. However, it must be remembered thatthe real experimental measurement is the quantity bh2.56SM/CHOL (1:1) 12.5Note: All tests done at 23°C unless stated otherwise.Each entry represents average of 5 to 6 vesicles.Maximum scatter from mean value: 15%5.5With regard to the initial area differenceA0, we have made estimates of this value on25 SOPC vesicles. It is noted that equation (3.6) can be written as (see (2.47)):KmhLAo —k 1/2— —— ( cmTves)2 A0 27rf° is measured as described in the previous section (in particular, see figure 3.4); themembrane tension Tves is a parameter we control through the suction pressure. Otherparameters in the above relation are assigned the following values: 2kcm 1.1 X 10—12 erg,Km = lOOdyn/cm and h = 4nm. Except in 4 cases, all values of /A0 are moreor less evenly distributed in the range —0.1 % to —1 %. The four remaining cases havepositive LAQ that are below 0.1 %. Thus, it appears that most bilayers are pre-loadedwith internal stresses. A negative value of iA0 means that the outer monolayer isstretched (from the stress free state) more than the inner monolayer. This suggests that,in the majority of the cases we examined, the stress free state of the bilayer is close tothe planar configuration.Table I: Summary of Results from Nanotether ExperimentLipid bh2 (106 dyn.s/cm) 2kcm (1012 erg)SOPC (15°C) 1.6 1.1SOPC (35°C) 1.75 1.1SOPC 1.75 1.1SOPC/CHOL 2.4 2.2MOPC 1.3 1.1MOPC/CHOL (1:1) 2.2 2.557Chapter 4Thermal Undulations of BilayerVesiclesIn this chapter, important effects of monolayer-coupling on the bilayer’s thermally excitedshape fluctuations are exposed. The dynamics of such a phenomenon has been consideredrecently for planar bilayers [Seifert and Langer, 1993], but a more complete treatment ofthe static and dynamic aspects for fluctuating vesicles has not appeared in the literature.Here, such an analysis is presented for the first time.From our measurements reported in the previous chapter, the bending rigidity of afluid bilayer is only 1 to 2 orders of magnitude larger than the ambient thermal energy kT(kT = 4.12 x i0—’ ergs at 298 K). Because of this flexibility, membrane conformationsare susceptible to thermal excitations, as evidenced from the “perpetual” undulations offlaccid bilayers seen under the light microscope. Microscopically, such motions are causedby the continual bombardments of molecules from the surrounding fluid at frequencies oftypically 1012 s [Chandrasekhar, 1943]. Collectively, this results in fluctuating forceswith correlation times that are much longer than the collision frequencies. However, itwill be assumed that the lifetimes of these forces are still much shorter than the relaxation times of membrane conformations; consequently, the fluctuating forces can only bedescribed statistically.Analyses of thermal undulations of bilayer vesicles based on the elastic model of Helfrich [1973] have been reported by several workers [Schneider et al., 1984; Milner andSafran, 1987; Faucon et al., 1989]. In all these works, the Hamiltonian has the formof a local curvature energy (see section 2.2.3). When perturbed from equilibrium, hilayer shapes recover at rates that are driven by the local bending rigidity and limited58by the viscosity of the suspending medium (water). Here, the widely neglected effectsof monolayer-coupling on such thermally driven motions are examined. For equilibriumcalculations, a Hamiltonian involving the sum of local and global contributions is used(section 2.2.5). With regard to dynamics, in addition to the conformational relaxationsas described, curvature effects generate fields of differential dilation (±) whose rates ofrecovery are driven by the membrane’s area elasticity and limited by interlayer drag. Thecoupling effect has been considered by Seifert and Langer [1993] for a planar bilayer. Here,we give theoretical predictions of the static (mean square amplitudes) and dynamic (time-dependent correlation function) features of fluctuating quasi-spherical vesicles based onthe formalism developed in chapter 2. Our calculations are extensions of the work bySchneider et al. [1984] and Mimer Safran [1987]. This analysis is not accompanied byexperimental work.Fluctuating bilayer vesicles have diameters of typically 2a 20gm. The constraintsof constant area and volume are again imposed. Here, because the membrane tensionis practically zero ( ‘ kcm /a2 lO6dyn/cm), these constraints are well justified. Assuch, it is important to allow for some amount of excess area in the vesicle 1 as a perfectsphere cannot undergo any shape changes. We will restrict our attention to quasi-sphericalvesicles that have only small amounts of excess area — typically a few percent. Theconformation of a quasi-sphere can be regarded, at any instant t, as the superpositionof a small normal displacement field 2 onto a spherical geometry. In spherical polarcoordinates, a quasi-sphere is described by the radial component r (r is not the quantitydefined in figure 2.1):r(,t) = a [1 + u(,t)] (4.1)where a is the radius of an imaginary sphere about which the vesicle undulates, and Qdenotes the polar angles 0 and qS. u(, t) is a non-dimensional field that characterizesthe normal displacement; for small excess area, it is much less than unity. Given u(1l, t),one can make expansions in terms of spherical harmonics with time-dependent amplitudesnm(t) as follows:u(fl,t) = unm(t)Ynm(fl) (4.2)n ,m1Excess area is the surplus of area the capsule has over that of a sphere of equivalent volume.2As shown in appendix E, tangential displacements have second order effects on shape perturbation.59With such an expansion, the mean square amplitude(U:m) (Unm(O)Unm(O)) (4.3)and the time autocorrelation functionCnm(t) (Unm(O) Unm(t)) (4.4)of each harmonic mode can be calculated. The notation ( ) denotes thermodynamicaverages. (U,m) and Cnm(t) are the equilibrium and dynamic properties of the randomsurface fluctuations; such quantities have in fact been constructed from vesicle imagesobserved using light microscopy. In the following, we will predict the mean square amplitudes and the correlation function of an undulating bilayer vesicle with interlayer effectstaken into account.4.1 Mean Square Amplitudes: StaticsThe approach here is to expand the free energy of deformation to second order in displacements. All first order terms representing the forces must vanish because the time-averagedshape (the sphere) corresponds to an equilibrium configuration. Equipartition of energyis then used to determine the mean square amplitude of each mode.The energetics of bilayer deformation are derived in section 2.2. As noted earlier,because it is energetically much more costly to stretch a bilayer membrane, the freeenergy associated with conformational changes is due entirely to curvature. Accountingfor stratification effects, we haveFcurv = Fiocat + F9106a (4.5)The general form of Fioaj is given by equation (2.38). For a bilayer that is symmetricabout the midplane, the spontaneous curvature vanishes. Also, because the integral ofthe gaussian curvature g over a closed surface is always a constant, we will not includesuch a term here. Equation (2.38) is therefore reduced toFiacai = (2kcm) J i5 dA (4.6)60where k denotes the bending rigidity of a monolayer. The global free energy is as givenby (2.46):Fglobal kA0 (* fJe dA —2(4.7)whereKh2 p LAO 4C— 2 ‘ — hA0.8Recall that /A0 is the initial area difference between the monolayers; this quantitymeasures the pre-loaded stress that is uniformly distributed within the bilayer.The curvature energy (equation (4.5)) must now be expanded in terms of the sphericalharmonic amplitudes Unm defined in (4.2). Leaving the calculations to appendix D, theresults are:Fiocat = l6kcm + kcm fl(fl+1)(fl+2)(fl1)71m (4.9)Fglobal = 87rk + k (+2)(fl1)m (4.10)n>1where k is the modified non-local bending rigidity defined as(1— aF)2(4.11)Since the curvature energy of a sphere is 8ir(2kcm + k), the excess free energy associatedwith deviations from spherical geometry is/Fcurv = Fiocai + Fg1obaz= (fl+2)(fl_1) {n(n+1)kcm + ] Un >1Following Mimer & Safran [1987], the area constraint is imposed by means of a Lagrangemultiplier 7 that fixes the surface area. The quantity 7 can alternatively be interpretedin two other ways: (a) From the viewpoint of mechanics, is the sum of the two lateraltensions in the top and bottom monolayers (see section 2.2.5); (b) when the excess areais variable, 7 can be considered the “chemical potential” that is conjugate to A. Thus,the shifted free energy is F’ Fcurv +7A; the amount of F’ in excess of the equilibriumvalue is= Fcurv + 7zA61Using the expression= (fl+2)(fl1)Um (4.12)n>1derived in appendix D, the expansion of tF’ is= (n +2)(n —1) [n(n + 1) 2kcm + 2k + 7a2] U:mn >1According to the equipartition theorem, every quadratic term in a Hamiltonian expressioncontributes an amount kT /2 to the energy. From the above expansion, we have thefollowing spectrum of mean square amplitudes for the undulatory modes:(Um)= (+2)(_1)[+1)2kcm + (2k/a2 +7) a2](4.13)A “compliant relation” between 7 and the excess area can also be obtained from(4.12) and (4.13)= a2 (2n+1)kT(4.14)2 n>i (+1)2kcm + (2/a2 + 7) a2where n corresponds to the microscopic cutoff; that is, the value a/ne should be aboutthe size of a constituent molecule.4.2 Correlation Function: DynamicsTo predict the dynamic features of an undulating bilayer vesicle, we start with the assumption that the correlation function, as defined in (4.4), decays according to the samedynamical laws that govern the relaxation of a non-fluctuating system [Landau and Lifshitz, 1993]. More precisely, we haveCnm(t) = (‘Un2m>fnm(t) (4.15)where fnm(t) represents the time dependence based on deterministic dynamics. Withoutloss of generality, we set fnm(O) = 1. As such, Cnm(0) is the mean square amplitude ofthe Ynm mode given by equation (4.13).To calculate fnm(t), we imagine a “zero temperature” situation in which there areno surface undulations. The bilayer vesicle is perturbed instantaneously at time t = 0to a shape away from equilibrium. At t = 0+, all perturbing forces are removed and62the vesicle is allowed to relax back to a sphere. In the absence of external forces,such a relaxation process is driven by the bilayer’s elasticity and its rate is limited bytwo dissipative mechanisms — the viscosity of water and interlayer drag. In terms of thedisplacement field, shape relaxation is characterized by the following expression:u(Q,i) = Unm(O)fnm(t) Ynm(fl) (4.16)n ,mHere, the same time dependence fnm(t) as in equation (4.15) is used. To solve for fnm(t),we must consider the problem of shell deformation in a viscous medium. In particular,kinematic and stress boundary conditions need be matched at the shell-fluid interface.Schneider et al. [1984] have solved the problem of an elastic bilayer vesicle suspendedin water. In their analysis, all dynamical variables are expanded to linear powers inthe displacement field u; what results is therefore the lowest order perturbative solutionabout the spherical shape. Here, we will extend the work of Schneider et al. to includemonolayer-coupling effects which give rise to non-local elasticity and interlayer dissipation.4.2.1 Equilibrium equations for the bilayerThe equations of equilibrium for a fluid shell are those given by (2.17); they are writtenhere in slightly more generalized forms to allow for non-axisymmetry:-- 2 112 ‘1= r c — V M — 2M— 9)] fl (4.17)— =Here, g is the gaussian curvature and is the unit normal to the surface. We will useand ó to denote the normal and tangential tractions on the membrane. These equilibriumequations can be simplified in the present problem as follows: The quantity (ë2— g) inthe first of equations (4.17) is of order 0(u2) and therefore can be neglected. With regardto the term (ëV8M— MVe) in the second equilibrium equation, it is shown in appendixF that, for an instantaneous initial condition, such a term vanishes identically. Thus, the3The equilibrium shape is a sphere of radius a. This presents an apparent contradiction since, giventhat the capsule has excess area, there is no spherical shape to return to! The solution to this dilemmais pointed out by Peterson [1985], in which he introduces the concept of the spherical limit. This is thelimit as the excess area vanishes. As a vesicle approaches the spherical limit, although the undulationamplitudes go to zero, the relaxation rates have well defined limiting values.63equilibrium equations simplify to= (e — V2M) (4.18)a-fs =4.2.2 Interlayer drag and bilayer relaxation functionThe moment resultant in the first of equations (4.18) contains the interlayer effects; generally, such a quantity can be written as (see equation (2.20))M = 2kcm + M (4.19)where M± has dynamic properties given by (2.70). For surface undulations, the tangential surface velocity v is negligible; we will therefore not include the convective terms.Neglecting also exchange of material across monolayers (c = 0), the dynamic equationfor M (equation (2.70)) becomes9M= DV82M + (4.20)In view of the mixed elastic and dissipative character of the bending moment, it is convenient to express it as a convolution integral. By convolving with the mean curvature,the bending moment isM(, t) = (t) e(, 0) + j (t — r) r) dr (4.21)1t(t) is the relaxation function of the bending moment M; physically, it is the transientmoment in response to a unit step change in mean curvature. (t) is obtained fromequations (4.19) and (4.20); as derived in appendix F, ii(t) has the form= 2k + i exp [_n(n + 1) D t /a2] (4.22)where the subscript n is included as a reminder of the wave number dependence. Thedecay rate varies as the square of the wave number; that is, inter-lamellar drag has strongereffects on modes with longer wavelengths.644.2.3 HydrodynamicsThe Reynold’s number that characterizes water flow around the vesicle is extremely low— typically of order 1O. As a result, all inertial effects can be neglected. The equationsof motion for such a situation are the “creeping motion” equations:= (4.23)0where i is the fluid viscosity and and p are respectively the velocity and pressurefields. Following Schneider et al. [1984], we will make use of the general solutions to thecreeping motion equations, and with boundary conditions tailored for slightly deformedspheres [Brenner, 1964].4.2.4 SolutionWe have discussed the mechanics of the bilayer and of the surrounding fluid separately.To obtain the deterministic time dependence fnm(t), both the kinematic (no slip) as wellas dynamic (force balance) conditions must be matched at the membrane-fluid interface.Leaving intermediate steps to appendix E, a general expression for fnm(t) is derived interms of an arbitrary relaxation function (t) that characterizes the viscoelastic properties of the shell. The result is given in the Laplace transformed space:Jo a3Z(n) n>2 (4.24)wn(n + 1)i2(w) + iiia3 Z(n) + To a2 ‘ —J(w) and it() are the Laplace transforms of f(t) and 1i(t) respectively. T0 is thezero order isotropic tension and the factor Z(n) isZn— (2n + 1)(2n2 + 2n — 1)(4 25)‘n(n+1)(n+2)(n—1)For large n, this is approximately 4/n. Equation (4.24) gives, to first order, the relaxationbehaviour of a general viscoelastic shell (characterized by the function1t(t)) submergedin a fluid of viscosity . For the bilayer, the transformed relaxation function, accordingto equation (4.22), is-= 2kcm+ (4.26)W W+L)D65where WD is the relaxation rate associated with the diffusivity of ; it is defined asn(n + 1) D/a2 (4.27)We also define here two other relaxation rates whose physical meanings will be discussedin the next section:n(n + 1)(2kcm + k) + r0a2(4.28)— n(n + 1) 2kcm + r0a2= ,7a3Z(4.29)The transformed relaxation function (equation (4.26)) is substituted into equation (4.24)and simplified. After an inverse transform, the final result isf(t)= 2 1[(wD—e1t— (WD— 2) e2t (4.30)The subscript m is dropped due to degeneracy. The two “bulk” rates are given by_____/_______— 1 i — i. —__ _ _ __\ 2 j v (wD+wC)2IWD+L)cN /_= 2\ 2 j V (D+w)Thus, we see that the shape recovery of a bilayer with interlayer drag involves two distincttime constants, namely j’ and f2’. These two time constants are in turn determinedby the three intrinsic recovery rates D, w and w. Note that the relation w < w,which follows directly from the definitions, implies4L?..’DWu< 1(WD + w) 2This in turn means that the time dependence, as given by (4.30), will never be oscillatory;as expected, shape recovery is always an overdamped process.4.3 DiscussionOur intention here is to examine the effects of monolayer coupling on the equilibriumundulations of bilayer vesicles. From the first part of our analysis, the spectrum of mean66square amplitudes has the form (cf. equation (4.13))2 kT(Um)rJ -2kn4 + (2k /a2 + r0)a2nIt is seen that the local and non-local bending rigidities affect the mean square amplitudesthrough different powers of the wave number n. More importantly, it is not possible todistinguish the mean tension -- from the non-local rigidity lc’ based on spectral analysis.Recall that ic, is scaled by a factor that reflects the initial area difference (equation (4.11))—( aZA022h A0)The quantity a /h in the above expression is of order io. For finite initial area differences(typically 0.1 %), this scaled non-local rigidity can be quite large. This may give theillusion of an anomalous membrane tension if non-local elasticity is not accounted for.In section 4.2, the role of interlayer drag in the equilibrium fluctuations of vesicles isexamined. Recovery dynamics of quasi-spherical bilayer vesicles have in fact been analysedby other workers, where the bilayer is treated as a single-layered structure with bendingrigidity kbj [Schneider et al., 1984; Milner and Safran, 1987]. In such situations, there isonly one recovery rate given byn(n + 1) kbl + r0a2Wsingleia3 ZThis recovery rate is driven by the bending elasticity and limited by the dissipation ofthe surrounding fluid. Because the intention here is to also account for inter-monolayercoupling, the above recovery rate will, in general, not be appropriate for describing thedynamics. However, there are two instances when two dynamically coupled monolayerscan be treated collectively as a single-layered structure: they are the limiting cases whenthe two layers are either held together rigidly or are completely free to slide past oneanother. In such cases, the corresponding bending rigidities areI 2kcm + icD for rigidly coupled monolayerskbl =2kcm for uncoupled monolayersFrom definitions (4.28) and (4.29), we see that w and w are precisely the recovery ratesin these two limits, corresponding respectively to the coupled and uncoupled situations.and w can be thought of as the decay rates of the mean curvature ë (in excessof the equilibrium value). Likewise, because WD is related to the inter-lamellar viscous67forces, it represents the decay rate of the differential dilation field c. It is natural tocompare the magnitudes of these three intrinsic rates. We first note that, assuming kand k are of the same order of magnitude, the ratio of w,. to will always be r 0(1).Next, we introduce the dimensionless parameterWDL — (4.31)wcwhich may be interpreted as the ratio of the following quantities:decay rate of a±L’Jdecay rate of ëWith the introduction of &, the notion of coupled and uncoupled monolayers can beregarded as limiting cases as this parameter approaches 0 and 00:lim f(t) = exp(—wt)lim f(t) = exp(—t)W-400In both cases, the time dependencies reduce to single decaying exponentials with theexpected time constants. By neglecting the mean tension, ‘ is given approximately by4i,aD(2kcm+ic)at short wavelengths (i.e., large wave numbers). As n increases, & approaches zero;this means the two monolayers will always appear rigidly coupled for sufficiently shortwavelengths.It is also interesting to see when the decay rates of c and ë are of equal magnitudes,which corresponds to a “cross-over” region between the extreme situations of coupled anduncoupled monolayers. By setting th to unity and using typical values of (2kcm + k)lOerg, i, 102 dyn.s/crn and D iO cm2/s, the cross-over length a/n is roughly0.1pm. Below this length (i.e., for larger n), decreases in magnitude, which impliesinterlayer drag is becoming more important.68Chapter 5SummaryThe theme of this thesis is on the important — but often neglected — effect of inter-monolayer coupling in a fluid bilayer structure. Monolayer coupling can further be catagorized according to its static and dynamic properties. While the static aspects have beenrecognized long ago [Evans, 1974; Evans, 1980], the dynamics of inter-monolayer coupling are discussed only recently in the literature [Evans et al., 1992; Seifert and Langer,1993]. In this thesis, a systematic development of the theory of monolayer coupling ispresented, with emphasis placed on the dynamics of such a phenomenon. The analysis isbased on principles of continuum mechanics. On the experimental side, I have developeda micromechanical technique for the measurement, for the first time, of the viscous dragbetween two unsupported surfactant monolayers. From this, the interlayer drag coefficientcan be determined.An introductory chapter is given on the molecular structure of the lipid bilayer. Emphasis is on (a) the weak coupling between monolayers which allows interlayer sliding tooccur; and (b) the highly dynamic character of the hydrocarbon chains in the hydrophobic interior. Of particular interest to us is the interpenetration of these chains into theopposite monolayer. Experimental evidences suggest large extents of interpenetration —up to 0.5 nm into a monolayer whose thickness is roughly 1.5 urn. Such processes occuron molecular time scales at typical frequencies of i0 s’ . On the time scale of our mechanical experiment (shear rates of 10 s’), the molecular collisions show up as a viscouseffect. More precisely, an interlayer drag is created as one monolayer slides past the otherone.The continuum theory of fluid surfactant membranes is developed in chapter 2, with69the coupling of monolayers accounted for. Formalisms are derived starting from ratherbasic principles. In particular, a dynamic equation (equation (2.69)) that describes thetransport of the coupling field (±) is derived.Interlayer drag is measured using the technique of micromanipulation. Such an investigation is described in detail in chapter 3. The particular experiment performed is calledthe nanotether experiment. This is a very versatile method that enables measurements bemade on the mesoscopic scale. Physical properties of molecularly thin surfactant membranes such as the bending rigidity, the non-local bending rigidity and the spontaneouscurvature can be measured by such a method. Here, the nanotether technique is usedto determine the interlayer drag within a bilayer structure. It is shown that the simplephenomenological model for interlayer drag (section 2.3) works very well for shear ratesup to, and probably beyond iO s—. Typical values of the drag coefficient are of orderlO7dyn . s/cm3 There has not been much work done on the measurement of interlayerviscous forces between fluid monolayers. The work by Merkel et aL[1989] on friction between a supported and a free monolayer has confirmed the order of magnitude of ourmeasurements. However, dependence of this drag coefficient on temperature is observed,which is contrary to our finding. The SFA (surface force apparatus) technique has alsobeen used to measure the dynamic drag between solid surfaces that are coated with surfactant monolayers [Yoshizawa et al., 1994]. In such cases, the values of b obtained appearto be 3 orders of magnitude larger than our measured value. Because such systems areusually under large normal loads, it is not clear if the above comparison is appropriate.Our anomalous result of b being independent of temperature can probably be rationalized by correlating interlayer drag to interpenetration. At high temperatures, althoughthe viscosity of the interior decreases, thermal motions become more vigorous. As a result,the hydrocarbon chains may penetrate deeper into the other half of the bilayer. The twoeffects may be compensatory, leading to a more or less constant value of b. Obviously,more modelling (such as molecular dynamics) needs be done in order to understand suchan effect.In chapter 4, our continuum model of the bilayer, including the effect of monolayercoupling, is applied to the equilibrium undulations of a bilayer vesicle. There are twointeresting findings: In the equilibrium spectrum of mean square amplitudes, the non—70local elasticity (reflecting static coupling) has the same wave number dependence as themembrane tension. This introduces another level of complexity to the experimental spectral analysis of bilayer shapes. In examining the dynamic recovery of coupled monolayersin a viscous medium, the relative importance of hydrodynamic dissipation and interlayerdissipation depends on the length scale at which the phenomenon occurs. 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Membrane Bending Energy and Shape Determinationof Phospholipid Vesicles and Red Blood Cells, European Biophysical Journal; 17:pp. 101-111.[68] Venable R.M., Zhang Y.H., Hardy B.J. and Pastor R.W. (1993). Molecular DynamicsSimulations of a Lipid Bilayer and of Hexadecane: An Investigation of MembraneFluidity, Science; 262: pp. 223-226.[69] Wennerström H. and Lindman B. (1979). Micelles, Physical Chemistry of SurfactantAssociation, Physical Reports; 52: pp. 1-86.[70] Wiener M.C. and White S.H. (1992). Structure of a Fluid DOPC Bilayer Determinedby Joint Refinement of X-ray and Neutron Diffraction Data, Biophysical Journal; 61:pp. 434-447.[71] Yeung A. and Evans E. (1994). Effects of Monolayer Coupling on the Thermal Undulations of Bilayer Vesicles, Journal de Physique; in preparation.[72] Yoshizawa H., Chen Y.L. and Israelachvili J. (1994). Fundamental Mechanisms ofInterfacial Friction I: Relation between Adhesion and Friction, Journal of PhysicalChemistry; in press.[73] Zaccai G., Büldt G., Seelig A. and Seelig J. (1979). Neutron Diffraction Studieson Phosphatidylcholine Model Membranes II: Chain Conformation and SegmentalDisorder, Journal of Molecular Biology; 134: pp. 693-706.76Appendix AFrom Virtual Work to EquilibriumEquationsIn this appendix, we will outline the algebraic steps that lead from the statement ofvirtual work to the equations of mechanical equilibrium for a thin shell. The virtual workof deformation is given by equation (2.8) in section 2.1.2:Sw JfdA [(Tm +CmMm) + (r+cM) + MmSCm + M6c]This equation can be written equivalently asSw = JJdA [Tm2 + + Mm(SO) + M°so] (A.1)wheredSO) = SCm+CmcosO60 =rIn terms of the virtual displacements Sx and 6x, virtual changes in the geometric quantities ared60 = Cm (6x) (A.2)6m/m = (6x) + Cm 6Xn (A.3)6A/= cos0+ c6x (A.4)SCm= c/Cm6— J2()— c,6x (A.5)Sc1k =— csO-(6x) — c6x (A.6)77We will evaluate equation (A.1) term by term. Using equation (A.3) and the fact thatdA = 27rr d.s, the first integral in (A.1) is6A SAmjJdA TmT 2irJ rrm—dsm Amd(6x)= 27rJ rrm +CmSXn dsdsIntegrating by parts, we getJJdA Tm = [2rrm6xt] + 2irJ[_-(rrm).6xt + rrmcm.Sxn] dsThe boundary terms are evaluated at N and S — the “north and south poles” of thecontour corresponding to r = 0. Note that this implies a closed surface. Because of thepresence of the radius r, the boundary terms in the above expression vanish; the resultingequation isJJdA TmE = fJdA[_’_(rrm).6xt + TmCm6Xnj (A.7)We now evaluate the second integral in (A.l). Using equation (A.4),fJdA = ffdA + rcxn) (A.8)Evaluation of the third term in (A.l) is slightly more complicated because it involves twointegrations by parts. Using equation (A.2) for 60, the first integration goes as follows:JfdA Mmd(60)= 27rJ rMmd(S0)ds= [2irrMm601 — 27rJ -(rMm) [mt—____Again, the boundary terms in the above expression vanish because of the factor r. Thesecond term in the integrand requires another integration by parts to recover Sx. However, the boundary terms from this integration do not vanish; the resulting expressionisffdA Mm’0 [2Mmcos0.Sxn1- JfdA [-(rMm).Sxt + j(rMm)6xn] (A.9)78Similarly, the fourth integral in (A.1) isffdA M°6O = 27rfMcosO6Ods= 2 J M COS 0 [m 8Xt — d(Sxn)] dsThe second term in the integrand containing the gradient of 6x needs be integrated; theresulting expression isJJdA MOSO60 = — [2Mcos0.6x]+ JJdA {Mcos0.6x + -(Mcos0).Sxfl] (A.1O)We now sum equations (A.7) to (A.1O) to obtain the virtual work. The two boundaryterms in (A.9) and (A.1O) cancel because Mm = Mçj, at r = 0; this is a consequence ofaxisymmetry. The virtual work of deformation isSW = JfdA {TmCm + TC — {(rMm) — M cosO]} Xn— {(rrm)— cosO+ Cm [(TMm) — M058]} .Sx (A.11)Equation (A.11) is then equated to the work done by the external forces; the latter isgiven bySWext = ffdA (pn SXn + PtThus, from Sw = SWeZt, we finally arrive at the equations of mechanical equilibrium fora thin shell:d dTmCm + T4C —jj-(rMm)—Mcos0id cos0 1 d cos0—Pt = ——(rrm) — r r+ Cm —--(rMm) — M4,rThese are equations (2.10) in section 2.1.2. Alternatively, they can be derived from thedirect balance of forces and moments on the shell [Evans and Skalak, 1980].79Appendix BAnalysis of Tether ExtrusionIn general, mechanical behaviour of the fluid bilayer are characterized by two “surfacefields”: The mean curvature and the differential area dilation a. is a more visiblefield that reflects bilayer conformations, while c accounts for the less obvious effectsdue to bilayer stratification. For the tether experiment, the overall vesicle shape is, to agood approximation, time-independent; consequently, there are no dynamics associatedwith e. On the other hand, although the a-j field is not visible, its dynamics is actuallyrather lively: The sharp bend at the vesicle-tether junction can be viewed as an intenseand localized “source” of a±; as the tether is pulled, this flux flows out from thesource and “diffuses” throughout the vesicle body. By neglecting the term correspondingto flip-flop in equation (2.69), the dynamics of this phenomenon is described by:2 (B.1)We will attempt to solve this equation on the entire vesicle surface, making necessarysimplificatons along the way.B.1 Dynamics of c in Vesicle-Tether JunctionB.1.1 Péclet number and order-of-magnitude argumentsAs shown in figure B. 1, the junction region that connects the spherical vesicle to the tetheris defined by end points a and b; these are the locations where the curvature gradientvanishes (spheres and cylinders have constant mean curvatures). Over this region, we80assume the a-i field is in steady state (or at least quasi-steady state) during tether pulling.If we further neglect flip-flop, the resulting dynamic equation is2v —-— — D V a± = hv -ç-(IS OS(convective) (diffusive) (source)We see that the field is created by convection of material through a curvature gradient.Introducing dimensionless variablesV:-_VtS ==the steady state equation becomesOa 1 2- O(rtë)V — —v ± VOs P Oswhere P is the Péclet number defined asP (B.2)For typical values of Tt 106cm, V 102cm/sec and D ‘-i 105cm2/sec (but ofcourse, we do not knowD yet), the Péclet number is of order iO.We will now show that, without making complicated calculations, important conclusions can be drawn from examining the magnitudes of various terms. Let z&± be thechange in &± over a distance of order rt; the dimensionless convective term will then beof order-v ‘Oswhile the diffusive term is of order1 2-___PWe can therefore neglect the convective term when P is small. The resulting equation is1 2- -___—Va=v (B.3)81Also, because the RHS of equation (B.3) is 0(1), it follows that z& P. Rememberingthat a, = &± h/rt, we arrive at the first important conclusion:(B.4)Since h/rt is typically of order 0.1 and we have shown that P is ‘•‘-‘ 10, the change ina over the junction is roughly 0.01%.Let us also compare the gradients of the two surface fields over the junction region. It iseasy to show that9ë 1 1Oc P— — and — —Os r? h OsIt follows that the gradients of the non-local and local bending moments have relativemagnitudes(OM±)(2kcm) =(2kcm) ‘ P (B.5)provided k/2 kcm “ 0(1).B.1.2 Integrating the low Péclet number equationThe low Péclet number equation is1 0 / 9a’\____D——jr————i+v =0rOs\ Os) OsFor incompressible membrane flow, the product rv is constant. The velocity field istherefore give by v = rV/r. With this relation, the above equation becomes+ 1r14O(he) — 0rOs\ Os) r D Os —This impliesr — + hP ë = constantOsNote both terms on LHS are of order P h/rt.Because of the above invariant relation, it is not necessary to know the exact geometryfrom a to b. Using this relation, we getra (- rt + P (B.6)\OSJa rt82B.1.3 KinematicsAssume convergent flow (r v = constant) for each individual monolayer. At b, Voutv, 4; we therefore can write— (rt + h/2) V — (rt — h/2) Vvout— , yin—r0At a, = r ra, hence= (B.7)From the equilibrium of tangential forces on the monolayers, we have the following relation(see equation (2.61)):D=v (B.8)Combining equations (B.7) and (B.8) gives(ôc±’\ hV h(B.9)\USJa .1) r,It follows from equation (B.6) that()b(B.1O)If we use the analogy of heat flow and consider c the “temperature”, then equation(B.1O) suggests that, during tether pulling, the cross sectional area at b is “insulated”;the field that is created at the sharp bend cannot enter the tether region and thereforemust diffuse onto the vesicle.B.2 Balance of ForcesEquilibrium equations for a fluid membrane are given by (2.17). It is more convenient,however, to define here a zero order stress resultant such that (c.f. equation(2.58))Ju(z) dz = —In terms of this quantity, the equilibrium equations becomed0 dM0 = — + e— (B.11)d.s ds+ 2Mcm — V2M83where we have set the tangential traction to zero and p = p is the pressure differenceacross the bilayer membrane. The bending moment can be written quite generally asM 2kcm + M = 2kcm + k0±/hB.2.1 Vesicle regionOn the vesicle surface, all moment terms can be neglected because the curvature is small;i.e., Tve$ >> k/R2. The equilibrium equations ared0 -— = 0 T0 Tuesds- - 2Tvesp = 0= DILVB.2.2 Vesicle-tether junctionAssuming the Péclet number is much less than unity, equation (B.5) leads to the conditiondM dc<< 2kcmds d.sThe tangential force balance in (B.l1) becomesd0 dc0 = — + 2kcmds dsd1 2=— + kedThe quantity o + kcm ë2 is an invariant. At point a, I Tves. Hence, at any otherlocation,To T — kcm 2Again, without knowing the geometry of this region, we can write(o)a (B.12)— cm(To)b Tves —B.2.3 Entering tether regionAt point b, Cm = 0 and c = 1/ri. Also, the termV2M vanishes because the gradientsof both ë and are zero (equation (B.10)). The normal force balance is, from (B.11)p =84Using the invariant relation from (B.12) and the expression p = 2 TVS/RV,2Tves ( kcm”\ 1— ITves —R rtJTtkern — Tves(1 2rtr? — rt ‘s. RNeglecting terms of order rt/R, the final expression isrt (k/Tves)”2 (B.13)which is equation (3.4).As shown in figure B.2, the condition of axial force balance at point b isft + rp = 2rcrt Tmwhere Tm 1S, in general, equal to To + Cc,Mm (see equation (2.6)). Here,Tm =.j;0 + c,M/ kcm’\ M + 2kcm/rt1Tves”1 +rj rjkern M= Tves + +The axial force balance therefore becomesft + rr2(2Tv:s)27rrt (Tves + +ft = 2Tr Tves (i—+ + 27rMcm2rr Tves + 2T— + 2M±rtSubstituting in T€5 kcm/r?, we finally getft = 2/Cern+ M (B.14)27r TtNote that all quantities in equations (B.13) and (B.14) are evaluated at b.B.3 Dynamics of a on SphereHere, the general dynamic equation is solved on the vesicle surface. We will ignore themembrane projection into the suction pipette; the vesicle is treated as a complete sphereexcept for a small opening of radius ra at the vesicle-tether junction. The sphere hasradius R that is much larger than ra.85B.3.1 Simplifying dynamic equation for aWith both the time and space derivatives of ë vanishing on the sphere, (B.1) becomes2—--— + v—--— = DV a±at asThe velocity v is again given by the convergent flow relation v = rt 14/r. We can thereforewrite2 ulöa±—=D Va—Pi-——rãswhere F is the Péclet number defined in (B.2). By arguing that/2 ‘löa±’(Va±).(-) 0(1)\r as jwe have, for small F, an expression that resembles the heat equation:2= DV a. (B.15)B.3.2 Boundary value problem and general solutionOne obvious solution to (B.15) is a± = —h I’, where F0 is a constant representing theinitial area difference between the monolayers (see (2.47)). This term will be added on atthe end.For a sphere that is “heated” by a localized source of radius Ta with boundary condition= x at r = Ta, the general solution isa(8,t) xg(O,t)where g(O, t) is some response function specific to the truncated sphere (for example, seesection 3.4 of Jackson [1975]).When x = X(t) is time dependent, since (B.15) is linear, we can write a(8, t) as aconvolution integral as followsa(O, t) = x(O) g(O, t) + j g(O, t — r) (r) dr (B.16)It is convenient to use Laplace transform in dealing with the above ODE. We first definethe “hat” notation as follows:J(w) £ [f(t)] = j e f(t) dt86The transform of equation (B.16) is then&(O,w) w (w) (O,w)while the gradient of this expression represents the flux boundary condition at a (in thew-space):()a’ () (B.17)Given that [Hall, 1949]eoTa ln(2Rv/ra)whereD2R ln(2Rv/ra)we can write its transform as1 1Ia — ra ln(2Rv/ra) W+Woand therefore boundary condition (B.17) becomes— LL)(LL) 1ô.s Ia — Ta ln(2Rv/ra) L1.)+WoFinally, we substitute in the transform of equation (B.9) and obtain a relation betweenand:h (2R LU+w0xQ’) = lnjb—)Notice that the dependence on ra has almost dropped out, except for the argument inthe logarithmic term. Because our analysis has relied on invariant relations, we do not(and need not) know the precise value of ra. In the above relation, ra can justifiably bereplaced by Tt in view of the weak logarithmic dependence. However, one can be a littlemore precise by guessing the following relation:ra 2rtWith such sophisticated modification, the above equation becomes(w) = ln()W+Wf4(w) (B.18)We can now put in different forms of w) and solve for (w) (and hence X(t)), or viceversa. This will be illustrated by the following two examples.87B.3.3 Constant rate of extrusionWe will derive here equation (3.5). Transform of the velocity is= constant = i(w) =Putting this expression into equation (B.18) and evaluating the inverse transform, theresult ishV /RV\ hVx(t) = —ln!--—1 + —tD \rtl 2RThe above expression can be used to evaluate the tether force using equation (B.14). Suchan approach is valid because the value of at a is practically the same as that at b(see equation (B.4)).Recall that, in general, the coupling moment can be written as M± = k ±/h. At b,M± has the form -M = (x — hF0)with h F0 being the initial area difference between the monolayers. Using (B.14), the finalexpression for the tether force is= 2kcm— kF0 + bh2ln(3L) V + (B.19)rt Tt 2Rwhere the relations- Kmh2 Km2 ;L=Vtare used.B.3.4 Constant tether forceThis situation is not directly related to our experiment; it is nevertheless included herefor completeness. Under constant tether force, we have the simple boundary condition ofX Xo = constant. The value Xo can be evaluated as follows: Substituting the relations2kcm—1/2‘cm Tves )M = (—hF0)88into (B.14), we haveXo=— 2(kcmTves)” +Here, the transform of x is simply xo/w. Putting this into equation (B.18), the solutionisV(t) XoDthln(R/rt)whereD= 2R ln(R/rt)B.4 Other Dissipative EffectsUp to now, only inter-monolayer viscous dissipation has been discussed. In addition tothis, more “conventional” hydrodynamic stresses can arise from the viscous propertiesof the surrounding water and the bilayer membrane; the latter gives rise to deviatorictensions that resist the rate of change of in-plane shearing [Evans and Skalak, 1980].Although nanotether extrusion represents a very severe case of in-plane shearing (squaresurface elements on the vesicle are mapped into highly elongated rectangular segmentson the tether), it is shown that even in such a situation, the conventional dissipativemechanisms are negligible in comparison to interlayer effects. Let us first consider thetension field in a spherical vesicle. In the absence of motion, the membrane tension isuniform. Deviations from this condition can be due to two transport processes; they are(a) flow of bulk fluid around the vesicle, and (b) flow of membrane material from thevesicle surface onto the tether. By neglecting all bending moment terms, the equation ofmechanical equilibrium in the tangential direction (second of equations (2.13)) readsTm T3 r——-+=—Pt (B.20)For a spherical object moving at velocity V in a fluid medium, the surface tangentialtraction is [Landau and Lifshitz, 1982]3Pt = —-— sinOVt (B.21)where is the fluid viscosity. The membrane deviatoric tension T3 is given by theconstitutive relation(dv vdr(?lmh) j —\as ras89where the product ‘7mh is the surface viscosity of the bilayer; 7m is the effective shearviscosity of the bilayer interior with units of dyn s/cm2 and h is the bilayer thickness.Assuming convergent flow such that v = rt Vt/r, the deviatoric tension isTt V drT = 2iimhr asand equation(B.20) becomesdTm = (377w sinO + 4’7mhrt cos2O)It follows from the above equation that the change in membrane tension due to conventional hydrodynamics is/Tm (377w — 27lm) (B.22)Typically, 10 dyn.s/cm2, 17m 1 dyn.s/cm2 and h/rt ‘—‘ 0.1. For extrusion ratesof 1001um/s (10—2 cm/.s), deviations in the membrane tension are of order iO dyn/cm,which is two orders of magnitude smaller than typical tension levels of 0.1 dyn/cm.It is also important to consider the relative contributions to the dynamic tether forcefrom interlayer drag and from surface hydrodynamics. From equation (3.7), the dynamictether force isfd(b) = 27rbh21n()The tether force due to surface hydrodynamics is [Hochmuth and Evans, 1982] isfc(71rn) 477mhVtThe ratio of these two quantities is2 77mbh 1n(R/rt)Typically, the product b Ii is 4 dyn . s/cm and the logarithmic term is around 6. Itfollows that surface hydrodynamics of the bilayer contributes to less than 10 % of thetotal dynamic tether force.90Appendix CForce Transducer: Analysis andCalibrationA transducer designed to measure microdyne forces is described in chapter 3. In thisappendix, we will outline the numerical procedure for calculating the stiffness of such atransducer. The assembly is again composed of an aspirated fluid bilayer vesicle withan attached bead. Such an arrangement is shown in figure C.1 ; relevant parametersare also defined in the figure. In this application, the bilayer is assumed to be an idealmembrane: Because the radii of curvature involved are orders of magnitude larger thanthe membrane thickness, all bending moments can be neglected. We will also impose theconstraints of constant area and volume; these are very good approximations because thebilayer is always under low tensions.Ci Force BalanceFigure C.1(b) illustrates the transducer under a load ft. Also shown in the drawing arethe pressures in the three regions. Since we are only interested in the differences betweenthese pressures, their absolute values are not important. We defineiPF0-F, ; pP-P0where LP is the pipette suction pressure and p the normal stress across the membrane.As will be shown, the “pipette angle” O, is an important parameter in determining mechanical equilibrium.Let us now consider equilibrium conditions on the membrane segment external to the91pipette. Neglecting all moment contributions, the equilibrium equations for a fluid membrane (equations (2.17) with Pt = 0 and Pm = p) are= r r = constantdsp = = 3 = p/r = constantThus, the external segment has uniform distributions of tension and mean curvature.Figures C.1(c) and C.1(d) show two separate segments of the vesicle and the equivalentforces acting on them (i.e., the free body diagrams); forces must be balanced on eachimaginary segment. At the pipette entrance, the membrane makes an abrupt bend tobecome a cylindrical segment. We will assume the membrane tension is continuous aroundthis bend — similar to the tension in a rope as it curves around a pulley. Thus, the sametension r is shown in the two sketches. In figure C.1(c), the equilibrium condition isirR2(F—F) = 27rRT= P—P, = (C.1)In figure C.1(d), the condition isirR (F, — F0) + ft = 27rRr sinO= F1 — F0 = sinO — (C.2)Combining equations (C.1) and (C.2), the pipette suction pressure can be written as= (F, — F) — (F, — F0)2r. ft= (1 — sinO) +orZFR -\r =. 1—ft) (C.3)2(1 — sin8)whereR2zF(C.4)For f << 1, the vesicle is approximately spherical; it follows that sin O Rp /R0. Insuch a case, the membrane tension isPRT2(1—R/R0)92Substituting (C.3) back into (C.2), and noting that p = — F0, we haveZ1FsinO / -• 1 — ft/sinO) (C.5)— sinRecall from above that the external membrane segment has uniform mean curvature givenby the ratio of p to r. From (C.5) and (C.3), the mean curvature is2sinO, (1_ t/sinO(C.6)R \ 1—ft IC.2 Numerical SolutionAlthough we have derived the conditions of equilibrium, it is difficult to obtain an analytical expression for the transducer stiffness. In this section, we resort to numerical means topredict the vesicle’s response to given loads. Using the above equations, the entire vesiclegeometry can be calculated. Conditions of constant area and volume are also accountedfor. In these regards, this present method is “exact”.The quantities that uniquely define the transducer are (R0 , R ,R) for the geometryand (ft, zF) for the forces; R0 is the vesicle radius at zero load. From these, thedimensionless groups necessary for our analysis are R0 /R, R /R and ft (defined in(C.4)). For actual calculations, we will implicitly let R = 1 so that all lengths aremeasured in units of “pipette radii”.The initial volume and area of the external segment (i.e., at ft = 0) areV0 =K [(1+z)2(2_z) + (1+z)22—z —4] (C.7)A0 = 27rR(z+z) (C.8)where= [i— (R/R0)2]; z = {i — (R/R0)2]In general, the mean curvature of an axisymmetric shape is- dO sinOc= —+——-ds rwhere 0 is the angle between the surface normal and the symmetry axis, and r is theradial distance from the symmetry axis to the meridian (see figure 2.1). The value , asgiven by equation (C.6), is uniform over the vesicle surface. When combined with (C.6),93the above relation can be integrated to obtain the vesicle shape. This is done as follows:Let s be the curvilinear distance along the meridian with s = 0 at the pipette entrance.The following variables are integrated along s, starting at s = 0, with the indicated initialvalues:0(0) = 0, ; dO/ds = ë — sin0/rr(O) = ; dr/ds = cosOz(0) = 0 ; dz/ds = sin0A(0) = 0 ; dA/ds = 2irrV(0) 0 ; dV/ds irr2 sinOThis is a set of simultaneous ODE’s that should be easily handled by the method ofRunge-Kutta. However, we are faced with the difficulty of not knowing the end point;i.e., the value of .s at the bead contact. For this purpose, a special Runge-Kutta routineis written that integrates to, instead of an end point, an end condition. Here, the endcondition is r = R with dr /d.s <0.The above integration is done based on a particular value of ë which in turn dependson j and 0, (see equation (C.6)). ] is an input parameter; the value of O, on theother hand, must be chosen to satisfy area and volume requirements. The procedure isas follows: The material in the pipette can be considered a reservoir of area and volume.For every Or,, values of the volume V and area A of the external segment will be obtainedfrom the above integration. In general, these are larger than their respective originalvalues V0 and A0 (equations (C.7) and (C.8)) because extra material is being drawn outof the pipette. The corresponding changes in projection length L (figure C.l) areZLV =7rR2irRThe conservation of area and volume requires that LL1 = This is satisfied byiterating on O, until the functionzL - L\L94is zero. The corresponding value of z is then used to calculate the transducer deflection6.C.3 Results and CalibrationTo calibrate the transducer, we need an independent measure of the applied axial force. Asimple method is shown in figure C.2. A human (the author’s) red blood cell is completelyaspirated into a suction pipette while still attached to a “transducer bead”. We candetermine the suction force on the red cell from the simple relation(prbc”\ A DJrbc — \.LGp ) rbcFor the above relation to apply, the red cell has to form a good seal in the orifice. Smallparticles are placed inside the red cell pipette during the experiment. The fact that theparticles are stationary implies there is no leak around the red cell.For this calibration procedure, two micropipettes are set up as shown in figure C.2. Keeping the pipettes stationary, /Prb is incremented in steps and the transducer deflection is recorded. A superposition of the theoretical curve and the actual data points isshown in figure C.3. Here, in order to facilitate comparison, experimental data are nondimensionalized by R and /F (pipette radius and suction pressure of the transducer)as indicated. This plot supports the validity of the analysis presented in this appendix.1The latex bead is designed to stick to both the vesicle and the red cell. This is accomplished byattaching two types of antibodies — those that are bilayer-specific and those that are red cell-specific — tothe bead surface.95Appendix DVesicle Undulations: EquilibriumCalculationsIn this appendix, we expand the curvature elastic energy in terms of normal displacementsabout the spherical shape. Because the spherical shape is at mechanical equilibrium, alllinear terms in such an expansion (representing the forces) vanish. Consequently, thelowest order expansion will necessarily be quadratic. This analysis is an extension of thework by Mimer and Safran [1987].Geometric descriptions will be done in spherical coordinates (r, 0, ), with (, ,) beingunit vectors in the respective directions. The distorted shape is parametrized by r(0, )as follows:r(0,) = a [1 + u(0,)]For quasi-spheres, u/a << 1. The surface is defined by the scalar functionr - a [1 + u(0,)]The unit normal vector i is given by==(r —— ) (1 + 2e2)_1l’2 (D.1)where: (1 + uY’ au 2 2() + Sfl20(O) 0(u)96Taking the divergence of the unit normal, we obtain the mean curvature of the surface:2 1 0. 1 Oflq5(r r) +r sine(sinOrio) +r sinO OqEquation (D.1) is substituted into the above expression and, after some algebra, thesecond order expansion of ë is= [i + (u—u) + (u2_2uu)] (D.2)whereü2 1 8/. 8u\ 1 82uVu _s1n8) +sin2Oand the expansion= (1 + u)’ 1 — u + u2is used.In general, for the parametrization r = r(O, q), an elemental surface has area given by= d r [r2 +2+sin2 () 211/2where df = sin 0 dO dq. Here,dA = dr2 (i + 2E2)h/2Again, to second order in u, it can be shown thatdA = da2 [i + 2u + (u2 +12)](D.3)An elemental volume is in general given bydV = (r.nidANoting that i= r, we get—1/2= r (i + 22)97Using expression (D.3) for dA, the differential volume is written asdV = d! (i + 3u + 3u2) (D.4)The differentials ëdA and ë2dA can now be evaluated from the above expressions.They are, to quadratic order,ëdA = 2ad [i + (u + u) +(12)]e2 dA = 4 d [1 + 2ü + (ü2 — 2uü + 2)]Using the identityfE2dc = Juüd1which can be shown by integration by parts, the above expressions are integrated:JedA 2a fd [1 + (u+u) + uü] (D.5)= 4 fd [i + 2 + (u2 — (D.6)Also,A= J dA a2 Jd [1 + 2u + (u2 + uü)] (D.7)V= f dV = fd (i + 3u + 3u2) (D.8)We now expand the normal displacement into orthogonal modes:= UnmYnm()n ,m= ZIG + ttnmYnm(2)n >0with denoting (0, S) and u0 u00Y. From the definition of U, it follows thatn(n+1)=2‘Unm YnmTh >0Equations (D.5) to (D.8) can now be expanded into modal components as follows:J e dA = 8a(1 + u0) + a n(n + 1) Ufl2m (D.9)n>098J2dA = 16 + fl(fl+1)(fl+2)(1)1m (D.1O)n>1A = fdA = 4a2(1+uo) + a2 [i +n(n+1)]Um (h1)n>0V= f dV (i + 3u0 + 3u) + a3 (D.12)ri >0The volume constraints is imposed at this point. From V = 4ira3/3, we geto = 4iruo(1+uo) +n>0This is a quadratic equation for u0. To the lowest order, that is, to second order in u,2= Un>0Substituting back into equations (D.9) to (D.11), the resulting expressions arefdA = 8a + (+2)(_1)m]fe2dA = 16 + fl(fl+1)(fl+2)(71)m (unchanged)Elastic energies of deformation can now be written easily from the above expressions asFiocai = (2kcm) f 2 dA= 87t(2kcm) + kcm fl(fl+l)(fl+2)(fll)Um (D.13)n>1Fgio&ai = A0(fc1A— )2= 8rk, + k (7+2)()1m (D.14)n>1where(aF)2(D.15)and the relation of area conservation A = A0 is used.99Appendix EVesicle Undulations: DeterministicDynamicsThe aim here is to obtain the response behaviour of a slightly perturbed sphere. Such aprocess is driven by curvature elastic forces; the recovery rate is limited by the bilayer’sinternal dissipation (interlayer drag) and the surrounding hydrodynamic effects. Theelastic restoring forces arise from the vesicle’s departure from spherical geometry. Tolowest order, these forces are expanded to first powers in the perturbative displacements;they will then be balanced against the viscous stresses. This analysis is an extension ofthe work by Schneider et al. [1984].E.1 Bilayer ShellE.1.1 GeometryBilayer surface is described by(O,q;t) = ae,. + a(O,q;t) ; << 1We will scale all lengths by a (a 1); hence,(E.1)and is written, quite generally, as= uë + + ,8sinO (E.2)u, a and 3 are all functions of (0, ; t) and of order e. We must now use methods ofdifferential geometry to evaluate quantities such as unit normal vector and curvatures (agood reference is chapter 1 of Green and Zerna [1968]).100Covariant base vectors are tangent to the surface and oriented along the coordinate curves;they areOr80 ‘Using the relations9er -.= e ; = 6r =0e= —(sin0 + cos0e)= sin0 = cos0 ;we get, to first order in ,IOu=— a) + (i + u + + (13 cosO + sin o) g/OaiOu=— 4) sin20) + — /3 sin9 cosO) e813 a+sinO(i+u++t9From these, the unit normal can be calculated:=(0x)/1 Ou\Ou \= er+(a_)eo+(/3sin0__)eThe surface metric tensor is simply = ä. To first order,8a= 1 + 2u + 2Oa 84)a8çf, = +sin2083 a\1a = sin2 0 [i + 2 ( + +tan)jand its determinant isOa adet = sin2 0 [1 + 2 (2u + +tan 0 ++ o (2)In general, an elemental surface has area dA =z1/2d0d. To ensure incompressibility,we must have sin2 0 dO dq = 1/2 dO dq. From this, it follows thatOa a 0/3(E.3)101is the condition for area conservation. Here, is the usual three dimensional gradientoperator.Contravariant form of the metric tensor is the inverse of a; that is,a66 = a/z ; a = ae9/ ; a6 = a6 = — ao17,//To first order,a66 = 1 — 2 ( +— ( 1 0 0/9— sin200+801 1 0/3___asin21 — 2 + +tan 0Next, we need to evaluate the curvature tensor b3. Its components are given bybe=n.ã_ ;After some straightforward algebra, we get/ On 02u= —1+u+2—— 02u 1 Ou Oa 1 03— OOOq tan 0 84 Oq sin2 0 002 / 1 Ou 1 02u 2o 0/9=— sin 0 1 + U— tan000 — sin20+tan0+ 2And finally, the local mean curvature is = — with repeated indices summed. Itis easy to show that=2—Vu—2u (E.4)Note that the tangential displacements c and /3 drop out of this first order expansion.They will, however, appear in higher order terms.E.1.2 KinematicsLet V (0, q) be the velocity field on the vesicle surface:(0,q5) =Ou 0o 0/3.= + + -b-- sinOe102Velocity of the bulk fluid adjacent to the membrane must equal (no-slip condition).The three components of V can therefore be velocity boundary conditions for the hydrodynamic problem. However, it is more convenient, for reasons that will become clear, tochoose a slightly different set of boundary conditions as follows:.T7=(E.5)== 0it. ( x =- [ (sin2 . = aThe first condition is simply the normal velocity component. The second is a measureof surface dilatation, while the last condition represents in-plane shearing. Note that thetangential components a and /3 only appear in the last condition.E.1.3 ForcesEquilibrium equations for the bilayer are, from equations (4.18)Un = (e_v2M)it= —where ö and ó are the normal and tangential traction (force/unit area) on the membrane surface. is the isotropic tension field. The moment resultant M embodies theelastic restoring forces and the bilayer’s internal dissipation due to interlayer drag. Itsviscoelastic behaviour is expressed, quite generally, in terms of a relaxation function (t)(equation (4.21)):M(, t) = j(t) (, 0) + j (t — t’) t’) dt’ (E.6)The total traction is simply= o + = (e_v2M)it_ ‘The quantities f, M, it and can be written as sums of a lead term plus a first orderperturbation:= ro+r ; r/r0<<1103M = m0 + m ; m/m0 < 1 c = 2 - V2u - 2u • u < 1 r0 represents a time-averaged tension. To first order in the perturbations, the surface traction vector is a ~ 2r0 + 2r - r0 ( v 2 u + 2v) - V + 2r0 2m L • a l -du\ ^ er fin \ - V r These components of a will become stress boundary conditions for the adjacent fluid. Again, we choose to represent these boundary conditions in a manner similar to above. v After gome algebra, one arrives at the following: n a = 2r0 + 2t - r0 (V2u + 2u) - V 2 m (E.7) * V • a = 2r0 c — V 2 r + 4r — 2r0 (V2u + 2u,) - 2 V2 72m In arriving at the second boundary condition (divergence of <r), equations (E.3) and (E.4) are used. Here, we see again that the tangential displacements a and (3 only appear in the'last equation. In general, boundary conditions that involve in-plane motions can be separated out (analogous to the separation of variables in solving PDE's) and will be ignored in the present analysis. E.1.4 Modal expansion We will proceed now to expand the boundary conditions into harmonic modes. Let u (n , t ) = u n n y n m ( f l ) / n m ( t ) ; /nm(0) = 1 . (E.8) n > 1 ' where / n m ( 0 is an undetermined time dependence; it is our goal to solve for this function. The above sum starts from n — 2 since the n = 0 mode violates volume constraint and n = 1 modes correspond to rigid body translations. We also let the perturbative tension be r(n,'0 = £ rnmVnm(Q)fntn(t) (E.9) n > 1 104 with the same time dependence. From equation (E.4) v 2 c ( n , o = - £ •n(n + i)(n + 2 ) ( n - i ) u n m y l i m ( n ) / n m ( < ) ; n > 1 The Laplacian of equation (E.6) can also be expanded in modal components as follows: / . v2M(n,t) = v 2 m ( n , t ) = fi(t)v2c(n,o) + j \ { t - t ' ) ^ [ v 2 c ( n , < ' ) ] dt' = - E n(n + l)(n + 2 ) ( n - l ) u „ m ^ n m ( 0 V „ m ( n ) ; (E.10) n > 1 where 0 - m ( O = M 0 / n m ( 0 ) + j ^ ( t - t ' ) ^ ( i ' ) d t ' (E . l l ) Using these relations, the first two kinematic boundary conditions (equations (E.5)) are n -V = £ u n m Y n m ^ - (E.12) n > l V-V = 0 and from equations (E.8), (E.9) and (E.10), the first two stress boundary conditions (equations (E.7)) are 1 n • a = 2r0 + V { 2rnm fnm i n > 1 + [r0 (n + 2)(n - l ) / n m + n(n + l)(n + 2)(n — l)firnm] } Km (E.13) V • a = 2t0 C + Y , { + n + 4) rnm fnm n > 1 + u n m [2r0(n + 2 ) ( n - l ) / „ m + 2n(n + l)(n + 2)(n - l)gn m 1 } Km E.2 Hydrodynamics The low Reynold's number velocity field is given by •qV'v = V p V • v = . 0 (incompressibility) where rj is the shear viscosity , v the velocity field, and p the pressure.; The general solution to these equations is given by La.mb [llappel and Brenner, 1973] in terms of three independent sets of solid spherical harmonics with coefficients pnm , if>ftm and Xhm > Based on this general solution, Brenner [1964] has formulated kinematic and stress boundary 105 conditions on a spherical surface. By letting v' be velocity field inside a unit sphere, the boundary conditions on its surface are n n-v' = £ n > 1 V-tT- = 2(2 n +J5)r] Pnm + Ynm(n)fnm(t) n > 1 n(n + 1) ,• . , ' , . ,, . (E.14) / Ynm(H) fnm(t) n • ( v x u = function of Xnm only Note that, again, the same time dependence is used. Corresponding boundary conditions for fluid outside the sphere are obtained by substituting—(n + 1 ) for n, p°m forpJ,m and for r n m . ; " Let Pr' be the stress vector on a surface of the interior fluid whose normal is in the r direction. The associated boundary conditions on the unit sphere are [Brenner 4 1964] n-p: (n2 — n — 3) n > 1 (2n-+ 3)77 « ' ( V x P r ' ) = function of Xnm only p'„m + 2 n ( n - l ) C , (n3 + 2n + 6) , ft ' i n 2 ,, ^ } P,nm + 2n(n — 1) ^ni K m ( n ) / n m ( 0 (E.15) Ynm(n)fnm{t) Here, p' denotes the hydrostatic pressure inside the vesicle. Corresponding conditions for fluid outside the vesicle are obtained by substituting —(n+1) for n , p°m for pj,m , t/>°m for xj}xnm and p° for p ' . The coefficients Xnm represent spheroidal harmonics associated with tangential motions on the vesicle surface and can therefore be ignored for our purposes.' They are, however, necessary for the complete solution of the flow field. E.3 Matching Boundary Conditions E.3.1 Laplace transform It is convenient to transform the convoluted time dependencies into frequency spacte, By letting C be the Laplace transform operator and using the "hat" notation (*) to denote a transformed variable, we have £ [ / n m ( 0 1 = /hm(«) 106 £[*!(«)] = m and from equation (E. 11), C[gnm(t)} = £(u;)/n m(0) + /i(w) [u)/n m(w) - / n m (0 ) ] We now transform boundary conditions on the shell. In w-space, the two kinematic conditions (equations (E.12)) are C (n • v) = £ u»m (ui - / J ) K„m(n) /„.»(«) ; (E.17) n > l £ ( V - K) = 0 • ^ Similarly, using equation (E.16), the transformed stress boundary conditions .(equations (E.13)) are /. • C ( r i - a ) = £ (2 r 0 ) + £ { 2 r n m n >1 + Unm [ r 0 (n + 2 ) ( n - 1 ) + wp. n{n + l)(n + 2)(n - 1)] } Ynm{U) / n m (w) (E.18) £ (V • A) = C(2T0C) + £ { (n2 + n + 4)r„m n > 1 + U n m [2r0(n + 2 ) ( n - l ) + w £ 2n{n + l)(n + 2)(n - 1) ] } Ynm{fl) fnm{u) i E.3.2 Matching kinematic boundary conditions: No-slip No-slip condition at the shell-interior fluid interface is expressed by the following equa-tions: ' C(n-V) = C{nv{) c ( v - v ) = C (V • t/'-) Equating (E.17) to the transforms of (E.14) leads to the following two equations: C J 2(2n + 3)t/ P ™ + = U n m ~ ' » ? ) 2(2 n + 3)77 The solutions are „(2n + 3 ) ( , , - l ) (E.19) = ^ tinm (w - /„-*) E.3.3 Matching stress boundary conditions: Force balance Substituting equations (E.19) into the transforms of (E,15), we ob conditions on the interior fluid with no-slip condition implicitly sati £ (n • PJ) = - £ (p') + r, £ ( n " 1 } f n + 3 ) -(« - fa) ain stress? boundary ifled to first order: n >1 £(V-Pr') = -C(p'c) + V E 3("+2Kn ^ - /n«7 «nmr-«(fl() n > l " 1 Similar stress boundary conditions are obtained for the exterior lluid by repla.cing n by — (n + 1): £ (n • Pr°) = - £ (p°) - r? J ] ( n + " 1 } - / J ) U n m y L ( p ) /n m(u,) n>l £ (V . p;) = -C(p°c) - v E 3(n n+(? + 2) - p ) / » » Defining the quantities APT = P* — P° and Ap = pl — p°, we have • £ ( n - A P r ) = - £ ( A p ) (2ra + l)(2n2 + 2n - 3) n(n + 1) £ ( V ' A P r ) = - £ ( A p c ) + f E n >1 (W " j ^ j "nra y„m(ft) (E.20) + 37/ E n > 1 ( n - l ) ( n + 2)(2n + l) , _ k n(n + 1) \ ^rim fnmi. u) Finally, the equilibrium condition at the shell-fluid interface is <7 + A PT = 0 In the transformed space, this condition is £ (n • S) + £ (n • A P r ) = 0 C(V-c) + S £ ( V - A P r ) = 0 Expressions from (E.18) and (E.20) are substituted into the above equations. Recognizing that Ap — 2t0 (on unit sphere), we have the following set of homogeneous equations for each mode: en r n m + c12 u n m = 0 cji r n m + C22 u n m = 0 108 where \ c n - 2 ... c« = t0 (n + 2)(n — 1) + u> ft n(n + l)(n + 2)(n — 1) . (2n + l)(2n2 + 2 n - 3 ) , _ . _ t ) + V n(n + l ) V J n m > c2i = n2 + n + 4 c22 = 2r0 (n + 2)(n - 1) + 2n(n + l)(n + 2)(n - 1) ( n - l ) ( n , + 2)(2n + l) / . + , .v (w - ; n m i n(n + 1 ) A ' For non-trivial solution, the determinant |c,j| is set to zero. It follows from this that /»m(w) = t I n v ^ j_ ( E - 2 1 ) ut]Z + n(n + l)u) fi{u)) + r0 where _ (2n + l)(2n2 + 2 n - l ) ' . Z ( n ) - n(n + l)(n + 2 ) ( » - l ) . . ^ 109 Appendix F Relaxation JPunction for the Bending Moment The linear relaxation function characteristic of the quasi-spherical bilayer vesicle is derived here. For a general viscoelastic shell, the bending moment can be expressed as the time convolution of a relaxation function n(t) with the mean curvature c(tt,t) : M ( n , 0 = / i ( t ) c ( f t , 0 ) -+ j f i i ( t - T ) ^ { S l , T ) d r . (F. l ) For a bilayer, the bending moment can be written quite generally as M = 2kcm c + kca± / h , •• (F.2) As discussed in section 4.2.2, the dynamic equation for a± (or equivalently, M±) is t (p-3) We now expand the fields a± and c in spherical harmonic modes: c(n,t) = £ cnmYnm(n) fnm(t) (F.4) n,m ' = <xnmYnm(a)ij>nrn(t) (F.5) n,m where the time dependent functions are as yet undetermined except for the initial condi-tions /nm(0) = 1 ; 0„™(0) 1 110 The bilayer is perturbed instantaneously at i = 0 , giving rise to some initial curvature field c ( 0 , 0 ) . The resulting ct± field is 1 a±(Q,0) = / i c ( n , 0 ) From the instantaneous condition, it follows that a n m = hcnm . Substituting into (F.5), we have , -a ± ( n , t ) = £ / i ^ y ^ ^ V n m C O r (F.6) n,m It is obvious from (F.6) that c V sa± — a± Vs c = 0 i It then follows from (F.2) and the above expression that c VSM - M Vj c = 0 Substituting (F.6) and (F.4) into (F.3), we arrive at an equation relating / n m ( i ) to 0„m(O'; # » » , rc(n + l ) £ _ dfnm ^ • + fl2 V™ ~ d t Using Laplace transformation, the above differential equation is converted into an alge-braic expression: where the hat notation denotes Laplace transform. The total bending moment (equation (F.2)) can now be written in the transformed space 6 as M(Sl,u) = 2fc e m c(n ,w) + kca±(n,u)/h = E W [cnmi«m(n) /nm(w)] ^ cm w u> + n(n + 1 )D/a2 (P.9) 1 Note: There is controversy over the appropriate initial conditions for the a± . field. It has been argued that, in accordance to the states described by the curvature energy (4.5), the starting conditions for the ' a± field should be * - "|(".o) a±(fi,0) =' 0(u3) » 0 However, it is shown that the time constants of the decaying modes are unaffected by th« particular initial conditions [Yeung and Evans, in preparation]. ' , . k1 * ^Aiiife 111 Equation (F. l ) can also be transformed as follows: M(n,u) = /t(u>)c (f2,0) + A H [wc(ft,w) - c (« ,0) ] -= u c (ft, w)fi(u)) ' (F.10) Comparing (F.9) with (F.10), we see that every modal component of M is a convolution of the corresponding component of c with a relaxation function whose Laplace transform is ' • '"'•'• - ' ' 2k » / \ cm' , Mnm[U>) = h kc. u u + n(n + l ) D / a 2 The inverse transform of /tnm(w) is fin(t) = 2kcm + kc exp — n(n + l) Dt/a 112 1 nm Polar Head Group Backbone Hydrocarbon Chains Figure 1.1: Molecular structure of a phospholipid molecule. Space filling model of a PC lipid molecule. Different regions within the molecule and typical dimensions are also illustrated. Here, the double bond is shown to cause a distinct bend in the unsaturated chain, while the saturated chain is in the all trans state. In reality, the two hydrocarbon chains are highly dynamic structures. 113 1 nm r Polar OH Group Rigid Ring Structur Short Hydrocarbon Chain Figure 1.2: Molecular structure of a cholesterol molecule. Space filling model of a cholesterol molecule. Different regions within the molecule and typical dimensions are also illustrated. Only the short hydrocar-bon tail has conformational freedom. Overall structure of cholesterol is much more rigid than phospholipids. 114 ( a ) 15 -V(4>) ( kT ) 1 0 " 5 -- 1 8 0 - 1 2 0 - 6 0 60 120 180 ( b ) <f> Figure 1.3: Torsional potential between two CHj groups. (a) Plot of torsional potential V(<f>) as a function of the rotational angle; <f> = 0 corresponds to the trans state. Energy is given here in terms of kT, where T is room temperature (298 K). Thus, 1 JIT = 0.59 kcal/mole. J (b) Schematic illustrations of the trans and the two gauche states. Note that the lowest energy trans state is the one where all four carbon atoms are co-planar. 115 Phase Molecular Shape Micelle n m m u m u i Bilayer > > h 2 O < Inverted Micelle Inverted Cone 0 » Cylinder a Cone Figure 1.4: Molecular shapes and condensed structures. Different molecular shapes and the resulting structures formed in aqueous environments. Molecular "shapes" represent the average space occupied by; the molecules. 116 2 6 10 14 Labeled Carbon Atom 4 Figure 1.5: Molecular order profiles. Typical order profiles for various lipids: Closed circles, DPPC; closed triangles, POPC; closed squares, DPPS. In general, orientationalorder decreases towards the centre of the bilayer. Reproduced from Seelig and Browning [1978]. 117 c Figure 2.1: Axisymmetric geometry. An axisymmetric shape is generated when the meridional curve, with curvi-linear coordinate s , is rotated about the symmetry axis £ . The coordinate z is oriented normal to the surface. 118 / / <v X / . 7 - / Figure 2.2: Stresses on a curved shell segment. The two principal stresses crm and cr^  are illustrated here. In general, they are functions of position. For clarity, the radius of curvature in the meridional direction (c" 1) is not shown. 119 Figure 2.3: Defining the origins in the thickness direction. Origins of the variables z , zu and z; are as illustrated here according to definitions (2.18). The bilayer thickness is denoted by H . 120 crs = b(v+ - v . ) Figure 2.4: Simple model for interlayer drag. Phenomenological model for interlayer viscous stresses that result from relative motions between the monolayers. 121 Figure 3.1: The nanotether experiment. Shown here schematically is the nanotether experiment as described in section 3.1. The nanotether is pulled at rate Vt and the tether tension is / , . 122 Figure 3.2: Experimental setup. Photograph of the microscope station where the nanotether experiment is per-formed. Some of the instruments are: (1) inverted microscope; (2) .camera; (3) inchworm motor; (4) inchworm power supply and control panel; (5) pres-sure controlling manometer; (6) position controller (joystick); (7) mercury arc lamp. 123 Figure 3.3: Photograph of nanotether experiment. Photograph of an actual experiment taken at a magnification (400 x ) that is lower than the "operating value" in order to show all relevant features. Although not visible, a nanotether of diameter ~ 40 nm is present between the bead and the vesicle on the right. During experiment, this vesicle is pulled: back at controlled rates. 124 f z (pdyn) . p o - r o r . I I 0 1 I I O-O-O - r -nonlocal I elasticity -rO f V U V I " o~~-o~ -o—o*-o— fo 0 -0.2 J L -I 1 I I 0.0 0.2 0.4 0.6 0.8 1.0 . 1 .2 ( f i m / s e c ) 220 0 L L——I—I—:—I 1 I I -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 time (sec) Figure 3.4: Tether pulling sequence. , \ ' • - - : • " ' . An SOPC vesicle is being pulled by a step change in velocity Vt. The force response (top) and extrusion speed (bottom) are plotted on a common time scale. Note that a threshold level of force exists prior to the dynamic extrusion; this initial force is set by the membrane tension and the bilayer's bending stiff-ness. Extrusion at constant rate produces a small elastic force that increases slowly in time plus a large dynamic force that is proportional to the rate of extrusion. . -125 = Figure 3.5: Determination of tether radiuii; V K % i Plotting the pipette projection length Lp vs. - the ftether length Lt for two „ different lipids at fixed membrane-tension^ (rbughlyO. 1 dynjcm). The tether t radius qan b$ determined fro^t^.sb^tof^a^h^lotsXfqvubtfon (3.2)). j The 1 linearity suggests that the tether radius is constant throughout the extrusion process; 126 Figure 3.6: Dynamic tether force vs. extrusion rate, The dynamic tether force fd, as illustrated in figure 3.4, is plotted against the extrusion rate for two types of lipids. From the slopes of these plots, the inerlayer drag coefficient b can be determined (equation (3.7)). 127 Figure B. l : Magnified view of the vesicle-tether junction. The vesicle-tether junction region is enclosed between points a and b as shown; these are the points where the membrane assumes spherical and cylindrical geometries, respectively. Membrane material flows along this fixed contour and has velocity Vt beyond point 6. 128 2 r, Figure B.2: Axial force balance on nanotether. ! Shown here are the various contributions to the net axial force on a nanotether. 129 Figure C.l: Mechanical analysis on force transducer. (a) and (b): Deformation of the force transducer from original configuration, (c) and (d): Free body diagrams of segments of the transducer, with the applied forces as shown. 130 (a) / / / / / / 7 7 APr 2Rp b o be (b) Figure C.2: Calibration of force transducer. (a) Schematic illustration of the calibration procedures. The transducer is subjected to'a known force created by the red cell "piston".; this force is simply the suction pressure times the cross sectional area. (b) Photograph of the actual calibration. Because of the high magnification and the type of suspending solution used, the vesicle can barely be seen. 131 -I — l 0 . 5 0 . 4 - - 0 . 4 0 . 3 -f t 0 . 3 0.2 •frbc irR*AP (•) V o. i- 0 .1 0 . 0 - 0 . 0 Figure C.3: Experimental and theoretical transducer characteristics Comparing the theoretical force-displacement curve ( / , vs. 8/FL) to experi-mental data. To facilitate comparison, frbc is non-dimensionaliz Jd as shown 132 

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