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Mechanicsof Inter-Monolayer Coupling in Fluid Surfactant Bilayers
by Anthony K.C. Yeung
B.A.Sc., University of British Columbia, 1983 M.A.Sc., University of British Columbia, 1987
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
THE FACULTY OF GRADUATE STUDIES
(Department of Physics)
We accept this thesis as conformin to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
June 1994
©Anthony K.C. Yeung, 1994 DE-6
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Fluid Abstract useful a effects conformational bilayer fluid measurements membranes. surfactant the as to spin-off monolayers 1 by to membranes. theoretical of an This one dissipation This as normal vesicle’s bilayer 100 application a another technique, from bilayers stratified thesis nanometers) A dynamics stresses. is stratification of Brownian continuum such basis then due inter-monolayer — examines The are giving of composite the which a measured for the composed Except phenomenological technique of are surrounding shape the present rise bilayer theory is such (monolayer-coupling) governed work developed for to structure. undulations experimentally of an “hidden” continuum viscous is viscous membranes for reported two the added hydrodynamics by 11 monolayer-coupling mono-molecular determination by the resistance, stresses coefficient Throughout degrees myself level are model, inter-monolayer here on by predicted of mesoscopic in are and a for of novel the complexity unsupported the of freedom very plays this for of this dynamic static monolayers sheets bilayer method present future with important. is only thesis, first length viscous and within that the to a (viscous) bending analyses bilayers. called developed, investigation, dynamic secondary the it membrane are scales are forces; the is For study weakly free shown nanotether composite rigidities. example, on (roughly coupling Further, features in to role. which of strat such slide that con held en bi 2 1 Contents 2.1 Mechanics Introduction Acknowledgements List 1.2 List 1.3 1.1 Table Abstract Equations 2.1.1 Structure Structure 1.3.4 1.3.3 1.3.2 1.3.1 1.2.1 Overview of of 1.3.6 1.3.5 1.2.3 1.2.2 of Figures Tables Contents The Interdigitation Disorder Motional The Molecular Hydrophobic Effects Components Stresses, of of of of Monolayer of four gel the Bilayers Amphiphilic Mechanical of and strains inside Thesis major freedom cholesterol motions fluid of effect the phospholipids phospholipids Coupling and phases Equilibrium of Molecules and fluid and hydrocarbon axisymmetry time condensed bilayer of 111 bilayers scales and chains interfacial cholesterol structures viii vii 17 20 20 10 13 11 15 14 iii x ii 1 8 8 5 5 3 7 6 2.1.2 Virtual work and the equilibrium equations 22
2.1.3 Equilibrium equations for a fluid membrane 25
2.1.4 Dividing one fluid membrane into two 27
2.2 Free Energy of Deformation 29 2.2.1 Elastic model for liquid crystalline membranes 29 2.2.2 Pure stretch 30
2.2.3 Pure bending and local curvature energy 31
2.2.4 Coupled layers and global curvature energy 33 2.2.5 Euler-Lagrange equations and mechanical equilibrium 35 2.3 Interlayer Viscosity 38 2.3.1 Phenomenological model and balance of tangential forces 38 2.3.2 Kinematics 39 2.3.3 Dynamic equation for a 41
3 Measurement of Interlayer Drag 42 3.1 General Description of Nanotether Extrusion 43
3.2 Summary of Equations and Experimental Strategy . 45 3.2.1 Two important equations 45
3.2.2 Experiment 1: Static measurement 46
3.2.3 Experiment 2: Dynamic measurement 47 3.3 Methods 48 3.3.1 Vesicles, solutions and beads 48
3.3.2 Micropipette setup 50 3.3.3 Experimental procedures 53 3.4 Data and Discussion 55
4 Thermal Undulations of Bilayer Vesicles 58
4.1 Mean Square Amplitudes: Statics 60 4.2 Correlation Function: Dynamics 62
4.2.1 Equilibrium equations for the bilayer 63
4.2.2 Interlayer drag and bilayer relaxation function . 64
4.2.3 Hydrodynamics 65
iv 4.2.4 Solution . 65
4.3 Discussion 66
5 Summary 69
Bibliography 72
APPENDICES
A From Virtual Work to Equilibrium Equations 77
B Analysis of Tether Extrusion 80 B.1 Dynamics of n in Vesicle-Tether Junction 80 B.1 .1 Péclet number and order-of-magnitude arguments 80
B.1.2 Integrating the low Péclet number equation 82
B.1.3 Kinematics 83
B.2 Balance of Forces 83 B.2.1 Vesicle region 84
B.2.2 Vesicle-tether junction 84
B.2.3 Entering tether region 84
B.3 Dynamics of on Sphere 85 B.3.1 Simplifying dynamic equation for c 86 B.3.2 Boundary value problem and general solution 86 B.3.3 Constant rate of extrusion 88 B.3.4 Constant tether force 88
B.4 Other Dissipative Effects 89
C Force Transducer: Analysis and Calibration 91
C.1 Force Balance 91
C.2 Numerical Solution 93
C.3 Results and Calibration 95
D Vesicle Undulations: Equilibrium Calculations 96
v E Vesicle Undulations: Deterministic Dynamics 100
E.1 Bilayer Shell •• . . 100
E.1.1 Geometry 100
E.1.2 Kinematics • . . . 102
E.1.3 Forces 103
E.1.4 Modal expansion 104
E.2 Hydrodynamics 105
E.3 Matching Boundary Conditions • • .. 106
E.3.1 Laplace transform • . . 106
E.3.2 Matching kinematic boundary conditions: No-slip • .. . 107
E.3.3 Matching stress boundary conditions: Force balance • . . 108
F Relaxation Function for the Bending Moment 110
vi List of Tables
I Summary of Results from Nanotether Experiment 57
vii
3.6 3.3
B.1 3.4
B.2 3.5 3.2 2.3
2.4
3.1 2.2
C.1 2.1
1.5
1.4
1.3
1.2
1.1
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129 126 125 120
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C.3 Experimental and theoretical transducer characteristics 132
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I I Acknowledgements also would would Wieslawa keeping being extend have and also like Ed the me research seriously comic emotional Auld my Rawicz, to like nicest company thank comic sincere to relief and person thank problem doubted who relief Evan Myer support gratitude of in has the a the and more and Evans, Bloom. and given the lab members throughout for lamb to for during completion innocent the me all my They his chops; the the liquid research of guidance off-hours; help have people x nature my this of nitrogen; to when supervisory been ordeal: this Susan supervisor, in and to the thesis; I genuine Ken most support. and Tha, to lab Ritchie, David the committee: to for needed for without scholars Hans-Günter list their introducing Knowles, who goes it; technical whose to and provides on Professors Andrew gentlemen. for Döbereiner, assurance me and, providing candies, to Leung, more Boye this I energetics properties forces” bility: susceptible relevance, the ones by surface munity ses and Being can ture condensed der The
1986;
1987; (discussed Introduction Chapter a of Waals configurational extend phase consists lipid clearly lateral Faucon Evans surface a Due and of undulations two bilayer the condensed [Helfrich, transitions attraction of later material omolecular to to to “fluctuation-induced visible tension and of a dimensional et undulations unique its macroscopic two liquid al., in Rawicz, is “thinness”, this under mono-molecular entropy that the 1989]. 1973]. occur rconfined or — matter properties crystal
[Silver, 1 resulting chapter) fundamental forms (thermal) the 1990]. have condensed More over The dimensions of physics — light the 1985; the spontaneously which been fluid a of recently, in in are In repulsions” bilayer wide microscope. the system, agitations. a a Gennis, surfactant addition, introduced reported [Leibler, membrane molecularly component material, narrow has bilayer (exceeding spectrum is considerable led a giving 1 1989]. when macroscopic gap equilibrium to 1989; as have [Schneider layers For Such the is into the surfactant of thin [Helfrich, of rise also In a the 10 stimulated bilayer all Bloom that surface flaccid “liquid an wavelengths, three composite im). work to biological undulations unique aqueous et are the statistical body dimensions, exhibits et molecules has al., 1978; membrane, undulations crystal Iii weakly notions al., for much addition been whose 1984; environment. (‘—i cell its 1991; Lipowsky with are approach” 4 a held and interest devoted extraordinary membranes. Milner nm) of of rich conformations either the thermally-driven Lipowsky, the the tend to “entropy-driven dynamic together whose set 2-D its macroscopic and right and in to suppressed to to biological of Its the fluid studying increase Leibler, Safran, surface bilayer phases shapes by analy It 1991]. struc flexi com van is has are a shape 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 of them 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. Held together by non-covalent forces, the monolayers are only prevented from separating in the transverse direction. Laterally, they are free to slide relative to one another; such motions are resisted only by dynamic (viscous) forces as the two “hydrocarbon brushes” (inner parts of the monolayers) undergo relative motions. The effect of stratification was recognized many years ago with introduction of the non-local curvature energy for bilay ers that form closed surfaces [Evans, 1974]. As will be shown in chapter 2, the non-local energy is a subtle contribution that complements the widely adopted Helfrich Hamilto han [Heifrich, 1973]; the two forms of curvature energy are of the same magnitude and therefore must be treated on equal footing. Through careful observation of shape trans formation sequences in fluid bilayer vesicles, it is becoming evident that the Hamiltonian based on the unit-membrane assumption cannot by itself account for all of the confor mational characteristics. It is then suggested that inclusion of the non-local term, which reflects monolayer coupling, is necessary for explaining the experimental findings [Seifert et al., 1992]. While elastic aspects of coupling give rise to the non-local curvature en ergy, dynamic (viscous) coupling between monolayers can be important in determining the time-dependent features of shape changes. As will be shown, interlayer drag is the dissipative mechanism that dominates over hydrodynamics on mesoscopic length scales (below 0.1 tim). Thus, coupling of monolayers in the bilayer membrane introduces a new level of complexity to the physics of condensed fluid interfaces. To address this issue, both static 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 monolayer coupling within a fluid bilayer. Such an approach neglects individual properties of the constituent molecules. Instead, the physical system is described by continuous fields such as density, velocity and stress; these variables can be considered local averages of many microscopic systems (see discussion at beginning of chapter 2). The continuum or macroscopic description complements other microscopic approaches: Based only on a few
2 constitutive parameters such as the elastic moduli, a continuum model can be used to predict collective behaviours of the physical system over very large length scales and very long times (relative to molecular scales). An example of such applications is the correct prediction 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 shear elastic modulus. However, the continuum approach offers no insight into why the shear modulus should vanish. Important characteristics associated with the membrane’s fluid state such as the “melting” of hydrocarbon chains and the sudden increase in lipid mobility are molecular features. Such microscopic information can be obtained, for example, from molecular dynamics simulations or from experiments such as NMR spectroscopy and X-ray diffraction. It would be hopeless, however, to model Brownian undulations over long distances using molecular dynamics calculations. In short, both microscopic and continuum approaches are essential as they complement each other in obtaining a more complete understanding of the physical phenomenon in question; this thesis concentrates on the latter in dealing with the mechanism of monolayer coupling.
1.1 Overview of the Thesis
In chapter 2, the continuum mechanical theory for fluid bilayers is developed starting from first principles. In particular, both the static and dynamic aspects of monolayer coupling are examined. It is generally true that mechanical theories for elastic bodies can be for mulated either from an energetic approach or from the direct balance of forces; we show the equivalence of the two formalisms in the present application to surfactant interfaces. A continuum description necessarily requires constitutive relations. It is shown in chapter 2 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 the isotropic stress within the bilayer. In addition, constitutive parameters describing static and dynamic coupling are introduced; they are the non-local bending rigidity Ic and the interfacial drag coefficient b. Our continuum model is considered “complete” if both these parameters are known. kc,is well characterized based on previous area compressibility measurements [Evans and Needham, 1987]. On the other hand, the interfacial drag coef ficient b is not known accurately — neither in magnitude nor in the way it changes with
3 other physical parameters such as temperature and chain structure. This leads logically to the next chapter.
Chapter 3 deals with the experimental work that constitutes a major portion of this thesis. It describes our measurement of interlayer viscous forces as a function of shear rates, from which the drag coefficient b is deduced. Bilayer test samples come in the form of vesicles (closed bags) with diameters typically around 20 ,um. Experiments on such vesicles are done with micromechanical techniques; i.e., the techniques of manipulat ing test samples using small suction pipettes. For this particular application, the use of micromanipulation involves pulling cylindrical bilayer tubes (diameters 50 nm) out of spherical 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 can simply be related to the tension force in the tether, with the latter on the order of micro dynes. Thus, to get at the interlayer viscous forces, one needs an ultra-sensitive force transducer that is capable of measuring tether tensions accurately. A convenient way of using micromechanical techniques to construct such a force transducer is described. Fur ther, a useful spin-off from the nanotether experiment is a method of measuring bending rigidities of molecularly thin surfactant layers. The interlayer drag coefficient is moni tored as a function of three parameters: temperature, difference in length between the two hydrocarbon chains of the surfactant molecules, and the incorporation of cholesterol into the bilayer. These features will be further discussed in this introductory chapter.
With the mechanical formalism of monolayer coupling developed in chapter 2 and the interlayer drag coefficient b measured in chapter 3, we have obtained a continuum description of the stratified fluid bilayer. As an application of this mechanical model, the role of monolayer coupling in the thermal undulations of flaccid bilayer vesicles will be examined. Chapter 4 presents such a theoretical analysis. The calculations are divided into two parts; they are (a) equilibrium statistics (spectrum of mean square amplitudes) based on the equipartition theorem, and (b) more complicated deterministic dynamics that require knowledge of the surrounding low Reynold’s number flow field. Similar calculations 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 coupling 1Microdyne (10 piconewton):Weightof 910gm; it is the typical force of a van derWaals“bond”. 4 effect can play a significant role in vesicle undulations. Finally, a general discussion and summary of this work is given in chapter 5. Before going into the continuum mechanical formalism, the remainder of this chapter will be spent on an introduction to the molecular structure of bilayer membranes. The discussion is very superficial and does not contain any new results. However, it does provide the non-biochemist reader (and the writer) a better understanding of the physical system under study.
1.2 Structure of Amphiphilic Molecules
Surfactants are more generally known as amphiphilic molecules or amphiphiles. Such molecules are characterized by their distinct hydrophobic (non-polar) and hydrophilic (polar) intramolecular regions. The two types of amphiphiles we discuss here are phos pholipids and cholesterol, which are the most abundant amphiphiles found in biological cell membranes. Because of the large number of possible variations in phospholipid struc ture, we will start by looking at its components.
1.2.1 Components of phospholipids
The molecular structure of a phospholipid can be represented by a simple sketch as shown in figure 1.1: Every phospholipid is composed of a polar head group that is connected to a stiff backbone structure at the mid-section. The backbone is in turn linked to two non polar hydrocarbon chains that have, in most cases, 12 to 24 carbon atoms. A typical phospholipid molecule has a length of 2 to 3 nm (with chains extended) and molecular weight of order 1000 Daltons. Hydrocarbon chains, also known as fatty acids, are series of 2CH groups linked by C—Csingle bonds. Occasionally, two carbon atoms are connected by C=C double bonds that are almost always in the cis configuration [Stryer, 1981]. One end of the chain is terminated by an unreactive 3CH group while the other contains a carboxyl group that is capable of forming covalent bonding (ester or amide) with other compounds. By convention, carbon atoms are numbered beginning with the one adjacent to the carboxyl group. Whereas carbon atoms connected by single bonds are free to rotate about the bond axis, such freedom is not allowed in C=C bonds. As a result, double bonds create
5 rigid kinks in a hydrocarbon chain that is otherwise much more flexible; these double bonds have very strong influence on a bilayer’s physical properties. Fatty acids with only single bonds are said to be saturated (with hydrogen). Generally, a fatty acid is identified by its length, the number of double bonds, and the locations of the double bonds. Some common 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 situated at 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, a phospholipid is formed by “plugging” two non-polar hydrocarbon chains and one polar head group into this socket. Another backbone structure is the sphingosine; it is a “two hole 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 fatty acid chain. The connection made with the fatty acid is an amide linkage. Chemical structures 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 be reproduced here. The three most common polar head groups are: phosphocholine (PC), phosphoetha nolamine (PE) and phosphoserine (PS). They are composed of a phosphate group (hence the term “phospholipid”) that is linked to another small hydrophilic compound — which can be choline, ethanolamine, or serine. The linkage is through a covalent bond known as phosphate 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 can be found elsewhere and will not be given here.
1.2.2 The four major phospholipids and cholesterol
Three of the four major phospholipids are formed from the glycerol backbone: The middle attachment 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 a polar head group. These three types of phospholipids are classified according to the head
6 group structure; they are the PC, PE and PS lipids. In addition, composition of the two fatty acids are specified in ascending “sn order” (sn-i followed by sn-2). Thus, a PC phospholipid with a stearic acid in the sn-i position and an oleic acid in sn-2 position is called an “SOP C” (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 are linked to a PC head group and a fatty acid chain. This second fatty acid is usually much longer than the 12-carbon chain (typically 20 carbons). Variation in structure of the sphingomyelin is due only to the composition of this fatty acid. The other type of amphiphilic molecules we introduce here is cholesterol. As shown in figure 1.2, the major portion of the molecule is a rigid planar ring structure. Attached to 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 region in the cholesterol molecule. Overall length of cholesterol is slightly less than 2 nm with the short hydrocarbon tail extended. Unlike phospholipids, there is no variation in the structure of cholesterol.
1.2.3 Motional freedom of hydrocarbon chains
Since 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 phos pholipids are known to have restricted internal motions [Seelig and Seelig, 1980]. By far the most exotic dynamics within phospholipids are the continual conformational changes in the hydrocarbon chains. Such motions are possible because azimuthal rotations about the C—Cbonds are allowed; for a free chain having 12 or more C—Cbonds, this can result in a large number of possible chain conformations. However, as illustrated in figure 1.3(a), not every angular position about the C—Cbond is equally comfortable for the 2CR group: 0 represents the trans (t) configuration as illustrated in figure 1.3(b); this energy minimum 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 min ima, which are called the gauche configurations (g+ and gj. The gauche states have potentials of approximately one kT (T = 298K) higher than that of the trans state. In
7 order to get to these states, the 2CR group needs to overcome a barrier of -‘ 6 kT. Thus, at any instant, rotational angles corresponding to the trans or gauche states are strongly favoured over the neighbouring positions. On the other hand, a barrier of 6 kT is not enough to prevent rapid interconversions amongst the energy minima; typical rates of this so-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 all its C—Cbonds in the trans configuration (the all-trans state); this is the lowest energy configuration for the chain. A gauche C—Cbond introduces a bend to the chain much like the 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 Bilayers
After a brief discussion on individual molecular structures, we turn now to the cooperative effects as large numbers of surfactants are present in aqueous environments. As we will see, 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 bilayer interior; we will discuss the time scales and extents of such internal motions. The special role of cholesterol within bilayers is also considered.
1.3.1 Hydrophobic effect and condensed interfacial structures
What makes water different from most other liquids is the dominance of hydrogen bond interaction amongst its molecules [Israelachvili, 1985]. Contrary to the more common van der Waals interaction, hydrogen bonding is highly directional. The tetrahedral structure (4 nearest neighbours) of ice is a consequence of this directionality. Even in the liquid state, a water molecule continues its attempts to maximize the number of hydrogen bond ings with its neighbours; it is this tendency that is believed to give rise to the hydrophobic effect. Consider the effect of having a foil-polar solute (such as an alkane) in water: Because water molecules adjacent to this solute cannot interact with it electrostatically, they must seek to establish hydrogen bondings elsewhere. This causes a reorientation of the adja
8 cent and nearby water molecules in a way that allows them to make the most hydrogen bonds with each other. However, this organization of water molecules around the solute is entropically very costly; the overall effect is a highly unfavourable free energy of solubi lization, which is commonly called the hydrophobic effect. The hydrophobic effect causes non-polar solutes to aggregate so as to minimize their contact with water. The forces that hold such condensed regions together are the familiar van der Waals forces, and it has been argued that the notion of “hydrophobic forces” is a misconception [Hilderbrand, 1979].
Aggregation of surfactants in aqueous solutions is due to the hydrophobic/hydrophilic character of the molecules. The resulting assemblies exhibit a large variety of forms, but they all share the common feature of using the polar head groups to shield the hydrophobic chains from water. A few such arrangements are shown in figure 1.4. Much work has been done on the thermodynamics of aggregation in attempts to predict the formation of these assemblies [Wennerström and Lindman, 1979]. However, a good intuition for self-aggregation can be obtained from just considering the molecular shapes and their packing 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 molecular shapes for forming the various structures are shown. Such arguments based on molecular shapes 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 can be explained by the fact that such head groups are somewhat larger than its adjacent glycerol backbone. Thus, with the two hydrocarbon chains slightly spread out and the lipid rotating about its long axis, the overall occupied space is indeed cylindrical (see figure 1.4). Similarly, the reason that unsaturated PE lipids self-assemble as inverted micelles is because PEs form considerably smaller head groups. This feature, together with chain unsaturation (chains with C=C kinks tend to flop around more and thus occupy more space), constitute a cone-shaped lipid molecule. Finally, lysolecithins, the one-legged PC lipids, are unable to form bilayers but aggregate quite readily into micelles. It is easy to see that, consistent with the scenario in figure 1.4, such lipids do have shapes of inverted cones.
9 In 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 and sphingomyelins).
1.3.2 The gel and fluid phases of bilayers
The 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. An attractive interaction exists at the hydrocarbon-water interface (where the hydrocarbon chains are attached to the glycerol backbone) due to the hydrophobic effect; this gives rise to an effective interfacial tension. Interactions in the headgroup region (i.e., from the glycerol backbone upward), on the other hand, are not as easily characterized. Being a hydrophilic environment, this region can soak up water and lead to hydrogen bond inter actions. In addition, complicated electrostatic forces arise because the head groups are dipolar, possibly charged, and likely to have free or adsorbed ions from the solution. There are also steric repulsion interactions among head groups. The combined effect of these forces is clearly one of cohesion. Depending on temperature and other thermodynamic parameters, molecules within a bilayer are ordered by intermolecular forces in distinctly different 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-trans configuration. The chains are oriented parallel to each other and are usually tilted with respect to the bilayer normal. As the temperature rises to the chain melting temperature T, van der Waals attraction and interfacial tension can no longer hold the hydrocarbon chains in parallel bundles; the chains undergo an abrupt transition to become “floppy” and spread out. This results in a shortening of projected chain lengths (along the molecular axes) and an increase in area occupied per molecule (by about 15 to 20 %); an average of 4 to 5 gauche conformations are estimated to be present in each chain. The collective tilt 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 at
10 temperatures above T are said to be fluid or liquid crystalline. Chain melting is due mainly to the interplay between thermal randomization and cohesive organization within the 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 at 24°C, 41°C and 55°C respectively [Mabrey and Sturtevant, 1976]. Lipids with even one double bond in one of the hydrocarbon chains have much lower T. Presumably, having rigid knots in the otherwise flexible chains makes it very awkward to pack the chains into parallel bundles (the gel state); the system is therefore much easier to disrupt. Typical chain melting temperatures for unsaturated lipids can be below 0°C.
For saturated PC lipids, there exists a “pre-transition” temperature T — roughly 10°C below 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 transi tion; 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 hy drocarbon chains remaining in the frozen (all-trans) state. Such an effect is believed to result from head group interactions; however, the phenomenon is not completely under stood. Bilayers composed of unsaturated lipids do not undergo pre-transitions [Evans and Needham, 1987]. Since we are only interested in the fluid bilayer, effects that occur below the main tran sition temperature will be of no concern. All we need to make certain is that experiments are done at temperatures well above T.
1.3.3 Disorder inside the fluid bilayer
The bilayer’s fluid state results when intermolecular cohesive forces are overwhelmed by thermal agitations. Hence, we expect a certain degree of randomness within such a fluid structure. As it turns out, experimental measurement of “disorder” reveals a very intuitive picture: Molecular motions are progressively more disordered towards the centre of the bilayer.
An important technique of measuring disorder in a fluid bilayer is nuclear magnetic resonance (NMR). In particular, the method of deuterium NMR, which has been instru mental in providing detailed information on a bilayer’s internal structure and dynamics,
11 involves selectively replacing protons in a lipid molecule by 211 atoms. The most common target groups are the 2CR segments along the hydrocarbon chains. With this practically non-perturbing probe in place, the deuterium quadrupole splitting is detected which in turn can be directly related to the order parameter Smoi of the deuterated 2group. The order parameter is defined such that it is unity for a frozen all-trans chain and zero for completely isotropic motions; it is a direct measure of the amount of orientational disorder at local positions. Figure 1.5 shows typical order profiles of several lipid membranes at the same reduced temperature of 0.061 [Seelig and Browning, 1978]. Here, the carbon atoms are numbered from the glycerol backbone; i.e., larger numbers correspond to C atoms closer to the bilayer centre. The important feature is that order in the first 6 to 8 carbon atoms are roughly equal, while the last atoms show rapid increase in motional disorder as they approach the bilayer centre. Another powerful method that provides structural information on the fluid bilayer’s interior is neutron diffraction. Accelerated neutrons (energies 0.1 eV, wavelengths of about 1 A) are scattered from bilayer samples to give diffraction patterns that reflect the bilayer’s internal structures. Because neutrons are scattered by atomic nuclei, transfor mation of the diffraction pattern amounts to a neutron scattering density profile across the bilayer. More interestingly, because protons and deuterons have very different scat tering properties, lipids that are selectively deuterated at different locations (as used in 211 NMR studies) have neutron density profiles that contain easily identifiable deuterated sites. Thus, the positions and positional fluctuations (locations and widths of the 211 peaks) of different segments along the hydrocarbon chain can be obtained. The posi tional fluctuations have been shown to correlate well with the order parameter measured by 2NMR techniques: For fluid DPPC (PC’s with two palmitic chains) bilayers, posi tional fluctuations of the 2CH segments are shown to increase by more than a factor of two towards the end of the chains [Zaccai et al., 1979]. In addition to experimental studies, numerical simulations have also confirmed the NMR 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 and temperature. More recent molecular dynamics calculations have also produced similar 2For a discussion on the technique, see Bloom et al.{1991]
12 order profiles across the bilayer [Pastor et al., 1988; Heller et al., 1993].
1.3.4 Interdigitation
It 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 the fully stretched value (the all-trans configuration) [Zaccai et al., 1979]. Because of the high degree of motional freedom near the bilayer centre, it is likely that hydrocarbon chains can extend beyond their monolayer region into the other half of the bilayer. This interdigitation effect may be very important to inter-monolayer dynamic coupling — an effect that will be studied experimentally in chapter 3. In the following, we will look at some evidences that point to such an occurence. Neutron scattering studies on fluid DPPC (two 16-carbon chains) bilayers have suc cessfully resolved the average positions of all 2CR chain segments [Zaccai et al., 1979]. Positional fluctuations of the segments are also given for partially hydrated lipids up to the twelfth 2CR segment. At the C-4 and C-12 segments, positional fluctuations are 1.5 A and 3.4 A respectively (C-12 is closer to the bilayer midplane). Such values measure the extent of thermal motions in the direction of the bilayer normal; they are expected to increase: (a) towards the centre of the bilayer, and (b) on approach to water saturation in the head group region (the case we are interested in). Unfortunately, no such data under these conditions are given. However, it is safe to assume that the terminal 3CR groups can have average positional fluctuations of at least 5 A. For a hydrocarbon interior that is 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 obtain structural information is introduced [Wiener and White, 1992]. Here, it is much clearer that the terminal methyl groups are able to penetrate deep into the other monolayer — again, up to a depth of 5 A. 2CR groups close to the end are also shown to be capable of penetrating into the opposite side. Molecular dynamics simulations have further confirmed the existence of chain inter penetration. Recent simulations results of Helter et al.[1993] agree well with the above mentioned data by Wiener and White [1992]. In addition, another study by de Loof et al.[1991] shows molecular motions that are consistent with our picture of interdigitation.
13 It also provides valuable insight into the dynamics of such motions.
1.3.5 Effects of cholesterol
So far, only single-component bilayers have been considered. We now discuss briefly the effects of adding cholesterol into a fluid bilayer membrane. Although cholesterol cannot form bilayers on its own, it incorporates very readily into a lipid bilayer structure. Like the 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 glycerol backbone, while the rigid ring structure extends down to the level of the eighth or ninth 2CH group. Whereas single-component bilayers are characterized by relatively sharp gel-to-fluid phase transitions, such features begin to disappear upon incorporation of cholesterol into the structure. With increasing cholesterol concentrations, “sharp features” corresponding to the main transition (for example, the peaks in calorimetric scanning plots) become broader and the associated transition enthalpies decrease. For cholesterol concentrations above 12.5 mol %, most PC bilayers behave as fluids even at temperatures well below chain melting [Evans and Needham, 1987]. Such a phenomenon can be attributed to the loss of positional order in the gel state due to lattice defects. With the presence of impurities (cholesterol), hydrocarbon chains in the gel state are forced to adopt more irregular conformations; partial chain disorder persists even at temperatures below T. The other equally dramatic effect of cholesterol is the strengthening of bilayers: Intro duction of cholesterol greatly increases a fluid bilayer’s area elastic modulus and rupture strength; at the same time, water permeability across the structure is diminished [Evans and Needham, 1987]. All these indications suggest a disordered system influenced by stronger cohesive forces. It appears that, whereas cholesterol has randomizing effects in the gel state, it actually helps to increase order in the hydrocarbon chains when the bi layer is fluid. This is supported by the fact that order parameters measured by 2H NMR are generally higher in the presence of cholesterol [Bloom et al., 1991]. The rigid ring structure of cholesterol probably acts as conformational constraints on the hydrocarbon chains, thus decreasing the number of gauche configurations and overall random motions. It is likely that cholesterol can also enhance chain interdigitation in fluid bilayers.
14 In particular, because cholesterol orders hydrocarbon chains by decreasing their average number of gauche configurations, the chains may lengthen and penetrate deeper into the opposite monolayer. Such an effect will be more pronounced for lipids with different chain lengths due to packing requirements.
1.3.6 Molecular motions and time scales
Even a semi-detailed discussion on molecular motions and the various methods of detection would be out of the scope of this introductory section. Here, we only wish to point out the various 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 employed in this broad field, the reader is referred to the comprehensive book on biomembranes by Gennis [1989]. Molecular motions in fluid bilayers can roughly be classified into three catagories: Lateral, rotational and conformational; conformational motions are primarily due to trans gauche isomerizations in the hydrocarbon chains. In addition, there are two other forms of lipid motion that are at the extremes of characteristic time constants: They are (a) vibrational modes of the 2CH groups with characteristic times r “-‘ 10’sec, and (b) exchange of lipids between monolayers — also known as lipid “flip-flop”4— which has r on the order of hours or even days. It should be noted that cholesterol flip-flop is usually thought 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 the form of diffusion constants. These constants are denoted Dtran and Drot which describe lateral translation and rotation; they have units of cm/sec and 1sec respectively. The relevant relations are 2 )2(r = 4Dtran t (02> = 2Drott where )2(r and (02) are the mean squares of displacement and angular rotation. It is up to the experimentalists to measure Dtra and t0Dr while the theorists attempt to derive them on the basis of different models. For phospholipids in a fluid bilayer at “-‘ 30°C,
15 Dtran has values ranging from 10—8to iO cm/sec. This means a lipid will diffuse to its nearest neighbour’s position in less than a microsecond2 (r 1nm). Typical values of Drot are lOsec’ or higher. Reciprocal of this number can be interpreted as the characteristic time8of lipid rotation; thus, a lipid takes less than 8lOsec to spin about its long axis. The rates of trans-gauche isomerizations vary with the position along the chain. Char acteristic times of such motions vary from 1010sec for C—Cbonds near the glycerol backbone to 1011sec 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 the chains. Such a rate is likely to be slower than trans-gauche isomerization (by one or two orders of magnitude) because chain extension is a cooperative effect that involves all C-C bonds. As shown by molecular simulations, characteristic time for interdigitation does appear to be around 9lOsec [de Loof et al., 1991]. For our experiment, which will be described in detail in chapter 3, the monolayers are made to slide past each other at relative velocities of 10cm/sec or less. Given that the chains are spaced roughly a few A’s apart in the 3bilayer plane, the time to cover this “lattice distance” is lOsec. During this time, the hydrocarbon chains will have penetrated the opposing monolayers5 at least iO times! This is effectively the rate of molecular collision that gives rise to macroscopic viscous effects.
16 because characterized have realistic systems; scopic the layers monolayers questionable: Such the thus ical theory chapter nm. are To approach Mechanics Chapter 2 lntermediate 1 Length A study of force-balance theoretical analyses lead negligible an continuum is order is and 1, of situations, approach developed the to needed. that the scales such inter-monolayer are mesoscopic N* 2 , internal validity on by conformational It follow regime basis larger assumed description a fluctuations. the is continuous mechanical approach. neglects Except the well eefrom here where mechanics than for dissipation. between of the 2 weakly known 2 continuum to analyzing wavelength rgnlformalism original regimes. for N individual of viscous be fields molecular description is Out is characteristics the At the rigidly bound that only of the Such first Monolayer balance of recent such lipid These drag. surfactant of theories number fractional valid such leaflets coupled visible properties and sight, a as bilayers non-conservative of work fields macroscopic of for necessity, a density, 17 the light; of is forces. of of treating in elastic (chemically membranes founded microscopic by deviations represent Heifrich lipid fluid a have of i.e., bilayer Seifert displacement the The unit bilayers, bilayer length been the the 100 molecules. on [1973]. work can averages membranes; and rim. bonded, from system bilayer that mechanics scales; the systems. developed is undergo Coupling an Langer in presented criterion are thermodynamic However, and roughly appropriate this must as over for Instead, internally stress For a relative of chapter from example). [1993], i.e., many continuum be that between stratified a in as analyzed where bilayer in the the this discussed microscopic all these mechanical the sliding establishes dissipative bilayer energetic 1 averages chapter. theoret In the macro and mono- whose seems fields more from and two 100 in is surface density is typically several million molecules per 2um the continuum assumption breaks down in the mesoscopic regime (at length scales, of lOnm). In the thickness dimension, the situation is much worse — there are only two molecules! However, it must be recognized that the above-mentioned processes are spatial averages. In considering time averages, one needs to compare intrinsic molecular time scales to the experimental “sampling time”. Owing to thermal effects, lipid molecules within each monolayer are constantly 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 for the lipid to spin about its long axis is less than 10—8 sec. Thus, if the time scale of an experiment is longer than, say, 10 sec, all one will see is a “smeared out” picture of the molecular motions. With large sampling times, continuum description for the monolayers can be valid down to length scales of 1 nm [Bloom et al., 1991]. Along the thickness direction, however, the bilayer consists of two mono-molecular layers that are weakly held together by van der Waals forces. Because exchange of lipid molecules between monolay ers (lipid “flip-flop”) necessarily requires the polar head groups to traverse the bilayer’s hydrophobic interior, such motions are strongly hindered. Typical times for such motions can 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 as distinct structures. In such situations, the appropriate continuum picture for the bilayer is that of two shell-like layers being held together by a pressure normal to the membrane surface. The two layers can slide laterally relative to one another, and the rate of such motions is limited by interlayer viscous forces. Although the lipid bilayer is possibly the thinnest membranous material, it still has elastic resistance to bending due to its finite thickness. In many situations where the fluid bilayer is flaccid and therefore cannot support any in-plane forces, higher order bending effects become dominant factors in determining mechanical equilibrium of the condensed structure. 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 shape of red blood cells [Fung, 1966; Fung and Tong, 1968]; however, the predicted behaviours were at odds with experimental findings. It was Canham [1970] who introduced the first successful energetic description of the lipid bilayer. He pointed out that, in view
18 of the molecular arrangement, it was incorrect to regard the lipid bilayer as “a slab of isotropic material”. Canham’s proposal was put on firm theoretical grounding by Heifrich [1973]. Adopting formalisms in the physics of liquid crystals, Heifrich expressed the bilayer bending energy in terms of invariants of the curvature tensor up to second order. This is the energy required to splay or “fan out” lipid molecules as the membrane is bent. In rather loose terms, it has the functional form 2),(c where c is local curvature of the bilayer and the symbol ()denotes averaging over the surface. Soon after, Evans [1974] introduced a more subtle but equally important curvature energy contribution to bilayers forming closed vesicles. This term accounts for the small difference in surface area between monolayers as the vesicle changes its shape; it has the form (c) 2• Because this energy depends on the overall conformation of the bilayer vesicle, it is called the global curvature energy. In contrast, the Hamiltonian that Heifrich proposed is known as the local curvature energy. As will be shown, these two contributions to the total bending energy are of the same magnitude and therefore must be considered simultaneously. Two decades after its developments, the above picture remains the accepted elastic theory for lipid bilayers. In the following, the surfactant bilayer is analyzed using the theory of thin shells. Our starting point is to evaluate the virtual work expended in an arbitrary displacement of such a structure. The principle of virtual work is “midway” between the energetic approach and the force-balance approach to mechanics; as such, it forms a connection between the 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 of such a derivation is that no restriction has yet been put on the nature of the forces; the equilibrium equations apply equally to elastic or dissipative systems. Pursuing in the other direction, when only conservative forces are involved, the statement of virtual work can be integrated to give the free energy of deformation. In section 2.1, ordinary equilibrium equations for thin shells are derived using the method of virtual work. The equations will then be specialized to liquid crystalline mate rials. This is the point of departure from conventional shell theories at which the bilayer will no longer be treated as “a slab of isotropic material”. These equations of equilibrium will be used in later chapters when dealing with bilayers that are intrinsically dissipative.
19 In section 2.2, the energetics of bilayer deformation are derived from a simple molecular model with two parameters (more precisely, parametric functions) [Heifrich, 1981; de Gennes, 1990]. We will show how the different forms of curvature energy can be obtained from this linear approach, and how elastic moduli are expressed in terms of the model parameters. 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. Tan gential motions between mono-molecular fluid layers represent hydrodynamics belowthe continuum limit. In such a situation, the conventional no-slip condition must necessarily break down — velocity is no longer a continuous function in space. The simple approach to this problem is to postulate an interlayer shear stress that is proportional to the rel ative velocity between the leaflets. It is shown that, for the lipid bilayer, interlayer drag manifests 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 whose rate is driven by membrane elasticity and limited by interlayer viscosity. A generalized dynamic equation will be derived, thus forming the theoretical framework for calculations presented in chapters 3 and 4. The experimental part of this thesis involves measurement of the dynamic coupling between fluid monolayers.
2.1 Equations of Mechanical Equilibrium
In this section, the principle of virtual work is used to derive equations of mechanical equilibrium for thin shells. The equations will then be specialized to shell materials that are of liquid crystalline nature. Equilibrium equations for stratified liquid crystalline structures will also be examined. Since we are interested in the lipid bilayer, only double- layered sheets are discussed.
2.1.1 Stresses, strains and axisymmetry
To simplify the analysis, only axisymmetric geometries are considered throughout this thesis. Although this is somewhat of a restriction, most physically important situations are represented. The coordinates of an axisymmetric surface are shown in figure 2.1: is the axis of symmetry about which a meridian is rotated to generate the shell surface, while
20 the 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. Somewhere inside 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 the surface z = 0. s and q are respectively the curvilinear distance along the meridian and the azimuthal angle; radial distance from the C-axis to the meridian is labelled r. If all the forces and displacements are rotationally symmetric about the C-axis, as will be the case 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 be made more precise here: Let c be a typical curvature of the shell surface and H the shell thickness; the shell is thin when the quantity e = Hc is much less than unity. In describing deformations of a thin shell, the basic assumptions adopted here are:
1. As the shell is bent, material points originally along the surface normal remain aligned 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 shell gets thinner. Our approach here is to express the strain at any location within the shell in terms of stretching and bending of the neutral surface. Let (Am, .Aqs)be the extension ratios of 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 the extension 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 is valid 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).
21 Turning now to the forces, we note that the internal stress (force per unit area) within the bilayer is a second order tensor [Landau and Lifshitz, 1986]; it is denoted by o’jj, where the subscripts (i,j) represent directions along the m, 5 or z coordinates. Because of axisymmetry, = 0zq5 = 0 For 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 the following developments. These stress components are, however, not negligible as they play 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 stress components are mm and o; they will be written as m and u from here on. 2.1.2 Virtual work and the equilibrium equations
Having considered stresses and deformations separately, we can now calculate the virtual work 6w required to displace a thin shell. We will then equate 6w to the work done on the body by external forces, from which the equations of mechanical equilibrium are derived. Before proceeding, it should be noted that the principle of virtual work is a means of obtaining mechanical equilibrium conditions for a unit structure; it is not applicable to two or more unconnected bodies. The power of the method lies in the fact that all internal forces within a mechanical system cancel each other and can therefore be ignored. Here, the bilayer is considered the “unit structure” whose internal components (the monolayers) are, owing to viscous effects, always in dynamic equilibrium. It is in this 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 even across the bilayer midplane. Virtual work is, in general, force x virtual displacement. For a thin continuum sat isfying Kirchoff’s conditions, 6w takes a seemingly more complicated but yet equivalent form [Evans and Skalak, 1980]: 6w = 0fJdA fdz (m A 6Am+ uAm 6A) (2.2) where is an elemental surface in the undeformed configuration. The notation 6( )
22 denotes 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, from equation (2.1),
6Am = 6,Xm(1 + z cm) + Z )‘m6cm (2.3) 6A, = 6A(1+zc) + zk6c, Using equations (2.1), (2.3) and the identity dA = ‘‘mA 0dA the expression for 6w becomes ,
6w = JJdA Jdz m [(i + z + z2g) + z (1 + z c) 6cm] (2.4) [(1+z3+z2g) + a + z(l+zcm) 6c] where and g are respectively the mean and gaussian curvatures; they are defined as
c—cm+cc ; g—c,c, (2.5)
The next step is to integrate across the thickness dimension z. Once the z depen dencies are “integrated out”, the shell is reduced to a two-dimensional surface coinciding with the shell’s neutral surface. The internal stresses are replaced with effective stress resultants (tensions) and moment resultants. Thus, the dimensionality of the problem has been reduced at the expense of introducing more surface fields. Let us first define the stress resultants and moment resultants as follows: Imagine exposing an edge of the shell and integrating the internal stress over the transverse cross sectional area to obtain the force. The stress resultant is, by definition, the integrated force per unit length of the neutral surface. Since the force is dependent on the orientation of the exposed edge, the stress resultant is a surface tensorial quantity; we will denote its principal values by Tm and r, with the subscript indicating the appropriate direction. Likewise, the moment resultant is the integrated bending moment about the neutral surface, per unit length of the neutral surface; the principal values are denoted by Mm and M. To write this exactly, a differential force in the meridional direction is
dfm dx(z)dz = JUrn as shown in figure 2.2. The azimuthal length ,1dx is dependent on z because of the curvature in the q direction; from simple geometry, dx(z) = 1dx,(O) (1 + z c). According 23 to 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 m(1=Ju +zc)dz ; T = Ja(1 +ZCm)dZ (2.6) Following an entirely similar argument, the moment resultants are
MmJmz(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 as
6w = JfdA [(Tm + mMm) + (T + cM) + Mm 6Cm + M 6c] (2.8)
From this equation, we see that, as expected, the moment resultants are conjugate to the virtual changes in curvatures. However, terms conjugate to the virtual changes in extension ratios are a sort of “generalized tensions” having the form of actual tension plus (curvature x moment resultant). This “complication” is a consequence of reducing a three dimensional body to a two dimensional surface. To arrive at the equilibrium equations, we will equate 6w to the work done by external forces, which, in general, may be due to a normal traction p and a tangential traction Pt; 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 this quantity is
6Wext = JJdA (pn 6x + Pt 6x) (2.9) where 6x and 6Xt are virtual displacements in the normal and tangential directions respectively. By writing 6w (equation (2.8)) in terms these same virtual displacements and equating to SWet, the conditions for mechanical equilibrium are obtained:
id d dr 7n = TmCm + TçtC — —(rMm)—M-- (2.10)
ld TdT 1 d M,dr Pt = TTm)rds — ——rds + Cm rds rd,s
Details of the calculations are given in appendix A. The first of equations (2.10) represents force balance in the normal direction; in the absence of bending moments, it is just the
24 familiar “law” of Laplace. The second equation is equilibrium condition in the tangential direction. The stress and moment resultants can be decomposed into isotropic and shear com ponents as follows:
1 (Tm + r) (2.11)
AI — (Mm + M) ; 3M (Mm M) (2.12)
With these definitions, equivalent forms of the equilibrium equations are
1 d / dM’\ 1 d 1dM 2M dr”\ T (Cm — — —— pn = + c) —— r— j — r 5 + 3 — I (2.13) r ds ds j r ds ds r ds j d’ 5dT 32T dr dII 1dM 2M dr —Pt = 5 3 d.s ds rds ds \ds r ds
Equations (2.10) and (2.13) can alternatively be derived from the direct balance of forces and moments on a differential shell segment [Evans and Skalak, 1980]. There is no restric tion on the origins of the forces involved; the equations are only statements of mechanical equilibrium and therefore can be applied to either elastic or dissipative materials. The only requirements for the equations to be valid are the Kirchoff hypotheses, which, for shells as thin as the bilayers, are well justified.
2.1.3 Equilibrium equations for a fluid membrane
It is necessary to have a precise mechanical characterization of the bilayer’s liquid crys talline state. For a two-dimensional surface having no resistance to bending, being fluid implies 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 in the case of liquid crystalline shells. We will postulate that in its fluid state, a bilayer has internal stresses that are rotationally symmetric about the surface normal. This is based on the following observations: At temperatures above the phase transition, lipid molecules are known to diffuse rapidly; in particular, the rotational frequency of a lipid about its long axis is of order i0 Hz. For observation times larger than the reciprocal of this frequency, all intermolecular forces will average out to give rotationally symmetric potentials about the molecular axes. Also, X-ray diffraction and neutron scattering stud
25 ies on fluid membranes have ruled out any collective tilt of molecular axes with respect to the bilayer normal. Expressed as an equation, the liquid crystalline bilayer has internal stresses such that Um(Z) = u1s(z) u(z) (2.14) Deviations from this relation can be attributed to dynamic effects. In particular, the deviatoric stress is Dv — qs — Dx where is a bulk viscosity characterizing the interior of the bilayer and Dv/Dx is the in-plane shear rate. Typically, is that of an oil, which is - 1 dyn.sec/cm The shear rate, under the rather “extreme” condition of varying the velocity2 by 100 gum/sec over a length of 10rim, is 10 sec’ . The resulting deviatoric stress is of order iO dyn/cm . On the other hand, the magnitude of cxis obtained by dividing the interfacial. tension2 by shell thickness; using typical values of 10dyn /cm for interfacial tension and 1nm for thickness, cx 108 is of order dyn/cm ; it is four orders of magnitude larger than its deviatoric part. Equation (2.14) is2therefore 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 become equal: 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. Using equations (2.6), (2.11) and (2.14),
= M (cmc) (2.16)
Substituting these expressions back into equations (2.13), the equilibrium conditions spe cific to a liquid crystalline shell are
pn — M(Cm_C)2 — 2VM (2.17) d lfdM de —Pt = —+--——M— ds 2 ds ds
26 where 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 well as viscous forces. The only requirement is that the internal stresses are rotationally symmetric about the surface normal.
2.1.4 Dividing one fluid membrane into two
The development in this subsection will appear to be no more than a mathematical exercise. 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 equations are written in terms of these quantities. Motivation for this is, of course, that the bilayer is composed of two separate surfactant layers. Physical significance of the results will be discussed in section 2.2.5. Up to now, integrations over the bilayer thickness are done with respect to z, with z ranging from —H/2 to H/2 and z = 0 corresponding to the neutral plane. Here, we will use the variables z and z to specify positions along the surface normal in the upper and lower layers, respectively. Origins of these coordinates are chosen at a distance h/2 away from the bilayer midplane as shown in figure 2.3; h is a length smaller than the bilayer thickness 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) 1Z z + h/2 ; — (H — h)/2 < zi < h/2 (lower half)
Integrations with respect to the above variables are understood to range over the given values; limits of integration will therefore not be specified. We will now calculate the bending moment resultant. Neglecting second moment of stress, we have (equation (2.15))
M = fdzzo
This can be written equivalently as 3We will show in section 2.2.4 that it is convenient to let the surfaces z = 2h/ and Z = 2_h/ coincide with the upper and lower neutral planes.
27 M = fdz (z + h/2) + fdzz (zi — h/2) a
= Jdzu z a + Jdz a — Jdz 1 + (fdz ) Defining M+ and M_ to be the moment resultants of the upper and lower halves, that is,
M+ = fdzu z a ; M_ = fdzj and letting M be the coupling moment given by M = (fdzu —1Jdz (2.19) we have ) M M + M + M (2.20)
Thus, the total moment is the sum of individual moments plus a coupling term. A similar expression can be written for the isotropic stress resultant . From the definitions of stress resultants (equation (2.6)) and using (2.14), the general expression for isotropic tension is = Jdza (i + ëz) (2.21) Following the same procedure of dividing the integral into upper and lower parts, the isotropic 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. Substitut ing (2.22) and (2.20) into equations (2.17), we get
— = (+ + ) e — (M + M_) (Cm —2c) 2V (M + M) (2.23) +2CmM± —2VM±
Pt (+ + -) + + M_) — (M + M_) +c dM ds
28 Compared to equations (2.17), we see that the equilibrium conditions are modified by terms containing the coupling moment M±.
2.2 Free Energy of Deformation
Equation (2.4) gives the work done in the virtual displacement of a thin shell. If the forces within the shell are elastic, Sw represents the reversible work of deformation; at constant 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 a derivation of the free energy must be done with caution because we are no longer dealing with virtual displacements. Discontinuities of the extension ratios at the bilayer midplane must be accounted for where necessary.
2.2.1 Elastic model for liquid crystalline membranes
Let us assume the shell is fluid and therefore equation (2.14) applies. In such a case, equation (2.4) for the virtual work simplifies to
Sw = JJdA fdz a(z) ( + z ( ë + Se J ‘\1+flJ + z2 ( g + 6g (2.24) \1+an J where o is the area dilation of the neutral plane defined as
an = — 1 (2.25)
We now need an elastic model for o(z). The area dilation a(z), which is a rotational invariant about the surface normal, is chosen as the measure of strain. Because areas at a distance z from the neutral surface are dilated by the factor (1 + zC+ z2g), a(z) is
a(z) = (1+a)(1+ze+z — 1 (2.26)
Note that a(O) = an as expected. g)For a linear elastic theory, the isotropic stress u(z) is expanded to first power in a(z).2 Here, the expansion is done about the planar and unstretched configuration (where a, e and g are all zero) which, in general, does not
29 correspond 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) where o0(z) and ic(z) are model parameters with the same dimensions as o (force per unit area). The zero order term u0(z) is included because the reference configuration is not necessarily stress free. However, we stipulate that the stress resultant in the unstretched state is zero:
Jcro(z) dz = 0 (2.28)
This condition suggests an equilibrium between interfacial effects that tend to draw the lipids 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 of u0(z) [Marelja, 1974; Israelachvili et al., 1980]. From the above considerations, we see that a0 must change sign along z.
i(z) is a variable compressibility along the thickness dimension. It is the coefficient in a harmonic 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 the lipid. Let us now look at the cases of pure stretch and pure bending separately.
2.2.2 Pure stretch
Here, all curvatures are zero (a = a) and the monolayers are assumed to undergo identical deformations. The bilayer can therefore be treated as a unit structure. Since we are dealing with elastic forces, the isothermal virtual work 6w becomes 36F where 8F is the Helmholtz free energy associated with stretching. Equation (2.24) simplifies, to 36F = 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 = 0JJdA Ka6a where we have used dA = (1 + a) 0dA K is the area elastic modulus defined as the zeroth moment of ic(z): . K = fi(z)dz (2.30)
30 Since ic(z) is a positive definite quantity, K > 0. Integrating the above expression for ,85F we arrive at the final result: 8F = 0ffdA Kc2 (2.31) For pure stretch, the isotorpic tension is simply
= fudz = Ka (2.32)
2.2.3 Pure bending and local curvature energy
In 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 becomes infinite. Similar derivation of the monolayer curvature energy can be done by replacing the 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 bent without 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 to
6F = 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 to second order in (z x curvature):
= JfdA fdz (z 0cr 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 gaussian curvatures; they are given by the second moments of i(z) and a0(z): = fz2ic(z)dz = k ; kg fz2ao(z)dz (2.35)
31 Since i(z) is a positive definite quantity, k > 0. Comparing the above expression to (2.30), it is noted that k ‘s-’ 2Kh On the other hand, the sign and magnitude of k9 are unclear. Strictly speaking, kg. cannot be considered an elastic modulus; together with the spontaneous curvature, they are the first and second moments of the initial stress 0o-(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 for the bilayer. To account for this, an initial stress u0(z) was introduced. We now minimize F to obtain the spontaneous mean curvature .c0 By differentiating equation (2.36) with respect to the mean curvature and setting it to zero, we get 0kc = — f 0zu(z) dz (2.37) For 0o-(z) symmetric about the neutral surface, the spontaneous curvature vanishes. In terms 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 curvature energy. Let us also derive an expression for the neutral plane. The condition for pure bending is that the lateral tension should vanish. For a cylindrically bent shell, the lateral tension is r = Judz 0f(u+ Kze) dz Recalling the assumption of zero initial tension (equation (2.28)), the neutral plane is calculated from the relation
fzk(z)dz = 0 (2.39)
A constitutive relation for the bending moment can also be obtained as follows:
M = JcTzdz = f(o + iczë) zdz
Using equations (2.37) and (2.35), the bending moment is
M = 1c (e — )c0 (2.40)
32 bottom monolayer The plane, last where variable the assumption Substituting surfaces Km from bottom will there smooth General to in tthe at The
(the 2.2.4 the diffuse upper term first have is developments bilayer the is normal c the Km added no layer: out surface expression compressibility zL, term is analogy to its
within Coupled layer exchange area (see = = given until that be vesicle). this distinct 0 effects on direction; 1<72 and zero. equation is modulus densities the øc, every into the of by
therefore J so for zj (see of heating is of a monolayer In equation far right uniformly (2.27), The =
particles layers dz the symmetric lipid
loosely J i(z). doing relative equation 0 have for (2.39)). will respectively coupling hand only = dz M molecule the a the assumed lead We
so, J bimetallic (2.27). holding
across and plane, kinematic distributed side tangential = monolayer. cr integral z, This a(z) are (2.30)). about to Jcrdz 1 Km dz moment = is
occupies, global therefore monolayers a (Judz We all the (see 0 two identically a gives + coupling = is the of strip. “irregularities” a+ fluid will 33 constraint chosen motion section A such surface = a+ a is For bilayer a ; in
similar J Kma_ (equation
definite curvature let on left + A shell layers — , the tc(z) Km moment; and zero more ëz to dz, the a 2.1.4 is density. fudzi) with midplane. upper to be permissible. expression is given symmetric average, = + because and together expression be rigorous the and that
(2.19)) f in a layer
this f cv_ K enough neutral surface unit Mismatch figure
dz the energy z the of be We can to , is derivation structure. can about layers relation By for form 2.3). same area time, will density surface be be allowing h between seen the dilations do also Area a area. in written each (2.28) closed is terms We not of (grad bilayer most require dilation as monolayer the Assuming molecules now separate top and follows: for surface a) on of upper (2.41) (2.42) easily mid look and will the the the the the in Thus, constitutive relation for the coupling moment is
= Krnh M (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 static situation 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 of u(z), virtual change in free energy is, from equations (2.24) and (2.15),
SF = JJMS3dA
The bending moment M is, according to (2.20), M=M+M+M
In evaluating the free energy, we will not be concerned with contributions from bending moments of individual layers (M+ and M_); each monolayer is a unit structure with local curvature energy given by equation (2.38) in the previous section. The global curvature energy is due to the coupling moment:
SFglobal M± S dA = Jf
Using expression (2.43) for M± and noting that this is uniform over the surface, we have
Kmh 11 - SFglobal Sc dA = 2 jj To evaluate the above integral, we recognize that, to first order in (h x curvature), area difference between the surfaces z = 0 and z1 = 0 (neutral planes of upper and lower monolayers) is zA = fJhedA
34 Allowing also for an initial area difference 0zA the differential area dilation is zA-zA , = 0 A = (ffhdA — )0A 0/A (2.45) For pure bending (6 = 0), variation of the above quantity is 6a = JJ6CdA
Substituting into the above expression for 6FglobaI, we get KmAo KmA SFglobal Fgioai 2 = 2 ± 0 = Finally, the expression for c from equation (2.45) is used. The resulting global curvature energy is: 2 Fglobal = )c 0A ( fJe dA — r) (2.46) where - Kmh p LAO lc — , io .47 2 — I. A UJlo k 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 equilibrium
Because stretching a bilayer is energetically much more costly than bending it, confor mations 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 + Fglobal We will assume the bilayer has no spontaneous curvature; the local and global energies are therefore
Fiocai (2kcm) JJe2 dA 2 Fglobal k 0A dA r) (*ff —
Here, kern is the bending rigidity of a monolayer. Since there are two monolayers, kern is multiplied by 2 (Note that k 2kcm; see equation (2.35)). The curvature energy Fcurv
35 is 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 retain its area and volume, we minimize instead the function G(,c;7÷,7_,p) Fcurv(i,c) + ff(++_)dA — fffpdV (2.48) 7+ and “y_are Lagrange mutipliers that ensure two dimensional incompressibility of the 7 ‘ upper and lower layers, while p is for volume conservation. Minimization of the above functional with respect to (A, c) leads to Euler-Lagrange equations that must agree with the equations of mechanical equilibrium.
Let us first minimize Fiocai. Proper use of Lagrange multipliers requires the minimiza tion to be done as if there was no constraint. Thus, writing dA = (1 + a,) 0dA the variation of Fiocaiis , 6Fiocai = (2kcm) Jf(2e + e2 dA In terms of virtual displacements 6Xt and 6x in the tangential and normal directions,
= 6x —2V (6x) — (c + c) 6x (2.49) = div(6x) + 6x (2.50) 1 + Qf where 2 ldfdf\ . ld V = —— ; div —— (rf) f rds r_)ds f = rds Using the relations
V2g) dA (g V dA ; divg) dA g dA JJ(f = Jf 2 f) JJ(f = ff(- it is easy to show that 6Fiocai = — JJdA [ (2kcm e)(Cm 2c) + 2V (2kcm )] sxn (2.51) Similarly, first variation of the non-local curvature energy is, from equation (2.46), 6Fglobal = M 6 Jf dA / \ = 6Q dA M±ff(Se+e 1+cv.j Note that this gives rise to an apparent contradiction: The two monolayers are required to be incom pressible,4 and yet they be must stretched/compressed slightly to create the a± field. This inconsistency can be resolved by recognizing that while area conservation is a global approximation, local variations in c± representhigher order perturbations.
36 where = Krnh M (ffhdA — )0LA 0/A Using the above relations for 6ë and 6o/(1 + an), the first variation of Fglobaj5
SFgio&ai= JJdA (2CmCM± SXn) (2.52)
Minimizing the remaining terms in equation (2.48) (with respect to ) and c) is now straightforward. The results are 7+JJ( + ) 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 summing equations (2.51) to (2.54). The resulting Euler-Lagrange equations are
P = (7+ + 7)3 — (2kcm)(m — 2c) — 32V (2kcmCj + 2CmCM± (2.55) o =
We now consider the equations of mechanical equilibrium (equations (2.23)). In the static situation, because the coupling moment M± is uniform, all terms involving dM±/ds drop out. Also, without spontaneous curvature, the bending moment in each monolayer is equal to kcmë; thus, M + M_ = 2kcm
Given these conditions, equations (2.23) reduce to
pn = (+ + ) — (2 kcme)(Cm— 2c) — 2V (2 kcm) + 2Cmc M —Pt =
For p,-, = p and Pt = 0, the equations of mechanical equilibrium are equivalent to the above Euler-Lagrange equations. It is interesting to note that the Lagrange multipliers for constant area are just the isotropic tensions.
37 equation with where, thesis tion, proportionality to Equations However, Figure that local we b compressibility 2.3.1 each distributed. considered In Lagrange 2.3 is an the will Let the embodies shear perturbations the monolayer interlayer is last according 2.4 us present interfacial to Interlayer can condition the Phenomenological equations examine of stresses section, depicts the develop This form be effective interlayer limiting a between modulus shear written will phenomenological to is given experienced drag the equations in the for the a by (2.21) diffuse time-dependent c stress shear situation neglecting tangential technique case consequence coefficient equivalently o by dynamics Km is — Viscosity and a 3 a and (2.57) o. as collectively tractions Pt of diffusive and = t by (2.15), mechanical The the = —* in for bv± — all limited the force can with within which — 00; ds model model di relative of simplest eM terms as measuring process monolayers process. at be \. such that dimensions balance until ; the written by relative the for — 38 containing equilibrium = velocity: is, a and top v± inter-monolayer whose —eM) model 2 1 interlayer the bilayer model fa(z) A given this — generalized and are for macroscopic motion the balance j dyn.sec/cm 3 rate v for parameter. enough equal each membrane. on bottom + dz grad second are — such drag is e— the between dM v_ monolayer. driven and shown ds M±. dynamic ie lipid time, a dynamics and viscosity. neutral of of phenomenon density opposite . equations Such show by tangential to The monolayers the reduce equation surfaces With a major that field of molecules In two-dimensional at situation a±. the what (2.17). to relaxation is is relative aim to the are midplane. uniformly We is gives forces follows, derived assume of can within Euler- scaled (2.56) (2.58) (2.57) begin This mo this rise be of by area as follows: + — 08 — — 0_s Pt , Pt — 1 + ha/2 — 1 — he/2 Using (2.56) and (2.58), tangential force balance (equation (2.57)) on the two monolayers are dM+] bv± (1 + e) [ (Ja dz) + e (upper monolayer) (2.59) dM_] — b v = (1 — e) [ (fa dz) + (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 the integrals, 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 is
v = Dgrada (2.61) where D is the mechanically driven diffusivity defined as
D (2.62) 2b 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+ — V
We will look at the kinematic relation between these quantities in the next section.
2.3.2 Kinematics
In equations (2.1), extension ratios (Am,A) at any location in the shell are given in terms of the extension ratios (sm, )) at the midplane. Here, because the monolayers are uncoupled, we allow for discontinuities in the extension ratios at the midplane z = 0:
z : At = 0 = ; = Atz=0:
39 ______
According to equation (2.50),
m = div(6x) + E6z (2.64) m
+ div(6x) + ëxn
Because normal displacements of the monolayers are equal, the same Sx is used in the above 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 define
Atz=li/2: Am —A— m Atz=—h/2: AmA
Using equation (2.1),
h — 11 — m , (i + (2.65) m+m )
= — (i c) ; A = (i - c)
To first order in (h x curvature), changes in extension ratios at the upper and lower neutral surfaces are
— — 6A + m + (2.66) m m
’A-A- 2 Now, from relation (2.63), c is associatedtm with differences between quantities at the upper and lower neutral surfaces. Using2 (2.66) from above, we have = 6Am 6A I m 5) h + = -p-- + + (2.67) 1+ + m m A; 6A — h = + 1+c- — The last step is to subtract the second of equations (2.67) from the first and substitute in equations (2.64). At the same time, infinitesimal changes in position are converted into time derivatives; i.e., we replace 6 by d/dt. The resulting relation is da dë = divv + (2.68)
40 2.3.3 Dynamic equation for c We are now ready to write down the dynamic equation for a±. Taking the divergence of equation (2.61) and using (2.68), we get ±DV2+h
In addition, we note that transfer of material between monolayers (lipid flip-flop) can also be a mechanism for relaxing the a± field. To first order, we assume such a rate be 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 a positive constant with units 1sec the reciprocal of c, is the characteristic flip-flop time. With the time derivative; d/dt separated into local and convective parts, the above equation is written as 8c + v5 = D 2V + h ( + v5 — (2.69) This dynamic equation for c is derived by Evans et al. [1992]. v is the velocity of the bilayer midplane; it is the average of v+ and v_. From the above equation, we see that changes in the mean curvature ë can be viewed as “sources” for the . field: Differential dilation is created by the local rate of change in curvature and/or by the convection of material along a curvature gradient. On the other hand, the diffusive term D V 2a± is a “sink” for smoothing out irregularities in the a distribution. The last term due to lipid flip-flop is also a sink for c; it is the only non-vanishing dynamic effect in the case where a± is evenly distributed. Because this is usually a very slow mechanism, relaxation by flip-flop is neglected in most situations. Since the coupling moment M± is proportional to a± by equation (2.43), we can also write the following dynamic equation for M:
vsa — (2.70) + = 2MDV + k ( + vs) k is the non-local bending rigidity defined in (2.47).
41 fluid illustrating rigidities useful variations, only extrusion interlayer introduced A quote measured chapter drag, curvature forces cous can This Measurement Chapter transducer Theoretical In Our • be ekyheld weakly chapter membrane a SOPC only the “by-product” applied transducer 2. method of values [Hochmuth stresses changes Results in within undergo the predicted We bilayers; bilayers discusses is section to analysis have important constructed of force of together, bilayers in sensitive a in section, b creating fluid from relative this at and unsupported chosen the behaviours 2.3.1. and 3 transducer the temperatures of the will bilayer; bilayer our of Evans, the technique results to representative they This to many using bending sliding interlayer also nanotether forces tether append will, and can method the fluid 1982; be will in of the different on which I this explained experiment be leads rigidity goal 15°C, as have be bilayers. very the drag these Bo created the Interlayer technique chapter. represents, described is plots 42 microdyne logically in and developed 25°C compositions. material to bilayer is are calculations turn in determine as Waugh, The of using summarized is and detail. follows: experimental Such based is gives to passes in procedures for we 35°C; level for a a our a novel 1989]. technique the experiment a measuring later rise on the to experimental treatment (weight through Because first the for the dynamic way to section. For are the formalism data viscous end ie measurements time, of Drag of known measuring now regions inter-monolayer following the with; measuring 10 9 gm) is (appendix are drag much strategy. Furthermore, established monolayers effects. as shown. this developed of coefficient nanotether situations: is interlayer clearer curvature so-called bending needed. B) Sharp Also, and and vis are in in of b a • MOPC bilayers at room temperature; and
• SOPC, MOPC and BSM (bovine brain sphingomyelin) bilayers — all containing 50 molar % cholesterol at room temperature.
3.1 General Description of Nanotether Extrusion
General features of the nanotether experiment are illustrated here. The purpose is to introduce the relevant variables and establish basic relations between them; details of experimental procedures will be postponed to a later section. Throughout this thesis, cgs units will be used. Bilayer samples come in the form of vesicles (closed bags) of various sizes. We choose to work with vesicles that are uni-lamellar and with excess area; i.e., those that have only 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 is 1_i extremely low mol(10’ or lower). For the duration of our experiment (r- 1sec) — and indeed for 4much longer times — the bilayer can be considered a closed system where the number of constituent particles is fixed. We will further assume that each vesicle has fixed surface area and volume; justifications for these last assumptions are as follows: The bilayer exhibits elastic behaviour when stretched isotropically; the compressibility relation is, from section 2.2.2, r = Ka r and K are respectively the isotropic tension and the area elastic modulus — both with units dyri/cm; a is the fractional change in area. Typical values of K are of order lOdyn/cm [Evans and Needham, 1987]. For the tether experiment, the applied tension is2 roughly 0.1 dyn/crn with deviations of less than 1% (lOdyn/cm). From the above expression, the resulting variation in area is estimated 3to be less than 1 part in iO; surface area of the bilayer vesicle can therefore be treated as constant. With regard to volume conservation, it is noted that the bilayer is a semi-permeable membrane that allows easy passage of water across its surface. Larger particles such as glucose, on the other hand, are effectively blocked (compared to water, glucose molecules take iO times longer to pass through). Our vesicles are suspended in a 0.2 M glucose solution with an equal
43 concentration of solutes trapped inside. Any net displacement of water into or out of a vesicle will amount to a difference in solute concentration across the bilayer membrane; the resulting osmotic stress can easily be as high as lOdyn/cm (0.1 atm). Because the pressures involved in our experiment are orders of magnitude lower (.-.-‘lOdyn/crn 52 than such osmotic stresses, no appreciable amount2 of water can be forced through the membrane and, consequently, the volume remains) unchanged. Thus, vesicle volume is maintained 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 vesicle is aspirated into the micropipette under a suction pressure The pipette size is typically 6 to 8 m in diameter and LFVS is a few hundred dyn/cm (100 dyn/cm 1 mm of water). A vesicle whose diameter is larger than the2pipette calibre2 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 is given by LFves Rv Tves = (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 %) and
can 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 estab lished, the bead is pulled back along the axis of symmetry as shown. Attached to the bead is a tubular piece of bilayer material (tether) whose radius rt is typically 30 rim (3 x 10cm). Except local to the two end regions, whose extents are of order ri, the tether6radius is uniform. 1 The small junction connecting the spherical vesicle segment to the nanotether is the region of most interest: Over the extent of this region, the bilayer curvature 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 dissipative stresses are concentrated; away from the junction, the curvature is effectively uniform. 11n principle, there may also be surface ripples on the tether. Because the tether is under tension, ripplingis not likelyto be a large effect.
44 The process of tether extrusion is characterized by the tether length L and the rate of extrusion V = dL/dt. Both of these are controllable parameters. The tension force in the tether is denoted by ft (units of dynes). Because of the surface area and volume constraints, an increase in tether length L must be compensated by a decrease in the pipette projection length L. The geometric relation between these two quantities leads to 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 Strat egy
Theoretical analysis of the tether experiment is based on the formalism developed in chapter 2. In particular, the general dynamic equation (equation (2.69)) is to be solved on the entire vesicle surface. The most important region is within the vesicle-tether junction where 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 dimensionless Péclet number P 2 defined as (3.3) where D = Km/2b is the mechanically driven diffusivity (equation (2.62)). The analysis in appendix B depends critically on the assumption that the Péclet number be small compared to unity. This condition will be verified at the end.
3.2.1 Two important equations
Based on the inequality F << 1, two equations that form the basis of our experimentation are derived. We will quote the results here and leave the details to appendix B. The first equation is one that expresses the tether radius in terms of membrane tension:
)1/2 rt = (kcm/Tves (3.4) 2The use of Péclet number is an idea borrowed from the field of transport phenomena [Landau and Lifshitz 1982]. It is useful in the analysis of diffusive processes much like the Reynolds number is useful in 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.
45 Referring to section 2.2.5, kern is the bending rigidity of a monolayer. The important feature of equation (3.4) is that the tether radius is independent of the rate of extrusion Vt; this is a direct consequence of the small Péclet number assumption. Conversely, a dependence of rt on V will mean P is not “small enough”, which in turn leads to the collapse of our entire analysis. The other important equation relates the tether force to the kinematics. It has the form - ft = 2kcm — kF + bh21n() V + (3.5) 27r rt 0 rt 22R where k,, I’ and Ii are defined in section 2.2.4. Again, this equation is only valid for
P << 1. We discuss each of these terms separately.
Using equation (3.4), the first term can be written as 2 kcrn/rt = 2 (kcmrves)” This term is due directly to the vesicle tension, which in turn is controlled2 by the pipette suction pressure LFVeS. With regard to the second term, recall that F is a measure of the initial area difference between the monolayers. This pre-loaded stress. shows up in the tether force and is actually quite significant. The sum of these first two terms on the RHS of (3.5), multiplied by 2r, represents the minimum force required to form a tether. This is the force corresponding to the situation where V = 0 and L is negligibly small; its magnitude is in the neighbourhood of microdynes. The last term is a non-local bending contribution to the tether force. For typical parameters of kc,r’-’erg10’ and R ‘—‘ 10cm, the tether force is a microdyne for 300tm. For our experiment, normally3 goes 50 1 up to giving rise a L L m, to tether force 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 relationship between the dynamic tether force and the rate of extrusion 4. The proportionality constant contains the parameter b that we wish to determine. Since this dynamic part of ft only shows up when there is motion, it can easily be separated from the other three contributions in equation (3.5).
3.2.2 Experiment 1: Static measurement
We must first verify that the Péclet number is small. This is done by checking the validity of equation (3.4) for the range of extrusion speeds that we will use for the next experimemt
46 (measurement of interlayer drag). In particular, tethers are pulled as shown in figure 3.1 at different speeds. For each pull, the values rj and Tves are determined and put into equation (3.4) to evaluate kcm. A constant value of kcm for the chosen range of extrusion speeds would justify the assumption of small Péclet number. The additional reward for this, of course, is an accurate measurement of kcm. The tether radius rt is measured according to the geometric relation (3.2) and the vesicle tension Tves is controlled by the suction pressure according to (3.1).
3.2.3 Experiment 2: Dynamic measurement
The next step is actual measurement of the interlayer drag coefficient b. Let us first define f° and fd as the initial-static and dynamic parts of the tether force:
cm — 2r — (3.6) f0 r kcfoj
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 for now that f can be measured with sufficient accuracy. As such, the contributions f° and fd can be determined separately: f0 is the tether force for V = 0 and L very short (less than 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 the bilayer “thickness” h (more accurately, the separation between neutral surfaces of the monolayers) is not known precisely, the experimental result is presented in terms of b.h2
47 be evenly to to always the groups to gene, called an is It the carried chiorofom/methanol all bilayer about the BSM in also tory Preparation (18:0, Bilayer (Alabaster, 3.3.1 3.3 already 3 The has allow the prepared. aqueous our aqueous sn-i To beads. same comes [Evans Oregon). 18:1) dinitrophenyl 19 PE make been on provided, are bilayers. notation and are by Vesicles is complete Methods carbons. Vesicles, in sample; a a head an the sn-2 attached in environments. The solution , AL) and clean a found 12-carbon of vesicles, inert MOPC dry glass chemically is chains; vesicles All group. Needham, DNP-labelled At (n 1 mixtures in evaporation teflon powder the surfactants desirable gas Cholesterol beaker). the dry containing (DNP), (2:1) to m 1 , m 1 about (14:0, solutions average chain such has the We other powder disc. n2 and labelled as of form. This 1987]. been put : bead 18:1) as The 40 m 2 ), a which m 2 while o“pre-hydrate” to specific end, solvent. of The are PE argon, length is the a t1 are form. is a where entire the Another surface and small purchased PE Phospholipids stored composition lipids well anti-DNP done of can disc desired h number the and molar organic lipids bovine lipid over of n1 Since assembly The established be amount is by to in are this and type beads then 48 recognized the solutes ratios with solution solution facilitate passing three from the solvent, brain obtained n2 proteins of other the disc of of placed double are (0.5 two molecular are is (e.g., phospholipids the Sigma types is dried sphingomyelin stored respectively technique surface chain purchased a (in form mol 16-carbon gently leaving adhesion other by that bonds stream from in SOPC/cholesterol chloroform/methanol) of lipids Chemicals specific %) at in a is, weights for bind lipids vacuum fatty poured in Molecular 35°C known of only in h number the according of the about between from chains. before these we Prof. specifically saturated antibodies, acid respective overnight. (BSM). used dry of onto concentrations incorporate (St. for Avanti 10 labelled the varies lipids introducing E. A Probes, here at the of mm. to the chemical surfactants Louis, Evans’ 1:1) One carbon chains. least water the Polar to vesicles on even are: During disc is After lipids can fatty is the attached Inc the supplier, into MO); 5 atoms labora (which spread within vapor, SOPC Lipids group easily hours using them DNP disc. (Eu this, acid into this and the are in it time, bilayer vesicles will form spontaneously at the pre-determined solute concentration. Solutions
Two different types of solutions are used in our experiment. The first type is used to hydrate the dry lipids (as described above); naturally, this will also be the solution that occupies the vesicle interior. Another type of solution is used for suspending the already formed vesicles. Because only water can pass through the bilayer, solutions on the two sides of the bilayer are distinct. The small difference in refractive index between the two media greatly enhances the visibility of the vesicle. In this experiment, we choose to have a 0.2 M sucrose solution inside the vesicles. On the outside, a solution consisting primarily of glucose is used for suspension; the osmolarity (a measure of osmotic activity) of this solution is adjusted to be slightly higher — by about 5% — than the interior solution. This causes the vesicles to deflate, thus giving us the desired excess area. Another advantage to such a combination is that the interior sucrose solution, being more dense than the outside medium, causes the vesicles to sink to the bottom of our working chamber, thus making the vesicles more accessible. In addition to glucose, it is essential to include small amounts of non-adsorbing polymer (polyethylene glycol, or PEG) and salt (NaC1) in the suspending solution. These solutes help to create non-specific attraction between the bilayer and the bead surface thus allowing antibody-antigen (DNP) reactions to take place [Evans, 1989].
The interior sucrose solution (0.2M) is made by dissolving 6.64 gm of sucrose (MW 342.30) into 100 ml of solution. The resulting osmolarity is 208 mOsm. The suspending solution 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) is dissolved into the solution. The final solution has an osmolarity of 215 mOsm. Bead Preparation Latex beads are purchased from IDC (Interfacial Dynamics Corp., Portland, OR). The particular type we buy have chemically active aldehyde groups present on the bead sur face. Bead diameters are 3.2 im and 5.4 ,um. Anti-DNP molecules are purchased from
Molecular Probes (Eugene, Oregon); it comes in solution form at 2 mg/ml. To attach the antibodies to the bead surface, latex beads are put into a dilute antibody solution (40
49 tg/ml) that is buffered at pH 7.4. The concentration of beads can vary, but is usually at ‘—0.l % solid. The mixture is left on a rocker for at least 2 hours. It is believed that the 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 setup
A picture of the micropipette setup is shown in figure 3.2. This is the microscope station at which all experiments discussed in this thesis are performed. Different components involved in such an experiment are discussed below: Making pipettes
Micropipettes are small cylindrical glass tubes. They have two important functions in our application: First, they are used as “micro-arms” to manipulate test samples on micron length scales. In addition, through precise control of the suction pressure, micropipettes can be used to exert well defined forces on the samples. Such pipettes are made in the laboratory as follows: Glass tubings with outside diameters of 1 mm and inner diameters of 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 tapers that go down to sizes of less than 0.1 m. The tube is then cut transversely at proper locations to give the desired inner diameters. For our experiment, the pipettes are cut with inner diameters of roughly 6 to 8 m. Because of the long taper, pipette diameters can be considered uniform (i.e. cylindrical) over axial lengths of 10 ,um. Glass tubings are pulled with a vertical pipette puller, manufactured by Kopf (Tujunga, CA), model number: 700C. The heat and pulling force are adjustable in such a device. Following this, the pipettes are filled by boiling them in a desired aqueous solution; this is usually a solution similar to that we use for suspending the vesicles. When not in use, micropipettes are always stored in an aqueous environment. Refrigeration is also recommended to slow down bacterial growth. Manipulating pipettes A micro-manipulation set includes a chuck for inserting the glass pipette, a receiver on which the pipette-chuck assembly is mounted, and a position controller (joystick) that is linked to the receiver through flexible hollow tubings. As such, the joystick is mechan
50 ically uncoupled from the receiver. Inside the receiver are three sets of bellows whose extensions/contractions control the xyz translations of the pipette; the bellows are in turn controlled pneumatically by the “remote” joystick via the hollow tubings. Using such a system, pipette motions up to 0.1 mm in all three directions can be made. Such motions are extremely smooth and free of hysteresis; fine manoeuvrings down to 0.1 m can be made easily. Because the bellows are driven by air pressure, there is no need to fill the system with special fluids. There are usually two micromanipulating systems at each microscope station — each controlling a pipette that inserts into either side of an open chamber containing the vesicles. Such a system is ideal for “manual” work. For our purposes, because one pipette has to be moved at well controlled speeds (for tether extrusion), the micromanipulating set is replaced by an inchworm motor. This is a piezo electric stepper motor that allows very accurate control of position and velocity (L and 14 in equation (3.5)) in the direction along the pipette axis; digital readouts of these values are also provided. The motor acquires the pre-set speed from resting position with effectively zero rise time. The range of motion for such a motor can be from 0.1 1um to 2.5 cm. Speeds are controllable from 0.004 1um/s (according to the manufacturer) to as high as 2000 nm/s. Our speed range is typically 10 to 300 pm/s. The micromanipulation systems we use are purchased from Microlnstruments (St. Louis, MO). The Inchworm motor is bought from Burleigh Instruments Inc. (Fishers, NY), model number 1W 710. Pressure control The pipette suction pressure is controlled by a home-built manometer set that consists of two water reservoirs. Pressure is created from simple hydrostatics; i.e., from the difference in elevation between the two reservoirs. Thus, a difference in elavation of 1 cm creates a hydrostatic pressure of 980 dyn/cm Reservoir elevation is controlled by a micrometer (the measuring apparatus) to2 the nearest 0.005 nun, which corresponds to pressure vari ations of 0.5 dyn/cm . To acquire. zero reference pressure, we observe the movement of a small particle2 deliberately placed inside a micropipette (note that the pipette is already placed in a chamber filled with water). Elevation of the entire manometer setup (includ ing both reservoirs) is adjusted until the small particle is stationary. Such a method is
accurate to 0.5 dyn/cm ; i.e., from this reference position, a difference in reservoir eleva tion of 0.005 mm2 will cause the particle to move. The pressure is monitored by a pressure
51 transducer connected in series within the manometer. It is purchased from Validyne En gineering Corporation (Northridge, CA). The particular model we use is DP 103, fitted with #10 diaphragm. Such an assembly has a linear range up to 880 dyn/crn with an accuracy of 0.5%. 2 Microscope, optics and image recording The 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). The ultraviolet light from this intense source is first filtered out through a glass piece. The remaining light is put through a band-pass filter centered at 546.1 nm with band width of 10 nm (supplier: Corion; Holliston, MA). The objective is a 40x lens that uses Hoffman modulation contrast microscopy (Modulation Optics Inc.; East Hills, NY). Such a system converts the otherwise undetected phase information into amplitude differences. For our vesicles, because there is a slight difference in refractive index between the interior su crose solution and the suspending glucose/salt solution, visibility is greatly enhanced by using such an optical setup. The experiment, as viewed from a 25x eyepiece, is recorded through a high contrast b/w camera (Dage-MTI Inc.; Michigan, IN) onto a SONY 3/4 inch video cassette recorder (model VO 5800). During playback, such a recorder is capa ble of making single frame forward and reverse search, thus facilitating detailed analysis of the experiment. Dimensions on the monitor are measured with a position analyser manufactured by Vista Electronics (LaMesa, CA), model number 305. This device pro duces digital readouts of distances according to the separation of parallel lines (vertical or horizontal) which are movable on the monitor; such a device is sometimes called the “video caliper”.
Force transducer The transducer for measuring tether forces is definitely home-made. It is composed of an aspirated vesicle attached to a bead — much like the nanotether assembly — but with the bead-vesicle contact made macroscopically large to prevent tethering. The situation is shown in figure C.1(a,b). Because the radii of curvature involved are much larger than the bilayer thickness (no tethers), we can make the membrane approximation in analysing the mechanics of this problem; i.e., all terms involving the bending moment M in equations
52 (2.17) can be dropped. It can be shown that the applied force ft scales as f rS where r is the membrane tension. For ft on the order of lOdyn and r 0.Oldyn/cm (an easily attained level), the deflection is 6 roughly a 1um. This is sufficiently large for measurement. Since we are close to the optical resolution, the error in length measure ments, as described above, is roughly ±0.1tm. This means the estimated forces will have errors of about 10%. The above relation is an equality if we included a factor of order unity on the right hand side. In appendix C, we will show how this factor can be calculated numerically. Also presented in the appendix is the experimental verification of the analysis.
3.3.3 Experimental procedures
The general experimental strategy is discussed in section 3.2. We will give here details of the actual procedures. In general, it is desirable to carry out micropipette experiments free of unwanted influences from other vesicles. For this reason, all our experiments involve two open chambers that are separated by a small air gap. One chamber contains the supply 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 vesicles across the air gap, they are transferred to the empty chamber by first inserting them in a large pipette. This large pipette, when taken across the air gap, retains the solution within and therefore the vesicles are protected. The chamber we use has temperature control 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 the chamber to the desired level. Vesicles in the empty chamber are set up as shown in figure 3.3. In this picture, the nanotether between the bead and the opposing vesicle cannot be seen because of its mesoscopic dimension (diameter ‘-‘ 40 rim). This picture is taken at a magnification lower than our “operating value” in order to show all the features. Static measurement A discussion on this experiment is outlined in section 3.2.2. Here, we want to measure the tether radius at different speeds of extrusion. According to equation (3.2), rt can be 4The accuracy can be improved significantly by more sophisticated image processing techniques.
53 pipette. is ranges extrusion of rated A tether; other transducer transducer, hence Dynamic tether extrusion a pressure value micron projection the by 3.3, by is length connected obtained manipulated much needed as discussion both adjusting touching The A tether is shown into than of tether the not length are L. for the range higher manometers 14 As (adjustable by measurement speeds are for the tether to inside needed. an carried is bead sphingomyelin This can speeds before, however, knowing nfigure in the the the with pulled by on increase the can taken of stationary interlayer be pneumatic is while speeds inchworm the pressure “sticky” this force) (adjustable analysis. be over a done obtained for the back up pipette. length obtained parameter), the are is experiment in 3.3. h test the pulling to tension to for by chosen set L pipette relation drag, recorded. at bead according the (SOPC 300 aspirating motor). In different controls) of We of to different similary. parameter), The following 50 this from 5 vesicle a rim/s. mco tethers sm/chol here in to extrusion want while ,um to lower between shortening the is the and the experiment, The 100 a to depends rates outlined During while a is speeds digital is to test pipette vesicle Values MOPC the equation Because value: vesicle microns. dynamic tension first pulled measure and speeds the of vesicle test holding 54 experiment, of readout while extrusion on pulled surface needed of in tether into diameter typically — back. vesicle L the in sphingomyelin/cholesterol are the (3.1); with course, measurements. are section Although is the the is the a the transducer set kept from type length very bead kept As on for and and vesicle is exact stationary deflection camera 14. at 0.03 (measured the held the shown 3.2.3. this of below obvious; the the lower without pull in not L Here, lipids dependence value dyn/cm. inchworm is the 0.1 by experiment camera vesicle; vesicle, visible back. and set is in to The dyn/crn. 80 the moving of pipette focussed the involved: section of to it cholesterol), avoid the off sum/s. the inchworm-controlled decreases the roughly experimental From on transducer is this as the Pressure pipette control transducer of focussed the suction pipette (i.e., breaking shown are B.4, Tension on (sm/chol) is monitor), L this For These monitor, 0.1 the easily the the by the on panel; projection speeds all position, readings in dyn/cm pressure (the about is suction pipette L. on bilayer effects in of speed figure setup lipids aspi done (and has one the the the the the the of a of “conventional hydrodynamics” (due to the bilayer’s surface viscosity and the external hydrodynamic drag) are not of importance. In particular, contribution of the bilayer’s surface viscosity to the measured tether force is typically a few %; the dynamic tether force is due almost entirely to interlayer drag.
As in the case for static measurements, a short tether of length ‘•‘-‘ 5 um is first pulled out. The force in this short tether is f0 as defined in section 3.2.3. Such a force is first measured. From this position, the test vesicle is pulled back at different speeds while deflection of the transducer is recorded. A typical plot of such a process is shown in figure 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 at the 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 of the transducer once the inchworm motor is stopped. This value is recorded and will be related to V in the analysis. The ranges of extrusion speeds V is the same as those used in the static experiment.
3.4 Data and Discussion
Typical plots of L vs. are shown in figure 3.5. Using equation (3.2), the tether radius
Vt can be determined from the slopes of such graphs. From the linearity of these plots, it is 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 a case, 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 %. The fundamental assumption of P << 1 is therefore justified (see section 3.2.1). For each type of lipid (i.e., each row in table I), 5 to 7 vesicles are tested. Each vesicle is pulled at four different rates to check the dependence (or independence) of the tether radius Vj on the extrusion rate 4. With the measured values of rt, bending rigidities are calculated using equation (3.4). Average values of the bilayer bending rigidity, quoted here as 2kcm, are summarized in table I.
55 For the dynamic experiment, typical plots of fd vs. V are shown in figure 3.6. It is remarkable that the resulting velocity dependence is solinear. This supports the validity of the simple phenomenological model introduced in section 2.3.1. From the slopes of these plots, 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 coefficient is given as b.h2 As expected, cholesterol acts to increase interlayer drag. This is probably due to cholesterol’s tendency to “straighten out” hydrocarbon chains, thus enhancing their interpenetrating ability into the opposite monolayer (sections 1.3.4 and 1.3.5). It is also seen that cholesterol has larger effects on the interlayer drag of MOPC (an increase of 75% in )2bh than it has on SOPC (an increase of 36% in 2).bh The difference between these 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 fact that cholesterol has different effects on these two lipids can be rationalized by arguing that 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 bh2 on temperature for SOPC bilayers. We have not investigated temperature effects on other types 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 discussed in chapter 5. Assuming h is 4 nm, the interlayer drag coefficient for SOPC is 10 dyn.s/crn Values of b for other lipids can be calculated similarly. However, it must be3 remembered that the real experimental measurement is the quantity b.h2 .
56 Table I: Summary of Results from Nanotether Experiment
Lipid 2bh (106 dyn.s/cm) 2kcm (1012 erg) SOPC (15°C) 1.6 1.1 SOPC (35°C) 1.75 1.1 SOPC 1.75 1.1 SOPC/CHOL 2.4 2.2
MOPC 1.3 1.1 MOPC/CHOL (1:1) 2.2 2.5
SM/CHOL (1:1) 12.5 5.5 Note: All tests done at 23°C unless stated otherwise. Each entry represents average of 5 to 6 vesicles. Maximum scatter from mean value: 15%
With regard to the initial area difference 0A we have made estimates of this value on 25 SOPC vesicles. It is noted that equation, (3.6) can be written as (see (2.47)): KmhLAo — — — 1/2 — (kcmTves) 2 0A 27r f° is measured as described in the previous section (in particular, see figure 3.4); the membrane tension Tves is a parameter we control through the suction pressure. Other parameters in the above relation are assigned the following values: 2kcm 1.1 X 10—12erg, Km = lOOdyn/cm and h = 4nm. Except in 4 cases, all values of /A are more or less evenly distributed in the range —0.1 % to —1%. The four remaining0 cases have positive LAQ that are below 0.1 %. Thus, it appears that most bilayers are pre-loaded with internal stresses. A negative value of 0iA means that the outer monolayer is stretched (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 to the planar configuration.
57 flaccid ihcorrelation with fluid typically will the recently frich ation by are of shape In described Here, layer (kT Safran, Vesicles Thermal Chapter this a the From Analyses susceptible static be = bilayer local [1973] times shapes such fluctuations bilayers continual 4.12 chapter, assumed 1987; for 1012 our and statistically. curvature an of x planar is have of recover measurements s i0—’ analysis dynamic only membrane Faucon seen to thermal important bombardments times that been thermal are [Chandrasekhar, bilayers 1 ergs under to energy at 4 the exposed. Undulations is et that reported aspects 2 undulations rates presented at orders conformations; al., lifetimes effects the excitations, [Seifert are 298 reported (see 1989]. that light for The much of of K). by of section magnitude fluctuating molecules of and are for microscope. monolayer-coupling 1943]. several dynamics of Because In these in longer as the bilayer driven Langer, all the consequently, 2.2.3). evidenced first 58 Collectively, forces workers these from previous than of larger of vesicles vesicles by time. this Microscopically, 1993], such When the works, the are the than from flexibility, [Schneider a of local chapter, has surrounding based tl much still but collision the on phenomenon perturbed this the the the not the a fluctuating bending Bilayer more on results ambient “perpetual” Hamiltonian appeared bilayer’s membrane the frequencies. et the such shorter complete al., fluid from bending elastic in rigidity has motions thermal forces 1984; fluctuating thermally in at equilibrium, been than undulations conformations the frequencies model has treatment rigidity n limited and can However, Milner energy considered are literature. the the only excited of caused forces relax form Hel of and kT hi be of of of it a by the viscosity of the suspending medium (water). Here, the widely neglected effects of monolayer-coupling on such thermally driven motions are examined. For equilibrium calculations, 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 relaxations as described, curvature effects generate fields of differential dilation (±) whose rates of recovery are driven by the membrane’s area elasticity and limited by interlayer drag. The coupling 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 on the formalism developed in chapter 2. Our calculations are extensions of the work by Schneider et al. [1984] and Mimer Safran [1987]. This analysis is not accompanied by experimental work. Fluctuating bilayer vesicles have diameters of typically 2a 20gm. The constraints of constant area and volume are again imposed. Here, because the membrane tension is practically zero ( ‘ kcm 2/a dyn/cm), these constraints are well justified. As lO 1 such, it is important to allow for some6 amount of excess area in the vesicle as a perfect sphere cannot undergo any shape changes. We will restrict our attention to quasi-spherical vesicles that have only small amounts of excess area — typically a few percent. The conformation of a quasi-sphere can be regarded, at any instant t, as the superposition of a small normal displacement field 2 onto a spherical geometry. In spherical polar coordinates, a quasi-sphere is described by the radial component r (r is not the quantity defined 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 Q denotes the polar angles 0 and qS. u(, t) is a non-dimensional field that characterizes the 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 amplitudes nm(t) as follows: u(fl,t) = unm(t)Ynm(fl) (4.2) n,m
Excess area is the surplus of area the capsule has over that of a sphere of equivalent volume. 2As1 shown in appendix E, tangential displacements have second order effects on shape perturbation.
59 With such an expansion, the mean square amplitude
(U:m) (Unm(O)Unm(O)) (4.3) and the time autocorrelation function
Cnm(t) (Unm(O) Unm(t)) (4.4)
of each harmonic mode can be calculated. The notation ( ) denotes thermodynamic averages. (U,m) and Cnm(t) are the equilibrium and dynamic properties of the random surface fluctuations; such quantities have in fact been constructed from vesicle images observed using light microscopy. In the following, we will predict the mean square ampli tudes and the correlation function of an undulating bilayer vesicle with interlayer effects taken into account.
4.1 Mean Square Amplitudes: Statics
The approach here is to expand the free energy of deformation to second order in displace ments. All first order terms representing the forces must vanish because the time-averaged shape (the sphere) corresponds to an equilibrium configuration. Equipartition of energy is 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 free energy associated with conformational changes is due entirely to curvature. Accounting for stratification effects, we have Fcurv = Fiocat + a1F9106 (4.5) The general form of Fioaj is given by equation (2.38). For a bilayer that is symmetric about the midplane, the spontaneous curvature vanishes. Also, because the integral of the gaussian curvature g over a closed surface is always a constant, we will not include such a term here. Equation (2.38) is therefore reduced to
Fiacai= (2kcm) Ji5 dA (4.6)
60 where k denotes the bending rigidity of a monolayer. The global free energy is as given by (2.46): 2 Fglobal kA fJedA (4.7) 0 (* — where Kh p LAO C 2 ‘ 4.8 — 2 — 0hA Recall that 0/A is the initial area difference between the monolayers; this quantity measures 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 spherical harmonic amplitudes Unm defined in (4.2). Leaving the calculations to appendix D, the results are:
Fiocat = l6kcm + kcm fl(fl+1)(fl+2)(fl1)71m (4.9)
Fglobal = 87rk + k (+2)(fl1)m (4.10) n>1 where k is the modified non-local bending rigidity defined as
— aF) (1 2 (4.11) Since the curvature energy of a sphere is 28ir(kcm + k), the excess free energy associated with deviations from spherical geometry is
/Fcurv = Fiocai + Fg1obaz = (fl+2)(fl_1) {n(n+1)kcm + U n >1 ] Following Mimer & Safran [1987],the area constraint is imposed by means of a Lagrange multiplier 7 that fixes the surface area. The quantity 7 can alternatively be interpreted in two other ways: (a) From the viewpoint of mechanics, is the sum of the two lateral tensions in the top and bottom monolayers (see section 2.2.5); (b) when the excess area is 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 equilibrium value is = Fcurv + 7zA 61 Using the expression = (fl+2)(fl1)Um (4.12) n>1 derived in appendix D, the expansion of tF’ is
= (n +2)(n —1) [n(n + 1) 2kcm + 2k + 7a2] U:m n >1 According to the equipartition theorem, every quadratic term in a Hamiltonian expression contributes an amount kT /2 to the energy. From the above expansion, we have the following spectrum of mean square amplitudes for the undulatory modes:
(Um) (4.13) = (+2)(_1)[+1)2kcm + (2k/a2 +7) a2]
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) a2 where n corresponds to the microscopic cutoff; that is, the value a/ne should be about the size of a constituent molecule.
4.2 Correlation Function: Dynamics
To predict the dynamic features of an undulating bilayer vesicle, we start with the as sumption that the correlation function, as defined in (4.4), decays according to the same dynamical laws that govern the relaxation of a non-fluctuating system [Landau and Lif shitz, 1993]. More precisely, we have
Cnm(t) = (‘Unm>fnm(t) (4.15) where fnm(t) represents the time dependence2 based on deterministic dynamics. Without loss of generality, we set fnm(O) = 1. As such, Cnm(0) is the mean square amplitude of the Ynm mode given by equation (4.13). To calculate fnm(t), we imagine a “zero temperature” situation in which there are no surface undulations. The bilayer vesicle is perturbed instantaneously at time t = 0 to a shape away from equilibrium. At t = 0+, all perturbing forces are removed and
62 the 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 by two dissipative mechanisms — the viscosity of water and interlayer drag. In terms of the displacement field, shape relaxation is characterized by the following expression:
u(Q,i) = Unm(O)fnm(t) Ynm(fl) (4.16) n,m
Here, 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 suspended in water. In their analysis, all dynamical variables are expanded to linear powers in the displacement field u; what results is therefore the lowest order perturbative solution about the spherical shape. Here, we will extend the work of Schneider et al. to include monolayer-coupling effects which give rise to non-local elasticity and interlayer dissipation.
4.2.1 Equilibrium equations for the bilayer
The equations of equilibrium for a fluid shell are those given by (2.17); they are written here 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 use and ó to denote the normal and tangential tractions on the membrane. These equilibrium equations can be simplified in the present problem as follows: The quantity 2(ë — g) in the first of equations (4.17) is of order 20(u and therefore can be neglected. With regard to the term — M Ve) in the second) equilibrium equation, is in 8M(ëV it shown appendix F that, for an instantaneous initial condition, such a term vanishes identically. Thus, the 3The equilibrium shape is a sphere of radius a. This presents an apparent contradiction since, given that the capsule has excess area, there is no spherical shape to return to! The solution to this dilemma is pointed out by Peterson [1985], in which he introduces the concept of the spherical limit. This is the limit as the excess area vanishes. As a vesicle approaches the spherical limit, although the undulation amplitudes go to zero, the relaxation rates have well defined limiting values.
63 equilibrium equations simplify to
— V2M) = (e (4.18)
a-fs = 4.2.2 Interlayer drag and bilayer relaxation function
The moment resultant in the first of equations (4.18) contains the interlayer effects; gen erally, 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 tangen tial surface velocity v is negligible; we will therefore not include the convective terms. Neglecting also exchange of material across monolayers (c = 0), the dynamic equation for M (equation (2.70)) becomes 9M = 82MDV + (4.20) In view of the mixed elastic and dissipative character of the bending moment, it is con venient to express it as a convolution integral. By convolving with the mean curvature, the bending moment is
M(, 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 transient moment in response to a unit step change in mean curvature. (t) is obtained from equations (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. The decay rate varies as the square of the wave number; that is, inter-lamellar drag has stronger effects on modes with longer wavelengths.
64 behaviour erties to zero terms
in J(w) as We For To 4.2.4 fields. Leaving creeping spheres where of —
The 4.2.3 typically motion a equation dynamic obtain large have fluid order Reynold’s of of and Following i [Brenner, intermediate the oinequations, motion an n, discussed
is Solution
of Hydrodynamics the for of isotropic of it() (force the (4.22), this arbitrary viscosity shell.
a Jo order such deterministic general number fluid is Schneider 1964]. are balance) approximately The a is the 1O. tension situation viscosity . steps wn(n relaxation the viscoelastic result mechanics that For Laplace As conditions et + to time and and Zn characterizes the a is 1) al. are appendix and result, given the with i 2 (w) ‘n(n+1)(n+2)(n—1) function dependence 4/n. bilayer, [1984], — the of transforms shell factor a 3 Z(n) (2n the boundary = in must “creeping all and Equation + (characterized the + E, we bilayer 2kcm inertial the iiia 3 (t) water W Z(n) 1)(2n 2 65 p a be will Laplace = fnm(t), transformed general are of matched Z(n) + conditions that motion” 0 and flow make is (4.24) f(t) effects + W+L)D respectively 2n - + transformed of both characterizes around expression use by and gives, — To the at can equations: 1) a 2 the relaxation the of the tailored 1 i(t) surrounding be to the the function kinematic the membrane-fluid neglected. first for general vesicle space: respectively. velocity the ‘ for order, fnm(t) function, n>2 1 t(t)) viscoelastic slightly — is fluid (no solutions The extremely the and is slip) submerged separately. derived relaxation according equations interface. deformed T 0 pressure as to (4.24) (4.26) (4 is (4.23) prop well the low 25) the in ______where WD is the relaxation rate associated with the diffusivity of ; it is defined as n(n + 1) 2D/a (4.27) We also define here two other relaxation rates whose physical meanings will be discussed in the next section:
n(n + 1)(2kcm + k) + ra02 (4.28)
— n(n + 1) 2kcm + ra02 (4.29) = a73Z The transformed relaxation function (equation, (4.26)) is substituted into equation (4.24) and simplified. After an inverse transform, the final result is
f(t) [(wD — e1t — (WD — 2) e2t (4.30) = 2 1 The subscript m is dropped due to degeneracy. The two “bulk” rates are given by
/ i. — — 1 i — \ 2 j v (wD+wC)2
IWD+L)cN = / 2 2 \ j V (D+w) Thus, we see that the shape recovery of a bilayer with interlayer drag involves two distinct time constants, namely j’ and f2’. These two time constants are in turn determined by the three intrinsic recovery rates D, w and w. Note that the relation w < w, which follows directly from the definitions, implies
L?..’DWu < 1 (WD4 + w) 2
This 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 Discussion
Our intention here is to examine the effects of monolayer coupling on the equilibrium undulations of bilayer vesicles. From the first part of our analysis, the spectrum of mean
66 square amplitudes has the form (cf. equation (4.13)) kT (Um)rJ2 - 42kn + (2k 2/a + r)0 a2n2 It is seen that the local and non-local bending rigidities affect the mean square amplitudes through different powers of the wave number n. More importantly, it is not possible to distinguish 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)) aZA — ( 2h02 )0A 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 the illusion 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 is examined. Recovery dynamics of quasi-spherical bilayer vesicles have in fact been analysed by other workers, where the bilayer is treated as a single-layered structure with bending rigidity kbj [Schneider et al., 1984; Milner and Safran, 1987]. In such situations, there is only one recovery rate given by
n(n + 1) kbl + Wsingle ra02 3ia Z This recovery rate is driven by the bending elasticity and limited by the dissipation of the surrounding fluid. Because the intention here is to also account for inter-monolayer coupling, the above recovery rate will, in general, not be appropriate for describing the dynamics. However, there are two instances when two dynamically coupled monolayers can be treated collectively as a single-layered structure: they are the limiting cases when the two layers are either held together rigidly or are completely free to slide past one another. In such cases, the corresponding bending rigidities are
I 2kcm+ icD for rigidly coupled monolayers kbl = 2kcm for uncoupled monolayers From definitions (4.28) and (4.29), we see that w and w are precisely the recovery rates in 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 excess of the equilibrium value). Likewise, because WD is related to the inter-lamellar viscous
67 forces, it represents the decay rate of the differential dilation field c. It is natural to compare the magnitudes of these three intrinsic rates. We first note that, assuming k and k are of the same order of magnitude, the ratio of w,. to will always be r 0(1). Next, we introduce the dimensionless parameter
WD L — (4.31) wc which may be interpreted as the ratio of the following quantities:
L’J decay rate of a± decay rate of ë
With the introduction of &, the notion of coupled and uncoupled monolayers can be regarded as limiting cases as this parameter approaches 0 and 00:
lim f(t) = exp(—wt) lim = W-400 f(t) exp(—t)
In both cases, the time dependencies reduce to single decaying exponentials with the expected time constants. By neglecting the mean tension, ‘ is given approximately by 4i,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 short wavelengths. 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 and uncoupled monolayers. By setting th to unity and using typical values of (2kcm+ k) lO i, 102 erg, dyn.s/crn and D iO 2/s,cm the cross-over length a/n is roughly 0.1pm. Below this length2 (i.e., for larger n), decreases in magnitude, which implies interlayer drag is becoming more important.
68 up occur; phasis effect. opposite bic one. on between a based pling recognized chanical can presented, monolayer rized The
1993].
Summary Chapter micromechanical molecular to An The interior. be theme are according 0.5 on and is More introductory determined. In continuum two experiment on monolayer. principles discussed nm this with coupling (b) long of (a) precisely, unsupported Of time into the thesis, this to emphasis the particular ago a its highly scales technique only theory of in weak thesis chapter monolayer (shear [Evans, Experimental
static 5 a continuum a an fluid recently systematic ttypical at interlayer placed coupling dynamic surfactant of is interest rates and is bilayer fluid on for 1974; given whose of dynamic on the in the mechanics. surfactant between character 10 to drag evidences the frequencies Evans, development the structure. monolayers. on measurement, important us thickness s’), literature dynamics the is is properties. created 69 1980], monolayers the molecular the of On membranes suggest Monolayer of the interpenetration is — molecular the From of [Evans of i0 the roughly but as hydrocarbon for the uha such experimental one While large s’ structure dynamics this, the often which theory et is monolayer . coupling collisions 1.5 first phenomenon. On al., extents developed the the neglected allows urn. the 1992; interlayer of time, of static of chains of monolayer side, the can time these of Such inter-monolayer show slides interlayer Seifert of interpenetration in aspects lipid further — in I the scale processes chapter have chains drag up The effect the past bilayer. viscous and as coupling of hydropho analysis be have coefficient developed sliding the a of into our Langer, catago viscous 2, occur inter- other been with drag Em cou me the to is is — the coupling of monolayers accounted for. Formalisms are derived starting from rather basic principles. In particular, a dynamic equation (equation (2.69)) that describes the transport of the coupling field (±) is derived. Interlayer drag is measured using the technique of micromanipulation. Such an inves tigation is described in detail in chapter 3. The particular experiment performed is called the nanotether experiment. This is a very versatile method that enables measurements be made on the mesoscopic scale. Physical properties of molecularly thin surfactant mem branes such as the bending rigidity, the non-local bending rigidity and the spontaneous curvature can be measured by such a method. Here, the nanotether technique is used to determine the interlayer drag within a bilayer structure. It is shown that the simple phenomenological model for interlayer drag (section 2.3) works very well for shear rates up to, and probably beyond iO s—. Typical values of the drag coefficient are of order 7lOdyn . 3s/cm There has not been much work done on the measurement of interlayer viscous forces. between fluid monolayers. The work by Merkel et aL[1989] on friction be tween a supported and a free monolayer has confirmed the order of magnitude of our measurements. However, dependence of this drag coefficient on temperature is observed, which is contrary to our finding. The SFA (surface force apparatus) technique has also been used to measure the dynamic drag between solid surfaces that are coated with sur factant monolayers [Yoshizawa et al., 1994]. In such cases, the values of b obtained appear to be 3 orders of magnitude larger than our measured value. Because such systems are usually 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 ratio nalized by correlating interlayer drag to interpenetration. At high temperatures, although the 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 two effects 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 such an effect. In chapter 4, our continuum model of the bilayer, including the effect of monolayer coupling, is applied to the equilibrium undulations of a bilayer vesicle. There are two interesting findings: In the equilibrium spectrum of mean square amplitudes, the non—
70
hydrodynamics
that,
membrane
in
dissipation tral local
a
analysis
viscous
elasticity
based
tension.
depends
on
medium,
of
(reflecting
bilayer
the
on
mesoscopic
measured
This
on
the
shapes.
the
introduces
relative
static
length
value
length
In
coupling)
examining
importance
scale
another
of
scales
b
from
at
has
71
which
(below
level
the
of
the
chapter
hydrodynamic
dynamic
of
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the
0.1
complexity
3,
phenomenon
gm).
wave
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interlayer
number
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to
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over the [10] [11]
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76 tities where In virtual This of
In
Equations
From Appendix deformation terms this equation Sw are work appendix, of Sw the JfdA to
can Virtual = is virtual the given JJdA be we equations 6m/m [(Tm 6A/ written will displacements by SCm Sc 1k +CmMm)
60 A [Tm2 equation outline of equivalently = = = = = cosO SO) d mechanical r Cm c/Cm 6
cos0 Work the 60 + (2.8) (6x) Sx algebraic = = + and in — as + — 77 + SCm+Cm (r+cM) equilibrium section d csO Cm J2() 6x, c6x + (6x) 6Xn
Mm(SO) to steps virtual -(6x) 2.1.2: that
— for Equilibrium changes c,6x a — lead thin + + c6x MmSCm M°so] shell. from in the the The + geometric M6c] statement virtual quan (A.1) (A.5) (A.6) (A.4) (A.2) (A.3) work of ____
We will evaluate equation (A.1) term by term. Using equation (A.3) and the fact that dA = 27rrd.s, the first integral in (A.1) is
6A SAm jJdA mT 2irJ rrm—ds T m Am d(6x) = 27rJ rrm +CmSXn ds ds
Integrating by parts, we get
JJdA Tm = [2rrm6xt] + 2irJ[_-(rrm).6xt + rrmcm.Sxn] ds
The boundary terms are evaluated at N and S — the “north and south poles” of the contour corresponding to r = 0. Note that this implies a closed surface. Because of the presence of the radius r, the boundary terms in the above expression vanish; the resulting equation is
JJdA 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 two integrations by parts. Using equation (A.2) for 60, the first integration goes as follows:
d(60) d(S0) JfdA Mm = 27rJ rMm ds
— = [2irrMm601 — 27rJ -(rMm) [mt
Again, the boundary terms in the above expression vanish because of the factor r. The second term in the integrand requires another integration by parts to recover Sx. How ever, the boundary terms from this integration do not vanish; the resulting expression is ffdA 0Mm’ [2Mmcos0.Sxn1 - JfdA [-(rMm).Sxt + j(rMm)6xn] (A.9)
78 Similarly, the fourth integral in (A.1) is
ffdA M°6O = 27rfMcosO6Ods
— d(Sxn)] = 2 J M COS 0 [m 8Xt ds The second term in the integrand containing the gradient of 6x needs be integrated; the resulting expression is
MOSO60 = JJdA — [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 boundary terms in (A.9) and (A.1O) cancel because Mm = Mçj,at r = 0; this is a consequence of axisymmetry. The virtual work of deformation is
SW = JfdA {TmCm + TC — {(rMm) — M cosO]} Xn
— cosO — — {(rrm) + Cm [(TMm) M058]} .Sx (A.11)
Equation (A.11) is then equated to the work done by the external forces; the latter is given by
SWext = ffdA (pn SXn + Pt
Thus, from Sw = SWeZt, we finally arrive at the equations of mechanical equilibrium for a thin shell: d d TmCm + 4TC — j j-(rMm)—Mcos0 id cos0 1 d cos0 —Pt = ——(rrm) — r + Cm —--(rMm) — M r ,4 r These are equations (2.10) in section 2.1.2. Alternatively, they can be derived from the direct balance of forces and moments on the shell [Evans and Skalak, 1980].
79 We vanishes to
is source rather with field B.1 simplificatons fields”:
As and B.1.1 good due
In
Analysis Appendix defined flip-flop general, shown will to localized
that e. approximation, and lively: bilayer The On attempt
(spheres Dynamics in reflects
by Péclet in “diffuses” mechanical the figure mean equation end The along “source” stratification. other to points and bilayer B. sharp
curvature number the solve 1, throughout time-independent;
hand, of cylinders the (2.69), of behaviour way. bend
a B conformations, this a±;
junction of and
although Tether For the at equation and
as
c and b; have the the the dynamics the of these region the tether the vesicle
vesicle-tether in the
constant order-of-magnitude tether differential on consequently, are fluid while
a-j Vesicle-Tether that 80 2 experiment, the of body. the is
field Extrusion bilayer this connects mean c pulled, entire locations is By area junction phenomenon accounts not curvatures). neglecting are vesicle there this the dilation the visible, characterized where spherical can overall are for surface, a. flux is be no its the the
the arguments
described Junction Over viewed vesicle dynamics dynamics term vesicle flows less curvature making is ytwo by this obvious a corresponding shape out as to more by: region, is an associated the necessary from gradient actually “surface is, intense visible effects tether (B.1) to the we a ______
assume 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 is
2 v —-— — 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 variables V:-_Vt
S = = the steady state equation becomes
Oa 1 O(rtë) — —v2- V V Os P ± Os where P is the Péclet number defined as
P (B.2) For typical values of Tt 10cm, V 10cm/sec and D ‘-i 10cm/sec (but of course, we do not yet),6 Péclet is of knowD the 2number order iO. 52 We will now show that, without making complicated calculations, important conclu sions can be drawn from examining the magnitudes of various terms. Let z&± be the change in &± over a distance of order rt; the dimensionless convective term will then be of order - v Os ‘ while the diffusive term is of order
1 2- P
We can therefore neglect the convective term when P is small. The resulting equation is
1 - —Va=v2- (B.3)
81 ____
Also, because the RHS of equation (B.3) is 0(1), it follows that z& P. Remembering that 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 in a over the junction is roughly 0.01%. Let us also compare the gradients of the two surface fields over the junction region. It is easy to show that 9ë 1 1Oc P — — and — Os r? —h Os It follows that the gradients of the non-local and local bending moments have relative magnitudes
(OM±) (2kcm (2kcm) ‘ P (B.5) ) = provided k/2 kcm “ 0(1).
B.1.2 Integrating the low Péclet number equation
The low Péclet number equation is
1 0 / 9a’\ D——jr————i+v =0 rOs\ Os) Os
For incompressible membrane flow, the product rv is constant. The velocity field is therefore give by v = rV/r. With this relation, the above equation becomes + 1r14O(he) —0 rOs\ Os) r D Os — This implies
r + hP ë = constant —Os Note both terms on LHS are of order P h/rt. Because of the above invariant relation, it is not necessary to know the exact geometry from a to b. Using this relation, we get
ra rt + (B.6) (-\OSJa Prt
82
must B.2 the however, Equilibrium If It In (B.1O) Combining From (see At v,
Assume B.1.3 follows we terms a, equation diffuse the
use 4; suggests field convergent of to
equilibrium Balance we from
the Kinematics equations this = define that equations onto therefore (2.61)): r analogy equation that, quantity, is the here created vout— ra, flow of vesicle. (B.7) during can for of a hence
tangential of — (B.6) (r zero the heat a (rt write at 0 v
and Forces fluid tether qiiru equations equilibrium the + = order that = flow r 0 h/2) (B.8) (ôc±’\ \USJa constant) sharp membrane forces — Ju(z) d0 d.s pulling, and stress V ()b D=v gives + + bend on consider
2Mcm e— dz resultant 83 for the dM = , the ds are cannot hV = .1) each monolayers, cross given yin— c — become individual — enter — the sectional r, h by such V 2 M (rt the “temperature”, (2.17). — we that h/2) tether have monolayer. area (c.f. It V the region at is equation(2.58)) more following b is then and At “insulated”; convenient, b, therefore equation relation (B.1O) (B.11) Vout (B.9) (B.8) (B.7) where we have set the tangential traction to zero and p = p is the pressure difference across the bilayer membrane. The bending moment can be written quite generally as
M 2kcm + M = 2kcm + k0±/h B.2.1 Vesicle region
On the vesicle surface, all moment terms can be neglected because the curvature is small; i.e., Tve$ >> k/R The equilibrium equations are 2 - —d0 = 0 0T Tues . ds - - Tves p = 2 D 0= ILV B.2.2 Vesicle-tether junction
Assuming the Péclet number is much less than unity, equation (B.5) leads to the condition dM dc << 2kcm ds d.s The tangential force balance in (B.l1) becomes
dc 0 = —d0 + 2kcm ds ds = d1 2 —d + ke The quantity o + kcmë2 is an invariant. At point a, I Tves. Hence, at any other location,
To T — kcm2 Again, without knowing the geometry of this region, we can write
(o)a (B.12)
— cm (To)b Tves —
B.2.3 Entering tether region At point b, Cm = 0 and c = 1/ri. Also, the term 2VM vanishes because the gradients of both ë and are zero (equation (B.10)). The normal force balance is, from (B.11)
p =
84 Using the invariant relation from (B.12) and the expression p = 2 TVS/RV, 2Tves ( kcm”\ 1 R — ITves rtJTt — kern — Tves( 2rt r? — 1rt ‘s. R Neglecting terms of order rt/R, the final expression is
rt (k/Tves)” (B.13) which is equation (3.4). 2 As shown in figure B.2, the condition of axial force balance at point b is
ft + rp = 2rcrt Tm where Tm 1S, in general, equal to To + Cc,Mm (see equation (2.6)). Here,
.j; Tm = 0 + c,M / kcm’\ M + 2kcm/rt 1Tves”1 + rj rj kern M = Tves + +
The axial force balance therefore becomes (2Tv:s) ft + 2rr 27rrt (Tves + +
ft = 2Tr Tves (i + + 27rM — cm 2rr Tves + 2T— + 2M± rt Substituting in 5T€ kcm/r?, we finally get ft = 2/Cern + M (B.14) 27r Tt Note that all quantities in equations (B.13) and (B.14) are evaluated at b. B.3 Dynamics of a on Sphere
Here, the general dynamic equation is solved on the vesicle surface. We will ignore the membrane projection into the suction pipette; the vesicle is treated as a complete sphere except for a small opening of radius ra at the vesicle-tether junction. The sphere has radius R that is much larger than ra.
85 B.3.1 Simplifying dynamic equation for a With both the time and space derivatives of ë vanishing on the sphere, (B.1) becomes
2 —--— + v—--—= DV at as a±
The velocity v is again given by the convergent flow relation v = rt 14/r. We can therefore write
2 ulöa± —=D Va—Pi-—— rãs where F is the Péclet number defined in (B.2). By arguing that / 2 ‘löa±’ (Va±).(-) 0(1) \r as j we have, for small F, an expression that resembles the heat equation:
= DV 2 a. (B.15) B.3.2 Boundary value problem and general solution
One obvious solution to (B.15) is a± = —hI’, where 0F is a constant representing the initial area difference between the monolayers (see (2.47)). This term will be added on at the end.
For a sphere that is “heated” by a localized source of radius Ta with boundary condition = x at r = Ta, the general solution is
a(8,t) xg(O,t) where g(O,t) is some response function specific to the truncated sphere (for example, see section 3.4 of Jackson [1975]). When x = X(t) is time dependent, since (B.15) is linear, we can write a(8, t) as a convolution integral as follows
a(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 define the “hat” notation as follows: J(w) £ [f(t)] = j e f(t) dt 86 The 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 the w-space): () (B.17) Given that [Hall, 1949] ()a’ eo Ta ln(2Rv/ra) where D 2R ln(2Rv/ra) we can write its transform as
1 1 Ia — ra ln(2Rv/ra) W+Wo and therefore boundary condition (B.17) becomes
— LL)(LL) 1 ô.s Ia — Ta ln(2Rv/ra) L1.)+Wo Finally, we substitute in the transform of equation (B.9) and obtain a relation between and: h (2R LU+w xQ’) = lnjb—) 0 Notice that the dependence on ra has almost dropped out, except for the argument in the 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 be replaced by Tt in view of the weak logarithmic dependence. However, one can be a little more precise by guessing the following relation:
ra 2rt
With such sophisticated modification, the above equation becomes W+W (w) = ln() f4(w) (B.18) We can now put in different forms of w) and solve for (w) (and hence X(t)), or vice versa. This will be illustrated by the following two examples.
87 B.3.3 Constant rate of extrusion
We 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, the result is hV /RV\ hV x(t) = —ln!--—1 + —t D \rtl 2R The above expression can be used to evaluate the tether force using equation (B.14). Such an 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 — )0hF with h 0F being the initial area difference between the monolayers. Using (B.14), the final expression for the tether force is
= 2kcm — kF + bh2ln(3L) V + (B.19) rt 0 Tt 2R where the relations
- Kmh Km 2 2 ; L=Vt are used.
B.3.4 Constant tether force
This situation is not directly related to our experiment; it is nevertheless included here for completeness. Under constant tether force, we have the simple boundary condition of
X Xo= constant. The value Xo can be evaluated as follows: Substituting the relations kcm — 1/2 2 ‘cm Tves ) M = 0(—hF )88 into (B.14), we have
Xo — 2(kcmTves)” + = Here, the transform of x is simply xo/w.2 Putting this into equation (B.18), the solution is D V(t) Xo t hln(R/rt) where D = 2R ln(R/rt) B.4 Other Dissipative Effects
Up to now, only inter-monolayer viscous dissipation has been discussed. In addition to this, more “conventional” hydrodynamic stresses can arise from the viscous properties of the surrounding water and the bilayer membrane; the latter gives rise to deviatoric tensions 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 (square surface elements on the vesicle are mapped into highly elongated rectangular segments on the tether), it is shown that even in such a situation, the conventional dissipative mechanisms are negligible in comparison to interlayer effects. Let us first consider the tension field in a spherical vesicle. In the absence of motion, the membrane tension is uniform. 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 the vesicle surface onto the tether. By neglecting all bending moment terms, the equation of mechanical equilibrium in the tangential direction (second of equations (2.13)) reads Tm 3T r ——-+=—Pt (B.20) For a spherical object moving at velocity V in a fluid medium, the surface tangential traction is [Landau and Lifshitz, 1982]
Pt = —-— 3 sinOVt (B.21) where is the fluid viscosity. The membrane deviatoric tension 3T is given by the constitutive relation (dv vdr (?lmh) j\as — ras 89 where the product ‘7mh is the surface viscosity of the bilayer; 7m is the effective shear viscosity of the bilayer interior with units of dyn 2s/cm and h is the bilayer thickness. Assuming convergent flow such that v = rt Vt/r, the deviatoric tension is
Tt V dr T = 2iimh r as and equation(B.20) becomes
dTm = (377w hrt sinO + 4’7m cos2O)
It follows from the above equation that the change in membrane tension due to conven tional hydrodynamics is
/Tm (377w — 27lm) (B.22)
Typically, 1 ‘—‘ 10 dyn.s/cm , 17m dyn.s/cm and h/rt 0.1. For extrusion rates of 100 1um/s (10—2cm/.s),2 deviations in the2 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 force from interlayer drag and from surface hydrodynamics. From equation (3.7), the dynamic tether force is fd(b) = 27rbh21n()
The tether force due to surface hydrodynamics is [Hochmuth and Evans, 1982] is fc(71rn) 7477mhVt The ratio of these two quantities is
2 77m bh 1n(R/rt)
Typically, the product bIi is 4 dyn . s/cm and the logarithmic term is around 6. It follows that surface hydrodynamics of the bilayer contributes to less than 10 % of the total dynamic tether force.
90 where the chanical these bilayer membrane: As constraints Let Figure the transducer. are an appendix,
A
Ci
Force
Appendix Calibration transducer will attached pressures membrane also us pressures, LP C.1(b) is now be defined
equilibrium. Force always we shown, is of Because consider The bead. in the constant will designed illustrates
thickness, Transducer: the their in under pipette assembly outline the
the Balance three Such the equilibrium absolute “pipette area figure. low to radii h transducer the suction all iPF 0 -F,
regions. C an the measure tensions. is and bending arrangement numerical again of values In angle” volume; curvature pressure this conditions Since microdyne composed moments are application, O, we under procedure these not is and is 91 involved are an on ; important.
shown Analysis p are can a of forces only important the load the pP-P 0 an very be the for interested are in membrane normal aspirated ft. neglected. is calculating bilayer good figure orders described We Also parameter stress approximations define C.1 is in shown of fluid segment We assumed the magnitude the across ; in
relevant and will bilayer in differences chapter in stiffness determining the also external the to drawing vesicle because be larger parameters membrane. impose 3. of between an In such to ideal than with me this the the the are a pipette. Neglecting all moment contributions, the equilibrium equations for a fluid mem brane (equations (2.17) with Pt = 0 and Pm = p) are
= r r = constant ds p = = 3 = p/r = constant
Thus, 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 equivalent forces acting on them (i.e., the free body diagrams); forces must be balanced on each imaginary segment. At the pipette entrance, the membrane makes an abrupt bend to become a cylindrical segment. We will assume the membrane tension is continuous around this bend — similar to the tension in a rope as it curves around a pulley. Thus, the same tension r is shown in the two sketches. In figure C.1(c), the equilibrium condition is irR(F—F) = 27rRT 2= P—P, = (C.1) In figure C.1(d), the condition is
irR (F, — )0F + ft = 27rRr sinO = 1F — 0F = sinO — (C.2)
Combining equations (C.1) and (C.2), the pipette suction pressure can be written as
= (F, — F) — (F, — )0F 2r . ft = (1 — sinO) + or ZFR -\ r = . 1—ft) (C.3) 2(1 — sin8) where RzF (C.4) For << 1, the vesicle is approximately spherical;2 it follows that sin Rp In f O 0/R such a case, the membrane tension is . PR T 2(1—R/R )0 92 Substituting (C.3) back into (C.2), and noting that p = — ,0F we have Z1FsinO / - • 1 — ft/sinO) (C.5) — sin
Recall from above that the external membrane segment has uniform mean curvature given by the ratio of p to r. From (C.5) and (C.3), the mean curvature is 2sinO, (1_ t/sinO (C.6) R \ 1—ft I C.2 Numerical Solution
Although we have derived the conditions of equilibrium, it is difficult to obtain an analyti cal expression for the transducer stiffness. In this section, we resort to numerical means to predict the vesicle’s response to given loads. Using the above equations, the entire vesicle geometry can be calculated. Conditions of constant area and volume are also accounted for. In these regards, this present method is “exact”.
The quantities that uniquely define the transducer are 0(R , R ,R) for the geometry and (ft, zF) for the forces; 0R is the vesicle radius at zero load. From these, the dimensionless groups necessary for our analysis are 0R /R, R /R and ft (defined in (C.4)). For actual calculations, we will implicitly let R = 1 so that all lengths are measured in units of “pipette radii”. The initial volume and area of the external segment (i.e., at ft = 0) are K 0V = [(1+z)2(2_z) + (1+z)(2—z) —4] (C.7) 0A = 27rR(z+z) 2 (C.8) where — = (R/R {i — [i )2] ; z = )2](R/R In general, the mean curvature02 of an axisymmetric shape is 02 - dO sinO c= —+——- ds r where 0 is the angle between the surface normal and the symmetry axis, and r is the radial distance from the symmetry axis to the meridian (see figure 2.1). The value , as given by equation (C.6), is uniform over the vesicle surface. When combined with (C.6),
93 the 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 initial values:
0(0) = 0, ; dO/ds = ë — sin0/r
r(O) = ; dr/ds = cosO
z(0) = 0 ; dz/ds = sin0
A(0) = 0 ; dA/ds = 2irr
V(0) 0 ; dV/ds 2irr sinO
This is a set of simultaneous ODE’s that should be easily handled by the method of Runge-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 routine is written that integrates to, instead of an end point, an end condition. Here, the end condition is r = R with dr /d.s <0. The above integration is done based on a particular value of ë which in turn depends on j and 0, (see equation (C.6)). ] is an input parameter; the value of O, on the other hand, must be chosen to satisfy area and volume requirements. The procedure is as 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 obtained from the above integration. In general, these are larger than their respective original values 0V and 0A (equations (C.7) and (C.8)) because extra material is being drawn out of the pipette. The corresponding changes in projection length L (figure C.l) are
ZLV = 7rR
2irR The conservation of area and volume requires that LL1 = This is satisfied by iterating on O, until the function zL - L\L
94 is zero. The corresponding value of z is then used to calculate the transducer deflection 6.
C.3 Results and Calibration
To calibrate the transducer, we need an independent measure of the applied axial force. A simple method is shown in figure C.2. A human (the author’s) red blood cell is completely aspirated into a suction pipette while still attached to a “transducer bead”. We can determine the suction force on the red cell from the simple relation
(prbc”\ A D Jrbc — \.LGp ) rbc
For the above relation to apply, the red cell has to form a good seal in the orifice. Small particles are placed inside the red cell pipette during the experiment. The fact that the particles 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. Keep ing the pipettes stationary, /Prb is incremented in steps and the transducer deflec tion is recorded. A superposition of the theoretical curve and the actual data points is shown in figure C.3. Here, in order to facilitate comparison, experimental data are non dimensionalized 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 by attaching two types of antibodies — those that are bilayer-specific and those that are red cell-specific — to the bead surface.
95 where The For unit as linear work Geometric about In
lowest
Vesicle
Appendix Calculations this follows: quasi-spheres, unit vectors by the terms order appendix, Mimer normal spherical descriptions in expansion in the and such vector we
u/a Undulations: respective shape. Safran expand = = an << will will i (r
expansion D 1. is be Because [1987]. necessarily the given The
—
: () directions. done r(0,) curvature surface by in (1 (representing the + spherical = + be — spherical r Sfl20(O) is uY’ The a elastic quadratic. - 96 defined [1 a distorted + coordinates [1 au energy u(0,)] the shape +
by
) Equilibrium 2 u(0,)] This forces) the (1 in shape is scalar terms analysis at + (r, 0(u) mechanical vanish. 2e2)_1l’2 0, is
2 ), function of parametrized normal is with an Consequently,
extension (, equilibrium,
displacements ,
by ) r(0, of being (D.1) the the
all ) Taking the divergence of the unit normal, we obtain the mean curvature of the surface:
1 1 Oflq (r 2 r) + 0. (sinOrio) + 5 r sine r sinO Oq Equation (D.1) is substituted into the above expression and, after some algebra, the second order expansion of ë is
= [i + (u—u) + (u2_2uu)] (D.2) where
ü 82u 2 1 8/. 8u\ 1 Vu _s1n8) + sin2O and the expansion
= (1 + u)’ 1 — u + u2 is used.
In general, for the parametrization r = r(O, q), an elemental surface has area given by
2 = d r [r2 + + 211/2 2sin () where df = sin 0 dOdq. Here, dA = 2dr (i + 2E2)h/2 Again, to second order in u, it can be shown that 12)] dA = 2da [i + 2u + (u2 + (D.3) An elemental volume is in general given by
dV = (r.nidA Noting that i= r, we get
—1/2 = r (i + 22)
97 Using expression (D.3) for dA, the differential volume is written as
dV = 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 identity fE2dc = Juüd1 which 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 dA a2 [1 + 2u + (u2 + uü)] (D.7) = J Jd V dV = (i 3u 3u2) (D.8) = f fd + +
We now expand the normal displacement into orthogonal modes:
= UnmYnm() n,m
= ZIG + ttnmYnm(2) n >0 with denoting (0, S) and u0 uY00 From the definition of U, it follows that n(n+1) =. ‘Unm Ynm 2 Th >0
Equations (D.5) to (D.8) can now be expanded into modal components as follows:
e dA = 8a(1 + u0 + a n(n + 1) mUfl (D.9) J ) n>0 2
98 where and Elastic Substituting This The the volume is energies a relation quadratic fe2dA constraints V A back fdA Fgio&ai Fiocai = = of J2dA of into
deformation f fdA area equation dV ======equations conservation is 8a 16 8rk, 87t(2kcm) A 0 = = imposed (2kcm) for o 4a 2 (1+uo) 2 16 + + can = u0. (fc1A (D.9) + k 4iruo(1+uo)
+ f + at (i now To n>1 2 n>1 = fl(fl+1)(fl+2)(71)m A + hspoint. this kcm to dA the = be 3u 0 (7+2)()1m — fl(fl+1)(fl+2)(1)1m (D.11), A 0 n>1 99 written lowest + + (+2)(_1)m] n>0 is 3u) )2 a 2 fl(fl+l)(fl+2)(fll)Um + used. From the n>0 (aF) 2 U order, n>0 easily + 2 resulting a 3 [i V that from ri = + >0 4ira 3 /3, n(n+1)] is, the expressions to above (unchanged) second we Um get expressions are order in (D.13) (D.15) (D.14) (D.12) (D.1O) (h1) u, as We the process Bilayer they lowest good E.1 elastic internal and differential E.1.1 u, The Vesicle Dynamics Appendix a will work aim will reference and is order, restoring surface is scale dissipation written, here Bilayer by then 3 driven Geometry geometry Schneider are these all is be is is to all lengths forces by chapter quite described balanced Undulations: forces obtain functions curvature (interlayer to (O,q;t) Shell et generally, arise evaluate by are al. 1 E the of against by a [1984]. expanded from of (a Green response elastic = drag) = (0, quantities as ae,. the uë 1); the and ; forces; and t) hence, vesicle’s + to viscous behaviour + Zerna and a(O,q;t) first 100 the such the of powers surrounding + stresses. [1968]). departure as order recovery ,8sinO of unit Deterministic a in slightly e. ; normal This the rate We from hydrodynamic perturbative analysis << must is perturbed vector spherical limited 1 now is and by use an sphere. displacements; geometry. curvatures effects. the extension methods bilayer’s Such (E.1) (E.2) The To of (a of a
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is the condition for area conservation. Here, is the usual three dimensional gradient operator. Contravariant form of the metric tensor is the inverse of a; that is,
66 = ; — a a/z a = ae9/ ; a6 = a6 = ,//ao17 To first order,
a66 = 1 — 2 ( +
1 0/9 — ( 0 + — 00sin 80 1 1 0/3 2 1 — 2 + + a 2sin tan 0 Next, we need to evaluate the curvature tensor .b3 Its components are given by
be=n.ã_ ;
After some straightforward algebra, we get
/ On = —1+u+2— 02u — 02u 1 Ou Oa 1 03 — OOOq tan 0 84 Oq 2sin 0 00 2 / 1 Ou 1 2o 0/9 = — sin 0 1 + U 02u + + — tan000 — sin20 tan0 2
And finally, the local mean curvature is = — with repeated indices summed. It is easy to show that u—2u=2—V (E.4) Note that the tangential displacements c and /3 drop out of this first order expansion. They will, however, appear in higher2 order terms. E.1.2 Kinematics
Let V (0, q) be the velocity field on the vesicle surface:
(0,q5) = Ou 0o 0/3. = + + -b--sinOe
102 Velocity 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 hydro dynamic problem. However, it is more convenient, for reasons that will become clear, to choose a slightly different set of boundary conditions as follows: .T7 = (E.5) = = 0 (sin2 = a it. ( x = - [ . The first condition is simply the normal velocity component. The second is a measure of surface dilatation, while the last condition represents in-plane shearing. Note that the tangential components a and /3 only appear in the last condition.
E.1.3 Forces
Equilibrium 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 mem brane surface. is the isotropic tension field. The moment resultant M embodies the elastic restoring forces and the bilayer’s internal dissipation due to interlayer drag. Its viscoelastic 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 order perturbation: = ro+r ; 0r/r<<1 103 M = 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
2 2 a ~ 2r0 + 2r - r0 (v u + 2v) - V m er fin \ L • a l -du\ ^ + 2r0 - Vr
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:
2 2 n a = 2r0 + 2t - r0 (V u + 2u) - V m (E.7)
2 2 722 * V • a = 2r0 c — V r + 4r — 2r0 (V u + 2u,) - 2 V m
In arriving at the second boundary condition (divergence of E.1.4 Modal expansion We will proceed now to expand the boundary conditions into harmonic modes. Let u(n,t) = unnynm(fl)/nm(t) ; /nm(0) = 1 . (E.8) n > 1 ' where /nm(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) 2 v c(n,o = - £ •n(n + i)(n + 2)(n-i)unmylim(n)/nm(<) ; n > 1 The Laplacian of equation (E.6) can also be expanded in modal components as follows: / . v2M(n,t) = v2m(n,t) = fi(t)v2c(n,o) + j\{t-t') ^ [v2c(n,<')] dt' ; = - E n(n + l)(n + 2)(n-l)u„m^nm(0V„m(n) (E.10) n > 1 where 0-m(O = M0/nm(0) + j^(t-t')^(i')dt' (E.ll) Using these relations, the first two kinematic boundary conditions (equations (E.5)) are n -V = £ unmYnm^- (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 = 2r + V { 2r f 0 nm nm i n > 1 + [r0 (n + 2)(n - l)/nm + n(n + l)(n + 2)(n — l)firnm] } Km (E.13) n 4 r V • a = 2t0 C + Y, { + + ) nm fnm n > 1 + unm [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 = Vp 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' = £ Ynm(n)fnm(t) (E.14) 2(2 n +J5)r] Pnm + n > 1 / n(n + 1) ,• . , ' ,. ,, . V-tT- = Ynm(H) fnm(t) n > 1 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 rnm. ; " 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) p'„m + 2n(n-l)C, Km(n)/nm(0 (E.15) 3 (n + 2n + 6}) , ft ' in2 ,, ^ P nm + 2n(n — 1) ^ni Ynm(n)fnm{t) n > 1 (2n-+ 3)77 «'(VxPr') = function of Xnm only 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 x xj} nm 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 £[/nm(01 = /hm(«) 106 £[*!(«)] = m and from equation (E. 11), C[gnm(t)} = £(u;)/nm(0) + /i(w) [u)/nm(w) - /nm(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(ri-a) = £(2r0) + £ { 2rnm n >1 + Unm [r0(n + 2)(n-1) + wp. n{n + l)(n + 2)(n - 1)] } Ynm{U) /nm(w) (E.18) 2 £ (V • A) = C(2T0C) + £ { (n + n + 4)r„m n >1 + Unm [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™ + = Unm ~ '»?) 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 ain stress? boundary conditions on the interior fluid with no-slip condition implicitly sati ifled to first order: £ (n • PJ) = - £ (p') + r, £ (n"1}fn + 3) -(« - fa) n >1 3( +2Kn £(V-Pr') = -C(p'c) + V E " ^ - /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 + 1} £ (n • Pr°) = - £ (p°) - r? J] " - /J) Unm yL(p) /nm(u,) n>l 3(n ( + 2) £ (V . p;) = -C(p°c) - v E n+ ? - p) /»» l Defining the quantities APT = P* — P° and Ap = p — p°, we have • £(n-APr) = -£(Ap) (2ra + l)(2n2 + 2n - 3) + f E (W " j^j "nra y„m(ft) (E.20) n >1 n(n + 1) £(V'APr) = -£(Apc) (n-l)(n + 2)(2n + l) , _ k + 37/ E ^rim fnmi.u) n > 1 n(n + 1) \ Finally, the equilibrium condition at the shell-fluid interface is <7 + A PT = 0 In the transformed space, this condition is £ (n • S) + £ (n • APr) = 0 S C(V-c) + £(V-APr) = 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 rnm + c12 unm = 0 cji rnm + C22 unm = 0 108 where \ cn - 2 ... c« = t0 (n + 2)(n — 1) + u> ft n(n + l)(n + 2)(n — 1) 2 . (2n + l)(2n + 2n-3) , _ ._t) + V n(n + l) V Jnm> 2 c2i = n + n + 4 c22 = 2r0 (n + 2)(n - 1) + 2n(n + l)(n + 2)(n - 1) (n-l)(n,+ 2)(2n + l) / . + , .v A(w - ;nm i n(n +1) ' For non-trivial solution, the determinant |c,j| is set to zero. It follows from this that /»m(w) = t In v ^ j_ (E-21) ut]Z + n(n + l)u) fi{u)) + r0 where _ (2n + l)(2n2 + 2n-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(ft,0)-+ jf ii(t-T)^{Sl,T)dr. (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 ' = /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) = /ic(n,0) From the instantaneous condition, it follows that anm = hcnm . Substituting into (F.5), we have , - a±(n,t) = £ /i^ y^^VnmCO r (F.6) n,m It is obvious from (F.6) that c Vsa± — 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 /nm(i) to 0„m(O' #»» , rc(n + l)£ _ dfnm ^ • + fl2 V™ ~ dt 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) = 2fcemc(n,w) + kca±(n,u)/h cm = E W [cnmi«m(n)/nm(w)] ^ 2 (P.9) w u> + n(n + 1 )D/a 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) + AH [wc(ft,w) - c(«,0)] - = uc (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' , kc. Mnm[U>) = u hu + n(n + l)D/a2 The inverse transform of /tnm(w) is fin(t) = 2kcm + kc exp — n(n + l) Dt/a 112 Polar Head Group Backbone 1 nm 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 r Polar OH Group Rigid Ring Structur 1 nm 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>) 10 " ( kT ) 5 - -180 -120 -60 60 120 180 Figure 1.3: Torsional potential between two CHj groups. (a) Plot of torsional potential V( 115 Phase Molecular Shape Micelle Inverted Cone nmm 0 umui » Bilayer Cylinder >> h2O < a Inverted Micelle 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 / / .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 nonlocal I elasticity po-ror. -rO I f V I U 0 1 fz I V I " I o~~-o~ -o—o*-o— (pdyn) . O-O-O -r- fo 0 J L -I 1 I I -0.2 0.0 0.2 0.4 0.6 0.8 1.0 .1.2 220 (fim/sec) 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; VK%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) / / / / / /77 2Rpbo APr 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- 0.3 •frbc ft irR*AP V 0.2 (•) 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