Advances in Colloid and Interface Science 208 (2014) 189–196

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Nonlocal membrane bending: A reflection, the facts and its relevance S. Svetina a,b,⁎, B. Žekš a,c a b c

Institute of Biophysics, Faculty of Medicine, University of Ljubljana, Ljubljana, Slovenia Jožef Stefan Institute, Ljubljana, Slovenia University of Nova Gorica, Nova Gorica, Slovenia

a r t i c l e

i n f o

Available online 24 January 2014 Keywords: Lamellar membranes Area difference elasticity Vesicle shape Membrane tethers Flip-flop Vesicle fusion Monocellular sheets

a b s t r a c t About forty years ago it was realized that phospholipid membranes, because they are composed of two layers, exhibit particular, and specific mechanical properties [1–3]. This led to the concept of nonlocal membrane bending, often called area difference elasticity. We present a short history of the development of the concept, followed by arguments for a proper definition of the corresponding elastic constant. The effects of the nonlocal bending energy on vesicle shape are explained. It is demonstrated that lipid vesicles, cells and cellular aggregates exhibit phenomena that can only be described in a complete manner by considering nonlocal bending. © 2014 Elsevier B.V. All rights reserved.

Contents 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical formulation of nonlocal and local bending energies . . . . . . . . . . . . History of the nonlocal bending concept . . . . . . . . . . . . . . . . . . . . . . . . Nonlocal bending constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlocal bending energy and vesicle shape . . . . . . . . . . . . . . . . . . . . . . . Various aspects of nonlocal bending . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Nonlocal bending ensures the stability of the tether pulled from the aspirated vesicle 6.2. Nonlocal bending enhances net transmembrane flow of phospholipid molecules . . 6.3. Stability of the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Vesicle fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Elastic properties of the red blood cell membrane . . . . . . . . . . . . . . . . 6.6. Elastic properties of closed monocellular sheets . . . . . . . . . . . . . . . . . 7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Envelopes of phospholipid vesicles, cellular organelles, cells and certain more complex biological entities with a closed surface are in ⁎ Corresponding author at: Institute of Biophysics, Faculty of Medicine, University of Ljubljana, Ljubljana, Slovenia. Tel.: +386 1 5437602; fax: +386 1 5437601. E-mail address: [email protected] (S. Svetina). 0001-8686/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cis.2014.01.010

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189 190 190 191 191 193 193 193 193 194 194 194 195 195 195

general lamellar membranes or their analogous layered structures. In many cases the distances between the individual layers of these structures are fixed. For example, the hydrophobic interaction between the two layers of a phospholipid membrane dictates that they are in close contact. A salient feature of this bilayer is, however, that these two layers are unconnected, meaning that they can, if forced to, slide one over the other. Consequently, lateral stresses on them are relaxed independently. Such a response is essentially nonlocal because the

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relaxation is spread over the whole closed surface. It has to be noted that, while the stretching deformation of a single-layered membrane in its liquid state is described by a single deformational mode, that of a membrane bilayer involves two independent deformational modes, one pertaining to the area expansivity and the other to the relative stretching of the two layers, which can be also viewed as nonlocal bending. Envelopes of cells can have, in addition to their phospholipid bilayer other accompanying layers, which implies their more complex behavior. However, if an additional layer is at a fixed distance from the already existing layers, its area is determined by their areas and it does not constitute a new degree of freedom. A single nonlocal bending energy term, of the same type as that derived for the bilayer, is thus applicable also to membranes with a number of layers larger than two. Nonlocal bending energy has been comprehensively discussed in several reviews [4–8]. However, there remain some motives for further discussion. One is to provide a more focused discussion, since none of those reviews were devoted solely to the nonlocal bending aspects. Further, there are some technical issues which still need to be settled: the energy term is known under different names (relative stretching, nonlocal bending, area difference elasticity), and there are different definitions for the corresponding material constant. Use of the concept may be hampered by the absence of well established, unified views. The nonlocal aspect of bending energy is also often left in the shade. For example, in an influential review on biological applications of membrane bending, the nonlocal bending contribution is scarcely mentioned [9]. An even stronger reason for a critical review is that, despite the comprehensive work of Miao et al. [10], it is still possible to find in the literature some disparate views about the effect of nonlocal bending energy on vesicle shape. A possible reason for some ambiguities in the use of this energy term and its neglect could be that some earlier developments of the related concepts are in widely scattered and not so readily accessible literature. It thus seems appropriate to round up the subject by collecting all the essential concepts together. The particular reason for this review to be part of this volume is the pioneering contribution of Wolfgang Helfrich [1] in identifying the problems of the mechanics of bilayer membranes. The review is divided into sections on the mathematical formulations of membrane nonlocal and local bending energies, on the history of the development of the concept of nonlocal bending, on the definition and measurement of the corresponding membrane material constant, and on the effects of the nonlocal bending energy on vesicle shape. We conclude with a selection of examples aimed at illustrating the role of nonlocal bending energy in vesicle and cell phenomena. 2. Mathematical formulation of nonlocal and local bending energies The nonlocal bending energy (Wr) is introduced by showing how the material parameters involved depend on those of the constituent layers. For a thin, closed membrane composed of any number of layers, it can be expressed as 2 1 kr
Wr ¼ C−C 0 2 A0

ð1Þ

where kr is the nonlocal bending constant, A0 is the membrane area, C is the integral of the sum of the principal membrane curvatures C1 and C2 over the membrane area (which, from here on, we call the integrated curvature) Z C¼

ðC 1 þ C 2 ÞdA0 ;

ð2Þ

and C 0 the preferred (equilibrium) value of this integral (the preferred integrated curvature). Phospholipid membranes in their liquid state at curvatures smaller than the reciprocal of few membrane thicknesses exhibit no shear.

The material properties that appear in Eq. (1) can be for such membranes expressed [11] in terms of the preferred areas (A0,i) and the area expansivity moduli (Ki) of the constituent n layers (i = 1,2,…n) and the distances between their neutral surfaces and the membrane neutral surface (hi) as n X

A0 ¼

i¼1 n X i¼1

kr ¼ A0

Ki

Ki A0;i

;

ð3Þ

n X K i h2i ; A0;i i¼1

ð4Þ

and C0 ¼

n A0 X Kh: kr i¼1 i i

ð5Þ

The membrane neutral surface is defined as the surface that determines the extension of the whole membrane relative to its preferred area A0. Its distance from a chosen layer is defined by the condition n X K i hi ¼0 A0;i i¼1

ð6Þ

from which its position can be obtained from the n − 1 distances between neighboring layers. In general, Eq. (1) is not limited to the description of closed surfaces; it also contributes to the mechanical behavior of any membrane whose layers are all laterally constrained at the membrane border. Why Eq. (1) is a convenient definition of the nonlocal bending energy term will be discussed in Section 4. The nonlocal bending energy is distinct from the ordinary membrane bending energy that represents local membrane properties and is given by the integral over the area densities of the local (Wb) and Gaussian (WG) bending energies Wb þ WG ¼

1 k 2 c

Z

2

Z

ðC 1 þ C 2 −C 0 Þ dA0 þ kG

C 1 C 2 dA0 ;

ð7Þ

where kc is the local bending constant, kG is the Gaussian bending constant, and C0 is the spontaneous curvature [12]. The nonzero spontaneous curvature C0 reflects transmembrane asymmetry — an unsupported piece of an asymmetrical membrane would assume mechanical equilibrium in a curved conformation with radius 2/C0. The local and Gaussian bending constants are sums over the corresponding constants of the constituent layers, whereas the spontaneous curvature is given in terms of these constants as n X

C0 ¼

kc;i C 0;i

i¼1

kc

;

ð8Þ

where kc,i is the bending constant and C0,i is the spontaneous curvature of the i-th layer [11]. 3. History of the nonlocal bending concept The nonlocal bending concept was established in three independent works published in 1974 to describe the elastic behavior of lipid bilayers. Helfrich [1] considered blocked lipid exchange between the monolayers of the bilayer and concluded that the effect of the consequent non-equilibrium lipid distribution goes hand in hand with membrane spontaneous curvature. Sheetz and Singer [2], by analogous reasoning, explained the shape transformations of a red blood cell arising from

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the asymmetric intercalation of drugs into the two layers of its bilayer. Thirdly, Evans [3] pointed out the distinction between the bending responses of connected and unconnected layers, and found for the latter that the free energy of the system includes terms which are, in addition to being linear, also quadratic in the surface average of the sum of the principal curvatures. He identified the quadratic term as the contribution to the nonlocal bending [13]. The bilayer elasticity was then derived in a systematic manner from the elastic properties of the composing layers ([14], see also [4] and Appendix A of [10]). It was assumed that the individual layers are fluid and that their elastic energy is the sum of the area expansivity term and the local bending term. By taking the two monolayers (or strictly their neutral surfaces) to be at a fixed distance and unconnected, it was possible to express the coefficients of the area expansivity of the bilayer, and of the local and nonlocal bending energies, in terms of the corresponding parameters of the monolayers. A quantitative description of bending moduli of multilayers was also developed [15] and subsequently derived in the manner outlined in the previous section [11]. Despite this theoretical background, up to the early nineties the consequences of the nonlocal bending had not really been explored. The concept became tangible when, by the use of the tether pulling experiment, it was possible to determine the nonlocal bending constant [16–18]. At about the same time it was also revealed that the nonlocal bending energy may play an important role in the establishment of vesicle shapes [19–23,10] and in membrane fluctuations [24]. Two different approaches already existed for shape determination, known as the spontaneous curvature model [25,26] and the bilayer couple model [27,26]. It became recognized that these two models are limiting cases of a more general model based on the minimization of the sum of the local and nonlocal bending energies [19–23,10]. Following the comprehensive study of Miao et al. [10], the concept of nonlocal bending received general recognition under the name of area difference elasticity. 4. Nonlocal bending constant The material constant of the nonlocal bending energy is defined variously in the literature. When considering the bilayer it was found appropriate to deal with what was called the relative stretching constant (Kr) [14]. Kr appears in the nonlocal bending energy term, when it is expressed as (Kr / 2A0)(ΔA − ΔA0)2, ΔA being the difference between the areas of the outer and inner monolayers and ΔA0 its preferred (i.e. equilibrium) value. Kr was thus aimed to indicate that the two monolayers of the bilayer are stretched differently, i.e. one is less and the other more extended than the whole membrane. The relative stretching constant is, for a symmetrical membrane, related to the area expansivity constant K by Kr = K / 4 [17]. It is thus a true material constant. The bending constants on the other hand depend in general on the material properties and the geometry of bent objects. By taking into account that, for a thin bilayer, ΔA = hC, (where h is the distance between the neutral surfaces of the two monolayers) the nonlocal bending constant kr (see Eq. (1)) is related to the relative stretching constant by kr = h2Kr. The nonlocal bending constant used by Evans [13] can be related to kr as B = kr / A0. In most recent literature on the subject the nonlocal bending constant used is related to kr as κ = kr / π (see [10]). These different definitions for the nonlocal bending constant call for unification. Here we collect some arguments for the choice of kr (see Eq. (1)). In general it can be expressed in terms of the area expansivity moduli of the constituting monolayers and the distances between their neutral surfaces (Eq. (4)). For a symmetrical bilayer, in which K r = K m / 2, with K m the area expansivity modulus of a monolayer, we have kr = h2Km / 2. This example can be used to show that the constant kr is related to the elastic properties of the constituting material and to its geometrical aspects in the same manner as the local bending constant kc. Assuming that the resistance to its area expansion (or compression) is uniformly distributed over its cross-

191

section [4], the local bending constant of a monolayer can be expressed as kc,m = Kmh2m / 12, where hm is the monolayer thickness. Since the bending constant of the bilayer is the sum of the bending constants of its monolayers and because, under the above assumption, the neutral surfaces of the monolayers are in their middle, we have hm = h, which gives kr/kc = 3. Because membrane layers are not homogeneous elastic material, the ratio kr/kc is expected to deviate from the value 3. This ratio is thus an indicator of the membrane structure. That kr is the consistently defined nonlocal bending constant will become apparent in the next section, where the ratio kr/kc will appear as a crucial parameter for the stability of some vesicle shapes. The different definitions of the nonlocal bending constant could be one of the reasons why reports of its measurement are scarce and that, consequently, there is a lack of any systematic overview of its values for different membranes, as is often carried out for the local bending constant [28]. The nonlocal bending constant has been measured by the tether pulling experiment in which a thin cylinder is formed under the application of a point force on an aspirated giant vesicle (Fig. 1) [16,18]. This method, originally developed for determining the local bending constant kc [29], is particularly suitable for determining kr because the pulling of a long thin tether leads to a large increase in the difference between the areas of the membrane layers and therefore a correspondingly large increase of the nonlocal bending energy. The analysis of the tether behavior under these experimental conditions [17] showed that the resistance to the force applied on the tether derives partly from the local and partly from the nonlocal bending energies. Because of this, the value obtained for the local bending constant for a 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine (SOPC) membrane [18] was smaller than that obtained in [29]. Despite the large variability in the mechanical behavior of different vesicles, the mean ratio kr/kc obtained (3.3) was close to 3 [18]. The mean values, obtained subsequently from analyses of the dynamic behavior of tethers, range from 3.1 [30] to 2.4 [31]. The value of the nonlocal bending constant has also been estimated indirectly through analysis of vesicle shape transformations. For the 1,2-dimyristoyl-sn-glycero-3-phosphatidylcholine (DMPC) membrane, Döbereiner et al. [32] estimated the ratio kr/kc to be between zero and 4.4. The value reported by Majhenc et al. [33] for SOPC was estimated to be between 1.5 and 3.5 and, by Sakashita et al. [34] for 1,2dioleoyl-sn-glycero-3-phosphatidylcholine (DOPC), below 2.5. 5. Nonlocal bending energy and vesicle shape Under flaccid conditions, i.e. when the vesicle volume (V) is smaller than Vs (Vs = 4πR3s / 3, where Rs = (A0 / 4π)1/2 is the radius of the spherical vesicle with membrane area A0), a vesicle attains the shape that corresponds to the minimum of the sum of the local and nonlocal bending energy terms [21–23,10]. The shape is to be sought under the constraints of constant membrane area (as A ≈ A0) and vesicle volume. In the minimization for shapes of a given topology, the Gaussian term can be omitted because for all these shapes it has the same value, e.g.

Fig. 1. Schematic (horizontal for a vertical experimental setup) representation of the tether pulling experiment. The vesicle was aspirated into a pipette under an aspiration pressure p0 − pp. A glass bead was attached to the vesicle and a tether formed because of its ! weight f . The vesicle shape was, in the analysis [17], described by four geometrical parameters, tether length (Lt), tether radius (Rt), length of the aspirated part of the vesicle (Lp) and radius of the spherical part of the vesicle (Rv).

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4πkG for the spherical topology. The shapes thus correspond to the minimum of the energy functional G ¼ W b þ W r þ λA−μV

ð9Þ

where λ and μ are the Lagrange multipliers ensuring the constancy of membrane area and vesicle volume, respectively. It is convenient to scale the system parameters relative to their value for a sphere with the area A0. In a correspondingly concise manner, the energy functional to be minimized, divided by the bending energy of the sphere for zero spontaneous curvature (g = G / 8πkc), reads as g¼

1 4

Z

2

ðc1 þ c2 −c0 Þ da þ

kr 2 e ðc−c0 Þ þ λa− μev kc

ð10Þ

where the reduced principal curvatures are c1 = C1Rs and c2 = C2Rs, the reduced spontaneous curvature is c0 = C0Rs, the reduced area differential is da = dA / 4πR2s , the reduced value of the integrated curvature is c¼

1 2

Z ðc1 þ c2 Þ da;

ð11Þ

and the reduced preferred integrated curvature is c0 = C 0 / 8πRs, with 8πRs the integrated curvature of the sphere with radius Rs. The reduced e = λR2s / 2kc and μe = μR3s / 6kc, and the reduced Lagrange multipliers are λ volume is v = V/Vs. The reduced area is a = 1. The variables with respect to which the minimization of Eq. (10) is to be carried out are the reduced principal curvatures, c1 and c2, everywhere on the membrane, and the reduced integrated curvature c. To minimize Eq. (10) first with respect to the reduced membrane principal curvatures c1 and c2, it is rewritten as e μev−c0 c þ g ¼ wb;0 þ λa−

kr 1 2 2 ðc−c0 Þ þ c0 4 kc

ð12Þ

where wb,0 is the local bending energy for c0 = 0 wb;0 ¼

1 4

Z

2

ðc1 þ c2 Þ da:

ð13Þ

To obtain the minimum of Eq. (12) with respect to the principal curvatures it is reasonable to minimize its first three terms at a fixed value of c.

By solving the corresponding shape equation [27], the shape and its bending energy wb,0 can be obtained for any pair of values of v and c. The stable shapes obtained may have different symmetries, depending on the values of v and c. The shape classes can be defined as the domains in the v − c phase diagram, where shapes of the same symmetry can be reached by continuous shape transformations caused by continuously varying v and c. Some, but not all, shape classes comprise axisymmetric shapes, and shapes of some axisymmetric classes in addition exhibit equatorial mirror symmetry. Class boundaries are either the symmetry-breaking lines or the so-called limiting shapes which are sections of spheres with two possible radii [27]. For several domains within the v − c shape phase diagram the shapes with the minimal wb,0 have been revealed, e.g. the oblate classes (discs and cups) by Svetina and Žekš [27] and Seifert et al. [26], the prolate classes (cigars and pears) by Svetina and Žekš [35] and Seifert et al. [26], and nonaxisymmetric classes by Heinrich et al. [23], Wintz et al. [36] and Ziherl and Svetina [37]. An example of the dependence of wb,0 on c for two prolate axisymmetric shape classes, one with (cigar class) and one without (pear class) equatorial mirror symmetry, is presented for v = 0.85 in Fig. 2a. In the second step of minimization of the energy functional g (Eq. (12)), it is minimized with respect to c. By replacing the first three terms by the already obtained dependences wb,0(c; v), we obtain  ∂wb;0 ðc; vÞ k
ef þ 2 r c−c0 ¼ 0 kc ∂c

ð14Þ

where the effective reduced preferred integrated curvature is defined as ef

c0 ¼ c0 þ

kc c : 2kr 0

ð15Þ

The material parameters that appear in Eq. (14) and determine the shape of a flaccid vesicle with reduced volume v are the reduced spontaneous curvature c0, the reduced integrated curvature c0 and the ratio between the nonlocal and local bending constants, kr/kc. As indicated, the shape of a vesicle depends on the combination of these three parameters — the effective preferred integrated curvature (Eq. (15)). It can be noted that the parameters c0 and c0 affect the vesicle shape in a synergistic manner. For a given ratio kr/kc and a given value of the effective reduced preferred integrated curvature c0 ef , it is not

Fig. 2. Effect of nonlocal bending on the stability of vesicle shapes (adapted from [21]). (a) The dependence of the local bending energy, wb,0, on the reduced integrated curvature,c; for cigar and pear class shapes at reduced vesicle volume v = 0.85. Also shown are some characteristic shapes (numbered 1 to 6). cs.b denotes the reduced integrated curvature at which the equatorial mirror symmetry of the cigar shapes is broken.cl.sh denotes the reduced integrated curvature of the limiting pear shape. (b) Graphical representation of the solutions of Eq. (14). Thick ef lines are the derivatives ∂w ðc; vÞ / ∂ c for the cigar and pear classes, and the thin straight full lines are −2kr(c − cef 0 ) / kc calculated for kr / kc = 6 and for c0 = 1.35, 1.45 and 1.55. The dashed line at cef = 1.42 corresponds to the effective reduced integrated curvature at which the system attains three solutions. (c) The domain of stability in the kr / kc − c shape phase diagram 0 (shaded area) of the pear class shapes at reduced volume v = 0.85.

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possible to know whether the corresponding stable shape is determined by the effect of c0 or of c0 . Eq. (14) can have a multitude of solutions, the consequence of the characteristic behavior of the function wb,0(c; v) and especially its derivative with respect to c. The analyses made for the cigar and pear shape classes are particularly instructive [21,10]. In the case of cigar shapes, the derivative ∂wb,0 / ∂c increases steeply and monotonically whereas, in the pear class, it is largest at the lower limit of the interval of its c values, attains a minimum and then increases until c reaches its limiting value cl.sh. (Fig. 2b). The graphically obtained solutions of Eq. (14) are the cross-sections between the ∂wb,0 / ∂c curve and the line − 2kr(c − c ef 0 ) / kc (Fig. 2b). Four examples of the thus obtained solutions are presented in Fig. 2b. The three full lines correspond to three different values of c0ef. At c0ef = 1.35 the only solution is the cigar shape. At c0ef = 1.45 the solutions are three fold: the middle one is unstable and one of the other two is stable locally. At c0ef = 1.55 the only solution is the asymmetric pear class shape. The dashed line corresponds to the value of c0ef = 1.42 at which for kr/kc = 6 there is a transition from the regime of a single cigar class solution to the regime of three solutions where, initially, the pear shape is stable only locally. The pear class shape becomes the absolute minimum shape when the energies of the shapes of the two classes are equal. The presented example (Fig. 2b) shows that, at continuously increasing cef 0 , at its value 1.42 there is a discontinuous change of the reduced integrated curvature from c = 1.09 to c = 1.18 which means that shapes with intermediate values of c are not stable. However, shapes with all values of c are stable if the ratio kr/kc is so large that the slope of the line −2kr(c − cef 0 )/kc is steeper than the second derivative of the wb,0(c; v) curve at the point of breaking the equatorial mirror symmetry, i.e. 2kr/kc N −∂2wb,0 / ∂c2 at c = cs.b.. In this case, by continuously changing c0ef, there is only a single solution of Eq. (14) for each of its values. For each reduced volume there is a critical value, (kr/kc)crit, of the ratio between the nonlocal and local bending constants, above which the shapes are stable for all values of c and the shape behavior is the same as that predicted by the bilayer couple model [27]. The region of unstable shapes in the shape phase diagram kr/kc – c is shown in Fig. 2c for reduced volume v = 0.85. The span of unstable shapes is largest at kr/kc = 0, which corresponds to the spontaneous curvature model [25,26]. For kr/kc b (kr/kc)crit, the interval of the unstable pear shapes depends on the second derivative of the curve wb,0(c; v) with respect to c. The critical ratio (kr/kc)crit is larger at larger values of the reduced volume [8]. The analogous behavior of oblate vesicle, i.e. disc and cup, shapes, has been also analyzed [38,39], and the dependence of the critical ratio (kr/kc)crit on the reduced volume v given in [40]. The nonlocal bending aspects of the oblate–prolate shape transitions have been analyzed in [23] and [32]. The presented analyses led to the following general view of the shape behavior of the unconstrained vesicles. The complexity of their shape behavior is contained in the function wb,0ðc; vÞ. All possible vesicle shapes can be characterized by two purely geometrical parameters, the reduced integrated curvature c and the reduced volume v, and can be represented by the v − c shape phase diagram of the bilayer couple model. All shape transformations within this shape phase diagram are continuous. When vesicle shapes correspond to the minimum of the sum of the nonlocal and local bending energies there are two types of vesicle shape behavior, one with continuous and the other with discontinuous shape transformations. The realizable shapes can be most generally represented by a three-dimensional v − kr/kc − c shape phase diagram. However, for the interpretation of different experimental observations it can be useful to derive from this diagram specific shape phase diagrams of the type v − c0 or v − c0. The latter, the spontaneous curvature shape phase diagram [26] is particularly important because it applies to the asymmetric single-layered membranes and in phospholipid bilayers to slow shape transformations in which the nonlocal bending energy is due to the lipid flip-flop (see Subsection 6.2) diminished.

193

6. Various aspects of nonlocal bending There are several vesicle and cellular phenomena that can only be interpreted by taking into consideration the nonlocal membrane bending. Here we attempt, by selected examples, mostly from the results of our research, to illustrate the variety of the nonlocal bending aspects rather than to give a comprehensive overview. 6.1. Nonlocal bending ensures the stability of the tether pulled from the aspirated vesicle The requirement of nonlocal bending for understanding vesicle behavior came to light very clearly through the analysis of the tether pulling experiment described in Section 4 (Fig. 1). Not only because this experiment enabled the nonlocal bending constant to be determined, as discussed there, but also for the reason that, without the corresponding energy term, the tether at the condition of constant lateral tension would not behave as it does. The balance of forces for a given tether configuration can also be obtained without considering this term [41,42], however, analysis of the stability of the system has shown that, at a constant applied force, the system is not stable, which means that its energy is maximal and not minimal [17]. It was demonstrated that, under the action of a constant force and at the value of the ratio kr/kc = 0, the tether would grow until the vesicle would be expelled from the pipette. Increasing the aspiration pressure at kr/kc b ~1 was shown to cause an increase of the tether length and vice versa. Only at kr/kc N ~1 does increased aspiration pressure cause an increase of tether length, ensuring the stability of the system; i.e. with increased aspiration pressure the tether length decreases. 6.2. Nonlocal bending enhances net transmembrane flow of phospholipid molecules The tether stability discussed in the previous subsection relies on the constancy of the preferred area difference, ΔA0. In phospholipid membranes this difference was believed to be relatively constant because the transmembrane movement of phospholipid molecules was found to be extremely slow, with relaxation times of the order of hours [43]. Improved techniques applied in performing tether pulling experiments have indicated that there is a flow of phospholipids from the denser to the less dense layer, at a rate that could not be reconciled with such long relaxation times [30]. An attempt was made to interpret this by taking into consideration the fact that the rate of transmembrane flow is larger because, due to the nonlocal bending energy, the transferring molecules experience an energy difference between their final and initial states [31]. However, the observed rates were still an order of magnitude larger than those predicted in this manner. They could have been explained by assuming that the nonlocal bending energy causes an advective flow of phospholipid molecules from the denser to the less dense layer via constantly formed transient membrane defects [44]. This flow is resisted by the friction between the two layers that gives rise to the effective coefficient of mechanical diffusivity characterized by the ratio Dm = Kr/b, where b is the ratio between the shear stress exerted at the center of the bilayer by one leaflet on the other and the differential velocity of leaflets relative to each other [45]. The idea that the nonlocal bending enhances the flip-flop of membrane constituents was explored more recently also by Bruckner et al. [46]. 6.3. Stability of the sphere Membranes are not ideally impermeable for the constituents of their environment, therefore vesicles are apt to slowly swell or shrink. When membrane elastic properties are described by the local bending energy (Eq. (7)) alone, a stable shape is the sphere but only for values of the reduced spontaneous curvature at which the slope of the dependence of the reduced local bending energy on the reduced volume is negative

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at v = 1. This slope was shown, analytically [47,48] and numerically by calculating w b (v) curves for some fixed values of c 0 [49], to be − 2 + c0/3, which requires, for a stable sphere, the reduced spontaneous curvature to be smaller than its critical value c0,crit = 6. Taking into consideration the nonlocal bending energy as well, the slope which for a stable sphere must be negative was shown to be − 2 + c0 / 3 − 2kr(1 − c0) / 3kc [40]. 6.4. Vesicle fusion Two vesicles fuse into a single vesicle when their membranes come into close proximity and merge into a single membrane. Analyses of the process, based on the mechanics of lipid membranes, have identified two important intermediate structures, the hemifusion structure and the fusion pore (reviewed in [50]). In the hemifusion state the two outer bilayer leaflets are merged, leaving the inner leaflets distinct while, in the fusion pore, both respective leaflets are merged. These intermediate states have, in general, higher energies than the initial and final states of the system and, because they therefore greatly affect the rate of the process, they are in the focus of the majority of the studies of the structural basis of the fusion process [50]. However, an important factor in the effectiveness of the fusion process is also the difference between the energies of the final and initial states of the system, to which the corresponding change of the nonlocal bending energy may contribute importantly. In a study of the difference between the final and initial energies of the fusion of two spherical vesicles, that included the contribution of the nonlocal bending energy, it was assumed that, after formation of the fusion pore, the two fusing spherical vesicles of the same size transform into a flaccid dumbbell vesicle with the reduced volume v = 1 / √2 [51]. The local bending energy of such a final vesicle is lower than that of the initial local bending energy by the sum of the local bending energy of the sphere (8πkc) and by how much the local bending energy of the dumbbell shape is lower than that of the sphere. In the case of fusion the Gaussian bending energy (see Eq. (7)) also changes because of the altered number of vesicles. Its contribution is expected to be positive because of the usually negative Gaussian bending constant. The contribution of the nonlocal bending energy to the energy difference depends on the value of the preferred area difference, ΔA0. Assuming that initially the preferred area difference is equal to the area difference, i.e. ΔA0 = ΔA, the nonlocal bending energy would contribute in a positive manner because the dumbbell shape has a lower value of ΔA than that of the two initial spheres together. However, it was reasoned [51] that the contribution of the nonlocal bending energy to the change of energy can be negative if ΔA0 is initially adjusted to the value that corresponds to the final dumbbell shape. Such an adjustment can, for example, be achieved by adding calcium to negatively charged membranes: the neutralization of the outer membrane surface charges causes the outer vesicle layers to shrink, which means that the sum of the ΔA0 values of the two fusing vesicles is reduced. At the same time the membrane is stretched asymmetrically, because the area of the inner layer cannot change in the case of a spherical vesicle. The fusogenic activity of calcium ions, viewed from the aspect of the nonlocal bending energy, is two-fold — their addition increases the energy of the initial state and concomitantly causes the necessary decrease of the preferred difference between the areas of the bilayer leaflets, thus ensuring that the final energy is sufficiently low [51]. 6.5. Elastic properties of the red blood cell membrane The red blood cell (RBC) membrane is composed of a lipid bilayer, decorated with integral membrane proteins, that is linked to a spectrin-based membrane skeleton. The role of nonlocal bending effects in RBC shape behavior was invoked implicitly through the bilayer couple mechanism of RBC shape transformations by Sheetz and Singer [2] who interpreted the shape transformations from discocytes to

echinocytes and stomatocytes on the basis of asymmetric changes of the areas of the individual leaflets of the RBC bilayer. They thus introduced the concept of preferred area difference — the formation of echinocytes was interpreted in terms of the intercalation of molecules into the outer bilayer layer, thus increasing its area and the difference ΔA0, while the formation of stomatocyte shapes was interpreted by their intercalation into the inner layer and thus a decreased ΔA0. The RBC membrane is essentially a trilayer, its two-dimensional membrane skeleton being a third layer in addition to the two layers of the bilayer. Its elastic behavior differs from that of lipid bilayers because the skeleton exhibits shear elasticity. However, the skeleton has a negligible bending constant, and contributes little to the nonlocal bending energy because its area expansivity is orders of magnitude smaller than those of the layers of the bilayer. The nonlocal bending constant of the RBC bilayer was estimated from the dependence of the equilibrium force by which a tether is pulled from the aspirated RBC on its length [52]. The ratio between the nonlocal and local bending constants obtained is around 2, comparable to that of phospholipid bilayers. In quantitative terms, a macroscopic description of the minimal contributing factors of RBC elasticity needed to describe the essential elastic properties of the RBC membrane has been proposed by Mukhopadhyay et al. [53]. It includes the local and nonlocal bending energy terms of the bilayer and the area expansivity and shear elasticity energy terms of the skeleton. This energy functional has enabled the same authors to explain why RBCs at increasing preferred area difference behave differently from phospholipid vesicles by forming echinocytes and not some prolate shapes [54]. The nonlocal bending energy of the RBC membrane [53] also proved to be essential in interpreting the observed aspiration of echinocytes into cell-sized micropipettes [55]. Echinocytes were formed that belong to different stages of echinocytosis that can be ascribed to different values of the RBC bilayer preferred area difference. The observation that it was more difficult to aspirate into the pipette the more advanced echinocytes could have been described solely on the basis of their larger preferred area differences [55]. 6.6. Elastic properties of closed monocellular sheets Monocellular sheets are two-dimensional aggregates of cells laterally held together by adhesion forces. Especially during the development of an embryo, various closed monocellular sheets exhibit different shape transformations. Some of their shapes look very similar to those of simple lipid vesicles or RBCs. For example, gastrulae assume typical stomatocytic shapes. It is therefore plausible to expect that the elasticity of monocellular sheets can be described in energy terms that are similar to the macroscopic energy terms that are exhibited by thin membranes. In this sense, it is possible, for example, to express the area expansivity and bending constant of the epithelial sheet on the basis of the properties of its constituent cells, their cortical tension and intercellular adhesion [56]. Nonlocal bending aspects also appear to be important. It has been shown, for example, that blastula wall invagination occurs by the support of the apical extracellular matrix [57]. Involvement of the effects of nonlocal bending in developmental processes is implicated by the gel swelling mechanism of the sea urchin epithelial invagination proposed by Lane et al. [58] which is based on the same principle as the bilayer couple mechanism of the red blood cell shape transformations of Sheetz and Singer [2]. The relevant layers considered were the hyaline layer (HL), the apical lamina (AL) and the monocellular sheet (MS) (Fig. 3). The hyaline layer consists of a network of hyaline and other filaments. Because filamentous networks are, in general, much less resistant to bending than to area extension it can be assumed that the main contribution of the hyaline layer to the epithelial elasticity is its area expansivity. The monocellular layer is expected to exhibit resistance to bending as well as to area expansivity [56]. The apical lamina is the region between the hyaline layer and the monomolecular sheet and is comprised of protein gel. Its thickness appears to be controlled by the length of the microvilli that keep the hyaline layer

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Fig. 3. A schematic of the blastula wall as a laminar structure composed of a monocellular sheet (MS), an apical lamina (AL) and a hyaline layer (HL). hHL and hMS are the distances between the neutral surfaces of the corresponding layers (adapted from [40]).

at a certain distance from the cell apical surfaces. It was proposed [58] that the epithelial sheet bends due to the swelling of the apical lamina caused by the cell's secreted hygroscopic substance. When the gel is swollen by absorbed water, one can suppose that the increased lamina volume manifests itself in increased preferred area of the apical lamina. In the light of nonlocal bending concepts, we are thus dealing with three layers of which the apical lamina at a given swollen state is inextensible and the expansivity constant of the hyaline layer is higher than that of the monocellular layer [40]. Increasing the area of the apical layer results in decrease of the effective preferred integrated curvature and thus in the shape transition into an oblate stomatocytic shape. Because these shape transformations occur at relatively high reduced volume they can be continuous only at high kr/kc ratios. In the described system this appears to be achieved by a division of labor — the monolayer sheet is responsible for the local bending of the system and the apical and hyaline layers for the effective kr [40]. 7. Summary Nonlocal bending, also termed area difference elasticity, is an inherent property of membranes that are composed of two or more layers. Nonlocal bending is a more general term because area difference elasticity implies that the property pertains only to bilayers. The nonlocal bending constant is properly defined when its definition accords with that of the local bending constant. On continuous variation of the reduced vesicle volume, reduced integrated curvature, or both, the shape transformations are either continuous or discontinuous, determined only by the vesicle reduced volume and the ratio between the nonlocal and local bending constants. The concept of nonlocal bending is required in order to understand the behavior of a variety of systems, from simple phospholipid vesicles to cell aggregates. Acknowledgments The authors thank Roger Pain for the critical reading of the manuscript. The work was supported by the Slovenian Research Agency through the research program P1-0055. References [1] Helfrich W. Blocked lipid exchange in bilayers and its possible influence on the shape of vesicles. Z Naturforsch 1974;29c:510–5. [2] Sheetz MP, Singer SJ. Biological-membranes as bilayer couples — molecular mechanism of drug-erythrocyte interactions. Proc Natl Acad Sci U S A 1974;71:4457–61. [3] Evans EA. Bending resistance and chemically induced moments in membrane bilayer. Biophys J 1974;14:923–31. [4] Svetina S, Žekš B. Elastic properties of closed bilayer membranes and the shapes of giant phospholipid vesicles. In: Lasic DD, Barenholz Y, editors. Handbook of nonmedical applications of liposomes. Boca Raton: CRC Press; 1996. p. 13–42. [5] Seifert U, Lipowsky R. Shapes of fluid vesicles. In: Lasic DD, Barenholz Y, editors. Handbook of nonmedical applications of liposomes. Boca Raton: CRC Press; 1996. p. 43–68.

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