CURVATURE AND LIPID MEMBRANES 2.798 TERM PAPER, FALL 2010

CURVATURE AND LIPID MEMBRANES 2.798 TERM PAPER, FALL 2010 NIRAJ INAMDAR 1. Introduction The cell membrane is the part of the cell that interacts directly with the extracellular environment, and which mediates communication between intracellular and extracellular agents. The cell membrane and lipid membranes in general play a considerable role in biological processes. If a lipid membrane is treated in as a continuum material, we may attempt to use physical variables from the classical theories of differential geometry and solid mechanics in order to characterize the mechanical state of the cell membrane. Hence, we will attempt to review some of the aspects related to the mechanics of lipid bilayers, and their relevance to cell biology. It is hoped that this short review will provide an adequate summary to the theory membrane curvature, and the application thereof to understanding of membrane curvature. 2. Lipid Membranes At the cell level, the primary membrane of relevance is the cell membrane. The cell membrane is the physical barrier that separates the interior of a cell from the extracellular environment. In general, the cell membrane may be said to be a subset of the larger class of biological membranes. Biological membranes are made from phospholipid bilayers, a structure which consists of two lipid molecule arrays arranged in such a way that the hydrophilic phosphate heads of the lipid molecule exist within the water-rich extracellular and intracellular environment, with the hydrophobic tails of the molecules oriented towards one another on the inside of the bilayer. The membrane typically has a net negative charge, and a large percentage of the membrane ( 50%) is made up of various membrane proteins that are crucial to cell-to-cell, cell-to-matrix, and extracellular to intracellular communication. 3. The Geometry of Surfaces The physical quantities that are used to describe the local geometry of a lipid bilayer may, under appropriate conditions, be those that are used to describe surfaces in the mathematical theory of differential geometry and in the theory of membranes and thin shells. Hence the language that will be used to characterize the geometry of membranes will be that from the general theory of surfaces. The primary quantity of interest in describing the local geometry of a surface is the curvature of the surface; and in qualifying what constitutes curvature, we will concentrate on surfaces embedded in three dimensional Euclidean space, E 3. In much the same way that the shape of a curve may be described locally by a circle, so may be the local shape of 1

2 NIRAJ INAMDAR Figure 1. The radii of curvature of the membrane bilayer, with individual monolayers shown. Thickness of the layers are greatly exaggerated. a surface at a given point be approximated by two mutually perpendicular circles; and the radii of these circles define the principal radii of curvature (figure 1). A comprehensive derivation of this fact may be found in any treatise on differential geometry 1. The radii of these circles R 1 and R 2 are equivalent to 1/κ 1 and 1/κ 2, respectively, where κ 1 and κ 2 are the principal curvatures. There exist two important algebraic expressions involving the principal curvatures [11]: the Gaussian curvature: (1) and the mean curvature: (2) K κ 1 κ 2 H 1 2 (κ 1 + κ 2 ) It is well-known that K and H are invariant under transformation of the local coordinates on the surface, so long as the transformation satisfies some basic conditions; hence, they will be taken to be the key geometrical quantities of interest. 4. Mechanics of the Thin Membrane Now that we have defined the geometrical quantities of interest in investigating surfaces, we may now consider some of the important mechanical aspects related to these. The theory here falls within the scope of the structural theory of thin shells and membranes, and appears to first have been applied to problems of biomembranes and cell biology by Helfrich (1973) [10]. It remains to be seen whether the lipid bilayer can indeed be treated as a continuum. 1 For instance, Levi-Civita (1950), Kreyszig (1959), and Lovelock and Rund (1975), among many others

CURVATURE AND LIPID MEMBRANES 2.798 TERM PAPER, FALL 2010 3 Indeed, for the adoption of such a treatment, we stipulate the following assumptions, consistent with Helfrich s theory and some of the other literature: (1) For the membrane surface area under consideration, the molecular density and intermolecular spacing is such that a physical deformation of the surface follows a stress-strain relationship that is Hookean. (2) The presence of proteins and other molecules embedded within the membrane outside those of particular interest (i.e. those which are curvature generating on a relevant scale) still permits local averaging of membrane characteristics, so that the local material homogeneity required for a continuum description holds. (3) The radius of curvature (R, say) at a given point is much greater than the membrane thickness d, i.e. R d, so as to permit application of the linear membrane or thin shell theory. Since a phospholipid bilayer is typically of thickness 50Å, it is typical to ignore the thickness d altogether from the point of view of elasticity manifest as membrane thickness change. (4) The tilt of the molecules is negligible, so that it may be assumed that the hydrocarbon chains that comprise the lipid bilayer are oriented normal to the surface of the membrane. The validity of this assumption is questionable (see section 4.2 below). 4.1. Stretching Elasticity. The lipid bilayer may be deformed in a purely tensional manner. If the bilayer is to be treated as an elastic material per assumption (1) above, we may write for the elastic energy of stretching w s (3) w s = 1 2 k s( a/a) 2 where k s is the elastic modulus of stretching (with units of energy per area) and a/a is the areal change over an initial area a. The corresponding stress is then (4) σ s = k s ( a/a)e where e is a unit vector perpendicular to a cut in the membrane and tangential to the surface. As Helfrich notes, a biomembrane is capable of exchanging its lipid molecules with the surrounding environment, and the elastic forces which are described in equation (19) may only last for a limited time. As discussed in assumption (3), the elasticity associated with membrane thickness change may be neglected. 4.2. Tilt. One aspect of the phospholipid membrane that should be considered is the tilt of the molecules. If we take as an axis the unit normal vector to the surface n at a given point, the lipid molecules, if assumed to be rod-like, may have an average orientation d not necessarily parallel to n. An elastic torque per unit area m t may be defined in the following manner: (5) m t = k t (n d) where k t is a tilt-elastic modulus (with units again of energy per area) and is the wedge product defined in the usual way. Equation (20) is equivalent to saying (6) m t = k t sin θ

4 NIRAJ INAMDAR if the deviation of d from n is given by an angle θ. The elastic energy associated with the tilt may then be written (7) w t = 1 2 k t(n d) 2 = 1 2 k t sin 2 θ since d and n are taken to be unit vectors, while the stress itself may be expressed as (8) σ t = m t e = k t (n d) e = k t e n d by the usual rules of exterior algebra. Far more comprehensive models of tilt have been developed (for instance, [9]), and it is envisioned that these may be directly applied for analysis to the biological problems discussed in section 5. 4.3. Bending Elasticity. Several quantities, in addition to those given above, may be defined for the analysis of membrane curvature. Among these, we may specify the spontaneous curvature of the membrane bilayer, Js B, which indicates the curvature of the surface in its non-deformed, unstressed state [4]. This will in general depend on the spontaneous curvatures of the individual monolayers that comprise the bilayers, Js out and Js in, for the outer and inner monolayers, respectively. At a given point in an undisturbed membrane, we have J s = H s, where H s is the mean curvature in the undisturbed state. The elasticity associated with bending may be characterized by the elastic bending moduli, k m and k b, for the monolayer and bilayer, respectively, while those associated with the Gaussian curvature are given by k m and k b, for the monolayer and bilayer respectively. In an arbitrary bending configuration, we may associate with it a corresponding bending energy, w b ; we may take as a definition ([10],[4]) for the monolayer bending modulus k m and the modulus of Gaussian curvature k m the following expression (9) w b = 1 2 k m(h J s ) 2 + k m K where J s is the spontaneous curvature for a monolayer. 4.4. Curvature Generation. The generation of membrane curvature is of particular interest, as we may be able to elucidate some of the energy expenditures and possibly the physical state of the membrane simply by observing the shape of the membrane. There are several methods by which curvature may be generated in a membrane. We may categorize them into those involving curvature generation by bending (the scaffold mechanism, local spontaneous curvature generation, and the bilayer couple mechanism) and those involving pulling by molecular motors. 4.4.1. Scaffold Mechanism. The scaffold mechanism refers to the condition where a membrane is compliant to a protein which binds to its surface. This requires the protein to have a predefined, rigid structure so as to counteract the tendency of the membrane to return to its non-deformed state, and for the protein to have a sufficiently strong interaction with the lipid surface of the membrane. In biological systems, clathrin-adaptorprotein complexes, proteins with BAR domains, and COPI and COPII are known to be implicated in scaffolding [2].

CURVATURE AND LIPID MEMBRANES 2.798 TERM PAPER, FALL 2010 5 4.4.2. Local Spontaneous Curvature Generation. Some proteins may possess moieties which are amphipathic, or have both hydrophilic and hydrophobic (or lipophilic) components. When this component of the protein becomes embedded in the membrane, a tilt will likely be induced, followed by the reorganization of the membrane to a tiltminimizing, curved state. Following Zimmerberg [4], we may describe this action as a wedge disrupting the structural organization of the lipid head groups of the membrane. Although the curvature generated by this mechanism is localized, the overall effect can be additive, so that, if multiple proteins along a section of membrane are inserted into membrane, an overall curvature may be generated. For instance, in order to create a liposome tubule (that is, a tube formation for which 2πR 1 could be equal to the length in one direction of a flat membrane with R 2 0), it has been shown that, in one instance, less than 30% of the surface needs to be covered by amphipathic moieties [3]. Two proteins that are suspected to play a role in local spontaneous curvature generation are epsin [7], amphiphysin [4], Sar1 [8]. Epsin, when added to liposomes, generates tubules. The structure of the epsin N-terminal homology (ENTH) domain indicates that epsin is not a scaffold protein. This domain does, however, seem to have a binding affinity for certain factors present in the membrane, and the insertion of the ENTH domain helix into bilayer may be responsible for instances of curvature generation. Likewise, the amphipathic helix of amphiphysin seems to perform the same function. In fact, if mutations that reduce the hydrophobic nature of amino acids in the helix are present, it is known that higher concentrations of amphiphysin are necessary in order to produce curvature in the membrane, corroborating the speculation. The insertion of the N-terminal α-helix of Sar1 generates curvature, and the replacement of these hydrophobic termini with alanine decreased the ability of Sar1 to generate tubular curvature. 4.4.3. Bilayer Couple Mechanism. Closely related to the phenomenon of local spontaneous curvature generation is the bilayer couple mechanism. If the amphipathic portion of the protein penetrates only through one monolayer, there will be an increase in the area of this monolayer. Subsequently, molecular interactions of the tail groups will induce curvature in response to this areal asymmetry. 4.4.4. Pulling. Suppose we have a lipid monolayer with a spontaneous curvature J s. It is possible to deform this monolayer simply by pulling, typically by the action of molecular motors. The pulling force will supply an energy to the system sufficient to generate stretching and curvature consistent with equations (3), (9), and (7) (if we consider tilt) with respect to the stretching energy w s, bending energy w b, and tilt energy w t. 4.5. The Energy Cost of Generating Curvature. We may assess the energy cost of transforming an undisturbed membrane into a particular shape. For instance, if we consider a cylindrical membrane (perhaps a lipid tubule) of radius R and length L, then the energy requirement E cyl is (neglecting tilt) evidently (10) E cylinder = 2πk b ( 1 2R J S )

6 NIRAJ INAMDAR If this energy is zero (E cyl = 0), then the membrane can form the tubule itself, and the resulting spontaneous curvature J s will be [4] (11) J s = 1 2R which is consistent with the identification of J s with H. For a spherical membrane of radius R, we have the energy cost of creation E sphere as [4] E sphere = 8π (k b + 12 ) (12) k b 8πk b RJ s Again, we may set this equal to zero to determine the spontaneous curvature J s : J s = 1 ( 1 + 1 ) k b (13) R 2 Some typical values of the curvatures may be found in the literature (e.g. [13]), with k b 6.4 10 20 J and k b 8 10 20 J. 5.1. Vesicle Fusion. 5. Biological Examples of Membrane Curvature 5.1.1. Introduction. Between neurons, chemical signals in the form of neurotransmitters are conveyed across synapses, or synaptic clefts, which characterize the gap between them. Within a neuron, neurotransmitters are carried within vesicles, spherical lipid bilayers which carry within them transmitter molecules. The vesicles are located at presynaptic terminals, that is, at specific locations adjacent to the synapse. When a neuron receives a particular signal, the vesicles fuse with the cell membrane, and release neurotransmitters into the immediate extracellular environment, viz. the synapse. Ca 2+ is known to be responsible for initiating this fusion event. It has been posited [3] that exact mechanism for this is linked to a deformation of the synaptic membrane by Ca 2+ sensing proteins such as synaptotagmin-1 and Doc2b, which may act in a manner similar to those proteins mentioned in section 4.4.2. Though in his seminal paper Helfrich states that he believes the phenomenon of tilting (see section 4.2) so minute as to be negligible, the presence of tilting does appear, according to [3] seem to be a fundamental mechanism in curvature generation as a result of protein implantation into the membrane during the fusion process. k b 5.1.2. Induction of Curvature on Membranes. In addition to proteins such as synaptotagmin-1 and Doc2b, the fusion process includes the SNARE class of proteins, proteins that are embedded in the membrane of both the vesicle and the target synaptic membrane. The primary functional component of synaptotagmin-1 is two C2 domains at carboxyl termini, at which Ca 2+ may bind. The presence of the positive calcium ion serves to eliminate the net negative charge of the membrane, and subsequently, the C2 domains may become bound and embedded into the membrane. The penetration extends through just the outer monolayer, approximately the length of the lipid backbone [1]. Since it is well-known that the insertion of amphipathic moieties into the membrane induces a curvature (see sections 4.4.2 and 4.4.3 above), it is certainly possible that the subsequent

CURVATURE AND LIPID MEMBRANES 2.798 TERM PAPER, FALL 2010 7 membrane displacement caused by the C2 domains of synaptotagmin-1 will induce a curvature in the manner consistent with the bilayer couple mechanism discussed above. Subsequently, the process of fusion, as conjectured by McMahon, et al., is as follows: A SNARE protein component on a vesicle binds with its target counterpart in the presynaptic cell membrane, bringing the cells in proximity to one another. The C2 domains of the synaptotagmin-1 will bind to the SNARE complex, and, if they have bound to Ca 2+, can become lodged in the membrane. The domains then pinch the membrane, buckling it and forming the so-called hemifusion stalk [5]. This high-curvature region will be unstable, and energy minimization will dictate the creation of a fusion pore linking the vesicle and the membrane, so that the vesicle may release its contents into the synaptic cleft. 5.2. Protein Localization by Recognition of Curvature; Sensing Curvature. 5.2.1. Introduction. The localization of proteins in the intracellular environment has been recognized to be crucial during cell morphogenesis. For instance, in certain bacteria, it has been found that a large percentage of encoded proteins ( 10%) exhibit particular subcellular localization [6]. Localization in bacteria can be important for processes such as cell division and chromosome organization. The standard model for how and why cells localize has typically been a diffusion and capture model, in which proteins diffuse either through the cytosol or, in the case of transmembrane proteins, through the membrane itself. In this scenario, a protein arrives at its destination by virtue of the fact that it is captured by some local target which recognizes it. This model, however accurate it may be in general, fails to explain how the target has been localized to begin with. One hypothesis, supported by experiments, is that proteins may localize at membranes in a non-biologically driven manner, that is, driven by geometrical cues purely [12]. 5.2.2. Protein Localization in Bacillus subtilis. Investigations of the developmental program of Bacillus subtilis have provided indications that localization of the protein SpoVM is dependent upon a slightly curved surface. The experimental evidence comes from the following set up: prior to starvation, B. subtilis sporulates, or creates an asymmetrical division in the cell, resulting in a mother cell and a forespore. Eventually the septum, or separation, between the forespore and the mother cell migrates around the forespore, breaking it off so that the forespore resides as an organelle within the mother cell. This spore will mature, and as it does, a protein coat will deposit on the surface of the spore. SpoVM is one the proteins which localizes on surface of the spore, and it is responsible for anchoring the base layer of the protein coat directly to the spore surface. It, too, forms an amphipathic α-helix, which is inserted into the spore membrane. A purely geometrical reason for SpoVM localization is supported by the fact that during mother cell-forespore separation, the only present positive curvature is that of the forespore surface, and it is here where SpoVM localizes; the lack of biological actuation of this process is suggested by the inability to identify a protein that was solely responsible for the localization of SpoVM. Several observations substantiate this [12]: (1) In experiments involving mutant cells in which the septum did not curve, SpoVM localized indiscriminately to all available surfaces, indicating that the presence of curvature is necessary for precise localization.

8 NIRAJ INAMDAR (2) SpoVM produced in mutant E. coli and S. cerevisiae localized to convex surfaces, while a mutant SpoVM mislocalized in B. subtilis, suggesting that it was not a B. subtilis-specific factor that caused mislocalization, while any convex surface would support localization. (3) In an experiment in which SpoVM was incubated with with population of variouslysized vesicles, the SpoVM preferentially localized to the smallest vesicles (those with the greatest curvature). Together, these findings suggest that, indeed, SpoVM is capable of recognizing the curvature of the membrane. Further examples of localization of proteins in bacteria may be found in the literature, e.g. [6]. 5.2.3. Sensing Curvature. The discussion above suggests that proteins are indeed capable of sensing the curvature of a lipid membrane surface. The question then arises as to what mechanism or mechanisms are involved in sensing surface curvature. There are several means through which this may be done [4]. For instance, proteins that partake in scaffolding or local spontaneous curvature generation sense curvature in the sense that they are affected by the energy costs associated with protein binding to the surface or the insertion of the amphipathic moiety into the membrane. Proteins that have a curvature but are not stiff enough to act as scaffolds can sense curvature via the energy expenditure that exists in bending the protein so that it conforms to the membrane surface. Finally, it is possible that pliable proteins may by bent by the membrane (via a collision or electrostatic interaction, for example), hence opening up cryptic sites in the protein which can then take part in various reactions. 6. Conclusion We have provided a short review of some of the mechanical aspects of the lipid membrane, and the possible role that it may play in the cell. The field is one of intense interest, and the speculation that purely mechanical responses can influence and be influenced by cellular processes is increasing based on promising experimental results. While we have outlined some of the basic quantities of interest in quantifying the cell membrane, more refined models are being and will continue to be developed so that, like any good model, predictions of membrane geometry-influenced behaviors may be predicted and ultimately verified experimentally. In particular, it is hoped that the speculations outlined above be given a definitive explanation in terms of membrane geometry.

CURVATURE AND LIPID MEMBRANES 2.798 TERM PAPER, FALL 2010 9 References [1] Edwin R. Chapman. How does synaptotagmin trigger neurotransmitter release? Annual Review of Biochemistry, 77:615 641, 2008. [2] L.V. Chernomordik and M.M. Kozlov. Protein-lipid interplay in fusion and fission in biological membranes. Annual Reviews of Biochemistry, 72:175 207, 2003. [3] Harvey T. McMahon et al. Membrane curvature in synaptic vesicle fusion and beyond. Cell, 140:601 605, 2010. [4] Joshua Zimmerberg et al. How proteins produce cellular membrane curvature. Nature Reviews: Molecular Cell Biology, 7:9 18, 2006. [5] L. Chernomordik et al. Lipids in biological membrane fusion. The Journal of Membrane Biology, 146:1 16, 1995. [6] L. Shapiro et al. Why and how bacteria localize proteins. Science, 326:1225 1228, 2009. [7] Marijn G.J. Ford et al. Curvature of clathrin-coated pits driven by epsin. Nature, 419:361 366, 2002. [8] Robert V. Stahelin et al. Contrasting membrane interaction mechanisms of ap180 n-terminal homology (anth) and epsin n-terminal homology (enth) domains. The Journal of Biological Chemistry, 278:28993 28999, 2003. [9] M. Hamm and M.M. Kozlov. Elastic energy of tilt and bending of fluid membranes. The European Physical Journal E, 3:323335, 2000. [10] Wolfgang Helfrich. Elastic properties of lipid bilayers. Z. Naturforsch, 28c:693 703, 1973. [11] Erwin Kreyszig. Differential Geometry. Dover, New York, 3rd edition, 1991. [12] Kumaran S. Ramamurthi. Protein localization by recognition of membrane curvature. Current Opinion in Microbiology, 13:1 5, 2010. [13] D.P. Siegel and M.M. Kozlov. The gaussian curvature elastic modulus of n-monomethylated dioleoylphosphatidylethanolamine: relevance to membrane fusion and lipid phase behavior. Biophysical Journal, 87:366 374, 2004.