Physical principles underlying the transduction of bilayer deformation forces during mechanosensitive channel gating

In mechanosensitive (MS) channels, gating is initiated by changes in intra-bilayer pressure profiles originating from bilayer deformation. Here we evaluated two physical mechanisms as triggers of MS channel gating: the energetic cost of protein-bilayer hydrophobic mismatches and the geometric consequences of bilayer intrinsic curvature. Structural changes in the Escherichia coli large MS channel (MscL) were studied under nominally zero transbilayer pressures using both patch clamp and EPR spectroscopic approaches. Changes in membrane intrinsic curvature induced by the external addition of lysophosphatidylcholine (LPC) generated massive spectroscopic changes in the narrow constriction that forms the channel 'gate', trapping the channel in the fully open state. Hydrophobic mismatch alone was unable to open the channel, but decreasing bilayer thickness lowered MscL activation energy, stabilizing a structurally distinct closed channel intermediate. We propose that the mechanism of mechanotransduction in MS channels is defined by both local and global asymmetries in the transbilayer pressure profile at the lipid-protein interface.     

The effect of local bending on gating of MscL using a representative volume element and finite element simulation

Many physiological processes such as cell division, endocytosis and exocytosis cause severe local curvature of the cell membrane. Local curvature has been shown experimentally to modulate numerous mechanosensitive (MS) ion channels. In order to quantify the effects of local curvature we introduced a coarse grain representative volume element for the bacterial mechanosensitive ion channel of large conductance (MscL) using continuum elasticity. Our model is designed to be consistent with the channel conformation in the closed and open states to capture its major continuum rheological behavior in response to the local membrane curvature. Herein we show that change in the local curvature of the lipid bilayer can modulate MscL activity considerably by changing both bilayer thickness and lateral pressure profile. Intriguingly, although bending in any direction results in almost the same free-energy cost, inward (cytoplasmic) bending favors channel opening, whereas outward (periplasmic) bending facilitates closing of the narrowest part of the MscL pore. This quantitative study using MscL as a model channel may have wide reaching consequences for the effect of local curvature on the physiological function of other types of prokaryotic and eukaryotic membrane proteins.     

The effect of local bending on gating of MscL using a representative volume element and finite element simulation

Many physiological processes such as cell division, endocytosis and exocytosis cause severe local curvature of the cell membrane. Local curvature has been shown experimentally to modulate numerous mechanosensitive (MS) ion channels. In order to quantify the effects of local curvature we introduced a coarse grain representative volume element for the bacterial mechanosensitive ion channel of large conductance (MscL) using continuum elasticity. Our model is designed to be consistent with the channel conformation in the closed and open states to capture its major continuum rheological behavior in response to the local membrane curvature. Herein we show that change in the local curvature of the lipid bilayer can modulate MscL activity considerably by changing both bilayer thickness and lateral pressure profile. Intriguingly, although bending in any direction results in almost the same free-energy cost, inward (cytoplasmic) bending favors channel opening, whereas outward (periplasmic) bending facilitates closing of the narrowest part of the MscL pore. This quantitative study using MscL as a model channel may have wide reaching consequences for the effect of local curvature on the physiological function of other types of prokaryotic and eukaryotic membrane proteins.     

Bending membranes

It is widely assumed that peripheral membrane proteins induce intracellular membrane curvature by the asymmetric insertion of a protein segment into the lipid bilayer, or by imposing shape by adhesion of a curved protein domain to the membrane surface. Two papers now provide convincing evidence challenging these views. The first shows that specific assembly of a clathrin protein scaffold, coupled to the membrane, seems to be the most prevalent mechanism for bending a lipid bilayer in a cell. The second reports that membrane crowding, driven by protein-protein interactions, can also drive membrane bending, even in the absence of any protein insertion into the bilayer.     

The separate profile structures of the functional calcium pump protein and the phospholipid bilayer within isolated sarcoplasmic reticulum membranes determined by X-ray and neutron diffraction

The detailed profile structure of the isolated sarcoplasmic reticulum membrane was studied utilizing a combination of X-ray and neutron diffraction. The water and lipid profile structures within the sarcoplasmic reticulum membrane were determined at 28 A resolution directly by neutron diffraction and selective deuteration of the water and lipid components. The previously determined electron density profile structure of the sarcoplasmic reticulum membrane at 12 A resolution was subjected to model refinement analysis constrained by the neutron diffraction results, thereby providing unique higher resolution calculated lipid and protein profile structures. It was found that the lipid bilayer profile structure of the isolated sarcoplasmic reticulum membrane is asymmetric, primarily the result of more lipid residing in the inner versus the outer monolayer of the sarcoplasmic reticulum lipid bilayer. The asymmetry in the lipid composition was necessarily coincident with a complimentary asymmetry in the protein mass distribution between the two monolayers in order to preserve the overall cross-sectional area of lipid and protein throughout the lipid bilayer region of the sarcoplasmic reticulum membrane profile structure. Approximately 50% of the mass of the total protein was found to be localized externally to the sarcoplasmic reticulum membrane lipid bilayer protruding from the outer lipid monolayer into the extravesicular medium. The structural features of the protein protrusion appear to be rather variable depending upon the environment of the sarcoplasmic reticulum membrane. This highly asymmetric structural organization of the sarcoplasmic reticulum membrane profile is consistent with its primary function of unidirectional calcium transport.     

How cholesterol is distributed between monolayers in asymmetric lipid membranes

The distribution of cholesterol in asymmetric lipid bilayers was studied by extensive coarse-grained molecular dynamics simulations. The effects of the lipid head group charge, acyl chain saturation, spontaneous membrane curvature and surface tension of the membrane were investigated. Four asymmetric bilayers containing DOPC, DOPS, DSPC or DSPS lipids were simulated on a time scale extended to tens of microseconds. We show that cholesterol strongly prefers anionic lipids to neutral and saturated lipid tails to unsaturated with a distribution ratio of ~0.7 in neutral/anionic bilayers and of ~0.4 in unsaturated/saturated bilayers. Multiple flip-flop transitions of cholesterol were observed directly, and their mean times ranged from 80 to 250?ns. It was shown that the distribution of cholesterol in the asymmetric membrane depends not only on the type of lipid, but also on the local membrane curvature and the surface tension. The membrane curvature enhances the influence of the lipid head groups on cholesterol distribution, while non-optimal surface tension caused by different areas per lipid in different monolayers increases the effect of the lipid tail saturation. It was clearly seen that the monolayers of asymmetric bilayers are interdependent. Mean distances from the bilayer center to cholesterol molecules depend not only on the type of the lipid in the considered monolayer but also on the composition of the opposite monolayer.     

Role of helix 0 of the N-BAR domain in membrane curvature generation

A group of proteins with cell membrane remodeling properties is also able to change dramatically the morphology of liposomes in vitro, frequently inducing tubulation. For a number of these proteins, the mechanism by which this effect is exerted has been proposed to be the embedding of amphipathic helices into the lipid bilayer. For proteins presenting BAR domains, removal of an N-terminal amphipathic α-helix (H0-NBAR) results in much lower membrane tubulation efficiency, pointing to a fundamental role of this protein segment. Here, we studied the interaction of a peptide corresponding to H0-NBAR with model lipid membranes. H0-NBAR bound avidly to anionic liposomes but partitioned weakly to zwitterionic bilayers, suggesting an essentially electrostatic interaction with the lipid bilayer. Interestingly, it is shown that after membrane incorporation, the peptide oligomerizes as an antiparallel dimer, suggesting a potential role of H0-NBAR in the mediation of BAR domain oligomerization. Through monitoring the effect of H0-NBAR on liposome shape by cryoelectron microscopy, it is clear that membrane morphology is not radically changed. We conclude that H0-NBAR alone is not able to induce vesicle curvature, and its function must be related to the promotion of the scaffold effect provided by the concave surface of the BAR domain.     

Phospholipid asymmetry in the isolated sarcoplasmic reticulum membrane

The total phospholipid content and distribution of phospholipid species between the outer and inner monolayers of the isolated sarcoplasmic reticulum membrane was measured by phospholipase A2 activities and neutron diffraction. Phospholipase measurements showed that specific phospholipid species were asymmetric in their distribution between the outer and inner monolayers of the sarcoplasmic reticulum lipid bilayer; phosphatidylcholine (PC) was distributed 48/52 +/- 2% between the outer and inner monolayer of the sarcoplasmic reticulum bilayer, 69% of the phosphatidyl-ethanolamine (PE) resided mainly in the outer monolayer of the bilayer, 85% of the phosphatidylserine (PS) and 88% of the phosphatidylinositol (PI) were localized predominantly in the inner monolayer. The total phospholipid distribution determined by these measurements was 48/52 +/- 2% for the outer/inner monolayer of the sarcoplasmic reticulum lipid bilayer. Sarcoplasmic reticulum phospholipids were biosynthetically deuterated and exchanged into isolated vesicles with both a specific lecithin and a general exchange protein. Neutron diffraction measurements directly provided lipid distribution profiles for both PC and the total lipid content in the intact sarcoplasmic reticulum membrane. The outer/inner monolayer distribution for PC was 47/53 +/- 1%, in agreement with phospholipase measurements, while that for the total lipid was 46/54 +/- 1%, similar to the phospholipase measurements. These neutron diffraction results regarding the sarcoplasmic reticulum membrane bilayer were used in model calculations for decomposing the electron-density profile structure (10 A resolution) of isolated sarcoplasmic reticulum previously determined by X-ray diffraction into structures for the separate membrane components. These structure studies showed that the protein profile structure within the membrane lipid bilayer was asymmetric, complementary to the asymmetric lipid structure. Thus, the total phospholipid asymmetry obtained by two independent methods was small but consistent with a complementary asymmetric protein structure, and may be related to the highly vectorial functional properties of the calcium pump ATPase protein in the sarcoplasmic reticulum membrane.     

High-resolution AFM of membrane proteins directly incorporated at high density in planar lipid bilayer

The heterologous expression and purification of membrane proteins represent major limitations for their functional and structural analysis. Here we describe a new method of incorporation of transmembrane proteins in planar lipid bilayer starting from 1 pmol of solubilized proteins. The principle relies on the direct incorporation of solubilized proteins into a preformed planar lipid bilayer destabilized by dodecyl-beta-maltoside or dodecyl-beta-thiomaltoside, two detergents widely used in membrane biochemistry. Successful incorporations are reported at 20 degrees C and at 4 degrees C with three bacterial photosynthetic multi-subunit membrane proteins. Height measurements by atomic force microscopy (AFM) of the extramembraneous domains protruding from the bilayer demonstrate that proteins are unidirectionally incorporated within the lipid bilayer through their more hydrophobic domains. Proteins are incorporated at high density into the bilayer and on incubation diffuse and segregate into protein close-packing areas. The high protein density allows high-resolution AFM topographs to be recorded and protein subunits organization delineated. This approach provides an alternative experimental platform to the classical methods of two-dimensional crystallization of membrane proteins for the structural analysis by AFM. Furthermore, the versatility and simplicity of the method are important intrinsic properties for the conception of biosensors and nanobiomaterials involving membrane proteins.     

Lipid composition and the lateral pressure profile in bilayers

The mechanisms by which variations in the lipid composition of cell membranes influence the function of membrane proteins are not yet well understood. In recent work, a nonlocal thermodynamic mechanism was suggested in which changes in lipid composition cause a redistribution of lateral pressures that in turn modulates protein conformational (or aggregation) equilibria. In the present study, results of statistical thermodynamic calculations of the equilibrium pressure profile and bilayer thickness are reported for a range of lipids and lipid mixtures. Large redistributions of lateral pressure are predicted to accompany variation in chain length, degree and position of chain unsaturation, head group repulsion, and incorporation of cholesterol and interfacially active solutes. Combinations of compositional changes are found that compensate with respect to bilayer thickness, thus eliminating effects of hydrophobic mismatch, while still effecting significant shifts of the pressure profile. It is also predicted that the effect on the pressure profile of addition of short alkanols can be reproduced with certain unnatural lipids. These results suggest possible roles of cholesterol, highly unsaturated fatty acids and small solutes in modulating membrane protein function and suggest unambiguous experimental tests of the pressure profile hypothesis. As a test of the methodology, calculated molecular areas and area elastic moduli are compared with experimental and simulation results.     

Inclusion-induced bilayer deformations: effects of monolayer equilibrium curvature

The energetics of protein-induced bilayer deformation in systems with finite monolayer equilibrium curvature were investigated using an elastic membrane model. In this model the bilayer deformation energy delta G(def) has two major components: a compression-expansion component and a splay-distortion component, which includes the consequences of a bilayer curvature frustration due to a monolayer equilibrium curvature, c(0), that is different from zero. For any choice of bilayer material constants, the value of delta G(def) depends on global bilayer properties, as described by the bilayer material constants, as well as the energetics of local lipid packing adjacent to the protein. We introduce this dependence on lipid packing through the contact slope, s, at the protein-bilayer boundary. When c(0) = 0, delta G(def) can be approximated as a biquadratic function of s and the monolayer deformation at the protein/bilayer boundary, u(0): delta G(def) = a(1)u(0)(2) + a(2)u(0)s + a(3)s(2), where a(1), a(2), and a(3) are functions of the bilayer thickness, the bilayer compression-expansion and splay-distortion moduli, and the inclusion radius (this expression becomes exact when the Gaussian curvature component of delta G(def) is negligible). When c(0) not equal 0, the curvature frustration contribution is determined by the choice of boundary conditions at the protein-lipid boundary (by the value of s), and delta G(def) is the sum of the energy for c(0) = 0 plus the curvature frustration-dependent contribution. When the energetic penalty for the local lipid packing can be ignored, delta G(def) will be determined only by the global bilayer properties, and a c(0) > 0 will tend to promote a local inclusion-induced bilayer thinning. When the energetic penalty for local lipid packing is large, s will be constrained by the value of c(0). In a limiting case, where s is determined only by geometric constraints imposed by c(0), a c(0) > 0 will impede such local bilayer thinning. One cannot predict curvature effects without addressing the proper choice of boundary conditions at the protein-bilayer contact surface.     

Mechanosensitive channel of large conductance

Microbial cells constitutively express the Large Conductance Mechanosensitive Channel which opens in response to stretch forces in the lipid bilayer. The channel protein forms a homopentamer with each subunit containing two transmembrane regions and gates via the bilayer mechanism evoked by hydrophobic mismatch and changes in the membrane curvature and/or transbilayer pressure profile. During the stationary phase and during osmotic shock the channel protein is up-regulated to prevent cell lysis. Pharmacological potential of MscL may involve discovery of new age antibiotics to combat multiple drug-resistant bacterial strains.     

Bilayer Thickness and Curvature Influence Binding and Insertion of a pHLIP Peptide

The physical properties of lipid bilayers, such as curvature and fluidity, can affect the interactions of polypeptides with membranes, influencing biological events. Additionally, given the growing interest in peptide-based therapeutics, understanding the influence of membrane properties on membrane-associated peptides has potential utility. pH low insertion peptides (pHLIPs) are a family of water-soluble peptides that can insert across cell membranes in a pH-dependent manner, enabling the use of pH to follow peptide-lipid interactions. Here we study pHLIP interactions with liposomes varying in size and composition, to determine the influence of several key membrane physical properties. We find that pHLIP binding to bilayer surfaces at neutral pH is governed by the ease of access to the membrane’s hydrophobic core, which can be facilitated by membrane curvature, thickness, and the cholesterol content of the membrane. After surface binding, if the pH is lowered, the kinetics of pHLIP folding to form a helix and subsequent insertion across the membrane depends on the fluidity and energetic dynamics of the membrane. We showed that pHLIP is capable of forming a helix across lipid bilayers of different thicknesses at low pH. However, the kinetics of the slow phase of insertion corresponding to the translocation of C-terminal end of the peptide across lipid bilayer, vary approximately twofold, and correlate with bilayer thickness and fluidity. Although these influences are not large, local curvature variations in membranes of different fluidity could selectively influence surface binding in mixed cell populations.     

Protein-Mediated Transformation of Lipid Vesicles into Tubular Networks

Key cellular processes are frequently accompanied by protein-facilitated shape changes in the plasma membrane. N-BAR-domain protein modules generate curvature by means of complex interactions with the membrane surface. The way they assemble and the mechanism by which they operate are largely dependent on their binding density. Although the mechanism at lower densities has recently begun to emerge, how membrane scaffolds form at high densities remains unclear. By combining electron microscopy and multiscale simulations, we show that N-BAR proteins at high densities can transform a lipid vesicle into a 3D tubular network. We show that this process is a consequence of excess adhesive energy combined with the local stiffening of the membrane, which occurs in a narrow range of mechanical properties of both the membrane and the protein. We show that lipid diffusion is significantly reduced by protein binding at this density regime and even more in areas of high Gaussian curvature, indicating a potential effect on molecular transport in cells. Finally, we reveal that the breaking of the bilayer topology is accompanied by the nematic arrangement of the protein on the surface, a structural motif that likely drives the formation of reticular structures in living cells.     

Protein-Mediated Transformation of Lipid Vesicles into Tubular Networks

Key cellular processes are frequently accompanied by protein-facilitated shape changes in the plasma membrane. N-BAR-domain protein modules generate curvature by means of complex interactions with the membrane surface. The way they assemble and the mechanism by which they operate are largely dependent on their binding density. Although the mechanism at lower densities has recently begun to emerge, how membrane scaffolds form at high densities remains unclear. By combining electron microscopy and multiscale simulations, we show that N-BAR proteins at high densities can transform a lipid vesicle into a 3D tubular network. We show that this process is a consequence of excess adhesive energy combined with the local stiffening of the membrane, which occurs in a narrow range of mechanical properties of both the membrane and the protein. We show that lipid diffusion is significantly reduced by protein binding at this density regime and even more in areas of high Gaussian curvature, indicating a potential effect on molecular transport in cells. Finally, we reveal that the breaking of the bilayer topology is accompanied by the nematic arrangement of the protein on the surface, a structural motif that likely drives the formation of reticular structures in living cells.     

Molecular mechanisms of spontaneous curvature and softening in complex lipid bilayer mixtures

Membrane reshaping is an essential biological process. The chemical composition of lipid membranes determines their mechanical properties and thus the energetics of their shape. Hundreds of distinct lipid species make up native bilayers, and this diversity complicates efforts to uncover what compositional factors drive membrane stability in cells. Simplifying assumptions, therefore, are used to generate quantitative predictions of bilayer dynamics based on lipid composition. One assumption commonly used is that “per lipid” mechanical properties are both additive and constant—that they are an intrinsic property of lipids independent of the surrounding composition. Related to this is the assumption that lipid bulkiness, or “shape,” determines its curvature preference, independently of context. In this study, all-atom molecular dynamics simulations on three separate multilipid systems were used to explicitly test these assumptions, applying methodology recently developed to isolate properties of single lipids or nanometer-scale patches of lipids. The curvature preference experienced by populations of lipid conformations were inferred from their redistribution on a dynamically fluctuating bilayer. Representative populations were extracted by both structural similarity and semi-automated hidden Markov model analysis. The curvature preferences of lipid dimers were then determined and compared with an additive model that combines the monomer curvature preference of both the individual lipids. In all three systems, we identified conformational subpopulations of lipid dimers that showed non-additive curvature preference, in each case mediated by a special chemical interaction (e.g., hydrogen bonding). Our study highlights the importance of specific chemical interactions between lipids in multicomponent bilayers and the impact of interactions on bilayer stiffness. We identify two mechanisms of bilayer softening: diffusional softening, driven by the dynamic coupling between lipid distributions and membrane undulations, and conformational softening, driven by the inter-conversion between distinct dimeric conformations.     

Nanoscale structural and mechanical effects of beta-amyloid (1-42) on polymer cushioned membranes: a combined study by neutron reflectometry and AFM Force Spectroscopy

The interaction of beta-amyloid peptides with lipid membranes is widely studied as trigger agents in Alzheimer's disease. Their mechanism of action at the molecular level is unknown and their interaction with the neural membrane is crucial to elucidate the onset of the disease. In this study we have investigated the interaction of water soluble forms of beta-amyloid Aβ(1-42) with lipid bilayers supported by polymer cushion. A reproducible protocol for the preparation of a supported phospholipid membrane with composition mimicking the neural membrane and in physiological condition (PBS buffer, pH=7.4) was refined by neutron reflectivity. The change in structure and local mechanical properties of the membrane in the presence of Aβ(1-42) was investigated by neutron reflectivity and Atomic Force Microscopy (AFM) Force Spectroscopy. Neutron reflectivity evidenced that Aβ(1-42) interacts strongly with the supported membrane, causing a change in the scattering length density profile of the lipid bilayer, and penetrates into the membrane. Concomitantly, the local mechanical properties of the bilayer are deeply modified by the interaction with the peptide as seen by AFM Force Spectroscopy. These results may be of great importance for the onset of the Alzheimer's disease, since a simultaneous change in the structural and mechanical properties of the lipid matrix could influence all membrane based signal cascades.     

Regulation of sodium channel function by bilayer elasticity: the importance of hydrophobic coupling. Effects of Micelle-forming amphiphiles and cholesterol

Membrane proteins are regulated by the lipid bilayer composition. Specific lipid-protein interactions rarely are involved, which suggests that the regulation is due to changes in some general bilayer property (or properties). The hydrophobic coupling between a membrane-spanning protein and the surrounding bilayer means that protein conformational changes may be associated with a reversible, local bilayer deformation. Lipid bilayers are elastic bodies, and the energetic cost of the bilayer deformation contributes to the total energetic cost of the protein conformational change. The energetics and kinetics of the protein conformational changes therefore will be regulated by the bilayer elasticity, which is determined by the lipid composition. This hydrophobic coupling mechanism has been studied extensively in gramicidin channels, where the channel-bilayer hydrophobic interactions link a "conformational" change (the monomer<-->dimer transition) to an elastic bilayer deformation. Gramicidin channels thus are regulated by the lipid bilayer elastic properties (thickness, monolayer equilibrium curvature, and compression and bending moduli). To investigate whether this hydrophobic coupling mechanism could be a general mechanism regulating membrane protein function, we examined whether voltage-dependent skeletal-muscle sodium channels, expressed in HEK293 cells, are regulated by bilayer elasticity, as monitored using gramicidin A (gA) channels. Nonphysiological amphiphiles (beta-octyl-glucoside, Genapol X-100, Triton X-100, and reduced Triton X-100) that make lipid bilayers less "stiff", as measured using gA channels, shift the voltage dependence of sodium channel inactivation toward more hyperpolarized potentials. At low amphiphile concentration, the magnitude of the shift is linearly correlated to the change in gA channel lifetime. Cholesterol-depletion, which also reduces bilayer stiffness, causes a similar shift in sodium channel inactivation. These results provide strong support for the notion that bilayer-protein hydrophobic coupling allows the bilayer elastic properties to regulate membrane protein function.     

Manipulating the folding of membrane proteins: using the bilayer to our advantage

The folding mechanisms of integral membrane proteins have largely eluded detailed study. This is owing to the inherent difficulties in folding these hydrophobic proteins in vitro, which, in turn, reflects the often apparently insurmountable problem of mimicking the natural membrane bilayer with lipid or detergent mixtures. There is, however, a large body of information on lipid properties and, in particular, on phosphatidylcholine and phosphatidylethanolamine lipids, which are common to many biological membranes. We have exploited this knowledge to develop efficient in vitro lipid-bilayer folding systems for the membrane protein, bacteriorhodopsin. Furthermore, we have shown that a rate-limiting apoprotein folding step and the overall folding efficiency appear to be controlled by particular properties of the lipid bilayer. The properties of interest are the stored curvature elastic energy within the bilayer, and the lateral pressure that the lipid chains exert on the their neighbouring folding proteins. These are generic properties of the bilayer that can be achieved with simple mixtures of biological lipids, and are not specific to the lipids studied here. These bilayer properties also seem to be important in modulating the function of several membrane proteins, as well as the function of membranes in vivo. Thus, it seems likely that careful manipulations of lipid properties will shed light on the forces that drive membrane protein folding, and will aid the development of bilayer folding systems for other membrane proteins.     

A Molecular Dynamics Simulation Study of Nanomechanical Properties of Asymmetric Lipid Bilayer

A very important part of the living cells of biological systems is the lipid membrane. The mechanical properties of this membrane play an important role in biophysical studies. Investigation as to how the insertion of additional phospholipids in one leaflet of a bilayer affects the physical properties of the obtained asymmetric lipid membrane is of recent practical interest. In this work a coarse-grained molecular dynamics simulation was carried out in order to compute the pressure tensor, the lateral pressure, the surface tension and the first moment of lateral pressure in each leaflet of such a bilayer. Our simulations indicate that adding more phospholipids into one monolayer results in asymmetrical changes in the lateral pressure of the individual bilayer leaflets. Interestingly, it has been observed that a change in phospholipid density in one leaflet affects the physical properties of unperturbed leaflet as well. The asymmetric behavior of the physical properties of the two leaflets as a result of a change in the contribution of the various intermolecular forces in the presence of additional phospholipids may be expressed formally.