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Membrane-Protein Interactions in Mechanosensitive Channels

Paul Wiggins ^* and Rob Phillips 

^* Department of Physics, and Division of Engineering and Applied Physics, California Institute of Technology, Pasadena, California

Correspondence: Address reprint requests to Paul A. Wiggins, Caltech, Physics, 1200 E. California Blvd., MC 103-33, Pasadena, CA 91106. Tel.: 626-395-3106; E-mail: pwiggins{at}caltech.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION FREE ENERGY OF THE... FREE ENERGY ESTIMATES AND... APPLICATION TO MscL GATING CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
In this article, we examine the mechanical role of the lipid bilayer in ion channel conformation and function with specific reference to the case of the mechanosensitive channel of large conductance (MscL). In a recent article we argued that mechanotransduction very naturally arises from lipid-protein interactions by invoking a simple analytic model of the MscL channel and the surrounding lipid bilayer. In this article, we focus on improving and expanding this analytic framework for studying lipid-protein interactions with special attention to MscL. Our goal is to generate simple scaling relations which can be used to provide qualitative understanding of the role of membrane mechanics in protein function and to quantitatively interpret experimental results. For the MscL channel, we find that the free energies induced by lipid-protein interaction are of the same order as the measured free energy differences between conductance states. We therefore conclude that the mechanics of the bilayer plays an essential role in determining the conformation and function of the channel. Finally, we compare the predictions of our model to experimental results from the recent investigations of the MscL channel by a variety of investigators and suggest a suite of new experiments.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION FREE ENERGY OF THE... FREE ENERGY ESTIMATES AND... APPLICATION TO MscL GATING CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The mechanosensitive channel of large conductance (MscL) is a compelling example of the interaction of a membrane protein and the surrounding lipid bilayer membrane. MscL is gated mechanically (Blount and Moe, 1999) and belongs to a growing class of proteins which have been determined to be mechanosensitive (Gillespie and Walker, 2001; Hamill and Martinac, 2001). In a recent short article (Wiggins and Phillips, 2004), we have argued that the mechanics of the bilayer is an important partner in the phenomena of mechanotransduction and channel function. In particular, we considered a simplified model where only the radius of the channel changes in transitions between the open and closed states. In this article, we present our free energy calculations in more generality and detail. Specifically, we have calculated the free energy due to the bilayer deformation as a result of the presence of a membrane protein using an analytic model developed for the study of bilayer mechanics (Canham, 1970; Helfrich, 1973; Evans, 1974). Many of the theoretical techniques exploited here have already been used with success in describing the role of the bilayer in the mechanics of the Gramicidin channel (e.g., Huang, 1986). In this article, we have applied asymptotic approximations to the exact solutions of this model, permitting many of the important results to be expressed, estimated, and understood with simple scaling relations. These scaling relations are then applied to estimate the relative importance of each and every term in the bilayer free energy budget. We find that the bilayer deformation free energy can be of the same order as the free energy differences between conformational states of the MscL channel as measured by Sukharev et al. (1999). These results strongly suggest that bilayer deformation plays an important role in determining the protein conformation, and therefore function, of transmembrane proteins in general, and MscL in particular. Although we have explicitly estimated the size of the bilayer deformation energy exclusively for the geometry of MscL, the results can easily be re-evaluated and reinterpreted in the context of other transmembrane proteins and mechanosensitive channels, in particular: MscS (Bass et al., 2002), Alamethicin (Opsahl and Webb, 1994), etc.

We emphasize that our goal in this current work is not to attempt a quantitative understanding of all of the degrees of freedom of the channel and bilayer, but rather to build a tractable model for the role of bilayer mechanics in the function of the MscL channel, while developing the model in more detail than in our previous article (Wiggins and Phillips, 2004).

The MscL channel is gated by membrane tension and has been studied extensively in patch-clamp experiments (Sukharev et al., 1999; Perozo et al., 2002a). Although several substates have been identified (Sukharev et al., 1999), the channel typically resides in one of two primary conductance states. At low tension the channel is almost exclusively closed (C). As the tension is increased the open state (O) becomes ever more prevalent, until it dominates at high tension. Rees and co-workers have solved the structure for one conformation using x-ray crystallography which appears to be the closed state (Chang et al., 1998). The open state has been modeled by a number of groups (Sukharev et al., 2001; Betanzos et al., 2002; Perozo et al., 2002b).

The outline of the article is as follows. In Free Energy of the Bilayer-Inclusion System, we briefly discuss the bilayer model, presenting a table of results showing the relative importance of different free energy penalties for bilayer deformation, and then define the generalized forces we use to discuss the effects of bilayer deformation induced by protein conformational change. In Free Energy Estimates and Physical Interpretation, we estimate the sizes of the bilayer deformation energy and forces for MscL, give a brief physical discussion of mechanisms that give rise to the bilayer deformation energy, and discuss the scaling of these bilayer deformation energies. In Application to MscL Gating, we compare our predictions for a two-state MscL model to experimental measurements made by Perozo et al. (2002a,b), Powl et al. (2003), Yoshimura et al. (2004), and others. In the Appendix, we provide a unit conversion table, bilayer parameters, the full names of the lipids, detailed derivations, and a discussion of the approximations used.

	FREE ENERGY OF THE BILAYER-INCLUSION SYSTEM

TOP ABSTRACT INTRODUCTION FREE ENERGY OF THE... FREE ENERGY ESTIMATES AND... APPLICATION TO MscL GATING CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
We begin by considering the free energy of the system as a whole: protein and bilayer. We assume that the system is in thermal equilibrium and define the free energy differences between states in the standard way as

(1)

where _i is the probability of state i and the free energy differences are defined with respect to the closed state. We can divide each of these free energies into two parts,

(2)

where G_P is the free energy associated with the protein's conformation and G is the free energy induced in the bilayer by the protein inclusion and includes both a deformation free energy from the bulk of the bilayer and an interaction energy at the interface between the inclusion and bilayer. For the sake of brevity we will usually refer to both of these bilayer-related contributions to the free energy as the bilayer deformation energy. Although a complete understanding of channel gating and function must encompass knowledge of both components of the free energy, G_P and G, our analysis in this article centers almost entirely on the bilayer deformation energy, G. Several groups have used molecular dynamics and related techniques to study G_P or G in its entirety (Gullingsrud et al., 2001; Gullingsrud and Schulten, 2002, 2003; Bilston and Mylvaganam, 2002; Elmore and Dougherty, 2003) but as is often the case for biological systems, these studies have been handicapped by the size of the MscL system. It is too complex for direct simulation on biologically relevant timescales. From an experimental standpoint, Sukharev et al. (1999) have measured the free energy differences between different states in MscL and have found differences of order 10 kT. These results reveal the energy scale associated with MscL gating and provide a reference by which different contributions to the free energy will be judged for their importance.

Our first goal in what follows is to persuade the reader that G is large enough to be of interest. That is, since it is clear that there are several distinct contributions to the overall free energy budget, we illustrate that the contribution due to the inclusion-induced bilayer deformation is comparable to the measured free energy differences between states. Since these contributions to G are of the same order of magnitude as G, we conclude that the effects of bilayer deformation are potentially interesting. Perozo et al. (2002a) have already answered this question experimentally, demonstrating that bilayer characteristics such as lipid acyl-chain length significantly affect the free energy.

The calculation of the bilayer free energy
The elastic deformation of the bilayer surrounding the channel is approximated with a model developed by Canham (1970), Helfrich (1973), and Evans (1974). Huang (1986) has applied this model to calculate the deformation energies induced by inclusions. These calculations have been elaborated upon by others: notably by Andersen and co-workers (Nielsen et al., 1998; Lundbæk et al., 1996; Lundbæk and Andersen, 1994; Goulian et al., 1998) and Dan and co-workers (Dan et al., 1994, 1993; Dan and Safran, 1995, 1998). Specifically, Goulian et al. (1998) have studied a similar model including applied tension. The bilayer deformation energy in this model is given by

(3)

where

(4)

and

(5)

G± are the free energies due to the curvature and the tension in the top and bottom surfaces of the bilayer and G_I is the interaction free energy between these two surfaces. Locally, the curvature of the top (or bottom) surface of the bilayer is described by the shape operator, , a rank-two tensor. The trace of this tensor is twice the mean curvature and its determinant is the Gaussian curvature. The energetic cost for increasing the mean curvature of the top (or bottom) surface of the bilayer is the bending modulus, K_B/2. The energetic cost for increasing the Gaussian curvature of the top (or bottom) surface is the Gaussian bending modulus, K_G/2. We have chosen this normalization so that the effective moduli for the bilayer as a whole are K_B and K_G.

The addition of certain surfactants and nonbilayer lipids results in the lowest energy conformation of a single layer of lipids being curved (Israelachvili, 1991; Gruner, 1989). This spontaneous curvature is introduced into the model through non-zero values of C_±, the spontaneous curvatures of the upper and lower layers of the bilayer, respectively. We define the composite and the mean spontaneous curvatures for the bilayer as

(6)

(7)

respectively. The energetic penalty associated with changes in the area of the top and bottom surfaces of the bilayer are the tensions, ±, respectively. We assume that the tensions in the two layers are equal since, on long timescales, the lipids can switch between the two leaflets to equalize the tension. The total tension, = 2_±, is an externally tunable parameter (see the Appendix for further discussion). The value 2u is the difference between the local thickness of the bilayer and the equilibrium thickness 2a. The energetic cost for changing the thickness of the bilayer is the compression-expansion modulus, K_A. For further discussion of the model, the reader is invited to view the extensive discussions in the literature (Helfrich, 1973; Huang, 1986; Dan et al., 1994; Goulian et al., 1998; etc.)

The presence of the channel will perturb the bilayer locally. To calculate the perturbation to the free energy due to the channel, we will assume that the radius of curvature corresponding to the vesicle or cell in which the inclusion is embedded is very large in comparison to the length scale of the inclusion itself and that the perturbation due to the bilayer inclusion is small enough to allow the equations to be linearized. In this approximation scheme, the bilayer deformation energy is

(8)

where is the expanded effective free energy density (written out in its expanded form in the Appendix) and ' is a Cartesian plane minus a circular inclusion of radius R. We can safely integrate out to infinity since the perturbation to the free energy density is localized around the inclusion. To construct the effective free energy density we describe the out-of-plane displacements of the upper and lower surfaces of the bilayer with the functions and , respectively, on ', as shown in Fig. 1. It is more transparent to work with the linear combinations of these two functions (Fournier, 1999), namely,

(9)

(10)

where h is the average position of the upper and lower surfaces of the bilayer which we will refer to as the midplane, and u is one-half the difference of the bilayer thickness and the equilibrium thickness. The overall structural picture is shown in Fig. 1 where the localized perturbation of the bilayer is depicted schematically.

View larger version (19K): [in this window] [in a new window]   FIGURE 1  A schematic picture of the bilayer-inclusion model. The geometry of the inclusion is described by four parameters: the radius R, the thickness W, and the radial slopes H'± of the top and bottom surfaces of the bilayer, respectively. If the surfaces of the bilayer are locally normal to the interface of the inclusion, as depicted above, H'± = ± in the small-angle approximation. The bilayer equilibrium thickness is 2a. The fields h ± (r) are the z displacements of the top and bottom surfaces of the bilayer, respectively. Their average is the midplane displacement, h(r), and half their difference is u(r) + a. The value u(r) is the local thickness deformation of a single leaflet of the bilayer. At the interface, twice this deformation, 2U, is the hydrophobic mismatch, W – 2a. The generalized forces on the inclusion induced by the bilayer are depicted for positive values. F is the expansion-compression force, is the tension, h is the midplane torque, and u is the shape torque.

(11)

(12)

which are again discussed at length in the literature (Huang, 1986; Nielsen et al., 1998; Fournier, 1999). The solution to these equations for the fields and can be written in terms of modified Bessel functions in cylindrical coordinates (Huang, 1986).

Due to the hydrophobic residues of the protein inclusion, we assume that the bilayer adheres to the external surface of the protein. As will be described in more detail below, the matching condition at this surface dictates half the boundary conditions for the bilayer (the remaining boundary conditions dictate that the bilayer is unperturbed at infinity). We consider proteins with azimuthal (cylindrical) symmetry. Although the MscL channel is not truly azimuthally symmetric, as a homo-pentamer, it is highly symmetric, at least in the closed state as the x-ray crystallography structure has demonstrated (Chang et al., 1998). To clearly distinguish values of the functions at the boundaries from the corresponding functions themselves, we will denote these parameters with capital letters. We fix the bilayer thickness, 2U + 2a, to match the hydrophobic thickness of the protein, W, at the interface, r = R,

(13)

2U is called the hydrophobic mismatch since it is the difference between the equilibrium thickness of the bilayer, 2a, and the thickness of the protein, W. For real proteins it is quite difficult to define exactly what one means by this region since real structures are not purely hydrophobic in the transmembrane region.

The closed state and a proposed model of the open state colored by hydrophobicity are depicted in Fig. 2.

View larger version (58K): [in this window] [in a new window]   FIGURE 2  Models of the closed and open states colored by hydrophobicity (Sukharev et al., 2001). Although the general region spanned by the membrane is evident from the hydrophobic regions on the protein interface, it is difficult to precisely define the thickness of this region. A closed-state thickness has been inferred from the data of Powl et al. (2003) and this region is schematically marked on the model of the closed state. Additional confirmation of this estimate for the hydrophobic thickness comes from the simulation of Elmore and Dougherty (2003).

(14)

or alternatively,

(15)

(16)

where ' is the derivative with respect to r, the radial distance from the inclusion. A physical interpretation of these slopes might be to assume the bilayer's surfaces are normal to the protein's surface at the boundary, although this need not be the case (Nielsen et al., 1998). At infinity we assume that the bilayer is unperturbed which may be cast in mathematical terms as

(17)

(18)

Solving the equilibrium equations for a given set of boundary conditions and plugging these solutions into the surface integral for the bilayer deformation energy results in the bilayer deformation energy for a given configuration of the protein (Huang, 1986). Each protein configuration corresponds to a different outcome for the bilayer deformation energy. This energy has been divided into several contributions based on the physical mechanism giving rise to it. In Table 1, we present a summary of these results. Brief derivations may be found in the Appendix. Generally, the bilayer deformation energies lend themselves to simple scaling laws, except for two cases: thickness and midplane deformation. In these cases the exact results to the model are somewhat complicated and the results that appear in the table are limits which are derived and discussed in the Appendix.

(19)

(20)

This can be recast verbally as the statement that the top and bottom surfaces of the bilayer have the same slope at the boundary and there is no bend in the inclusion interface. In the small-angle limit, H' can be replaced by the angle away from normal of the interface. If we do introduce a bend in the middle of the interface, the orientations of the upper and lower interfaces are independent. Assuming that the interface of the bilayer is normal to the protein surface, we can replace the slopes with the angles away from normal, ± (Dan and Safran, 1998), in the small-angle limit, as pictured schematically in Fig. 1.

Forces, torques, and tensions
The physical effects of bilayer deformation on the inclusion conformation can be recast in a more intuitive form by appealing to forces, tensions, and torques rather than free energies. For example, most of the bilayer deformation energies will generate a tension on the interface due to their radial dependence. The applied tension, , is not the whole story! The generalized forces are obtained by differentiating the bilayer deformation energy with respect to bilayer excursions. Implicitly, these generalized forces are defined through

(21)

where A R² is the area of the protein, H'± are the slopes of the bilayer surfaces at the interface, and W is the thickness of the hydrophobic region of the protein. Explicitly, these generalized forces may be written as

(22)

(23)

(24)

Since we have already used to denote the applied tension, we use to denote the net radial tension on the inclusion interface: the sum of the applied tensions and other bilayer deformation induced contributions. When the tension is positive, it is tensile. F is the compression-expansion force, normal to the plane of the bilayer, acting on the inclusion. When the compression-expansion force is positive, it acts to induce inclusion-thickness expansion. The values _± are cylindrical torques acting on the top and bottom surfaces of the inclusion around the midplane. It will usually be more convenient to work with the torques complimentary to H' and U' rather than H'_±. We define the midplane torque as the cylindrical torque on the interface as a whole:

(25)

When the midplane torque is positive, it acts to induce increases in the midplane slope. This cylindrical torque is generated by bending stresses alone and is therefore related to the principal curvatures at the boundary (Landau and Lifshitz, 1986) through the relation

(26)

where is the circumference of the inclusion, and and are the principal curvatures at the boundary of the midplane, in the directions parallel and perpendicular to the boundary, respectively. (For azimuthally symmetric surfaces the principal curvatures are always radial and azimuthal, and furthermore the azimuthal curvature is where r is the cylindrical radius, and _N is the angle of the upward surface normal away from vertical. For example, see Boal, 2002.) We can define the shape torque as the cylindrical torque complimentary to U',

(27)

When the shape torque is positive, it induces radial expansion at the midplane and radial compression at the outer surfaces of the bilayer. When the shape torque is negative, it induces radial compression at the midplane and radial expansion at the outer surfaces of the bilayer. The bending stress picture of the shape torque is somewhat more complicated than for the midplane torque due to the interaction between the two layers. The generalized forces are depicted in Fig. 1 for positive values and their physical interpretation and size are discussed in Free Energy Estimates and Physical Interpretation.

Relation between pressure gradients and generalized forces
Another way to recast the interaction between the membrane protein and the surrounding bilayer is by introducing the notion of pressure gradients. Cantor (1997, 1999) has made calculations of the out-of-plane pressure gradients in the bilayer. He has shown that the pressure is compressive in the middle of the bilayer and expansive near the surface. Cantor (1997) and de Kruijff (1997) have discussed the effects of this gradient on protein conformation. If the -helices (MscL's transmembrane domains M1 and M2) can be interpreted (to a first approximation) as rigid, the effects of this pressure gradient are to produce a tension and cylindrical torques. The tension on the interface is the integrated bilayer pressure,

(28)

where z is the position in the bilayer, running from –a to a. This integrated tension must be the net tension . If we allow the inclusion to have a hinge at z = 0, cylindrical torques about this circumference are induced on each section of the inclusion (see Fig. 1). In the small-angle limit, these cylindrical torques are

(29)

(30)

where the torques have been defined to match our previous definitions in Forces, Torques, and Tensions, when the angles made by two surfaces of the interface, ±, are defined such that

(31)

The ±_P must correspond to our ±. The effects of the pressure gradient on our constrained system are neatly reduced to three of the generalized forces we have already discussed. The fourth force, F, is just the integrated shear stress.

	FREE ENERGY ESTIMATES AND PHYSICAL INTERPRETATION

TOP ABSTRACT INTRODUCTION FREE ENERGY OF THE... FREE ENERGY ESTIMATES AND... APPLICATION TO MscL GATING CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
In Free Energy of the Bilayer-Inclusion System, we summarized the bilayer model and presented the lowest order contributions to the bilayer deformation energy in Table 1. The aim of the present section is to revisit each of these individual contributions to the overall free energy, estimate its magnitude for MscL, and discuss the scaling and physical mechanism giving rise to the bilayer deformation energy. To estimate the bilayer deformation energies, we need structural information for MscL. From x-ray crystallography data (Chang et al., 1998), in the closed state, MscL appears to have an external radius of 23 Å. Sukharev et al. (2001) have speculated that the open state's external radius is 35 Å. We use typical bilayer elastic parameters as summarized in the Appendix. In addition, the Appendix contains a brief discussion of the scaling of these parameters with bilayer thickness. Numerical results are multiplied by scaling relations to remind the reader what values have been used in their computation and how the free energies scale with changes in inclusion geometry, tension, etc.

Areal deformation
The areal deformation free energy is the dominant tension-dependent term and typically provides the mechanism for opening the MscL channel. The physical interpretation of this term is shown schematically in Fig. 3. The form of this contribution is well understood (for example, see Hamill and Martinac, 2001) and is analogous to the –PdV term for an ideal gas in three dimensions. For areal deformations, the bilayer lipids act like a two-dimensional gas with a free energy change given by

(32)

where is the tension. At high tension, the open state is favored due to its larger area. Sukharev et al. (1999) have measured the opening tension to be _* = 1.2 pN Å^–1. (The value _* is the tension at which the channel is open half the time. This tension will depend on the bilayer in which the channel is reconstituted but we use this number as the typical size of the opening tension.) The areal deformation energy is

(33)

and is plotted as a function of applied tension in Fig. 4. The way in which this free energy is expressed is to normalize the tension in units of the opening tension, _*, and the area in terms of the closed-state area, A_C. As we expect, the typical free energies generated by radial changes are large. This is no surprise since the tension acts as the switch between the closed state and the larger open state. The most striking feature of this energy in comparison with those we will discuss below is its areal dependence. This free energy scales as the square of the channel radius, whereas almost all other contributions will approximately scale as the circumference. This scaling difference has important consequences for the stability of the conductance states and will give rise to a picture of the tension-induced opening of the channel much like the picture used to discuss nucleation of second phases. We have gone to some length to develop the importance of this scaling difference in our previous article (Wiggins and Phillips, 2004).

View larger version (11K): [in this window] [in a new window]   FIGURE 3  A cartoon of areal deformation. Tension, represented by the arrows, is transmitted through the bilayer to the inclusion. For positive biaxial tension, radial expansion of the inclusion reduces the free energy of the bilayer. The vesicle or cell can be viewed as a bilayer reservoir where tension is the energetic cost per unit area of bilayer in the local system.

View larger version (10K): [in this window] [in a new window]   FIGURE 4  The theoretical areal deformation free energies for the open (dashed line) and closed state (solid line) as a function of applied tension.

Gaussian curvature
Gaussian curvature normally contributes to the free energy topologically (independent of the local shape of the bilayer). However, at the inclusion, the bilayer has a boundary which will allow non-topological contributions to the free energy (E. Evans, personal communication). In the small-angle limit, the Gaussian curvature free energy is

(34)

as demonstrated in the Appendix. Measurements of the Gaussian curvature modulus are compatible with a wide range of values: K_G <– K_B/2 (see references in Boal, 2002). We estimate that for MscL, the free energy contribution from the midplane slope is

(35)

where the deformation energy has been written in a dimensionless form in terms of the bending modulus, K_B, the closed-state radius, and a modest interface angle of 0.1. (We expect the contribution from U' to be of the same order.) We have chosen this small angle since a large tilt angle for the interface is not evident from the closed-state structure or the modeled open state (see Fig. 2).

As indicated above, the free energy is typically fairly small unless H' or U' are large. Since G_G is radially independent, it induces no tension. On the other hand, Gaussian curvature does induce a torque of the form

(36)

which points toward H'_± = 0 if K_G < 0. The induced midplane torque is exactly what is expected from the bending stresses in Eq. 26.

Spontaneous curvature
Spontaneous curvature arises from the addition of detergents and non-bilayer-forming lipids to the bilayer. These molecular additions cause the lowest energy configuration of a single layer of lipids to be curved. The general phenomenon of spontaneous curvature in lipid structures is reviewed by Gruner (1989). In general, measurements of the spontaneous curvature, C, have been for H_II-phase-forming molecules where the positive spontaneous curvature can be deduced from the lattice structure (Gruner, 1989; Chen and Rand, 1997; Keller et al., 1993). Values of C^–1 = 20 Å^–1 (DOPE) are experimentally attainable (Keller et al., 1993). Less is known about negative spontaneous curvature, induced by micelle-inducing detergents and lysophospholipids. See Fig. 5 for a brief discussion of molecular shape and spontaneous curvature. To induce a composite bending modulus for the bilayer, the layers must be asymmetrically doped, although the molecules can exchange between the leaflets and move within a leaflet to energetically favorable locations caused by localized regions of high complementary curvature (de Kruijff et al., 1977; Kumar et al., 1989). For the sake of making an explicit estimate, we ignore these complications.

View larger version (22K): [in this window] [in a new window]   FIGURE 5  A schematic depiction of molecular shapes which influence spontaneous curvature (Israelachvili, 1991). Molecules with a cylindrical shape, such as phospholipids, will assemble into bilayers. Cone shaped molecules, such as lysophospholipids will assemble into micelles, the lowest energy configurations. For our sign conventions, these cone-shaped molecules induce negative spontaneous curvature. Inverted cone-shaped molecules, such as cholesterol, DOPC, and DOPE assemble into HII phases (Gruner, 1989) and induce positive spontaneous curvatures. The size of the spontaneous curvature is thought to be related to the difference in size between the polar headgroup and the acyl tails. Figure adapted from Lundbæk and Andersen (1994).

(37)

These symbolic results are equivalent to those in Dan and Safran (1998) and are plotted in Fig. 7. To estimate the size of this contribution for MscL, we have written the free energy in a dimensionless form using the large positive spontaneous curvature of a DOPE monolayer (C^–1 = 20 Å) (Keller et al., 1993), a relatively modest tilt angle (H' = 0.1), and the closed-state radius. The prime notation is meant to denote that this estimate is for either these values of C and H' or and U'. The resulting free energy can be the same order of magnitude as the areal deformation energy, implying it may play an important role in channel function.

View larger version (15K): [in this window] [in a new window]   FIGURE 6  A schematic depiction of spontaneous curvature induced by several species of lipids in the bilayer. The shaded lipids depict a nonbilayer lipid which induces positive spontaneous curvature. A tilted inclusion interface can lead to a reduction in the stress caused by the nonbilayer lipids as depicted above. Spontaneous curvature induces both torques and tension at the interface. For energetically favorable tilt, the tension acts to open the channel. The torque on the inclusion from a bilayer leaflet with positive spontaneous curvature acts to increase tilt by expansive pressure at the surface and compressive pressure at the midplane. When only one leaflet of the bilayer is doped, both a midplane and a shape torque are induced but they cancel for the undoped leaflet.

View larger version (21K): [in this window] [in a new window]   FIGURE 7  The spontaneous curvature free energy as a function of the composite spontaneous curvature C for various midplane slopes. At the top we have shown the corresponding concentration ratio for the DOPE/DOPC system of Keller et al. (1993). For positive C, the bottom leaflet consists of pure DOPC and the top leaflet is a DOPE/DOPC mix with mole fraction x of DOPE. For negative C, the top leaflet consists of pure DOPC and the bottom leaflet is a DOPE/DOPC mix with mole fraction x' of DOPE. We have plotted the free energy for a range of spontaneous curvatures that are larger than those that can be realized for DOPC/DOPE bilayers, since they may be relevant for other lipids or detergent-lipid bilayers.

Spontaneous curvature gives rise to both a tension, due to the radial dependence of the free energy, and torques, due to the dependence of the free energy on H' and U'. The tension on the boundary of the protein is

(38)

where we have estimated the size of the induced tension by writing it in a dimensionless form using the same parameters as the deformation energy described above. This induced tension can have either sign, resulting in contributions that are either tensile or compressive. If curvature stress is relieved by spontaneous curvature, it is energetically favorable to increase the radius and the tension tends to open the channel, whereas if the curvature stress is increased by the spontaneous curvature, the tension will be compressive. The midplane torque is

(39)

which is again written in a dimensionless form as described above. The torque induces inclusion conformations that would allow energetically favorable bending as explained above and depicted in Fig. 6. The midplane torque is non-zero only for asymmetrically doped bilayers and its symbolic form matches the spontaneous curvature term deduced from bending stress in Eq. 26. The shape torque is

(40)

which, for positive mean spontaneous curvature, acts to compress the midplane and expand the outer surface region of the inclusion. (Again, we have written the torque in a dimensionless form, as described above.)

Keller et al. (1993) have studied the Alamethicin channel reconstituted into DOPC/DOPE bilayers. This is a particularly beautiful system since the spontaneous curvature of the mixed bilayer interpolates linearly with the relative concentration of the components, allowing a continuous range of spontaneous curvatures. The bilayers of Keller et al. (1993) are symmetric, implying that C = 0. In this case, the spontaneous curvature free energy is

(41)

(Dan and Safran, 1998), which would predict free energy differences between states to be linear in , which Keller et al. (1993) have shown experimentally.

Bilayer interface energy
The bilayer and protein are glued together by hydrophobic-hydrophilic interaction forces which are strong enough to hold the protein in the bilayer at a typical cytoplasmic pressure of several atmospheres. It is natural to assume that in addition to the internal protein and bulk bilayer energies there will be an interaction term from the interface. There are many complicated scenarios which might be dreamed up, but the simplest is to assume that there is free energy proportional to the area of protein and bilayer in contact, resulting in a free energy

(42)

where W is the thickness of the hydrophobic region. The constant of proportionality, , is the interface energy and has units of energy/area. Thus far, we have concentrated exclusively on the bilayer bulk for two reasons: 1), the continuum model is almost certainly a reasonable approximate model for the processes of interest; and 2), the material parameters for the bilayer are known from earlier experiments (Rawicz et al., 2000). In contrast, little is known about the validity of this model for the interface nor is there any estimate for the size of , the interface energy. This class of interface terms gives rise to a tension and a compressive-expansive force:

(43)

(44)

The effects of the tension and compressive force depend on the sign , the interface energy density. When is positive, the interface is minimized, leading to compressive forces. When is negative (the affinity of lipid and protein are high), the interface is maximized and the forces are expansive.

We have introduced this energy as a sanity check for our boundary conditions. We have somewhat naively assumed that the membrane adapts to an arbitrary protein shape. This assumption certainly fails when the adhesive forces attaching the membrane to the protein are not large enough to sustain the strain in the membrane. It is therefore useful to develop an approximate expression for these adhesive forces. We know the interface energy for a typical hydrophobic-hydrophilic mismatch (Hamill and Martinac, 2001) is

(45)

which is large compared to the other tensions in the problem. The compressive force countering the creation of this interface is

(46)

where we have used the closed-state radius to write the force in a dimensionless form. This force can be interpreted as the critical force required to strip the protein from the bilayer. As we have reasoned above, this force will be important when we consider the large deformation limit on forces and energies due to thickness deformation.

Midplane deformation
The free energy associated with the deformation of the midplane of the bilayer is another contribution in the overall free energy budget. These constant thickness deformations like those pictured in Fig. 8, are induced by conically shaped proteins. Midplane deformation contributes to the bilayer deformation energy through both bending of the bilayer and from a corresponding increase in bilayer area. The exact result to the linearized model is derived in the Appendix, but the dominant contribution at high applied tension is given by

(47)

where the parameters used to write the deformation energy in a dimensionless form are the closed-state radius, the opening tension, and a modest interface tilt angle (H' = 0.1). The H'² dependence of the midplane deformation energy is as one would expect since no bending corresponds to H' = 0 and results in the minimum energy (in the absence of spontaneous curvature). Dan and Safran (1998) have discussed deformation energies with a similar dependence on H', but with a different size and physical origin. Note that midplane deformation scales differently with the applied tension (^1/2) from the other contributions and can therefore be distinguished from the other bilayer deformation energies by measuring the tension dependence of the free energy.

View larger version (13K): [in this window] [in a new window]   FIGURE 8  A conically shaped protein induces bilayer bending. To match a conical inclusion interface, the bilayer must deform. The deformation leads to energetic contributions both from an increase in bilayer area and from bilayer bending. Midplane deformation induces both a midplane torque and a tension. The tension is always compressive. The midplane torque acts to reduce interface tilt and restore the bilayer to its undeformed configuration. We estimate that the midplane deformation energy is probably not important for MscL gating.

(48)

where we have estimated the typical size of the decay length by writing it in a dimensionless form using the opening tension. (The size and scaling of the bending moduli are described in the Appendix.) This decay length is approximately the same size as the channel radius. At high tension this length scale is reduced thus improving the asymptotic result and also increasing the size of the energy. For MscL, unless the bending modulus is significantly softened, we are unlikely to be able to access this regime since the lysis tension for bilayers is typically not much more than _* (Olbrich et al., 2000). The scaling result we have derived overestimates the bilayer deformation energy. (See the Appendix for further discussion.) Both the exact result and asymptotic result are plotted as a function of applied tension in Fig. 9. Despite this overestimate, the energy is still small compared with the areal deformation, so we conclude that midplane deformation is probably not a key player in the free energy budget for MscL. This effect has also been explored in a recent article by Turner and Sens (2004).

View larger version (13K): [in this window] [in a new window]   FIGURE 9  The midplane deformation energy is illustrated above as a function of tension. We have plotted the approximate scaling result (dashed line) discussed below, the exact result to the model (solid line) discussed in Calculation of Free Energy (see Appendix), as well as the areal deformation energy for the closed state, the opening tension * (dotted line). All the energies are computed for the closed state using an unrealistically large midplane slope (H = 0.5) to exaggerate the effect. Although the scaling result is several kT larger than the exact result, it accurately reflects the scaling at high tension, and provides a limit for the exact result. The 1/2 dependence of the midplane deformation energy has not been observed experimentally.

(49)

which acts to inhibit channel opening. For the typical constants chosen here, _H is about a hundredth of the opening tension, again confirming that the midplane deformation is probably not important for MscL conformation or function. Like the spontaneous and Gaussian curvature contributions, the midplane deformation also places a torque on the protein,

(50)

which we have written in a dimensionless form as described above. This is a restoring torque toward the lowest energy configuration H' = 0 (in the absence of spontaneous curvature).

Thickness deformation
The free energy contribution from thickness deformation results from changes in the separation between the upper and lower surfaces of the bilayer induced by the hydrophobic mismatch between the inclusion and the bilayer. This effect is depicted schematically in Fig. 10. The energetic contribution from this mismatch can be quite significant. Bilayer thickness deformation has been studied by many authors: e.g., Mouritsen and Bloom (1984) and Huang (1986), and more recently by Nielsen et al. (1998) and Goulian et al. (1998). These authors have all solved the model exactly, but we introduce a large radius asymptotic expansion to significantly simplify our results. Expanding the exact solution of the model in powers of the radius gives

(51)

where For MscL, the only important terms are the first two. In the Appendix we have plotted both the approximate and exact solutions to demonstrate that the interesting physics is captured by our approximations. Ignoring higher order terms, the resulting contribution is

(52)

where

(53)

View larger version (17K): [in this window] [in a new window]   FIGURE 10  Bilayer thickness deformation due to a hydrophobic mismatch. To match the inclusion's hydrophobic boundary, the bilayer thickness must be deformed. Microscopically, the lipid tails are deformed as illustrated schematically above. The modulus for these deformations is KA. For large mismatches, the energy contribution from thickness deformation can be quite significant. We estimate that this energy is important for MscL gating. Thickness deformation induces a compression-expansion force, a tension, and a shape torque, which are also depicted above. The compression-expansion force acts to reduce the mismatch. The shape torque acts to induce interface tilt to reduce the bilayer bending. The tension generated by a mismatch is always compressive.

(54)

which is approximately satisfied for the critical tension measured by Sukharev et al. (1999). (See the Appendix for details about the scaling and size of the elastic moduli. The tension above has been put into a dimensionless form using the parameters for a PC lipid of acyl length 18.) One might worry that for small bilayer thickness the small tension limit would no longer be satisfied, but we will show that the opening tension is also reduced in this case. In the low tension limit, ß± is

(55)

We will refer to ß as the inverse decay length since it defines the length scale over which the thickness deformation perturbation decays. This length scale is given by

(56)

This decay length defines the large radius limit, which is satisfied even for the closed state of MscL since R_C > ß^–1.

The dominant contribution at large radius is which corresponds to ignoring the curvature of the interface entirely (Dan et al., 1993). To estimate the typical size of this contribution, we set U' = 0,

(57)

where we have written the deformation energy in a dimensionless form using the closed-state radius, a small mismatch (U = 1 Å), and the effective elastic modulus , defined

(58)

This is the result listed in Table 1. Since large mismatches are possible and the deformation energy grows as the square of the mismatch, this contribution can be very significant. This U² dependence, analogous to a linear spring, is exactly what we expect since the minimum energy occurs for a perfect thickness match between the protein and bilayer (U = 0). Mouritsen and Bloom (1984, 1993) were the first to discuss this dependence. Its phenomenological significance has been stressed by Lundbæk et al. (1996). The thickness deformation energy is a function of both K_A, the local thickness deformation modulus, and the bending modulus, K_B. Physically, K_B provides a compatibility condition for adjacent lipids which sets the size of the deformed region. The thickness deformation energy is also approximately linear in R since the area of the bilayer deformed is approximately proportional to the channel circumference. The size of thickness deformation energy and its radial dependence imply that is almost certainly important in the energetics of MscL. Unless both U' and U are zero or cancel, this term will contribute due to the radial expansion of the channel between the closed and open states. Even if the height of the hydrophobic region were to remain unchanged, this contribution would still be very significant (see Wiggins and Phillips, 2004). Let us mention, as a brief aside, that the functional form of is a very pleasing result since, although the prefactor appears to depend on the bilayer width, a, it is approximately independent of a! Please see the Appendix for a brief argument. Because this scaling is not obvious and we will often use scaling arguments, we write the result in terms of to alleviate the temptation of thinking a^–3/2.

One of the difficulties in implementing this model is the uncertainty concerning boundary conditions and in particular what slopes should be assigned for the bilayer-inclusion interface. One of the possibilities studied by other authors (Helfrich and Jakobsson, 1990), is to treat U' as a free parameter and minimize the free energy with respect to it. In the asymptotic limit this calculation becomes very simple. Taking the low tension limit ( = 0), and choosing U' to minimize gives a free energy one-half that which is obtained by naively assuming U' = 0, namely,

(59)

As a result, we argue that the qualitative conclusions—the importance of this correction—are indifferent to the particular choice made for this boundary condition, but quantitatively the choice of boundary conditions can have a significant effect.

Although dominates at large radius, for MscL-like geometries, is approximately as large. which is radially independent, is

(60)

for U' = 0, where the energy has been put in a dimensionless form using the closed-state radius, and a small mismatch (U = 1 Å). As can be seen above, for the closed state, this energy is almost as large as the dominant scaling term and is also proportional to U². In general, the effects of this term on channel gating are not as pronounced since it is radially independent and will not contribute a term to the free energy difference between the open and closed states proportional to R. Likewise, it will not contribute to the tension. The asymptotic expressions for the thickness deformation energy are compared with the exact results to the linearized theory in Fig. 16 in the Appendix. We plot the thickness deformation energy for the closed state in Fig. 12, in the next section.

View larger version (9K): [in this window] [in a new window]   FIGURE 16  Validity of asymptotic approximation for dimensionless thickness deformation free energy. The curves above depict the difference between the exact result (Eq. 116, solid curve), the asymptotic expansion (Eq. 118, dashed curve), and the dominant scaling result (shown in Table 1, dotted curve). There is excellent agreement between the approximate result and exact result for radii relevant for MscL:

View larger version (17K): [in this window] [in a new window]   FIGURE 12  Interface and thickness deformation energy of the closed state compared to experimental data from Powl et al. (2003) as a function of lipid bilayer thickness. The dot-dashed curve is the hydrophobic interface energy (GW) without thickness deformation (the limit is ). The dotted curve is the exact thickness deformation energy (GU exact) without saturation. The dashed curve is the asymptotic thickness deformation energy (GU) without saturation (the limit is * ). The solid curve is the saturating thickness deformation energy (GUW, see the Appendix for details). The o and + symbols are the experimental values measured by Powl and co-workers for TbMscL and EcoMscL, respectively. We have chosen the closed-state thickness of the channel (WC = 37.5 Å) to match the thickness of the bilayer at the minimum of the experimental bilayer deformation energy. This thickness is compatible with the known closed-state structure. For small mismatches there is a much better qualitative agreement between the thickness deformation energy than the hydrophobic interface theory. For large mismatch, the experimental data points are significantly smaller than the energy predicted by theory. We discuss this apparent discrepancy in the next section.

View larger version (13K): [in this window] [in a new window]   FIGURE 11  Bilayer thickness deformation saturates when the energy required to further deform the membrane is equal to the interface energy required to create a hydrophobic-hydrophilic interface. This failure of the bilayer to conform to the protein is depicted schematically above.