INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY REFERENCE SYSTEMS RESULTS DISCUSSION APPENDIX REFERENCES

Lipid bilayers are self-assembled structures of amphipathic molecules with material properties similar to those of smectic liquid crystals (Helfrich, 1973; Evans and Hochmuth, 1978). Changes in bilayer shape (lipid packing) therefore will incur an energetic cost (Helfrich, 1973, 1981). This is important because the hydrophobic bilayer-spanning domains of integral membrane proteins (Deisenhofer et al., 1985; Henderson et al., 1990; Doyle et al., 1998) couple the proteins to the surrounding bilayer (Owicki et al., 1978). Consequently, when membrane proteins undergo conformational changes that involve the protein-lipid boundary (Unwin and Ennis, 1984; Unwin, 1995; Kaback and Wu, 1997; Sakmar, 1998; Perozo et al., 1998), the structure of the surrounding bilayer will be perturbed, and the free energy difference between two protein conformations will vary with the difference in bilayer deformation energy associated with the different bilayer perturbations (Gruner, 1991). The bilayer deformation energies can be evaluated using the theory of elastic liquid-crystal deformations (Huang, 1986), and, because the bilayer mechanical properties vary as a function of the lipid composition (Evans and Needham, 1987; Needham, 1995), the energetics of bilayer-protein interactions provide for a mechanism by which the bilayer lipid composition can be a determinant of protein conformation and function.

The bilayer component of biological membranes contains lipids that in isolation form nonbilayer structures (Luzzati and Husson, 1962) (see Epand (1997) for a recent summary), and isolated lipid monolayers at equilibrium may be nonplanarthey may have a curvature (Cullis and deKruijff, 1979; Gruner, 1985; Seddon, 1990; Lundbæk et al., 1997; Andersen et al., 1999). This propensity to form nonbilayer structures is likely to be important. First, many cells regulate their bilayer lipid composition such that optimal cell growth occurs close to, but below, the bilayernonbilayer phase transition temperature (Lindblom et al., 1993; Rilfors et al., 1993; Rietveld et al., 1993) (see Hazel (1995) for a recent summary). Second, changes in monolayer equilibrium curvature modulate the function of many integral membrane proteins (cf. Epand (1997) for a review), as well as well-defined model systems (Keller et al., 1993; Lundbæk and Andersen, 1994; Bezrukov et al., 1995, 1998; Lundbæk et al., 1996), suggesting that the monolayer equilibrium curvature could be a modulator of biological function (Gruner, 1985; Hui, 1997).

The monolayer equilibrium curvature is determined by the effective "shapes" of the monolayer-forming lipids, which in turn are determined by the variation of the lateral stress or pressure profile t(z) through the monolayer (see Fig. 1 a). For an isolated, planar monolayer at equilibrium, the integral of the profile t(z) over the monolayer thickness is zero (Seddon, 1990), and the average molecular shape of the lipids is cylindrical. If the (unperturbed) lipid molecules are not cylindrical, the positive and negative stresses are not symmetrical about a neutral surface (a surface where the area does not change with changes in monolayer curvature; Rand et al., 1990; Templer et al., 1994), and there will be a bending moment, or torque, around this surface. A nonzero bending moment means that the monolayer will tend to curve away from a planar geometry, toward its equilibrium curvature c₀ (Fig. 1 b).

View larger version (23K): [in this window] [in a new window]   FIGURE 1   Intermolecular forces, lipid shape, monolayer curvature, and bilayer stress. (a) Effective lipid shape (left) together with intermolecular interactions (center) determines the lateral pressure profile in a monolayer (right). (b) The spontaneous radius of curvature R0 together with an (arbitrary) assignment of a surface normal determines the monolayer equilibrium curvature c0. (c) Monolayers with equilibrium curvature c0  0 change their effective lipid molecular shape from cones to cylinders to form a (frustrated) planar bilayer.

Whatever the monolayer equilibrium curvature, the two monolayers must adapt to one another to form a bilayer. In the case of symmetrical bilayers, the bilayer curvature will be zero. Thus, for lipid molecules that form curved monolayers, the adaptation involves a change in the effective lipid shape, from noncylindrical to cylindrical (Seddon, 1990) (Fig. 1 c). This change in shape means that energy is stored in the bilayerthe so-called curvature frustration energy (Gruner, 1985; Sadoc and Charvolin, 1986). Inclusions (lipids or proteins) that perturb the bilayer will alter the local energy density; conversely, inclusions may be affected by the deformation energy, which will affect protein function (Andersen et al., 1999).

	THEORY

TOP ABSTRACT INTRODUCTION THEORY REFERENCE SYSTEMS RESULTS DISCUSSION APPENDIX REFERENCES

Continuum analyses of bilayer configurations are based on the concept of bilayer elasticity. Any planar bilayer configuration is endowed with a potential (elastic) energy. A change in bilayer configuration causes a reversible change in energy, and configurations with the lowest energy are the most likely to occur. The symbols used in this article are defined in Table 1.

Formulating the model

A length mismatch between the thickness of the hydrophobic core of an unperturbed bilayer, d₀, and the length, l, of the hydrophobic exterior surface of a bilayer inclusion, an integral membrane protein, will introduce an elastic deformation of the bilayer in the vicinity of the inclusion (Fig. 2 a). When the strength of the hydrophobic interactions between the bilayer-spanning part of the protein and the bilayer core is strong enough to ensure that there is no exposure of hydrophobic residues to water, when there is strong hydrophobic coupling (Andersen et al., 1999), the bilayer deformation at the inclusion/bilayer boundary will be d₀  l.

View larger version (19K): [in this window] [in a new window]   FIGURE 2   Inclusion-induced bilayer deformations and local curvature. (a) When d0  l, hydrophobic matching at the inclusion/bilayer boundary will cause the two monolayers to bend and thin or thicken, which gives rise to a bilayer deformation energy. For symmetrical bilayers and symmetrical cylindrical deformations, the problem can be reduced to a radially varying deformation of a monolayer with an unperturbed thickness d0/2, where z = u(r) denotes the perturbation in monolayer thickness at distance r from the inclusion axis. At the inclusion/bilayer boundary (at r0), the deformation is u0. The slope of the deformation at the contact surface, du/dr|r0, is denoted by s. (b) Local curvature. The position of a point P on the surface is given by  = (x, y, u(x, y)); the associated area element normal is . The two directors whose curvatures are extrema are the principal directions; the corresponding principal curvatures are c1 = 1/R1 and c2 = 1/R2.

The ensuing bilayer deformation energy arises from contributions due to changes in bilayer thickness (with an associated energy density K_a(2u/d₀)², where K_a is the compression-expansion modulus and u is the local perturbation in monolayer thickness) and changes in monolayer curvature (with an associated energy density K_c(c₁ + c₂  c₀)²/2, where K_c is the mean splay distortion modulus and c₁ and c₂ are the principal monolayer curvatures) (Helfrich, 1973; Huang, 1986) (Fig. 2 b). In addition to these major contributions, there are two minor contributions: a surface-tension term, which previous analyses have shown to be negligible (Huang, 1986; Helfrich and Jakobsson, 1990; Nielsen et al., 1998), and a Gaussian curvature energy term with associated energy density (c₁c₂)²/2, which also is negligible (see Appendix).

Besides the above energy contributions, there also may be an energetic cost associated with packing the lipids in immediate contact with the inclusion, which arises because the presence of the inclusion will decrease the range of motion of the bilayer lipids (Chiu et al., 1991, 1999; Woolf and Roux, 1996). The total deformation energy therefore is

(1)

where G_continuum is the continuum contribution to G_def, due to the K_a(2u₀/d₀)²/2 and K_c(c₁ + c₂  c₀)²/2 energy densities, and G_packing denotes the (local) energetic cost due to the inclusion-induced packing constraints, which we will incorporate through the choice of boundary conditions used to solve the continuum problem.

In the case of uniform single component bilayers that are symmetrical about an unperturbed bilayer midplane, the continuum contribution to the bilayer deformation energy induced by a cylindrical inclusion with radius r₀ is obtained by integrating the energy densities over the perturbed area:

(2)

where K_cc₀²/2 is the curvature frustration energy density in the unperturbed bilayer. The material constants, K_a and K_c, have been determined in "macroscopic" continuum measurements (Evans and Hochmuth, 1978; Evans et al., 1995); it is not clear, however, whether these values are appropriate for describing bilayer deformations (cf. Helfrich, 1981).

To solve Eq. 2, which also will establish the deformation profile, one needs four boundary conditions. The first two are straightforward, as they describe the unperturbed bilayer far from the inclusion:

(3a)

and

(3b)

where u(r) denotes the monolayer perturbation as a function of r. The last two boundary conditions describe the perturbed bilayer at the inclusion/bilayer boundary and are subject to uncertainty.

For the third boundary condition, we assume that there is strong hydrophobic coupling, in which case the initial monolayer deformation u₀, at r = r₀, will be determined by the mismatch between l and d₀:

(3c)

Equation 3c will not hold generally, as the bilayer deformation may be so large that the incremental change in the deformation energy may exceed the energetic penalty for exposing hydrophobic residues to water (Andersen et al., 1999; Lundbæk and Andersen, 1999).

The energetic consequences of lipid packing adjacent to the inclusion are introduced through the choice of the fourth boundary condition. If G_packing = 0, then G_def = G_continuum, and the minimum value of G_continuum is attained when (Landau and Lifshitz, 1986)

(3d)

or, equivalently, when G_continuum/s = 0, where s = u/r|_r0. That is, if one can neglect any molecular detail at the inclusion/lipid boundary, then s will relax toward the value for which G_continuum is a minimum (Helfrich and Jakobsson, 1990), which we denote by s = s_min. We refer to Eq. 3d as the relaxed boundary condition and use the superscript rel whenever Eq. 3d applies.

The liquid-crystalline characteristics of lipid bilayers generally will make G_packing  0, in which case it is necessary to introduce molecular detail to describe the constraints on the lipid packing (Ring, 1996). Given the known variation of G_continuum with s (Huang, 1986; Helfrich and Jakobsson, 1990), we introduce the lipid packing constraints by constraining the value of s. For example, if a rigid cylindrical inclusion is imbedded in a bilayer composed of effectively cylindrical molecules, s will be close to zero because there can be no voids in the bilayer core at the lipid-protein boundary. We therefore choose the fourth boundary condition to be

(3e)

This boundary condition is in concordance with experimental results on the variation in gramicidin channel lifetime with bilayer thickness (Huang, 1986; Lundbæk and Andersen, 1999). Its physical significance is that the acyl chain movement adjacent to the inclusion will be constrained (cf. Chiu et al., 1999). If the lipid molecules in successive rings around the inclusion were free to slide relative to each other, the acyl chains in each monolayer would tilt with respect to the monolayer surface, and the lipid director would no longer be parallel to the surface normal, or s  0. In the limit where the energetic penalty for tilt vanishes, s will become equal to s_min.

If the lipid shape is changed, from cylindrical to cone-shaped, but the penalty for tilt remains, a void-free alignment of the lipids around a cylindrical inclusion would mean that

(3f)

where R_head is the effective radius of the lipid headgroup. Equation 3f is an approximation, as it is assumed that the inclusion, or the inclusion-induced bilayer deformation, does not perturb the lipid shape. Accepting this, Eq. 3f is accurate to within 1% for 0.3  R_headc₀  0.3. (Equation 3e describes the special case where c₀ = 0.) We refer to Eq. 3f as the constrained boundary condition, and use the superscript con whenever Eq. 3f applies. (One can similarly assign the value of u/r|_r0 for noncylindrical inclusions.)

Because of the uncertainties about the lipid packing around an inclusion, which has an impact on the choice of s, we examine how G_def varies for different choices of s.

Solution to the model

Examination of Eq. 2 shows that G_continuum, which from now on is equivalent to G_def (subject to the value of s), is composed of two terms that formally are independent of c₀ and a term that explicitly depends on c₀. This distinction between (formally) c₀-dependent and c₀-independent terms becomes useful when the solution to the problem is formulated, as it turns out to be advantageous to evaluate separately the value of G_def for c₀ = 0, which will be denoted G_def,c0=0, and then add the explicitly c₀-dependent contribution.

When c₀ = 0 the bilayer deformation energy can be written as

(4)

where G_CE,c0=0 is the compression-expansion component

(5)

and G_SD,c0=0 is the splay-distortion component

(6)

(The c₁c₂-dependent (or Gaussian curvature) term is negligible compared to the other c₀-independent terms (see Appendix).) The c₀-dependent term in Eq. 2 depends on the fourth boundary condition only and can be written in closed form (Ring, 1996):

(7)

Combining Eqs. 4-7, G_def can be written as

(8)

The general solution to Eq. 4 is quadratic in u₀ and s (Nielsen et al., 1998):

(9)

where the coefficients a₁, a₂, and a₃ are functions of the mechanical moduli (K_a and K_c), r₀ and d₀, the parameters that describe the bilayer-inclusion system (scaling relations that allow these coefficients to be determined for any choice of K_a, K_c, r₀, and d₀ will be described in the Results section). Not only G_def,c0=0, but also the component energies (G_CE,c0=0 and G_SD,c0=0) are biquadratic functions of u₀ and s:

(10a)

and

(10b)

which is important when evaluating the various contributions to G_def.

For the constrained boundary condition and c₀ = 0, s = 0 and

(11a)

The bilayer deformation energy thus is equivalent to the energy stored in a linear spring, and it is convenient to define a bilayer spring constant as

(11b)

For the relaxed boundary condition and c₀ = 0, G_def,c0=0/s = 0 and

(12)

or

(13a)

which again is equivalent to the energy stored in a linear spring with the bilayer spring constant

(13b)

Equations 8, 9, 11a, b, and 13a, b provide a basis for describing the energetic consequences of inclusion-induced bilayer deformations. For either boundary condition used here, G_def,c0=0 can be described by a linear spring model with a characteristic bilayer spring constant,

(14)

The magnitude of the spring constant varies with the choice of boundary conditions (Eq. 3d or 3e) used to describe the lipid packing at the inclusion/lipid contact surface (cf. Eqs. 11b and 13b).

When c₀  0, the expression for G_def (Eq. 8) contains, in addition to the quadratic terms describing G_def,c0=0 (cf. Eq. 9), a G_MEC term that is linear in s (Eq. 7), which has important consequences for the G_def(u₀) relations.

	REFERENCE SYSTEMS

TOP ABSTRACT INTRODUCTION THEORY REFERENCE SYSTEMS RESULTS DISCUSSION APPENDIX REFERENCES

Bilayer material constants

To evaluate the quantitative importance of the inclusion-induced deformation energy, we use experimental values of K_a and K_c for 1-stearoyl-2-oleoyl-phosphatidylcholine (SOPC), alone and with cholesterol; dioleoylphosphatidylcholine (DOPC); and glycerolmonooleate (GMO). SOPC is the reference phospholipid because its 18:0/18:1 chain composition approximates the average acyl chain composition of biological membranes (Marsh, 1990). To illustrate how the results can be extended to other systems, we use scaling relations to estimate G_def in different systems. The scaling relations were evaluated using, first, bilayers composed of an equimolar SOPC and cholesterol mixture, which increases K_a and K_c by three- to fourfold relative to SOPC; second, bilayers composed of DOPC, in which K_c is decreased by fourfold with little change in K_a, which reduces the relevant length scale by 1/2 (Nielsen et al., 1998); and third, bilayers composed of GMO, which decreases K_a/K_c by twofold and for which there is an experimental estimate for H_B (Lundbæk and Andersen, 1999). The material constants for the four systems are listed in Table 2. There is variability among the values of material constants obtained by different investigators (cf. Needham, 1995; Nielsen et al., 1998). The values in Table 2 serve as reference points only; one can use the scaling relations to evaluate the bilayer deformation energy for any choice of material constants.

Protein models

The effects of lipid composition (bilayer mechanical characteristics) on the conformational equilibrium in membrane proteins were evaluated using, first, the transmembrane dimerization of gramicidin (gA) channels, and, second, the closeopen transition in gap junction channels. The channels are treated as rigid cylinders with the dimensions listed in Table 3.

	RESULTS

TOP ABSTRACT INTRODUCTION THEORY REFERENCE SYSTEMS RESULTS DISCUSSION APPENDIX REFERENCES

Given the structures of Eqs. 8 and 9, it is useful to start out by exploring the consequences of the biquadratic relation between G_def,c0=0, u₀, and s (Eq. 9). The reference system will be a membrane-spanning protein with r₀ = 3.0 nm (corresponding to a gap junction channel) in a bilayer with properties similar to those of a SOPC bilayer with d₀ = 3.0 nm; the reference deformation will be a hydrophobic mismatch of 0.2 nm (=2u₀ = d₀  l).

The biquadratic nature of the deformation energy

Fig. 3 shows numerical evaluations of Eq. 2 for the reference system and c₀ = 0. Fig. 3 a shows how s_min varies as a linear function of u₀. The compression-expansion and splay-distortion components of G_def,c0=0^rel, taken together, lead to a surprising simplicity (Eq. 12). Fig. 3 b shows the corresponding relation between u₀ and G_def,c0=0^rel, which is described by a linear spring formalism (cf. Eq. 13a). Fig. 3, c and d, shows solutions of Eq. 2 as functions of u₀ (for three fixed values of s) and s (for three fixed values of u₀). In each case s  0 or u₀  0 preserves the shape of the quadratic curve but shifts the position of the minimum. The importance of the boundary conditions at r₀ is seen by comparing Fig. 3 b with Fig. 3, c and d.

View larger version (19K): [in this window] [in a new window]   FIGURE 3   Bilayer deformations and deformation energies. (a) The relation between smin and u0 (Eq. 12) for a SOPC bilayer. (b-d) Numerical evaluation of Gdef,c0=0 (Eq. 4). The curves can be described by Eq. 9, using the a*1  a*3 values from Table 4. (b) Gdef,c0=0 for the relaxed boundary condition (Eq. 3d) as a function of the initial deformation u0. (c) Gdef,c0=0 as a function of u0 for constrained values of s = +0.25 (- -), s = 0 (), and s = 0.25 (··· ···). (d) Gdef as a function of s for constrained values of u0 = 0.1 (- -), u0 = 0 (), and u0 = 0.1 (··· ···) nm.

The coefficients a₁, a₂, and a₃, which describe the system, are listed in Table 4, together with the coefficients a₁^CE  a₃^CE and a₁^SD  a₃^SD. Given these values, s_min = 0.86u₀ (where u₀ is in nm); the two spring constants are H_B^con = 88.8kT/nm² (Eq. 11b) and H_B^rel = 35.6kT/nm² (Eq. 13b). For a given deformation, the bilayer deformation energy varies by a factor of 2.5 for the constrained as compared to the relaxed boundary condition.

The relaxed boundary condition

Combining Eqs. 7 and 9, G_def can be expressed as a function of u₀ and s:

(15)

where (=2K_cr₀c₀) incorporates the G_MEC contribution to G_def. For the relaxed boundary condition and c₀  0, the value of s for which G_def is a minimum is

(16)

Substituting Eq. 16 into Eq. 15,

(17)

Fig. 4 shows G_def^rel as a function of c₀ for fixed u₀, and vice versa. In either case, a u₀ (or c₀) different from zero will translate the G_def^rel versus u₀ (or c₀) relation in the plane; but the basic relation, as exemplified by the spring constant, is invariant.

View larger version (14K): [in this window] [in a new window]   FIGURE 4   Effect of c0 and u0 on Gdef for the s = smin boundary condition. (a) Gdefrel(c0) for u0 = 0 (), +0.1 (- -), +0.2 (··· ···), and +0.3 (-·-·-) nm. (b) Gdefrel(u0) for c0 = 0 (), +0.1 (- -), +0.2 (··· ···), and +0.3 (-·-·-) nm1. When u0 < 0, the situation is similar, with the sign of c0 reversed (results not shown).

For any choice of u₀ or c₀, the value of G_def^rel is that which minimizes the sum of the three component energies. To understand the interplay between these components, we analyze first the situation where c₀ is a free parameter (Fig. 4 a), then the situation where u₀ is a free parameter (Fig. 4 b).

For a given u₀, how will the monolayer equilibrium curvature effect the deformation energy? For a fixed u₀, G_def^rel(c₀) goes through a global maximum. That is, G_def^rel(c₀) will have two balance points where G_def^rel(c₀) = 0. At these points, G_def,c0=0^rel is exactly balanced by the release of curvature frustration energy due to the monolayer bending. For a fixed u₀ > 0 (Fig. 4 a), a small positive c₀ can make G_def^rel(c₀) = 0; somewhat surprisingly, a large negative c₀ also can make G_def^rel(c₀) = 0.

For a fixed u₀, s = 0 at the global maximum for G_def^rel(c₀) because G_def^rel/ = s. Using Eq. 16, the curvature at the maximum is

(18a)

and, combining Eqs. 17 and 18a,

(18b)

which is formally identical to G_def,c0=0^con (Eq. 14 with the spring constant given by Eq. 11b). The similarity is apparent, however, because c₀|_max is a function of u₀ (Eq. 18a); but the result highlights the interactions between the bilayer material constants and the boundary conditions in determining G_def.

For a given c₀, how will a u₀  0 effect G_def? For a fixed c₀, G_def^rel(u₀) will go through a global minimum (Fig. 4 b); when c₀  0, G_def^rel(u₀)|_min < 0. For c₀ > 0, a large positive u₀ (and a negative u₀ of more modest magnitude) can make G_def^rel(u₀) = 0 (Fig. 4 b). These balance points arise from the exact match between the release of curvature stress and G_def,c0=0^rel. The situation is similar for c₀ < 0, but the sign of u₀ is reversed (results not shown).

The minimum of G_def^rel(u₀) denotes how much energy can be released by an inclusion-induced deviation from a planar bilayer geometry. The deformation at the minimum is given by

(19a)

and

(19b)

When c₀  0 the minimum for G_def occurs at u₀  0. That is, a bilayer inclusion can relieve the local bilayer curvature stress, or, alternatively, the potential energy density associated with the bilayer curvature stress can drive a protein conformational change. The energy release is

(20)

For the reference deformation, and c₀ = 0.1 nm¹, this energy is 2.4kT. It should be compared with the curvature frustration energy: ~3.1kT if the curvature frustration energy density, K_cc₀²/2, is integrated over the inclusion area, and ~5.3kT if the energy density is integrated over the area of the inclusion plus the first annulus of lipid molecules surrounding the inclusion. Only ~75% of the frustration energy (<50% if we include the first lipid annulus in the appropriate area) is tapped by the 0  u₀|_min release.

To further understand how c₀  0 affects the bilayer deformation profile and energy, it is helpful to decompose G_def^rel(c₀) using an expression similar to Eq. 8:

(21)

G_CE^rel(c₀, u₀) and G_SD^rel(c₀, u₀) are biquadratic functions of u₀ and s (Eq. 10a, b), and they can be written using Eq. 16 as

(22a)

and

(22b)

Similarly, G_MEC^rel(c₀) can be written as

(23)

Fig. 5 shows G_CE^rel(c₀), G_SD^rel(c₀), G_MEC^rel(c₀), and G_def^rel(c₀) for the reference deformation. G_CE^rel(c₀) and G_SD^rel(c₀) are always positive: G_SD^rel(c₀) has a minimum for c₀ < 0 and G_CE^rel(c₀) has a minimum for c₀ > 0. G_MEC^rel(c₀) has a maximum (> 0) and becomes negative for large negative and positive values of c₀. G_MEC^rel(c₀) = 0 when either c₀ = 0 or s = 0 (cf. Eq. 7).

View larger version (16K): [in this window] [in a new window]   FIGURE 5   Effect of c0 on Gdefrel for a fixed u0 (=0.1 nm): Gdefrel(c0) () and its components. - - - -, GCErel(c0); ··· ···, GSDrel(c0); -·-·-, GMECrel(c0).

The maximum value of G_def^rel(c₀) is > 0, and it is important to understand the behavior at the two balance points, where G_def^rel(c₀) = 0, where the system has "tapped" the potential energy stored in the curvature frustration energy. The balance points occur when the discriminant of Eq. 15 is zero:

(24)

which is the case when

(25)

Equation 25 can be solved for u₀ (at a fixed c₀) or c₀ (at a fixed u₀):

(26a)

(26b)

Combining Eqs. 15 and 25, the s values that satisfy the G_def^rel(c₀) = 0 condition are

(27)

To understand the two solutions, consider a hypothetical situation where c₀ is varied by pharmacological manipulations, with no change in the other material constants. When c₀ = 0, there will be a finite bilayer deformation energy when u₀  0. For a fixed u₀, it is possible to change c₀ such that the local relief of curvature stress around the inclusion will balance exactly the deformation energy at c₀ = 0. This balance can occur for two different values of c₀. The origin of the two balance points is seen in Fig. 6 a, which shows how the c₀-dependent translation of the G_def,c0=0^rel(s) curve gives rise to two different solutions for G_MEC^rel(c₀), where c₀ is determined by Eq. 26a. The solution for c₀ > 0 makes intuitive sense because u₀ > 0. A positive curvature will facilitate the dimpling needed to satisfy the demand for hydrophobic matching. The counterintuitive solution for c₀ < 0 arises because it is the sum of the CE, SD, and MEC contributions to G_def^rel that is minimized. The bilayer can relieve its curvature stress by assuming another positive value of s_min, which leads to a different profile for the component energies. Fig. 6 b shows the two u₀ versus c₀ relations (Eq. 26b), and Fig. 6, c and d, shows the monolayer deformation profiles for the two solutions. For either solution, the profile is nonmonotonic. As expected, the nonmonotonic shape is most pronounced for c₀ < 0 (Fig. 6 d).

View larger version (14K): [in this window] [in a new window]   FIGURE 6   Effect of c0 on Gdef and the deformation profile. (a) Effect of monolayer curvature on Gdef(s) for a fixed u0 (=0.1 nm). , Gdef,c0=0. - - and ··· ··· are the Gdef(s) relations that satisfy the Gdefrel(c0) = 0 condition (where c0 is determined by Eq. 26a). The corresponding GMEC contributions are shown as dotted dashed lines (labeled (1) and (2)). (b) The two solutions for u0 as function of c0 (Eq. 26b). The two Gdef(s) = 0 solutions from a are labeled (1) and (2). (c) The monolayer deformation profile for c0 = 0.035 nm1 and s = 0.111 (solution (1)). (d) The monolayer deformation profile for c0 = 0.271 nm1 and s = 0.111 (solution (2)).

To understand the relationship between G_def^rel and u₀ (for a fixed c₀  0), we examine the underlying energy components (Fig. 7). G_CE^rel(u₀) and G_SD^rel(u₀) (Eq. 22a, b) are always positive (Fig. 7 a); G_SD^rel(u₀) has a minimum for u₀ < 0, whereas the minimum for