INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Membrane fusion requires transient structural reorganization of at least some lipids (for recent reviews, see Jahn and Südhof, 1999; Lentz et al., 2000; Epand, 2000; Burger, 2000; Chernomordik and Zimmerberg, 1995). Experimental (Chanturiya et al., 1997; Chernomordik et al., 1995a; Zimmerberg et al., 1993) evidence points to the existence of so-called hemifusion structures, which are relatively long-living intermediates appearing during the early stage of fusion. Such intermediate structures are found for fusion events that do or do not involve specialized fusion proteins (Chernomordik et al., 1997), suggesting their generic and lipid-based nature.

Fusion intermediates are likely to set energy barriers that the fusion process has to overcome. Thus, they are essential determinants of the fusion rate. Fusion intermediates may also have played a role for the evolution of the fusion protein machinery (Lee and Lentz, 1997). Yet, until now little structural information is available because experimental methods that probe structural aspects of the highly localized fusion intermediates are rather limited. Therefore, various theoretical models of the fusion scenario have been suggested (Siegel, 1993; Noguchi and Takasu, 2001; Müller et al., 2002). Of particular interest is the formation of an initial, hemifused; intermediate structure in which only the lipids of the proximal monolayers of two fusing bilayers mix. So far, the so-called stalk model (Kozlov and Markin, 1983) is the most consistent hypothesis for the hemifusion intermediate (Gaudin, 2000; Basánez et al., 1997, 1998). The stalk consists of a lipidic tether that connects the two proximal monolayers of the fusing membranes¹, as shown in Fig. 1 A. The stalk represents a metastable state. That is, even in its optimal conformation, it entails a positive free energy penalty (compared with an unperturbed lipid bilayer). The magnitude of this energy is important for the rate of the initial fusion events.

View larger version (15K): [in this window] [in a new window]   FIGURE 1   Schematic illustration of the fusion stalk model (A), and the new suggestion according to Kozlovsky and Kozlov (2002) (B). Both structures are rotationally symmetric around the z axis and exhibit mirror symmetry with respect to the equatorial plane. The broken lines correspond to the bilayer midplanes. Some lipids are shown schematically. The stalk in A consists of three different parts: distal monolayer (D), proximal monolayer (P), and tether (T). The tether contains all those lipids whose hydrocarbon tails point, on average, onto the z axis. The stalk in B lacks the tether-like connection between the fusing bilayers. Moreover, the profile of the bilayer interfaces exhibits sharp edges and corners.

In previous models (Siegel, 1993, 1999), the free energy of a fusion stalk was calculated based upon the energy needed to bend the involved lipid monolayers. A second energy contribution has to be included in this kind of models. This so-called interstitial energy accounts for the formation of a void region within the hydrophobic core of a fusion stalk (a similar interstitial energy appears in the inverse-hexagonal, H_II, phase). It turned out that particularly due to the interstitial energy the predicted energies of the fusion stalks were too high to account for the experimental observation of hemifusion intermediates. Stalks with energies of considerably more than 100 k_BT (in which k_B is the Boltzmann constant and T the absolute temperature) are very unlikely to serve as the hemifusion intermediate. In fact, Kuzmin et al. (2001) have estimated that the stalk energy should not exceed a value of about 40 k_BT to appear within the experimentally observed time scale. Hence, the apparent energetic discrepancy calls for either an alternative structural model of the fusion stalk or a new concept in calculating its energy.

Very recently, the "energy crisis" was solved by an elegant work of Kozlovsky and Kozlov (2002). They have suggested a modified structural model of the fusion stalk (Fig. 1 B) in which the two proximal monolayers of fusing membranes are directly connected without the formation of a lipidic tether. The key point of their model is the addition of a new internal degree of freedom (that is, another so-called order parameter) into the calculation of the stalk energy. This new degree of freedom is the lipid tilt. It expresses the ability of the hydrocarbon chains to adjust their average orientation. In fact, it is the lipid tilt degree of freedom that allows a fusion stalk to exist without the formation of a void region inside. In the past, the possibility of lipid tilt was frequently recognized (Helfrich, 1973; MacKintosh and Lubensky, 1991; Seifert et al., 1996). Lipid tilt was also investigated in connection with formation of the inverse hexagonal, H_II, phase (Hamm and Kozlov, 1998; May and Ben-Shaul, 1999) and lipid-protein interactions (May, 2000; Fournier, 1998, 1999). Kozlovsky and Kozlov (2002) have added the lipid tilt to the ability of the lipid layers to undergo splay deformations. In absence of lipid tilt the splay deformation becomes equivalent to a bending deformation of a lipid layer. Using both these order parameters (splay and tilt), the modified stalk was predicted to have few times smaller energy compared with previous models.

It should be noted that the new stalk structure (that of Fig. 1 B) has a property that is absent in previous models: the interface between the hydrocarbon core and the polar environment is no longer smooth at every point. In particular, there is a sharp corner at each side of the stalk, located at the axis of rotational symmetry (the z-axis in Fig. 1 B). In addition to that, a sharp edge is formed at its waist, where the stalk exhibits mirror symmetry. As discussed by Kozlovsky and Kozlov (2002), the possibility of nonsmooth interfaces arises from the lipid tilt degree of freedom. It allows the formation of a sharp membrane interface without a concomitant divergence of the splay energy.

Let us shortly discuss two other recent approaches to calculate the free energy of fusion stalks. Kuzmin et al. (2001) have suggested a theoretical model that includes, besides bending, tilt of the lipid molecules. The model starts from preformed "nipples" that decrease the local distance of two fusing membranes. It was estimated that this "nipple"-containing bilayers are separated from a fusion stalk structure by an energy barrier of somewhat less than 40 k_BT, in agreement with their estimate (see above). However, creation of the "nipple"-containing bilayers also requires an energy, which in fact is quite high (the authors estimated 200 k_BT). Hence, also the model of Kuzmin et al. (2001) requires an extraordinary high energy to form a stalk out of two apposed, planar bilayers.

In another recent approach, Markin and Albanesi (2002) have postulated a "stress free stalk." The key point of their model is the optimization of the cross-sectional shape of the stalk's neck in terms of its bending energy (rather than being a circular arc, the authors suggest it to be a surface of constant mean curvature). The approach of Markin and Albanesi (2002) indeed leads to very small (in some cases even negative) bending energies of a stalk. However, their model does not solve the principal problem of the large interstitial energy. Indeed, the authors use a reduced interstitial energy and justify this assumption in terms of impurities that fill the interstices and reduce their energy. The lipid tilt degree of freedom is not included in the model of Markin and Albanesi (2002), but the authors discuss its possible influence. They find that, even if using the reduced interstitial energy, a simple tilt deformation of the lipids could give rise to a collapse of the void. This clearly suggests that tilt deformations should be taken into account self-consistently in the calculation of the stalk structure.

In the present work we recalculate the structure and energy of fusion stalks using membrane elasticity theory. One aim of our study is to show that (unlike in the modified structural model suggested by Kozlovsky and Kozlov (2002)) a fusion stalk is likely to contain a tether-like lipidic connection between the two fusing membranes (similarly to that shown in Fig. 1 A). To this end, we shall incorporate a tether into our calculations and minimize the stalk free energy with respect to its length. (This implies that the presence of a tether will only be predicted if this is energetically favorable). The main objective of the present study is to clarify the role of the lipid tilt versus the ability of a membrane to form edges (and corners). In particular, we shall argue that the ability of the lipids to tilt is not sufficient to substantially lower the free energy of a fusion stalk. However, it is a necessary condition to allow for the formation of membrane edges. In fact, it is the existence of membrane edges that brings about a remarkable reduction of the stalk's free energy. We shall show this by using exactly the same model to compare lipid stalks with and without the ability to form edges (and corners) included. In the former case, the interfacial region of the stalk can be nonsmooth, whereas in the latter case it is smooth everywhere. We show that the bare presence of a "smoothness-constraint" leads to a severalfold increase of the stalk's free energy. To this end, we use membrane elasticity theory to optimize the size and shape of the fusion stalk without any further structural assumptions. To summarize, we suggest that, in contrast to most previous calculations of the fusion stalk structure, the interfacial region of a fusion stalk is nonsmooth. Instead, it exhibits edges and corners that crucially influence the stalk's free energy.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

The fusion stalk structure shown in Fig. 1 A is cylindrically symmetric around the z axis and exhibits mirror symmetry with respect to the equatorial plane. It consists of three different regions. Two of them, denoted by D and P, form a lipid bilayer with D and P assigned to the distal and proximal monolayer, respectively. The third region, denoted by T, is a tether-like lipidic connection between the two opposite proximal monolayers. The difference between the bilayer part and the tether can clearly be understood in terms of the packing properties of the corresponding lipid molecules. The flexible hydrocarbon chains of the bilayer lipids (that is, those residing in regions D and P) point, on average, onto a two-dimensional surface, namely the midplane of the bilayer (broken lines in Fig. 1 A). In contrast, the packing of the lipid chains in the central, tether-like region (T) is similar to that of a cylindrical micelle. Here, the hydrocarbon chains are, on average, directed to a single one-dimensional line. Due to the cylindrical symmetry, this line is straight and corresponds to the z axis in Fig. 1 A. Consequently, in absence of a tether-like lipidic connection (as is the case in Fig. 1 B) all lipids are subject to a bilayer-like packing, and the midplanes of the two apposed bilayers (broken lines in Fig. 1 B) touch each other at a single point on the z axis.

It is important to realize that the lipids residing within (and close to) a fusion stalk have different physical properties compared with those in a planar, unperturbed bilayer membrane. We refer to these differences as a structural perturbation of the lipids in a fusion stalk. This structural perturbation entails a free energy penalty, F, of the stalk. The extent of perturbation is determined by several requirements. First, the lipid chains must be able to fill out the hydrophobic core of the stalk. That is, no void is allowed to form within the entire hydrocarbon chain region. Second, the lipid head groups must protect the hydrocarbon chains from the unfavorable contact with the aqueous environment. Finally, the actual lipid perturbation must minimize the overall free energy, F, of the stalk structure.

Free energy of fusion stalk

The hydrophobic core of a fusion stalk (like that shown in Fig. 1 A) consists of hydrocarbon chains that are chemically linked to the corresponding polar head groups of the lipids (often through a glycerol backbone, but the exact molecular details do not enter into the present model). The interface between the hydrophobic lipid chains and the polar environment can be specified by a describing surface A that we express mathematically by a vector x. (For example, in Fig. 1, A and B, x would describe the thick solid lines.) To each position x we can assign a unit vector n = n(x), pointing along the average direction of the hydrocarbon chain that originates at x. (Note that we treat all hydrocarbon chains as identical.) Besides by their direction, the lipid tails are also characterized by their (effective) length, b, which results from an average over a sufficiently large number of different chain conformations. Most conveniently, we identify b as the distance between x and the bilayer midplane (or the z axis if x belongs to the tether part), measured along n(x). Using these notations, the structure of the fusion stalk is fully determined by x and the corresponding directors b(x) = b n(x).

Because x describes a surface, we can associate with each position at x a unit vector N that points normal to the surface. Of course, n and N need not point into the same direction at any given x. In general, there may be an angle, , between these two vectors. Mathematically,  is related to the dot product N · n = cos.

The next step is to write down for any given structure of the fusion stalk, as characterized by b(x), a reasonable expression for the elastic free energy, F. This will finally allow us to find the structure of the fusion stalk such that F = F[b(x)] adopts a minimum. Within the framework of continuum elasticity, we shall express the free energy, F, of the fusion stalk as an integration over the entire surface A

(1)

Assuming sufficiently small local perturbations, we shall use for the elastic free energy density (energy per unit area), , the quadratic expression

(2)

The first term in Eq. 2 describes the local compression (or expansion) of the lipid layer; K is the corresponding compression modulus (being equivalent to the lateral compression modulus of a monolayer), and s = b/b₀  1 is the relative chain dilation with respect to the equilibrium chain length b₀.

The second and third terms in Eq. 2 account for the splay energy of the lipid chains in which  is the bending stiffness, c₀ is the spontaneous curvature, and  · n denotes the divergence of the unit vector n. This energy contribution is related to the effective "molecular shape" (Israelachvili, 1992; Evans and Wennerström, 1994) of the lipid molecules. Depending on their chemical structure, and in particular on the bulkiness of the head group versus that of the hydrocarbon chains, lipid molecules may preferentially adopt the shape of a cone, an "inverted" cone, or intermediately that of a cylinder. Deviations of the actual packing geometry of a lipid layer from the preferred one cause an energetic penalty, namely, the splay energy.

The last term in Eq. 2 is the tilt energy of the lipid molecules; k_t is the corresponding tilt modulus. The tilt energy accounts for the angular deviation, , of the lipid director b (or, equivalently, n) from the normal direction N. We finish this section with five remarks. 1) In their recent approach, Kozlovsky and Kozlov (2002) have used an elastic free energy that is identical to the last three terms of Eq. 2. The only difference of the present approach is, thus, the consideration of the compression/expansion term Ks²/2. Yet, it is exactly this term that is required for a comparison of fusion stalks with and without membrane edges. 2) Generally, there is an additional energetic contribution present in Eq. 2 that accounts for a saddle-splay deformation of the lipid layer. Yet, this term is irrelevant for membrane shape optimization because of the Gauss-Bonnet theorem. Similarly, in the present case, a saddle-splay term would contribute only a constant to the actual minimization of the fusion stalk structure and hence is omitted. 3) The free energy, F, is an excess free energy with respect to a planar and unperturbed lipid layer (for which s =  =  · n = 0). The present work, thus, assumes thermal equilibrium of the lipids residing in the fusion stalk with those far away from it. 4) Generally, the molecular lipid volume remains constant during an elastic deformation of a lipid layer. This conservation would enter the free energy density in Eq. 2 at higher order than quadratic (unless c₀  1/b₀ (May, 2000), but we shall consider sufficiently small spontaneous curvature in the present work). Hence, even though the molecular lipid volume is conserved, it does not enter into the free energy . 5) The expression for the free energy density in Eq. 2 refers to a specific choice of the describing surface, namely the so-called neutral surface. For lipid layers, it is generally reasonable to assume that the neutral surface is located near the interface between the hydrophobic lipid chains and the polar environment (Szleifer et al., 1990).

In the following we shall use the expression for F, defined in Eqs. 1 and 2, to calculate the free energy of a fusion stalk. It is convenient to rewrite the free energy, F, of the entire stalk as a sum of the corresponding free energies of all involved subregions

(3)

in which F_bl, F_P, F_D, and F_T denote the free energy contributions of the bilayer, proximal monolayer, distal monolayer, and tether, respectively.

Internal degrees of freedom of a fusion stalk

To find the optimal stalk structure and its free energy, F, we shall not rely on any specific assumption on the size and shape of the fusion stalk. Rather, we seek to fully optimize the stalk structure with respect to all relevant degrees of freedom. However, we shall adopt the two following evident structural assumptions. 1) There is an axis of rotational symmetry and a plane of mirror symmetry. This assumption is justified because there is no energetic incentive in our free energy expression that could give rise to a break of symmetry. Conveniently, we use cylindrical coordinates {r, , z} and identify the axis of rotational symmetry with the z axis (see Fig. 1 A). 2) The topological structure of the fusion stalk corresponds to that in Fig. 1 A. That is, a single lipidic tether connects two perturbed bilayers. Of course, the length of the tether can vary such as to minimize F (for vanishing tether length, the structure in Fig. 1 B would be recovered).

Due to the symmetry, we only need to consider, say, the upper one-half of a cross-section of the fusion stalk, which is schematically shown in Fig. 2. The structure of the stalk shown in its cross-section consists of three regions: distal monolayer, proximal monolayer, and tether. The length of the tether is 2l; proximal and distal monolayers range from r_A  r  r_B and 0  r_D  r, respectively (the radial coordinate of the distal monolayer is denoted by r_D).

View larger version (20K): [in this window] [in a new window]   FIGURE 2   Cross-section of a stalk in the r, z-plane; two lipids are shown schematically. Because of the mirror symmetry only the part above the equatorial plane is shown. The structure is rotationally symmetric with respect to the z axis. The thick lines mark the describing surfaces in each of the three regions: distal monolayer, proximal monolayer, and tether. The describing surfaces are located in proximity to the interface between the hydrophobic lipid chains and the polar environment. The broken line between the distal and proximal monolayer marks the midplane of the perturbed bilayer. The proximal monolayer is characterized at its radial coordinate r (with rA  r  rB) by the height h = h(r) of the describing surface, by the tilt angle  = (r) of the lipid director with respect to the z axis, and by the director length, b = b(r). Similarly, the distal monolayer is characterized at its radial coordinate rD (with 0  rD  r) by the corresponding functions hD = hD(rD), D = D(rD), and bD = bD(rD). Note the angle  between the bilayer wing and the r axis. Finally, we describe the tether (which has half-length l) with respect to its z coordinate by the tilt angle, T = T(z), of the lipid director (measured with respect to the r axis), and by the director length bT = bT(z).

The shape and structure of each of the three hydrophobic regions is fully characterized by the corresponding function b(x). As introduced above, x specifies the shape of the describing surface, A, that separates the hydrophobic interior from the polar environment, and b = b n is the corresponding lipid director (of direction n and length b). Within the z, r-plane, the stalk structure, b(x), can be represented in each region (proximal monolayer, distal monolayer, and tether) by a number of conveniently chosen functions. In the following, we shall introduce these functions (see also Fig. 2).

The shape vector x of the proximal monolayer is fully characterized by the local height function, h = h(r), of the describing surface. The corresponding lipid director at position r (with r_A  r  r_B) is given by the tilt angle with respect to the z axis,  = (r), and by the director length, b = b(r).

Analogously to the proximal monolayer, the structure of the distal monolayer, as a function of the radial coordinate r_D (with 0  r_D  r), is given by the local height function of the describing surface, h_D = h_D(r_D), by the tilt angle with respect to the z axis, _D = _D(r_D), and by the director length b_D = b_D(r_D).

Whereas the two perturbed monolayers are both described in terms of the radial coordinate, it is more convenient to characterize the tether region with respect to the z axis. One needs two functions, we choose first the tilt angle, _T = _T(z), of the lipid director, measured with respect to the r axis, and second the director length b_T = b_T(z).

Fig. 2 visualizes the geometrical meaning of the functions h, , b, h_D, _D, b_D, _T, and b_T; mathematical definitions of the parameterization, b(n), in terms of these internal degrees of freedom are given in the Appendix.

So far, the perturbed bilayer (consisting of the proximal and distal monolayer) is characterized by the six functions h, , b, h_D, _D, and b_D. However, these functions cannot be chosen independently because the two monolayers are structurally coupled. That is, the hydrocarbon chain regions of the respective monolayers have to fit each other without leaving any void. This requirement can easily be put into a simple mathematical condition: For two lipids, whose headgroups are anchored at opposite leaflets of the bilayer, one at position r and the other one at position

(4)

the constraint

(5)

must be fulfilled. Consequently, we refer to Eqs. 4 and 5 as the hydrophobic matching condition. Eq. 5 implies that the perturbed bilayer has not six but only five independent internal degrees of freedom, say h(r), (r), b(r), _D(r_D), and b_D(r_D). Hence, together with the tether region, characterized by _T(z), and b_T(z), we are able to express the structure and free energy of the entire stalk in terms of seven independent functions. In the Appendix, we show how to calculate the separate contributions to the free energy F = F_P + F_D + F_T in terms of the seven independent functions.

Free energy minimization

Generally, the mathematical method of functional minimization provides a convenient way to find the unknown functions, h(r), (r), b(r), _D(r_D), b_D(r_D), _T(z), and b_T(z), such that F adopts a minimum. To this end, seven differential equations (Euler equations) for the seven independent functions can be derived and must be solved with respect to appropriate boundary conditions. However, in the present case this is a formidable task because the Euler equations are nonlinear with respect to the unknown functions. The situation simplifies considerably if we linearize the Euler equations with respect to both an unperturbed membrane (in which b = b_D = b₀,  = _D = 0 and h = const) and a uniform tether (in which _T = 0 and b_T = b₀). In the present work we use the linearized Euler equations to obtain the conformation of the fusion stalk. This, however, requires us to ensure that the deformation of the stalk is sufficiently small compared with both an unperturbed, planar membrane and a uniform tether. In particular, it must be (r) /2 and _T(z) /2. On the other hand, at the transition region between the proximal monolayer and the tether (that is at r = r_A and at the corresponding z_B = z(r_A)) compactness of the stalk's hydrophobic region can only be achieved if the relation (r_A) + _T(z_B) = /2 is fulfilled (see also Fig. 2). Hence, smallness of (r_A) would go at the expanse of _T(z_B) and vice versa. The optimal compromise is where (r_A)  _T(z_B)  /4. In the Results section we show that, indeed, the energetic minimum of the stalk is adopted for the situation where neither (r_A) nor _T(z_B) deviates much from /4.

The Euler equations must be solved with respect to certain boundary conditions. Two boundary conditions reflect the compactness of the stalk structure: _D(r_D = 0) = 0 and the above-mentioned (r_A) + _T(z_B) = /2. Some other boundary conditions can be formulated if one requires the proximal and distal monolayers of the fusion stalk to join an unperturbed bilayer (for which according to Eq. 2 vanishes everywhere). This implies certain conditions for the lipid directors at positions r_B and r. If in addition to that we demand r_B = r then the unperturbed bilayer, joining the stalk, is parallel to the r axis (or, equivalently, to the plane of mirror symmetry). This case is obviously relevant if a stalk is formed starting from two, apposed, parallel bilayers. Yet, this situation need not be the energetically most favorable one. We may, alternatively, allow the bilayer wings to adjust their angle,  (see Fig. 2), with respect to the r axis such that F adopts a minimum. The difference of the free energy F for both cases reflects the energetic cost of transforming an energetically optimal stalk into one that connects two parallel membranes. In the Results section we shall consider both cases.

There are two more boundary conditions to be chosen. In fact, these two boundary conditions play a crucial role for the purpose of the present study. One of them specifies the length of the lipid director, b_D, originating at position r_D = 0 of the distal monolayer. If the fusion stalk has a smooth interface, then this length must be chosen such that the condition

(6)

is fulfilled. Alternatively, if the stalk is not required to adopt a smooth interface, b_D(r_D = 0) is allowed to adjust such as to minimize F. In the latter case, the distal monolayer may exhibit a corner at its symmetry axis. The second remaining boundary condition specifies the length, b(r = r_A), of the lipid director at the transition region between the proximal monolayer and the tether. Again, this length may be chosen such as to ensure smoothness of the stalk structure. Alternatively, it can be allowed to adjust to minimize F. In the latter case, the interfacial region of the stalk may exhibit an edge at r = r_A. These two different sets of boundary conditions, one ensuring a smooth interface of the stalk and the other one minimizing F at the expansive of forming a nonsmooth interface, are the key point of our present investigation. They allow a direct comparison of the free energies, F, for smooth versus nonsmooth stalks.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

We present calculations for the following set of material parameters: K = 0.35 k_BT/Ų,  = 6.8 k_BT, c₀ = 0, k_t = 0.1 k_BT/Ų, and b₀ = 14 Å. The material constants of lipid bilayers obtained by different investigators do often vary, depending on the experimental method and the system (compare Niggemann et al., 1995). Hence, our present choice for K and , which follows Evans and Rawicz (1990), is not meant to represent a specific system. Rather, we use it to illustrate the mechanism of how membrane edges reduce the stalk energy. The uncertainty about the correct magnitude is even larger for the tilt modulus, which has not yet been determined experimentally. The value k_t = 0.1 k_BT/Ų follows from a simple estimate that directly relates the tilt modulus to the chain stretching rigidity of the lipid chains (May and Ben-Shaul, 1999; Hamm and Kozlov, 1998).

Structure of smooth stalk

We first demand the stalk to have smooth interface everywhere. In this case we must use appropriate boundary conditions for the Euler equations as discussed in the Free Energy Minimization section (the exact boundary conditions are outlined in the Appendix; see Eqs. 19 and 20).

Recall that the bilayer wing of the stalk at r_B either optimizes its orientation (angle  in Fig. 2) or is forced to be flat ( = 0). In Fig. 3, A and B, we present calculations of the corresponding stalk structures for r_B = 70 Å. In both cases, the tether half-length l is optimized. The interfacial regions of both structures are smooth everywhere. The corresponding free energies of the stalks are similar but very high. This is in agreement with previous calculations (Siegel, 1993, 1999). The main difference of the present study is that no structural assumptions on the shape and conformation of the stalks is made. We only assume its topological structure and interfacial smoothness everywhere.

View larger version (30K): [in this window] [in a new window]   FIGURE 3   Calculated structure of a smooth fusion stalk (with the corresponding free energies, F = FP + FD + FT, indicated in units of kBT) for K = 0.35 kBT/Å2,  = 6.8 kBT, c0 = 0, kt = 0.1 kBT/Å2, and b0 = 14 Å. At rB = 70 Å the bilayer adopts its optimal orientation (A) or joins a flat, unperturbed membrane (B).

The stalk in Fig. 3 B is forced to join a flat bilayer (at r_B). Yet, its energy, F, is only marginally higher than the one for optimal bilayer wing orientation displayed in Fig. 3 A. This shows that bending of the bilayer at r_B from its optimal orientation into a flat conformation is energetically inexpensive. This allows the conclusion that the stalk energy is relatively insensitive with respect to the membrane-membrane distance of the fusing bilayers. Similarly, we conclude that the stalk energy only weakly depends on the choice of r_B. That is, for any r_B > 70 Å, the stalk free energy, F, would be found between the values given in Fig. 3, A and B. Only a considerably smaller choice of r_B < r_A +  would substantially increase F. Here  is the decay length of the membrane tilt angle perturbation, which is given by (May, 2002). Typically for lipid bilayers (and consistent with our present choice of the material parameters)   10 Å. Hence, our present results would be similar for any choice of r_B r_A + . Similar considerations also apply for nonsmooth stalks.

Structure of stalk with membrane edges

A stalk structure of much lower free energy forms if sharp membrane edges are allowed to occur. Two corresponding calculations are presented in Fig. 4, A and B. The stalk in Fig. 4 A is optimized with respect to the wing orientation of the bilayer (expressed by the angle ; see Fig. 2); for Fig. 4 B, it is  = 0. The two structures in Fig. 4 differ from the corresponding ones in Fig. 3 only in the choice of those two boundary conditions that specify whether the stalk is forced to be smooth at r = r_A and r_D = 0 or not (as discussed in the Free Energy Minimization section).

View larger version (33K): [in this window] [in a new window]   FIGURE 4   Structure of the fusion stalk (with the corresponding free energies, F = FP + FD + FT, indicated in units of kBT) for K = 0.35 kBT/Å2,  = 6.8 kBT, c0 = 0, kt = 0.1 kBT/Å2, and b0 = 14 Å. At rB = 70 Å, the bilayer adopts its optimal orientation (A) or joins a flat, unperturbed membrane (B).

The stalk structures clearly exhibit edges at the joint between the proximal monolayer and the tether. Moreover, there is a sharp corner located at the intersection between the z axis and the interface of the distal monolayer. This considerably helps the lipid chains to fill up the hydrocarbon region at r = z = 0 without causing large lipid layer perturbations. As a consequence, the free energies of the nonsmooth stalks are several times lower compared with the corresponding smooth stalks.

The stalks shown in Fig. 4 have a tether length of 2l = 12 Å. Instead of minimizing F with respect to l one can also impose a certain l to the stalk and calculate the corresponding F. We have performed this calculation for the stalk displayed in Fig. 4 A; the result is shown in Fig. 5.

View larger version (9K): [in this window] [in a new window]   FIGURE 5   Dependence of the stalk free energy on the half-length, l, of the tether. The calculation corresponds to the nonsmooth stalk shown in Fig. 4 A with variable l.

The minimum appears at some intermediate l for the following reason. For large l, the length of the tether does not affect the structure of the two perturbed bilayers. Moreover then, the free energy F_T of the cylindrical micelle-like tether is proportional to its length, implying F ~ l. The slope of F(l) must be positive because the lipid bilayer is energetically preferred compared with a uniform tether. For small l, the tether nearly vanishes and so does its free energy F_T. Yet, the hydrophobic core of the stalk must be compact, implying _T(z_B) = 0 (recall z_B = z(r = r_A)). The corresponding lipids have thus a very high tilt angle , which results in high free energies of the perturbed lipid bilayers. A compromise is offered for intermediate l in which F_T is still small and _T(z_B) can relax to values for which the corresponding lipid tilt deformations are not too high.

Note that l = 0 corresponds to the model suggested by Kozlovsky and Kozlov (2002). We shall not attempt to directly compare with their calculated energies for two reasons: First, our approach allows the local lipid chain lengths to adjust and hence contains an additional order parameter. Second, and perhaps more important, our method to calculate F assumes small values of  (this was needed to linearize the Euler equations). Yet, for (r = r_A) = /2 as required by l = 0 the profile (r) becomes large near r_A, which may induce a substantial overestimation of F. Still, despite our uncertainty regarding l = 0, the existence of an energetic minimum for intermediate l is a reliable prediction of the present model. Thus, it is very likely that fusion stalks contain a tether as shown in Fig. 4.

Comparison of smooth with non-smooth stalks

The free energy of the stalks depends strongly on the assumption whether interfacial smoothness is imposed or not. The main structural difference is the degree of chain stretching for those lipids that fill the interstices at r = z = 0. This is shown in Fig. 6 for the smooth stalk (broken lines) and nonsmooth stalk (solid lines) displayed in Figs. 3 A and 4 A, respectively. The left diagram of Fig. 6 shows the relative chain dilation s(r) = b(r)/b₀  1 for the proximal monolayer. The corresponding relative chain dilation for the distal monolayer, s_D(r_D) = b_D(r_D)/b₀  1, has the same functional dependence as s(r) (as following from the Euler equations; see Appendix). That is s_D(r_D) = s(r) in which r_D = r_D(r) relates "corresponding" directors according to Eq. 4. Note in particular that s(r = r_A) = s_D(r_D = 0). The right diagram in Fig. 6 displays the tilt angles, (r) and _D[r_D(r)] where, again, the director at r (belonging to the proximal monolayer) and the "corresponding" one at r_D = r_D(r) (belonging to the distal monolayer) point onto the same midplane position as expressed by Eq. 4. The large difference of s(r) at small r between smooth and nonsmooth stalks expresses the different degree of chain stretching, causing substantial differences in the corresponding free energies. What is the reason for the pronounced chain stretching of smooth stalks near the point z = r = 0? This stretching results from the packing of the respective lipid chains. On one hand, the lipids must not leave any voids in the hydrophobic interior of the stalk. That means they must (on average) be able to reach the point z = r = 0. The lipid packing, on the other hand, becomes very unfavorable if the monolayer's interface is strongly bent towards the point z = r = 0. This would create high splay energies for the corresponding lipids. The resulting compromise is an only moderately curved interface, as the cross-sections in Fig. 3 show. Yet, this leads to the large distance between the lipid head groups and the point z = r = 0, implying pronounced chain stretching. This situation changes drastically if the constraint of interfacial smoothness is abandoned. Then, the two lipid layers, forming regions P and T (see Fig. 1), can pack separately. An edge is created between regions P and T, but this allows the lipid heads to come close enough to the point z = r = 0 to avoid high chain stretching. At the first glance, the presence of an edge might indicate a divergence of the free energy. However, this is not the case as was already pointed out by Kozlovsky and Kozlov (2002). All relevant lipid deformations (those taken into account in Eq. 2) remain finite. Bending of the describing surface itself is not connected with an energetic cost. Hence, there is no reason for an edge (or corner) not to form if this lowers the free energy of the stalk.

View larger version (13K): [in this window] [in a new window]   FIGURE 6   Relative chain dilation, s(r) = b(r)/b0  1, and tilt angles with respect to the z axis, (r) and D[rD(r)], of a smooth (broken lines) and nonsmooth (solid lines) stalk. The profiles correspond to Figs. 3 A and 4 A.

Dependence on tilt modulus

Dependence on spontaneous curvature

View larger version (37K): [in this window] [in a new window]   FIGURE 7   Dependence of F and its components (FD, FP, and FT) on the spontaneous curvature, c0, of the lipid layer (all energies are in units of kBT) for smooth (left) and nonsmooth stalks (right). The bilayer is allowed to adopt its optimal wing orientation at rB = 70 Å (upper diagrams) or is forced to be flat ( = 0) at rB = 70 Å (lower diagrams). The structures corresponding to c0 = 0 are shown in Figs. 3 and 4.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

We have directly compared smooth and nonsmooth fusion stalks. The corresponding free energies are remarkably different. It appears that membrane edges dramatically reduce the energy stored in a fusion stalk. Membrane edges become possible through the ability of the lipids to undergo tilt deformations. The corresponding tilt modulus, k_t, is currently not well known. Even though the principal conclusions of the present study remain the same for different choices of k_t, the need for more reliable values of k_t is apparent.

The reduction of the stalk energy through the formation of membrane edges is very similar to the recent findings of Kozlovsky and Kozlov (2002). The main difference is that the present model predicts the existence of a lipidic tether-like connection between the proximal monolayers, whereas this part is absent in the work of Kozlovsky and Kozlov (2002).