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Department of Physics, University of Washington, Seattle, Washington
Correspondence: Address reprint requests to M. Schick, E-mail: schick{at}phys.washington.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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The initial stages of the sequence are relatively clear (1
,2). The membranes to be fused must be brought sufficiently close to one another, within a few nanometers. To do so, water must be removed, which takes energy. Presumably, this is provided by fusion proteins in biological systems, but can, in laboratory samples in which such proteins are absent, be provided simply by ordinary depletion forces (3). As a result of the decrease of water, the free energy per unit area of the system increases; in other words, the system is now under tension. The free energy can be reduced if the system sheds area. Fusion, which accomplishes this, is one possible response of the system to that tension. The next stage in the process is that, locally, some lipid tails in the membrane leaves which are closest to one another, i.e., the cis leaves, flip over and embed themselves in the hydrophobic environment of the cis leaf of the other bilayer, thereby forming a "stalk" (4), as depicted in Fig. 1 a. This process is consistent with experimental evidence ((1) and references therein), and has been seen directly in simulations of coarse-grained, microscopic, models of membranes (5–9).
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) was that the stalk expands radially from an axis perpendicular to the bilayers, as in Fig. 1 b. The cis layers retract, leaving a hemifusion diaphragm, which consists only of the leaves of the two membranes that were initially furthest from one another, the trans leaves. Note that membrane area has been reduced as the hemifusion diaphragm now consists only of two, trans, leaves in place of the original four, two cis and two trans. The appearance of a hole in this hemifusion diaphragm completes the formation of the fusion pore, Fig. 1 c. On the basis of phenomenological modeling similar to that employed earlier (4), a second scenario was suggested: that the pore forms without significant radial expansion of the stalk (10,11). A third possibility was revealed by simulations of coarse-grained, microscopic models (5,6,12). In this, which we denote the first stalk-hole mechanism, the stalk does not expand radially, but elongates asymmetrically. Its presence makes more favorable the formation nearby of a hole in either bilayer by reducing the line tension of the hole (13). The stalk then surrounds the hole, which also produces a hemifusion diaphragm, as in the standard stalk mechanism, but one consisting of a cis and trans layer of one of the original bilayers. The appearance of a second hole, this in the hemifusion diaphragm, then completes the fusion pore. A hemifusion diaphragm is also consistent with experimental evidence ((14) and references therein). In a variant of this mechanism, denoted the second stalk-hole mechanism, the second hole appears before the first is surrounded. The mobile stalk then surrounds them both forming the fusion pore. After formation of the fusion pore, the pore expands to further eliminate area and thus reduce the system's free energy. Simulations of coarse-grained, microscopic, models have observed the original mechanism (8,9,12), and also the stalk-hole mechanism (5–9,12). If the path to fusion is not well established, neither is the limiting free energy barrier of the process. It had been thought, on the basis of phenomenological calculations, that the free energy to form the initial stalk was so large that its formation could well be the barrier to fusion. Improvements in the way the stalk was modeled (15), and in the phenomenological free energy describing the elastic properties of the membrane (16), which forms the stalk, resulted in a marked reduction in the estimate of the free energy of formation of the stalk. For a bilayer with symmetric leaves characterized by a spontaneous curvature appropriate to dioleoylphosphatidylcholine (DOPC), this quantity is estimated by Kozlovsky and Kozlov (16) to be 43 kBT, and by Kuzmin et al. (11) to be 
25 kBT. In contrast to phenomenological theories, self-consistent field theory has been applied to a coarse-grained microscopic model of a symmetric membrane (17), resulting in an even lower estimate of 13 kBT. Irrespective of the particular number, it would not appear that the formation of the stalk presents the largest barrier to fusion.
If stalk formation is not the rate-limiting process in fusion, what is? In the standard picture in which the stalk expands radially into a hemifusion diaphragm, it is the formation of this structure, which takes a great deal of energy. For a symmetric bilayer of DOPC, a diaphragm of modest radius of 2.5 nm costs on the order of 80 kBT, if one uses the estimate of Kozlovsky and Kozlov (16) for the diaphragm line tension. How large the diaphragm must become before a pore forms is not clear from this calculation. Kuzmin et al. (11) consider a modified stalk and a different radial symmetric intermediate, a pre-pore. They find its energy, 60 kBT, to be less than that of a hemifusion diaphragm, and the largest along the fusion pathway. Self-consistent field calculations examined both the classical pathway (17) and the first stalk-hole mechanism (13), and located the barriers to fusion for symmetric bilayers. In the former, the largest barrier occurred when the hemifusion diaphragm expanded to a radius, which was of the same order of the hydrophobic thickness of a bilayer. Pore formation followed. The value of the barrier ranged from 25–65 kBT, depending upon the tension and the architecture of the amphiphiles. The barrier decreases with increasing tension and as the architecture tends toward dioleoylphosphatidylethanolamine (DOPE), and away from DOPC. Calculated barriers in the stalk-hole mechanism tended to be somewhat smaller than in the standard mechanism, but only by a few kBT. Thus, the two mechanisms seem to be comparable in terms of their energetics, at least for the symmetric membranes examined.
Biological membranes are not symmetric, however. In human red blood cell membranes, for example, most of the cholinephospholipids, sphingomyelin (SM) and phosphatidylcholine (PC), are found in the outer, ectoplasmic, leaf, and most of the aminophospholipids, phosphatidylethanolamine (PE) and phosphatidylserine (PS), are found in the inner, cytoplasmic, leaf (18,19). In particular the mol % of PC in the outer/inner leaf is 22:8, of SM is 20:5, of PS is 0:10, and of PE is 8:27 (18). To maintain this imbalance costs energy (20), therefore it is reasonable to assume that it plays some physiological function. One suggestion is that this imbalance promotes fusion in intracellular events (21–23). The reasoning is as follows. Of the four major lipid groups cited above, three of them, SM, PC, and PS (24) form bilayers under physiological conditions. They make up 65% of the total bilayer, but 84% of the outer, trans, leaf. PE, however, does not form lamellae, but rather an inverted hexagonal phase (25). It has often been noted (26) that regions of this nonbilayer phase resemble the nonbilayer configurations posited to occur in fusion. Furthermore, PE resides predominantly in the inner, cytoplasmic, leaf of the plasma membrane. While it makes up only 35% of the total bilayer composition of human red blood cell membranes, it comprises 54% of the inner leaf. It is presumed to also reside predominantly in the outer leaf of a bilayer vesicle within the cell, as the outer leaf of such a vesicle would make contact with the inner leaf of the plasma membrane during fusion of the vesicle and plasma membrane, and thereby have the opportunity to exchange lipid content. However, it is precisely the inner leaf of the plasma membrane and the outer leaf of a vesicle which would be closest to one another during fusion (i.e., would be the cis leaves), and would undergo the largest deviation from a planar configuration. Hence, the enhanced concentration of PE in these leaves would presumably promote fusion.
There is much experimental evidence to support the view that the presence of hexagonal-forming lipids in the cis leaves enhances fusion. In particular, model membranes (i.e., which have equal composition in both leaves) fuse readily when composed of a mixture of PE and PS approximating that of the inner leaf of the erythrocyte membrane (27), while those consisting of PC and SM do not. Asymmetric membranes were investigated by Eastman et al. (21), who utilized dioleoylphosphatidic acid (DOPA), a lipid with a headgroup smaller even than PE, which they could move from cis to trans layers by applications of a pH gradient. They found fusion to be correlated with the amount of DOPA in the cis leaf. With DOPA present in the cis leaf in modest amounts, 5 mol %, fusion of large unilamellar vesicles occurred readily on the addition of Ca2+. However, when DOPA was sequestered in the trans leaf, little or no fusion was observed. Conversely, if one adds to the cis leaf lauroyl lysophosphatidylcholine, which has a large headgroup when compared to its single tail, fusion is inhibited dramatically (23).
As important as this asymmetry appears to be to the process of fusion, it is little addressed in theoretical calculations. In phenomenological ones, it has been accounted for by allowing the inner and outer leaves to be characterized by different spontaneous curvatures. In this way, Kozlovsky and Kozlov (16) predict that the free energy of stalk formation depends essentially only on the spontaneous curvature of the cis leaves, and decreases rapidly as this curvature is made more negative, (i.e., as one proceeds from the lamellar formers toward the hexagonal formers). A similar calculation and result follows for the free energy of formation of the hemifusion diaphragm (28). That the free energies of the stalk and hemifusion diaphragm depend essentially only on the properties of the cis leaf is in accord with the experimental observations (21). There are no direct results for the effect of the asymmetry on the largest barrier to fusion, however. Further, by treating the entire cis layer as having the same spontaneous curvature, the calculation cannot capture the ability of hexagonal-forming lipids to respond locally to an environment in which the leaves are locally deformed, which will result, in general, in their distribution being nonuniform (29). Simulations of fusion have not yet considered the effects of asymmetry, presumably because the asymmetric distribution represents a constrained equilibrium, a situation more difficult to handle than an unconstrained one.
In this article, we consider two important conditions noted above; that the bilayer leaves consist of at least two classes of lipids, lamellar formers and hexagonal formers, and that these lipids are distributed asymmetrically with respect to the cis and trans layers. We do so by extending the application of self-consistent field theory to microscopic models of membranes initiated earlier (13,17). The basic assumption of this approach is that the self-assembly into bilayer vesicles and the processes which these vesicles can undergo, such as fusion, are common to systems of amphiphiles, of which lipids are but one example. Recent work on vesicles consisting of diblock copolymers serves to illustrate this point (30). It follows that these processes can be explored in whatever system of amphiphiles proves to be most convenient. For the application of self-consistent field theory, that system is one of block copolymers in a homopolymer solvent. While the processes that amphiphiles undergo are presumably universal, the energy scales of these processes are system-dependent, and thus it is necessary to be able to compare the energy scale in a biological bilayer with the scale in a system of block copolymers. From the comparison of the value of a dimensionless ratio in the former system to the same ratio calculated in the latter, it was determined that the energies in the biological system were greater than those in the polymer system by a factor of 2.5. The ratio chosen for comparison was that of two energies, the thermal energy, kBT, and the surface free energy per unit area of the bilayer multiplied by the square of its thickness.
We examine in two stages the effect on the energies of fusion intermediates caused by the asymmetric distribution of amphiphiles in multicomponent bilayers. First, we consider the effect on the barriers to fusion due to the presence of two kinds of amphiphiles in leaves of identical composition, as in artificial membranes. We do this for the standard mechanism, and for both the first and second stalk-hole fusion mechanisms. We find that the barriers are reduced appreciably because the hexagonal-forming amphiphiles can go to the regions where they relieve the most strain (29). The barriers in the two variants of the stalk-hole mechanism are not very different from one another. We then consider the same overall composition, but redistribute the two amphiphiles asymmetrically, with the hexagonal formers being more concentrated in the cis leaves. The barriers in the standard fusion mechanism and in the second stalk-hole mechanism are calculated. The overall effect of having two such different amphiphiles distributed unequally between the two leaves is dramatic. The major barriers to fusion in the two scenarios are reduced to such an extent that they are now comparable to the rather small initial barrier to stalk formation. This barrier is not affected appreciably by the addition of the hexagonal formers nor by their asymmetric distribution, and remains on the order of 13 kBT. As a result, the fusion pathway consists of two small barriers. Once bilayers are brought sufficiently close to initiate the process, fusion should therefore proceed rapidly.
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THE MODEL |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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,17), so we will only discuss here the necessary extensions. We consider a system of two different amphiphiles, which are each AB block copolymers and are denoted 1 and 2, and a solvent of A homopolymer. The volumes occupied by a solvent chain of N segments, and of a chain of amphiphile 1, also taken to be of N segments, are Nv, where v is the volume of each segment. The volume occupied by a chain of amphiphile 2 of 
segments is 
The fraction of hydrophilic, A, monomers in amphiphile 1 is f1 and that in amphiphile 2 is f2. In our subsequent calculations we shall take f1 = 0.4, close to the value of 0.43 which would characterize DOPC, and f2 = 0.294, approximately the value characterizing DOPE (17). In order that the hydrophobic length of the two different amphiphiles be the same, we require 
so that 
Thus we have two amphiphiles with the same hydrophobic length, but different hydrophilic lengths. Amphiphile 2 is a hexagonal former with a smaller hydrophilic headgroup than amphiphile 1, which is a lamellar former. We denote the local volume fraction of hydrophilic elements of amphiphile 1 to be 
A, 1(r), of amphiphile 2 to be A, 2(r), and of the solvent to be A, s(r). The total local volume fraction of hydrophilic elements is denoted
 | (1) |
Similarly the total local volume fraction of hydrophobic elements is
 | (2) |
The amounts of each of the components are controlled by activities, 
1, 2, and s. The system is taken to be incompressible and of volume V. Because of the incompressibility constraint, only two of the activities are independent. Within the self-consistent field approximation, the excess free energy, 
sym(T, A, 1, 2, s), of the bilayer system of area A, is given by
 | (3) |
, and functionals of the fields wA and wB. These fields, and the Lagrange multiplier 
(r), which enforces the local incompressibility condition, are determined by the self-consistent equations
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
), and the barriers to fusion are calculated for the standard and for the stalk-hole mechanisms as in the previous two articles (13,17). The free energy in the self-consistent field approximation, 
is obtained by inserting into the free energy of Eq. 3 the functions satisfying the self-consistent Eqs. 4–8 with the result
 | (9) |
With the excess free energy known, the surface free energy per unit area, or equivalently, the surface tension, 
, follows from
 | (10) |
).
Just before formation of the stalk-hole complex, the elongated stalk was treated as if it were in the shape of a circular arc with a fractional angle, 0 
1, and radius R, as shown schematically in Fig. 2 a. Its free energy is
 | (11) |
= 1, and Fs is the free energy of a stalk. This is because it is the sum of the energies of the two end caps of a structure for which 
1, and these two end caps together make a stalk.
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–35), which is partially surrounded by the elongated stalk. This intermediate is approximated by the configuration shown in Fig. 2 b whose free energy is
 | (12) |
= 1 in which the stalk would have completely surrounded the hole forming a hemifusion intermediate, FH(R – ) is the free energy of a hole of radius R – in a bilayer, and Fd is the free energy of the defects at the end of the arc. Equality of the free energies of Eqs. 11 and 12 defines a ridge line in the space of parameters and R, and the minimum of this ridge defines a saddle point along this fusion path.
In the second stalk-hole mechanism, just after formation of the stalk-hole complex, there are two holes, one in each bilayer, partially surrounded by the elongated stalk. Again, the picture is as in Fig. 2 b, but now the circular object in the center of the figure represents the two holes, rather than the one as previously. Thus, the figure at the extreme right now represents a fusion pore. The free energy of this configuration is
 | (13) |
) = 2FH(R – ) is the free energy of two holes, each of radius R – , one above the other, and F'd the energy of the two defects at the end of the arc. Again equality of Eqs. 11 and 13 defines a ridge line in the space of parameters and R. The minimum along this ridge defines the fusion barrier along this second stalk-hole pathway. |
RESULTS FOR SYMMETRIC BILAYERS |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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0.5 Rg, we find no stable stalk solution of the self-consistent equations. We have taken the volume Nv which appears in the free energy, Eq. 9, to be 
as in our previous work (13,17). The four solid curves in Fig. 3 correspond to volume fractions of the hexagonal former of 0, 0.04, 0.11, and 0.17 from top to bottom. The first thing to be noted is that, at small radii, there are no solutions that show a local minimum of the free energy as a function of radius. Such solutions would correspond to a metastable stalk. That there are no metastable stalks in a bilayer composed of amphiphiles with f = 0.4, close to the value f = 0.43, which would characterize DOPC, had been noted earlier (17). One consequence of this observation is that fusion of such bilayers would have to take place via one large thermal excitation due to the lack of a metastable stalk intermediate. As a consequence, the timescale for fusion would be expected to be rather long, certainly longer than if the intermediate were metastable. The results of Fig. 3 show that the addition of hexagonal formers up to volume fractions of 0.17 equally in each leaf does not bring about the existence of a metastable stalk.
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/0 = 0.1, where 0 is the interfacial free energy per unit area between coexisting solutions of hydrophobic and hydrophilic homopolymers at the same temperature. At larger values of the radius of the hemifusion diaphragm than shown in the figure, the free energy of the diaphragm decreases due to the tension. One sees from Fig. 3 that the barrier to fusion does indeed decrease with the addition of hexagonal formers. As can be seen in the figure, this reduction comes about both because of the reduction in energy of the fusion pore and of the hemifusion diaphragm. The reduction in the pore energy is due to the effect of the hexagonal formers, which can go to the sharp bend of the cis leaf existing in the pore. Similarly, the reduction in the energy of the hemifusion diaphragm is due to the hexagonal formers concentrating at the rim of the diaphragm. This is shown in Fig. 4. In Fig. 4 a, the volume fractions of the heads (dashed line), and tails (solid line) of the hexagonal-forming amphiphile far from the hemifusion diaphragm are shown as a function of z/Rg. In Fig. 4 b, we show the volume fractions of the hexagonal-forming amphiphile in a cut through the hemifusion diaphragm itself in the plane of reflection symmetry, the z = 0 plane, as a function of the radial coordinate, 
/Rg. (Recall that such a hemifusion diaphragm is shown in Fig. 1 b.) One sees that the diaphragm has an approximate radius of 5 Rg. A comparison of the plots in Fig. 1, a and b, shows that the local volume fraction of tails of the hexagonal former at the diaphragm rim increases by 20%, and that the local density of heads of this amphiphile increases there by almost 50%.
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). As seen there, for f = 0.35, and /0 = 0.1, the barrier to fusion is somewhat lower in the first stalk-hole mechanism than in the standard mechanism, and the barrier in the latter varies more rapidly with architecture, f, than in the stalk-hole mechanism. In the upper panel of Fig. 6, we compare the fusion barriers in the first and second stalk mechanisms. The former is shown in solid circles and the latter is shown in solid triangles. Defect energies are taken to be 4 kBT. One sees that there is not a great deal of difference in the energy barriers in the two mechanisms. One also notes that the second stalk-hole mechanism has a lower energy than that of the first when the fraction of hexagonal-forming amphiphiles is low. The situation is reversed as the fraction increases. This is to be expected as the second stalk-hole intermediate consists of portions of a fusion pore and of two holes. Both of these structures are disfavored by the hexagonal-forming amphiphiles. On the other hand, the first stalk-hole intermediate consists of portions of a hemifusion diaphragm, and only one hole. The hemifusion diaphragm is favored by the hexagonal-forming amphiphiles.
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RESULTS FOR ASYMMETRIC BILAYERS |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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1, 2, and s and the incompressibility condition. Therefore we want to calculate the excess free energy 
where 
is the number of hexagonal-forming amphiphiles in the cis leaf of the bilayer:
 | (14) |
A,2(r), and therefore of 
This adds to the system's internal energy a term of the form
 | (15) |
The field h(r) is taken to be non-zero only in the cis leaf. Our choice is
 | (16) |
). The field is non-zero only in the region shown in Fig. 7. From the coupling term of Eq. 15, one sees that the greater the strength of the field, h0, the larger will be the volume fraction of the hexagonal formers in the cis layer. Once the excess free energy of this asymmetric system, 
is obtained, differentiation of it with respect to the field strength h0 yields A, 2(r). The number of hexagonal formers in the cis layer, 
is then obtained by integration of this quantity, Eq. 14. The value of the field strength h0 can then be adjusted to obtain the desired concentration of hexagonal formers in the cis layer.
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 | (17) |
The self-consistent equations, Eqs. 4–8, are unaffected. Again, the free energy in the self-consistent field approximation is obtained by substituting the functions that satisfy the self-consistent equations into the free energy of Eq. 17 with the result
 | (18) |
is now obtained by a Legendre transform
 | (19) |
 | (20) |
Because the system is constrained to have a different concentration of hexagonal formers in the cis leaf than in the trans leaf, its free energy will clearly be greater than if it were not so constrained. This is also true of the free energies of the various intermediates, like the stalk, hemifusion diaphragm, and pore. For the fusion process, however, we are interested in differences in free energies between the intermediates and the flat bilayers, and these differences can certainly be less in the constrained system.
Standard mechanism
The calculations for the standard mechanism are relatively straightforward due to the axial and reflection symmetry of the stalk, the hemifusion diaphragm, and the pore.
In Fig. 8 we show results for a bilayer under a tension /0 = 0.1 composed of the lamellar-former comprising a fraction 1 = 0.650 of the bilayer by volume, and the hexagonal former comprising a fraction 2 = 0.350 by volume. Results are presented for the excess free energy of the hemifusion diaphragm (solid lines) and of the fusion pores (dashed lines) for different volume fractions in the cis leaf of the hexagonal-forming amphiphile. In the upper set of curves, there is no asymmetry, so that the volume fraction of hexagonal former in the cis leaf, 
is the same as in the whole bilayer. In the middle curve, the volume fraction of the hexagonal-former in the cis leaf has been increased to 
Its volume fraction in the trans leaf is concomitantly reduced to 
and the volume fractions of the lamellar former in the cis and trans leaves are 0.605 and 0.695, respectively. In the lowest curve, we have set 
so that 
and the volume fractions of the lamellar former in the cis and trans leaves are 0.569 and 0.731. The barrier to fusion is reduced from 11 kBT to 8.5 kBT, to 5 kBT, as the asymmetry increases. For the largest asymmetry shown, the barrier to fusion is essentially no greater than the barrier to formation of the initial stalk itself. Furthermore for this asymmetry, the intermediate stalk is finally a metastable structure. As noted earlier, this has a large effect on the timescale of fusion by permitting the process to occur in two stages rather than one.
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Our results for the barrier to fusion of asymmetric bilayers within the second stalk-hole mechanism are shown in Fig. 9. We have calculated them for bilayers in which the average volume fraction of hexagonal formers in the entire bilayer is kept fixed at 2 = 0.350, while the fraction of hexagonal formers in the cis layer, 
takes the values 
(i.e., no asymmetry), 
and 
The barrier to fusion for this second stalk-hole pathway is shown by the triangles. The values of at the saddle point in the fusion pathway are = 0.073 for = 0.174 when and = 0.18 when These barriers to fusion are compared to those calculated in the standard mechanism and shown in squares. These values were shown previously in Fig. 8. Finally, we also compare them with the free energies of the stalk, shown in circles.
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in the stalk-hole mechanism imply that the stalk does not have to elongate very much to nucleate the formation of the two holes which, when surrounded by the stalk, will become the fusion pore. (We recall that in surrounding the holes, the energy of the system is reduced as the line tensions of the bare holes are replaced by the lower line tension of a hole next to a stalk (13).) Hole formation is enhanced, and the barrier to fusion reduced, because the majority amphiphile, 1 = 0.65, is a lamellar-former with f = 0.4. Furthermore, the actual volume fraction of the lamellar former near the rim of a hole will be larger than this because the amphiphiles are free to move within a leaf to that region where they will reduce the energy most. The increase of with the fraction of hexagonal formers in the cis leaf is readily understood. As the fraction of hexagonal former in the cis leaf increases, the energy of a pore decreases, as noted previously. It follows from Eq. 13 that , the fraction of the stalk-hole intermediate that resembles a pore, will increase. |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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We first considered bilayers whose leaves have identical compositions, and added hexagonal formers to each leaf equally. We examined the effect of this addition on the barrier to fusion as calculated in the standard mechanism, and the first and second stalk-hole mechanisms. We noted that the stalk was not a metastable intermediate in these systems in which DOPC-like amphiphiles were the dominant constituent. Nonetheless, we considered the barrier energy and found it to be reduced significantly in the standard mechanism from 24 kBT with no hexagonal-formers to 11 kBT with a volume fraction of 0.35 hexagonal formers. This is seen in Figs. 5 and 9. As noted earlier (17), we expect that the energies in biological, lipid, membranes are higher by a factor of 2.5 than in the block copolymer membranes we are considering. Thus the above barrier values would correspond to one of 60 kBT being reduced to 28 kBT. The reduction in the fusion barrier of the standard mechanism is due to a reduction in the energy of the hemifusion intermediate, partly because the average number of hexagonal formers has increased (16), and partly because the hexagonal-forming amphiphiles preferentially go to the edge of the hemifusion diaphragm, as seen in Fig. 4. Fig. 5 shows that the greatest rate of decrease comes about when the hexagonal formers are first added to the pure bilayer of lamellar formers. This rapid decrease occurs because the hexagonal formers go to the regions where they can most readily reduce the free energy. The distribution of the different amphiphiles is not spatially uniform when there are fusion intermediates. The reduction of the barrier energy in the stalk-hole mechanism is more modest, but is not insignificant. In the second stalk-hole mechanism, it is reduced from 8.3 kBT when there are no hexagonal-formers to 6.8 kBT with a volume fraction of 0.350 hexagonal-formers. Again this would correspond to a reduction from 21 kBT to 17 kBT in a biological system.
We then examined the effect on the barrier to fusion of an unequal distribution of hexagonal and lamellar formers in the two leaves. We considered a system in which the hexagonal formers make up a volume fraction of 0.350 of the whole system, much as they do in human red blood cell membranes. This brings about a further significant reduction in the barrier to fusion in both mechanisms. In the standard mechanism, this is due to the reduction in energy of the hemifusion diaphragm, while in the stalk-hole mechanism it is due primarily to the reduction in energy of the elongated stalk.
The energy of the initial stalk itself is not affected very much either by the addition of hexagonal formers to each leaf equally, as seen in Fig. 3, or by the redistribution of the hexagonal formers between the two leaves, as seen in Fig. 9. The former result is in contrast to the prediction of phenomenological theories of a sensitive dependence upon the amount of hexagonal formers (16). While the absolute energy is little affected by the addition of hexagonal formers, we found that their asymmetric distribution caused the stalk to become a metastable intermediate, which would allow fusion to become a two-step, rather than one-step, thermally activated process.
Certainly the most important result of our calculation is the following: although the fusion process remains one with two barriers, one due to stalk formation and another that depends upon the specific mechanism, the second barrier is rapidly reduced by the addition of hexagonal former of greater abundance in the cis layer to a value comparable to that of the initial stalk itself. As emphasized earlier, the calculated energy of the stalk is rather small, 5 kBT in our copolymer system, corresponding to 13 kBT in a biological membrane.
We note that the volume fraction of hexagonal former in the cis leaf at which the two barriers become approximately equal occurs in our model at a value of 
The average fraction of hexagonal formers in the bilayer is 0.35. Under the assumption of equal molecular weights for the A and B components of the diblock, these volume fractions correspond to a mole fraction of 0.47 in the cis leaf of a bilayer whose average mole fraction is 0.39. Again, the mole fractions of hexagonal formers in the membrane of human red blood cells are 0.54 in the cis leaf and 0.35 when averaged over both leaves of the bilayer. Thus equality of the two barriers occurs in our model at a somewhat smaller asymmetry between leaves than occurs in red blood cell membranes. As the asymmetry increases, the second barrier to fusion continues to decrease and eventually becomes negative. When this occurs in a bilayer under zero surface tension, the bilayer is unstable. In the system shown in Fig. 9, this instability occurs at a mole fraction of hexagonal former in the cis layer of 0.50.
We examined both the standard, hemifusion diaphragm pathway to fusion, and the more recently proposed stalk-hole pathway. In the system with the mixture of lamellar and hexagonal formers similar to that of red-blood cell membranes, we found that the barriers to fusion in the two mechanisms did not differ greatly, with those in the new mechanism being slightly lower. This would indicate that fusion could proceed by either pathway. In the standard mechanism, fusion is nonleaky. In the stalk-hole mechanism it can be leaky. The small values of , the fraction of hole surrounded by the stalk when fusion occurs, which were obtained in the preceding section certainly would bolster the possibility of leakage. However, as Eastman et al. (21) already noted, even though fusion between model membranes is generally leaky, "...systems exhibiting asymmetric transbilayer distributions of lipid clearly have the potential to be self-regulating and possibly to exhibit leak-tight fusion.... It will be of particular interest to determine the leakiness of fusion events in such systems." This work strongly reinforces that observation.
Our results predict that the rate of fusion in asymmetric systems depends nonlinearly on the volume fraction of hexagonal formers in the cis layer. This results from at least two effects. First, even linear changes of energy barriers with volume fraction of hexagonal former translate into nonlinear changes of fusion rates because fusion is a thermally activated process. This nonlinear behavior is in accord with the results of Eastman et al. (21). Second we found in our system resembling DOPC, DOPE mixtures with a fixed average composition that the stalk intermediate became metastable only when the asymmetry attained a certain minimum value. At that point, the fusion rate is expected to increase significantly.
To reiterate, our major result is that the two barriers to fusion are comparable and small for an amount of hexagonal former found in the cis layer, which does not differ greatly from that found in red blood cell membranes. One important implication of this result is that fusion should proceed readily once external sources have brought the membranes sufficiently close to initiate the process. Our results have been obtained by examining planar bilayers, and for the study of the fusion of endocytotic vesicles with the plasma membrane, the large curvature of the vesicles should probably be taken into account (36,37), something that can be done within the self-consistent field theory we have employed. However, it is known that such a curvature only enhances the fusion rate (38), and so the process should again proceed readily once vesicle and membrane are brought to an optimum distance.
The observation that fusion should proceed quickly once the membranes are brought sufficiently close naturally leads to the question of how the energies of fusion intermediates depend upon the distance between the two tense membranes, which might fuse. This is an issue we shall address in a later publication.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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This work was supported by the National Science Foundation under grant No. 0503752.
Submitted on September 28, 2006; accepted for publication December 4, 2006.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONTHE MODELRESULTS FOR SYMMETRIC BILAYERSRESULTS FOR ASYMMETRIC BILAYERSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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and 
From top to bottom, the volume fractions of type 2 diblocks in the bilayers are 0.00, 0.04, 0.11, and 0.17.
