INTRODUCTION

Top Abstract Introduction Theoretical results Discussion Appendix 1 Appendix 2 References

Lamellar (L), inverted cubic (Q_II), and inverted hexagonal (H_II) phases have very different topologies. The transitions between these phases have to make substantial changes in topology. In particular, they must make or break extensive connections between lipid/water interfaces. In this respect, the transition mechanisms resemble the first steps in membrane fusion. Many researchers have speculated that the first intermediates in these two different processes, fusion and lamellar/inverted phase transitions, are similar. A structure known as a "stalk" has been proposed as the first intermediate to form in the process of membrane fusion (Markin et al., 1984; Chernomordik et al., 1985, 1987; Leikin et al., 1987; Kozlov et al., 1989; Siegel, 1993). The stalk hypothesis is fairly successful in rationalizing many observations concerning fusion in model and biomembrane systems (Chernomordik et al., 1995a; Siegel, 1993; Basáñez et al., 1998; see Chernomordik et al., 1995b, and Chernomordik and Zimmerberg, 1995 for reviews). It is therefore important to see if a phase transition mechanism based on stalk intermediates can also explain the observed dynamics of lamellar/inverted phase transitions.

Siegel (1993) developed a method for estimating the energies of hypothetical intermediate structures with respect to planar bilayers, which is an elaboration of the technique of Markin et al. (1984). The method is based on studies of the relative energies of lipids in lamellar and inverted phases by Gruner, Parsegian, Rand, and others (for reviews, see Gruner (1990), Lindblom and Rilfors (1989), Seddon (1990), and Tate et al. (1991)). Kozlov et al. (1994) have used a method with the same principal elements to reproduce a complex portion of the phase diagram of water/dioleoylphosphatidylethanolamine (DOPE). This demonstrates that the method yields fairly accurate results. Siegel (1993) used the method to propose a modification of the original stalk theory of membrane fusion.

Here the same method is used to find the lowest-energy sequence of intermediate structures capable of generating H_II and Q_II phases from an L phase. The calculations are made for the lipid DOPE, using the same bending elastic modulus and the same temperature-dependent H_II unit cell dimensions in excess water as Kozlov et al. (1994). The energies of the structures can be calculated as a function of temperature, so we can also estimate how the relative stability of the different structures changes with temperature.

The first intermediates to form during lamellar/inverted phase transitions should be the same as in the stalk theory (Markin et al., 1984). In the process of fusion, stalks were originally proposed to expand radially to form extensive areas of single-bilayer diaphragm between apposed liposomes (e.g., Markin et al., 1984). However, Siegel (1993) showed that, in lipid systems without apolar oils or alkanes, radial expansion should be limited, and stalks should expand radially into smaller structures known as hemifusion intermediates or transmonolayer contacts (TMCs). TMCs are critical intermediates in the fusion process, because they can decay into fusion pores. Here it is shown that TMCs should also play an important role in the lamellar (L)/inverted hexagonal (H_II) phase transition. TMCs can aggregate within the planes of apposed bilayers to form a structure that can elongate directly into a domain of H_II phase. In a recent study of the L/H_II transition mechanism via time-resolved cryo-transmission electron microscopy (TRC-TEM), Siegel and Epand (1997) presented evidence for the existence of this aggregate of TMCs. A transition mechanism based on TMCs is also more compatible with observations than mechanisms based either on inverted micellar intermediates (Siegel, 1986a) or on "conical LIPs" (Hui et al., 1983) for another reason. Hui et al. (1983) speculated that stalk-like intermediates could elongate directly into line defects, which are structures resembling part of an H_II phase unit cell. A similar structure was invoked by Siegel (1986a). Here it is shown that this process is not spontaneous.

It was proposed (Siegel, 1993) that individual TMCs can rupture to form fusion pores (also called interlamellar attachments or ILAs; Siegel, 1986a,b). ILAs are structural elements that assemble into Q_II phases (Siegel, 1986c; Siegel et al., 1989c; Frederik et al., 1991). Here it is shown that, as a lipid system is heated toward T_H, thermodynamically stable ILAs and Q_II phases should form. The Q_II phase should then form H_II phase at higher temperatures. This behavior is observed in many systems with thermotropic lamellar/inverted phase transitions.

It is clear that formation of ILAs and Q_II phases is very slow and hysteretic in some systems. Presumably, the rate of ILA formation from TMCs determines whether the L phase in a given system forms Q_II phases (by accumulation of many ILAs) or H_II phases (by accumulation of a large steady-state population of TMCs) on the experimental time scale. Composition-dependent factors other than the curvature elastic energy and interstice energies may affect the rate of fusion pore (ILA) formation from TMCs. Some of these factors, particularly the composition-dependent bilayer rupture tension, are discussed here. The results suggest an important new means by which low levels of particular lipids like lysolipids, or certain types of peptides, could substantially enhance the rate of fusion in model membranes and biomembranes. A summary of some of the results of this theoretical analysis has been presented elsewhere (Siegel and Epand, 1997).

	THEORETICAL RESULTS

Top Abstract Introduction Theoretical results Discussion Appendix 1 Appendix 2 References

Principles of the method for calculating intermediate energies

The energies of the intermediates are treated as the sum of the curvature elastic energies of the lipid monolayers and the formation energies of hydrophobic interstices within the structures. Details are given in Siegel (1993) and Appendix 1. Briefly, the curvature elastic energy (G_c) of a lipid monolayer is calculated by the method of Helfrich (1973): the energy is of the form where k_m and C₀ are lipid composition-dependent properties that can be measured by appropriate x-ray diffraction experiments, C₁ and C₂ are the principal radii of curvature of the monolayer, and the integral is taken over the area of the monolayer surface. The curvatures and area are evaluated at a defined plane within the thickness of the monolayer. Hydrophobic interstices are found in the H_II phase and have a positive free energy of formation. The energy can be calculated from results of structural studies of these phases (Siegel, 1993).

The first intermediates to form during bilayer/nonbilayer transitions are probably stalks, which transform into TMCs

It was previously proposed that the first intermediate to form between two apposed membranes (Fig. 1 A) during membrane fusion is the stalk (Markin et al., 1984; Chernomordik et al., 1985, 1987; Leikin et al., 1987; Kozlov et al., 1989). The calculations of Siegel (1993) also support this view. Of the hypothetical intermembrane intermediates that have been proposed so far, stalks have the lowest energy. A stalk is depicted in Fig. 1 B. The structure is cylindrically symmetrical about the dashed vertical axis and has a shape like a thread spool or the center of an hourglass.

View larger version (21K): [in this window] [in a new window]   FIGURE 1   The modified stalk theory of membrane fusion and inverted phase. (A) Planar L phase bilayers. (B) Stalk. The stalk is cylindrically symmetrical about the dashed vertical axis. It is composed only of lipids in the apposed (cis) monolayers of the two bilayers. (C) Trans monolayer contact (TMC) or hemifusion intermediate. (D) TMCs can form two different types of structures. If the bilayer diaphragm in the middle of the TMC ruptures, it forms a fusion pore (also referred to as an interlamellar attachment or ILA). Left: A cross section through a perspective view of an ILA. If ILAs accumulate in sufficient numbers, they form ILA lattices, which can rearrange to form QII phase. Right: For systems close to the L/HII phase boundary, TMCs can also aggregate to form HII phase precursors, which will be discussed later (Fig. 7). In C and D the edges of lipid monolayers are stippled. Figure adapted from Siegel and Epand (1997).

A stalk can either revert to the original bilayer structure or form a trans monolayer contact (TMC), depicted in Fig. 1 C. The high-curvature region of the cis monolayer of the stalk expands radially, and the trans monolayers of the original bilayers dimple inward to contact each other in the center. This forms a bilayer diaphragm composed only of lipids originally found in the trans (nonapposed) monolayers of the two apposed bilayers. This diaphragm is surrounded by a linear hydrophobic interstice. The unfavorable energy of interstice creation prevents the bilayer diaphragm from growing in diameter by more than a nanometer or so. A TMC can greatly reduce its energy by diaphragm rupture, which produces an ILA (Fig. 1 D). An ILA is the same as a fusion pore, so formation of an ILA between two apposed liposomes corresponds to membrane fusion.

The energies of stalks, TMCs, and ILAs change as a function of size. In Fig. 2 A, the energies of these intermediates in dioleoylphosphatidylethanolamine (DOPE) are plotted as a function of r, the marginal radius of the original stalk, under conditions corresponding to T = T_H, with  = 0. Stalks can always reduce their energy by shrinking (decreasing r). However, stalks that form with a large enough value of r will be able to spontaneously form a TMC. For comparison, similar plots are given for the energies of the intermediates in systems with C₀ = 0.25 and 0.1 nm¹ in Fig. 2, B and C. These values are the values expected for DOPE/DOPC  3/2 mol/mol, and pure DOPC; respectively. The energies of TMCs and ILAs increase with increasing C₀.

View larger version (18K): [in this window] [in a new window]   FIGURE 2   Plots of the free energy of stalks, TMCs, and ILAs as a function of marginal radius, r, for systems with different values of C0. (A) C0 = 0.3484 nm1, corresponding to DOPE at T = TH. (B) (B) C0 = 0.250 nm1. This corresponds to a mixture of DOPE/DOPC  3/2 mol/mol at around room temperature. (C) C0 = 0.100 nm1, corresponding roughly to pure DOPC at room temperature. Energies are in units of kBT, where kB is Boltzmann's constant and T = 298 K.

How might these intermediate structures mediate lamellar/inverted phase transitions? We wish to test whether stalks, TMCs, and ILAs can generate H_II and Q_II phases from the L phase. Let us discuss the mechanism of the L/Q_II phase transition first, because it arises very naturally from the results already obvious in Fig. 2. Then we will return to the mechanism of the L/H_II phase transition, which is more subtle.

ILAs and inverted cubic phases are thermodynamically stable in a temperature interval around T_H

Previous work indicates that ILAs are Q_II phase precursors (Siegel, 1986c; Siegel et al., 1989c). Therefore, if the modified stalk theory generates large numbers of ILAs in systems in the appropriate temperature range, then it is capable of producing Q_II phase formation. Q_II phases should also be stable through at least part of the temperature interval in which ILAs are stable.

An important and obvious feature of the results in Fig. 2 is that ILAs are much more stable than planar bilayers at T_H. T_H is defined as the temperature at which the free energies of lipid in the L and H_II phases are equal. If formation of no other structure intervenes, this will be the temperature at which the L/H_II phase transition occurs. However, the model predicts that there should also be polymorphism (ILA and Q_II phase formation) in the vicinity of T_H. In fact, ILAs are predicted to be thermodynamically stable in general for a substantial temperature region around T_H.

We can estimate the temperature range for ILA stability by calculating the free energy of ILAs as a function of C₀. C₀ is a function of temperature. As T increases, C₀ decreases (becomes more negative). At some value of C₀, the free energy of the ILA will be the same as an equivalent area of planar L phase bilayer. This should correspond to the lowest temperature at which ILAs will form in significant numbers. The highest temperature for ILA formation will be the temperature where the free energy of the ILA equals the free energy of an equivalent amount of H_II phase lipid. Above that temperature, at equilibrium, H_II phase will form in preference to ILAs. The temperature dependence of C₀ has been obtained from the temperature dependence of the unit cell dimensions of the H_II phase (Tate and Gruner, 1989; Kozlov et al., 1994). It is convenient to express C₀ as its inverse, R₀. For small temperature intervals around T = T_H (Kozlov et al., 1994),

(1)

In Fig. 2, it is obvious that the free energy of ILAs decreases (they become more stable) as the temperature increases toward T_H. However, it is also clear from Fig. 2 that the ILA energy at any given C₀ or T depends on the value of the marginal radius r. In fact, the equilibrium energy of an ILA is sensitive to both the marginal radius and the axial radius of the final ILA structure. To compute the energy of an ILA as a function of temperature, we need to know the equilibrium dimensions of an ILA as a function of C₀. We will attack this problem first and then return to the problem of determining the temperature range of ILA stability.

Equilibrium dimensions of ILAs

Once formed from a TMC, an ILA should be able to expand or contract to reach dimensions at which it is more stable. Chizmadzhev et al. (1995) have previously shown this for fusion pores, evaluating the curvature energy on a bilayer basis. Here, similar calculations are made, evaluating the curvature energy of the structure as the sum of the energies of both monolayers, which gives a more accurate account of the energy at small ILA dimensions and a more accurate account of the C₀ dependence of the energy. The energy of an isolated ILA is plotted as a function of size in Fig. 3 A, for the value of C₀ at T = T_H  10 K. The radii are displayed as the values at the midplane of the bilayer surface of the ILA, to ease interpretation in terms of the overall dimensions of the ILA. The marginal radius r_b is the radius in the plane of the ILA axis, and R_b is the axial radius of the ILA. The lowest energy region is a trough that extends to the upper right of the diagram. An isolated ILA would expand to large values of r_b and R_b, although the slope of the surface in this trough is very small at large radii, and the rate of expansion would eventually become trivial. This dependence is qualitatively very similar to that described by Chizmadzhev et al. (1995). If an ILA can expand almost indefinitely, as Fig. 3 A implies, what other constraints will limit this expansion and determine the true energy?

View larger version (33K): [in this window] [in a new window]   FIGURE 3   (A) The free energy of an ILA as a function of the marginal radius rb and axial radius Rb, both evaluated at the bilayer midplane. C0 = 0.3311 nm1 (DOPE at T = TH  10 K). An ILA with small values of rb and Rb will increase in size in both dimensions to lower the curvature energy (in kBT). (B) The free energy per lipid molecule (kBT/molecule) of an ILA as a function of rb and Rb. There is a unique size that minimizes the energy per lipid (rb = 4.1 nm and Rb = 2.66 nm).

The free energy of an individual ILA is lower than an equivalent area of planar bilayer, but the energy decreases more and more slowly as the ILA expands. Therefore the free energy of the system as a whole can be minimized by producing many ILAs with more modest dimensions. We calculate this area-minimized energy by dividing the total ILA energy by the area of bilayer in the ILA. The result is expressed as a free energy per lipid molecule, using the area per lipid molecule a = 0.65 nm² for DOPE (Kozlov et al., 1994). The area of bilayer in an ILA is

(2)

The energy per lipid in an ILA is plotted in Fig. 3 B, for a C₀ value corresponding to T = T_H  10 K. Note that now there is a minimum-energy dimension, although the minimum is rather broad. The most stable ILA at T_H has r_b and R_b values of ~4.1 and ~2.66 nm, respectively. If a bilayer is 4 nm thick, this corresponds to an ILA "waist" diameter of ~9 nm. The same procedure can be used to calculate the equilibrium dimensions of ILAs at any value of C₀.

The ILA dimensions calculated in this manner are in fairly good agreement with observations. The observed "waist" diameter of ILAs in similar systems varies across a wide range, as expected on the basis of the broad minima in Fig. 3 B, but is usually 12-15 nm (Siegel et al., 1989c; Frederik et al., 1991). This is somewhat larger than the predicted value. However, the only ILAs whose dimensions can be accurately measured via cryoelectron microscopy are those that are comparatively isolated (i.e., not in multilamellar arrays that are densely packed with ILAs, where superposition effects obscure the features of individual structures). Isolated ILAs are still free to expand to larger dimensions, as indicated in Fig. 3 A: they have not come up against the constraint of maximizing the number of ILAs per unit area. Thus they may be larger than the area-minimized dimensions calculated above.

The upper temperature limit of ILA stability

As the temperature, T, increases, the H_II phase should eventually become more stable than ILAs. To find this temperature, we need expressions for the free energies of lipid in the H_II phase as a function of T. The free energy per molecule of lipid in the H_II phase with respect to the L phase is given by (Siegel, 1993)

(3)

where the interstice energy has been expressed in terms of k_m and r_H (equation 4 of Siegel, 1993). r_H(T) is the radius of the H_II phase monolayers as a function of T, r_H(T_H) is the value of this radius at T = T_H, and a is the area per lipid molecule at the monolayer midplane. In this derivation, it is assumed that r_H(T) is always equal to R₀(T), which appears to be an excellent approximation (Kozlov et al., 1994), and that a is constant throughout the relevant temperature regime. When the area-minimized energies of ILAs (obtained from plots like Fig. 3 B) and H_II phase (Eq. 3) are plotted as a function of C₀, the plots intersect at a value of C₀ corresponding to T = T_H + 13°C (Fig. 4).

View larger version (29K): [in this window] [in a new window]   FIGURE 4   ILA and HII phase energy as a function of temperature. Filled squares: HII phase. Filled circles: ILA with only geometric mean curvature energy. The dotted vertical line indicates the temperature at which HII phase becomes more stable than an array of ILAs. Circles: ILA with chain-packing energy included. Squares: ILA with Gaussian curvature energy included. See Appendix 2 for a discussion of the chain packing and Gaussian curvature energies.

The lower temperature limit for ILA stability

The lower temperature limit for ILA stability should be the temperature at which the free energy of ILA lipid is the same as an equivalent amount of L phase lipid. The free energy per lipid molecule in ILAs and H_II phase is plotted versus temperature in Fig. 4. These energies are referenced to the free energy of L phase lipid, which is defined as zero at all temperatures. The free energy of ILA lipid decreases asymptotically to that of L phase lipid as the temperature decreases. Therefore, it is difficult to make an accurate prediction of the lower temperature limit of ILA stability: the slope of the ILA energy versus temperature plot is small, and small imperfections in the model produce large differences in the estimated intersection temperature. For instance, at 70 degrees below T_H, the free energy per lipid molecule in an ILA is 0.011k_BT less than the free energy of L phase lipid. This difference is only halved by a further 30 degree reduction in temperature.

The true onset temperature is probably higher. TMCs are precursors to ILAs, and the free energy required to produce TMCs increases with decreasing temperature. TMCs may be too rare for there to be any significant rate of ILA production at very low temperatures (see below). Moreover, the calculation assumes that R₀ depends linearly on temperature (Eq. 1), and that the curvature elastic modulus k_m is independent of temperature. Both of these assumptions are unrealistic for such a large temperature range of 100 degrees. As T decreases, R₀ may increase faster than indicated by Eq. 1, and the bilayers probably become more rigidly planar as they approach the L/L phase transition temperature, T_c, where the acyl chains freeze. In fact, the T_c of DOPE is only 26 degrees below T_H (Marsh, 1990). Since gel phase bilayers are apt to have larger values of R₀, ILAs are not expected to form from gel phases. Therefore, our rough estimate for the onset temperature for ILA formation is that ILAs can start to appear at T_c but should become numerous only at higher temperatures. This is consistent with the observation by many authors (e.g., Cullis et al., 1978; Ellens et al., 1986, 1989) that ILAs formed at high temperature only revert to bilayer structure when the system is cooled to low temperatures (below T_c), and Q_II phases have recently been observed to transform directly to the gel phase at T_c (Tenchov et al., 1998). The effects of including Gaussian curvature and chain-packing energies on the temperature range of ILA stability are estimated in Appendix 2.

How well does the predicted temperature range of ILA stability agree with observed behavior?

Therefore, if we consider only the geometric mean curvature elastic energy of ILAs, the present model predicts that ILAs are stable in a temperature region extending from T_c to ~13 K above T_H. Q_II phases, which assemble from ILAs, should be stable through at least part of this temperature range. The possible mechanisms of transformation of ILA arrays into Q_II phase, and of Q_II phase into H_II phase, are discussed at the end of this Results section.

The range of ILA stability is predicted relative to the value of T_H, so one can determine how well the model predicts the temperature range of ILA stability only in systems in which the L/Q_II transition is so hysteretic that one can accurately measure T_H. T_H also has to be far enough above the ice point and far enough above the L/L phase transition for the onset of ILA formation to be easily observable. N-methylated DOPE (DOPE-Me) is a system that meets both of these requirements, because it has a T_c of 12.5°C (Gruner et al., 1988) and a T_H of 66°C (Ellens et al., 1989; Siegel et al., 1989a,c). DOPE-Me has H_II phase structural dimensions at T_H that are similar to those of DOPE and probably has a similar value of k_m, because of the similar structure of the acyl chains and headgroups. ILAs appear in DOPE-Me between 30 and 40 K below T_H, as observed by cryoelectron microscopy and ³¹P NMR (Gagné et al., 1985; Ellens et al., 1989; Siegel et al., 1989a,c). ILAs must form in significant numbers to be visible by either technique, so it is possible that small numbers of ILAs form at even lower temperatures. For instance, 1 or 2% of the lipid in the system must exist as ILAs to be detected via ³¹P NMR (Ellens et al., 1989). Siegel and Banschbach (1990) found that the Q_II phase in DOPE-Me is stable to ~6-13 K above T_H. ILAs, detected as an isotropic ³¹P NMR resonance, are present at temperatures at least 14 K above T_H (Gagné et al., 1985). Therefore, the observed range of ILA stability is reasonably close to the predictions of this simple theory.

Do all thermotropic systems form the Q_II phase at lower temperatures than the H_II phase, as predicted?

Many thermotropic systems form Q_II phases in a temperature interval between L and H_II phases, as required by the present model. Examples include the monoglycerides (e.g., Caffrey, 1987; Briggs and Caffrey, 1994) and branched-chain phosphatidylcholines (Lewis et al., 1994). However, there are also many systems in which L/Q_II transitions are either very slow and hysteretic, or absent. Principal examples are pure phosphatidylethanolamines and some N-alkylated phosphatidylethanolamines. Q_II phases form in DOPE at around T_H, but they must be induced by temperature-cycling across T_H (Shyamsunder et al., 1988). While DOPE-Me spontaneously forms Q_II phases in this interval, the transition is very slow (Gruner et al., 1988; Siegel and Banschbach, 1990). Tenchov et al. (1998) recently reported formation of Q_II phases by temperature cycling in four PEs and one PE derivative. Other N-alkylated phosphatidylethanolamines show signs of the same behavior (Leventis et al., 1991). Finally, although many glycolipids readily form Q_II phases (e.g., Mannock et al., 1992), others only show signs of hysteretic Q_II phase formation (Mannock et al., 1994).

These observations imply that Q_II phases could be stable in all systems, but that there is a kinetic barrier to the L/Q_II transition in some systems, as has been suggested previously (Gruner et al., 1988; Shyamsunder et al., 1988; Anderson et al., 1988; Siegel and Banschbach, 1990). The possible nature of this kinetic barrier is treated later in this section (see the text beginning with the subsection, titled What determines whether ILAs and Q_II phases form spontaneously at T < T_H?). The fact that Q_II phases are not observed throughout the observed range of ILA stability (Gruner et al., 1988; Siegel and Banschbach, 1990) indicates that there are additional factors that determine the thermodynamic stability of these phases. These most likely include a chain-packing effect (Anderson et al., 1988), which is thought to be temperature dependent (Lewis et al., 1989; Siegel, 1993).

The effectiveness of temperature cycling in producing Q_II phases (Shyamsunder et al., 1988; Veiro et al., 1990; Tenchov et al., 1998) is easily understood in light of the stability of ILAs. If a system forms small numbers of ILAs each time it is heated through T_H, the ILAs will persist when it is cooled below T_H. After enough cycles, a sufficiently large fraction of the lipid will exist as ILAs to rearrange into a Q_II lattice (Siegel, 1986a; Siegel and Banschbach, 1990).

How could L/H_II phase transitions proceed?

Now let us return to the question of how intermediates in the modified stalk theory might mediate the L/H_II phase transition. To transform into the H_II phase, all of the lipid monolayers in the lamellar phase have to be rolled up into tubes. As discussed previously (Siegel and Epand, 1997), the lowest energy, fastest pathway between the two phases would be to make a few small connections between apposed lipid/water interfaces in the lamellar phase and then lengthen them in the plane of the apposed bilayers into single units or domains of H_II phase, by diffusion of lipid from adjoining planar bilayers. Stalks and TMCs seem to be the lowest energy interbilayer structures that can form (Siegel, 1993), so we use these as models of the initial connections. Cross sections of stalks and a TMC are shown in Fig. 5, A and B, respectively. Note that the cross sections of the stalks and TMC all contain a region (dashed diamond) that is very similar to the cross section of an H_II phase unit cell (Fig. 5 C).

View larger version (25K): [in this window] [in a new window]   FIGURE 5   Comparison of the cross sections of (A) stalks and (B) a TMC, with the cross section of a unit cell of HII phase (C). The cross section of a unit cell of the HII phase is outlined in dashed lines in A and B. The regions of monolayer that are cross-hatched in A and B have unfavorable curvature energy. (Adapted from Siegel and Epand, 1997.)

In L/H_II transition mechanisms like those proposed by Hui et al. (1983), Caffrey (1985), Siegel (1986a), and Siegel et al. (1994), stalks or TMCs could elongate in the plane parallel to the two apposed bilayer membranes to form a prism of H_II-like structure. These structures were referred to as "line defects" (Siegel, 1986a). The reasoning was that lipid in such a structure would have a chemical potential very nearly equal to that of lipid in the H_II phase, so at a temperature slightly above T_H, lipid would diffuse into the structure from the adjoining bilayer, lengthening it. These prisms would tend to align with other prisms forming between the same pair of bilayers (Siegel, 1986a) to assemble H_II phase domains.

The L/H_II phase transition does not occur via formation of line defects from individual stalks or TMCs

However, a transition mediated by line defects from isolated stalks or TMCs cannot rapidly form large domains of H_II phase. The analysis of Siegel (1986a) showed that line defects formed between the same pair of apposed bilayers could elongate rapidly and would tend to align with each other to form two-dimensional rafts of H_II tubes. However, there is no obvious mechanism by which the rafts formed between different pairs of bilayers could align with each other on a time scale of seconds or less to form three-dimensional arrays of H_II tubes. Thus it is hard to see how the transition could proceed via growth of line defects from individual stalks or TMCs.

More importantly, it was subsequently argued (Siegel and Epand, 1997) that individual stalks or TMCs cannot spontaneously grow into H_II-like line defects at temperatures near T_H and hence cannot be responsible for the rapid transitions observed under those circumstances. A detailed analysis of this second point is given here.

Let us consider the two types of stalks that might evolve into line defects. In Fig. 5 A, the stalk at the top left has flat-topped interstices, and the stalk at the bottom left has dimpled interstices (see the Theoretical Methods). However, there are important differences between the cross sections of these structures and the cross section of the H_II phase unit cell (Fig. 5 C). A line defect structure with the cross section depicted at top left in Fig. 5 A has hydrophobic interstices that are nearly three times as large in cross section as in the H_II phase. The reduction in curvature free energy obtained by forming the curved interfaces in the H_II phase is more than offset by the energetically unfavorable creation of these larger interstices. Using the surface area scaling method to evaluate the energies of the interstices (Eq. A3), the free energy per unit length of such a line defect relative to the L phase, g_LD, would be

(4)

where the second bracketed term is the product of the perimeter of the interstice and the interstice "surface tension." At T = T_H, the value of r is very close to /R₀/ (Kozlov et al., 1994). Thus, using Eq. A3, Eq. 4 becomes

(5)

The free energy per lipid molecule in such a structure, relative to the free energy in the L phase, is g_LD divided by the number of lipid molecules entering this structure per unit length of extension, which is n_LD,

(6)

Dividing Eq. 5 by Eq. 6 yields the free energy per lipid molecule in the interstice with flat-topped interstices with respect to the lamellar phase.

The free energy of a line defect growing from the stalk at the bottom left in Fig. 5 A can be calculated in similar fashion. In this case, the major difference in cross section between the stalk and the H_II phase unit cell is the positive-curvature regions of monolayer at the margins (cross-hatched regions in Fig. 5 A, bottom left). These positive-curvature regions are energetically unfavorable. The sum of the curvature elastic and interstice energy terms for a line defect of this second geometry is

(7)

where g_tsv is the free energy of interstices bounded by monolayers of radius r (Siegel, 1993):

(8)

The first, second, and third terms on the right side of Eq. 7 are the curvature energy of the curved trans monolayer margins of the cross section (bottom left of Fig. 5 A), the two dimples in the trans monolayer, and the two cis monolayer segments, respectively. Assuming r = r_H = |R₀|, as above, Eq. 7 becomes

(9)

and (by simple geometry) n_LD is

(10)

Inspection of Eqs. 5 and 9 shows that g_LD for both hypothetical structures is positive with respect to the L phase, indicating that spontaneous growth of these structures will not occur at T = T_H. Using Eqs. 4-10, the free energies per lipid molecule in the line defects with cross sections at the top and bottom left of Fig. 5 A are ~0.14 and ~0.12k_BT at T = T_H. Although these energies are small for every lipid molecule, the energies of the macroscopic lengths of such line defects (which would be needed to achieve the phase transition) would be very large indeed.

The only circumstances in which isolated stalks can elongate spontaneously into line defects at T = T_H is when the local angle between the apposed interfaces (, right-hand side of Fig. 5 A) is 30°. However, this cannot be true at all places within the closely apposed bilayers of a bulk L phase: the bilayers would have to be deformed into 30° bends along the entire periphery of the growing line defect and be maintained in that configuration. This requires the input of additional energy and would prohibit the side-to-side aggregation of line defects that is necessary to form bulk H_II phase (Siegel, 1986a).

Although the cross section of a TMC (Fig. 5 B) also contains the cross section of a unit cell of H_II phase (dashed lines), the regions of bent bilayer at the periphery of the structure (cross-hatched areas) also have a large, unfavorable curvature energy. These regions are larger than the corresponding regions in stalk cross sections (Fig. 5 A, left), and it is easy to show that line defect growth from isolated TMCs takes even more energy than from isolated stalks. Therefore, elongation of TMCs also cannot be the mechanism of the L/H_II phase transition at T near T_H.

Experimental results support the analysis given here. In a previous temperature-jump time-resolved cryoelectron microscopy (TRC-TEM) study of bilayer/inverted phase transition mechanisms (Siegel et al., 1994), only very rare examples of line defects were observed. Interestingly, these rare examples occurred at the rims of contacts between apposed unilamellar liposomes, where the apposed bilayers met at angles of almost 30° (Figure 10 of Siegel et al., 1994). The present analysis shows that this is the only condition in which line defect formation is possible at T  T_H. Siegel et al. (1994) speculated that the absence of line defects at T  T_H was due to a fast reversion of the line defects on the time scale of specimen cooling and vitrification (~0.1 ms). In light of this subsequent analysis, isolated line defects are simply too unstable to form at T  T_H.

Formation of H_II phase precursors from aggregates of TMCs

Siegel and Epand (1997) previously suggested that TMCs could aggregate and form nuclei that could elongate into H_II phase domains. This proposal will be analyzed in more detail here.

The energy of isolated stalks and TMCs would be substantially decreased if the curvature energy of regions of monolayer at the periphery of the structures (cross-hatched regions in Fig. 5, A and B) could be reduced. For example, for a system like DOPE near T_H, two-thirds of the energy needed to form a TMC from L phase lipid goes into producing the curved bilayer regions at the periphery of the TMC. Side-to-side aggregation of two TMCs can reduce the free energy of these curved bilayer regions, lowering the free energy of the pair below the total of the energies of the isolated TMCs.

The aggregation process is depicted in Fig. 6. Fig. 6 A shows isolated TMCs forming within a stack of multilayers. A TMC (Fig. 6 A) is characterized by three parameters: the marginal radius r, the dimple radius r₃, and the marginal angle . The highest energy portion of the TMC is the "skirt" of bilayer corresponding to a range of  of 60°-90°. It will be shown below that small reductions in the angle  at any value of r substantially reduce the TMC energy, whereas small changes in r and r₃ have little effect on the energy of the TMC (Siegel, 1993). If two TMCs aggregate side by side as shown in Fig. 6 B, the area of "skirt" is reduced in the region of TMC contact. More importantly, the local value of  at the TMC-TMC contacts is reduced from 90° to a figure close to 60° (stippled area of bilayer in Fig. 5 B). The local values of r and r₃ have to increase and decrease, respectively, to accommodate the local change in . However, the local spacing between bilayers around the periphery of the TMCs need not change to accommodate this local change in TMC geometry, since the sizes of r and r₃ can change in a complementary fashion around the circumference of the TMC. The aggregation process should continue in this pairwise fashion until an extended aggregate forms with a body-centered primitive tetragonal symmetry. Within this aggregate, each TMC has eight nearest neighbors: four arranged in a square around the "top" of the central TMC, and four around the "bottom." A cross section through this structure in the 110 plane is shown in Fig. 6 C. The shape of the array is complex, but it is similar to the geometry of the ILA array postulated as an intermediate in L/Q_II phase transitions (figure 3 A of Siegel, 1986c; figure 3 C of Siegel and Banschbach, 1990) and later observed via freeze fracture EM (Ellens et al., 1989; Siegel et al., 1989c). The major difference is that in the TMC array the catenoidal interbilayer connections contain bilayer dimples instead of the water channels of the ILA array.

View larger version (22K): [in this window] [in a new window]   FIGURE 6   TMC aggregation into HII phase precursors. (A) A pair of isolated TMCs in a stack of planar bilayers. The axis of each TMC is shown as a dashed vertical line. The marginal radius r, dimple radius r3, and marginal angle  are indicated. (B) Sideways aggregation of two TMCs reduces  and increases r where they are in contact (stippled area of bilayer). The values of r and  change in complementary fashion in the vicinity of the contact, so that the local spacing between bilayers around the periphery of the TMCs does not change. The process is driven by the large reduction in TMC energy achieved by locally reducing the value of  (see text and Fig. 7). (C) The aggregation process proceeds to form a TMC aggregate with body-centered or primitive tetragonal symmetry. A cross section of this structure in the 110 plane is shown here. This cross section consists of closed cylinders of monolayer packed in quasihexagonal fashion (i.e., each cylinder has six nearest neighbors). At T > TH, this structure could acquire lipid molecules by diffusion from adjoining bilayers and extend directly into a domain of HII phase. (D) Cross section of a bundle of HII phase tubes, to illustrate the similarity of the geometry in C. (Adapted from Siegel and Epand, 1997.)

Energetics of TMC aggregation

The resulting shape of the TMC aggregate, or even of a pair of aggregated TMCs, is complex and hard to evaluate with our simple treatment (Siegel, 1993). However, it is shown here that two TMCs, in the process of diffusing toward each other, will spontaneously form the structure shown in Fig. 6 B. We compare the size of the gradients in free energy associated with changing each of , r, and r₃. This shows whether the reduction in free energy due to a local decrease in  will be much more than enough to "pay" for the energy required to locally change r and r₃. The gradients are calculated for an isolated, undeformed TMC with  = 90° at the minimum-energy dimensions in DOPE at t = T_H (r = 3.7 nm, r₃ = 2.7 nm). The extent of the required changes in r and r₃ is estimated by assuming that the local spacing between bilayers does not change around the periphery of the aggregated TMCs. This is sensible, because it minimizes the amount of distortion necessary to make a stack of bilayers commensurate with the faces of a TMC array. Then it can be shown that the value of r at the sites of TMC-TMC contacts, r_c, is approximately

(11)

and the value of r₃ at TMC-TMC contacts, r_3c, will be approximately

(12)

Fig. 7 shows plots of the energy of an isolated TMC subjected to changes in each of the three dimensions. Fig. 7 A shows that the energy of a TMC decreases rapidly and continuously with decreasing . The full line shows the curve for the equilibrium values of r and r₃. Reducing  to ~60° reduces the energy by almost 45k_BT. The size of this gradient is insensitive to the values of r and r₃. For example, Eqs. 11 and 12 were used to calculate the local values of r and r₃ necessary to achieve  = 60° at the TMC-TMC contacts. Then the energy of an isolated TMC was evaluated as a function of  with these values of r and r₃ around the entire periphery of a TMC. The results are the same to within 0.5k_BT at any . Fig. 7 B is a plot of the energy of an isolated TMC as a function of changes in r and r₃, respectively, given in terms of the values of r or r₃ needed to achieve a given value of  between two aggregated TMCs. It is obvious that reductions in  yield much greater reductions in TMC energy than the required changes in r and r₃: ~45k_BT is obtained by reducing  to 60°, but only ~5 or ~6k_BT is needed to deform r and r₃ enough for that to happen. This shows that the aggregation process depicted in Fig. 6 B and C, will be spontaneous. Note that all of the curves in Fig. 7, A and B, are continuous and monotonic. This means that as two TMCs diffuse together, they continuously reduce their mutual energy as they form an aggregated pair: no activation energy is necessary for aggregation to occur. However, note that the TMC aggregate structure depicted in Fig. 6 C is energetically metastable. The energy of each TMC in such an aggregate is substantially lower than the energy of an isolated TMC but is still greater than an equivalent amount of lipid in the L phase.

View larger version (19K): [in this window] [in a new window]   FIGURE 7   Energetics of TMC aggregation. (A) Plot of the change in energy of a TMC as the value of  is decreased from 90° (i.e., from the value for an isolated TMC with  = 0). (B) For the local value of  to be less than 90° between two aggregated TMCs, the local values of r and r3 have to change. (B) Plot of the energy required to change r (circles) and r3 (squares) for an entire TMC by the amount required to achieve the local value of  indicated on the horizontal axis between two aggregated TMCs. Reductions in  release much more energy than is required to achieve the required changes in r and r3.

The driving force for aggregation of TMCs into clusters should also be affected by an entropic factor. To calculate the size of this factor, one needs accurate values for the difference in energy between aggregated and isolated TMCs. For the reasons given above, the present model gives only crude estimates of this energy. In addition, although stalks and TMCs seem to be the lowest energy intermembrane intermediates that can form, the energies of isolated TMCs calculated by this model are too large for a structure that must form in significant numbers to mediate the transition (see Discussion). This suggests that the energies for isolated TMCs may be inaccurate, and hence use of them to calculate TMC abundances and aggregation thresholds is inappropriate. The sole aim of the calculations here is to establish that there is a substantial driving force for TMC aggregation.

Consequences of spontaneous TMC aggregation

This array has two important properties. First, note that individual TMCs within the array cannot revert to planar bilayer structure without also changing the structure of all of the surrounding TMCs: the value of  for all of the neighbors must increase to 90°, and the data in Fig. 7 A show that this increases the energy of all of the neighboring TMCs. This means that the activation energy for reversion of each TMC in the array is larger (by 10-20k_BT) than for reversion of isolated TMCs. Moreover, reversion of a TMC within the array involves coordinated motion in each of the neighboring TMCs, involving many more lipids in the reversion process than for reversion of an isolated TMC. Therefore, once an aggregate of TMCs forms, it should be kinetically metastable: it will endure for a much longer period of time than isolated TMCs. Even though the individual TMCs are unstable, the TMC array could endure for a macroscopic length of time.

The second important property of the TMC aggregate is its symmetry. Fig. 6 C shows that the cross section of the aggregate in the 110 plane has quasihexagonal symmetry. The cross section consists of closed, roughly cylindrical segments of monolayer packed in a hexagonal array (i.e., each cylinder has six nearest neighbors). This cross section resembles the cross section of a domain of H_II phase (Fig. 6 D). If the TMC aggregate is heated through T_H, lipid diffusion within the adjoining bilayers could cause this cross section to extend out of the plane of the paper, collapsing into a more perfectly hexagonal geometry as it extends, growing directly into a domain of H_II phase. The cross section in Fig. 6 C is essentially prisms of H_II phase unit cell connected by flat regions of L phase bilayers. Elongation of the structure at the expense of adjoining L phase bilayers should be spontaneous close to T_H. Therefore, Siegel and Epand (1997) proposed that this is the mechanism of the L-to-H_II phase transition: aggregates of TMCs constitute nuclei that grow directly into H_II phase domains by lipid diffusion.

The theoretical methods used to arrive at this mechanism (Siegel 1993) involve approximations. To simplify the mathematics, the intermediates are considered to have simplified geometries (i.e., to be made up of monolayer segments that are portions of spheres, cylinders, cones, or circular toroids of revolution). Related, but more complex, geometries like catenoids of ellipsoids of revolution may be involved, so that the predicted shape of the TMC aggregate is only approximate. The TMC aggregate may actually have a structure even more similar to the H_II phase than the structure depicted in Fig. 6 C.

Temperature dependence of TMC and TMC aggregate formation

Fig. 8 is a plot of the minimum TMC energy in the temperature interval below T_H. The energy changes by only 30% between T_H 20 K and T_H. If TMCs are numerous at T_H, then substantial numbers of TMCs should form starting at temperatures well below T_H. Therefore, TMC aggregates should also be able to exist (at least transiently) at T < T_H. This is as observed (see below).

View larger version (15K): [in this window] [in a new window]   FIGURE 8   Energy of TMCs as a function of temperature below TH.

TMCs and TMC aggregates should be more common in LUV than in MLV systems

As will be discussed below, there is some evidence for the TMC aggregation process depicted in Fig. 6. However, this evidence was obtained using a system in which the L/H_II transition was initiated in a suspension of large unilamellar vesicles (LUVs), as opposed to the multilamellar vesicle (MLV) systems in which the transition usually takes place (see below). Hence, before we consider this evidence, let us consider how the concentration or structure of the transition intermediates might be different in LUV versus MLV systems undergoing this phase transition.

TMC aggregates are expected (Siegel and Epand, 1997) to be larger and more numerous when the L/H_II phase transition begins with the lipid dispersed as LUVs. There are two reasons for this. First, the bilayers at the periphery of LUV-LUV contacts are curved, locally reducing  for LUVs forming in these regions, which reduces the amount of energy necessary to form TMCs. Second, the value of the marginal radius, r, for TMCs (see Fig. 1 C) of minimum energy at T = T_H is ~3.7 nm (Fig. 2 A). This results in an interbilayer spacing at the periphery of the stalk of between 5 and 6 nm. In contrast, the equilibrium water layer thickness between PE bilayers is only ~1 nm or less (Rand et al., 1988). The energy of a TMC that would "fit" between the lamellae within MLVs is more than twice as high as the energy of a TMC with larger values of r. MLVs are usually microns in extent. To form TMCs of minimal-energy dimensions, a lot of water would have to be transported through the many bilayers of the MLV, and within the planes between the bilayers. This would be a very slow process. In contrast, in aggregates of small (0.1 µ) LUVs, there are many water passages around the LUVs that reach the exterior of the aggregates, and the spacing between the apposed LUVs is initially much larger than in MLVs. For both of these reasons, TMCs should be more numerous and TMC aggregates larger and more plentiful when the L/H_II phase transition starts within aggregates of LUVs. Experimentally, it is not difficult to start the transition in dispersions of LUVs. PE LUVs can be made at high pH, where the PE is charged, the LUVs do not aggregate, and the L/H_II transition cannot occur (Ellens et al., 1986, 1989). Acidification of the suspending buffer induces rapid aggregation of the LUVs, followed by intermediate formation and H_II phase formation if T  T_H (Siegel et al., 1989c, 1994; Siegel and Epand, 1997).

There is experimental evidence that some types of long-range inverted phase order can be achieved more easily in LUV dispersions than in MLVs, as is proposed here for TMCs. The L/Q_II phase transition is more rapid in acidified LUV dispersions of DOPE-Me than in MLVs (Ellens et al., 1989). Freeze-fracture electron microscopy shows that acidified LUV dispersions form large domains of Q_II phase within 10 min or less after acidification, whereas MLVs incubated under similar conditions show only numerous ILAs and few if any Q_II inclusions. Moreover, light scattering and fluorescence data (Ellens et al., 1989) suggest that Q_II domain formation is more cooperative in LUVs than in MLVs and occurs within only 1-2 min. In contrast, Siegel and Banschbach (1990) showed that the L/Q_II phase transition in MLV samples takes ~1 h or more.

Experimental evidence for TMC aggregates and TMC aggregate-mediated L/H_II transitions in LUVs

Siegel and Epand (1997) studied the evolution of microstructure with time in acidified dispersions of DiPoPE LUVs via TRC-TEM. Their results show morphology that is consistent with the transition mechanism described in detail in the present work. Specifically, starting at temperatures as much as 38° below T_H, Siegel and Epand (1997) noted the appearance of many interbilayer structures of the same overall shape and size as predicted for TMCs (diameter ~10 nm). These structures tended to occur in large aggregates, which were usually disordered at lower temperatures. Unfortunately, the superposition of so many structures in the aggregates and the low contrast generated by such small structures in cryo-TEM (Siegel et al., 1994) did not permit demonstration of the detailed structure of the individual intermembrane structures, although they were clearly not ILAs. This is compatible with the prediction that TMC aggregates should exist well below T_H. With increasing temperature, the intermediates formed domains with quasihexagonal order that were tenths of a micron in diameter. In some projections, these domains resembled the cross sections of H_II phase domains, and arrays of bilayers with larger interbilayer spacings in others. These quasihexagonal domains formed at temperatures as much as 21° below T_H. These results suggest that the quasihexagonal domains correspond to the transition intermediate depicted in Fig. 6 C. Importantly, no evidence for isolated line defect-like structures was detected. As the temperature increased above T_H, the lipid aggregates showed a smooth transition to H_II-like order: the domains of intermediates increased in size and could not be distinguished from H_II phase. This is compatible with the suggested mechanism of H_II domain evolution from the quasihexagonal structure. Siegel and Epand (1997) also showed that the TMCs and TMC arrays were transient. ³¹P NMR spectra were obtained from PE LUV dispersions within 1-2 min after acidification at temperatures several degrees below T_H. At these temperatures, quasihexagonal domains made up a large fraction of the specimens observed seconds after acidification by TRC-TEM. The NMR spectra showed only lamellar phase patterns, with occasional small isotropic components that decayed within minutes. Large numbers of TMCs or domains of quasihexagonal structure would have produced isotropic resonances. Therefore, the intermediate morphology observed via TRC-TEM decays within minutes or less. This is as expected, because the equilibrium phase under these circumstances is still the L phase.

However, the data in Siegel and Epand (1997) are subject to two caveats. First, the LUVs in Siegel and Epand (1997) had to be produced at high pH for technical reasons, where PE is unstable with respect to hydrolysis (S. Burgess, personal communication). Some, but not all, of the morphology observed by Siegel and Epand (1997) might correspond to H_II phases formed at lower temperature than in pure PE due to formation of some hydrolysis products (D. P. Siegel, work in progress). Second, there is some reason to believe that the effective T_H is lower for LUVs than for MLVs.

The free energy benefit of adsorption of one PE bilayer to another is substantial. When the bilayers can approach each other very closely, as in MLVs, the free energy of the MLV assembly is lower than that of well-separated isolated membranes, like those in LUV dispersions. The short-range interactions between PE bilayers (van der Waals forces and interbilayer hydrogen bonding) account for much of the adhesion energy (McIntosh and Simon, 1986, 1996; Rand et al., 1988). If the bilayers are maintained more than ~1 nm apart (as in LUV dispersions), the effective adhesion energy of the bilayers would be nearly zero (Rand et al., 1988), and the free energy per unit area of DiPoPE bilayers in LUV dispersions will be higher than for DiPoPE in MLVs. Formation of transition intermediates should require less activation energy and occur at slightly lower temperatures in LUV aggregates than in MLVs. Moreover, according to the present theory, H_II phases form from aggregates of TMCs, which keep the average interbilayer separation equal to 3 or 4 nm. Thus it is conceivable that for a small range of temperature below T_H, an H_II phase could be more stable than the bilayers in aggregates of LUVs but less stable than the bilayers in MLVs. This hypothetical H_II phase would form from the LUVs and then immediately decay back into L-phase MLVs. It is possible that some of the quasihexagonal morphology observed via TRC-TEM at T < T_H actually corresponds to a metastable H_II phase. This possibility would also be compatible with the absence of nonlamellar ³¹P NMR spectra 1 min after acidification of LUVs at T < T_H, because the metastable H_II phase formed under these circumstances would rapidly revert to more stable MLVs. Therefore it is important to calculate the extent to which the free energy of bilayer-bilayer interaction could lower the transition temperature in LUVs.

Let the free energy of adhesion per unit area of one pair of PE interfaces be G_ads. The free energy of adhesion per lipid molecule is then aG_ads/2, where a is the area per lipid molecule at the bilayer/water interface. The free energy per lipid molecule in the bilayer of an LUV is then higher than that of equilibrium L phase (MLV) lipid by aG_ads/2. The free energy difference between H_II phase and L phase (MLV) lipid, G_H, can be estimated by writing

(13)

where H_H and S_H are the enthalpy and entropy of the transition per molecule, evaluated at T = T_H. The approximation used in the first equality in Eq. 13 is that the entropy change in forming the H_II phase at a temperature T is essentially the entropy change at T = T_H. This is a good approximation for a small temperature range around