INTRODUCTION

TOP ABSTRACT INTRODUCTION MODELING THE FUSION PORE THEORY DISCUSSION APPENDIX A REFERENCES

The growth of fusion pores is poorly understood. Fusion proteins play a major role in creating a pore, but their importance to pore expansion is unclear. Regardless of the specific mechanisms that are controlled by the proteins, the membranes that comprise the pore wall must deform as a pore grows. It is even possible that fusion proteins exert their effect on pore growth by regulating these deformations. Independent of the contribution of fusion proteins to pore expansion, the underlying physics of the membrane deformations must be a critically important aspect of the process. The work required to deform membranes depends strongly on membrane tensions, and therefore the tension of fusing membranes must affect the rate of pore growth. This is observed (Solsona et al., 1998; Markosyan et al., 1999). In general, the tensions of two membranes will be different, and after the membranes fuse the tensions will equilibrate to a common intermediate value. In cellular situations, tensions of plasma membranes can be substantial (Waugh and Bauserman, 1995). Membrane tensions of intracellular compartments may be larger than those of plasma membranes, and the membrane tensions of these compartments can be different: exocytotic granule membranes are thought to be under significantly more tension than plasma membranes (Monck et al., 1990; Solsona et al., 1998). Some secretory granules swell upon formation of a fusion pore (Zimmerberg et al., 1987; Curran and Brodwick, 1991; Marszalek et al., 1997) and may thereby create an additional and substantial tension. Moreover, postfusion convective flow of Golgi into endoplasmic reticulum membrane appears to be driven by tension differences (Sciaky et al., 1997). In model systems, the tensions and their differences can be even greater. For fusion of two planar membranes made from different lipids (Chernomordik et al., 1987), the tension differences will not change over time because each planar membrane tension is maintained by its Gibbs-Plateau border. To fuse liposomes to planar membranes, the liposomes are routinely swelled (Zimmerberg et al., 1980; Cohen et al., 1980) to increase their membrane tension. This promotes both fusion (Cohen and Niles, 1993) and pore expansion (Chernomordik et al., 1995; Chanturiya et al., 1997).

We previously derived equations that describe lipid flow through a fusion pore of any fixed size that connects two membranes of different tensions (Chizmadzhev et al., 1999). The current paper extends these equations to investigate the dynamics of pore growth. We considered pore growth as movement of the pore wall caused by two forces. The first are the tension and bending forces and the second are the viscous forces derived by standard membrane mechanics (Evans and Skalak, 1980). Fusion pore dynamics and lipid flux were both calculated using Lagrange's equations with dissipation (Goldstein, 1950). The dissipation was described as a shear friction within monolayers and a relative friction due to lipids moving past each other in different monolayers. Because an initial pore may form within a hemifusion diaphragma bilayer that continues to separate aqueous contents after the contacting monolayer leaflets have mergedwe considered lipid flux through these pores and pore growth as well. Our equations are clearly applicable to the fusion of pure lipid bilayers. They are also directly applicable to biological fusion pores once they have grown beyond their initial state because their walls should have characteristics typical of biological membranes. The results of our calculations show that pore widening can be promoted by pore lengthening. If the fusion proteins regulated pore length, they would be able to control the process of pore growth via that single parameter.

	MODELING THE FUSION PORE

TOP ABSTRACT INTRODUCTION MODELING THE FUSION PORE THEORY DISCUSSION APPENDIX A REFERENCES

The geometry of the system

In general, as long as a pore's radius is much smaller than the size of the fusing objects, the two membranes can be considered planar and parallel to each other, connected by the fusion pore. We thus conceptualize a fusion pore as being of toroidal shape, connecting two parallel planar bilayers each of thickness 2h, whose neutral surfaces (the interfaces between the two monolayers) are separated by 2H (Fig. 1 A). This geometry is exactly as described previously (Chizmadzhev et al., 1999), but in the present study we allow the pore radius to vary. As previously, H is kept constant. The system is cylindrically symmetrical about the z axis. The pore radius R is defined as the distance from the z axis, which passes through the center of the pore, to the junction between the toroidal and planar surfaces. The radius of the narrowest portion of the lumen of the pore is r_p = R  (H + h). The radius of the fusing objects (e.g., a planar membrane or cell) is given by R_m R. ₁ and ₂ designate the tensions of single monolayers in the upper (1) and lower (2) membranes (Fig. 1 A). The two bilayer tensions are different, 2₁ > 2₂, and are kept constant at R_m. Cylindrical coordinates (r, z, ) describe the geometry of the planar membranes (Fig. 1). For the toroidal portion, we use the specialized coordinates (, , ), where  takes on values within the interval H + h >  > H  h, the angle  is confined to the interval [/2, /2] (Fig. 1 A), and the azimuthal angle  lies in the interval [0, 2] (Fig. 1 B). H and h remain constant, but the pore radius (R or r_p) is time dependent.

View larger version (21K): [in this window] [in a new window]   FIGURE 1   A toroidal fusion pore connecting two planar membranes, 1 and 2, at different tensions, 21 and 22, with 21 > 22. (A) Cross-sectional side view of the system in x, z coordinates. The membrane-solution interfaces are represented by bold solid lines, and the surfaces of constant lipid density (CLD), for each monolayer, are shown as dashed lines. The interfaces between monolayers are denoted by the thin solid lines. (B) Top view of the system in x, y coordinates. The walls of the toroidal pore meet the planar membranes at radius R. The radius of the narrowest portion of the water-filled pore lumen is given by rp. Thus the toroidal part of the membrane lies between rp and R. The coordinate systems (x, y, z), (r, , z), and (, , ) illustrated here are described in the main text and Appendix A.

Simplifying monolayers as two-dimensional surfaces

In deriving the equations, we treat the fused membranes as homogeneous lipid bilayers. We extend the results to biological membranes in the Discussion.

For a pore to expand, lipids must redistribute between the pore wall and planar membranes, and hence pore expansion and lipid movement are intimately associated. In this paper we derive equations for the rate of growth of a toroidal fusion pore when the tensions of the membranes, 2₁ and 2₂, are different. Because the tensions are different, lipid will flow from one membrane to the other (Chizmadzhev et al., 1999). For any given pore radius, the distribution of these lipid velocities within the membranes will quickly reach steady state. Thus, at any moment, the work performed by membrane tension in causing lipid flow and pore growth is equal to the dissipation of energy due to viscosity. Two types of viscosity are involved in lipid movement. The first originates from lipid-lipid and lipid-protein interactions within each monolayer. These shear deformations, described by a shear viscosity _s, are present in both the planar membrane and toroidal pore when lipid moves. The second viscosity arises from friction between lipid monolayers as they move past each other, described by a relative viscosity, _r. The viscous friction between a monolayer and the bathing water is negligibly small (Chizmadzhev et al., 1999).

Lipid flow within a curved toroidal pore is complicated by the fact that the areas available to the lipid headgroups and acyl chains within a monolayer are different. Within the inner monolayer (the monolayer lining the pore lumen, Fig. 1), a greater area is available to a lipid headgroup than to the acyl chains (i.e., the distance , Fig. 1, is greater in the headgroup region than for the acyl chains). As a consequence, the region occupied by the headgroups is expanded relative to the portion filled by the acyl chains, which is compressed. For the outer monolayer (the monolayer contacting the extracellular space), the opposite situation pertains. We avoid the mathematical complexities of treating a curved monolayer of finite thickness with nonconstant density by choosing within each monolayer a surface of constant lipid density (CLD) that lies between the polar headgroups and the hydrophobic acyl chains (Chizmadzhev et al., 1999). The lipid density within this surface is the same as that of the planar membranes. (The neutral surface, often used as a referent, is defined as the surface on which deformations of bending and area extension are independent of each other (Kozlov and Winterhalter, 1991). The surface of CLD and the neutral surface, defined differently, are not necessarily the same. But the pivotal plane (Leikin et al., 1996)a surface where the area per lipid does not change with membrane deformationsis a surface of CLD.) In this way the fluid mechanical problem of lipid flow is reduced to a two-dimensional problem of flow of an incompressible liquid, with the two surfaces of the CLD interacting with each other through the relative viscosity. To allow explicit calculations, we assume that a surface of CLD is located in the middle of its monolayer (i.e., at  = H ± h/2).

	THEORY

TOP ABSTRACT INTRODUCTION MODELING THE FUSION PORE THEORY DISCUSSION APPENDIX A REFERENCES

Velocity distributions

We will consider separately the lipid velocities in the upper planar membrane (Fig. 1 A, 1), the lower planar membrane (Fig. 1 A, 2), and the toroidal surface of the pore. We will then match the velocities at the junctions of the toroidal pore with both planar membranes (r = R). For definiteness, we choose the positive direction of velocity as motion away from the z axis for membrane 1 and toward the z axis for membrane 2. In other words, velocity is positive for flow from membrane 2 to membrane 1. By reason of symmetry, lipid flow is radial in the planar portions of the membranes.

From conservation of area for any element of the membrane, lipid velocity is

(1)

where '_r and _r are the velocities in the two respective monolayers at r. '_1,2 and _1,2, which need to be determined, are the linear velocities of lipids in the two monolayers at r = R. The subscripts 1 and 2 correspond to membranes 1 and 2, and the superscripts ' and " denote the surfaces of CLD of the inner and outer monolayers, respectively. Double superscripts are used when the equations have the same form for both monolayers.

To obtain the lipid velocity on the toroidal portion of the pore, we again employ the principle that the area of any element of membrane is conserved as it moves through the pore. We temporarily drop the superscripts ' and " because expressions for lipid velocity are the same for each monolayer. For a pore to expand, for any portion of a toroidal surface of CLD enclosed between angles 0 and , a net influx of lipid from the planar membrane into the toroidal pore must occur because the area of the surface of CLD within the toroid increases. This is a redistribution of lipid between the planar and toroidal portions of the membranes and not a net flux of lipid from one planar membrane to the other. There is, however, a net flux of lipids between planar membranes through the pore, referred to as "transpore" flux, because of differences in membrane tensions. In other words, pore growth leads to an accumulation of lipid within the walls of the pore; transpore flux does not.

To calculate the lipid flow, we use a moving coordinate system that is fixed to and moves with the CLD surface. The area enclosed between angles 0 and  is

(2)

Only R is time-dependent, yielding

(3)

where _R = dR/dt is the translational velocity of the pore (i.e., the pore velocity). Because the area of the toroidal portion of a pore increases when the pore expands (or decreases when it shrinks) and because there is transpore flux, lipid flows at the boundaries 0 and  according to

(4)

where u is the lipid velocity in the moving coordinate system (e.g., u(0) is the lipid velocity at the equatorial circumference,  = 0). Because the velocity at the junction between the planar and toroidal portions of the membrane is to be determined (Eq. 1), it proves convenient to introduce the parameter  as

(5)

From Eqs. 3-5 we obtain for lipid velocity u

(6)

Ultimately we require the velocity of an element of membrane in the fixed coordinate system. This is obtained by adding _R sin  to Eq. 6 (_R sin  is the projection of pore velocity (_R is parallel to the planar membranes) onto the tangent of the CLD surface at any given ). Reintroducing the superscripts ' and " for inner and outer monolayers, we obtain

(7)

where H'^," are the distances between surfaces of CLDs of corresponding monolayers (Fig. 1). By matching the lipid velocities (Eqs. 1 and 7) at the junction of the planar and toroidal portions (r = R and  = ±/2), we eliminate the unknown constants ' ^,"_1,2 and obtain the velocity distribution in the planar portions as

(8)

(9)

The velocity distributions, Eqs. 7-9, depend on the three independent parameters ', ", and _R, which will be determined below. It is worth noting the physical meaning of the three terms in these three equations. The first term of each of them is the lipid velocity of transpore lipid flow and is identical to that obtained for an immobile, fixed pore (Eq. 3 of Chizmadzhev et al., 1999). It is symmetrical relative to the equatorial plane and thus is the same for the two membranes. The second term is lipid velocity due to simple lateral movement of the pore wall (translation) caused by pore expansion. The third term is the velocity of lipid that redistributes between the planar membranes and the toroidal pore when the pore expands or contracts. It is independent of transpore flux and thus does not vanish even when transpore lipid flow is zero. The influx of lipid into the toroid occurs because of lipid redistribution; the velocity of lipid influx is small for R H but is greater than the second term of lipid velocity (velocity due to pore translation) at R  H because H/2R > 1. As a result, lipid velocity can even be negative at r = R when an expanding pore is small. In contrast to the first term, the second and third are antisymmetrical relative to the equatorial plane and  = 0. As will be seen, an appreciation of the symmetries is important for understanding lipid flow and pore movement.

Pore dynamics

We use Lagrange's equations with dissipation (Goldstein, 1950) to describe the viscous motion in the system. Because the velocities (and fluxes) quickly reach steady state, Lagrange's equations in our notation have the form

(10)

where W is obtained from the bending energy of a curved pore wall, W_b, and the elastic energy, W, which is computed as the work done by the externally applied tensions 2₁ and 2₂. F is the dissipation function of the system that accounts for the frictional forces, and _i are generalized coordinates describing the state of the system.

The choice of natural generalized coordinates becomes apparent by considering a single monolayer with total area A_m and external radii R_m1 and R_m2 in membranes 1 and 2, respectively. The work dW done by tensions ₁ and ₂ to cause infinitesimal variations of R_m1 and R_m2, with A_m remaining constant, is

(11)

This equation can be rewritten as

(11′)

where ₊ = ₁ + ₂ and  = _1  2.

The area A_m is given by

(12)

The first term in this expression is the area of the monolayer within the toroidal pore of radius R (Eq. 2), the second term is the area of the monolayer within the upper (1) planar membrane, and the third term is the area of the monolayer within the planar portion of the lower membrane (2). From Eq. 12 and dA_m = 0 and dH = 0 we obtain

(12′)

We introduce the variable A as

(13)

Substituting Eqs. 12' and 13 into Eq. 11', we obtain

(12'')

The right-hand side of Eq. 12" is an exact differential, and, hence, W is a state function (or a potential) of the system in the coordinates {A, R}. {A, R} are natural generalized coordinates of the system: from Eq. 13 it is clear that the coordinate A describes lipid redistribution within the monolayer between membranes 1 and 2, while the coordinate R determines pore dynamics. The corresponding generalized velocities are , which provides transpore flux, and  = _R, which gives the velocity of the pore. is related to the lipid flow parameter  (Eqs. 7-9) by

(14)

By reintroducing the superscripts ' and " and remembering that the coordinate R is the same for the two monolayers, we obtain from Eq. 12" that

(12‴)

where A₊ = A' + A". It is useful to define a dual coordinate to A₊ as A = A'  A". ₊ and are related to ' and " by expressions similar to Eq. 14:

(14′)

where ₊ = ' + " and  = '  ".

For reference, we use the fact that W_b depends only on R to rewrite Eq. 10 in coordinates {A₊, A, R} as

(10′)

We obtain directly explicit expressions for W and the derivatives of W on the left side of Eq. 10' from Eq. 12 as

(15)

We compute the derivative of W_b from the expression for bending energy of a membrane with zero spontaneous curvature (Helfrich, 1973), given as (Kozlov and Markin, 1983; Markin et al., 1984)

(16)

where B is the membrane bending modulus.

To calculate the dissipation function, F, we use the relation (Goldstein, 1950)

(17)

and separate the dissipation rate into two terms, one for dissipation due to shear (s) intramonolayer friction and the other due to relative (r) intermonolayer friction. That is,

(18)

Shear dissipation for an incompressible fluid is calculated (Landau and Lifshitz, 1987) for each monolayer as

(19)

where '_jk is the viscous stress tensor. Clearly, only radial and angular deformations are nonzero in the planar portions of the membranes. The corresponding elements of the viscous stress tensor are

(20)

In the toroidal portion, the only two nonzero components are given by (Appendix A)

(21)

The same expressions as Eqs. 20 and 21 hold for the outer monolayers, except that the index ' is replaced by ".

Substituting Eqs. 20 and 21 into Eq. 19 yields after integration

(22)

where _s^f is the dissipation rate of transpore lipid flow and does not depend on the rate of pore dilation or contraction. It can be shown that by using ₊ and as given by Eq. 14', _s^f is the same as the dissipation rate of lipid flow for a fixed pore (it is the sum of Eqs. B1 and B2 of Chizmadzhev et al., 1999). The second term in Eq. 22 provides the rate of energy dissipation due to pore movement.

The intermonolayer dissipation rate is (Evans and Hochmuth, 1978; Chizmadzhev et al., 1999)

(23)

where

(24)

Integration of Eq. 23 yields

(25)

where _r^f is the rate of dissipation caused by lipid flow through a pore of fixed size (Chizmadzhev et al., 1999). As occurs for shear friction (Eq. 22 for _s), pore movement here additively contributes a term to relative friction (Eq. 25). Thus a most important conclusion has resulted from these calculations: the dissipation rates caused by both shear and relative friction separate into dissipation caused by transpore lipid flow (which is the same for an enlarging and a fixed size pore of the same size) and dissipation caused by pore growth. This separation is a consequence of the symmetry properties of the lipid velocity distributions (see discussion following Eq. 9).

We utilize this separation to rewrite the last equation of the system of Lagrange's equation (Eq. 10'), which depends on the variables _R and R but not on A₊ and A, in the form

(26)

where F_R is the portion of the dissipation function that depends only on _R. From Eqs. 17, 22, and 25 we obtain

(27)

Using Eq. 15 for W/R, Eq. 16 for W_b, and Eq. 27 for F_R, we obtain from Eq. 26 that the pore expansion velocity _R is

(28)

where  = 2₊ = 2(₁ + ₂). Therefore, the rate of pore expansion depends only on the sum of the tensions (Eq. 28).

The velocity of the migration of the pore in radius space (i.e., the pore velocity) in response to a force is given by

(29)

where is the potential of the force field, which governs pore movement. From Eq. 28, is given by

(30)

where is obtained by integrating the second term of the numerator of Eq. 28 with respect to R, and ₀ is an integration constant that is independent of R. The pore mobility, u_R, is defined by the effective viscosity,

(31)

By comparing Eqs. 12 and 30, we see that (R) is the work necessary to form a pore of radius R at constant A₊ and A. Consequently, (R) is effectively the "partial free energy" of the pore and determines pore dynamics. We have thus rigorously shown that a toroidal fusion pore can be considered to be a quasiparticle that migrates in R-space with mobility u_R under the force field of d(R)/dR and that both the mobility and force field can be explicitly calculated. Substituting Eqs. 30 and 31 into Eq. 29 yields the pore velocity, _R = dR/dt, in the form

(32)

where (R) is the effective line tension of the fusion pore,

(33)

Equation 32 is formally the same as the expression for the velocity of a pore within a single bilayer membrane with effective two-dimensional viscosity of 4 (Deryaguin and Gutop, 1962; Deryaguin and Prokhorov, 1981). The factor 4 appears because we assigned a two-dimensional viscosity to each monolayer and there is a total of four monolayers. Whereas line tension of a pore within a single bilayer is usually assumed to be independent of pore radius, in our treatment the line tension  is explicitly calculated and is dependent on pore radius, R.

Because the bending energy, Eq. 16, is a nonlinear function of R, the differential equation (Eq. 28) must be solved numerically rather than analytically. In the case of a large pore, R > 2H, the bending energy W_b varies linearly with R and is given as

(34)

Hence the line tension  becomes independent of R:

(35)

In this case Eq. 29 can be solved analytically as

(36)

where R₀ is the initial pore radius and R_c is a critical radius. For R₀ > R_c the pore expands and for R₀ < R_c the pore contracts. The characteristic time for pore evolution (expansion or contraction) is

(37)

Thus, for large pores, the radius increases exponentially with total tension, , as the driving force.

The lipid velocity distribution is defined by Eqs. 7-9, which contain the parameters ' and " (or, equivalently, ₊ and ) and _R. _R is given by Eq. 28. The variables ' and ", which determine the transpore flux, are obtained from the first two Lagrange equations (Eq. 10'). Because neither ₊ nor cross-multiplies with _R, the first two Lagrange equations are independent of _R (but dependent on R(t)). Therefore, the lipid velocities ' and " are independent of _R and depend on the tension difference 2(_1  2), but not on the overall tension 2(₁ + ₂). These lipid velocities are the same as those for a fixed pore as calculated previously (Chizmadzhev et al., 1999, Eqs. B2, B11, and B12). These equations for velocity provide a full and rigorous solution of lipid flow in the model. In conclusion, the tension gradient induces a transpore lipid flux, but it does not affect pore evolution; the sum of the tensions induces pore expansion but does not affect transpore lipid fluxthat is, transpore lipid flux and pore expansion are independent of each other. Physically this occurs because lipids that move from one planar membrane to the other do not alter the number of lipids within the pore wall. Only the accumulation (or depletion) of lipids within the pore wall affects pore growth.

Pore dynamics is determined by the partial free energy, , and mobility (Eqs. 28 and 31). We therefore examine the basic features of and W_b, the two components of , as a function of r_p = R  H  h. We illustrate the energies with the reasonable values B = 10¹² erg (Niggemann et al., 1995), spontaneous membrane curvature, K_s = 0, H = 10 nm, and we pick  = 1 dyn/cm for definiteness (Fig. 2). The work required to bend membranes into the toroidal shape of the pore wall varies with r_p. The bending energy, W_b, at first decreases steeply with increasing r_p, passes through a minimum, and then rises with a constant slope at r_p > 10 nm. W_b decreases for small r_p because the equatorial curvature decreases as r_p increases (i.e., the naturally flat, zero-spontaneous-curvature membrane has to bend less). Hence, if  = 0, W_b does not reach a maximum. That is, the pore would not enlarge indefinitely without membrane tension, even though the pore wall could bend. Because dW_b/dr_p is much larger than |d/dr_p|, does not greatly influence the shape of the curve of total energy (r_p) at r_p < 10 nm; only causes a displacement of the entire curve along the energy axis for small r_p. Eventually the increases in area of the toroidal pore wall with r_p become the dominant effect on W_b, and thus W_b rises linearly with r_p (Eq. 34). But for r_p greater than ~10 nm, declines as r_p² (Eq. 31); the competition between the asymptotically increasing linear function of W_b(r_p) and the decreasing parabolic function (r_p) results in a maximum in (r_p) (Fig. 2, (r_p^max) = _max). The energy barrier between _min and _max is rather high (~60kT). For the toroidal pore the line tension depends on R and can be explicitly calculated from the total tension,  = 2(₁ + ₂); the bending modulus of the membranes, B; and the separation of the membranes, H (Eq. 33). If  = 0,  = W_b and the energy of the toroidal pore keeps rising with increasing r_p (Fig. 2). In other words, the pore could never expand if  = 0. The energy barrier is finite for K_s = 0 only if membrane tension is nonzero. As the tension becomes larger, the barrier height is lowered.

View larger version (19K): [in this window] [in a new window]   FIGURE 2   Toroidal pore partial free energy  = Wb   and its components and Wb as functions of pore lumen radius rp = R  (H + h) for h = 2 nm, H = 10 nm, B = 1012 erg, Ks = 0, and  = 1 dyn/cm.

So far, it has been assumed that the spontaneous curvature K_s = 0. Increasing K_s from zero (Fig. 3, curve 2) to a positive value (curve 1) makes the slope of W(r_p) steeper and the energy barrier becomes higher. Decreasing K_s to a negative value reduces the barrier, and if K_s becomes sufficiently negative, the barrier disappears completely (curve 3). In principle, fusion proteins could promote pore growth by effecting decreases in K_s. But K_s is determined by many molecular interactions and configurations such as local waters of hydration; we consider it unlikely that all fusion proteins could control pore growth, because it would be difficult to control a parameter that is affected by so many variables, each having its own regulating factors. To allow further consideration of how pore growth is controlled by membrane tension and pore length, we continue to use K_s = 0 in the remainder of this paper. (The displacement of all energy curves from zero is arbitrary, as is their placement with respect to each other. That is, the absolute values of energy shown in the graphs are not meaningful; only the shapes and therefore the differences in energy within each curve have numerical significance. This occurs because we choose the unfused state as the reference state to calculate energy differences, and this reference state varies with the conditions analyzed. As examples: in the unfused state, two parallel membranes under different tensions have different energies; the energies of unfused planar membranes change as K_s is varied.)

View larger version (15K): [in this window] [in a new window]   FIGURE 3   Toroidal pore partial free energy as a function of rp = R  (H + h) for different values of Ks. The values of Ks are 0.02 nm1 (curve 1), 0 (curve 2), and 0.02 nm1 (curve 3). Other values of the parameters are as in Fig. 3. The curves were calculated according to Eq. 31 and expressions for Wb given by Markin et al. (1984).

Whether a toroidal pore expands or contracts depends on its initial radius, r_p⁰. If 0 < r_p⁰ < r_p^max (where r_p^max is r_p of the energy barrier) the fusion pore will settle to r_p = r_p^min, while if r_p > r_p^max the pore expands indefinitely, r_p  . Spontaneous contraction or expansion of an object, depending on whether it is smaller or larger than a critical size, is common to all theories of nucleation. For r_p > r_p^max the line tension  becomes independent of r_p (Eq. 35). If, because of thermal fluctuations, the barrier has been surmounted, r_p increases exponentially with a time constant given by Eq. 37,  = 8/.

Growth of pores in a hemifusion diaphragm

Hemifusion, the merger of outer but not inner leaflets, is conjectured to be an intermediate of full fusion. At this intermediate, a single lipid bilayer, referred to as a hemifusion diaphragm, continues to separate aqueous phases. Lipid flows along the continuous outer monolayers because of the tension gradient. We consider the case in which the hemifusion diaphragm has extended somewhat to a radius R_d, but the diaphragm is still small compared to the size, R_m, of the fusing objects (Fig. 4, shown with a pore of radius R in the diaphragm). Because the two membranes are initially at different tensions, 2₁ > 2₂, the two monolayers that comprise the hemifusion diaphragm are also under different tensions: the monolayer contributed by membrane 1 (monolayer (1)) is under tension 2₁, and monolayer (2) is under tension 2₂ (Fig. 4). We consider the pore while its radius is small compared to that of the diaphragm, R R_d (Fig. 4). The formation of this pore creates a continuous path for lipid flow from monolayer (2) to monolayer (1). We denote the velocities of lipids within monolayer (1) and (2) as ₁ and ₂, respectively (Fig. 4). We assume that lipid flow quickly becomes stationary and velocities are small. Characterizing the pore with a constant line tension  (Deryaguin and Gutop, 1962; Deryaguin and Prokhorov, 1981) reduces the problem to a two-dimensional cylindrically symmetrical flow along two parallel planes. There is friction between the two flows. While flow rates can be determined by equating the work performed by tension with the dissipation due to friction (as was done above for the toroidal pore), it is more conveniently presented by locally balancing the tension gradient against intermonolayer friction as described by the Navier-Stokes equations,

(38)

(39)

where '(r) and "(r) are the tensions at an arbitrary point r in monolayers (1) and (2). Obviously, the velocities ₁ and ₂ are not equal if the pore enlarges. Letting the tensions at the border of the diaphragm be equal to 2₁ and 2₂, the boundary conditions for Eq. (39) are

(40)

(41)

(42)

Equation 42 is a balance of forces at the edge of the pore (Deryaguin and Gutop, 1962), with _s describing the intramonolayer shear friction.

View larger version (12K): [in this window] [in a new window]   FIGURE 4   Schematic representation of a pore with radius R in a hemifusion diaphragm of radius Rd. The tensions of membranes 1 and 2 are given by 21 and 22. The tension at the border of the diaphragm must equal the sum of the tensions at the boundary of the system, 21 + 22. The lipid velocity of each monolayer is given by 1 and 2. Monolayers 1 and 2 of the hemifusion diaphragm are indicated by numerals. The arrows designate the positive directions for lipid velocity within each monolayer.

We introduce the variables

(43)

where V is twice the pore velocity dR/dt and U characterizes the rate of lipid exchange, through the pore, between monolayers. The lipid flux through the pore is equal to 2R · U. The solution of Eqs. 38 and 39 in conjunction with the continuity equation,  · v = 0, yields

(44)