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Initiation and dynamics of hemifusion in lipid bilayers

One approach to the understanding of fusion in cells and model membranes involves stalk formation and expansion of the hemifusion diaphragm. We predict theoretically the initiation of hemifusion by stalk expansion and the dynamics of mesoscopic hemifusion diaphragm expansion in the light of recent experiments and theory that suggested that hemifusion is driven by intra-membrane tension far from the fusion zone. Our predictions include a square root scaling of the hemifusion zone size on time as well as an estimate of the minimal tension for initiation of hemifusion. While a minimal amount of pressure is evidently needed for stalk formation, it is not necessarily required for stalk expansion. The energy required for tension induced fusion is much smaller than that required for pressure driven fusion.


I. INTRODUCTION
Membrane hemifusion is a possible pathway (see Müller et al. (2002) for an alternative view) to the complete fusion of membranes (Chernomordik et al., 1995). Current theories associate the initiation of hemifusion with the formation of a contact zone between the membranes in which the two proximal monolayers are connected by a stalk-shaped neck. The stalk then expands and a region is formed (region C in Fig. 1), in which the two distal monolayers form a single bilayer. In general, the energetic cost of the splay of the lipid chains in the stalk, prohibits its spontaneous expansion. However, the presence of additional, external forces (e.g. pressure, surface tension gradients, electrostatic effects) can lead to expansion of the stalk into a 'hemifusion region' and to the growth of this zone. Clear evidence for the existence of these two distinct pre-fusion stages, stalk formation and hemifusion, was found for PEG mediated fusion of vesicles (Lee and Lentz, 1997).
A recent theoretical paper (Safran et al., 2001) suggested that the flow of lipids from region B to region A can be caused by an increase of the surface tension in region A due to the presence (in that region only) of additional polymer in solution. The tension gradient between these regions induces a flow of lipids, that leads to the growth of region C.
A different scenario, where hemifusion can be an alternative pathway to fusion was found in influenza hemagglutinin-mediated fusion (Chernomordik et al., 1998;Leikina and Chernomordik, 2000). The initial local stalk may evolve to a fusion pore (Müller et al., 2002), or it may expand to hemifusion. In the latter case, no fusion occurs.
In this paper, we predict the dynamics of the expansion of the initial stalk and its role in the growth of a mesoscopic hemifusion diaphragm. The nucleation of a stalk by thermal fluctuations was recently shown to be thermally accessible (Kozlovsky and Kozlov, 2002;Markin and Albanes, 2002). A detailed description of the kinetics of this nucleation event (that typically describes the formation of a stalk of several nanometers in extent) is outside the scope of our work. Instead, we focus on estimates of the conditions that facilitate stalk expansion into hemifusion. We discuss the implications of our theory on biological fusion mechanisms and on in-vitro experiments. In addition, we predict the growth of the hemifusion region (e.g. from nanometers to microns) as a function of time and discuss the physical parameters that can be used to control the time scale for hemifusion. This dynamic part is relevant mainly to in-vitro experiments, since biological fusion events generally remain at the microscopic scale of the stalk.

If hemifusion is an intermediate state of fusion then
it is important to contrast the time scales of hemifusion diaphragm expansion and pore formation, in order to determine the rate limiting step. Chizmadzhev et al. (2000) predicted that pore expansion is exponential in time, with a time scale of η m /δp < 1 sec, where η m is the membrane viscosity and δp is the surface tension difference (both are estimated below). However, if pore nucleation is slow enough significant expansion of the hemifusion diaphragm can occur before pore formation. This is the case considered here, where we predict that the hemifusion diaphragm expands as the square root of time.
Our theoretical model is motivated by and consistent with the experiments described by Kuhl et al. (1996), where two bilayers supported on mica surfaces were brought into contact in the presence of a PEG-water solution. Hemifusion, that eventually extended over a distance of 50µ was observed in a time of about 10 minutes, while the time it took the initial stalk to form was less then 3 minutes. This suggest that, at least in this experiment, the rate limiting step for hemifusion is the expansion of the fusion zone, as opposed to stalk formation.
This paper presents a simple theoretical model relevant to this experimental system (Kuhl et al., 1996), and predicts the time dependence of hemifusion expansion. The overall time scale we find is comparable with the measurements of Kuhl et al. (1996) while the details of the predicted temporal dependence have yet to be tested experimentally.
1 Illustration of the experimental geometry, adopted from Kuhl et al. (1996). Regions A, B and C are defined in the text. R and RB are the inner and outer radii of region B, respectively.

II. PHYSICAL MODEL
Our theoretical model is illustrated in Fig. 1 that is a simplification of the experimental system of Kuhl et al. (1996) wherein two bilayers deposited on mica cylinders are brought together in a solution of PEG and water. The lipids of the distal monolayers are physisorbed on the mica; this fixes their lateral density. From here on in this paper, the term 'lipid density' relates to the lateral density of the proximal monolayers (see fig. 1).
We assume that the lipids are in local equilibrium, so at a particular location r, the free energy per lipid (in the proximal monolayers) µ( r), does not depend on the lipid microstate, but only on the lipid density σ( r). This assumption of local equilibrium is consistent with our results that predict an overall time scale for hemifusion expansion that is much larger than the local diffusion time of a single lipid molecule.
The experimental system we consider is macroscopically cylindrically symmetric and we therefore assume cylindrical symmetry of all the physical quantities at mesoscopic length scales. This is justified because all flows (of water and lipids) are laminar, and there are no mechanisms that might induce angular fluctuations or instabilities.
We distinguish between three regions, illustrated in Fig. 1: • Region A -where the distance d between the bilayers is typically much larger than the polymer correlation length ξ (Safran et al., 2001). In this region, the outer lipid monolayer is in contact with the PEG in the solution. The free energy per molecule in this region is given byμ(σ( r)), and is different (in its functional form) from the free energy µ(σ) of the monolayer in the absence of PEG.
• Region B -where d ≪ ξ. For these values of d, the PEG density near the bilayers is negligible and our model assumes that there is no PEG in contact with the bilayers in this region. The free energy per lipid in this zone is µ(σ( r)). In addition we assume that the distance between the mica surfaces is constant (the mica surfaces in the experiment are deformed and flattened under pressure), and that this region is ring-shaped with an outer radius R B , and an inner radius R.
• Region C -the region where the distal bilayers are in contact.
The bilayers are Langmuir-Blodgett deposited in water, without PEG, which is added later. The energy per lipid when the monolayers are in contact with water is µ(σ) and the proximal monolayers are initially Langmuir-Blodgett deposited with a density σ 0 that minimizes µ. When PEG is added, it induces an effective attraction between the polar heads (Safran et al., 2001), and changes the functional form of the energy as function of the lipid density toμ(σ).
The effect of lipid condensation in the presence of PEG (Bartucci et al., 1996;Maggio and Lucy, 1978;Tilcok and Fisher, 1979) hasa been discussed in terms of the the dehydration of the bilayer by the PEG (Mishima et al., 1997). This dehydration affects the lipids in region A that are in microscopic proximity to the PEG, but has no effect on the lipids in region B. In Sec. VI we demonstrate that the osmotic pressure induced by the PEG is too small to induce hemifusion. This stands in contrast to the surface tension effects that are the main focus of our work.
If equilibrium could be reached, the lipid density in region A would tend to increase in the presence of PEG. However, the number of lipids in the monolayers cannot increase to any significant degree within the time scale of the experiments, since the concentration of lipids in the bulk solution is negligible and the number of lipids that can be transported from region B to region A is much smaller than that of region A. Thus, the lipid density is unchanged and the energy per lipid in region A is nowμ(σ 0 ) > µ(σ 0 ), with the derivativeμ ′ (σ 0 ) < 0 due to the induced head attraction. This condensation effect thus leads to a negative tension in the proximal monolayers that ideally would cause them to contract in extent. They cannot do this without exposing the chains of the inner monolayers to the water and this is energetically prohibitive. The outer monolayers are therefore stressed and one way of relieving that stress is for additional lipid to enter this region; this will allow the local lipid density to increase, while still covering the original area occupied by the outer monolayer.
The PEG concentration near the outer monolayers in region B is given by c B = c A (d/ξ) 2 , where c A is the PEG concentration near the outer monolayers in region A, ξ is polymer correlation length, and d is the distance between the bilayers in region B (Safran et al., 2001). Since by the definition of region B, the bilayer spacing in that region is small, d ≪ ξ, we have c B ≪ c A and the PEG concentration in region B is negligible; we thus take this concentration to be zero. The energy per lipid in region B is initially given by µ(σ 0 ), where σ 0 is the lipid density in the absence of polymer. Since the free energy per lipid, µ, is minimized when the density σ = σ 0 and the tension in region B initially vanishes, since either expansion or compression of the lipids will increase their energy. The tension gradient between regions A (initially at negative tension) and B (initially at zero tension) induces a flow of lipids from region B to region A. Since region A is much larger than region B, we can treat it as a reservoir, and assume that even though lipid is flowing from region B to region A, the lipid density in region A is not changed from its initial value of σ 0 . The system is a dynamical one and the chemical potential (equivalent in our single component system to the free energy per lipid, µ) is not constant in all of space at the mesoscopic scale; this results in lipid flow and dynamics. However, since local equilibrium is maintained, we must have equal chemical potentials at any given point in the system. In particular, at the boundary between regions A and B, the chemical potentials of the lipids must be equal: is the lipid density at the edge of region B. We note that this equality of chemical potentials determines the lipid density at the boundary of region B, σ b ; the functional form of the two free energies µ(σ) and µ(σ) are not the same, since in region A, the lipids are in contact with polymer.
The initial lipid density in region B (σ 0 , which is the density at which the lipids self-assemble in water in the absence of polymer) is higher than the lipid density at the AB boundary: σ 0 > σ b . This inequality is a consequence of the fact that the tension at the boundary is negative, as shown in section IV. More intuitively, the negative tension in region A tends to pull in additional lipids from the boundary region of region B into region A as explained above. This lipid flow reduces the lipid density at the boundary r = R B from σ 0 to σ b . In turn, the reduced lipid density at the boundary of regions A and B, (σ b < σ 0 ) induces a flow of lipids from the rest of region B towards the boundary. This is because the minimum energy state in region B is one where σ = σ 0 > σ b ; thus lipids from the entirety of region B flow to the boundary in an attempt to restore the lipid density there to values closer to σ 0 . This flow, in turn, reduces the lipid density at the boundary between regions B and C (the hemifusion region) at r = R, and lead to a negative tension that tends to expand region C.
At the boundary of regions B and C, the lipid density is determined by a force balance between the membrane negative tension (arising from the lipids flowing to the AB boundary), that tends to expand region C, and the force exerted by the boundary ring around region C that tends to shrink it. The main contribution to the energy of this ring is of the tilt of the lipid tails imposed by the toroidal geometry. This tilt is needed in order to form the three-way junction of the boundary ring cross section while avoiding an intra-membrane void, which has a much higher energetic cost (Kozlovsky and Kozlov, 2002). The energetic cost of the tilt can be considered through the related intra-membrane strain and the adjacent stress tensor (Hamm and Kozlov, 2000).
We assume that for R ≫ d the energetic cost f t for a cross section of the BC boundary ring is independent of R. Thus, the ring energy is given by E r (R) = 2πRf t . The force per unit length that the ring exerts on region B of the membrane tends to shrink region C and pull region B in the −r direction. This force (per unit length) is and tends to shrink the boundary ring; that is, the expansion of region C is energetically costly. In local equilibrium, this force is balanced by the surface tension p, which may be considered as a two dimensional lateral lipid pressure , in region B of the monolayer that tends to expand the ring: Negative tension in region B tends to cause this region to contract and thus provides a force in ther direction, balancing the force due to the BC boundary.

III. MONOLAYER DYNAMICS
In this section we derive the dynamics that govern the expansion of the hemifusion region and predict the flow of lipids within the monolayer as a function of the lipid density and of time.
There are three local, dissipative forces that oppose any lipid motion.
• The stress, or force per unit area due to the viscosity of the water that is moved along with the lipids is given by η∂v w /∂z, where v w is the water velocity and η = 0.01 erg s/cm 3 is the viscosity of water.
2 Enlargement of region B (between the dashed lines) shown in Fig. 1. The lipid density in region A is the initial density σ0. In our model we assume a step in the density profile, so the lipid density at RB is σ b < σ0. At R it is approximated by σ0. The normalized lipid density profile ρ = (σ − σ0)/σ0 as a function of the radius r for R/RB = 0.2 is plotted using Eq. 9. The free energy per lipid in region B is µ(σ) which is a function of the local lipid density. In region A, the free energy per lipid µ(σ0) everywhere.
The stress is of order ηv/d, where v is the lipid velocity and d is the spacing between the bilayers in region B.
• The stress, or force per unit area due to the monolayer viscosity is given by η m ∇ 2 v, where η m is the monolayer friction coefficient (Seifert and Langer, 1993). For a laminar flow we estimate this stress as η m v/R B 2 ; that is, the relevant dimension is the size of region B in which there is monolayer flow.
• The stress, or force per unit area that is due to the friction between the monolayers is given by bv, where b is the friction coefficient. This stress depends only on the motion of the outer relative to the inner monolayer where there is no flow; there is therefore no dependence of the length scale related to the geometry of the different regions.
The friction between a DMPC monolayer and a supporting HTS 1 monolayer at T = 45 • C is b = 7 · 10 6 erg s/cm 4 , while for a supported OTS 2 monolayer the friction is b = 2.9 · 10 8 erg s/cm 4 (Merkel et al., 1989). The experiments of Kuhl et al. (1996) were carried out at 25 • C. It has been observed that the diffusion coefficient of a molecule in a DMPC monolayer increases about three folds when T is increased from 25 • C to 45 • C (Haibel et al., 1998;Merkel et al., 1989;Vaz et al., 1985), which suggest a corresponding decrease in b. In this work we use an estimated value of b = 10 8 erg s/cm 4 . For DMPC bilayers at T = 25 • C the bilayer viscosity is η m ∼ 3 · 10 −7 erg s/cm 2 (Merkel et al., 1989). The values relevant to the experiments of Kuhl et al. (1996) are 1 trichlorosilanes with hexadecyl chains 2 trichlorosilanes with octadecyl chains d = 2 · 10 −7 cm and R B = 5 · 10 −3 cm. With the estimates for the stress given above, we find that the frictional force due to relative motion of the two monolayers is much larger than either the lipid or water viscosity contributions to the stress. We thus neglect these latter two effects and predict the dynamics for a system where the only relevant dissipation is due to the relative friction between the monolayers.
The lipid flow is induced by the tension gradient ∇p, and is opposed by the frictional bv. The force balance equation is In Appendix A we derive the lipid local dynamics using Eq. 3 and the continuity equation. We consider the dynamics only to first order in the lipid density variations ρ = (σ − σ 0 )/σ 0 , which is known from experiments to be small. In Kuhl et al. (1996) a variation of |ρ| ≈ 0.05 was measured.
To first order in ρ the local dynamics has the form of a diffusion equation where α = σ 0 3 µ ′′ (σ 0 ) is the harmonic 'spring constant' of the monolayer. For a small density variation |ρ| ≪ 1 the surface energy cost is δg = 1 2 αρ 2 , and the related tension difference is δp = αρ. We note that the surface energy g = σµ(σ) is the Gibbs free energy per unit area, and is different then the surface tension p, which has the thermodynamic role of the two dimensional pressure.
We estimate α using the phenomenological form where γ is the effective surface tension of the hydrocarbon-water interface (Ben-Shaul, 1995). The second term in Eq. 5 accounts for the (electrostatic) effective head-group repulsion, while the first term represents the effective hydrocarbon-water repulsion. We note that this effective repulsion is smaller than the repulsion of the bare hydrocarbon-water interface, and has been estimated as γ ∼ 20 erg/cm 2 (Israelachvili, 1991). From Eq. 5 we obtain α = 2γ ∼ 50 erg/cm 2 . For b = 10 8 erg s/cm 4 the effective 'diffusion constant' is 5 · 10 −7 cm 2 /sec. This quantity is larger than the actual, microscopic diffusion constant measured for free liquid bilayers above the gel transition, that are of the order of 10 −8 − 10 −7 cm 2 /sec (Haibel et al., 1998;Sonnleitner et al., 1999;Vaz et al., 1985). The Einstein relation is not applicable in our case, since the flow (that happens to scale like diffusion) of the lipids from the high to low density regions is not due to the random motion of the molecules, but due to the tension gradient α∇ρ. Indeed, for a characteristic molecular area a = 10 −14 cm 2 we find that the related energy per molecule is αa ∼ 10k B T .

IV. BOUNDARY CONDITIONS AND GLOBAL DYNAMICS
The boundary conditions for the lipid density were already discussed in section II and we review them here for convenience. The local tension equilibrium at the boundary with region A determines the local lipid density σ b at R B . In Appendix A we show that the tension in the monolayer is given by Since the tension in region A is negative, from the tension equality at the boundary we see that p(R B ) = σ b 2 µ ′ (σ b ) is negative. Moreover, because the function µ(σ) has a minimum at σ 0 it is convex in a neighborhood of σ 0 . If σ b is in that neighborhood, then the condition µ ′ (σ b ) < 0 yields that σ b < σ 0 .
The boundary ring near the hemifusion region at R exerts a force that opposes hemifusion expansion; this is because the boundary energy of the hemifusion region is increased as this region grows. This force is locally balanced by the negative tension in region B where lipids are flowing towards region A. As lipids pass from region B to A the lipid density in region B decreases; the tension in region B, and in particular near its boundary with region C, becomes more negative and pulls on region C causing its expansion.
The density of lipids in region B at the boundary R is determined from the force balance Eq. 2. Using Eq. 6 we may write Eq. 2 as Before the flow begins, the initial lipid density in region B is σ 0 , which implies that ρ = 0. For this value of the lipid density there is zero tension in region B, the stalk does not expand and hemifusion does not develop. Due to the tension gradient between region B and A, lipids flow out of region B and a negative tension is built up. If at a certain time the lipid density at r = R is low enough so that Eq. 7 is satisfied, the stalk begins to expand.
After the flow of lipids is initiated, lipids are removed from region B as they flow towards region A and the lipid density in region B is lower than σ 0 . The lipid density in region B cannot, however, be smaller than the value of σ b , because when σ = σ b the free energies per lipid in regions A and B are equal, and the flow stops. Thus, we require σ 0 ≥ σ ≥ σ b in all of region B if there is to be flow and stalk expansion that leads to hemifusion. At an early time after the stalk formation, while the stalk does not expand, the lipid density in all of region B approaches the equilibrium density profile σ(r) = σ b . Using Eq. 7, the condition for the stalk to begin to expand with a finite amount of time is: where ρ b = (σ b − σ 0 )/σ 0 and R 0 is the radius of the stalk. In our model, we consider the process for R much larger than the molecular size R 0 that characterizes the size of the stalk. The tilt energy f t is in general positive. From Eqs. 6 and 8, for R ≫ R 0 we have |ρ(R)| ≪ |ρ b |. Since we consider all quantities only to first order in ρ b we use the approximation ρ(R) = 0. In Appendix B we use the integral continuity equation, that expresses the conservation the lipid number in the system, in order to obtain a dynamic equation for the hemifusion radius R. In Appendix C we show that the time scale that governs the local dynamics is much faster then the rate of change of R. We use an adiabatic approximation in order to solve the dynamics. First, we fix R and find the asymptotic (t → ∞) lipid density profile We use this density profile to obtain the dependence of the hemifusion radius R on the time t to find: This predicts an approximately square root dependence of the hemifusion region size on time (with logarithmic corrections). The same temporal dependence was obtained by Kumenko et al. (1999) under the assumption of constant lateral lipid density. However, their result is quantitatively different from ours since they have considered the monolayer viscosity as the main dissipative force, while we have showed that it is negligible compared to the friction b. From Eq. 10 we find that the time it takes the hemifusion region to evolve from the initial stalk radius R = R 0 ≪ R B to a final radius of R = R B is ∆t = −bR B 2 /2αρ b . With ρ b = −0.05 and α/b = 5 · 10 −7 cm 2 /sec, we predict that the time for expansion of the hemifusion zone to a scale of R B = 50 µm is ∆t ≈ 500 sec. This is consistent with the experiment of Kuhl et al. (1996) where a time of ∆t = 600 sec was measured.
The time ∆t found here can also be derived (up to a numerical factor) from a simple scaling argument, that does not depend on the specific details of our model. As hemifusion is initiated, the tension difference between the bulk (at R B ) and the hemifusion front (at R) is −αρ b . When R B ≫ R the average tension gradient is ∇p ≈ −αρ b /R B . For a fully damped flow with a friction coefficient b the average lipid velocity isv = ∇p/b. The hemifusion front (BC boundary) advances with the velocity ∼v of the lipids near it. The time to advance a distance of R B with a velocityv is

V. INITIATION OF HEMIFUSION
The change in the monolayer surface energy due to the presence of PEG in region A is δg = σ 0 (μ(σ 0 ) − µ(σ 0 )), where µ(σ 0 ) is the free energy per lipid in the absence of PEG, andμ(σ 0 ) is the free energy per lipid in the presence of PEG. Since we have defined σ b by the condition µ(σ b ) =μ(σ 0 ), we can expand µ around its minimal value σ = σ 0 , and find that to lowest order in ρ b the surface energy difference δg and the tension difference δp induced by the PEG are In Kuhl et al. (1996) a change of ρ b ≈ −0.05 in lipid density was deduced from the measured thinning of the bilayer. Using the value α = 50 erg/cm 2 we estimate δg ≈ 0.06 erg/cm 2 ; δp ≈ 2.5 erg/cm 2 . Initiation of stalk expansion is relevant not only to events of mesoscopic fusion, but also to in-vivo fusion events, where a fusion pore is formed soon after stalk expansion. In many cases of biological interest, the fusion process is regulated by fusion proteins that promote stalk formation and expansion. One hypothesized bio-molecular mechanism that promote expansion is the penetration of hydrophobic fusion protein domains into the membrane and its subsequent destabilization (Bentz and Mittal, 2000). The protein domains may increase the membrane surface energy by inducing an effective attraction of the hydrophobic head groups, similar to the effect of PEG (Safran et al., 2001); they may also penetrate the membrane, increasing the intra-membrane tension. Our theory suggests that the former mechanism, which work to increase in δg, may be more effective energetically than the latter, which increases δp. That is, for a given change in lipid density, ρ b , a smaller energy is involved (Eq. 11). SNARE 3 proteins that promote exocytosis in nerve synapses are thought to induce stalk expansion through a conformational change by which the protein pull on the stalk to widen it (Scales et al., 2001). Another possible cause for stalk expansion is calcium ions induced membrane tension (Arnold, 1995). We conclude from our theory that the latter mechanism may be more effective energetically.
In section IV we found that in order for expansion of the hemifusion region to occur, the driving force due to the negative tension in region B must be large enough to overcome the tendency of the boundary of region C to shrink. We thus deduced that the normalized lipid density at R B must obey From this condition, we estimate the minimum stalk radius R 0 for which the lateral tension in the monolayer can induce expansion. The energy of the lipid tails tilt at the hemifusion front is estimated by Markin and Albanes (2002) as f t = 2 ·10 −6 erg/cm. For the values of α and ρ b given above, we find that the mechanism described here is sufficient to cause hemifusion for R 0 ≥ 8nm, which is of the order of the typical radius of a thermally nucleated stalk (Yang and Huang, 2002). Note that if ρ b vanishes (that is, no polymer is present in region A) hemifusion will not be initiated for any finite (reasonable) stalk radius.

VI. THE ROLE OF PRESSURE
Experiments have demonstrated that hemifusion may be caused by sufficiently large normal pressure (Helm et al., 1989) or by negative pressure in the water layer (MacDonald, 1985;Yang and Huang, 2002). We shall now determine the conditions under which pressure induced in region B can in and of itself (i.e. with no surface tension effects as induced by the added polymer) cause hemifusion expansion by forcing water to flow out of the contact zone. We do this by using the simplifying assumption that the water in region B is under a constant pressure p w = p n + p o , where p n is the normal pressure on the bilayers and p o is the osmotic pressure induced by the solute in the bulk. The finite thickness of the water layer in region B (whose thickness is on the order of a nanometer) is always maintained because of hydration forces: the water molecules are organized around the polar head groups of the lipids in order to partially cancel their electric dipole; removing the water layer would increase the free energy because of the energetic cost of these electric dipoles whose normal components, in general, point to the same direction due to the hydrophobic nature of the lipid layer. Thus the water flow out of region B and into region A is possible only by the expansion of region C.
The energy (per unit area) difference associated with a pressure difference of p w is p w d, where d is the distance between the two proximal monolayers. This should be compared with the energy difference δg associated with the free energy gradient in the monolayer. In the experiment of Kuhl et al. (1996) that yield p w d ≈ 0.08 erg/cm 2 , which is of the same order of δg. Nevertheless, we show below that the external normal pressure has only a minor effect on the pressure in the monolayer and on its density. We will thus show that under the experimental conditions of Kuhl et al. (1996), the external pressure is insufficient to cause hemifusion expansion.
In the experiment of Kuhl et al. (1996) the applied normal pressure is p n = 0.3 atm and the osmotic pressure is p o ∼ 0.1 atm, so the total pressure between the bilayers is p w ≈ 0.4 atm. We now estimate the contribution of this pressure to the lipid density variation in the experiment. For a fluid membrane, the relation between the tension p -the two dimensional pressure in the membrane -to the three dimensional pressure p w , is p w = p/h, where h is the thickness of the monolayer. In order to induce the observed density variation ρ b = 0.05 the tension needed is |p| = 2.5 erg/cm 2 . For h = 5 nm the pressure required to induce such tension is 5 atm -much larger than the actual pressure in the experiment. Thus, the contribution of the normal and the osmotic pressures to the density variation is negligible compared with the surface tension effects due to the PEG-lipid interactions that result in densification of the lipids. This result underscores the point made in Sec. V: changes in the pressure are much less effective than surface energy variation for the initiation of stalk expansion.
We now estimate the pressure p w needed to initiate hemifusion, without a lipid density gradient (that is, with ρ b = 0). The radial force per unit length on the boundary at R due to the external normal pressure is From Eq. 1, the condition for spontaneous fusion is p w d > f t /R 0 . For the values given above, we require p w ≥ 10 7 dyne/cm 2 = 10 atm. Experimental results in different conditions are within that range. The pressure needed for the hemifusion of bilayers directly supported on mica (with no added polymer or other mechanisms that give rise to lipid density gradients) was found by Helm et al. (1989) to be p w ∼ 40 atm. Wong et al. (1999) used a surface forces apparatus to apply pressure on DMPC bilayers supported on polymer layers. The polymer layer allowed the bilayers some lateral conformational freedom, thus permitting more freedom for the adjustment of stalk shape and size (Kozlovsky and Kozlov, 2002;Markin and Albanes, 2002). In that case, where the stalk geometry could easily adjust, the cost for forming the stalk was reduced and hemifusion was observed at a much lower pressure of p w = 2 atm. In the experiment of Kuhl et al. (1996) the pressure p w ≈ 0.4 atm is too low to be the driving force for hemifusion. Pressure in itself is not enough to cause hemifusion, but it is sometimes necessary 4 . Leckband et al. (1993) showed that the amount of pressure needed for hemifusion is directly related to the lipid density near the contact area. In that experiment, two bilayers were brought into contact using a surface forces apparatus. When Ca ++ ions were introduced, there was a phase separation in the bilayers. The density of lipids in the bilayer regions that were brought into contact was characterized by the hydrophobic adhesion energy. When thinner regions were brought together (characterized by adhesion energy of E ad = 3.8 erg/cm 2 ) they either hemifused spontaneously, or required only a small amount of pressure (p n ≤ 1 atm) to induce hemifusion. For denser bilayers (E ad = 0.15 erg/cm 2 ) a pressure of p n = 4 atm was required for hemifusion.
Yang and Huang (2002) induced negative osmotic pressure on the water layer between the bilayers by lowering the relative humidity of the environment of DPhPC 5 . At 80% humidity the lipids were at the lamellar phase. As the relative humidity was decreased the water were expelled from between the bilayers by the osmotic pressure and the lamella were connected by stalks, directly observed by x-ray diffraction. In this experiment the dehydration was due to negative pressure of the water layer induced by the reduced relative humidity, and not by normal pressure, but the physical effect of the two is similar.

VII. SUMMARY
In this paper we used a model based on lipid density gradients induced by surface energy variation that occur far from the hemifusion zone, to predict the the conditions for the initiation of hemifusion by stalk expansion and the dynamics of mesoscopic hemifusion. Our theory was motivated by the experiments of Kuhl et al. (1996). However, the quantitative scheme presented here can be generalized to any system of two lipid bilayers initially connected by a stalk, where a perturbation in region A, mesoscopically far from the stalk, causes tension in the membrane in that region. For example one can apply our results to tension induced by the electrostatic interactions caused by calcium ions (Leckband et al., 1993), tension induced by laser tweezers (Bar-Ziv and Moses, 1994;Moroz et al., 1996), or the effective tension induced by the attraction of oppositely charged bilayers 4 In the experiments it is difficult to distinguish between applied pressure and time in contact effects (T. Kuhl, private communication). 5 diphaytanoyl phosphaditylcholine (Pantazatos and MacDonald, 1999).
We have compared the effect of the friction of the two monolayers, the the water viscosity and the intramonolayer viscosity on the two dimensional lipid motion and showed that the friction dominates. Thus, the lipid dynamics depend on the friction and not on hydrodynamics. This means that the spacing between the two layers is irrelevant for the lipid dynamics.
Experiments similar to those of Kuhl et al. (1996) can test the predictions of the model for the time scales as functions of the lipid density and friction as well as the value of the driving force due to the tension induced in region A. One could vary each of the parameters ρ b (the relative change in lipid density), α (related to the induced tension) and b (the interlayer friction) independently, and measure the 'hemifusion radius' R(t), the final radius R B and the time to complete the process ∆t as functions of these parameters.
In particular, the friction b can be varied independently of α by changing the composition of the distal bilayers while maintaining the same composition of the proximal bilayers. The friction can be varied by changing the interactions between the chains that is responsible for most of the friction, via chain length changes or temperature changes (Yoshizawa et al., 1993).
Once an empirical, temporal profile for the hemifusion expansion, R(t), is measured for systems with known parameters, one can use the same experiment to estimate the effective diffusion constant for the lipid flow, α/b, for different lipid bilayers. One can easily vary the lipid density at the boundary, σ b by changing the polymer (or calcium ions) concentration since the density σ b is determined by the equality of the chemical potentials of the lipids exposed to the polymer and those exposed only to the water.
The static part of our theory deals with the initial conditions required for stalk expansion. We have evaluated the necessary density variation ρ b = −f t /αR 0 and demonstrated that the related surface energy 1 2 αρ b 2 is much smaller then the surface tension αρ b . This result is not surprising, since it is a general result of a first order expansion around an energetic minimum. Still, it does give a new insight regarding biological fusion mechanisms. It suggest mechanisms working through the change of the surface energy δg are much more effective than mechanisms that exert force or normal pressure on the stalk.
The predicted dependence of stalk expansion on the lipid density can be tested by measuring the critical density ρ b at which stalk expansion occurs. The results may serve to learn more about the stalk structure and energetics.
We expect that near the end of the process of hemifusion expansion, when R(t) ≈ R B experimental results may differ from our predictions, since the density profile of the polymer (or calcium ions in the case of Leckband et al. (1993)) may vary in a gradual manner around R B ; in our theory we assumed a sharp ('step function') decrease of the polymer density at R B . We also expect a deviation from our theory when the radius R(t) of the hemifusion region is close to its initial, molecular stalk radius R 0 , due to microscopic details of the lipid structure in the stalk.
We distinguish between hemifusion induced by surface tension gradients, which we consider in our model, and hemifusion induced by pressure. Hemifusion may be induced by normal pressure on the bilayers (Helm et al., 1989;Wong et al., 1999) or by dehydration which induces negative pressure in the water layer between them (Yang and Huang, 2002). We showed that this pathway to hemifusion requires much more energy (per unit area) than fusion that is induced by surface tension gradients.
We have shown that the induced pressure p w in the experiment of Kuhl et al. (1996) cannot be the primary direct cause of hemifusion. Still, pressure does play an important role in stalk formation. It may also effect stalk expansion through its effect on the lipid tilt energy f t and on the initial stalk radius R 0 .

APPENDIX A: Local lipid dynamics
We bring here the full calculation of the local lipid dynamics. Note that though in our final result we leave only the terms linear in ρ, one may also calculate in the same framework the non linear terms in the case ρ is not small.

APPENDIX B: Global lipid dynamics
In Sec. IV we consider the boundary conditions for the lipid density. In order to fully predict the dynamics of hemifusion expansion, we also need to determine the flow at the boundaries. For this we use the integral form of the continuity equation: The left hand side of Eq. B1 describes the rate of change of the lipid number in region B while the right hand side gives the flow of lipids through the boundary R B . We assume cylindrical symmetry, so v = v rr . From Eqs. A1 and A3 we obtain v r (r) = − 1 b (2µ ′ (σ) + σµ ′′ (σ)) σ ∂σ ∂r .
We now use Eq. A5 to calculate the left hand side of Eq. B1: If we take only terms linear in ρ, Eq. B1 gives:

APPENDIX C: Adiabatic solution
Equations B4 and A5 along with the boundary conditions completely determine the time evolution of the monolayers to first order in ρ. From these equations we can calculate R(t), and predict the temporal profile of hemifusion expansion. We write these two equations using dimensionless variables and scale the spatial variables so that they are of order of unity, in order to get an estimate of the time scales. The natural spatial scale is the final size of the hemifusion region, R B . We thus define: x = r/R B ,R = R/R B ,ρ = ρ/ρ b as well as two time variables: a "fast" time τ = αt/bR B 2 at which the local lipid flow occurs, and a "slow" (since ρ b is small) timē τ = |ρ b |τ which is the scale over which the hemifusion region expands. Eqs. B4 and A5 become Since all the variables that appear on the right hand side of Eqs. C1 and C2 are of order unity, the units of τ andτ suggest the time scales of the processes described by the equations. For |ρ b | ≪ 1 we have τ ≫τ , which implies that we can use an adiabatic approximation: the local lipid flow occurs quickly so that the lipid density is instantaneously given by the asymptotic equilibrium solution of Eq. C1 for τ → ∞. We then use this solution to determine the slower time evolution of the hemifusion radius R from Eq. C2.
At asymptotically long times, both sides of Eq. C1 vanish. The adiabatic density profile reached is Plugging this solution into Eq. C2 we obtain .