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We calculate the characteristic energies of fusion between planar bilayers asafunction of the distance between them, measured from thehydrophobic/hydrophilic interface of one of the two nearest, cis, leaves to theother. The two leaves of each bilayer are of equal composition; 0.6 volumefraction of a lamellar-forming amphiphile, such as dioleoylphosphatidylcholine,and 0.4 volume fraction of a hexagonal-forming amphiphile, such asdioleoylphosphatidylethanolamine. Self-consistent field theory is employed tosolve the model. We find that the largest barrier to fusion is that to createthe metastable stalk. This barrier is the smallest, about 14.6 $k_BT$, when thebilayers are at a distance about 20 percent greater than the thickness of asingle leaf, a distance which would correspond to between two and threenanometers for typical bilayers. The very size of the protein machinery whichbrings the membranes together can prevent them from reaching this optimumseparation. For even modestly larger separations, we find a linear rate ofincrease of the free energy with distance between bilayers for the metastablestalk itself and for the barrier to the creation of this stalk. We estimatethese rates for biological membranes to be about 7.1 $k_BT$/nm and 16.7$k_BT$/nm respectively. The major contribution to this rate comes from theincreased packing energy associated with the hydrophobic tails. From this weestimate, for the case of hemagglutinin, a free energy of 38 k_BT for themetastable stalk itself, and a barrier to create it of 73 k_BT. Such a largebarrier would require that more than a single hemagglutinin molecule beinvolved in the fusion process, as is observed.

Dependence of the energies of fusion on the intermembrane separation: optimal and constrained