  We calculate the characteristic energies of fusion between planar bilayers as
afunction of the distance between them, measured from the
hydrophobic/hydrophilic interface of one of the two nearest, cis, leaves to the
other. The two leaves of each bilayer are of equal composition; 0.6 volume
fraction of a lamellar-forming amphiphile, such as dioleoylphosphatidylcholine,
and 0.4 volume fraction of a hexagonal-forming amphiphile, such as
dioleoylphosphatidylethanolamine. Self-consistent field theory is employed to
solve the model. We find that the largest barrier to fusion is that to create
the metastable stalk. This barrier is the smallest, about 14.6 $k_BT$, when the
bilayers are at a distance about 20 percent greater than the thickness of a
single leaf, a distance which would correspond to between two and three
nanometers for typical bilayers. The very size of the protein machinery which
brings the membranes together can prevent them from reaching this optimum
separation. For even modestly larger separations, we find a linear rate of
increase of the free energy with distance between bilayers for the metastable
stalk itself and for the barrier to the creation of this stalk. We estimate
these 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 the
increased packing energy associated with the hydrophobic tails. From this we
estimate, for the case of hemagglutinin, a free energy of 38 k_BT for the
metastable stalk itself, and a barrier to create it of 73 k_BT. Such a large
barrier would require that more than a single hemagglutinin molecule be
involved in the fusion process, as is observed.
