  A question of considerable interest to cell membrane biology is whether phase
segregated domains across an asymmetric bilayer are strongly correlated with
each other and whether phase segregation in one leaflet can induce segregation
in the other. We answer both these questions in the affirmative, using an
atomistic molecular dynamics simulation to study the equilibrium statistical
properties of a 3-component {\em asymmetric} lipid bilayer comprising an
unsaturated POPC (palmitoyl-oleoyl-phosphatidyl-choline), a saturated SM
(sphingomyelin) and cholesterol with different composition ratios. Our
simulations are done by fixing the composition of the upper leaflet to be at
the coexistence of the liquid ordered ($l_o$) - liquid disordered ($l_d$)
phases, while the composition of the lower leaflet is varied from the phase
coexistence regime to the mixed $l_d$ phase, across a first-order phase
boundary. In the regime of phase coexistence in each leaflet, we find strong
transbilayer correlations of the $l_o$ domains across the two leaflets,
resulting in {\it bilayer registry}. This transbilayer correlation depends
sensitively upon the chain length of the participating lipids and possibly
other features of lipid chemistry, such as degree of saturation. We find that
the $l_o$ domains in the upper leaflet can {\em induce} phase segregation in
the lower leaflet, when the latter is nominally in the mixed ($l_d$) phase.
