  We set up the statistical mechanics for a nearly flat, thermally equilibrated
fluid membrane, attached to an elastic network through one of its sides. We
predict that the resulting structural (inversion) asymmetry of the membrane,
notably due to the elastic network attached to one of its sides, can generate a
local spontaneous curvature $C_0$, that may in turn destabilize the otherwise
flat membrane. As $C_0$ rises above a threshold at a fixed temperature, a flat
tethered membrane in the thermodynamic limit becomes structurally unstable,
signaling {\em crumpling} of the flat membrane. In-vitro experiments on red
blood cell membranes after depletion of adenosine-tri-phosphate molecules and
artificial deposition of spectrin filaments on lipid bilayers may be used to
verify our results.
