  Effects of the mismatch between the hydrophobic length, d, of transmembrane
alpha helices of integral proteins and the hydrophobic thickness, D_h, of the
membranes they span are studied theoretically utilizing a microscopic model of
lipids. In particular, we examine the dependence of the period of a lamellar
phase on the hydrophobic length and volume fraction of a rigid, integral,
peptide. We find that the period decreases when a short peptide, such that
d<D_h, is inserted. More surprising, we find that the period increases when a
long peptide, such that d>D_h, is inserted. The effect is due to the
replacement of extensible lipid tails by rigid peptide. As the peptide length
is increased, the lamellar period continues to increase, but at a slower rate,
and can eventually decrease. The amount of peptide which fails to incorporate
and span the membrane increases with the magnitude of the hydrophobic mismatch
|d-D_h|. We explicate these behaviors which are all in accord with experiment.
Predictions are made for the dependence of the tilt of a single trans-membrane
alpha helix on hydrophobic mismatch and helix density.
