  Integral membrane proteins deform the surrounding bilayer creating
long-ranged forces that influence distant proteins. These forces can be
attractive or repulsive, depending on the proteins' shape, height, contact
angle with the bilayer, as well as the local membrane curvature. Although
interaction energies are not pairwise additive, for sufficiently low protein
density, thermodynamic properties depend only upon pair interactions. Here, we
compute pair interaction potentials and entropic contributions to the
two-dimensional osmotic pressure of a collection of noncircular proteins. In
contrast to direct short-ranged interactions such as van der Waal's,
hydrophobic, or electrostatic interactions, both local membrane Gaussian
curvature and protein ellipticity can induce attractions between two proteins
at distances of up to ten times their typical radii. For flat membranes,
bending rigidities of $\sim 30k_{B}T$, and moderate ellipticities, we find
thermally averaged attractive interactions of order $\sim 2k_{B}T$. These
interactions may play an important role in the intermediate stages of protein
aggregation.
