  We model membrane proteins as anisotropic objects characterized by
symmetric-traceless tensors and determine the coupling between these
order-parameters and membrane curvature. We consider the interactions between
transmembrane proteins that respect up-down (reflection) symmetry of bilayer
membranes and that have circular or non-circular cross-sectional areas in the
tangent-plane of membranes. Using a field theoretic approach, we find
non-entropic $1/R^{4}$ interactions between reflection-symmetry-breaking
transmembrane proteins with circular cross-sectional area and entropic
$1/R^{4}$ interactions between transmembrane proteins with circular
cross-section that do not break up-down symmetry in agreement with previous
calculations. We also find anisotropic $1/R^{4}$ interactions between
reflection-symmetry-conserving transmembrane proteins with non-circular
cross-section, anisotropic $1/R^{2}$ interactions between
reflection-symmetry-breaking transmembrane proteins with non-circular
cross-section, and non-entropic $1/R^{4}$ many-particle interactions among
non-transmembrane proteins. For large $R$, these interactions might provide the
dominant force inducing aggregation of the membrane proteins.
