  We calculate the induced elastic-interaction between pointwise membrane inclusions that locally interact up to quadratic order with the membrane curvature tensor. For isotropic inclusions, we recover the usual interaction proportional to the inverse fourth power of the separation, however with a prefactor showing a non-trivial dependence on the rigidity $\Gamma$ of the quadratic potential. In the large $\Gamma$ limit, corresponding to ``hard'' inclusions, we recover the standard prefactor first obtained by Goulian et al. [Europhys. Lett. \textbf{22}, 145 (1993)]. In the small $\Gamma$ limit, corresponding to "soft" inclusions, we recover the recent result of Marchenko and Misbah [Eur. Phys. J. E \textbf{8}, 477 (2002)]. This shows that the latter result bears no fundamental discrepancy with previous works, but simply corresponds to the limit of soft inclusions. We discuss how the same inclusion can be depicted as hard or soft according to the degree of coarse-graining of the pointwise description. Finally, we argue that conical transmembrane proteins should be fundamentally considered as hard inclusions. 
Elastic interaction between "hard'' or "soft" pointwise inclusions on   biological membranes