Multibody interactions between protein inclusions in the pointlike curvature model for tense and tensionless membranes

The pointlike curvature constraint (PCC) model and the disk detachment angle (DDA) model for the deformation-mediated interaction of conical integral protein inclusions in biomembranes are compared in the small deformation regime. Given the radius of membrane proteins, which is comparable to the membrane thickness, it is not obvious which of the two models should be considered the most adequate. For two proteins in a tensionless membranes, the PCC and DDA models coincide at the leading-order $\sim r^{-4}$ in their separation but differ at the next order. Yet, for distances larger than twice the proteins diameter, the difference is less than $10\%$ . Like the DDA model, the PCC model includes all multibody interactions in a non-approximate way. The asymptotic $\sim r^{-4}$ many-body energy of triangular and square protein clusters is exactly the same in both models. Pentagonal clusters, however, behave differently; they have a vanishing energy in the PCC model, while they have a non-vanishing weaker $\sim r^{-6}$ asymptotic power law in the DDA model. We quantify the importance of multibody interactions in small polygonal clusters of three, four and five inclusions with identical or opposite curvatures in tensionless or tense membranes. We find that the pairwise approximation is almost always very poor. At short separation, the three-body interaction is not sufficient to account for the full many-body interaction. This is confirmed by equilibrium Monte Carlo simulations of up to ten inclusions.