Universal amplitudes of the Casimir-like interactions between four types of rods in fluid membranes

Abstract

The fluctuation-induced, Casimir-like interaction between two parallel rods of length L adsorbed on a fluid membrane is calculated analytically at short separations d L. The rods are modeled as constraints imposed on the membrane curvature along a straight line. This allows to define four types of rods, according to whether the membrane can twist along the rod and/or curve across it. For stiff constraints, all the interaction potentials between the different types of rods are attractive and proportional to L/d. Two of the four types of rods are then equivalent, which yields six universal Casimir amplitudes. Repulsion can occur between different rods for soft constraints. Numerical results obtained for all ranges of d/L show that the attraction potential reaches kBT for d/L 0.2. At separations smaller than dc ≈ L(L/ p) 1/3 , where p is the rod persistence length, two rods with fixed ends will bend toward each other and finally come into contact because of the Casimir interaction.

Membrane structure formation induced by two types of banana-shaped proteins

“…It was ignored in the present work since it is normally sub-dominant for small inclusions imposing large membrane deformations. This Casimir force is known however to induce an effective attraction between long straight rods [52,53]. We have investigated straight rods in our simulations, and found that the latter have too weak Casimir interactions to induce rod clusters for the present length (N sg = 10).…”

“…We have exhibited the leading order term ∝ 1/R 2 , which describes the interaction quantitatively well for R larger than a few times r rod . Note that this interaction is of longer range than the ∝ 1/R 4 fluctuation-induced Casimir interaction between straight rods and it has a different angular dependence [52,53]. Graphs of this interaction are represented in Fig.…”

Membrane-Mediated Interactions

“…This approximation was used in the case of disk-shaped inclusions in Refs. [37,58], showing that the fluctuation-induced interaction potential then scales as 1/d 1/2 . In Ref.…”

“…In addition, the interaction energy was studied numerically versus d/L, thanks to a discretization scheme [59], showing the transition between the asymptotic behaviors at large d/L [27] and at small d/L [58] were recovered. Finally, the bending and coming into contact of the rods due to the fluctuation-induced interaction was discussed: it was predicted to occur below a certain value of d [58].…”

“…This situation was further studied in Ref. [58]. Rods were modeled as constraints imposed on the membrane curvature along a straight line, allowing to define four types of rods, according to whether the membrane can twist along the rod and/or curve across it (see Fig.…”