On the fluctuations of the force exerted by a lipid nanotubule

After deriving the projected stress tensor in cylindrical geometry for a fluid membrane described by the Helfrich Hamiltonian, we calculate the average force f exerted by a thermally fluctuating nanotubule of radius R, and its standard deviation Delta f. We obtain f and Delta f in terms of the internal membrane tension sigma, the bending rigidity kappa, the temperature k(B)T and a molecular cutoff Lambda. We find for f a shift approximately 1/square root(sigma) with respect to the mean field behavior approximately square root(sigma). We obtain (Delta f)(2) approximately R ln(R/b) where b is a molecular length, Delta f being typically small compared to f. Taking into account the difference between the internal tension sigma and the actual mechanical tension tau applied to the membrane from which the tubule is drawn, we discuss the amplitude of the fluctuation-induced corrections to the average force. Our results, obtained in the harmonic approximation, hold for tubules with aspect ratio not larger than approximately = 200, of radius significantly smaller than 100 nm, that are connected to a large membrane reservoir, e.g., a giant vesicle.