A method to compute elastic and entropic interactions of membrane inclusions

Curvature mediated elastic interactions between inclusions in lipid membranes have been analyzed using both theoretical and computational methods. Entropic corrections to these interactions have also been studied. Here we show that elastic and entropic forces between inclusions in membranes can compete under certain conditions to a yield a maximum in the free energy at a critical separation. If the distance between the inclusions is less than this critical separation then entropic interactions dominate and there is an attractive force between them, while if the distance is more than the critical separation then elastic interactions dominate and there is a repulsive force between them. We assume the inclusions to be rigid and use a previously developed semi-analytic method based on Gaussian integrals to compute the free energy of a membrane with inclusions. We show that the critical separation between inclusions decreases with increasing bending modulus and with increasing tension. We also compute the projected area of a membrane with rigid inclusions under tension and find that the trend of the effective bending modulus as a function of area fraction occupied by inclusions is in agreement with earlier results. Our technique can be extended to account for entropic effects in other methods which rely on quadratic energies to study the interactions of inclusions in membranes.