Kinetics of self-assembly of inclusions due to lipid membrane thickness interactions

Self-assembly of proteins on lipid membranes underlies many important processes in cell biology, such as, exo- and endo-cytosis, assembly of viruses, etc. An attractive force that can cause self-assembly is mediated by membrane thickness interactions between proteins. The free energy profile associated with this attractive force is a result of the overlap of thickness deformation fields around the proteins. The thickness deformation field around proteins of various shapes can be calculated from the solution of a boundary value problem and is relatively well understood. Yet, the time scales over which self-assembly occurs has not been explored. In this paper we compute this time scale as a function of the initial distance between two inclusions by viewing their coalescence as a first passage time problem. The first passage time is computed using both Langevin dynamics and a partial differential equation, and both methods are found to be in excellent agreement. Inclusions of three different shapes are studied and it is found that for two inclusions separated by about hundred nanometers the time to coalescence is hundreds of milliseconds irrespective of shape. Our Langevin dynamics simulation of self-assembly required an efficient computation of the interaction energy of inclusions which was accomplished using a finite difference technique. The interaction energy profiles obtained using this numerical technique were in excellent agreement with those from a previously proposed semi-analytical method based on Fourier-Bessel series. The computational strategies described in this paper could potentially lead to efficient methods to explore the kinetics of self-assembly of proteins on lipid membranes. Author summarySelf-assembly of proteins on lipid membranes occurs during exo- and endo-cytosis and also when viruses exit an infected cell. The forces mediating self-assembly of inclusions on membranes have therefore been of long standing interest. However, the kinetics of self-assembly has received much less attention. As a first step in discerning the kinetics, we examine the time to coalescence of two inclusions on a membrane as a function of the distance separating them. We use both Langevin dynamics simulations and a partial differential equation to compute this time scale. We predict that the time to coalescence is on the scale of hundreds of milliseconds for two inclusions separated by about hundred nanometers. The deformation moduli of the lipid membrane and the membrane tension can affect this time scale.