An example of a process that can be studied with these methods is the diffusive motion of proteins on the surface of the red blood cells.  Single particle tracking experiments have shown that proteins are trapped in corrals (formed by the cytoskeleton beneath the membrane) with sizes of about a hundred nanometers for periods of about a hundred microseconds.  Here, our method is suitable for analyzing the role of thermal undulations of the membrane as a mechanism for carrying the protein over the cytoskeleton (see figure).

Our membrane model consists of a curvature free energy plus additional arbitrary potentials, hydrodynamic coupling, and thermal fluctuations through a Langevin random force. All of the equations are most easily handled in Fourier space due to either decoupling of modes for analytical work or for reasons of computational efficiency for simulations. Our Fourier Space Brownian Dynamics (FSBD) algorithm is an efficient method for evolving membrane configurations in simulations that can be used for general systems where membrane undulations are important.

The elastic membrane model is, to some extent, analytically tractable for harmonic potentials. Specifically, we study localized harmonic interactions that pin the membrane to the underlying cytoskeleton and calculate the probability that a thermal undulation is large enough and persists long enough for the passage of the protein out of the corral (see figures). This escape probability is completely characterized by the height autocorrelation function which can be computed without simulation. A diffusion constant can then be estimated for the protein diffusing in the plane of the membrane. The results suggest that thermal fluctuations are a possible mechanism for promoting the passage of proteins over the cytoskeletal barrier.

To extend our model to include interactions between the cytoskeleton and the membrane, we can add a repulsive potential at the edges of the corrals (see figures). This interaction is nonharmonic and therefore requires simulation over small time steps. The results for the diffusion constant are modified somewhat from the previous case, but still suggest that thermal fluctuations may be a mechanism that contributes to the mobility of proteins on the surface of the red blood cell. More detail is presented in the following papers.