A peridynamic approach to computation of elastic and entropic interactions of inclusions on a lipid membrane

The physics behind the interaction of inclusions on lipid membranes is relevant to the self-assembly of proteins that mediate exo- and endo-cytosis in cells. These interactions can be quantified in terms of free energy changes that can be calculated using Gaussian integrals since the potential energy of a lipid membrane with inclusions can be expressed as a quadratic form in terms of a stiffness matrix that includes the coupling of in-plane tension and out-of-plane deflections. The free energy calculation requires the evaluation of the determinant of a stiffness matrix appearing in the energy expression. However, computing the determinant of an extremely large stiffness matrix whose size is dictated by the characteristic length scale of the membrane and its molecular constituents poses computational challenges. This study presents a peridynamic (PD) approach to construct the stiffness matrix and evaluate its determinant accurately without any computational challenges. It specifically employs the PD differential operator for accurate evaluation of the determinant of the stiffness matrix which is a key step in the calculation of the partition function. The simultaneous use of fine and coarse grids along with PD interpolations eliminates the computational challenges for decreasing grid spacing. This is illustrated by reproducing the worm-like-chain force-extension relation for a fluctuating elastic rod. In the case of a membrane without any inclusions, the PD predictions converge and approach an analytical solution for the tension-area relation of a membrane as the grid spacing decreases. In the presence of inclusions, the PD predictions capture the expected deformation of the membrane as well as the behavior of relative free energy for varying spacing, shape and position of the inclusions. The magnitude of relative free energy increases with increasing number of inclusions, thus indicating higher degree of interaction in larger clusters. This approach is general and enables the exploration of the effect on membrane-inclusion interactions of different membrane geometries with curvature, boundary conditions and the contact angle between the membrane and inclusion.