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.

脂质膜上的内含物相互作用背后的物理学与介导细胞内外胞吞作用的蛋白质的自组装有关。这些相互作用可以通过使用高斯积分来计算的自由能变化来量化，因为带有夹杂物的脂膜的势能可以用包括面内张力耦合的刚度矩阵表示为二次形式。和平面外偏转。自由能计算需要评估在能量表达式中出现的刚度矩阵的行列式。但是，要计算一个非常大的刚度矩阵的行列式，其大小取决于膜及其分子成分的特征长度尺度，这给计算带来了挑战。这项研究提出了一种围动力学（PD）方法，以构造刚度矩阵并准确评估其行列式，而没有任何计算挑战。它特别采用PD微分算子来精确评估刚度矩阵的行列式，这是计算分区函数的关键步骤。同时使用细网格和粗网格以及PD插值消除了减小网格间距的计算难题。通过再现波动的弹性杆的蠕虫状链力-延伸关系可以说明这一点。在没有任何夹杂物的膜的情况下，当网格间距减小时，PD预测会收敛并接近针对膜的拉伸面积关系的解析解。如果有夹杂物，PD预测捕获膜的预期变形以及相对自由能的行为，以改变夹杂物的间距，形状和位置。相对自由能的大小随夹杂物数量的增加而增加，因此表明较大簇中的相互作用程度更高。这种方法是通用的，并且能够探索不同膜几何形状与曲率，边界条件以及膜与夹杂物之间的接触角对膜-夹杂物相互作用的影响。