Fast algorithms for accurate molecular dynamics simulations of membrane proteins

为了压缩生物分子的分子动力学模拟的成本，通常使用快速算法来计算库仑相互作用，例如快速多极方法（FMM）和粒子网格埃瓦尔德（PME）。另一方面，在生物膜模拟中，通常在z轴方向上重复膜，以构成三维周期性，然后可以直接由PME和周期FMM处理。然而，膜的人工重复可能导致模拟上的不期望的伪影。在这项工作中，我们最初的目标是开发一个周期性的FMM，将规避这些文物。我们首先在FMM理论的框架下推导了定压模拟所必需的维里公式。我们发现，多极子的局部扩展，这是FMM的关键操作的转换，影响维里在两个方面：通过晶格依赖的转换矩阵，并通过多极子。后一项贡献以前没有报告过。因此，FMM的计算时间复杂度为O（NlogN），而不是O（N）。数值试验的结果还表明，单胞的净偶极子对库仑相互作用的精度产生了严重的影响，因此发展一种有效的消除这种影响的方法也是同样重要的，尽管这一发展还有待于以后的工作。另一方面，周期性的FFT通常遭受关于相邻小区之间的交互的大量计算。在这项工作中，一个集群计算系统，这是基于合作的FMM算法和专用硬件的库仑力评估，开发和改进。目前，对精确维里计算的进一步研究正在进行中，这对于产生精确的FMM计算以及减少可能由周期性引起的伪影是必要的。

To compress the costs of the molecular dynamics simulations of biomolecules, Coulomb interactions are usually calculated using fast algorithms, such as the fast multipole method (FMM) and particle mesh Ewald (PME). On the other hand, in biomembrane simulations, it is usual to repeat membranes in the z-axis direction so as to constitute a three-dimensional periodicity, which can then be directly handled by PME and periodic FMM. However, the artificial repeat of the membrane may cause undesirable artifacts on the simulations. In this work, we originally aimed at developing a periodic FMM that will circumvent such artifacts. We first derived the formula for virials in the framework of the FMM theory, which are necessary for constant-pressure simulation. We found that the conversion of the multipole to local expansion, which is the key operation of FMM, affects the virials in two ways : through lattice-dependence of the conversion matrix and through that of the multipoles. The latter contribution has not been reported previously. The FMM computation is then O(NlogN) instead of O(N). The results of numerical tests also suggest that the net dipole of the unit cell produces a serious effect on the precision of the Coulomb interactions and hence it is equally important to develop an effective strategy for removing the effect, although the development is left for future work. On the other hand, periodic FMMs generally suffer from great deal of computation regarding interaction between neighboring cells. In this work, a cluster computing system, which is based on the cooperation of FMM algorithm and dedicated hardware for the Coulomb-force evaluation, was developed and improved. Currently, further investigation on the accurate virial calculation is being undertaken, which is necessary for producing accurate FMM calculation as well as for reducing artifacts that may be caused by the periodicity.