Fast algorithms for accurate molecular dynamics simulations of membrane proteins

为了压缩生物分子动力学模拟的成本，库仑相互作用通常使用快速算法计算，如快速多极法（FMM）和粒子网格埃瓦尔德法（PME）。另一方面，在生物膜模拟中，通常在z轴方向上重复膜，从而构成三维周期性，然后可以直接通过PME和周期性FMM进行处理。然而，膜的人工重复可能会在模拟中引起不希望的伪影。在这项工作中，我们最初的目标是开发一个周期性的FMM，它将绕过这些工件。我们首先在FMM理论的框架下推导出了恒压模拟所必需的维里数公式。我们发现多极向局部展开的转换是FMM的关键操作，它通过转换矩阵的格依赖性和多极的格依赖性两种方式影响病毒。后者的贡献以前没有报道过。然后FMM计算是O（NlogN）而不是O(N)。数值试验的结果还表明，单位晶胞的净偶极子对库仑相互作用的精度产生严重影响，因此制定消除这种影响的有效策略同样重要，尽管发展留给未来的工作。另一方面，周期性fmm通常需要大量计算邻近细胞之间的相互作用。本文开发并改进了基于FMM算法与专用硬件相结合的库仑力计算集群系统。目前，对精确的维数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.