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成本，通常使用快速多极子方法(FMM)和粒子网格Ewald(PME)等快速算法来计算库仑相互作用。另一方面，在生物膜模拟中，通常需要在z轴方向上重复膜层以形成一个三维周期，然后可以直接用PME和周期FMM来处理。然而，膜的人工重复可能会在模拟上造成不希望看到的伪影。在这项工作中，我们最初的目标是开发一个周期性的FMM，它将绕过这些人工产物。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公式。我们发现，多极向局域展开的转换是FMM的关键操作，它通过两种方式影响维里：通过转换矩阵的晶格依赖和通过多极的晶格依赖。后一种贡献以前没有报道过。因此，FMM计算是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有严重的影响，因此制定一种有效的策略来消除这种影响是同样重要的，尽管这一发展有待于进一步的工作。另一方面，周期性的FMM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，开发并改进了一个基于FMM算法和库仑力计算专用硬件的集群计算系统。目前，对准确的维里计算进行了进一步的研究，这是产生准确的FMM计算以及减少周期性可能造成的伪影所必需的。

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.