Samanta Amit, Chen Ming, Yu Tang-Qing, Tuckerman Mark, E Weinan
Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA.
Department of Chemistry, New York University, New York 10003, USA.
J Chem Phys. 2014 Apr 28;140(16):164109. doi: 10.1063/1.4869980.
Many problems in biology, chemistry, and materials science require knowledge of saddle points on free energy surfaces. These saddle points act as transition states and are the bottlenecks for transitions of the system between different metastable states. For simple systems in which the free energy depends on a few variables, the free energy surface can be precomputed, and saddle points can then be found using existing techniques. For complex systems, where the free energy depends on many degrees of freedom, this is not feasible. In this paper, we develop an algorithm for finding the saddle points on a high-dimensional free energy surface "on-the-fly" without requiring a priori knowledge the free energy function itself. This is done by using the general strategy of the heterogeneous multi-scale method by applying a macro-scale solver, here the gentlest ascent dynamics algorithm, with the needed force and Hessian values computed on-the-fly using a micro-scale model such as molecular dynamics. The algorithm is capable of dealing with problems involving many coarse-grained variables. The utility of the algorithm is illustrated by studying the saddle points associated with (a) the isomerization transition of the alanine dipeptide using two coarse-grained variables, specifically the Ramachandran dihedral angles, and (b) the beta-hairpin structure of the alanine decamer using 20 coarse-grained variables, specifically the full set of Ramachandran angle pairs associated with each residue. For the alanine decamer, we obtain a detailed network showing the connectivity of the minima obtained and the saddle-point structures that connect them, which provides a way to visualize the gross features of the high-dimensional surface.
生物学、化学和材料科学中的许多问题都需要了解自由能表面上的鞍点。这些鞍点起着过渡态的作用，是系统在不同亚稳态之间转变的瓶颈。对于自由能仅依赖于少数几个变量的简单系统，可以预先计算自由能表面，然后使用现有技术找到鞍点。对于自由能依赖于多个自由度的复杂系统，这是不可行的。在本文中，我们开发了一种算法，用于在无需事先了解自由能函数本身的情况下“即时”找到高维自由能表面上的鞍点。这是通过应用宏观尺度求解器（这里是最平缓上升动力学算法），采用非均匀多尺度方法的一般策略来实现的，所需的力和海森矩阵值通过诸如分子动力学等微观尺度模型即时计算得出。该算法能够处理涉及许多粗粒度变量的问题。通过研究与以下内容相关的鞍点来说明该算法的实用性：（a）使用两个粗粒度变量，即拉马钱德兰二面角，研究丙氨酸二肽的异构化转变；（b）使用20个粗粒度变量，即与每个残基相关的全套拉马钱德兰角对，研究丙氨酸十聚体的β-发夹结构。对于丙氨酸十聚体，我们获得了一个详细的网络，展示了所得到的极小值的连通性以及连接它们的鞍点结构，这为可视化高维表面的总体特征提供了一种方法。
