Department of Chemistry, Kings College London, London, England.
Laboratory of Theoretical Chemistry, ELTE Eötvös Loránd University, Budapest, Hungary.
J Chem Phys. 2019 Apr 7;150(13):134107. doi: 10.1063/1.5083924.
Markov state models (MSMs) provide some of the simplest mathematical and physical descriptions of dynamical and thermodynamical properties of complex systems. However, typically, the large dimensionality of biological systems studied makes them prohibitively expensive to work in fully Markovian regimes. In this case, coarse graining can be introduced to capture the key dynamical processes-slow degrees of the system-and reduce the dimension of the problem. Here, we introduce several possible options for such Markovian coarse graining, including previously commonly used choices: the local equilibrium and the Hummer Szabo approaches. We prove that the coarse grained lower dimensional MSM satisfies a variational principle with respect to its slowest relaxation time scale. This provides an excellent framework for optimal coarse graining, as previously demonstrated. Here, we show that such optimal coarse graining to two or three states has a simple physical interpretation in terms of mean first passage times and fluxes between the coarse grained states. The results are verified numerically using both analytic test potentials and data from explicit solvent molecular dynamics simulations of pentalanine. This approach of optimizing and interpreting clustering protocols has broad applicability and can be used in time series analysis of large data.
马尔可夫状态模型（MSMs）为复杂系统的动力学和热力学性质提供了一些最简单的数学和物理描述。然而，通常情况下，所研究的生物系统的大维度使得它们在完全马尔可夫状态下的工作变得非常昂贵。在这种情况下，可以引入粗粒化来捕捉关键的动力学过程——系统的慢度，并降低问题的维度。在这里，我们介绍了几种可能的马尔可夫粗粒化选择，包括以前常用的选择：局部平衡和 Hummer-Szabo 方法。我们证明，粗粒化的低维 MSM 满足其最慢松弛时间尺度的变分原理。这为之前证明的最优粗粒化提供了一个极好的框架。在这里，我们表明，对于两到三个状态的这种最优粗粒化，可以根据平均首次通过时间和粗粒化状态之间的通量来给出简单的物理解释。使用解析测试势和戊丙氨酸的显溶剂分子动力学模拟数据对结果进行了数值验证。这种优化和解释聚类协议的方法具有广泛的适用性，可以用于大数据的时间序列分析。
