Markov State Models (MSM) and related techniques have gained significant traction as a tool for analyzing and guiding molecular dynamics (MD) simulations due to their ability to extract structural, thermodynamic, and kinetic information on proteins using computationally feasible MD simulations. The MSM analysis often relies on spectral decomposition of empirically generated transition matrices. Here, we discuss an alternative approach for extracting the thermodynamic and kinetic information from the so-called rate/generator matrix rather than the transition matrix. Although the rate matrix itself is built from the empirical transition matrix, it provides an alternative approach for estimating both thermodynamic and kinetic quantities, particularly in diffusive processes. We particularly discuss a fundamental issue with this approach, known as the embeddability problem and offer ways to address this issue. We describe eight different methods to overcome the embeddability problem, including a novel approach developed for this work. The algorithms were tested on data from a one-dimensional toy model to show the workings of these methods and discuss the robustness of each method in terms of its dependence in lag time and trajectory length.

中文翻译：

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解决过渡速率估计中的可嵌入性问题

马尔可夫状态模型（MSM）和相关技术作为分析和指导分子动力学（MD）模拟的工具已获得了广泛的关注，这是因为它们具有使用计算上可行的MD模拟提取蛋白质的结构，热力学和动力学信息的能力。MSM分析通常依赖于凭经验生成的跃迁矩阵的频谱分解。在这里，我们讨论了从所谓的速率/发电机矩阵而不是过渡矩阵中提取热力学和动力学信息的另一种方法。尽管速率矩阵本身是根据经验转换矩阵构建的，但它提供了一种估算热力学和动力学量的替代方法，尤其是在扩散过程中。我们特别讨论使用这种方法的一个基本问题，称为可嵌入性问题，并提供解决此问题的方法。我们描述了八种克服嵌入性问题的方法，包括为此工作开发的一种新颖方法。该算法对一维玩具模型的数据进行了测试，以显示这些方法的工作原理，并根据滞后时间和轨迹长度的依赖性讨论了每种方法的鲁棒性。