分子动力学中慢动力学模式的变分交叉验证
Variational cross-validation of slow dynamical modes in molecular kinetics
摘要：
马尔可夫状态模型（MSMs）是一种广泛使用的方法，用于近似分子动力学传播子的本征谱，从而深入了解生物分子系统的长时间尺度统计动力学和慢动力学模式。然而，缺乏一个统一的理论框架来选择替代模型阻碍了进展，特别是对于非专家将这些方法应用于新的生物系统。在这里，我们考虑交叉验证与一个新的目标函数的估计这些缓慢的动力学模式，广义矩阵瑞利商（GMRQ），它衡量的能力的秩- $m$投影算子捕捉系统的慢子空间。它示出的变分定理的边界GMRQ从上面的第一个$m$特征值的系统的传播的总和，但这个界可以被违反时，所需的矩阵元素估计受到统计不确定性。这种过拟合可以通过交叉验证来检测和避免。这些结果使得有可能构建马尔可夫状态模型的蛋白质动力学的方式，适当地捕捉系统和统计误差之间的权衡。
英文摘要：
Markov state models (MSMs) are a widely used method for approximating the eigenspectrum of the molecular dynamics propagator, yielding insight into the long-timescale statistical kinetics and slow dynamical modes of biomolecular systems. However, the lack of a unified theoretical framework for choosing between alternative models has hampered progress, especially for non-experts applying these methods to novel biological systems. Here, we consider cross-validation with a new objective function for estimators of these slow dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which measures the ability of a rank-$m$ projection operator to capture the slow subspace of the system. It is shown that a variational theorem bounds the GMRQ from above by the sum of the first $m$ eigenvalues of the system's propagator, but that this bound can be violated when the requisite matrix elements are estimated subject to statistical uncertainty. This overfitting can be detected and avoided through cross-validation. These result make it possible to construct Markov state models for protein dynamics in a way that appropriately captures the tradeoff between systematic and statistical errors.
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