Caves Leo S D, Verma Chandra S
Structural Biology Laboratory, Department of Chemistry, University of York, York, United Kingdom.
Proteins. 2002 Apr 1;47(1):25-30.
Central to the study of a complex dynamical system is knowledge of its phase space behavior. Experimentally, it is rarely possible to record a system's (multidimensional) phase space variables. Rather, the system is observed via one (or few) scalar-valued signal(s) of emission or response. In dynamical systems analysis, the multidimensional phase space of a system can be reconstructed by manipulation of a one-dimensional signal. The trick is in the construction of a (higher-dimensional) space through the use of a time lag (or delay) on the signal time series. The trajectory in this embedding space can then be examined using phase portraits generated in selected subspaces. By contrast, in computer simulation, one has an embarrassment of riches: direct access to the complete multidimensional phase space variables, at arbitrary time resolution and precision. Here, the problem is one of reducing the dimensionality to make analysis tractable. This can be achieved through linear or nonlinear projection of the trajectory into subspaces containing high information content. This study considers trajectories of the small protein crambin from molecular dynamics simulations. The phase space behavior is examined using principal component analysis on the Cartesian coordinate covariance matrix of 138 dimensions. In addition, the phase space is reconstructed from a one dimensional signal, representing the radius of gyration of the structure along the trajectory. Comparison of low-dimensional phase portraits obtained from the two methods shows that the complete phase space distribution is well represented by the reconstruction. The study suggests that it may be possible to develop a deeper connection between the experimental and simulated dynamics of biomolecules via phase space reconstruction using data emerging from recent advances in single-molecule time-resolved biophysical techniques.
对复杂动力系统进行研究的核心是了解其相空间行为。在实验中，几乎不可能记录系统的（多维）相空间变量。相反，是通过一个（或少数几个）发射或响应的标量值信号来观测系统。在动力系统分析中，系统的多维相空间可以通过对一维信号进行处理来重构。关键在于利用信号时间序列上的时间滞后（或延迟）构建一个（高维）空间。然后可以使用在选定子空间中生成的相图来检查这个嵌入空间中的轨迹。相比之下，在计算机模拟中，人们面临着丰富数据的困扰：可以以任意时间分辨率和精度直接获取完整的多维相空间变量。在这里，问题是如何降低维度以便于进行分析。这可以通过将轨迹线性或非线性投影到包含高信息含量的子空间中来实现。本研究考虑了来自分子动力学模拟的小蛋白质胰凝乳蛋白酶原的轨迹。使用主成分分析对138维笛卡尔坐标协方差矩阵来检查相空间行为。此外，从表示结构沿轨迹的回转半径的一维信号重构相空间。比较从这两种方法获得的低维相图表明，重构能够很好地表示完整的相空间分布。该研究表明，利用单分子时间分辨生物物理技术的最新进展所产生的数据，通过相空间重构，有可能在生物分子的实验动力学和模拟动力学之间建立更深入的联系。
