Fortuitous Correlations in Molecular Dynamics Simulations: Their Harmful Influence on the Probability Distributions of the Main Principal Components.

Nonsense correlations frequently develop between independent random variables that evolve with time. Therefore, it is not surprising that they appear between the components of vectors carrying out multidimensional random walks, such as those describing the trajectories of biomolecules in molecular dynamics simulations. The existence of these correlations does not imply in itself a problem. Still, it can present a problem when the trajectories are analyzed with an algorithm such as the Principal Component Analysis (PCA) because it seeks to maximize correlations without discriminating whether they have physical origin or not. In this Article, we employ random walks occurring on multidimensional harmonic potentials to evaluate the influence of fortuitous correlations in PCA. We demonstrate that, because of them, this algorithm affords misleading results when applied to a single trajectory. The errors do not only affect the directions of the first eigenvectors and their eigenvalues, but the very definition of the molecule's "essential space" may be wrong. Additionally, the main principal component's probability distributions present artificial structures which do not correspond with the shape of the potential energy surface. Finally, we show that the PCA of two realistic protein models, human serum albumin and lysozyme, behave similarly to the simple harmonic models. In all cases, the problems can be mitigated and eventually eliminated by doing PCA on concatenated trajectories formed from a large enough number of individual simulations.