Principal Component Analysis and Long Time Protein Dynamics

Abstract

It has been suggested that principal component analysis can identify slow modes in proteins and, thereby, facilitate the study of long time dynamics. However, sampling errors due to finite simulation times preclude the identification of slow modes that can be used for this purpose. This is demonstrated numerically with the aid of simulations of the protein G-actin and analytically with the aid of a model which is exactly recoverable by principal component analysis. Although principal component analysis usually demonstrates that a set of a small number of modes captures the majority of the fluctuations, the set depends on the particular sampling time window and its width.

Algorithmic dimensionality reduction for molecular structure analysis

“…The difficulty arises from two factors: ͑1͒ the intrinsic dimensionality for most problems is unknown, and ͑2͒ there is debate for some problems as to whether simulation approaches can provide sufficient sampling of the phase space to facilitate an accurate analysis of dimensionality reduction. 3,13,24,25,27,42 In this paper, we investigate the ability of well-known nonlinear dimensionality reduction algorithms to identify accurate, low-dimensional substructures in the conformation space for an eight-membered ring. We chose this particular molecule for several reasons.…”

A theoretical model for the collective motion of proteins by means of principal component analysis

“…[38,61] That is, the initial state points of the trajectory obtained during the equilibration process of a system by MD simulations may represent a nonphysical process, thus they should not be taken into account during PCA. [61] From the statistical point of view, [63] the elements of the covariance matrix are greatly influenced by the state of statistical control of the data upon which the matrix is based.…”

“…In other words, the subspaces spanned by each half are similar to each other and to the one obtained from the full trajectory analysis. In addition, the eigenvectors obtained from the two halves independently were cross projected and the following matrix is defined [38] P αβ = e (1) α · e (2) β (10) where e ( ) α is the α-th eigenvector obtained from the -th trajectory window ( = 1 2) and α β = 1 · · · M. The results presented in Figure 2B show that the projections of the principal modes are nearly diagonal, indicating that all modes of one half have high projections on the modes of other half. Thus, the directions in the configurational space of both halves of the trajectory are ordered similarly with respect to the amount of fluctuation.…”

“…In particular, one may determine a small number of important modes by PCA and project equations of motion on the resulting lowdimensional vector space. However, an important question arises at this point, whether the correlation functions obtained by molecular dynamics trajectory are accurate enough to capture the correlations that exist between the slow and fast modes [38] (and the references therein). Recently, Lange and Grubmüller [39] have shown that it is possible to derive the equations of motion in the essential degrees of freedom subspace in order to describe long time protein dynamics.…”

Applications of Principal Component Analysis (PCA) in Materials Science

“…The latter route is usually advocated since in biomolecular MD simulations, since it is well known that PCA presents difficulties with respect to proper sampling (Balsera et al 1996;Caves et al 1998). An excellent analysis about reliability of PCA with respect to sampling issues can be found in the work of Skjaerven et al (Skjaerven et al, 2011).…”