New insights into the meaning and usefulness of principal component analysis of concatenated trajectories

GUSTAVO PIERDOMINICI-SOTTILE; JULIANA PALMA

JOURNAL OF COMPUTATIONAL CHEMISTRY

JOHN WILEY & SONS INC

Lugar: New York; Año: 2015 vol. 36 p. 424 - 432

0192-8651

A comparison between different conformations of a given protein, relating both structure and dynamics, can be performed in terms of combined principal component analysis (combined-PCA). To that end a trajectory is obtained by concatenating molecular dynamics trajectories of the individual conformations under comparison. Then, the principal components are calculated by diagonalising the correlation matrix of the concatenated trajectory. Since the introduction of this approach in 1995 it has had a large number of applications. However, the interpretation of the eigenvectors and eigenvalues so obtained is based on intuitive foundations, because analytical expressions relating the concatenated correlation matrix with those of the individual trajectories under consideration have not been provided yet.  In this article we present such expressions for the cases of two, three and an arbitrary number of concatenated trajectories. The formulas are simple and show what is to be expected and what is not to be expected from a combined-PCA. Their correctness and usefulness is demonstrated by discussing some representative examples. The results can be summarised in a simple sentence: the correlation matrix of a concatenated trajectory is given by the average of the individual correlation matrices plus the correlation matrix of the individual averages. From this it follows that the combined-PCA of trajectories belonging to different free energy basins provides information that could also be obtained by  alternative and more straightforward means.