We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.

It has recently been suggested by Mu et al. [Proteins 58, 45 (2005)] to use backbone dihedral angles instead of Cartesian coordinates in a principal component analysis of molecular dynamics simulations. Dihedral angles may be advantageous because internal coordinates naturally provide a correct separation of internal and overall motion, which was found to be essential for the construction and interpretation of the free energy landscape of a biomolecule undergoing large structural rearrangements. To account for the circular statistics of angular variables, a transformation from the space of dihedral angles {phi(n)} to the metric coordinate space {x(n)=cos phi(n),y(n)=sin phi(n)} was employed. To study the validity and the applicability of the approach, in this work the theoretical foundations underlying the dihedral angle principal component analysis (dPCA) are discussed. It is shown that the dPCA amounts to a one-to-one representation of the original angle distribution and that its principal components can readily be characterized by the corresponding conformational changes of the peptide. Furthermore, a complex version of the dPCA is introduced, in which N angular variables naturally lead to N eigenvalues and eigenvectors. Applying the methodology to the construction of the free energy landscape of decaalanine from a 300 ns molecular dynamics simulation, a critical comparison of the various methods is given.

A systematic approach to construct a low-dimensional free energy landscape from a classical molecular dynamics (MD) simulation is presented. The approach is based on the recently proposed dihedral angle principal component analysis (dPCA), which avoids artifacts due to the mixing of internal and overall motions in Cartesian coordinates and circumvents problems associated with the circularity of angular variables. Requiring that the energy landscape reproduces the correct number, energy, and location of the system's metastable states and barriers, the dimensionality of the free energy landscape (i.e., the number of essential components) is obtained. This dimensionality can be determined from the distribution and autocorrelation of the principal components. By performing an 800 ns MD simulation of the folding of hepta-alanine in explicit water and using geometric and kinetic clustering techniques, it is shown that a five-dimensional dPCA energy landscape is a suitable and accurate representation of the full-dimensional landscape. In the second step, the dPCA energy landscape can be employed (e.g., in a Langevin simulation) to facilitate a detailed investigation of biomolecular dynamics in low dimensions. Finally, several ways to visualize the multidimensional energy landscape are discussed.

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