Construction of the free energy landscape of biomolecules via dihedral angle principal component analysis

Abstract

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.

Accurate free energy calculation along optimized paths

“…By considering these two aspects, we only include dihedral angles in the restraint potential. Very recently, Stock and coworkers 29 showed that full-dimensional free energy landscape could be reproduced accurately by dihedral-angle principal component analysis. In our work, we only restrain two dihedral angles: phi and psi, for the backbone of the peptide and the nonhydrogen atom dihedral angles for side chains.…”

Exploring the Energy Landscape of Small Peptides and Proteins by Molecular Dynamics Simulations

“…The characterization of the free energy landscape of larger systems is less straightforward and therefore needs to be carried out in a systematic manner. With this end in mind, in the following we consider the free energy landscape of hepta -alanine (Ala 7 ), which was obtained from an 800 -ns MD simulation in aqueous solution at 300 K [5] . To get a fi rst impression, Figure 2.3 shows two -dimensional representations of the free energy landscape of Ala 7 , as obtained from an dPCA of the backbone dihedral angles.…”

“…Doing so, we found that k = 23 are a suitable number of clusters (see Ref. [5] for details). This fi ndings are reconfi rmed by the results of a kinetic clustering of the Ala 7 trajectory, that is, a clustering that defi nes its states through their metastability rather than through geometric similarity.…”

“…Now states 2 and 4 are direct neighbors connected by a single barrier, and states 1 and 2 are only connected via state 4 . The energies ΔG 24 and ΔG 12 of these spurious barriers and the corresponding transition rates k 24 and k 12 may be smaller or larger than in full dimensionality [5] .…”

Collective Estimation of Multiple Bivariate Density Functions With Application to Angular-Sampling-Based Protein Loop Modeling

“…Although most problems in the protein structure field depend on coordinate-based methods, backbone-angle-based methods have provided an attractive alternative approach in various protein structure-related problems, such as protein structure prediction (Simons et al, 1999;Hamelryck et al, 2006;Boomsma et al, 2008;Zhao et al, 2010), protein loop modeling (Ting et al, 2010), model quality assessment (Benkert et al, 2008;Gao et al, 2009;Archie and Karplus, 2009), prediction server ranking (Qiu et al, 2008;Maadooliat et al, 2013a), protein structure alignment (Miao et al, 2008;Challis and Schmidler, 2012), free energy function learning (Mu et al, 2005;Altis et al, 2008;Riccardi et al, 2009), and molecular dynamics simulation (Altis et al, 2007). In this paper, we focus on statistical modeling of the bivariate distribution of protein backbone angles.…”