title: Euclidean sections of protein conformation space and their implications in dimensionality reduction.
abstract: Dimensionality reduction is widely used in searching for the intrinsic reaction coordinates for protein conformational changes. We find the dimensionality-reduction methods using the pairwise root-mean-square deviation (RMSD) as the local distance metric face a challenge. We use Isomap as an example to illustrate the problem. We believe that there is an implied assumption for the dimensionality-reduction approaches that aim to preserve the geometric relations between the objects: both the original space and the reduced space have the same kind of geometry, such as Euclidean geometry vs. Euclidean geometry or spherical geometry vs. spherical geometry. When the protein free energy landscape is mapped onto a 2D plane or 3D space, the reduced space is Euclidean, thus the original space should also be Euclidean. For a protein with N atoms, its conformation space is a subset of the 3N-dimensional Euclidean space R(3N). We formally define the protein conformation space as the quotient space of R(3N) by the equivalence relation of rigid motions. Whether the quotient space is Euclidean or not depends on how it is parameterized. When the pairwise RMSD is employed as the local distance metric, implicit representations are used for the protein conformation space, leading to no direct correspondence to a Euclidean set. We have demonstrated that an explicit Euclidean-based representation of protein conformation space and the local distance metric associated to it improve the quality of dimensionality reduction in the tetra-peptide and β-hairpin systems.
authors.affiliation: Gustaf H. Carlson School of Chemistry and Biochemistry, Clark University, Worcester, Massachusetts, 01610.
authors.forename: Mojie
authors.initials: M
authors.lastname: Duan
authors.forename: Minghai
authors.initials: M
authors.lastname: Li
authors.forename: Li
authors.initials: L
authors.lastname: Han
authors.forename: Shuanghong
authors.initials: S
authors.lastname: Huo
mesh_terms: D001426:Bacterial Proteins / Q000737:chemistry*; D004735:Energy Transfer; D008958:Models, Molecular; D056004:Molecular Dynamics Simulation; D009842:Oligopeptides / Q000737:chemistry*; D010446:Peptide Fragments / Q000737:chemistry*; D025341:Principal Component Analysis; D011487:Protein Conformation; D017510:Protein Folding; D054730:Protein Interaction Domains and Motifs; D017433:Protein Structure, Secondary; D058767:Protein Unfolding; D013223:Statistics as Topic; D013499:Surface Properties; D009626:Terminology as Topic
chemical_list: D001426:Bacterial Proteins; C034284:IgG Fc-binding protein, Streptococcus; D009842:Oligopeptides; D010446:Peptide Fragments; C026593:alanyl-alanyl-alanyl-alanine
keywords: dimensionality reduction; free energy landscape; isomap; principal component analysis; protein conformation space