## Abstract

Restricted motions in proteins (e.g., N-H bond dynamics) are studied effectively with NMR. By analogy with restricted motions in liquid crystals (LC), the local ordering has in the past been primarily represented by potentials comprising the L = 2, |K| = 0, 2 spherical harmonics. However, probes dissolved in LCs experience nonpolar ordering, often referred to as alignment, while protein-anchored probes experience polar ordering, often referred to as orientation. In this study we investigate the role of local (site) symmetry in the context of the polarity of the local ordering. We find that potentials comprising the L = 1, |K| = 0, 1 spherical harmonics represent adequately polar ordering. It is useful to characterize potential symmetry in terms of the irreducible representations of D_{2h} point group, which is already implicit in the definition of the rotational diffusion tensor. Thus, the relevant rhombic L = 1 potentials have B_{1u} and B_{3u} symmetry whereas the relevant rhombic L = 2 potentials have A_{g} symmetry. A comprehensive scheme where local potentials and corresponding probability density functions (PDFs) are represented in Cartesian and spherical coordinates clarifies how they are affected by polar and nonpolar ordering. The Cartesian coordinates are chosen so that the principal axis of polar axial PDF is pointing along the z-axis, whereas the principal axis of the nonpolar axial PDF is pointing along ±z. Two-term axial potentials with 1 ≤ L ≤ 3 exhibit substantial diversity; they are expected to be useful in NMR-relaxation-data-fitting. It is shown how potential coefficients are reflected in the experimental order parameters. The comprehensive scheme representing local potentials and PDFs is exemplified for the L = 2 case using experimental data from ^{15}N-labeled plexin-B1 and thioredoxin, ^{2}H-, and ^{13}C-labeled benzenehexa-n-alkanoates, and nitroxide-labeled T4 lysozyme. Future prospects for improved ordering analysis based on combined atomistic and mesoscopic approaches are delineated.

Original language | English |
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Pages (from-to) | 2886-2898 |

Number of pages | 13 |

Journal | Journal of Physical Chemistry B |

Volume | 120 |

Issue number | 11 |

DOIs | |

State | Published - 31 Mar 2016 |

### Bibliographical note

Funding Information:This work was supported by the Israel-U.S.A. Binational Science Foundation (Grant No. 2010185 to E.M. and J.H.F.), and the Israel Science Foundation (Grants No. 437/11 and No. 369/15 to E.M.). This work was also supported by NIH/NIGMS grant P41GM103521 to J.H.F.

Publisher Copyright:

© 2016 American Chemical Society.