Local Ordering at Mobile Sites in Proteins from Nuclear Magnetic Resonance Relaxation: The Role of Site Symmetry

Abstract

Restricted motions in proteins (e.g., N–H bonds 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 LC’s experience non-polar 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 D2h point group, which is already implicit in the definition of the rotational diffusion tensor. Thus, the relevant rhombic L = 1 potentials have B1u and B3u symmetry whereas the relevant rhombic L = 2 potentials have Ag 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 non-polar 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 non-polar 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 15N-labeled plexin-B1 and thioredoxin, 2H-, and 13C-labeled benzenehexa-n-alkanoates, and nitroxide-labeled T4 lysozyme. Future prospects for improved ordering analysis based on combined atomistic and mesoscopic approaches are delineated.

Conformational Entropy from Restricted Bond-Vector Motion in Proteins: The Symmetry of the Local Restrictions and Relation to NMR Relaxation

“…In Figure 3 a,b S k is the same for positive and negative c 0 1 ( S 0 1 ), i.e., for the primary polar axis pointing along + z and − z . 23 A single branch is featured by ( S 0 1 ) 2 ( Figure 3 c).…”

“…For c 1 1 > 0 ( c 1 1 < 0), the primary polar axis is tilted in the + zx (− zx ) plane of the local ordering frame. 4 , 23 …”

“…Order parameters are ensemble averages of real Wigner functions defined in terms of restricting potentials. Here, we are using the order parameters 4 , 23 The normalized probability density function required to calculate these ensemble averages is given by where u (θ,φ) is the restricting potential.…”

“… 16 , 19 In accordance with typical experimental data, the lowest even L terms, and in some cases the lowest odd L terms, have been kept, yielding u = – c 0 2 D 00 2 – c 2 2 ( D 02 2 + D 0–2 2 ) for nonpolar ordering 17 , 18 , 20 − 22 and u = −c 0 1 D 00 1 – c 1 1 ( D 0–1 1 – D 01 1 ) for polar ordering. 23 − 26 …”

“…4 , 5 Polar ( L = 1) ordering prevails when there is no inversion symmetry with respect to the origin of the director frame. This is the case for bond-vectors anchored at the protein 23 , 24 (or any other relaxation probe anchored at the entity that represents the local director). In view of the various approximations, admixtures are most appropriate.…”