Search

In Paper I [Polimeno et al., J. Chem. Phys. 150, 184107 (2019)], we proposed a general approach for interpreting relaxation properties of a macromolecule in solution, derived from an atomistic description. A simple scheme (the semiflexible Brownian, SFB, model) has been defined for the case of limited internal flexibility, but retaining full coupling with external degrees of freedom, inclusion of all of the momenta, and dissipation. Here we discuss the application of the SFB model to the practical evaluation of orientation spectral densities, based on two complementary computational treatments.

A framework for the stochastic description of relaxation processes in flexible macromolecules, including dissipative effects, is introduced from an atomistic point of view. Projection-operator techniques are employed to obtain multidimensional Fokker-Planck operators governing the relaxation of internal coordinates and global degrees of freedom and depending upon parameters fully recoverable from classic force fields (energetics) and continuum models (friction tensors). A hierarchy of approaches of different complexity is proposed in this unified context, aimed primarily at the interpretation of magnetic resonance relaxation experiments. In particular, a model based on a harmonic internal Hamiltonian is discussed as a test case.

Long-lived imbalances of spin state populations can circumvent fast quadrupolar relaxation by reducing effective longitudinal relaxation rates by about an order of magnitude. This opens new avenues for the study of dynamic processes in deuterated molecules. Here we present an analysis of the relaxation properties of deuterated methyl groups CD 3 . The number of coupled equations that describe cross-relaxation between the 27 symmetry-adapted spin states of a D 3 system can be reduced to only 2 non-trivial "lumped modes" by (i) taking the sums of the populations of all states that equilibrate rapidly within each irreducible representation of the symmetry group, and (ii) by combining populations that have similar relaxation rates although they belong to different irreducible representations. The quadrupolar relaxation rates of the spin state imbalances in CD 3 groups are determined not by the correlation time of overall tumbling of the molecule, but by the frequency of jumps of methyl groups about their three-fold symmetry axis. We access these states via dissolution dynamic nuclear polarization (D-DNP), a method that allows one to populate the desired long-lived states at cryogenic temperatures and their subsequent detection at ambient temperatures after rapid dissolution. Experimental examples of DMSO-d 6 and ethanol-d 6 demonstrate that long-lived deuterium spin states are indeed accessible and that their lifetimes can be determined. Our analysis of the system via "lumped" modes allows us to visualize different possible spin-state populations of symmetry A, B, or E. Thus, we identify a long-lived spin state involving all three deuterons in a CD 3 group as an A/E imbalance that can be populated through DNP at low temperatures.

We report the generation and observation of long-lived spin states in deuterated methyl groups by dissolution DNP. These states are based on population imbalances between manifolds of spin states corresponding to irreducible representations of the C 3v point group and feature strongly dampened quadrupolar relaxation. Their lifetime depends on the activation energies of methyl group rotation. With dissolution DNP, we can reduce the deuterium relaxation rate by a factor up to 20, thereby extending the experimentally available time window. The intrinsic limitation of NMR spectroscopy of quadrupolar spins by short relaxation times can thus be alleviated.

We present novel means to hyperpolarize deuterium nuclei in 13 CD 2 groups at cryogenic temperatures. The method is based on cross-polarization from 1 H to 13 C and does not require any radio-frequency fields applied to the deuterium nuclei. After rapid dissolution, a new class of long-lived spin states can be detected indirectly by 13 C NMR in solution. These long-lived states result from a sextet-triplet imbalance (STI) that involves the two equivalent deuterons with spin I = 1. An STI has similar properties as a triplet-singlet imbalance that can occur in systems with two equivalent I = 12 spins. Although the lifetimes T STI are shorter than T 1 (C z ), they can exceed the life-time T 1 (D z ) of deuterium Zeeman magnetization by a factor of more than 20.

We propose a fractional Brownian dynamics model for time correlation functions characterizing the internal dynamics of proteins probed by NMR relaxation spectroscopy. The time correlation functions are represented by a broad distribution of exponential functions which are characterized by two parameters. We show that the model describes well the restricted rotational motion of N-H vectors in the amide groups of lysozyme obtained from molecular dynamics simulation and that reliable predictions of experimental relaxation rates can be obtained on that basis.

In this article, a formal expression for the conformational entropy of a bond vector in a protein is derived using the networks of coupled rotators model for the description of internal dynamics. Analytical relationships between NMR order parameters and conformational entropies are derived, and the possibility to extract the latter from NMR experiments is discussed. These results are illustrated in the case of the calcium-binding protein calbindin.

Nuclear magnetic resonance (NMR) has proven to be the most valuable tool for investigating internal dynamics of proteins. In this perspective, the interpretation of NMR relaxation data eventually relies on a model of the motions. In this article, we propose to compare two radically different approaches that aim at describing internal dynamics in proteins. It is shown that the correlation functions predicted by a network of coupled rotators can be interpreted in terms of a heuristic approach based on fractional Brownian dynamics for each of the vectors in the network. Our results are interpreted in terms of the probability distributions of relaxation modes in both processes, the median of which turns out to be the relevant quantity for the comparison of both models.

Internal motions in proteins, such as oscillations of internuclear vectors u(N(i)H(i) (N)) of amide bonds about their equilibrium position, can be characterized by a local order parameter. This dynamic parameter can be determined experimentally by measuring the longitudinal and transverse relaxation rates of (15)N(i) nuclei by suitable NMR methods. In this paper, it is shown that local variations of order parameters S(ii) (2) can be predicted from the knowledge of the structure. To this effect, the diffusive motion of the internuclear vector u(N(i)H(i) (N)) is described in a potential that takes into account the deviations of the angles theta(ij) between u(N(i)H(i) (N)) and neighboring vectors u(N(j)H(j) (N)) from their average value and similarly of deviations of the angles subtended between u(N(i)H(i) (N)) and u(X(j)Y(j)), where X(j) and Y(j) are heavy atoms in the vicinity of the u(N(i)H(i) (N)) vector under investigation. It is shown how the concept of vicinity can be defined by a simple cutoff threshold, i.e., by neglecting vectors u(X(j)Y(j)) with distances d(N(i),X(j))>7.5 A. The local order parameters S(ii) (2) can be predicted from the structure using a limited set of coordinates of heavy atoms. The inclusion of a larger number of heavy atoms does not improve the predictions. Applications to calmodulin, calbindin, and interleukin 4 illustrate the success and limitations of the predictions.

In this paper some effects of dynamical coupling between two interaction-carrying vectors on the internal auto- and cross-correlation functions are investigated in the limit of small amplitude motions. A linearized Langevin approach allows the derivation of explicit expressions for these correlation functions and for the corresponding order parameters.