Noise and Functional Protein Dynamics

Abstract

The magnetic field dependence of the proton-spin-lattice relaxation rate in rotationally immobilized proteins shows that the one-dimensional character of the protein primary structure causes a dramatic increase in the population of low-frequency motions from 10 kHz to 20 MHz. As a consequence, the probability and rate at which functionally critical conformational states are thermally sampled in a protein are dramatically increased as well, when compared with a three-dimensional lattice structure. Studies of protein dynamics often focus on time periods far shorter than those associated with catalytic function, but we show here that the magnetic field dependence of the proton nuclear spin-lattice relaxation rate in rotationally immobilized proteins reports unambiguously the structural fluctuations in the frequency range from 10 kHz to 20 MHz. This relaxation rate decreases with increasing Larmor frequency according to a power law that derives from the distribution of dynamical states, the localization of the structural disturbances, and the spatial distribution of hydrogen atoms in the structure. The robust theoretical foundation for the spin-relaxation process, loosely characterized as a direct spin-phonon coupling, shows that the disturbances propagate in a space of reduced dimensionality, essentially along the stiff connections of the polypeptide chain. The reduced dimensionality traps the disturbance and changes the efficiency for energy redistribution in the protein and the processes that drive nuclear spin relaxation. We also show that the Larmor frequency dependence of the protein-proton-spin-lattice relaxation rate constant is related to the frequency dependence of force constants and mean-square displacement commonly observed or calculated for proteins. We believe that these approaches give additional physical insight into the character of the extremely low-frequency protein dynamics.

Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation

“…Here, extension of the correlation function g x (s) from an exponential to either a power law or a Mittag-Leffler function would generalize the free induction decay signal G(t) and the Hahn echo as described by Callaghan [55]. In a similar manner Bryant and others [56][57][58][59] have studied power law behavior of the T 1 relaxivity Table 2 Comparison of the pulse gradient spin echo diffusion attenuation curves for different models…”

Fractal symmetry of protein interior: what have we learned?

“…In fact, information obtained with MFD and SD complement each other; while MFD provides information on the arrangement of the atoms of the protein, the SD, although lacking direct kinetic information, relates to the dynamics of the system and the density of the vibrational states. For somewhat more technical (yet highly illuminating) discussions on the inter-relationship between MFD and SD, interested readers can immensely benefit from information provided in various publications (see references [64,[143][144][145][146][147]154]). …”

Nuclear magnetic relaxation dispersion study of the dynamics in solid homopolypeptides

“…[9][10][11][12] The essential idea behind the relaxation mechanism is that the structural fluctuations in the protein propagate along the primary polypeptide chain which causes transient fluctuations in the proton-dipole-dipole couplings, hence, relaxation. [4][5][6]13 The efficiency of this fracton based relaxation mechanism is dramatically increased by the reduced dimensionality for the disturbance propagation in folded polypeptides or proteins, which is close to 1.3 rather than 3. In this study, we extend this nuclear relaxation theory to describe the 1 H MRD of other systems such as dry, neutral, nonpolar homopolypeptides at various temperatures.…”

“…The spectral dimension characterizes the dimensionality of the disturbance propagation, and the fractal dimension characterizes the distribution of protons in space. [4][5][6]13,14 The exponent in the experimental power law is related to these dimensions;…”

“…Thus, we conclude that while a distribution of local motions is likely, it is relatively narrow compared with earlier estimates based on a simpler Lorentzian model for the spectral density functions. [32][33][34][35][36][37][38] Solid-state 13 C CP-MAS NMR study of valine methyl groups in 13 C fully-labeled antamanide 39 have shown that two methyl groups of the valine amino acid coexist in different environmental conformations that influence the frequencies of the side-chain group motion. Deuterium solid-state relaxation measurements on N-acetyl-DL-valine 40 indicate that two methyl groups of valine undergo a three-site jump process with different activation energies and the spin-lattice relaxation times that differ by an order of magnitude.…”