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.

FIGURE 1

The proton spin-lattice relaxation rate recorded as a function of the magnetic field strength plotted as the proton Larmor frequency for samples of dry lysozyme and BSA at room temperature. The solid line presents the best fit of the data with Eqs. 3a and 3b and b = 0.78. This value of b leads to a value d_f = 3 from Eq. 3b, which indicates a uniform distribution of protons. The peaks at 2.8, 2.4, and 0.8 MHz in the relaxation rate profile are caused by proton relaxation coupling to the amide nitrogen when the ¹⁴N energies match the proton Zeeman levels.

FIGURE 2

The proton spin-lattice relaxation rate recorded as a function of the magnetic field-strength plotted as the proton Larmor frequency for lysozyme samples hydrated to various degrees (weight %) at room temperature. The solid lines are the best fits to the data using Eqs. 3a and 3b. The two parameters adjusted are b and the exchange rate constant between the protein protons and the water proton populations. The value d_f is obtained from b according to Eq. 3b.

FIGURE 3

Frequency dependence in MHz of the normalized density of vibrational distribution of states σ(ω)/3N calculated from Eq. 1 for different dimensionalities d (, d_S = 4/3 and 1). The range of frequency (10⁴ − 2 × 10⁷ Hz) probed by magnetic relaxation dispersion is indicated by an arrow. The frequency, Ω ≈ 10¹³ Hz, corresponds to the highest vibrational mode. One notes a dramatic increase (18 orders of magnitude) of the density of vibrational states at low-frequency when reducing the dimensionality from d = 3 to d = 1.

FIGURE 4

The fractal dimension d_f of the proton distribution obtained from Fig. 2 with use of Eqs. 3a and 3b and plotted against the ratio F between the protein-protons and the water-protons at equilibrium. The continuous line is a logarithmic fit to the data.

FIGURE 5

The computed variation of the exponent of the mean-square displacement of Eq. 7 versus the fractal dimension, d_f, is shown as a solid line. The exponent is 1 for d_S/d_f = 1/2. The dashed line is a guide for the eye and indicates a linear dependence.