Analytic treatment of nuclear spin-lattice relaxation for diffusion in a cone model.

We consider nuclear spin-lattice relaxation rate resulted from a diffusion equation for rotational wobbling in a cone. We show that the widespread point of view that there are no analytical expressions for correlation functions for wobbling in a cone model is invalid and prove that nuclear spin-lattice relaxation in this model is exactly tractable and amenable to full analytical description. The mechanism of relaxation is assumed to be due to dipole-dipole interaction of nuclear spins and is treated within the framework of the standard Bloemberger, Purcell, Pound-Solomon scheme. We consider the general case of arbitrary orientation of the cone axis relative the magnetic field. The BPP-Solomon scheme is shown to remain valid for systems with the distribution of the cone axes depending only on the tilt relative the magnetic field but otherwise being isotropic. We consider the case of random isotropic orientation of cone axes relative the magnetic field taking place in powders. Also we consider the cases of their predominant orientation along or opposite the magnetic field and that of their predominant orientation transverse to the magnetic field which may be relevant for, e.g., liquid crystals. Besides we treat in details the model case of the cone axis directed along the magnetic field. The latter provides direct comparison of the limiting case of our formulas with the textbook formulas for free isotropic rotational diffusion. The dependence of the spin-lattice relaxation rate on the cone half-width yields results similar to those predicted by the model-free approach.

Spin-lattice NMR relaxation by anomalous translational diffusion.

A model-free theoretical framework for a phenomenological description of spin-lattice relaxation by anomalous translational diffusion in inhomogeneous systems based on the fractional diffusion equation is developed. The dependence of the spin-lattice relaxation time on the size of the pores in porous glass Vycor is experimentally obtained and found to agree well with our theoretical predictions. We obtain nonmonotonic behavior of the translational spin-lattice relaxation rate constant (it passes through a maximum) with the variation of the parameter referring to the extent of inhomogeneity of the system.

Modeling the "glass" transition in proteins.

A model of a protein as a disordered system of identical spherical particles (which imitate protein side chains) interacting with each other via a repulsive soft sphere potential U(r) infinity r(-beta) is constructed. The particles undergo the conformational motion (CM) within their own harmonic conformational potentials around some mean equilibrium positions ascribed by the tertiary structure of the protein. A first principles calculation of the positional correlation functions for the CM is carried out. The general analysis is exemplified by the case in which the mean equilibrium positions of the particles form a cubic tightly-packed (face- centered) lattice (each particle has 12 nearest neighbors) with the step b(hydr) =6.6 A (the average distance between the centers of mass of hydrated protein subunits). The model yields dramatic slowing down of the relaxation with the decrease of temperature followed by a sharp glass transition at some crossover temperature T(c) < 200 K. At the transition the liquid-like dynamic behavior (the correlation functions tend to zero with time) is altered by the glass-like one (the correlation functions tend with time to some non-zero limit). In the liquid-like region above the crossover temperature the relaxation exhibits distinct alpha-process following the beta-one. The glass transition results from the interaction of the particles. Thus the model suggests that namely direct interactions of the fragments of protein structure rather than protein-solvent interactions are the origin of the phenomenon of the glass transition. The known increase of T(c) up to 300 K at dehydration of the protein is attributed to the known concomitant compression of the globule upon drying by about 4-6% so that positions of individual atoms displace by about 0.6 A (modeled by the decrease of the step of the lattice b by 0.6 A so that b(dehydr)=6 A). The model suggests that the solvent influences the phenomenon of the glass transition indirectly determining the tertiary structure of the protein rather than via own freezing. In the model the transition from the liquid-like dynamic behavior to the glass-like one can be obtained even in a cluster containing a few particles. Thus the results of the model can be considered as an argument in favor of the point of view that the transition to the glassy behavior can take place for a very small domains of the protein comprising only several constituting fragments of its structure. The model predicts that for the dehydrated protein the alpha-relaxation process is strongly repressed.

Modeling the correlation functions of conformational motions in proteins.

A first principles calculation of the correlation function for conformational motion (CM) in proteins is carried out within the framework of a microscopic model of a protein as a heterogeneous system. The fragments of the protein are assumed to be identical hard spheres undergoing the CM within their conformational potentials about some mean equilibrium positions assigned by the tertiary structure. The memory friction function (MFF) for the generalized Langevin equation describing the CM of the particle is obtained on the basis of the direct calculation which is feasible for the present model of the protein due to the existence of a natural large parameter, viz. the ratio of the minimal distance between the mean equilibrium positions of the particles (approximately 7A) to the amplitude of their CM (<1A). A relationship between the MFF and the correlation functions of the CM of the particles is derived which makes their calculation to be a self-consistent mathematical problem. The general analysis of the MFF is exemplified by a simple model case in which the mean equilibrium positions of the particles form a regular lattice so that the correlation functions for all particles are the same. In this case the MFF is shown to be an infinite series of the powers of the auto-correlation function whose coefficients are independent on temperature. The latter is a result of the abstraction of the interaction potential by that of hard spheres which actually corresponds to the high temperature limit. On the examples of cubic and triangular lattices the coefficients are shown to be non-negative values which increase with the increase of the packing density of the particles and quickly tend to zero with the increase of their index. Thus the MFF can be approximated by a polynomial of the correlation function and the resulting mathematical equation is analogous to the one from the dynamic theory of liquids. The correlation function of the CM is obtained by numerical solution of the equation. At realistic packing densities for proteins it exhibits transparent non-exponential decay and includes two relaxation processes: the first one on the intermediate timescale (tens of picoseconds) and the second on the long timescale (its characteristic time is about tens of nanoseconds at small values of the friction coefficient and increases by orders of the magnitude with the increase of the latter). Thus the present approach provides the microscopic basis for previous phenomenological models of cooperative dynamics in proteins.

Mathematical modelling of electrostatic fluctuations in subtilisin active site.

The intensities of electrostatic fluctuations in subtilisin active site generated by conformational motion of charged side chains, polar side chains and peptide bonds of the main chain are calculated. The comparative analysis of all these fragments reveals that there are few of them which make the main contribution to the total value of the intensity, which has been found to be approximately 10(7) g.cm-1.s-2. These are Ser 125, Thr 220 and peptide bonds of aminoacids 125-126, 218-219. Our present analysis enables us to compare the relative contribution of different fragments but we do not pretend to obtain precise absolute values. The reason for this is the lack of the detailed selective information on the mean-square amplitudes and correlation times of conformational motion of the fragments and on the values of local dielectric constants in the interior of subtilisin. The possibility for electrostatic fluctuations in enzyme active site to be an efficient nonspecific source of substrate activation is discussed.

Conformational dynamics of protein side chains and enzyme-substrate interaction.

The electric interaction of charged and polar protein side chains with dipole moments of the bonds of substrate molecules bound in enzyme active sites is considered. The conformational motion of the side chains leads to the fact that the electric interaction, besides a constant (electrostatic) part, contains a fluctuating one, which is a random force (noise) exerted on the substrate molecule. On the time scale of enzyme turnovers this noise can be considered as the white one with good approximation. The noise is external, and the explicit expression for its intensity has been obtained. The possible functional role of the noise as an activating factor in enzyme catalysis is discussed.