Motion, internal local

The motion of a protein on its PES can be described as anharmonic motions near local minima (i.e. conformations), with rare hops between conformations. While the system executes this motion, we can record, for example, the distance Q(t) between two residues. If the Fourier transform of Q(t) is relatively peaked, then the distance between these residues varied in time like a damped harmonic motion. The quantity Q(t) is not an oscillator with energy levels, that is embedded in the enzyme, rather it is an internal distance of, for example, residues that participate in the equilibrium fluctuations of the enzyme. 

The molecular motions responsible for the relaxation include (1) overall translational or rotational motions, (2) local motions such as internal rotations around C—C bonds or molecular axis of symmetry or segmental motions of polymer chains, and (5) exchange between two different distinct chemical sites within the same molecule or different molecules. 

To nnderstand the internal molecnlar motions, we have placed great store in classical mechanics to obtain a picture of the dynamics of the molecnle and to predict associated patterns that can be observed in quantum spectra. Of course, the classical picture is at best an imprecise image, becanse the molecnlar dynamics are intrinsically quantum mechanical. Nonetheless, the classical metaphor mnst surely possess a large kernel of truth. The classical stnichire brought out by the bifiircation analysis has accounted for real patterns seen in wavefimctions and also for patterns observed in spectra, snch as the existence of local mode doublets, and the... 

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

NMRrelaxation and diffusion experiments provide important insights into both the internal molecular dynamics and the overall hydrodynamic behavior of unfolded and partly folded states. Local variations in backbone dynamics are correlated with propensities for local compaction of the polypeptide chain that results in constriction of backbone motions (Eliezer et al., 1998, 2000). This can occur through formation of... 

The forces that exist within a fluid at any point may arise from various sources. These include gravity, or the weight of the fluid, an external driving force such as a pump or compressor, and the internal resistance to relative motion between fluid elements or inertial effects resulting from variation in local velocity and the mass of the fluid (e.g., the transport or rate of change of momentum). 

While the assumption of an isotropic rotational motion is reasonable for low molecular weight chelates, macromolecules have anisotropic rotation involving internal motions. In the Lipari-Szabo approach, two kinds of motion are assumed to affect relaxation a rapid, local motion, which lies in the extreme narrowing limit and a slower, global motion (86,87). Provided they are statistically independent and the global motion is isotropic, the reduced spectral density function can be written as ... 

The overall tumbling of a protein molecule in solution is the dominant source of NH-bond reorientations with respect to the laboratory frame, and hence is the major contribution to 15N relaxation. Adequate treatment of this motion and its separation from the local motion is therefore critical for accurate analysis of protein dynamics in solution [46]. This task is not trivial because (i) the overall and internal dynamics could be coupled (e. g. in the presence of significant segmental motion), and (ii) the anisotropy of the overall rotational diffusion, reflecting the shape of the molecule, which in general case deviates from a perfect sphere, significantly complicates the analysis. Here we assume that the overall and local motions are independent of each other, and thus we will focus on the effect of the rotational overall anisotropy. 

Yamakawa and co-workers have formulated a discrete helical wormlike chain model that is mechanically equivalent to that described above for twisting and bending/79111 117) However, their approach to determining the dynamics is very different. They do not utilize the mean local cylindrical symmetry to factorize the terms in r(t) into products of correlation functions for twisting, bending, and internal motions, as in Eq. (4.24). Instead, they... 

Figure 4.8. Calculated value of the rms amplitude or local polar libration (<5e2> /2) that satisfies Eq. (4.60) or Eq. (4.61) versus the assumed equilibrium polar angle (e0). The solid lines are the solutions of Eq. (4.60) for the indicated values of the reduced linear dichroism (LDr). The dashed lines are the solutions of Eq. (4.61) for the indicated values of A when the local angulaT motion of the transition dipole is assumed to be isotropic. The dotted lines are the solutions of Eq. (4.61) for the indicated values of A when the local angular motion of the transition dipole is assumed to be purely polar. The intersection of pairs of curves defines the region allowed" by a particular pair of LDr and A values and a particular assumption about the degree of anisotropy of the local angular motion of the transition dipole. If the LDr lies between -0.92 and -1.02, as indicated by experiment, then for isotropic internal motion, e0 = 70.5°, and 1/2 = 0.122 (7°) fall in the allowed region.

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