Rotary diffusion constant

Ellipsoidal or rod-shaped molecules have two different rotary diffusion constants while, if the dimensions of the molecules are different along all three axes, three constants must be specified.36... 

Rocky Mountain spotted fever 7 Rods (visual receptor cells) 390 Root hairs, dimensions of 30 Roseoflavin 788, 789s Rossmann fold. See Nucleotide-binding domain Rotamases 488 Rotary diffusion constant 463 Rotation of molecules 462,463 Rotational barrier 44 Rotifers 24, 25... 

Rotary Diffusion Constants and Relaxation Times Their Significance for... 

Measurements of dielectric constant as a function of frequency. These dielectric dispersion measurements permit the estimation of the relaxation times or rotary diffusion constants which characterize the rotary Brownian movement of the protein molecule. 

Double refraction of flow measurements. These also give a measurement of rotary diffusion constant and are especially adapted to the study of highly asymmetrical molecules. 

In what follows, we shall consider rather briefly the evidence from sedimentation and diffusion constants, from viscosity measurements, and from X-ray diffraction, while the determination of rotary diffusion constants and relaxation times from the measurements of dielectric dispersion and double refraction of flow will be presented in more detail. 

Here, F is the volume fraction of the solute. The Einstein coefficient V is 2.5 for spheres and is always larger for molecules vhich deviate from the spherical shape, v is of course a function of the parameter a. It approaches a lower limit at very high velocity gradient and an upper limit v, at zero velocity gradient. Most relatively small proteins with absolute dimensions less than 500 or 600 A in any direction have sufficiently high rotary diffusion constants so that the measured value of V may be taken as equal to for the velocity gradient in almost any capillary viscometer. The rest of this discussion of viscosity will be... 

For many purposes, it is more convenient to characterize the rotary Brownian movement by another quantity, the relaxation time t. We may imagine the molecules oriented by an external force so that the a axes are all parallel to the x axis (which is fixed in space). If this force is suddenly removed, the Brownian movement leads to their disorientation. The position of any molecule after an interval of time may be characterized by the cosine of the angle between its a axis and the x axis. (The molecule is now considered to be free to turn in any direction in space —its motion is not confined to a single plane, but instead may have components about both the b and c axes.) When the mean value of cosine for the entire system of molecules has fallen to ile(e — 2.718... is the base of natural logarithmus), the elapsed time is defined as the relaxation time r, for motion of the a axis. The relaxation time is greater, the greater the resistance of the medium to rotation of the molecule about this axis, and it is found that a simple reciprocal relation exists between the three relaxation times, Tj, for rotation of each of the axes, and the corresponding rotary diffusion constants defined in equation (i[Pg.138]

Generalizing the hydrod3mamical equations derived by Stokes for spheres, Edwardes (35) calculated the coefficients fj, Ca and for ellipsoids as a function of their axial ratios. The general equations are complicated but for ellipsoids of revolution, which may be characterized by only two values of f, they assume a simpler form, and have been employed by Gans 47) and F. Perrin 92) to evaluate the rotary diffusion constants of molecules which may be treated as ellipsoids of revolution. The formulas of Gans and Perrin are not identical, but the numerical values of 0 calculated from them are nearly so, so that the formulas of either author may be used in practice. In the following discussion we shall employ Perrin s equations. 

The rotary diffusion constant o and relaxation time of a sphere of the same volume would be given by the relations... 

Consider first the case of an elongated ellipsoid of revolution a >b). Rotary Brownian movement of the a axis about the b axis is characterized by the relaxation time and the corresponding rotary diffusion constant 0 = 1/2t [see Equation (19a)]. These constants are conveniently expressed by their values relative to those for a sphere of the same volume. Denoting by q the ratio bja, Perrin s equation reads... 

Thus, for a given value of ajb, the rotary diffusion constant of an elongated ellipsoid is inversely proportional to the cube of its length. 

Thus for a flattened ellipsoid, as bja becomes infinite, both relaxation times become infinite, and the corresponding rotary diffusion constants approach zero. When bja is very large, the relaxation times are proportional to the cube of the b semi-axis, and do not depend at all on the length of the a semi-axis. 

These phenomena can be interpreted in terms of molecular orientation by the velocity gradient in the flowing liquid, opposed by the rotary Brownian movement which produces disorientation and a tendency toward a purely random distribution. The intensity of this Brownian movement is charaterized by the rotary diffusion constants, 0, discussed in the preceding section. The fundamental treatment of this problem, for very thin rod-shaped particles, was given by Boeder (5) the treatment has been generalized, and extended to rigid ellipsoids of revolution of any axial ratio, by Peterlin and STUARTi 56), [98), (99) and by Snell-MAN and Bj5knstAhl (J9J). The main features of their treatment are as follows 1 ... 

If the rotating cylinder could be brought suddenly and smoothly to rest, the second term on the right would vanish, and the resulting equation would describe the process of disorientation, by free rotary diffusion, of the molecules which had previously been partly oriented by the flow. The speed of disorientation is proportional to the rotary diffusion constant, 0, and the final steady state of random distribution corresponds to the equation q = const. = Nj4 n. 

If the values of co and co from (33) and (34) are substituted in (32), a differential equation for the function q is obtained, for which a general solution by expansion in an infinite series has been obtained by Peterlin 96), 98), 99). The solution is best expressed with the aid of the parameter a = GjB, the ratio of velocity gradient to rotary diffusion constant. The general solution is very complex, and the terms converge slowly but for low values of [Pg.146]

The possibility of making use of dielectric measurements for the study of the relaxation times is largely dependent on the very polar nature of amino acids, peptides and proteins. We must therefore discuss briefly the relation between dielectric constant and dipole moment in polar liquids, the discussion being for the moment restricted to static fields, or fields of frequency small compared to the rotary diffusion constants of the molecules. 

The dielectric di persion of DNA solutions was measured with various samples. The dielectric increment and the relaxation time of helical DNA are proportional to the square of the length of the molecule, hut values for coil DNA are distinctly smaller than for helical DNA. The rotary diffusion constant is measured simultaneously with the dielectric measurement. The agreement of both relaxation times is fair in a region of low molecular weight, hut the disparity becomes pronounced when DNA is larger. Theories on the mechanism of ionic electric polarization are reviewed. Currently, counter ion polarization for a cylindrical model seems to account most reasonably for the dielectric relaxation of DNA. 

Parameter a should not be confused with the Cole-Cole parameter.) The value of a is tabulated by Edsall et al. (19) for various axial ratios. The rotary diffusion constant is related to a by the following formula,... 

Normal stress measurements on concentrated solutions of helical polypeptides were conducted by lizuka [1,42]. However he used these to calculate extinction angles, from which the rotary diffusion constant was deduced, and thence an apparent particle size from tables given by Scheraga [43]. In a personal communication to Kiss and Porter, lizuka commented that he had observed negative normal stresses in solutions of PBLG + Ch Br with concentrations of greater than 10% (i.e. probably liquid crystalline) however he ascribed this to the adhesive force of the solution (E. lizuka, personal communication, April 1977). 

Many polymer scientists have used homogenous analysis for the determination of molecular size via limiting viscosity numbers and translational and rotary diffusion constants but few recognize that these techniques and investigations of birefringence of polymer solutions were pioneered by Charles Sadron. 

Btoetsma, S. Rotary diffusion constant of a cylindrical particle, J. Chem. Phys. 32,1626—31 (1960). 

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