Effective hydrodynamic ellipsoid

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

The value of P is a measure of the effective hydrodynamic ellipsoid (table 4). 

In the hydrodynamic theories for any of the quantities in the last column of Table I, each quantity depends on two parameters, the size and the shape. For example, for an ellipsoid of revolution these two parameters could be the volume and axial ratio, F and p, respectively, of the effective hydrodynamic ellipsoid. Therefore, we may draw a first very important conclusion, viz., that a determination of only one quantity, e.g., /, cannot provide us with a value of Ve or p. It is clear that two quantities are required, e.g., f and [r ] if two such quantities are available, then both F, and p may be computed. Of course, if one has information in advance as to the value of F or p (e.g., if the particle behaves as a sphere, with p = 1), then a single hydrodynamic quantity suffices to give the value of the other parameter. Unfortunately, one never has this advance information and, therefore, must carry out two different kinds of hydrodynamic measurements in order to obtain a pair of hydrodynamic quantities. 

Again, fo refers to a sphere of the same volume as the effective hydrodynamic ellipsoid and 1/J is a function of p only (Perrin, 1934). 

In summary, the foregoing theory permits the determination of the dimensions a and b of the effective hydrodynamic ellipsoid without arbi-... 

Additional checks can be made by comparing the results of hydro-dynamic measurements with those from light scattering and electron microscopy. If data from the various methods agree, then there is an indication that the effective hydrodynamic ellipsoid does not differ too greatly from the actual molecule. However, in the absence of such a comparison, the use of hydrodynamic methods to obtain molecular size and shape involves the risk that the various factors disemssed here may lead to arti-... 

In summary, a pair of hydrodynamic quantities is required in order to obtain the dimensions of the effective hydrodynamic ellipsoid. While the relationship between this ellipsoid and the actual particle is not clear yet, it seems safe to assume that there is an intimate connection between the two. Comparison of hydrodynamic data with those from light-scattering and other techniques may provide a clue as to how close this relationship is. These kinds of investigations have provided us with an idea of the compactness and shape of globular proteins in solution, and do provide us with a basis for setting up models for studying the internal configura-... 

For the native albumin the axes of the effective ellipsoid are listed. All other values in this column are the radii of the respective hydrodynamic spheres. 

The compaction behavior is also indicated by the effective volume fraction of an aggregate (pejf (Fig- 5), calculated from the aspect ratio of an equivalent ellipsoid. The shear-rale dependence on effective volume fraction was small so that a maximum compaction was not observed even at high shear rates. Again, higher shear rates required for higher compaction may violate model assumptions, such as Stokes regime for the hydrodynamics and the conditions for the overdamped motion. 

If E is measured as volts per meter, then dx/dt is called the mobility of the ion for the chosen experimental conditions. The coefficient / has a theoretical foundation in hydrodynamics, and a functional relationship between / and the coordinates of the particle can be derived for a few regular shapes (for a sphere it is known as the Stokes equation, but there are mathematical solutions for ellipsoids and cylinders). It is also known that / is inversely proportional to the randomizing effects of diffusion of a large number of ions (Einstein-Sutherland relationship). Essentially, Eq. (6) describes the movement of a single ion under the influence of an electrical field. Rarely, if ever, can one ion be studied experimentally, because at finite concentrations of ions there are > 10 ions per liter (a 10 ° M solution of 0.1 /xg liter for an ion of relative mass 10 contains 10 ions per liter). Diffusion of this population of ions spreads the boundary about an elec-trophoretically transported point called the centroid, and it is the velocity of this point that is described by Eq. (6) for experimental situations. The centroid or first moment (x) can be evaluated from Eq. (7) using a set of rectangular coordinates determined experimentally over an elec-trophoresing boundary. 

One of the difficulties in theoretical studies of the hydrodynamic effects on vesicle dynamics is the no-slip boundary condition for the embedding fluid on the vesicle surface, which changes its shape dynamically under the effect of flow and curvature forces. In early studies, a fluid vesicle was therefore modeled as an ellipsoid with fixed shape [194]. This simplified model is still very useful as a reference for the interpretation of simulation results. 

The theory of Keller and Skalak [194] describes the hydrodynamic behavior of vesicles of fixed ellipsoidal shape in shear flow, with the viscosities t)in and qo of the internal and external fluids, respectively. Despite of the tqtproximations needed to derive the equation of motion for the inclinalion angle 6, which measures the deviation of the symmetry axis of the ellipsoid from the flow direction, this theory describes vesicles in flow surprisingly well. It has been generalized later [197] to describe the effects of a membrane viscosity Tjmb-... 

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