Ultracentrifuge equation

In a two-component system of an electrostatically neutral solute and a solvent in a centrifuge cell at a distance r from the center of rotation of the cell, Eisenberg (1976) showed that the solution of Equation (7) yields the well-known equilibrium ultracentrifugation equation ... 

Combination of Equations (9-88)-(9-91) gives what is called the Lamm differential ultracentrifuge equation ... 

The evolution with time of the concentration profile x (s> D, r, t) in a sector shaped ultracentrifuge cell is given by the Lamm [45] equation ... 

Equations suitable for simulation of molecular weight distributions for any initial distribution and chosen values of G(scission) and G(crosslinking) have been developed and demonstrated. The molecular weight distributions may be obtained by GPC (with the limitation of changes in relative hydrodynamic volumes) and by sedimentation velocity in the ultracentrifuge. 

Most probable settling velocity from sedimentation data Particle-size determination from sedimentation equation Sedimentation in an ultracentrifuge Solvation and ellipticity from sedimentation data Diffusion and Gaussian distribution Temperature-dependence of diffusion coefficients... 

The method of equilibrium ultracentrifugation described by Yphantis (28) was used in these studies. A double-sector centerpiece with two 2 1/2° sectors 12 mm. in length was used with standard Schlieren optics. Apparent molecular weights, M, were determined from the equation ... 

It would be of considerable interest to extend the technique just presented to problems involving nonlinear equations because there are many situations in ultracentrifugation where nonideality is a dominant feature. Furthermore, it is known (4, 14) that even for two-component systems with nonideality the theory for estimating the sedimentation constant based on a diffusion-free (c = 0) approximation can lead to systematic error. Therefore, the development of an approximate procedure for nonlinear equations would be useful for further progress in analytical separation methods. 

Basic Sedimentation Equilibrium Equation. Sedimentation equilibrium experiments are performed at constant temperature. The condition for sedimentation equilibrium is that the total molar potential, m, for all components i be constant everywhere in the solution column of the ultracentrifuge cell. Mathematically this can be expressed as... 

So far the equations that have been developed are independent of the shape of the cell. In order to make Equation 17 more useful, it is necessary to relate Ci( — 0) to the initial concentration of component i, Coi. To do this one applies conservation of mass, which states that in a closed system (the solution column in the ultracentrifuge cell) the total amount of solute is constant at all times. Since mass equals concentration times volume, it is necessary to know the volume of the centerpiece, and this depends on the shape of the centerpiece. Two types of centerpieces are currently used. One is a sector-shaped centerpiece, and the other is a six-channel, equilibrium [or Yphantis (24)] centerpiece. Top views of both centerpieces are shown in Figure 1. In either centerpiece, one side is reserved for solvent and the other for solution. 

Here BLS is the second virial coefficient of the polymeric solute in the original solution before ultracentrifugation. BLs is a quantity which can be obtained in light-scattering experiments (17, 25, 30) or in Archibald experiments (31), provided it is calculated from a plot of l/MWapp° vs. c. Here 1/Mw pp° is obtained from values of Mw pp (at rm or rb) that have been extrapolated to zero time. The reason for using Equation 75 is that it leads to a simple method of estimating the MWD in nonideal solutions. 

In obtaining Equation 11 it has been assumed that the partial specific volumes v of the associating species are equal we have also assumed that the specific refractive index increments of the associating solutes are equal. In Equation 12 R is the universal gas constant (8.314 X 107 ergs/deg-mole), p is the density of the solution (gram/ml), and T is the absolute temperature. Equation 11 is also valid for the Archibald experiment but only at rm or rb, the radial positions (in the solution column of the ultracentrifuge cell) of the air-solution meniscus and of the cell... 

From Rayleigh and schlieren optics, one could calculate cMwa, and hence one could calculate A (cMwa) at every r in the solution column of the ultracentrifuge cell and use the analysis used with the other methods (see Equation 76 and the discussion following it). This could also be done even if only one of the reactants had an absorption spectrum, since c = cA° + cB°. The other possibility is that the complex AnBm has a different absorption spectrum from the reactants, or that in the formation of AnBw some chromophoric groups are buried so that the absorbance of A and B represents a quantity proportional to cA and cB. In either case one could then calculate... 

Ultracentrifugation techniques (Kabanov et al., 1995) may also provide the molecularM)ass ( of the polymeric micelles through the Svedberg equation ... 

The expansion behavior of carboxylic latex particles can be studied by several methods (10). The present comparison was made using a sedimentation method which involved the measurement of particle sedimentation rates in an ultracentrifuge at various degrees of neutralization. Assuming the change in particle volume is equal to the volume of water absorbed, an expanded particle settles slower, as its density decreases, according to the equation ... 

Schuck, P. (2000). Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and Lamm equation modeling. Biophys.J. 78(3), 1606-1619. 

At high speeds greater than 40,000 rpm in the ultracentrifuge, macromolecules settle towards the rotor periphery. Under these conditions the sedimentation coefficient s, is determined from the speed of sedimentation divided by the angular acceleration. The sedimentation coefficient is related to molecular weight using Equation (4.15). 

S-FFF has been compared with analytical ultracentrifugation (AUC) with respect to the fractionation of a 10-component latex standard mixture with narrow particle size distribution, known diameters (67-1220 nm) and concentration [ 127]. With an analytical ultracentrifuge, the particle sizes as well as their quantities could be accurately determined in a single experiment whereas in S-FFF deviations from the ideal retention behavior were found for particles >500 nm resulting in smaller particle size determination in the normal as well as in the programmed operation. It was concluded that, without a modified retention equation which accounts for hydrodynamic lift forces and steric exclusion effects, S-FFF cannot successfully be used for the size characterization of samples in that size range. 

Sedimentation equilibrium analytical ultracentrifugation of isolated ZDD was used to determine its solution molecular weight. These experiments revealed the presence of a single species in solution corresponding to the molecular mass of the dimeric ZDD fragment. Global analysis of the equilibrium data using equation 1... 

The molecular weight of a protein or other macromolecule can be obtained from ultracentrifugation studies. The principles involved are rather simple a high-molecular-weight molecule sediments faster and diffuses slower than a lower-molecular-weight molecule of the same density. This is expressed in the Svedberg equation ... 

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