Sedimentation equilibrium experiment

Apart from tliese mainstream metliods enabling one to gain a comprehensive and detailed stmctural picture of proteins, which may or may not be in tlieir native state, tliere is a wide variety of otlier metliods capable of yielding detailed infonnation on one particular stmctural aspect, or comprehensive but lower resolution infonnation while keeping tlie protein in its native environment. One of tlie earliest of such metliods, which has recently undergone a notable renaissance, is analytical ultracentrifugation [24], which can yield infonnation on molecular mass and hence subunit composition and their association/dissociation equilibria (via sedimentation equilibrium experiments), and on molecular shape (via sedimentation velocity experiments), albeit only at solution concentrations of at least a few tentlis of a gram per litre. 

The centrifugal field in the sedimentation equilibrium experiment is the analog of the membrane in an osmometer. 

Alternatively, in sedimentation equilibrium experiments, the ultracentrifuge is operated at slow speeds for longer time to permit the solute molecules to attain equilibrium between sedimentation and diffusion. If the sample is monodisperse, its concentrations cb c2 can be measured at 2 positions xh x2 in the cell. 

SEC-MALLS and SEC-LALLS Coupled chromatography and light-scattering photometry that allows the determination of a number of important values along with chain length distribution, sedimentation equilibrium experiment Ultracentrifugation technique that allows chain length information to be determined. 

Almost simultaneously with the first attempt to determine molecular weight from equilibrium sedimentation, Rinde tried to widen this method to include determination of the molecular weight distribution (MWD) of a polydisperse system (3). Unfortunately, this attempt proved to be more complicated and did not result in establishment of a reliable routine. Since the appearance of Rinde s dissertation in 1928, many investigators have tried to determine MWD. Most of these efforts, however, did not provide a successful comprehensive technique (4-16). This objective has been accomplished only in a few cases under very limited conditions, such as in case of a Gaussian or near Gaussian MWD, in which only characterizing parameters had to be determined. Scholte (17, 18) determined MWD by performing an experimental procedure based on several equilibrium experiments. 

Sedimentation velocity experiments were performed using a 12-mm Kel-F double-sector capillary synthetic boundary cell. Each equilibrium experiment was performed with aid of two separate 12-mm, 4° singlesector aluminum cells, one containing solvent and the other solution. In all cases the schlieren angle was 65°. 

Molecular Weight Distributions from Sedimentation Equilibrium Experiments... 

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... 

Method 2. Follow the procedure of Albright and Williams (25) and do sedimentation equilibrium experiments on the same solution at different speeds. Then calculate Mw app(cell) from... 

The analysis of mixed associations by light scattering and sedimentation equilibrium experiments has been restricted so far to ideal, dilute solutions. Also it has been necessary to assume that the refractive index increments as well as the partial specific volumes of the associating species are equal. These two restrictions are removed in this study. Using some simple assumptions, methods are reported for the analysis of ideal or nonideal mixed associations by either experimental technique. The advantages and disadvantages of these two techniques for studying mixed associations are discussed. The application of these methods to various types of mixed associations is presented. 

Evaluation of Mweq. Sedimentation Equilibrium Experiments and the Archibald Method. At constant temperature the condition for sedimentation equilibrium is that the total potential jZi for each associating species i (i = A, B, or AnBw) be constant at every radial position r in the solution column of the ultracentrifuge thus... 

The quantity cMweq can also be evaluated from sedimentation equilibrium experiments done at different speeds on the same solution. Here one gets to sedimentation equilibrium with a given solution at one speed the necessary information (concentration and/or concentration gradient) is recorded. Then the speed is changed once sedimentation equilibrium is attained at the second speed, c and/or dc/dr are recorded. This procedure is repeated at two or more additional speeds. Then one calculates Mw (cell mass) or Mw (cell vol) from the following equations ... 

Archibald Experiments, Sedimentation Equilibrium Experiments at Different Speeds, and Light Scattering Experiments The analysis of a mixed association described by Equation 1 is similar for these three... 

With sedimentation equilibrium experiments at different speeds one can also use... 

Analysis of Mixed Associations from Conventional Sedimentation Equilibrium Experiments. In these experiments one measures a quantity Mieq (14, 28) instead of Mweq. The basic sedimentation equilibrium equation for each reactant is... 

Analysis from Archibald Experiments or from Sedimentation Equilibrium Experiments at Different Speeds. This discussion here is illustrated by the association described by Equation 1. When vA vB and 7 b, analogs of Equations 17 and 18 will yield Mieq when extrapolated to zero time. Similarly, Mieq will also be obtained from the analog of Equation 13 when values of Mi, are extrapolated to zero time. For this situation... 

The analysis of nonideal, mixed associations from conventional sedimentation equilibrium experiments is a very difficult matter, and at present seems to be an impasse. These difficulties arise because the redistribution of the reactants is combined with nonideal behavior. We can illustrate the difficulties with the following example. The equation for reactant A can be written as... 

A similar equation results for cBrm. Now a priori we do not know (d In Ja/dcB), cAr, cBr, or K we do not know BAa and Bbb (see Equations 67 and 68) from measurements on pure A and pure B. We have no way of knowing ir a priori, so we cannot use equations similar to 28 or 31. Similar considerations apply if vA vB and = b- Thus, it appears at present that we are forced to use the Archibald experiment or sedimentation equilibrium experiments at different speeds to analyze mixed associations when one is restricted to Rayleigh and/or schlieren optics. 

It would then be possible to obtain K as well as < i and 2. The dye toluidene blue has been reported to change color when it is bound to heparin (36) undoubtedly other examples are known. Here the sedimentation equilibrium experiment could be used to study dye binding such studies have been reported by Steinberg and Schachman (37). 

Analysis by the Archibald Method or by Sedimentation Equilibrium Experiments at Different Speeds. Instead of using Mweq here, one uses Mwa, the apparent weight-average molecular weight. For the Archibald experiment one obtains Mwa,t at rm or r6 by the application of Equations 13-16. The extrapolation of Mwa>t to zero time gives Mwo. For sedimentation equilibrium experiments at different speeds, one can evaluate Mwa by two different methods here one uses either Equations 17 or 18. For a mixed association such as A + B AB, the basic sedimentation equilibrium equation can be written as... 

For the sedimentation equilibrium experiments at different speeds one could use A(cMwMz)app, the nonideal analog of A(cMwMz) or A(Sc Mi2),... 

We have shown some of the ways that mixed associations could be analyzed by light scattering, Archibald, or sedimentation equilibrium experiments. It might be appropriate at this point to make a comparison of some advantages and disadvantages of each technique. 

The sedimentation equilibrium experiment requires much smaller volumes of solution, about 0.15 ml. With six-hole rotors and multichannel centerpieces (41) it is potentially possible to do fifteen experiments at the same time. For situations where the photoelectric scanner can be used one might (depending on the extinct coefficients) be able to go to much lower concentrations. Dust is no problem since the centrifugal field causes it to go to the cell bottom. For conventional sedimentation equilibrium experiments, the analysis of mixed associations under nonideal conditions may be virtually impossible. Also, sedimentation equilibrium experiments take time, although methods are available to reduce this somewhat (42, 43). For certain situations the combination of optical systems available to the ultracentrifuge may allow for the most precise analysis of a mixed association. The Archibald experiment may suffer some loss in precision since one must extrapolate the data to the cell extremes (rm and r6) to obtain MW(M, which must then be extrapolated to zero time. Nevertheless, all three methods indicate that it is quite possible to study mixed associations. We have indicated some approaches that could be used to overcome problems of nonideality, unequal refractive index increments, and unequal partial specific volumes. 

In this paper we have used the quantity (1 — vp0) in writing equations for sedimentation equilibrium experiments. Some workers prefer to use the density increment, 1000(dp/dc)Tfn, instead when dealing with solutions containing ionizing macromolecules. This procedure was first advocated by Vrij (44), and its advantages are discussed by Casassa and Eisenberg (39). Nichol and Ogston (13) have used the density increment in their analysis of mixed associations. The subscript p. means that all of the diffusible solutes are at constant chemical potential in the buffer... 

Received January 17, 1972. Work supported by grants GM 15551 and GM 17611, National Institutes of Health. It is based on the MS theses submitted by Allen H. Pekar (nonideal, sedimentation equilibrium experiments and Archibald method) and by Peter J. Wan (light scattering) to the Graduate School of the Illinois Institute of Technology. 

Indirect methods for obtaining information on the kinetics of the associa-tion/dissociation equilibrium include sedimentation velocity and GPC experiments. The application of these techniques is based on comparison of sedimentation or GPC elution curves with model curves based on theories for separation of unimers and micelles during a sedimentation velocity (Gilbert 1955) or GPC (Ackers and Thompson 1965 Coll 1971 Prochazka et at. 1988, 1989) experiment. Experiments have been performed that demonstrate several of the qualitative model predictions (Prochazka et at. 1989). The main conclusions were that GPC curves with two well-separated peaks can only result from a slow dynamic molecule micelle equilibrium, and that no simple interpretation of elution curves in terms of relative concentrations of unimer and micelles is possible (Prochazka et at. 1989). Thus no quantitative information on the kinetics of the molecule micelle equilibrium can be obtained from sedimentation velocity or GPC data. 

Information on polydispersity of PDADMAC samples synthesized under different conditions could be obtained further by the Mz/Mw -ratio from low-speed sedimentation equilibrium experiments in 0.5 m NaCl [35]. 

From a theoretical analysis of the intermediate stages during which the solute is being redistributed, Archibald136 has developed a technique which involves measurements at intervals during the early stages of the sedimentation equilibrium experiment and so does not entail a long wait for equilibrium to be established. The ratio s/D can be calculated from the expression... 

The usual method, then, is to perform a sedimentation equilibrium experiment, in which a solution containing a solute of known behavior is spun at a much lower speed until the concentration gradient remains invariant with time. The distribution of solute for an ideal, two-component system should correspond to the equation, where c is the concentration of solute with... 

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