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Department of Structural Biology, Weizmann Institute of Science, Rehovot 76100, Israel
Reprint requests to: Henryk Eisenberg, Department of Structural Biology, Weizmann Institute of Science, Rehovot 76100, Israel; e-mail: henryk.eisenberg{at}weizmann.ac.il; fax: 972-8-934 4136
Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.0235803.
After we published a review on modern analytical ultracentrifugation by Jacob Lebowitz, Marc S. Lewis, and Peter Schuck(
Protein Science 11: 2067–2079 [2002]
—Mark Hermodson
Editor–In–Chief
Protein Science
Abstract
The reasons for replacing the classical strictly two-component velocity and equilibrium analytical ultracentrifugation Svedberg equations by multicomponent equations, applicable in the study of biological macromolecular systems, are given.
Keywords: Analytical ultracentrifugation; Svedberg equations; multicomponent systems; thermodynamic equations; protein and nucleic acid solutions; detergents
A recent "tutorial" review (Lebowitz et al. 2002) emphasizes the new role of analytical ultracentrifugation, reborn by the development of novel instrumental design and the advanced application of computer analysis. A weak point, not affecting the renewed use of analytical ultracentrifugation, is the fact that current authors in this topical field of study essentially refer to the classical Svedberg analysis, which was the basis for the development of the technology, and has been known for a long time to apply strictly to two-component systems only. This is correct for synthetic no-charge-carrying organic polymers in single organic solvents; however, it is often not applicable to charge-carrying proteins or nucleic acids in aqueous salt, sugar, detergent, or denaturant solutions. For the development and use of the thermodynamic theory applicable to multicomponent systems, the reader is referred to the following papers: Casassa and Eisenberg (1964), Fujita (1975, Fujita 1994), Eisenberg (1976, 1994, 1999, 2000, 2003), and Ebel et al. (2000), where references to work by additional workers will be found. Consideration of the multicomponent nature of biological macromolecular solutions is essential in approaching this problem and evaluating errors and loss of information connected with its neglect.
To keep matters simple in this presentation, a three-component system containing a cosolvent component 3 only in addition to the solvent component 1 and the macromolecular component 2 will be considered. This does not limit extension to any number of low-molecular or macromolecular components. For finite macromolecular concentrations, a powerful connection exists between the equilibrium sedimentation differential equation and the osmotic pressure concentration derivative (d
/dc2)
 | (1) |
is the angular velocity of the ultracentrifuge, c2 is in grams per milliliter, and the density increment (
/c2)µ is at constant chemical potentials µ of solutes diffusible through a semipermeable membrane and measurable at low concentrations by the Kratky (Kratky et al. 1973) Paar DMA 5000 densitometer accurate to six significant figures. A 1-mL dialyzed reusable solution at c2 
1 mg/mL is required.
The osmotic pressure concentration derivative (d/dc2) can be expanded in a virial series, yielding the molar mass in grams per mole at vanishing macromolecular concentrations,
 | (2) |
In the limit of vanishing concentration c2 equation 1
reduces to
 | (3) |
In examining the dimensionality of equation 3, the ratio M2/c2 is in units of milliliters per mole and M2 will be given in whatever units are used for the concentration (Eisenberg 1994, 2000, 2003).
The ratio of the velocity sedimentation and diffusion coefficients s and D in the limit of vanishing component 2 concentration is given in similar fashion by
 | (4) |
The density increment (/c2)µ for the three-component system is given by
 | (5) |
2 and 3 are the partial specific volumes of components 2 and 3, 
3 = (w3/w2)µ is an interaction parameter indicating the change in gram molality w3 with the change in gram molality w2 at constant chemical potentials of components 1 and 3 diffusible through a semipermeable membrane, and is the density of the solvent in the absence of component 2. Protein volumes and hydration are well defined (Ebel et al. 2000; Eisenberg 2000).
For symmetry reasons equation 5 can also be written as
 | (6) |
1 = (w1/w2)µ and 1 and 3 are related by
 | (7) |
The interaction coefficients 1 and 3 cannot be associated with specific interaction with only either component 1 or 3, but should each be considered as relating to both components 1 and 3. A zero value of either 1 or 3 does not indicate lack of interaction with component 1 or 3. Furthermore a positive value of 1 yields a negative value of 3 or vice versa by equation 7.
The density increment (/c2)µ is substituted sometimes by (1 - 
'), mimicking the two-component buoyancy equation (Eisenberg 1976). This should be avoided as ' has no defined meaning and should therefore not be used (Eisenberg 2003).
It is now seen that in the absence of component 3 (/c2)µ in equations 5 and 6 reduces to (1 - 2), and equations 3 and 4 reduce to the classical two-component Svedberg equations at vanishing macromolecular concentrations. Neglect of the consideration of the multicomponent nature of protein and nucleic acids solutions leads to errors of 9% for DNA/NaCl (1 mole/L) solutions, 0.4% for BSA/NaCl (1 mole/L) solutions, 16.7% for BSA/GdnCl (4 moles/L) solutions, and 4.9% for aldolase/sucrose (0.399 moles/L) solutions in molecular weight determinations (Table 1 in Eisenberg 2000). Furthermore, additionally important information relating to solvent interactions, hydration, cosolvent interactions, and fractal probing of the macromolecular particles are not considered in this neglect (Eisenberg 1994, 2003). Analytical ultracentrifugation powerfully connects in a broad complementary sense to light-, X-ray-, and neutron-scattering experiments (Eisenberg 1999), providing additional structure, shape, and interaction information.
There are no unusual difficulties or material requirements in the determination of the density increments (/c2)µ in most systems.
In a newly proposed, simplified experimental approach requiring less material (Courtenay et al. 2000; Anderson et al. 2002), water vapor pressure osmometry in an attractive procedure replaces equilibrium dialysis density increment measurements, comparing with electromotive-force determination and isopiestic distillation (Eisenberg 1976). Interaction parameters are obtained that, combined with partial specific volumes and density data, enable the calculation of the density increments required in analytical ultracentrifugation. Additional work is required to establish the significance of this new procedure.
References
Anderson, C.F., Courtenay, E.S., and Record Jr., M.T. 2002. Thermodynamic expressions relating different types of preferential interaction coefficients in solutions containing two solute components. J. Phys. Chem. B 106: 418–433.[CrossRef]
Casassa, E.F. and Eisenberg, H. 1964. Thermodynamic analysis of multicomponent systems. Adv. Protein Chem. 19: 287–395.[Medline]
Courtenay, E.S., Capp, M.W., Anderson, C.F., and Record Jr., M.T. 2000. Vapor pressure osmometry studies of osmolyte–protein interactions: Implications for the action of osmoprotectants in vivo and for the interpretation of "osmotic stress" experiments in vitro. Biochemistry 39: 4455–4471.
Ebel, C., Eisenberg, H., and Ghirlando, R. 2000. Probing protein–sugar interactions. Biophys. J. 78: 385–393.
Eisenberg, H. 1976. Biological macromolecules and polyelectrolytes in solution. Clarendon Press, Oxford.
———. 1994. Protein and nucleic acid hydration and cosolvent interactions: Establishment of reliable baseline values at high cosolvent concentrations. Biophys. Chem. 53: 57–68.[CrossRef][Medline]
———. 1999. Ultracentrifugation, light, x-ray and neutron scattering of peptides, proteins and nucleic acids in solution. In Peptide and drug analysis (ed. R.E. Reid), pp. 825–859. Marcel Dekker, New York.
———. 2000. Analytical ultracentrifugation in a Gibbsian perspective. Biophys. Chem. 88: 1–9.[CrossRef][Medline]
———. 2003. Adair was right in his time. European Biophys. J. 32: 406–411.
Fujita, H. 1975. Foundations of ultracentrifugation analysis. Wiley, New York.
———. 1994. Notes on the derivation of sedimentation equilibrium equations. In Modern analytical ultracentrifugation (eds. T.M. Schuster and T.M. Laue), pp. 3–14. Birkhäuser, Boston.
Kratky, O., Leopold, H., and Stabinger, H. 1973. The determination of the partial specific volume of proteins by the mechanical oscillator technique. Methods Enzymol. 27: 98–110.[Medline]
Lebowitz, J., Lewis, M.S., and Schuck, P. 2002. Modern analytical ultracentrifugation in protein science: A tutorial review. Protein Sci. 11: 2067–2079.
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