Theoretical Background

The analytical ultracentrifuge is used to perform two different types of experiments, referred to as sedimentation velocity and sedimentation equilibrium. Sedimentation velocity is a hydrodynamic technique and is sensitive to both the mass and shape of a macromolecule. It can be used qualitatively to

Figure 1 Sedimentation velocity of HIV-1 integrase catalytic core domain. Protein concentration 5.4 mg mL_1 in 20 mmol L_1 Tris, pH 7.5, 500 mmol L_1 NaCl, 1 mmol L_1 EDTA, 5 mmol L_1 DTT. Data obtained at 40C at a rotor speed of 50000 rpm. Radial absorption scans are recorded at 250 nm at 5 min intervals. The rate of movement of the boundary is determined by the sedimentation coefficient, s, and the spreading of the boundary reflects the diffusion constant, D.

(1 - vp )T,b where 2 is the partial specific volume of the solute, p is the density of the solvent and y is the viscosity of the solvent. The subscript 20,w refers to properties measured at 20°C in water, and subscript T,b refers to properties measured at temperature T in a buffer solution b.

The sedimentation coefficient is related to molecular properties according to the following equation:

Figure 1 Sedimentation velocity of HIV-1 integrase catalytic core domain. Protein concentration 5.4 mg mL_1 in 20 mmol L_1 Tris, pH 7.5, 500 mmol L_1 NaCl, 1 mmol L_1 EDTA, 5 mmol L_1 DTT. Data obtained at 40C at a rotor speed of 50000 rpm. Radial absorption scans are recorded at 250 nm at 5 min intervals. The rate of movement of the boundary is determined by the sedimentation coefficient, s, and the spreading of the boundary reflects the diffusion constant, D.

characterize sample homogeneity and quantitatively to define mass and shape parameters of the molecular species present in a sample. The experiments are based on simple physical principles. Application of a strong centrifugal field (high rotational velocity) leads to the net movement of solute molecules away from the air-solvent interface (the meniscus) and towards the bottom of the cell, giving rise to a moving boundary (Figure 1). Radial scans are recorded at regular time intervals, and the data are analysed to determine both the rate of movement and broadening of the boundary as a function of time. For a homogeneous sample, a single boundary forms; for mixtures, either a single or multiple boundaries may be resolved. In quantitative terms, the rate of sedimentation of a macromolecule, v = dr/dt, is proportional to the force w2r, where r is the radial distance from the centre of rotation, t is time, w is the rotational velocity. The ratio v/w2r is defined as the sedimentation coefficient, s. The sedimentation coefficient has the units of time, and is expressed in Svedberg (S) units (1S = 10~13s). The sedimentation coefficient may depend on concentration so it is customary to extrapolate s to zero concentration, to give s0. In addition, to allow comparison of sedimentation coefficients obtained in different solvents and at different temperatures, s0 is usually corrected to standard conditions (pure water at 20°C) using the following equation:

where M is the molecular mass, f is the frictional coefficient (which is related to macromolecular shape and size), and N0 is Avogadro's number. The solvent parameters p and y are experimentally measurable or can be calculated from the solvent composition using tabulated data. For proteins, 2 can be calculated with reasonable accuracy from the amino acid composition. Any further interpretation of the sedimentation coefficient requires an independent way to measure either M or f. Fortunately, the frictional coefficient is available from the sedimentation velocity data itself. During a veloity run the boundary not only moves towards the cell bottom but also becomes broader due to diffusion. Thus, in addition to measurement of the sedimentation coefficient, s, sedimentation velocity data can also be analysed to obtain the diffusion constant, D. According to the Einstein relationship, the diffusion constant is inversely proportional to the frictional coefficient:

where k is the Boltzmann constant and T is the absolute temperature. Combining eqns [3] and [4] one obtains the Svedberg equation:

where R is the gas constant. Thus, measurement of both s and D for a homogeneous sample in a sedimentation velocity experiment provides an independent method of obtaining the molecular mass. Given the mass, the frictional coefficient contains information about the shape and hydration of the molecule. Traditionally, frictional properties have been interpreted by modelling a macromolecule as a hydrated ellipsoid. However, more detailed, structure-based hydrodynamic calculations of frictional properties can now be readily performed using bead models.

In contrast, to sedimentation velocity, sedimentation equilibrium is a thermodynamic technique that is

Figure 2 Sedimentation equilibrium of a 13 base pair DNA sequence. Continuous line, single-stranded; dotted line, double-stranded. 10mmolL~1 Tris, pH 7.5, 50mmolL~1 NaCl, 15 mmol L~1 KCl, 0.1 mmol L~1 EDTA, 2 mmol L~1 Spermidine. Data obtained at 4°C at a rotor speed of 40000 rpm. Radial absorption scans recorded at 260 nm. The molecular mass of the duplex DNA is twice that of the single-stranded form.

Figure 2 Sedimentation equilibrium of a 13 base pair DNA sequence. Continuous line, single-stranded; dotted line, double-stranded. 10mmolL~1 Tris, pH 7.5, 50mmolL~1 NaCl, 15 mmol L~1 KCl, 0.1 mmol L~1 EDTA, 2 mmol L~1 Spermidine. Data obtained at 4°C at a rotor speed of 40000 rpm. Radial absorption scans recorded at 260 nm. The molecular mass of the duplex DNA is twice that of the single-stranded form.

sensitive to the mass but not the size or shape of a macromolecule. Equilibrium sedimentation is a rigorous and very accurate method of determining the molecule mass and association state of macro-molecules. It is also one of the best methods of defining reversible interactions of macromolecules in solution. Sedimentation equilibrium is performed at lower rotor speeds than sedimentation velocity experiments. When the centrifugal force is sufficiently small, the process of diffusion significantly opposes the process of sedimentation and a stable, smooth, equilibrium concentration distribution of macromolecules will eventually be obtained throughout the cell (Figure 2). For an ideal, homogeneous macromolecule, the radial equilibrium distribution is a simple exponential function of the buoyant mass of the macromolecule, M(1 — vp):

where c(r) is the sample concentration at radial position r and c2 is the sample concentration at an arbitrary reference radial distance r2. Deviations from the simple exponential behaviour described by eqn [5] can result from the presence in the sample of either multiple noninteracting or interacting macro-melecular species, or thermodynamic nonideality. For heterogeneous, polymeric systems, various molecular weight averages (Mn, Mw and Mz) are obtained by appropriate transformations of the data and are used to assess polydispersity and self-association behaviour. In the context of protein biochemistry, the data are usually analysed in terms of discrete oligomeric species, and equilibrium AUC is an excellent method to determine the native association state of proteins. In the case where discrete oligomeric species are in reversible equilibrium, the stoichiometries (N), equilibrium constants (Keq) and even the thermodynamic parameters (AH, AS) that define the interactions can be obtained using appropriate data analysis methods.

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Solar Panel Basics

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