Box 9.1 (continued)
are centrifuged and the centrifugation stopped before the particles, which are denser than the
densest layer in the gradient, have reached the bottom of the tube (see Figs. 9.2, 9.3, 9.4, 9.5,
and 9.6). As particles migrate, they slow down, because the difference between their density and
that of the solution diminishes, while viscosity increases. This creates a stacking effect, opposing
the broadening effect of translational diffusion. The position reached at any given time is related to
the mass, the density, and the hydrodynamic radius (R H ) of the particles. Given proper abaci
(McEwen 1967), it is possible to derive the sedimentation coefficient s 20,w that the particle would
exhibit in pure water at 20
C, from which, knowing the specific volume
v (inverse of the density)
and R H , the molecular mass can be deduced (see below). The fractions can be collected and their
composition analyzed. SGSV is mostly used to examine the distribution and composition of
particles in samples and as a preparative method, rather than for quantitative determination of
solution properties.
• Equilibrium (isopycnic) sedimentation in sucrose gradients
Denser gradients, e.g. 20–40% w/w sucrose, are used to separate particles according to their
density, the centrifugation being carried out until the particles have reached that position in the
gradient where their density matches that of the solvent. The particles can be either sedimented or
floated. This approach is used, for instance, to separate one from another membrane fragments
whose density differs due to different protein/lipid ratios, such as bacterial inner and outer
membrane fragments. It will not be considered in the present chapter.
• Sedimentation velocity analytical ultracentrifugation (SV-AUC) in the absence of a
density gradient (§ 9.3.7; Fig. 9.12)
Alternatively, a homogeneous solution can be subjected to a gravity field. Those molecules
closest to the bottom of the tube will rapidly pellet, whereas those close to the top will progressively
sediment while diffusing. The distribution of the particles in the gradient is followed throughout the
experiment by measuring the absorbance of the solution and its refractive index at each point in the
buffer column as a function of time (see Fig. 9.12). Analysis of the distribution profiles yields
the sedimentation coefficient s and the translational diffusion coefficient D t of the particles, or
of each of the particles in a mixture, a situation in which this approach is particularly powerful
(cf. Figs. 9.12 and 9.14). Molecules with different absorbance spectra, such as protein and
surfactant or protein and nucleic acid, can be followed at different wavelengths.
For an ideal solution, the sedimentation coefficient of a particle, s, in Svedberg (1 S ¼ 10
À13 s),
is related to its buoyant molar mass, M b ; its Stokes radius, R S ; and Avogadro’s number, N A , by the
Svedberg equation:
s ¼ M b = 6πN A η 0 R S
ð
Þ
where η 0 is the viscosity of the solvent (in PaÁs
À1
). (Note: the Stokes radius or StokesEinstein radius of a solute, R S , is the radius of a hard sphere that diffuses at the same rate as the
solute. In biophysics, R S and the hydrodynamic radius, R H , tend to be used as synonyms of each
other.)
M b is the mass of the particle minus the mass of the solvent it displaces. It is related to the
particle molar mass, M, and its partial specific volume,
v (in mLÁg
À1
), by the following equation:
M b ¼ M
À
1 À ρ 0
v
Á
,
where ρ 0 is the density of the solvent.
• Equilibrium sedimentation analytical ultracentrifugation (Eq-AUC) in the absence of a
density gradient (§ 9.3.7; Fig. 9.13)
412
9 Solution Studies of Membrane Protein/Amphipol Complexes
are centrifuged and the centrifugation stopped before the particles, which are denser than the
densest layer in the gradient, have reached the bottom of the tube (see Figs. 9.2, 9.3, 9.4, 9.5,
and 9.6). As particles migrate, they slow down, because the difference between their density and
that of the solution diminishes, while viscosity increases. This creates a stacking effect, opposing
the broadening effect of translational diffusion. The position reached at any given time is related to
the mass, the density, and the hydrodynamic radius (R H ) of the particles. Given proper abaci
(McEwen 1967), it is possible to derive the sedimentation coefficient s 20,w that the particle would
exhibit in pure water at 20
C, from which, knowing the specific volume
v (inverse of the density)
and R H , the molecular mass can be deduced (see below). The fractions can be collected and their
composition analyzed. SGSV is mostly used to examine the distribution and composition of
particles in samples and as a preparative method, rather than for quantitative determination of
solution properties.
• Equilibrium (isopycnic) sedimentation in sucrose gradients
Denser gradients, e.g. 20–40% w/w sucrose, are used to separate particles according to their
density, the centrifugation being carried out until the particles have reached that position in the
gradient where their density matches that of the solvent. The particles can be either sedimented or
floated. This approach is used, for instance, to separate one from another membrane fragments
whose density differs due to different protein/lipid ratios, such as bacterial inner and outer
membrane fragments. It will not be considered in the present chapter.
• Sedimentation velocity analytical ultracentrifugation (SV-AUC) in the absence of a
density gradient (§ 9.3.7; Fig. 9.12)
Alternatively, a homogeneous solution can be subjected to a gravity field. Those molecules
closest to the bottom of the tube will rapidly pellet, whereas those close to the top will progressively
sediment while diffusing. The distribution of the particles in the gradient is followed throughout the
experiment by measuring the absorbance of the solution and its refractive index at each point in the
buffer column as a function of time (see Fig. 9.12). Analysis of the distribution profiles yields
the sedimentation coefficient s and the translational diffusion coefficient D t of the particles, or
of each of the particles in a mixture, a situation in which this approach is particularly powerful
(cf. Figs. 9.12 and 9.14). Molecules with different absorbance spectra, such as protein and
surfactant or protein and nucleic acid, can be followed at different wavelengths.
For an ideal solution, the sedimentation coefficient of a particle, s, in Svedberg (1 S ¼ 10
À13 s),
is related to its buoyant molar mass, M b ; its Stokes radius, R S ; and Avogadro’s number, N A , by the
Svedberg equation:
s ¼ M b = 6πN A η 0 R S
ð
Þ
where η 0 is the viscosity of the solvent (in PaÁs
À1
). (Note: the Stokes radius or StokesEinstein radius of a solute, R S , is the radius of a hard sphere that diffuses at the same rate as the
solute. In biophysics, R S and the hydrodynamic radius, R H , tend to be used as synonyms of each
other.)
M b is the mass of the particle minus the mass of the solvent it displaces. It is related to the
particle molar mass, M, and its partial specific volume,
v (in mLÁg
À1
), by the following equation:
M b ¼ M
À
1 À ρ 0
v
Á
,
where ρ 0 is the density of the solvent.
• Equilibrium sedimentation analytical ultracentrifugation (Eq-AUC) in the absence of a
density gradient (§ 9.3.7; Fig. 9.13)
412
9 Solution Studies of Membrane Protein/Amphipol Complexes