Proteins and other macromolecules can be separated by size...

Proteins and other macromolecules can be separated by size using centrifugation. The idea is to spin a sample containing proteins of different size in solution. The spinning produces a centrifugal force per unit mass $g_{c},$ which leads to diffusion with a drift velocity that depends on the protein size. We assume that a protein in the sample can be approximated as a ball of radius $R$ (a) Following the discussion in the chapter, fill in the steps leading up to the formula for the drift velocity of the protein as a function of its radius (Equation 12.45 ), $v_{\text {drift }}=\frac{2\left(\rho_{\text {protein }}-\rho_{\text {solvent }}\right) g_{c} R^{2}}{9_{\eta}}$ where $\rho_{\text {protein }}$ and $\rho_{\text {solvent }}$ are the densities of the protein and the solvent and $\eta$ is the solvent viscosity. (b) Estimate the drift velocity for hemoglobin in water in an ultracentrifuge with $g_{c} \approx 10^{5} g$, where $g=10 \mathrm{m} / \mathrm{s}^{2}$ is the acceleration of freely falling objects in Earth's gravitational field. Assume a typical protein density of $1.2 \mathrm{g} / \mathrm{cm}^{3}$ (c) We would like to separate two similar proteins, having the same density, $\rho^{\prime}=1.35 \mathrm{g} / \mathrm{cm}^{3} .$ They have diameters of $4 \mathrm{nm}$ and $5 \mathrm{nm}$, respectively. The two protein species start out mixed together in a thin layer at the top of a $1 \mathrm{cm}$ long centrifuge tube. How large should the centrifuge acceleration $q_{c}$ be so that the two proteins are separated before they drift to the end of the tube?

Proteins and other macromolecules can be separated by size using centrifugation. The idea is to spin a sample containing proteins of different size in solution. The spinning produces a centrifugal force per unit mass $g_{c},$ which leads to diffusion with a drift velocity that depends on the protein size. We assume that a protein in the sample can be approximated as a ball of radius $R$
(a) Following the discussion in the chapter, fill in the steps leading up to the formula for the drift velocity of the protein as a function of its radius (Equation 12.45 ),
$v_{\text {drift }}=\frac{2\left(\rho_{\text {protein }}-\rho_{\text {solvent }}\right) g_{c} R^{2}}{9_{\eta}}$
where $\rho_{\text {protein }}$ and $\rho_{\text {solvent }}$ are the densities of the protein and the solvent and $\eta$ is the solvent viscosity.
(b) Estimate the drift velocity for hemoglobin in water in an ultracentrifuge with $g_{c} \approx 10^{5} g$, where $g=10 \mathrm{m} / \mathrm{s}^{2}$ is the acceleration of freely falling objects in Earth's gravitational field. Assume a typical protein density of $1.2 \mathrm{g} / \mathrm{cm}^{3}$
(c) We would like to separate two similar proteins, having the same density, $\rho^{\prime}=1.35 \mathrm{g} / \mathrm{cm}^{3} .$ They have diameters of $4 \mathrm{nm}$ and $5 \mathrm{nm}$, respectively. The two protein species start out mixed together in a thin layer at the top of a $1 \mathrm{cm}$ long centrifuge tube. How large should the centrifuge acceleration $q_{c}$ be so that the two proteins are separated before they drift to the end of the tube?

Key Concepts

Centrifugation

Centrifugation is a separation technique that uses a rapidly spinning rotor to generate a centrifugal force, which acts on particles in a fluid. This force dramatically exceeds the standard gravitational force, enabling the acceleration of separation processes based on differences in particles’ mass, size, or density. In macromolecular separation, such as with proteins, these enhanced forces make it possible to drive sedimentation of the molecules through a medium, leading to their spatial separation over time.

Stokes' Law

Stokes’ Law governs the drag force experienced by a spherical particle moving through a viscous fluid. It states that the frictional force is directly proportional to the particle’s radius, the fluid's viscosity, and the velocity of the particle. This law is foundational for calculating the resistance against motion that the particle experiences, and, when balanced with the driving force from the centrifugal field, it leads to the derivation of the drift or sedimentation velocity of the protein.

Buoyancy and Density Difference

The net force acting on a particle in a fluid under centrifugation is determined by the difference between the particle’s density and the solvent’s density. This density difference leads to a buoyant force that partially counteracts the centrifugal force. Understanding this concept is essential because it allows the determination of the effective force that drives the sedimentation of particles, with the greater the density difference, the faster a particle will drift.

Viscosity

Viscosity is a measure of a fluid's resistance to flow and deformation, which directly impacts the drag force acting on a particle moving through it. In the context of centrifugation, the solvent viscosity plays a critical role in determining the rate at which particles settle because it opposes the motion induced by the centrifugal force. The interplay between the driving force (net centrifugal force) and the resistive force (viscous drag) ultimately defines the sedimentation or drift velocity.

Drift (Sedimentation) Velocity

The drift, or sedimentation, velocity is the steady state velocity reached by a particle when the net driving force (after accounting for buoyancy) is exactly balanced by the viscous drag acting on it. This velocity is derived by equating the centrifugal force-induced net force to the drag force from Stokes’ Law, resulting in an expression that depends on the particle’s radius squared, the density difference between the particle and the surrounding fluid, the centrifugal acceleration, and the fluid's viscosity. This concept is crucial for designing separation protocols, as it dictates how quickly different particles can be resolved in a centrifuge.

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