Question

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?

Short answer

The required centrifuge acceleration can be calculated by using the equation for drift velocity, the known protein radii, and the definition of acceleration. From there, plug into the given information, rearrange and solve for \( g_c \).

Step-by-step solution

Derive the drift velocity formula

To derive the drift velocity formula, use the Stokes' Law which shows the force experienced by small spherical particles suspended in a medium. We modify this for the case of a centrifuge by replacing gravitational acceleration (\( g \)) with the centrifugal acceleration (\( g_c \)). The formula expresses the drift velocity as \(v_{drift} = \frac{2}{9} \cdot \frac{{(\rho_{protein} - \rho_{solvent}) \cdot g_c \cdot R^2}}{\eta}\).

Calculate the drift velocity for hemoglobin

Substitute the given values in the formula to find the drift velocity of hemoglobin in water in an ultracentrifuge. Given \(\rho_{protein} = 1.2 g/cm^3\), \(g_{c} = 10^{5} g\), \(g = 10 m/s^2\), and \(R\) is the radius of the hemoglobin molecule. Since it's not given, the radius of a hemoglobin molecule can be looked up in a reference material and it is approximately 2.8nm or \(2.8 \times 10^{-7} cm\).

Determine the centrifuge acceleration for protein separation

To determine how large the centrifuge acceleration should be so that the two proteins are separated before they drift to the end of the tube, you need to use the formula for the drift velocity and set it equal for both proteins given their different radii. Then solve for \( g_c \).

Key concepts

Centrifugation in Biology

Centrifugation is a powerful technique routinely used in biology to separate components of complex mixtures, such as cells, organelles, or macromolecules like proteins and nucleic acids. By spinning a sample at high speeds, a centrifugal force is applied, causing heavier particles to move outward towards the bottom of the tube, while lighter particles remain closer to the top.

The force exerted on particles in the centrifuge is expressed in terms of relative centrifugal force (RCF) or 'g-force', which is much greater than Earth's gravitational force, allowing for the effective separation of molecules with minute differences in mass or density. Different rotor types, like fixed-angle or swinging bucket, can be used depending on the desired outcome, such as pelleting or creating a density gradient for separation. The use of centrifugation is integral in macromolecular studies, diagnostics, and vaccine production, making it a fundamental technique in both research and clinical labs.

Macromolecule Analysis

Macromolecules, which include proteins, nucleic acids, carbohydrates, and lipids, play vital roles in biological processes. Due to their significance, the analysis and separation of these molecules are crucial in understanding their function and structure. Centrifugation allows researchers to isolate and purify specific macromolecules based on size, shape, density, and solubility.

In the context of analyzing proteins, techniques like ultracentrifugation can be used to study protein-protein interactions, molecular weights, and conformations. By spinning a sample under controlled conditions, proteins can be separated into distinct bands or layers, each corresponding to molecules of different densities or sizes. Scientists can then extract these layers to further analyze protein content using biochemical methods such as SDS-PAGE, Western blotting, or mass spectrometry, providing detailed insights into the protein's characteristics.

Protein Density and Separation

Understanding protein density is crucial for their separation using centrifugation. Proteins with different densities will sediment at different rates when subjected to a centrifugal force. Separating proteins based on density is particularly useful when they have similar sizes but different molecular compositions.

The key factors determining the rate of protein movement, or drift velocity, in a centrifuge are the protein and solvent densities, the size of the protein (approximated as a spherical radius), and the viscosity of the solvent. The relationship between these factors is expressed in the Stokes’ law-derived formula mentioned in the solution steps. Higher protein density relative to the solvent density results in a faster sedimentation rate, allowing dense proteins to be separated from less dense proteins and other components in the mixture. By altering the centrifugal force or the viscosity of the solution, researchers can fine-tune the separation process for detailed and precise macromolecular analysis.