Question

Ribonuclease has a partial specific volume of \(\left(0.707 \mathrm{~cm}^{3}\right.\) \(\mathrm{g}^{-1}\) ) and diffusion coefficient of \(\left(13.1 \times 10^{-7} \mathrm{~cm}^{2} \mathrm{sec}^{-1}\right)\) corrected to water at \(20^{\circ} \mathrm{C}\). The density of water at \(20^{\circ} \mathrm{C}\) is \(0.998\left(\mathrm{~g} / \mathrm{cm}^{3}\right) .\) Calculate the molecular weight of ribonuclease using the Svedberg equation. Assume that the sedimentation coefficient for ribonuclease is \(2.02 \times 10^{-13}\) \(\mathrm{sec}\).

Short answer

The molecular weight of ribonuclease, calculated using the Svedberg equation, is approximately \(36.1845 \times 10^{29}\) g/mol.

Step-by-step solution

List the given values

We have the following given values: - Partial specific volume (ν): \(0.707 \ \mathrm{cm^{3}g^{-1}}\) - Diffusion coefficient (D): \(13.1 \times 10^{-7} \ \mathrm{cm^{2}sec^{-1}}\) - Density of water (ρ): \(0.998 \ \mathrm{g/cm^{3}}\) - Sedimentation coefficient (s): \(2.02 \times 10^{-13} \ \mathrm{sec}\) - Avogadro's number (N_A): \(6.022 \times 10^{23} \ \mathrm{mol^{-1}}\)

Substitute the values in the Svedberg equation

Substituting the given values into the Svedberg equation, we get: \[ M = \dfrac{N_A \rho \nu D}{s} = \dfrac{(6.022 \times 10^{23}\ \mathrm{mol^{-1}})(0.998\ \mathrm{g/cm^{3}})(0.707\ \mathrm{cm^{3}g^{-1}})(13.1 \times 10^{-7}\ \mathrm{cm^{2}sec^{-1}})}{2.02 \times 10^{-13}\ \mathrm{sec}} \]

Calculate the molecular weight

Now, we need to calculate the molecular weight: \[ M = (6.022 \times 10^{23} \ \mathrm{mol^{-1}})(0.998\ \mathrm{g/cm^{3}})(0.707\ \mathrm{cm^{3}g^{-1}})(13.1 \times 10^{-7} \ \mathrm{cm^{2}sec^{-1}}) \div (2.02 \times 10^{-13}\ \mathrm{sec}) \] \[ M = (6.022 \times 0.998 \times 0.707 \times 13.1) \times 10^{23 - 7 + 13} \] \[ M = 36.1845 \times 10^{29} \] Thus, the molecular weight of ribonuclease is approximately \(36.1845 \times 10^{29}\) g/mol.

Key concepts

Molecular Weight Calculation

Calculating the molecular weight of a substance is a fundamental technique in biochemistry and molecular biology. It's essential to determine the molecular weight to understand the size and composition of molecules like proteins and enzymes. In this exercise, we calculate the molecular weight by employing the Svedberg equation. This powerful equation uses various physical measurements concerning the substance in solution. It takes into account the sedimentation coefficient, diffusion coefficient, partial specific volume, and the density of the solvent. The molecular weight is denoted by the letter \( M \) and calculated using:\[ M = \dfrac{N_A \rho u D}{s} \]where:

• \( N_A \) is Avogadro's number.
• \( \rho \) is the density of the solvent, in this case, water.
• \( u \) is the partial specific volume.
• \( D \) is the diffusion coefficient.
• \( s \) is the sedimentation coefficient.

This step-by-step application of the Svedberg equation helps us derive the molecular weight of ribonuclease, essential for further analysis and experiments.

Sedimentation Coefficient

The sedimentation coefficient is a key factor in determining the molecular weight using the Svedberg equation. It represents how fast a particle sediments in a centrifugal field and is measured in Svedbergs (S), which is equivalent to \( 10^{-13} \) seconds. It can be influenced by the size, shape, and density of the molecule, as well as the viscosity of the medium. A bigger sedimentation coefficient indicates a heavier or more sedimentable particle. Knowing the sedimentation coefficient helps to distinguish between different molecules such as Ribonuclease, which, in this exercise, has a known sedimentation coefficient of \(2.02 \times 10^{-13} \) s. The sedimentation coefficient contributes to the formula as a divisor, highlighting its role in making the calculated molecular weight an estimation that aligns with empirical data.

Diffusion Coefficient

The diffusion coefficient (\( D \)) is a parameter that describes how quickly molecules diffuse through a medium or solution. It's calculated in units of \( \mathrm{cm^2s^{-1}} \), providing information about the mobility of molecules or particles. In the context of the Svedberg equation, the diffusion coefficient helps balance the sedimentation coefficient. While sedimentation describes how molecules settle, diffusion illustrates how they spread out, providing a comprehensive picture of molecular behavior in solution. For Ribonuclease, the given value of \( D \) is \( 13.1 \times 10^{-7} \ \mathrm{cm^2s^{-1}} \), corrected to water at \( 20^{\circ} \text{C} \). This information is crucial for deriving accurate molecular weights through the interplay of molecular forces in the solution.

Partial Specific Volume

Partial specific volume represents the volume occupied by one gram of a protein or polymer when dissolved. It plays an important role in calculating molecular weight using the Svedberg equation. This measure is unique for every molecule based on its structure and how it interacts with the solvent. For ribonuclease, the partial specific volume is given as \( 0.707 \ \mathrm{cm^3g^{-1}} \). It indicates how much physical space each gram of ribonuclease occupies in water, giving insights into the protein's size and intermolecular interactions. When incorporated into the Svedberg equation, it provides a necessary conversion factor between mass and volume, ensuring the output—the molecular weight—is accurate and reflective of the protein solution's real behavior.