The rate of sedimentation of a recently isolated protein was monitored at \(20^{\circ} \mathrm{C}\) and with a rotor speed of 50000 r.p.m. The boundary receded as follows: $$ \begin{array}{llllllll} t / \mathrm{s} & 0 & 300 & 600 & 900 & 1200 & 1500 & 1800 \\ r / \mathrm{cm} & 6.127 & 6.153 & 6.179 & 6.206 & 6.232 & 6.258 & 6.284 \end{array} $$ Calculate the sedimentation constant and the molar mass of the protein on the basis that its partial specific volume is \(0.728 \mathrm{~cm}^{3} \mathrm{~g}^{-1}\) and its diffusion coefficient is \(7.62 \times 10^{-11} \mathrm{~m}^{2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\), the density of the solution then being \(0.9981 \mathrm{~g} \mathrm{~cm}^{-3}\). suggest a shape for the protein given that the viscosity of the solution is \(1.00 \times 10^{-3} \mathrm{~kg} \mathrm{~m}^{-1} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\).

The Svedberg equation plays a crucial role in the field of biochemistry, particularly in understanding the sedimentation characteristics of macromolecules like proteins and nucleic acids. It is a fundamental equation used to determine the sedimentation coefficient, often denoted as 's'. The sedimentation coefficient represents how quickly a particle sediments in a centrifugal field and is a measure of a particle's size and shape.

The sedimentation velocity of a particle is given by \( \frac{\text{dr}}{\text{dt}} \) which represents the rate at which the particle moves toward the bottom of the centrifuge tube under the influence of centrifugal force. This rate is impacted by the angular velocity (converted to radians per second), \( \omega \), and the distance, r, from the axis of rotation. The Svedberg equation for the coefficient is \( s = \frac{\text{dr}}{\text{dt}} \times \frac{1}{\omega^2 r} \) where \( \omega^2 r \) defines the angular velocity and radius product squared.

This coefficient has units of time, typically expressed in Svedberg units (S), where 1 Svedberg (1S) is equal to 1x10^-13 seconds. The sedimentation coefficient not only provides insight into the molecular weight of a particle but also reflects its shape and density, as these factors influence how a particle sediments in a centrifuge.

Molar mass calculation is an essential application of sedimentation data and can be achieved through the use of the Svedberg equation mentioned earlier. Once the sedimentation coefficient (s) is established, the next step is to relate it to the molar mass (M) of the molecule. The Svedberg equation leverages the principles of sedimentation and diffusion to yield the molar mass as follows: \( M = \frac{sRT}{D(1-v\rho)} \) where 'R' is the universal gas constant (8.314 J/mol·K), 'T' is the temperature in Kelvin, 'D' is the diffusion coefficient, 'v' is the partial specific volume, and '\rho' is the solution's density.

The importance of harmonizing units in the equation cannot be overstated—this ensures accuracy in the calculation. The diffusion coefficient expresses how a particle spreads out over time due to Brownian motion, while the partial specific volume represents the volume occupied by one gram of the particle in question. Given these parameters and constants, the molar mass of a molecule such as a protein can be precisely determined, allowing scientists to better understand the molecule's abundance and significance in biological systems.

The analysis of sedimentation data can extend beyond basic size and molar mass to provide clues about the shape of a protein. Protein shape determination is particularly useful in biophysical characterization and understanding how proteins function in a biological context. One key aspect of this analysis is the frictional ratio (f/f0), which compares the frictional coefficient of the particle (f) to that of a perfectly spherical particle of the same volume (f0).

The frictional coefficient encompasses both the size and shape of the protein and can be deduced from the ratio of the sedimentation coefficient to the diffusion coefficient (s/D). When the frictional ratio is approximately 1, it suggests the protein is spherical; values greater than 1 indicate an elongated or irregular shape. These deductions are vital, as the shape of a protein affects its stability, interactions, and function within cells. By effectively coupling the Svedberg equation with diffusion measurements, researchers can propose models of protein structure that inform our understanding of their role and dynamics within a biological system.