Protein Chemistry - Part 1 - General Protein Chemistry - Ashmarin I. P. 1968

Determination of the Size and Shape of Protein Molecules
Analytical Ultracentrifugation
Sedimentation Velocity Method

In this method, the rotor speed is about 60,000 rpm (typically corresponding to 250,000 g). This leads to The formation of a boundary between the solvent and the solution, which migrates toward the bottom of The Cell. This method makes it possible to assess the purity of a preparation from the number of boundaries and to calculate the sedimentation coefficient in order to use this data for determining the Molecular Weight of a substance, as well as to evaluate changes in conformation and molecular weight during various molecular transformations, and so on.

The migration velocity of the boundary depends on the balance between the applied centrifugal force and the frictional resistance of the medium. Another essential condition for the movement of solute molecules relative to the solvent is the density difference between them. Let us consider the motion of a protein molecule at a distance x from the center of rotor rotation at an angular velocity w (rad/s). The centrifugal acceleration in this case is equal to w²x. If the molecular weight of the protein is M and v is its partial specific volume, then the centrifugal force acting on the molecule will be equal to the product of the centrifugal acceleration and its effective mass. The latter represents the difference between the true mass of the molecule m = M/N and the mass of the solvent displaced by the molecule (where p is the solvent density):

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At the beginning of the experiment, the velocity of the protein molecules dx/dt is zero. As the velocity increases, the frictional resistance of the medium grows until it balances the centrifugal force. Upon reaching this steady state, the molecule begins to move through the medium at a constant velocity. The frictional drag force, equal to the centrifugal force, is determined by the equation

where f is the frictional coefficient. It is directly proportional to the thermal energy of the molecule RT (where R is the gas constant and T is the absolute Temperature) and inversely proportional to the diffusion coefficient D:

At steady state

Multiplying both sides of the equation by the Avogadro number converts m into M:

Rearranging this equation, we obtain:

The expression (dx/dt)/w²x represents the sedimentation velocity per unit of centrifugal acceleration and is called the sedimentation coefficient S. Since w has dimensions reciprocal to time (rad/s), S has the dimension of time (seconds). The values of S for protein solutions are typically on the order of 10^-13–10^-12 s. A value of S equal to 10^-13 s is taken as the unit of the sedimentation coefficient and is designated as 1 Svedberg unit, or 1 S. Substituting the value of S into equation (39), we can write:

This equation forms the basis for determining the molecular weights of macromolecules by the sedimentation velocity method. The method involves determining not only the sedimentation coefficient and diffusion coefficient at a given temperature in a medium of known viscosity, but also determining the partial specific volume.

To find the sedimentation coefficient, the positions of the sedimentation boundaries on the sedimentation diagram are measured at specific centrifugation time intervals, and the coefficient is calculated using the formula below:

where x_n and x_m are the positions of the sedimentation boundary at times t_n and t_m, respectively. The values of x and t can be obtained experimentally; it only remains to calculate the angular velocity w. Angular velocity is expressed in radians per second. It is known that a radian is the angle subtended by an arc equal to the radius, and that one full revolution equals 2π radians. Consequently, multiplying the rotor rotation speed (rev/s) by 2π gives the value of w.

The magnitude of the sedimentation coefficient depends on the density, viscosity, and temperature of the solvent. Therefore, to compare different Proteins, it is standard practice to normalize these coefficients to standard conditions (Water, 20°C). In this case, the sedimentation coefficient is denoted as S_20,w and is calculated using the formula

where η_t, , and p_t are the viscosity, partial specific volume, and density of the salt solution at temperature , while represent the same quantities for water at 20°C. The values of these parameters can be obtained from appropriate reference tables.

Since aggregation processes can occur in protein solutions—the extent of which depends on the protein concentration—it is desirable to eliminate The Effect of concentration on the sedimentation velocity. This is achieved by measuring the sedimentation coefficient at various protein concentrations and graphically extrapolating the reduced S_20,w values to zero concentration. This yields the sedimentation coefficient value, which is denoted as