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

Determination of the size and shape of protein molecules
Protein diffusion rate

When a protein solution is brought into contact with a protein-free solvent, protein molecules diffuse into the solvent. The rate of this process can be expressed as The amount of protein crossing the interface per unit time, or, in differential form, dm/dt. Obviously, the rate of diffusion must be proportional to the interfacial area A and the concentration gradient of the protein in the direction of diffusion dc/dx, i.e.,

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where D is the proportionality constant, known as the diffusion coefficient, which is determined by the Specific characteristics of the given protein and solvent, as well as the Temperature. Naturally, D is higher when the protein molecules are smaller, the solvent viscosity η is lower, and the temperature T is higher. For spherical molecules, whose size is characterized by a single parameter—the radius r—this relationship is expressed by the formula

where N is Avogadro's number and R is the gas constant. The diffusion coefficient for Proteins is on the order of 10^-6—10^-7 cm²∙sec^-1, whereas for Amino Acids it is 10^-5 cm²∙sec^-1. Since the Molecular Weight of a spherical protein particle equals the volume of a sphere of radius r multiplied by the particle density σ and Avogadro's number, formula (9) can be transformed into the following

Thus, knowing the diffusion coefficient and the density of the protein particle, one can determine its molecular weight.

At the same time, experimentally obtained values of the diffusion coefficient can also be used to determine molecular shape—specifically, to calculate its axial ratio. Since protein molecules are largely non-spherical, they diffuse more slowly than a spherical molecule of the same molecular weight. In this case, by substituting the experimentally determined diffusion coefficient into formula (9), one can calculate the "apparent" radius r of the protein particle. On the other hand, by determining the molecular weight of the given protein through other Methods, one can calculate r₀, the radius of an ideal spherical particle with the same density, using the formula

Where is the volume of a single molecule. A non-spherical protein molecule can be approximately described as an ellipsoid of revolution. Therefore, once The ratio of the "apparent" radius to the ideal radius r/r₀ is established, special formulas can be used to calculate the ratio b/a, where b is the equatorial semi-axis of the ellipsoid and a is the polar semi-axis of revolution. For prolate ellipsoids, b/a is less than 1, while for oblate ones, it is greater than 1.

Experimentally, the diffusion constant is determined by measuring the rate at which a sharp boundary—formed in a vertical Cell between the protein solution and the solvent—spreads out. This measurement is carried out by recording Changes in the refractive index gradient, followed by calculating the concentration gradient, since the two are linearly related.

However, establishing a sharp boundary between two solutions is very difficult. Therefore, the two solutions are more often separated by a porous disc, and the diffusion rate is assessed by measuring the amount of protein that has passed through a unit area in a given time. The diffusion cell is calibrated using a protein solution of a known diffusion coefficient. If the molecular weight of this protein is M₁ and its diffusion coefficient is D₁, then the molecular weight of the unknown protein M₂ can be determined by the formula

If the "apparent" radii (r₁ and r₂) and particle densities (σ₁ and σ₂) are equal, the formula simplifies considerably:

Molecular weights calculated from diffusion rates yield values of the same magnitude as those obtained by other methods. However, deviations from spherical shape and molecular Hydration, both of which affect the diffusion rate, mean that this method is rarely used for direct molecular weight calculations. More frequently, it is combined with the Sedimentation Velocity Method.