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

Determination of the size and shape of protein molecules
Analytical ultracentrifugation
Sedimentation equilibrium method

If an ultracentrifuge is run for a sufficiently long time at speeds around 8,000–15,000 rpm, The rate of mass transfer due to sedimentation becomes low and comparable to the rate of diffusion. Consequently, after a certain period, an equilibrium concentration distribution is established throughout the entire Cell. The rate of dissolved Protein Transport Across the cross-sectional area A driven by the centrifugal force is given by c∙A∙dx/dt. According to Fick's law, the rate of diffusion across the same surface is — D∙A∙dc/dx. As follows from equation (35), the velocity dx/dt equals the centrifugal force divided by the frictional coefficient (φ/f). Hence, the net velocity of protein movement is:

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Since f=kT/D, then

At equilibrium, the net transport of protein at any point in The Cell is zero, and dc/dx = φc/kT. Substituting the value of φ from equation (34) and multiplying both the numerator and the denominator by Avogadro's number, we obtain

Rearranging the equation by moving the terms containing c to the left-hand side and those containing χ to the right-hand side, and then integrating, we get

where с₂ and c₁ are the concentrations, and х₂ and х₁ are the distances from the rotor center to a given pair of points within the cell. Equation (45) serves as the fundamental equation for Sedimentation Equilibrium.

Sedimentation equilibrium experiments are typically conducted at relatively low rotor speeds, such as 8,000 rpm, which corresponds to a modest centrifugal acceleration (around 5,000 g). Therefore, the sedimentation pattern does not exhibit a distinct peak; instead, it shows a characteristic distribution of concentrations or concentration gradients (Fig. 41). From these sedimentation patterns, one can determine the distances х₂ and x₁ as well as the concentrations с₂ and c₁. At the same time, because of the low centrifugation speed, achieving sedimentation equilibrium takes a very long time, often spanning several days. Aside from the inconvenience of such prolonged ultracentrifugation runs, There is a risk of Denaturation of unstable Proteins during the experiment. Therefore, to accelerate the attainment of equilibrium, it is advisable to establish a protein concentration gradient in the cell prior to centrifugation by layering solutions of different concentrations.

As a result, despite being built on a rigorous theoretical foundation, this method is used much less frequently than sedimentation velocity. To overcome these limitations, Archibald proposed a theoretical framework for calculating molecular weights from the rate of approach to sedimentation equilibrium without waiting for true equilibrium to be reached. Although innovative, his method also involves certain experimental challenges and is still relatively rarely applied in practice.