Sedimentation & Ultracentrifugation

Ultracentrifugation

Ultracentrifuges spin samples at 10,000–100,000 rpm, generating forces up to 600,000 × g. This separates macromolecules by size, shape, and density.

Sedimentation Coefficient

The sedimentation coefficient s describes how fast a particle moves per unit centrifugal force:

$s = \frac{m(1 - \bar{v}\rho)}{f}$

Where:

• m — particle mass (kg)
• v̄ — partial specific volume (~0.73 mL/g for proteins)
• ρ — solvent density (g/mL; 1.0 for water)
• f — translational friction coefficient (f = 6πηr for a sphere)
• η — solvent viscosity (1×10⁻³ Pa·s for water at 20°C)

The term (1 − v̄ρ) is the buoyancy factor — it accounts for solvent displacement.

Svedberg Unit

1 S (Svedberg) = 10⁻¹³ s

Typical values: ribosomes (70S, 80S), proteins (1–20S).

Svedberg Equation

Combining sedimentation (s) and diffusion (D) gives molecular weight:

$M_{\text{g/mol}} = \frac{s \cdot R \cdot T}{D \cdot (1 - \bar{v}\rho)} \times 1000$

where s is in seconds, R = 8.314 J·mol⁻¹·K⁻¹, D in m²/s. The result in kg/mol × 1000 = g/mol = Da.

Pelleting Time

Time to move a particle from radius r₁ to r₂:

$t = \frac{\ln(r_2/r_1)}{s \cdot \omega^2}$

Where ω = 2π·rpm/60 (rad/s) is the angular velocity.

Example Values

Functions to Implement

• sedimentation_coefficient_S(M_Da, r_nm, v_bar=0.73, rho=1.0, eta=1e-3, T_K=293)
• svedberg_molecular_weight_Da(s_S, D_m2_s, v_bar=0.73, rho=1.0, T_K=293)
• pelleting_time_s(s_S, r1_m, r2_m, rpm)