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Analytical ultracentrifugation observes concentration profiles during rotation, rather than only recovering a separated pellet. Sector-shaped sample cells rotate past optical detectors. Absorbance scans measure attenuation at selected radii. A Rayleigh interferometer compares a sample sector with a solvent reference, converting concentration-dependent refractive-index changes into fringe shifts. Repeating scans follows a sedimentation boundary or the approach to a stationary distribution.^[ref]

For a dilute solute of buoyant mass $m_b$ and translational friction $f_h$, radial force balance gives $s=m_b/f_h$, where $s$ is the sedimentation coefficient. Diffusion supplies $D=k_{B}T/f_h$. A velocity experiment resolves boundary motion and broadening; a sedimentation equilibrium experiment instead measures $d\ln c/d(r^2)=m_b{\omega }^2/(2k_{B}T)$. This follows from the Boltzmann distribution in centrifugal potential $-m_b{\omega }^2{r}^2/2$. The two approaches separate frictional size information from buoyant mass information; shape, hydration and solvent properties must be specified to infer a unique geometric dimension.

Centrifugation separates suspended material by its motion relative to a rotating liquid. A centrifuge rotor creates an outward centrifugal acceleration in the rotating reference frame; material denser than its surroundings tends to move outward, opposed by drag force. Collected material can form a centrifugation pellet, leaving a supernatant, or remain in separated zones within a density gradient. The useful question is which physical difference the separation exploits: a difference in sedimentation coefficient or a difference in buoyant density.

Centrifugation makes otherwise slow settling useful for studying biological cells, organelles, proteins and DNA. Separation and identification are different steps: a centrifugation pellet is a collected fraction, not proof that it contains only one kind of organelle. The lysozyme wall-removal experiment illustrates a complementary readout: testing material released into the separated liquid can be more informative than inspecting the retained solids.

Density-gradient centrifugation uses a liquid whose density varies along the separation direction. A band does not by itself establish which mechanism produced it: rate-zonal centrifugation separates by migration rate before equilibrium, whereas isopycnic centrifugation uses equilibrium positions set by buoyant density.

The distinction is between a moving clock and a restoring balance. In rate-zonal centrifugation, two species may occupy different positions because their sedimentation coefficients differ, even when their buoyant densities agree. In isopycnic centrifugation, two differently sized species can converge to the same band because their buoyant densities agree. Conversely, equal mass does not ensure equal migration velocity: shape affects drag force, and medium density affects buoyancy.^[ref]

The worked finite-time rate-zonal separation and linear-gradient isopycnic focusing examples isolate these mechanisms in chosen mathematical models. The actual DNA experiment described under isopycnic centrifugation illustrates the second mechanism; its initial collection of biological cells into centrifugation pellets is a different operation from the later molecular banding. Collecting liquid fractions from a tube can recover material at different positions, but the position alone does not identify its molecules or establish purity.

Why does an isopycnic centrifugation band return after displacement? Choose a dilute mathematical model with outward coordinate $z$, constant local centrifugal acceleration $a>0$, fixed displaced volume $V>0$, linear friction coefficient $f>0$, and an imposed liquid density ${\rho }_f(z)={\rho }_0+Gz$ with $G>0$. Treat the species' buoyant density ${\rho }_b$ as constant in this medium. The sedimentation coefficient force balance gives the ordinary differential equationIts exponential function solution is $z(t)=z_{\ast }+(z(0)-z_{\ast })e^{-kt}$. The sign of the velocity reverses across $z_{\ast }$, proving stable equilibrium when $z_{\ast }$ lies inside the liquid. Changing $V/f$ changes the approach time $1/k$, while $z_{\ast }$ depends on buoyant density and the imposed density gradient. A real centrifuge rotor has radially varying centrifugal acceleration; the constant-$a$ approximation describes a short radial interval. Here the density gradient is held fixed, rather than modeling its formation.

For an original dimensionless example, let $u=z/L$ and $\tau =t/T$, where $L$ is column length and $T$ a chosen time scale. Write $q=z_{\ast }/L$ and $\lambda =kT$, so $u(\tau )=q+(u(0)-q)e^{-\lambda \tau }$. Species A and B have the same $q=0.45$ but different friction-dependent rates: $({\lambda }_A,u_A(0))=(0.4,0.1)$ and $({\lambda }_B,u_B(0))=(1.6,0.95)$. A moves outward and B inward, both toward $u=0.45$. Species C has $(q_C,{\lambda }_C,u_C(0))=(0.8,0.8,0.1)$ and approaches a separate band at $u=0.8$. At $\tau =4$, their positions are approximately $0.3793$, $0.4508$ and $0.7715$: a finite-time photograph would still show A short of its limiting position.

An equilibrium band has finite width because diffusion opposes focusing. In the same local model, let $c(z)$ be concentration and let the constant diffusion coefficient be $D>0$. Combining the restoring drift with Fick's laws, zero net flux requiresDividing by $c$ and integrating proves the Gaussian function profile, whose standard deviation is ${\sigma }_z=\sqrt{D/k}$. This local result assumes that the band is narrow relative to its distance from the boundaries and that $D$, $k$ and the density gradient vary negligibly across it. Equal buoyant densities therefore imply coincident centers in this model, not identical widths. A band outside the available density range has no such interior restoring point.

Let $M$ be molar mass in $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, $\bar{v}$ partial specific volume, ${\rho }_s$ solvent density, and $N_A$ the Avogadro constant. The buoyant molecular mass is $m_b=M(1-\bar{v}{\rho }_s)/N_A$. Combining sedimentation coefficient $s=m_b/f_h$ and the thermal diffusion-friction relation $D=k_{B}T/f_h$ eliminates the unknown friction and proves the displayed formula, with $R_{\mathrm{gas}}=N_A{k}_B$. Once friction is known, $R_h=f_h/(6\pi \eta )$ defines a hydrodynamic radius using Stokes's law. Mass and radius are different inferences: an extended molecule need not occupy the volume of that friction-equivalent sphere.

For a dilute suspended sphere in Stokes flow, the translational friction coefficient is $f_h=6\pi \eta R$, with dynamic viscosity $\eta$ and hydrodynamic radius $R$. To derive the diffusion-friction relation, put particles in a weak conservative force $F=-U^{\prime }$. Their drift velocity is $F/f_h$, while the diffusive flux is $-Ddc/dx$. At equilibrium the Boltzmann distribution gives $c^{\prime }=-c{U}^{\prime }/(k_{B}T)=cF/(k_{B}T)$. Zero total flux therefore gives $cF/f_h-DcF/(k_{B}T)=0$, hence the Stokes–Einstein relationHere $k_B$ is the Boltzmann constant, $T$ absolute temperature and $D$ the translational diffusion coefficient. The sphere, no-slip continuum-fluid and dilute thermal-equilibrium assumptions enter through the drag law and flux balance.

At $T=293.15\mathrm{K}$ and the supplied viscosity, the predicted product is $DR=2.1472\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The derived predictions and radii inferred from the measured diffusion coefficients are:

The last three agree closely at the precision of the supplied data. The ribonuclease diffusion coefficient is about $16\%$ below the spherical prediction, corresponding to a larger frictional radius than the stated radius. Hydration, shape, a different operational radius and coarse measurement precision can matter; molar mass is not an independent parameter in the spherical diffusion law. A compact constant-density globule would have $R\propto M^{1/3}$, whereas collagen's large frictional radius relative to its mass is consistent with an extended nonspherical molecule. The inferred Stokes radius is a friction-equivalent radius, not automatically a geometric half-length.

In analytical ultracentrifugation, scan the radial protein concentration while a rotor spins at known angular speed $\omega$. Optical absorbance or refractive-index interference records a moving sedimentation boundary and its diffusive broadening.^[ref] With solvent density ${\rho }_s$, partial specific volume $\bar{v}$ and molar mass $M$ in $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, the buoyant mass is $m_b=M(1-\bar{v}{\rho }_s)/N_A$. Force balance gives $f_{h}v=m_b{\omega }^{2}r$, so the sedimentation coefficient is $s=v/({\omega }^{2}r)=m_b/f_h$. A boundary trajectory gives $s={\omega }^{-2}d\ln r/dt$; diffusion gives $f_h=k_{B}T/D$.

Combining them gives the molar mass from sedimentation and diffusion:Known mass and buoyancy can instead determine friction from $s$ alone, and comparison with a compact sphere tests shape or hydration. In sedimentation equilibrium, the stationary radial profile obeys $d\ln c/d(r^2)=m_b{\omega }^2/(2k_{B}T)$, providing buoyant mass without a friction model. This is a sedimentation-diffusion balance in a uniform solvent, rather than density-gradient banding at an isopycnic point.

Rate-zonal centrifugation resolves an initially narrow sample zone by differences in sedimentation coefficient during a finite interval. Its bands remain transient: extending the interval can carry material into a centrifugation pellet. Thus a density gradient need not imply an equilibrium buoyant density measurement.^[ref]

A typical preparation layers a narrow sample above a preformed density gradient, often containing sucrose. The initial zone must be narrow enough that differences in starting position do not overwhelm the later separation. The run ends while useful bands remain inside the liquid. In the simple case where all particles are denser than every part of the column, none has an interior isopycnic centrifugation position.^[ref][ref]

The sedimentation coefficient $s=(m-{\rho }_{f}V)/f$ explains why separation is not an ordering by mass alone. At fixed local centrifugal acceleration, increasing displaced volume can increase buoyancy, while a less compact shape can increase the linear friction coefficient and reduce velocity. Even for a sphere, Stokes's law gives $s=2b^2({\rho }_p-{\rho }_f)/(9\eta )$, so both the density difference and the dynamic viscosity matter. The resulting sedimentation coefficient can change along a density gradient.

The sedimentation coefficient $s$ is radial drift velocity divided by local centrifugal acceleration. It has units of time and combines particle properties with those of the surrounding liquid. In a dilute, deterministic mathematical model with negligible particle inertia, an effective displaced volume $V$, particle mass $m$, local liquid density ${\rho }_f$ and linear friction coefficient $f>0$ giveThe subtraction represents buoyancy. Equal mass does not imply equal $s$: changing shape changes drag force, and changing the medium changes both buoyancy and viscosity.^[ref]

For an isolated sphere of radius $b$ and uniform density ${\rho }_p$, Stokes's law supplies $f=6\pi \eta b$, where $\eta$ is dynamic viscosity, and $m=(4\pi /3)b^3{\rho }_p$. ThereforeThis original force-balance check predicts fourfold $s$ on doubling $b$ with everything else fixed, and zero drift when ${\rho }_p={\rho }_f$. It applies to dilute isolated spheres in Stokes flow, not arbitrary folded proteins or crowded suspensions. If $s$ and $\omega$ remain constant in a uniform medium, the ordinary differential equation $dr/dt=s{\omega }^{2}r$ has the exponential function solution $r(t)=r(0)e^{s{\omega }^{2}t}$. Arrival at an outer boundary therefore depends on starting position as well as $s$; a separation has no universal size cutoff.