Isopycnic centrifugation

In isopycnic centrifugation, a species concentrates near the position where the liquid's density matches its buoyant density. In a density gradient increasing outward, material displaced inward encounters less-dense liquid and tends to move outward; material displaced outward encounters denser liquid and tends to move inward. Diffusion broadens the resulting band. If the density gradient never reaches the species' buoyant density, a matching equilibrium position need not exist within the vessel.^[ref]

The 1958 paper The replication of DNA in Escherichia coli by Matthew Meselson and Franklin W. Stahl (pp. 672–676, Figures 1–4) makes the distinction concrete. Its separation apparatus was a Spinco Model E ultracentrifuge with an Escherichia coli lysate in a cesium chloride density gradient. Ultraviolet wave absorption photographs of the rotating sample were scanned with a recording microdensitometer: peak position reported buoyant density, while signal above baseline reported DNA concentration.^[ref]

Cultures received labeled nitrogen; chilled biological cells were collected as centrifugation pellets and disrupted before molecular banding.^[ref]

Mixed lysates labeled fully with the nitrogen isotope ${}^{15}\mathrm{N}$ and unlabeled lysates supplied reference bands; a mixture including intermediate material checked its position relative to those references. Figure 1 followed band formation over time, helping distinguish a migrating distribution from an approximately stationary one. These controls address resolution, calibration and approach to equilibrium. They do not make every ultraviolet-absorbing component identifiable as DNA, or turn a band into a direct image of individual strands. The DNA replication article develops the copying inference; here the key result is a separation by buoyant density, not a claim that heavier-labeled DNA is physically larger.^[ref]

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.

Buoyant density is the surrounding liquid density at which a species has no net centrifugal drift under the stated conditions. It depends on interactions with the medium, including associated solvent, rather than necessarily equaling the dry substance's density. Equal buoyant densities can give overlapping bands even for differently sized proteins or DNA fragments.^[ref]

A cesium chloride density gradient forms when the salt $\mathrm{CsCl}$ redistributes in a rotating chemical solution. It can supply matching-density positions for DNA during isopycnic centrifugation.^[ref]