The relative isotopic abundances in a sample.

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.

An encoded instruction representation for a software execution mechanism. Compilation can produce bytecode that an interpreter executes.

An IP3-sensitive calcium store contains releasable calcium behind a membrane bearing functional IP3 receptors. The endoplasmic reticulum is an important example. Pumping calcium into it or exporting calcium across the cell membrane lowers free cytosolic calcium; intracellular redistribution and loss from the whole biological cell are different balances.^[ref]

Finite availability matters: a release pathway cannot deliver more calcium than its accessible store contains unless replenishment occurs. A falling cytosolic calcium response can reflect reduced release, increased removal, or both. Store content, release rate and free cytosolic concentration therefore answer different questions. Binding to proteins can also change the free fraction without changing the total number of calcium ions.

An IP3 receptor is an intracellular ligand-gated ion channel, rather than the extracellular signal's cell-surface receptor. Its opening is regulated by both IP3 and cytosolic calcium: IP3 binding is not a command that fixes one constant release rate. Mammalian IP3 receptors have three subtypes, with possible mixed assemblies and splice variants; these need not have identical regulation or cellular roles. The 2010 authored account distinguishes established channel behavior from then-unresolved structural explanations.^[ref]

Phospholipase C catalyzes hydrolysis of membrane PIP2, producing soluble IP3 and membrane-associated diacylglycerol. This consumes a phospholipid substrate, rather than manufacturing calcium. G protein-regulated and receptor tyrosine kinase-regulated forms of PLC provide different routes to this reaction.^[ref]

Two conjugate optical planes are object and image planes related by an imaging system: points in one are imaged onto corresponding points in the other in the paraxial approximation. For a converging thin lens forming a real image in air, their distances $u,v>0$ obey $1/u+1/v=1/f$. Exchanging $u$ and $v$ preserves this relation; this symmetry gives the two settings of the Bessel lens displacement method.

The hydrodynamic radius is the radius of a hypothetical no-slip sphere with the same translational diffusion coefficient as the dissolved species. In the dilute continuum regime the Stokes–Einstein relation giveswhere $k_B$ is the Boltzmann constant, $T$ the absolute temperature, $\eta$ the dynamic viscosity of the solvent, and $D$ the translational diffusion coefficient. This effective radius need not equal the radius of gyration or a geometric molecular radius. It is a useful descriptor for SEC molar mass calibration, rather than a complete specification of how every shape enters every pore.

Toluene is methylbenzene, ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{C}{\mathrm{H}}_3$. It combines an aromatic ring with a benzylic methyl group, which creates reaction sites with different electronic environments.

It is a common organic solvent. Oxidation and substitution can affect either the side chain or the ring depending on reagents and mechanism, so identifying the reacting position is central to describing a toluene transformation.

Para-terphenyl contains three benzene rings connected in a line through para positions, with formula ${\mathrm{C}}_{18}{\mathrm{H}}_{14}$. The rings are linked by single bonds rather than fused along shared edges.

Inter-ring torsion affects conjugation and optical behavior, making it distinct from fused polycyclic aromatic hydrocarbons. Specifying para connectivity distinguishes it from the ortho and meta terphenyl isomers.

Pentacene is a linearly fused five-ring aromatic hydrocarbon, ${\mathrm{C}}_{22}{\mathrm{H}}_{14}$. Its extended conjugation and crystal packing make it relevant to organic-semiconductor research.

Charge transport in a solid depends on interactions between molecules as well as the electronic structure of an isolated molecule. Pentacene therefore connects molecular orbital ideas with the properties of an ordered molecular material.