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In 1918, Irving Langmuir measured adsorption using freshly split mica strips, handled at their edges, separated by iron spacers and held with brass fasteners. A glass bulb contained the bundle; a similar blank contained matching iron and brass but no mica. Evacuated glass connections, Mercury seals, liquid air traps, and McLeod gauges provided controlled gas handling. Calibrated compartment volumes and pressure readings before and after expansion gave free-gas amounts; their deficit indicated uptake. Lowering pressure and measuring released gas tested desorption.^[ref]

Hydrogen blank runs calibrated cooling-related pressure differences. Irving Langmuir pre-cooled the bulbs to reduce thermal delay; he suggested slow diffusion between strips could control the remaining approach to equilibrium. Critically, blank uptake was not subtracted: he estimated the mica-bulb results were 10–30% too high. Liquid air temperature drift was unrecorded. These limits constrain inference of intrinsic binding from the uptake curve. The paper's pressure unit called “bar” was $1\mathrm{dyn}\mathrm{c}{\mathrm{m}}^{-2}=0.1\mathrm{Pa}$, unlike the modern bar used above.^[ref]

Lateral mobility of membrane proteins is movement along a membrane surface. It differs from membrane transport through the boundary and from reversing membrane topology. Attachment, crowding and barriers can restrict diffusion; the fluid mosaic model does not require every membrane protein to wander freely.^[ref]

An original local model treats a small, approximately flat patch as an unbounded plane with uniform diffusion coefficient $D$, no drift and no trapping. Represent the two displacement coordinates by independent Brownian motions, each with zero expected value and variance $2Dt$. Since $r^2=x^2+y^2$, addition of the two coordinate expectations givesThis is a root-mean-square displacement, not distance traveled along a jagged trajectory or the displacement of every protein. For a stipulated $D=0.08\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $t=5\mathrm{s}$, $⟨r^2⟩=1.6\mu {\mathrm{m}}^2$ and $r_{\mathrm{rms}}\approx 1.265\mu \mathrm{m}$. Reaching a stipulated root-mean-square scale $L=2\mu \mathrm{m}$ gives $t=L^2/(4D)=12.5\mathrm{s}$. These model values are not measured diffusion coefficients or mixing times from the historical experiment.

Multiplying the target displacement scale by two multiplies the model time by four. This quadratic scaling is a useful way to see why local encounters and redistribution over an entire biological cell need not happen on the same timescale. A curved, bounded cell membrane, restricted paths and different mobile populations require more than this flat-patch model; disappearance of an initial pattern alone does not determine $D$.

A limiting rate of charge transfer per area imposed by reactant transport or another bottleneck. In a simple diffusion film, $j_L=nFD{c}_b/\delta$ and a consuming electrode has $c_s/c_b=1-j/j_L$. Increasing internal roughness does not necessarily increase external mass-transfer flux proportionately.

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.

At a sharp mush–liquid edge with linear liquidus $T_L=T_m-mC$, marginal equilibrium makes the liquid temperature gradient tangent to the local liquidus: $T_z=-m{C}_z$. Together with $T=T_L$ at the edge and $T\ge T_L$ in the exterior liquid, it supplies the usual absence-of-constitutional supercooling closure. When solute diffusion is neglected inside the mushy layer but retained outside it, interfacial solute conservation can require a small solid fraction jump; imposing unit porosity as an additional finite-diffusivity condition generally overdetermines this sharp-edge reduction.

A mass spectrometry instrument with a photographic readout. In the 1919 apparatus described in Chapter V: The Mass-Spectrograph, a gas discharge produced positive ions. Two narrow slits formed a ribbon beam in a vacuum; plates applied an electric field to spread it, an aperture selected part of it, and a magnetic field bent it oppositely. Focusing reduced the spread caused by differing ion speeds, concentrating a given mass-to-charge ratio into a line on a photographic plate.

The observed quantity was line displacement from a fiducial spot. Known reference lines calibrated displacement against mass-to-charge ratio; repeated exposures with changed magnetic field tested the calibration. Evacuation limited collisions. Comparing line shapes with narrow and wide apertures helped distinguish geometric broadening from photographic spreading. Finite slit widths, plate response, and overlapping molecular or multiply charged ions limited interpretation. See Chapter V: The Mass-Spectrograph (§§33–40: apparatus and photographic procedure; §§43 and 45–46: calibration, aperture comparisons, and accuracy).

In Mass spectra and isotopes by Francis William Aston (Nobel lecture, 12 December 1922), the reported neon lines at 20 and 22 have companions at 10 and 11 from doubly charged ions. That paired pattern and comparison with reference lines support different atomic masses within one chemical element, rather than an atomic mass concentrated at the bulk average. The lecture also describes an earlier failed separation by distillation and partial separation by diffusion, showing why the focused mass spectrum supplied stronger evidence. These numbers used a historical oxygen-based reference scale. The lecture's whole-number rule was an empirical approximation at its stated accuracy, not the modern exact definition $A=Z+N$.^[ref]

Movement of material by diffusion, convection or migration.

The mean displacement is the expected value of a particle's displacement over a time interval. It describes systematic transport, while the mean squared displacement also measures fluctuations. For one component with finite second moment,For constant drift speed $u$ and ordinary diffusion of coefficient $D$, the mean displacement is $ut$ and its centered variance is $2Dt$, so the uncentered mean squared displacement is $u^2{t}^2+2Dt$. With fixed initial velocity $v_0$ and free inertial Langevin dynamics, the mean displacement is $v_0{\tau }_v(1-e^{-t/{\tau }_v})$, where ${\tau }_v$ is the velocity relaxation time. Its fluctuation variance is given by the integrated Ornstein-Uhlenbeck displacement.

A mean squared displacement is the expected value of a squared position change over a specified time lag. For isotropic free diffusion in $d$ dimensions, it equals $2dD\tau$. A nonzero mean drift contributes its squared displacement as well as the displacement variance; specifying the tracked coordinates and whether the mean is subtracted is essential.

membrane permeability describes passage of a specified substance through a membrane under stated conditions. It is distinct from lateral diffusion within the membrane and from the mechanical force needed to rupture the membrane.

A membrane potential is an electric potential difference across a cell membrane. We use the conventionTaking the external electric potential as zero makes $V_m<0$ mean that the inside is at lower electric potential. “Negative inside” describes a difference, not an absolute electric potential, and does not imply that the entire cytosol has a large net negative electric charge.

Consider a cell membrane permeable only to potassium ions, with higher potassium concentration inside. Diffusion initially favors outward transport. If balancing negative ions cannot follow, that loss of positive electric charge produces a small inner excess of negative electric charge and an outer positive excess. The resulting electric field favors inward potassium transport. At the Nernst equilibrium potential, chemical and electrical contributions to electrochemical potential balance; ions still cross both ways, with zero net transport.^[ref]

In living biological cells, several ions can permeate and pumps maintain their concentration differences through active transport. Pumps transporting net electric charge also contribute electric current. Thus $V_m=E_{\mathrm{K}}$ is a special limit, not a universal rule; membrane potential need not always be negative.^[ref] The existing reversal potential example shows how opposing electric currents can balance without each ion reaching thermodynamic equilibrium. This makes selective permeability a way to convert chemical differences into electrical signals: changing the available ion channels changes which electrochemical gradients influence the voltage.

A morphogen is a signal whose spatial concentration helps specify different biological cell fates or behaviors during biological development. A morphogen gradient provides a positional cue: biological cells at different distances from a source experience different local concentrations and can follow different developmental pathways. A spatial concentration variation alone does not establish that a substance is a morphogen; its graded developmental effects matter. See Alberts et al., Molecular Biology of the Cell, 4th edition (2002): morphogens as graded developmental signals.

The beautiful idea is that one distributed signal can help organize several regions of a developing body. A continuous morphogen gradient can supply the input for distinct domains of gene expression, rather than requiring a separate signal for every position. A morphogen response threshold is a useful idealization of that interpretation: high, intermediate and low concentrations are assigned different outcomes. The morphogen reaction-diffusion equation describes one way to generate a morphogen gradient; the equation is a model of transport and reaction, while the morphogen is the signal being transported.

Detection and interpretation also depend on the receiving biological cell. For an extracellular morphogen, receptors and signal transduction connect the local concentration to a response; receptor binding occupancy is only one step in that connection. The same morphogen gradient can be interpreted differently in different tissues because their developmental context differs. Thus a morphogen response threshold summarizes a particular response in a particular context, rather than being a universal property of the signal. See Scott F. Gilbert, Developmental Biology, 6th edition (2000): positional information and context-dependent interpretation.

An important experimental example came in 1988. Wolfgang Driever and Christiane Nüsslein-Volhard used antibodies to examine the bicoid protein in early fruit-fly embryos. They reported a concentration profile approximately described by an exponential function, highest at the front, with the protein concentrated in cell nuclei. Their abstract proposed diffusion from localized messenger RNA and dispersed degradation as a likely origin of the morphogen gradient, rather than proving a unique transport mechanism. ^[ref]

In a companion study, Driever and Nüsslein-Volhard genetically changed the amount and distribution of bicoid messenger RNA, measured the resulting protein morphogen gradient, and compared developmental maps. Increased local bicoid protein levels shifted prospective anterior structures toward the rear; decreased levels shifted them toward the front. This linked a measured concentration profile to positional effects in biological development. The toy calculations below illustrate that logic with arbitrary numbers, not measured bicoid parameters. ^[ref]

A morphogen gradient is a spatially varying concentration profile of a morphogen. The profile $c(x)$ specifies the local amount; its mathematical gradient, $c^{\prime }(x)$ in one dimension, specifies how rapidly that amount changes with position. In the simplest positional interpretation, a biological cell responds to its local concentration, without having to measure the gradient across itself. Production, transport and removal together determine a morphogen gradient.

For a worked example, take a stationary one-dimensional tissue occupying $x\ge 0$, constant diffusion coefficient $D>0$, and uniform first-order reaction removal at rate $k>0$. At steady state, this specialization of the morphogen reaction-diffusion equation gives the ordinary differential equation and boundary conditionsHere $D$ has units of length squared per time, $k$ has units of inverse time, and $c_0$ is a prescribed source concentration. The Dirichlet boundary condition fixes the concentration at $x=0$. It does not prescribe a source flux; any required flux follows from the solution. These assumptions define a toy morphogen gradient, not a universal account of transport during biological development.

Define the diffusion length $\ell =\sqrt{D/k}$. Substitution into the ordinary differential equation shows that its general solution is $A{e}^{x/\ell }+B{e}^{-x/\ell }$. The vanishing boundary condition at infinity forces $A=0$, and the source Dirichlet boundary condition forces $B=c_0$. HenceOver every additional diffusion length, the concentration falls by the same factor $e^{-1}$. Faster diffusion increases $\ell$; faster removal decreases $\ell$. This makes the diffusion length a useful bridge between microscopic transport and the spatial reach of a morphogen gradient.

Choose $c_0=100$ arbitrary concentration units and $\ell =100\mu \mathrm{m}$, where $1\mu \mathrm{m}=10^{-6}\mathrm{m}$. Then $c(x)=100e^{-x/(100\mu \mathrm{m})}$: at $x=0$, $100\mu \mathrm{m}$ and $200\mu \mathrm{m}$, the concentrations are $100$, approximately $36.79$ and approximately $13.53$. The following diagram combines this morphogen gradient with the idealized responses calculated under morphogen response threshold.

An idealized model balancing localized production, diffusion and degradation. In one dimension at steady state, away from the source, $D{c}^{\prime \prime }-kc=0$ gives $c(x)=c_0{e}^{-x/\lambda }$ with $\lambda =\sqrt{D/k}$. Tissue geometry and time-dependent source distributions can invalidate a simple point-source interpretation.

For dilute, independently migrating carriers of number density $n$, charge $q$ and diffusion coefficient $D$, balancing drift with diffusion gives mobility $D/(k_{B}T)$ per applied force. Multiplying drift velocity by charge density yields $\sigma =n{q}^{2}D/(k_{B}T)$. State which species defines $n$ and $D$: a vacancy diffusivity must be combined with vacancy density. Correlated motion requires a correction rather than automatic use of a tracer diffusivity.

Neurotransmitter clearance removes released neurotransmitter from the region where it can activate receptors, through diffusion, cellular uptake or enzyme-mediated breakdown. Its routes and rates help delimit a chemical synapse's response in space and time; see the mechanisms and source under chemical synapse.

By Michael Cross. The two-component chemical-kinetics linear stability calculation supplies the reaction Jacobian and finite-wavenumber diffusion instability. Only generic mathematics and chemical physics are used here; biological applications are excluded.

A nuclear pore complex is an assembly of proteins spanning the nuclear envelope and regulating passage between the cell nucleus and cytosol. It is a selective gateway, not simply an unregulated hole: messenger RNA export and import of nuclear proteins require appropriate transport machinery. See Alberts et al.: nuclear pore complexes.

The nuclear pore complex occupies a junction of both nuclear envelope membranes. Its protein scaffold surrounds an aqueous passage connecting cytosol to the cell nucleus, rather than the intermembrane space. Diffusion permits small soluble cargo to pass; selective receptors transport other cargo. Molecular mass alone is not a universal cutoff.^[ref]

Primary paper, in the English translation collected in Investigations on the Theory of the Brownian Movement: sites.pitt.edu/~jdnorton/lectures/Rotman_Summer_School_2013/Einstein_1905_docs/Einstein_Brownian_English.pdf

Sections 3–5 connect equilibrium drift and diffusion, the spherical-particle drag coefficient, and mean-squared displacement. The linked scan includes the 1926 translation reprinted in the 1956 Dover edition; the research paper itself is dated 1905.

Optical molasses surrounds slow chemical atoms with opposing red-detuned laser beams, usually three pairs along perpendicular axes. In the elementary Doppler cooling picture, each pair damps one component of velocity. The result resembles viscous drag, even though the atoms are moving through a vacuum rather than a liquid.

Uniform balanced optical molasses supplies velocity damping, without a restoring force towards a chosen position. An atom at rest has zero mean force wherever it sits within the uniform illuminated region. Without noise or external forces, $m\dot{v}=-\alpha v$ givesThe atom approaches a position determined by its initial condition, not the beam intersection. Recoil-driven diffusion and gravity eventually remove atoms from finite beams. A long residence time in optical molasses is therefore compatible with the absence of position trapping. An optical trap provides spatial confinement by a different force mechanism; a magneto-optical trap adds a position-dependent magnetic field and suitable beam electromagnetic polarization to create a restoring force.