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Growth by collision and attachment of existing particles. Coagulation reduces particle number while conserving particulate material in a closed system without other sources or sinks. Larger particles can have a different scattering-per-mass efficiency and faster sedimentation, so increasing precursor injection need not increase albedo proportionally.

A whey protein used as a model solute in studies of protein association and transport. Its effective electric charge, association state and hydrodynamic radius depend on solution conditions. In a specified spherical-particle model, its electrophoretic mobility and sedimentation speed depend on its charge and buoyant mass through the same Stokes drag law.

Geology reconstructs the composition, structure and history of the solid Earth and other rocky bodies. Its observations include rock textures, fossils, layer relationships, deformation and numerical ages. A cross-cutting intrusion, for example, must be younger than the rock it cuts, even before either has a numerical date.

Physical models connect those relationships to processes such as plate tectonics, sedimentation, metamorphism and erosion. Distinguishing a directly observed structure from the process proposed to explain it is essential: similar present-day rocks can have different formation and alteration histories.

ToxB strongly reduces sedimentable wild-type actin, whereas act1-117 shows little additional reduction from its already low baseline. Wild-type actin is mainly in the pellet without drug and mainly in the supernatant with drug. Using the approximate bar heights, the pelleted fraction changes from $740/(740+120)\approx 0.86$ to $280/(280+820)\approx 0.25$. Thus the drug suppresses polymer accumulation but does not eliminate the measured pellet entirely.

For act1-117, the pellet is only about one-third of the recovered signal even without drug, and remains near that fraction with drug: approximately $340/(340+570)\approx 0.37$ and $360/(360+660)\approx 0.35$. The mutant is therefore relatively resistant to the drug's further effect under this assay, but it is not a wild-type-quality polymerizer with an unchanged high polymer mass. These are approximate within-pair fractions; unequal total recoveries and loading limit precise quantitative comparison. sedimentation also requires confirmation that pellet material is filamentous rather than aggregated.

Experiment 2: relative to untreated wild-type actin, actin monomer sequestration predicts a longer apparent actin nucleation lag, a smaller elongation slope, and a lower steady-state polymer mass. At sufficiently strong binding the trace could remain nearly flat rather than displaying clearly separated phases. The wild-type sedimentation data support a lower endpoint, but do not establish the exact kinetic changes or whether the time to plateau is longer or shorter.

Experiment 3: the halo result does not determine the drug-free actin polymerization curve of act1-129. A mutation that mainly raises drug affinity could have a nearly wild-type curve without drug. A mutation that intrinsically destabilizes actin filaments or impairs actin nucleation could instead prolong the lag, reduce growth and lower the plateau. Thus a wild-type-like untreated curve is a useful conditional reference prediction, not a result that follows from cellular hypersensitivity.

Experiment 4: act1-129 is about sixfold more sensitive in the cellular assay. If that reflects a stronger direct effect of ToxB on its actin monomers, predict greater suppression than in experiment 2: a longer or effectively absent actin nucleation phase, slower or undetectable elongation, and a still lower plateau relative to its own drug-free control. It could be almost flat. However, cellular hypersensitivity could also arise from reduced functional reserve or altered interactions with other proteins; it does not guarantee stronger inhibition of purified actin. These alternatives are exactly what the comparison with experiment 3 resolves. No numerical lag, slope or plateau can be calculated from halo sizes alone.

Injected sulfur dioxide is oxidized, principally through hydroxyl radical chemistry, to sulfuric acid, which nucleates or condenses into aqueous sulfate particles. A suitably sized, weakly absorbing particle population scatters some incoming sunlight back to space, increasing planetary albedo and decreasing absorbed solar energy. The sulfur dioxide molecules are precursors, not the persistent reflective population.

The albedo lifetime reflects aerosol formation, growth/aerosol coagulation, sedimentation, stratospheric transport and eventual return to efficiently scavenged tropospheric air. The stratosphere has little atmospheric precipitation, so the effect can outlast analogous tropospheric aerosol. The assumed year is an effective persistence scale, not a universal lifetime of sulfur dioxide oxidation or an exact clock for every particle.

A small Reynolds number, $\mathrm{Re}=\rho UL/\eta \ll 1$, means that inertia is small compared with viscous forces. It can arise from small speed, small length or large dynamic viscosity. Examples are the locomotion of a microswimmer and the slow sedimentation of a small particle in a viscous liquid.

To neglect local acceleration as well, an externally imposed time scale $T$ must satisfy $\rho L^2/(\eta T)=L^2/(\nu T)\ll 1$. This holds for a steady flow or a sufficiently slowly varying quasi-steady Stokes approximation. Small Reynolds number alone is insufficient for that extra step. For an incompressible flow with constant material coefficients and uniform gravity, the Navier-Stokes equation then reduces to the Stokes equation:Taking the divergence and using commuting spatial derivatives givesTaking the curl, which annihilates the pressure gradient and uniform gravitational force, givesThese are the Harmonic pressure and vorticity in Stokes flow results.

A concrete counterexample to inferring both solely from small Reynolds number is Stokes' second problem: an oscillating plate with arbitrarily small velocity amplitude has negligible inertia, but ${\partial }_t\mathbf{\omega }=\nu {\nabla }^2\mathbf{\omega }$ is nonzero. Its vorticity is not a harmonic function. The quasi-steady qualification is therefore necessary.

Increase particle volume fraction slowly by controlled evaporation or gentle concentration until ordering becomes favourable, while preserving sufficient colloidal repulsion to prevent irreversible sticking. Hard-sphere-like monodisperse suspensions can crystallize by packing entropy; weak, reversible attractions can also favour ordered assembly if they allow defect correction. A controlled drying front or patterned surface may guide the orientation, and gentle sedimentation can create a concentration gradient that crosses the ordering range.

The practical requirement is slow structural change relative to Brownian motion, with uniform temperature, well-characterized ionic strength and a narrow size distribution. An abrupt salt addition or rapid drying can remove the repulsive barrier and trap aggregates instead of growing a crystal. Scattering or microscopy should distinguish crystalline peaks/order from merely increased turbidity.

A useful microscopic screening estimate compares thermal energy with the buoyant gravitational potential energy over one particle radius. Write $\Delta \rho =50\mathrm{kg}{\mathrm{m}}^{-3}$ and $m_b=(4\pi /3)a^3\Delta \rho$. The sedimentation length is ${\ell }_g=k_{B}T/(m_{b}g)$, and the gravitational Peclet number isUsing $T=298\mathrm{K}$ and requiring ${\mathrm{Pe}}_g\lesssim 1$ givesEquivalently, combining Stokes law speed $v_s=2\Delta \rho g{a}^2/(9\eta )$ and diffusion coefficient $D=k_{B}T/(6\pi \eta a)$ gives $v_{s}a/D={\mathrm{Pe}}_g$. Buoyancy, not the full polystyrene density, is essential to the estimate. Taking a diameter rather than radius as the comparison length changes the numerical cutoff by an order-one factor.

There is no universal largest particle that stays uniformly suspended indefinitely. If uniformity over a vessel height $H$ is required, use ${\ell }_g\gtrsim H$, not merely ${\ell }_g\gtrsim a$. For example $H=1\mathrm{cm}$ gives a much smaller radius, about 58 nm. Observation time, density matching, convection, particle polydispersity and shape also matter. Independent of sedimentation, electrostatic/steric barriers must prevent aggregation; low ionic strength, suitable surface chemistry and clean solvent can be as important as size.

Diffusion and mobility. For a dilute isolated sphere in a Newtonian fluid, Stokes's law gives the drag magnitude $F_{\mathrm{drag}}=6\pi \eta Rv=\xi v$, where $\eta$ is the dynamic viscosity and $R$ is the hydrodynamic radius. The mechanical mobility is therefore $1/\xi$; the translational hydrodynamic mobility matrix is ${\xi }^{-1}I$.

To relate mobility to diffusion, put the particles in a weak external potential energy $U(x)$. Their drift speed is $-U^{\prime }(x)/\xi$. For number density $n(x)$, the sum of drift and Fick's first law fluxes isAt thermal equilibrium, the Boltzmann distribution gives $n\propto e^{-U/(k_{B}T)}$, and the net flux is zero. Substituting $n^{\prime }=-n{U}^{\prime }/(k_{B}T)$ into $J=0$ yields $D=k_{B}T/\xi$. Thus the Stokes–Einstein relation isHere $k_B$ is the Boltzmann constant and $T$ is absolute temperature. The assumptions include a no-slip approximately spherical particle, a continuum fluid, a small particle Reynolds number, and negligible hydrodynamic interactions with other particles or boundaries.

Gravitational sedimentation. The buoyant force must be included, since the mass of displaced water is comparable to the protein mass. Using $1\mathrm{Da}=1.66054\times 10^{-27}\mathrm{kg}$ givesso the buoyant mass in the stated spherical model isThe drag coefficient is $\xi =6\pi (10^{-3})(2.16\times 10^{-9})=4.0715\times 10^{-11}\mathrm{kg}{\mathrm{s}}^{-1}$. At terminal speed, $\xi v_g=m_{b}g$, soIgnoring buoyancy would give $1.47\times 10^{-11}\mathrm{m}{\mathrm{s}}^{-1}$ and would not use the supplied water density. This calculation treats the stated radius as both the hydrodynamic radius and the radius determining displaced volume; a real hydrated protein can require a separately measured partial specific volume.

Electrical drift. In the specified simple electrophoresis model, terminal force balance is $Ze\mathbf{E}=\xi \mathbf{v}$. Thereforewhere $\mu$ is the electrophoretic mobility and $e$ the elementary electric charge. For the given positive beta-lactoglobulin charge,in the direction of the electric field. This exceeds the gravitational sedimentation speed by about $8.6\times 10^6$.

Separation time. The two populations have equal drag coefficients because their radii are equal. Their difference in drift speed is therefore set by one elementary charge:Starting with coincident band centres, their separation is $\Delta vt$. HenceAt that time the original band has travelled $20\mathrm{mm}$ and the mutant $21\mathrm{mm}$. Their masses do not enter this ideal electrical force balance: equal-radius particles with different masses but the same charge would have equal electrophoretic mobility in this model.

Practical discrimination. A one-charge difference is enough for the stipulated separation, so the method is sensitive to mutations that change effective charge or friction. It cannot distinguish every mutation: substitutions that preserve charge and hydrodynamic size need not separate. Buffer pH changes protein ionization, and realistic electrophoresis also involves the ionic atmosphere, electro-osmosis, walls or gel interactions, and finite band widths.

For example, the Stokes–Einstein relation at $293.15\mathrm{K}$ gives $D=9.94\times 10^{-11}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. For an initially narrow band, diffusive broadening of electrophoretic bands givesat the calculated separation time. Thus the centres are about $3.15$ diffusion standard deviations apart, rather than forming infinitely sharp bands. The stated $1\mathrm{mm}$ centre-separation criterion is achievable in the model, but actual resolution also depends on the loading width and other dispersion mechanisms.

Equivalent centrifugal acceleration. For the original protein to attain its electrical drift speed by sedimentation, its buoyant centrifugal force must equal $20eE$:ConsequentlyThis is the acceleration at the protein's position in the rotating solution; no rotation rate can be assigned without a centrifuge radius. If the intended comparison is instead the relative separation speed of the two bands, use $\Delta m_b=600\mathrm{Da}$, since their displaced volumes are equal. Matching $|\Delta m_b|a_{\mathrm{eff}}=eE$ then gives $8.04\times 10^7\mathrm{m}{\mathrm{s}}^{-2}$. In that separation experiment the heavier original protein moves faster centrifugally, whereas the more highly charged mutant moves faster electrically.

A colloidal grain in a liquid is large enough to see with a microscope, yet small enough for thermal motion to compete with its tendency to settle. In the Perrin sedimentation experiment, the equilibrium number density falls with height: a visible miniature atmosphere. Counting grains at different heights measures an energy scale; tracking their Brownian motion measures a transport rate. Connecting both to the same Avogadro constant makes the experiment a quantitative test of kinetic theory. The Wiener measure describes an idealized path law; these measurements instead test a physical model and its material parameters.

Jean Baptiste Perrin's 1909 memoir traces his sedimentation experiments to May 1908. It reports relative counts $100$, $47$, $22.6$, $12$ at heights $5$, $35$, $65$, $95\mu \mathrm{m}$, and compares radius estimates from Stokes settling velocity, counted grain mass, and rows of touching grains.^[ref]

The 1926 lecture describes resin precipitated from alcohol into water, washed and fractionated by centrifugation to obtain nearly monodisperse gamboge or mastic grains. Diameter and density were measured separately. A sealed slide-and-cover-glass chamber prevented evaporation; a vertical microscope resolved roughly $2\mu \mathrm{m}$ layers through a roughly $0.1\mathrm{mm}$ depth. Refocusing and photographing compared layer populations after equilibrium; changing temperature reversibly changed the distribution.^[ref]

In Les Atomes (1913), Jean Baptiste Perrin restricts the eyepiece view to fewer than five or six grains and alternates observations between heights. Focus displacement requires a refractive index correction to recover actual layer separation. Horizontal positions were marked with a camera lucida, usually every $30\mathrm{s}$, and distances calibrated with a micrometer; the polygon joins sampled positions rather than recording the intervening trajectory.^[ref][ref]

The fraction of measured material assigned to polymer under a stated separation or detection criterion. For corrected pellet and supernatant measurements, $f_P=P/(P+S)$; the assignment is valid only if sedimentation separates the intended species.

Removal of reactive nitrogen from an air mass through formation and sedimentation of nitric-acid-containing particles. This can delay reformation of chlorine nitrate and prolong active chlorine chemistry. It is a physical/chemical atmospheric removal process, distinct from biological denitrification.