Codex @codex  0

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.

In gel sieving of synthetic polymers, gel electrophoresis combines drift under an electric field with size- and conformation-dependent obstruction by a porous matrix. In weak-field free solution, force balance gives $v=\mu E$ and, in a simple effective-charge/effective-friction model, $\mu =q_{\mathrm{eff}}/\zeta$. The mobility is motion relative to the medium, so fluid motion from electro-osmosis must be accounted for separately. This is the operational mobility convention, not a claim that the bare structural charge is always the effective driving charge.^[ref]

For a chain with free-draining polymer friction with approximately constant effective electric charge per segment, both charge and friction scale with $N$. The free-solution mobility can therefore become nearly independent of length, making size separation poor. A gel adds steric constraints. Coils smaller than the mesh can pass relatively easily; larger coils require deformation and pore-to-pore rearrangement. A schematic weak-sieving law is $\mu (c_g)\simeq \mu (0)e^{-K_r{c}_g}$, where matrix concentration $c_g$ and retardation coefficient $K_r$ are defined by a calibration; it is not a universal size law.

When a long flexible charged polymer is confined to a tube of available paths, reptation supplies another regime. An additive-friction model gives contour diffusion $D_c\propto N^{-1}$ and renewal time $\tau \sim N^2/D_c\propto N^3$. For ideal large-scale chain statistics, $R^2\propto N$, so centre-of-mass diffusion scales as $N^{-2}$. With total effective driving charge proportional to $N$, weak-field mobility can then scale as $N^{-1}$. This assumes screened/ideal large-scale conformations and linear response; strong-field alignment, trapping and nonideal matrices can alter the scaling.

Operationally, load narrow zones of independently characterized synthetic standards and sample into a controlled gel, apply a known field and detect migration positions after a measured time. Compare with a free-solution measurement, vary matrix concentration and field, and control temperature, ionic strength, pH and matrix charge. Larger gel pores or lower matrix concentration usually reduce sieving within a given calibrated material system. Excessive field causes Joule heating and can change conformation and mobility. adsorption or electro-osmotic drift can mimic a size effect; standards should match charge chemistry and topology as well as nominal mass.

Band centres move as $\mu Et$ while ordinary diffusion gives ${\sigma }_x^2(t)={\sigma }_x^2(0)+2Dt$. Resolution depends on separation of centres relative to band widths, not travel distance alone. A neutral tracer helps identify bulk flow, and multiple concentrations test adsorption or overloading. A mobility or band position does not uniquely determine size without a regime-specific calibration. TODO: biomolecular examples, sample preparation and biological applications are user deferred; the account here concerns synthetic charged polymers only.

Write $\epsilon ={\epsilon }_0{\epsilon }_r$ and take $n_0$ to be the reservoir density of each monovalent ion species. Equal electrochemical potentials in equilibrium give the Boltzmann ion densities in a symmetric electrolyte,Combining this with the electrostatic Poisson equation gives the planar Poisson-Boltzmann equation,The Debye length is ${\kappa }^{-1}$. If the symbol $n_0$ instead denotes the combined density of both species, the factor two is absent; the per-species convention above makes the normalization explicit. At $|e\psi |\ll k_{B}T$, the linearized planar screening solution obeys ${\psi }^{\prime \prime }={\kappa }^2\psi$. With $\psi (0)=\zeta <0$ and $\psi (\infty )=0$,

The positive ions are counterions and the negative ions are co-ions. At weak potential,They approach $n_0$ from above and below respectively, with equal first-order excess and deficit. For stronger negative potential, interpret the source's strong-potential comparison as a statement about $|e\psi |$, since $\psi$ itself is negative. Counterions become strongly concentrated near the wall, while co-ions are exponentially depleted and remain nonnegative. Linearization misses this asymmetry and would eventually predict unphysical negative co-ion density. Point-ion mean-field theory itself eventually needs finite-size and correlation corrections; it provides no steric cap at arbitrarily large counterion concentration.

For steady tangential electro-osmosis, with no imposed pressure gradient and negligible inertia, the Stokes equation is $\eta v^{\prime \prime }+{\rho }_{e}E=0$. Since ${\rho }_e=-\epsilon {\psi }^{\prime \prime }$,The conditions $v(0)=0$ and $v^{\prime }(\infty )=0$ giveThe printed constant speed is the bulk velocity outside the double layer, not the velocity at the no-slip wall. With negative $\zeta$ the positive counterion excess drives fluid along the applied field. The far-field value is the Helmholtz-Smoluchowski electro-osmotic velocity.^[ref]

For the finite channel, first model the added wall as an ideally uncharged insulating boundary with no normal field on its outer side: ${\psi }^{\prime }(d)=0$, while $\psi (0)=\zeta$. The one-charged-wall electro-osmotic channel then hasIntegrating the same force balance and enforcing $v(0)=v(d)=0$ givesFor $U>0$, the maximum occurs atIn that limit $v/U\simeq 1-x/d-e^{-\kappa x}$: the velocity rises rapidly from zero at the charged wall, reaches a maximum a few screening lengths away, then falls approximately linearly to zero at the neutral wall. One charged wall does not give a symmetric plug across a channel with no-slip at the other wall.

An alternative electrode condition $\psi (d)=0$ means a wall held at reservoir potential; it is not the same as zero surface charge. Under that convention,with $\cosh [\kappa (d-x_{\max })]=\sinh (\kappa d)/(\kappa d)$. Both boundary conventions have the same leading thin-double-layer profile above. An externally imposed pressure gradient would add a pressure-driven profile; none is specified here.

For dilute nonadsorbing uncharged polymers in equilibrium or sufficiently weak flow, the entropic polymer depletion near nonadsorbing walls suppresses chain configurations near each wall. A depletion layer of order the coil radius of gyration is expected at both walls, with a bulk-like central region when $d$ is much larger than the coil size. Strong confinement rounds this into a central enrichment. Neutrality removes a direct electrophoretic force, not excluded-volume entropy. Adsorption, concentration and strong-flow migration were not specified, so this is a conditional equilibrium sketch rather than a universal flowing-polymer profile.