Electrophoresis

Electrophoresis is motion of charged dispersed species relative to a surrounding medium under an applied electric field. The drift depends on electrical driving and friction, including the surrounding ionic atmosphere. Gel electrophoresis adds a porous matrix that makes migration sensitive to molecular size and conformation.

Capillary electrophoresis separates species by their motion in an electric field inside a narrow liquid-filled tube. In its free-solution form, different electrophoretic mobilities produce different drift speeds; a detector at a fixed position records the arriving bands as an electropherogram. The liquid itself can also move by electroosmotic flow, so arrival order is not determined by molecular charge alone.

A simple apparatus places the ends of a capillary in two buffer reservoirs containing electrodes. An applied electric potential difference drives migration along the tube; a detection window interrogates the passing material. In the uniform-field model, with total capillary length $L$, detector distance $L_d$ and signed applied potential $U$,The last expression requires net motion towards the detector. Thus a neutral marker can move with electroosmotic flow even though its electrophoretic mobility is zero. The instrument's distinction between total length and detector distance matters when converting migration times into apparent mobility.^[ref]

For an original ideal example, take $L=0.50\mathrm{m}$, $L_d=0.40\mathrm{m}$ and $U=20\mathrm{kV}$. Two net mobilities $(4,5)\times 10^{-8}{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ give speeds $1.6$ and $2.0\mathrm{mm}{\mathrm{s}}^{-1}$ and migration times 250 and 200 s. These are model values, not assigned chemical identities. Band widths from diffusion, injection and other dispersion mechanisms determine whether the time difference is actually resolved.

Narrow tubes also help remove Joule heating. For a uniform cylindrical liquid conductor, heat generation per volume is $\sigma E^2$, with electrical conductivity $\sigma$, while wall area per liquid volume is $2/a$ for tube radius $a$. This geometric scaling motivates small bores, although temperature control still depends on heat transfer through the wall and surroundings. Increasing the electric field indefinitely is therefore not a free route to perfect separation.

A primary 1981 demonstration used a 1 m glass capillary with 75 micrometre internal diameter, a 30 kV applied potential and a fluorescence detector. It resolved fluorescent derivatives of amino acids, peptides and urinary amines. The observed result was separated fluorescent zones; the need to derivatize the samples was an explicit detection limitation.^[ref]

The capillary electrophoresis heat-dissipation tradeoff is the balance between rapid electrically driven separation and the temperature rise caused by Joule heating. Narrow capillaries offer a large wall area per liquid volume, while high electric fields increase migration speed and electrical heat production. The 1981 primary experiment demonstrated efficient separations in a 75-micrometre-bore, one-metre glass capillary at 30 kV; it does not imply that arbitrarily high fields are thermally harmless.^[ref]

A simple model makes the competing scalings explicit. For a cylindrical liquid column of radius $a$, uniform electrical conductivity $\sigma$ and axial field $E$, current and heat production per unit length areAt fixed conductivity and bore, doubling the field doubles drift speed for constant electrophoretic mobility but quadruples generated heat. Halving the bore radius reduces heat per length by four at fixed field.

For steady radial heat conduction, let the liquid's thermal conductivity be $\kappa$ and impose a fixed wall temperature $T_w$. With uniform volumetric heating $q^{\prime \prime \prime }=\sigma E^2$, the cylindrical heat equation givesRegularity at the axis and the wall condition fix the integration constants. The center-to-wall rise is $\Delta T_{\max }=\sigma E^2{a}^2/(4\kappa )$. The symbol $\kappa$ here denotes thermal conductivity, not an optical decay constant.

For stipulated $\sigma =0.1S/m$, $E=30kV/m$, $a=25\mu \mathrm{m}$ and $\kappa =0.6W/(\mathrm{m}\mathrm{K})$, the model gives $P^{\prime }=0.1767$ W/m and center-to-wall rise 0.02344 K. Doubling the radius or field multiplies that rise by four. This ideal calculation holds the wall temperature fixed; it does not include heating through the capillary wall, instrument housing or cooling system.

Temperature gradients can change liquid viscosity, migration and band dispersion. Conversely, reducing bore size also reduces sample volume and optical path through the sample. A useful instrument experiment therefore records current and separation behavior while controlling cooling conditions, rather than judging success only by reduced migration time. The thermal model predicts what to test; it is not a measured temperature profile from the cited experiment.

An electropherogram is a recorded trace of an electrophoresis separation. In capillary electrophoresis, it commonly plots detector signal against migration time: separated populations reach a detection window at different times and produce peaks. Peak position, width, height and integrated area describe different features of that trace.

The position depends on the migration mechanism and geometry. In a simple uniform-field model with detector distance $L_d$, electric field $E$, electrophoretic mobility ${\mu }_{\mathrm{ep}}$ and electroosmotic flow represented by mobility ${\mu }_{\mathrm{eo}}$,when the resulting drift is towards the detector. A changed buffer-flow contribution can therefore shift migration times without changing the analyte's identity. The instrument's actual timing and mobility conventions must be specified; the first-party software documentation gives its explicit migration-time and apparent-mobility definitions.^[ref]

An electropherogram peak area is the integral above an assigned baseline. For an original Gaussian-component model,The component area is $A_i$, while its height above baseline is $A_i/({\sigma }_i\sqrt{2\pi })$. Choose $(A_1,{\sigma }_1,t_1)=(1,0.4,5)$ and $(A_2,{\sigma }_2,t_2)=(2,0.8,10)$ in compatible signal/minute units. The two component heights are equal, although the second has twice the integrated signal. These are synthetic components, not experimental analyte concentrations.

The cited software defines time-corrected area as $A_i/t_i$.^[ref] In this example, both corrected areas are $1/5=2/10=0.20$. This is another observable, not permission to equate every corrected area with an absolute amount without response calibration and an appropriate measurement model.

A peak can also contain unresolved co-migrating species. Its apparent simplicity does not establish chemical purity. Useful interpretation therefore combines the trace with baseline selection, migration references, detector response and independent identification evidence, rather than treating each visible peak as a complete chemical explanation.

An electropherogram peak area is the time integral of detector signal above an assigned baseline over a specified peak interval:Its units are signal multiplied by time. It differs from peak height and from the amount of material unless a detector-response and transport model connects them. Broadening a peak can lower its height while preserving its area.

For an isolated Gaussian distribution of signal, $s-b=H\exp [-(t-t_m{)}^2/(2{\sigma }_t^2)]$, integration over the complete peak gives $A_p=H{\sigma }_t\sqrt{2\pi }$. An original peak with $H=2$ signal units and ${\sigma }_t=3$ s has area $15.04$ signal-seconds. Doubling the width at unchanged area halves the height. Integrating only a finite interval loses the corresponding tails; integrating overlapping peaks can include another species' response.

Why can migration time enter quantification? Suppose a concentration-sensitive detector gives $s-b=kc$ for a band moving at constant speed $v$ through a capillary of internal area $S$. The amount passing the detector isSince $v=L_d/t_m$, $n\propto A_p/t_m$ at fixed $S$, $L_d$ and response coefficient $k$. A slow band spends longer in the detection window and can have more raw time-integrated signal for the same amount. This motivates time-corrected electropherogram peak area; the first-party software explicitly calculates it as raw area divided by migration time.^[ref]

For stipulated raw areas 120 and 180 signal-seconds at migration times 200 and 300 s, both corrected values are 0.60 signal units. Equal corrected values imply equal modeled amounts only if the relevant response coefficients and sampling conditions agree. Different species need not have equal detector responses, and a detector measuring a different physical observable requires its own model.

Baseline error has an especially simple consequence. If the estimated baseline exceeds the true one by $\delta b$ throughout a window of duration $T$, the area error is $-\delta bT$. A 0.01 signal-unit offset over 60 s subtracts 0.60 signal-seconds. Blank measurements, explicit integration limits and inspection of unresolved neighbors therefore matter even when the numerical integration itself is exact.

A time-corrected electropherogram peak area is a raw electropherogram peak area divided by its migration time,It compensates for the longer passage time of a slower band through a concentration-sensitive detector in capillary electrophoresis. The first-party processing documentation defines this quantity explicitly and computes a percentage by dividing each corrected area by the sum of corrected areas.^[ref]

The transport model explains the correction. If the baseline-subtracted detector signal is $kc(t)$, internal capillary area is $S$ and band speed through the window is constant $v$, thenWith detector distance $L_d$ and $v=L_d/t_m$,Thus dividing the time integral by migration time removes one speed dependence. It does not determine the response coefficient $k$, injected sample amount or chemical identity.

For an original example, two peaks have raw areas 90 and 150 signal-seconds and migration times 150 and 300 s. Their corrected areas are 0.60 and 0.50 signal units. With equal detector response coefficients, their modeled relative amounts are $6:5$, giving $54.55\%$ and $45.45\%$ of the corrected total. Raw-area normalization would instead give $37.5\%$ and $62.5\%$.

Species-dependent detector response remains important. If the first coefficient is twice the second, amounts are proportional to $0.60/2$ and $0.50/1$, giving $37.5\%$ and $62.5\%$ instead. Equal normalized corrected areas therefore need not mean equal amounts when the detector responds differently to the species. A measurement calibration must match the response and sample conditions being interpreted.

The reduction also assumes each band travels at a representative constant speed. If the electric field or electroosmotic flow changes substantially during a run, $L_d/t_m$ is an overall migration speed and need not equal the speed during detection. Baseline error, unresolved neighbors and nonlinear detector response are likewise not repaired merely by dividing by time. The software's chosen migration-time definition is part of the calculation: its CE mode uses the peak apex, while its CIA mode uses the midpoint of the start and end times.^[ref]

A porous gel changes the electrophoretic mobility of charged synthetic polymers through confinement, deformation and pore-to-pore dynamics. Free-draining charge and friction can both grow proportionally to chain length, leaving free-solution mobility nearly length independent; matrix obstruction adds size sensitivity. reptation supplies one weak-field long-chain regime, not a universal calibration. Matrix concentration, effective charge, solvent conditions, conformation, topology, adsorption, electro-osmotic drift and heating can all alter migration. Size standards should match the relevant chemistry and operating regime.

A narrow population band undergoing uniform electrophoresis and ordinary diffusion has mean position $x_0+\mu Et$ and variance ${\sigma }_0^2+2Dt$. This follows from the advection-diffusion equation by changing to the coordinate $x-\mu Et$. For two species with mobility difference $\Delta \mu$, their centre separation is $|\Delta \mu |Et$, whereas diffusion widths grow only as $\sqrt{t}$. Starting from point-like bands with common diffusion coefficient $D$, a required separation of $q$ standard deviations needsA prescribed distance between centres alone is a resolution convention; finite loading widths and other dispersion mechanisms must be included when interpreting actual electrophoresis data.

The electrophoretic mobility $\mu$ is the proportionality coefficient between drift velocity and a weak applied electric field. Its sign specifies motion relative to the field. When only travel distances are used, the mobility magnitude is often intended. For long linear DNA in a weak-field reptation regime, $|\mu |$ can scale inversely with chain length, whereas free-solution mobility is approximately length-independent. Neither law supplies a universal gel electrophoresis size calibration for arbitrary gel concentration, field strength or molecular topology.

The small-particle, thick-double-layer approximation ${\mu }_e=2ϵ\zeta /(3\eta )$ under its linear electrokinetic assumptions.

The thin-double-layer approximation ${\mu }_e=ϵ\zeta /\eta$, neglecting important relaxation and surface-conduction corrections.