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370 Electrophoresis

experimental conditions so that no net transport of ionic dm(e)/dt =−ACqV/Nf. (15)
species occurs. It is obvious that transport of a solute can
Equating Eq. (14) with Eq. (15) and eliminating A gives
occur only if the density of the ion is greater than that of
the solvent. Despite the considerable difference between d ln C/dx =−qV/DN f. (16)
electrical and gravitational forces, it would not be possi-
The net charge q of the polyampholyte is a function of pH,
ble to generate a sufficiently large electrical field in polar
and if we assume a linear pH gradient, then the charge at
solutions that would force a particle to settle in a solvent
position x[q(x)] is defined as
of higher density than itself. The current would be suffi-
cient to boil the solution. Hence, it is possible to design q(x) = q(0) + (x − x 0 ) dq/dx, (17)
an experiment where at the beginning a band of solution
where dq/dx is determined by dq/dpH and dpH/dx,
containing macroions is placed on top of a solvent column
the experimental variate. The reference charge q(0) in
consisting of a preformed density gradient. This gradient
Eq. (17) is chosen as that found at the isoelectric point
can be formed by varying the concentration of a neutral
of the macroion (x 0 ), which is zero. Thus, substituting
molecule down the column. (Sucrose might be used since
−3
it is neutral and has a density of ∼1.6gcm ; protein den- Eq. (17) into Eq. (16) gives (F ≡ Nf )
−3
sities are ∼1.3gcm .) If the band is now electrophoresed d ln c dq V
=−(x − x 0 ) . (18)
down the column, a point occurs where the density of the dx dx FD
macroion is less than that of the solvent; transport stops When values of x − x 0 are small, dx can be replaced by
and a stable boundary forms in this plane, which is often 1 d(x − x 0 ) . Thus, retaining the experimental variable
2
2
called its isopycnic point. The charge on the macroion has dpH/dx, Eq. (18) becomes
not been neutralized by the gradient, so this is not an elec-
dq dpH V
trical equilibrium but an equilibrium between two equal 2
d ln c =− d(x − x 0 ) . (19)
but opposite forces on the macroion. 2dpH dx FD
A different situation can be generated for poly- Integration between the limits of c and x by defining c 0 as
amopholytes where, instead of electrophoresing a band the concentration at x 0 gives
along a density gradient, one forces the band to travel 2
along a pH gradient. In this case the net charge of the −(dq/dpH (dpH/dx) V (x − x 0 )
C = C 0 exp . (20)
polyampholyte decreases as it moves toward its isoelec- 2FD
tric point until it reaches the pH where it carries no net
By analogy with the Gaussian probability relation-
charge and a stationary boundary forms as a result of the
ship, the width of the profile equals [FD/(dq/dpH)
equilibrium between electrical and diffusive forces. An
(dpH/dx)V ], where (dpH/dx) and V are two experi-
infinitely thin zone is not formed at equilibrium for ei-
mental variables, so the only molecular parameters are
ther of these conditions because diffusion disperses the
(dq/dpH), F, and D. The product FD equals RT (R be-
zone and the zone is stable only as long as the electri-
ing the gas constant and T being the temperature). Hence,
cal field is applied. The resulting shape is approximately
Eq. (20) becomes
Gaussian. A functional relationship can be derived for the

isolectric equilibrium that relates the concentration at any −(dq/dpH (dpH/dx) V (x − x 0 ) 2
point within the zone to the molecular properties of the C = C 0 exp , (21)
2RT
macrion.
The electrophoretic force qV on the polyampholyte in inwhichthenumeratorisanenergyterm.Asimilarexpres-
an electric field of V volts per unit distance is negative sion can be derived for the isopycnic experiment. How-
because it moves against the voltage gradient and is ever, the width of the band would be inversely proportional
to the density gradient and density of the macroion instead
−qV = (dx/dt) Nf. (13) of being inversely proportional to the pH gradient and dif-
ferential charge for equilibrium at the isoelectric point; the
The opposing force at equilibrium is given by Fick’s first
latter is called isoelectric focusing.
law of diffusion for the mass flux dm/dt through an area
A:
E. Electrophoretic Mobility, Sedimentation
dm(D)/dt =−DA dc/dx. (14) Coefficient, and Diffusion Coefficient
for Macroions
At equilibrium the net transport across any plane is zero,
so the electrophoretic flux equals that of diffusion. The Three common mobilities are used to describe the size
electrophoretic mass flux dm(e)/dt is and shape of a macroion: (1) electrophoretic mobility,