DISC ELECTROPHORESIS—I
BACKGROUND AND THEORY*
Leonard Ornstein
Cell Research Laboratory, The Mount Sinai Hospital, New York, N. Y.
Although electrophoresis is one of the most effective methods for the separation of ionic
components of a mixture, the resolving power of different electrophoretic methods is quite
variable. To separate two component ions, it is necessary to permit migration to continue until
one of the kinds of ions has traveled at least one thickness of the volumes that it initially occupied
(the starting zone) further than the other. However, the sharpness, and therefore the resolution, of
the zones occupied by each ion diminishes with time because of the spreading of the zones as a
result of diffusion. Remarkable resolution has been achieved when advantage is taken of the
frictional properties of gels to aid separation by seiving at the molecular level (see Smithiesl). A
new method, disc electrophoresis,† has been designed that takes advantage of the adjustability of
the pore size of a synthetic gel and that automatically produces starting zones of the order of 10
microns thickness from initial volumes with thicknesses of the order of centimeters. High
resolution is thus achieved in very brief runs.
With this technique, over 20 serum proteins are routinely separated from a sample of whole
human serum as small as one microliter in a 20-minute run (see FIGURE 1). Direct analysis of
even very dilute samples becomes routine because the various ions are automatically concentrated
to fixed high values at the beginning of the run just prior to separation. Preliminary laboratory
studies and theoretic considerations provide evidence of the applicability of this technique to a
wide range of ionic species for both analytic and large-scale preparative purposes.
Theory has also provided the basis for a simple application of disc electrophoresis to the
simultaneous determination of both the free mobility and the aqueous diffusion constant of a
protein.
This report will detail some mechanisms that provide a rationale for the resolution afforded by
zone electrophoresis in many gels; will develop the theory of some new modifications of zone
electrophoresis that have been designed to take maximum advantage of these mechanisms; and
will provide some examples of the results that disc electrophoresis has produced.
BACKGROUND
This study was first stimulated by the revolutionary results of the starch gel technique of
Smithies 2 and the application of this technique by Hunter3,4 for producing “zymograms” of
enzymes of histochemical interest. Our recent cytochemical studies,5-7 which were then well
under way, promised to be substantially clarified by the application of Hunter's technique.8 We
had heard of the variability of the starch gel and of difficulties in its preparation. We also had had
______________________________________________________________________________
* The present manuscript is an expanded and updated version of a paper that was first made available to
the scientific public in preprmt form in January, 1962, through the generosity of the Distillation Products
Division of Eastman Kodak Company, Rochester, N. Y.
† The name was derived from the dependence of the new technique on discontinuities in the
electrophoretic matrix and, coincidentally, from the discoid shape of the separated zones of ions in the
standard form of our technique.
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Ornstein: Disc Electrophoresis
323
experience with polyacrylamide gels because, some years earlier, they had been recommended to
the author by G. Oster9-13 as a potential embedding medium for sectioning tissues. In contrast to
starch gels, polyacrylamide gels are thermostable, transparent, strong, relatively inert chemically,
can be prepared with a large range of average pore sizes, and are non-ionic (even starch carries a
few anionic groups which are, in part, responsible for the backwards endosmotic flow noted with
its use). It seemed that attempts to duplicate the desirable properties of starch gels with
acrylamide might be rewarding. The first attempt, designed and executed by B. J. Davis, was so
successful that it encouraged Davis and me through almost a year of successive failures until an
understanding of mechanism and a reasonable degree of reproducibility began to be achieved. It
was about that time that we first reported our preliminary results.8,14 Soon afterwards, an
independent report from the Pepper Laboratory of the University of Pennsylvania also appeared
recommending the use of polyacrylamide as a substitute for starch gels.15
DIFFERENCES BETWEEN ELECTROPHORESIS IN GELS AND OTHER M ETHODS
Two striking differences had characterized the behavior of serum proteins during separation by
zone electrophoresis in gels when compared to their behavior during paper or moving-boundary
electrophoresis:
(1) The measured mobilities are different in magnitude and even the order of the mobilities is,
in some instances, reversed. Smithies had proposed that a sieving effect in the gel, based on
molecular size of a protein relative to the gel pore size might account for this difference. 2 Such
sieving possibilities had been first considered, but without notable success, by Synge and
Tiselius. 16
(2) More fractions are resolved in gels and the zones are usually narrower than in paper or free
electrophoresis for equal separations from the origin and for equal running times.
MECHANISMS OF D IFFERENCES
There are two primary mechanisms responsible for these differences:
(1) The viscous properties of gels and solutions of very long chain polymers are unique. A
particle moving through a gel experiences a frictional resistance, f, which is a complex function
of r, the particle's radius. The viscosity of a gel is low when 2r is small compared to the average
pore of a gel, and is virtually infinite when 2r is very large compared to the pore size (see
Appendix A). In the case of a solution of very long chain polymers, the situation would be the
same except that as 2r approaches the length of the polymer chains, the rate of change of viscosity
will again decrease somewhat as the particles begin to be able to transfer sufficient momentum to
the chains to displace them, to “enlarge” pores, and therefore to “slide” through the network of
linear chains (see FIGURE 2A). These cases contrast with the case for both Newtonian and nonNewtonian liquids, where viscosity, defined by Stoke's Law, η = ƒ/6πr, is constant and
independent of r because in this case ƒ is directly proportional to r.
(2) The thinner the starting zone in the direction of the electric field, the higher will be the
resolution (until the point is reached where diffusion spreading of a zone's edges becomes large
colnpared to the starting dimensions) .
Pore Size
We will now examine some aspects of (1):
On the basis of a crude model (see Appendix B), it can be computed that a 71/2 per cent
Ornstein: Disc Electrophoresis
325
TABLE 1
PARAMETERS OF S OME PROTEINS*
______________________________________________________________________________________
Molecular
Protein
Mobility, mw
weight
Length
Diameter
________________________________________________________________________
Albumin
Transferrin
ß 1 lipoprotein
γ globulin
Fibrinogen
α2 macroglobulin
-6.1
-3.3
-3.0 (approx.)
-1.0 (approx.)
-2.1
-4.2
69,000
90,000
1,300,000
156,000
400,000
850,000
150
190
185
235
700
—
38
37
185
44
38
—
___________________________________________________________________________________________________________
* Data from Oncley l7 and Schultz.l8 mw in mobility units, length and diameter in Angstrom
units. 1 mobility unit = 10-5 cm.2/volt-sec. Approximate mobilities for 0°C.
polyacrylamide gel (or solution of linear polymer) will have an “average pore size” of about 50
Angstroms (the diameter of the hydrated chain is about 10 Angstroms). TABLE 1 gives some
dimensions of a few plasma proteins. We would predict that a 71/2 per cent polyacrylamide
should exhibit extreme frictional resistance to the migration of fibrinogen, , ß1 lipo-protein (and
perhaps the α2 macroglobulin and γ globulin), and that the other proteins should be able to pass
through, though with substantially more difficulty than in a simple aqueous system. In FIGURE 1
(a 7 per cent gel) we find these expectations corroborated. Thus, with a synthetic polymer like
polyacrylamide, because the average pore size of a gel depends on the concentration of polymer
(e.g., a 30 per cent gel produces about a 20 Angstrom pore), we can tailor the pore size to the
dimensions of the molecule to be separated.
Diffusion Constant and Free Mobility
This opportunity to prepare gels of different pore sizes suggests a simple method for the
simultaneous determination of the “free mobility” and aqueous diffusion constant of a protein. If
two gels are prepared with different pore sizes (gel 1 and gel 2) but identical ionic components
(X 1 = X2, see Appendix C), the distances traveled by a protein in the two gels, dl and d2, in
parallel runs at the same voltage gradient, V, and for a fixed time, t, will yield mobilities mll = d l
/tV and m2 = d2/tV. In addition, m l = QX1/f1and m2 = QX 2/f2, where Q is the charge of the
protein, and fl and fl are the frictional resistances of the two gels, and since the diffusion constants
___________________________________________________________________________________________________________
FIGURE 2. (A) Hypothetical viscosity [as defined by Stokes' Law, η = ƒ/6πr, using spherical test particles (e.g.,
proteins) of different r ] as a function of the particle radius, for a polyacrylamide gel and solutions of long chain linear
polymers of different average chain lengths, all "prepared" from 71/2% acrylamide monomer solutions. (B) Ordinate
and abscissa as in A. ........."71/2%" gel; __.__, "3%" gel; -------, "20%" solution of linear polyacrylamide chains with
RMS "coiled-chain lengths" of about 1000 Angstroms and molecular weights of about 106;––––, "Combination" gel
consisting of a "3%" gel "filled" with the solution of linear polymer. Two ß1 proteins, Transferrin C, • , and ß1 lipoprotein, °, are located on the "71/2" and "Combination" gel curves. (C) Hypothetical "calibration curves" for
determining the diffusion constant in water, Dw and the "free mobility" in water, mw , from the ratio of the measured
mobilities of any unknown protein in two standard gels of different average pore size. The value of m1/m2 for the
unknown protein is determined from measurement and is located (as indicated) on the appropriate curve. The abscissa
gives the value of Dw .The value of the ordinate for the point on the second curve with the same abscissa value
provides the required value of mw//m1 for this unknown protein. Since m1 is a measured value mw/ can be computed
directly.
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in the gels, D1 = KT/ fl and D2 = KT/f2, where K is Boltzmann's constant and T is the absolute
temperature, then
m l/m 2 = dl/d 2 = f2/f1 = D 1/D.
From a series of molecules of known mobilities in water, mw, and known diffusion constants in
water, Dw “calibration curves” of Dw against ml/m 2 and mW /m l can be constructed (see FIGURE
2C). From the curves, a measured value of d1/d 2 for an unknown protein permits the
determination of the diffusion constant of the protein in water, Dw, and therefore the calculation
of its “free mobiliy" in water, m w = m1(m w/m1).
Thin Starting Zones
We will now examine an approach for capitalizing on the potentials of (2):
As early as 1897, Kohlrausch 10 observed that, under conditions set by his “regulating
function” (derived in Appendix D), if two solutions of ions were layered one over the other, such
that a solution with ions of substance γ of high mobility were placed below a solution of a slow
ion, α, of the same sign of charge, then the boundary between the two ionic species would be
sharply maintained as they migrated in an applied electric field because the two species (on each
side of the boundary) would have been “arranged” to move at the same speed (provided that the
lower solutions were the denser of the two, and that the potential were applied with such polarity
that the ions of α and γ moved downwards).
If concentrations different from those specified by this regulating function existed when the
potential was applied, it was shown that the concentrations at the boundary would, in time,
“regulate” automatically to those required by the regulating function. Large departures from
“regulating” concentrations would, however, lead rapidly to conventive instabilities, preventing
the ready formation of the moving boundary. On the other hand, in gels and porous media (which
prevent convection), given sufficient time, a sharp moving boundary will form and be maintained
independent of the densities of the starting solutions and their initial concentrations, provided that
the faster ions precede the slower.
Equation 9 from Appendix D,
(A)
x γcα
x αcγ
m αzγ(mγ – mβ)
m γzα(mα – mβ)
— = —— = —————–
(Γ)
where x is the fraction of dissociation, c is the concentration, m is the mobility, and α, γ, and ß
are the ions (with ß of opposite charge to α and γ and common to both solutions) provides us with
the ratio of the total concentrations of α substance (A) to γ substance (Γ) for initiating and
maintaining a stable moving boundary.
For the purpose of illustrating how Equation 9 can be used to produce thin starting zones, α
will be the glycinate ion, ß the potassium ion, and γ the chloride ion. Then,
zα = -1
zγ = -1
m α = -15 mobility units20 (see footnote to TABLE 1 ),
m β = +37 mobility units (see footnote to TABLE 1 ),
m γ = -37 mobility units (see footnote to TABLE 1),
and therefore (glycine)/(chloride) = (A)/(r) = 0.58 (essentially independent of pH of either
solution from pH 4 to pH 10, within which range, neither the hydrogen nor hydroxyl ions will
appreciably contribute to conductivity, provided that the chloride concentration is greater than
10-3 M).
Ornstein: Disc Electrophoresis
327
Above pH 8.0, most serum proteins have free mobilities in the range from –0.6 to –7.5 units. If
the effective mobility of glycine, mαx α, were less than –0.6, the mobilities of the serum proteins
would fall between that of the glycine and that of chloride. This requirement is satisfied when x α
= 1/30, since the glycinate ion has a mobility of –15 units. The pH at which this degree of
dissociation of glycine occurs, can be calculated. From Equation l0a of Appendix D,
pH = pKa–log10 [ (1/ x α) – 1 ] .
Therefore the pH of the glycine solution must be 8.3 if xα = 1/30.
If a protein molecule with a mobility of –1.0 units is placed in the glycine solution (pH 8.3)
one centimeter above the glycine-chloride boundary, by the time the boundary (or a glycine
molecule) has migrated one centimeter, the protein will have migrated two centimeters and will
then be located at the boundary. It would continue to run at the boundary because the mobility of
the chloride is greater than that of the protein. If a very large number of albumin molecules
(mobility approximately –6.0 units) are placed in the glycine solution (pH 8.3), they will
concentrate at the boundary between the chloride and the glycine at a concentration satisfying
equation 9 where α is now the serum albumin, β, the potassium ion, and γ, the chloride ion. Then,
zα = –30 (approx. charge of albumin molecule at pH 8.32l ),
zγ = –1,
m α = –6.0 mobility units,
m β = + 37 mobility units,
m γ = –37 mobility units,
and therefore (albumin)/(chloride) = 9.3 x 10-3 . If (chloride) = 0.06 M, then (albumin) = 0.00056
M. That is, the albumin (M.W. 68,000) will automatically concentrate to about 3.8 per cent
behind the chloride and would then stay at constant concentration. If the initial concentration of
albumin had been 0.01 per cent in the glycine buffer, and if 1 milliliter of this mixture is placed
on top of the chloride solution in a cylinder of one square centimeter cross section, then after the
chloride boundary has moved about one millimeter, all of the albumin will have concentrated into
a disc (right behind the chloride) which would be about 25 microns thick. (In practice, this might
be done by using a porous anticonvection medium all through the volume occupied by the
chloride and through the one centimeter height of the glycine column.) In this manner, a 380-fold
increase in concentration can be achieved in a few minutes and the protein is reduced to a very
thin lamina or disc. If the original concentration had been 0.0001 per cent, the same final
concentration would result, but the total change would now be 38,000-fold. If the column were
100 cm. in length, the same amount of protein would have been concentrated. If, instead of a
single protein, a one milliliter mixture with mobilities ranging from –1.0 to –6.0 units is placed
over the chloride solution, by the time the boundary has migrated one centimeter in the applied
electric field, all the proteins will have concentrated into very thin discs, one stacked on top of the
other in order of decreasing mobility, with the last followed immediately by glycine. We will call
this process “steady-state stacking.” If the chloride boundary (and the following stack of discs)
is permitted to pass into a region of smaller pore size such that the mobility of the fastest protein
drops below that of the glycine, the glycine will now overrun all the protein discs and run directly
behind the chloride, and the proteins will now be in a uniform linear voltage gradient, each
effectively in an extremely thin starting zone, and will migrate as in “ordinary zone
electrophoresis.” (However, the rate of increase of frictional resistance with molecular diameter
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Annals New York Academy of Sciences
.
is so large with “ordinary gels” (see FIGURE 2A) that an ordinary small-pore gel that would slow
the albumin to less than –0.5 mobility units would completely exclude most proteins with
molecular weights above 150,000. A specially formulated gel will permit us to overcome this
restriction. However, discussion of this problem will be deferred to page 330.
Alternatively, if the boundary (and the following stack of discs) is permitted to pass into a
region of higher pH, e.g., a pH of 9.8 (the pKa of glycine), at which x gly equals one half and
therefore mglyx gly equals –7.5 mobility units, the glycine will now overrun all the protein discs
and run directly behind the chloride, and the proteins will now be in a uniform linear voltage
gradient, each effectively in an extremely thin starting zone, and will migrate as in "free
electrophoresis." The required stationary pH boundary is quite easily established (see Appendix
E) subject to degradation only by diffusion.
Combining Pore-size Control and Thin Starting Zones
We will now combine the mechanisms of 1 and 2 (page 323):
An electrophoretic matrix can be prepared into which discontinuities in pH and gel pore sizes
as well as Kohlrausch conditions are incorporated (FIGURE 3) .
The protein sample is placed as shown in FIGURE 3A and, as a voltage is applied, instead of
the proteins running ahead of the glycine to catch up with the chloride (as described above), the
chloride overtakes the proteins which then "sort" out according to their mobilities into highly
concentrated discs of protein, stacked exactly as described above (see FIGURE 3B). A "largepore" (approximately 3 per cent acrylamide) gel is usually used as the porous anticonvection
medium in this region as well as in the "spacer" region (see FIGURE 3). The protein sample is
mixed into the "spacer mixture" and is usually gelled in place on top of the spacer. As the glycine
following the stack of proteins moves through these regions, the pH is maintained at 8.3 (see
Appendix E).
At some time after stacking is complete (the time depending on the thickness of the spacer gel)
the proteins reach the small-pore gel where changes in their mobilities occur. Because of the
special viscous properties of the gel, proteins of equal free mobility but of appreciably different
molecular weight (different diffusion constant) will migrate with markedly different mobilities
and will easily be separated (see FIGURE 3C). (Five per cent to 10 per cent acrylamide gels have
proved to effect useful separations of human serum proteins.) In a 71/2 per cent gel, the fastest
pre-albumin has a mobility less than –5.0 units. We therefore arrange for a “running pH” of about
9.5 where the effective mobility of glycine (mglyx gly) is about –5.0 units in the 71/2 per cent gel,
due to both the lower value of xgly and the slightly higher “viscosity” of a 71/2 per cent gel
(compared to water) for such a small ion.
Given the concentration of chloride (Γ), and the value of xα = 1/30, it is possible to compute
from Equations 9, 18, 19, and 20 of Appendices D and E, the concentration of base (B) L1 for the
large-pore gel (3 per cent), and the concentration of base(B) L2 for the small-pore gel (71/2 per
cent). In calculating (B)L2 a pH U' of 9.5, the “running pH,” and xα' = 1/3 [the value of xα at pH U'
see Appendix E] rather than 8.3 and are used in Equations 19 and 20 in order to program the
desired pH change. The particular base used will usually be chosen so that (14 – pKb) falls
between pH Ll and pHL2 in order to provide some buffering action in both solutions. The
concentration of base (B)U, for the upper buffer, is computed from Equation 20.
Ornstein: Disc Electrophoresis
329
FIGURE 3. Disc electrophoresis (see also Appendices D and E).
(The calculations for the upper buffer from Equations 9 and 20 specify what actual values of
concentrations of glycine and base will automatically develop in the specimen and spacer gels
behind the protein stack because of the base and chloride concentrations introduced into those
gels. Therefore, so long as glycine is the only major anionic component in the upper buffer
reservoir, and the base provides the only major cationic component in the lower buffer reservoir,
and, in addition, if both reservoir solutions are reasonably well buffered and electrically
conductive, the actual concentrations in these reservoirs are not at all critical. It will, however,
usually be the case that the calculated upper buffer will coincidentally also satisfy all these
conditions.) Thus the thin starting zones plus the sieving effect of the gel together provide high
resolution. The “preconcentrating” step, permitting the use of extremely dilute samples, is an
extra bonus.
CHANGES IN CONCENTRATION IN A DISC
AS IT ENTERS THE SMALL-PORE GEL
As the first protein disc following the chloride enters the small-pore gel, its mobility decreases
but the regulating function predicts that the resulting increase in voltage gradient will be
sufficicnt to keep the velocity of the protein in the gel just equal to the velocity of the chloride
ahead of it. However, since the voltage gradient also depends upon the cations in the system, the
effect of the small-pore gel on their mobility must be taken into account. If the percentage change
in the mobility of the cation were the same as that of the protein (if the two had equal diffusion
constants), the resulting increase in voltage gradient would exactly match the decrease in
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Annals New York Academy of Sciences
.
mobilities of both ions and there would be no change in the concentration within the disc as it
entered the small-pore gel.
On the other hand, if the cation were smaller than the protein (the usual case), its mobility
would hardly decrease relative to that of the protein, and the rise in voltage gradient would not be
sufficient to maintain the speed of the protein in the disc equal to the chloride ahead of it. The
necessary additional voltage gradient increases must come from a decrease in protein and cation
concentration, which means that as the disc of protein enters the small-pore gel it will dilute and
become somewhat thicker. The larger the cation, the smaller will this change in concentration be.
(This contrasts with the usual sharpening of bands that occurs in the absence of “stacking” in
starch gels.2)
After the last protein disc has entered the small-pore gel, the glycine follows and also enters.
Because of the programmed pH increase from 8.3 to 9.5 at this boundary, the effective mobility
of glycine increases to a value somewhat higher than the fastest protein. Consequently, the
glycine overruns all of the protein discs and follows immediately behind the chloride.
VOLTAGE GRADIENTS AND pH WITHIN ZONES, AND ZONE S PREADING
Thus far no consideration has been given to the voltage gradient and pH within a disc as
compared to that of the buffered regions ahead of and behind the disc. Let us now examine this
point.
The pH and the voltage gradient within a disc, pHi and Vi , will not be the same as in the
glycine buffer ahead of and behind it. F IGURE 4 gives (see Appendix G) pHi and Vo/Vi as a
function of m p, the mobility of the protein in the gel for three different values of mobilities of the
base cation. Measured mobilities must be corrected by the factor V i /Vo and will be for the pH i
indicated (rather than pH 9.5) .
If the conductivity in a disc is higher than that of the pH 9.5 buffer, the voltage gradient in the
disc will be lower than in the solution ahead of it and behind it, and the front edge of the disc will
spread faster than by diffusion alone and the rear edge will spread more slowly. The converse
holds if the disc has a lower conductivity. Similarly, if the pH of a disc is lower than that of the
buffer, the front will spread more rapidly than it would as a result of diffusion and the rear less
rapidly.
As can be seen in FIGURE 4, the effect of both the pH and voltage gradient differences are
additive in accelerating the spreading of the front of a disc. However the lower the mobility of the
protein in the gel, the smaller will be these differences and therefore the more nearly will the disc
spread as predicted from diffusion alone.
Therefore the actual total spreading of a disc for a fixed ionic environment, voltage gradient,
and distance of migration in gels of different pore size diminishes with pore size (and therefore
with protein mobility in the gel) and will approach that predicted from diffusion alone as a limit.
Here we find a second strong motive for attempting to “unstack” the proteins by using only a
change in pore size (no pH change), because, as is clear from FIGURE 4, if all the protein
mobilities are reduced to less than –0.5 units in the gel we will have approached much closer to
“ideality.” FIGURE 2B shows the viscosity curves for a very large-pore gel and a relatively smallpore solution of “medium-length” linear chains (broken curves). Clearly, by combining these
(solid curve), the required frictional and macromechanical (anticonvection) properties can be
Ornstein: Disc Electrophoresis
331
FIGURE 4. pH inside a disc, pHi, and ratio of voltage gradient outside a disc to voltage gradient inside a
disc, V o/Vi as a function of mobility of protein in the gel, mp, and the mobility of the cation, mp for a
running pH of 9.5 (standard system).
fabricated. Such "combination gels" have been successfully prepared and yield both improved
resolution as well as a number of substantial gains in convenience and reproducibility. The details
of their preparation and performance will be reported elsewhere.22
ZONE SPREADING DUE TO DIFFUSION
Disc electrophoresis produces discs with very high concentrations at to (the time the glycine
enters small-pore gel and first overruns the disc). The concentration within the disc at t o is
independent of the actual thickness, To, at that time (which depends only upon the amount of that
protein in the original sample and the conditions set by the regulating function [Equation 9]) .
It might be argued that since diffusion will cause a disc to spread at a rate proportional to t l/2
(see Appendix F) and since electrophoretic migration separates ions at a rate proportional to time,
t, one would expect that the longer the running time the greater the resolution. However, this
argument neglects the effects of spreading on the concentration of protein in a disc and therefore
on our ability to detect a disc. A disc for which (T – To) / To is less than 1 at the end of a run is
easily detected because there has been very little change in concentration. For proteins present in
low amounts in the sample (To very thin), T will be of the order of 2(2 Dt) l/2 (see Appendix F)
after a very short running time. For
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Annals New York Academy of Sciences
concentration will immediately begin to drop and resolution will decrease continuously because
of the loss in detectivity that this concentration drop represents.
Therefore the shorter the running time and the higher the voltage gradient, the higher will be
the resolution of trace components.
It can also be shown (see Appendix F) that for a given protein and ionic environment and a
fixed voltage gradient, if a protein is permitted to migrate the same distance in gels of different
pore size, the diffusion spreading will be the same in all the gels independent of pore size.
OHMIC HEATING
The scale of disc electrophoresis is limited by the ohmic heating that occurs during a run.
This is controlled by keeping the product of the voltage gradient and the smallest cross section of
the column less than approximately 20 volts in the region occupied by the proteins for runs in
which (chloride) = 0.06M, without need to restort to forced cooling. If the concentration of
chloride (and therefore all the other ionic substances) is reduced by a factor H, the voltage
gradient can be increased by Hl/2 to give the same ohmic heating. The limit to how far this
approach can be conveniently carried is set by the facts that To increases linearly with H and also,
as the concentrations are lowered, the relative contributions of H + and OH- to conductivity can no
longer be neglected (see bottom of page 326).
STEADY-STATE STACKING FOR BOTH LARGE-SCALE PREPARATIVE
“SEPARATIONS” 23 AND ULTRAMICROANALYSIS24
As mentioned on pages 327 and 331, the thickness of each disc when the steady-state of the
stacking step has been reached (e.g., somewhere in the spacer gel) is proportional to the amount
of the particular protein in the original sample.
If very large quantities of protein are used (e.g., a 100 cm. column [of any cross section] of
whole human serum, desalted and diluted to about 3.8 per cent protein [see page 327] with the
glycine buffer) and are gelled in place on top of another very long column of, for example,
“regular spacer gel,” then, if a current is passed, when the protein front has migrated sufficiently
far, all the proteins will have stacked; but now the individual elongated discs will have
thicknesses about equal in centimeters to the percentage of the individual proteins in the serum. A
longer column of the same concentration serum would give even thicker discs. Except for the
individual concentration gradients across the boundaries between two discs (which will produce
zones of mixed proteins of the order of microns in thickness, see page 333), the elongated discs
will contain pure proteins in high concentration and can, for example, be permitted to
“electrophoresis”out of the lower end of the column into a properly designed fraction collector.
If one is dealing with a mixture of proteins that have identical mobilities, these will not be
separated. If their free mobilities are identical or almost identical (e.g., the haptoglobins of type 22), a system arranged to produce “steady-state stacking” in a relatively small-pore gel will resolve
these components except for the very rare case where both mw1 = mw2, and Dl = D 2.
(In very long columns run for very long periods of time, net electrical neutrality of the gel is,
for “mechanical reasons,” much more important than in the short runs illustrated in FIGURES 1
and 5.)
Ten micron gel fibers, prepared for “steady-state stacking” and immersed in a nonpolar
Ornstein: Disc Electrophoresis
333
medium, provide sufficient sensitivity for the analysis of proteins and peptides in the picogram
range (10-l2 grams). The separated components are detected and measured in situ in the gel fiber
with an interference microscope. (Further development of this technique will be reported
elsewhere.24 )
FIXED BOUNDARY WIDTH DURING S TEADY-STATE STACKING
The concentration gradients of the individual components across the boundary between two
discs during steady-state stacking will be a function of the difference in effective mobilities
between the two components, the voltage gradients on each side, and the diffusion constants of
each substance.
Let us determine the width of the diffusion zone, measured from the plane of the boundary
(were there no diffusion) to the “edge” of the diffusion zone.
If we consider the moving boundary in a coordinate system that moves with it, the diffusion of
a molecule of component 1, the trailing component, into the region occupied by component 2, the
leading component, along the axis of the column can be described by Einstein's diffusion
equation, x22 = 2Dlt, where x2 is the root-mean-square (RMS) distance of migration of
component 1 with diffusion constant D l, away from the boundary into the disc of component 2
along the axis in time t.
The velocity of migration due to diffusion alone is therefore
dx 2/dt = D1/x2.
The apparent velocity of electrophoretic migration of a molecule of component 1 in disc 2 (in
the absence of diffusion) would be,
V2 (mlx l – m 2x 2)
(where, as before, xi is the degree of disociation of the ith component.)
We will define the “edge”of the diffuse zone as located at that distance from the plane of the
boundary at which the two velocities are equal and opposite.
D1/x2. = V2 (mlx l – m 2x 2) where m 2 > m l and therefore
x2 = D l/V2 (mlx l – m2x 2) = KT/QlXlV2(1 – m2x 2/m lx l )
and by symmetry,
x1 = D 2/V1 (m2x 2 – m1x 1) = KT/Q2X2V1(1 – m1x 1/m 2x 2).
Since both “edges” have diffused, the voltage gradient across the diffuse zones will be
different from either V l or V 2. An RMS “half width” of 2x takes this into account. (However, for
an explicit solution to the problem of the concentration distribution function across such a
boundary, see, for example, MacInnes and Longsworth. 25)
For industrial scale separations, it is important to know the cost of separation of two
components, using steady-state stacking, in terms of the work necessary to accomplish the
separation of a given amount of a pure substance as a function of its abundance ratio in the
mixture, the differences in effective mobilities, and the minimum usable voltage gradient.
Work = (no. molecules)(effective charge per molecule)(total distance of migration) (voltage
gradient).
The work to separate nl molecules of component 1 from a mixture with n2 molecules of 2,
where the abundance ratio of component 1, rl = nl / (nl + n2) using steady-state stacking is
Work = (nl /rl )(QlXl)[2x/r l (m1x 1/m 2x 2)](Vl); therefore,
Work = 2KTnlQlXl/r12Q2X2(1—m1x 1/m 2x 2) 2.
If rl is initially less than 0.5 (e.g., the HU235O3+ ion as compared to HU238O3+ in solutions of
uranyl salts), a more economical procedure will involve first arranging for the enrichment of
component 1 by separation of pure component 2 so that there will be l consecutive enrichment
334
.
Annals New York Academy of Sciences
steps with a 50 per cent yield of pure 2 per step, and enrichment of 1 to about r l = 0.5 after l steps,
and the work expended in this operation will be,
l
Work = 2KTn2Q2X2/Q1X1(1 – m2x 2/m 1x 1 Σ 2l [2l (r2 – 1) + 1]2, where
)2
0
l = –log2(1—r2).
This latter calculation has not included the work expended in transporting the common ion, β,
or any of the ions in the solutions ahead of and behind the two discs in question. It will usually be
quite simple to keep the total work below 10 times the above value.
RESULTS, HISTORICAL NOTES, AND CONCLUSION
FIGURE 5 illustrates typical runs on human serum with a system similar to the model outlined
above, using chloride, glycine, tris(hydroxymethyl) aminomethane, and a 7 per cent
polyacrylamide gel. Proteins with isoelectric points below pH 8.9, with free mobilities at pH 8.9
and 9.5, which fall between –7.5 and –2.0 units, and with minimum diameters of less than 200
Angstroms and maximum diameters of less than about 400 Angstroms are sharply separated in
this system. Details of the procedures are reported in Part II, page 404.26 By using the same
equations (interchanging acids for bases), a system in which a moving boundary consisting of a
fast cation (viz., K + ) followed by a weak base can be used to stack and separate such basic
proteins as histones in this same pH range {e.g., pHU = 8.3, pHU' = 6.6 (2,6, lutidene+ , K + , and
glycinate– ) } Likewise, the system can be adapted to lower pH's where most proteins are cationic
{e.g., pHU, = 4.0, pHU' = 2.35 (glycinium+ , K+ , acetate–}.
We are at present engaged in the development of a spacer-separation gel in a glass tube, which
will be stable on storage and ready for the simple addition of sample, photopolymerization of the
sample in the top of the glass tube, and attachment to the electrode reservoirs.* Such units, it is
hoped, will directly provide highly reproducible runs, especially useful for routine clinical
diagnostic purposes. At present, other premixed reagents prepared in accordance with our detailed
procedures26 as well as appropriate auxiliary equipment are available.†
A flying-spot scanner* (see FIGURES 6, 7, and 8) is nearing completion. This instrument
measures the concentration of protein in a disc to within 5 per cent over a dynamic concentration
range of from 0.03 per cent to 6 per cent from a three microliter sample of serum. It measures the
concentration at 400 points along the pattern (thinnest resolved discs in a one inch pattern are
about 50 microns thick).
A preliminary analytical technique27 for the inexpensive identification and classification of
large numbers of such patterns for diagnostic purposes (of the order of 104 or more per week) had
been designed and has now been reworked to increase its sophistication (account to be published
in the near future28), and will be programmed for a large-scale digital computer.
The first appreciation of the possibilities of using the “steady-state stacking” potentiality
provided by the Kohlrausch Regulating Function19 for separations of ionic species was recorded
by James Kendall while at Columbia Univer-
* The work on standardizing the gel and the development of both the scanner and the
analytical computer program have been supported by the Diagnostic Research Branch of the
National Cancer Institute of the U. S. Public Health Service, Contract Number 3096.
† CANALCO, Bethesda, Md.
336
Annals New York Academy of Sciences
Ornstein: Disc Electrophoresis
337
sity. 29-34 In the period from 1923 to 1926, he reported a number of unsuccessful attempts to
separate the natural isotopes of Cl–,29,30,3l but successfully separated traces of Radium from
Mesothorium I and Barium,34 and also successfully separated a number of rare earth ions from
one another.32,33 all on agar-agar gel columns. For reasons unknown to the author, Kendall
appears to have discontinued this work after leaving Columbia in 1926. His last publications on
this method appear in three review articles, 35–37 and in an article in Science35 he even suggested
the use of the method to separate proteins. In all cases, Kendall and his co-workers used
completely dissociated salts and did not appear to be aware of the advantages of working with the
wider range and more easily programmed, steady-state stacking that can be designed by using a
weak acid or base within a few pH units of its pKa to provide the trailing ion and a properly
chosen counter ion in proper concentration as a buffer. Surprisingly, Kendall's work lay
essentially forgotten except for brief references to his 1923 work29 as the first application of agaragar gels to electrophoretic separations. In 1953 Longsworth reviewed Kendall's work and
provided an heroic modern demonstration of steady-state stacking for the separation of a mixture
of essentially completely dissociated salts in a Tiselius apparatus, with out an anticonvection
medium. 38 In 1957 Poulik 39 observed that by replacing the borate buffer in starch gel with citrate
buffer, while maintaining borate in his electrode reservoirs, a moving anionic boundary passed
FIGURE 7. Block diagram of a mechanical flying-spot scanner: Io, background signal; I, sample
signal; O.D., optical density; lN916, “logarithmic” diodes. The flying-spot is generated in the following
manner: The arc of a very high brightness 100 watt high-pressure xenon arc87 (Duro Test Corp., North
Bergen, N.J.) is imaged by a N.A. 0.3 microscope objective, M.O., onto the end of a clad, 50 micron
diameter, 20-inch-long glass fiber (N.A. 0.6). This end is locked in position. The other end of the fiber is
carried on the “pen” of a high-speed Offner rectilinear pen galvanometer (Offner Division, Beckman
Instruments Inc., Chicago, Ill.) driven through a power amplifier, P.A., by the oscilliscope sweep generator.
We are thus provided with a high-brightness, high N.A., low inertia source, permitting fairly high speed
flying-spot operation.
338
Annals New York Academy of Sciences
FIGURE 8. The absorption spectrum of the anion of “Dibromo-Trisulpho-Fluorescein,” a newly
synthesized dye that binds strongly to the cationic groups of acid denatured protein. By measuring the
“background” intensity at wavelengths longer than 610 mµ and the intensity transmitted by the sample
with a narrow spectral band of green light, a reproducible “doublebeam” technique is easily instrumented.
Substitution of a narrow-band violet filter for F2 permits measurements of discs that have excessively
high O.D. at the absorption peak.
through his system. After staining his gels, he found appreciably improved resolutoin of some
proteins. At that time, he recommended the use of “discontinuous buffer systems” to increase
resolution. While correctly recognizing that a discontinuity in voltage gradient was probably
responsible for his improved results, that he did not appear to recognize that the phenomena of
the Kohlrausch Regulating Function were involved is indicated by the following statement: “The
nature of the process which causes this system to give improved resolution is being investigated
with the view of finding a continuous system of the same resolving power [author's italics].”
Shortly thereafter we also noticed a similar phenomenon in our polyacrylamide gels. In these
the persulfate “catalyst” concentrations were quite high (i.e., the concentration recommended by
the American Cyanamid Corporation for the chemical initiation of polymerization of solutions of
acrylamide monomers 40) and a moving boundary between borate and persulfate or sulfate (the
breakdown product of persulfate) was found to be responsible.
Unaware at the time of Kendall's work or of the relevance of Poulik's observations to our own,
we were nonetheless fortunately able to unravel the details of the mechanism of steady-state
stacking presented above, with explicit solutions for the cases using the ions of weak acids or
bases as the trailing ion.
It is also interesting to note, parenthetically, that the classic asymmetries between ascending
and descending boundaries in the Tiselius apparatus were known to be due to the Kohlrausch
Ornstein: Disc Electrophoresis
339
phenomena and were the subject of intensive study (see Longsworthl9 for a review of this
subject), but the majority of such studies were aimed only at understanding and eliminating the
artifacts of the Tiselius method. Although some of these artifacts were clearly recognized to
involve boundary sharpening, the intentional use of these phenomena to design an electrophoretic
method of increased resolution appears to have never before been suggested, except as perhaps
might be inferred from Longsworth's review37 of Kendall's work.
Our disc electrophoresis procedure has already been successfully applied to a wide range of
research and clinical problems. (See, for example, References 41 through 78 and the articles in
this monograph.)
The combination of high resolution, sensitivity, reproducibility, simplicity, versatility, and
speed made possible by the design of disc electrophoresis will be of interest in diverse fields,
including enzymology and immunology; it will be of interest in the analysis of blood sera, body
fluids, and tissue extracts for physiological research and medical diagnostic purposes, in the study
of protein struclure and the “genetic code,” in the study of embryological induction and
differentiation, and in the study of plant, animal, and human genetics and evolution. We hope that
both the potentials of disc electrophoresis and the insight into mechanisms that have grown out of
our experience in developing this technique will stimulate a substantial increase in the use of
electrophoresis for the separation and identification of ionic substances.
The author wishes to acknowledge the contribution of his colleague, B. J. Davis, who shared
equally in the conceptual and practical development of these methods and sine quo non.
APPENDICES
Included in these appendices are a set of arguments that (a) provide an explicit basis for some
of the statements in the main body of the text and (b) present estimates of the magnitudes of some
effects that might have been expected to severely restrict the scope of disc electrophoresis.
(A ) Non-Newtonian Viscosity
It is to be expected that for molecules somewhat larger than the average pore size the viscosity
will, in addition, be non-Newtonian. At low particle velocity, the pore would be almost
impermeable to the molecule, but at sufficiently high velocity such molecules may be able to pass
through the pores. The force exerted locally on the gel structure by the high energy molecule
would cause the pore to stretch by transfer of momentum, permitting the molecule to “tunnel”
through a pore that was initially smaller than the molecular diameter. A second analogous nonNewtonian effect can be expected with rod-shaped molecules. At high velocity, the frictional
resistance of the gel will tend to keep the rods aligned parallel to the electric field presenting their
smallest cross sections to the pores.
By comparing mobilities at high and low voltage gradients (100 volts/cm. to 5 volts/cm.), we
have not yet found evidence for non-Newtonian effects. This is not surprising since it is not until
the electrophoretic pressure approaches the pressure exerted on the gel chains and proteins by the
thermal agitation of water molecules that such effects would be expected to be measurable (i.e.,
that the “directed momentum” transferred to the “effectively massive” gel chains by
“electrophoretic collisions” will not be thermally randomized between such collisions). The
electrophoretic pressure depends linearly on voltage gradient spacing of chains of a gel or
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Annals New York Academy of Sciences
and the thermal pressure on absolute temperature. It can be shown that, at 300°K, these pressures
will not be equal until the voltage gradient reaches the order of 106 volts per centimeter.
(B ) “Pore Size” of Gels and Solutions of Long Chain Polymers
The dependence of the effective pore size of a gel or solution of long chain polymer on
concentration can be crudely estimated as follows: We assume that the length of an average chain
is very large compared to the diameter of the chain molecule and that, in a gel, the distance
between crosslinks is also very large. (For the purpose of zone electrophoresis and the arguments
presented here, the crosslinks in the gel serve mainly to stabilize the mass of intertwined chains
against convection.)
Consider all the chains arrayed to form a cubic lattice of n x n squares per face (see FIGURE
9). The lattice has an edge of unit length and therefore a “pore” has a “diameter,” σ of (l/n) – d (1
+ l/n) which, for large values of n approaches (l/n) – d where d is the diameter of the chain. There
will be 2(n+1) unit lengths of chain defining a lattice face and a total of 3(n+1)2 unit lengths to
form the entire lattice.
If we increase the concentration of the polymer by a factor J, we multiply the total length of
chain per unit volume by J. If we were to re-form a new cubic lattice of unit volume, the number
of squares per edge would increase to (n+l)J l/2 – 1, which for large values of n is approximately
nJl/2, and the new pore diameter would decrease to approximately (l/nJl/2) – d. Of course the
FIGURE 9. Two by two lattice with edge of unit length, “pore diameter,” σ, and chain diameter d (see
Appendix B). In this model we have assumed that, at the corners of the elementary cubes, the intersecting
chains occupy the same volume and also that the minimum “inside diameter” (σ) of the elementary cube is
the average “inside diameter” of that cube. These assumptions result only in a small error in σ so long as σ
is at least two times larger than d.
Ornstein: Disc Electrophoresis
341
solution of polymer chains will tend to be random rather than regular as in a cubic lattice.
Because of thermal perturbation, at 300°K, the size of the individual pores of a flexible cubic
lattice of the above type, or of a random lattice of the same polymer content, would be very
nearly the same, even averaged over short time intervals. The above considerations are therefore
adequate to indicate the dependence of average pore size on the concentration of polymer (which,
assuming complete conversion, also indicates the dependence of pore size on monomer
concentration ) .
(C) Definition of Electrophoretic Mobility
and the Effect of the Ionic Environment
In zone electrophoresis a sample of a mixture of ions is placed in a starting zone (see FIGURE
10) in a linear conducting matrix, and a potential, v, is applied across the length, 1, setting up a
potential gradient, V = v/l, along the matrix, After a time, t, cations will have migrated towards
the cathode and anions towards the anode for distances proportional to their electrophoretic
mobilities (see FIGURE 10B). The mobility is defined by m = d/tV = QX/f, where d is the
distance of migration of an ion in time t, Q is the net charge of the ion, f is the frictional resistance
of the medium, and X is a dimensionless factor that changes the effective charge of the ion and
depends on the ionic environment and size of the ion.*
FIGURE 11 shows the approximate value of X as a function of ionic strength, I, for spherical
molecules of radius where r = 5 Angstroms, 25 Angstroms, and 100 Angstroms. For ellipsoidal
molecules, X is close to the value for a sphere of radius equal to the minor radius of the ellipse.
(D) The Regulating Function for Weak Electrolytes
The electrical conductivity, λ, of a solution of ions is a function of the concentration of the ith
ion, ci its mobility, mi, and its elementary charge, z i, such that
λ = E Σci mi z i
.
,
(l)
.
where E is the charge of the electron.
*
(1 + Wrb)f(Wr)
W = ——————
1 + W(r + rb)
where
W
I
r
rb
ci
zi
E
C
K
T
.
(modifieded from Gorin79),
(8πE2I/CKT)1/2 and
=
the ionic strength,
= 1/2Σcizi2,
= radius of particle (protein),
= radius of "buffer" ion (e.g., cation opposite anionic protein),
= concentration of the ith ion
= elementary charge of the ith ion
= charge of the electron,
= dielectric constant of the medium
= Boltzmann's constant, and
= absolute temperature.
f(Wr) is a function derived by Henry80 and is tabulated in FIGURE 11 (from Abramson et al. 79) I/W is the
"thickness" of the Debye-Huckel "double layer" surrounding a charged particle in an ionic medium. This thickness, at
300,° absolute, in aqueous systems (C=80) is equal to 3 X 10 –8 I–1/2 cm. Wr is the ratio of the particle radius to this
thickness.
X, as calculated here will be overestimated by the amount of the third order correction of Gronwall, La Mer, and
Sandved,8l which depends upon both charge and Wr (see Gorin82). The broken line in FIGURE 11 includes this
correction for a particle with r = 25 Angstroms and z = –25 (e.g., a protein like albumin).
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Annals New York Academy of Sciences
FIGURE 10. Zone electrophoresis: After time, t, cations have moved toward the cathode and anions
toward the anode, and the zones have spread in thickness as a result of diffusion (see Appendix C).
If we consider acids and bases at pH's near the pKa or (14—pKb), only part of the population
of molecules will be charged at any one time. If xi is the fraction of dissociation (i.e., the ratio of
charged molecules to the sum of the charged and uncharged forms), then each molecule can be
viewed as being charged xi of the time and uncharged the rest of the time. The average velocity of
migration, s i , of the molecule in the voltage gradient V, will be
si = Vmi xi ,
(2)
If two solutions, L (the lower) containing substance γ, and U (the upper) containing substance
α (ions of α and γ of like sign of charge with m γx γ greater than mαx α), are layered U over L
FIGURE 11. Variation in “effective” charge, XQ (where Q is the true charge), of an ion as a function
of ionic strength, I, and the radius of the ion (see Appendix C).
Ornstein: Disc Electrophoresis
343
in a cylinder, and a potential is placed across the cylinder (from end to end), then the
concentration of substance in these solutions for the velocity, sα to equal sγ may be derived as
follows:
Let
sα = V Um αx α = sγ = V Lm γx γ .
(3)
Since the current, Y, through both solutions (which are electrically in series) is the same, it
follows from Ohm's Law that V L = Y/λLS and V U = Y/λUS, where S is the cross sectional area of
the cylinder; therefore, from (l) and (3),
mαx α
————
ΣciUm iUziU
=
mγx γ
————
ΣciLm iLziL
(4)
Equation 4 is a modified form of the Kohlrausch Regulating Function. l9 When the conditions
specified in the equation are satisfied, substances α and γ will migrate down the cylinder with
equal velocity and the boundary between them will be maintained. If a molecule of α were to find
itself in the bulk of solution L (where, from Equation 3, VL is less than V U), it would migrate
more slowly than the molecules of γ (and therefore more slowly than the boundary) and would be
overtaken by the boundary. Conversely, a molecule of γ in the bulk of U will move faster than the
boundary and will overtake it, thereafter migrating at the same velocity as the boundary. Let us
now consider two solutions with one common ion, β, and the two ions α and γ with charge of
opposite sign to β (from Equation 4):
mαx α
m γx γ
––––––––––––– = ––––––––––––
cαm αzα + cβUm βzβ
cγm γzγ + cβLm βzβ
(5)
The condition of net macroscopic electrical neutrality in each solution requires that
cαzα = – cβUzβ and cγzγ = – cβLzβ
(6)
.
Therefore, from (5) and (6)
mαx α
–––––––––––––
cαzα(m α – m β)
=
m γx γ
––––––––––––
cγzγ(m γ – m β)
.
(7)
Let the total concentration of molecular species i, be
(I) = c i/x i
then,
(A)
x γcα
x αcγ
,
m αzγ(mγ – mβ)
m γzα(mα – mβ)
(8)
— = —— = —————–
(Γ)
.
(9)
The relationship in this equation is relatively insensitive to temperature. Mobilities change
Annals New York Academy of Sciences
344
with temperature mainly as a result of the sensitivity of the frictional resistance of the medium to
temperature (the temperature coefflcient of viscosity). Proportional changes in all mobilities
cancel out in Equation 9. (This analysis can rather easily be extended to cover polyvalent ions. )
From the Henderson-Hasselbalch Equation for pH, where
pH = pKa + logl0ci/(iH) ,
where iH is an acid, and
pH = (14—pKb) + logl0(i)/ci ,
where i is a base, and from (8),
xα = c α/(A) = cα/[c α +(αH)] = 1/(1 + l0 (pKa – pH)) ,
xβ = c β/(B) = cβ/[c β + (β)] = 1/(1 + 10–[(14 – pKb) – pH])
(10a)
.
(10b)
(E) Moving and Stationary pH Boundaries*
Whereas Equation 9 explicitly estab!ishes the relationship between (A) an (Γ), our discussion
of the application of steady-state stacking in disc electrophoresis (page 327) also requires, in
general, that a particular pH be established and maintained behind the moving boundary. This
must be programmed by providing a particular concentration, (B)L,of a weak base (or acid, if α
and γ are bases) in the solution containing (Γ).
The following considerations permit us to calculate (B)L for the case of monovalent acids and
bases:
Since in any region of the column, from Ohm's Law and Equation 1,
Y = VSλ = VSEΣcim izi
,
(11)
then the fraction of the current carried by the β ion in any region (which is known as the
transference number20) is
Yβ/Y = c βm βzβ/Σcim izi .
(12)
The difference in the β ion current on the two sides of the moving boundar provides a measure
of the net rate of trapsport of the β ion.
YβL – YβU = EYmβzβ[(cβL/λL) – (cβU/λU)]
.
(13)
* While this manuscript was in press, T. Jovin and A. Chrambach of Johns Hopkins University brought a discrepancy between our earlier version of Appendix E and the literature to our attention. In previous developments
of the moving boundary equations (see, for example,l9,20) conservation of mass of constituents on passage of a
moving boundary was introduced as an explicit condition for the solution of the moving boundary equation.
While our Equation 9 can be derived as above without explicitly considering this restriction, we have re-examined
our previous Equations 11, 12, 14, 15, and 16 (see footnote, page 1) and find that they were in fact implicitly in
violation of the Law of Conservation of Mass with respect to the β component. The corrected version is presented in Appendix E; however, the method described in Part II,26 and illustrated in FIGURES 1 and 5, was prepared
before this error was detected, an in these cases xα and pHU (but not x α‘ and pHU‘) are different from those used
in the theoretical model (i.e., they are 1/8 and 8.9 rather than 1/30 and 8.3 respectively).
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Annals New York Academy of Sciences
Since all β that is transported must be transported as β ion, the Law of Conservation of Mass
also requires that
YβL – YβU = ESszβ[(B)L – (B) U]
,
(14)
where s = sα = sγ from Equation 3; therefore
[(cβL/λL) – (cβU/λU)]EYm β/Ss = (B)L – (B) U
.
(15)
From Equation 4, λU = λLm αx α/m γx γ , from Equations 3 and 11, VL/S = l/m γx γ, and
Y/SλL = V L; therefore
(cβL – c βUm γx γ/m αx α)mβ/m γx γ = (B)L – (B) U .
(16)
From Equations 6 and 10a, cβL = – (Γ)xγzγ/zβ , and cβU = – (A)xαzα/zβ , therefore
(– (Γ)z γ + (A)zαm γ/m α)mβ/m γzβ = (B) L – (B)L .
(17)
Dividing by (Γ), introducing the value for (A)/(Γ) from Equation 9, and simplifying,
(B) L – (B) U
––––––––––
(Γ)
mβzγ(mγ – mα)
= –––––––––––––
mγzβ(mα – mβ)
This equation has a form similar to that of Equation 9.
Given xα, from Equation 10a,
pHU = pKa – log10[(1/xα) – 1]
From Equations 6 and 10b,
(18)
.
(B) U = – (A)xα (1 + 10–[(14 – pKb) – pH U] )z α/zβ
From Equations 6 and 10b,
(B) L = – X(Γ)xγzγ/zβ
(19)
.
.
where X ≥ 1. (From, 10b, when X = 2, the pH of the lower solution, pHL = (14 – pKb), and the
lower solution will be maximally buffered. In general, less than maximal buffering is tolerable.)
From Equation 2l, [(B)L – (B) U/(Γ) = [(–X(Γ)z γx γ/zβ) – (B) U]/(Γ), therefore, from Equations 6
and lOb,
(14 – pKb) = pHU – logl0{[(Γ)/(A)][(Xzγx γ/zβ)+[(B)L – (B)U]/(Γ)](zβ/zαx α) – l} . (22)
Having chosen a base satisfying Equation 22, using the values of (Γ)/(A) from Equation 9 and
of [(B)L – (B) U]/(Γ) from Equation 18, then from Equations 18 and 20, (B)L can be explicitly
determined.
If, in addition, we wish to set up a stationary pH boundary, with pH U above and pHU’ below,
that will remain at a boundary between L l and L2, then pH U' and x α' (xα at pHU'), rather than pH U
and xα. are used in calculating (B)L2 . When the potential is applied and the moving boundary
passes the boundary between Ll and L 2, the pH of the solution above that boundary will remain
equal to pHU, but the pH between that boundary and the moving boundary will equal pHU'. (see
FIGURE 3C).
Ornstein: Disc Electrophoresis
346
(F) Diffusion Spreading of a Disc
The increase in thickness due to diffusion alone is (T–To) = 2x, where, from Einstein's
equation, x = (2Dt)l/2, x is the root-mean-square displacement of a particle as a result of
Brownian motion after a time t, where D is the diffusion constant in the gel, T is the thickness of a
disc after time t, and To is the actual thickness of the disc just after the glycine has overrun it
(which defines time zero for disc electrophoresis). To is directly proportional to the amount of
protein in the original sample (see page 327). Any spreading in excess of T that is not directly
attributable to differences in the voltage gradients and pH's inside and outside of the disc (see
page 331), will be due to heterogeneity of charge and/or size of the proteins of the disc.
Since m = d/tV = QX/f (from Appendix C) and D = KT/f, then from Einstein's equation,
x = (2KTd/VQX)1/2. It can be concluded that for a given protein and ionic environment, a fixed
distance of separation from the origin, and a fixed voltage gradient, the difJusion spreading of the
disc is fixed, independent of the frictional resistance of the medium.
(G) The pH and Voltage Gradients Inside and Outside of a Disc
For equilibrium between the glycinate ion in the disc and that outside it, the Law of Mass
Action requires that the product of the concentration of cation and glycinate ion outside a disc
equal that inside.
cαocβo = cαicβi
,
(23)
where sub-subscript o refers to regions outside a disc and i, to regions inside. Net electrical
neutrality requires that
cβizβ = – cαizα – cpzp
,
(24)
where subscript p refers to protein.
These two conditions define the Gibbs-Donnan equilibrium and permit us to calculate the
concentration of glycinate ion inside a disc.
Solving (23) for cβi , and substituting in (24),
(cαi) 2zα/zβ + cαicpzp/zp + cαocβo = 0 .
(25)
The solution to this quadratic equation is,
– cpzp/zp ± [(cpzp/zp) 2 – 4(cαocβozα/zβ)]1/2
cαi = –––––––––––––––––––––––––––––––––––
2z α/zβ
(26)
For a base with zβ = + 1 and glycine in the small-pore gel at pH 9.5 (when xα' = 1/ 3 ) this
reduces to
pzp ± [(c pzp) 2 + (A)2 4/9] 1/2
cαi = (A)i x αi = c––––––––––––––––––––––––
2
(27)
We can now calculate the pH within the disc as well as the ratio of the voltage gradient inside
the disc to that outside. From (9) we compute cpzp, and from (6), (10a), (12) and (25)
pHi= 9.5 + log10(A)xα'/[(A)xαi – cpzp,]
From (1),
.
(28)
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Annals New York Academy of Sciences
– cpzp(– mp + mβ) + (A) i x αi (15 + mβ)
Vo/Vi = ––––––––––––––––––––––––––––––––
(A) (15 + mβ)/3
(29)
During “steady-state-stacking,” as well as during separation in the “smallpore” gel, the GibbsDonnan equilibrium specified above implies Donnan potentials and diffusion potentials across the
two boundaries of a disc. These also effect the spreading, relative to that expected from diffusion
alone, but will not be considered here. Their effects diminish as the mobility of the protein in the
gel, mp, diminishes.
(H) Dissociation of Complexes During Electrophoresis
It has been shown by Ogston83 (see also Boyak84 and Mysels85) that complexes of ionic
species (in cases where the mobilities of the complexes and the components are different from
one another) will dissociate during an electrophoretic separation to an extent proportional to the
square of the voltage gradient.
In a system such as our model for disc electrophoresis (stacking pH, 8.3; running pH, 9.5), the
voltage gradients to which a complex may be exposed during stacking are up to 10 times greater
than during running (on the order of up to 40 volts/cm.). As a result, if a stack spends sufficient
time in the sample and spacer gels (e.g., if the spacer is sufficiently long) such complexes (e.g.,
enzyme and prosthetic group) may become completely dissociated even before they reach the
“origin” (the junction between the spacer and small-pore gels). If the spacer and sample gels are
reduced in length, or are eliminated (i.e., conditions approaching those used in Smithies' starch
gel technique2 or Raymond's polyacrylamide technique15), the complexes may, by comparison,
remain almost completely undissociated even to the very end of the run.
The degree of dissociation of complexes in disc electrophoresis will therefore generally be
greater than in other methods for equal voltage gradients and times of migration past the origin.
The degree of dissociation among a set of disc electrophoresis runs may also vary if the amount
of complex (and therefore the thickness of the relevant discs in the stack) and/or the length of the
spacer gels are varied.
When such phenomena are observed, increasing the sample “load” per run will result in the
observation of a relative increase in the complex disc in the separated pattern and a relative
decrease in the discs of its components. Conversely, increasing the length of the spacer results in
a decrease in the concentration of the complex disc and an increase in the concentration of its
components. In this way the elements of such a system are easily identified.
It has been our observation that such systems are relatively uncommon among mixed protein
solutions so far studied. One serum post-albumin (apparently a seromucoid) and some prealbumin serum “polypeptides” (hormones?) seem to be involved in such an associationdissociation equilibrium.86
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