Cell Research Laboratory, The Mount Sinai Hospital, New...

DISC ELECTROPHORESIS—I

BACKGROUND AND THEORY*

Leonard OrnsteinCell Research Laboratory, The Mount Sinai Hospital, New York, N. Y.

Although electrophoresis is one of the most effective methods for the separation of ioniccomponents of a mixture, the resolving power of different electrophoretic methods is quitevariable. To separate two component ions, it is necessary to permit migration to continue untilone 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, ofthe zones occupied by each ion diminishes with time because of the spreading of the zones as aresult of diffusion. Remarkable resolution has been achieved when advantage is taken of thefrictional properties of gels to aid separation by seiving at the molecular level (see Smithiesl). Anew method, disc electrophoresis,† has been designed that takes advantage of the adjustability ofthe pore size of a synthetic gel and that automatically produces starting zones of the order of 10microns thickness from initial volumes with thicknesses of the order of centimeters. Highresolution is thus achieved in very brief runs.

With this technique, over 20 serum proteins are routinely separated from a sample of wholehuman serum as small as one microliter in a 20-minute run (see FIGURE 1). Direct analysis ofeven very dilute samples becomes routine because the various ions are automatically concentratedto fixed high values at the beginning of the run just prior to separation. Preliminary laboratorystudies and theoretic considerations provide evidence of the applicability of this technique to awide 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 thesimultaneous determination of both the free mobility and the aqueous diffusion constant of aprotein.

This report will detail some mechanisms that provide a rationale for the resolution afforded byzone electrophoresis in many gels; will develop the theory of some new modifications of zoneelectrophoresis that have been designed to take maximum advantage of these mechanisms; andwill 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 ofSmithies2 and the application of this technique by Hunter3,4 for producing “zymograms” ofenzymes of histochemical interest. Our recent cytochemical studies,5-7 which were then wellunder way, promised to be substantially clarified by the application of Hunter's technique.8 Wehad 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 tothe scientific public in preprmt form in January, 1962, through the generosity of the Distillation ProductsDivision of Eastman Kodak Company, Rochester, N. Y.

† The name was derived from the dependence of the new technique on discontinuities in theelectrophoretic matrix and, coincidentally, from the discoid shape of the separated zones of ions in thestandard form of our technique.

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Ornstein: Disc Electrophoresis 323

experience with polyacrylamide gels because, some years earlier, they had been recommended tothe author by G. Oster9-13 as a potential embedding medium for sectioning tissues. In contrast tostarch 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 afew anionic groups which are, in part, responsible for the backwards endosmotic flow noted withits use). It seemed that attempts to duplicate the desirable properties of starch gels withacrylamide might be rewarding. The first attempt, designed and executed by B. J. Davis, was sosuccessful that it encouraged Davis and me through almost a year of successive failures until anunderstanding of mechanism and a reasonable degree of reproducibility began to be achieved. Itwas about that time that we first reported our preliminary results.8,14 Soon afterwards, anindependent report from the Pepper Laboratory of the University of Pennsylvania also appearedrecommending the use of polyacrylamide as a substitute for starch gels.15

DIFFERENCES BETWEEN ELECTROPHORESIS IN GELS AND OTHER METHODS

Two striking differences had characterized the behavior of serum proteins during separation byzone electrophoresis in gels when compared to their behavior during paper or moving-boundaryelectrophoresis:

(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 onmolecular size of a protein relative to the gel pore size might account for this difference.2 Suchsieving possibilities had been first considered, but without notable success, by Synge andTiselius.16

(2) More fractions are resolved in gels and the zones are usually narrower than in paper or freeelectrophoresis for equal separations from the origin and for equal running times.

MECHANISMS OF DIFFERENCES

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 functionof r, the particle's radius. The viscosity of a gel is low when 2r is small compared to the averagepore of a gel, and is virtually infinite when 2r is very large compared to the pore size (seeAppendix A). In the case of a solution of very long chain polymers, the situation would be thesame except that as 2r approaches the length of the polymer chains, the rate of change of viscositywill again decrease somewhat as the particles begin to be able to transfer sufficient momentum tothe chains to displace them, to “enlarge” pores, and therefore to “slide” through the network oflinear chains (see FIGURE 2A). These cases contrast with the case for both Newtonian and non-Newtonian liquids, where viscosity, defined by Stoke's Law, η = ƒ/6πr, is constant andindependent 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 theresolution (until the point is reached where diffusion spreading of a zone's edges becomes largecolnpared 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


TABLE 1

PARAMETERS OF SOME PROTEINS*______________________________________________________________________________________

MolecularProtein Mobility, mw weight Length Diameter

________________________________________________________________________

Albumin -6.1 69,000 150 38Transferrin -3.3 90,000 190 37ß1 lipoprotein -3.0 (approx.) 1,300,000 185 185γ globulin -1.0 (approx.) 156,000 235 44Fibrinogen -2.1 400,000 700 38α2 macroglobulin -4.2 850,000 — —

___________________________________________________________________________________________________________

* Data from Oncleyl7 and Schultz.l8 mw in mobility units, length and diameter in Angstromunits. 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 50Angstroms (the diameter of the hydrated chain is about 10 Angstroms). TABLE 1 gives somedimensions of a few plasma proteins. We would predict that a 71/2 per cent polyacrylamideshould exhibit extreme frictional resistance to the migration of fibrinogen, , ß1 lipo-protein (andperhaps the α2 macroglobulin and γ globulin), and that the other proteins should be able to passthrough, 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 likepolyacrylamide, 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 thedimensions 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 thesimultaneous determination of the “free mobility” and aqueous diffusion constant of a protein. Iftwo gels are prepared with different pore sizes (gel 1 and gel 2) but identical ionic components(X1 = X2, see Appendix C), the distances traveled by a protein in the two gels, dl and d2, inparallel runs at the same voltage gradient, V, and for a fixed time, t, will yield mobilities mll = dl

/tV and m2 = d2/tV. In addition, m l = QX1/f1and m2 = QX2/f2, where Q is the charge of theprotein, 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 linearpolymers of different average chain lengths, all "prepared" from 71/2% acrylamide monomer solutions. (B) Ordinateand abscissa as in A. ........."71/2%" gel; __.__, "3%" gel; -------, "20%" solution of linear polyacrylamide chains withRMS "coiled-chain lengths" of about 1000 Angstroms and molecular weights of about 106;––––, "Combination" gelconsisting of a "3%" gel "filled" with the solution of linear polymer. Two ß1 proteins, Transferrin C, •, and ß1 lipo-protein, °, are located on the "71/2" and "Combination" gel curves. (C) Hypothetical "calibration curves" fordetermining the diffusion constant in water, Dw and the "free mobility" in water, mw , from the ratio of the measuredmobilities of any unknown protein in two standard gels of different average pore size. The value of m1/m2 for theunknown protein is determined from measurement and is located (as indicated) on the appropriate curve. The abscissagives the value of Dw .The value of the ordinate for the point on the second curve with the same abscissa valueprovides the required value of mw//m1 for this unknown protein. Since m1 is a measured value mw/ can be computeddirectly.

326 Annals New York Academy of Sciences

in the gels, D1 = KT/ fl and D2 = KT/f2, where K is Boltzmann's constant and T is the absolutetemperature, then

ml/m2 = dl/d2 = f2/f1 = D1/D. From a series of molecules of known mobilities in water, mw, and known diffusion constants inwater, Dw “calibration curves” of Dw against ml/m2 and mW/ml can be constructed (see FIGURE

2C). From the curves, a measured value of d1/d2 for an unknown protein permits thedetermination of the diffusion constant of the protein in water, Dw, and therefore the calculationof its “free mobiliy" in water, mw = m1(mw/m1).

Thin Starting Zones

We will now examine an approach for capitalizing on the potentials of (2):As early as 1897, Kohlrausch10 observed that, under conditions set by his “regulating

function” (derived in Appendix D), if two solutions of ions were layered one over the other, suchthat a solution with ions of substance γ of high mobility were placed below a solution of a slowion, α, of the same sign of charge, then the boundary between the two ionic species would besharply maintained as they migrated in an applied electric field because the two species (on eachside of the boundary) would have been “arranged” to move at the same speed (provided that thelower solutions were the denser of the two, and that the potential were applied with such polaritythat the ions of α and γ moved downwards).

If concentrations different from those specified by this regulating function existed when thepotential 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, preventingthe ready formation of the moving boundary. On the other hand, in gels and porous media (whichprevent convection), given sufficient time, a sharp moving boundary will form and be maintainedindependent of the densities of the starting solutions and their initial concentrations, provided thatthe faster ions precede the slower.

Equation 9 from Appendix D,

(A) xγcα mαzγ(mγ – mβ)

(Γ) xαcγ mγzα(mα – mβ)— = —— = —————–(A) xγcα mαzγ(mγ – mβ)

(Γ) xαcγ mγzα(mα – mβ)

where x is the fraction of dissociation, c is the concentration, m is the mobility, and α, γ, and ßare the ions (with ß of opposite charge to α and γ and common to both solutions) provides us withthe ratio of the total concentrations of α substance (A) to γ substance (Γ) for initiating andmaintaining a stable moving boundary.

For the purpose of illustrating how Equation 9 can be used to produce thin starting zones, αwill be the glycinate ion, ß the potassium ion, and γ the chloride ion. Then,

zα = -1zγ = -1

mα = -15 mobility units20 (see footnote to TABLE 1 ),mβ = +37 mobility units (see footnote to TABLE 1 ),mγ = -37 mobility units (see footnote to TABLE 1),

and therefore (glycine)/(chloride) = (A)/(r) = 0.58 (essentially independent of pH of eithersolution from pH 4 to pH 10, within which range, neither the hydrogen nor hydroxyl ions willappreciably contribute to conductivity, provided that the chloride concentration is greater than10-3M).


Above pH 8.0, most serum proteins have free mobilities in the range from –0.6 to –7.5 units. Ifthe effective mobility of glycine, mαxα, were less than –0.6, the mobilities of the serum proteinswould fall between that of the glycine and that of chloride. This requirement is satisfied when xα= 1/30, since the glycinate ion has a mobility of –15 units. The pH at which this degree ofdissociation of glycine occurs, can be calculated. From Equation l0a of Appendix D,

pH = pKa–log10 [ (1/ xα) – 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 glycinemolecule) has migrated one centimeter, the protein will have migrated two centimeters and willthen be located at the boundary. It would continue to run at the boundary because the mobility ofthe 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 willconcentrate at the boundary between the chloride and the glycine at a concentration satisfyingequation 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,mα = –6.0 mobility units,mβ = + 37 mobility units,mγ = –37 mobility units,

and therefore (albumin)/(chloride) = 9.3 x 10-3. If (chloride) = 0.06 M, then (albumin) = 0.00056M. That is, the albumin (M.W. 68,000) will automatically concentrate to about 3.8 per centbehind the chloride and would then stay at constant concentration. If the initial concentration ofalbumin had been 0.01 per cent in the glycine buffer, and if 1 milliliter of this mixture is placedon top of the chloride solution in a cylinder of one square centimeter cross section, then after thechloride boundary has moved about one millimeter, all of the albumin will have concentrated intoa disc (right behind the chloride) which would be about 25 microns thick. (In practice, this mightbe done by using a porous anticonvection medium all through the volume occupied by thechloride and through the one centimeter height of the glycine column.) In this manner, a 380-foldincrease in concentration can be achieved in a few minutes and the protein is reduced to a verythin lamina or disc. If the original concentration had been 0.0001 per cent, the same finalconcentration would result, but the total change would now be 38,000-fold. If the column were100 cm. in length, the same amount of protein would have been concentrated. If, instead of asingle protein, a one milliliter mixture with mobilities ranging from –1.0 to –6.0 units is placedover the chloride solution, by the time the boundary has migrated one centimeter in the appliedelectric field, all the proteins will have concentrated into very thin discs, one stacked on top of theother in order of decreasing mobility, with the last followed immediately by glycine. We will callthis 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 proteindrops below that of the glycine, the glycine will now overrun all the protein discs and run directlybehind the chloride, and the proteins will now be in a uniform linear voltage gradient, eacheffectively in an extremely thin starting zone, and will migrate as in “ordinary zoneelectrophoresis.” (However, the rate of increase of frictional resistance with molecular diameter

328 Annals New York Academy of Sciences .

is so large with “ordinary gels” (see FIGURE 2A) that an ordinary small-pore gel that would slowthe albumin to less than –0.5 mobility units would completely exclude most proteins withmolecular weights above 150,000. A specially formulated gel will permit us to overcome thisrestriction. 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 aregion of higher pH, e.g., a pH of 9.8 (the pKa of glycine), at which xgly equals one half andtherefore mglyxgly equals –7.5 mobility units, the glycine will now overrun all the protein discsand run directly behind the chloride, and the proteins will now be in a uniform linear voltagegradient, each effectively in an extremely thin starting zone, and will migrate as in "freeelectrophoresis." The required stationary pH boundary is quite easily established (see AppendixE) 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), thechloride overtakes the proteins which then "sort" out according to their mobilities into highlyconcentrated discs of protein, stacked exactly as described above (see FIGURE 3B). A "large-pore" (approximately 3 per cent acrylamide) gel is usually used as the porous anticonvectionmedium in this region as well as in the "spacer" region (see FIGURE 3). The protein sample ismixed into the "spacer mixture" and is usually gelled in place on top of the spacer. As the glycinefollowing the stack of proteins moves through these regions, the pH is maintained at 8.3 (seeAppendix 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 thespecial viscous properties of the gel, proteins of equal free mobility but of appreciably differentmolecular weight (different diffusion constant) will migrate with markedly different mobilitiesand will easily be separated (see FIGURE 3C). (Five per cent to 10 per cent acrylamide gels haveproved to effect useful separations of human serum proteins.) In a 71/2 per cent gel, the fastestpre-albumin has a mobility less than –5.0 units. We therefore arrange for a “running pH” of about9.5 where the effective mobility of glycine (mglyxgly) 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 computefrom Equations 9, 18, 19, and 20 of Appendices D and E, the concentration of base (B)L1 for thelarge-pore gel (3 per cent), and the concentration of base(B)L2 for the small-pore gel (71/2 percent). In calculating (B)L2 a pHU' of 9.5, the “running pH,” and xα' = 1/3 [the value of xα at pHU'see Appendix E] rather than 8.3 and are used in Equations 19 and 20 in order to program thedesired pH change. The particular base used will usually be chosen so that (14 – pKb) fallsbetween pHLl and pHL2 in order to provide some buffering action in both solutions. Theconcentration of base (B)U, for the upper buffer, is computed from Equation 20.


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 ofconcentrations of glycine and base will automatically develop in the specimen and spacer gelsbehind the protein stack because of the base and chloride concentrations introduced into thosegels. Therefore, so long as glycine is the only major anionic component in the upper bufferreservoir, 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 electricallyconductive, 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 theseconditions.) Thus the thin starting zones plus the sieving effect of the gel together provide highresolution. The “preconcentrating” step, permitting the use of extremely dilute samples, is anextra bonus.

CHANGES IN CONCENTRATION IN A DISCAS IT ENTERS THE SMALL-PORE GEL

As the first protein disc following the chloride enters the small-pore gel, its mobility decreasesbut the regulating function predicts that the resulting increase in voltage gradient will besufficicnt to keep the velocity of the protein in the gel just equal to the velocity of the chlorideahead of it. However, since the voltage gradient also depends upon the cations in the system, theeffect of the small-pore gel on their mobility must be taken into account. If the percentage changein the mobility of the cation were the same as that of the protein (if the two had equal diffusionconstants), the resulting increase in voltage gradient would exactly match the decrease in

330 Annals New York Academy of Sciences .

mobilities of both ions and there would be no change in the concentration within the disc as itentered the small-pore gel.

On the other hand, if the cation were smaller than the protein (the usual case), its mobilitywould hardly decrease relative to that of the protein, and the rise in voltage gradient would not besufficient to maintain the speed of the protein in the disc equal to the chloride ahead of it. Thenecessary additional voltage gradient increases must come from a decrease in protein and cationconcentration, which means that as the disc of protein enters the small-pore gel it will dilute andbecome 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” instarch 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 mobilityof glycine increases to a value somewhat higher than the fastest protein. Consequently, theglycine overruns all of the protein discs and follows immediately behind the chloride.

VOLTAGE GRADIENTS AND pH WITHIN ZONES, AND ZONE SPREADING

Thus far no consideration has been given to the voltage gradient and pH within a disc ascompared to that of the buffered regions ahead of and behind the disc. Let us now examine thispoint.

The pH and the voltage gradient within a disc, pHi and Vi , will not be the same as in theglycine buffer ahead of and behind it. FIGURE 4 gives (see Appendix G) pHi and Vo/Vi as afunction of mp, the mobility of the protein in the gel for three different values of mobilities of thebase 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 willspread faster than by diffusion alone and the rear edge will spread more slowly. The converseholds if the disc has a lower conductivity. Similarly, if the pH of a disc is lower than that of thebuffer, the front will spread more rapidly than it would as a result of diffusion and the rear lessrapidly.

As can be seen in FIGURE 4, the effect of both the pH and voltage gradient differences areadditive in accelerating the spreading of the front of a disc. However the lower the mobility of theprotein in the gel, the smaller will be these differences and therefore the more nearly will the discspread 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 thereforewith 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 achange in pore size (no pH change), because, as is clear from FIGURE 4, if all the proteinmobilities 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 small-pore solution of “medium-length” linear chains (broken curves). Clearly, by combining these(solid curve), the required frictional and macromechanical (anticonvection) properties can be


FIGURE 4. pH inside a disc, pHi, and ratio of voltage gradient outside a disc to voltage gradient inside adisc, Vo/Vi as a function of mobility of protein in the gel, mp, and the mobility of the cation, mp for arunning pH of 9.5 (standard system).

fabricated. Such "combination gels" have been successfully prepared and yield both improvedresolution as well as a number of substantial gains in convenience and reproducibility. The detailsof 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 glycineenters small-pore gel and first overruns the disc). The concentration within the disc at to isindependent of the actual thickness, To, at that time (which depends only upon the amount of thatprotein 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 tl/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, thisargument neglects the effects of spreading on the concentration of protein in a disc and thereforeon our ability to detect a disc. A disc for which (T – To) / To is less than 1 at the end of a run iseasily detected because there has been very little change in concentration. For proteins present inlow 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


concentration will immediately begin to drop and resolution will decrease continuously becauseof the loss in detectivity that this concentration drop represents.

Therefore the shorter the running time and the higher the voltage gradient, the higher will bethe resolution of trace components.

It can also be shown (see Appendix F) that for a given protein and ionic environment and afixed voltage gradient, if a protein is permitted to migrate the same distance in gels of differentpore 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 ofthe column less than approximately 20 volts in the region occupied by the proteins for runs inwhich (chloride) = 0.06M, without need to restort to forced cooling. If the concentration ofchloride (and therefore all the other ionic substances) is reduced by a factor H, the voltagegradient can be increased by Hl/2 to give the same ohmic heating. The limit to how far thisapproach 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 nolonger 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 thestacking step has been reached (e.g., somewhere in the spacer gel) is proportional to the amountof 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] ofwhole human serum, desalted and diluted to about 3.8 per cent protein [see page 327] with theglycine 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 sufficientlyfar, all the proteins will have stacked; but now the individual elongated discs will havethicknesses about equal in centimeters to the percentage of the individual proteins in the serum. Alonger column of the same concentration serum would give even thicker discs. Except for theindividual concentration gradients across the boundaries between two discs (which will producezones of mixed proteins of the order of microns in thickness, see page 333), the elongated discswill 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 beseparated. If their free mobilities are identical or almost identical (e.g., the haptoglobins of type 2-2), a system arranged to produce “steady-state stacking” in a relatively small-pore gel will resolvethese components except for the very rare case where both mw1 = mw2, and Dl = D2.

(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 1and 5.)

Ten micron gel fibers, prepared for “steady-state stacking” and immersed in a nonpolar


medium, provide sufficient sensitivity for the analysis of proteins and peptides in the picogramrange (10-l2 grams). The separated components are detected and measured in situ in the gel fiberwith an interference microscope. (Further development of this technique will be reportedelsewhere.24 )

FIXED BOUNDARY WIDTH DURING STEADY-STATE STACKING

The concentration gradients of the individual components across the boundary between twodiscs during steady-state stacking will be a function of the difference in effective mobilitiesbetween the two components, the voltage gradients on each side, and the diffusion constants ofeach 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 ofa molecule of component 1, the trailing component, into the region occupied by component 2, theleading component, along the axis of the column can be described by Einstein's diffusionequation, x2

2 = 2Dlt, where x2 is the root-mean-square (RMS) distance of migration ofcomponent 1 with diffusion constant Dl, away from the boundary into the disc of component 2along the axis in time t.

The velocity of migration due to diffusion alone is thereforedx2/dt = D1/x2.

The apparent velocity of electrophoretic migration of a molecule of component 1 in disc 2 (inthe absence of diffusion) would be,

V2 (mlxl – m2x2)(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 theboundary at which the two velocities are equal and opposite.

D1/x2. = V2 (mlxl – m2x2) where m2 > ml and thereforex2 = Dl/V2 (mlxl – m2x2) = KT/QlXlV2(1 – m2x2/mlxl )

and by symmetry, x1 = D2/V1 (m2x2 – m1x1) = KT/Q2X2V1(1 – m1x1/m2x2).

Since both “edges” have diffused, the voltage gradient across the diffuse zones will bedifferent from either Vl or V2. An RMS “half width” of 2x takes this into account. (However, foran explicit solution to the problem of the concentration distribution function across such aboundary, see, for example, MacInnes and Longsworth.25)

For industrial scale separations, it is important to know the cost of separation of twocomponents, using steady-state stacking, in terms of the work necessary to accomplish theseparation of a given amount of a pure substance as a function of its abundance ratio in themixture, the differences in effective mobilities, and the minimum usable voltage gradient.

Work = (no. molecules)(effective charge per molecule)(total distance of migration) (voltagegradient).

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/rl (m1x1/m2x2)](Vl); therefore,Work = 2KTnlQlXl/r1

2Q2X2(1—m1x1/m2x2)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 ofcomponent 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 – m2x2/m1x1)2 Σ 2l [2l (r2 – 1) + 1]2, where 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 bequite 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 outlinedabove, using chloride, glycine, tris(hydroxymethyl) aminomethane, and a 7 per centpolyacrylamide gel. Proteins with isoelectric points below pH 8.9, with free mobilities at pH 8.9and 9.5, which fall between –7.5 and –2.0 units, and with minimum diameters of less than 200Angstroms and maximum diameters of less than about 400 Angstroms are sharply separated inthis system. Details of the procedures are reported in Part II, page 404.26 By using the sameequations (interchanging acids for bases), a system in which a moving boundary consisting of afast cation (viz., K+) followed by a weak base can be used to stack and separate such basicproteins as histones in this same pH range {e.g., pHU = 8.3, pHU' = 6.6 (2,6, lutidene+, K+, andglycinate– ) } 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, whichwill be stable on storage and ready for the simple addition of sample, photopolymerization of thesample in the top of the glass tube, and attachment to the electrode reservoirs.* Such units, it ishoped, will directly provide highly reproducible runs, especially useful for routine clinicaldiagnostic purposes. At present, other premixed reagents prepared in accordance with our detailedprocedures26 as well as appropriate auxiliary equipment are available.†

A flying-spot scanner* (see FIGURES 6, 7, and 8) is nearing completion. This instrumentmeasures the concentration of protein in a disc to within 5 per cent over a dynamic concentrationrange of from 0.03 per cent to 6 per cent from a three microliter sample of serum. It measures theconcentration at 400 points along the pattern (thinnest resolved discs in a one inch pattern areabout 50 microns thick).

A preliminary analytical technique27 for the inexpensive identification and classification oflarge numbers of such patterns for diagnostic purposes (of the order of 104 or more per week) hadbeen designed and has now been reworked to increase its sophistication (account to be publishedin 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” potentialityprovided by the Kohlrausch Regulating Function19 for separations of ionic species was recordedby James Kendall while at Columbia Univer-

* The work on standardizing the gel and the development of both the scanner and theanalytical computer program have been supported by the Diagnostic Research Branch of theNational Cancer Institute of the U. S. Public Health Service, Contract Number 3096.

† CANALCO, Bethesda, Md.


sity.29-34 In the period from 1923 to 1926, he reported a number of unsuccessful attempts toseparate the natural isotopes of Cl–,29,30,3l but successfully separated traces of Radium fromMesothorium I and Barium,34 and also successfully separated a number of rare earth ions fromone another.32,33 all on agar-agar gel columns. For reasons unknown to the author, Kendallappears to have discontinued this work after leaving Columbia in 1926. His last publications onthis method appear in three review articles,35–37 and in an article in Science35 he even suggestedthe use of the method to separate proteins. In all cases, Kendall and his co-workers usedcompletely dissociated salts and did not appear to be aware of the advantages of working with thewider range and more easily programmed, steady-state stacking that can be designed by using aweak acid or base within a few pH units of its pKa to provide the trailing ion and a properlychosen counter ion in proper concentration as a buffer. Surprisingly, Kendall's work layessentially forgotten except for brief references to his 1923 work29 as the first application of agar-agar gels to electrophoretic separations. In 1953 Longsworth reviewed Kendall's work andprovided an heroic modern demonstration of steady-state stacking for the separation of a mixtureof essentially completely dissociated salts in a Tiselius apparatus, with out an anticonvectionmedium.38 In 1957 Poulik39 observed that by replacing the borate buffer in starch gel with citratebuffer, 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, samplesignal; O.D., optical density; lN916, “logarithmic” diodes. The flying-spot is generated in the followingmanner: The arc of a very high brightness 100 watt high-pressure xenon arc87 (Duro Test Corp., NorthBergen, N.J.) is imaged by a N.A. 0.3 microscope objective, M.O., onto the end of a clad, 50 microndiameter, 20-inch-long glass fiber (N.A. 0.6). This end is locked in position. The other end of the fiber iscarried on the “pen” of a high-speed Offner rectilinear pen galvanometer (Offner Division, BeckmanInstruments 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 speedflying-spot operation.


FIGURE 8. The absorption spectrum of the anion of “Dibromo-Trisulpho-Fluorescein,” a newlysynthesized 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 samplewith 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 excessivelyhigh O.D. at the absorption peak.

through his system. After staining his gels, he found appreciably improved resolutoin of someproteins. At that time, he recommended the use of “discontinuous buffer systems” to increaseresolution. While correctly recognizing that a discontinuity in voltage gradient was probablyresponsible for his improved results, that he did not appear to recognize that the phenomena ofthe Kohlrausch Regulating Function were involved is indicated by the following statement: “Thenature of the process which causes this system to give improved resolution is being investigatedwith 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 thesethe persulfate “catalyst” concentrations were quite high (i.e., the concentration recommended bythe American Cyanamid Corporation for the chemical initiation of polymerization of solutions ofacrylamide monomers40) and a moving boundary between borate and persulfate or sulfate (thebreakdown 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-statestacking presented above, with explicit solutions for the cases using the ions of weak acids orbases as the trailing ion.

It is also interesting to note, parenthetically, that the classic asymmetries between ascendingand descending boundaries in the Tiselius apparatus were known to be due to the Kohlrausch


phenomena and were the subject of intensive study (see Longsworthl9 for a review of thissubject), but the majority of such studies were aimed only at understanding and eliminating theartifacts of the Tiselius method. Although some of these artifacts were clearly recognized toinvolve boundary sharpening, the intentional use of these phenomena to design an electrophoreticmethod of increased resolution appears to have never before been suggested, except as perhapsmight be inferred from Longsworth's review37 of Kendall's work.

Our disc electrophoresis procedure has already been successfully applied to a wide range ofresearch and clinical problems. (See, for example, References 41 through 78 and the articles inthis monograph.)

The combination of high resolution, sensitivity, reproducibility, simplicity, versatility, andspeed 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, bodyfluids, and tissue extracts for physiological research and medical diagnostic purposes, in the studyof protein struclure and the “genetic code,” in the study of embryological induction anddifferentiation, and in the study of plant, animal, and human genetics and evolution. We hope thatboth the potentials of disc electrophoresis and the insight into mechanisms that have grown out ofour experience in developing this technique will stimulate a substantial increase in the use ofelectrophoresis for the separation and identification of ionic substances.

The author wishes to acknowledge the contribution of his colleague, B. J. Davis, who sharedequally 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 someof the statements in the main body of the text and (b) present estimates of the magnitudes of someeffects 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 viscositywill, in addition, be non-Newtonian. At low particle velocity, the pore would be almostimpermeable to the molecule, but at sufficiently high velocity such molecules may be able to passthrough the pores. The force exerted locally on the gel structure by the high energy moleculewould 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 non-Newtonian effect can be expected with rod-shaped molecules. At high velocity, the frictionalresistance of the gel will tend to keep the rods aligned parallel to the electric field presenting theirsmallest cross sections to the pores.

By comparing mobilities at high and low voltage gradients (100 volts/cm. to 5 volts/cm.), wehave not yet found evidence for non-Newtonian effects. This is not surprising since it is not untilthe electrophoretic pressure approaches the pressure exerted on the gel chains and proteins by thethermal 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). Theelectrophoretic pressure depends linearly on voltage gradient spacing of chains of a gel or

.340 Annals New York Academy of Sciences

and the thermal pressure on absolute temperature. It can be shown that, at 300°K, these pressureswill 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 onconcentration can be crudely estimated as follows: We assume that the length of an average chainis very large compared to the diameter of the chain molecule and that, in a gel, the distancebetween crosslinks is also very large. (For the purpose of zone electrophoresis and the argumentspresented here, the crosslinks in the gel serve mainly to stabilize the mass of intertwined chainsagainst convection.)

Consider all the chains arrayed to form a cubic lattice of n x n squares per face (see FIGURE9). 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. Therewill be 2(n+1) unit lengths of chain defining a lattice face and a total of 3(n+1)2 unit lengths toform the entire lattice.

If we increase the concentration of the polymer by a factor J, we multiply the total length ofchain per unit volume by J. If we were to re-form a new cubic lattice of unit volume, the numberof squares per edge would increase to (n+l)J l/2 – 1, which for large values of n is approximatelynJl/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 (seeAppendix B). In this model we have assumed that, at the corners of the elementary cubes, the intersectingchains occupy the same volume and also that the minimum “inside diameter” (σ) of the elementary cube isthe average “inside diameter” of that cube. These assumptions result only in a small error in σ so long as σis at least two times larger than d.


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 cubiclattice of the above type, or of a random lattice of the same polymer content, would be verynearly the same, even averaged over short time intervals. The above considerations are thereforeadequate 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 monomerconcentration ) .

(C) Definition of Electrophoretic Mobilityand the Effect of the Ionic Environment

In zone electrophoresis a sample of a mixture of ions is placed in a starting zone (see FIGURE10) in a linear conducting matrix, and a potential, v, is applied across the length, 1, setting up apotential gradient, V = v/l, along the matrix, After a time, t, cations will have migrated towardsthe cathode and anions towards the anode for distances proportional to their electrophoreticmobilities (see FIGURE 10B). The mobility is defined by m = d/tV = QX/f, where d is thedistance of migration of an ion in time t, Q is the net charge of the ion, f is the frictional resistanceof the medium, and X is a dimensionless factor that changes the effective charge of the ion anddepends 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 sphericalmolecules of radius where r = 5 Angstroms, 25 Angstroms, and 100 Angstroms. For ellipsoidalmolecules, 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 ithion, ci its mobility, mi, and its elementary charge, zi, such that

.

λ = E Σci mi zi , (l).

where E is the charge of the electron. * .

W = ——————(1 + Wrb)f(Wr) 1 + W(r + rb)

(modifieded from Gorin79), where W = (8πE2I/CKT)1/2 and the ionic strength,

I = 1/2Σcizi2,

r = radius of particle (protein),rb = radius of "buffer" ion (e.g., cation opposite anionic protein),ci = concentration of the ith ionzi = elementary charge of the ith ionE = charge of the electron,C = dielectric constant of the mediumK = Boltzmann's constant, andT = 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, at300,° 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 thisthickness.

X, as calculated here will be overestimated by the amount of the third order correction of Gronwall, La Mer, andSandved,8l which depends upon both charge and Wr (see Gorin82). The broken line in FIGURE 11 includes thiscorrection for a particle with r = 25 Angstroms and z = –25 (e.g., a protein like albumin).


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 populationof molecules will be charged at any one time. If xi is the fraction of dissociation (i.e., the ratio ofcharged molecules to the sum of the charged and uncharged forms), then each molecule can beviewed as being charged xi of the time and uncharged the rest of the time. The average velocity ofmigration, si , 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 substanceα (ions of α and γ of like sign of charge with mγxγ greater than mαxα), are layered U over L

FIGURE 11. Variation in “effective” charge, XQ (where Q is the true charge), of an ion as a functionof ionic strength, I, and the radius of the ion (see Appendix C).


in a cylinder, and a potential is placed across the cylinder (from end to end), then theconcentration of substance in these solutions for the velocity, sα to equal sγ may be derived asfollows:

Let sα = VUmαxα = sγ = VLmγxγ . (3)

Since the current, Y, through both solutions (which are electrically in series) is the same, itfollows from Ohm's Law that VL = Y/λLS and VU = Y/λUS, where S is the cross sectional area ofthe cylinder; therefore, from (l) and (3),

———— = ———— mαxα mγxγ

ΣciUmiUziU ΣciLmiLziL (4)

Equation 4 is a modified form of the Kohlrausch Regulating Function. l9 When the conditionsspecified in the equation are satisfied, substances α and γ will migrate down the cylinder withequal velocity and the boundary between them will be maintained. If a molecule of α were to finditself in the bulk of solution L (where, from Equation 3, VL is less than VU), it would migratemore slowly than the molecules of γ (and therefore more slowly than the boundary) and would beovertaken by the boundary. Conversely, a molecule of γ in the bulk of U will move faster than theboundary and will overtake it, thereafter migrating at the same velocity as the boundary. Let usnow consider two solutions with one common ion, β, and the two ions α and γ with charge ofopposite sign to β (from Equation 4):

––––––––––––– = –––––––––––– mαxα mγxγ

cαmαzα + cβUmβzβ cγmγzγ + cβLmβzβ

(5)

The condition of net macroscopic electrical neutrality in each solution requires that

cαzα = – cβUzβ and cγzγ = – cβLzβ . (6)

Therefore, from (5) and (6)

––––––––––––– = –––––––––––– mαxα mγxγ

cαzα(mα – mβ) cγzγ(mγ – mβ)

. (7)Let the total concentration of molecular species i, be

(I) = ci/xi , (8)then,

(A) xγcα mαzγ(mγ – mβ)

(Γ) xαcγ mγzα(mα – mβ)— = —— = —————–(A) xγcα mαzγ(mγ – mβ)

(Γ) xαcγ mγzα(mα – mβ). (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 totemperature (the temperature coefflcient of viscosity). Proportional changes in all mobilitiescancel 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 = cα/(A) = cα/[cα +(αH)] = 1/(1 + l0(pKa – pH)) , (10a)

xβ = cβ/(B) = cβ/[cβ + (β)] = 1/(1 + 10–[(14 – pKb) – pH]) . (10b)

(E) Moving and Stationary pH Boundaries*

Whereas Equation 9 explicitly estab!ishes the relationship between (A) an (Γ), our discussionof the application of steady-state stacking in disc electrophoresis (page 327) also requires, ingeneral, that a particular pH be established and maintained behind the moving boundary. Thismust be programmed by providing a particular concentration, (B)L,of a weak base (or acid, if αand γ are bases) in the solution containing (Γ).

The following considerations permit us to calculate (B)L for the case of monovalent acids andbases:

Since in any region of the column, from Ohm's Law and Equation 1,

Y = VSλ = VSEΣcimizi , (11)

then the fraction of the current carried by the β ion in any region (which is known as thetransference number20) is

Yβ/Y = cβmβzβ/Σcimizi . (12)

The difference in the β ion current on the two sides of the moving boundar provides a measureof 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 dis-crepancy between our earlier version of Appendix E and the literature to our attention. In previous developmentsof the moving boundary equations (see, for example,l9,20) conservation of mass of constituents on passage of amoving 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-examinedour previous Equations 11, 12, 14, 15, and 16 (see footnote, page 1) and find that they were in fact implicitly inviolation of the Law of Conservation of Mass with respect to the β component. The corrected version is present-

ed in Appendix E; however, the method described in Part II,26 and illustrated in FIGURES 1 and 5, was preparedbefore this error was detected, an in these cases xα and pHU (but not xα‘ and pHU‘) are different from those usedin the theoretical model (i.e., they are 1/8 and 8.9 rather than 1/30 and 8.3 respectively).


Since all β that is transported must be transported as β ion, the Law of Conservation of Massalso requires that

YβL – YβU = ESszβ[(B)L – (B)U] , (14)

where s = sα = sγ from Equation 3; therefore

[(cβL/λL) – (cβU/λU)]EYmβ/Ss = (B)L – (B)U . (15)

From Equation 4, λU = λLmαxα/mγxγ , from Equations 3 and 11, VL/S = l/mγxγ, andY/SλL = VL; therefore (cβL – cβUmγxγ/mαxα)mβ/mγxγ = (B)L – (B)U . (16)

From Equations 6 and 10a, cβL = – (Γ)xγzγ/zβ , and cβU = – (A)xαzα/zβ , therefore

(– (Γ)zγ + (A)zαmγ/mα)mβ/mγzβ = (B)L – (B)L . (17)

Dividing by (Γ), introducing the value for (A)/(Γ) from Equation 9, and simplifying,

(B)L – (B)U mβzγ(mγ – mα)

(Γ) mγzβ(mα – mβ) –––––––––– = –––––––––––––

(18)

This equation has a form similar to that of Equation 9.Given xα, from Equation 10a,

pHU = pKa – log10[(1/xα) – 1] . (19)From Equations 6 and 10b,

(B)U = – (A)xα (1 + 10–[(14 – pKb) – pHU])zα/zβ . From Equations 6 and 10b,

(B)L = – X(Γ)xγzγ/zβ .

where X ≥ 1. (From, 10b, when X = 2, the pH of the lower solution, pHL = (14 – pKb), and thelower solution will be maximally buffered. In general, less than maximal buffering is tolerable.)

From Equation 2l, [(B)L – (B)U/(Γ) = [(–X(Γ)zγxγ/zβ) – (B)U]/(Γ), therefore, from Equations 6and lOb,

(14 – pKb) = pHU – logl0{[(Γ)/(A)][(Xzγxγ/zβ)+[(B)L – (B)U]/(Γ)](zβ/zαxα) – l} . (22)

Having chosen a base satisfying Equation 22, using the values of (Γ)/(A) from Equation 9 andof [(B)L – (B)U]/(Γ) from Equation 18, then from Equations 18 and 20, (B)L can be explicitlydetermined.

If, in addition, we wish to set up a stationary pH boundary, with pHU above and pHU’ below,that will remain at a boundary between Ll and L2, then pHU' and xα' (xα at pHU'), rather than pH U

and xα. are used in calculating (B)L2 . When the potential is applied and the moving boundarypasses the boundary between Ll and L2, the pH of the solution above that boundary will remainequal to pHU, but the pH between that boundary and the moving boundary will equal pHU'. (seeFIGURE 3C).


(F) Diffusion Spreading of a Disc

The increase in thickness due to diffusion alone is (T–To) = 2x, where, from Einstein'sequation, x = (2Dt)l/2, x is the root-mean-square displacement of a particle as a result ofBrownian motion after a time t, where D is the diffusion constant in the gel, T is the thickness of adisc 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 ofprotein in the original sample (see page 327). Any spreading in excess of T that is not directlyattributable to differences in the voltage gradients and pH's inside and outside of the disc (seepage 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 fixeddistance of separation from the origin, and a fixed voltage gradient, the difJusion spreading of thedisc 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 MassAction requires that the product of the concentration of cation and glycinate ion outside a discequal that inside.

cαocβo = cαi

cβi , (23)

where sub-subscript o refers to regions outside a disc and i, to regions inside. Net electricalneutrality requires that

cβizβ = – cαi

zα – cpzp , (24)where subscript p refers to protein.

These two conditions define the Gibbs-Donnan equilibrium and permit us to calculate theconcentration of glycinate ion inside a disc.

Solving (23) for cβi , and substituting in (24),

(cαi)2zα/zβ + cαicpzp/zp + cαo

cβo = 0 . (25)The solution to this quadratic equation is,

cαi = –––––––––––––––––––––––––––––––––––– cpzp/zp ± [(cpzp/zp)2 – 4(cαocβozα/zβ)]1/2

2zα/zβ (26)

For a base with zβ = + 1 and glycine in the small-pore gel at pH 9.5 (when xα' = 1/ 3 ) thisreduces to

cαi = (A)i xαi = ––––––––––––––––––––––––

cpzp ± [(cpzp)2 + (A)2 4/9]1/2

2 (27)

We can now calculate the pH within the disc as well as the ratio of the voltage gradient insidethe 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,] . (28)From (1),


– cpzp(– mp + mβ) + (A)i xαi (15 + mβ) (A) (15 + mβ)/3

Vo/Vi = –––––––––––––––––––––––––––––––– (29)

During “steady-state-stacking,” as well as during separation in the “smallpore” gel, the Gibbs-Donnan equilibrium specified above implies Donnan potentials and diffusion potentials across thetwo boundaries of a disc. These also effect the spreading, relative to that expected from diffusionalone, but will not be considered here. Their effects diminish as the mobility of the protein in thegel, mp, diminishes.

(H) Dissociation of Complexes During Electrophoresis

It has been shown by Ogston83 (see also Boyak84 and Mysels85) that complexes of ionicspecies (in cases where the mobilities of the complexes and the components are different fromone another) will dissociate during an electrophoretic separation to an extent proportional to thesquare of the voltage gradient.

In a system such as our model for disc electrophoresis (stacking pH, 8.3; running pH, 9.5), thevoltage gradients to which a complex may be exposed during stacking are up to 10 times greaterthan during running (on the order of up to 40 volts/cm.). As a result, if a stack spends sufficienttime 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 arereduced in length, or are eliminated (i.e., conditions approaching those used in Smithies' starchgel 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 begreater 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 amountof complex (and therefore the thickness of the relevant discs in the stack) and/or the length of thespacer gels are varied.

When such phenomena are observed, increasing the sample “load” per run will result in theobservation of a relative increase in the complex disc in the separated pattern and a relativedecrease in the discs of its components. Conversely, increasing the length of the spacer results ina decrease in the concentration of the complex disc and an increase in the concentration of itscomponents. 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 proteinsolutions so far studied. One serum post-albumin (apparently a seromucoid) and some pre-albumin serum “polypeptides” (hormones?) seem to be involved in such an association-dissociation equilibrium.86

REFERENCES

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