Molecular Weight Determination of Protein-Dodecyl Sulfate Complexes by Gel Electrophoresis in a Discontinuous Buffer System

This report describes methods and results obtained by combining the techniques of sodium dodecyl sulfate (SDS) gel electrophoresis and electrophoresis in discontinuous buffer systems. The SDS gel system utilizes a sulfateborate discontinuity which stacks and unstacks protein-SDS complexes over a range of 2,300 to 320,000 daltons, providing high resolution fractionation. The properties of protein-SDS complexes are investigated by calculating retardation coefficients and apparent free mobilities from Ferguson plots. Apparent free mobilities are approximately constant, establishing a linear relationship between the logarithm of the relative mobility and the retardation coefficient. The retardation coefficient is shown both empirically and theoretically to be a uniform function of molecular weight of protein-SDS complexes over specified ranges, providing a rationale for determining molecular weight from plots of the negative logarithm of relative mobility against molecular weight.

From the Section on Physical Chemistry, Laboratory oj Neurochemistry, National Institute of Mental Health, Bethesda, Maryland WOO1 4 SUMMARY This report describes methods and results obtained by combining the techniques of sodium dodecyl sulfate (SDS) gel electrophoresis and electrophoresis in discontinuous buffer systems. The SDS gel system utilizes a sulfateborate discontinuity which stacks and unstacks protein-SDS complexes over a range of 2,300 to 320,000 daltons, providing high resolution fractionation.
The properties of protein-SDS complexes are investigated by calculating retardation coefficients and apparent free mobilities from Ferguson plots. Apparent free mobilities are approximately constant, establishing a linear relationship between the logarithm of the relative mobility and the retardation coefficient.
The retardation coefficient is shown both empirically and theoretically to be a uniform function of molecular weight of protein-SDS complexes over specified ranges, providing a rationale for determining molecular weight from plots of the negative logarithm of relative mobility against molecular weight.
This report describes an electrophoretic system capable of stacking and fractionating protein-dodecyl sulfate complexes over a range of 2,300 to 320,000 daltons.
The system combines the advantages of electrophoresing protein-SD% complexes pioneered by Shapiro,Viiiuela,and Maize1 (1) and the advantages of achieving thin starting zones by use of discontinuous buffers discovered by Ornstein (2) and Davis (3). This SDS discontinuous system was developed to fractionate plasma membrane proteins solubilized in SDS and does provide high resolution patterns of solubilized membranes, resolving over 40 discrete bands (4,5).
In the course of accumulating data with this SDS system, it became apparent that the usual type of calibration curve used in SDS gel electrophoresis was only approximately linear and that, the actual curves of log molecular weight (M) versus relative mobility (RF) were sigmoidal in nature. It also became apparent that certain protein markers could not be fitted on linear 1 The abbreviation used is: SDS, sodium dodecyl sulfate. log M uersus RF plots unless 25% deviat,ions were tolerated. It was not apparent whether this was due to the alkaline nature of the discontinuous system, the lower ionic strength, or the particular proteins used. In order to investigate the sources of these deviations, it became necessary to explore the theoretical foundations of determining molecular weights from relative mobilities in gels. This subject had been investigated Cur proteins, but at the time of this study only empirical correlations of molecular weight and relative mobility had been reported for protein-SDS complexes.
Recently, however, Chrambnch and Rodbard (6, 7) have outlined a theoretical treatment for this subject which is similar to the one presented here.
In this report I explore the relationship between relative mobility and molecular weight of protein-SDS complexes by calculating retardation coefficients and free mobilities from gels of differing acrylamide concentrations.
The finding that protein-SDS complexes have nearly identical free mobilities provides a theoretical justification for calculating molecular weight,s from relative mobility values obtained at a single gel concentration. The assumptions behind these calculations illustrate the sources of error inherent in the method, and ways of minimizing these errors are suggested.

MATEHlALS AND METIIOljS
The principle used in obtaining a disc system t)o operate in SDS was to find a system capable of stacking and unstacking charged, highly mobile polymers.
After trying several systems designed for mlcleic acids, we discovered that borat.e-sulfate bufiers gave excellent resolution.
While attempting to opt,imize this system, we discovered that this system was one of the 4269 multiphasic buffer systems calculated from theory by Jovin,Dante,and Chrambach (8). The recipe given below comes from their computer output.
The gels are described by the notation of Hjerten (9), in which the first mrmeral (T) denotes the total weight of monomer Issue of October 25, 1971 Il. M. Neville, .JT. 6329 (acrylamide plus N,N'-methylenebisa~rplamide) per 100 ml of solvent, and the second numeral (C) denotes the amount of N, h"methylenebisacrylamide expressed as a perccnt,age (w/w) of the total amount of monomer.
Upper gels are all 3.2 x 6.25. Lower gels vary between 5 x 2 and 25 x 0.1. The standard gel (11.1 X 0.9 and ammonium persulfate are the minimal amounts required to produce polymerization at 10 min. If either catalyst is present in excess, polymerization time will be increased and distortion of the surface will occur. (c) Isobutyl alcohol (30 ~1) is layered over the lower gel solution after filling tubes and is removed after polymerization prior to casting t.he upper gel.3 (d) If gels 10 cm or longer are cast in narrow tubes (10 x 0.5 cm), contraction of the volume on polymerizat.ion can cause a doll.m\-ard curvature of the upper surface. Substituting th' m vinyl sheet,ing tied with dental floss or rubber bands for rubber stoppers allows t'he curvature to take place at the tube bottom thus minimizing the problem. Gels 5 mm in diameter were run at 1.5 ma l)er tube at 25" in a jacketed reservoir with circulation of buffer around the gels by means of a magnetically driven st,irring bar in the lower reservoir. With large samples (0.2 to 0.4 ml), current was reduced to 0.5 to 0.1 ma until the sample entered the upper gel.
Staining or destaining was done with Coonlassie blue as described bgr Weber and Osborn (lo), except that staining time was increased to 15 hours and the dye concentration was reduced to 0.007% to 0.07%.
ReMive mobilities were calculated relative to the borate-sulfate front which stacked the marker dye bromphenol blue (Canalco, Rockville, Maryland) or a red impurit,y of high mobility in the high pH or high gel concentration runs. Dye fronts were marked by insertion of a section of stainless steel surgical wire.
Proteins used as markers are listed in Table I along with their sources and references for molecular weights.
Individual protein concentrations generally ranged between 0.1 and 5 pg per sample. When available, chromatograplIically purified proteins were purchased.
F'roteins, except as noted, were dissolved in 0.05 M Na2C03 and exposed to 8 mg of SDS per mg of protein for 1 min before the addition of 10% by volume &mercaptoethanol. They were then dialyzed against upper gel buffer containing 0.1% SDS, 0.05y0 dithiothreitol, 2% sucrose, and a trace of bromphenol blue. When the dimera of y-plobulill and /Zgalactosidase were desired, these proteins were dissolved in 0.1 y0 SDS without reduction or heating.
Experiments designed for the calculation of retardation coefficients utilized a constant ratio of bisacrylamide to acrylamide of 1:90.

RESULTS
Log JI Versus RF Plofs-The sulfate-borate system stacks protein-SDS complexes over a molecular weight, range of 2,300 to 320,000, providing very sharp bands even when sample column heights of 2 cm are used. Under condit,ions of these experiments, 20 ng of protein in the 40,000 to 100,000 molecular weight range could be resolved by the C'oomassie blue stain. Values of relative mobility (RF) obtainctl n-ith the sulfat,eborate system in a gel (11.1 x 0.9) are plotted against molecular weight (X) on a semilog scale (Fig. 111). This relat,ionship, first introduced by Shapiro, Vifiuela, and Maize1 (1 j, has been found to be empirically valid for at least 40 different protein-SW complexes (10,11). The results shown in Fig. ln are similar t,r> t,hose reported in neutral continuous systems in that, ia,j a relationship exists between RF and III, (b) the curve connecting the high SI points is hyperbolic, and (c) t'he low -11 region is the region of maximum scatter from a smooth curve.
These results differ from previously reported data in that in the low M regic~ill it is possible to fit points with two different straight lines. In the range of 70,000 to 15,000, a straight line can he drawn through the points for albumin and hemoglobin and nine marka. lie on this line (deviations in M less than 5y0). Thr points for DSase, chgmotrypsinogen, cytochrome c, and chymotrypsin B chain all show deviations of more than 20% of &Z from this line. Homever, a steeper line will fit these points and four others. 1 shallow sigmoidal curve will provide the least deviation of points from a line. When data from i, 9,13, and 15% gels are plotted in the same manner, the results arc similar, and the imprej'ion is gained that the true nature of the function is sigmoidai. (RF) for a variety of protein-SDS complexes subjected to electrophoresis at pH 9.5 on gel (11.1 X 0.9). In the upper molecular weight region the curve is hyperbolic while in the lower regions the scatter obscures t,he nature of the relatiol,ship.
In B, theoretical curves are constructed for RF in Fig. 2. l'his type of plot, first described by Ferguson (16) (17) to adequately describe the behavior of 17 globular proteins on acrylamide gels varying in M from 45,000 to 500,000.
The protein-SDS complexes display two properties not seen with proteins.
Complexes of JI greater than 60,000 do not shop a linear relationship between log RF and gel concentration. In addition, the molecular weight at which nonlinearity appears is del,endent on gel conceutration, high gel concentrations augmenting the effect.
The most int,eresting feature on the Ferguson plot is the fact that the values for the Y intercept are nearly identical for the 10 different complexes.
The Y intercept value represents the relative mobility at zero gel concentration, and for proteins carrying different charges, different free mohilities are observed (17). Although it has been previously shown that proteins in SDS bind a constant amount, of SDS per unit weight of protein (18, 19), this fact alone woulcl not' lead to free mobilities independent of X.
In order to have free mobilities independent of .2 .4 .6 .8 and M on the same scale from values of RF computed from Equation 1 by assuming a constant value of free mobility, 100 log 100 Yo = 230, and a linear dependency of KR on M (see Table II).
M, the ratio of the effective charge to frictional coefficient must be independent of -11. In other words, the complex must behave under free electrophoresis as a free draining structure so that t,he mobility of a large molecule is identical with the mobility of a segment or smaller molecule (20). It is interesting to note that the free electrophoretic mobility of DNA, both native and denatured, is independent, of dl between one-quarter million and 130 million (20).
Although the T intercepts are nearly identical in Fig. 2, the variations are larger than that due to measuring error (0.02 RATI for RF = 0.5). In Fig. 3, the slope of the lines connecting points (log RF and T) for identical proteins (retardation coefficient) is plotted against log RF to determine whether any kend with molecular weight can be detected.
(Plots are done only in regions where log RF Dersus T plots are linear.) There is no apparent trend with molecular weight.
However, it is interesting that the points for DNase and hemoglobin show a significant deviation from the line. These deviations must be ascribed to an apparent free mobility different from the average free mobility. Fro. 2. The logarithm of the relat,ive mobility for 10 protein-SDS complexes is plotted ZJ~TSUS the gel monomer concentration (% 2') and the plots are extrapolated t,o T = 0. All of the complexes have a nearly identical free mobility.
Note that the highest molecular weight complexes show deviations from linear plots at high 2'. Numbers at right refer to protein code in Table I.  It is possible that certain complescs exhibit more complicated interactions with the gel, such as adsorption, resulting in nonlinear Ferguson plots as the gel concentration approaches zero. This situat,ion would lead to an apparent free mobility different from the average free mobility when all measurements are made at high gel concentrations.
Eflecfs of Varying pR-In Fig. 4, the relationship between -log RF and X for gels (11.1 X 0.9)  4. Molecular weight is plotted versus the negative logarithm of the relative mobility for various protein-SDS complexes in gels (11.1 X 0.9) run at running pHs between 8.5 and 10 (see "Materials and Methods"). As the pH is lowered, more and more proteins travel within the stack in the separation gel. The influence of pH on Yo has not been investigated but the relationship between some points on this plot appear significantly dih'erent for the pH 9.5 points VETSUS the pH 10 points.
ing the apparent velocity of the stack (2,8). Consequently, some proteins traveling within the stack at low-pH are excluded from the stack at high PH. In a gel (11.1 x 0.9) at pH 8.5 (the running pH of the stacking gel), subunits having M <50,000 are in the stack, although at pH 10, X of 5,000 and above are excluded from the stack.
The results of this study of SDS gel electrophoresis show that the empirical relationship observed between molecular weight and relative mobility in neutral continuous systems is also observed in alkaline discontinuous systems. In addition, these results show that the apparent free relative mobilities (YO) of protein-SDS complexes are nearly constant. When data obtained with discontinuous systems are plotted in the usual manner, log Jll versus RF, considerable scatter is observed in the low molecular weight regions.
In an attempt to understand the causes of this scatter, it became necessary to investigate the theoretical foundation for obtaining molecular weights from a relative mobility value on a single gel. Previously, it had been shown that molecular weights of proteins could be estimltted if relative mobilities were obtained at more than one gel concentra Con by use of the Ferguson equation (16). ,l log Rp = -I"Rl + log Yo (1) where RF is the relative mobilit,y at gel concentration T, Y0 is the relative mobility at zero gel concentration (t)he apparent free relative mobility), and Kn is the ret,ardation coefficient a.nd is a function of molecular size and the percentage of cross-linking. If the percentage of cross-linking is held con%mt, K, can be determined from values of RF taken at various values of T. For 17 globular proteins, K, is directly proportional to II1 (17). These results, which show that. log Y0 is a constant, establish a linear relationship between log RF and Kn at any given 7' (Fig.  3). Once the average value of log Y, has been determined for a given buffer system and percentage of cross-linking, the retardation coefficient can be calculated from a single log RF value at 6332 XDX Gel Electrophoresis Vol. 246, (chymotrypsinogen) show'large deviations from the line. any single value of T. Retardation coefficients calculated in this manner are represented Kd. K,' will differ from K, to the extent, that the particular apparent, free mobility differs from the average free mobility.
In order to determine molecular weight from KRf, it is only necessary to determine that there exists a uniform dependence of 111 on Knt. The theory of gel electrophoresis developed by Ogston (21) and Morris (22) and recently extended by Rodbard and C'hrambach (23) provides a theoretical basis for the dependence of K, on effective molecular radius. These workers COIIsider a gel to be made up of spaces or pores and that for any molecule of effective radius, R,, a fractional volume, f, of spaces is available.
This model interconnects both gel filtration, where t,he distribution coefficient K,, is equivalent to f, and gel electrophoresis, where The dependence off on R, is exponential, with the exponent a being determined by assumptions involving pore geometry or the type of distribution of spaces.
The most likely exponent is considered to be 2, and data consistent with this value have been reported for acrylamide in gel fiNration (24) and gel electrophoresis (23). However, other relntiomships have not been excluded, and the nature of the data is such that many different dependencies can be fit. kl is a constant including gel fiber length per unit of volume, and T is the fiber radius, which has a value of 5 A for 1% cross-linking ('34). Taking logarithms log 12.~ -log I', = -kz(R, + T)~ Since T is proportional to kz (24), this term may be eliminated by division.

KR = ka(Ra + T)~ (7)
and where R, > r KR 'u kr(RJa Reynolds and Tanford (18) and Fish,Reynolds,and Tanford (19) have shown t,hat the hydrodynamic properties of protein-SDS complexes det.ermined from viscosometric and gel filtration data are a unique function of molecular chain length.
Their data are expressed in t,erms of the Stokes radius, R,.
R, = kg(M)* By assuming that R, can be substituted for R,, KR N k,[k,(M)bl" log KR s ab log M + log kg ks = k,(kg)* (10) (11) Equation 11 shows that plots of log Kn or log K,' against log X will be linear with a slope ab proportional to the relationship between R,% and M. Deviations from linearity will occur when the assumptions underlying Equation 11 break down, for example, when the condition R, > r is not satisfied. When a = 2, deviations in ab great'er than 10% do not occur until R, I: 15 A or R, 5 25 A for a = 1. The decrease in slope in Figs. 5 and 6 below M of 15,000 could be due to the influence of the r term. Equation 11 provides a way of relating relative mobility data to molecular weight by means of a plot which is linear under a set of known assumptions.
When the values of Fish et al. (19) for lig and b are used with our K,' data in Equation 11, we calculate the value of a as 1.
As mentioned previously, the preferred theoretical value is 2.
If it is assumed that a = 2, then b = 0.34 and R, varies as MOJ4 rather than as Xo.73, as determined by Reynolds and Tanford (18). This would mean that the effective radius in gel electrophoresis is less than the effective radius in gel filtration. When the data of Hedrick and Smith (17) for globular proteins (R, varies as X0.3?) are plotted according to Equation 11, ab = 0.5, and again assuming a = 2, the effective radius is less than the Stokes radius.
Although the values for a and b in Equation 11 are in doubt for gel electrophoresis, the equation is useful in that alterations in the relationship of R, on M will be apparent by a change in slope. Such a change occurs below 15,000 daltons and above 70,000 daltons. The decrease at 15,000 daltons was also noted for protein-SDS complexes in gel filtration (19) and may be due to the change in geometry from a prolate ellipsoid to a sphere in this region.
The effect of the r term will also tend to decrease the slope in this region.
Above 70,000 (Fig. 7), the slope decreases, indicating a decrease in effective radius. Fisher and Dingman (25) have presented data on electrophoretic behavior of nucleic acids in gels by means of log K, versus log X plots. Rod-shaped nucleic acids3 show a smaller slope ab than random coil nucleic acids of the same 111. In addition, the RF of rod- From 18,000 to 70,000 daltons the slope is 0.68 and decreases below and above this region.
shaped nucleic acids increased with increasing voltage gradient. These authors speculate t'hat the highly asymmetrical rodshaped molecules may be capable of orienting in a manner which minimizes their frictional resistance to the gel (25 If within the region of constant dependency of k',' on M any t,wo markers of known X have their RF determined, the cotlst.ants k.7 and log Y0 are fixed, and M can be determined directly from values of -log RF for any unknown. This type of plot is shown in Fig. 7. The standard error is ~3000 M. Comparing Figs. 5 and 7 shows that, it is not necessary to have the exact relationship of KE' on M for estimating M. Fig. 5 shows chymotrypsinogen and DBase to be anomalous points. However, this fact is not as obvious from the plot in Fig. 7. The commonly used plot, log M versus RF, is not as useful since it is sigmoidal in shape when there is a constant dependency of KR on X and when Y0 is a constant (see Fig. IB).
From the foregoing discussion it can be seen that the assumptions involved when M is calculated from a value of RF in SDS gel electrophoresis are (a) a constant value of Y. for marker and unknown protein-SDS complexes and (b) a constant dependency of R, on X within some range of M. These assumptions are imposed by t#he use of the marker proteins used to form the calibrat,ion line when M is plotted as some function of RF. If the markers used have significantly different values of Y0 than the unknown protein-SDS complexes, errors in X will result. Both DBase and hemoglobin are examples of this type of error.
Each shows deviations in plots of log RF against KRf and deviations in log RF versus X plots. That the latter deviations are due to the incorrect estimate of Y0 is seen by the fact that the deviations are not present when K, rather than k',' is plotted against M (see Table II).
Therefore, the Ferguson plot provides a means of detecting this type of error.
Similarly, if the markers used to form the calibration line have a different dependency of J4 on R, than the unknowns, errors will result. Therefore, extrapolations into ranges not covered by markers should be avoided, and multiple markers should be used in transition regions.
The deviation of chymotrypsinogen in -log RF versus X plots appears t,o reflect the second type of error. The deviation in M is +15% in a Kn versus M plot and j-17% in a -log RR vewus M plot. C'ytochrome c and chymotrypsin C chain lie in a region of altered dependency of R, on M. An explanation for the scat- Circles represent proteins whose KR have been determined.
Lines fitted to all points or circles alone are essentially identical although the standard error drops from 3500 to 3000 when a11 points are used. ter of the dat.a is thus provided.
Whether or not the conditions of the alkaline buffer system or specific effects of the borate ion aggravate these problems as compared to neutral systems remains to be seen.
Gel electrophoresis of protein-SDS complexes with the use of other discontinuous buffer systems is being investigated and those studies may answer some of these questions (6,26).