Part
11:
Analysis and Theory
ROBERT A. ALBERTY University of Wisconsin, Madison, Wisconsin
T H E MOST important application of electrophoresis a t t,hepresent time is to the analysis of naturally occurring mixtures of colloids, such as proteins, polysaccharides, and nucleic acids, and of the products obtained in the course of fractions to obtain purified components. Electrophoresis often offers the only available method for the quantitative analysis of such systems. The interpretation of electrophoretic patterns including the determination of electrophoretic mobility and the analysis of colloid mixtures requires knowledge of the t,heory of moving boundaries to which this article give a brief introduction. The Electrophoretic Analysis of Plasma. Human and animal plasma and sernm and derived proteins have been widely stndied by electrophoresis (1,4,9,11, 15). Tiselius ($0) showed t,hat the protein called serum globnlin by earlier workers was in reality made up of three proteins identifiable by Blectrophoresis which he nanied a, (3, and 7 globulins. Longsworth (11) lat,er found that in pH 8.6 sodinm rliethylharhiturat,e (veronal) buffer of 0.10 ionic strength anot,her prot,ein appeared whirh he cksignat,ed a,. Figure 1 shows t,he electrophoret,ir pat,t,ernsfor a sample of normal human plasma i n this I~ufferafter 150 minutes at a. potent,ial gradient of 6.0 volt,s/rm. In addit,ion to these globulins, t,here is albumin, which makes np over half of normal hnman plasma, and fibrinogen, $. The 6 and B boundaries. which remain near the initial boundary position, do not represent additional components, as was originally snpposed, so they are ignored in analysis. Owing to the incomplete resolution of the prokin ~ e a k s it . is necessarv to make a more or less arhit-rarv separation in order to determine the area due to a given component. This is usually accomplished by constructing Gaussian-shaped curves for each peak such that their ordinates at, every point add up to give the experimental curve. In some cases these curves may not be symmetrical because of gradients of pH and conductivity within the boundaries, and the proteins included in one component are not necessarily homogeneous with respect to size, shape, or isoelectric point,. If A , ,k the total area of the electrophoretic pattern exclusive of the r boundary in the descending pattern or the 6 boundary in the ascendmg- -pattern, and A, is the ~
The following table gives the relative and absolute concentrations of proteins in the plasma of normal young male adults as determined by electrophoresis in pH 8.6,0.10 ionic strength sodium diethylbarbiturate hnffer (6). The plasma was diluted with buffer (1 volume plasma, 2 volumes buffer) and was then dia l y ~ against d bnffer in the cold for two or more days. TABLE I Electrophoretic Analysis and Mobilities for Normal Human Plasma in 0.10 Ionic Strength Diethylbarbiturate Buffer (pH 8.6) Relaliue
Mobility,
Cowentralion,
%
Comvonat
m.* set.-' uolt-'
G./lW cc.
Albumin a ,-globulin a rglobulin 4-globulin Fibrinogen 7-globulin
The electrophoret,ic analysis of plasma varies in a marked way in pat,hological cases (9). Figure 2
~
' Part I, Methods and Cdculntions, was published in August,, 1948, page 426.
area of the peak corresponding to one component,, the percentage of this component in the mixture is
Alb
-
D6':~ccnoing
.
Ascending
Figun 1. Elsct.ophartic Pat,of Norrnll xoni. stnnpth ~ i . t h ~ l b . . b i t ~ ~ ~ t~. " f f.f tdinut- .t 6.0 volt./Crn.
o m
619
numan Plasma in PH 8.6 after 150
JOURNAL OF CHEMICAL EDUCATION
620
shows the plasma pattern of a patient with multiple myeloma.' For more description of the clinical application of electrophoresis, the reader is referred to recent reviews (8, 19). The areas in the electrophoretic pattern are directly proportional to the changes in refractive index across the boundaries hut are not directly proportional to the concentrations of the various proteins unless their refractive index increments are equal and the concentration of only the one component changes across a boundary. One of the purposes of this article is to review methods for reducing both of these uncertainties in electrophoretic analysis. In order to allow for differences in refractive index increments, the areas in the pattern should be divided by the refractive index increments of the corresponding proteins before calculating the percentage composition. The refractive index increments for a protein may he defined either as the differencebetween the refractive index of a solution containing 1 g. of protein per liter and the pure solvent (An/AW) or the difference for a solution containing 1 g. of protein nitrogen per liter (An/AN). The refractive index increments on a weight of protein basis vary less than those on a protein nitrogen basis. For example, An/AW for plasma proteins using D sodium light varies from 1.88 X for 7-globulin to 1.71 X for 8-lipoprotein while An/AN varies from 1.17 X l O P for 7-globulin to 4.05 X lo-' for 8-lipoprotein ( 2 ) . The choice of bufferfor the electrophoretic analysis of a particular type of mixture is extremely important. In the case of human plasma, for example, orl-globulin is not resolved from albumin, and 7-globulin is not resolved from the 6 and e boundaries in pH 7.4, 0.02 M sodium phosphate and 0.15 N NaCl buffer. Diethylbarbiturate (veronal) buffer of 0.10 ionic strength a t pH 8.6 gives perhaps the best resolution for human plasma, while horse plasma shows a more satisfactory pattern in 0.008 M NaHzP04to 0.064 M NaaHPOd buffer a t pH 7.7 than in diethylbarbiturate buffer. No general statement may be made regarding the optimum protein concentration or the optimum duration of the exoeriment because these deoend unon the mo' This pattern was provided through the courtesy of Dr. P. P. Cohen.
gi5 rnnn!ng
Figure
2.
Asce,--i'~g
Elntrophoretic pattern of the Plasm. of a patient with Multipl. Mydoma in th. Same Bufter . a Figure 1
bilities and relative amounts of the components and the question to be decided by the analysis. The Ideal Case in the Electroph,mesis of Mixtures. The methods used above for calculaCmg the relative analysis of plasma and the actual concentrations of the components assume that the electrophoresis experiment has been carried out under what Longsworth (14) has referred to as "ideal" conditions. In the ideal case electrophoresis should be carried out with very dilute protein solutions in buffers of rather high electrolyte concentration so that the conductivity of the solution will be determined almost entirely by the buffer ions because of the low mobility and low equivalent concentration of the protein ions. Thus the buffer ions insure throughont the U-tube a uniform pH and a uniform electric field through which the protein ions migrate. In the electric field the different proteins move away from the initial boundary with different velocities, each forming a moving bomdary in which the protein concentration varies from a constant value below the boundary to zero above the boundary. In this ideal case of electrophoresis the pattern obtained in one limb of the U-tube would be exactly a mirror image (enantiograph) of the pat,tern from the other limb, and the velocities of the protein boundaries would he the same in the two limhs. In Figures 1 and 2 we may note a number of differences between the ascending and descending patterns. (1) The distances moved by the protein boundaries are greater in the ascending limb than in the descending m l (2) The rising albumin boundary is sharper than the descending albumin boundary. (3) The area of t,he 6 boundary is greater than that of the s boundary. (4) The areas under any given peak are not the same in the ascending and descending patterns, although the total areas of the ascending and descending patterns including the stationary boundaries are equal. These differences are a result of the fact that the contribution of the protein ions to the conductivity of the solution is not negligible and causes small gradients of conduct,ivity, pH, and protein concentration across the moving boundaries in addition to the gradient of the protein constituent which disappears in the boundary. Since these effects are of different magnitude in solutions of different protein concentration and buffer salt concentration t,he apparent analysis of plasma varies with both of the latter (1, 16, 18). It has been sugrested that the true composition of a protein mixture may be obtained by extrapolation of the apparent composition either to zero protein concentration a t constant ionic strength or to infinite salt concentration at constant protein content (14). A useful type of graph for this ext.rapolation is a plot of the apparent analysis obtained from the schlieren diagram against the ratio of protein concentrat,ion (%P) to ionic strength (1). Figure 3 shows such plots for three protein mixture which have been studied at, several ionic strengths or protein concentrations. In each case the apparent analysis is the average of those obtained from the ascending and descending patterns. The normal human
.
NOVEMBER. 1948
plasmas were analyzed by Perlmann and Kaufman in pH 8.6 diethylbarbiturate buffer with various amounts of sodium chloride added to raise the ionic strength above 0.10 and a t a protein concentration of 1.87% (16). The hog serums were analyzed by Svensson in 0.068 N phosphate buffer of pH 7.7 with various amounts of added sodium chloride (17). The synthetic mixture of 8-globulm and albumin made up to contain 65 per cent albumin by refractive index increment was analyzed by Armstrong, et al., in pH 8.6 diethylbarbiturate buffers (1). The magnitude of the error in electrophoretic analysis indicated by Figure 3 suggests that in more critical work two or more experiments in which either the protein concentration or salt concentration is varied should be performed so that an extrapolation may he made. Introduction to the Theory of Moving Boundaries. In order to understand the deviations of electrophoretic patterns from ideality it is necessary to examine the theory of moving boundaries in general. Moving boundaries are also used for the determination of the mobilities and transference numbers of inorganic ions and the bwsic theory is well developed for simple mixtures. Important additions to the theory of moving boundaries have been made recently by Dole (6), Longsworth (IS, l4), and Svensson (IS),and this article will serve as an elementary introduction to the theory as applied to the electrophoresis of proteins. In order to derive the relationship between ion mobility and transference number, we must first look a little more closely a t the physical constant specific conductivity, K. Specific conductivity may be defined by equation (2)
where R is the resistance in ohms of a column of conductor of cross-sectional area q and length 1. Thus the units of specific conductivity are ohm-' cm.-I. Substituting E/i for R by Ohm's law
From this equation we see that specific conductivity is the current (in amperes) carried through a 1-cm. cube of conductor between two opposite faces diiering in potential by one volt. In the case of solutions of electrolytes, the current is equal to the summation of the rates of transport of electric charge by the different ionic species. If there are C, Faraday equivalents of an ion per liter of solution and this ion constituent has a mobility of UJ cm.' volt-' see.-', the electric charge carried by the ion through a square centimeter crosssectional area perpendicular to the electric field in one secondunderunit potential gradient is u,CJ/lOOO equivalents. In order to obtain the current in amperes (coulombsper second) we must multiply by the number of coulombs in an equivalent, which is 96,500 (Faraday's constant).
35
0
10
20
30
% P+ r/2 Figure 3. Extrapolation of Apparent El.ctroph-ti0 And* It Vauioua Protein con cent ratio^ and Salt Concsntrations to "Ideal Condition."
Here, n is the total number of different ionic species. In writing this equation we adopt a convention which is convenient for the discussion of electrolytic solutions, and that is, we give the equivalent ionic concentrations C1, CZ,C8, . . . Cnthe sign of the ion so that the product C,uJ is always positive and ZC, = 0 for an electrically neutral solution. The transference number of an ion is defined ws the fraction of the total current carried by that particular ion so that
This equation gives the relation between mobility and transference number. The second form of the equation is obtained from equation (4): The fundamental law in the study of moving boundaries is the moving boundary equation which relates the displacement of a separated single boundary and the concentrations and transference numbers of the ions in the homogeneous solutions on either side.= The
'
I n the ease of proteins or other weak electrolytes this e q u s tion is applicable only in so far as the buffer maintains a constant pH.
JOURNAL OF CHEMICAL EDUCATION
SOLN
OLI
anion. The boundary moved through 5.04 liters (corrected for electrode reaction) per Faraday a t O°C., and so the transference number of the chloride ion is Tcr
=
VQ8Cm- = (5.04)(0.1)= 0.504
To calculate the mobility of the chloride ion a t 0°C. the specific conductivity of the 0.10 N KC1 solution must be determined and is 0.007138 ohm-' cm.-'
=
following derivation is given by Longsworth (IS). Consider the boundary, a of Figure 4, between the solutions a and 6 that moves against the current, i, through a volume Va8 (liters per Faraday) (corrected for the electrode reaction) to the position b on passage of one Faraday equivalent of electricity. If CJmis the equivalent concentration of an anion constituent j in the solution a and CJ8its concentration in the 6 solntion, the number of equivalents of this species present and after the pasinitially in the volume V"@is CJmV"@ sage of the current, CJWm'"B. Moreover, a number of equivalents of the j ion equal to its transference number, T f , in the solution simultaneously enter this volume through the plane a t a while TJaequivalents leave through the plane a t b. Conservation of mass for the j ion in the volume V"8 then gives Tim
- Tip = VDLB(Cjm - C#)
=
V,"S C,
(7)
This is the usual case encountered with proteins. Substituting eqnation (7) for TJin equation (5) and solving for u we obtain the following:
where v, is the volume in cm.' moved through by the j boundary per coulomb. This equation is identical with equation (8) of Part I. The application of equations (7) and (8) may be illustrated by consideration of a two-salt moving boundary such as that illustrated in Figure 5 where the initial boundary was formed between 0.1 N KIOa and 0.1 N KC1. In this case the KIOa solution is placed in the bottom of the U-tube since it is more dense. The K + ion is common to both solutions, and C1- is the faster ("leadimg") anion and 10,- is the slower ('Lindicator") a By definition the current flows in the same direction as the positive ions.
The stationary boundary in Figure 5 is a boundary between two differentconcentrations of KIOa and in this respect is similar to the e boundary in the electrophoresis of proteins. This salt concentration gradient is a consequence of the fact that the KIOa solution behind the moving boundary adjusts itself to a lower conductance so that the iodate ions (which have a lower mobility than the chloride ions) will move as rapidly as the chloride ions ahead of the boundary. In this case the KIOJ solution between thestationary boundary and the moving boundary has been analyzed and found to have a concentration of 0.0640 N (12). In order to discuss this concentration change a t the position of the initial boundary, it is convenient to introduce an additional function called the "regulating function," w, which was discovered by Kohlrausch (7).
( 6)
The volume VQ@ is positive if the boundary moves with the current and negative if it moves against the current,' and the same equation is obtained f o r a cation constituent if C, is always taken with a sign corresponding to the charge on the ion. If one ion constituent is absent on one side of the boundary and may therefore be said to disappear in the boundary, equation (6) for that ion is: T,
-37.3 X 10-6 om.' see.-' volt-'
The very unique property of this quantity is that it remains a constant for any given level in the electrophoresis cell regardless of the number of boundaries which.pass by that level in the cell. This means that in the case of a moving boundary, the regulating function must have the same value in the two solutions on either side of the boundary. If a stationary boundary exists, the solutions on either side must have diierent values for the regulating function. It may be shown
INITIAL
FINAL SCHLIEREN PATTERN
0 1 N XCL
Figure 5. D.termin.tion of the Tmn.f...nc. Number .nd Mobility of Chlorida ron Usins tha Morins Boundary Method
The final sehlieren psttern ia ahown to the right of the diagrammatio repmentation of the boundaries in the cell.
NOVEMBER. 1948
that the ratio of the values of the regulating function for the two solutions separated by the stationary houndary is equal to the ratio of the concentrations across the boundary. I n the case of constant relative ion mobilities, all ionic species are diluted by this same factor a t the stationary boundary. By constant
relative ion mobilities we mean that the ratios of the mobilities of the ions to the mobility of one of the ions are the same throughout the electrophoretic cell although the absolute ion mobilities vary somewhat because of diierences in salt concentration and viscosity. Longsworth and MacInnes (10) have shown that if the protein solution is diluted with the uu-ionized part of the buffer by this factor before electrophoresis, the 6 and e boundaries disappear because of the equality of the regulating functions (10). A complete theoretical description of the boundary displacements and concentration changes through the boundaries could, in principle, be obtained from the compositions of the original solutions, the differential equations of continuity, the electroneutrality requirement, and a specification of ion mobilities as a function of composition. However, because of the interrelation of diffusion and electrical migration, only the case of three ion species, which is the simplest possible in an ordinary electrophoretic experiment, is open to an exact mathematical treatment, and then only if the mobilities can be regarded as constants. In order to avoid mathematical complexity the equations may be developed in a form independent of the particular path by which an ion concentration changes between phases. Dole (6) has recently developed a general solution for the moving boundary equation which assumes only that the relative ion mobilities are constant. Accordimg to Dole's theory, a system that contains n ions will, in general, form a maximum of n - 1 boundaries, one of which is a stationary boundary. If the system contains p anions and q cations, there will generally be p - 1 boundaries with negative velocities, and p - 1 boundaries with positive velocities. Since only relative ion mobilities, rt, n, . . . r*, are used in Dole's theory, it is convenient to define a "relative specific conductivity," u, analogous to specific
conductivity. The Vu products (where V is, as usual, the volume moved through by the boundary per Faraday) are obtained as solutions of polynomials of the type
The values of x which satisfy this equation, if ordered from the extreme negative to the most positive, corresponds to the Vu products for the boundaries from the one with the greatest negative velocity to the one with the most positive.
The concentration changes of each ion species across each moving boundary may be obtained from
in which r, is the relative mobility of the ion species absent in the solution P. Using the concentrations of the various ion species calculated from these equations, the analysis which would be obtained for an assumed mixture may be computed and compared with the result expected for "ideal" electrophoresis. A Discussion of the Deviations from Ideality. Although Dole's theory is, in general, not strictly applicable to proteins because of small variations in pH in the U-tube, which causes the protein mobilities to vary somewhat, the application of this theory to the electrophoresis of a mixture of two proteins illustrates the nature and magnitude of the deviations of electrophoresis from the usually assumed "ideal case." Let us consider a mixture containing equal weights of two proteins, S a n d T, on the alkaline side of their isoelectric points which have relative mobilities of -0.30 and -0.15 (relative to the mobility of Naf) and net charges of 0.00036 and 0.00018 Faraday equivalents per gram of protein. The properties of these proteins correspond to those of serum albumin ( S ) and serum globulin (T). Each is present a t a concentration of 1 per cent andsuch a solution could represent serum in which the alhuminglobulin ratio is unity diluted with two or three volumes of buffer (14). The buffer in this case is 0.05 N sodium diethylbarbiturate a t pH 8.6. Figure 6a shows the concentrations of the two proteins ( S and T) and the sodium diethylbarbiturate ion (NaV) throughout the electrophoretic cell after 0.001 Faraday has been passed through the cell. The initial boundary between protein solution (below) and the buffer (on top) against which it was dialyzed to equilibrium is formed a t 0. It is noted that on the ascending side the S and T boundaries have moved through 4.24 and 2.24 cc., respectively, while in the descending side the volumes are 3.95 and 1.90 cc. Figure 6b gives the difference between the refractive index, n, of the solution a t various levels in the electrophoretic cell and the refractive index of the buffer, no. The refractive indices were calculated by adding up the contributions by the sodium proteinates and the sodium diethylbarbiturate having the concentrations given in Figure 6a. (The refractive index increments for the proteins are assumed to be 0.00186 per 1 g./100 cc. and for the sodium diethylbarbiturate 0.04055 per equivalent/liter.) Figure 6c is a plot of the slope, dn/dz, of Figure 6b against position in the cell, and so it is the schlieren diagram which would have been obtained in this hypothetical experiment. The areas in the schlieren diagram are proportional to the refractive index diierences between the homogeneous solutions on the two sides of the boundaries. Since the diethylbarbiturate ion concentration is different in the various regions of the electrophoretic cell while the diethylbarbituric acid concentration is constant (14), the pH is not constant in the electrophoretic
JOURNAL OF CHEMICAL EDUCATION
624 conc. prorein
n-n.-
a+
dv
Fiw0. Conantr.tiolu .nd Boundary Displac.m.nte for 1 Hypothetical Electrophoretia Experiment with a Solution Containing Equal Amount. of Two Protei-. S .nd T.
The sodium diethylbarbiturate buffer is of pH 8.8. r/2 = 0.05. (a) Con~entratima. The concentration scale for the proteins ia at the to9 and for the NaV s t the bottom. (b) Refractive index as a function of height in the oell. (c) Sohlieren pattern, which is refractive index gradient as s function of height in the oell. ( d ) pH and electric field atrength ( E , voltdom.) between the various boundsriea.
cell. The pH of the vazious phases was calculated from the corresponding diethylbarbiturate ion concentrations by using the Hendersou-Hasselbach equation and is tabulated to the right of Figure 6c. It should be noticed that the pH of the dilyzed protein solution is not the same as the buffer, pH 8.60, but is lower because of the Donnan effect. To calculate the pH of the protein solution, the first order Dounan theory was used to determine the diethylbarbiturate ion concentration in the protein solution (Cv-") from its concentration in the bnffer ( C v - 3 . C p is the protein concentration in Cv-= = Cv-7
- '/;Cp
(13)
equivalents per liter and all the concentrations are signed quantitie~. The specific conductivity of the various phases is different and may be calculated from the concentrations of the various ions and their mobilities if the effect of the slightly varying viscosity of the phases is ignored. When the current is adjusted to give a potential gradient of 10 volts/cm. in the protein solution, the potential gradients in the other phases will have the values given in the last column of Figure 6. This figure will now be used to discuss the deviations of the usnal electrophoretic analysis from ideality. (1) The Buffer and Protein Concent~ationBoundaries. The buffer concentration gradient ( e ) on the descending side a t the position of the initial boundary is a boundary between the initial buffer solution (0.05 N ) above the boundary and a buffer solution of higher concentration below the boundary. The concentration of the buffer solution immediately below the e boundary is determined by the value of the Koblrausch regulating
function for the urotein solution and in this case has a concentration o i 0.0560 N. In the case of constant relative ion mobilities the concentration gradients do not move with respect to the solvent. However, in practice the boundary is not stationary but moves somewhat under the influence of the current because of the volume change taking place a t the electrode and the differences in bnffer ion transport numbers a t the two concentrations. One of the advantages of diethylbarbiturate buffers is that the e boundary moves slowly in the opposite direction from the proteins of plasma, so that better resolution is obtained between the e boundary and r globulin. The protein concentration gradient (6) on the ascending side is a result of a dilution of the protein components and the barbiturate ion a t the position of the initial boundary. According to Kohlrausch, all the ionic components are diluted by the same factor a t this boundary, and in this case the dilution factor is 0.8831. At the e boundary, the sodium diethylbarbiturate concentration is 1/0.8831 times greater below the boundary than above. (8) Volumes Swept through by the Moving Boundaries. It may be noted that the ascending boundaries sweep through larger volumes than the descending boundaries. Thus, the ascendmg S boundary swept through 4.24 cc. while the descending S boundary swept through 3.95 cc. This is generally true because the field strength is greater between the ascending boundary and the 6 boundary than it is in the original protein solution. As shown by Longsworth and MacInnes ( l o ) , the electrophoretic mobility may be determined most directly from the volume swept through by the descending boundary and the conductivity of the protein solution. This gives the correct mobility for the faster component (r, = -0.30), but the mobility calculated using the conductivity of the original protein solution is not quite correct for the slower component (r, = -0.145 rather than -0.150). Thus the mobilities of the globulins and fibrinogen in plasma are subject to small errors and may be expected to vary somewhat with the protein concentration and the relative amounts of the various components. The calculation of mobilities from ascending patterns is more complicated (10). '(5) Areas in the Electrophoretic Pattern. The total area of the electrophoretic pattern is the same on both sides and is proportional to the refractive index d i e r ence between the protein solution and bnffer. The total area is independent of time of electrophoresis provided none of the boundaries leave the limb of the cell, because
Since As > A,, the area of the moving peaks is less in the ascending pattern. Since our mixture contains equal concentrations of two proteins with equal refractive index increments, the areas of the two moving peaks in the ideal case would be the same and each would correspond to An =
NOVEMBER. 1948
0.00186. Becake of the superimposed barbiturate ion and protein gradients, the refractive index change across the faster moving boundary is too great in both ascendmg and descending patterns, yieldmg 55.2y0 on the ascending and 51.3% on the descending for the S protein instead of the expected 50%. The nature of this error was fist discovered by Svensson (18). It should he noted that the inverted salt gradients are not proportional to the change in protein concentration a t each boundary expressed in grams per unit volume, but are proportional to the change in protein concentration expressed in equivalents per unit volume, which will differ widely for proteins of widely differing charge. The error in the analysis caused by the superimposed gradients is less on the descending side. Experiments by Armstrong, Budka, and iblorrison (1) indicate that the experimentally observed deviations are somewhat larger than those predicted by Dole's theory. This error in analysis caused by superimposed gradients would be reduced by carrying out the experiment at.higher ionic strength and lower protein concentration, and as mentioned above, the correct analysis may be obtained by extrapolation. The error becomes more serious if buffer ions of higher mobility, as phosphate are used (14). Most electrophoretic experiments are carried out a t 0.10 ionic strength, but this calculation was carried out for 0.05 ionic strength where the deviations are slightly larger and more easily represented graphically. When the absolute concentrations of the protein components in a mixture are calculated from the areas of the corresponding peaks, assuming that the area is due to that protein alone, an error is made because of the superimposed buffer and protein gradients. A first approximation to the absolute concentration of any protein in a mixture may be obtained by ~nultiplying the total protein concentration by its relative percent. The absolute concentration of any protein in a mixture may also be obtained if the apparatus constant K and the refractive index increment An/AW of the protein are known by using equation (15). If A, is the total area of the pattern on the descending side exclusive of the e peak and AA.is the area of the latter peak (3, lo), K C; = An/ A W G ( A ;
+ 2 8.)
(15)
The approximate error of this method for determining protein concentration in a mixture may be judged from a calculation of the concentration of the T protein in Figure 6c.
=
0.945% (true value = 1.000%)
(4)Shape of the Peaks. None of the present theories give quantitative information concerning the shape of the peaks in the electrophoretic pattern if there are superimposed pH and conductivity gradients. The rate of boundary spreading is determined partially by
these gradients and partially by diffusion and inhomogeneity of the protein. In Figure 6c the shapes of the peaks have been drawn arbitrarily to represent the case usually encountered in the electrophoresis of proteins, that is, the descending peaks are broader and shorter than the correspondmg ascending peaks. This may be explained qualitatively by reference to the potential gradients in the last column of Figure 6 because the electric field strength is greater on the leading edge of the descending boundary than on the trailing edge so that molecules which are in the leading edge of the boundary move more rapidly than those in the trailing edge, and the'houndary becomes broader than expected for diffusion alone (conductivity effect). On the ascending side the situation is reversed so that the boundary is sharpened. The broadening of the descending peaks and the sharpening of the ascending p e a k is partially compensated by a differencein the pH of the solutions on either side of the moving boundary. On the descending side, for example, the pH is lower on the leading edge of the boundary so that molecules in the leading edge have lower mobilities because of the lower pH (since they are on the alkaline side of their isoelectric points). In some cases the pH effect may predominate so that the boundaries are sharper on the descending side and broader on the ascending, as in the case of the electrophoresis of glutamic acid (14). As a result of the superimposed pH and conductivity gradients, the moving protein boundaries may deviate from the symmetrical Gaussian shape which would be expected if all the spreading were caused by diffusion. LITERATURE CITED (1) ARMSTRONG, S. H., JR., M. J. E. UUDKA,AND K. C. MORRISON,J . Am. Chem. Soe., 69,416 (1947). (2) ARMSTRONG, S. H., JR., M. J. E. BUDXA,K. C. MORRIBON, n m M. HASSON,69, 1747 (1947). (3) BRIGGS, D. R.,AND R. HALL,ibid., 67,2007 (1945). (4) DEUTSCH,H.F.,AND M. GOODLOE, J . B i d . Chem., 161, 1 11946). (5) DOLE,V. P.,AND E. BRAWN, J . Clili. Invest., 23,708 (1944). (6) DOLE, V. P.,J . Am. Chem. Soc., 67, 1119 (1945). (7) KOHLRAUSCH, F.,Ann. Physik, 62, 209 (1897). (8) LUETSCHER, J. A., JR., Ph?piological R w i e w ~27, . 621 (1947). (9) LONGSWORTH, L: G.; T.-SHEDLOVSKY, AND D . A. MACINNES.J . Emtl. Med.. 70. 399 11939). 110) . . LONGSW~RTH. L. G.. ANDD. A. MACINNES. J . Am. Chem. Soc., 62, 705 (194b). (11) LONGSWORTH, L. G., Chem. Rev., 30,323 (19421. (12) LONGSWORTH, L.G., 3. Am. Chem. Soc., 66,449 (1044). (13) LONGSWORTH, L. G., ibid., 67, 1109 (1945). (14) LONGSWORTH, L. G., J.Phy8. andColl. Chem.,51,171 (1947). (15) MOORE,D.H.,J. Biol. Chem., 161,21 (1945). (16) PERLMANN, G.E.,AND D. KAUPMAN, J. Am. Chem. Soc., 67, 638 (1945). H., AT!& Kemi, M i n e d Geol., 17A, No. 14, 1 (17) SVENSSON, 1194.1). ~-.-.,. (18) SVENSSON, H.,ibid., 22A, No. 10, 1 (1946). (19) STERN,K. G., AND M. REINER,Yale J . Biol. Med., 19, 67 (1947). (20) TISELIUS,A., Biochmn. J., 31,1464 (1937); 31,313 (1937). (21) TlsE~rns,A., Trans. Faraday Soe., 33,524 (1937).
.-~--,~
