Interpretation of zonal asymmetry in gel filtration

ARCHIVES

OF

BIOCHEMISTRY

interpretation

AND

BIOPHYSICS

117, 184-191(1966)

of Zonal M. V. TRACEY

Asymmetry AND

in Gel Filtration

D. J. WINZOR

C.S.I.R.O. Wheat Research Unit, North Ryde, New South Wales, Australia

ReceivedMarch 10, 1966 In gel filtration zonal asymmetry reflects a combination of the effects of solute heterogeneity and of concentration dependence of migration rate. A simple, empirical method of analysis is described whereby the effects of variations in velosity are eliminated, thus permitting an estimate of solute purity. The analysis is in terms of the apparent diffusion (spreading) coefficients at different concentration levels throughout the zone, heterogeneity being detected by differences between these values. Constancy of the apparent diffusion coefficients is a necessary but insufficient condition for homogeneity. Although developed for chromatographic systems the procedure may be applied to all forms of zonal transport data. Results of gel filtration experiments on several proteins are reported to demonstrate its use.

In gel filtration experiments asymmetry of from corrected height-area ratios, could aIso the eluted zone is usually attributed to het- be adapted to either advancing or trailing erogeneity of the solute under investigation fronts in gel filtraton. However, these methods all require prior determination of the ve(e.g., ref. 1). However, this interpretation implicitly assumesa unique velocity for each locity-concentration relationship fromaseries solute, whereas the detection of concentra- of frontal runs at different protein concentration dependence of elution volume for oval- tions. The present study concerns t’he develbumin (2, 3), trypsin (3), and bovine serum opment of a procedure to obtain information albumin (3) implies that this assumption is on heterogeneity from zonal transport patlikely to be justified only at low levels of pro- t,erns without such additional data. The tein concentration. Furthermore, the demon- method is empirical since the crucial step stration (3) that the skewness of ovalbumin involves a mathematically unsupported asprofiles results from this concentration de- sumption, the justification of which is propendence of migration rate indicates the im- vided by application of the procedure to portance of considering such effects before numerical examples of zonal distributions, zonal asymmetry is interpreted in terms of calculated from an analytical expression deheterogeneity. It was therefore felt that a veloped for chromatographic systems exhisimple procedure for distinguishing between biting linear concentration dependence (Eq. the effects of concentration dependence and (15) of ref. 9). Details of the method’s apheterogeneity could find practical application plication to gel filtration data on several particularly in preparative studies, where commercially available protein preparations higher column loadings render the above as- are also reported. sumption suspect. ANALYSIS OF ZONAL DISTRIBUTIONS Comparison of experimental datawith theFormulation of Procedure oretical patterns, as in Fig. 2 of ref. 3, undoubtedly provides the most rigorous test of The model system on which this analysis homogeneity, while the procedures used in of zonal distributions is based comprises a electrophoresis (4) and sedimentation (5-8) packed column of uniform cross section in for analyzing boundary distributionsin terms which a zone of a single solute is subjected to of apparent diffusion coefficents, calculated elution development chromatography at a 184

INTERPRETATION

OF ZOKAL

1x.7

ASYMMETRY

c*onstant flow rate. Initially t,he concentration of solute in the mobile phase is considered to be (‘0 throughout the zone, it being zero cllsewhere, and equilibrium between phases is assumed at) all times. Theoret,ical consideration is Iimit.ed to the transient stage of the nGgr;ition process during which a pla,tenu region of (~onwntration Co persists, so that the zow may be envisaged as two separate boundary systems. The chromatographic analog of the diffusion coefficknt in free transport cspwiments is assumed to be indrpcndnlt of conccutration. Although this quantity is not thP diffusion coefficient, its nlagnitudc ckycnding largely upon the value of hhc pnrt;it.ion coefficient (lo), the word diffusion will be uscld for effects it is used to &>scribc. In gel filtration the few solutes investigaled h:we exhibited linear concentration dep(~ndww of clut.ion volume (2, 3), indicat,ing :I partition isotherm of the type n = KIC + K.‘P, whmc n is I-hc concentration of adsorbed solute. For such systems Houghton (9, II:IS derived the following continuity cqmt ion :

in which I7 is t,he velocity of a solute obeying the isotherm r~ = KIC, and X is a function of KZ and I’. (A and c’ are defined in Eq. (4) of ref. 9.) D is the diffusion coefficient and 2 is measured relative to an origin moving with a velocity c’. The derivation of this differential cquat.ion is based on t,he convention that flow is positive in the direction solution -+ solvent, and Ey. (1) therefore applies directly to t)hrt cwending or advancing front. Since velocity k a vector quantity it follows that I,: is rwgatiw for a descending or t,raiIing boundary. Thus the continuit’y equations describing the ascending and descending sides both contain the same velocity and diffusion terms, which are added in the former cast and subtracted in the latter. It is assumed that a similar situation exists after integration of the differential equation, i.e., that the spread of a boundary may be envisaged as the resultant of a velocity and a diffusion cont,ribution, t.he absolute magnitudes of the respective contributions being duplicatcld in t,hc composite front..

MIGRATION

DESCENMNG

‘d

‘d Dt STANCE

Diagram distribution before plateau region. FIG.

1.

of a represent.ativc the disappearance

of

xonal 1he

186

TRACEY

AND WINZOR

fieient of the model solute, the value so obtained being independent of the concentration level selected. Calculation of diffusion coefficients from values provides a qualitative basis ~diffusion for detecting heterogeneity of solute since half of the additional boundary spreading resulting therefrom is incorporated in the value of xdiffusion, thus leading to the calculation of an apparent diffusion coefficient larger than the actual value. Further information on heterogeneity may be obtained by repeating the analysis at several concentration levels within the zone, where the extent of spreading due to heterogeneity will generally differ and give rise to a consequent variation in calculated apparent D values. Such analyses of distributions are greatly facilitated by the use of probability graph paper, which transforms a Gaussian diffusion curve into a linear plot with slope related to the quantity 42Dt. Thus the ultimate criterion of homogeneity by the present method is attainment of a linear probability plot, the slope of which agreeswith that expected on the basis of the known diffusion coefficient. The lack of a reliable estimate of this latter quantity in gel filtration reduces the sensitivity of this analysis, but additional information is provided by repeating the experiment with different column lengths but constant cross section and flow rate, agreement between D values so obtained also being a necessary condition for homogeneity. It should be noted that a nonlinear diffusion plot merely indicates the inadequacy of Eq. (1) to describe the system under test, possibly due to neglect of terms to cover a finite rate of solute exchange between phases, or a concentration-dependent diffusion coefficient, in addition to heterogeneity. The assumption of constant D in the above analysis requires comment, as the velocity and the diffusion coefficient are largely governed by the partition ratio (lo), which in turn depends upon C. However, inspection of the two quantities reveals a much greater importance of the velocity variation compared with changes in the diffusion term. This neglect of B-C dependence is paralleled in analyses of boundary distributions in electrophoresis (4) and sedimentation (58), where

relatively minor variations in D are also disregarded. Validity

of the Procedure

As mentioned earlier the assumptions made in the postulation of Eqs. (Z-4) have no mathematical basis, since a general solution to t,he analytical expression for concentration resulting from integration of Eq. (1) has not been obtained because of its complexity. However, this expression (Eq.(15) of ref. 9) is readily solved for particular numerical examples, and confidence in the pragmatic validity of Eqs. (2-4) has come from the successful application of the suggested procedure to computed zonal distributions. Figure 2 summarizes data from zonal distributions calculated to simulate the gel filtration behavior of ovalbumin (3). The decided asymmetry of the zonesat the higher concentration levels, evident from Fig. 2 of ref. 3, is reflected by the difference between the corresponding probability plots of the ascending and descending boundary distributions, the curvature being a measure of the deviation from Gaussian spreading. At the lowest concentration (1.64 mg per milliliter) the non-Gaussian nature of the two boundaries, and the consequent zone skewness, are clearly evident from the probability plot, although the elution profiles are symmetrical to casual inspection. However, the good agreement between the ideal diffusion plot (X = 0 in Eq. (1)) and the experimental points obtained by Eq. (4) is of most interest from the present viewpoint. It should be noted that this latter observation only establishes the validity of any assumptions or approximations inherent in the procedure for the selected sets of parameters, but from analyses of other computed zonal distributions we believe that similar conclusions apply for combinations of these variables likely to be significant in gel filtration experiments. Zones without Plateau Regions

The foregoing considerations have all been restricted to a transient stage of zonal migration, and attention is now directed toward the more general phenomenon of a zone containing no plateau region. In this case the

INTERPRETATION

21 2

I

I 0

I

I

OF ZONAL ASYMMETRY

I

2 BOUNDARY

I

I

0 SPREAD

I

2 (ml)

I

1s;

I 0

I

J 2

plots of boundary spreading from the tonal distributions in Fig. 2 FIG. 2. Probability of ref. 3, patterns (a), (b), and (c) being the respective counterparts of the upper, center, and lower diagrams. In each case the solid line represents the spread (X - w) due to dif?usion alone; and the dashed and dotted lines, the actual spread of the ascending and descending boundaries, respectively: the circles are values of z~~ff,,~,~~~ calculated from 1Cq. (4). Superimposition of the ascending and descending diffusion plots necessit:ltes :I rcversa1of the direction of the abscissaaxis for one boLmd:try.

zone represents the resultant of two overlapping boundaries, and its descript’ion consequently requires a far more complex summation of the ascending and descending boundary systems than the simple addition suitable in frontal experiments. However, for a solute migrating independently of concentration such zones are symmetrical (e.g., ref. 9), indicating equal contributions of the two systems to the overall spread, (Xd - .%) (Xa - Xa), at all concentration levels, and consequently Eqs. (24) also apply. The lack of values of a, and a, presents no difficulties, since inspection of Eq. (4) shows that use of t*he quantity (X, - X=)/2 instead of Zdiffusion in the probability plot merely transposes the origin of t’he abscissa by (& - x=)/2. In such circumstances it is more convenient for t)his analysis to use Gaussian graph paper (13), on which the probability scale is expressed as a percentage of the maximal rat,her than the total value, i.e., C/C,,, instead of C/l: C&L Application

to Elution ProJiles

The usual experimental record in gel filtration is an elution profile, viz., C = f(t) at

,I insteatl of t,hr recluiretl i’ornl, (’ = at constant t. Consequently, rigorous

constant

f(X)

application to elutiori profiles nc~ccssit,at,rs revision of the method to take into :~c:ount t,ht:

different time fac%orx pertaining to migrat’ional and diffusional spread :~t t~h concentration level. To this end it’ has been shown for diffusion-free systems ~h:tt, &t.ion volumes should be substituted for vtalocitics in the conbinuity equation (14, 15). Howovcr, t,here are several experiment8al obs(trvations which suggest that, little error nl:~y IW introduced by dir& application of I
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to the mirror image of the C-X distribution at a time corresponding to the mean elution volume, the greatest justification of this assumption being provided by the agreement between profiles and calculated C-X distributions (3). It should be noted that C is a mobile phase concent8ration, and that all of the material applied is accounted for in the elution profile, in cont’rast with the situation within the column, where a considerable proportion of solute would not be included in the value of C.l However, from the viewpoint of mass conservation there is no objection to considering the entire migration process to be confined to the mobile phase, although the values of the velocity and diffusion coefficient are, of course, governed by partition. It is to this hypothetical C-X distribution that the elution profile refers. EXPERIMENTAL

PROCEDURE

Lysozyme (three-times crystallized) and ovalhumin (five-times crystallized) were supplied by Sigma Chemical Co., and cu-chymotrypsin (threetimes crystallized) by Worthington Biochemical Corp. Five&mea crystallized soybean t,rypsin inhibitor was provided by Nutritional Biochemical Co. and the crystalline bovine serum albumin was a product of Commonwealth Serum Laboratories, Melbourne, Australia. These proteins were used without further purification m solutions in the appropriate buffer at a nominal concentration of 15 mg per milliliter. Three ml of such a solution was applied to a 1.7 X 26 cm column of Sephadex G-100 previously equilibrated with the buffer to be used. A continuous record of protein concentration in the column effluent was given by a Technicon AutoAnalyzer using a modification of the biuret method (21). A port’ion of the protein solution was also sampled by the apparatus at the beginning of the run to give a directly comparable measure of the protein concentration applied to the column. It was assumed that the proteins all gave the same color yield. RESULTS

The results of gel filtration experiments on four proteins in phosphate buffer (0.07 M NaHzP04, 0.01 M Na2HP04), pH 6, are 1 We are grateful to Dr. G. A. Gilbert of Birmingham University for bringing this point to our attention in a personal communication to Dr. L. W. Nichol concerning refs. 3 and 20.

WINZOR

summarized in Fig. 3. Data on bovine serum albumin (Fig. 3a) have been included to demonstrate the effect of asymmetry due to impurity (dimer) upon the probability plot, which indeed shows distinct departure from linearity. From Fig. 3b it is evident that the asymmetry of the elution profile obtained with soybean trypsin inhibitor is also due to heterogeneity. These results may be contrasted with those given by ovalbumin (Fig. 3c), where the skewness of the elution pattern is not reflected in the diffusion plot, a result which serves to illustrate the necessity of considering effects of velocity-concentration dependence in the interpretation of eonal asymmetry. The data obtained with lysozyme (Fig. 3d) apparently satisfy the requirements of a homogeneous solute, an unexpected result since this protein is known to polymerize under similar conditions (22). Inspection of Eqs.. (2-4) shows that effects of positive as well as negative concentration dependenceof migration rate are eliminated, and consequently detection of rapidly reacting systems by curvature of the probability plot relies upon sufficient variation of the mean diffusion coefficient due to progressive changes in the amount of polymer present at the various concentration levels. In order to test the application of the method to such systems, experiments were performed on a-chymotrypsin under conditions where its polymerization has been studied in detail. Unfortunately excessive tailing of the zones under these conditions (Fig. 4; also Fig. 1 of ref. 2) complicates interpretation at low levels of concentration, but this interference appears to be restricted to the region in which C/C,,, < 15%, and such data have therefore been omitted from the diffusion plots. At pH 7.9 (0.01 M NaHPO*), where a monomer--\hexamer equilibrium seemslikely (23), the probability plot shows decided curvature (Fig. 4a). However, in phosphate buffer, pH 6.2 (0.114 M NaH2P04, 0.029 M NazHPOk), the degree of association is considerably less (24), and the variation of the diffusion coefficient was undetected (Fig. 4b). The latter result establishes a definite weakness of the present analysis aa applied to systems in which the magnitude of the diffusion coefficient is unknown, and conse-

INTERPRETATION

OF ZONAL ASYMME;TI:Y

98 \

n

‘\

(a)

\

1 (b)

q;&

60

50

40

V (ml )

2’ 0

I

I 5

I

t

I I 10 0 (vd-va)/2

Cd)

I

I 5

I

\I 10

FIG. 3. EluGon profiles (insets) and diffusion plots from gel filtration experiments 011 proteins in phosphate buffer, pH 6: (a) bovine serum albumin (C,, = 14.3 mg/ml’l; ib) soybean trypsin inhibitor (12.8 mg/ml); (c) ovalbumin (14.6 mg/ml); id) lysozymr i14.1 mg/ml).

quently it, is important to eliminate the possibility of rapid associaCon equilibrium before assigning homogeneity to a solute. In gel filtration studies of proteins this distinct,ion is usually made fairly readily, since skewness resulting from polymerization is the reverse of that normally encountered (compare Figs. 3c and 4b). Accordingly, a protein zone possessing a more diffuse trailing edge is taken to indicate heterogene&y of t’he solute under investigation, due either to the presence of slowly migrating impurities (Fig. 3~1)or to rapid chemical interconversion (Icig. 4). Such a clear-cut dist’inction may not be possible in other forms of chromatography, or indeed in gel filtration studies on solutes ot’her than proteins, since physical interactions could with equal probability lead to positive concentration dependence and hence asymmetry in t.he reverse sense (11). Although the degree of polymerization in lysozyme solutions at pH 6 is almost cer-

bainly sufficiently small for the cYrv:lturo of the probability plot t,o be undrtrc~le(l, the shape of the zone is at first sight cwntradictory to such a postulate, its ~ylrlmc%ry suggesting an ideal solute with no wncwttration dependewe of velocity. Howver, normal (-UP) concentraztion depcndrnw has been observed with lysozymc undw conditions of simkr pH but lower ionic st wngth (Xichol, L. W., and Winzor, D. .J., rmpublished data), and it is therefore assurnc~tlthat Fig. 3d has resukd from transporl of :I rapidly polymerizing system in whic*h t hc: effects of chemical react’ion upon migrnl.ion rate are counterbalanced by t,how :I rising from physical intcrwtion. The inapplkabilitjy of t)he clneGc~:~lin tcrprctation of zonal asymmetry to tmnsport data on solutes exhibiting caoncentration dcpenderwe of ve1ocit.y has ofbcn twn denlonstratcd (e.g., wfs. 3, 9, rind 2.;). Whik

TRACEY

190

homogeneity, evidence of which is provided only by failure to detect heterogeneity. However, as demonstrated in the Resulls section, the present analysis as applied to gel filtration is particularly vulnerable on this ground because of the lack of an independent estimate of the diffusion (spreading) coefficient. Although the procedure has been developed for analysis of chromatographic data, the fact that the continuity equation is e,xpressed in terms of a velocity and a diffusion coefficient in the mobile phase suggests that the method should also apply to zonal electrophoretic and ultracentrifugal data. The continuity equation expressing the condition of mass balance in a single phase transport system differs from Eq.-( 1j, it being

90

60

(a)

::

AND WINZOR

as=& at

8x2

10

(5) t (bl

21 0

-I2

4 (vd-v,)/2

6

0

voh(Co +

kzccf - 2C -

3kzC”) ;z

for asolute whose velocityisdehnedby the relationship v = ~(1 - kd’ - klkd?), wherevO FIG. 4. Elution profiles (insets) and diffusion is the velocity at zero concentration. Equaplots from gel filtration experiments on rapidly tion (5) has been obtained by direct analogy polymerizing systems: (a) cu-chymotrypsin (Co = with expressions derived by Hoch (4), who 14.9 mg/ml) in 0.03 I phosphate, pH 7.9; (b) CYalso used the flow convention whereby aschymotrypsin (15.1 mg/ml) in 0.20 I phosphate, cending velocities are positive. From Eq. pH 6.2. (5) it is evident that the argument invoked in the derivation of Eqs. (24) relates to all this study also serves to re-emphasize this negative conclusion its main contribution is zonal transport data, and that linear conthe development of a procedure by which the centration dependence does not appear to required information on solute purity may be be a necessary condition for its validity. obtained from such experiments. The pres- Advantage has in fact already been taken of this latter observation in that the velocityent analysis of zonal distributions for homoconcentration relationships for the polymgeneity in terms of constancy of diffusion coefficient throughout the zone is similar in erizing systems (Fig. 4) were almost cerprinciple to that adopted in boundary anal- tainly nonlinear. The possibility of applying this procedure ysis of systems undergoing concentrationdependent diffusion (26, 27), in which the to single phase zonal transport data seems more likely in ultracentrifugal work than in diffusion coefficient may also be evaluated by pairing of X values so that effects of electrophoresis, since the density gradient, required for gravitational stability of the concentration dependence are eliminated. While the procedure described certainly has zone in both cases, will probably give rise to fluctuations of field strength in the latter the advantage of simplicity as a recommedasystem, whereas sedimentation behavior is t,ion, a limitation is that a linear probability apparently unaltered (28, 29). A method for plot is a necessary but insufficient condition evaluating the sedimentation coefficient from for homogeneity. In this connection it might be noted that similar criticisms apply to all such data has already been described (29), met’hods, transport or otherwise, of assessing and accordingly the ability of an analysis

1NTEKPI:ETATION

OF ZONAL

:dong t,he present lines to provide a value of 1he diffusion coefficient is possibly a more appealing prospect’ than t’he information on hct’erogeneity. However, it should be borne in mind that both of these quantities will geucrally differ from those required for rigorous application of the Svedberg equatiw.

A disadvantage of the present procedure is thch lack of a mathemat.ical basis, without which the validity of its application to all systems cannot be unequivocally established, :md analysis under conditions permitting classical interpretation is therefore still to be profcrrrcl on theoretical grounds. From the practical viewpoint a more serious objection is the inadequacy of the selected model to tlfhxcribe the actual experiment’al system. The approximalions involved in the direct appliwtion of the method to elution profiles have already been mentioned, and a somewhat similar situation would exist, in its use with zonal ultr:tc~ent~rif~lgal data because of the dependence of velocity upon radial distance, S. However, the problems created by inhomogrnc~ous fields and radial dilution resulting from centrifugation in sector-shaped ~11s have rcc*eivrtl chonsiderable theoretical att~rnt,ion (e.g., ref. SO), and modification of the procedure to account for these effects should consequently be possible. While there is support for application of the above proredure to zonal data it should be st,ressed t.hut, the evidence inno waylessens the need of a mathematically $roved analysis which pertains to a model system meeting the exact experimental requirenlents. In t’he meantime the results of this study have established that the method described could prove particularly useful in preparative gel filtration studies, where it is important, to know whet,her any skewness of the zone of solute under investigat,ion can be justifiably ignored from the viewpoint of heterogeneity. ACKNOWLEDGMENT We wish to thank Australia11 National

Professor A. G. Ogston of the University, and Dr. L. W.

ASYMMETRY

Nichol, University this manuscript.

I!,1 of Melbourne,

for

criticizitlg

REFERENCES 1. PEDERSEN, K. O., Arch. Bioc,hr,t,. Hiopkys. Suppl. 1, 157 (1962). 2. WINZOR, D. J., 9ND ScHERAo.4. 11. A., I$;,>chemistry 2, 1263 (1!%3). 3. WINZOIL, D. J.> AND NICHoL, L. W.. Biochim. Biophys. Acta 104, 1 (19G.5). 4. HOCH, H., Biochem. J. 46, 199 (1950). 5. FUJITA, H., J. Chem. Phys. 24, 1084 (1956). 6. BALD~~IN, R. L., Biochem. J. 66, 503 (1957). 7. FUJITA, H., J. Pkys. Ch,ern.63, 1092 (1959). 8. CREETH, J. M., Proc. Roy. Sot. (London) Akr. A 282, 403 (1964). 9. HOUGHTON, G., J. Phys. Chem. 67, 84 (1963). 10. BETHUNE, J. L., AND KEGELES, G., .1. Ph?is. Chem. 66, 1761 (1061). 11. NICHOL, L. W., AND OGSTON, A. G., I'm-. 130:q. Xoc. (London) Ser. B163,343 (1965). 12. CREETH, J. M., .I. .la~. Chcm. Snc. 77, fi-l28 (1955). 32, 1253 (1961 J. 13. ONNO, P., Rev. Sci. In&. 14. ACKERS, G. Ii., AND THOMPSON, T. E., Proc. Natl. Bcad. Sci. I/S. 63, 342 (19%). 15. GILBERT, L. M., AXI) (:ILBER'I., (:.A., Iiiuchenz.

J. 97, 7C (1965). 16. ANDREWS, P., Biochem. J. 91, 222 (1964). 17. WHITAKER, J. 1: ,, -tna,l. Chem. 36, l!kXl (1963). 39, 193 18. PORATH. ,J., Biwhim. Biophyh. .I&

119GO). 19. GELowE, B., J. Chro?natog. 3, 330 ilOfiOj. 20. NICHOL, L. W., AND WINZOR, I). J.. Riochim. Biophus. rlcta 94, 591 (1965). C. P., AND 21. GORN~LL, ‘4. G., BARDAXTLL, DAVID, M.&f., J. Biol. Chem. 177, 751 ilY49). 22. SOPHIAX~P~LOUS, A. .J., AND VAN HOLDE, K. E., J. Hiol. Chew. 239, 2516 (1964). 23. GILBERT, G. A., Disclrssion.? Faradall $0~. 20, 68 (1955). 24. RAO, hf. S. N., AND KEGELES. G., ,/. .Lm. Chem. Sot. 80, 5724 (1958). 25. BETHUXE, J. L., AND KEGELES, G., J. Phys. Chem. 65, 433 (1961). 26. CREE~TH, J. M., ASD GOSTING, L. J., J. Phys.

Chem. 62, 58 (1958). 27. CREETH, J. M., J. Phys. Chem. 62. 66 (1958). 28. MARTIN, Lt. Cr., ASD ARIES, B. N., J. Biol. Chem. 236, 1372 (1961). 29. SCHUMAKER, V. K.,AND ROSENBLOOM, J., Biochemistry 4, 1005 (1965). Theory of Sedi30. FUJITA, H., “Mat,hematical mentation Analysis.” Academic Press, New York (1962).