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A relationship between the molecular weights of macromolecules and their elution volumes based on a model for Sephadex gel filtration
SRCHIVES
OF
BIOCHEMISTRY
A Relationship and Their
AND
BIOPHYSICS
between Elution
471-478
107,
the Molecular
Volumes
Based
(1964)
Weights
of Macromolecules
on a Model
for Sephadex
Gel
Filtration PHIL Front the Hormone
Research Laboratory,
G. SQUIRE University
Received
April
of California,
Berkeley,
California
4, 1964
Equations relating the elution volume in Sephadex gel filtration to the molecular weight and the molecular radius of a given solute have been derived on the basis of a model in which the elements of volume available to solvent within the gel are approximated by a combination of cones, cylinders, and crevices. The equations are based on the assumption that the volume of liquid passing through the column between the applmatron of a macromolecular solute and emergence of the solute in maximal concentration is equal to the volume within the Sephadex bed which is available to macromolecules of a given size. The volume which for steric reasons is unavailable t.o t,he macromolecular solute is explicitly described in terms of the model. These equations are tested and two parameters are evaluated by means of data relat,ing molecular weights and elution volumes taken from the literature. One of the parameters can also be evaluated from the ratio of the bed volume to the void volume. The good agreement between values calculated in these two ways provides support for the notion that the simple model is essentially equivalent to the Sephadex gel. Molecular weights calculated from the elution volume by means of the equations derived here in most instances agree rather well with values calculated by conventional means. A few serious discrepancies in the data, however, suggest that caution be used in applying this or perhaps any other simple relationship between molecular weights and elution volume to the estimation of molecular weights.
It is widely recognized (5) that gel fi.ltration through Sephadex is a powerful methdo for the fractionation of proteins according to size, since fractions emerge from the column in the order of decreasing molecular size. This observation has stimulated interest in finding a simple relationship between the molecular weights of macromolecules and their elution volumes in the hope that such a relationship might be useful in obtaining a rough estimate of the molecular weight of a protein from its elution
volume.
Conversely,
tionship should be useful predicting the resolution known molecular weight obtain by gel filtration on Wit’h this objective in
such
a
rela-
as a means of of proteins of that one would Sephadex. mind, Andrews
(1) has studied the gel filtration of proteins of known molecular weight through columns of agar gel. From the results of these experiments he prepared standard curves in which the volume passing through the column before the protein emerges in maximum concentration, I’, is plotted as a function of the logarithm of molecular weight. When molecular weights of several other proteins were then estimated from their elution volumes by means of the standard curve, the agreement wit’h molecular conventional determined by weights met,hods was usually quite good. Andrews and Folley (2) have also est#inlated the molecular weights of bovine, ovine, and porcine pituitary growth hormones from their
471
elution
volu~ncs
on
Sephadex
G-7.5
472
SQUIRE
by means of a standard curve prepared for this gel. Molecular weights calculated in this manner are approximately one half of the values obtained by conventional means. Quite recently, Whitaker (3) has reported the results of the gel filtration of several standard proteins through Sephadex G-75 and G-100. He found that the ratio of the elution volume to the void volume, V/V, , is a linear function of the logarithm of the molecular weight, and again the scatter of points around the regression line is usually rather small. Certain exceptions were noted, however, and reasonable explanations were advanced. THEORETICAL
Although the empirical equations of Whitaker (3) provided a useful relationship between molecular weights and elution volumes throughout the range for which these equations were tested, it was thought worthwhile to derive an expression of this relationship based upon a reasonable model for Sephadex gel filtration. It seems likely that the equation derived in this manner might apply over a wider range of molecular weights than that of Whitaker. It will be noted that in the treatment described herein no consideration is made of rates of diffusion or of adsorption processes. Instead it is based entirely on excluded volumes, t’hat is to say, elements of volumes within the Sephadex gel which are unavailable to protein molecules of a given size. The argument of Porath (4, 5) to the effect that, at the flow rates that are experimentally accessible, the elution volume is independent of flow rate, and hence apparently not dependent upon diffusion or other rate processes, seems convincing. The notion that there are within the Sephadex gel certain forbidden regions which vary in volume with the size of the molecular species, and that this is the basis of fractionation, has been frequently expressed (4-7). In this paper a quantitative treatment of this idea is given and the basis of these forbidden or excluded volumes is explicitly described.’ r It was my impression that this had not been previously done, but during the preparation of
Reversible adsorption is considered as an additional phenomenon which can frequently be avoided, but which, where present, may be considered as superimposed upon the more general process that depends upon excluded volumes. Initially the proteins will be considered as spheres of equal hydration, but variations in shape and hydration will be considered subsequently. It is assumed that the volume of liquid, V, that passes through the column between the application of the sample and emergence of a protein from the column in maximum concentration is equal to the volume within the column, which is available to that protein. Thus it remains only to calculate the total volume available to solvent, and the volume that for steric reasons is unavailable to the protein. The highly complex matrix of the Sephadex gel interior is treated as a combination of cones, cylinders, and crevices. The principle of excluded volumes will be illustrated for the cone, but the analogous expressions for cylinders and crevices (rectangular parallelopipeds) may be derived in a similar manner. The volume outside the gel (between the spherical beads) is taken as I’, , the void volume. The volume accessible to solvent, within the cone is K=~&=kVo,
(1)
where (Y is the angle between the side of the cone and the vertical to the apex. The volume inside the cone which is available to a protein of radius r is the volume of a similar cone having a radius at the base equal to R - r. This is more readily visualthis manuscript, a paper by Porath (5) came to my attention in which a Sephadex model containing conical cavities was treated, and in a manner similar to the one presented here. Whereas Porath’s equation has a somewhat different form than the one derived here, it also provided an adequate basis for the correlation of the molecular weight of dextran fractions with the elution volume data of Granath and Flodin (9). There is also a manuscript by T. C. Laurent and J. Killander, in press (J. Chromatog.), in which a theory of gel filtration based upon a model consisting of a threedimensional network of fibers is developed and tested experimentally.
SEPHADEX
GEL
ized if the ent’ire mass of the protein is considered to be located at its center, but the nearest distance of approach of the center of the molecule to the edge of the cone remains the dist.ance T. The vohnnc inside the cone which is available to the centers of the protein molecules is then given by Eq. (2). As pointed out by Schachman and Lauffcr (10) in quite a different application of the same steric effect, statistical analysis of the concentration profile would also lead to t,he same conclusion concerning the extent of t’he excluded volume. This is given by t’he equation
Combining
these equations we have
p;, =
kVo
CRi3y)3.
The total volume available to the protein, which, according to the principle upon which t’his treatment is based, is taken as the elution volume, T’, , is then given by the expression 17, = vo + live
1 - i (
)
3
(cones>. (4)
-4 similar treatment for cylinders of radius Iz and crevices of width 2R leads to the corresponding equations. ’
(cylinders)
(5)
(crevices).
(6)
and
There remains the problem of assigning the proper fractions of the total volume inside the gel to cones, cylinders, and crevices. Perhaps a rational way of doing this could be found, but we make an arbitrary distribution that leads to an exceptionally convenient result. If we specify that l?’ = 9g, k’ = 9gZ, and 1; = 3g3, then VP = vo + 3gvo
+3gq1
(
1 - a
>
-L"~+g~vo(l
-g
O'
473
FILTRATION
and
+[l+g(l
-;)1”.
C8)
In this treatment the proteins are considered as spheres. If we also assume that r is proportional to the cube root of the molecular weight, we can also write
g.= [l+g(l -q,
(9)
where C corresponds to the molecular weight of the smallest protein that cannot enter the gel. The constants of g and P3 are to be evaluated from experimental results, but two requirements might be pointed out at this time. If the model upon which Eq. (8) was derived is consistent with the analytical resuhs, it is required that as the radius 1’ of the macromolecule approaches R, V, becomes equal to I’. It is also required that when 1‘ becomes very small, i.c., approaches the size of the water molecule, VP/V0 should approach I’JT’, , where l’, is approximately equal to bed volume less the weight of Sephadex. We shall see t#hat both of these requirements are satisfied. APPLICATIOK OF EQUATlOS CURRENT DiZTd
TO
In order to test Eq. (9) and evaluate the constants, data from the recent. literature have been tabulated in Table I. The calculated values, JP3 and (I’/ll-O)l!B, are selfexplanatory. The constant’s g and Cl13 as evaluated from plots of (l’,/l’,)l;R - 1 vs. JP3 are recorded beneath the tables. The value of the molecular weight was then calculated from the elution volumes by means of Eq. (9), solved for JP3: Mf~,“, = $[l
+ g - (q”].
(10)
the deviations between Mcalc Finally, and the molecular weight obtained by conventional means are recorded in the final column with the sign of the deviation included to facilitat’e analysis of t,he deviations. In Table II are recorded the result’s of experiments by Granath and Flodin (9) on the fractionation of dextran by gel filtration
TABLE
I
RELATIONSHIP BETWEEN THE MOLECULAR WEIGHT OF PROTEINS AND ELUTION VOLUME IN GEL FILTRATION ON SEPHADEX Proteins
M X 10-a
Y"3
A. Sephadex BSA BSA BSA BSA
tetramera trimer” primer<& monomer’
268 201 143 67
9 = 0.480, C”3 = 73.0
64.5 58.6 51.2 40.7
Mtil”, = 152[1.480 -
v/v0
(v/vO)"z
_. Ml/3 ca,e M X 10-atalc 70Error
G-200 1.18 1.33 1.46 1.82
1.060 1.102 1.136 1.221
63.8 57.5 53.8 39.4
260 190 156 61
1-3.0 -5.5 t-9.1 -9.0
1.050 1.075 1.118 1.213
63.5 59.6 53.2 38.8
256 212 150 58
-4.5 t5.5 t-5.0 -13
(V/VQ)~‘~]
B. Sephadex G-150 BSA BSA BSA BSA
tetramere trimera dimera monomera
268 201 143 67
&J= 0.470, Cl’3 = 71.0
ll!!f$
64.5 58.6 51.2 40.7
= 151.0 [1.470 -
1.16 1.23 1.39 1.78
(VlVoi4
_
C. Sephadex G-100 BSA trimera r-Globulin BSA dimera p-Hydroxysteroid dehydrogenase* BSA monomera BSA monomer’ a-Hydroxysteroid dehydrogenaseb A5 - 3 Ketosteroid
isomeraseb
Pepsin/ Trypsinf ol-Chymotrypsin Cytochrome cf q = 0.447, C”3 = 56.5
201 156 134 100
58.6 53.8 51.2 46.5
1.07 1.07 1.14 1.29
1.022 1.022 1.048 1.090
55.2 55.2 51.8 46.4
67
40.7
1.130 1.111 1.160
41.2 43.7 37.3
83 51.8
$24 +10
1.258 1.185 1.225 1.236 1.271
24.6 34.0 28.8 27.4 22.8
14.8 39.2 23.8 20.5 11.8
-64 +10.4 0 -8.9 -9.2
1.032 1.034 1.080 1.120 1.197 1.153 1.185 1.195 1.213 1.259 1.248 1.440
40.9 40.7 36.5 32.0 25.8 29.8 27.0 26.0 24.4 20.2 21.2 3.65
68 67 48.5 35.2 17.2 26.4 19.6 17.5 14.5 8.2 9.5 0.049
+1.5 0.0 $38.4 -0.1 -28.6 $13.3 -4
47.1
36.2
1.43 1.37 1.56
41.08 35.5 23.8 22.5 13.0
43.5 32.9 28.8 28.3 23.5
1.97 1.66 1.83 1.88 2.05
Mtj13, = 130.0 r1.447 -
168 168 138 100 69
-16.5 +7.7 +3.0 0.0 $3.0
(VIV0)l’~l
D. Sephadex G-75 BSA monomerc BSA monomerf p-Lactoglobulinc Pepsi& Trypsinf Ovine prolactind c+Chymotrypsin Bovine RTVAse” Bovine RSAse” Cytochrome cf Bacillus subtilis RNAse’ iYaC1’ $7 = 0.480, Cl’3 = 43.8
M%
67
40.7
35.0 35.5 23.8 23.3 22.5 13.8
32.8 32.9 28.8 28.6 28.3 24.0
13.0 10.7 ,059
23.5 22.0 3.11
= 87.5 r1.480 -
1.09 1.10 1.26 1.40 1.71 1.53 1.66 1.70 1.78 1.99 1.93 2.97
+27 +5.0 -37 -11.2 -17
(V/Vo)1’3]
Q Reference (7), 0.1 M tris buffer of pH 8.1 in0.2 M h‘aC1. * Reference (14)) 0.05 M tris buffer of pH 8.2 in 0.1 M NaCl. c Reference (II), 0.5 N ammonium acetate buffer. d Reference (15), in 0.1 M NaHCOa. eReference (16)) buffer consisting of 0.5 M XaCl and 0.01 M NH,HCO,, pH 8.6. The void volume was not given, but bovine plasma albumin was included as a marker. We have calculated V/V,, for bovine RNAse and R. subtilis RN&e from the relation V/V, = V/Vllb X V&VI by using the value V&V0 = 1.09, taken from reference (11). f Reference (3), pH 6.0, ionic strength 0.494. 474
SEPHADES GEL FILTRATION TABLE II RELATIOSSHIPBETWEENTHEMOLECULARWEIGHT OF DEXTR.~N AND ELUTIOX JsOLCME ON SEPHADEX G-85” -
5% Mcslo Error
~ v/ vo ( -I5080 3930 3180 2G90 2270 1790 1350 1110 800
17.20 1.790 2.015 15.80 2.140 14.71 13.91 2.275 2.402 13.15 12.16 ~ 2.618 11.06 2.825 10.35 I 3.018 9.28 ~ 3.210
1.210 1.265 1.290 1.318 1.340 1.380 1.415 1.449 1.478
17.00 15.52 14.80 14.00 13.35 12.18
11.13 10.13 9.29
y = 0.794, (‘1’3 = 23.3 u:~,“, = 29.4 [1.794 -
4900 -3.5 3720 3220 2740 2370 1800 1370 1030 800
-5.3 f1.3
f1.9 +4.-l f0.5 +1.5 - 7.2 0.0
(v/lio)q
a Reference (9). on Sephadex G-87. The treatment of data is the same as in Table I. It is gratifying that the results from different laboratories can be correlated so well. This constitutes justification for the notion that Eq. (9) can be used with reasonable accuracy after the introduction of values for the constants g and C1j3 that correspond to the water regain value of the Sephadex used in the experiment, and that calibration with standard proteins of the specific column to be used is not essential. It is likely, however, that some increase in accuracy would result from this practice since, for example, most of the data from Whitaker (3) recorded in Table I-D show a negative deviation while the results of King and Norman (11) all show a positive deviation. Also, it may be observed that t#he deviations in Tables I-A and I-R and Table II, where the results of a single investigator using a single column and the same solvent are correlated, the deviations are somewhat smaller than in Table I-C and I-D in which results from several laboratories are compiled. Variations in temperature and ionic strength, which we have not taken into account, have been shown bv Whitaker (3) to result in small but significant variations in the relative elution volume T’/I’, . It should also be pointed out, that although the elution volunles
475
taken from the literature may have been determined with sufficient accuracy for the purpose intended by the original authors, they may not be accurate enough for the determination of molecular weights. Some of t’he deviations reported here might well result from this cause, again suggesting that better values of the constant C1j3 might be desirable. Even so, the error in estimating the molecular weight is less than 10% in most cases, with certain exceptions, notably the i15-3-ketosteroid isomerase in Table I-C and p-lactoglobulin, trypsin, and cytochrome c in Table I-D. DISCUSSION In the considerations leading to Eq. (l), we have neglected the possible role that variations in adsorption, shape, and hydrodynamic volume might play. Here we are not concerned with irreversible adsorption which would merely result in decreased recovery, but with reversible adsorption which would result in an additional retardation superimposed upon the excluded volume effect. Proof for the premise that reversible adsorption is a separate phenomenon which can frequently be avoided would be difficult to provide. In studies on gel filtration, adsorption has been clearly identified in certain instances. Most of these examples of adsorption on Sephadex are either cases of adsorption of small aromatic compounds (4, 18) or adsorption of basic proteins from acidic buffers of low ionic strength (19).2 With the application of this method restricted to the gel filtration of macromolecules in buffers as specified here, these t,ypes of adsorption would not come into consideration. In addition, Whitaker (3) has interpreted a retardation of lysozyme in terms of adsorption resulting from the formation of a weak complex between dextran and the active site of the enzyme, which is &elf a mucopolysaccharidase. He also remainds us of a * In order to minimize adsorption due to ion exchange, the pH of the buffer should be well above the isoelectric points of the proteins under investigation. The negative deviation reported in Table I for cyt,ochrome c and a-chymotrypsin may be due to their high isoelectric points.
476
SQUIRE
similar delayed elution of pancreatic ar-amylase from Sephadex G-75 observed by Gelotte (20). But these examples of adsorption of enzymes having carbohydrate as a substrate might be considered as exceptional cases. With these restrictions in mind, we are left with the observation that among the proteins that have been examined by Sephadex gel filtration in buffers meeting the requirements we have specified, a good correlation between molecular weights and elution volumes can be obtained for many proteins, by equations based upon a mechanism other than adsorption. This, it would seem, is the principle justification for relegating adsorption to an occasional supplementary role. That some proteins are additionally retarded by a process that is clearly adsorption would not appear to make this basic premise untenable, but it does introduce a distinct hazard into the general application of this method to the determination of molecular weights of macromolecules. The possibility that shape factors might play an important role and even that one might be able to draw conclusions regarding shape from gel filtration experiments has intrigued others (1, 6) as well as this author, but certain observations of the results correlated here appear to indicate otherwise. Certainly considerable changes in over-all shape must occur when one considers the series of protein components, BSA monomer, dimer, trimer, and tetramer; and yet there is no consistent trend in the deviations recorded in the final column of Table I-A, -B, and -C for Sephadex G-200, 150, and 100. It is t,rue that on Sephadex G-150 and G-200 the deviation for the dimer is 18% greater than the deviation for the monomer, and an argument might be made that this reflects the change in shape factor. Yet on Sephadex G-100, the BSA monomer and dimer both show a positive deviation of 3 %. Thus it appears that, considering all the data on BSA and its polymers on the three grades of Sephadex, the deviations are random or very nearly so. That is to say, the relative elution volumes of BSA and its polymers on all three grades of Sephadex can be predicted, within a certain random
error, purely on the basis of molecular weight, and no effect of variations in shape is observed. However, in view of the potential importance of the possibility of drawing conclusions regarding shape from gel filtration experiments, this conclusion is drawn tentatively pending further data. In addition to variations in shape we might also consider variations in the degree of swelling that macromolecules might undergo in solution. The hydrodynamic radius of a spherical protein in solution can be estimated from the relation r = gq3(1
+ TJ3,
(11)
where w represents the number of grams water bound per gram of protein of molecular weight M and partial specific volume v. If, as is frequently done (cf., for example, reference 12, p. 647), we assume that w is 0.3 gm water per gram of protein, the factor [I + (w/fi~]“~ is 1.12. It will also be convenient to evaluate the term = 0.67.
(12)
It seems possible that even among proteins variations in the factor w might result in deviations in their relative elution volume from that predicted by Eq. (9), since the assumption that r is proportional to M1j3, upon which Eq. (9) is based, would not be correct. A clear example of this effect may be seen in a comparison of the constants P/3 derived from the gel filtration of proteins with the value calculated from the dextran experiments. As can be seen in Table III, the values for V3 show the increase that would be expect,ed with the water regain of the Sephadex, but the value 23.3 for Sephadex G-87 is not consistent with the values obtained from the protein experiments. Rather, a value of about 50 would be expected. This discrepancy would be explained if the hydrodynamic radius of dextran is 2.15 times as great as that of a “typical” protein of the same molecular weight. From the results of light-scat,tering measurements, Senti et al. (13) have estimated the root mean square radius of several fractions
SEPHADES TABLE
GEL
III
1.ALUES FOR THE CONSTANTS q AXD 01’3 CORREI SPOSDING TO DEXTRAN PREPARATIOKS HAVING DIFFERENT WATER REGAIX VALUES
G200 G-150 G-100 G-75 G-87
Proteins Prot,eins Proteins Proteins Dextrsn
0.480 0.470 0.417 0.480 0.794
74.0 71.0 56.5 43.8 23.3
of dextran in the molecular weight range of 422,000-2,700,OOO. Their data are expressed by the relation R
=
0 66
jjjO.43
(13)
By combining Eqs. (11-13) we can make a rough estimate of the relative radial swelling of dextran as compared with a “typical” protein. l’d -zx 1’P
0.66 MO.43 = 0.88 MO.lO. (14) 1.12 x 0.67 Mo.33
If we take M = 3000, an intermediate value among those recorded in Table II, we calculate ?‘d - = 0.88 X 2.23 = 1.96. I’,
(15)
The agreement between the calculated value 1.96 and the required factor 2.15 is probably adequate considering the assumptions that have been made, notably the use of Eq. (13) to calculate the root mean square radius corresponding to a molecular weight far outside the range where Eq. (13) was shown to be valid. We have evaluated the constants g and C113 from experimental results as though they were empirical constants, but they both have a physical significance in terms of the model from which Eq. (8) was derived, and unless the experimentally determined values of these constants agree well with the values they should have in terms of the model, then the model is inadequate. When 1’ becomes very small, Eq. (8) reduces to the form -vi, = (1 + g)’ = 1.4703 = 3.18, V”
(16)
FILTRATIOS
477
where Ir, is the total volume of the Sephadex bed available to water. (The value 0.470 for g corresponds to Sephadex G-150, but the value is approximately the same for G-200, and G-75 as well; cf. Table III.) The values for 17,,/T7~ calculated from the bed volumes and void volumes recorded by Pedersen (7) are 3.40 for Sephadex G-100, 3.16 for G-150, and 2.94 for G-200, an agreement which seems quite satisfactory. Also in terms of the model, macromolecules of radius 1’ equal to, or greater than R, would not enter the gel and would be eluted with the void volumes. Equation (8) does not apply when 1’ is greater than R, but it may be seen by inspection that as Y approaches R, I/, approaches V. , and from Table I, it can be seen that in most cases the error in Alcitlc remains rather small even when V/V0 approaches unity.3 As a consequence we might expect that Eqs. (8) or (9) might apply throughout the entire range of molecular weights where V, is greater than V. , and in fact, the value of 41 calculated for IYaCl (Table 1-D) is in reasonable agreement with the true molecular weight (although perhaps the mean ionic radius would have been a more pertinent parameter rather than X113). The wide range of applicability of these equations might be important in some cases since, as Whitaker (3) has pointed out, his logarithmic relationship fails as V,/Bo approaches unity. (It would also be expected to fail for very small molecules since the value of T’/I’, calculated from the equation of Whitaker, for water, is 5.05 instead of a predicted value of about 3.2.) The many instances of good correlation between molecular weights and elution volumes recorded by other investigators as well as the present author provide, it would seem, ample justification for the viewpoint bhat tentative conclusions concerning the 3 In one respect the model most certainly does not correspond to the Sephadex beads. The highly complicated matrix of the interior of the Sephadex gel obviously bears little resemblance to the combination of cones, cylinders, and crevices which characterizes the model. In this respect the model can only be considered as equivalent to the Sephadex beads in its gel-filtration behavior.
478
SQUIRE
molecular weight of proteins may be drawn from measurements of their elution volume in Sephadex filtration, and the possibility of doing so by such a simple method which in fact may even be part of the purification procedure is most attractive. Yet from this discussion we see that at least two assumptions must be made: that adsorption is negligible, and that the ratio between the molecular weight and the cube of the radius is “typical.” If the buffer is well chosen, it seems likely that in most cases these assumptions might not result in serious error, in many cases less than 10%. We have seen, especially in the discussions of Whitaker (3), that the anomalous behavior of several well-characterized proteins can be convincingly explained in terms of the known properties of these proteins, but it is not clear how one could predict, for example, in the case of a newly isolated protein, whether its gel filtration behavior would be typical of its molecular weight or would be anomalous; thus, confirmation of the molecular weight by classical physicochemical methods which are not subject to these uncertainties would be most desirable before the molecular weight can be considered firmly established. Furthermore, in those instances where there is a serious discrepancy between molecular weights calculated from gel-filtration data and those calculated from physicochemical methods, it is the opinion of the present author that this discrepancy does little more than point to an area requiring further study, but even so, it might in some instances serve a most useful purpose. ACKNOWLEDGMENTS This work was 6097 from the U. author would like laboratory for his
supported in part by grant AM S. Public Health Service. The to thank Dr. C. H. Li of this interest, the other members of
this laboratory for fruitful discussions, and Dr. J. Porath of the Institute of Biochemistry, University of Uppsala, Sweden, for his comments on t,he
manuscript. REFERENCES 1. ANDREWS, P., Nature 196, 36 (1962). 2. ANDREM, P., AND FOLLEY, S. J., Biochem. J. 87, 3p (1963). 3. WHITAKER, J. R., Anal. Chem. 36, 1950 (1963). 4. PORATH, J., Biochem. Biophys. Acta 39, 193 (1960). Chem. 17, 209 5. PORATH, J., Advan. Protein (1962). doctoral disserta6. FLODIN, P., Unpublished tion, University of Uppsala (1962). 7. PEDERSEN, K. O., Arch. Biochem. Biophys. Suppl. 6, 157 (1962). 8. PORATH, J., Pure Appl. Chem. 6, 233 (1963). 9. GRANATH, K. A., AND FLODIN, P., Nakromol. Chem. 48, 160 (1961). 10. SCHACHMAN, H. K., AND LAUFFER, M. A., J. Am. Chem. Sot. 71, 536 (1949). 11. KING, T. P., AND ~TORMAN, P. S., Biochemistry 1, 709 (1962). (H. Neurath 12. EDSALL, J. T., in “The Proteins” and K. Bailey, eds.), p. 8. Academic Press, New York, 1953. 13. SENTI, F. R., HELLMAN, N. N., LUDWIG, N. H., BABCOCK, G. E., TOBIN, R., GLASS, C. A., AND LAMBERTS, B. L., J. Polymer Sci. 17, 527 (1955) . 14. SQUIRE, P. G., DELIN, S., AMI PORATH, J., abstract, Vol. VII, No. 146, p. 597, Intern. Congr. Biochem. 6th, 1964. 15. SQUIRE, P. G., STARMAN, B., APU‘DLI, C. H., J. Biol. Chem. 238, 1389 (1963). 16. HARTLEY, R. W., JR., RUSHIZKY, G. W., GRECO, A. E., AND SOBER, H. A., Biochemistry 2, 794 (1963). 17. KAWAHARA, F. S., WANG, S.-F., AND TALALAY, P., J. Biol. Chem. 237, 15C0 (1962). 18. GELOTTE, B., J. Chem. 3, 330 (1960). 19. GLAZER, A. N., AND WELLNER, D., Nature 194, 862 (1962). 20. GELOTTE, B., quoted by P. FLODIN in “Dextran Gels and Their Applications in Gel Filtration.” Pharmacia, Uppsala, Sweden, 1962.
OF
BIOCHEMISTRY
A Relationship and Their
AND
BIOPHYSICS
between Elution
471-478
107,
the Molecular
Volumes
Based
(1964)
Weights
of Macromolecules
on a Model
for Sephadex
Gel
Filtration PHIL Front the Hormone
Research Laboratory,
G. SQUIRE University
Received
April
of California,
Berkeley,
California
4, 1964
Equations relating the elution volume in Sephadex gel filtration to the molecular weight and the molecular radius of a given solute have been derived on the basis of a model in which the elements of volume available to solvent within the gel are approximated by a combination of cones, cylinders, and crevices. The equations are based on the assumption that the volume of liquid passing through the column between the applmatron of a macromolecular solute and emergence of the solute in maximal concentration is equal to the volume within the Sephadex bed which is available to macromolecules of a given size. The volume which for steric reasons is unavailable t.o t,he macromolecular solute is explicitly described in terms of the model. These equations are tested and two parameters are evaluated by means of data relat,ing molecular weights and elution volumes taken from the literature. One of the parameters can also be evaluated from the ratio of the bed volume to the void volume. The good agreement between values calculated in these two ways provides support for the notion that the simple model is essentially equivalent to the Sephadex gel. Molecular weights calculated from the elution volume by means of the equations derived here in most instances agree rather well with values calculated by conventional means. A few serious discrepancies in the data, however, suggest that caution be used in applying this or perhaps any other simple relationship between molecular weights and elution volume to the estimation of molecular weights.
It is widely recognized (5) that gel fi.ltration through Sephadex is a powerful methdo for the fractionation of proteins according to size, since fractions emerge from the column in the order of decreasing molecular size. This observation has stimulated interest in finding a simple relationship between the molecular weights of macromolecules and their elution volumes in the hope that such a relationship might be useful in obtaining a rough estimate of the molecular weight of a protein from its elution
volume.
Conversely,
tionship should be useful predicting the resolution known molecular weight obtain by gel filtration on Wit’h this objective in
such
a
rela-
as a means of of proteins of that one would Sephadex. mind, Andrews
(1) has studied the gel filtration of proteins of known molecular weight through columns of agar gel. From the results of these experiments he prepared standard curves in which the volume passing through the column before the protein emerges in maximum concentration, I’, is plotted as a function of the logarithm of molecular weight. When molecular weights of several other proteins were then estimated from their elution volumes by means of the standard curve, the agreement wit’h molecular conventional determined by weights met,hods was usually quite good. Andrews and Folley (2) have also est#inlated the molecular weights of bovine, ovine, and porcine pituitary growth hormones from their
471
elution
volu~ncs
on
Sephadex
G-7.5
472
SQUIRE
by means of a standard curve prepared for this gel. Molecular weights calculated in this manner are approximately one half of the values obtained by conventional means. Quite recently, Whitaker (3) has reported the results of the gel filtration of several standard proteins through Sephadex G-75 and G-100. He found that the ratio of the elution volume to the void volume, V/V, , is a linear function of the logarithm of the molecular weight, and again the scatter of points around the regression line is usually rather small. Certain exceptions were noted, however, and reasonable explanations were advanced. THEORETICAL
Although the empirical equations of Whitaker (3) provided a useful relationship between molecular weights and elution volumes throughout the range for which these equations were tested, it was thought worthwhile to derive an expression of this relationship based upon a reasonable model for Sephadex gel filtration. It seems likely that the equation derived in this manner might apply over a wider range of molecular weights than that of Whitaker. It will be noted that in the treatment described herein no consideration is made of rates of diffusion or of adsorption processes. Instead it is based entirely on excluded volumes, t’hat is to say, elements of volumes within the Sephadex gel which are unavailable to protein molecules of a given size. The argument of Porath (4, 5) to the effect that, at the flow rates that are experimentally accessible, the elution volume is independent of flow rate, and hence apparently not dependent upon diffusion or other rate processes, seems convincing. The notion that there are within the Sephadex gel certain forbidden regions which vary in volume with the size of the molecular species, and that this is the basis of fractionation, has been frequently expressed (4-7). In this paper a quantitative treatment of this idea is given and the basis of these forbidden or excluded volumes is explicitly described.’ r It was my impression that this had not been previously done, but during the preparation of
Reversible adsorption is considered as an additional phenomenon which can frequently be avoided, but which, where present, may be considered as superimposed upon the more general process that depends upon excluded volumes. Initially the proteins will be considered as spheres of equal hydration, but variations in shape and hydration will be considered subsequently. It is assumed that the volume of liquid, V, that passes through the column between the application of the sample and emergence of a protein from the column in maximum concentration is equal to the volume within the column, which is available to that protein. Thus it remains only to calculate the total volume available to solvent, and the volume that for steric reasons is unavailable to the protein. The highly complex matrix of the Sephadex gel interior is treated as a combination of cones, cylinders, and crevices. The principle of excluded volumes will be illustrated for the cone, but the analogous expressions for cylinders and crevices (rectangular parallelopipeds) may be derived in a similar manner. The volume outside the gel (between the spherical beads) is taken as I’, , the void volume. The volume accessible to solvent, within the cone is K=~&=kVo,
(1)
where (Y is the angle between the side of the cone and the vertical to the apex. The volume inside the cone which is available to a protein of radius r is the volume of a similar cone having a radius at the base equal to R - r. This is more readily visualthis manuscript, a paper by Porath (5) came to my attention in which a Sephadex model containing conical cavities was treated, and in a manner similar to the one presented here. Whereas Porath’s equation has a somewhat different form than the one derived here, it also provided an adequate basis for the correlation of the molecular weight of dextran fractions with the elution volume data of Granath and Flodin (9). There is also a manuscript by T. C. Laurent and J. Killander, in press (J. Chromatog.), in which a theory of gel filtration based upon a model consisting of a threedimensional network of fibers is developed and tested experimentally.
SEPHADEX
GEL
ized if the ent’ire mass of the protein is considered to be located at its center, but the nearest distance of approach of the center of the molecule to the edge of the cone remains the dist.ance T. The vohnnc inside the cone which is available to the centers of the protein molecules is then given by Eq. (2). As pointed out by Schachman and Lauffcr (10) in quite a different application of the same steric effect, statistical analysis of the concentration profile would also lead to t,he same conclusion concerning the extent of t’he excluded volume. This is given by t’he equation
Combining
these equations we have
p;, =
kVo
CRi3y)3.
The total volume available to the protein, which, according to the principle upon which t’his treatment is based, is taken as the elution volume, T’, , is then given by the expression 17, = vo + live
1 - i (
)
3
(cones>. (4)
-4 similar treatment for cylinders of radius Iz and crevices of width 2R leads to the corresponding equations. ’
(cylinders)
(5)
(crevices).
(6)
and
There remains the problem of assigning the proper fractions of the total volume inside the gel to cones, cylinders, and crevices. Perhaps a rational way of doing this could be found, but we make an arbitrary distribution that leads to an exceptionally convenient result. If we specify that l?’ = 9g, k’ = 9gZ, and 1; = 3g3, then VP = vo + 3gvo
+3gq1
(
1 - a
>
-L"~+g~vo(l
-g
O'
473
FILTRATION
and
+[l+g(l
-;)1”.
C8)
In this treatment the proteins are considered as spheres. If we also assume that r is proportional to the cube root of the molecular weight, we can also write
g.= [l+g(l -q,
(9)
where C corresponds to the molecular weight of the smallest protein that cannot enter the gel. The constants of g and P3 are to be evaluated from experimental results, but two requirements might be pointed out at this time. If the model upon which Eq. (8) was derived is consistent with the analytical resuhs, it is required that as the radius 1’ of the macromolecule approaches R, V, becomes equal to I’. It is also required that when 1‘ becomes very small, i.c., approaches the size of the water molecule, VP/V0 should approach I’JT’, , where l’, is approximately equal to bed volume less the weight of Sephadex. We shall see t#hat both of these requirements are satisfied. APPLICATIOK OF EQUATlOS CURRENT DiZTd
TO
In order to test Eq. (9) and evaluate the constants, data from the recent. literature have been tabulated in Table I. The calculated values, JP3 and (I’/ll-O)l!B, are selfexplanatory. The constant’s g and Cl13 as evaluated from plots of (l’,/l’,)l;R - 1 vs. JP3 are recorded beneath the tables. The value of the molecular weight was then calculated from the elution volumes by means of Eq. (9), solved for JP3: Mf~,“, = $[l
+ g - (q”].
(10)
the deviations between Mcalc Finally, and the molecular weight obtained by conventional means are recorded in the final column with the sign of the deviation included to facilitat’e analysis of t,he deviations. In Table II are recorded the result’s of experiments by Granath and Flodin (9) on the fractionation of dextran by gel filtration
TABLE
I
RELATIONSHIP BETWEEN THE MOLECULAR WEIGHT OF PROTEINS AND ELUTION VOLUME IN GEL FILTRATION ON SEPHADEX Proteins
M X 10-a
Y"3
A. Sephadex BSA BSA BSA BSA
tetramera trimer” primer<& monomer’
268 201 143 67
9 = 0.480, C”3 = 73.0
64.5 58.6 51.2 40.7
Mtil”, = 152[1.480 -
v/v0
(v/vO)"z
_. Ml/3 ca,e M X 10-atalc 70Error
G-200 1.18 1.33 1.46 1.82
1.060 1.102 1.136 1.221
63.8 57.5 53.8 39.4
260 190 156 61
1-3.0 -5.5 t-9.1 -9.0
1.050 1.075 1.118 1.213
63.5 59.6 53.2 38.8
256 212 150 58
-4.5 t5.5 t-5.0 -13
(V/VQ)~‘~]
B. Sephadex G-150 BSA BSA BSA BSA
tetramere trimera dimera monomera
268 201 143 67
&J= 0.470, Cl’3 = 71.0
ll!!f$
64.5 58.6 51.2 40.7
= 151.0 [1.470 -
1.16 1.23 1.39 1.78
(VlVoi4
_
C. Sephadex G-100 BSA trimera r-Globulin BSA dimera p-Hydroxysteroid dehydrogenase* BSA monomera BSA monomer’ a-Hydroxysteroid dehydrogenaseb A5 - 3 Ketosteroid
isomeraseb
Pepsin/ Trypsinf ol-Chymotrypsin Cytochrome cf q = 0.447, C”3 = 56.5
201 156 134 100
58.6 53.8 51.2 46.5
1.07 1.07 1.14 1.29
1.022 1.022 1.048 1.090
55.2 55.2 51.8 46.4
67
40.7
1.130 1.111 1.160
41.2 43.7 37.3
83 51.8
$24 +10
1.258 1.185 1.225 1.236 1.271
24.6 34.0 28.8 27.4 22.8
14.8 39.2 23.8 20.5 11.8
-64 +10.4 0 -8.9 -9.2
1.032 1.034 1.080 1.120 1.197 1.153 1.185 1.195 1.213 1.259 1.248 1.440
40.9 40.7 36.5 32.0 25.8 29.8 27.0 26.0 24.4 20.2 21.2 3.65
68 67 48.5 35.2 17.2 26.4 19.6 17.5 14.5 8.2 9.5 0.049
+1.5 0.0 $38.4 -0.1 -28.6 $13.3 -4
47.1
36.2
1.43 1.37 1.56
41.08 35.5 23.8 22.5 13.0
43.5 32.9 28.8 28.3 23.5
1.97 1.66 1.83 1.88 2.05
Mtj13, = 130.0 r1.447 -
168 168 138 100 69
-16.5 +7.7 +3.0 0.0 $3.0
(VIV0)l’~l
D. Sephadex G-75 BSA monomerc BSA monomerf p-Lactoglobulinc Pepsi& Trypsinf Ovine prolactind c+Chymotrypsin Bovine RTVAse” Bovine RSAse” Cytochrome cf Bacillus subtilis RNAse’ iYaC1’ $7 = 0.480, Cl’3 = 43.8
M%
67
40.7
35.0 35.5 23.8 23.3 22.5 13.8
32.8 32.9 28.8 28.6 28.3 24.0
13.0 10.7 ,059
23.5 22.0 3.11
= 87.5 r1.480 -
1.09 1.10 1.26 1.40 1.71 1.53 1.66 1.70 1.78 1.99 1.93 2.97
+27 +5.0 -37 -11.2 -17
(V/Vo)1’3]
Q Reference (7), 0.1 M tris buffer of pH 8.1 in0.2 M h‘aC1. * Reference (14)) 0.05 M tris buffer of pH 8.2 in 0.1 M NaCl. c Reference (II), 0.5 N ammonium acetate buffer. d Reference (15), in 0.1 M NaHCOa. eReference (16)) buffer consisting of 0.5 M XaCl and 0.01 M NH,HCO,, pH 8.6. The void volume was not given, but bovine plasma albumin was included as a marker. We have calculated V/V,, for bovine RNAse and R. subtilis RN&e from the relation V/V, = V/Vllb X V&VI by using the value V&V0 = 1.09, taken from reference (11). f Reference (3), pH 6.0, ionic strength 0.494. 474
SEPHADES GEL FILTRATION TABLE II RELATIOSSHIPBETWEENTHEMOLECULARWEIGHT OF DEXTR.~N AND ELUTIOX JsOLCME ON SEPHADEX G-85” -
5% Mcslo Error
~ v/ vo ( -I5080 3930 3180 2G90 2270 1790 1350 1110 800
17.20 1.790 2.015 15.80 2.140 14.71 13.91 2.275 2.402 13.15 12.16 ~ 2.618 11.06 2.825 10.35 I 3.018 9.28 ~ 3.210
1.210 1.265 1.290 1.318 1.340 1.380 1.415 1.449 1.478
17.00 15.52 14.80 14.00 13.35 12.18
11.13 10.13 9.29
y = 0.794, (‘1’3 = 23.3 u:~,“, = 29.4 [1.794 -
4900 -3.5 3720 3220 2740 2370 1800 1370 1030 800
-5.3 f1.3
f1.9 +4.-l f0.5 +1.5 - 7.2 0.0
(v/lio)q
a Reference (9). on Sephadex G-87. The treatment of data is the same as in Table I. It is gratifying that the results from different laboratories can be correlated so well. This constitutes justification for the notion that Eq. (9) can be used with reasonable accuracy after the introduction of values for the constants g and C1j3 that correspond to the water regain value of the Sephadex used in the experiment, and that calibration with standard proteins of the specific column to be used is not essential. It is likely, however, that some increase in accuracy would result from this practice since, for example, most of the data from Whitaker (3) recorded in Table I-D show a negative deviation while the results of King and Norman (11) all show a positive deviation. Also, it may be observed that t#he deviations in Tables I-A and I-R and Table II, where the results of a single investigator using a single column and the same solvent are correlated, the deviations are somewhat smaller than in Table I-C and I-D in which results from several laboratories are compiled. Variations in temperature and ionic strength, which we have not taken into account, have been shown bv Whitaker (3) to result in small but significant variations in the relative elution volume T’/I’, . It should also be pointed out, that although the elution volunles
475
taken from the literature may have been determined with sufficient accuracy for the purpose intended by the original authors, they may not be accurate enough for the determination of molecular weights. Some of t’he deviations reported here might well result from this cause, again suggesting that better values of the constant C1j3 might be desirable. Even so, the error in estimating the molecular weight is less than 10% in most cases, with certain exceptions, notably the i15-3-ketosteroid isomerase in Table I-C and p-lactoglobulin, trypsin, and cytochrome c in Table I-D. DISCUSSION In the considerations leading to Eq. (l), we have neglected the possible role that variations in adsorption, shape, and hydrodynamic volume might play. Here we are not concerned with irreversible adsorption which would merely result in decreased recovery, but with reversible adsorption which would result in an additional retardation superimposed upon the excluded volume effect. Proof for the premise that reversible adsorption is a separate phenomenon which can frequently be avoided would be difficult to provide. In studies on gel filtration, adsorption has been clearly identified in certain instances. Most of these examples of adsorption on Sephadex are either cases of adsorption of small aromatic compounds (4, 18) or adsorption of basic proteins from acidic buffers of low ionic strength (19).2 With the application of this method restricted to the gel filtration of macromolecules in buffers as specified here, these t,ypes of adsorption would not come into consideration. In addition, Whitaker (3) has interpreted a retardation of lysozyme in terms of adsorption resulting from the formation of a weak complex between dextran and the active site of the enzyme, which is &elf a mucopolysaccharidase. He also remainds us of a * In order to minimize adsorption due to ion exchange, the pH of the buffer should be well above the isoelectric points of the proteins under investigation. The negative deviation reported in Table I for cyt,ochrome c and a-chymotrypsin may be due to their high isoelectric points.
476
SQUIRE
similar delayed elution of pancreatic ar-amylase from Sephadex G-75 observed by Gelotte (20). But these examples of adsorption of enzymes having carbohydrate as a substrate might be considered as exceptional cases. With these restrictions in mind, we are left with the observation that among the proteins that have been examined by Sephadex gel filtration in buffers meeting the requirements we have specified, a good correlation between molecular weights and elution volumes can be obtained for many proteins, by equations based upon a mechanism other than adsorption. This, it would seem, is the principle justification for relegating adsorption to an occasional supplementary role. That some proteins are additionally retarded by a process that is clearly adsorption would not appear to make this basic premise untenable, but it does introduce a distinct hazard into the general application of this method to the determination of molecular weights of macromolecules. The possibility that shape factors might play an important role and even that one might be able to draw conclusions regarding shape from gel filtration experiments has intrigued others (1, 6) as well as this author, but certain observations of the results correlated here appear to indicate otherwise. Certainly considerable changes in over-all shape must occur when one considers the series of protein components, BSA monomer, dimer, trimer, and tetramer; and yet there is no consistent trend in the deviations recorded in the final column of Table I-A, -B, and -C for Sephadex G-200, 150, and 100. It is t,rue that on Sephadex G-150 and G-200 the deviation for the dimer is 18% greater than the deviation for the monomer, and an argument might be made that this reflects the change in shape factor. Yet on Sephadex G-100, the BSA monomer and dimer both show a positive deviation of 3 %. Thus it appears that, considering all the data on BSA and its polymers on the three grades of Sephadex, the deviations are random or very nearly so. That is to say, the relative elution volumes of BSA and its polymers on all three grades of Sephadex can be predicted, within a certain random
error, purely on the basis of molecular weight, and no effect of variations in shape is observed. However, in view of the potential importance of the possibility of drawing conclusions regarding shape from gel filtration experiments, this conclusion is drawn tentatively pending further data. In addition to variations in shape we might also consider variations in the degree of swelling that macromolecules might undergo in solution. The hydrodynamic radius of a spherical protein in solution can be estimated from the relation r = gq3(1
+ TJ3,
(11)
where w represents the number of grams water bound per gram of protein of molecular weight M and partial specific volume v. If, as is frequently done (cf., for example, reference 12, p. 647), we assume that w is 0.3 gm water per gram of protein, the factor [I + (w/fi~]“~ is 1.12. It will also be convenient to evaluate the term = 0.67.
(12)
It seems possible that even among proteins variations in the factor w might result in deviations in their relative elution volume from that predicted by Eq. (9), since the assumption that r is proportional to M1j3, upon which Eq. (9) is based, would not be correct. A clear example of this effect may be seen in a comparison of the constants P/3 derived from the gel filtration of proteins with the value calculated from the dextran experiments. As can be seen in Table III, the values for V3 show the increase that would be expect,ed with the water regain of the Sephadex, but the value 23.3 for Sephadex G-87 is not consistent with the values obtained from the protein experiments. Rather, a value of about 50 would be expected. This discrepancy would be explained if the hydrodynamic radius of dextran is 2.15 times as great as that of a “typical” protein of the same molecular weight. From the results of light-scat,tering measurements, Senti et al. (13) have estimated the root mean square radius of several fractions
SEPHADES TABLE
GEL
III
1.ALUES FOR THE CONSTANTS q AXD 01’3 CORREI SPOSDING TO DEXTRAN PREPARATIOKS HAVING DIFFERENT WATER REGAIX VALUES
G200 G-150 G-100 G-75 G-87
Proteins Prot,eins Proteins Proteins Dextrsn
0.480 0.470 0.417 0.480 0.794
74.0 71.0 56.5 43.8 23.3
of dextran in the molecular weight range of 422,000-2,700,OOO. Their data are expressed by the relation R
=
0 66
jjjO.43
(13)
By combining Eqs. (11-13) we can make a rough estimate of the relative radial swelling of dextran as compared with a “typical” protein. l’d -zx 1’P
0.66 MO.43 = 0.88 MO.lO. (14) 1.12 x 0.67 Mo.33
If we take M = 3000, an intermediate value among those recorded in Table II, we calculate ?‘d - = 0.88 X 2.23 = 1.96. I’,
(15)
The agreement between the calculated value 1.96 and the required factor 2.15 is probably adequate considering the assumptions that have been made, notably the use of Eq. (13) to calculate the root mean square radius corresponding to a molecular weight far outside the range where Eq. (13) was shown to be valid. We have evaluated the constants g and C113 from experimental results as though they were empirical constants, but they both have a physical significance in terms of the model from which Eq. (8) was derived, and unless the experimentally determined values of these constants agree well with the values they should have in terms of the model, then the model is inadequate. When 1’ becomes very small, Eq. (8) reduces to the form -vi, = (1 + g)’ = 1.4703 = 3.18, V”
(16)
FILTRATIOS
477
where Ir, is the total volume of the Sephadex bed available to water. (The value 0.470 for g corresponds to Sephadex G-150, but the value is approximately the same for G-200, and G-75 as well; cf. Table III.) The values for 17,,/T7~ calculated from the bed volumes and void volumes recorded by Pedersen (7) are 3.40 for Sephadex G-100, 3.16 for G-150, and 2.94 for G-200, an agreement which seems quite satisfactory. Also in terms of the model, macromolecules of radius 1’ equal to, or greater than R, would not enter the gel and would be eluted with the void volumes. Equation (8) does not apply when 1’ is greater than R, but it may be seen by inspection that as Y approaches R, I/, approaches V. , and from Table I, it can be seen that in most cases the error in Alcitlc remains rather small even when V/V0 approaches unity.3 As a consequence we might expect that Eqs. (8) or (9) might apply throughout the entire range of molecular weights where V, is greater than V. , and in fact, the value of 41 calculated for IYaCl (Table 1-D) is in reasonable agreement with the true molecular weight (although perhaps the mean ionic radius would have been a more pertinent parameter rather than X113). The wide range of applicability of these equations might be important in some cases since, as Whitaker (3) has pointed out, his logarithmic relationship fails as V,/Bo approaches unity. (It would also be expected to fail for very small molecules since the value of T’/I’, calculated from the equation of Whitaker, for water, is 5.05 instead of a predicted value of about 3.2.) The many instances of good correlation between molecular weights and elution volumes recorded by other investigators as well as the present author provide, it would seem, ample justification for the viewpoint bhat tentative conclusions concerning the 3 In one respect the model most certainly does not correspond to the Sephadex beads. The highly complicated matrix of the interior of the Sephadex gel obviously bears little resemblance to the combination of cones, cylinders, and crevices which characterizes the model. In this respect the model can only be considered as equivalent to the Sephadex beads in its gel-filtration behavior.
478
SQUIRE
molecular weight of proteins may be drawn from measurements of their elution volume in Sephadex filtration, and the possibility of doing so by such a simple method which in fact may even be part of the purification procedure is most attractive. Yet from this discussion we see that at least two assumptions must be made: that adsorption is negligible, and that the ratio between the molecular weight and the cube of the radius is “typical.” If the buffer is well chosen, it seems likely that in most cases these assumptions might not result in serious error, in many cases less than 10%. We have seen, especially in the discussions of Whitaker (3), that the anomalous behavior of several well-characterized proteins can be convincingly explained in terms of the known properties of these proteins, but it is not clear how one could predict, for example, in the case of a newly isolated protein, whether its gel filtration behavior would be typical of its molecular weight or would be anomalous; thus, confirmation of the molecular weight by classical physicochemical methods which are not subject to these uncertainties would be most desirable before the molecular weight can be considered firmly established. Furthermore, in those instances where there is a serious discrepancy between molecular weights calculated from gel-filtration data and those calculated from physicochemical methods, it is the opinion of the present author that this discrepancy does little more than point to an area requiring further study, but even so, it might in some instances serve a most useful purpose. ACKNOWLEDGMENTS This work was 6097 from the U. author would like laboratory for his
supported in part by grant AM S. Public Health Service. The to thank Dr. C. H. Li of this interest, the other members of
this laboratory for fruitful discussions, and Dr. J. Porath of the Institute of Biochemistry, University of Uppsala, Sweden, for his comments on t,he
manuscript. REFERENCES 1. ANDREWS, P., Nature 196, 36 (1962). 2. ANDREM, P., AND FOLLEY, S. J., Biochem. J. 87, 3p (1963). 3. WHITAKER, J. R., Anal. Chem. 36, 1950 (1963). 4. PORATH, J., Biochem. Biophys. Acta 39, 193 (1960). Chem. 17, 209 5. PORATH, J., Advan. Protein (1962). doctoral disserta6. FLODIN, P., Unpublished tion, University of Uppsala (1962). 7. PEDERSEN, K. O., Arch. Biochem. Biophys. Suppl. 6, 157 (1962). 8. PORATH, J., Pure Appl. Chem. 6, 233 (1963). 9. GRANATH, K. A., AND FLODIN, P., Nakromol. Chem. 48, 160 (1961). 10. SCHACHMAN, H. K., AND LAUFFER, M. A., J. Am. Chem. Sot. 71, 536 (1949). 11. KING, T. P., AND ~TORMAN, P. S., Biochemistry 1, 709 (1962). (H. Neurath 12. EDSALL, J. T., in “The Proteins” and K. Bailey, eds.), p. 8. Academic Press, New York, 1953. 13. SENTI, F. R., HELLMAN, N. N., LUDWIG, N. H., BABCOCK, G. E., TOBIN, R., GLASS, C. A., AND LAMBERTS, B. L., J. Polymer Sci. 17, 527 (1955) . 14. SQUIRE, P. G., DELIN, S., AMI PORATH, J., abstract, Vol. VII, No. 146, p. 597, Intern. Congr. Biochem. 6th, 1964. 15. SQUIRE, P. G., STARMAN, B., APU‘DLI, C. H., J. Biol. Chem. 238, 1389 (1963). 16. HARTLEY, R. W., JR., RUSHIZKY, G. W., GRECO, A. E., AND SOBER, H. A., Biochemistry 2, 794 (1963). 17. KAWAHARA, F. S., WANG, S.-F., AND TALALAY, P., J. Biol. Chem. 237, 15C0 (1962). 18. GELOTTE, B., J. Chem. 3, 330 (1960). 19. GLAZER, A. N., AND WELLNER, D., Nature 194, 862 (1962). 20. GELOTTE, B., quoted by P. FLODIN in “Dextran Gels and Their Applications in Gel Filtration.” Pharmacia, Uppsala, Sweden, 1962.