Structure parameters of molecules and media evaluated by chromatographic partition

Journal oj chromatography,

410 (1987) 233-248 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands CHROM. 19 956

STRUCTURE PARAMETERS OF MOLECULES BY CHROMATOGRAPHIC PARTITION II. GEOMETRICAL

EXCLUSION

AND MEDIA EVALUATED

IN GELS

HENRIK WALDMANN-MEYER Fysisk-Kemisk

Institut, Technical University of Denmark, DK 2800 Copenhagen-Lyngby

(Denmark)

(Received June 22nd, 1987)

SUMMARY

In continuation of the geometrical exclusion model for porous glasses [J. 350 (1985) 11, a gel model based on cavities approaching conical shape is described. The partition coefficient K emerges as a function of the Stokes’ radius R,, polymer volume fraction, chain and cavity radius. The K(R,) equations were tested in twelve different gels with proteins, dextrans and Ficoll fractions. The model is self-consistent, inter aEia since calculated chain radii agree with physical measurements. The correlation between gel parameters appears also to be valid for electrophoresis. Conversely, it is seen that the Ogston model does not apply to gels. From general theory, R, is expressed as AM” where A and x are structurespecific constants. The resulting linearized K(M) plots therefore clearly distinguish between random coils, rods and globular molecules. From such plots the effective bond length, degree of ideality and range of axial ratios are directly determined. Moreover, calibration is set on a non-empirical basis and thus greatly simplified. Chromatogr.,

INTRODUCTION

In a recent paper1 we described the dependence of the partition coefficient (Kn) on pore shape and size, as well as on molecular radius and weight, for rod-like and random-coiled molecules in controlled pore glasses. On the basis of a straightforward geometrical exclusion model (GEM) for cylindrical pores, it was shown that the pore radii calculated from the linear J&, KS.molecular Stokes’ radius (R,) correlations were practically identical with the values determined by means of mercury-intrusion porosimetry in eleven different glasses. This fact, together with a re-evaluation of Casassa’s result?, emphasized the unique role of the Stokes’ radius and thus confirmed the concept of chromatographic partition as a diffusive quasi-equilibrium process3. The partition coefficient therefore reflects not only molecular parameters, but structural features of the chromatographic medium as well. In the case of gels, a precise evaluation of matrix characteristics and their interplay is evidently more complicated than for porous glasses. Two main theoretical 0021-9673/87/$03.50

0

1987 Elsevier Science Publishers B.V.

234

H. WALDMANN-MEYER

approaches have been propounded and tested by means of actual experiments, viz. Ogston’s equation for the free space available to a sphere in a random network of rigid fibres4, and Porath’s expression for a network chiefly constituted by voids that become narrower with depth and thus can be visualized as conical cavities5. An equation formulated by Squire6 for cavities of mixed shapes could not be corroborated7. Ogston’s formulation describes partition as a function of the molecular radius, the matrix-polymer concentration, and the chain radius, whereas in the Porath equation Kn varies with R, and the radius of the conical void. The latter approach has been re-formulated and extensively tested by us*. The new GEM equation defines Kn as a function of R,, cavity radius, polymer concentration and chain radius. In the present paper, the relation between the three matrix parameters, as predicted by the equation, is examined by means of fifteen data sets obtained from the literature and our laboratory. The same data are employed for an evaluation of the Ogston theory. Special emphasis will be given to partition as a function of the molecular weight. In fact, by replacing R, in the GEM equation by AM”, where A and x are structure-specific constants, it is possible not only to distinguish between globular, rod-like and random-coiled molecules, but also to calculate a number of fundamental molecular parameters that so far could only be determined by means of more sophisticated techniques. This is entirely analogous to the GEM theory for porous glassesI, with the advantage that more precise linearizations will be obtained for most types of gels. THEORY

Consider a random network of polymer chains in a swollen gel bead. The probability of a parallel chain orientation, giving rise to either cylindrical or lamellar cavities, appears as rather remote, even more so when the presence of entanglements and cross-linkages is taken into account. Likewise, the existence of spherical voids is difficult to envisage. In short, the free spaces will assume highly irregular shapes beyond those regions where chains are physically or chemically connected. According to Floding, because of the restricted chain mobility at the connections it is possible to visualize localized polymer density gradients within the gel, with the density highest around the connecting sites and gradually diminishing with distance from the sites. In our view, possible local gradients would be subordinate to the fact that filaments of hydrophilic polymers swell anisotropically, in such a way that the degree of swelling is highest at the side facing the solution, whence permeability decreases with the distance from the solution boundary l”rl l. The change of the permeability coefficient with film thickness was ascribed by Ito 12*13to stresses resulting from differential swelling. When considering a spherical gel bead, therefore, it is probable that the approach to swelling equilibrium is concomitant to an overall density gradient within the bead, with the concentration of loops and thus of chains increasing towards the centre. A balance between internal forces and swelling pressure may contribute to maintain the spherical shape. These findings lend physical support to Porath’s conception of an array of voids, which statistically can be regarded as cones. This is, to a first approximation, equivalent to a gel bead in which the free space available to a permeating molecule decreases with the molecular size, as described below.

EVALUATION

OF STRUCTURE

PARAMETERS.

235

II.

The geometrical exclusion model for gels

We define the total bed volume as V,, equal to the sum of Vi (imbibed solvent), V, (polymer matrix) and V, (excluded solvent). V, denotes the elution volume. Since the volume of a cone is given by (rc/3)hR: where h is the height and R, the radius,’ we represent Vi by N conical cavities as Vi = (NX/3)hR;

(1)

Assuming a cylindrical shape for the polymer chain and denoting its diameter as 2r0, the gel volume is Vi + V, = (Nn/3)(h

+ rO/sin 8)(R,

+ r,jcos 0)’

(2)

where 0 stands for one-half of the solid angle subtending the cone, and r. is used because the polymer chain is shared by two adjacent cavities. Consider now a molecule within the gel. Since the smallest molecules will be able to permeate the total gel phase (Vi + V,), the volume available to the centre of mass is equal to the reduced elution volume, viz: h + ”

- R,

R, + L

- R,

2

(3)

cos 8 ,. where R, is the Stokes’ radius of the pertinent molecule. Although from theory Stokes’ law does not apply to radii smaller than some 0.5 nm, practice has shown that the limit may lie significantly lower. Since tan

o

5

=

= Rx + r&s

h

sin 8

8 =

h + ro/sin 8

R, +

r,/cos8 -

h + rO/sin B -

R&OS 8

R,,!sin 8

(4)

the sine term of eqn. 3 can be replaced, and by combination with eqn. 1 the expression for the classical partition coefficient is obtained as KD =

ve- vo Vi

R, cos 8 + r. =

R, 3

(5)

R, cos 9

Analogously, the partition coefficient KAv is readily calculated from eqns. 2 and 3, and replacing the sine terms by means of eqn. 4 to

KAv ZE

ve- vo ve- vo V, -

Vo = Vi + Vm

R, R, cos 0 + r.

3

!

(6)

Taking third roots and defining the ordinate intercept in eqn. 5 as k = 1 + expressions become

ro/R, cos 13,the,linearized

KAj3 = k - R,/RX cos 8

(7)

H. WALDMANN-MEYER

236

and K;$ = 1 - R,/(kRx cos 0)

(8)

It can be seen that since KD/KAv = 1 + Y,/Vi = constant, both coefficients have the same exclusion limit. The experimental maximum is reached at R, = ro, as distinct from R, = 0 as generally assumed. The extrapolated ordinate intercepts are of considerable interest. The fact that KAv extrapolates to unity greatly facilitates the analysis of partition as a function of the molecular weight described below. On the other hand, from the intercept k we derive the fundamental relation between cavity radius and polymer concentration as follows. Let R, = 0 in eqn. 3, whence eqns. 2 and 3 become equivalent and V, - V. = Vi + I’,, such that KD = 1 + V,/Vi. Hence, the intercept value of eqn. 7 is k

E

1 +

ro/(Rx COS6)

= (1 + VJI’i)1’3 = (1 -

~0))~‘~ = (1 - C/d)-1’3

(9)

where rp is the polymer volume fraction Vm/(Vi + V,), c the polymer concentration in gram per cm3 of gel and d the density of the dry polymer. It is readily apparent that for a given chain radius r. the cavity radius R, cos 0 is obtained directly from cp = c/d. Thus, the partition coefficient is defined by three parameters, viz. the effective hydrodynamic radius R,, the polymer volume fraction cp and the chain radius ro. This means that molecules of identical radius, but different molecular weights or structures, cannot be separated by gel permeation chromatography. The crucial role of the radius R, has been theoretically and experimentally substantiated in the previous paper dealing with controlled pore glassesl. It is precisely this role that establishes the basis for a quantitative, non-empirical, correlation between chromatographic partition and molecular weight as described in the following. For any molecule subject to Stokes’ law, R, = AM”

(10)

where A and x are well-defined constants that reflect fundamental specific properties of molecular structure. This equation is derived from general physico-chemical theory and has been developed in detail for random-coiled and rod-like molecules in the previous articlel. Here, it will be extended to prolate ellipsoids which represent the great majority of proteins. Table I summarizes A and x values for different kinds of molecule. For coils’. eqn. 10 becomes R e = A&f%

(11)

where for large M/MO values, Ati = 0.2714 ~/I%#’ and B denotes the effective bond length, MO the monomer molecular weight, and o! is the expansion coefficient equal to 1.00 for ideality.

EVALUATION

OF STRUCTURE

PARAMETERS.

237

II.

TABLE I THE RELATION

R, = AM” BETWEEN RADIUS AND MOLECULAR

WEIGHT

See text. The units of A are cm(mol/g)“2 for coils and cm(mol/g)‘/3 for rods and ellipsoids.

Mm.xl103 Random coils* (ideal solutions) Dextrans Polystyrenes

-

70 >2.50

Prolate ellipsoids Globular proteins*

Fibrous proteins**

x

A = Ae

Rods* SDS-proteins

A/IO-@

100 800 9000 1000

0.2 0.179

0.50 0.50

A = p&/pm

x = l/3 +m

0.022

0.73 max. 0.82

A = /?A0

x = 1/3+fi

0.423 0.478 0.551 0.654

0.395 0.383 0.370 0.393

l Derived in the previous paper]. ** Calculated from Felgenhauer’s critical compilation of 64 proteins14. (Correlation Y = 0.99). ** Calculated from SoberIs for 34 proteins (r = 0.89).

For ellipsoids and rods’, on the other hand, R, =

AoM”3flf,

(12)

where A0 = (3P/47~nN~)l’~,which for an average partial specific volume for proteins of P = 0.725 cm3/g gives A0 = 0.66 . lo-*. The frictional coefficient can be correlated with the axial ratio asfif = p(a/b)“, equal to p(M/p)” for homologous rodsl, where (a/b) increases linearly with M. Hence, for such rods, A = p&/pm and x = l/3 + m. In contrast, globular proteins evidently have little or no tendency to become more elongated with the molecular weight. Lettingflfo = p(Q)” = F(M)“, by combination with eqn. 12, A = jk4, and x = l/3 + ti. In spite of significant variation in P and the fact that f/f0 may reflect both asymmetry and hydration, the overall picture emerging for M < lo6 is j&

N 0.724 . hPo5 (globular proteins)

and fife z APo6 (fibrous proteins) Thus, for any molecule whatsoever eqn. 10 can be combined with eqns. 7 or 8 as K;j3 = k - M”A/R, cos 0

(13)

H. WALDMANN-MEYER

238

or K;$ = 1 - APA/k R, cos 8

(14)

where x = 0.4 for proteins in the usual range and x = 0.5 for random coils in idea1 solutions. When x is unknown it may be determined1 from -In (k - Kg3) = --x In M - ln(A/R, cos 0)

(15)

-In (1 - K#

(16)

or = --x In A4 - ln(A/k R, cos 8)

Note however, that precise x and A values will be obtained only when the actual ordinate intercepts from eqns. 7 or 8 are used, since a small difference will be greatly magnified in the logarithmic term by the use of k or 1.00. The GEM equations will be tested by application to data obtained with twelve different gels in this laboratory and from the literature. Special emphasis will be given to an evaluation of the Ogston model4 by means of the same data, and to the concentration function of the gel cavity radius. EXPERIMENTAL

Chromatography was performed on a precision-bore column of 1.60 cm I.D. containing 40-60 cm3 of gel. The column was thermostated to 2O.O”Cand effluents were measured with a 2138 Uvicord-S (LKB, Bromma, Sweden) monitor at 206 hm. IJ’, was obtained from the effluent weight at maximum absorbance and the buffer density. The total volume V, was calculated from the bed height measured by means of a vernier height gauge, and corrected for dead volume. The excluded volume Y0 was determined with tobacco mosaic virus (TMV), kindly donated by the Institute for Plant Pathology, Danish Research Centre for Plant Protection. In Sephadex experiments, blue dextran (Pharmacia, Uppsala, Sweden) was used. The elution volume of glycine was taken as (Vi + V,). Unless otherwise mentioned, phosphate buffer (pH 7.00, 1 = 0.145 mol/dm3) was employed because of its low absorbance at 206 nm. Samples of 100 mm3 were used at l-3% (w/w). Proteins were highest grade products (Sigma) and were used without further treatment. Fluorescein-labelled dextrans 3, 20, 40, 70 and 150, as well as Dextran-500 (Batch Nr. FDR 876) were purchased from Pharmacia. PARTITION AS FUNCTION

OF PORE SIZE AND MOLECULAR

STRUCTURE

In order to test the model, we began by analysing fifteen sets of experimental data, six of which are taken from the literature. Since the latter are tabulated as KAv values, eqn. 8 is applied throughout the compilation given in Table II. The analysed substances were proteins and fluorescent dextrans, the radii of which are listed in

EvALuATI~N~FSTRUCTUREPARAMETERS.~I.

239

TABLE11 PARTITIONASFUNCTIONOFRADIUS IV = number of permeants; r = linear correlation coefficient; R,, = abscissa intercept (cf: eqn. 17); b * ordinate intercept; c = g polymer per cm3 gel. Densities used in calculating c: agarose, 1.695; Sephacryl, 1.38; Sephadex, I .64; hyaluronic acid, 1.45 g/cm3. N

Gel

GEM theory (eqn. 8) --P

Ogston model (eqn. 21)

R<$

R,,lJ

h

r,!A

c

1.006 1.006 0.983 1.021 1.012 0.986

26.31 22.7 54.03 23.18 16.94 39.32

2.90 1.28 19.65 14.92 9.37 27.52

16 16 ** 16 l* **

0.998 1.006 0.989

26.48 17.45 20.58

21.00 15.10 24.43

16 17 **

I.071

102

4* 10 8 7 3 13 10 6

0.994 0.992

950.50 362.73 412.60 186.34 208.66 208.88 173.92 134.88

Sephacryl S-200

11

0.999

101.62

Sephadex

5 4 6 IO*** 5

0.997 1.000 0.999 0.998

44.10 43.04 80.15 74.38 71.09

1.110 1.090

5.08 3.88 6.81

11.20 6.71 6.33

** ** t*

1.067 1.077

7.72 7.54

9.30 9.34

** f*

10

0.988

113.61

1.008

8.74

5.44

18

Agarose, 2% 4% 4B-CL 6% 6B 6B-CL (Agar)

i12

G50-f GSO-f GIOO-M GIOO-M GlOO-Sf

Hyaluronic acid (CL)

0.990 0.966 0.997 0.991 1.OOo

1.000

I .ooo

KAv < 0.94 computed. * This investigation. *++ I = 0.445 mol/dm3. l

l

Table III, as well as Ficoll fractions. Radii of the latter, determined by gel permeation chromatography, were taken from Laurent l 6. In Table II, R,, denotes the exclusion radius given by the abscissa intercept in eqns. 7 or 8, i.e.: R,, = kR, cos 9 = R, cos 9 + r.

and’b is the ordinate intersection (eqn. 8). Table II shows that the linear correlation required by eqn. 8 is satisfied since, in all but one case, r > 0.99. Moreover, for all gels except Sephadex, b = 1.0015 f 0.012

compared with 1.00 predicted by eqn. 8. The Sephadex anomaly is ascribed to hydration and adsorption effects, and will be treated elsewherei9. Molecular weight limits were calculated from the R, values for &v = 0.05, considered to be the smallest &v measurable with precision, and the R,(M) functions given in Table I. The limits are in good agreement with the manufacturer’s values. Cross-linking does not affect the exclusion limit of agarose gels, as also noted by

EVALUATION

0

OF STRUCTURE

20

40

PARAMETERS.

60

II.

241

80

100

8,

R.

Fig. I. Partition as function of Stokes’ radius in five gels according to GEM eqn. 7. (0) (A) dextrans. Exclusion radii and correlation coefficients as given in Table II.

0.8

-

0.8

-

0

20

40

80

80

100

yo.a

Proteins;

120

Fig. 2. The M” function for globular proteins (eqn. 13 and Table I). For numerical results, see Table IV.

242

II. WALDMANN-MEYER

.

Fig. 3. Random-coiled dextrans in ideal solution (eqn. 13 and Table I). The results are given in Table IV.

2.0.

l.o.~~~

1

0 8.0

9.0

I

I

10.0

11.0

J 12.0

In M

Fig. 4. The double logarithmic GEM plot (eqn. IS) for proteins (0) and dextrans (A) in two gels. M = @s,o for dextrans as required. The resulting molecular parameters x and A appear in Table IV.

EVALUATION

OF STRUCTURE

PARAMETERS,

243

II.

TABLE IV PARTITION

AS FUNCTION

OF MOLECULAR

WEIGHT

p.4, values are given in cm(mol/g)‘i3 and A0 in cm(mol/g)l’*.

--r

Eqn.

Fig.

Gel

Proteins 13

2

4B-Cl 6B-Cl s-200 G-IOOM G-50f

0.991 0.987 0.997 0.994 0.967

6B-Cl s-200

0.993 0.979

15

4

Intercept

IS~~pel

1.0226 1.0217 1.084 1.1109 1.147

1.1899, 2.0009. 4.6288. 6.1273. 10.1709~

6.1656 5‘4301

0.3997 0.3951

IO9

jiAo

1o-3 lO-3 1O-3 1O-3

1o-3

4.888 4.128 4.480 4.197 3.831 4.333 4.242

Dextrans

As

1P

13

3

4B-Cl 6B-Cl s-200

0.996 1.000 0.999

1.0051 1.0034 I .045.5

0.5673 lO-3 1.0878. 1O-3 2.3740. 1O-3

2.331 2.244 2.298

15

4

6B-Cl s-200

1.000 0.999

6.8669 5:9143

0.5038 0.4874

2.149 2.614

that (jiA’o) remains unchanged, whereas (AB) shows a slight increase. The fundamental feature, however, is the precision with which the structure-specific slope can be determined. In fact, the average slope is 0.397 for proteins, which is identical with the value for low-molecular-weight specimens in Table I, and 0.496 for dextrans as compared with 0.500. In contrast to the intercepts, slopes are independent of the gel, as required by eqn. 15. This type of linear correlation may prove to be extremely useful inasmuch as different hydrodynamic structures can readily be distinguished. While prolate proteins in, for example, Sephadex 6B-CL give ffKn) N 6.17 - 0.4 In M (c$ Table IV), the same light proteins would yield flKn) z 9.16 - 0.73 In M when brought into rod-like form by denaturation since, as previously derivedl, A changes to 0.0218 . IO-’ and x to 0.73. Analogously, if random coils such as dextrans were analysed under non-ideal conditions, both the slope and the intercept would increasel. The consequences for calibration of gel media will be discussed in the last section. PORE SIZE AS FUNCTION

OF POLYMER

CONCENTRATION

AND CHAIN RADIUS

We have previously shown that in porous glasses the chromatographically determined R, values are practically identical with the pore radii measured by means of mercury-intrusion porosimetry l. No such test is feasible in gels. However, the GEM theory predicts a stringent relation between R, and the polymer concentration. This relation is given by eqn. 9, which in the limit qn = c/d = 0 gives k = 1 and R,costI-+c~,whileforc=d,k + a3 and R, cos 9 = 0. By combination with eqn. 7, for all molecules Kn = 1 or zero, respectively, as expected.

II. WALDMANN-MEYER

244 TABLE V AGAROSE EXCLUSION

RADII -

ADDITIONAL

% Agarose

R,XIA

Determination

1.9v 2.95’ 3.91* 9** 12*

974.9 668 509 75.7 61.9

Electroosmotic Electroosmotic Electroosmotic M(&, = 0) N M(Ko = 0) 2

DATA Ref.

migration migration migration 5.61 10s 4.22 lo5

of unlabelled dextrans of unlabelled dextrans of unlabelled dextrans (proteins). Chromatography (proteins). Chromatography

22 22 22 20 20

* Litex LSL-agarose. ** Beads prepared from Reactifs IBF agarose.

Table II lists R,, values for agarose of different concentrations. Since R,, = R, cos 8 + ro = kR, cos 8 (eqn. 17), and from eqn. 9, R, cos s/r0 = l/(k - l), it is seen that

Rex=

ro[kl(k

-

111

(1’3)

A more tangible expression is obtained by expanding eqn. 9 in various ways and taking the mean as R,, II ~(34~

-

1)

(19)

The error introduced by this approximation is 0.08% for cp = 0.1 (x 17% agarose) and 0.02% for cp = 0.05. The equation was applied to the seven gels for which Y > 0.99 (Table II), together with the results given in Table V. Fig. 5 depicts R,, vs. l/c according to eqn. 19. The linear correlation is surprisingly good in view of the fact that the exclusion radii were determined from both chromatography and electrophoresis and that four different gels are covered, viz. beads and compact gels from three sources, and agar particles. Moreover the polymer concentration of Sepharose was taken directly from the rather approximate w/v percentage given by the gel number. On the other hand, the effect of small differences in pH and temperature may be neglected. Also, an ionic strength variation from 0.02 to 0.08 mol/dm3 does not seem to affect the value of R,, as measured by electrophoresis22. As can be seen, R,, becomes zero at c N 0.17. Though beads of c = 0.2 have been described20, most types of agarose appa’rently lose their gel properties in this region. Note that exactly the same concentration limit is obtained by application of eqn. 18. In a recent investigation by two-dimensional electrophoresisz3 the linear R,, vs. l/c relationship is substantiated for c > 0.013, whereas R,, -+ 0 for c -+ 00. As also observed from another electrophoretic study24, at concentrations below 0.01 the slope decreases. Evidently, the geometrical exclusion model is no longer valid at higher dilutions, as the polymer chains become increasingly more mobile. In fact, chromatography can apparently not be carried out in media containing less than l-2% agarose.

EVALUATION

OF STRUCTURE

PARAMETERS.

II.

245

AGAROSE /

800 -

600-

400-

200-

Fig. 5. The relation between exclusion radius and gel concentration as given by eqn. 19. R,, determined by chromatography: l (Table II) and A (Table V). R,, determined by electrophoresis: H (Table V). R,, = 22.157/c - 132.86 (r = 0.9932), shown in the inset as R,,:c,where c is grams of polymer per cm3 of gel.

Polymer chain radii and the Ogston model

A fundamental result of eqn. 19 is the size of the chain radius ro. For the data illustrated in Fig. 5 the slope is 22.157 A g/cm” which for d = 1.695 g/cm3 gives r. = 4.36 A. From both physico-chemical measurements and the chemical structure, Hickson and Polsonz5 found values of r. N 5.8 A, whereas they obtained N 7.8 %, from electron microscopy. By X-ray diffraction, optical rotation and molecular building, Arnott et al. 26 arrived at a helix diameter of N 15 A (ref. 27) and a cross-section of the intrahelical cavity extending along the axis of > 4.5 A. It seems therefore reasonable to assign 2: 5 A to the agarose chain radius. This value is in sharp contradiction to r. calculated from Ogston’s model4 and given in Table II, namely (ro) N 24.6 f 7.5 A for seven agarose gels. The situation is identical for hyaluronic acid, since a chain radius of 8.7 A is obtained from eqn.

H.W.~LDMANN-M~~YER

246

21 (cf: Table II), as compared with 3.3 8, given by Ogston and Phelps29. According to Ogston KAV = exp[ -nL(R,

+ r0)2]

(20)

where L is the polymer chain length per cm3 of gel. For a chain of cylindrical shape, as assumed in his model, the mass/length ratio is c/L = &vi. We therefore replace the adjustable parameter TCLin eqn. 20 by c/d& and obtain the plotting expression (-ln

K*v)r’2 = (c/@‘~ + (c/~J”~ Re/rO

(211

A correlation of the logarithmic term vs. R, should therefore intersect the ordinate at (c/@” and the abscissa at -yo. Agarose radii of 25 A were explained by LaurentI on ground of strong hydration. However, NMR analysis2* has proven that the amount of irreversibly bound water does not exceed 3-8 mg/g agarose. Significant hydration would also lead to b values greatly differing from unity lg. Table II shows that this is not the case with agarose. Table II shows that agarose concentrations calculated by means of eqn. 21 may be up to four times higher than the real values and bear no relation to them. The same difference is found for cross-linked hyaluronic acid of c 0.0145 when using Ogston and Phelps’ d 1.45 (ref. 29). These discrepancies have not been observed before, since eqn. 21 appears in terms of nL throughout the literature. The differences with Sephadex concentrations calculated from solvent regain values are much smaller, although these values appear to be highly overestimated19. Also, plots according to eqn. 21 give in all cases linear correlations identical with those obtained from eqns. 7 and 8. On the other hand, the equation does not lead to a reasonable expression off(K*v) as log M. Since the radii of the only gel parameter that can be assessed are incorrect, it is concluded that the Ogston model does not apply, to gels. ’ CONCLUSIONS

It is evident that any expression involving partition coefficients will be strictly valid only in absence of adsorption. In principle, solute adsorption could be a function of the molecular size, affect the excluded or the fully included test substance alone, or be present as a combination thereof. Simulation shows that these effects produce anomalous intercepts in the GEM plots and can therefore be elucidated. The problem will be treated in connection with Sephadex gels19. As shown by the relation between gel concentration, exclusion and polymer chain radii, the clearly simplified modei from which the theory derives appears to be entirely self-consistent. Partition is a function of molecular size as expressed by the effective hydrodynamic radius R,, and of cavity shape and size. When R, is substituted by the frictional factorf = 6qoR, obtained from diffusion or sedimentation, all molecules, whatever their structure, still give the same correlation with the partition coefficient. For random coils’ R, can be replaced by R,/x = K([fjA4)1’3, as shown by Benoit et aL3*. When R, is expressed in terms of molecular weight the linearized K(M) functions make it possible not only to distinguish between different

EVALUATION

OF STRUCTURE

PARAMETERS.

II.

247

-In(l-

Fig. 6. The sigmoidal curve, known from empirical K “3. log M plots, as a result of replacing -In(l - K1’3) in eqn. 16 by K.

molecular structures, but also to determine effective bond lengths, radii of gyration, degrees of ideality and approximate axial ratios. The advantage of linear expressions that contain well-defined physical parameters seems quite considerable. Thus, if the exclusion radius for a given gel has been determined once for all, calibration only requires a minimum of samples and may, in principle, even be unnecessary when the molecular parameters A and x are known, as in the case of globular proteins. It is revealing that the empirical K vs. log M plots derive directly from the double logarithmic equations 15 and 16. This is illustrated in Fig. 6, where the wellknown sigmoidal curve, characterized by a long quasi-linear portion, results from a -log(L - P3) vs. K correlation. An analogous curve was obtained from -log(f - Kl/‘), required for cylindrical pores’. In view of the information obtained from the geometrical exclusion model for gels, it appears that separations achieved with non-porous glass spheres31s32 may be ascribed to a similar process. These and other chromatographic media are being investigated. ACKNOWLEDGEMENTS

I am indebted to Dr. F. Yssing Hansen for most valuable discussions and to Mrs. H. Birch for her expert technical assistance.

248

H. WALDMANN-MEYER

REFERENCES 1 2 3 4 5 6 7 8 9

10 11 12 13

14 15 16 17 18 19

20 21 22

23 24 25 26

H. Waldmann-Meyer, J. Chromatogr., 350 (1985) 1. E. F. Casassa, J. Polym. Sci., Part B, 5 (1967) 773. E. F. Casassa and Y. Tagami, Macromolecules, 2 (1969) 14. A. G. Ogston, Trans. Faraday Sec., 54 (1958) 1754. J. Porath, Pure Appl. Chem., 6 (1963) 233. P. G. Squire, Arch. Biochem. Biophys., 107 (1964) 471. L. M. Siegel and K. J. Monty, Biochim. Biophys. Acta, 112 (1966) 346. H. Waldmann-Meyer, in R. Epton (Editor), Chromatography of Synthetic and Biological Polymers, Vol. 1, Ellis Horwood, Chichester, 1978, Ch. 19, p. 289. P. Flodin, Dissertation, Uppsala University, 1962. F. L. Warburton, Trans. Faraday Sot., 42B (1946) 151. J. A. Barrie, in J. Crank and G. S. Park (Editors), Diffusion in Polymers, Academic Press, London and New York, 1968, Ch. 8. Y. Ito, J. Sot. High.Polym. Jpn., 17 (1960) 397. Y. Ito, J. Sot. High Polym. Jpn., 17 (1960) 489. K. Felgenhauer, Hoppe-Seyler’s Z. Physiol. Chem., 355 (1974) 1281. H. A. Sober (Editor), Handbook of Biochemistry, CRC Press, Cleveland, OH, 1968. T. C. Laurent, Biochim. Biophys. Acta, 136 (1967) 199. P. Andrews, Biochem. J., 91 (1964) 222. T. C. Laurent, Biochem. J., 93 (1964) 106. H. Waldmann-Meyer, in preparation. S. Hjerttn and K. 0. Eriksson, Anal. Biochem., 137 (1984) 313. K. A. Granath, J. Colloid Sci., 13 (1958) 308. H. Waldmann-Meyer and T. Jacobsen, in M. J. Dunn (Editor), Electrophoresis ‘86, VCH Verlagsges., Weinheim, 1986, p. 259. P. Serwer and S. J. Hayes, Anal. Biochem., 158 (1986) 72. P. G. Righetti, B. C. W. Brost and R. S. Snyder, J. Biochem. Biophys. Meth., 4 (1982) 347. T. G. L. Hickson and A. Polson, Biochim. Biophys. Acta, 165 (1968) 43. S. Arnott, A. Fulmer, W. E. Scott, I. C. M. Dea, R. Moorhouse and D. A. Rees, J. Mol. Biol., 90

(1974) 269. 27 A. Amsterdam, Z. Er-El and S. Shaltiel, Arch. Biochim. Biophys., 171 (1975) 673. 28 J. Andrasko, Biophys. J., 15 (1975) 1235. 29 A. G. Ogston and C. F. Phelps, Biochem. J., 78 (1960) 827. 30 H. BBnoit, Z. Grubisic, P. Rempp, D. Decker and J. G. Zilliox, J. China. Phys., 63 (1966) 1507. 31 K. 0. Pedersen, Arch. Biochem. Biophys., Suppl. 1 (1962) 157. 32 H. Small, J. Colloid Interface Sci., 48 (1974) 147.