Effect of Solute Shape or Conformation in Size Exclusion

10 Effect of Solute Shape or Conformation in Size Exclusion Chromatography Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 20, 2017 | http://pubs.acs.org Publication Date: November 26, 1980 | doi: 10.1021/bk-1980-0138.ch010

W. W. Y A U and D. D. B L Y Central Research and Development Department, Experimental Station, Ε. I. Du Pont de Nemours & Company, Wilmington, D E 19898

ABSTRACT The dependence on molecular weight of the size of macromolecules or other l a r g e - p a r t i c l e solutes in s o l u t i o n varies as a function of the shape of the solute molecule or p a r t i c l e . Solute conformation or shape, therefore, affects the slope of the c a l i b r a t i o n curve and range of molecular weight separations a v a i l a b l e i n s i z e - e x c l u s i o n chromatography (SEC). Various theories which have been derived for d i f f e r e n t solute conformations are u n i f i e d i n t h i s work by ex pressing the s i z e of the solute i n terms of a reduced s i z e parameter, R , the radius of g y r a t i o n . I t i s pre dicted according to the u n i f i e d theory, that a SEC c o l umn containing a single pore-size packing will have about two decades of MW spearation range for random -coil polymers, three decades of MW range for s o l i d sphere solutes and only one decade of MW range for rod - l i k e p a r t i c l e s or molecules. This indicates that anal y s i s of the slope of a SEC-MW c a l i b r a t i o n curve can be used to study the conformation of solute macromolecules. g

0-8412-0586-8/80/47-138-197$05.00/0 © 1980 American Chemical Society Provder; Size Exclusion Chromatography (GPC) ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 20, 2017 | http://pubs.acs.org Publication Date: November 26, 1980 | doi: 10.1021/bk-1980-0138.ch010

198

SIZE EXCLUSION CHROMATOGRAPHY (GPC)

As the scope of s i z e exclusion chromatography (SEC, or gel permeation chromatography, GPC) expands, for example, into p a r t i c l e measurements and biopolymer, gel filtration chromatographic (GFC) separ a t i o n s , increased needs for i n t e r p r e t a t i o n of solute shape are developing (e.g. hard-sphere for p a r t i c l e s and c o l l o i d s , r i g i d - r o d for biopolymers, and random-coil for f l e x i b l e polymers). The SEC retention chara c t e r i s t i c s for solutes with these d i f f e r e n t conformations have been studied separately i n the theoretical treatments by Casassa, Giddings, Ackers and others.(1-4) However, these are b a s i c a l l y mechanistic theories, developed to provide fundamental understanding of SEC retention i n terms of solute s i z e s ; there remains a need to r e l a t e these theories to p r a c t i c a l SEC calibration procedures which are based on solute molecular weight (MW). In t h i s communication are summarized some of the important p r a c t i c a l insights which theory provides about SEC-MW c a l i b r a t i o n ; predictions also have been derived to show the e f f e c t of solute conformation on the slope and molecular weight range of SEC c a l i b r a tion. For s i m p l i c i t y , the radius of gyration, R , was chosen as a common reduced solute s i z e parameter to compare various theories concerned with d i f f e r e n t solute conformations. A more rigorous treatment would require the use of molecular projections as the common SEC s i z e parameter.(5) However, s i g n i f i c a n t insights are still gained through the s i m p l i f i e d approach. These provide a t i e between theory and experimentally observed solute shape e f f e c t s , and lead to a better understanding of SEC c a l i b r a t i o n p r a c t i c e s . g

Background E x i s t i n g SEC r e t e n t i o n t h e o r i e s h a v e b e e n independently developed f o r each o f the m o l e c u l a r s h a p e m o d e l s shown i n F i g u r e 1. The deep h o l l o w c y c l i n d r i c a l p o r e i n t h e f i g u r e ( A , B, a n d C) i l l u s t r a t e s t h e SEC e x c l u s i o n e f f e c t o n t h r e e t y p e s o f s o l u t e molecules, hard-sphere, r i g i d - r o d , and random-coil, r e spectively. The i n d i v i d u a l t h e o r i e s a n d t h e i r bases o f c o m m o n a l i t y a r e now r e v i e w e d b r i e f l y . In the o r i g i n a l works, t h e o r e t i c a l models w e r e d e v e l o p e d t o e x p l a i n t h e SEC c a l i b r a t i o n c u r v e

Provder; Size Exclusion Chromatography (GPC) ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


10.

YAU AND BLY

Effect of

S o l u t e

S h a p e

or

Conformation

199

Journal of Physical Chemistry

Figure 1. Exclusion effect in cylindrical cavities (I) ((A) hard sphere of radius r; (B) thin rod of length L , in two orientations in the plane of the cross section; (C) random-flight chain with one end at point 0 showing allowed conformation ( ) and forbidden conformation ( ))


200

SIZE EXCLUSION CHROMATOGRAPHY

(GPC)


shape as a function of s i z e and shape of the solute and column packing pore. The t h e o r e t i c a l considerations i n t h i s paper are based p a r t i c u l a r l y on the c y l i n d r i c a l pore shape model f o r s i m p l i c i t y i n argument. Actual SEC packings generally have pore s t r u c tures of a random or i r r e g u l a r shape, but i t i s expected that the r e s u l t s of t h i s study as regards solute shape e f f e c t s should be generally applicable using the c y l i n d r i c a l pore shape model. In fundamental SEC studies retention i s often described i n terms of a d i s t r i b u t i o n c o e f f i c i e n t . The t h e o r e t i c a l d i s t r i b u t i o n c o e f f i c i e n t K i s defined as the r a t i o of solute concentration inside and outside of the packing pores under s i z e exclusion conditions. The experimental d i s t r i b u t i o n c o e f f i c i e n t Kg , as defined i n Equation 1, i s a measurable quantity that can be used to check the theory. E

K

=

SEC

i V W V

(1)

where V i s the retention volume f o r any p a r t i c u l a r solute and V Q and are the respective i n t e r s t i t i a l volume and the t o t a l permeation volume of the packed SEC columns tëO R

The exclusion e f f e c t of hard-spheres i s i l l u s t r a t e d i n Figure ΙΑ., which shows a s p h e r i c a l solute of radius r inside an i n f i n i t e l y deep c y l i n d r i c a l cav i t y of radius a . Here the exclusion process can be described by straightforward geometrical considera tions , namely, solute exclusion from the walls of the c a v i t y . Furthermore, i t can be shown that c

K

E

=

(1-

(2)

c The exclusion e f f e c t of a r i g i d - r o d i n the same c y l i n d r i c a l pore i s shown i n Figure IB., where the length of the rod i s L,. Quantifying the ex c l u s i o n process here i s much more complicated than i n the hard sphere case. Exclusion of the rod depends on the rod o r i e n t a t i o n i n three dimensions and s t a t i s t i c a l methods must be used f o r the evaluation. For r i g i d - r o d s i t has been shown that K i s described bycJ F


10. Kj,

Effect of Solute Shape or Conformation

Y A U AND B L Y

=

1 - ~ [ ( 1 - β ) Ε ( | , β ) - (l-β ) F ( | , 3 ) ] 2

where β = L^/2a

2

, w i t h β£ 1, a n d E , F a r e

201

(3) elliptical


integrals. F i g u r e 1C. i l l u s t r a t e s two c o n f o r m a t i o n s o f a f l e x i b l e polymer c h a i n w i t h one end f i x e d i n s i d e the c a v i t y . Even w i t h one end f i x e d , t h e c h a i n s t i l l c a n assume a l a r g e number o f c o n f o r m a t i o n s . The presence o f t h e c a v i t y w a l l , however, does e x c l u d e some c o n f o r m a t i o n s , f o r e x a m p l e , t h e d a s h e d o n e s h o w n in thesketch. This restraint o f conformational free dom c a u s e s a d e c r e a s e i n b o t h t h e e n t r o p y a n d t h e s o l ute concentration i n s i d e t h e c a v i t y . F o r t h i s case i t h a s b e e n shown t h a t :

K

E

=

4

\ m=l

B " expt-(^—) 1 2

m

(4)

c

where t h e n u m e r i c a l c o n s t a n t β i s the m-th r o o t o f the B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f order zero, a n d Rg i s t h e s o l u t e r a d i u s o f g y r a t i o n . E a c h o f t h e a b o v e e x i s t i n g SEC r e t e n t i o n t h e o r i e s i s u n i q u e l y r e l a t e d t o o n l y one p a r t i c u l a r solute-shape model. Because o f the d i f f e r e n c e s i n t h e b a s i c s o l u t e - s i z e p a r a m e t e r , r , L]_, a n d Rg i n t h e i n d i v i d u a l c a s e s , t h et h e o r y o f each s o l u t e - s h a p e model stands alone. S i n c e , under these c o n d i t i o n s the d i f f e r e n t t h e o r i e s are n o td i r e c t l y comparable, i t i s d i f f i c u l t t o make i n t e g r a t e d o b s e r v a t i o n s o r t o u n d e r stand various p r a c t i c a l i m p l i c a t i o n s i nthese t h e o r i e s . To g a i n m o r e i n s i g h t s i n t o SEC c a l i b r a t i o n , a common s i z e p a r a m e t e r i s n e e d e d t o u n i f y t h e SEC t h e o r i e s . U n i f i c a t i o n o f Theory S i n c e t h e p a r a m e t e r Rg i s known t o b e a b a s i c SEC s i z e p a r a m e t e r f o r r a n d o m - c o i l t y p e m o l e c u l e s (Equation 4 ) , and s i n c e i t i s a l s o a w e l l - d e f i n e d s t a t i s t i c a l average s i z e parameter, a p p l i c a b l e t o s o l u t e s o f any shape i n c l u d i n g t h e sphere and r o d l i k e m o l e c u l e s , Rg h a s b e e n c h o s e n i n t h i s w o r k t o s e r v e a s a common, r e d u c e d , s o l u t e - s i z e p a r a m e t e r f o r d e s c r i b i n g t h e t h e o r y o f SEC s e p a r a t i o n . B y d e f i n i t i o n :



202

SIZE EXCLUSION C H R O M A T O G R A P H Y

(GPC)

w h e r e Ν i s t h e number o f m a s s e l e m e n t s i n a s t r u c t u r e a n d X^ i s t h e dd ii s t ca n c e f r o m t h e i - t h m a s s e l e m e n t t o t h e c e n t e r o f mma s s . , F o r s o l i d s p h e r e s and r i g i d r o d s , E q u a t i o n (5) g i v e s : R

g

R

1 / 2

(sphere)

=

(3/5)

r

(rod)

=

(1/12) L . .

(6)

1 / 2

(7)

The c o m p o s i t e p l o t s o f t h e t h e o r e t i c a l K c u r v e s ( E q u a t i o n s 2, 3 a n d 4) f o r t h e t h r e e s o l u t e s h a p e d ) i n t e r m s o f t h e common r e d u c e d s i z e p a r a m e t e r Rq ( E q u a t i o n s 5, 6, a n d 7) a r e s h o w n i n F i g u r e 2. This p l o t shows t h a t on an R b a s i s a l l s o l u t e s o f d i f f e r e n t c o n f o r m a t i o n s s h o u l d b e h a v e v e r y s i m i l a r l y i n a n SEC experiment. (This r e s u l t i s c o n s i s t e n t w i t h the u n i v e r s a l SEC c a l i b r a t i o n c o n c e p t .(S) E

q

The m o l e c u l a r w e i g h t M o f a s o l i d - s p h e r e i s p r o p o r t i o n a l t o t h e v o l u m e o f t h e s p h e r e , i . e . , M 1/2, b u t f o r a n i d e a l i z e d r a n d o m - f l i g h t p o l y m e r , α = 1/2.



10.

YAU AND BLY

Effect of Solute Shape or

Conformation

203

The values of the exponent f o r M i n Equa tions 8, 9, and 10 determine the rate at which the s i z e of the respective macromolecular shapes change with the molecular weight. For example, a tenfold i n crease i n molecular weight roughly corresponds to a 10 χ L p 2 χ r, or 3 χ R change i n molecular s i z e for the three d i f f e r e n t solute conformations under study. I t i s expected, therefore, that the l i n e a r portion of the SEC-MW c a l i b r a t i o n curve (see Figure 3) w i l l be steepest f o r the sphere-like solutes ( i . e . the l e a s t e f f e c t i v e separation or resolution) and the c a l i b r a t i o n slope f o r spheres w i l l be 3/2 of that f o r c o i l e d molecules. The l i n e a r portion of the curve f o r the r o d l i k e solutes i s expected to have the shallowest slope ( i . e . most r e s o l u t i o n ) , only h a l f that of c o i l e d solutes. As expected and i l l u s t r a t e d here i n Figure 3, an SEC column containing a single-pore-size packing should have a MW separation range of about one decade for r o d l i k e molecules and three decades f o r spheres, compared to the usual, approximately two-decade MW separation range f o r random-coil solutes. Experimental Support f o r the U n i f i e d Theory Published data i n support of the above ob servations are shown i n Figures 4 and 5. Figure 4 i s the c a l i b r a t i o n curve f o r the SEC analysis of s i l i c a sol p a r t i c l e s as a function of s i z e XU This curve i l l u s t r a t e s the hard-sphere case where about one decade of p a r t i c l e diameter separation i s observed; note t h i s i s equivalent to about 3 decades of MW. The c a l i b r a t i o n curve slopes f o r polybenzyl-L-Glutamate (PBLG) and polystyrene (PS) are compared i n Figure 5. The l e s s e r slope f o r the polybenzyl-L-Glutamate i s con s i s t e n t with the expected trend discussed above .(2) This d i f f e r e n c e i n c a l i b r a t i o n curve slope i s expected because of the molecular conformation difference be tween PBLG (rod) and PS ( c o i l ) . Conclusions A s i m p l i f i e d analysis of the e f f e c t of p a r t i c l e shape or molecular conformation on SEC c a l i b r a t i o n has l e d to the p r e d i c t i o n that the more open structure of r i g i d rod shaped solutes gives a r e l a t i v e l y f l a t SEC-MW c a l i b r a t i o n curve. As the solute conformation becomes more compact (random-coil to solid-sphere), the SEC-MW c a l i b r a t i o n curve becomes increasingly steep


204


SIZE E X C L U S I O N C H R O M A T O G R A P H Y

7

SPHERE

6

4

1

1

Ο Κ

Figure 3.

SEC

Theoretical SEC calibration curves for various shapes


(GPC)

10.

YAU AND BLY

Effect of Solute Shape or Conformation

205

100 cr

10


1

SILICA SOL SIZE, nm

5.0

5.2

5.4

5.6

5.8

6.0

V , ml R

Journal of Chromatography

Figure 4. SEC calibration curve for silica sol separation (hard sphere particles, single pore size column) (1). Column: PSM-1500 (8.9 μ/η), 30 χ 0.78 cm; mobile phase: 0.1M Na HPO -NaH PO pH 8.0 Ë

Figure 5.

k

t

llt

SEC calibration curves: random-coil vs. rigid-rod (&) (SEC column set of several pore sizes, Ν,Ν-dimethylacetamide solvent at 80°C)


SIZE EXCLUSION CHROMATOGRAPHY (GPC)

206

a n d c o v e r s a l a r g e r MW s e p a r a t i o n r a n g e . I t should now b e p o s s i b l e t o u s e t h e s l o p e o f t h e SEC-MW c a l i b r a t i o n curves, generated from s i n g l e pore s i z e c o l umns, t o s t u d y t h e c o n f o r m a t i o n o f t h e s o l u t e m o l e c u l e s . Acknowledgment


We t h a n k J . J . K i r k l a n d f o r h i s i n t e r e s t a n d h e l p f u l d i s c u s s i o n s on t h i s work.

References 1.

Ε. F. Cassassa, J. Phys. Chem., 75, 3929 (1971).

2.

J . C. Giddings, Ε. Kucera, C. P. Russell, and M. Ν. Myers, J. Phys. Chem., 72, 4397 (1968).

3.

G. K. Ackers, Biochemistry, 3, 723 (1964).

4.

W. W. Yau and J. J. Kirkland, and D. D. Bly, "Modern Size-Exclusion Liquid Chromatography," Wiley, New York, 1979, Chapters 2 and 9.

5.

E . F . Casassa, Macromplecules, 9, 182 (1976).

6.

C. Tanford, "Physical Chemistry of Macromolecules," Wiley, New York, 1961, Chapters 3 and 5.

7. J. J. Kirkland, J. Chromatography, 185, 273 (1979). 8.

J . V. Dawkins and M. Hemming, Polymer, 16, 554 (1975).

RECEIVED May 20, 1980.


Effect of Solute Shape or Conformation in Size Exclusion

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