28 The Charge Fraction of Ionic Polysaccharides PAUL ANDER
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Department of Chemistry, Seton Hall University, South Orange, N J 07079
Equations are d e r i v e d from the Manning theory o f p o l y e l e c t r o l y t e s o l u t i o n s that permit the c a l c u l a t i o n o f the charge f r a c t i o n of i o n i c polysaccharides using experimental counterion and coion d i f f u s i o n c o e f f i c i e n t s . Results are reported f o r the sodium s a l t s o f h e p a r i n , dextran s u l f a t e , a l g i n a t e and the c a l cium s a l t o f h e p a r i n . The p r o p e r t i e s of b i o l o g i c a l p o l y e l e c t r o l y t e s i n s o l u t i o n depend on the charge f r a c t i o n o f the p o l y e l e c t r o l y t e , i.e., the f r a c t i o n o f s t o i c h i o m e t r i c charge on the p o l y e l e c t r o l y t e uncompensated by bound counterions. The Manning theory of p o l y e l e c t r o l y t e s o l u t i o n s emphasizes that counterion condensation onto the p o l y e l e c t r o l y t e occurs for p o l y e l e c t r o l y t e s with ξ > ξ , where ξ = e / e k T b , b i s the average a x i a l distance between charges on the p o l y e l e c t r o l y t e and e / e k T i s the Bjerrum l e n g t h , with 1-4 ξc = | z | , z being the charge on the counterion (subscript 1) More r e c e n t l y , Manning has extended h i s theory to include those counterions that are "territorially bound" or trapped i n the do main o f the p o l y e l e c t r o l y t e , but are somewhat free t o move along the p o l y i o n . 5 Counterions that are n e i t h e r condensed site-bound nor t e r r i t o r i a l l y - b o u n d are i n the ion atmosphere, along with the c o i o n s , if simple s a l t i s added. The i d e a of counterion conden s a t i o n has r e c e n t l y r e c e i v e d s u b s t a n t i a t i o n from t h e o r e t i c a l in v e s t i g a t i o n s o f the Poisson-Boltzmann equation f o r p o l y e l e c t r o lyte solutions. c
2
2
-1
1
1
6,7
Since i o n i c polysaccharides are s t i f f e r than most s y n t h e t i c p o l y e l e c t r o l y t e s , they were thought t o be appropriate experiment a l models to t e s t t h e o r i e s based on c y l i n d r i c a l symmetry. System a t i c determinations of counterion and coion d i f f u s i o n c o e f f i c i e n t s and a c t i v i t y c o e f f i c i e n t s have been made i n the presence o f ionic p o l y s a c c h a r i d e s ; these measurements p e r t a i n to the i n t e r a c t i o n o f small ions i n the i o n i c atmosphere with the p o l y e l e c t r o l y t e . Here, r e s u l t s are presented for estimating the f r a c t i o n of ions d i s s o c i a t e d from the i o n i c polysaccharide i n simple s a l t s s o l u t i o n s ,
0097-6156/81/0150-0405$05.00/0 © 1981 American Chemical Society Brant; Solution Properties of Polysaccharides ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
406
SOLUTION
PROPERTIES
OF
POLYSACCHARIDES
i n c l u d i n g t h e sodium s a l t s o f h e p a r i n , d e x t r a n s u l f a t e and a l g i n and t h e c a l c i u m s a l t o f h e p a r i n . The r e s u l t s f o r h e p a r i n a n d d e x t r a n s u l f a t e a r e compared w i t h t h o s e f o r s o d i u m p o l y s t y r e n e s u l f o n a t e , whose c h a r g e d e n s i t i e s a r e c o m p a r a b l e . Experimental: The d e t e r m i n a t i o n o f s m a l l i o n t r a c e r d i f f u s i o n c o e f f i c i e n t s a n d t h e i o n i c p o l y s a c c h a r i d e s e m p l o y e d have b e e n d i s c u s s e d e l s e w h e r e . - ' - ί?
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5
R e s u l t s and D i s c u s s i o n : Manning*s f o r m u l a t i o n t r e a t s a l l uncondensed c o u n t e r i o n s and a l l c o i o n s i n t h e Debye-Huckel approxima tion. I n o r d e r t o be c o r r e l a t e d t o Manning's t h e o r y , t h e e x p e r i m e n t a l d a t a ought t o be e x t r a p o l a t e d t o zero c o n c e n t r a t i o n o f a l l i o n i c s p e c i e s because t h e t h e o r y y i e l d s l i m i t i n g laws. I f one d e n o t e s Zp a s t h e v a l e n c e o f a c h a r g e s i t e o n t h e p o l y i o n , (z^)p as t h e v a l e n c e o f t h e c o u n t e r i o n o r i g i n a l l y p r e s e n t w i t h t h e p o l y i o n p r i o r t o t h e a d d i t i o n o f s a l t , and ( z i ) and ( z 2 ) the v a l e n c e s o f t h e c o u n t e r i o n s and c o i o n s p r o v i d e d b y t h e added s a l t , r e s p e c t i v e l y , t h e n i n t h e common c o u n t e r i o n c a s e , where t h e added c o u n t e r i o n a n d c o u n t e r i o n o r i g i n a l l y p r e s e n t a r e t h e same, t h e M a n n i n g t h e o r y yieldsI»2>i a
s
f . [ l - ζΑ(ξ
D /D° = 1
1
1
1
where f . i s a c o n d e n s a t i o n counterions i t i s f. ι
λ
(ξ ξ χ
=
s
s
_1
,ξ Χ)/.3]
(1) c term which i s u n i t y f o r coions and f o r
+ ΐ)/(χ + 1 ) f o r ξ > ξ
(2)
with 2
Α(ξ,Χ) = Σ Σ [π(ξζ m n=~ m +n =0
+ η ) + |ζ_ |
Ρ
1
Ρ
r
+ ( | ζ. | 1 s
+
)Χ
_ 1
]"
2
2 s (3)
w i t h Χ = η /η , t h e r a t i o o f t h e n o r m a l i t y o f p o l y e l e c t r o l y t e t o the m o l a r i t y simple s a l t . I t w i l l now b e shown how c o u n t e r i o n a n d c o i o n d i f f u s i o n r e s u l t s can be used t o e s t i m a t e t h e degree o f d i s s o c i a t i o n o f a p o l y e l e c t r o l y t e i f t h e e x p e r i m e n t a l d i f f u s i o n c o e f f i c i e n t s approx imate those p r e d i c t e d from t h e Manning t h e o r y . The M a n n i n g t h e o r y gives ( D
+
D
Na / Sa
+ )
=
"Α(ΐ,Λ)/3] _1
( D - / D ° _ ) = 1 - Α(ΐ,ξ Χ)/3 C 1
C 1
and
Brant; Solution Properties of Polysaccharides ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
M (5)
28.
ANDER
Charge
Fraction
of Ionic
407
Polysaccharides
2/D° 2-) = 1 - kAUXh)/3
(D
(6)
0
k
h
where f*™ i s t h e f r a c t i o n o f s o d i u m i o n s i n t h e s o l u t i o n w h i c h a r e c o n d e n s e d . The f r a c t i o n o f s o d i u m i o n s w h i c h a r e c o n d e n s e d is f ^ = 1 r e a r r a n g i n g eqs h 5 and 6 , +
f
a
+
a
N
W
-
a
λ
n
d
9
+
- (D
c l
_/
D 8 i
_)
™
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and
f! . , ^Ha+'O Na+ = 13
+
( D
so
2 u
-
/ D
(8) sof
where i t i s u n d e r s t o o d t h a t b o t h d i f f u s i o n r a t i o s a p p e a r i n g a r e t o b e d e t e r m i n e d a t t h e same X v a l u e . A s r n e q u i v a l e n t s o f Na+ o f t h e t o t a l ( n ^ + n ) e q u i v a l e n t s o f Na+ presumed b o u n d , one h a s g
f
Na
+
=
r
n
p
/
(
n
p
+
n
s> =
r
X
/
(
X
+
(
D
9
)
C o u n t e r i o n c o n d e n s a t i o n o n t o p o l y e l e c t r o l y t e s h a s b e e n opera t i o n a l l y d e f i n e d as a s s o c i a t i o n such t h a t t h e t o t a l f r a c t i o n o f p o l y i o n s i t e s compensated f o r w i t h c o u n t e r i o n r e m a i n s i n v a r i a n t over a wide range o f X v a l u e s . I f t h e i n t e r a c t i o n o f Na+ w i t h p o l y e l e c t r o l y t e s i s p r o g e r l y d e s c r i b e d as " c o u n t e r i o n condensa t i o n " , then a p l o t o f f ^ ( X + l ) v s . X should be l i n e a r w i t h s l o p e r . Note t h a t t h i s w o u l d i n d i c a t e t h a t t h e f r a c t i o n o f s o d i um i o n s d i s s o c i a t e d f r o m t h e p o l y e l e c t r o l y t e i s c o n s t a n t a n d i n dependent o f t h e c o n c e n t r a t i o n s o f p o l y e l e c t r o l y t e a n d o f s i m p l e salt. I t i s c l e a r t h a t r_ i s t h e f r a c t i o n o f t h e c o n d e n s e d o r bound Na+ i o n s w h i c h a r e o r i g i n a l l y on t h e p o l y e l e c t r o l y t e a n d ( l - r ) i s t h e charge f r a c t i o n o f t h e p o l y e l e c t r o l y t e . The s o d i u m i o n d i f f u s i o n r a t i o s / aqueous Na^SO^ s o l u t i o n s a r e compared t o t h e r e s u l t s p r e d i from t h e Manning theory. Since t h e molecular weight o f heparin i s r a t h e r low ^ ( a b o u t 1 2 , 0 0 0 ) , i t s d i f f u s i o n c o n s t a n t was a p p r o x i m a t e d f r o m C a i o n d i f f u s i o n r e s u l t s ^ , where t h e h e p a r i n d i f f u s i o n c o e f f i c i e n t i s t h a t o f D +2 f o r X>5 b e c a u s e t h e added t r a c e r c a l c i u m i o n i s condensed onto t h e h e p a r i n b y exchanging w i t h condensed sodium i o n . Note i n F i g u r e 1 t h a t b e t t e r a c c o r d w i t h t h e o r y i s o b t a i n e d when t h e s o l i d M a n n i n g l i n e i s r e c t i f i e d f o r t h e h e p a r i n d i f f u s i o n F i g u r e 2 shows t h a t t h e M a n n i n g t h e o r y ( s o l i d l i n e ) p r e d i c t s t h e c h l o r i d e i o n d i f f u s i o n c o e f f i c i e n t s over t h e whole c o n c e n t r a t i o n r a n g e o f h e p a r i n a n d o f s o d i u m chloride.§ I t h a s b e e n s h o w n — t h a t t h e counterion d i f f u s i o n c o e f f i c i e n t i s independent o f t h e a +
D
D
N a +
i
N
n
+
+
Brant; Solution Properties of Polysaccharides ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
408
SOLUTION
PROPERTIES
OF
POLYSACCHARIDES
1.0
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Figure 1. Sodium ion diffusion ratio in aqueous sodium heparin solutions con taining sodium sulfate. The theoretical solid line is rectified to account for the diffusion of heparin to give the broken line (S).
4
8
12
16
20
X
1.0 - q p —
0.8\-
0.6
Dc.-
D°
c
• Ο 3 Ο
0.2
0.0100 N NaCl 0.0050 0.0010 0.0005
Figure 2. Chloride ion diffusion ratio in aqueous sodium heparin solutions con taining sodium chloride. The solid line is predicted from the Manning theory (&).
10
Table I. A Comparison of the Experimental Charge Fraction with Those Predicted from the Manning Theory for Several Ionic Polysaccharides Charge Fraction Slope Sodium Heparin Sodium Dextran Sulfate Sodium Polystyrenesulfonate Sodium Alginate Sodium D N A Calcium Heparin Calcium D N A 1 0
1 0
3.0 2.9 2.6 1.4 4.2 3.0 4.2
0.43 0.65 0.66 0.36 0.61 0.41 0.43
± 0.01 0.02 ± 0.01 ± 0.02 ± 0.05 ± 0.03 ± 0.04 db
d-r) 0.43 0.35 0.34 0.64 0.39 0.18 0.14
Brant; Solution Properties of Polysaccharides ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
0.33 0.34 0.38 0.70 0.24 0.17 0.12
ANDER
28.
Charge
Fraction
of Ionic
409
Polysaccharides
n a t u r e o f the c o i o n . U s i n g t h e data f o r DN /DN and D Q -2/D -2 f o r s o d i u m h e p a r i n , fN was c a l c u l a t e d f r o m e q 8 andUa p l o t U w a s made u s i n g e q 9, w h i c f i i s shown i n F i g u r e 3. The l i n e i n F i g u r e 3 was drawn v i s u a l l y t h r o u g h t h e o r i g i n and t h e s l o p e r_ g i v e n i n T a b l e 1, was computed f r o m t h e p o i n t s o n l y . The e x p e r i m e n t a l c h a r g e f r a c t i o n ( l - r ) f o r s o d i u m h e p a r i n o f 0.U3 i s i n good agreement w i t h t h e t h e o r e t i c a l c h a r g e f r a c t i o n ( z - N ) ' o f 0.33 c o n s i d e r i n g t h e c o r r e c t i o n s employed f o r t h e p o l y e l e c t r o l y t e diffusion. From t h e d a t a f o r s o d i u m d e x t r a n s u l f a t e i n 0.0005N Na2SO4 i n r e f e r e n c e 8 and u s i n g eqs 8 and 9, f c ( l + X) was p l o t t e d a g a i n s t X· The r e s u l t i n g l i n e a r i t y gave a s l o p e o f 0.65 ± 0.02, which i s the f r a c t i o n o f s i t e s on sodium d e x t r a n s u l f a t e h a v i n g bound s o d i u m i o n s . The c h a r g e f r a c t i o n on 0.35 i s i n e x c e l l e n t agreement w i t h t h e t h e o r e t i c a l v a l u e o f 0.3*+, a s i s l i s t e d i n T a b l e 1. N o t e t h a t t h e l i n e a r i t y shown i n F i g u r e s 1 and 2 i n d i c a t e t h a t t h e c h a r g e f r a c t i o n o f e a c h p o l y s a c c h a r i d e i s a constant v a l u e and i n d e p e n d e n t o f t h e i o n i c s t r e n g t h o f t h e s o l u t i o n . I t was o f i n t e r e s t t o compare t h e s e r e s u l t s w i t h t h o s e o b t a i n e d f o r a s y n t h e t i c p o l y e l e c t r o l y t e o f c o m p a r a b l e c h a r g e d e n s i t y . The r e p o r t e d d i f f u s i o n r e s u l t s U f o r sodium p o l y s t y r e n e s u l f o n a t e o f ξ = 2.6l i n 0.001N N a C l and eqs 7 and 9 gave t h e l i n e a r p l o t o f f°(l + X ) v s . X shown i n F i g u r e 3. The s l o p e o f 0.66 ± 0.01 r e s u l t s i n a c h a r g e f r a c t i o n o f 0.3*+, w h i c h compares e x c e l l e n t l y w i t h t h e t h e o r e t i c a l v a l u e o f 0.38. Then t h e c h a r g e f r a c t i o n o f p o l y e l e c t r o l y t e s as c a l c u l a t e d from t r a c e r d i f f u s i o n r e s u l t s and the Manning t h e o r y appear t o be c o n s t a n t , independent o f i o n i c s t r e n g t h and depend o n l y on t h e l i n e a r c h a r g e d e n s i t y p a r a m e t e r ξ. A s i m i l a r t r e a t m e n t was t r i e d u s i n g t h e d a t a f o r s o d i u m a l g i n a t e , an i o n i c p o l y s a c c h a r i d e o f l o w e r c h a r g e d e n s i t y (ξ = 1.U3) a n d whose i o n i c m o i t i e s a r e c a r b o x y l g r o u p s compared t o a m i x t u r e o f s u l f a t e and c a r b o x y l g r o u p s o n h e p a r i n and s u l f a t e g r o u p s o n d e x t r a n s u l f a t e . U s i n g the d i f f u s i o n data i n r e f e r e n c e 9 f o r sodium a l g i n a t e i n 0.0005N N a C l f o r X
