The Charge Fraction of Ionic Polysaccharides - ACS Symposium

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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

+



_ 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.-



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