Ordered Fluids and Liquid Crystals - American Chemical Society

17 Mechanism of Dielectric Relaxation of Deoxyribonucleic A c i d SHIRO TAKASHIMA

Downloaded by COLUMBIA UNIV on March 15, 2013 | http://pubs.acs.org Publication Date: January 1, 1967 | doi: 10.1021/ba-1967-0063.ch017

Electromedical Division, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, P a .

The dielectric dispersion of DNA solutions was measured with various samples. The dielectric increment and the relaxation time of helical DNA are proportional to the square of the length of the molecule, but values for coil DNA are distinctly smaller than for helical DNA. The rotary diffusion constant is measured simultaneously with the dielectric measurement. The agreement of both relaxation times is fair in a region of low molecular weight, but the disparity becomes pronounced when DNA is larger. Theories on the mechanism of ionic electric polarization are reviewed. Currently, counter ion polarization for a cylindrical model seems to account most reasonably for the dielectric relaxation of DNA.

T he dielectric constant of d e o x y r i b o n u c l e i c a c i d ( D N A ) s o l u t i o n was first measured b y A l l g e n (1) a n d J u n g n e r , J u n g n e r a n d A l l g e n (9), w h o r e p o r t e d t h a t D N A e x h i b i t e d a n anomalous dispersion i n t h e 100-kc. region w i t h a large dielectric i n c r e m e n t . T h e y c a l c u l a t e d t h e dipole m o m e n t a n d t h e dielectric r e l a x a t i o n t i m e a n d observed a dipole m o m e n t of a p p r o x i m a t e l y 10 t o 1 0 D e b y e u n i t s a n d a r e l a x a t i o n t i m e of a b o u t 10~~ second. T h e y a t t e m p t e d to a c c o u n t for t h e dielectric r e l a x a t i o n of D N A i n t e r m s of t h e D e b y e t h e o r y . r

l

3

4

7

H o w e v e r , t h e r e l a x a t i o n t i m e t h e y observed was w i d e l y different f r o m the r e l a x a t i o n t i m e of r o t a r y d i f f u s i o n ( r . ) of a b o u t 10~ second o b s e r v e d b y E d s a l l (4). I f dielectric p o l a r i z a t i o n is caused b y t h e o r i e n t a t i o n of a p e r m a n e n t d i p o l e , t h e r e l a x a t i o n t i m e m u s t be s i m i l a r to t h a t for r o t a r y diffusion. T h e r o t a r y d i f f u s i o n of elongated particles u s u a l l y represents t h e r o t a r y m o t i o n of t h e w h o l e b o d y a r o u n d t h e short axis. I f D N A has a p e r m a n e n t dipole i n t h e transverse d i r e c t i o n , t h e w h o l e molecule w o u l d rotate a r o u n d t h e m a j o r axis a n d t h e dielectric r e l a x a t i o n t i m e w o u l d n o t necessarily be t h e same as t h a t of r o t a r y diffusion. T h u s t h e y concluded t h a t the difference between t h e r o t a r y a n d dielectric r e l a x a t i o n t i m e s o b r o

3

232 In Ordered Fluids and Liquid Crystals; Porter, R., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

17.

TAKASHiMA

Dielectric

Relaxation

of

DNA

233


served was caused b y t h e fact t h a t D N A h a d a p e r m a n e n t d i p o l e across t h e m a j o r axis. L a t e r , J e r r a r d a n d S i m m o n s (8) measured the dielectric dispersion of D N A a n d also observed a dispersion i n t h e 30-kc. region. T h e frequency region used b y A l l g e n et al. a n d J e r r a r d et al. was r e s t r i c t e d t o h i g h f r e quencies. M o r e o v e r , the D N A samples used b y A l l g e n et al. seem t o h a v e l o w m o l e c u l a r w e i g h t . T h e dielectric properties depend r a t h e r s t r o n g l y o n t h e degree of p o l y m e r i z a t i o n , a n d t h e results o b t a i n e d w i t h a D N A of l o w m o l e c u l a r w e i g h t cannot be generalized. R e c e n t l y , T a k a s h i m a (27) measured t h e dielectric dispersion of D N A i n t h e l o w frequency region w i t h D N A samples of v a r y i n g m o l e c u l a r weights. H e extended the measurements d o w n t o 50 c.p.s. a n d observed a dielectric dispersion of D N A i n a m u c h lower frequency region t h a n those observed b y p r e v i o u s w o r k e r s a n d a dielectric i n c r e m e n t m u c h larger t h a n t h e v a l u e s p r e v i o u s l y o b t a i n e d . F u r t h e r m o r e , he observed t h a t t h e dielec tric increment and the relaxation time depend strongly on molecular weight — i . e . , degree of p o l y m e r i z a t i o n i n t h i s case. H e suggested o n the basis of these observations t h a t D N A h a d a l o n g i t u d i n a l i n s t e a d of a transverse m o m e n t as c o n c l u d e d b y A l l g e n et al., a n d i n d i c a t e d t h e p o s s i b i l i t y t h a t t h e m a j o r axis of D N A is o r i e n t e d to some extent i n t h e d i r e c t i o n of t h e electric field after t h e c r e a t i o n of a n i n d u c e d dipole. T h i s conclusion was c r i t i c i z e d b y P o l l a c k (18), w h o concluded o n t h e basis of his t h e o r y t h a t t h e dielectric r e l a x a t i o n of D N A m a y be e x p l a i n e d i n t e r m s of a s i m p l e M a x w e l l - W a g n e r t h e o r y (11, 29). I n t h i s w o r k , t h e size dependence of the dielectric i n c r e m e n t a n d t h e r e l a x a t i o n t i m e is r e i n v e s t i g a t e d (28). T h e hydrodynamic length, which is e s t i m a t e d f r o m t h e r o t a r y d i f f u s i o n constant, is used i n s t e a d of m o l e c u l a r w e i g h t since t h e l a t t e r b y no means represents the a c t u a l l e n g t h i n t h e s o l u t i o n . A l s o a careful c o m p a r i s o n of dielectric r e l a x a t i o n t i m e a n d r o t a r y r e l a x a t i o n t i m e is a t t e m p t e d . A c t u a l l y , the c o m p a r i s o n is meaningless u n less i t is m a d e o n t h e same D N A s a m p l e because b o t h r e l a x a t i o n t i m e s , p a r t i c u l a r l y t h e r o t a r y r e l a x a t i o n t i m e , are s t r o n g l y dependent o n t h e size of t h e molecule. Therefore, t h e a t t e m p t b y A l l g e n et al. t o c o m p a r e t h e v a l u e of dielectric r e l a x a t i o n t i m e t h e y o b t a i n e d w i t h t h e v a l u e of r o t a r y r e l a x a t i o n t i m e o b t a i n e d b y E d s a l l w i t h a different D N A s a m p l e is n o t p a r t i c u l a r l y significant. F i n a l l y , a t t e m p t s are m a d e o n a t h e o r e t i c a l basis t o e x p l a i n t h e u n u s u a l l y large dielectric increments a n d r e l a x a t i o n t i m e s of D N A . T h e d i s cussion is l i m i t e d t o i o n i c - t y p e p o l a r i z a t i o n s i n t h i s report. T h e a v a i l a b l e theories, s u c h as t h e M a x w e l l - W a g n e r t h e o r y (29) a n d t h e surface c o n d u c t i v i t y t r e a t m e n t , are reviewed a n d a n a l y z e d . T h e s e theories do n o t ex p l a i n t h e dielectric r e l a x a t i o n of D N A s a t i s f a c t o r i l y . F i n a l l y , t h e c o u n t e r i o n p o l a r i z a t i o n t h e o r y is described, a n d i t is d e m o n s t r a t e d t h a t i t explains most reasonably the dielectric r e l a x a t i o n of D N A .

In Ordered Fluids and Liquid Crystals; Porter, R., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1967.

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FLUIDS A N D LIQUID CRYSTALS

Experiments A W h e a t s t o n e bridge designed b y S c h w a n (21) was used for dielectric measurements, w h i c h were carried o u t between 50 c.p.s. (cycles per second) a n d 200 k c . T h e bridge w a s designed for measurement w i t h c o n d u c t i v e m a t e r i a l s a n d is suitable for a D N A s o l u t i o n . T h e conductance of a d i l u t e D N A s o l u t i o n is u s u a l l y 20 to 50 ^mhos at a c o n c e n t r a t i o n of 0.01 to 0.03%. T h e m a g n i t u d e of t h e e x p e r i m e n t a l error i n lossy solutions has been d i s cussed i n d e t a i l b y S c h w a n (21). T h e m a j o r source of e x p e r i m e n t a l error w i t h c o n d u c t i v e solutions is electrode p o l a r i z a t i o n . T w o methods are used to e l i m i n a t e t h i s effect. I n t h e first, t h e p l a t i n u m electrodes are v e r y c a r e f u l l y p l a t e d w i t h p l a t i n u m b l a c k a c c o r d i n g t o t h e m e t h o d of M a c z u k a n d S c h w a n (21). T h e measured c a p a c i t y of t h e s o l u t i o n c a n be expressed b y E q u a t i o n 1 (21), where C is t h e true c a p a c i t y of t h e s o l u t i o n , C is t h e electrode p o l a r i z a t i o n capacitance, ω is t h e a n g u l a r frequency, a n d R is t h e measured resistance. T o m a k e t h e second t e r m of E q u a t i o n 1 s m a l l ,


s

v

C = C+ s

(1)

l/\C*Ru]

either R o r C m u s t be m a d e large. A n increase i n electrode c a p a c i t y is t h e easier m e t h o d for decreasing t h e second t e r m . T h e effect of electrode p o l a r i z a t i o n c a n s t i l l be considerable w i t h c o n d u c t i v e solutions, even w i t h a good p l a t i n g . T h i s gives rise t o a d i f f i c u l t y i n d e t e r m i n i n g t h e l o w fre q u e n c y dielectric constant. T h e dispersion of D N A depends l a r g e l y o n t h e l e n g t h of t h e molecule. T h e effect of electrode p o l a r i z a t i o n is n o t serious for s m a l l D N A molecules because t h e dielectric dispersion is i n a r e l a t i v e l y h i g h frequency region. F o r large molecules, however, f u r t h e r correction is essential. A dielectric cell w a s constructed i n w h i c h t h e distance between the electrodes w a s v a r i a b l e . M e a s u r e m e n t s were repeated t w i c e a t t w o p

Table I.

Experimental Error of Measurement of Capacity and Conductivity at Low Frequencies

Frequency, C.p.s. ' 200 100 70 50

Capacity (AC),

/W±0.2 ±0.3 ±0.8 ±2.0

Conductivity (AG), μηιΗο ±0.0015 ±0.003 ±0.003 ±0.003

Dielectric Const. ± 4.28 ± 6.24 ± 8.58 ±21.42

electrode distances, s a y 10 a n d 1 c m . E l e c t r o d e p o l a r i z a t i o n is i n d e pendent of t h e electrode distance a n d c a n be e l i m i n a t e d b y u s i n g E q u a t i o n 2 (21). C is t h e t r u e c a p a c i t y a t 10 c m . , a n d C i a n d Ci are t h e capacities a n d R\ a n d R are t h e resistances a t electrode distances of 10 a n d 1 c m . , respectively. s

2

T h e c a p a c i t y a n d c o n d u c t i v i t y of t h e s o l u t i o n c a n be measured d o w n t o 100 c.p.s. w i t h reasonable a c c u r a c y ( T a b l e I ) . H i g h m o l e c u l a r weight D N A , however, has a n anomalous d i s p e r s i o n i n a l o w frequency region. T h e l o w frequency p l a t e a u of t h e dielectric dispersion of these samples seems t o appear e v e n below 100 c.p.s. T h e c a p a c i t y of a D N A s o l u t i o n



17.

Dielectric

TAKASHiMA

Relaxation

of

235

DNA

loo kc Figure 1.

Dielectric dispersion of salmon sperm DNA

1. Dielectric constant 2. Conductivity Concentration. 0.01% Horizontal line, t , indicates low frequency dielectric constant obtained from ColeCole's plot 0

becomes m u c h m o r e difficult to measure below 100 c.p.s. O n e cannot o b t a i n the l o w frequency p l a t e a u b y e x t r a p o l a t i n g the d i s p e r s i o n c u r v e w i t h o u t considerable a r b i t r a r i n e s s . I n a case l i k e t h i s , the use of a C o l e - C o l e p l o t (3) is h e l p f u l . A s s h o w n i n F i g u r e 2, the C o l e - C o l e p l o t of D N A is s y m m e t r i c a l . T h u s w e c a n estimate the l o w frequency dielectric constant f r o m t h e intersection of the circle w i t h the abscissa. T h e v a l u e of t h e l o w frequency dielectric constant o b t a i n e d b y t h i s m e t h o d is m u c h m o r e r e l i a b l e . I n the present experiment, the low frequency dielectric constant is always obtained b y this method. T h e i m a g i n a r y p a r t of t h e dielectric constant (dielectric loss e") is calculated from the formula e" =

(κ-

(3)

κ.)/2φ

τ

where κ is t h e l o w frequency c o n d u c t i v i t y i n m i c r o m h o s , a n d e is t h e a b solute v a l u e of t h e dielectric constant of free space. T h e d i s p e r s i o n of c o n d u c t i v i t y is s h o w n i n F i g u r e 1. I t is evident f r o m E q u a t i o n 3 t h a t a s m a l l error i n t h e c o n d u c t i v i t y measurement c a n cause a considerable error i n t h e dielectric loss at l o w frequencies. C o n d u c t i v i t y is measured w i t h a n error of ± 0 . 0 0 1 jumho. A l t h o u g h t h e t e m p e r a t u r e of the s o l u t i o n is c o n t r o l l e d b y c i r c u l a t i n g t h e r m o s t a t e d w a t e r , the fluctuation of c o n d u c t i v i t y cannot be p r e v e n t e d . T h e correction m e t h o d for the d r i f t of c o n d u c t i v i t y has been g i v e n (10, 27). T h e r a n d o m error of the c o n d u c t i v i t y r e a d i n g is suf ficiently s m a l l for d e t e r m i n i n g the dielectric loss. T h e e x p e r i m e n t a l errors i n the c a p a c i t y a n d c o n d u c t i v i t y readings i n the l o w frequency region are s u m m a r i z e d i n T a b l e I . I n the last c o l u m n , the m a g n i t u d e of the error i n t h e dielectric constant is s h o w n . T h e error of ± 2 1 . 4 2 at 50 c.p.s. i n the dielectric constant seems l a r g e ; however, t h e t o t a l dielectric increment is so large t h a t t h i s error is not r e a l l y serious. 0

r


236

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LIQUID CRYSTALS

T h e flow birefringence a n d t h e e x t i n c t i o n angle of D N A s o l u t i o n are m e a s u r e d w i t h a R a o birefringence a p p a r a t u s M o d e l - B - 2 2 . T h e e x t i n c t i o n angle, χ, is related to a p a r a m e t e r , a, i n t h e e q u a t i o n of B o e d e r (2) a n d t h a t of P e t e r l i n a n d S t u a r t (17); χ = \

tan" -1 = 1

-

i | [ 1 - f( ,a,b)]

(4)

a

( P a r a m e t e r a s h o u l d n o t be confused w i t h t h e C o l e - C o l e parameter.) v a l u e of a is t a b u l a t e d b y E d s a l l et al. (19) for v a r i o u s a x i a l ratios. r o t a r y d i f f u s i o n constant is related to a b y t h e f o l l o w i n g f o r m u l a ,

(5)

a = β/θ


The The

where β is t h e v e l o c i t y gradient. T h e r o t a r y r e l a x a t i o n t i m e is c a l c u l a t e d f r o m t h e r o t a r y d i f f u s i o n constant b y t h e f o r m u l a r = 0/2 T h e l e n g t h of D N A is c a l c u l a t e d b y u s i n g t h e P e r r i n e q u a t i o n

(6) (16),

where 0& is t h e r o t a r y d i f f u s i o n constant a r o u n d t h e m i n o r axis, η is the v i s c o s i t y of t h e solvent i n poises, a n d a a n d b are t h e s e m i m a j o r a n d m i n o r axes i n centimeters. A x i s a is n o t necessarily t h e f u l l y stretched l e n g t h of D N A , a n d b is n o t necessarily t h e r a d i u s of the double helix. Since the P e r r i n e q u a t i o n c o u l d n o t be s o l v e d a n a l y t i c a l l y , a n I B M 1710 c o m p u t e r was used t o o b t a i n t h e s o l u t i o n for t h e l e n g t h of t h e D N A molecule. C a l f t h y m u s a n d s a l m o n s p e r m D N A were used i n t h i s experiment. T h e l e n g t h of calf t h y m u s ranges f r o m 10,500 t o 1500 A . a n d t h a t of s a l m o n s p e r m D N A f r o m 7400 t o 1300 A . A 2 0 - k c . sonic o s c i l l a t i o n is a p p l i e d to p r o d u c e s m a l l e r D N A samples. V i s c o s i t y is measured w i t h a R a o couettet y p e v i s c o m e t e r w i t h v a r y i n g s h e a r i n g stress. I t is o b t a i n e d b y e x t r a p o l a t i n g t h e consistency c u r v e t o zero shear. D N A is d i s s o l v e d i n freshly deionized w a t e r . T h e p H of t h e w a t e r is e x a m i n e d each t i m e since d i s t i l l e d w a t e r becomes acidic o n storage. Since D N A is u n s t a b l e at l o w i o n i c s t r e n g t h , t h e p H of t h e w a t e r m u s t be m a i n t a i n e d close t o n e u t r a l t o a v o i d d e n a t u r a t i o n . I f D N A is dissolved i n a c i d i c w a t e r , i t does n o t e x h i b i t flow birefringence n o r large dielectric i n crement. T h i s indicates d e n a t u r a t i o n , p r o b a b l y s t r a n d separation. C a l f t h y m u s D N A is m o r e stable t h a n s a l m o n s p e r m D N A . T h e c o n c e n t r a t i o n of D N A is 0 . 0 1 % unless otherwise s t a t e d . Results Native D N A . T h e dielectric dispersion of s a l m o n testes D N A is s h o w n i n F i g u r e 1. I t is o b v i o u s t h a t t h e dielectric constant of a D N A s o l u t i o n rises f a r above t h e dielectric constant of w a t e r a n d is s t i l l increasing at 50 c.p.s. U n l e s s we h a v e a n e n t i r e l y different measurement t e c h n i q u e , w e cannot hope t o e x t e n d the frequency t o w a r d t h e l o w e r frequency region. T h u s at present i t is n o t possible to o b t a i n t h e complete dispersion c u r v e to estimate t h e l o w frequency dielectric constant.


17.

T A K A S H i M A

Dielectric

Relaxation

of

237

DNA

T h e use of the C o l e - C o l e p l o t is often h e l p f u l for o b t a i n i n g the l o w a n d h i g h frequency dielectric constant. A t y p i c a l example of the C o l e - C o l e p l o t of a D N A s o l u t i o n is s h o w n i n F i g u r e 2. T h e intersections between t h e arc a n d t h e abscissa give v a l u e s of 620 a n d 70, r e s p e c t i v e l y . The h o r i z o n t a l l i n e designated e i n F i g u r e 1 is t h e v a l u e o b t a i n e d b y t h i s 0

300r-


200

C

700

Figure 2.

Cole-Cole plot of salmon sperm DNA

Ordinate. Imaginary part Abscissa. Real part of dielectric constant Numbers in figure are frequencies A,B. High and low frequency dielectric constant

m e t h o d , a n d seems to be i n reasonable agreement w i t h t h e possible v a l u e w h i c h w o u l d h a v e been o b t a i n e d b y t h e e x t r a p o l a t i o n of t h e d i s p e r s i o n curve. T h e dielectric measurements are c a r r i e d o u t w i t h D N A solutions w i t h w i d e l y different m o l e c u l a r dimensions. T h e dielectric i n c r e m e n t as w e l l as the r e l a x a t i o n t i m e decreases w i t h t h e decrease of t h e l e n g t h . T h e decrease i n the dielectric i n c r e m e n t is p a r t i c u l a r l y p r o n o u n c e d . Curve 1 i n F i g u r e 3 shows t h e r e l a t i o n s h i p between t h e dielectric i n c r e m e n t a n d t h e l e n g t h of the molecule. T h e f o l l o w i n g e m p i r i c a l r e l a t i o n s h i p between t h e l e n g t h a n d the dielectric i n c r e m e n t is o b t a i n e d : Ae = 0.29 X 10~ X L 5

2

= AL

2

(8)

T h e r e l a x a t i o n t i m e is c a l c u l a t e d b y u s i n g the e q u a t i o n ' τ = 1/2τ/

(9)

β

where r is t h e r e l a x a t i o n t i m e , a n d f is t h e c r i t i c a l frequency. T h e rela t i o n s h i p between r e l a x a t i o n t i m e a n d l e n g t h is s h o w n b y c u r v e 2 i n F i g u r e 3, a n d is expressed b y the f o l l o w i n g e m p i r i c a l f o r m u l a , c

τ = 1.6 X 10~ XL 5

2

= BL

2

(10)

Comparison between Rotary and Dielectric Relaxation Time. The conclusion d r a w n b y A l l g e n et al. (1) was based o n the fact t h a t the d i e l e c t r i c r e l a x a t i o n t i m e t h e y observed was w i d e l y different f r o m the r o t a r y r e l a x a -



238

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LIQUID CRYSTALS

LENGTH A. Figure 3.

Dependency of dielectric increment and dielectric relaxa tion time of DNA on length in solution

Ordinate (left). Dielectric increment Ordinate (right). Relaxation time Abscissa. Length obtained from Equation 7, expressed in A. Ο (curve 1). Dielectric increment (scale left ordinate) • (curve 2). Relaxation time (scale right ordinate)

t i o n t i m e . T h e dielectric r e l a x a t i o n t i m e observed i n t h i s experiment is considerably larger t h a n those observed b y A l l g e n et al. M o r e o v e r , the dielectric a n d r o t a r y r e l a x a t i o n times were p r e v i o u s l y o b t a i n e d i n d e p e n d e n t l y w i t h different D N A samples. I t is n o w evident t h a t the c o m p a r i s o n is almost meaningless unless i t is m a d e o n the same D N A sample since b o t h r e l a x a t i o n times depend o n the size of D N A . I n t h i s experiment, dielectric dispersion a n d r o t a r y d i f f u s i o n constant were measured s i m u l t a n e o u s l y w i t h the same D N A sample ( F i g u r e 4). T h e r e l a x a t i o n times are p l o t t e d w i t h the ordinate o n a l o g a r i t h m i c scale. T h e difference between the t w o r e l a x a t i o n times is a p p r o x i m a t e l y t w o f o l d w h e n t h e D N A molecule is s m a l l — i.e., i n the range of 2000 to 3000 A . C o n s i d e r i n g the error i n v o l v e d i n t h e d e t e r m i n a t i o n of r e l a x a t i o n t i m e s , the s i m i l a r i t y is m u c h closer t h a n ex pected. H o w e v e r , t h e d i s p a r i t y becomes more a n d more p r o n o u n c e d as the D N A molecule gets larger. T h e r o t a r y r e l a x a t i o n t i m e is about 20 times larger t h a n the dielectric r e l a x a t i o n t i m e w i t h t h e largest D N A used i n t h i s e x p e r i m e n t (length a b o u t 10,000 Α . ) . T h i s result is as expected


17.

T A K A S H i M A

Dielectric

Relaxation

239

of DNA

since t h e r o t a r y r e l a x a t i o n t i m e is p r o p o r t i o n a l t o t h e cube of t h e l e n g t h a n d t h e dielectric r e l a x a t i o n t i m e is p r o p o r t i o n a l t o t h e square of t h e l e n g t h , a c c o r d i n g t o E q u a t i o n 9. Therefore, r o t a r y r e l a x a t i o n t i m e increases m u c h faster t h a n dielectric r e l a x a t i o n t i m e . A l t h o u g h t h e difference between the t w o r e l a x a t i o n t i m e s is s u b s t a n t i a l , the d i s p a r i t y is m u c h s m a l l e r t h a n t h a t observed b y A l l g e n et al. : a 1 0 0 0 f o l d difference w i t h a D N A of m o l e c u l a r weight of about 10 . U n d o u b t e d l y the discrepancy between t h e results of A l l g e n et al. a n d t h e present results is greater t h a n t h e e x p e r i m e n t a l error a n d n o t m e r e l y o w i n g t o the difference i n t h e size of D N A . R e c e n t l y , O ' K o n s k i et al. (25) r e p o r t e d t h a t D N A h a d t w o dielectric dispersions, one i n t h e m e g a c y c l e region a n d t h e o t h e r i n the f e w - k i l o c y c l e region. O b v i o u s l y , A l l g e n et al. based t h e i r c o n c l u s i o n Downloaded by COLUMBIA UNIV on March 15, 2013 | http://pubs.acs.org Publication Date: January 1, 1967 | doi: 10.1021/ba-1967-0063.ch017

6

O.lL

Ο

Figure 4»

1

5000

\

LENGTH A.

10.000

Dielectric relaxation time and rotary relaxation time of DNA against length of DNA molecule 1. Dielectric relaxation time 2. Rotary relaxation time Ο Calf thymus DNA Ordinate on logarithmic scale


240

ORDERED

FLUIDS AND

LIQUID

CRYSTALS

o n t h e h i g h frequency dispersion a n d T a k a s h i m a observed o n l y t h e l o w frequency dispersion.


D e n a t u r e d D N A . F i g u r e 5 illustrates the dielectric dispersion of h e a t d e n a t u r e d D N A . D N A undergoes a t r a n s i t i o n f r o m a d o u b l e h e l i c a l c o n figuration to a s i n g l e - s t r a n d r a n d o m c o i l configuration, caused b y h e a t i n g o r b y a c i d o r a l k a l i . T h e d i s p e r s i o n c u r v e s h o w n here represents a d i e l e c t r i c dispersion of h e a t - d e n a t u r e d c o i l D N A . T h e d e n a t u r a t i o n is confirmed b y the decrease of v i s c o s i t y a n d t h e disappearance of flow birefringence.

FREQUENCY Figure 5.

Dielectric dispersion of heat-denatured DNA 1. Dielectric constant 2. Conductivity in μηι1ιθ8 Concentration 0.08%

A l t h o u g h the m a g n i t u d e of dielectric i n c r e m e n t is m u c h s m a l l e r t h a n t h a t of h e l i c a l D N A (the scale of the o r d i n a t e is g r e a t l y enlarged), one c a n s t i l l observe considerable i n c r e m e n t a n d d i s t i n c t d i s p e r s i o n i n t h e 5-kc. region. U n d o u b t e d l y the dispersion of d e n a t u r e d D N A is different f r o m t h a t of n a t i v e h e l i c a l D N A . F i g u r e 6 shows the change i n the d i e l e c t r i c i n c r e m e n t as a f u n c t i o n of the t e m p e r a t u r e of h e a t i n g . T h e measurements were c a r r i e d out at 25°C. after q u i c k cooling. The changes i n t h e m a g n i t u d e of birefringence a n d specific v i s c o s i t y are also p l o t t e d i n F i g u r e 6. T h e p a r a l l e l i s m a m o n g these three q u a n t i t i e s is r e m a r k a b l e . T h i s c l e a r l y indicates t h a t t h e dipole m o m e n t is associated w i t h t h e m a j o r axis of D N A a n d hence i t disappears w i t h the loss of sec ondary structure on denaturation. Discussion T h e above results s t r o n g l y i n d i c a t e t h a t D N A has a l o n g i t u d i n a l d i p o l e moment. Since there is a s u b s t a n t i a l difference between t h e dielectric a n d


17.

T A K A S H i M A

Dielectric

Relaxation

of

241

DNA


r o t a r y r e l a x a t i o n t i m e s , we c a n h a r d l y e x p l a i n t h e dielectric p o l a r i z a t i o n of D N A w i t h a p e r m a n e n t d i p o l e m o d e l . D N A m a y h a v e a transverse p e r m a n e n t d i p o l e w h i c h arises m a i n l y f r o m t h e group m o m e n t of base p a i r s . H o w e v e r , i t does n o t seem t o g i v e rise t o a large net m o m e n t because of p a r t i a l c a n c e l l a t i o n of these m o m e n t s w i t h each other as a result of t h e spiral structure. Since D N A is a h i g h l y charged m a c r o m o l e c u l e s u r r o u n d e d b y a l a y e r of counter ions, i t is m o r e p r o b a b l e t h a t t h e dielectric p o l a r i z a t i o n of D N A arises f r o m the p o l a r i z a t i o n of the i o n atmosphere. Various mechanisms of i o n i c p o l a r i z a t i o n h a v e been proposed. T h e theories of ionic p o l a r i z a t i o n for a s p h e r i c a l p a r t i c l e suspension were r e v i e w e d a n d c a r e f u l l y d i s cussed b y S c h w a n (20, 21). S i n c e D N A is a t h i n elongated molecule, those theories m u s t be modified s u b s t a n t i a l l y . V a r i o u s theories for ellipsoids are briefly reviewed here.

M a x w e l l - W a g n e r T y p e T h e o r y . I f a sphere w i t h a d i e l e c t r i c constant €2 a n d c o n d u c t i v i t y k is suspended i n a m e d i u m w i t h a dielectric constant €i a n d a c o n d u c t i v i t y k\, t h e dielectric constant of t h e suspension m a y be expressed b y t h e M a x w e l l e q u a t i o n (11), 2

€

!

*

-

€

2 * + € € l

_

61* 5

-

€2*

/

2€!*+€ *

K

2

where q is the v o l u m e f r a c t i o n of s p h e r i c a l particles.

n

x

}

I f we insert t h e f o l -


242

O R D E R E D FLUIDS AND

LIQUID

CRYSTALS

l o w i n g relations for the complex dielectric constants a n d rearrange t h e equation, ι

€i* = €i

(12) *

€2

47ΤΑ'2 .

I

= €

2

T h e dielectric constant of t h e s p h e r i c a l suspension becomes Ae


+ 1T-h+ ΐωτ A

(13)

^6l/C2 — €2^ΐ)

η

2

€i(2ei + e?)(2/ci +

(Λ

Λ

χ

κ*)

T h e h i g h frequency dielectric constant is

— *[ ·-£τΐ]

21

2

Dei + e

2

(

2

2

)

4^2

A g a i n , e a n d r are the h i g h frequency dielectric constant a n d r e l a x a t i o n t i m e . I n contrast t o t h e spherical suspension, t h e dielectric increment a n d the r e l a x a t i o n t i m e depend o n t h e shape a n d dimensions of t h e molecule because of t h e presence of p a r a m e t e r η i n t h e n u m e r a t o r of E q u a t i o n s 21 a n d 22. A c c o r d i n g t o E q u a t i o n 18, l/η becomes v e r y large, as t h e molecule is elongated. A c c o r d i n g l y , b o t h dielectric increment a n d r e l a x a t i o n t i m e c a n be large for elongated molecules. T h i s t h e o r y appears p r o m i s i n g i n e x p l a i n i n g t h e u n u s u a l l y large dielectric i n c r e m e n t a n d t h e r e l a x a t i o n t i m e of D N A . I f w e assume t h a t t h e a x i a l r a t i o of D N A is 200, η becomes a b o u t 0.00013. L e t us assume t h a t ei a n d e are 80 a n d 5, respectively. A l s o i f we assume 1 0 t o Ι Ο μΐηΐιο for fc , E q u a t i o n 22 gives a v a l u e of 4.3 X 10~ to 4.3 X 10~ second. T h e s e values are c o m p a r a b l e w i t h t h e observed v a l ues. H o w e v e r , E q u a t i o n 21 gives a v a l u e for t h e dielectric i n c r e m e n t of about 10.8 for t h e v o l u m e f r a c t i o n 0.0001, w h i c h is t h e c o n c e n t r a t i o n d i v i d e d b y t h e specific g r a v i t y . T h i s is f a r t o o s m a l l c o m p a r e d w i t h t h e observed v a l u e . T h e difference between t h e t h e o r e t i c a l a n d e x p e r i m e n t a l values is so great t h a t n o i m p r o v e m e n t of t h i s t h e o r y is expected. More over, t h e a s s u m p t i o n k = 0 used here is e v i d e n t l y u n r e a l i s t i c f o r aqueous suspensions. F o r t h e present case, E q u a t i o n s 21 a n d 22 m u s t be d e r i v e d b y a s s u m i n g a finite v a l u e for t h e c o n d u c t i v i t y of t h e s o l v e n t . T h e d e r i v a t i o n of these equations w i t h a finite v a l u e f o r k\ is n o t s t r a i g h t - f o r w a r d , a n d the f o r m s u c h as E q u a t i o n 19 cannot be reached w i t h o u t a n a s s u m p t i o n w h i c h is n o t generally acceptable. I n a d d i t i o n , electrolyte molecules are s u r r o u n d e d b y a counter i o n l a y e r w h i c h has a c o n d u c t i v i t y a n d a dielectric constant different f r o m those of t h e solute o r m e d i u m . T h u s t h e y f o r m a shell s u r r o u n d i n g t h e molecule. Therefore, t h e s i m p l e m o d e l u s e d a b o v e is n o t a p p r o p r i a t e f o r p o l y e l e c t r o l y t e s . R e c e n t l y , P o l l a c k d e r i v e d , b y a d a p t i n g a s i m p l e procedure, a M a x w e l l W a g n e r t y p e of e q u a t i o n f o r a h i g h l y elongated e l l i p s o i d of r e v o l u t i o n (18). A l t h o u g h h i s procedure i s c o n s i d e r a b l y different f r o m those of F r i c k e a n d S i l l a r s , t h e final f o r m is essentially t h e same. H e d e r i v e d t h e f o l l o w i n g equations f o r t h e r e l a x a t i o n t i m e : rz = 2o e/Cb% (23) T = 2e/k (24)


œ

2

-4

- 6

4

2

2

2

t

t


244

ORDERED FLUIDS A N D LIQUID CRYSTALS

w h e r e η is t h e r e l a x a t i o n t i m e i n t h e l o n g i t u d i n a l d i r e c t i o n a n d r t h a t i n t h e transverse d i r e c t i o n , fa a n d k are t h e l o n g i t u d i n a l a n d t r a n s v e r s e c o n d u c t i v i t i e s , a a n d b are t h e m a j o r a n d m i n o r axes, a n d C = 2 + In ( 4 a + 6 ). T h e resemblance between E q u a t i o n s 22 a n d 23 a n d 24 is r a t h e r o b v i o u s . E q u a t i o n 23 gives a v a l u e for r e l a x a t i o n t i m e w h i c h is c o m p a r a b l e w i t h t h e observed one. H o w e v e r , P o l l a c k d i d n o t d e r i v e a n e q u a t i o n for d i e l e c t r i c i n c r e m e n t , a n d t h e test of h i s t h e o r y c a n n o t be complete. M o r e o v e r , t h e m o d e l he used is t h e same as t h a t of F r i c k e a n d i s n o t a p p r o p r i a t e f o r electrolyte molecules. I n c o n c l u s i o n , t h e M a x w e l l - W a g n e r t y p e theories for elongated m o l e c u l e do n o t seem t o a c c o u n t for t h e d i e l e c t r i c p r o p e r t i e s of D N A s a t i s f a c t o r i l y . t

t


2

2

S h e l l M o d e l . T h e case of a sphere s u r r o u n d e d b y a l a y e r of a s h e l l w i t h a c o m p l e x dielectric constant,

€ 3

*

=

€

3

is t r e a t e d b y M i l e s a n d R o b e r t s o n (12). r e l a x a t i o n t i m e as f o l l o w s :

- ^ i ω

(25)

T h e y a r r i v e d at a n expression for

* + * ±*(2fa + fa)

(26)

€ = e + 2d/ae3

(27)

fa = k + 2d/afa

(28)

2 6

2

where 2

2

2

where subscripts 1, 2, a n d 3 refer t o t h e m e d i u m , t h e p a r t i c l e , a n d t h e shell, respectively. T h e presence of t h e shell increases e a n d fa i n t h e M a x w e l l W a g n e r e q u a t i o n b y t h e a m o u n t s of 2(d/a)ez a n d 2(d/a)fa, b u t does n o t change t h e f o r m of t h e e q u a t i o n . T h e shell m o d e l , i n c l u d i n g t h e c a l c u l a t i o n of dielectric i n c r e m e n t , was t r e a t e d m o r e c o m p l e t e l y b y P a u l y a n d S c h w a n (14). 2

T h i s m o d e l is p e r t i n e n t for p o l y e l e c t r o l y t e s w h i c h h a v e a c o u n t e r i o n l a y e r s u r r o u n d i n g t h e molecules. H o w e v e r , t h e m a t h e m a t i c a l difficulties are considerable for t h e e l l i p s o i d a l shape w i t h a shell. F r i c k e (5) d e r i v e d a n e q u a t i o n for a n e l l i p s o i d w i t h a n o n c o n d u c t i n g shell. S i n c e p o l y e l e c t r o l y t e s are s u r r o u n d e d b y a c o n d u c t i n g counter i o n atmosphere, his t r e a t m e n t m a y n o t be r e l e v a n t t o t h e present case. S u r f a c e C o n d u c t i v i t y . T h e basic difference b e t w e e n t h e p r e v i o u s t r e a t m e n t s a n d t h i s t r e a t m e n t is t h a t a fluctuation of c o u n t e r ions is c o n sidered i n t h i s case. A p a r t i c l e w i t h a p u r e c o n d u c t i n g shell ( w i t h fa, e = 0) is t r e a t e d b y O ' K o n s k i (13) for s p h e r i c a l a n d e l l i p s o i d a l p a r t i c l e s . O ' K o n s k i assumed t h e existence of a f r e q u e n c y - i n d e p e n d e n t surface c o n 3


17.

Dielectric

TAKASHiMA

Relaxation

of

245

DNA

d u c t a n c e , fc , a r o u n d t h e p a r t i c l e w h i c h a c o m p l e x dielectric c o n s t a n t , 3

*

€2

=

€2

—

47Γ&2

I

.

CO

i n a c o n t i n u o u s m e d i u m w i t h a complex d i e l e c t r i c c o n s t a n t , *

47Γ&1

ei

.

ζ

= €i

ω H e also assumed t h a t t h e surface charge d e n s i t y undergoes a t a n g e n t i a l as w e l l as a v e r t i c a l v a r i a t i o n w h e n a n electric field is a p p l i e d .

B y solving

t h e c o n t i n u i t y e q u a t i o n w i t h a p p r o p r i a t e b o u n d a r y c o n d i t i o n s , he o b t a i n e d


a n e q u a t i o n for a n e l l i p s o i d w h i c h has t h e same f o r m as F r i c k e ' s E q u a t i o n 17 except for t h e m a g n i t u d e of t h e excess c o n d u c t i v i t y . «-«!*

= q(ej* -

€ 1

* ) / [ 1 + n(ej* -

where η is g i v e n b y E q u a t i o n 18.

e i

*)/ *J

(29)

e i

T h e c o m p l e x d i e l e c t r i c constants, ei*

a n d e / , are n o w g i v e n b y

€ l

* =

€

l

(30)

- ^ f c i ; ω

4r(fey + —

e i

3

~ *

3

ω

2k /J)

—

(31)

S

whereas ey* is g i v e n b y ey* = ey — 4:π1ΰμ/ω. i n F r i c k e ' s o r i g i n a l e q u a t i o n . J represents axes a , b, a n d C., d e p e n d i n g o n t h e d i r e c t i o n of

field.

I t is o b

v i o u s t h a t t h e presence of surface c o n d u c t i v i t y , fc , m e r e l y increases t h e 3

c o n d u c t i v i t y , 7c , b y t h e a m o u n t of

2kz/J.

2

T h e dielectric i n c r e m e n t at l o w frequencies is g i v e n b y t h e f o l l o w i n g expression;

Α β 1

" Τ

6 1

1 + (*y/Jfci " l ) n

h

ny is a g a i n a p a r a m e t e r g i v e n b y E q u a t i o n electric

field.

refers to t h e d i r e c t i o n of t h e

L i k e w i s e r e l a x a t i o n t i m e is g i v e n b y _1_ ey + Τ

The

( 3 2 )

y

numerical values

6i(l/ny ~ 1)

m

4π *y + * i ( l / w y - 1) of

dielectric i n c r e m e n t

K

were

calculated

x }

by

O ' K o n s k i o n t h e basis of E q u a t i o n 32, for v a r i o u s a x i a l r a t i o s a n d surface conductivities.

T h e l e n g t h of t h e largest D N A s a m p l e u s e d i n t h i s e x p e r i

m e n t is a p p r o x i m a t e l y 1 Χ 10 A . 4

I f w e assume t h e r a d i u s of t h e c y l i n d e r

w h i c h contains t h e D N A m o l e c u l e is a b o u t 50 Α . , t h e a x i a l r a t i o w o u l d be about

100.

(Ae/q)

of a b o u t 8200 for a p a r a l l e l field for t h i s a x i a l r a t i o w i t h t h e surface

E q u a t i o n 32 gives a v a l u e for specific d i e l e c t r i c i n c r e m e n t


246

ORDERED

FLUIDS AND LIQUID

CRYSTALS

c o n d u c t i v i t y 100 t i m e s greater t h a n t h e c o n d u c t i v i t y of t h e m e d i u m .

This

v a l u e is too s m a l l c o m p a r e d w i t h t h e observed v a l u e of 1500 at a c o n c e n t r a t i o n of 0 . 0 1 % (this w i l l give 0.375 v o l u m e %

o n t h e basis of t h e a b o v e

d i m e n s i o n a n d t h e specific dielectric i n c r e m e n t of 40 Χ

10 ). 4

I n t h e a b o v e c a l c u l a t i o n , t h e v o l u m e f r a c t i o n is e s t i m a t e d w i t h o u t t h e counter i o n l a y e r .

I f we i n c l u d e t h e c o u n t e r i o n l a y e r ( w i t h t h e D e b y e -

H u c k e l r a d i u s a p p r o x i m a t e l y 200 A . at t h e i o n i c s t r e n g t h used for m e a s u r e m e n t s ) , t h e v o l u m e f r a c t i o n w i l l be larger t h a n t h a t used a b o v e b y a f a c t o r of 15 t o 20.

T h e r e f o r e , t h e v o l u m e f r a c t i o n of t h e s o l u t i o n used for t h e

measurement is 6.0 i n s t e a d of 0 . 3 7 5 % .

T h u s t h e observed specific d i e l e c

t r i c i n c r e m e n t w o u l d be 2.5 Χ 10 , w h i c h is s t i l l c o n s i d e r a b l y larger t h a n 4


the theoretical value.

A s we h a v e seen, there are great u n c e r t a i n t i e s c o n

c e r n i n g t h e e s t i m a t i o n of t h e v o l u m e f r a c t i o n of p o l y electrolytes.

Under

these c o n d i t i o n s , i t m a y be b e t t e r to discuss t h e p r o b l e m o n a q u a l i t a t i v e basis a n d not t a k e t h e n u m e r i c a l agreement as conclusive. F r o m F i g u r e 4 a n d T a b l e V I I I of R e f . 13, O ' K o n s k i ' s t h e o r y

pre

d i c t s t h a t t h e i n c r e m e n t increases w i t h t h e increase of a x i a l r a t i o w i t h a slope of less t h a n 1 a n d t h a t t h e d i e l e c t r i c i n c r e m e n t reaches a m a x i m u m v a l u e at t h e a x i a l r a t i o of 30 a n d decreases a g a i n w i t h t h e f u r t h e r increase of a x i a l r a t i o .

These predictions contradict the experimental observations.

R e l a x a t i o n t i m e c a n be c a l c u l a t e d f r o m E q u a t i o n 33. cule is h i g h l y elongated, l/n

a

of 200.

has a v a l u e of ( 0 . 0 0 0 1 3 3 )

-1

W h e n the mole for a n a x i a l r a t i o

T h e r e f o r e , t h e v a l u e i n t h e b r a c k e t s is represented o n l y b y t h e

t e r m 1/ny.

U s u a l l y t h e d i e l e c t r i c constant of t h e p a r t i c l e is t a k e n to be

m u c h s m a l l e r t h a n t h a t of w a t e r , a n d t h e n u m e r a t o r is represented

by

Therefore,

ei/n . a

TJ = -— Χ τ — Τ Ί Γ Ί — 4?r kj + ki/ria Since k « j

v**J

2 / c / a , (k + 2fc /a) w o u l d be p r a c t i c a l l y e q u a l t o 2fc /a. 5

3

3

3

s u m e t h a t if 2 / c / a = 4.0 X 10~ m h o c m . 3

3

- 1

W e as

a n d fci = 10~ m h o c m . , t h e 6

- 1

r e l a x a t i o n t i m e w o u l d be a b o u t 0.5 X 10~ second, a b o u t one o n e - h u n d r e d t h 5

of t h e v a l u e o b t a i n e d b y m e a s u r e m e n t .

A l t h o u g h t h e agreement m a y be

i m p r o v e d s o m e w h a t b y choosing different v a l u e s for v a r i o u s p a r a m e t e r s , i t does n o t seem possible to r e a c h s a t i s f a c t o r y agreement. Counter Ion Polarization.

S c h w a n (20) a t t e m p t e d to e x p l a i n t h e d i

electric d i s p e r s i o n of a s p h e r i c a l p a r t i c l e suspension i n t e r m s of c o u n t e r i o n p o l a r i z a t i o n , a n d S c h w a r z c a r r i e d out t h e m a t h e m a t i c a l f o r m u l a t i o n

(23),

a n d f o u n d t h a t t h e d i s p l a c e m e n t of t h e c o u n t e r i o n i n t h e d o u b l e l a y e r is e q u i v a l e n t t o t h e existence of c o m p l e x surface c o n d u c t i v i t y , fc.-r^fr.. where k

so

(eua)

by O'Konski.

is t h e f r e q u e n c y - i n d e p e n d e n t

(35) surface c o n d u c t i v i t y defined

S c h w a r z f o u n d t h a t t h e presence of t h i s complex


surface

17.

Dielectric

TAKASHiMA

Relaxation

of DNA

247

c o n d u c t i v i t y gives rise t o a surface capacitance w h i c h rises f a r a b o v e t h e capacitance of w a t e r . H e d e r i v e d equations f o r t h e d i e l e c t r i c i n c r e m e n t and relaxation time,

Τ

(1 + P / 2 ) ~kT 2

W

(37)

K

2ukT


where Ρ is t h e v o l u m e f r a c t i o n , e is t h e electronic charge, R is t h e r a d i u s of t h e p a r t i c l e , σ is t h e surface charge d e n s i t y , a n d u is t h e m o b i l i t y of t h e counter i o n . T h i s t h e o r y is extended t o a n e l l i p s o i d a l p a r t i c l e b y T a k a s h i m a b y u s i n g e l l i p s o i d a l coordinates. T h e procedure is described i n d e t a i l i n t h e f o l l o w i n g discussion. [Pennock (15) t r e a t e d t h i s p r o b l e m b y u s i n g s p h e r o i d a l ordinates a n d reached t h e same conclusion.] I f one suspends a n ellipsoid w i t h complex dielec

G E N E R A L ELLIPSOID.

t r i c constant, * ei = ei

4π&ιΖ co

i n a m e d i u m w i t h complex d i e l e c t r i c c o n s t a n t , *

\-Kkii

62

= 6

2

CO

a n d applies a p a r a l l e l electric field a l o n g t h e m a j o r axis, t h e field causes a flux of c o u n t e r ions JE = -βησνψ

(38)

8

where u is t h e m o b i l i t y of c o u n t e r i o n , a n d ψ is t h e surface p o t e n t i a l . T h e 8

flux is opposed b y a d i f f u s i o n - c o n t r o l l e d flux of c o u n t e r ions of m a g n i t u d e , J

= -ukTVa

D

(39)

Therefore, t h e e q u a t i o n of c o n t i n u i t y is g i v e n b y = euaV t 2

s

+ ukTV\

(40)

at T h e L a p l a c i a n V ^ , i n e l l i p s o i d a l coordinates is 2

_

2

v

ψ

°

=

4

( r - { ) ( « - n ) Ο ι - r)

x

Ordered Fluids and Liquid Crystals - American Chemical Society

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