Ordering and Gelation in DNA Solutions - ACS Publications

Chapter 13

Ordering and Gelation in DNA Solutions Victor A. Bloomfield

Reversible Polymeric Gels and Related Systems Downloaded from pubs.acs.org by IMPERIAL COLLEGE LONDON on 11/27/18. For personal use only.

Department of Biochemistry, University of Minnesota, St. Paul, MN 55108

We present a review of the experimental f i n d i n g s regarding the g e l a t i o n and ordinary-extraordinary (slow mode) d i f f u s i o n t r a n s i t i o n s in s o l u t i o n s of short DNA fragments, and propose some mechanistic explanations. I t is suggested that DNA rods may a s s o c i a t e end-to-end by stacking o f terminal base p a i r s , with g e l a t i o n a r i s i n g from the entanglement of the l i n e a r multimers thus formed. The slow c o l l e c t i v e d i f f u s i o n i n the extraordinary phase may r e s u l t from formation of an e l e c t r o s t a t i c a l l y s t a b i l i z e d c o l l o i d a l c r y s t a l . Numerical estimates support the plausibility of these mechanisms of phase t r a n s i t i o n ; but the lack of s e n s i t i v i t y of l i g h t s c a t t e r i n g t o g e l a t i o n , and the f a i l u r e of t r a c e r d i f f u s i o n experiments t o r e f l e c t slow mode d i f f u s i o n , are puzzles that remain t o be explained. Solutions o f short, r o d l i k e fragments o f DNA can e x i s t i n at l e a s t four d i f f e r e n t states or "phases": a normal, i s o t r o p i c f l u i d at low DNA concentration and moderate s a l t ; a c l e a r , nonbirefringent, v i s c o e l a s t i c g e l at moderate DNA concentrations (D ; a biréfringent l i q u i d c r y s t a l at higher concentrations (2.) ; and an extraordinary phase with a very low t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t i n low s a l t Q ) . Higher molecular weight DNA undergoes monomolecular and paucimolecular condensation i n t o t o r o i d s i n the presence of multivalent cations (4-6). F o r s y n t h e t i c DNA with an a l t e r n a t i n g purine-pyrimide sequence, the condensation i s often accompanied by a t r a n s i t i o n from a normal right-handed B-DNA h e l i x t o a left-handed Z-DNA h e l i x or t o some other h e l i c a l v a r i a n t (2) T r a n s i t i o n s among these states depend strongly on i o n i c strength as w e l l as on other s o l u t i o n v a r i a b l e s such as polymer concentration and molecular weight, temperature, and s p e c i f i c i o n e f f e c t s . In t h i s paper, we s h a l l review work from our laboratory on the g e l a t i o n and ordinary - extraordinary t r a n s i t i o n s observed with r o d l i k e DNA fragments, and discuss some ideas about the mechanisms and intermolecular forces underlying these t r a n s i t i o n s , p a r t i c u l a r l y as they appear t o r e l a t e t o i o n i c e f f e c t s . Comparisons w i l l be made with some other p o l y e l e c t r o l y t e systems that undergo the ordinary-extraordinary t r a n s i t i o n . 0097-6156/87/0350-0199S06.00/0 ©1987 American Chemical Society

REVERSIBLE POLYMERIC GELS AND RELATED SYSTEMS

200 Materials

and

Methods

The materials and methods used i n t h i s work have been described i n d e t a i l i n the o r i g i n a l p u b l i c a t i o n s Il 3). B r i e f l y , the DNA used i n the g e l a t i o n work was prepared by extensive s o n i c a t i o n of c a l f thymus DNA; i t had a degree of polymerization of 200 ± 30 base p a i r s (bp). The DNA used i n the ordinary-extraordinary t r a n s i t i o n studies was prepared by nuclease d i g e s t i o n of chromatin from chicken erythrocytes; i t was an equimolar mixture of two s i z e s , 140 bp and 160 bp, averaged to 150 bp. A Rheometrics cone-and-plate rheometer was used to obtain the storage (G ) and l o s s (G") moduli and the complex v i s c o s i t y of the DNA g e l s . Q u a s i e l a s t i c l i g h t s c a t t e r i n g (QLS) was used to monitor g e l formation (by the magnitude of the dynamic s c a t t e r i n g at zero c o r r e l a t i o n time r e l a t i v e to the s t a t i c s c a t t e r i n g background), t o obtain d i f f u s i o n c o e f f i c i e n t s and hydrodynamic r a d i i of d i f f u s i n g molecules and aggregates. QLS a l s o was used to monitor the onset and extent of the ordinary to extraordinary t r a n s i t i o n , through the appearance of d i s t i n c t l y non-single-exponential decay of the a u t o c o r r e l a t i o n function. T o t a l l i g h t s c a t t e r i n g i n t e n s i t y , determined by photon counting methods, was a l s o used to c h a r a c t e r i z e DNA gels and to follow the appearance of the extraordinary phase. r

1

Gelation

Under appropriate conditions of s a l t and temperature, short DNA fragments, about 700 Â i n length, form c l e a r , non-birefringent, thermally r e v e r s i b l e gels at low concentrations, ca 1-2% by weight (1). Such gels, though they have the o p t i c a l p r o p e r t i e s of normal s o l u t i o n s , w i l l not pour out of the tubes i n which they are formed. The lowest g e l a t i o n concentrations are i n the range ( L d ) , where L i s the length and d the e f f e c t i v e diameter of the rod, at which i n t e r - r o d c o l l i s i o n s become important ( i . e . , at which the excluded volume per rod becomes comparable to the s o l u t i o n volume per rod). Rheometric measurements show normal viscous behavior f o r the ungelled s o l u t i o n s , but once the g e l has formed, G dominates G" by about 8-fold over the e n t i r e range of DNA concentrations. This i s the behavior expected f o r an e l a s t i c g e l . Contrary to the expectations of Doi-Edwards theory (&), the complex v i s c o s i t y extrapolated to zero shear v a r i e s with DNA concentration C rather than C . G i s also l i n e a r i n C i n the gel region, rather than v a r y i n g as the 1.6-1.7 power of ( C - C ^ ) as found f o r r o d l i k e s u b s t i t u t e d polydiacetylene (jfc). To monitor the g e l a t i o n t r a n s i t i o n as a function of s o l u t i o n conditions, we u t i l i z e d the fact that i n a QLS experiment the r a t i o of the dynamic s i g n a l S due to mobile s c a t t e r e r s , to the s t a t i c s c a t t e r i n g background B, i s expected to vary as the square of the concentration of mobile s c a t t e r e r s . Thus as s o l turned to g e l , S/B decreased markedly; the midpoint of the curve was taken as the g e l t r a n s i t i o n temperature T . The t o t a l s c a t t e r i n g i n t e n s i t y from DNA s o l u t i o n s at a given C d i d not change during the t r a n s i t i o n , and the t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t of the mobile s c a t t e r e r s i n the g e l phase was the same as that of DNA molecules i n s o l u t i o n , i n d i c a t i n g that the g e l s t r u c t u r e presents l i t t l e impediment to the motion of ungelled molecules. As C increases, the thermal t r a n s i t i o n range narrows, and Tg increases i n a roughly l i n e a r fashion. Assuming that the e f f e c t of Na ions on the t r a n s i t i o n could be expressed i n the mass-action form 2

- 1

1

1

cr

g

+

DNA

sol

+ nNa

+

== D N A

gel

'Na

+ n

(1)

t

13.

201

Ordering and Gelation in DNA Solutions

BLOOMFIELD

we found that η was 1.7 (per DNA molecule of 200 bp), independent of T. For Mg , η was about 1.0. M g was roughly 10 times more e f f e c t i v e than N a i n provoking g e l a t i o n . A n a l y s i s of the temperature dependence of g e l a t i o n gave an apparent ΔΗ of -14.3 kcal/mol i n NaCl, and -27.8 kcal/mol i n MgCl / f o r the e q u i l i b r i u m implied by Equation 1. See below f o r an estimate o f the thermodynamic parameters f o r the elementary a s s o c i a t i o n step i n a p a r t i c u l a r model of the g e l a t i o n . The length dependence of g e l a t i o n has not yet been c a r e f u l l y i n v e s t i g a t e d , since i t was studied with a s i n g l e average s i z e o f DNA. Some p r e l i m i n a r y gel e l e c t r o p h o r e s i s observations on g e l l e d DNA i n d i c a t e s that longer molecules were not p r e f e r e n t i a l l y trapped i n the g e l , but these require more d e t a i l e d confirmation. The DNA concentration dependence of the g e l a t i o n midpoint was approximately l i n e a r between 7.5 and 17 mg/mL (with a negative i n t e r c e p t , corresponding t o the fact that a c r i t i c a l concentration i s required t o form a g e l ) . This does not conform with the C** dependence p r e d i c t e d by Doi and Edwards (£) . ++

++

+

2

Mechanistic Ideas. Our observations show that DNA concentration and ion e f f e c t s have dominant e f f e c t s i n g e l a t i o n . General long-range Coulombic i n t e r a c t i o n s c e r t a i n l y play a r o l e , but the strong e f f e c t i v e n e s s o f M g compared t o N a i n inducing g e l a t i o n i n d i c a t e s that s p e c i f i c a t t r a c t i v e ion e f f e c t s are a l s o important. The small number o f ions that appear t o be "bound" upon g e l a t i o n suggests a small number o f contacts between the rods; t h i s i s consistent with the p e r c o l a t i o n ideas o f S i n c l a i r et a l Q ) . The s i z e a b l e ΔΗ of the s o l - g e l t r a n s i t i o n i n d i c a t e s that some s i g n i f i c a n t exothermal process helps t o s t a b i l i z e the g e l . A mechanism that appears t o be q u a l i t a t i v e l y consistent with a l l o f these observations i s a s s o c i a t i o n , e i t h e r base p a i r i n g or base stacking, between residues at the ends of separate DNA molecules. ++

+

End-to-end a s s o c i a t i o n . The simplest model i s one i n which each a s s o c i a t i o n step has the same e q u i l i b r i u m constant K. This i s equivalent t o the f a m i l i a r F l o r y most-probable polymerization scheme. Thus we write the recursion r e l a t i o n s i

1

Ci = KCiCi.! = K ~ C

i 1

,

i=2,3,...

(2)

Summation o f the geometric s e r i e s over a l l species f o r the t o t a l concentration C and rearrangement gives t

1

Ci = K " ( K C / ( l + K C ) ) t

i

t

(3) Ci = C /(l+KC ) t

t

Although the theory of g e l a t i o n of an entangled s o l u t i o n of p o l y d i s p e r s e r o d l i k e molecules i s poorly understood, i t seems reasonable that entanglements would become important when the average center-to-center distance between molecules i s s i g n i f i c a n t l y smaller than t h e i r length. This occurs f o r molecules o f degree of polymerization i when i

L

l /

(N C fi/1000)" A

t

1 / 3

> 1

(4)

where Li i s the monomer length (680 Â f o r 200 bp B-form DNA). Since there i s a wide range o f degrees of polymerization, i t


202

should be an i n i t i a l l y adequate approximation to replace Equation 4 with ï

L / w

1

(N C f'/1000)~ A

1/3

> 1

t

(5)

where ' i s the weight-average degree of polymerization, and f the mole f r a c t i o n , of molecules with i > 2. This i m p l i c i t l y assumes that the s o l i s monomeric DNA fragments, while molecules with i > 2 are incorporated i n the g e l . f i s simply KP^./(1+ K P ) , and the sum f o r ' i s e a s i l y evaluated. The value of the l e f t - h a n d s i d e of Equation 5 i s p l o t t e d against KC i n Figure 1 f o r Cj- = 10 mg/mL = 7.8xl0" M. It i s obvious that the i n e q u a l i t y i s r e a d i l y s a t i s f i e d f o r KC*. > 1. To proceed further, we need a molecular mechanism that w i l l enable us to estimate K. w

1

t

w

t

5

Base Stacking at Ends. One p o s s i b i l i t y i s that i n t a c t duplex molecules a s s o c i a t e end-to-end by base s t a c k i n g . In that case can w r i t e K

At T=Tgel, K

app = r

a p p

D N A

gell/(

D N A

solJ

=

f

,

/

f

l

=

K C

we

6

t·

= 1, so In Κ(T=Tg) = -In C

t

= -AH/RT + AS/R.

(7)

g

From data on T vs C^ i n NaCl s o l u t i o n s , (Figure 4 i n réf. (1)), i t i s p o s s i b l e to solve Equation 7 f o r the thermodynamic parameters: ΔΗ = -10.9 kcal/mol and AS = -16.6 cal/mol-deg. The f i t to the g e l a t i o n data i s shown i n Figure 2. These values compare reasonably w e l l with values measured f o r s t a c k i n g of bases i n d i n u c l e o s i d e phosphates (10 11). They are i n the same range, but d i f f e r somewhat from those given i n Table II of r e f . (1) f o r NaDNA (ΔΗ = -14.3 kcal/mol and AS = -38.8 cal/mol-deg). The discrepancy a r i s e s i n part because the values i n (1) were obtained from van't Hoff a n a l y s i s of Κ vs Τ at a p a r t i c u l a r DNA concentration, while those given here are from a n a l y s i s of T vs C and i n part because the f i t here i s to the parameters of a p a r t i c u l a r model i n which a l l species beyond monomer are i n the gel form. g

f

g

t ;

Base P a i r i n g at Ends. The process of breakage during s o n i c a t i o n might w e l l produce dangling s i n g l e - s t r a n d ends, which could then p a i r with complementary sequences on other molecules, l e a d i n g to extensive three-dimensional networks. However, we found that treatment of the sonicated DNA preparation with SI nuclease, which removes s i n g l e stranded regions, had no e f f e c t on the g e l a t i o n process. On the other hand, i t might be conjectured that normally p a i r e d bases at the ends of DNA molecules might t r a n s i e n t l y unpair, and then form i n t e r m o l e c u l a r l y base-paired complexes. Since the DNA s t u d i e d i s short, melting i n the middle of the duplex (loop formation) i s extremely u n l i k e l y , and the s i n g l e sequence approximation, with melting from the ends only, i s w e l l satisfied. I f a molecule i s L base p a i r s long, with the two strands h e l d i n r e g i s t e r by at l e a s t a s i n g l e bond, and the e q u i l i b r i u m constant f o r converting a s i n g l e base p a i r from unbonded to bonded i s s, then the s t a t i s t i c a l weight of a c o n f i g u r a t i o n with m bases p a i r e d i s p r o p o r t i o n a l to (L-m+1)s , where the p r e f a c t o r i s the number of ways m consecutive bonded bases can be arranged among L p o s s i b l e l o c a t i o n s . Thus the p r o b a b i l i t y P that at l e a s t ν base p a i r s are open i s m

v

13. BLOOMFIELD

203


0.001 λ

0.01

1

1

1

1

1

0.1

1

10

100

1000

KC

t

Figure 1. V a r i a t i o n o f (•) mole f r a c t i o n of dimeric and l a r g e r molecules, (0) weight-average degree o f polymerization f o r dimers and l a r g e r , and (•) r a t i o o f weight-average length t o average center-to center distance o f polymerized molecules, as function of KC . t

20 Η

1

1

1

1

1

»

5

7

9

11

13

15

17

1

[DNA], mg/mL Figure 2. Dependence o f temperature of g e l a t i o n midpoint on DNA concentration i n 50 mM DNA (points) , compared with p r e d i c t i o n s of end-stacking model ( l i n e ) . Data from (1).


204 L-V

P

which may P

v

=

v

-

Σ m=l

(L-m+l) s

m

L / Σ (L-m+1) s m=l

m

(8)

be summed to give [s

( L + 1

~

v )

- (L+1-V)s + L-V]

/ [s

( L + 1 )

- (L+l)s + L] . (9)

With ΔΗ = -7 kcal/mol, a f a i r l y t y p i c a l value f o r base p a i r formation, and a melting temperature (where s=l) of 80°C, P i s t y p i c a l l y of order 10"^ or l e s s (Figure 3). I f f o r s i m p l i c i t y at l e a s t V base p a i r s must be open to form an intermolecular base p a i r , and that bond i t s e l f has V base p a i r s (V i s 3-4 at room temperature from studies of o l i g o n u c l e o t i d e s ) then the intermolecular a s s o c i a t i o n constant i s K = s . Assuming equal p r o b a b i l i t y of occurrence of a l l four bases at any p o i n t of the sequence, the p r o b a b i l i t y that any given open end w i l l f i n d i t s complement i s (1/4) . Then, i f the t o t a l molar concentration of monomers i n a l l s t a t e s of aggregation i s C Q , the f r a c t i o n of m a t e r i a l i n the dimer form (assuming only a small amount i s dimerized) i s v

v

v

v

f

2

= C /C 2

0

=

v

(s/4) C

2

0

P .

(10)

v

For 200 bp fragments at 10 mg/mL, C Q = 7.6 χ 10"" M. With the values f o r ΔΗ and T proposed above, s ranges from 18.6 at 273°K to 3.6 at 313°K, the experimental temperature range. Thus f i s always « 1 (Figure 4) , and the hypothesis that end melting leads to intermolecular b a s e p a i r i n g and subsequent g e l a t i o n through network formation appears to be untenable. Regardless of whether end-to-end a s s o c i a t i o n of r o d l i k e DNA fragments occurs by base s t a c k i n g or base p a i r i n g , no theory e x i s t s that would enable confident p r e d i c t i o n of the s a l t dependence. Q u a l i t a t i v e l y , i t i s c l e a r that i n c r e a s i n g s a l t concentration should f a c i l i t a t e the c l o s e approach of l i k e - c h a r g e d p o l y i o n s , and that M g which i n t e r a c t s s t r o n g l y with the backbone phosphates, should be more e f f e c t i v e than Na . Both of these p r e d i c t i o n s are i n agreement with experiment. The end-to-end model f o r g e l a t i o n s u f f e r s from the obvious d i f f i c u l t y that s c a t t e r i n g experiments should be able to detect the increased average molecular weight and decreased d i f f u s i o n c o e f f i c i e n t attendant on multimolecular a s s o c i a t i o n before gelation. It i s conceivable that only very small extents of a s s o c i a t i o n are required to produce g e l a t i o n , and that the g e l i t s e l f has the same s c a t t e r i n g power as the free molecules. Further work i s c l e a r l y needed to a s c e r t a i n the p l a u s i b i l i t y of the model. 5

m

2

++

+

Ordinary-Extraordinary

Transition

At s a l t concentrations above about 0.01 M, the l i g h t s c a t t e r i n g behavior of d i l u t e s o l u t i o n s (1.5 mg/ml) of mononucleosomal DNA fragments accords with t h e o r e t i c a l expectation Q ) . That i s , as the i o n i c strength i s lowered, the t o t a l s c a t t e r i n g i n t e n s i t y (or apparent molecular weight) decreases, and the apparent translational diffusion coefficient D increases i n r e c i p r o c a l fashion. These trends are expected from r e p u l s i v e p o l y e l e c t r o l y t e i n t e r a c t i o n s and the inverse r e l a t i o n between apo * scattering structure factor. a p p

D

a n c

t

h

e

13.

275

205


BLOOMFIELD

280

285

290

295

300

305

310

315

Figure 3. P r o b a b i l i t y of f i n d i n g at l e a s t ν bases unpaired f o r 200 bp DNA with T - 353°K and ΔΗ « -7 kcal/mol. v: *,4; v,5; 1, 6. m

10

8 +

c /c 2

X10

t

9

2 +

273

Η 278

h 283

288

293

H 298

h 303

308

313

Τ Figure 4. Mole f r a c t i o n o f dimers computed f o r ν « 4 with the end-melting model as a f u n c t i o n of T.


206

As the s a l t concentration continues to decrease, however, matters change d r a m a t i c a l l y (2). The t o t a l s c a t t e r i n g i n t e n s i t y decreases more abruptly, and the QLS a u t o c o r r e l a t i o n function, which has been a simple single-exponential decay, becomes markedly two-exponential. The two decay rates d i f f e r by as much as two orders of magnitude. The f a s t e r continues the upward trend of D p from higher s a l t , and i s thus assigned the term "ordinary". The slower, which i s about 1/10 of D pp 9 s a l t , and appears to r e f l e c t a new mode of s o l u t i o n dynamics, i s termed "extraordinary". A n a l y s i s of the r e l a t i v e c o n t r i b u t i o n s of the slow and f a s t decays to the t o t a l i n t e n s i t y , by "peeling" o f f the slower from the l i n e a r region of a semi-log p l o t of the a u t o c o r r e l a t i o n function, i n d i c a t e s that i t i s the ordinary s c a t t e r i n g that decreases i n i n t e n s i t y . The extraordinary c o n t r i b u t i o n remains roughly constant once i t appears. The s o l u t i o n i n the extraordinary region i s not biréfringent, shows no evidence of p h y s i c a l phase separation, and shows no maxima or minima i n the s t r u c t u r e f a c t o r w i t h i n the a c c e s s i b l e range of s c a t t e r i n g angles (30 to 135 deg) . a

p

a t

n i

n

a

Mechanistic Ideas. The ordinary-extraordinary t r a n s i t i o n has a l s o been observed i n s o l u t i o n s of dinucleosomal DNA fragments (350 bp) by Schmitz and Lu (12) . Fast and slow r e l a x a t i o n times have been observed as functions of polymer concentration i n s o l u t i o n s of s i n g l e - s t r a n d e d p o l y ( a d e n y l i c acid) (13 14) but these experiments were conducted at r e l a t i v e l y high s a l t and are i n t e r p r e t e d as a t r a n s i t i o n between d i l u t e and semidilute regimes. The ordinary-extraordinary t r a n s i t i o n has a l s o been observed i n low-salt s o l u t i o n s of p o l y ( L - l y s i n e ) (lu), and poly(styrene sulfonate) (16 17). In p o l y ( L - l y s i n e ) , which i s the b e s t - s t u d i e d case, the t r a n s i t i o n i s detected only by QLS, which measures the mutual d i f f u s i o n c o e f f i c i e n t . The 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 (12) , e l e c t r i c a l c o n d u c t i v i t y (lu.) , e l e c t r o p h o r e t i c m o b i l i t y (18 20 21) and i n t r i n s i c v i s c o s i t y (22.) do not show the same profound change. It appears that the t r a n s i t i o n i s a manifestation of c o l l e c t i v e p a r t i c l e dynamics mediated by long-range forces; but the mechanistic d e t a i l s of the phenomenon are quite obscure. It i s worth noting that the ordinary-extraordinary t r a n s i t i o n i n s o l u t i o n s of DNA fragments d i f f e r s i n at l e a s t one respect from the same t r a n s i t i o n observed i n p o l y ( l y s i n e ) or poly(styrene s u l f o n a t e ) . For these l a t t e r polymers, D r i f f o r d and Dalbiez (12) have formulated the simple r u l e C b/2Ibj = Ζ r e l a t i n g the monomer concentration C , charge spacing b, i o n i c strength I, Bjerrum length b j , and s a l t valence Z. For double-stranded DNA at 1.5 mg/ml, t h i s p r e d i c t s that the t r a n s i t i o n w i l l occur at I = 0.002 M, while i n fact i t s midpoint i s at 0.01 M NaCl. The reason f o r t h i s discrepancy i s not c l e a r ; one obvious d i f f e r e n c e i s that DNA i s a s t i f f wormlike chain, while the other polymers are f l e x i b l e . Both the g e l a t i o n and ordinary-extraordinary t r a n s i t i o n s take place i n the general range of concentrations p r e d i c t e d by the simple C = (L^d)~* argument based on the onset of rod-rod collisions. S t i g t e r (22) has shown how, f o r a p o l y e l e c t r o l y t e i n low s a l t , the r e p u l s i v e diameter may be strongly augmented by the d i f f u s e ion atmosphere extending outward from the rod surface. However, even with t h i s augmentation, the ordinary-extraordinary t r a n s i t i o n takes place at a concentration that i s s e v e r a l - f o l d lower than p r e d i c t e d f o r the i s o t r o p i c - a n i s o t r o p i c t r a n s i t i o n t r e a t e d by S t i g t e r . This i s apparent from Figure 5, which shows r

r

r

f

m

m

r

13.

BLOOMFIELD

207

Ordering and Gefotion in DNA Solutions

100 ι

1

r

v. ι

0.0001

0.001

0.01

0.1

1.0

S a l t Cone, M

Figure 5. Regions of s t a b i l i t y of i s o t r o p i c and a n i s o t r o p i c s o l u t i o n s o f 150 bp DNA, c a l c u l a t e d according t o S t i g t e r (23.) . The l i g h t band corresponds t o the coexistence region f o r f u l l y charged DNA, the dark band t o DNA with 76% of charge n e u t r a l i z e d by counterion condensation. The salt/DNA concentration regions where the g e l a t i o n and ordinary-extraordinary t r a n s i t i o n s were s t u d i e d are i n d i c a t e d by brackets.

208


the ranges of s a l t and DNA concentration i n which i s o t r o p i c and a n i s o t r o p i c s o l u t i o n s are expected to c o e x i s t . Separate coexistence bands are computed f o r DNA with f u l l charge, and f o r DNA with charge reduced to 24% o f the formal value by counterion condensation. C o l l o i d a l C r y s t a l . One i n t r i g u i n g p o s s i b i l i t y i s the formation of a c o l l o i d a l c r y s t a l s t a b i l i z e d s o l e l y by r e p u l s i v e e l e c t r o s t a t i c forces, as proposed o r i g i n a l l y by Wigner (24) f o r e l e c t r o n s . The s t r u c t u r e and dynamics o f such e l e c t r o s t a t i c a l l y s t a b i l i z e d ordered suspensions have been s t u d i e d e x t e n s i v e l y with polystyrene latexes with l i t t l e or no added s a l t (e.g. 25), where they give r i s e t o Bragg s c a t t e r i n g with v i s i b l e l i g h t . A maximum i n the s c a t t e r i n g s t r u c t u r e f a c t o r , with concomitant minimum i n trans' ^ °een seen i n low s a l t s o l u t i o n s o f poly (lysine) (26.) and p o l y ( s t y r e n e sulfonate) (22); t h i s i s c o n s i s t e n t with formation o f an ordered suspension. A t h e o r e t i c a l a n a l y s i s o f the s t a b i l i t y o f such c o l l o i d a l c r y s t a l s o f s p h e r i c a l l a t e x p a r t i c l e s has been c a r r i e d out by Marcelja et a l (28.) . They employ the Lindemann c r i t e r i o n that a c r y s t a l w i l l be s t a b l e i f the rms thermal displacement o f the p a r t i c l e s about t h e i r e q u i l i b r i u m p o s i t i o n s i s a small f r a c t i o n f of the l a t t i c e spacing R. Comparison with Monte Carlo simulations shows that f i s about 0.1 f o r "hard" c r y s t a l s , and 0.08 f o r " s o f t " c r y s t a l s s t a b i l i z e d by long-ranged e l e c t r o s t a t i c f o r c e s . This l a t t e r c r i t e r i o n t r a n s l a t e s i n t o a c r i t i c a l r a t i o D

a

s

Γ - (z

2

e f f

e ) /£Rk T = z B

e f f

2 (bB

j e r r u m

/R)

(11)

where z ff i s the e f f e c t i v e p a r t i c l e charge i n u n i t s o f the proton charge e, ε i s the d i e l e c t r i c constant, k i s the Boltzmann constant, Τ the temperature, and bRjerrum the Bjerrum length i n the s o l u t i o n . Monte Carlo simulations snow that a c o l l o i d a l c r y s t a l w i l l be s t a b l e i f Γ > 155 ± 10. A s i m i l a r a n a l y s i s has been c a r r i e d out f o r f l e x i b l e p o l y e l e c t r o l y t e s which can be s t i f f e n e d by e l e c t r o s t a t i c i n t e r a c t i o n s , by De Gennes et a l (22.) . We have used the procedure o f Marcelja et a l (2&) t o compute Γ f o r s p h e r i c a l molecules with the same charge and volume as 200 bp r o d l i k e DNA. Each molecule i s at the center o f a Wigner-Seitz c e l l o f volume (4/3)rtR? with bulk s a l t concentrations spanning the experimental range Q ) . The nonlinear Poisson-Boltzmann equation i s solved numerically with appropriate boundary conditions at the p a r t i c l e surface and the c e l l boundary. The r e s u l t s are that Γ = 155 at about 3 mg/mL DNA (twice the experimental concentration) with no added s a l t , but Γ i s always < 155 f o r added s a l t i n the experimental range. For NaPSS, with dp = 3800 at 1-4 χ 10" mg/mL, Γ > 300, c o n s i s t e n t with the observation o f a s t r u c t u r e f a c t o r maximum. While these r e s u l t s f a l l short of p r e d i c t i n g s t a b i l i t y f o r our DNA s o l u t i o n s , they are c l o s e enough t o encourage f u r t h e r e x p l o r a t i o n . The obvious next step i s t o model the p a r t i c l e s as rods with various o r i e n t a t i o n s at the l a t t i c e p o s i t i o n s , t o compare the t o t a l e l e c t r o s t a t i c r e p u l s i o n between l i n e a r and s p h e r i c a l ( e f f e c t i v e l y point) charge d i s t r i b u t i o n s . Q

B

2

D i f f u s i o n a l Dynamics. I f the c o l l o i d a l c r y s t a l i s s t a b l e , does i t e x p l a i n the e x t r a o r d i n a r i l y slow d i f f u s i o n a l behavior? While some d e t a i l e d t h e o r i e s o f the dynamics o f c o l l o i d a l c r y s t a l s have been constructed (25 30), they are d i f f i c u l t t o apply t o our DNA s o l u t i o n s . Therefore, i t i s o f i n t e r e s t t o present a much simpler semiquantitative approach. f

13.

BLOOMFIELD


209 z

We s t a r t with the f a m i l i a r E i n s t e i n d i f f u s i o n equation, =2Dt, which i s t o be solved f o r D = D i n the c o l l o i d a l crystal. The rms displacement can be estimated from the Lindemann c r i t e r i o n as < x > l ' = fR i f t i s the p e r i o d of a l a t t i c e v i b r a t i o n , 2TC/C0Q . The c h a r a c t e r i s t i c frequency CÛQ can be estimated from the standard expression CÛQ = V(k/m), where k i s the force constant and m the e f f e c t i v e p a r t i c l e mass. For charged p a r t i c l e s on a l a t t i c e i n t e r a c t i n g through screened Coulomb forces, d i f f e r e n t i a t i o n of the force law with respect t o i n t e r p a r t i c l e distance gives x t a l

2

2

f

k

.

Ordering and Gelation in DNA Solutions - ACS Publications

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