Theoretical Estimates of Helical Structure in Polynucleotides

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Theoretical Estimates of Helical Structure in Polynucleotides WILMA K. OLSON Department of Chemistry, Rutgers University, New Brunswick, NJ 08903

P o t e n t i a l energy s t u d i e s over the last decade have supported the hypothesis that the p r i n c i p a l conformations o f the n u c l e i c acids a r e implicit i n t h e i r chemical a r c h i t e c t u r e ( 1 - 1 0 ) . Because the p o t e n t i a l s a r e f u n c t i o n s o f molecular s t r u c t u r e , these studies have helped to e l u c i d a t e the i n t e r r e l a t i o n s h i p between the base, sugar, and phosphate moieties o f the n u c l e o t i d e repeating u n i t and the gross morphology o f the p o l y n u c l e o t i d e c h a i n . The energies u s u a l l y r e f l e c t short-range nonbonded i n t e r a c t i o n s i n s i n g l e - s t r a n d e d n u c l e i c a c i d fragments and, i n principle, apply only to the treatment of i d e a l unperturbed randomly coiling chains (11). The d a t a , however, provide useful s t a r t i n g conformations in c o n s t r u c t i n g models o f s i n g l e - s t r a n d e d p o l y n u c l e o t i d e h e l i c e s (12, 13, 14, 15) and i n a n a l y z i n g X-ray c r y s t a l l o g r a p h i c data (16). In order to extend these conformational energy s t u d i e s t o the a n a l y s i s o f m u l t i - s t r a n d e d n u c l e i c a c i d systems, it is necessary t o devise a procedure to i d e n t i f y the arrangements o f the p o l y n u c l e o t i d e backbone t h a t can accomodate double, triple, and higher order h e l i x f o r m a t i o n . As a first step to t h i s end, a computational scheme is o f f e r e d here to i d e n t i f y the double h e l i c a l s t r u c t u r e s compatible with given base p a i r i n g schemes. The f e a s i b i l i t y o f a duplex i s estimated on the b a s i s o f semie m p i r i c a l energy estimates o f base s t a c k i n g and hydrogen bonding i n a m i n i a t u r e double h e l i x ( i . e . , complementary d i n u c l e o s i d e monophosphates) o f s p e c i f i e d conformation. The computed energies, however, a r e simply a measure o f the geometric a c c e p t a b i l i t y of the s t r u c t u r e rather than a r e l i a b l e measure o f h e l i x s t a b i l i t y . The method i s a p p l i c a b l e to p o l y n u c l e o t i d e duplexes generated by any p l a u s i b l e base p a i r i n g arrangement and i s thus useful i n t e s t i n g and comparing the v a r i o u s hydrogen bonding schemes t h a t may s t a b i l i z e a duplex. A d i r e c t approach l i k e t h i s complements t r a d i t i o n a l DNA model b u i l d i n g s t u d i e s ( 1 7 , 18, 19) that i d e n t i f y i n d i r e c t l y the combinations o f backbone parameters t h a t f i t a p a r t i c u l a r s e t o f h e l i c a l parameters o r s t r u c t u r a l c o n s t r a i n t s . The present method a l s o o f f e r s a means t o e x t r a p o l a t e t h e o r e t i c a l and experimental 0-8412-05 89-2/ 80/47-141 -251 $05.00/0 © 1980 American Chemical Society French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

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s t u d i e s o f low molecular weight n u c l e i c a c i d analogs to p o l y n u c l e o t i d e systems. Since the l o c a l parameters can be c o n t i n uously v a r i e d , t h i s scheme f u r t h e r provides a means to study h e l i x f l e x i b i l i t y and to i n t e r p r e t macroscopic p r o p e r t i e s o f the n u c l e i c acid helices i n s o l u t i o n . Double Strand Formation Double h e l i c a l s t r u c t u r e s may be constructed from complementary s i n g l e - s t r a n d e d p o l y n u c l e o t i d e chains sharing a common h e l i c a l axis according to the procedure o u t l i n e d below. The two strands o f the complex are assumed t o be r e g u l a r h e l i c e s defined by a common s e t o f backbone and g l y c o s y l t o r s i o n a n g l e s . The data presented here are l i m i t e d to model p o l y ( d A ) p o l y ( d T ) double h e l i c e s s t a b i l i z e d by Watson-Crick base p a i r s between a n t i p a r a l l e l strands. H e l i c a l parameters (n,h) and c y l i n d r i c a l atomic coordinates {r-j, e-j, z-j} a s s o c i a t e d with each s i n g l e strand o f the p o l y n u c l e o t i d e complex may be obtained from the s i x backbone r o t a t i o n angles ( • ' , ( ( J , a) , a), , and the g l y c o s y l r o t a t i o n (x) using the v i r t u a l bond scheme i l l u s t r a t e d i n Figure 1. The h e l i c a l repeat n and step height h may be expressed as simple f u n c t i o n s o f the v i r t u a l bond length v and the angles e and * that describe the o r i e n t a t i o n of adjacent chain r e s i d u e s : #

1

1

v

cos(^)

= - cos(^)

h sin(^)

= v cos(^)

v

sin(^)

(1)

cos(^-)

(2)

The v i r t u a l bond parameters may be obtained from the s i x f i x e d chemical bond l e n g t h s , the s i x f i x e d valence a n g l e s , and the s i x v a r i a b l e backbone r o t a t i o n angles f o l l o w i n g procedures o u t l i n e d elsewhere (12^, 2 0 ) . Atoms comprising each n u c l e o t i d e repeating u n i t may a l s o be transformed i n t o a common v i r t u a l bond coordinate system f o l l o w i n g published methods ( 2 0 ) . The f i n a l coordinate system assigned each s i n g l e - s t r a n d e d h e l i x i s chosen so t h a t the z - a x i s c o i n c i d e s with the h e l i x axis and so that the x - a x i s passes through the o r i g i n o f the i n i t i a l v i r t u a l bond. The orthogonal matrix J T ( e , e , ^ ) t h a t e f f e c t s the coordinate transformation from the frame o f the i n i t i a l v i r t u a l bond to that o f the h e l i x i s o b t a i n e d , f o l l o w i n g Shimanouchi and Mizushima ( 2 1 ) , from the expression: v

v

T ( e , e

v

T ( e

v

y )

= 1(e)

T(e,e ,* ) v

v

(3)

where T ^ e , ^ ) = X,(* -2TT) Z ( - e ) i s the matrix relating coordinate systems'of adjace'nt virtuaT bonds with 7

v

v

French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

15.

OLSON

Helical Structure in Polynucleotides

0

"1

X(t ) V

253

0

cos*

0

sini^

cos*

v

v

- s i nip

v

v

(4) v

and

Z(e ) v

cose

-sine

sine

cose

0

(P (5)

0

The parameter e i n E q . 3 i s the c y l i n d r i c a l repeating angle o f the h e l i x i n radians and i s given by 2ir/n. The r e l a t i v e s p a t i a l l o c a t i o n s o f the two chains forming the t h e o r e t i c a l duplex are determined on the basis o f the geometry o f a " p e r f e c t " Watson-Crick base p a i r . In this i d e a l s i t u a t i o n the three atoms A - H ^ - B comprising each hydrogen bond are v i r t u a l l y c o l l i n e a r (as measured by C O S Y ^ ^ / 2 with y = TT-'^-23° f r e q u e n t l y reported i n f i b e r d i f f r a c t i o n s t u d i e s {17_ 2 3 ) . According to p o t e n t i a l energy estimates o f pentose pseudorotation i n model nucleosides ( O l s o n , W.K., unpublished d a t a ) , the C2'-endo s t a t e is lower i n energy than the C3'-exo s t a t e . The C2 -endo pucker i s a l s o observed with g r e a t e r frequency i n the known X-ray c r y s t a l s t r u c t u r e s of low molecular weight n u c l e i c a c i d analogs ( 2 4 ) . 9

,

D e t a i l s o f the t h e o r e t i c a l l y p r e d i c t e d double h e l i c a l conformations are evident from the composite contour diagrams o f base s t a c k i n g and hydrogen bonding energies i n Figure 3. The four f i x e d backbone r o t a t i o n angles are here s e t a t the f o l l o w i n g values defined with respect to trans = 0°: = - 4 3 ° , ' = - 3 5 ° , < > f = 45°, i|> = -140°. The g l y c o s y l t o r s i o n x i s maintained a t 75° with respect t o trans arrangements o f o r - C l ' - N 9 - C 4 i n the dA residues and of 0 r - C r - N l - C 2 i n the dT u n i t s . The energies are calculated at increments o f 5° between 85° and 135° i n both u>' and u> and the contours i n the r e s u l t i n g two dimensional g r i d are l o c a t e d by interpolation. The dashed and s o l i d contours i n Figure 3 are i n d i c a t i v e o f h e l i c e s with f a v o r a b l e base s t a c k i n g and hydrogen bonding e n e r g i e s , r e s p e c t i v e l y . These contours are l o c a t e d 4 kcal/mole above the minimum Vs value denoted by + and the minimum V ^ B value l o c a t e d a t * . The surface i s a l s o d i v i d e d i n t o l e f t and right-handed h e l i c a l f i e l d s by the dotted contour o f h = 0 A . A noteworthy f e a t u r e o f F i g u r e 3 i s the l a r g e area o f conformation space a s s o c i a t e d with f a v o r a b l e s i n g l e - s t r a n d e d base stacking. The 4 kcal/mole V $ contour c o i n c i d e s approximately with the s e t o f g e o m e t r i c a l l y acceptable base s t a c k i n g conformations determined p r e v i o u s l y ( 2 5 ) . The low energy base s t a c k i n g occurs e x c l u s i v e l y i n right-handed s t r u c t u r e s f o r the choice o f f i x e d parameters chosen here. T h i s preference f o r right-handed h e l i c e s p e r s i s t s upon minor v a r i a t i o n s i n the B-DNA backbone (26) and occurs a l s o i n the A-DNA h e l i c e s reported p r e v i o u s l y ( 2 2 ) . The


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DIFFRACTION

METHODS

minimum base s t a c k i n g s t a t e of the a) u> p a i r i n Figure 3 appears at (115°, 105°), corresponding to a r e l a t i v e l y t i g h t l y wound h e l i x with n = 9.5 and h = 3.2 A. Adjacent bases i n t h i s c o n f o r mation adopt an almost i d e a l o r i e n t a t i o n , the angle A between the overlapping base planes being 6° and the s e p a r a t i o n of base planes being 3.4 A. According to the V H B contours i n Figures 3, only a small p r o p o r t i o n of p o l y n u c l e o t i d e conformations meet the somewhat rigorous c r i t e r i a of acceptable hydrogen bonding i n DNA double helices. The A^T bases on complementary strands a s s o c i a t e in acceptable Watson-Crick base p a i r i n g arrangements when the backbone i s confined to two narrow ranges centered along the u>' = 10? axis. Conformations i n the smaller domain centered at (105°, 85°), however, introduce severe s t e r i c contacts between consecutive bases i n each strand that r u l e out double h e l i c a l s t r u c t u r e s . Both hydrogen bonding and base s t a c k i n g achieve acceptable values in the l a r g e r area centered a t (105°, 112°). The minimum energy double h e l i x l o c a t e d a t ( 1 0 5 ° , 115°) i s an unusual 1 3 - f o l d s t r u c t u r e with h = 3.6 A. A s l i g h t displacement of the bases away from the h e l i x axis (denoted by * i n Figure 4) introduces a small hole down the core of t h i s complex. As a consequence of molecular s i z e , consecutive A residues in the poly(dA) chain e x h i b i t greater s t a c k i n g overlaps than consecutive T bases of the complementary strand. The planes of complementary A and T bases describe an angle x = 5 i n d i c a t i v e o f a s l i g h t p r o p e l l e r t w i s t i n g of base pairs. 1

o

Minor v a r i a t i o n s of the backbone and g l y c o s y l r o t a t i o n s from the f i x e d values used i n the sample computations above produce a v a r i e t y of t h e o r e t i c a l l y acceptable double h e l i c e s . As evident from the p a r t i a l l i s t of s t r u c t u r e s i n Table I , these s t r u c t u r e s include several 1 0 - f o l d duplexes s i m i l a r to the B-DNA models from f i b e r d i f f r a c t i o n s t u d i e s as well as the l a r g e r 1 3 - f g l d complex. Despite the l a r g e f l u c t u a t i o n s i n h from 1.7 to 4.3 A, the bases a s s o c i a t e at standard s e p a r a t i o n distances (3 A < < 4 A and 2.8 K < < 3.0 A ) and o r i e n t a t i o n s (A < 30° and < 30°) i n a l l cases. In order to avoid severe s t e r i c contacts at small values o f h, the bases may t i l t up to values o f n = 45° with respect to the standard o r i e n t a t i o n (n = 90°) perpendicular to the h e l i x a x i s . Flexible

Helices

The large number of t h e o r e t i c a l l y acceptable duplexes described above suggests that DNA may adopt a v a r i e t y of c l o s e l y related helical structures in solution. A conformational blend of such h e l i c e s w i t h i n a s i n g l e complex o f f e r s a more r e a l i s t i c i n t e r p r e t a t i o n o f configuration-dependent p r o p e r t i e s o f DNA i n s o l u t i o n than the various r i g i d molecular models c u r r e n t l y i n use (27, 28). Chemical exchange s t u d i e s (29, 30, 31, 32, 33) incTicate, however, t h a t the DNA duplexes do not remain p e r f e c t l y


15.

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257

130

120

| M 0

100

90 90

100

110 u)\

120

130

deg

Figure 3. Contour diagram of the base stacking potential energy V of sequential adenine bases and the hydrogen bonding potential energy V B of the complementary A - T base pairs as a function of the phosphodiester rotation angles «/ and o>. The energy contours enclose conformations within 4 kcal/mol of the minima marked by (+) for V and (X) for V . The dotted contour of h = 0 A divides the space into fields according to chirality. 8

H

R

Hn

Figure 4. Detailed molecular representation of the theoretically predicted base stacking and base pairing of A (single lines) and T (solid lines) bases in the low energy o/o> = 105°, 115° helix of Figure 3. The view is drawn perpendicular to the helix axis represented by (+).


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FIBER

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Table 1 Comparative Geometric Parameters of Selected B-DNA Duplexes Duplex

Conformation x , deg

*' , deg

\ deg

a)', deg

a), deg

, deg

*» deg

I

75

-43

-35

105

115

45

-140

II

75

-43

-45

115

120

45

-140

III

95

-43

-25

100

95

45

-140

IV

70

-43

-30

95

130

29

-131

V

90

-43

-30

95

125

29

-131

Parameters Helical n

Base Stacking h, A

, A

A,

deg

Base P a i r i n g l>, A

, deg

I

13.0

3.6

3.4

1.5

2.9

4.5

II

10.3

3.4

3.0

13.2

2.9

9.4

III

10.5

1.7

3.7

26.2

2.9

7.5

IV

10.2

4.3

3.8

16.6

2.8

3.6

V

10.6

7.7

3.1

11.5

3.9

4.0

i n t a c t in s o l u t i o n . The p o l y n u c l e o t i d e backbone a l s o undergoes "breathing" motions that expose the protons involved i n hydrogen bonding to the s o l v e n t . According to the p o t e n t i a l energy surface i n Figure 3, such d i s r u p t i o n s from ideal hydrogen bonding may i n v o l v e only minor conformational changes of the duplex geometry. Furthermore, these s l i g h t changes do not d i s r u p t the base stacking known to p e r s i s t i n a "breathing" DNA system (31). The well-known f l e x i b i l i t y of DNA i n s o l u t i o n very l i k e l y r e f l e c t s both a v a r i e t y of double h e l i c a l s t r u c t u r e s and a number o f c l o s e l y - r e l a t e d nonb a s e - p a i r i n g conformers. As a f i r s t approximation to the treatment of h e l i c e s i n s o l u t i o n , DNA f l e x i b i l i t y i s assumed to a r i s e from the ideal duplex conformer at w'w = (105°, 115°) i n Figure 3 and from the surrounding conformations on t h i s two-dimensional energy surface that preserve base stacking within the l i m i t s 3 A < < 4 A and A < 12°. Using t h i s model, i t i s p o s s i b l e to reproduce the r a d i i of g y r a t i o n observed i n DNA molecules at 25°C ranging between 400 and 5000 base p a i r s i n length (25). These l i m i t e d motions are f u r t h e r c o n s i s t e n t with the observed frequency o f loop formation and c y c l i z a t i o n in DNA of various lengths (34). The large proportion of non-base-paired s t a t e s i n the model ( c f . Figure 6 ) ,


15.

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259

however, i s not c o n s i s t e n t with the measured f r a c t i o n (1/20) o f open states o f DNA at 0°C ( 3 3 ) . An improved model that includes several ideal duplexes and a smaller number o f r e l a t e d non-basep a i r i n g s t r u c t u r e s i s expected to reproduce the known proportion of breathing as well as the chain dimensions i n DNA. The f l e x i b l e h e l i x modeled here i s best described by the e n t i r e array o f conformations i t can assume. A comprehensive p i c t u r e o f t h i s array i s provided by the three-dimensional s p a t i a l p r o b a b i l i t y density f u n c t i o n Wg^r) o f a l l p o s s i b l e end-to-end vectors (25^, 35.). T h i s f u n c t i o n ^ s equal to the p r o b a b i l i t y per u n i t volume i n space that the f l e x i b l e chain terminates a t vector p o s i t i o n £ r e l a t i v e t o the chain o r i g i n 0, as r e f e r e n c e . An approximate p i c t u r e of t h i s d i s t r i b u t i o n f u n c t i o n i s provided by the three f l e x i b l e s i n g l e - s t r a n d e d B-DNA chains o f 128 residues i n Figure 5 ( a ) . The conformations o f these molecules are chosen a t random by Monte Carlo methods (3J5, 36) from the conformations a c c e s s i b l e t o the duplex model. The three molecules are drawn i n a common coordinate system defined by the i n i t i a l v i r t u a l bond o f each s t r a n d . For c l a r i t y , the sugar and base moieties are omitted and the segments are represented by the v i r t u a l bonds connecting successive phosphorus atoms. The three f l e x i b l e chains i n Figure 5(a) possess pseudohelical backbones and pseudohelical axes t h a t change d i r e c t i o n continuously. The i n d i v i d u a l turns o f each pseudohelix vary considerably i n s i z e as a consequence o f the range o f l o c a l nucleotide motions. The gradual bending o f the pseudohelical backbones describes t r a j e c t o r i e s with large r a d i i o f c u r v a t u r e . The end-to-end separations are less than that o f the r e g u l a r h e l i x o f the same length represented i n Figure 5 ( b ) . The small group of random molecules crudely describe a s p a t i a l d i s t r i b u t i o n f u n c t i o n . The f i r s t 2-3 turns o f each pseudohelix are roughly superimposable and a l s o are only s l i g h t l y a l t e r e d from the r e g u l a r h e l i x . A t these chain l e n g t h s , the molecules are r o d l i k e and the d i s t r i b u t i o n of r i s confined to a small domain ( 3 4 ) . Segments more removed from the chain o r i g i n , however, are found to deviate appreciably from the regular structure. The 128-segment s t r u c t u r e s are best described as wormlike c h a i n s . The d i s t r i b u t i o n o f end-to-end vectors o f chains o f t h i s s i z e i s confined t o an umbrella- or mushroom-shaped volume ( 2 5 , 3 4 ) . At longer chain l e n g t h s , the d e v i a t i o n s between the various chain conformers become much more pronounced. Eventually at lengths o f 8,000 to 10,000 n u c l e o t i d e s , the d i s t r i b u t i o n f u n c t i o n i s an i d e a l Gaussian ( 2 5 ) . The p r o b a b i l i t y of occurrence o f end-to-end vectors i s completely random and a l l values of r are e q u a l l y probable. F l e x i b l e double h e l i c e s are expected to f o l l o w approximately the t r a j e c t o r i e s of the DNA-B s i n g l e strands o u t l i n e d above. According to the measurements o f DNA b r e a t h i n g , at l e a s t 95% of the motions o f a double-stranded u n i t are l i m i t e d to s t a t e s that preserve Watson-Crick base p a i r i n g . The extent to which r o t a t i o n s of the p a i r about the 1 3 - f o l d h e l i x f u l f i l l t h i s c r i t e r i o n can CD'CD


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METHODS

Figure 5. Computer generated perspective representation of single-stranded BDNA chains. The 128-residue chains are represented by the sequence of virtual bonds connecting successive phosphorus atoms, (a) Three representative flexible helices generated by Monte Carlo methods; (b) the regular co'o> = 105°, 115° helix predicted by potential energy methods.


15.

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be estimated from the contour surface i n Figure 6. Here the two strands o f the duplex a r e not required t o assume i d e n t i c a l conformations. The s o l i d l i n e s i n the f i g u r e are contours o f the d i s t a n c e o f s e p a r a t i o n d between C T atoms on adjacent residues within d i f f e r e n t single-stranded helical units. Upon v a r i a t i o n of a)' and a) over the range o f phosphodiester r o t a t i o n s used above t o d e s c r i b e f l e x i b l e DNA s i n g l e s t r a n d s , the parameter d i s found to vary between 4 and 5 K. In c o n t r a s t to the gradual changes i n d , the dashed contours d ' are more c l o s e l y spaced and thus are more s t r o n g l y dependent upon the u>' and u> r o t a t i o n s . The l a t t e r p a r a meter i s the C l s e p a r a t i o n d i s t a n c e between two adjacent f r e e bases t h a t are hydrogen bonded with i d e a l Watson-Crick geometry to the s i n g l e - s t r a n d e d h e l i c a l backbone. Duplex formation i s p o s s i b l e i f the complementary chain can assume a backbone arrangement t h a t l i n k s these two f r e e bases. In Figure 6 t h i s complement a r y s t r a n d conformation i s estimated as a s t a t e with a value o f d i d e n t i c a l to the d ' value r e q u i r e d by the s t a t e o f the f i r s t strand. The s i n g l e p o i n t denoted by x where d = d = 4 . 8 A i n both strands i s the i d e a l duplex p r e d i c t e d i n Figure 3. The motions o f a f l e x i b l e duplex a r e thus l i m i t e d i n Figure 6 t o the shaded area where d = d ' = 4-5 A . 1

1

Summary The above p o t e n t i a l energy method provides a convenient way to i d e n t i f y d i r e c t l y those conformations o f the n u c l e i c a c i d backbone and base that can p a r t i c i p a t e i n double h e l i x f o r m a t i o n . While these "energies" are n e c e s s a r i l y approximate, they a f f o r d a basis f o r c l e a r d i s c r i m i n a t i o n between s t e r i c a l l y allowed and s t e r i c a l l y forbidden s t r u c t u r e s . The "energy" approach a l s o o f f e r s a means to e x t r a p o l a t e experimental s t u d i e s (nmr, X - r a y , e t c . ) on the conformation o f small model compounds to the p o l y n u c l e o t i d e l e v e l and to t e s t the relevance o f the data i n a h e l i c a l complex. In a d d i t i o n , the method provides a s t a r t i n g point f o r a r e f i n e d p o t e n t i a l energy a n a l y s i s of double h e l i c a l conformation and s t a b i l i t y . The contour surfaces that relate l o c a l motions o f the backbone to h e l i c a l "energies" provide a d d i t i o n a l i n s i g h t i n t o the c o n f o r mational nature o f h e l i c e s i n s o l u t i o n . The continuous v a r i a t i o n of r o t a t i o n angles permits a d i r e c t molecutar treatment o f chain f l e x i b i l i t y and o f various s o l u t i o n properties o f helices ( i . e . , l i g h t s c a t t e r i n g dimensions, c y c l i z a t i o n and loop c l o s u r e p r o b a b i l i t i e s , extent o f " b r e a t h i n g " , e t c . ) . E a r l i e r models o f DNA i n s o l u t i o n i n v o l v e e i t h e r s p e c i f i c r e g u l a r h e l i c a l s t r u c t u r e s (27, 28) or approximate a r t i f i c i a l d e s c r i p t i o n s such as r i g i d r o d , wormlike c o i l (37, 3 8 ) , o r Gaussian. The computations d e s c r i b e d here a l s o c l a r i f y why the various d e s c r i p t i o n s o f DNA apply a t d i f f e r e n t chain l e n g t h s . The configuration-dependent p r o p e r t i e s of DNA o f a l l s i z e s r e f l e c t very l i m i t e d l o c a l molecular motions. With i n c r e a s e i n chain length the motions o f the DNA as a whole


262

FIBER

90

100

110

120

DIFFRACTION

METHODS

130

a/, deg.

Figure 6. Contour diagram of the distances d and d' between successive Cl' atoms in a DNA duplex as a function of the phosphodiester angles. The (X) denotes the conformation of the ideal theoretical helix of Figure 3 with d = d' = 4.8 A. The shaded area on the diagram describes the local motions that preserve base stacking in a flexible duplex. See text for further explanation.


15.

OLSON


magnify g r a d u a l l y and approach e v e n t u a l l y random c o i l .

263

the behavior o f an i d e a l

L i s t o f Symbols 1.

Mathematical < >

The s t a t i s t i c a l mechanical average o f the q u a n t i t y e n c l o s e d , taken over a l l c o n f i g u r a t i o n o f the c h a i n . The matrix X o f order 3 x 3 . The chain v e c t o r connecting the ends of the p o l y nucleotide. The o r i g i n o f the c h a i n .

T 7 *~ 0, 2.

Roman L e t t e r Symbols d

Distance between C l atoms on adjacent residues i n a particular helical unit, Distance between the C l ' atoms t h a t are attached to two f r e e bases bound to a p a r t i c u l a r h e l i c a l u n i t . Deoxyadenosine. Deoxythymidine. Distance between the terminal heavy atoms o f a hydrogen bond, H e l i c a l step h e i g h t , H e l i c a l repeat = 2-ir/e. C y l i n d r i c a l radius a s s o c i a t e d with atom i . Length o f the v i r t u a l bond connecting s u c c e s s i v e P atoms o f the p o l y n u c l e o t i d e . P o t e n t i a l energy o f hydrogen bonding. P o t e n t i a l energy of base s t a c k i n g . D i s t r i b u t i o n f u n c t i o n o f the vector jr, i n three dimensions with respect to the chain o r i g i n as reference, Cartesian axis, Cartesian axis, Cartesian a x i s . C y l i n d r i c a l v e r t i c a l displacement a s s o c i a t e d with atom i . Separation of an atom i n base i + 1 from the plane o f base i . 1

d" dA dT D h n r.| v VHB Vs Wo(r) w

x y z z-j Z

3.

Conventions

Greek L e t t e r Symbols Y e e.-

Angle described by the three atoms comprising a hydrogen bond. The c y l i n d r i c a l repeating a n g l e , i n r a d i a n s , o f the polynucleotide h e l i x , C y l i n d r i c a l r o t a t i o n angle a s s o c i a t e d with atom i .


264

FIBER

A T < > f ' x •

1

,

,

,

,

^

METHODS

The complement to the pseudovalence angle formed by successive v i r t u a l bonds. Angle described by the planes o f adjacent bases i n the same p o l y n u c l e o t i d e s t r a n d . Angle described by the planes o f two bases arranged i n a hydrogen-bonding geometry. Torsion angle about atoms P - 0 5 ' - C 5 ' - C 4 . Torsion angle about atoms C 4 - C 3 - 0 3 ' - P . Glycosyl t o r s i o n angle between base and pentose. Torsion angle about atoms 0 5 - C 5 ' - C 4 - C 3 . T o r s i o n angle about atoms C 5 - C 4 - C 3 ' - 0 3 ' . The pseudo r o t a t i o n angle defined with respect to a 0° = trans arrangement of three successive v i r t u a l bonds. T o r s i o n angle about atoms 0 3 - P - 0 5 - C 5 " . Torsion angle about atoms C S ' - O S ' - P - O S ' .

V

0

DIFFRACTION

v

< * > w

,

1

,

l

,

,

Acknowledgment The author i s g r a t e f u l t o the National I n s t i t u t e s o f Health (USPHS Grant GM 20861) and the Charles and Johanna Busch Memorial Fund of Rutgers U n i v e r s i t y f o r laboratory support and to the Center f o r Computer and Information S e r v i c e s of Rutgers U n i v e r s i t y f o r computer time. A Career Development Award from the USPHS (GM 00155) and an A. P. Sloan Fellowship to W.K.O. are a l s o g r a t e f u l l y acknowledged. Literature

Cited

1. 2.

O l s o n , W. K.; F l o r y , P. J. Biopolymers, 1972, 11, 25-56. Sasisekharan, V . , i n "Conformation of B i o l o g i c a l Molecules and Polymers"; Bergmann, E. D.; Pullman, B . , Eds. Fifth Jerusalem Symposia on Quantum Chemistry and Biochemistry, 1973; pp. 247-260. 3. Y a t h i n d r a , N . ; Sundaralingam, S . , i n " S t r u c t u r e and Conformat i o n o f N u c l e i c Acids and P r o t e i n - N u c l e i c A c i d I n t e r a c t i o n s " ; Sundaralingam, M.; Rao, S. T., Eds. U n i v e r s i t y Park P r e s s : B a l t i m o r e , MD, 1975; pp. 649-676. 4. Thornton, J. M.; Bayley, P. M. Biochem. J., 1975, 149, 585-596. 5. Broyde, S. B . ; W a r t e l l , R. M.; S t e l l m a n , S. D.; H i n g e r t y , B.; Langridge, R. Biopolymers, 1975, 14, 1597-1613. 6. Pullman, B.; Saran, A. Prog. N u c l e i c Acids Res. and Molec. Biol., 1976, 18, 216-326 and references c i t e d t h e r e i n . 7. G o v i l , G. Biopolymers, 1976, 15, 2302-2308. 8. T o s i , C.; Clementi, E . ; Matsuoka, O. Biopolymers, 1978, 17, 67-84. 9. T h i y a g a r a j a n , P.; Ponnuswamy, P. K. Biopolymers, 1978, 17, 533-553 and 2143-2158. 10. Broch, H . ; V a s i l e s c u , D. Biopolymers, 1979, 18, 909-930.


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265

11. O l s o n , W. K. Biopolymers, 1975, 14, 1775-1795. 12. O l s o n , W. K. Biopolymers, 1976, 15, 859-878. 13. Y a t h i n d r a , N . ; Sundaralingam, M. N u c l e i c Acids R e s . , 1976, 3, 729-747. 14. Fujii, S . ; Tomita, K. N u c l e i c Acids R e s . , 1976, 3 , 1973-1984. 15. H i n g e r t y , B . ; Broyde, S. N u c l e i c Acids R e s . , 1978, 5, 127-137. 16. S t e l l m a n , S . D.; H i n g e r t y , B . ; Broyde, S . ; Subramanian, E . ; S a t o , T.; Langridge, R. Biopolymers, 1973, 12, 2731-2750. 17. A r n o t t , S . ; Chandresekaran, R.; S e l s i n g , E . , i n " S t r u c t u r e and Conformation o f N u c l e i c Acids and P r o t e i n - N u c l e i c A c i d I n t e r a c t i o n s " ; Sundaralingam, M . ; Rao, S . T., E d s . U n i v e r s i t y Park Press: B a l t i m o r e , MD, 1975; pp. 577-596 and references cited therein. 18. Z h u r k i n , V. B . ; Lysov, Yu. P . ; Ivanov, V. I . Biopolymers, 1978, 17, 377-412. 19. Miller, K. J. Biopolymers, 1979, 18, 959-980. 20. O l s o n , W.. K. Macromolecules. 1975, 8 , 272-275. 21. Shimanouchi, T.; Mizushima, S. J. Chem. P h y s . , 1975, 23, 707-711. 22. O l s o n , W. K. Biopolymers, 1978, 17, 1015-1040. 23. A r n o t t , S . ; S e i s i n g , E. J. M o l . Biol., 1974, 88, 509-521. 24. Murray-Rust, P . ; Motherwell, S. Acta C r y s t . , 1978, B34, 2534-2546. 25. O l s o n , W. K. Biopolymers, 1979, 18, 1213-1233. 26. The two-dimensional ω'ω energy surfaces of each o f the t h e o r e t i c a l duplexes listed in Table I e x h i b i t the same general f e a t u r e s o f Figure 3. 27. Hogan, M . ; Dattagupta, N . ; C r o t h e r s , D. M. P r o c . N a t l . Acad. Sci. USA, 1978, 75, 195-199. 28. L e v i t t , M. P r o c . N a t l . Acad. Sci. USA, 1978, 7 5 , 640-644. 29. P r i n t z , M. P . ; von H i p p e l , P. H. P r o c . N a t l . Acad.Sci.USA, 1965, 53, 363-370. 30. Frank-Kamenetskii, M. D . ; L a z u r k i n , Yu. S. Ann. Rev. Biophys. B i o e n g . , 1974, 3 , 127-150. 31. Teitelbaum, H . ; Englander, S. W. J. M o l . Biol., 1975, 92, 55-78 and 79-92. 32. McGhee, J. C.; von H i p p e l , P. H. Biochem., 1975, 14, 12811296 and 1297-1303. 33. K a l l e n b a h , N. R.; Mandal, C.; Englander, S. W., i n " S t e r e o dynamics o f Molecular Systems"; Sarma, R. H . , E d . Pergamon Press: New York, 1979; pp. 271-282. 34. O l s o n , W. K., i n "Stereodynamics o f Molecular Systems"; Sarma, R. H . , E d . Pergamon P r e s s : New York, 1979; pp. 297-314. 35. Y e v i c h , R.; O l s o n , W. K. Biopolymers, 1979, 18, 113-145. 36. J o r d a n , R. C.; B r a n t , D. A.; Cesaro, A . Biopolymers, 1978, 17, 2617-2632. 37. Kratky, O.; Porod, G. Rec. T r a v . Chim. Pays-Bas, 1949, 6 8 , 1106-1122. 38. Yamakawa, H . ; Shimada, J.; Fujii, M. J. Chem. P h y s . , 1978, 68, 2140-2150. RECEIVED June 2 4 , 1 9 8 0 .


Theoretical Estimates of Helical Structure in Polynucleotides

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