The Variable Virtual Bond - ACS Symposium Series (ACS Publications)

13 The Variable Virtual Bond Modeling Technique for Solving Polymer Crystal Structures PETER ZUGENMAIER

Downloaded by UNIV OF NEW SOUTH WALES on April 16, 2016 | http://pubs.acs.org Publication Date: November 17, 1980 | doi: 10.1021/bk-1980-0141.ch013

Institute of Macromolecular Chemistry, University of Freiburg, D-7800 Freiburg i.Br., West Germany ANATOLE SARKO Department of Chemistry, State University of New York, College of Environmental Science and Forestry, Syracuse,NY13210

Although various procedures are available for the model analysis of fibrous polymers, methods based on the virtual bond representation of the asymmetric residue may be of advantage in many cases. In the following, we describe one such method that began with simple procedures applied to polysaccharides, but has now been refined into a flexible and powerful model analysis tool that is simple to use with any class of polymer. Its use i n the present case, however, is illustrated with examples drawn from the structure analysis of polysaccharides. The Virtual Bond Method The earliest attempts at model analysis of polysaccharides typified by the x-ray crystal structure analysis of amylose tri-acetate - were usually conducted i n three steps (1). In the first step, a model of the chain was established which was i n agreement with the fiber repeat and the lattice geometry, as obtained from diffraction data. In the second step, the invariant chain model was packed into the unit c e l l , subject to constraints imposed by nonbonded contacts. This was followed, i n the third step, by efforts to reconcile calculated and observed structure factor amplitudes. It was quickly realized that helical models of polysaccharide chains could be easily generated and varied using the virtual bond method. Figure 1 illustrates the generation of a two-fold helical model of a {±+k)-linked polysaccharide chain. The virtual bond* VB, i s the vector linking successive glycosidic (bridge) oxygens. The starting point of this vector has coordi(VB^ - h^)l/2 nates X, \ »* 1 = 0;* whereas for the end1 = 0,' v, ^1 = - ~r—."" 2 sin \| 'A/21 point they are = -yj_ sin A, 1/2 = y± cos A., 3 = h. (VB = length of the virtual bond; h = axial rise per residue, here o/2\ n = number of residues per helix repeat, here 2; A = 27ft/n; ±t = number of turns i n repeat (+ right- and - left-handed). Provided a suitable model exists for the monomer residue, a reasonably correct conformation of the chain is obtained simply by rotating the K /rs

2

0-8412-0589-2/80/47-141-225$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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Figure 1. Construction of a two-fold helical model of a polysaccharide with the virtual bond method. Increasing the length VB of the virtual bond is shown by the dashed line.

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


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e n t i r e r e s i d u e a b o u t t h e virtual bond u n t i l t h e b r i d g e a n g l e x i s w i t h i n t h e e x p e c t e d r a n g e . F o r most p o l y s a c c h a r i d e s , s u c h r o t a t i o n y i e l d s t w o c o n f o r m a t i o n s w i t h t h e p r o p e r a n g l e x , a s shown i n F i g u r e 2. One o f t h e t w o c h o i c e s i s g e n e r a l l y r u l e d o u t b e c a u s e o f e x c e s s i v e , s h o r t nonbonded c o n t a c t s w i t h i n t h e c h a i n . A l t h o u g h o t h e r methods a r e a v a i l a b l e t o c o n s t r u c t m o d e l s o f p o l y s a c c h a r i d e h e l i c e s , such a s b y r o t a t i o n s about t h e two bonds l e a d i n g t o t h e b r i d g e o x y g e n ( t h e § a n d r o t a t i o n s ) , t h e virtual bond method p o s s e s s e s s e v e r a l a d v a n t a g e s . W i t h i t , a h e l i x w i t h g i v e n n a n d h c a n b e c o n s t r u c t e d e a s i l y , a n d o n l y one v a r i a b l e r o t a t i o n a b o u t t h e virtual bond - i s n e e d e d f o r g r o s s changes o f conformation. The c o n s e q u e n c e s o f c h a n g i n g t h e l e n g t h o f VB s u c h as c h a n g e s i n t h e h e l i x d i a m e t e r a n d t h e b r i d g e a n g l e x , a r e e a s i l y p r e d i c t a b l e , a s shown i n F i g u r e 1. M o s t i m p o r t a n t l y , m o d e l r e f i n e ment w i t h t h i s method i s s i m p l e , a s d e s c r i b e d i n t h e f o l l o w i n g sections. 9

M o d e l B u i l d i n g a n d R e f i n e m e n t w i t h t h e V i r t u a l Bond M e t h o d As shown i n F i g u r e 1 , o n l y t h e p o s i t i o n s o f t h e r e p e a t atoms o f t h e monomer r e s i d u e a r e s p e c i f i e d b y t h e virtual bond. W i t h i n t h i s constraint, considerable latitude i s available f o rthep o s i t i o n s o f a l l o t h e r atoms o f t h e r e s i d u e . T h e s e atoms c a n b e d e s c r i b e d i n t w o a l t e r n a t e w a y s , a s shown i n F i g u r e 3. I n t h e f i r s t m e t h o d , a s t r i n g o f c o n n e c t e d atoms e x t e n d s f r o m t h e l o w e r t o t h e u p p e r atom o f t h e virtual bond (i.e., f r o m Oh t o 0 1 i n F i g u r e 3 A ) . A l l o t h e r atoms o f t h e r e s i d u e n o t i n t h i s m a i n s t r i n g a r e p l a c e d i n separate s t r i n g s , w h i c h a r e a t t a c h e d as branches t o t h e main string. ( F i x e d h y d r o g e n s , e.g., t h o s e a t t a c h e d t o r i n g c a r b o n s , could be t r e a t e d i d e n t i c a l l y . However, i t i s s i m p l e r t o c a l c u l a t e t h e i r p o s i t i o n s when n e e d e d , i n a c c o r d a n c e w i t h p r e s e l e c t e d C-H b o n d l e n g t h s a n d a s s o c i a t e d b o n d a n g l e s ) . The p o s i t i o n o f e a c h atom i n a s t r i n g i s e x p r e s s e d b y p o l a r c o o r d i n a t e s r , 0, (j>, w h e r e r i s t h e b o n d l e n g t h , 6 i s t h e b o n d a n g l e , a n d i s t h e c o n f o r m a t i o n a n g l e , a l l r e l a t i v e t o p r e v i o u s atoms. (.These c o o r d i n a t e s a r e i l l u s t r a t e d f o r atom 05 i n F i g u r e 3 A ) . C o n v e r s i o n b e t w e e n t h e p o l a r a n d c a r t e s i a n c o o r d i n a t e s i s e a s i l y a c c o m p l i s h e d , whenever n e e d e d . When a b o n d a n d i t s a n g l e s c a n n o t b e e x p r e s s e d b y p o l a r c o o r d i n a t e s a s s o c i a t e d w i t h a n a t o m , s u c h a s t h e " o p e n " b o n d shown b y a d a s h e d l i n e i n F i g u r e 3A, i t s l e n g t h a n d a l l d e s i r e d a n g l e s can s t i l l b e e x p l i c i t l y d e f i n e d . A l l b o n d l e n g t h s a n d a n g l e s , i n c l u d i n g t h e l e n g t h o f t h e virtual bond a n d t h e a n g l e s a s s o c i a t e d w i t h i t , c a n now b e t r e a t e d a s v a r i a b l e s d u r i n g r e f i n e m e n t . The m o d e l o f t h e r e s i d u e c a n a l s o b e d e s c r i b e d b y a s e c o n d p r o c e d u r e , shown i n F i g u r e 3B. Two s t r i n g s o f atoms a r e u s e d , b e g i n n i n g a t s e p a r a t e ends o f t h e virtual bond. T h e s t r i n g s a r e n o t c o n n e c t e d t o one a n o t h e r , l e a v i n g t w o " o p e n " b o n d s . T h i s m e t h o d i s u s e f u l when t h e l e n g t h o f t h e virtual bond i s t o r e m a i n f i x e d during refinement. The g o a l o f m o d e l b u i l d i n g i s t o p r o d u c e a p o l y m e r c h a i n t h a t


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Figure 3. Two alternate methods of describing an a-D-glucose residue using the variable virtual bond method. The bond length, bond angle, and conformation angle for atom 05 are shown as r , 0 , . 5

5

5


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p o s s e s s e s s o u n d s t e r e o c h e m i c a l f e a t u r e s a n d , a t t h e same t i m e , i s i n g o o d agreement w i t h d i f f r a c t i o n d a t a . T h e s e r e q u i r e m e n t s c a n be met, a s shown i n F i g u r e h b y p r o p e r u s e o f i n f o r m a t i o n a v a i l a b l e from e x p e r i m e n t , and model r e f i n e m e n t b a s e d o n t h e o r e t i c a l p r i n c i p l e s (2 3). I n t h e refinement o f t h e model, a l l bond l e n g t h s , bond angles and conformation angles a r e o p t i m i z e d r e l a t i v e t o a set o f standards, simultaneously with the p o s i t i o n o f the chain i n the u n i t c e l l . The r e f i n e m e n t i s c a r r i e d o u t b y m i n i m i z i n g t h e f u n c t i o n (^,5.): N Y = T STD"? (A. - A . ) + W~ T w ^ C d . , - d . . ) (1) i=l 5=1 J=l where t h e f i r s t t e r m r e p r e s e n t s t h e b o n d e d a n d t h e s e c o n d t e r m t h e nonbonded i n t e r a c t i o n s . I nt h i s equation, i s any c a l c u l a t e d bond l e n g t h , bond o r t o r s i o n a n g l e ; A ^ i s an average o r s t a n d a r d v a l u e o f A^; STD ^_ i s a w e i g h t o r s t a n d a r d d e v i a t i o n o f A ^ N is t h e number o f b o n d e d i n t e r a c t i o n s i n t h e r e f i n e m e n t ; dQ±^ i s t h e nonbonded e q u i l i b r i u m d i s t a n c e b e t w e e n atoms i a n d j; d^ i s t h e a c t u a l nonbonded d i s t a n c e b e t w e e n atoms i, Q; n i s t h e number o f nonbonded c o n t a c t s a n d i s t h e o v e r a l l weight f a c t o r which b a l a n c e s t h e b o n d e d a n d nonbonded i n t e r a c t i o n s . The s t a n d a r d v a l u e s A ^ f o r t h e b o n d l e n g t h s , b o n d a n g l e s a n d conformation angles can be o b t a i n e d b y a v e r a g i n g from s i n g l e - c r y s t a l s t r u c t u r e s o f carbohydrates (6). T h i s a l s o y i e l d s t h e c o r r e s p o n d i n g s t a n d a r d d e v i a t i o n s STD i. The e q u i l i b r i u m nonbonded d i s t a n c e s d ±j a n d t h e i r a s s o c i a t e d w e i g h t s have l i k e w i s e been d e t e r m i n e d f r o m known c r y s t a l s t r u c t u r e s o f c a r b o h y d r a t e s (7.). The a c t u a l values o f t h e constants used f o r p o l y s a c c h a r i d e s are given i n T a b l e s I a n d I I . F o r a good b a l a n c e b e t w e e n t h e t w o t e r m s o f eq. (.1), a v a l u e o f 0.5 i s a p p r o p r i a t e f o r t h e o v e r a l l w e i g h t W. 9

9

n


2

0

1

1

0

2

2

0

1

1

J

Q

Q

Q

9

0

0

Q

Table I C o n s t a n t s f o r t h e nonbonded r e p u l s i o n t e r m o f E q . ( l ) . (When ^ i j > d o i j , w = 0 e x c e p t f o r t h e h y d r o g e n b o n d ) . 9

Interaction

type

C C C 0 C • • • .H

0 . . . .0 0 H H • •. .H 0 0 (H-bond)

d

Q9

A

3.70 3.60 3.30 3.60 3.25 3.20 2.80

W

3.00 3.00 1.35 3.00 1.U0 0.50 20.00

I t should be c l e a r t h a t a refinement based on t h e minimizat i o n o f t h e f u n c t i o n Y r e s u l t s i n a s t r u c t u r e o f minimum s t e r i c


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Table II Average bond l e n g t h s , bond angles and t o r s i o n angles f o r an a - D glucose residue shown i n Figure 3. Included are lower and upper l i m i t s and average standard deviations ( 6 ) .


Bond Lengths (&) 1.U15 (1.U05-1.U35)

C(l)-OU) C(l)-0(5) C(5)-0(5) C(U)-0(U) C(3)-C(U) C(2)-C(3) CC2)-0(2} C(3)-0(3) C(5)-C(6) C(6)-0(6) 0(U)..0(l)

l.UlU

(1.392-1.^28)

1.U36 1.U26 1.523 1.521

Cl.U25-l.U6U) Cl.U09-l.UU6) Cl.509-1.537) Cl.508-1.536)

I.51U

Cl.U95-l.53U)

1.U23 C l . U l l - l . U U o l I.U29 Cl.Ul0-l.UU6)

I.U27 C l . U i 5 - l . U U 2 ) v a r i a b l e CU.10-U.6o) 0.01 A

Ave. STD Bond Angles (degrees)

o(U)-od). •cCi o ( i ) - c d ) . •0(5

7 ».0 111.6 llU.O 1*5.5 105.5 110.5 110.8 109.7 106.9 111.8 1

C(1)-0C5)- •C(5 OCl)-O(U).

•c(U

oCU)-c(U).•C(3 C(U)-C(3)- •C(2 C(3)-CC2)- •0(2 C(10-C(3)- •0(3 0(5)-C(5)- •CC6 C(5)-C(6)- •0(6 Gycosidic bond angle

variable 1.5°

Ave. STD Torsion Angles

8,

(degrees)

COO-OCU). • 0 ( l ) - C ( l ) ° 0(U)-0(1). • C ( l ) - 0 ( 5 ) 0 ( 1 ) - C ( 1 ) . •OC5)-CC5). •C(U)-C(3) •C(3)-C(2) OCU)-C(U).•C(2)-0(2) C(U)-C(3)- •C(3)-0(3) OCU)-C(U)- •C(5)-C(6) C ( l ) - 0 ( 5 ) - •C(6)-0(6) 0(5)-C(5)-

b

od)-o(U).

(71.0.'-77.0) (109. 8-112.7) (113 2-11h.7) -U8.5) (103.6-112.lt) (106. 0-113.6) (106. U-113.2) (106. 5 - H 2 . 5 ) (106. 8-107.9) (109. U-113.8)

b

- 2 . 0 (±5) - 5 7 . 6 (±5) 57-7 (±5) - 6 0 . 1 (±5) 168.0 (±5) -177.9 (±5) -69.O (±5) -nh.k (±5) variable 0


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Table Torsion Angles

231


I I Ccontinued)

3 ,

3.0°

A v e . STD

Convention f o r t o r s i o n angles: 0° when b o n d s A-B a n d C-D exeois; c l o c k w i s e r o t a t i o n o f b o n d C-D r e l a t i v e t o A-B i s p o s i t i v e .


I n v o l v e s t h e virtual

bond.

'Any d e s i r e d v a l u e c a n b e u s e d ; h o w e v e r , t h e t h r e e s t a g g e r e d c o n f o r m a t i o n s , d e n o t e d gg tg a n d gt have t o r s i o n a n g l e s -60°, 180° and 60°, respectively. y

CRYSTAL

9

AND

MOLECULAR

DETERMINATION

STRUCTURE

OF POLYMERS

EXPERIMENT

THEORY

X-ray fiber diagram

molecular data of oligomer

potential

fiber

single

calculations

repeat

crystals

energy

CONFORMATION ANALYSIS

size and symmetry

possible

of the unit

limited by fiber repeat

cell

PACKING

conformations

ANALYSIS

diffraction

stereochemical^

intensities

models

reasonable

REFINEMENT

AGAINST

DIFFRACTION

DATA

|crystal and m o l e c u l a r

Figure 4.

potential energies of contact pairs

structure!

The strategy of determining the crystal and molecular structure of polymers based on model refinement


232

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e n e r g y . T h i s i s t r u e e v e n f o r t h e f u n c t i o n a s w r i t t e n i n e g . Cl) w h i c h y i e l d s a n e m p i r i c a l , u n i t - l e s s v a l u e o f t h e " e n e r g y " Y. A t r u e energy can be o b t a i n e d b y s u b s t i t u t i n g proper f o r c e c o n s t a n t s f o r STD ^ u s i n g t o r s i o n a l p o t e n t i a l s f o r c o n f o r m a t i o n a n g l e s , a n d s u b s t i t u t i n g Lennard-Jones o r Buckingham p o t e n t i a l s f o r t h e quadr a t i c nonbonded t e r m . Even though such f u n c t i o n s add r i g o r t o t h e p r o c e d u r e , e x p e r i e n c e h a s shown t h a t t h e y a d d l i t t l e t o t h e r e f i n e m e n t , w h i l e i n c r e a s i n g demands f o r c o m p u t e r t i m e . The a s s u m p t i o n u n d e r l y i n g t h e p r e d i c t i o n o f a minimum-energy s t r u c t u r e , o f g i v e n n and h i s t h a t i t i s i d e n t i c a l w i t h t h e c r y s t a l s t r u c t u r e . T h i s i s g e n e r a l l y t r u e , a l t h o u g h agreement may n o t b e p r e s e n t i n a l l d e t a i l s . A t t i m e s , more t h a n one minimume n e r g y s t r u c t u r e may e x i s t . T h e r e f i n e m e n t s h o u l d , t h e r e f o r e , continue w i t h b r i n g i n g t h e c a l c u l a t e d and observed s t r u c t u r e f a c t o r a m p l i t u d e s i n t o a g r e e m e n t , a s shown i n F i g u r e k. T h i s i s done by r e f i n i n g t h e same p a r a m e t e r s o p t i m i z e d i n t h e s t e r e o c h e m i c a l refinement, except t h a t t h e c r i t e r i o n o f refinement i s the m i n i m i z a t i o n o f t h e r e s i d u a l s R = £ | | F | | - | | F | | / I | F | o r /?" = ( > ( | F | l ol) /Z l ol ' • "these e q u a t i o n s , F a n d F a r e t h e c a l c u l a t e d and observed s t r u c t u r e f a c t o r amplitudes, r e s p e c t i v e l y , and W are the weights assigned t o i n d i v i d u a l r e f l e c t i o n s . The r e s i d u a l R" i s p r e f e r r e d o v e r R, b e c a u s e i t a l l o w s t h e u s e o f r e f l e c t i o n weights. F i n a l l y , t h e r e a r e c a s e s where a c o m b i n e d r e f i n e m e n t b a s e d on s i m u l t a n e o u s m i n i m i z a t i o n o f Y a n d R" Cor R) may b e n e c e s s a r y . F o r e x a m p l e , when t h e number o f r e f l e c t i o n s i s s m a l l , p u r e x - r a y r e f i n e m e n t may i n t r o d u c e u n a c c e p t a b l e s t e r e o c h e m i c a l f e a t u r e s . T h i s i s guarded a g a i n s t by m i n i m i z i n g a l i n e a r combination o f Y and R" i n t h e f o r m o f a f u n c t i o n $ = f R " + ( l - f ) Y , where t h e f r a c t i o n a l weight / i s chosen t o b a l a n c e t h e two terms. Good r e s u l t s have b e e n o b t a i n e d w i t h / r a n g i n g f r o m 0.9 t o O.985, w h i c h u s u a l l y w e i g h t s t h e R" t e r m a t l e a s t e q u a l l y w i t h t h e Y t e r m . The m o d e l d e s c r i p t i o n a n d r e f i n e m e n t b a s e d o n t h e virtual bond method n e e d n o t b e r e s t r i c t e d t o a s i n g l e monomer r e s i d u e . Any number may b e u s e d , w i t h o n l y one virtual bond n e e d e d t o s p a n a l l of theresidues. 0

9


3

F

w

F

J

C

I

O

q

C

n

c

Q

Method o f C o n s t r a i n e d O p t i m i z a t i o n The c o n s t r a i n e d o p t i m i z a t i o n p r o c e d u r e , o r i g i n a l l y d e v e l o p e d f r o m t h e s i m p l e x method a n d f i r s t d e s c r i b e d b y Box, i s i d e a l l y s u i t e d t o m o d e l r e f i n e m e n t C8). I t i s a s e a r c h method t h a t s e a r c h es f o r t h e minimum o f a m u l t i d i m e n s i o n a l f u n c t i o n w i t h i n g i v e n i n tervals. I t p o s s e s s e s a l l t h e a d v a n t a g e s o f s e a r c h m e t h o d s , among them t h a t c a l c u l a t i o n o f d e r i v a t i v e s i s n o t n e c e s s a r y , a t e s t t o assure t h e independence o f v a r i a b l e s can be o m i t t e d , and d i v e r s e v a r i a b l e s c a n be e a s i l y i n c l u d e d . These a r e e x a c t l y t h e r e q u i r e ments o f m o d e l r e f i n e m e n t where b o n d l e n g t h s , b o n d a n g l e s , t o r s i o n a n g l e s , and o t h e r parameters a r e used w i t h i n e x p e r i m e n t a l l y defined limits.


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The p r i n c i p l e s o f t h e method c a n b e u n d e r s t o o d w i t h t h e h e l p o f F i g u r e 5. The minimum o f a f u n c t i o n Fix) i s d e t e r m i n e d w i t h i n the l i m i t s and # . The v a r i a b l e x r e p r e s e n t s a s e t o f n v a r i a b l e s (x^_ ) t o which a d d i t i o n a l c o n s t r a i n t s other t h a n t h e l i m i t s may a p p l y . F o r i n s t a n c e , X]_ may b e one p o i n t f u l f i l l i n g a l l t h e c o n d i t i o n s , t h u s F(x±) i s t h e v a l u e o f t h i s f u n c tion at A d d i t i o n a l k p o i n t s (#2* 2* •••> ^ k ) generated i n a random manner w i t h i n t h e g i v e n l i m i t s a n d t h e v a l u e s o f t h e f u n c t i o n F(x2)> FCx^) a r e c a l c u l a t e d . The l a r g e s t f u n c t i o n v a l u e , F(x±) i n F i g u r e 5, i s r e p l a c e d b y a new F(x^) f o r a t r i a l p o i n t w h i c h i s a t x + a[x -x±), where x i s t h e c e n t r o i d o f t h e r e m a i n i n g p o i n t s ( a good v a l u e f o r a i s 1 . 3 ) . I f t h i s t r i a l p o i n t r e p r e s e n t s no i m p r o v e m e n t , i t i s moved h a l f w a y t o w a r d s t h e c e n t r o i d t o g i v e a new t r i a l p o i n t xj_. The p r o c e d u r e i s t h e n r e p e a t ed. I f t h e t r i a l p o i n t i s r e f l e c t e d o u t s i d e an i n t e r v a l l i m i t , i t is reset t ojust the inside ofthe l i m i t . As long as t h e p o i n t s h a v e n o t c o l l a p s e d i n t o t h e minimum, t h e p r o c e d u r e i s r e p e a t e d with a l l points i n turn. The u s e o f k > n+1 p o i n t s e n s u r e s t h a t t h e c o m p l e x does n o t c o l l a p s e i n t o a subspace. F a l s e minima a r e n o r m a l l y e l i m i n a t e d t h r o u g h t h e p r o c e d u r e o f r e f l e c t i n g a p o i n t about t h e c e n t r o i d , and b e c a u s e t h e s e t o f p o i n t s i s d i s t r i b u t e d o v e r t h e w h o l e i n t e r val. I f a f a l s e minimum p r e s e n t s p a r t i c u l a r d i f f i c u l t i e s , i t c a n u s u a l l y be e l i m i n a t e d b y r e p e a t i n g t h e o p t i m i z a t i o n w i t h d i f f e r e n t sets o ft r i a l points. In terms o f t h e refinement i l l u s t r a t e d h e r e , t h e f u n c t i o n Y i s t h e f u n c t i o n F(x). The f i r s t p o i n t , Xj_, i s r e p r e s e n t e d b y a l l v a r i a b l e bond l e n g t h s , bond a n g l e s , c o n f o r m a t i o n a n g l e s , c h a i n p o s i t i o n parameters, coordinates o f t h e solvent o f c r y s t a l l i z a t i o n , etc., o f t h e i n i t i a l m o d e l . A l l o t h e r p o i n t s #2, •••» ^ k * r e p r e s e n t t r i a l v a l u e s f o r t h e same n v a r i a b l e s w i t h i n t h e d e s i r e d i n t e r v a l l i m i t s and s u b j e c t t o any o t h e r c o n s t r a i n t s , such as coupl i n g o f v a r i a b l e s o r hydrogen bond f o r m a t i o n . C l e a r l y , t h e number and t y p e o f v a r i a b l e s , a n d t h e i r l i m i t s a n d c o n s t r a i n t s a r e e a s i l y c h a n g e d i n t h i s p r o c e d u r e , a s i s t h e f o r m o f t h e f u n c t i o n . The s e a r c h p r o c e d u r e i s a l s o r e l a t i v e l y r a p i d a n d does n o t s u f f e r f r o m a slowdown i n t h e v i c i n i t y o f t h e minimum, a s may o c c u r i n s t e e p e s t - d e s c e n t methods. m a x

9

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s

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"PS79" Computer P r o g r a m

The m o d e l r e f i n e m e n t p r o c e d u r e s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s have b e e n a s s e m b l e d , o v e r a p e r i o d o f y e a r s 0+> £ , 1 0 ) , i n t o a c o m p u t e r p r o g r a m , known a s "PS79" i n i t s c u r r e n t v e r s i o n . A l t h o u g h t h e p r o g r a m was p r i n c i p a l l y d e v e l o p e d f o r u s e w i t h p o l y saccharide c r y s t a l s t r u c t u r e s , i t i s equally applicable t o other polymers. I t can b e used t o r e f i n e a model w i t h r e s p e c t t o s t e r e o c h e m i s t r y o n l y , u s i n g e q . (l)» o r w i t h r e s p e c t t o d i f f r a c t i o n d a t a only, using the residuals R o rR o r as a combination o f t h e two, u s i n g t h e f u n c t i o n $. I n a d d i t i o n , t h e r e f i n e m e n t s t r a t e g y c a n b e u

9



234 FIBER DIFFRACTION


METHODS


13.

ZUGENMAIER

A N D SARKO


235

very f l e x i b l e . F o r example, o n l y the c h a i n conformation c a n be r e f i n e d b y o m i t t i n g a l l i n t e r m o l e c u l a r c o n t a c t s f r o m t h e nonbonded t e r m o f eq. ( l ) . I n s u c h c a s e s , t h e r a n g e o f v a r i a b l e s c o u l d e x t e n d f r o m a s i n g l e one o f r o t a t i o n o f t h e r e s i d u e a b o u t t h e virtual bond t o a l l b o n d l e n g t h s , v a l e n c e - b o n d a n g l e s , a n d c o n f o r m a t i o n a n g l e s o f t h e monomer r e s i d u e . C o n v e r s e l y , o n l y t h e c h a i n packing i n t h e u n i t c e l l c o u l d be r e f i n e d b y completely e l i m i n a t i n g t h e f i r s t t e r m o f eq. Cl)» a n d u s i n g o n l y i n t e r m o l e c u l a r n o n bonded c o n t a c t s i n t h e second term. A d d i t i o n a l l y , any d e s i r e d combination o f the p r e v i o u s extremes c o u l d be used, extending t o t h e c a s e where a l l c o n f o r m a t i o n a l a n d p a c k i n g v a r i a b l e s a r e s i m u l t a n e o u s l y r e f i n e d . The same a p p l i e s r e g a r d l e s s o f w h e t h e r t h e r e f i n e m e n t c r i t e r i o n i s Y, R" ( o r i ? ) , o r $. The i n i t i a l d e s c r i p t i o n o f t h e m o d e l i s s i m p l e , a s shown i n F i g u r e 3. The a t o m i c c o o r d i n a t e s o f a n y s u i t a b l e s t r u c t u r e c a n s e r v e a s t h e i n p u t t r i a l s t r u c t u r e , e v e n i n c l u d i n g a wrong monomer r e s i d u e . The p o l a r c o o r d i n a t e s a r e c a l c u l a t e d from t h e t r i a l s t r u c t u r e , a d j u s t e d and m o d i f i e d as necessary, and then s u b j e c t e d to refinement i n accordance w i t h the s e l e c t e d l i s t o f v a r i a b l e s , l i m i t s a n d c o n s t r a i n t s . A n y s e t o f s t a n d a r d v a l u e s a n d nonbonded p o t e n t i a l f u n c t i o n p a r a m e t e r s c a n b e u s e d . H y d r o g e n bonds c a n be d e f i n e d a s d e s i r e d , v a r i a b l e s c a n b e c o u p l e d , a n d t h e p o s i t i o n s o f s o l v e n t molecules c a n be i n d i v i d u a l l y r e f i n e d . S i n g l e and mult i p l e h e l i c e s a r e e q u a l l y e a s i l y handled, as a r e a v a r i e t y o f space groups. The c a l c u l a t i o n c a n b e t e r m i n a t e d when: Ca) t h e minimum o f t h e f u n c t i o n does n o t i m p r o v e w i t h i n a g i v e n a c c u r a c y , Cb) a c e r t a i n t i m e h a s e l a p s e d , Cc) t h e number o f d e s i r e d i t e r a t i o n s i s e x c e e d e d , o r Cd) when no improvement i s o b t a i n e d a f t e r 20 c a l c u l a t i o n a l s t e p s . The p r o g r a m p r o v i d e s v a r i e d o u t p u t , i n c l u d i n g c o o r d i n a t e s w r i t t e n on f i l e t h a t c a n b e used a s i n p u t t o s u c c e e d i n g runs. The "PST9 p r o g r a m i s c u r r e n t l y i n o p e r a t i o n o n s e v e r a l m a j o r c o m p u t e r s - IBM 370 s e r i e s , UNIVAC 1100 s e r i e s , CDC 6000 s e r i e s , and DEC-10 - a n d i n t h e m a j o r i t y o f c a s e s r e f i n e m e n t r u n s c a n b e completed w i t h i n t h e f a s t turnaround job l i m i t a t i o n s o f i n d i v i d u a l s h o p s . U s u a l l y , o n l y t h e f i n a l x - r a y r u n s w i l l demand more t i m e . ,f

Conclusions S i n c e i t s i n t r o d u c t i o n s e v e r a l y e a r s ago, t h e virtual bond c o n s t r a i n e d o p t i m i z a t i o n method h a s p r o v e d v e r y u s e f u l i n s t u d i e s o f p o l y s a c c h a r i d e c r y s t a l s t r u c t u r e . N o t a b l e among t h e s u c c e s s e s t h a t can be a s c r i b e d t o i t a r e t h e s t r u c t u r a l determinations o f the d o u b l e - h e l i c a l amylose Cll)» t h e c e l l u l o s e polymorphs o f d i f f e r e n t c h a i n p o l a r i t i e s C l 2 , 1 3 ) , a n d o f a number o f o t h e r p o l y s a c c h a r i d e s and t h e i r d e r i v a t i v e s . As d e s c r i b e d i n a review o f amylose s t r u c t u r e s elsewhere i n t h i s volume, t h e use o f t h i s r e f i n e m e n t method h a s p r o d u c e d s t r u c t u r a l d e t a i l t h a t h a s p r e v i o u s l y b e e n u n a v a i l a b l e C l l ) . T h e s e r e s u l t s have p r o v i d e d much-needed 9



236

FIBER DIFFRACTION

METHODS

i n s i g h t i n t o how p o l y s a c c h a r i d e s c r y s t a l l i z e , and i n t o s u c h a s p e c t s o f s t r u c t u r e as symmetry, r e l a t i o n s h i p s b e t w e e n h e l i x c o n f o r m a t i o n and p a c k i n g , h y d r o g e n - b o n d i n g , w a t e r and o t h e r s o l v e n t s o f c r y s t a l l i z a t i o n , and e f f e c t s o f c h e m i c a l s u b s t i t u t i o n on t h e s t r u c t u r e . An a d d e d b e n e f i t has b e e n t h e r e a l i z a t i o n t h a t t h e c r y s t a l s t r u c t u r e c a n i n many i n s t a n c e s be p r e d i c t e d f r o m t h e stereochemistry alone. As i s t o be e x p e c t e d , t h i s o p t i m i z a t i o n method p o s s e s s e s b o t h some a d v a n t a g e s and d i s a d v a n t a g e s . Among t h e a d v a n t a g e s a r e : ( l ) M o l e c u l a r m o d e l s a r e g e n e r a t e d and r e f i n e d w i t h i n d e s i r e d l i m i t s o f b o n d l e n g t h s , b o n d a n g l e s and t o r s i o n a n g l e s . ( 2 ) The g e n e r a t i o n o f m o d e l s i s s i m p l e and f l e x i b l e , and i s a p p l i c a b l e t o d i f f e r e n t polymers. ( 3 ) Virtual bonds c a n be u s e d t o d e s c r i b e one o r more monomer r e s i d u e s , o r e v e n atoms i n a b r a n c h s t r i n g . F o r exa m p l e , t h e p l a n a r i t y o f an a c e t y l g r o u p - 0 - C ( A ) - C C M ) - H ^ c a n be k e p t i n t a c t b y p l a c i n g t h e atoms i n t h e f o l l o w i n g s t r i n g : 0 -•C(A) •> 0 ( A ) •> C ( M ) -> H and b y k e e p i n g a l l d i s t a n c e s b e t w e e n t h e atoms and a l l c o n f o r m a t i o n a n g l e s r e s p o n s i b l e f o r p l a n a r i t y c o n s t a n t . (k) The c o n s t r a i n e d o p t i m i z a t i o n p r o c e d u r e i s f a s t , as l o n g a s t h e c a l c u l a t i o n of the function i s f a s t . T h i s i s t r u e f o r b o t h conf o r m a t i o n and p a c k i n g r e f i n e m e n t u s i n g e q . t l \ e v e n w i t h a l a r g e number o f v a r i a b l e s . ( 5 ) V a r i a b l e s and c o n s t r a i n t s c a n be c h o s e n a t w i l l , w i t h o u t r e g a r d t o t h e i r number o r t y p e . Among t h e c h i e f d i s a d v a n t a g e s i s t h e f a c t t h a t when t h e c a l c u l a t i o n of the f u n c t i o n i s slow, the refinement proceeds s l o w l y . F o r i n s t a n c e , i n x - r a y r e f i n e m e n t t h e c o m p u t a t i o n o f i ? " ( o r R) i s l e n g t h y , p a r t i c u l a r l y when t h e number o f r e f l e c t i o n s and t h e numb e r o f v a r i a b l e s a r e b o t h l a r g e . T h i s disadvantage i s , however, n o t s e r i o u s as t h e i n c r e a s e d demand on c o m p u t e r t i m e i s s t i l l w i t h i n r e a s o n a b l e l i m i t s e s t a b l i s h e d b y most c o m p u t e r s h o p s . A more s e r i o u s l i m i t a t i o n p l a c e d on t h i s method may o c c u r when t h e d i f f r a c t i o n d a t a a r e p o o r . F o r example, a c o r r e c t u n i t c e l l i s a n e c e s s a r y p r e r e q u i s i t e f o r any r e f i n e m e n t , y e t i n many c a s e s i t s d e t e r m i n a t i o n f r o m f i b e r x - r a y d a t a may be q u e s t i o n a b l e . T h i s l i m i t a t i o n may be a v o i d e d b y o b t a i n i n g g o o d e l e c t r o n d i f f r a c t i o n diagrams from polymer s i n g l e c r y s t a l s . Similar limitations a r i s e f r o m an i n a b i l i t y t o r e c o r d d i f f r a c t i o n i n t e n s i t i e s c o r r e c t l y , r e s u l t i n g i n p o o r agreement o f x - r a y and s t e r e o c h e m i c a l r e f i n e m e n t s . However, as d e s c r i b e d b y o t h e r a u t h o r s i n t h i s v o l u m e , t w o - d i m e n s i o n a l r e c o r d i n g t e c h n i q u e s h o l d o u t a g r e a t d e a l o f hope f o r improving the q u a l i t y of the i n t e n s i t y data. With t h i s imp r o v e m e n t , t h e s t r u c t u r e a n a l y s i s o f c r y s t a l l i n e p o l y m e r s may y e t approach the r e l i a b i l i t y of s i n g l e - c r y s t a l s t r u c t u r e determinations. 9

Acknowledgment s T h i s w o r k has b e e n s u p p o r t e d b y N a t i o n a l S c i e n c e F o u n d a t i o n g r a n t CHETT2T7U9 ( t o A.S.) a n d a g r a n t f r o m D e u t c h e F o r s c h u n g s -


13.

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gemeinschaft ( t o P.Z.). C o o p e r a t i v e e f f o r t s o f t h i s work have a l s o b e e n s u p p o r t e d b y a NATO R e s e a r c h G r a n t No. 1386, t o b o t h authors.

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1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

Sarko, A . ; Marchessault, R.H., J. Amer. Chem. Soc., 1970, 89, 6454-6462. Kitaigorodskii, A.I., Acta Crystallogr., 1965, 18, 585-590. Williams, D.E., Science, 1965, 147, 605. Zugenmaier, P . ; Sarko, A . , Biopolymers, 1976, 15, 2121-2136. Zugenmaier, P . ; Kuppel, A . ; Husemann, E . , in "Cellulose Chemistry and Technology", J.C. Arthur, Jr., Ed. ACS Symposium Series No. 48, American Chemical Society: Washington, D.C., 1977, pp. 115-132. Arnott, S.; Scott, W.E., J. Chem. Soc. Perkin Trans. 2, 1972, 324-335. Zugenmaier, P . ; Sarko, A . , Acta Crystallogr., 1972, B28, 31583166. Box, M . J . , Comput. J., 1965, 8, 42-52. Zugenmaier, P . ; Sarko, A . , Biopolymers, 1973, 12, 435-444. Zugenmaier, P . , Biopolymers, 1974, 13, 1127-1139. Sarko, A . ; Zugenmaier, P . , This symposium. Sarko, A . , Tappi, 1978, 61, 59-61. Woodcock, C.; Sarko, A . , Macromolecules, to be published.

RECEIVED

February 19, 1980.


The Variable Virtual Bond - ACS Symposium Series (ACS Publications)

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