Modeling of Interactions of Polysaccharide Chains - American

Chapter 17

Modeling of Interactions of Polysaccharide Chains Application to Crystalline Polymorphism of Starch Granules 1

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Downloaded by CORNELL UNIV on July 26, 2012 | http://pubs.acs.org Publication Date: July 6, 1990 | doi: 10.1021/bk-1990-0430.ch017

Serge Pérez , A. Imberty , and Raymond P.

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Scaringe

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Laboratoire de Physicochimie des Macromolécules, Institut National de la Recherche Agronomique, B.P. 527, 44026, Nantes, Cédex 03, France Research Laboratory, Eastman Kodak Company, Rochester, NY 14650 2

This paper describes a method f o r p r e d i c t i n g the packing r e l a t i o n s h i p of two polysaccharide chains and its use i n studying polymorphism i n starch. Given a r i g i d model of an i s o l a t e d double h e l i x , its i n t e r a c t i o n with a second double h e l i x is studied at varied h e l i x - a x i s t r a n s l a t i o n s and mutual r o t a t i o n a l orientations while keeping the h e l i c e s i n van der Waals contact. The s t a b i l i t y of each structure i s evaluated by an energy calculated using atom -atomp o t e n t i a l s that includes compensation f o r hydrogen bonding. Models f o r starch were based on the f i b e r repeat distance on f i b e r d i f f r a c t i o n patterns and are double-helices composed of left-handed single strands r e l a t e d by two-fold r o t a t i o n a l symmetry. Two stable r e l a t i o n s h i p s were found f o r both the p a r a l l e l and a n t i p a r a l l e l models. The structure predicted to be most stable corresponds to a duplex of p a r a l l e l double-helices as found i n both the c r y s t a l l i n e A and Β allomorphs. From t h i s r e s u l t , an explanation of the t r a n s i t i o n from Β to A is proposed. Over the years, modeling of carbohydrates has emphasized intramolecular rather than intermolecular structures. The same holds true i n the study of synthetic polymers and polypeptides. Only one such study f o r carbohydrates comes to mind (1) where the u n i t c e l l dimensions and symmetry were not used. Even there, a volume constraint was used, l i m i t i n g the possible structures. When such constraints are used, one does not obtain an explanation f o r why the c r y s t a l structure i s the stable form. 0097-6156/90/0430-0281$06.00A) © 1990 American Chemical Society In Computer Modeling of Carbohydrate Molecules; French, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


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COMPUTER MODELING OF CARBOHYDRATE MOLECULES

We hope t o understand and develop general r u l e s f o r the s t a b i l i t y of some i n t e r - h e l i x arrangements. Methods f o r i n v e s t i g a t i n g the i n t e r - h e l i x s t r u c t u r e and energy through non-bonded f o r c e s have been suggested by a number of workers (2-7). Those procedures minimize the i n t e r h e l i x energy. Our method (8.9) moves the h e l i c e s as c l o s e t o each other as i s p o s s i b l e without causing i n t e r p e n e t r a t i o n of the van der Waals r a d i i of atoms of the two d i f f e r e n t h e l i c e s . A f t e r the h e l i c e s are p o s i t i o n e d t o the s h o r t e s t i n t e r h e l i c a l d i s t a n c e f o r a g i v e n r o t a t i o n and h e l i x - h e l i x t r a n s l a t i o n , the energy i s c a l c u l a t e d . T h i s technique takes c o n s i d e r a b l y l e s s computer time than methods i n v o l v i n g energy minimization. In the present work, we extend the method t o compensate f o r the hydrogen bonds present i n carbohydrates. The hydroxylated c h a r a c t e r of carbohydrate polymers i n f l u e n c e s between-chain i n t e r a c t i o n s through networks of hydrogen bonds t h a t occur d u r i n g c r y s t a l l i z a t i o n . Frequently, s e v e r a l p o s s i b l e a t t r a c t i v e i n t e r a c t i o n s e x i s t t h a t l e a d t o d i f f e r e n t packing arrangements, and s e v e r a l allomorphic c r y s t a l l i n e forms have been observed f o r p o l y s a c c h a r i d e s such as c e l l u l o s e , c h i t i n , mannan and amylose. The s i t u a t i o n i s even more complex when water or other guest molecules are present i n the c r y s t a l l i n e domains. Another c o m p l i c a t i o n i s t h a t p o l y s a c c h a r i d e polymorphism i n c l u d e s d i f f e r e n t h e l i x shapes as w e l l . For the present work, we s t u d i e d the polymorphism of s t a r c h with our extended method. S t a r c h , an energy r e s e r v e f o r green p l a n t s and a major food carbohydrate, has many p r a c t i c a l a p p l i c a t i o n s . Native s t a r c h e x h i b i t s two d i f f e r e n t d i f f r a c t i o n p a t t e r n s t h a t depend on the b o t a n i c a l o r i g i n : A-type i n c e r e a l s t a r c h e s and B-type i n tuber s t a r c h e s (10). In both, d i f f r a c t i o n i s thought t o a r i s e mainly from the s h o r t chains t h a t are connected at branch p o i n t s of the amylopectin component o f s t a r c h (11). The s h o r t chains have 12 t o 20 D-glucose r e s i d u e s l i n k e d a ( l - 4 ) . F i b e r d i f f r a c t i o n s t u d i e s (12.13) demonstrated t h a t both forms have the same 1.05 nm repeat d i s t a n c e along the h e l i x a x i s . Recent r e i n v e s t i g a t i o n of t h e i r c r y s t a l s t r u c t u r e s (14.15) e s t a b l i s h e d t h a t the same conformation of the i n d i v i d u a l amylosic strands (a n e a r l y p e r f e c t left-handed, s i x - f o l d h e l i x r e p e a t i n g i n 2.1 nm) e x i s t s i n both allomorphs. Through the r o t a t i o n a l symmetry of the d o u b l e - h e l i x , the repeat d i s t a n c e i s halved t o 1.05 nm. Such d o u b l e - h e l i c e s were f i r s t proposed i n 1972 by Kainuma and French (16). The recent r e i n v e s t i g a t i o n s showed t h a t the doubleh e l i c e s of A and Β s t a r c h are packed i n a p a r a l l e l f a s h i o n . They are s t a b i l i z e d mainly by numerous van der Waals i n t e r a c t i o n s and by hydrogen bonding. The important d i f f e r e n c e s between the two s t r u c t u r e s l i e i n the amount of water present, and p o s i t i o n i n g of the h e l i c e s t o accomodate the d i f f e r e n t amounts of water. Under c e r t a i n

In Computer Modeling of Carbohydrate Molecules; French, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

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Modeling ofInteractions ofPolysaccharide Chains

c o n d i t i o n s o f heat and moisture, the Β form can be converted t o the A form. T h i s s o l i d - s t a t e conversion has been observed i n f i b e r s (12) and i n i n t a c t s t a r c h granules (12).


Methods Our stategy i s t o f i r s t c o n s t r u c t models o f s i n g l e stranded h e l i c e s t h a t repeat i n 2.1 nm and decide whether they are l e f t - o r right-handed. T h i s i s done with a t r a d i t i o n a l Ramachandran p l o t o f energy vs. Φ and Ψ t o r s i o n angles, o v e r l a i d whith contours o f i s o - n and i s o - h . A s i n g l e s t r a n d i s then used t o generate the second s t r a n d o f a d o u b l e - h e l i x through two-fold r o t a t i o n (for every atom with coordinates o f x, y and ζ there i s a new one a t -x, -y and z ) . The r e s u l t i n g d o u b l e - h e l i x i s then p l a c e d i n our new program which generates a second d o u b l e - h e l i x and t e s t s the i n t e r a c t i o n s between the two double-helices. Nomenclature. A fragment o f amylosic c h a i n (maltose) i s shown i n F i g u r e 1, with l a b e l s on the atoms and t o r s i o n angles o f i n t e r e s t . The r e l a t i v e o r i e n t a t i o n o f two contiguous (1-4) l i n k e d a-D-glucose r e s i d u e s i s given by t o r s i o n a l angles Φ and Φ which are d e f i n e d by the 4 atom sequences 0-5 - C - l - 0-1 - C-4· and C - l - 0-1 - C-4 C-5 , r e s p e c t i v e l y . Other conformational parameters are the o r i e n t a t i o n s o f the primary hydroxyl groups around C-5 - C-6 bonds. T h i s o r i e n t a t i o n i s r e f e r r e d t o as e i t h e r gauche-trans. gauche-gauche o r trans-gauche. In t h i s terminology, the t o r s i o n angle 0-5 - C-5 - C-6 - 0-6 i s s t a t e d f i r s t , followed by the t o r s i o n angle C-4 - C-5 - C-6 - 0-6 (18). The s i g n o f the t o r s i o n angles agrees with the r u l e s recommended by the IUPAC-IUB Commission o f Biochemical Nomenclature (19.20). H e l i c a l arrangements are d e s c r i b e d i n terms o f a s e t o f h e l i c a l parameters (ϋ/h); η i s the number o f r e s i d u e s ( i . e . backbone glucose u n i t s ) per t u r n o f the h e l i x , and h i s the t r a n s l a t i o n along the h e l i x a x i s . The c h i r a l i t y o f the h e l i x i s d e s c r i b e d by the s i g n o f h. A r b i t r a r i l y , a right-handed h e l i x w i l l have p o s i t i v e h value; conversely, negative v a l u e s o f h w i l l designate left-handed h e l i c e s . Whenever the values h = 0 o r η = 2 are i n t e r c r o s s e d , the screw sense o f the h e l i x changes t o the opposite s i g n . 1

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Modeling the S i n g l e - S t r a n d H e l i x . S t a r t i n g geometry f o r the glucose r e s i d u e i n i t s ^ conformation was taken from the c r y s t a l s t r u c t u r e o f α-D-Glucose monohydrate (21). The reported p o s i t i o n s f o r hydrogen atoms were not used s i n c e they are known t o be p o o r l y determined by x-ray d i f f r a c t i o n s t u d i e s . Instead, the p o s i t i o n s o f hydrogen atoms t h a t are attached t o carbons were generated u s i n g a C-H bond length o f 0.1 nm and a bond v e c t o r r e l a t e d a p p r o p r i a t e l y t o the C-C and C-0 bond v e c t o r s . H y d r o x y l i c hydrogen atoms were not considered. 4


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Program PFOS (22) c a l c u l a t e d the energy of the maltose d i s a c c h a r i d e a t the values o f Φ and Ψ, u s i n g the f o r c e - f i e l d described i n reference 23.. The value o f the C - i - o - l - C-4 valence angle was 117° and Φ and Φ were stepped i n 5° increments. Iso-energy contours a r e drawn at 1 kcal/mol spacings with respect t o the minimum energy. The contours o f i s o - n and i s o - h were d e r i v e d with the algorithm reported by Gagnaire e t a l . (24). T h i s map, based only on a d i s a c c h a r i d e , i s not accurate near h=0 f o r a p o l y s a c c h a r i d e because h e l i c e s with small values of h would s u f f e r s t e r i c c o n f l i c t s between adjacent t u r n s . Since the h e l i c e s i n the n a t i v e forms o f s t a r c h are very extended, t h i s i s not a problem i n t h i s case. Because of the exact geometry of the s e l e c t e d glucose residue, h e l i c e s with l a r g e values of η and h are favored. Other r e s i d u e geometries must be used t o c o n s t r u c t s a t i s f a c t o r y models of known allomorphs such as V amylose.


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Chain P a i r Modeling. In the f o l l o w i n g a n a l y s i s , we assume t h a t the chains are r e g u l a r h e l i c e s , i . e . t h a t they have screw symmetry, with a repeat d i s t a n c e , t . In a p e r f e c t c r y s t a l , such chains must e i t h e r be p a r a l l e l o r a n t i p a r a l l e l . Four i n t e r h e l i c a l parameters are r e q u i r e d t o d e f i n e t h e geometric o r i e n t a t i o n o f chain A r e l a t i v e t o chain Β (see Figure 2). The parameters and t h e i r ranges are : /xA : r o t a t i o n of A about i t s a x i s (0° t o 360°) μΒ : r o t a t i o n o f Β about i t s a x i s (0° t o 360°) Δχ : normal d i s t a n c e between t h e h e l i x axes o f A and Β (no l i m i t ) Δζ : t r a n s l a t i o n along the h e l i x a x i s o f one chain r e l a t i v e t o the other (0 t o t , nm) Such a s e t o f i n t e r h e l i c a l parameters r e l a t e s d i r e c t l y t o the symmetry operations which are found i n c r y s t a l structures. : μΑ ψ βΒ, represents the case where chain A and chain Β a r e not r e l a t e d by any symmetry operation. Both independent chains would be needed t o d e f i n e the asymmetric u n i t of a c r y s t a l . : μΑ = μΒ, represents the case where chain Β i s d e r i v e d from chain A by a pure t r a n s l a t i o n a l symmetry element. : μΑ = μΒ + 180° and Δζ = 0, represents the s i t u a t i o n where the two chains are p a r a l l e l and r e l a t e d by a two-fold operation. A two-fold screw-axis would be d e s c r i b e d by μΑ = μΒ + 180° and Δζ = t / 2 . : βΑ = -μΒ + 180° and Δζ = 0, represents the s i t u a t i o n where the two chains are a n t i p a r a l l e l and r e l a t e d by a two-fold operation. For a two-fold screw a x i s , the r e l a t i o n s h i p i s βΑ = -μΒ + 180° and Δζ = t / 2 .


17. PÉREZ ET AL.

Modeling of Interactions ofPolysaccharide Chains


0(2)

F i g u r e 1. Schematic r e p r e s e n t a t i o n of two contiguous a-(1-4) l i n k e d glucopyranose r e s i d u e s (maltose), along with the l a b e l l i n g of the atoms and the t o r s i o n angles of i n t e r e s t .

F i g u r e 2. I n t e r h e l i c a l parameters r e q u i r e d t o d e f i n e the geometric o r i e n t a t i o n of chain A r e l a t i v e t o chain B.


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Other r e l a t i o n s h i p s between c r y s t a l l o g r a p h i c elements of symmetry and i n t e r h e l i c a l parameters can be e a s i l y derived. Contacting Procedure- As shown i n Figure 3, f o r a given v a l u e s of βΑ, βΒ and Δζ, our program determines a normal t r a n s l a t i o n Δχ t h a t p l a c e s one or more atoms i n van der Waals contact without any interpénétration as described e a r l i e r (8). The s u r f a c e of the chain i s d e f i n e d by c i r c u m s c r i b i n g a hard-sphere of the appropriate van der Waals r a d i u s R i , around each c o n s t i t u e n t atom. In g e n e r a l , the f i n a l p o s i t i o n of the two polymeric chains i s c h a r a c t e r i z e d by the f o l l o w i n g : 1/ For a t l e a s t one atom p a i r ( i , j ) , the i t h atom of the chain A i s separated from the j t h atom of the c h a i n Β by the sum of R i and R j . The atom p a i r i , j which s a t i s f i e s t h i s c o n d i t i o n i s r e f e r r e d t o as the determining contact. 2/ There i s no atom p a i r i n g between the two chains t h a t has a d i s t a n c e c l o s e r than the appropriate van der Waals r a d d i i sum. Obviously, c o n d i t i o n (2) cannot be f u l f i l l e d f o r an atom p a i r i n v o l v e d i n an hydrogen bond. Since the h y d r o x y l i c hydrogens are not e x p l i c i t l y considered here, hydrogen bonds are d e f i n e d i n terms of the d i s t a n c e between the hydrogen donor oxygen atom and the oxygen which accepts the hydrogen atom. A l l p o t e n t i a l couples of atoms e l i g i b l e t o p a r t i c i p a t e i n an i n t e r c h a i n hydrogen bond are i d e n t i f i e d and ommited from the c o n t a c t i n g procedure. T h i s i m p l i c i t e l y means t h a t hydrogen bonding w i l l not v i o l a t e p r i n c i p l e (1) f o r the van der Waals bonded atoms. I n t e r c h a i n Energy C a l c u l a t i o n s . I f a c o n t a c t i n g procedure i s used, chain-chain c o n s t r u c t i o n r e q u i r e s only geometric information, and i n p r i n c i p l e , one can subsequently c a l c u l a t e the energy of the r e s u l t i n g i n t e r a c t i o n s ( E ) t o any degree of approximation. For a f o r m a l l y i n f i n i t e chain, the expression f o r the i n t e r c h a i n i n t e r a c t i o n energy i s : A B

Na

Ε AB

Nb

ω

= Σ Σ Σ i=i

j = l =1 η

Ε, 4

iJ

1 1

where Na i s the number of atoms per i d e n t i t y p e r i o d of c h a i n A, Nb i s the number of atoms per i d e n t i t y p e r i o d of chain B, and ω i s the number of r e p e a t i n g u n i t s . Atom-atom p o t e n t i a l s have been used e x t e n s i v e l y f o r the study of molecular c r y s t a l s , and many u s e f u l e m p i r i c a l parameters s e t s have been designed. The i n t e r a c t i o n energy of the two chains i s considered t o be the sum over a l l pairwise i n t e r a c t i o n s . In the present work, such i n t e r a c t i o n i s considered according t o the 6-12 p o t e n t i a l f u n c t i o n s . The energy of an atom p a i r i s given by an expression of the form :


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Modeling of Interactions of Polysaccharide Chains 287

PÊREZ ET AL "

A

/r

12 i

j

n

6 "

C

r

/ ijn

where r ^ j i s the d i s t a n c e between the i t h atom i n the r e f e r e n c e i d e n t i t y p e r i o d of chain A and the j t h atom i n the nth i d e n t i t y p e r i o d of chain B. These terms incorporate a short-range r e p u l s i v e i n t e r a c t i o n and a short-range a t t r a c t i v e i n t e r a c t i o n , r e s p e c t i v e l y . To these may be added coulombic i n t e r a c t i o n s . As f o r the energy s t a b i l i z a t i o n a r i s i n g from hydrogen bonding, an e x t r a term has t o be included. In the present work, we l i m i t ourselves t o i n v e s t i g a t i n g whether the information provided by short-range i n t e r a c t i o n s alone i s of u t i l i t y f o r i d e n t i f y i n g s t r u c t u r a l assemblies of polymer chains. In performing the i n t e r c h a i n energy c a l c u l a t i o n , we have used a c u t o f f d i s t a n c e such that dij0) and left-handed (h

Modeling of Interactions of Polysaccharide Chains - American

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