N-H Mapping for Polymers - ACS Symposium Series (ACS Publications)

14 N-H Mapping for Polymers A L F R E D D. F R E N C H

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Southern Regional Research Center, P.O. Box 19687, New Orleans, L A 70179 WALTER A. FRENCH Interactive Data Corporation, 350 California St. Suite 1450, San Francisco, CA 94104

Helical shapes of polymers are c o n v e n i e n t l y , if only approximately, described by n, the number of monomeric residues per h e l i x p i t c h p, and h, the r i s e per residue (Figure 1 ) . The present paper addresses the problem of determining which shapes, or values of n and h can be attained by d i f f e r e n t polymers. Determination of allowed values of n and h i s important in the e a r l y stages of d i f f r a c t i o n a n a l y s i s , because a l l appropriate trial models should be i n v e s t i g a t e d . Such knowledge i s also important f o r understanding the nuclear magnetic resonance spectra of polymers, the i n t e r a c t i o n s of polymers with other m a t e r i a l s , and the behavior of polymers in s o l u t i o n . We propose that the study of regular h e l i c e s a l s o applies t o i r r e g u l a r h e l i c e s by extension of Natta's monomer equivalence postulate (1). From a review of the geometric features of monomeric residues of c y c l i c oligomers of glucose, i t seems that the average geometric features of the monomeric units o f an irregular helix are approximately equal t o the monomeric geometry of the most equivalent regular h e l i x . Previously, the Ramachandran technique ( 2 J was used t o learn the range of allowed molecular shapes. We now propose a new representation of the allowed shapes of a polymer. The new representation solves serious problems inherent i n the Ramachandran r e p r e s e n t a t i o n . In a d d i t i o n , our method expands the u t i l i t y of conformational a n a l y s i s . The idea f o r n-h mapping occurred i n Japan, and the f i r s t p r i n t e d n-h map was discussed in Japanese ( 3 ) · The n-h approach was a l s o discussed i n a review paper, in English ( 4 J . The present paper i s more complete treatment of the n-h approach and an announcement of the impending a v a i l a b i l i t y of a new computer program, NHMAP.

0-8412-0589-2/80/47-141-239$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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Problems with the Ramachandran Method In the Ramachandran method, allowed conformations are studied by r o t a t i n g the monomers through the $ and Y angles about the bonds to the inking atom (Figure 2 ) . At each increment of monomer r o t a t i o n , the model i s tested for c o n f l i c t s between nonbonded atoms. Models with no serious c o n f l i c t s or with low potential energy are considered to be stereochemically feasible. The stereochemical energy, or f e a s i b i l i t y , i s then reported on a grid of $ and Y values (Figure 3 ) . A f t e r n and Ji are determined f o r each g r i d p o i n t , contours of i s o - n and i s o - h are l a i d over the allowed zones and the g r i d to produce the Ramachandran map (Figure 4 ) . However, representing allowed shapes on a g r i d has fundamental l i m i t s . These l i m i t s a r i s e when one converts from the l i n k i n g conformation, expressed in terms of $ and Y, to the polymer conformation, described by the values of n and h. The l i m i t s r e s u l t because c a l c u l a t i o n of _n and h values depends on the exact monomeric geometry and l i n k i n g bond angle used f o r the stereochemical c a l c u l a t i o n s . In the o r i g i n a l work, monomers were assumed to be rigid. Now, however, a large number of s i n g l e - c r y s t a l studies have shown that monomers are more f l e x i b l e in crystalline environments than was o r i g i n a l l y thought. T h e r e f o r e , the $ and Y values of a Ramachandran map do not give unique values of ji and if the necessary v a r i e t y of monomeric geometries is considered. The only remedy possible, if grids are r e t a i n e d , i s to prepare enough $-Y maps to allow each variant of monomeric geometry and l i n k i n g bond angle to be combined and tested. This c o l l e c t i o n of maps would be less useful than an o v e r a l l summary of the allowed conformations. a-Glucose, a f l e x i b l e monomer of considerable i n t e r e s t , has a c h a i r form that was thought to be r i g i d (Figure 5 ) . However, s i n g l e - c r y s t a l studies show a range of 0.6 A f o r the distances between the 0(1) and 0(4) atoms in various c r y s t a l s containing the glucose molecule. This range r e s u l t s from the d i f f e r e n t angles at which the l i n k i n g C ( l ) - O l ) and C(4)-0(4) bonds depart from the bulk of the r i n g ( 5 ) . As we show l a t e r , t h i s monomeric v a r i a b i l i t y , when coupled with a smaller v a r i a b i l i t y in l i n k i n g bond a n g l e , permits f e a s i b l e h e l i c e s with s i m i l a r values of $ and Y but values of n ranging from 5 to 10. The n-h Map A simple summary map i s a g r i d of h v s . j i . As shown in F i g u r e 6, we i n d i c a t e h e l i x c h a i r a l i t y by p o s i t i v e and negative values of n. C u r r e n t l y , we are r e l y i n g on hard-sphere a n a l y s i s for determining feasibility, giving allowed and disallowed zones. The atomic r a d i i are given in Table I .

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

FRENCH AND FRENCH

n_=6 . if _n integral

Figure 2.

N-H

Mapping for Polymers

241

Figure 1. The helical parameters n, h, p, and r (crystallographic repeat) for a right-handed helix. If the helix is integral, p = r, and it is always true that p = n X h-

The axes for & and * rotation


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Figure 4. Combination of stereochemical^ allowed zones (Figure 3) and contours of iso-n and iso-h values. Adapted from Ref. 11. The positions of the contour lines vary substantially with different monomeric geometries.


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243

0(5)

4.6A

Figure 5. Illustration of monomeric flexibility. In this a-pyranosic ring, substantial variation occurs but the C(1)-C(D) distance stays nearly constant. The virtual angles 0(1)-C(1)-C(4) and C(l)-C(4)-0(4) each have 15° ranges and change equally for a given change in 0(4)-0(l) distance.

6 -

_J -8

I i i I i i I i i i -6

-4

left handed

2 —

4

i — i — i

6

8

right handed

Figure 6. The n-h map for cellulose, determined by hard-sphere criteria (see text and Table I). Note the zones of helices of small h and large n as well as the more usual conformations with small n and large h. The short contacts found in models for intermediates between the allowed zones would not be so severe as to prevent interconversion of the two types. Increments for n and h were 1 and 0.2 A, respectively.


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TABLE I

Atom P a i r

F u l l y Allowed* Distance (A)

Minimally Allowed** Distance (A)

C-C

3.2

2.96

C-0

2.9

2.66

C-H

2.4

2.16

0-0

2.8

2.56

0-H

2.4

2.16

H-H

2.1

1.86

* If six or more contacts between f u l l y allowed and minimally allowed are found, the model is r e j e c t e d . ** If one or more contacts smaller than minimally allowed are found, the model i s r e j e c t e d . F e a s i b l e models were required 1130 20. a

n

d

to have g l y c o s i d i c

angles

between

1 2

Each ji-ji point could be examined f o r f e a s i b i l i t y by a variety of methods. Although we discuss below several techniques f o r these examinations, our primary point i s not an evaluation of these various methods. Our main point is that the ranges of polymeric shape must be reported only a f t e r a thorough t e s t of the important v a r i a b l e s : monomeric geometry and l i n k i n g bond angle. P h y s i c a l , s p a c e - f i l l i n g models are reasonable t o o l s in the search f o r allowed shapes. Computer methods, however, permit bond lengths and angles to be e a s i l y a l t e r e d . One computer method would be to survey the c o l l e c t i o n of maps mentioned above. Other f e a s i b l e methods include molecular mechanics and i t s r e l a t e d v a r i a n t s , v a r i a b l e v i r t u a l bond (6) and linked-atom, l e a s t squares (7) modeling. When only a few observed structures are a v a i l a b l e to represent the monomer, a technique s i m i l a r to these can be used to provide a reasonable range of monomeric shapes. This range of monomeric geometries could then be used in a rigid-monomer, computer-modeling method. Modeling of a range of d i f f e r e n t monomeric shapes is discussed in another contribution (8).


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If a f l e x i b l e , but c o n s t r a i n e d , modeling technique i s used f o r the e n t i r e polymer, the shapes depicted as f e a s i b l e should represent a l l the conformations that are a c t u a l l y allowed f o r regular h e l i c e s . The d i f f i c u l t y with these methods i s , however, that we do not yet know p r e c i s e l y the l i m i t s of monomeric flexibility. That point i s the subject of some other current research {9). Program NHMAP We are c u r r e n t l y developing a computer program f o r n-h mapping. Previous experience led us to the rigid-monomer, virtual-bond method (10). Because virtual-bond modeling operates d i r e c t l y with the values of n and Ji, i t i s the simplest method, and i t is rapid. A l s o , modeling with a v a r i e t y of observed monomers has a number of advantages. For one, it answers the q u e s t i o n : What are the allowed shapes of a given polymer, based on the known geometries? Our program i s c u r r e n t l y w r i t t e n to run with an IBM l e v e l G compiler. Deviations from ANSI Fortran IV are limited to character representation s p e c i f i c to the IBM 360 computer. The system i s command d r i v e n ; users input appropriate monomeric c o o r d i n a t e s , atomic r a d i i , desired l i m i t s on angles, and ranges of _n and h t o be t e s t e d . A f t e r a run, some of the input values may be changed while other input remains the same. Because the program is designed to be used i n t e r a c t i v e l y by the researcher, the user i n t e r f a c e i s as f l e x i b l e as p o s s i b l e . Input values may be entered e i t h e r by the minimum, maximum, and increment s i z e , or by the actual desired v a l u e s . A l l commands are in free format. Incorrect input syntax i s detected and i t s l o c a t i o n marked without a f f e c t i n g other input or the success of the run. The purpose of i n t e r a c t i o n i s to enable the user to t e s t in c l o s e r detail around critical regions. Capabilities

of ji-ji Maps

We have structured the computer program for n-h mapping to test for effects on the allowed zones caused by different monomeric geometries, d i f f e r e n t ranges of l i n k i n g bond a n g l e , and d i f f e r e n t ranges of $ and Y. Figure 7 shows the domain of allowed left-handed shapes of the a-1,4 glucan, amylose. The smaller allowed zones r e s u l t i n g from two d i f f e r e n t monomers are superimposed on the t o t a l allowed zone. Inadequate coverage of the t o t a l allowed zone r e s u l t s from using only one or two monomeric geometries. In p a r t i c u l a r , F i g u r e 7 shows that only a


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small fraction of conformational space i s a v a i l a b l e to the monomer with an 0 ( 4 ) — 0 ( 1 ) distance of 4.25 A. Previously, Goebel, D i m p f l , and Brant reported the two maps r e s u l t i n g from these two monomers (11). In Figure 8 , the smaller overlays on the allowed zones depict our guesses at the e f f e c t s of r e s t r i c t i n g the range of l i n k i n g bond angle or the ranges of * and v . The e f f e c t of a narrowed range of l i n k i n g bond angle i s r e l a t i v e l y s m a l l ; most of the allowed zone i s s t i l l found. Holding * and ¥ constant substantially reduces the range of allowed n and h v a l u e s . Because of the range of monomeric geometries and lirllcing bond a n g l e s , however, a number of d i f f e r e n t h e l i c e s can s t i l l be formed. The computer program i s a l s o s t r u c t u r e d so that other t e s t s , such as f o r the existence of a p a r t i c u l a r hydrogen bond, can be e a s i l y added. This approach enables the researcher to e x t r a p o l a t e the r e s u l t s of a s i n g l e - c r y s t a l study of an oligomer to polymers. Even i f monomeric geometry and l i n k i n g bond angle were i n v a r i a n t , n-h maps would s t i l l have some advantages over maps. One advantage i s that n-h maps f o r d i f f e r e n t polymers are different (compare Figures 6 and 9 ) , whereas maps have allowed zones that look s i m i l a r . Another problem with fc-y maps i s that small changes in $ and Y sometimes r e s u l t in large changes in n and h. Such a coarse (in terms of ji and Ji) g r i d search might cause some f e a s i b l e n-Ji combinations to be inadvertantly disallowed. More importantly, n-h maps can represent the allowed shapes for polymers that are more complicated than the simple homopolymers discussed h e r e i n . That advantage of n-h maps a r i s e s because the chemically repeated unit i s allowed to have various shapes. In the r\-h mapping approach, a geometric change in the chemical repeating unit of a heteropolymer is, in p r i n c i p l e , no d i f f e r e n t from an i n t e r n a l change in the simpler monomers of a homopolymer. The r e s u l t s from a stereochemical study of each can s t i l l be placed on n-h maps. The n-h maps are a l s o useful in themselves. Relative molecular f l e x i b i l i t y can be assessed by comparing the allowed areas f o r two d i f f e r e n t polymers. We think that 1/n v s . h p l o t s would have advantages f o r t h i s use (3). Once a l l allowed shapes of a given polymer are represented on a s i n g l e map, some fundamental questions can be r a i s e d . Compare, i n F i g u r e 9, the large range of allowed shapes with the observed shapes of a n y l o s e , indicated by the dots in the drawing. (Both right-handed and left-handed models are shown f o r the observed polymorphs because the experimental evidence i s usually not clearly in favor of either chairality.) the observed forms seem to be grouped. Why are some shapes p r e f e r r e d over others? This question is discussed in references 4 and 12.


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247

h(A)

-10

-8

-6

-4

2

Figure 7. Map for left-handed amylose, showing the regions allowed by two different monomers having virtual bond (0(4)-0(l)) lengths of 4.57 A and 4.25 A. A similar experiment was reported in Ref. 11, except that the two zones were on different maps. Here, the two allowed zones are superimposed on the total allowed zone resulting from consideration of all residues available from single-crystal studies of maltose. Increments for n and h were 1 and 0.2 A, respectively.

MA)

Figure 8. Restricted zones on n-h maps of left-handed amylose resulting from limits on: (left) the linking bond angle: estimated region allowed by 117° < T < 119° compared with 113° < r < 722°; fright; linking torsion angles: estimated region allowed when 20 < < 0, —10

N-H Mapping for Polymers - ACS Symposium Series (ACS Publications)

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