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MULTIPLE SCATTERING CALCULATIONS FOR BIOLOGICAL CATALYSTS
R. Pettifer, D. Foulis, C. Hermes
To cite this version:
R. Pettifer, D. Foulis, C. Hermes. MULTIPLE SCATTERING CALCULATIONS FOR BI- OLOGICAL CATALYSTS. Journal de Physique Colloques, 1986, 47 (C8), pp.C8-545-C8-550.
�10.1051/jphyscol:19868102�. �jpa-00225997�
JOURNAL D E
PHYSIQUE
C o l l o q u e C8, s u p p l 6 m e n t a u n o 1 2 , Tome
47,dbcembre 1986
MULTIPLE SCATTERING CALCULATIONS FOR BIOLOGICAL CATALYSTS
R.F.
PETTIFER*, D . L . FOULIS*a n d
C. HERMESEMBL O u t s t a t i o n Hamburg, C/o DESY, N o t k e s t r a s s e
85,0-2000 Hamburg,
F.R.G." a n d D e p a r t m e n t of P h y s i c s , U n i v e r s i t y o f Warwick, GB-Coventry
CV4 7 A L ,G r e a t - B r i t a i n
A b s t r a c t
A
r e s t r i c t e d m u l t i p l e s c a t t e r i n g c a l c u l a t i o n b a s e d on t h e p l a n e wave a p p r o x i c ~ a t i o n b u t r e t a i n i n g IIankel t u n c t i o n t e r m s i s p r e s e n t e d f o r z i n c bound t o f o u r i m i d n z o l e l i g a n d s v h i c h i s a n e x c e l l e n t n o d e l f o r many enzyme a c t i v e s i t e s . The d a t a show t h a t n u l t i p l e s c a t t e r i n g c a l c u l 2 t i o n s a r e e s s e n t i a l t o u n d e r s t a n d t h e EXAFS s p e c t r u m .
I ti s a l s o shown t h a t F o u r i e r t r a n s f o r m s of t h i s d a t a a r e v e r y s e n s i t i v e t o r o t a t i o n s a b o u t a n a x i s p e r p e n d i c u l a r t o t h e i m i d a z o l e p l a n e , a n d l e s s s o f o r o u t of p l a n e r o t a t i o n s . The F o u r i e r t r a n s f o r m beyond s h e l l one
i sn o t s i m p l y i n t e r p r e t a b l e i n t e r m s of m e t a l l i g a n d d i s t a n c e s owing t o t h e i n t e r f e r e n c e of many n u l t i p l e s c a t t e r i n g p a t h s .
I t has b e e n a p p r e c i a t e d f o r some t i n e t h a t EXAFS
n u l t i p l e s c a t t e r i n g e f f e c t s m a n i f e s t t h e m s e l v e s p a r t i c u l a r l y i n s y s t e m s i n which t h e e m i t t e r atom and two s c a t t e r e r s a r e c o - l i n e a r I l l . F u r t h e r work [2,31 h a s e x t e n d e d t h e s e i d e a s t o t h e non-coline.ar s i t u a t i o n i n d i c a t i n g t h e i m p o r t a n c e of m u l t i p l e s c a t t e r i n g t o d e t e r m i n e t h r e e and f o u r body c o r r e l l a t i o n f u n c t i o n s . R e c e n t l y I41 l i n e a r c a r b o n y l g r o u p s h a v e been s t u d i e d u s i n g n u l t i p l e s c a t t e r i n g t e c h n i q u e s which i l l u s t r a t e t h e i x t p o r t a n c e of t h i s e f f e c t f o r m a t e r i a l s of i n t e r e s t i n c a t a l y t i c r e s e a r c h . I n t h i s c o n t r i b u t i o n we a r e p r i n c i p a l l y i n t e r e s t e d i n m u l t i p l e s c a t t e r i n g from r.iolecules i n v o l v e d i n b i o l o g i c a l c a t a l y s i s . iIost m e t a l atoms i n t h e c a t a l y t i c a l l y a c t i v e c e n t r e s of enzymes a r e c o o r d i n a t e d by h i s t i d i n e g r o u p s , e.g. c a r b o n i c a n h y i i r a s e , carbo:cypeptidase. The bonding g r o u p of t h i s amino a c i d i s t h e f i v e membered i n i i u a z o l e r i n g . The r i n g
i sf l a t and e x t r e n e l y r i g i d . Elo n o d e s of t!le r i n g c a n be e x c i t e d below
6 0 0 cm-'[51. C o n s e q u e n t l y , dar.~ping or' n ~ u l t i p l c s c a t t e r i n g by i n t e r n a l x o d e s s h o u l d be a ninimur;~.
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o fa s i n g l e compound Z n ( I n ) , ( C 1 0 4 which was measured a t
20 I;.C a l c u l a t i o n s a r e p e r f o r m e d up t o t r i p l e s c a t t e r i n g f o r a l l r e l e v a n t p a t h s .
Byp l o t t i n g t h e p o l a r s c a t t e r i n g d i a g r a m ,
w ec a n u s e t h i s t o p r e d i c t t h o s e p a t h s which c o n t r i b u t e s i g n i f i c a n t l y t o t h e s p e c t r u m .
The air.1 of t h i s p a p e r
i st o d i s c o v e r t h e e x t e n t t o which d o u b l e and t r i p l e s c a t t e r i n g n f f e c t t i l e C?:APS spectrur.1 f r o n t h e i n i d a z o l e l i g a n d . F o l l o w i n g t h i s , we perfor13 coinputer e x p e r i r . ~ c n t s t o i n v e s t i g a t e t h e n a t u r e of t h e t r i p l e peaked s t r u c t u r e o f t e n o b s e r v e d
byu s a t E1;JL i n
ev a r i e t y of e n z y n e s a n u i n p a r t i c u l a r by o t h e r s i n c a r b o n i c a n h y d r a s e
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19868102
JOURNAL DE PHYSIQUE
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s c a l e u drawing of t h e p l a n a r i m i d a z o l e r i i i g i n Zn(Im14
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comparison of t h e k-cubed weighted f i n e s t r u c t u r e s . The J o t t e d l i n e
i s t h e measured spectrum ( r a v d a t a ) and t h e s o l i u l i n e i s t h e
c a l c u l a t e d s p e c t r u n u s i n g t h e c r y s t a l s t r u c t u r e
a n dt h e r n a l p a r a m e t e r s
g i v e n i n T a b l e
1.163.
I n p a r t i c u l a r , t h e l a t t e r p a p e r s h o v e d some m y s t i f y i n g c h a n g e s i n t h e F o u r i e r t r a n s f o r n p e a k s b e l o n g i n g t o t h e r i n g s y s t e m s u r r o u n d i n g a z i n c m e t a l i o n when t h e pH of t h e s o l u t i o n was v a r i e d .
It i sc l e a r l y of g r e a t i n p o r t a n c c t o u n d e r s t a n d t h e o r i e n t a t i o n of t h e s e r i n g s a s a f i r s t s t e p i n e x p l a i n i n g c a t a l y t i c a c t i v i t y i n t h e s e s y s t e m .
C a l c u l a t i o n s alere p e r f o r n e d u s i n g a n i n t e r m e d i a t e a p p r o x i m a t i o n between t h e a s y m p t o t i c p l a n e wave c a l c u l a t i o n and t h e f u l l c u r v e d wave c a l c u l a t i o n .
I twas r e c o g n i s e d e a r l i e r 111 t h a t t h e a p p r o x i m a t i o n s u s e d i n d e r i v i n g t h e f a m i l i a r plane-wave e x p r e s s i o n i n v o l v i n g a s i n e t e r n f e l l i n t o two p a r t s . F i r s t l y , we h a v e t h e n e g l e c t of t h e c u r v a t u r e of t h e i > h o t o - e l e c t r i c v a v e f r o n t and s e c o n d l y t h e r e p l a c e m e n t Of Hankel f u n c t i o n s by t h e i r a s y m p t o t i c forms. The f o r m e r a p p r o x i c a t i o n i s v e r y s e r i o u s when s c a t t e r i n g t a k e s p l a c e from heavy atoms,
e.g.T e l l u r i u m 171. However, f o r a t o m s w i t h
smalli o n c o r e r a d i i , u s u a l l y a c c o n p a n i e d
4ys m a l l i n t e r a t o m i c d i s t a n c e s , t h e c o n d i t i o n
k R * 1i s n o r e s e r i o u s . Our c a l c u l a t i o n s t h u s u s e a n t o d i f i c a t i o n o f Lee a n d P e n d y ' s [11 e q t n .
3 . 8 ,which
i sw r i t t e n f o r a n g u l a r l y a v e r a g e d d a t a
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where
ir u n s o v e r a l l a p p r o p r i a t e p a t h s a n d
Ztal:es o n t h e t h r e e f o r m s shown below f o r s i n g l e d o u b l e and t r i p l e s c a t t e r i n g
Z ( s i n g l e 1
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e,, i s t h e a n g l e between v e c t 0 r s . A Debye-llaller t e r m was a d d e d t o e a c h p a t h t o a c c o u n t i n a v e r y c r u d e way f o r t h e e f f e c t s of t h e r m a l n o t i o n . I n t h e s e c a l c u l a t i o n s t h e e n t i r e e x p r e s s i o n was m u l t i p l i e d
bya c o n s t a n t f a c t o r of
0.8t o a c c o u n t f o r s h a k e e f f e c t s . F u r t h e r , a c o n s t a n t i m a g i n a r y component of t h e s e l f e n e r g y of
4eV was i n c l u d e d u s i n g a complex argument f o r t h e Bankel f u n c t i o n s .
To c o m p l e t e t h e i n p u t t o t h e s e c a l c u l a t i o n s s e t s o f p h a s e - s h i f t s were c a l c u l a t e d u s i n g f r e e atom l i k e c h a r g e d e n s i t i e s w i t h l a r g e a t o m i c r a d i i . The exchange p o t e n t i a l was added u s i n g a n e n e r g y d e p e n d a n t H a r a - D i r a c t e r m 181 and t h e m u f f i n - t i n d i s c o n t i n u i t y was removed by
r e n o r m a l i s i n g t h e p o t e n t i a l .
A
rjood X-ray d i f f r a c t i o n c r y s t a l s t r u c t u r e of zn(Im),
( C 1 0 4h a s been
p u b l i s h e d 191. T h i s shows t h a t two of t h e f o u r i m i d a z o l e l i g a n d s a r e
c o p l a n a r w i t h t h e m e t a l atom, w h e r e a s f o r t h e o t h e r two r i n g s t h e
m e t a l atom l i e s
0.26a o u t of t h e r i n g p l a n e . F i g .
1shows a
p r o j e c t i o n of t h e i m i d a z o l e r i n g t o g e t h e r w i t h t h e m a g n i t u d e of t h e
(28-548 JOURNAL DE PHYSIQUE
s c a t t e r i n g a m p l i t u d e f (
B 1d i s p l a y e d a s a p o l a r diagram. Ire a r e f o r t u n a t e i n t h i s s y s t e m a s t h e s c a t t e r i n g f o r m s two c o n e s i n t h e f o r w a r d and backward d i r e c t i o n . T h i s i s n o t a l w a y s t h e c a s e p a r t i c u l a r l y when heavy m e t a l s c a t t e r i n g i s i n v o l v e d .
Iti s t h u s c l e a r t h a t 6 o u b l e s c a t t e r i n g p a t h s s u c h a s
2-3o r
4-5c a n be e l i m i n a t e d .
A l lc o n t r i b u t i n g p a t h s a r e g i v e n i n T a b l e
1t o g e t h e r w i t h a h a l f p a t h v a r i a n c e which i s e q u i v a l e n t t o t h e mean s q u a r e r e l a t i v e d i s p l a c m e n t f o r s i n g l e s c a t t e r i n g . These v a l u e s t o g e t h e r w i t h t h e c r y s t a l s t r u c t u r e a r e t r e a t e d a s a n A n s a t z owing t o t h e f a c t t h a t we a n t i c i p a t e t h a t r o c k i n g modes of t h e r i n g w i l l d o m i n a t e p a t h v a r i a t i o n s . F u r t h e r , w e a p p r e c i a t e t h a t o u r t r e a t m e n t of t h e r m a l e f f e c t s i s a c r u d e a p p r o x i m a t i o n a n d a p a r t from c o n f i g u r a t i o n a l a v e r a g i n g l i t t l e c a n be a c h i e v e d by o p t i n i s i n g t h e s e v a l u e s .
F i g .
2shows t h e k3 w e i g h t e d e x p e r i m e n t a l s p e c t r u n and t h e r e s u l t s o f t h e c a l c u l a t i o n . The t h e o r e t i c a l s p e c t r u n h a s been s h i f t e d s u c h t h a t i t s o r i g i n l i e s a t 9650.0
eV,w h e r e a s t h e h a l f e d g e h e i g h t e n e r g y l i e s a t 9663.5 eV. We c a n s e e i m m e d i a t e l y t h a t t h e c a l c u l a t i o n r e p r o d u c e s a l l f e a t u r e s of t h e s p e c t r u m , however, t h e f e a t u r e s a r e over-emphasised i n t h e
5t o
9fi-I r e g i o n . T h i s i s p r i n c i p a l l y due t o t h e o v e r e s t i m a t e of t r i p l e s c a t t e r i n g f r o n t h e r e a r e n d of t h e r i n g . T h i s h y p o t h e s i s i s t e s t e d i n F i g .
3 ,which shows t h e F o u r i e r t r a n s f o r m t a k e n o v e r t h e r a n g e shown i n F i g .
2.Lie c a n s e e t h e t r i p l e t s t r u c t u r e between
2.5and
4.08, which was a l s o o b s e r v e d f o r c a r b o n i c a n h y d r a s e
[61,a l t h o u g h t h e c a l c u l a t i o n s o v e r e s t i m a t e i t s s i z e .
W ea l s o n o t e t h a t t h e t h r e e main p e a k s a l s o c o i n c i d e w i t h t h o s e of t h e e x p e r i m e n t t o w i t h i n 0.03 a. The d i s c r e p a n c y between t h e o r y and e x p e r i m e n t c a n be c a u s e d by many f a c t o r s which
w i l lbe t h e s u b j e c t of f u r t h e r i n v e s t i g a t i o n .
I t
i s o f i n t e r e s t t o compare t h e s i n g l e s c a t t e r i n g c a l c u l a t i o n w i t h t h e r e s t r i c t e d m u l t i p l e s c a t t e r i n g c a l c u l a t i o n shown i n P i g .
4 .T h i s d i a g r a n c l e a r l y i n d i c a t e s t h a t m u l t i p l e s c a t t e r i n g i s e s s e n t i a l t o u n d e r s t a n d t h e s e s p e c t r a , i n p a r t i c u l a r t h e d o u b l e t a t
4 8i s n o t r e p r o d u c e d .
F i n a l l y , w e t u r n t o
acomputer e x p e r i m e n t where we r o t a t e t h e i m i d a z o l e r i n g a b o u t t h e f i r s t n i t r o g e n
N 1 . I ti s known from t h e c y s t a l s t r u c t u r e t h a t t h e m e t a l atom c a n be d i s p l a c e d o u t of t h e i m i d a z o l e p l a n e by
0.26A.T h i s c o r r e s p o n d s t o a r o t a t i o n of approx.
The m a g n i t u d e of t h e F o u r i e r
Ac o m p a r i s o n o f t h e s i n g l e t r a n s f o r m of t h e d a t a shown i n s c a t t e r i n g ( d o t t e d l i n e ) Fig.
2.S o l i d l i n e t h e o r y t h e o r y w i t h t h e r e s t r i c t e d and d o t t e d l i n e e x p e r i m e n t s . m u l t i p l e s c a t t e r i n g t h e o r y
( s o l i d l i n e ) .
1
P a t h s u s e d f o r c a l c u l a t i o n i n c l u d i n g t h e h a l f v a r i a n c e of t h e p a t h l e n g t h . u 2i s
t h e v a r i a n c e of t h e i n t e r a t o m i c d i s t a n c e f o r s i n g l e s c a t t e r i n g .Type P a t h u2 Type P a t h ff2 Type
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8' a b o u t a n a x i s i n t h e r i n g p l a n e p e r p e n d i c u l a r t o t h e m e t a l t?1 bond.
T h e r e
i s
no r e a s o n , however, t h a t p r e v e n t s r o t a t i o n s of t h e same o r d e r of m a g n i t u d e a b o u t a n a x i s p e r p e n d i c u l a r t o t h e p l a n e of t h e i m i d a z o l e r i n g . Tle h a v e i n v e s t i g a t e d b o t h c a s e s .P i g . 5 shows t h e s e n s i t i v i t y of t h e F o u r i e r t r a n s f o r m s f o r t h e l a t t e r c a s e
.
I n p a r t i c u l a r ,i t
s h o u l d be n o t e d t h a t t h e peak a t 2.5A i s
v e r y s e n s i t i v e t o t h i s movement, I ti s
c l e a r t h a t a n 8' r o t a t i o n n e r e l y s p l i t s t h e Zn-C d i s t a n c e s when viewed from a s i n g l e s c a t t e r i n ga0 L I 1.1 I., LO L S 1 0 1% (.I L5 %D
E M W RADIUS A
E i s l 5
hD L S 1.0 1.5 LO i s 3.4 IS LO & S 5 Q
EDKEO RADIUS A
E j g , h
The F o u r i e r t r a n s f o r m of t h e F o u r i e r t r a n s f o r m s of t h e c a l c u l a t e d s p e c t r u m f o r t h e c a l c u l a t e d s p e c t r u m which p l a n a r r i n g a f t e r r o t a t i o n i l l u s t r a t e s r o t a t i o n o f t h e a b o u t a n a x i s p a s s i n g t h r o u g h r i n g o u t of t h e p l a n e by D o l r l p e r p e n d i c u l a r t o t h e p l a n e . ( d o t t e d l i n e ) compared t o z e r o a t n o r o t a t i o n , b) r o t a t i o n by r o t a t i o n ( s o l i d l i n e ) .
8 t o b r i n g C4 c l o s e r t o t h e Zn-IIJ1 d i r e c t i o n , c ) r o t a t i o n by -8' t o b r i n g N 5 c l o s e r t o t h e Zn-N1 d i r e c t i o n .
framework. However, by decomposing t h e c o n t r i b u t i o n s i n t h i s a r e a t h i s peak c o n t a i n s c o n t r i b u t i o n s from d o u b l e a n d t r i p l e s c a t t e r i n g p a t h s a s w e l l
as
t h e s i n g l e s c a t t e r i n g . Thus, i ti s
q u i t e f a l s e t o a s s i g n a n y s i n g l e s c a t t e r i n g c o n t r i b u t i o nt o
t h i s peak.I f t h e Z i n c a t o n d o e s n o t l i e i n t h e r i n g p l a n e t h e s i t u a t i o n
i s
much s i m p l e r , b e c a u s e m a i n l y t h e r e a r end r i n g s c a t t e ri s
a f f e c t e d . T h i si s
shown i n F i g . 6 .JOURNAL DE PHYSIQUE
We h a v e shown t h a t m u l t i p l e s c a t t e r i n g
i s
i n p o r t a n t t o i n t e r p r e t t h e E:!AFS s p e c t r a and t h e i r F o u r i e r t r a n s o r m s . I n p a r t i c u l a r , we h a v e c o n f i r m e d t h e s p e c u l a t i o n [61 t h a t s c a t t e r i n g from t h e r e a r end o f t h e s y s t e m i n v o l v e s s u b t l e m u l t i p l e s c a t t e r i n g p e a k s . The o n l y way t o d e a l w i t h m u l t i p l e s c a t t e r i n gi s
t o u s e a b i n i t i o c a l c u l a t i o n s , F o u r i e r f i l t e r i n gi s
i n a p p r o p r i a t e t o d e a l w i t h t h i s problem.The r e s u l t s
a r e
p a r t i c u l a r l y e n c o u r a g i n g i n t h a t t h e y i n d i c a t e t h a t t h e o r i e n t a t i o n o f t h e r i n g c a n be e s t a b l i s h e d . T h i si s
p a r t i c u l a r l y i m p o r t a n t i n b i o l o g i c a l r e s e a r c h a s t h e r i n g s y s t e m whichi s
a t t a c h e d t o t h e mass of t h e enzyrne c a n f l e x t h e n e t a l n i t r o g e n bond anci c a u s e t i n y p u r t u r b a t i o n s t o t h e e l e c t r o n i c s y s t e m v i t a l f o r c a t a l y t i c a c t i v i t y . The chanGes of a m p l i t u d e of t h e F o u r i e r p e a k s n o t e d i n c a r b o n i c a n h y d r a s e when chariqinq t h e pH of t h e s y s t e m 161 a r e s i m i l a r t o t h o s e p r o d u c e d when t h e m e t a l a t o ni s
b e n t o u t of t h e i n i d a z o l e p l a n e .t?e w i s h t o t h a n k U. S i m o n i s (Univ. I!uenster/FRG) f o r a l l o w i n g r e p r o d u c t i o n of t h e s p e c t r u m of Z i n c i m i d a z o l e p e r c h l o r a t e [ l o ] . D.L.F. t h a n k s EIlEL f o r a p r e - d o c t o r a l f e l l o w s h i p a n d R.F.P. t h e U n i v e r s i t y of Tiariiiclc f o r l e a v e of ab: ence.
111 LEE, P.A. and PEIIDRY, J . B . , Phys. Rev. I?.
11.
(1975) 2735-2811.121 TEO, B.R., J. Am, Chea. Soc. (1981) 3930-4001.
(31 SOLATID, J.J., CPAIJE, S.C. and SALDESC:IlrIGLCR, J . D . , J. Chem.
Phys.
LZ
(1382) 162.141 GUlIAII, S.J., BIiISTED,
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and ROSS, I., J. PFiYS. C.U
1845-1861.
151 BGDGSOP?, J . U . , PERCY, G.C. and Y:!ORL,iTOII, D.A., T r a n s i t i o n i l e t e l Chcn.
3
(1978) 302-304.I 6 1 YACIINJISE:', V., POTTERS, L. and SF'IRO, T. G., J.
.
Chen. Soc.J&5
(1983) 6596-66012.[71 PETTIFER, R.F., " I n n e r - S h e l l and X-ray P h y s i c s o f Atoms a n d S o l i d s " Ed. : FA?3IAR, D. J., XLIE;IPOPPEtI, K. and WP,TSON, L.II., Plenum (1981) 653-666.
181 PCTTIFERI R.P.
and
COX, A.D., EXAFS and Hear Edge S t r u c t u r e , S p r i n g e r v e r l a g , Ed. : RIAZiCOIlI, A., Ii!COCCIA, L. and STIPICII, S. (1983) G6-72.191 BEAR, C.A., DUGGAIT, K.A. and FXEZiiCEJ, I-I.C., Acta C r y s t . - (1975) 2713-2715.
[ l o ] SIEIOtTIS, U., Ph.D. Univ. Huenster/F:G ( 1 3 8 5 ) .
