1 Twenty Years Hard Labor as a Fiber Diffractionist STRUTHER ARNOTT
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Department of Biological Sciences, Purdue University, West Lafayette, IN 47907
X-ray diffraction can be used to help determine the molecular geometry of polymers that prefer to be long helices rather than more complexly folded structures. It is usually possible to prepare specimens i n which such helical molecules are aligned with their long axes parallel. Often further lateral organization occurs, but rarely to the degree of a three-dimensionally ordered single crystal. Potentially this is an advantage, since there is more information (about the Fourier transform (1, 2, 3) of a molecular structure) i n the continuous intensity distribution i n the diffraction pattern from a less well-ordered system than there i s the "sampled" distribution characteristic of a single crystal. But, since "sampling" also implies local amplification of the molecular transform (at reciprocal lattice points), i t s absence results i n much weaker diffraction signals and the theoretical advantage of knowing the continuous intensity variation is offset by the experimental difficulty of recording it accurately. A further complication is that there are a great many kinds of partially-ordered systems of helical molecules, each giving rise to different types of diffraction pattern i n which both continuous intensity and Bragg maxima occur. If we wish quantitatively to analyze a diffraction pattern we, of course, have to succeed i n modelling not only the molecular structure but also the molecular packing. This i s true
0-8412-0589-2/80/47-141-001$07.50/0 © 1980 American Chemical Society French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
2
FIBER DIFFRACTION METHODS
for
any
d i f f r a c t i o n p a t t e r n , but
for fiber diffraction
patterns
t h e r e i s a d d i t i o n a l c o m p l e x i t y b e c a u s e t h e modes o f p a c k i n g more v a r i e d and
complex than i n s i n g l e
Fibrous biopolymers
crystals.
are a f f l i c t e d
a l s o by
the problem of
p h a s e d e t e r m i n a t i o n common t o a l l X - r a y a n a l y s e s and
by
are
of s t r u c t u r e ,
t h e same l i m i t a t i o n s o f r e s o l u t i o n t h a t a f f e c t
diffraction
a n a l y s e s o f m o s t m a c r o m o l e c u l e s e v e n when ( l i k e g l o b u l a r enzymes) they are o r g a n i z e d
i n single crystals.
The
ways i n w h i c h
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problems have been overcome f o r f i b r o u s systems a r e commonplace, a l t h o u g h biopolymers
t h e e m p h a s i s may
enough f o r f a c i l e p h a s e d e t e r m i n a t i o n . h i g h symmetry and
Nor,
These
atoms h e a v y
because of
their
tendency to d i s o r d e r , i s i t easy to o b t a i n
i s o m o r p h o u s heavy-atorn occupancy.
quite
be u n f a m i l i a r .
do n o t u s u a l l y h a v e c o v a l e n t l y - b o u n d
these
d e r i v a t i v e s without m u l t i p l e s i t e
T h e r e f o r e , many o f
the w e l l - t r o d d e n paths t h a t l e a d
from s e t s of d i f f r a c t i o n i n t e n s i t i e s to a unique s o l u t i o n molecular
s t r u c t u r e are not a v a i l a b l e .
More u s u a l l y one
of builds
a s t e r e o c h e m i c a l l y p l a u s i b l e model o f a r e s i d u e t h a t f i t s a h e l i x w h i c h has
t h e d i m e n s i o n s and
symmetry
determined from the l a y e r - l i n e spacings a b s e n c e s and pattern.
is
from the
general i n t e n s i t y d i s t r i b u t i o n i n the
T h e r e a f t e r t h e p r o b l e m i s one
fundamentally
and
of refinement.
(4J As
If
of,
t a s k of r e f i n i n g each p o s s i b i l i t y
a d j u d i c a t i n g among o p t i m i z e d m o d e l s o f e a c h k i n d by tests
systematic
diffraction
d i f f e r e n t i n i t i a l models are conceived
then the a d d i t i o n a l
into
characteristics
there and
appropriate
. I see i t , s t r u c t u r a l b i o c h e m i s t s
and
polymer
u s i n g f i b e r d i f f r a c t i o n data should s t r i v e to mimic
scientists classical
c r y s t a l l o g r a p h i c s t u d i e s so as t o a r r i v e a t s i m i l a r l y c r e d i b l e s o l u t i o n s o f s t r u c t u r e s by
s i m i l a r l y n o n c o n t r o v e r s i a l methods o f
procedure that are s i m i l a r l y r e p r o d u c i b l e i n other We
laboratories.
a r e o b v i o u s l y on t h e t h r e s h o l d o f g r e a t l y i m p r o v i n g
accuracy
o f i n t e n s i t y measurement b u t
I will
the
leave d i s c u s s i o n of
French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
1.
ARNOTT
3
A Fiber Diffractionist
t h i s t o o t h e r s and d i s c u s s i n t u r n t h e d i f f e r e n t k i n d s o f packing arrangements a v a i l a b l e t o f i b r o u s molecules, scheme f o r d e t e r m i n i n g
a general
t h e i r s t r u c t u r e s and p a c k i n g s , and
e x a m p l e s o f a r b i t r a t i o n among c o m p e t i n g m o d e l s .
Types o f D i s o r d e r and Consequent D i f f r a c t i o n E f f e c t s
Although
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will be
somewhat i d e a l i z e d ,
s e r v e t o i n d i c a t e t h e v a r i e t y o f p a c k i n g modes t h a t may
encountered.
molecules
The m o d e l h a s p a r a l l e l a r r a y s o f h e l i c a l
with their
l o n g axes i n t e r s e c t i n g a p l a n e
t o them a t p o i n t s f o r m i n g present not
t h e f o l l o w i n g g e n e r a l model
d i s c u s s i o n we w i l l
infinite,
although
a ( r e g u l a r rhombic) n e t . I n t h e ignore the f a c t that these nets a r e
f i n i t e n e t a r e a has t h e important
consequence o f broadening
diffraction signals,
a g g r a v a t i n g problems o f i n t e n s i t y measurement. be r e c o g n i z e d
thereby I t should
also
t h a t f i b e r s t y p i c a l l y c o n s i s t o f many s m a l l d o m a i n s
l i k e o u r m o d e l and t h a t t h e s e a r e p a r a l l e l h e l i x axes'
perpendicular
d i r e c t i o n b u t no o t h e r .
i n respect o f the
T h i s means ( f o r e x a m p l e )
t h a t when t h e d o m a i n s a r e f u l l y c r y s t a l l i n e t h e d i f f r a c t i o n the f i b e r i s l i k e
from
that from a r o t a t e d s i n g l e c r y s t a l , w i t h t h e
penalty of overlapping d i f f r a c t i o n signals f o rreciprocal
lattice
p o i n t s w i t h t h e same r e c i p r o c a l s p a c e c y l i n d r i c a l p o l a r r a d i u s (R i n F i g . 1 ) . However, f o r t h e moment we w i l l
discuss only
the consequences o f d i f f e r e n t types o f d i s o r d e r i n g o f m o l e c u l a r p a c k i n g within An
each s m a l l domain.
i s o l a t e d h e l i c a l molecule
i s i n essence a
c r y s t a l " because o f i t s a x i a l p e r i o d i c i t y . transform
"one-dimensional
I t sFourier
(]L, 2> 3 ) i s t h e r e f o r e c o n f i n e d t o l a y e r l i n e s a n d o n
each l a y e r l i n e i t i s a continous
f u n c t i o n p r o p o r t i o n a l t o Twhere
French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
4
FIBER DIFFRACTION METHODS
T = EE nj
f. J J
( 2 i r R r . ) e x p [ i { n ( < M .+ir/2)+2Tr*,z .} ]
n
J
J
(i)
J
If G
= E f. J ^ Z i r R T j ) exp[i(2Trilz -n(|) )],
n
j
(ii)
j
and t
= G
n
exp[in(*-hr/2)],
n
( i i i )
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then T = E t n is
(iv)
n
an a b b r e v i a t e d
S y m b o l s and
form of
definitions:
( i ) that w i l l
be
o = a x i a l repeat
found u s e f u l below. along
length
S^ i s t h e r e c i p r o c a l ( i . e . d i f f r a c t i o n ) s p a c e v e c t o r c a r t e s i a n components ( S , n , C ) ; C = ^/o
9
i s an
i n t e g e r ; S^ = R + _C v e c t o r i a l l y ;
c y l i n d r i c a l polar coordinates coordinates coordinates
Z
C
j/ >
(R,I|J,0 a r e
of
2 i r ) ; Az the a x i a l th ^
the p
displacement
h e l i c a l molecule; A
i n t e g e r d e t e r m i n e d by
f o r an
is a
vector
N-fold
the s e l e c t i o n r u l e
w h e r e m = 0,+l,+2, e t c .
That T i s a s e r i e s of B e s s e l r a t h e r than
trigonometric
i s merely a consequence of u s i n g c y l i n d r i c a l
coordinates for
the
the f i r s t
the net on w h i c h the h e l i c e s a r e a r r a y e d ;
i n t e g r a l h e l i x , n i s an
functions
cartesian
k i n d o f o r d e r n and ^ the r e l a t i v e o r i e n t a t i o n of the p helical
( a s a f r a c t i o n o f o)
n = £-Nm,
the
f j i s the s c a t t e r i n g f a c t o r of
m o l e c u l e (as a f r a c t i o n of
in
has
I ( t h e l a y e r l i n e number)
o f p o i n t t h a t has
(Y) = B e s s e l f u n c t i o n o f
n argument Y;
that
(£,n,C); (r.,(|>.,Z.) a r e t h e c y l i n d r i c a l p o l a r tfo 3 3 3 of the j atom o f one r e s i d u e o f t h e h e l i c a l
m o l e c u l e ; z^ = atom; J
helix;
( r , _., oz/)
f o r atoms i n r e a l s p a c e and
points i n r e c i p r o c a l space.
Not
only
is this a
framework f o r d e s c r i b i n g a h e l i c a l m o l e c u l e , but e c o n o m i e s i n c o m p u t i n g T.
For
polar (R, ^,
convenient
i t can l e a d
h e l i c e s , only Bessel
£/
