structure factor calculations for some helical polypeptide

7/28/2019 Structure Factor Calculations for Some Helical Polypeptide

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H. I=AUPTMAN AND J. KA RL E 97

~ ma y be chosen to be a ~ag. Then the values of allphases ~ugg and ~gua, being li near ly semi -dependenton ~, are determined. Next, F~ may be chosen to bea q~uu. Then the values of all phases q~uu and ~u,linearly semi-dependent on ~, are determined. Finallythe values of all phases q~uguand q~guu,linearly semi-dependen t on the pair qh, ~2, are determined.

6.2. Ty p e 2 P 1The first paragraph of 6.1 carries over verbatim for

Type 2P~ with the single exception that the semin-vari ant phases are now of the form ~ggg and q~uuu.

As an illustration of the specification of origin,~1 may be chosen to be a ~u~g. The values of all phasesq~uug and ~ea~, linearly semi-dependent on ~, aredetermined. Next, ~ may be chosen to be a ~a~.Then the values of all phases q~guuand ~ugg, linearlysemi-dependent on ~2, are determined. Finally thevalues of all phases q)ugu and ~g~a, linea rly semi-dependent on the pair ~, ~, are determined.

6.3. Ty p e 3P 2

The phases which are the structure seminvariantsare of the form ~ggg, T~ug, ~ug~ and q~guu.The value ofany phase ~, not of this form, may be specifiedarbitrarily. Once this is done, the values of all phases,of necessity linearly semi-dependent on ~1, are deter-mined. For example, T1 may be chosen to be ~ggu.

6.4. Ty p e 3 P 3The phases which are the structure seminvariants

are of the form q~ggg,q~ggu, q)uug,and q)uuu. The valueof any phase ~1, not of this form, may be specifiedarbitrarily. Once this is done, the values of all phases,

of necessity linearly semi-dependent on qh, are deter-mined. For example, ~1 may be chosen to be ~uga.

6.5. Ty p e 4 PEvery phase is a structure seminvariant and its

value is determined by the observed intensities. Thevalue of no phase may be specified arbitrarily. In thistype, the choice of the functional form of the structurefactor is equivalent to the unique selection of theorigin.

7 . C o n c lu d i n g r e m a r k s

Monograph I (1953) and this paper prese nt a detailedprocedure for specifying the origin in any centro-symmetric space group. This has been done bydemonstrating the existence of relationships betweenthe observed intensities and values of the phases viathe structure seminvariants. With the specific state-ment of the nature of these relationships, it is possibleto go directly from observed intensities to the valuesof phases. It will be the aim of future publications toemploy the formulas of our two recent papers (1958)

to obtain specific procedures for phase determinationfor all the space groups.

R e f e r e n c e s

I4_AUPTHAI~, H. & KARLE, J. (1953). Solut ion of the PhasProblem. I . The Centrosymmetric Crystal .A.C.A. Mono-graph No. 3. New York: Polycrystal Book Service.

HAUPr~_AN, H. & I~aL E, J. (1956). Acta Crys t .9, 45.I-IAut


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98 S T R U C T I J R E F A C T O R C A L C U L A T I O N S F O R S O M E H E L I C A L P O L Y P E P T I D E M O D E L S

i n t e n s it i e s . A q u i c k m e t h o d o f o b t a i n i n g c a l c u l a t e dh e l i c a l i n t e n s i t i e s i s t h e r e f o r e d e s i r a b l e s o t h a t t r i a ls t r u c t u r e s o b t a i n e d f r o m m o d e l b u i l d in g m a y b ec h e c k e d b y c o m p a r i n g t h e c a l c u l a t e d w i t h t h e o b -s e r v e d i n t e n s i t i e s .

T h e c a l c u l a t e d i n t e n s i t y o f d i f f r a c t i o n b y a p r o p o s e df i b e r s t r u c t u r e c a n b e o b t a i n e d e i t h e r b y c o m p u t a t i o n ,u s i n g t h e s t r u c t u r e - f a c t o r e x p r e s s i o n o f C o c h r a ne t a l .( 19 5 2) ; o r i t c a n b e o b t a i n e d o p t i c a l l y w i t h t h e o p t i c a ld i f f r a c t i o n s p e c t r o m e t e r ( H u g h e s & Ta y l o r , 1 9 5 3 ).S t o k e s ( 1 9 5 5 ) h a s d i s c u s s e d t h e t h e o r e t i c a l b a c k g r o u n df o r t h e u s e o f t h e o p t i c a l d i f f r a c t i o n s p e c t r o m e t e r f o rp r o d u c i n g t h e o p t i c a l t r a n s f o r m o f a he l i ca l s t r u c t u r e .T h i s m e t h o d p r o v i d e s o n l y a p p r o x i m a t e s t r u c t u r ef a c t o r s f o r c o m p a r i so n w i t h t h e o b s e r ve d X - r a y d if -f r a c t i o n p h o t o g r a p h , b e c a u s e o f t h e d i f f i c u l t y o f d e a l -i n g w i t h o v e r l a p p i n g a t o m s a n d w i t h t h e d i f f e r en c e si n s c a t t e r i n g f a c t o r s . I n a d d i t i o n , a n u m b e r o f p r o -j e c t e d v i e w s m u s t b e s u m m e d i f o n e w i sh e s to p r o d u c ea t r a n s f o r m i n w h i c h t h e r e is r a n d o m a n g u l a r o r i en t a -t i o n o f t h e m o l e c u l e s a b o u t t h e h e l i c a l ax i s . T h e t i m es p e n t i n c a l c u l a t i n g a n d p l o t t i n g t h e s e v i e w s c a n b ec o n s i d e r a b l e .

We h a v e t h e r e f o r e p r e p a r e d a p u n c h e d - c a r d m e t h o df o r c a l c u l a t i n g f i b e r d i f f r a c t i o n p a t t e r n s f o r he l i c als t r u c t u r e s w i t h r a n d o m a n g u l a r o r i e n ta t i o n .

T h e r e h a v e b e e n s e v e r a l h e l i c a l c o n f i g u r a t i o n s p r o -p o s e d f o r p o l y p e p t i d e c h a i n s , i n c l u d i n g t h e a a n d yh e l i c e s ( P a u l i n g e t a l . , 1 9 5 1 ) , t h e ~ - h e l i x ( L o w &B a y b u t t , 1 9 52 ) a n d s e v e r a l o t h e r s . Wi t h t h e e x -c e p t i o n o f t h e ~ h e l ix , F o u r i e r t r a n s f o r m s h a v e n o tb e e n c a l c u l a t e d f o r t h e s e m o d e l s . I n o r d e r t o s h o w t h ed i f f e r e n c e s i n d i f f r a c t i o n p a t t e r n s t o b e e x p e c t e df r o m t h e m , w e h a v e c o m p u t e d t h e h e li ca l t r a n s fo r m sf o r s e v e r a l s u c h h e l i c e s .

T h e a v e r a g e d i n t e n s i t y e x p r e s s i o n

T h e F o u r i e r t r a n s f o r m o f a h el i c a l m o l e c u l e w i t hp e r i o d i c i t y c a l o n g i t s a x i s m a y b e e x p r e s s e d a s t h es u m o f a s e r i e s o f B e s s e l f u n c t i o nJ n f o r e a c h l a y e rl i n e 1 ( C o c h r a n e t a l . , 1 9 5 2 ) . T h u s , a t a p o i n t i n r e c i p -r o c a l s p a c e w i t h c y l i n d i i c a l c o o r d i n a t e s ( R , F,1/c) ,t h e F o u r i e r t r a n s f o r m i s

F ( R , v /, 1 /c ) = ~ v_ ~F j j ~ , ( 2 ~ r ~ R )n j

Stru cture (1) 7 helix

w h e r e r ~, ~ , z~ a r e t h e c y l i n d r i c a l c o o r d i n a t e s o f t h ej t h a t o m o f t h e a s y m m e t r i c u n i t w h i c h h a s a t o m i cs c a t t e r i n g f a c t o r f i , a n d t h e s u m m a t i o n o v e r n i s f o ra l l t h e a l l o w e d v a l u e s o f n o n t h e l a y e r l i n e o f i n d e x 1.

T h i s e x p r e s s i o n c a n b e w r i t t e n

F ( R , y~, l /c ) = ~ ' ( A n + i n n )e x p in~p , (2 )n

w h e r e

A ~ - - ~ f i J . ( 2 ~ r~ R ) c o s n ( ~ - ~ - ~v~) + - - ,/

B n = .-~ f , J n ( 2 ~ r , R ) s in { n ( ~ - c f , ) + 2 ~ z ~ } .(3 )J

T h e i n t e n s i t y i s t h e n

F F * = ~ ~ { ( A n A m + B n B m )n m

+ i ( A m B n - A n B r n ) }e x p i ( n - m ) v / . (4 )

T h i s e x p r e s s i o n , b e i n g a f u n c t i o n o f F, i s n o tc y l i n d r i c a l l y s y m m e t r i c a l . I n a f i b e r d i a g r a m , w h e r et h e m o l e c u l e s a r e p a r a l l el , b u t o r i e n t e d a t r a n d o ma b o u t t h e f i b e r a x i s , t h e r e l e v a n t i n t e n s i t y i s c y l i n -d r i c a l l y a v e r a g e d ; t h a t i s , a v e r a g e d o v e r a l l v a l u e so f ~v. E q u a t i o n ( 4 ), w h e n a v e r a g e d , i s z e r o u n l e s sn = m , w h e n

= (A n , Bn ) . (5 )n

A n e q u i v a l e n t e x p r e s s i o n h a s b e e n d e r i v e d b yF r a n k l i n & K l u g ( 1 95 5 ), b u t t h e f o r m o f (5 ) i s m o r es u i ta b l e f o r m a c h i n e c o m p u t a t i o n .

T h e o r d e r s n , o f t h e ] 3 e s se l f u n c t i o n s o n e a c h l a y e rl i n e a r e d e f i n e d b y t h e e q u a t i o n

1 /c = n / P + m / p , (6 )

w h e r e P i s t h e p i t c h o f t h e h e l i x , p i s t h e a x i a l t r a n s l a -

~ io n p e r a s y m m e t r i c u n i t , a n d m i s a n i n t e g e r. F o rm o l e c u l e s w h e r e t h e r e a r e ]c r e s i d u e s p e r r e p e a t u n i ta l o n g t h e f i b e r a x i s , s u c c e s s i v e o r d e r s o f B e s s e l f u n c -t i o n s o n a g i v e n l a y e r l i n e w i l l d i f f e r i n o r d e r b y ] c .I n g e n e r a l , e x c e p t f o r h e li c e s w i t h s m a l l i n t c g r a ls c r e w s , ]c i s l a rg e e n o u g h s o t h a t t h e m a j o r c o n t r i b u -t i o n t o t h e i n t e n s i t y o n a g i v e n l a y e r l in e i s d u e t ot h e l o w e s t - o r d e r B e s s e l f u n c t i o n w h i c h s a t i s f i e s e q u a -t i o n ( 6) . U n d e r t h e s e c i r c u m s t a n c e s , i t is n o t n e c e s s a r yt o c a l c u l a t e m o r e t h a n t h e c o n t r i b u t i o n o f t h e l o w e s t -o r d e r B e s s e l f u n c t i o n o n e a c h l a y e r li n e. H o w e v e r ,

Ta b l e 1

(2) 3.0 residue

helix

(5) Modified

(3) ~z hel ix (4) c~ heli x ~ hel ixResidues per turn

c-axis (A)

Interval k between successive Besselorders on each layer line

Num ber of Bessel funct ions per layerline used in calculation

36 residues 3 residues 22 residues 18 residues 23 residuesin 7 turns per turn in 5 turns in 5 turns in 5 turns

35.28 6.00 25.3 26-64 26-64

36 3 22 18 23

i 4 1 1 I


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D A V I D R . D A V I E S A I X T D A L E X A N D E R : R I C H

2 3 4 ~ 1 7 ~

211 15 - - " - -

I0 XX 1419 ,

~ 1~ . . . .18 317 1 2 - " - - - "

1 51 4 - / ~ - - ~ - ~ - - - " ~ " " " - " " ~ - - ~, ~ , 10 J1 312 211 /" " , ~ / ,

9 , _ _ _ / . - -

I , , 7 V

-

~ ( A - ' ) I 0 . 1 0 ' , , , / ' ~ 2

T i T. i1 1 I ,

0 0 . 1 0 0 . 2 0 0 - 3 0 0 . 4 0 0 . 5 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 0 . 1 0 0 . 2 0 0 - 3 0 0 . 4 00 - 5 0( a ) R ( A - ' ) ( b ) R ( / ~ - b ( c ) R ( A - b

17 ~ - '

1 6 - -

10 i '

7 / ' \ ~ I ~ . . . . . . . .i I / /

_ ~ - _ _ i . / : ,~ - ~ - , ,_ _ _ .

0 0"10 0"20 0 .30 0 .40 0 0"10 0"20 0"30 0"40 0"50

( d ) R ( A 11 (e ) R ( A 1 )

F i g . 1 . T h e c a l c u l a t e d a v e r a g e d i n t e n s i t i e s f o r p o l y p e p t i d e m o d e l s : ( c t) ~ h e l i x , ( b) 3 . 0 r e s i d u e h e l i x , ( c) ~ h e l i x , ( d ) ( e) m o d i f i e d ~ h e l i x . T h e r e c i p r o c a l l a t t i c e s c a l e is t h e s a m e f o r a l l f i v e g r a p h s . T h e C s c a l e i s s h o w n i n ( b ) . T h e sr e p r e s e n t s t h e a v e r a g e d i n t e n s i t y o f a r i g h t - h a n d e d h e l i x o f L - a m i n o a c i d s ( o r a l e f t - h a n d e d h e l i x o f ] ) - a m i n o a c i d sd a s h e d l i ne r e p r e s e n t s a l e f t - h a n d e d h e l i x o f L - a m i n o a c i d s .


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I0 0 STRUCTU RE FACTOR CALCULATIONS FOR SOME HELICA L POLYPEPTI DE MODELS

if k is small, as in the 3.0 residue helix (see Table 1,p. 98), then more than one Bessel function makes asubstantial contribution to a given layer line. Theseare simply added, as shown in equation (5).

M e t h o d o f c o m p u t a t i o n

A punched-card method has been prepared whichuses the IBM 604 computer. In this procedure a detailcard is prepared for each point in reciprocal space foreach atom of the asymmetric unit of the helix. Eachdetail card contains the atomic coordinates, r~., cp, zj,a punch to distinguish the various atomic species, thereciprocal-space coordinates 1 and R, and t he ap-propriate value of n for each I.

Values of sin 0, 2 ~ r ~ R , cos {n(~r-cpj )+2: t lz~/c}ands in {n (z~-q ) j )+2zdz j / c}are then computed with theuse of a simple power series and are punched on tothe detail cards. Appropriat e values of fi (sin 0) andJn (2 : r r~R) ,taken from master decks, are also punchedon to the detail cards. Finally, the quantities An, Bn,and 9. 2n + B , ~ of equation (5) are calculated and

punched on to a deck of summary cards which arethen listed and plotted in the form shown in Fig. 1.The transforms were evaluated at intervals of

0.02 A -1 along R on all lay er lines. Th~ Bessel-function master deck contains Jn (2x~rcR) to the nearest0"001 over the range 0.00 _< 27~rR < 25.00 at intervalsof 0.05. These were originally obtained from theTa b l e s o f t h e B e s s e l F u n c t i o n s(1947).

For these calculations, the asymmetric unit consistsof five atoms, N, C~, Cfl, and carbo xyl C and O. Theatomic scattering factors were those obtained bygraphical interpolation from the tables of McWeeny(1951) and contain no correction for thermal vibra-tion. Values of f (sin 0) were chosen to t he nearest

0.01, using intervals of sin 0 equal to 0.005 in th erange 0.00_< sin 0_< 0.600 and 0.01 in the range0.600 < sin 0 < 1.000.

For helical molecules which crystallize ~4th onehelix in the unit cell, the appropriate intensity ex.pression becomes

( A n + B n )F * (R , v /, l / c) = ~ ~ 2n

n> m

+2 ~ .~, { ( A n A m + B n B m )cos ( n - m ) v2m

- ( A m B n - A n B m )sin (n-m)y~}. (7 )

This expression is a function of ~p and it is thereforeincorrect to use the cylindrically averaged intensityexpression for crystalline material. For helices like the~ and ~ helix, however, where ( n - m ) is 18 and 22respectively, the yJ-dependent part of (7) will be smallfor the intense layer lines, and the cylindrically aver-aged expression (5) will provide a suitably accurateintensity pattern. However, computation of the trans-

form for a crystalline 3.0 residue helix would requirethe use of the expression (7).

R e s u l t s

Fig. l(a-d ) shows the avera ged intensities for righ tand left-handed helices of L-amino acids in the (a)y helix , (b) 3.0 residue helix (Donohue, 1953), (c)

helix and (d) a helix. These are all calculated with

the fl-carbon atom present, hence they all representpossible models for poly-L-alanine. Various charac-teristics of these helices are listed in Table 1. The resultsfor the a helix agree well with those of Pauling,Corey, ake l & Marsh (1955), when allowance ismade for the fact that Pauling et a l. appear to haveincluded a temperature factor in their calculation.

It is interesting to note that all of these transformshave an intense layer line near ~ = 0.2 /~-1. Further-more, this layer line is a sensitive index of the handed-ness of the helix, since there are marked differencesbetween right- and left-handed members of the sameseries.

Comparison of the transforms for the c and

helices shows that they have a remarkably similargeneral intensity distribution. This similarity can begreat ly increased by adjusti ng the 7~ helix to give 23residues in 5 turns with c = 26.64 A, as shown inFig. l(e). Examinati on of such a structure by themethod of Low & Grenville-Wells (1953) shows aC-Co-N angle of 117 with a hydrogen-bond lengthof 2.86 A a nd with t he hydro gen atoms displaced 5 from the N-O line. However, this method for con-structing a polypeptide helix throws all the distortioninto the a-carbon angle. By introducing small distor-tions in the rest of the molecule, e.g. removing thestrict planarity of the peptide group, it is possibleto effect a considerable reduction in the co-carbonangle distortion. For example, a 5 deviation from theplanarity of the peptide group would reduce thea-ca rbon angle to 113 . Donohue (1953) points outthat there might exist certain conditions under whichthe 7~ helix would be favored, a ltho ugh i t is at fir stsight less stable than the a helix. We have, therefore,re-examined the data on synthetic polypeptidesbearing in mind the striking similarity between thetran sfor ms of the a and 7~ helices.

The main differences between the calculated dif-fraction patterns of the a and ~ helices occur on theequator and on the 18th layer llne. The 18th layer llneof the ~ helix is described by a first-order Bessel func-tion and is therefore non-meridional in character, incontrast to the zero-order Bessel function of the~-helix, which is meridional. In addition, the peakintens ity of the 18th layer line of the ~ helix is betweenone-quarter and one-third that of the a-helix. Theequatorial patterns, however, differ greatly; in par-ticular, the a-helix transform has its first minimumat R = 0.16 f i x 1 .


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D A V I D R . D A V I E S A l ~ D A L E X A I ~ D E R 1 %IC H

T h e s y n t h e t i c p o l y p e p t i d e ~ - p o l y - L - a l an i n e h a s b e e np r e p a r e d a s h i g h l y o r i e n te d a n d c r y s t a l li n e f i b e rs b yB a m f o r d et al. ( 1 95 4 ), a n d d i f f r a c t i o n d a t a o b t a i n e df r o m t h e s e f i b e r s a r e p r e s e n t e d b y B r o w n & T r o tt e r( 1 95 6 ). B e c a u s e t h e s i d e c h a i n s o f p o l y - n - a l a n i n e c o n -s i s t o n l y o f m e t h y l g r o u p s w h o s e p o s i t io n s a r e k n o w n ,i t s d i f f r a c t i o n p a t t e r n i s p a r t i c u l a r l y s u i t a b l e f o rc o m p a r i s o n w i t h t h e c a l c u l a t e d t r a n s f o r m s o f p o l y -p e p t i d e m o d e l s . B r o w n & T r o t t e r h a v e o b s e r v e dt h a t t h e 1 .5 A r e f l e c t i o n i n c ~ - p o ly - L - a l an i n e c a n b e s tb e i n d e x e d a s ( 0 ,0 , 0, 4 7) . T h i s a p p e a r s a s a w e llr e s o l v e d i n t e n s e r e fl e c ti o n , a n d i s a c c o m p a n i e d b y t h e( 1 ,0 , 1, 4 7) r e f l e c t i o n , w h i c h i s m u c h w e a k e r. T h eo r i e n t a t i o n o f t h e f i b e r i s s u c h t h a t t h i s 1 .5 A r e f le c -t i o n c a n o n l y b e m e r i d i o n a l . I n a d d i t i o n , c o n s i d e r a t io no f t h e r e l a t i v e i n t e n s i t i e s o f t h e e q u a t o r i a l r e f l e c t i o n so f ~ - p o l y - L - a l a n i n e c l e a r l y f a v o r t h e ~ h e l i x m o d e l .T h u s , a l t h o u g h t h e r e i s a s t ri k i n g s i m i l a r i t y b e t w e e nt h e t r a n s f o r m s o f t h e ~ a n d g h e l i c es , t h e d i f f e r e n c e sa r e s t i l l s u f f i c i e n t s o t h a t a c l e a r d i s t i n c t i o n c a n b em a d e b e t w e e n t h e m f o r c o m p a r i s o n w i t h w e l l o r i e n t e dd i f f r a c ti o n d a t a . I t s h o u l d b e p o i n t e d o u t , h o w e v e r ,t h a t s u c h a c le a r d i s t i n c t io n c a n n o t b e m a d e b e t w e e na n ~ a n d a 7~ h e l i x f r o m a n i n s p e c t i o n o f a p o o r l yo r i e n t e d f i b e r p a t t e r n .

R e f e r e n c e s

SAM:FORD, C. H., BROWl~, L. , ELLIOTT, A. , HA~BY, W. E& TROTTER, I . F . (1954).N a t u r e , L o n d . 173 , 27 .

BROWN, L. & TROTTER, I . F. (1956).Tr a n s . F a r a d a y S o c52 , 537 .

COCURAN, W ., CRICK, F. H. C. & VA ND, V. {1952).A c tCryst . 5, 581.

DONOHUE, J . (1953).P r o c . N a t . A c a d . S c i . , Wa s h .3 9470 .

FRAN~I~, 1%. & I~VG, A. (1955) .Acta Crys t . 8, 777.HVGHES, W . & TAYLOR, C. A . (1953).J . S c i . I n s t r u m

30, 105.Low , B. & BAYBUTT, R .B . (1952) .J . A m e r. C h e m . S o

74 , 5806 .Low , B .W . & GRE~CV~T.E-WELLS, H .J . (1953) .P r o c

N a t . A c a d . S c i . , Wa s h .39 , 785 .MCWEENY, R. (1951).Aeta Crys t . 4, 513.PAULING, L. , COREY, 1%. B. & BRANSON, H . R . (19

P r o c . N a t . A c a d . S e i ., Wa s h .37 , 205 .PAULING, L . , COREY, R .B . , YAKE L, H .L . & MAR

R . E . ( 1 9 5 5 ) . Acta Crys t . 8, 853.RIc ~ , A. & CRICK, F. H. C . (1955) .N a t u r e , L o n d . 1 7 6

915.STOKES, A. R. (1955).Acta Crys t . 8, 27.Tables o f Besse l Func t ions{ 19 47 ). C a m b r i d g e : H a r v a

U n i v e r s i t y P r e ss .WATSON, J . D . & CRICK, F. H. C . (1953) .N a t u r e , L o n d171 , 737 .

Acta Crys t . (1959). 12, 101

A n I m p r o v e d M e t h o d f or D e t e r m i n i n g t h e R e l a t i v e P o s i t io n s o f M o l e c u l e s

BY C. A. TAYLOR A~D K. A. MORLEY

P h y s i c s D e p a r t m e n t , C o ll eg e o f S c i e n c e a n d Te c h n o l o g y, M a n c h e s t e r1, E n g l a n d

(Received17 September 1958)

T h e m o l e c u l a r - l o c a t i o n m e t h o d d e s c r i b e d b y Ta y l o r ( 19 54 ) h a s b e e n f o u n d t o f a i l u n d e r c e r t a i nc o n d i ti o n s . T h e m e t h o d d e s c r i b e d i n t h e p r e s e n t p a p e r w a s d e s i g n e d to o v e r co m e t h e s e l i m i t a t i o n sa n d p r o v e s t o b e m o r e s a t i s f a c t o r y i n o t h e r r e s p e c t s a s w e l l ; i t i n v o l v e s n o a p p r o x i m a t i o n s a n d t h er e s u l t s a r e p r e s e n t e d i n t h e f o r m o f a c o n t o u r e d g r a p h w h i c h i s e a si e r t o a s s e s s t h a n t h e p a t t e r n o fb a n d s p r o d u c e d i n t h e e a r li e r m e t h o d .

I n t r o d u c t i o n

I n u s i n g t h e F o u r i e r o r o p t i c a l- t r a n s f o r m a p p r o a c h t oc r y s t a l - s t ru c t u r e d e t e r m i n a t i o n , t h e s h a p e a n d o r i e n-t a t i o n o f a m o l e c u l e o r g r o u p o f a t o m s i s d e t e r m i n e dl a r g e l y b y c o n s i d e r i n g t h e d i s p o s i t i o n o f t h e s t r o n g e rr e f l ex i o n s ; c o n s i d e r a t io n o f t h e w e a k e r a n d a b s e n tr e f le x i o n s m a y t h e n g i v e i n f o r m a t i o n a b o u t t h ep o s i t i o n o f t h e m o l e c u l e o r g r o u p r e l a t i v e t o o t h e r s i nt h e u n i t c e ll ( H a n s o n , L i p s o n & Ta y l o r , 1 9 53 ) .

A s y s t e m a t i c m e t h o d o f s o l v in g t h e p o s i ti o n p r o b l e mw a s s u g g e s t e d b y Ta y l o r ( 19 5 4) a n d h a s b e e n s u c c e s s-f u l l y u s e d o n a n u m b e r o f s t r u c tu r e s . D u r i n g i t sa p p l i c a t i o n t o o n e p a r t i c u l a r s t r u c t u r e , h o w e v e r ,c e r t a i n l i m i t a t i o n s w e r e d i s c o v e r e d ( C r o w d e r, M o r l e y

& Ta y l o r , 1 9 5 7 ). T h e e s s e n t i a l r e q u i r e m e n t f o r s u ci n t h e e a r l i e r m e t h o d i s t h a t t h e r e f l e x i o n s t o b e us h o u l d b e c h o s e n t o s a t i s f y c e r t a in c o n d i t i o n s w hb e c o m e m o r e s t r i n g e n t a s th e s y m m e t r y o f t h e i n c r e a s e s .

I n t h i s s t r u c tu r e , h o w e v e r , a n a c c id e n t a l s y m m er e l a t i o n s h i p i n t h e m o l e c u l e m a d e i t i m p o s s i b l ec h o o s e re f l e x i o n s s a t i s f y i n g t h e s e c o n d i t i o n s . T h e m e t h o d , b r i e f ly o u t l i n e d a t t h e M o n t r e a l C o n f e r e( Ta y l o r , 1 9 5 7 ) a n d n o w t o b e d e s c r i b e d i n d e tw a s d e v e l o p e d t o o v e r c o m e t h i s d i f fi c u l ty. I t h a s bf o u n d t o h a v e a n u m b e r o f a d d i t i o n a l a d v a n t a g e s ot h e e a r l i e r m e t h o d . I n p a r t i c u l a r , t h e r e s u l ts p r e s e n t e d i n a m o r e e l e g a n t f o r m . T h e f i n a l p o s i

structure factor calculations for some helical polypeptide

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