UBC Theses and Dissertations

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UBC Theses and Dissertations

Electron paramagnetic resonance study of cytochrome C solutions and single crystals Mailer, Colin 1971

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ELECTRON PARAMAGNETIC RESONANCE STUDY OF CYTOCHROME C SOLUTIONS AND S I N G L E CRYSTALS b y • C O L I N MAILER B . S c . U n i v e r s i t y o f S t . A n d r e w s , S c o t l a n d , 1960 M.Sc. U n i v e r s i t y o f L o n d o n , E n g l a n d , 1963 A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA A u g u s t , 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia, Vancouver 8, Canada. Date i ABSTRACT E l e c t r o n p a r a m a g n e t i c r e s o n a n c e (EPR) s i g n a l s f r o m t u n a f e r r i c y t o c h r o m e c s o l u t i o n s were o b t a i n e d between 4 o 2 0 K and 77°K, w i t h g - v a l u e s g 1 = 1.25, g 2 = 2 C 2 5 , g 3 = 3.05« The g 3 l i n e i s 380 gauss wide between 4<,2°K and 50°K w i t h G a u s s i a n shape, b u t has become 700 gauss wide w i t h L o r e n t z i a x i shape a t 77°K. The t e m p e r a t u r e i n d e p e n d e n t shape and w i d t h a r e b e s t e x p l a i n e d by a d i s t r i b u t i o n o f r h o m b i c c r y s t a l f i e l d p o t e n t i a l s ( r . m . s . d e v i a t i o n » 1 1 % ) . The L o r e n t z i a n shape a r i s e s f r o m a s h o r t ( 1 0 ~ s e c . ) s p i n - l a t t i c e r e l a x a t i o n t i m e . EPR s p e c t r a f r o m h o r s e h e a r t f e r r i c y t o c h r o m e c s i n g l e c r y s t a l s were a n a l y s e d t o o b t a i n t h e o r i e n t a t i o n o f t h e g-axes r e l a t i v e t o t h e c r y s t a l l o g r a p h i c a x e s . The g^-a x i s was 76° f r o m t h e c r y s t a l c - a x i s , c l o s e t o t h e heme no r m a l (71.5° t o c - a x i s ) d e t e r m i n e d f r o m t h e 3 - d i m e n s i o n a l X - r a y s t r u c t u r e by D i c k e r s o n . The o t h e r 2 g-axes l a y a p p r o x i m a t e l y a l o n g t h e N-Fe-N d i r e c t i o n s i n t h e heme r i n g . An amended v e r s i o n o f E i s e n b e r g e r and P e r s h a n ' s t h e o r y was u s e d t o e x p l a i n t h e a n g u l a r v a r i a t i o n o f t h e b r o a d l i n e s (300-2000 gauss) seen i n t h e c r y s t a l s — b e s t f i t was o b t a i n e d w i t h t h e d i s t r i b u t i o n o f l i g a n d f i e l d s f r o m t h e s o l u t i o n s t u d y p l u s a 1.5° v a r i a t i o n i n g - a x i s o r i e n t a t i o n . The u n d i f f e r e n t i a t e d a b s o r p t i o n l i n e shapes o b s e r v e d a t 4„2°K i n b o t h s o l u t i o n s and s i n g l e c r y s t a l s were e x p l a i n e d by t h e P o r t i s t h e o r y o f r a p i d a d i a b a t i c p a ssage i n s o l i d s * T h i s t h e o r y was t e s t e d w i t h a model s y s t e m o f c h a r r e d d e x t r o s e , and f o u n d t o be v a l i d . U s i n g t h e t h e o r y t h e r e l a x a t i o n t i m e ( t ) o f t h e cytochrome c s y s t e m was f o u n d t o be, f r o m t h e phase l a g o f t h e EPR s i g n a l r e l a t i v e t o t h e m a g n e t i c f i e l d m o d u l a t i o n , 3.8 x 10~^ s e c . a t 4.2°K. T was o b t a i n e d between 4.2°K and 18°K f r o m t h e r a p i d p a ssage s i g -n a l s , and between 50°K and 70°K f r o m t h e l i n e w i d t h o f t h e s p e c t r a . The t e m p e r a t u r e dependence o f T below 20°K c o u l d 9 a r i s e f r o m a c o m b i n a t i o n o f a T Raman s p i n - l a t t i c e r e l a x a -t i o n p r o c e s s w i t h a t e m p e r a t u r e i n d e p e n d e n t s p i n - s p i n —8 r e l a x a t i o n t i m e o f o r d e r 10 seconds ( w h i c h might a r i s e f r o m d i p o l a r i n t e r a c t i o n s between n e i g h b o r i n g i r o n a t o m s ) . PROLOGUE THE THIRD DERIVATIVE T u r n i n g and t u m b l i n g i n t h e l a t t i c e , The i r o n atom c a n n o t b e a r t h e c o l d ; T h i n g s f a l l a p a r t ; t h e c r y s t a l c a n n o t h o l d ; D i s o r d e r i s l o o s e d upon t h e w o r l d , The b l o o d - r e d b l o c k i s l o o s e d , and everywhere The symmetry o f s o l i d s t a t e i s drowned; The b e s t l a c k o f a l l c o h e s i o n , w h i l e t h e w o r s t A r e f u l l o f p a s s i o n a t e i n t e n s i t y . S u r e l y some r e v e l a t i o n i s a t hand; A f t e r D i c k e r s o n e t a l . (1967) TABLE OF CONTENTS /vHS'XFw-A^1!* « e o e e o 0 « « « o o « o e o e e o o o o c o c e a c c e a c e e « o o o o e o o TA.£LJE Of 1 OON'XENTS » o * o o o » « e o e e 0 o * c » o o « o & c « o o c t o e > e c * o c c XJXST 0 iJ* *P./V]BXj£t3 e o o o o e o e o « * > » G O O o « o o o o e o o < j o o e o o e o e e o o • XJXSI* O F jPXGUF^ES » o o e « e e o * e c » f i i o « » « « > e o c c o o « o o o o o e c e e o A.CKNO WLEDGEMENTS o o c o * » o « € > * « « o o « o c > « o o o o « o o o o o o « « « o o © CHAPTER X * G X N T R O D C T C T X O N X © X P l T S f cLCS o e o o o o c i e c o £ » c e o « e o o # o « a o e o e o l o 2 EPR i n B i o c h e m i s t r y ........ ». 1.3 EPR i n T r a n s i t i o n M e t a l Compounds.. 1.4 B i o l o g i c a l a n d C h e m i c a l P r o p e r t i e s 3.1*1 tOCl"llTOmS C » o * » « > « o « o o e e « e e * o < > e CHAPTER 2.0 EXPERIMENTAL APPARATUS 2.1 I n t r o d u c t i o n .o.....*.... 0 . 0 0 0 0 . . . 2.2 H e m o p r o t e i n P r o p e r t i e s r e l e v a n t ; t O e o * o o o o * o c > 9 c « « e t i o c > « o o « o o e * c > » 2.3 T h e EPR s p e c t r o m e t e r .............. 2.4 Low t e m p e r a t u r e a p p a r a t u s ......... 2.5 T e m p e r a t u r e m e a s u r e m e n t ........... 2.6 S a m p l e m o u n t i n g ................... 93 96 CHAPTER 3„0 EPR OF CYTOCHROME C I N SOLUTION 3.1 I n t r o d u c t i 3 o 2 ' X ' i ' l S O i r ^ O 4 f > e c > o o o e o o o o o e * o n e c a s o o o e o c o o c 3 o 3 R & S V l I t S o o e e o e e e o e o o c o e c o e * e e e e e c o e e o A 3«4 D i s c u s s i o n o f l i n e s h a p e s ............ 82 3 « C O n C i \ l S X O n S e > e o o « » o e f > e > o c o & o Q < » o o « o * * > o o o ^0 CHAPTER 4.0'SINGLE CRYSTAL EPR - ORIENTATION OF g-AXES 4.1 I n tjCOdUC t l On o e o o . . o o t t « c . o o « e o . * o e o o o o 92 4.2 C r y s t a l P r o p e r t i e s ................... 92 4 e 3 T e c h n i q u e . o f M e a s u r e m e n t . . . » » » . o . . . » . 4 e 4 M e t h o d o f a n a l y s i s . . . . . . . . . . . . s o . . . . . . 4.6 D i s c u s s i o n o f o r i e n t a t i o n r e s u l t s ...« 126 4o 7 C O l l C l V I S i O F I S o e » o o o o o o » o « o o e c > o © « o o « « * o o 1-50 CHAPTER 5.0 S I N G L E CRYSTAL EPR - LINEWIDTH V A R I A T I O N AS A FUNCTION OF g-VALUE 5 « X IntlTOCiUCtlOn o > o o o « » o o « » o o a o o o o o o o e e e c o 5.2 M a t e r i a l s a n d M e t h o d s ................ 152 5.3 S o u r c e s o f l i n e b r o a d e n i n g ........... 153 5.4 T h e o r y o f l i n e b r o a d e n i n g ............ 155 5 « 5 R G S L l l t S o e o e o * o o « e « o * e o e o o « o » e o o t f O o e « « S e S O X S Cl-1 S S X On c » « » e « © © © * © o < » © © © o © © © * © 0 * o o o * * * 1_S 3 5 0 7 Cone3.visxons © © © © * © © o * © e * « > © © » © © o © © © © e © « > 165 174 o o e * t > o o o o e e <> o « » « * CHAPTER 6 0 0 CYTOCHROME C EPR - LINESHAPES AND RAPID PASSAGE 6 . 1 I n t r o d u c t i o n 6 0 2 The n a t u r e o f t h e p r o b l e m . 6 . 3 R a p i d a d i a b a t i c p assage i n a s i n g l e S p X H . j p 3 . C ] C © " C o e o o o o o o o e o o e o o o o o c o o o o o o o 6 . 4 R a p i d a d i a b a t i c p a s s a g e i n a d i s t r i b u t i o n o f s p i n p a c k e t s ......... 6 . 5 T e s t o f P o r t i s e q u a t i o n s w i t h model 6 . 6 R e s u l t s f o r cytochrome c ... 6 . 7 C o n c l u s i o n s CHAPTER 7 * 0 RELAXATION TIMS STUDY OF CYTOCHROME C 7 © X XntiTOd\J.CtXOri e e f t o o o c o o o o e o o o e e e o e o e c e o 7*2 C a l c u l a t i o n o f r a p i d p a s s a g e s i g n a l as a f u n c t i o n o f r e l a x a t i o n t i m e 7 . 3 R e l a x a t i o n t i m e f r o m f a s t p a s s a g e , between 4.2°K and 20°K ...... 7.4 R e l a x a t i o n t i m e f r o m EPR l i n e w i d t h s 7 . 5 Review o f t h e o r i e s o f r e l a x a t i o n t i m e s . . . . . . . . « . o o < . 7 . 6 D i s c u s s i o n ....... r. 7 . 7 C o n c l u s i o n s . « o © © o o « « o « o o o o e o o e o « e < » o o o o o o ( k « o o » o e « * o o o * f » P a q e •11 'MBTicArrf jot A P P E N D I X - A3. - CALCULATION OF EPR L I N E BROADENING DUE TO VARIATIONS I N RHOMBIC AND A X I A L AX © JL Xrx "b.xroo.\jio tiL ox* ©©©•©©©#>«?©*>*©©©©©©«*•©©©« 257 A1.2 The g r o u n d s t a t e K r a m e r s D o u b l e t , «. . . . 257 A l , 3 E v a l u a t i o n o f R h o m b i c C o n t r i b u t i o n . e 26L A l ,,4 E v a l u a t i o n o f A x i a l C o n t r i b u t i o n . <>.. 26U L I S T OF TABLES Page T a b l e I Dependence o f EPR s i g n a l h e i g h t on microwave f r e q u e n c y f o r v a r i o u s t y p e s o f samples •<.•.„• 29 I I T h e r m o e l e c t r i c p o t e n t i a l d i f f e r e n c e , E, w i t h r e s p e c t t o 0°K and thermopower, d E / d t . M7 I I I C o m p a r i s o n o f l i n e h e i g h t s a t g-extrernes .... 81 I V L i n e s h a p e s as a f u n c t i o n o f t e m p e r a t u r e o o . O B 85 V O r i e n t a t i o n I - R e s u l t s o f l e a s t s q u a r e s f i t t O w C[\l CL'GX OX"! C ^  O 2 ) C t l > e « « r « » « « « > « i & < » A O O O « > * 0 *'© • ft © © * « 1-09 V I O r i e n t a t i o n I - D i r e c t i o n c o s i n e s and a n g l e s o f p r i n c i p a l g-axes r e l a t i v e t o r o t a t i o n cDCJL S © • o o © © © • © © * © « » * © * * o « o © o © o © o o © © o « o o c > © o © e © 1-1-3 V I I O r i e n t a t i o n I I - R e s u l t s o f l e a s t s q u a r e s f i t t O eqUatXOn ( 4 © 2 ) • e t » e o o o « * e o a f r A o a o o » « o o c t o * o o 117 V I I I O r i e n t a t i o n I I - D i r e c t i o n c o s i n e s and a n g l e s o f p r i n c i p a l g-axes r e l a t i v e t o r o t a t i o n CL2CX 3 o » © f > © « « ? f t o © © c > o © o » « © o © < > o « » c > o o © e © o c > » © o o © © o © o 11-8 I X E r r o r s i n l e a s t s q u a r e s f i t t o e q u a t i o n (4<,2) 133 X P r i n c i p a l g - a x i s d i r e c t i o n s r e l a t i v e t o t h e C J T ^ S t clX clJC© S . « o c o o. e - « * c o o o o o o o © a 4 > o o * o © o o < * o o o o o 1-37 T a b l e X I S i g n c o n v e n t i o n s u s e d f o r o r b i t a l s « « e o 145 X I I D a t a f o r t h e o r e t i c a l c a l c u l a t i o n o f X I I I S u m m a t i o n o f L o r e n t z i a n l i n e s w i t h a G a u s s x a n d x s t r i j o u t x o n a e o e e o s » « o » o « o » « o o « * o < » o d 233 s L I S T OF FIGURES CHAPTER 1.0 F e + i n c y t o c h r o m e c .......... 9 O O O *> * P a g e 1.1 E l e c t r o n i c e n e r g y l e v e l s o f t h e 3+ . _, l e 2 D i a g r a m s o f t h e a t o m s c o - o r d i n a t e d t o CHAPTER 2.0 2.1 B l o c k d i a g r a m o f 24 GHz EPR . S p e c t r o m e t e r coecooc.scsooooococ..oc-coo 26 2.2 D i a g r a m o f m i c r o w a v e s e c t i o n o f t h e EPR S p e C t r O m e t e r ».«.«. oo«o».oo»eooe*o»o 28 2.3 B l o c k d i a g r a m o f a u t o m a t i c f r e q u e n c y COntjTOX S y S t ^ m o « * e e e © o * c > e 9 0 * o » © e o f * o e « « 35 2.4 M e a s u r e d a n d c a l c u l a t e d c u r v e s o f minimum d e t e c t a b l e number o f s p i n s .... no 2.5 D i a g r a m o f s i n g l e v a c u u m s p a c e l o w t e m p e r a t u r e d e w a r . . . . . . . . . . 44 2„6 D i a g r a m o f EPR c a v i t y s h o w i n g t h e r m o c o u p l e .........eoo... . . . . . o . o 5L 2.7 D i a g r a m o f s a m p l e c h a n g i n g s y s t e m ..... 55 2.8 D i a g r a m o f m o u n t i n g s y s t e m f o r S O l l - l t l O n , ScHTijpX3S OQO©O6>OGOGO©9O©C9©6«©0 2.9 D i a g r a m o f m o u n t i n g s y s t e m f o r s i n g l e C J T ^ S t S l i S c o o c o o o e o e o e o o o o o & c o o « o c o o © © © © CliAPTER 3.0 3 e l EPR s p e c t r a f r o m r a n d o m l y o r i e n t e d m o l e c u l e s p o s s e s s i n g 3 g - v a l u e s 3.2 M i c r o w a v e p o w e r d e p e n d e n c e o f t u n a c y t o c h r o m e c EPR s i g n a l 3c3 T e m p e r a t u r e d e p e n d e n c e o f a r e a o f t u n a c y t o c h r o m e c EPR s i g n a l . . . . a . . . . . . . . . . 3.4 EPR s p e c t r a o f f r o z e n s o l u t i o n s o f c y t o c h r o m e c a t X — b a n d . . . • • • o . e e . e e c e * 3.5 EPR s p e c t r a o f f r o z e n s o l u t i o n s o f c y t o c h r o m e c a t t w o d i f f e r e n t tGffljpSJTci'fcU.irG S « o o « « © p o © o © c > o « > © o o 0 © o © « © © e o 3o6 EPR s p e c t r a o f f r o z e n s o l u t i o n s o f c y t o c h r o m e c a t t w o m i c r o w a v e p o w e r X6V6.XS « e o * « « o p o c i o e * e r ® « o e e « > c a o « a o * o o o « * 3.7 T e m p e r a t u r e d e p e n d e n c e o f w i d t h o f c y t o c h r o m e c EPR a b s o r p t i o n l i n e ...... 3.8a E x a m p l e s o f h i g h t e m p e r a t u r e c y t o c h r o m e c EPR a b s o r p t i o n l i n e s ..... 3.8b N o t a t i o n u s e d i n t a b l e I V . © e © © e © © c o e e o c v e o © © © © © © O © O « 9 CHAPTER 4.0 • 4.1 L a b e l l i n g o f t h e a, b a n d c a x e s i n a f e r r i c y t o c h r o m e c s i n g l e c r y s t a l 4«2 EPR s p e c t r u m f r o m a s i n g l e c r y s t a l o f h o r s e h e a r t f e r r i c y t o c h r o m e c a t e £ J\ o o o o o o e o c « Q O O O O o o e « « o © o © o o e e o o e c c e 4.3 C o - o r d i n a t e s y s t e m u s e d t o d e s c r i b e t h e p o s i t i o n s o f t h e p r i n c i p a l g - a x e s r e l a t i v e t o t h e l a b o r a t o r y m a g n e t i c f x s l d dijrscta-on © o « © © © © © « « © e © • © « © o © © © o © © 4 . 4 S t e r e o g r a p h i c p r o j e c t i o n o f . c y t o c h r o m e C C r y s t a l . o . . . o e o . . . . . . e o e ^ o o . . © o . . . . . . 4©5 T h e W u l f f n e t »«•««•..#.«...*..«*..»..**.. 4©6 g - v a l u e a s a f u n c t i o n o f a n g l e o f f i e l d r o t a t i o n f o r O r i e n t a t i o n 1 4.7 S t e r e o g r a m o f O r i e n t a t i o n I ............ 4.8 g - v a l u e a s a f u n c t i o n o f a n g l e o f f i e l d r o t a t i o n f o r O r i e n t a t i o n I I . 4.9 S t e r e o g r a m o f O r i e n t a t i o n I I 4.10 F u l l s t e r e o g r a m o f O r i e n t a t i o n I ....... 4.11 C o m p a r i s o n o f O r i e n t a t i o n I a n d O i T X & n t c t t i o n XX © © © o © o # © © © © © * o © © © © © © « © © © © 4.12 P r o j e c t i o n o f g - a x e s o n t o heme p l a n e ... 4.13 g - v a l u e s a s a f u n c t i o n o f r h o m b i c f i e l d a n d J a h n - T e l l e r c o u p l i n g • ••.•«-......... o • « © i o o © o o * « o o e o XI CHAPTER 5c0 5 © 2 3 e 5.4 5.5 5.6 CHAPTER 6.0 6.1 i . 2 N o t a t i o n u s e d f o r l i n e w i d t h c a l c u l a t i o n s . © o © * o © © © © © © « ' © © 4 i © © © a © © o e 9 « » © C o - o r d i n a t e s y s t e m u s e d t o d e s c r i b e t h e p o s i t i o n s o f t h e p r i n c i p a l g - a x e s r e l a t i v e t o t h e l a b o r a t o r y m a g n e t i c f l 3 l c L d l C t l Oil e e e © © « © o G r © o e © o © e © e o f r © a © o V a r i a t i o n i n w i d t h o f l i n e s 1 a n d 2 o f O r i e n t a t i o n I I a s c r y s t a l i s r o t a t e d r e l a t i v e t o d . c . m a g n e t i c f i e l d ........ T h e o r e t i c a l v a r i a t i o n i n . w i d t h o f l i n e 2 o f O r i e n t a t i o n I I s h o w i n g i n d i v i d u a l c o n t r i b u t i o n s o s e e . o o c c e o o o s . V a r i a t i o n i n w i d t h o f l i n e 1 o f O r i e n t a t i o n I a s c r y s t a l i s r o t a t e d r e l a t i v e t o d . c . m a g n e t i c f i e l d T h e o r e t i c a l v a r i a t i o n i n w i d t h o f l i n e 1 o f O r i e n t a t i o n I s h o w i n g i n d i v i d u a l c o n t r i b u t i o n s ...... » © o a © a o © i EPR d i s p e r s i o n l i n e f r o m a s i n g l e c r y s t a l o f h o r s e h e a r t f e r r i c y t o c h r o m e C c l t 4 © 2 K. o © * e e © o o M i c r o w a v e p o w e r d e p e n d e n c e o f s i n g l e c r y s t a l d i s p e r s i o n s i g n a l h e i g h t 159 164 171 173 176 178 e © © © o v 188 190 P a g e 6 a 3 M a g n e t i c f i e l d m o d u l a t i o n d e p e n d e n c e o f s i n g l e c r y s t a l d i s p e r s i o n s i g n a l 1*1 <5X Cjl*lfc c » o o e o > o o o o o i o o o « t F o e o o o G c o e o c o o o 3 e 1.92 6 . 4 EPR s p e c t r a f r o m c h a r r e d d e x t r o s e S c U T l J p X © e o e c o o o o o c © a a e c c e © e o o « e e o o o » o o e 20(3 6 . 5 M i c r o w a v e p o w e r d e p e n d e n c e o f c h a r r e d d e x t r o s e d i s p e r s i o n s i g n a l h e i g h t c c . . * 208 6 0 6 . M a g n e t i c f i e l d m o d u l a t i o n a m p l i t u d e d e p e n d e n c e o f c h a r r e d d e x t r o s e d i s p e r s i o n s i g n a l h e i g h t . . . o . ; . c o . . c e . « 210 6 . 7 M a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y d e p e n d e n c e o f p h a s e l a g o f c h a r r e d d e x t r o s e d i s p e r s i o n s i g n a l ............ 213 6 . 8 EPR d i s p e r s i o n s p e c t r a o f c h a r r e d d e x t r o s e f o r a s p e c i a l c a s e o f t h e P O i r t l S tl"l60J*r^ © © © e © e * » © © < p o e © © « © © » © o © & © e o 21_7 6 . 9 M a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y d e p e n d e n c e o f p h a s e l a g o f c y t o c h r o m e c s i n g l e c r y s t a l d i s p e r s i o n s i g n a l . 0 . 0 221 CHAPTER '7.0 7.1 R a p i d p a s s a g e s i g n a l h e i g h t a s a f u n c t i o n o f r e l a x a t i o n t i m e . . . • « . . . . . 227 7.2 R a p i d p a s s a g e s i g n a l h e i g h t a s a f u n c t i o n o f a b s o l u t e t e m p e r a t u r e ...... 230 • Pag^e 7 « 3 P l o t o f i n v e r s e r e l a x a t i o n , t i m e a g a i n s t t e m p e r a t u r e between. 4„2°K 3.1*1(3. 20 IC o « o o e o o o o * s o o o < » * o * e * « 9 o e e o « o o « > 232 7 0 4 I n d i v i d u a l c o n t r i b u t i o n s t o t o t a l l i n e w i d t h between 30°K and 7 7 ° K ..„.„<, <> 237 7 0 5 P l o t o f i n v e r s e s p i n - l a t t i c e r e l a x a t i o n t i m e as a f u n c t i o n o f t e m p e r a t u r e between 4 „ 2 0 K and 7 7 ° K c » . » o 245 ACKNOWLEDGEMENTS Many p e o p l e have h e l p e d i n t h e c o m p l e t i o n o f t h i s work. My e s p e c i a l t h a n k s go tos Dr. C.P.S. T a y l o r , my s u p e r v i s o r , whose c l e a r s c i e n t i f i c t h i n k i n g and e x c e l l e n t e x p e r i m e n t a l a b i l i t y was a c o n s t a n t example t o f o l l o w . I t has been a p l e a s u r e t o work u n d e r him. My w i f e , K a t h l e e n , who n o t o n l y d i d t h e d i a g r a m s and e q u a t i o n s b u t a l s o gave c o n s t a n t s u p p o r t and e n c o u r a g e -ment. Dr. M.R. Roach and t h e members o f t h e U.W.O. B i o p h y s i c s Department who made, t h e i r " H onorary V i s i t i n g G r a d u a t e S t u d e n t " welcome. D r s . J.M. B o l t o n , P.W. Whippey and M.D. Owen who a c t e d as an u n o f f i c i a l " e x a m i n i n g c o m m i t t e e " h e r e a t U.W.O. The N o r t h A t l a n t i c T r e a t y O r g a n i z a t i o n and t h e M e d i c a l R e s e a r c h C o u n c i l f o r f i n a n c i a l s u p p o r t . CHAPTER 1.0 INTRODUCTION 1.1 PREFACE The a i m s o f t h i s c h a p t e r a r e , f i r s t l y , t o d e s c r i b e t h e m a i n u s e s o f e l e c t r o n p a r a m a g n e t i c r e s o n a n c e ( E P R ) i n b i o c h e m i s t r y ( s e c t i o n 1 . 2 ) . T h e n , t o r e v i e w t h e EPR w o r k d o n e o n p r o t e i n s c o n t a i n i n g t r a n s i t i o n m e t a l i o n s , w i t h e m p h a s i s o n c y t o c h r o m e c ( s e c t i o n 1 . 3 ) . F i n a l l y , we g i v e a b r i e f i n t r o d u c t i o n t o t h o s e p h y s i c a l a n d c h e m i c a l a s p e c t s o f EPR w h i c h t h e r e m a i n d e r o f t h e t h e s i s w i l l t r e a t i n d e t a i l ( s e c t i o n 1 . 4 ) . 1.2 EPR I N BIOCHEMISTRY The b a s i c p r o p e r t y o f EPR i s i t s a b i l i t y t o d e t e c t u n p a i r e d e l e c t r o n s p i n s i n a compound. The p r e s e n c e o f s u c h u n p a i r e d s p i n s i n b i o l o g i c a l s y s t e m s a r i s e f r o m v a r i o u s c a u s e s a n d t h e s e a r e d e t a i l e d b e l o w . EPR i s u s e f u l i n t h e f o l l o w i n g c a s e s w h e r e u n p a i r e d s p i n s a r e f o u n d : ( i ) i n b i o l o g i c a l m o l e c u l e s w h i c h h a v e b e e n e x p o s e d t o r a d i a t i o n , w h e r e t h e y a r i s e m a i n l y f r o m b r o k e n b o n d s . ( i i ) i n e n z y m e - s u b s t r a t e r e a c t i o n s , a s s o c i a t e d w i t h t h e 2 i n t e r m e d i a t e s ( u s u a l l y u n s t a b l e ) w h i c h a r e f o r m e d i n t h e r e a c t i o n p r o c e s s . ( i i i ) i n t h o s e m o l e c u l e s w h i c h c o n t a i n t r a n s i t i o n m e t a l i o n s , i n w h i c h t h e y a r e a s s o c i a t e d w i t h t h e e l e c t r o n s a r o u n d t h e m e t a l i o n . The w o r k c a r r i e d o u t i n t h i s t h e s i s s t u d i e s t h e b e h a v i o u r o f c y t o c h r o m e c, a p r o t e i n c o n t a i n i n g i r o n i n a heme g r o u p , a n d t h e r e f o r e f a l l s i n t o c a t e g o r y ( i i i ) . We s h a l l n o t d i s c u s s a n y EPR s t u d i e s o u t s i d e o f t h i s c a t e g o r y , a s c o n s i d e r a b l e i n f o r m a t i o n i s a v a i l a b l e o n ( i ) a n d ( i i ) i n s e v e r a l r e c e n t b o o k s ( P o o l e , 1967; A l g e r , 1968; a n d e s p e c i a l l y I n g r a m , 1 9 6 9 ) . 1.3 EPR I N TRANSITION METAL COMPOUNDS 1.3.1 I n t r o d u c t i o n The o n l y t r a n s i t i o n m e t a l s t h a t h a v e b e e n s t u d i e d by EPR i n b i o l o g i c a l s y s t e m s a r e c o b a l t , v a n a d i u m , m o l y b d e n u m , m a n g a n e s e , c o p p e r , z i n c , a n d i r o n . V e r y l i t t l e w o r k h a s b e e n d o n e o n t h e f i r s t t w o . C o b a l t i s f o u n d i n v i t a m i n b u t o n l y g i v e s s i g n a l s a f t e r i r r a d i a t i o n (Hogenkamp e t a l . , 1 9 6 3 ) . V a n a d i u m EPR s p e c t r a h a v e b e e n o b t a i n e d i n a m e t h a n o l e x t r a c t f r o m t h e mushroom A m a n i t a m u s e a r i a b y M u s s o ( u n p u b -l i s h e d r e s u l t s q u o t e d i n B e i n e r t a n d P a l m e r , 1 9 6 5 ) . The v a l e n c e s t a t e o f t h e m e t a l i s u n k n o w n . M o l y b d e n u m h a s b e e n e x t e n s i v e l y s t u d i e d due t o i t s p r e s e n c e i n x a n t h i n e o x i d a s e , p a r t i c u l a r l y b y B r a y (1964) a n d c o l l a b o r a t o r s . T h e s e w o r k e r s h a v e u s e d t h e a p p e a r a n c e o f t h e m o l y b d e n u m s i g n a l t o m o n i t o r t h e r e a c t i o n k i n e t i c s o f t h e e n z y m e . The manganese i o n h a s a l s o b e e n u s e d t o p r o b e enzyme a c t i o n b y i t s e f f e c t o n t h e s p i n - l a t t i c e r e -l a x a t i o n t i m e o f w a t e r p r o t o n s i n e n z y m e - m e t a l - s u b s t r a t e c o m p l e x e s ( C o h n , 1 9 7 0 ) . C o p p e r g i v e s EPR s i g n a l s a l s o , a n d t h e b i n d i n g o f t h i s i o n h a s b e e n e x a m i n e d i n a w i d e r a n g e o f p r o t e i n s ( e . g . , M a l m s t r o m a n d V a n n g a r d , 1 9 6 0 ; P e i s a c h e t a l . 1 9 6 7 ) . T h e mode o f b i n d i n g o f c o p p e r i o n s i n t o i n s u l i n s i n g l e c r y s t a l s h a s b e e n s t u d i e d b y B r i l l a n d V e n a b l e ( 1 9 6 4 ) a n d V e n a b l e ( 1 9 6 5 ) . ( T h e l a t t e r r e f e r e n c e , V e n a b l e ' s Ph.D. T h e s i s , p r o v i d e d much e x p e r i m e n t a l g u i d a n c e o n c r y s t a l m o u n t i n g a n d d a t a a n a l y s i s i n t h e e a r l y s t a g e s o f t h i s p r o -j e c t , a l t h o u g h t h e m e t h o d s f i n a l l y u s e d t o o b t a i n t h e r e s u l t o f c h a p t e r 4 d i f f e r e d f r o m h i s ) . T h e o n l y c l a s s o f p r o t e i n s w h i c h h a v e b e e n s t u d i e d b o t h i n ( f r o z e n ) s o l u t i o n a n d a s s i n g l e c r y s t a l s a r e t h o s e c o n t a i n i n g i r o n a t o m s i n t h e f o r m o f heme. Non-heme i r o n p r o t e i n s ( S a n P i e t r o , 1 9 6 5 ) h a v e b e e n e x a m i n e d b y EPR, b u t o n l y i n f r o z e n s o l u t i o n — s e e f o r e x a m p l e P a l m e r e t a l . ( 1 9 6 7 ) . 1.3.2 EPR o f heme p r o t e i n s , i n c l u d i n g c y t o c h r o m e c T h i s s e c t i o n ' g i v e s a h i s t o r i c a l s u r v e y o f EPR s t u d i e s o f heme p r o t e i n s s The f i r s t p u b l i s h e d r e s u l t s w e r e t h o s e o f I n g r a m a n d B e n n e t t ( 1 9 5 5 ) on p a s t e s o f m e t - h e m o g l o b i n ( m e t - H b ) , m e t - m y o g l o b i n ( m e t - M b ) , h e m o g l o b i n f l u o r i d e ( H bF) a n d myo-g l o b i n f l u o r i d e ( M b F ) . T h e s e p r o t e i n s w e r e i n t h e h i g h s p i n f o r m (S - 5/2) w i t h g - v a l u e s o f 5.95. The same p a p e r a l s o g a v e r e s u l t s f o r t w o l o w s p i n (S = 1/2) c o m p o u n d s , H b - a z i d e a n d Mb-a.zide, w h i c h h a d g - v a l u e s s p r e a d b e t w e e n 2.2 a n d 2.8. T h i s w o r k was e x t e n d e d t o s i n g l e c r y s t a l s o f Mb an d met-Hb t w o y e a r s l a t e r ( B e n n e t t e t a l . . , 1 9 5 7 ) . I n t h i s c l a s s i c p a p e r t h e s e w o r k e r s o b t a i n e d t h e o r i e n t a t i o n o f t h e p r i n c i p a l g - a x e s i n t h e m o l e c u l e r e l a t i v e t o t h e c r y s t a l -l o g r a p h i c a x e s . A s t u d y o f s i n g l e c r y s t a l s was a l s o r e p o r t e d o n t h e a z i d e c o m p l e x ( G i b s o n a n d I n g r a m , 1 9 5 7 ) . The d a t a o b t a i n e d o v e r t h e p e r i o d 1 9 5 5 - 5 7 was r e v i e w e d b y G i b s o n e t a l . ( 1 9 5 8 ) . G o r d y a n d R e x r o a d ( 1 9 6 1 ) p u b l i s h e d t h e f i r s t EPR s p e c t r a o f c y t o c h r o m e c a t a t e m p e r a t u r e o f 4.2°K. T h e s e w e r e o b t a i n e d o n a c o m m e r c i a l p o w d e r s a m p l e o f h o r s e h e a r t c y t o c h r o m e , a n d g a v e t h e g - v a l u e s 5.9, 2.8, 2.2 a n d 2.0. The f i r s t a n d l a s t o f t h e s e p r e s u m a b l y a r i s e f r o m h i g h s p i n d e n a t u r e d c y t o c h r o m e c a n d / o r f r o m Hb o r Mb i m p u r i t i e s . The g = 2.8 a n d 2.2 r e s o n a n c e s h a d u n u s u a l s h a p e s — t h a t o f u n d i f f e r e n t i a t e d a b s o r p t i o n l i n e s — d e s p i t e a f i r s t d e r i v a t i v e d i s p l a y b e i n g u s e d . I t i s p o s s i b l e t h a t t h e s e w o r k e r s o b s e r v e d f a s t p a s s a g e d i s p e r s i o n s i g n a l s s i m i l a r t o t h o s e i n c h a p t e r s 3 a n d 6 o f t h i s t h e s i s . T a y l o r ( u n p u b l i s h e d r e s u l t s , 1 9 6 1 ) o b s e r v e d an EPR d e r i v a t i v e a b s o r p t i o n s p e c t r u m a t 77°K i n t u n a c y t o c h r o m e c w h i c h h a d g - v a l u e s o f 3,05 a n d 2,24 i n a g r e e m e n t w i t h r e s u l t s p u b l i s h e d s e v e n y e a r s l a t e r ( v i d e i n f r a ) . T a y l o r ' s r e s u l t s w e r e i n f a c t , t h e m a i n s t i m u l u s f o r t h e w o r k on c y t o c h r o m e c p r e s e n t e d i n t h i s t h e s i s . G o r d y a n d R e x r o a d ( 1 9 6 1 ) a l s o p u b l i s h e d p o w d e r EPR s p e c t r a o f a c y t o c h r o m e c - — n i t r i c o x i d e c o m p l e x — t h e y o b s e r -v e d a t r i p l e t w i t h s p l i t t i n g o f 18 g a u s s w h i c h t h e y i n t e r p r e t e d a s due t o t h e n i t r o g e n a t o m o f t h e n i t r i c o x i d e . B e n n e t t e t a l . ( 1 9 6 1 ) p u b l i s h e d t h e heme p l a n e o r i e n t a t i o n s o f c r y s t a l s o f Mb t y p e s B, C, D a n d F ( f r o m t h e m u s c l e s o f d i v i n g mammals) d e t e r m i n e d f r o m t h e i r EPR s p e c t r a a n d c o m p a r e d t h e r e s u l t s w i t h m e a s u r e m e n t s o n t h e o p t i c a l d i c h r o i s m o f t h e s e c r y s t a l s ( B r a g g a n d P i p p a r d , 1 9 5 3 ; P e r u t z , 1 9 5 3 ) . Due t o t h e s y s t e m o f d o u b l e b o n d s i n t h e p l a n a r heme r i n g , t h e e l e c t r i c v e c t o r o f p o l a r i s e d l i g h t s h o u l d a b s o r b more s t r o n g l y when i t l i e s p a r a l l e l t o t h e p l a n e t h a n when p e r p e n d i c u l a r . T h e o b s e r v e d d i c h r o i s m a g r e e d w e l l w i t h t h a t c a l c u l a t e d f r o m t h e EPR d e t e r m i n a t i o n o f t h e heme p l a n e o r i e n t a t i o n . E h r e n b e r g ( 1 9 6 2 ) u n s u c c e s s f u l l y a t t e m p t e d t o o b t a i n EPR s p e c t r a f r o m f e r r i c y t o c h r o m e c s o l u t i o n s a t 77°K, b u t d i d r e p o r t l o w s p i n s p e c t r a f r o m s e v e r a l m y o g l o b i n c o m p l e x e s (Mb~0H, - NH^, e t c . ) whose g - v a l u e s a g r e e d w i t h t h o s e g i v e n i n t h e e a r l i e r w o r k ( G i b s o n e t a l • , 1 9 5 8 ) . I n 1 9 64 ( S t r y e r et. a l . , 1 9 6 4 ) was p u b l i s h e d t h e X - r a y s t r u c t u r e o f M b - a z i d e a n d t h e c o r r e l a t i o n b e t w e e n p r i n c i p a l g - a x i s d i r e c t i o n s a n d t h e m o l e c u l a r o r i e n t a t i o n was made f o u r y e a r s l a t e r ( H e l c k e e t a l . , 1 9 6 8 ) . We s h a l l d i s c u s s t h e H e l c k e p a p e r more f u l l y i n c h a p t e r 4, f o r t h e r e a r e s i g n i f i c a n t d i f f e r e n c e s b e t w e e n t h e i r r e s u l t s o n a z i d e a n d o u r s o n c y t o c h r o m e c , Y o n e t a n i a n d S c h l e y e r ( 1 9 6 8 ) a l s o r e p o r t e d f a i l i n g t o d e t e c t EPR s i g n a l s f r o m c y t o c h r o m e c a t 77°K, i n c o n t r a s t t o t h e r e s u l t s o f R e i n e t a l . ( 1 9 6 8 ) who r e p o r t e d g - v a l u e s o f 3.0, 2.26 a n d 2.0 f o r s o l u t i o n s o f n a t i v e c y t o c h r o m e c . S a l m e e n a n d P a l m e r ( 1 9 6 8 ) o b t a i n e d g - v a l u e s o f 3.06, 2.24 a n d 1.24 a t 20°K. The s m a l l p e a k a t g = 1.24 was n o t d e t e c t e d b y R e i n e_t a l . . p r o b a b l y b e c a u s e t h i s i s one o f t h e l o w e s t g -v a l u e s r e p o r t e d i n heme p r o t e i n s a n d t h e y d i d n o t c o v e r a w i d e e n o u g h r a n g e o f m a g n e t i c f i e l d . R e i n e t a!L. ( 1 9 6 8 ) a l s o p u b l i s h e d g - v a l u e d a t a f o r c y t o c h r o m e c a t e x t r e m e s o f pH ( a l k a l i n e pH - g v a l u e s = 2 . 7 3 , 2.14 a n d 1.77; a t a c i d p H - g - v a l u e s o f 6.3 a n d 2 . 1 ) , f o r f e r r i c y t o c h r o m e c y a n i d e ( 3 . 0 4 , 2.3, 2 . 0 1 ) , f o r f e r r i c y t o c h r o m e a z i d e ( 2 . 7 7 , 2.27 a n d . 1.85) a n d f o r f e r r i c y t o c h r o m e f l u o r i d e ( 6 . 1 , 1 . 9 7 ) . Y o n e t a n i a n d S c h l e y e r ( 1 9 6 8 ) a l s o r e p o r t e d c y t o c h r o m e c EPR s p e c t r a a t e x t r e m e s o f pH t h a t w e r e s i m i l a r t o t h o s e o f R e i n e t a l . T h e s e p u b l i c a t i o n s on s o l u t i o n s o f n a t i v e c y t o c h r o m e c a n d d e r i v a t i v e s — j u s t a t t h e t i m e we o b t a i n e d o u r p r e -l i m i n a r y r e s u l t s a t 4 . 2 ° K — c a u s e d a c h a n g e i n e m p h a s i s o f t h i s t h e s i s p r o j e c t f r o m s o l u t i o n s t o s i n g l e c r y s t a l s . We w i s h e d t o o b t a i n t h e o r i e n t a t i o n s o f t h e g - a x e s , an e x p l a n a -t i o n f o r t h e b r o a d l i n e w i d t h s s e e n ( i n s o l u t i o n ) a n d t h e l i n e s h a p e s s e e n a t l o w t e m p e r a t u r e s . E i s e n b e r g e r a n d P e r s h a n ( 1 9 6 7 ) p u b l i s h e d t h e f i r s t l i n e w i d t h s t u d y o f p r o t e i n s on s i n g l e c r y s t a l s o f met-Hb a n d m e t ~ H b - a z i d e . T h e y p o s t u l a t e d t h a t t h e l i n e s s e e n i n m e t-Hb w e r e b r o a d b e c a u s e t h e p r i n c i p a l g - a x e s w e r e n o t s t r i c t l y p a r a l l e l f r o m one heme g r o u p t o a n o t h e r * F o r t h e H b - a z i d e , t h e y d e m o n s t r a t e d t h a t t h e l i n e w i d t h s c o u l d b e l a r g e l y e x p l a i n e d b y t h e c o m b i n a t i o n o f t h i s g - a x i s m i s -o r i e n t a t i o n w i t h a v a r i a t i o n i n t h e p r i n c i p a l g - v a l u e s f r o m on e m o l e c u l e t o a n o t h e r . ( T h i s l a t t e r e f f e c t c a n a r i s e f r o m a n u n u s u a l l y w i d e d i s t r i b u t i o n o f t h e r h o m b i c p o t e n t i a l o f t h e l i g a n d f i e l d a r o u n d i t s mean v a l u e ) . H e l c k e e t a l . ( 1 9 6 8 ) p o s t u l a t e d t h a t t h e w i d t h s t h e y o b s e r v e d i n a n o t h e r l o w s p i n c o m p l e x , m e t - M b - a z i d e , w e r e t o b e e x p l a i n e d s o l e l y b y t h e m i s o r i e n t a t i o n m e c h a n i s m . We w i l l d i s c u s s t h e s e r e s u l t s i n c o n s i d e r a b l e d e t a i l i n c h a p t e r 5, w h e r e we w i l l s how t h a t b o t h t h e " E i s e n b e r g e r " a n d " H e l c k e " t h e o r i e s p u b l i s h e d m u s t be c o r r e c t e d s l i g h t l y a n d e x t e n d e d b e f o r e t h e y c a n e x p l a i n o u r l i n e w i d t h r e s u l t s s a t i s f a c t o r i l y . K o n ( 1 9 6 9 ) s t u d i e d t h e EPR o f n i t r i c o x i d e c y t o -c h r o m e c a n d p r e s e n t e d g o o d e v i d e n c e , f r o m t h e n i t r o g e n h y p e r f i n e s p l i t t i n g o b s e r v e d , t h a t t h e NO r a d i c a l r e p l a c e s t h e s u l p h u r a t o m o f m e t h i o n i n e - 8 0 i n t h e n a t i v e p r o t e i n . K o n was u n a b l e t o d u p l i c a t e t h e r e s u l t s o f G o r d y a n d R e x r o a d ( 1 9 6 1 ) ; t h e n i t r o g e n s p l i t t i n g s he o b s e r v e d b e i n g 24 g a u s s a n d 6.8 g a u s s . O t h e r e x a m p l e s o f h y p e r f i n e s t r u c -3+ t u r e f r o m a t o m s b o u n d t o t h e c e n t r a l F e i o n a r e t h o s e o f K o t a n i a n d M o r i m o t o ( 1 9 6 7 ) o n t h e s p l i t t i n g p r o d u c e d b y t h e f l u o r i d e a t o m i n a m y o g l o b i n f l u o r i d e c r y s t a l . S c h o l e s ( 1 9 6 9 ) o b t a i n e d t h e f i r s t EPR s p e c t r u m o f t h e h y p e r f i n e l i n e s f r o m t h e f o u r p y r r o l e n i t r o g e n s o f t h e heme p l a n e , b y c o -c r y s t a l l i s a t i o n o f h e m i n w i t h p e r y l e n e . An EPR s i n g l e c r y s t a l s t u d y o f h e m o g l O b i n - N O was p u b l i s h e d b y C h i e n ( 1 9 6 9 ) i n w h i c h he o b t a i n e d t h e o r i e n t a t i o n o f t h e b o u n d NO r a d i c a l . R e c e n t l y EPR s p e c t r a o f c y t o c h r o m e c a t 77°K h a v e b e e n p u b l i s h e d by M o r t o n a n d B o h a n ( 1 9 7 1 ) who e x a m i n e d t h e s p e c t r a o f h y o p h i l i s e d p o w d e r s a n d s o l u t i o n s o f c y t o c h r o m e c, a n d i n d i c a t e d t h a t t h e m o l e c u l e i s c o n s i d e r a b l y d i s t o r t e d o n l y o p h i l i z a t i o n . T h e i r r e s u l t s a r e c o n s i s t e n t w i t h t h e u n -p u b l i s h e d w o r k o f T a y l o r m e n t i o n e d a b o v e . The o b s e r v a t i o n s o n l y o p h i l i z a t i o n a l s o a g r e e w i t h a n EPR s t u d y o f Y o n e t a n i a n d S c h l e y e r ( 1 9 6 7 ) on m y o g l o b i n a n d c y t o c h r o m e c p e r o x i d a s e who d e m o n s t r a t e d t h a t t h e s e p r o t e i n s w e r e h i g h l y s u s c e p t i b l e t o c h a n g e s i n s t a t e , s u c h a s f r e e z i n g , t h a w i n g a n d d r y i n g . O u r own p u b l i s h e d w o r k on s o l u t i o n s ( M a i l e r a n d T a y l o r , 1 9 7 1 ) i s r e p o r t e d i n d e t a i l i n c h a p t e r 3. T h e r e h a s b e e n on e EPR s t u d y o f s i n g l e c r y s t a l s o f c y t o c h r o m e c r e p o r t e d — t h i s was a p r e l i m i n a r y n o t e o n t h e heme p l a n e o r i e n t a t i o n i n b o n i t o c y t o c h r o m e c b y H o r i a n d M o r i m o t o ( 1 9 7 0 ) . T h e s e w o r k e r s , h o w e v e r , made no m e n t i o n e i t h e r o f l i n e w i d t h c h a n g e s o r o f t h e f a s t p a s s a g e e f f e c t s we r e p o r t i n t h e m a j o r s e c t i o n s o f t h i s t h e s i s ( c h a p t e r s 3, 5, 6 a n d 7) s i n c e t h e y o b t a i n e d a l l t h e i r r e s u l t s a t o n e t e m p e r a t u r e — 2 0 . 3 ° K — t h e b o i l i n g p o i n t o f l i q u i d h y d r o g e n . T h e b o n i t o c y t o c h r o m e h a s a d i f f e r e n t u n i t c e l l f r o m h o r s e a n d t u n a c y t o c h r o m e a n d d e t a i l e d X - r a y s t u d i e s h a v e n o t b e e n r e p o r t e d o n i t . I - 4 BIOLOGICAL AND CHEMICAL PROPERTIES OF CYTOCHROME C 1.4.1 I n t r o d u c t i o n We s u r v e y v e r y b r i e f l y t h e b i o l o g i c a l p r o p e r t i e s o f c y t o c h r o m e c ( s e c t i o n 1.4.2) a n d t h e n d e s c r i b e t h e c h e m i s t r y o f t h e f e r r i c , i o n ( s e c t i o n 1.4.3) a n d o f t h e p a r a -m a g n e t i c c e n t r e o f t h e m o l e c u l e ( s e c t i o n 1 . 4 . 4 ) . 1.4.2 C y t o c h r o m e c C y t o c h r o m e c i s a l o w m o l e c u l a r w e i g h t ( 1 2 , 5 0 0 ) p r o t e i n w i d e l y d i s t r i b u t e d i n t h e a n i m a l a n d p l a n t k i n g d o m . T h i s i r o n - p o r p h y r i n - c o n t a i n i n g p r o t e i n i s f o u n d i n t h e t e r m i n a l c h a i n i n t h e m i t o c h o n d r i a o f a l l a e r o b i c o r g a n i s m s . C y t o c h r o m e c t r a n s f e r s e l e c t r o n s b e t w e e n t w o c y t o c h r o m e c o m p l e x e s — c y t o c h r o m e r e d u c t a s e a n d c y t o c h r o m e o x i d a s e , t h e i r o n a t o m c y c l i n g b e t w e e n t h e f e r r i c ( F e + ) a n d t h e f e r r o u s 2 ( F e + ) v a l e n c e s t a t e . The p r o p e r t i e s o f one o f t h e s e s t a t e s , 10 t h e f e r r i c , c a n be s t u d i e d b y EPR a s i n t h i s s t a t e t h e r e i s a n u n p a i r e d e l e c t r o n p r e s e n t . C y t o c h r o m e c i s u n i q u e among t h e c y t o c h r o m e s i n t h a t i t c a n be e a s i l y e x t r a c t e d f r o m t i s s u e s w i t h o u t a p p r e c i a t a b l e d e n a t u r a t i o n ( e . g . , M a r g o l i a s h a n d W a l a s e k , 1 9 6 7 ) . I n c o n s e q u e n c e , i t h a s b e e n v e r y w i d e l y s t u d i e d ( s e e t h e r e v i e w o f M a r g o l i a s h a n d S c h e j t e r , 1 9 6 6 ) . S i n c e c y t o c h r o m e c c a n t r a n s f e r e l e c t r o n s , t h e e l e c t r o n i c s t r u c t u r e o f t h e a c t i v e c e n t r e ( t h e heme g r o u p ) i s o f p a r t i c u l a r i n t e r e s t , a n d EPR i s an i d e a l t e c h n i q u e t o i n v e s t i g a t e t h i s s i n c e i t i s s p e c i f i c f o r t h e e l e c t r o n i n t e r -a c t i o n a t t h e F e a t o m, t h e a c t i v e c e n t r e . 3 1.4.3 C h e m i s t r y o f t h e F e * f r e e i o n 3 T h e f e r r i c i r o n ( F e + ) i o n h a s 23 e l e c t r o n s o f w h i c h 18 r e s i d e i n c l o s e d s h e l l s ; t h e f i v e r e m a i n i n g h a v e t h e o r b i t a l c o n f i g u r a t i o n ( 3 d ) " * . T h e s e 5 e l e c t r o n s a r e e a c h c h a r a c t e r i z e d b y an o r b i t a l a n g u l a r momentum q u a n t u m number / ( e q u a l t o 2) a n d s p i n q u a n t u m number s ( = i l / 2 ) . T h e r e a r e t h u s a l t o g e t h e r ( 2 2 + 1 ) o r b i t a l s t a t e s a n d ( 2 s + l ) s p i n s t a t e s . The t o t a l n u m ber o f a l l o w e d s t a t e s a v a i l a b l e t o t h e e l e c t r o n s b e i n g (2 I + l ) . ( 2 s + 1) - e q u a l t o 10 f o r t h i s s y s t e m . I n t h e R u s s e l l - S a u n d e r s c o u p l i n g s cheme, t h e e l e c t r o n s a r e c o n s i d e r e d t o b e c o u p l e d t o f o r m a s y s t e m w i t h t o t a l o r b i t a l a n g u l a r momentum L ( = X^^ a n c * t o t a l s p i n a n g u l a r momentum S ( X s j ) . T h e r e a r e s e v e r a l p o s s i b l e com-b i n a t i o n s o f L a n d S t o f o r m t h e t o t a l a n g u l a r momentum, J . T h e s e , when e l e c t r o s t a t i c i n t e r - a c t i o n s a r e t a k e n i n t o a c c o u n t , g i v e r i s e t o a s e r i e s o f s t a t e s w i t h d i f f e r e n t e n e r g i e s . 4 - I / T y p i c a l e n e r g y s e p a r a t i o n s a r e 10 cm o r more ( s e e f i g u r e 1 . 1 ) . C o n s e q u e n t l y i n EPR we o n l y c o n s i d e r t h e p r o p e r t i e s o f t h e l o w e s t s t a t e — E P R e n e r g i e s b e i n g o f o r d e r 1 c r a " \ The g r o u n d s t a t e o f (3d)~" i s c o r r e s p o n d i n g t o a s t a t e w i t h 5 , e l e c t r o n s , e a c h o f s p i n +1/2 a n d z e r o o r b i t a l a n g u l a r momentum ( J = 5 / 2 ; L=0; S = 5 / 2 ) . The 2S+1 s t a t e s a r e d e g e n e r a t e i n t h e a b s e n c e o f a m a g n e t i c f i e l d , b u t i n a m a g n e t i c f i e l d , Ho, t h e e n e r g y o f e a c h s t a t e i s g i v e n b y g ^ i ^ . H o . M j . Kj i s c a l l e d t h e m a g n e t i c q u a n t u m number a n d t a k e s i n t e g r a l o r h a l f i n t e g r a l v a l u e s o f J — f r o m +J t o - J . f3 i s t h e B o h r m a g n e t o n a n d g L i s t h e L a n d e g - f a c t o r g i v e n b y ; L J ( J + I) - L(L + I) + S(S+ I) 2J(J + I) F o r t h e f r e e i o n g L = 2 ( f i g u r e 1 . 1 a ) . I f o n e p e r f o r m s a n EPR e x p e r i m e n t b y i n d u c i n g t r a n s i t i o n s b e t w e e n t h e s p l i t l e v e l s w i t h r . f . e n e r g y o f f r e q u e n c y v ( e n e r g y h Z/)» t h e n u n d e r t h e s e l e c t i o n r u l e A M j = -1 we h a v e r e s o n a n c e i f : hi/ = g L . /3 • H [<Mj* I) - M, FIGURE 1.1 E l e c t r o n i c e n e r g y l e v e l s o f t h e F e i o n . ( a ) S p l i t t i n g o f g r o u n d s t a t e o f a f r e e F e + i o n i n a n a p p l i e d m a g n e t i c f i e l d . ( b ) S p l i t t i n g o f t h e g r o u n d s t a t e o f a n F e + i o n b o u n d i n t o c y t o c h r o m e c i n a n a p p l i e d m a g n e t i c f i e l d . F o r e x p l a n a t i o n o f s y m b o l s s e e t e x t ( s e c t i o n 1.4.4) (a) (b) • • 9 4 -•!/2 -5/ 2 2/3H "g r i l '29 • electron states system states |zx> |yz> Free Ion Cubic Field Axial Field Rhombic Reid Spin Orbit (Kramer's Doublet) Magnetic Field I n f a c t we a r e n o t c o n c e r n e d w i t h t h e f r e e i o n b u t w i t h a c o m p l e x a n d m u s t t a k e a c c o u n t o f t h e i o n ' s e n v i r o n m e n t . The i r o n a t o m o f a l l heme p r o t e i n s i s c o -o r d i n a t e d t o t h e n i t r o g e n a t o m o f 4 p y r r o l e g r o u p s — f o r m i n g t h e p l a n a r heme r i n g ( f i g u r e 1 . 2 ) . I n c y t o c h r o m e c , t h e r e i s f u r t h e r c o - o r d i n a t i o n t o t h e S - n i t r o g e n atom o f t h e h i s t i d i n e a m i n c - a c i d i n t h e p r o t e i n b a c k b o n e a n d t o t h e s u l p h u r a t o m o f m e t h i o n i n e - 8 0 ( b y c o n v e n t i o n t h e s e a r e l a b e l l e d p o s i t i o n s 5 a n d 6 r e s p e c t i v e l y ) , s e e f i g u r e 1.2 . 1.4.4 C h e m i s t r y o f t h e F e + i n c y t o c h r o m e c The s i x n e i g h b o u r s ( o r l i g a n d s ) f o r m an o c t a h e d r a l a r r a n g e m e n t a r o u n d t h e i r o n ; t h e i r e l e c t r o s t a t i c i n t e r a c t i o n w i t h t h e c e n t r a l i o n w i l l r e m o v e , p a r t i a l l y o r c o m p l e t e l y , t h e d e g e n e r a c y o f t h e i r o n d - o r b i t a l s . The p a t t e r n a n d m a g n i t u d e o f t h e s p l i t t i n g o f t h e s e o r b i t a l s d e t e r m i n e s t h e m a g n e t i c p r o p e r t i e s o f t h e m o l e c u l e . T h e s t u d y o f t h i s t y p e o f i n t e r a c t i o n i s c a l l e d L i g a n d F i e l d T h e o r y a n d a w i d e l i t e r a t u r e e x i s t s o n t h i s s u b j e c t . B a l l h a u s e n ( 1 9 6 2 ) a n d G r i f f i t h ( 1 9 6 1 ) a r e s t a n d a r d t e x t s ; G r i f f i t h ( 1 9 5 6 , 1 9 5 7 , 1 9 5 8 , 1 9 6 4 , 1965) a n d K o t a n i ( 1 9 6 1 , 1 9 6 4 ) h a v e d o n e much t o p r o v i d e a c o h e r e n t t h e o r e t i -c a l f r a m e w o r k t h a t c o v e r s m o s t o f t h e e x p e r i m e n t a l d a t a o n h e m o p r o t e i h s . H a r r i s - L o e w ( 1 9 7 0 ) a n d W e i s s b l u t h ( 1 9 6 6 ) h a v e p r o v i d e d r e c e n t r e v i e w s . The W e i s s b l u t h r e v i e w i s t h e b e s t s i n g l e s o u r c e , f o r t h e p h y s i c i s t w i s h i n g t o u n d e r s t a n d t h e FIGURE 1.2 D i a g r a m o f t h e a t o m s c o - o r d i n a t e d t o F e c y t o c h r o m e c . F o r e x p l a n a t i o n , s e e t e x t ( s e c t i o n 1 . 4 . 3 ) . S (Methionyl) p : p N (Histidyl) 17 c h e m i s t r y o f heme c o m p o u n d s ; he d e v e l o p s t h e t h e o r y i n a l e i s u r e l y f a s h i o n ( e x p l a i n i n g t h e many d i f f e r e n t s y s t e m s o f n o t a t i o n u s e d b y o t h e r s ) a n d c o n c e n t r a t e s on t h e EPR, mag-n e t i c s u s c e p t i b i l i t y a n d M o s s b a u e r p r o p e r t i e s o f h e m o g l o b i n , . T h e o c t a h e d r a l l y c o - o r d i n a t e d l i g a n d s p r o d u c e a c u b i c e n v i r o n m e n t a b o u t t h e i r o n — t h e s y m m e t r y e l e m e n t s o f t h i s a r r a n g e m e n t b e i n g t h e same a s t h o s e o f a c u b e . G r o u p t h e o r e t i c a l t e c h n i q u e s a r e u s e d t o d e s c r i b e how t h e d e -g e n e r a c y o f t h e d - o r b i t a l s w i l l be l o w e r e d b y t h i s e n v i r o n m e n t . U n d e r c u b i c s y m m e t r y t h e f i v e 3 d - o r b i t a l s s e g r e g a t e i n t o t w o s e t s - — o n e t h r e e - f o l d a n d t h e o t h e r t w o - f o l d s p a t i a l l y d e g e n e r a t e . F o r e l e c t r o n e g a t i v e l i g a n d s ( N , S ) , t h e t h r e e - f o l d d e g e n e r a t e s e t ( i n g r o u p t h e o r y n o t a t i o n -t h e t 2 s e t ) i s l o w e r i n e n e r g y t h a n t h e o t h e r ( t h e e- s e t ) . T h e e n e r g y g a p b e t w e e n t h e t 2 g a n d e^ s e t s l a r g e l y d e t e r m i n e s t h e m a g n e t i c p r o p e r t i e s o f t h e c o m p l e x . Two l i m i t i n g c a s e s c a n be d i s t i n g u i s h e d : ( i ) h i g h s p i n - when t h e e n e r g y g a p i s much s m a l l e r t h a n t h e e l e c t r o n p a i r i n g e n e r g y ( p a i r i n g e n e r g y b e i n g t h e r e s u l t o f e l e c t r o s t a t i c a n d e x c h a n g e i n t e r a c t i o n s w h i c h t e n d t o a l i g n t h e e l e c t r o n s w i t h t h e same v a l u e o f s p i n ) t h e n t h e f i v e e l e c t r o n s d i s t r i b u t e t h e m s e l v e s o v e r a l l o r b i t a l s 3 2 ( t 0 e ) t o y i e l d maximum s p i n (S = 5 / 2 ) . ( i i ) l o w s p i n - when t h e s p l i t t i n g e n e r g y i s g r e a t e r t h a n t h e p a i r i n g e n e r g y t h e f i v e e l e c t r o n s e n t e r t h e 5 t - l e v e l s , h a v e t h e c o n f i g u r a t i o n t ~ , a n d minimum s p i n ( S = 1 / 2 ) e M a g n e t i c s u s c e p t i b i l i t y m e a s u r e m e n t s i n f e r r i c y t o c h r o m e c ( T a s a k i e t a l . j 1 9 6 7 ) ha v e shown t h a t i t i s a l o w s p i n c o m p l e x o f s p i n 1/2, a n d t h e r e f o r e h a s t h e f i v e d - e l e c t r o n s i n t h e s i x ( 3 s p a t i a l x 2 s p i n ) ^ o r b i t a l S o T h e t o t a l - c a p a c i t y o f t h e t 2 g o r b i t a l s i s 6 e l e c -t r o n s , a n d h e n c e t h e 5 e l e c t r o n s y s t e m b e h a v e s a s a s i n g l e p o s i t i v e h o l e . T h i s e n a b l e s u s t o d e a l w i t h a s i m p l e (3d)"*" h o l e s y s t e m ; r a t h e r t h a n t h e c o m p l e x ( 3 d ) s y s t e r r u T h e s p i n H a m i l t o n i a n c o n c e p t i s u s e d t o d e a l w i t h t h e EPR o f t h i s s y s t e m . The g r o u n d s t a t e i s a s s u m e d t o be a d o u b l e t w i t h an e f f e c t i v e s p i n , S, o f 1/2 a n d t h e e n e r g i e s o f the l e v e l s i n a n a n n l i e d f i e l d Ho are a-B.H-M (M = £ 1/2 T h e g - f a c t o r i s n o l o n g e r t h e L a n d e g - f a c t o r , g L , b u t i s a p a r a m e t e r t o be d e t e r m i n e d i n t h e EPR e x p e r i m e n t f r o m t h e v a l u e s o f D a n d H t h a t g i v e r e s o n a n c e . G e n e r a l l y g w i l l b e h a v e l i k e a s y m m e t r i c t e n s o r , a n d , w i t h r e s p e c t t o a n a x i s s y s t e m s u i t a b l y o r i e n t e d i n t h e m o l e c u l e , c a n be c h a r a c -t e r i z e d b y a t m o s t 3 v a l u e s . I t i s f o u n d f o r c y t o c h r o m e c t h a t t h e r e a r e 3 g - v a l u e s ( S a l m e e n a n d P a l m e r , 1 9 6 8 ) a l l d i f f e r e n t f r o m t h e f r e e s p i n v a l u e o f 2. The h o l e t h e r e f o r e d o e s n o t b e h a v e a s a f r e e s p i n ; a c o n t r i b u t i o n f r o m o r b i t a l a n g u l a r momentum i s p r e s e n t ( i . e . , t h e r e e x i s t s s p i n - o r b i t c o u p l i n g ) . The g - v a l u e s a l l d i f f e r e n t i m p l y t h a t t h e e n v i r o n m e n t a r o u n d t h e i r o n h a s s y m m e t r y l o w e r t h a n c u b i c , i . e . , m u s t c o n t a i n a x i a l 19 a n d r h o m b i c e l e m e n t s w h i c h r e m o v e t h e d e g e n e r a c y o f t h e t 2 o r b i t a l s ( f i g u r e 1 . 1 b ) . I n t h e a b s e n c e o f s p i n - o r b i t c o u p l i n g t h e t 2 o r b i t a l s w o u l d f o r m a c l o s e d s h e l l e x c e p t f o r a h o l e i n J d 2 ^ > ( f i g u r e 1 . 1 b ) . S p i n - o r b i t c o u p l i n g m i x e s t h e s e s t a t e s s o t h a t t h e j d z x ^ > h o l e i s d i s t r i b u t e d o v e r a l l t h e o r b i t a l s , r e s u l t i n g i n a s e t o f t h r e e K r a m e r s d o u b l e t s , e a c h a l i n e a r c o m b i n a t i o n o f t h e e l e c t r o n i c w a v e -f u n c t i o n s (.see A p p e n d i x 1 ) . The l o w e s t e n e r g y d o u b l e t i s t h e o n l y o n e a p p r e c i a b l y o c c u p i e d a t l o w t e m p e r a t u r e s — a p p l i c a t i o n o f a m a g n e t i c f i e l d t o t h i s d o u b l e t c a n a c c o u n t f o r t h e EPR s p e c t r a o b s e r v e d . One c a n w o r k b a c k f r o m t h e o b s e r v e d g -v a l u e s t o o b t a i n t h e s p l i t t i n g o f t h e t 2 o r b i t a l s . A r h o m -b i c i t y (V/D) o f 0.55 a n d t e t r a g o n a l ! t y ( o r a x i a l i t y ) ( D / X ) o f 2.64 a c c o u n t f o r t h e g - v a l u e s o f 3.06, 2.24 a n d 1.25 o b s e r v e d f o r c y t o c h r o m e c ( S a l m e e n a n d P a l m e r , 1 9 6 8 ; B l u m b e r g , 1 9 6 8 ) . ( \ i s t h e s p i n - o r b i t c o u p l i n g c o n s t a n t ) . A p a r t f r o m o u r d i s c u s s i o n i n c h a p t e r 4, we do n o t c a r r y o u t a n a n a l y s i s o f t h e g - v a l u e s i n t e r m s o f e n e r g y l e v e l s , s i n c e o u r m a i n i n t e n t i s t o d e t e r m i n e t h e r e l a t i o n -s h i p b e t w e e n t h e d i r e c t i o n s o f t h e p r i n c i p a l g - a x e s a n d t h e m o l e c u l a r a x e s , a n d t o s t u d y t h e d e p e n d e n c e o f l i n e w i d t h o n o r i e n t a t i o n , t e m p e r a t u r e a n d i n s t r u m e n t a l c o n d i t i o n s i n o u r EPR e x p e r i m e n t s . CHAPTER 2.0 EXPERIMENTAL APPARATUS 2.1 INTRODUCTION . -The a i m o f t h i s c h a p t e r i s t o d e s c r i b e t h e a p p a r a t u s u s e d t o o b t a i n t h e EPR r e s u l t s . We b e g i n w i t h a l i s t i n g o f t h e h e m o p r o t e i n p r o -p e r t i e s t h a t a r e r e l e v a n t t o EPR s p e c t r o s c o p y , a n d w h a t l i m i t s t h e s e i m p o s e on t h e s p e c t r o m e t e r ( s e c t i o n 2 . 2 ) . A f t e r t h i s we d e s c r i b e t h e EPR a p p a r a t u s , w i t h e m p h a s i s o n t h o s e f e a t u r e s t h a t r e q u i r e s p e c i a l comment ( s e c t i o n 2 . 3 ) . We t h e n g i v e t h e d e t a i l s o f t h e l o w t e m p e r a t u r e a p p a r a t u s ( s e c t i o n 2.4) a n d o f t h e t e m p e x - a t u r e m e a s u r e m e n t s y s t e m ( s e c t i o n 2 . 5 ) . F i n a l l y , we p r e s e n t t h e m e t h o d o f m o u n t i n g s a m p l e s ( s e c t i o n 2 . 6 ) . 2.2 HEMOPROTEIN PROPERTIES RELEVANT TO EPR 2.2.1 C o n c e n t r a t i o n The s m a l l amount o f I r o n i n m o s t h e m o p r o t e i n s ( 0 . 5 % b y w e i g h t i n c y t o c h r o m e c ) means t h a t t h e number o f u n p a i r e d s p i n s i s l o w . F o r c y t o c h r o m e c t h e r e i s . o n l y one i r o n a t o m p e r m o l e c u l e , s o t h a t a one m o l a r s o l u t i o n c o n t a i n s 6 x 10"" s p i n s p e r 3 . i t r e 0 T y p i c a l c o n c e n t r a t i o n s o f hemo-p r o t e i n s u s e d a r e 10 rnillimolar» w h i c h i n a n EPR s a m p l e 18 v o l u m e o f 0.2 m l . i s a b o u t 10 s p i n s . F o r s i n g l e c r y s t a l s t h e m o l e c u l a r c o n c e n t r a t i o n i s an o r d e r o f m a g n i t u d e h i g h e r , b u t t h e s a m p l e volume, i s v e r y much l e s s — c y t o c h r o m e c t u n a c r y s t a l s h a v e a v o l u m e o f a b o u t 2 x 10 * m l . ( n e e d l e s 2 mm x 0.1 mm x 0.1 mm) c o n t a i n i n g a p p r o x i m a t e l y 1 0 ^ s p i n s . 2 . 2 c2 S o l v e n t N a t i v e p r o t e i n s s u r v i v e o n l y i n a q u e o u s s o l u t i o n , a n d a t r o o m t e m p e r a t u r e t h e l i q u i d c a u s e s h e a v y d a m p i n g o f t h e m i c r o w a v e p o w e r . I t i s p o s s i b l e t o o v e r c o m e t h i s p r o b l e m b y u s i n g v e r y s m a l l s a m p l e s , b u t t h i s r e d u c e s t h e number o f s p i n s c o n s i d e r a b l y . F r e e z i n g o f t h e s a m p l e w i l l a l s o s o l v e t h e p r o b l e m a n d i s one r e a s o n f o r o p e r a t i n g a t t e m p e r a t u r e s b e l o w t h e f r e e z i n g p o i n t o f w a t e r . 2.2.3 L i n e w i d t h The m a i n r e a s o n f o r o p e r a t i n g a t l o w t e m p e r a t u r e s i s t o n a r r o w t h e EPR l i n e s s o t h a t t h e a b s o r p t i o n s i g n a l s c a n b e s e e n a b o v e t h e n o i s e b a c k g r o u n d . I n l o w s p i n h e m o p r o t e i n s t h i s c o n d i t i o n i s v e r y n e c e s s a r y f o r t h e e l e c t r o n s p i n s i n t e r -a c t s t r o n g l y ( v i a s p i n - o r b i t c o u p l i n g ) w i t h t h e r m a l l a t t i c e v i b r a t i o n s g i v i n g l i n e s h u n d r e d s t o t h o u s a n d s o f g a u s s w i d e a t t e m p e r a t u r e s o f 77°K a n d a b o v e . V e r y f e w w o r k e r s h a v e b e e n a b l e t o o b s e r v e c y t o c h r o m e c l i n e s a t 77°K 5 " a l t h o u g h i n o t h e r l o w s p i n h e m o p r o t e i n s EPR c a n e a s i l y be s e e n a t t h i s t e m p e r a t u r e ( e . g . E h r e n b e r g , 1 9 6 2 ) . T h i s i m p l i e s we have t o work a t o r n e a r l i q u i d h e l i u m t e m p e r a t u r e ( 4 . 2 ° K ) . EPR s t u d i e s o f o t h e r low s p i n h e m o p r o t e i n s ( e . g . , h e m o g l o b i n a z i d e — - E i s e n b e r g e r and P e r s h a n , 1967) have shown . t h a t even a t l i q u i d h e l i u m t e m p e r a t u r e s t h e l i n e s r e m a i n b r o a d , b e i n g s e v e r a l hundreds o f gauss w i d e . The a r e a o f an a b s o r p t i o n l i n e i s p r o p o r t i o n a l t o t h e number o f s p i n s c o n t r i b u t i n g t o t h e l i n e , and i s c o n s t a n t f o r a c o n s t a n t number o f s p i n s . F o r a f i r s t d e r i v a t i v e s p e c t r u m , t h e a r e a o f t h e ( i n t e g r a t e d ) a b s o r p t i o n l i n e i s a p p r o x i m a t e l y e q u a l t o t h e s i g n a l h e i g h t x ( t h e p e a k - t o - p e a k 2 l i n e w i d t h ) — P o o l e , 1967, p. 551. T h e r e f o r e t h e s i g n a l h e i g h t i s i n v e r s e l y p r o p o r t i o n a l t o the s q u a r e o f the l i n e w i d t h . F o r a d e r i v a t i v e l i n e w i d t h o f 100 gauss i t c a n be c a l c u l a t e d t h a t t h e s y s t e m must be a b l e t o d e t e c t 1 0 1 1 s p i n s o r l e s s . 2 . 2 . 4 S a t u r a t i o n U n f o r t u n a t e l y a l o n g s p i n - l a t t i c e r e l a x a t i o n t i m e l i m i t s t h e r a t e a t w h i c h t h e upper s p i n l e v e l c a n l o s e e n e r g y t o t h e l a t t i c e . A p p l i c a t i o n o f t o o much power t o t h e s p i n s y s t e m c a u s e s t h e i n d i v i d u a l a b s o r p t i o n l i n e s t o b r o a d e n and t o d e c r e a s e i n h e i g h t — t h i s e f f e c t i s c a l l e d dynamic s a t u r a -t i o n ( A b r a g a m a n d B l e a n e y , 1 9 7 0 ) . Working a t v e r y low m i c r o -b e __c wave powers—-10" t o 10" w a t t s — s h o u l d overcome t h i s p r o b l e m . 2.2.5 D i s p e r s i o n D u r i n g t h e c o u r s e o f t h i s w o r k i t became n e c e s s a r y t o e x a m i n e t h e d i s p e r s i o n s i g n a l f r o m c y t o c h r o m e c , a n d t h i s r e q u i r e d m o d i f i c a t i o n o f t h e c o n v e n t i o n a l a u t o m a t i c f r e q u e n c y c o n t r o l s y s t e m , a s d e s c r i b e d b e l o w ( s e c t i o n 2 . 3 . 4 ) . 2.2.6 Summary The r e q u i r e m e n t s , t h e n , o f t h e s p e c t r o m e t e r a r e : a n a b i l i t y t o d e t e c t 10^ u n p a i r e d s p i n s ( o r f e w e r ) i n a o n e g a u s s l i n e b o t h b y d i s p e r s i o n a n d b y a b s o r p t i o n , a t a n i n c i -d e n t m i c r o w a v e p o w e r o f lO""** w a t t s o r l e s s , w i t h s a m p l e t e m p e r a t u r e s b e t w e e n 4.2°K a n d 77°K. 2.3 THE EPR SPECTROMETER 2.3.1 B a s i c d e s i g n - H e n n i n q ( 1 9 6 1 ) s p e c t r o m e t e r The b a s i c d e s i g n o f t h e s p e c t r o m e t e r f o l l o w s t h a t o f H e n n i n g ( 1 9 6 1 ) a n d F a u l k n e r ( 1 9 6 2 ) a n d i s a homodyne b a l a n c e d m i x e r s y s t e m . T h e m a i n a d v a n t a g e s o f t h i s t y p e o f s p e c t r o m e t e r a r e : ~. ( i ) t h e s i g n a l d e t e c t i o n s y s t e m i s o p e r a t e d i n d e p e n d e n t o f t h e m i c r o w a v e p o w e r i n c i d e n t o n t h e p a r a m a g n e t i c s a m p l e ; ( i i ) t h e s p e c t r o m e t e r c a n be t u n e d t o d e t e c t e i t h e r t h e a b s o r p t i o n o r d i s p e r s i o n component, o f a n EPR s i g n a l , w i t h o u t r e q u i r i n g t h a t an EPR s i g n a l be p r e s e n t ; ( i i i ) b a l a n c e d m i x e r d e t e c t i o n m i n i m i s e s k l y s t r o n n o i s e ; ( i v ) o n l y a s i n g l e k l y s t r o n i s u s e d . The b l o c k d i a g r a m o f t h e c o m p l e t e s p e c t r o m e t e r i s shown i n f i g u r e 2,1, a n d t h e d e t a i l s o f t h e m i c r o w a v e s e c t i o n i n f i g u r e 2.2. The mode o f o p e r a t i o n i s now d e s c r i b e d , f o l -1 o w i n g H e n n i n g ( 1 9 6 1 ) . The s a m p l e i s p l a c e d i n a c a v i t y w h i c h f o r m s one o f t h e s i d e arms ( 2 ) o f a m i c r o w a v e b r i d g e . The o t h e r a r m ( 1 ) c o n t a i n s a v a r i a b l e l o a d w h i c h c a n b a l a n c e t h e b r i d g e . The k l y s t r o n p o w e r i s f e d i n t o t h e b r i d g e t h r o u g h a r m ( E ) . When t h e m a g n e t i c f i e l d i s s w e p t t h r o u g h r e s o n a n c e t h e r e f l e c t i o n c o e f f i c i e n t o f t h e c a v i t y i s a l t e r e d a n d a n o u t o f b a l a n c e s i g n a l a p p e a r s i n arm ( H ) , w h i c h i s f e d i n t o o n e arm o f t h e b a l a n c e d m i x e r d e t e c t o r v i a a 60 d b . i s o l a t o r . T h e b i a s a r m o f t h e m i x e r ( E ) i s c o n n e c t e d t o t h e m a i n w a v e -g u i d e v i a a d i r e c t i o n a l c o u p l e r , a n d p r o v i d e s b i a s p o w e r t o t h e m i x e r c r y s t a l s ; t h e a m p l i t u d e a n d p h a s e o f t h e b i a s c a n b e a d j u s t e d b y means o f a n a t t e n u a t o r a n d a p h a s e s h i f t e r . T h e a n t i p h a s e a u d i o f r e q u e n c y s i g n a l s f r o m b o t h m i x e r c r y s t a l s a r e f e d i n t o a b a l a n c e d t r a n s f o r m e r c o n n e c t e d t o a c o n v e n t i o n a l e l e c t r o n i c d e t e c t i o n s y s t e m o f p r e a m p l i f i e r p h a s e s e n s i t i v e d e t e c t o r a n d c h a r t r e c o r d e r . T h e m a g n e t i c f i e l d i s m o d u l a t e d a t a n a u d i o f r e q u e n c y , w h i c h i s a l s o s u p p l i e d t o t h e p h a s e s e n s i t i v e d e t e c t o r . When t h e d . c . f i e l d i s s w e p t s l o w l y t h r o u g h t h e r e s o n a n c e l i n e t h e f i r s t h a r m o n i c o f t h e m o d u l a t e d s i g n a l i s d e t e c t e d , f i l t e r e d a n d d i s p l a y e d . FIGURE 2.1 B l o c k d i a g r a m o f 24 GHz s p e c t r o m e t e r . Power Supply Klystron AFC AFC Pre Amplifier < Microwave Circuit Signal Detectoi Chart Recorder Phase Sensitive Detector K V Power Meter Q A D.C. Magnet Coils A.C. Modulation Coils Coil Drive Amplifier Varian Magnet SUPP v X-Drive ro en FIGURE 2 . 2 S c h e m a t i c d i a g r a m o f m i c r o w a v e s e c t i o n o f t h e EPR s p e c t r o m e t e r . Vorian EM II38V Klystron (E> 15 db 2 0 db Bias Arm ^ Attenuator A ? Phase {/) Shifter 20 db v : Det Dio ector I I d e s UuuJ n To RA.R. > C.R.O. l | — * • ~(^) Wavemeter Cavity 20 db I Signal Arm Precision Attenuator 20 db _> To Power Meter Bolometer > 60 db Attenuator — v ^ - CE Precision Short 2 Variable Coupler 20 db -D- o EPR Cavity •=*=- To A.F.C. oo T A B L E I D e p e n d e n c e o f EPR s i g n a l h e i g h t o n m i c r o w a v e f r e q u e n c y f o r v a r i o u s t y p e s o f s a m p l e s ( a f t e r A l g e r , 1 9 6 8 , p . 98) L i m i t e d D i e l e c t r i c C a s e S a m p l e l o s s S a t u r a t i o n Q ( P c r F i l l i n g F a c t o r s T o t a l F r e q u e n c y D e p e n d e n c e 1 y e s 2 n o 3 y e s n o n o n o n o n o y e s OJ -1 OJ ~% OJ -% OJ OJ OJ -3/4 OJ OJ OJ 7/2 OJ 11/4 4 5 6 n o n o n o n o y e s y e s y e s n o y e s OJ -H OJ'% (jj~% OJ -3/4 OJ OJ UJ - 3 / 4 w (€") ( The audio frequency used by Henning was 30 Hz, and 13 with t h i s system he estimated he could detect 2 x 10 spins of s o l i d DPPH (linewidth 2 gauss) with unity s i g n a l to noise -3 r a t i o at a power of 10 watt. This i s about an order of magnitude larger than the t h e o r e t i c a l minimum value of about 12 10 spins at t h i s power l e v e l at room temperature. At l i q u i d helium temperatures t h i s would be improved by a f a c t o r of 150 due to an increased spin population d i f f e r e n c e ( i n the r a t i o s T room/T helium = 300/4.2 = 60X) and to improved coupling and higher c a v i t y Q(2-3X). Small protein c r y s t a l s frozen i n a minimal amount of s o l u t i o n have very l i t t l e d i e l e c t r i c l o s s . I f the sample does not saturate at high microwave power l e v e l s , then we have case I of table I, and i f i t does we have case 3. The former case gives an 11 f o l d increase i n s i g n a l f o r a 2 f o l d increase i n microwave frequency, and the l a t t e r a 7 f o l d increase f o r the same change i n frequency. In solutions where the sample s i z e i s l i m i t e d by the c a v i t y dimensions and by the radio frequency f i e l d d i s -t r i b u t i o n , there i s l i t t l e to be gained by increasing the microwave operating frequency (cases 2 and 4 of table I ) . Since our main aim was to observe EPR from s i n g l e c r y s t a l s of cytochrome c, we chose a frequency of 24-25 G Hz f o r operation. This i s approximately two and a h a l f times greater than the more commonly used X-band frequency of 9 G Hz and leads to a 25 times increase i n s i g n a l f o r case 1. The o n l y d r a w b a c k o f t h i s i n c r e a s e d f r e q u e n c y i s t h a t t h e m a g n e t u s e d m u s t be c a p a b l e o f p r o d u c i n g h i g h e r f i e l d s t o o b t a i n t h e same r a n g e o f g - v a l u e s a t 25 GHz a s a t 9 GHz. 2.3.3 M o d i f i c a t i o n ( i i ) - i n c r e a s e d m a g n e t i c f i e l d  m o d u l a t i o n f r e q u e n c y We o p e r a t e w i t h a h i g h m a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y i n o r d e r t o r e d u c e t h e n o i s e a p p e a r i n g a t t h e i n p u t o f t h e p h a s e s e n s i t i v e d e t e c t o r . The s i l i c o n d i o d e s u s e d t o d e t e c t t h e m o d u l a t e d m i c r o w a v e p o w e r f r o m t h e c a v i t y a d d n o i s e t o t h e EPR s i g n a l . The amount o f n o i s e a d d e d d e p e n d s u p o n t h e p o w e r i n c i d e n t o n t h e d i o d e a n d t h e f r e q u e n c y o f r n e m a g n e t i c f i e l d m o d u l a t i o n ( w h i c h b e c o m e s t h e s i g n a l m o d u l a t i o n f r e q u e n c y ) . The m o s t i m p o r t a n t s o u r c e i s " 1 / f " n o i s e ( s o c a l l e d b e c a u s e o f i t s f r e q u e n c y d e p e n d e n c e ) , a s i t d o m i n a t e s t h e n o i s e p o w e r s p e c t r u m a t a l l f r e q u e n c i e s b e l o w a b o u t 100 k H z . A b o v e t h i s f r e q u e n c y , o n l y f r e q u e n c y i n d e p e n d e n t t h e r m a l a n d s h o t n o i s e r e m a i n . By u s i n g a v a r i a b l e f r e q u e n c y p h a s e s e n s i t i v e d e t e c t o r , a n d a w i d e b a n d p o w e r a m p l i f i e r , we c a n m o d u l a t e t h e m a g n e t i c f i e l d a t f r e q u e n c i e s f r o m 100 Hz t o 100 k H z . The " l / f " n o i s e i s c l e a r l y s e e n a t t h e l o w e r f r e q u e n c i e s . S i n c e we d e t e c t v o l t a g e s ( p r o p o r t i o n a l t o t h e s q u a r e r o o t o f t h e p o w e r ) t h e n o i s e , N, v a r i e s w i t h t h e s q u a r e r o o t o f t h e f r e q u e n c y a n d t h e r e f o r e N ( 1 0 0 k H z ) i s 1.7 x 1 0 ~ 2 . N ( 3 0 H z ) . 32 The large range of detection frequencies i s necessary f o r some of the ra p i d adiabatic passage experiments on cyto-chrome c i n chapter 6. 2.3.4 Modification ( i i i ) - improvements to AFC c i r c u i t s * The f a s t passage experiments also required that we operate the spectrometer i n the dis p e r s i o n mode. Tuning the det e c t i o n system f o r dispersion i s done by changing the phase of the power i n the bias arm by 90° from that used f o r absorption detection (Henning, 1961). In order to maintain a low s i g n a l to noise r a t i o i n t h i s mode of operation the automatic frequency control system (AFC) had to be modified. Wilhmshurst (1967), i n a recent book, has given a d e s c r i p t i o n of the requirements that an AFC system of an EPR spectrometer must f u l f i l . A spectrometer operated i n the dispersion mode i s Inherently n o i s i e r than one operated i n the absorption mode. The phenomenon of dispersion r e s u l t s i n a s h i f t of the f r e -quency of the c a v i t y as the f i e l d moves through the magnetic resonance. In the dispersion mode the instrument i s s e n s i t i v e t o t h i s frequency change, i n the absorption mode i t i s not (providi n g the differ e n c e between the k l y s t r o n and the c a v i t y resonance frequencies i s small). In the dispersion mode the Thi s section was written i n c o l l a b o r a t i o n with Dr. C. P. S. Taylor, who designed and b u i l t the f i n a l version of the AFC. spectrometer converts the frequency modulation noise of the kl y s t r o n to amplitude noise i n the output. In addition, any noise or d r i f t i n the natural resonant frequency of the c a v i t y w i l l appear as output noise. S t a b i l i z i n g the k l y s t r o n to an external c a v i t y w i l l reduce the k l y s t r o n noise but not that caused by random d r i f t i n the resonant frequency of the experimental c a v i t y . I f the AFC i s locked on the experimental c a v i t y resonance, the AFC w i l l act to compensate f o r the s h i f t of frequency caused by dis p e r s i o n and thus reduce the dis p e r s i o n s i g n a l . However, t h i s e f f e c t can be n u l l i f i e d by modulating the magnetic f i e l d , and hence the true dispersion s i g n a l , at a higher frequency than the AFC c a r r i e r . At the higher frequency the loop gain of the AFC i s zero and the dispersion s i g n a l i s not reduced. In such an arrangement the k l y s t r o n and the c a v i t y track each other over the frequency band pass of the AFC. An important point must be made about an AFC that derives i t s e r r o r signal from the mismatch of k l y s t r o n and ca v i t y resonance frequencies (class 0 type AFC—Wilmshurst, 1967). To our knowledge i t has not been made before. The true c o n t r o l point i n the loop i s not at the resonant c a v i t y but at the input to the DC am p l i f i e r that provides the cor-r e c t i o n voltage f o r the k l y s t r o n r e f l e c t o r (see f i g u r e 2 . 3 ) . The feedback loop operates i n such a fashion as to reduce the voltage at the input of the DC operational a m p l i f i e r to zero. This means that i f the o f f s e t voltage of the a m p l i f i e r FIGURE 2.3 Block diagram of automatic frequency c o n t r o l (AFC). Reflector Voltage < ® EMII38V Klystron D.C. Correcti cjn Voltage Integrator Error Pre Amplifier kHz Tuned Amplifier * 1 II kHz Phase Detector EPR Cavity II kHz to Klystron II kHz Oscillator B LP LP d r i f t s , then the frequency dif f e r e n c e between the k l y s t r o n and the resonant ca v i t y must d r i f t o f f i n the opposite sense so that the s i g n a l coming from the phase-sensitive detector to the amp l i f i e r w i l l j u s t oppose the change i n the o f f s e t voltage and maintain the amp l i f i e r input at zero. The basic conclusion i s that d r i f t s at the a m p l i f i e r input, of whatever sort, w i l l r e s u l t i n compensatory displacements of the k l y s t r o n frequency from the ca v i t y resonance. In s i m i l a r fashion noise a r i s i n g i n the DC a m p l i f i e r w i l l be r e f l e c t e d onto the k l y s t r o n as FM noise. . In the absorption mode such noise i s discriminated against, but the dispersion mode e f f i c i e n t l y t r a n s f e r s i t to the output of the spectrometer. I f the gain of the DC am p l i f i e r i s kept at unity i n the frequency region where i t exercises proportional c o n t r o l only, then the noise from any high grade operational a m p l i f i e r i s only a few microvolts, and w i l l not cause trouble. How-ever, operational amplifiers d i f f e r i n the s t a b i l i t y of t h e i r input o f f s e t currents and voltages. Those not s p e c i a l l y s t a b i l i z e d s u f f e r v a r i a t i o n s of mV/°C, and mV over 24 hr, and ••: nA f o r the current, while chopper s t a b i l i z e d operational a m p l i f i e r s s u f f e r changes i n the order of y}J and pA. C l e a r l y only the l a t t e r c l a s s are s u i t a b l e i n an AFC f o r a di s p e r s i o n mode instrument. The AFC used i n i t i a l l y was based on the proportional c o n t r o l l e r of George and Teaney (1960). This c i r c u i t 37 modulated the k l y s t r o n r e f l e c t o r at 11 kHz producing f r e -quency modulated microwave power in c i d e n t on the c a v i t y . Performance was improved by adding an integrator i n t o the feedback loop (Berry and Benton, 1965), which provides i n t e g r a l plus proportional c o n t r o l , and reduces the steady state error to zero (class I AFC). Our c i r c u i t r y went through several versions; the f i n a l and most successful one used integrated c i r c u i t s f o r a m p l i f i c a t i o n , and a very high q u a l i t y chopper s t a b i l i s e d operational a m p l i f i e r (Philbrick/Nexus 1701) as i n t e g r a t o r . The AFC c i r c u i t i s schematized i n f i g u r e 2.3. Experimentally we found that t h i s AFC added no noise to the spectrometer output, and as i t operates at about 11 kHz i t had no e f f e c t on dispersion s i g n a l s modulated at higher than that frequency; and because of the bandwidth of the i n t e g r a t o r , no e f f e c t below that frequency u n t i l 100 to 200 Hz. I t s s t a b i l i z i n g e f f e c t on the EPR d i s p e r s i o n s i g n a l was e v i -dent and s a t i s f a c t o r y . 2.3.5 Performance of the EPR Spectrometer Hyde (1961) states that "of a l l the measurements one can make with EPR equipment, the determination of absolute spin concentrations i s the most d i f f i c u l t . " We have sidestepped t h i s problem of d i r e c t measure-ment by using a secondary s t a n d a r d — C r ^ + ions i n MgO. This i s c a l i b r a t e d against gaseous oxygen l i n e s by the manufacturers (Strand Labs. Inc., Cambridge, Mass., USA). They claim that the e f f e c t i v e spin concentrations are accurate to within 5%. The chromium ion, i n the MgO host L a t t i c e , gives a narrow l i n e , one gauss wide, with g-value 1.97. Our sample i s i n powder form sealed i n t o a 2 mm o.d. quartz tube, con-14 t a i n i n g an e f f e c t i v e spin concentration of 2 x 10 spins per centimetre of tube. At room temperature t h i s standard does not show saturation below 10 mW i n c i d e n t power, but below 77°K i t saturates at microwatt power l e v e l s . As a r e s u l t , our s e n s i t i v i t y measurements were c a r r i e d out at room temperature. A t y p i c a l measurement of s i g n a l height (converted to e f f e c t i v e number of spins) over a wide range of c a v i t y power i s shown i n fig u r e 2.4 ( O ) . These were obtained with 100 kHz f i e l d modulation of 1 gauss amplitude. For comparison the t h e o r e t i c a l s e n s i t i v i t y , which was c a l c u l a t e d using a treatment s i m i l a r to that given i n Poole (1967, chapter 14), i s also presented i n f i g u r e 2.4. Our measured s e n s i t i v i t y at room temperature i s within a f a c t o r of 2 of the t h e o r e t i c a l one. The 150x improve ment estimated i n going from room to helium temperature gives a f i n a l s e n s i t i v i t y of between 1 0 1 0 and 1 0 1 1 spins at 10"^ watts of microwave power i n c i d e n t — o u r design goal. I t should be mentioned, however, that the t h e o r e t i -c a l minimum number of detectable spins may be somewhat l e s s FIGURE 2.4 Measured and c a l c u l a t e d curves of minimum detectable number of spins as a function of microwave power, at room temperature. # of Spins Detectable with S-U of Unity and Linewidth of 1 Gauss o o o o than that given, because of measurement u n c e r t a i n t i e s i n the values of such q u a n t i t i e s as the Q of the ca v i t y , the f i l l i n g f a c t o r of the sample and the noise c o n t r i b u t i o n from the detection diodes. Both Poole (1967) and Alger (1968) have lengthy discussions on t h i s subject. 2.4 LOW TEMPERATURE APPARATUS 2.4.1 Introduction The purpose of the low temperature system i s to enable any temperature between 4.2°K ( l i q u i d helium b o i l i n g point) and 77°K ( l i q u i d nitrogen b.p.) to be maintained i n the EPR sample. The dewar that encloses the EPR c a v i t y must be able to f i t i n s i d e the 2.25 inch magnet gap, and contain an EPR c a v i t y of about 1.6 inches diameter. These inner and outer diameter r e s t r i c t i o n s mean that we must use a dewar with a si n g l e vacuum space between room and l i q u i d helium temperature We b r i e f l y consider the sources of heat leak i n t o t h i s type of dewar and describe i t s construction i n the next three s e c t i o n s . The sources of heat leak are: ( i ) r a d i a t i o n across the single vacuum space (se c t i o n 2.4.2) ( i i ) conduction through the r e s i d u a l gas i n the s i n g l e vacuum space (section 2.4.3); ( i i i ) conduction down the waveguide and other connections to the EPR c a v i t y (section 2.4.4). 2.4.2 T h e r m a l R a d i a t i o n T h e r m a l r a d i a t i o n f r o m t h e ' h o t ' o u t e r w a l l a c r o s s t h e v a c u u m s p a c e i s l a r g e e n o u g h t h a t a r a d i a t i o n s h i e l d c o o l e d t o 77°K m u s t be i n s e r t e d i n t o t h e v a c u u m s p a c e . ( I n t h e more common tw o d e w a r s y s t e m s , an o u t e r d e w a r c o n t a i n i n g l i q u i d n i t r o g e n s u r r o u n d s an i n n e r d e w a r h o l d i n g l i q u i d h e l i u m . R a d i a t i o n i n t o t h e h e l i u m d e w a r o n l y comes from, t h e v a c u u m w a l l c o o l e d t o 7 7 ° K ) . :• V a r i o u s s i n g l e w a l l d e w a r s h a v e b e e n s u g g e s t e d , e . g . K r o o n ( 1 9 6 6 ) o r F r i n d t a n d S t u a r t ( 1 9 6 8 ) , w h i c h d i f f e r o n l y i n t h e w a y s b y w h i c h t h e h e a t s h i e l d i s c o n n e c t e d t o t h e n i t r o g e n b a t h . M o s t d e s i g n s h a v e c o m p l i c a t e d g l a s s / m e t a l s e a l s w h i c h a r e d i f f i c u l t t o f a b r i c a t e . O u r g l a s s b l o w e r * c i r c u m v e n t e d t h i s s e a l i n g p r o b l e m b y s h r i n k i n g one w a l l o f t h e n i t r o g e n c o n t a i n e r o n t o a c o p p e r h e a t s h i e l d — a s shown i n f i g u r e 2.5. A l o w e r n i t r o g e n b a t h i s p l a c e d a r o u n d t h e p o r t i o n o f t h e d e w a r t a i l w h i c h p r o j e c t s b e l o w t h e p o l e p i e c e s o f t h e m a g n e t t o make s u r e t h a t t h e h e a t s h i e l d i s c o o l e d a s c l o s e t o 77°K a s p o s s i b l e a l o n g i t s w h o l e l e n g t h . A p r e ~ c o o l i n g p e r i o d o f s e v e r a l h o u r s w i t h l i q u i d n i t r o g e n was f o u n d t o be e s s e n t i a l f o r s u c c e s s f u l h e l i u m t r a n s f e r . M r . R. E b e r h a r d t o f t h e D e p t . o f P h y s i c s , U n i v e r s i t y o f W i n d s o r , O n t . FIGURE 2.5 Schematic diagram of single vacuum space dewar containing copper heat s h i e l d . 45 2.4.3 Conduction through r e s i d u a l gas In order to reduce thermal conduction through the r e s i d u a l gas, the dewar vacuum space had to be pumped to l e s s than 10~^ t o r r (1 t o r r - 1 mm mercury pressure). For most of the experiments i n t h i s t h e s i s , the pumping system was a 2" diameter Veeco o i l d i f f u s i o n pump, w i t h a Veeco ro t a r y o i l pump f o r backing. When the d i f f u s i o n pump col d trap was cooled with l i q u i d nitrogen, the required pressure could be reached. However, since t h i s system was designed f o r other experiments that d i d not require such low pressures, the attainment of l e s s than 10~^ mm was never c e r t a i n , as frequent unsuccessful t r a n s f e r s showed. This s i t u a t i o n was r e l i e v e d by the purchase of our own pumping system (from Edwards High Vacuum, Canada, Ltd., O a k v i l l e , O n t a r i o ) . This system could e a s i l y a t t a i n 10""^ t o r r when cooled with l i q u i d nitrogen. I f one kept pumping while trans--7 f e r r i n g l i q u i d helium, the dewar pressure dropped to 4.10 t o r r due to the combined e f f e c t of the d i f f u s i o n pump and the cryopumping e f f e c t on the col d helium. Gaseous heat conduction i s therefore n e g l i g i b l e . 2.4.4 Conduction down EPR c a v i t y connections ~- The amount of heat conducted i n t o the EPR c a v i t y — v i a the s t a i n l e s s s t e e l waveguide and other connections—was Pumping system of Dr. P. W. Whippey, Dept. of Physics, U n i v e r s i t y of Western Ontario, London, Ontario, to whom our thanks. estimated to cause a b o i l o f f of 0.5 to 1.0 l i t r e per hour of l i q u i d helium. Another source of heat i n the system i s the jo u l e heating e f f e c t i n the magnetic f i e l d modulation c o i l s f i x e d to the c a v i t y . This heating was seldom a problem because the EPR sig n a l s observed when l i q u i d helium was i n the dewar were those of f a s t passage, and one of the features of t h i s s i g n a l i s that the maximum s i g n a l height i s obtained at a f i e l d modulation of 1 gauss. At t h i s modulation amplitude the joule heating was n e g l i g i b l e . 2.4.5 Performance of the cryogenic system Under the conditions mentioned above, v i z . good dewar vacuum, several hours precooling and low magnetic f i e l d modulation, the helium from a single transfer.—about one l i t r e — t o o k from one to two hours to b o i l down to the l e v e l of EPR c a v i t y . Once below the cavity, b o i l o f f slowed down as the major source of heat input was removed from the l i q u i d . However, s u f f i c i e n t cold gas was evolved (by r e s i d u a l heat rad i a t i o n ) that the c a v i t y and sample could be maintained at 4.2°K f o r a further 30 minutes. The best performance used 5 l i t r e s of l i q u i d helium i n 2 tr a n s f e r s to cool the EPR cavi t y from 77°K to 4.2°K and maintain the helium l e v e l above the c a v i t y for 5 hours. The warm up time from 4.2°K to 77°K was approxi-mately 3 hours. The i n i t i a l temperature r i s k was 1°K per 47 ' TABLE I I  f r o m W h i t e (1968) p . 371 T h e r m o e l e c t r i c p o t e n t i a l d i f f e r e n c e E w i t h r e s p e c t  to 0°K a n d t h e r m o p o w e r dE/tLT ( a f t e r P o w e l l , B u n c h , a n d C o r r u c c i n i , 1961) C o n s t a n t a n v e r s u s Cu T E dE/dT (°K) (fJV) (/xV/degK) 1 0.17 0.33 2 C 6 6 0.66 3 1.48 0.98 4 2.62 1.30 5 4.07 1.61 6 5.83 1.92 7 7.90 v. 2.22 8 10.26 2.52 9 12.92 2.81 10 15.88 3.10 12 22.64 3.66 14 30.50 4.20 16 39.43 4.73 18 49.40 5.25 20 60.40 5.77 22 72.42 6.27 24 85.43 6.76 26 99.43 7.24 28 114.4 7.71 30 130.3 8.17 32 147.1 8.62 34 164.7 9.05 T a b l e I I c o n t i n u e d C o n s t a n t a n v e r s u s C u T E dE/dT (°K) (^.V) (^V/degK) 36 183.3 9.48 38 202.7 9.90 40 222.9 10.31 45 276.8 11.28 50 335.6 12.20 55 398.8 13.07 60 466.2 13.89 65 537.5 14.66 70 612.7 15.38 75 691.2 16.03 80 773.0 16.69 85 858.1 17.37 90 946.7 18.04 100 1133.7 19.36 120 1546.4 21.90 140 2009.5 . 24.41 160 2522.7 26.86 180 3083.1 29.16 200 3688.6 31.37 250 5388.0 36.51 300 7330.2 41.09 minute from 4 . 2 ° to 10 degrees, slowing to 0.5°K by 20°K. By 70°K the ra t e had slowed to 0.2°K per minute. The r a p i d i n i t i a l r i s e made EPR measurements d i f f i -c u l t f o r the bridge balance and ca v i t y resonant frequency were changing. Attempts were made to maintain the temperature j u s t above 4 .2°K by a very slow t r a n s f e r of l i q u i d helium f o r the storage dewar. These were not very successful as the flow of l i q u i d could not be regulated s u f f i c i e n t l y c l o s e l y . The temperature simply f e l l to 4.2°K and remained there. However, above 20°K i t was possible to s t a b i l i s e the tem-perature to within - 1°K by such a slow tra n s f e r of helium. 2.5 TEMPERATURE MEASUREMENT The temperature i n the EPR c a v i t y was monitored by a thermocouple mounted i n the wall as i n fig u r e 2.6. This thermocouple was constructed of copper and constantan wire (C-V Instruments Co., Wallingford, Conn. #3006). The c a l i b r a t i o n of the thermocouple was made by using the data published i n White (1968) f o r a Cu-constantan thermocouple, i n table I I . With one junction i n l i q u i d helium and the other i n l i q u i d nitrogen, the voltage dif f e r e n c e was le s s than 2yxV from the 720^t.V c a l c u l a t e d from table I I . We have therefore assumed that t h i s scale i s cor r e c t f o r our thermocouple. We sought the answer to the q u e s t i o n — i s the sample temperature the same as the thermocouple temperature?—by FIGURE 2*6 S c h e m a t i c d i a g r a m o f EPR c a v i t y s h o w i n g m e t h o d o f m o u n t i n g t h e r m o c o u p l e . Sample Inlet Tube Sample Tube Holder LP h-1 making another thermocouple and mounting i t i n s i d e a sample holder. At temperatures between 4.2°K and 77°K the tem-perature d i f f e r e n c e between the two thermocouples (wall and tube) was not greater than about 0.1°K, even when the temperature was changing at l°K/minute. (This i s a r e s u l t of the low s p e c i f i c heat of metals at low temperatures and the high thermal conductivity of helium gas; these two f a c t s ensure that thermal equilibrium i s attained very r a p i d l y throughout the apparatus). We therefore concluded that the sample temperature i s given by the wall thermocouple readings between 4.2 and 77°K, to within 0.1°K. 2.6 SAMPLE MOUNTING 2.6.1 Introduction In the next sections we describe the sample changing apparatus (2.6.2), the arrangements f o r mounting solutions (2.6.3) and f o r mounting c r y s t a l s (2.6.4). 2.6.2 Sample changing system In order to conserve l i q u i d helium, we wanted to be able to change samples while the EPR c a v i t y was at 4.2°K. The main problems to be overcome were: ( i ) how to prevent the entry of a i r i n t o the apparatus, which could freeze and prevent further samples being mounted; ( i i ) t o have sample h o l d e r s r u gged enough t o s u r v i v e c y c l i n g between room and l i q u i d h e l i u m t e m p e r a t u r e s . The f i r s t p r o b l e m was s o l v e d by u s i n g t h e a i r l o c k sample c h a n g i n g s y s t e m o f E s t l e and W a l t e r s ( 1 9 6 1 ) . The a p p a r a t u s i s shown i n f i g u r e 2.7. These a u t h o r s gave a sequence o f o p e r a t i o n s i n c l u d i n g e v a c u a t i o n o f the a i r l o c k t o e n s u r e t h a t no a i r l e a k e d i n t o t h e dewar. However, our e x p e r i e n c e was, t h a t p r o v i d e d one k e e p s t h e k n u r l e d cap t i g h t when i n s e r t i n g t h e sample r o d , and e n s u r i n g t h a t t h e p l u g v a l v e was c l o s e d when t h e r e was no tube i n t h e c a v i t y , t h e r e was no p r o b l e m o f b l o c k a g e due t o f r o z e n a i r . The second p r o b l e m i s more d i f f i c u l t as we c o u l d n o t use t e f l o n o r n y l o n sample t u b e s , due t o t h e i r EPR s i g n a l s a t g = 4 . 3 and g = 2. These were s t r o n g enough t o o b s c u r e t h e c ytochrome c EPR l i n e s . Even q u a r t z t u b i n g o f v e r y h i g h p u r i t y ( S p e c t r o s i l , Thermal A m e r i c a n F u s e d Q u a r t z , New J e r s e y , USA) o f t e n gave l a r g e s i g n a l s a t 4.2°K. However, by s e a r c h i n g t h r o u g h t h e b a t c h o f q u a r t z t u b i n g , p o r t i o n s c o u l d be f o u n d t h a t gave a r e d u c e d EPR s i g n a l , and t h e s e were u s e d as o f t e n as p o s s i b l e . We had a number o f p r o b l e m s w i t h sample t u b e b r e a k a g e , e s p e c i a l l y upon w i t h d r a w a l , b o t h f r o m t h e r m a l s t r e s -s e s due t o l a r g e t e m p e r a t u r e g r a d i e n t s , and f r o m m e c h a n i c a l f o r c e s on t h e t u b e . C a r e f u l a n n e a l i n g o f t h e t u b i n g h e l p e d r e d u c e t h e f i r s t c ause o f b r e a k a g e . M o u n t i n g th e q u a r t z t u b e on t h e end o f a s h o r t n y l o n r o d i n s t e a d o f d i r e c t l y t o t h e FIGURE 2.7 Schematic of sample changing system ( a f t e r E s t l e and Walters, 1961) taken from Alger (1968, p. 168). s t a i n l e s s s t e e l tube used f o r sample changing helped reduce breakages from the second cause. Even with these precautions, breakages did occur which caused constant a t t r i t i o n of our " s i g n a l f r e e " quartz sample tubes. 2.6.3 Mounting cytochrome c solutions The cytochrome c solutions were mounted i n a S p e c t r o s i l quartz tube—3 mm o.d., 2 mm i . d . — g l u e d to a nylon rod which was screwed i n t o a 1/8" o.d. s t a i n l e s s s t e e l tube. Figure 2.8 shows the arrangement. The glue used was a 50:50 mixture by volume of Ivory soap and glycerine (Alger, 1968, p. 241). This pro-duces a waxy, soapy s o l i d that forms good seals down to 4.2°K. This glue worked well, although very o c c a s i o n a l l y the sample was l e f t behind on withdrawal of the rod because the bond across the small area of contact was not strong enough. 2.6.4 Mounting cytochrome c s i n g l e c r y s t a l s The apparatus i s shown i n f i g u r e 2.9. The sample tube i s a 3 mm o.d., 2 mm i . d . S p e c t r o s i l quartz tube 5 cm. long, closed o f f i n the middle. The c r y s t a l was placed i n the top h a l f of the tube as shown. By s u i t a b l e t i l t i n g of the tube, and teasing with a f i n e h a i r , the cytochrome c r y s t a l could be oriented more or l e s s as desired. The angles of the c r y s t a l faces r e l a t i v e to the v e r t i c a l could be measured with FIGURE 2.8 M o u n t i n g s y s t e m f o r s o l u t i o n s a m p l e s . Stainless Steel Tube Fiduciary Marker Knurled Cap Nylon Rod Glue Quartz Tube (Solution) EPR Cavity FIGURE 2.9 M o u n t i n g s y s t e m f o r s i n g l e c r y s t a l s . Knob Bearing r ystal 73 Stainless Steel Tube Pointer 3 M Protractor 0 - 360° 7.5 mm EPR Cavity Teflon Holder p r o t r a c t o r and s t r a i g h t edge when v i e w e d i n a b i n o c u l a r m i c r o s c o p e , F o r maximum s i g n a l , t h e c r y s t a l must be p o s i t i o n e d -An—the- r e g i o n o f l a r g e s t microwave m a g n e t i c f i e l d i n t h e c a v i t y . The c r y s t a l was i n s e r t e d t o a p o s i t i o n 7.5 mm - 1 mm below t h e t o p o f t h e EPR c a v i t y . T h i s was t h e r e g i o n o f maximum ( > 9 0 % ) r . f . f i e l d s t r e n g t h a l o n g t h e c e n t r a l a x i s . When t h e sample was r o t a t e d u s i n g t h e k n o b — p o i n t e r — p r o t r a c t o r s y s t e m shown, f i g u r e 2.9, s l i g h t d e v i a t i o n s f r o m p e r f e c t s t r a i g h t n e s s i n the s t a i n l e s s s t e e l tube c a u s e d t h e sample t o move s e v e r a l m i l l i m e t e r s l a t e r a l l y . T h i s d i s -p l a c e m e n t was r e d u c e d t o l e s s t h a n 2 nun by h a v i n g t h e e x t r a l e n g t h o f sample tube e x t e n d i n t o a t e f l o n c o l l e t i n t h e c a v i t y base, t h e r e b y k e e p i n g t h e cytochrome c c r y s t a l i n t h e maximum r e g i o n o f t h e a x i a l r . f . f i e l d . In p r a c t i c e t h e s y s t e m worked w e l l , w i t h no p r o b l e m on i n s e r t i o n ; however, as me n t i o n e d , b r e a k a g e o f t e n o c c u r r e d on w i t h d r a w a l . CHAPTER 3 o 0 EPR OF CYTOCHROME C I N SOLUTION* 3 . 1 INTRODUCTION I n t h i s c h a p t e r we p r e s e n t t h e r e s u l t s o f a n EPR s t u d y o f t u n a c y t o c h r o m e c i n s o l u t i o n , b e t w e e n a t e m p e r a t u r e o f 4 . 2 ° K a n d 7 7 ° K . The a i m i s t o e x p l a i n t h e s i g n a l s s e e n i n t e r m s o f c u r r e n t t h e o r i e s o f t h e EPR o f heme c o m p o u n d s . We g i v e a b r i e f g e n e r a l i n t r o d u c t i o n ( 3 . 2 ) t h e n s e c t i o n s o n t h e p h y s i c a l p r o p e r t i e s o f t h e s i g n a l s s e e n ( 3 . 3 ) . A f t e r t h i s s p e c i a l a t t e n t i o n i s f o c u s s e d o n t h e l i n e w i d t h v a r i a t i o n a s a f u n c t i o n o f t e m p e r a t u r e ( 3 . 4 ) a n d o u r c o n -c l u s i o n s a r e d r a w n i n s e c t i o n 3 . 5 * 3 . 2 THEORY The t h e o r y o f EPR s i g n a l s f r o m l o w s p i n f e r r i c heme co m p o u n d s , s u c h a s c y t o c h r o m e c , h a s b e e n w o r k e d o u t b y G r i f f i t h ( 1 9 5 7 ) a n d K o t a n i ( 1 9 6 4 ) . T h e r e a r e r e c e n t r e v i e w s b y H a r r i s - L o e w ( 1 9 7 0 ) a n d " W e i s s b l u t h ( 1 9 6 6 ) . T h i s t h e o r y The r e s u l t s a n d much o f t h e s u b s t a n c e o f t h i s c h a p t e r h a v e b e e n p u b l i s h e d ; C. M a i l e r a n d C. P. S. T a y l o r , C a n . J . B i o c h e m . 4 9 , 6 9 5 - 6 9 9 , ( 1 9 7 1 ) . p r e d i c t s t h r e e g - v a l u e s a r i s i n g f r o m a x i a l a n d r h o m b i c d i s t o r t i o n s o f t h e o c t a h e d r a l e n v i r o n m e n t o f t h e heme i r o n . P o o l e ( 1 9 6 7 ) a n d K n e u b C i h l ( 1 9 6 1 ) h a v e shown how, i n s o l u t i o n , t h e EPR a b s o r p t i o n i s d i s t r i b u t e d o v e r t h e s e g - v a l u e s . F o r d e l t a f u n c t i o n l i n e s h a p e s t h e a b s o r p t i o n s p e c t r u m i s a s shown i n f i g u r e 3 . 1 a ; f o r l i n e s o f f i n i t e w i d t h t h e s p e c t r u m b e c o mes a s i n f i g u r e 3 . 1 b a n d i n t h e u s u a l d e r i v a t i v e d i s p l a y o b t a i n e d f r o m t h e EPR d e t e c t i o n s y s t e m , t h e s p e c t r u m o f f i g u r e 3 . 1 b a p p e a r s a s i n f i g u r e 3 . 1 c . o f t h e d i s t r i b u t i o n a r e g o o d a p p r o x i m a t i o n s t o t h e u n d i f f e r e n -t i a t e d a b s o r p t i o n l i n e t h a t w o u l d be s e e n i n a s i n g l e c r y s t a l a t t h e same m a g n e t i c f i e l d v a l u e . T h i s i s b e c a u s e t h e s o l u t i o n l i n e s h a p e a s a f u n c t i o n o f m a g n e t i c f i e l d , F ( H ) , i s o b t a i n e d b y summing t h e l i n e s h a p e o f a n i n d i v i d u a l e l e c t r o n s p i n a b s o r p t i o n , f ( H 0 - H ) , o v e r a d i s t r i b u t i o n , g ( H q ) . The f u n c t i o n g ( H Q ) t a k e s a c c o u n t o f t h e v a r i a t i o n i n t h e r e s o n a n c e p o s i t i o n ( H Q ) a n d i n t h e t r a n s i t i o n p r o b a b i l i t y a r i s i n g f r o m t h e r a n d o m o r i e n t a t i o n o f t h e m o l e c u l e s w i t h r e s p e c t t o t h e m a g n e t i c f i e l d ( B l e a n e y , 1 9 6 0 ) . T h u s we h a v e h o w e v e r , n e a r t h e e n d s o f t h e d i s t r i b u t i o n g ( H g ) i s a p p r o x i -m a t e l y c o n s t a n t , w i t h t h e r e s u l t t h a t The p e a k s i n t h e d e r i v a t i v e s p e c t r u m a t t h e e x t r e m e s F ( H ) 3 FIGURE 3.1 EPR spectra from randomly oriented molecules possessing 3 g-values. (a) T h e o r e t i c a l spectrum f o r the case where the i n d i v i d u a l absorption l i n e s are i n f i n i t e s i m a l l y narrow. This curve thus represents the function g(Hg). (b) The absorption spectrum f o r r e a l molecules with f i n i t e absorption line-width. This curve i s modified from a passage e f f e c t dispersion spectrum of ferricytochrome c at 5°K by a r t i f i c i a l l y narrowing the line-width near g^ so as to produce a more pronounced shoulder. The g-values obtained from the actual experimental trace were g^ = 1.26, g 2 = 2.25, g 3 = 3.06 to within about 5%. (c) A hand sketched d e r i v a t i v e of curve (b). The usual output of EPR spectrometers i s a d e r i v a t i v e such as t h i s . In the regions of g^ and g^ the curve has the shape of i n d i v i d u a l absorption l i n e s , as discussed i n the te x t . 65 f(H 0-H).dH 0 hence dF(H) dH = g(H 3).f(H 3-H) so that the d e r i v a t i v e spectrum i s proportional to the actual absorption l i n e at H^. We make use of t h i s i n our analysis of l i n e widths. 3.3 RESULTS 3.3.1 q-values s i m i l a r to the r e s u l t s of Salmeen and Palmer (1968) on beef heart cytochrome c. The d e r i v a t i v e of the absorption spectrum was as predicted i n f i g u r e 3.1 c , but the g = 1.25 l i n e seen by Salmeen and Palmer was not d i s c e r n i b l e as the sig n a l - t o - n o i s e r a t i o was too low. The presence of a g - 1.25 l i n e could be i n f e r r e d from spectra run at 4.2°K where saturation and f a s t passage e f f e c t s gave large s i g n a l s . In order to determine the properties of the cytochrome c paramagnetic resonance we confined our a t t e n t i o n to the g = 3.05 low f i e l d absorption l i n e , where the l i n e width was narrowest. 3.3.2 Signals obtained above 20°K were seen and we s h a l l consider t h i s 'high' temperature region f i r s t . Above 20°K, we obtained g-values of 3.05 and 2.24, At temperatures above 20°K no saturation e f f e c t s A t 2 2 ° K we v a r i e d t h e m i c r o w a v e p o w e r f r o m 3 0 m i c r o w a t t s t o 1 0 m i l l i w a t t s , a n d p l o t t e d t h e h e i g h t o f t h e g = 3 o 0 5 l i n e v e r s u s m i c r o w a v e p o w e r i n f i g u r e 3 . 2 . The s t r a i g h t l i n e o b t a i n e d o f s l o p e 1 / 2 shows t h a t h e i g h t i s p r o p o r t i o n a l t o t h e s q u a r e r o o t o f p o w e r , t h e e x p e c t e d b e -h a v i o r f o r a n u n s a t u r a t e d EPR a b s o r p t i o n ( A b r a g a m a n d B l e a n e y , 1 9 7 0 ) . To a v o i d s a t u r a t i o n l i n e b r o a d e n i n g we d i d n o t e x c e e d 1 0 m i l l i w a t t s o f m i c r o w a v e p o w e r , i n c i d e n t o n t h e c a v i t y . The p o w e r a b s o r b e d b y t h e s a m p l e i s p r o p o r t i o n a l t o t h e d i f f e r e n c e o f p o p u l a t i o n i n t h e t w o e n e r g y l e v e l s b e t w e e n w h i c h t h e r e s o n a n c e t a k e s p l a c e . I n a s y s t e m i n e q u i l i b r i u m t h i s p o p u l a t i o n d i f f e r e n c e i s g i v e n b y t h e B o l t z m a n n e q u a t i o n . F o r h V v e r y much l e s s t h a n k T ( h e r e h» = 0 . 7 cm" 1) t h e p o p u l a t i o n d i f f e r e n c e i s p r o p o r t i o n a l t o t h e p o w e r a b s o r b e d a n d t h e number o f a b s o r b e r s , we e x p e c t a g r a p h o f p o w e r a b s o r b e d b y t h e l i n e v e r s u s 1/T s h o u l d be a s t r a i g h t l i n e p a s s i n g t h r o u g h t h e o r i g i n . To t e s t t h i s , t h e t e m p e r a t u r e o f t h e s a m p l e was v a r i e d f r o m 2 2 ° K t o 7 7 ° K a n d t h e v a r i a t i o n o f a r e a o f t h e g = 3 p e a k d e t e r m i n e d — f i g u r e 3 . 3 shows t h e r e s u l t s . The a r e a was t a k e n t o be p r o p o r t i o n a l t o l i n e h e i g h t X h a l f w i d t h . The e x p e r i m e n t a l p o i n t s l i e n e a r a n a p p r o p r i a t e s t r a i g h t l i n e , t h e m a j o r e r r o r i n t h e v a l u e o f t h e a r e a b e i n g t h e m e a s u r e m e n t o f t h e l i n e w i d t h . The r e s u l t s s u m m a r i z e d i n f i g u r e s 3 . 2 a n d 3 . 3 e s t a b l i s h e d t h a t t h e EPR l i n e r e s u l t s f r o m a t w o - l e v e l s y s t e m . ( I t i s i m p l i c i t i n FIGURE 3.2 H e i g h t o f t h e a b s o r p t i o n l i n e a t g = 3.05 o f a f r o z e n s o l u t i o n o f t u n a cytochrome c a t 22°K as a f u n c t i o n o f t h e microwave power i n c i d e n t on t h e sample. Signal Height - Arbitrary Units FIGURE 3.3 Area of the absorption l i n e at g = 3.05 of a frozen s o l u t i o n of tuna cytochrome c as a function of the inverse temperature. (The area i s proportional to the d i f f e r e n c e i n e l e c t r o n i c population of the two l e v e l s between which the resonance occurs.) 71 the theories of G r i f f i t h (1957) and Kotani (1964) that the EPR si g n a l s a r i s e from a p a r t i c u l a r two-level system c a l l e d a Kramers doublet.) 3.3.3 Signals obtained below 20°K In the region below 20°K, the shape of the EPR spectrum changed from the d e r i v a t i v e form (fig u r e 3.1 c ) to a l i n e shape that resembled the u n d i f f e r e n t i a t e d l i n e shape of f i g u r e 3.1b • With 1 mW of i n c i d e n t power the amplitude of t h i s broad l i n e increased with decreasing temperature, u n t i l at 4.2°K the g = 3 region had increased i n amplitude approximately 150 f o l d . The same l i n e shape, with lower amplitude was recorded with as l i t t l e as 10 uW of power. To decide whether t h i s unexpected r e s u l t was an instrumental f a u l t of the K-band spectrometer, t h i s e x p e r i -ment was repeated on the X-band equipment of Dr. C. Schwerdtfeger (U.B.C. Physics Department); the r e s u l t s are shown i n f i g u r e 3.4 f o r temperatures of 4.2°K and 1.7°K. The 4.2°K r e s u l t , s i m i l a r to that of Salraeen and Palmer at 2 0 ° K , shows that our spectrometer was operating d i f f e r e n t l y . However, the eventual appearance of the anomalous s i g n a l at the lower temperature confirmed that the e f f e c t was not s o l e l y an a r t i f a c t of our apparatus. Weger (1960) systematically examined how the-micro-wave power l e v e l , magnetic f i e l d modulation frequency and magnetic f i e l d sweep rate a f f e c t the EPR s i g n a l s obtained FIGURE 3.4 EPR s p e c t r u m o f f r o z e n s o l u t i o n o f f e r r i c y t o c h r o m e a t X - b a n d ( 9 G H z ) . M a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y i s 4 0 0 H z . ( a ) T = 4.2°K, s h o w i n g a b s o r p t i o n d e r i v a t i v e . ( b ) T = 1.7°K, s h o w i n g f a s t p a s s a g e d i s p e r s i o n . from a spectrometer. This study suggested to us that the behavior of the signal i n the K-band spectrometer, and at low temperatures at X-band arises from a combination of dynamic saturation and f a s t passage e f f e c t s due to a long s p i n - l a t t i c e r e l a x a t i o n time, and that what i s observed i s a dispersion s i g n a l and not absorption. S i m i l a r e f f e c t s have been reported i n a l k a l i halides by Hyde (1960) and were used by him to estimate the value of s p i n - l a t t i c e and spin-spin r e l a x a t i o n times. The explanation of the d i f f e r e n t r e s u l t s on the two spectrometers i s therefore as follows: At 20°K, the spin system i s not saturated and the expected d e r i v a t i v e absorption s i g n a l i s seen. At 4.2°K the spin system is. saturated, the absorption l i n e i s broadened and the dispersion signal appears. Normally, a large pro-portion of any dispersion s i g n a l i s removed by the automatic frequency control (AFC) system. In the X-band experiment ( f i g u r e 3.4) t h i s i s what happened, and only the absorption d e r i v a t i v e s i g n a l was seen. In the K-band apparatus, however, the bandwidth of the AFC system i s too small, and the d i s -persion si g n a l i s not removed. At 1.7°K the dispersion s i g n a l was s u f f i c i e n t l y large that the X-band AFC could not remove i t a l l . To t e s t t h i s hypothesis, we v a r i e d the sample temperature between 4.2°K and 20°K and observed the s i g n a l s at constant microwave power—figure 3.5 gives the r e s u l t s . FIGURE 3.5 EPR s p e c t r a o f f r o z e n s o l u t i o n s o f t u n a f e r r i c y t o c h r o m e c a t t w o d i f f e r e n t t e m p e r a t u r e s : ( a ) T = 20°K - a b s o r p t i o n d e r i v a t i v e s p e c t r u m . ( b ) T = 4.2°K - d i s p e r s i o n f a s t p a s s a g e s p e c t r u m . Field Kilogauss W i t h t h e t e m p e r a t u r e f i x e d a t 18.9PK m i c r o w a v e p o w e r l e v e l s o f 10 a n d 1 mW g a v e t h e s p e c t r a o f f i g u r e 3.6. T h e r e f o r e , e i t h e r l o w e r i n g t h e t e m p e r a t u r e ( t o i n c r e a s e t h e r e l a x a t i o n t i m e ) o r i n c r e a s i n g t h e m i c r o w a v e p o w e r ( t o i n c r e a s e t h e s a t u r a t i o n ) g i v e s t h e a n o m a l o u s s i g n a l , t h u s s u p p o r t i n g t h e h y p o t h e s i s . We h a v e p u t t h e a p p e a r a n c e o f t h e s e f a s t p a s s a g e s i g n a l s t o g o o d u s e , a s t h e r e s u l t s o f t h e s i n g l e c r y s t a l s t u d i e s show. ( C h a p t e r 4, 5, a n d 6) 3.3.4 S h a p e s o f s o l u t i o n EFR s p e c t r u m A s m e n t i o n e d i n t h e T h e o r y s e c t i o n , b o t h P o o l e (1967) a n d K n e u b u h l (1961) h a v e a t t e m p t e d t o e x p l a i n t h e b r o a d EPR s p e c t r a s e e n i n s o l u t i o n . The P o o l e t r e a t m e n t — o r i g i n a l l y d e r i v e d i n 1958 ( K o h i n a n d P o o l e , 1958) a n d q u o t e d i n h i s r e c e n t b o o k (1967)-c a l c u l a t e s t h e l i n e s h a p e b y a s s u m i n g a r a n d o m d i s t r i b u t i o n o f o r i e n t a t i o n s o f t h e m o l e c u l e s w i t h r e s p e c t t o t h e m a g n e t i c f i e l d , t h e n a v e r a g i n g t h e r e s o n a n c e m a g n e t i c f i e l d o v e r a l l o r i e n t a t i o n s a s s u m i n g 3 u n e q u a l g - v a l u e s g ^ , g^, g ^ ( g 3 > g 2 > g x ) -H i s t r e a t m e n t was a p p l i e d t o s y s t e m s whose g - v a l u e s a r e v e r y c l o s e t o g e t h e r ( e . g . f o r c a r b a z y l , g ^ g ? i s a p p r o x i m a t e l y 0.06) a n d n e g l e c t s t h e v a r i a t i o n i n t r a n s i t i o n p r o b a b i l i t y t h a t becomes i m p o r t a n t when t h e g - v a l u e s a r e f a r a p a r t ( B l e a n e y , 1960). FIGURE 3.6 EPR s p e c t r a o f f r o z e n s o l u t i o n s o f t u n a f e r r i c y t o c h r o m e c a t t w o m i c r o w a v e p o w e r l e v e l s : ( a ) P o w e r i n c i d e n t on c a v i t y = 10 m i l l i w a t t s - s h o w i n g f a s t p a s s a g e s i g n a l . ( b ) P o w e r i n c i d e n t on c a v i t y = 1 m i l l i w a t t - s h o w i n g a b s o r p t i o n d e r i v a t i v e s i g n a l . Field Kilogauss TABLE I I I Comparison of l i n e heights at q-extremes g-value g x = 1.25 g £ = 2.25 g 3 = 3.06 Kneubuhl Poole Observed (height at g 3) . 5.7 13.3 5.5 (height at Kneubuhl (1961) included t h i s v a r i a t i o n of proba-b i l i t y i n his c a l c u l a t i o n s (which are considerably more complex than Poole's), the general e f f e c t being to r a i s e the i n t e n s i t y of the low g-value (high f i e l d ) region of the spectrum. Doing t h i s f o r cytochrome c gives the r e s u l t s shown i n table I I I . •• This agreement with the more general Kneubuhl theory i s encouraging, since i t supports our i m p l i c i t assump-t i o n that we are dealing with a sing l e molecular species with the three p r i n c i p a l g-values quoted. However, we s h a l l see how t h i s assumption must be modified when we consider our l i n e width r e s u l t s i n the next s e c t i o n . 3.4 DISCUSSION OF L I N E S H A P E S 3.4.1 Explanation of lineshapes below 50°K The l i n e width at g = 3.05 i s p l o t t e d as a fun c t i o n of temperature i n f i g u r e 3.7 which shows that the l i n e width has a constant value of about 380 gauss below 50°K but broadens to a value of 700 gauss at 77°K. The usual i n t e r -p r e t a t i o n f o r such behavior i s that at lower temperatures spin spin r e l a x a t i o n has taken over from s p i n - l a t t i c e r e l a x a t i o n as the dominant energy removal process. However, i f t h i s were so the l i n e shape would remain L o r e n t z i a n — t h e appropriate shape f o r a spin system r e l a x i n g with a s i n g l e time constant. FIGURE 3,7 W i d t h o f t h e a b s o r p t i o n l i n e a t g = 3.05 o f a f r o z e n s o l u t i o n o f t u n a c y t o c h r o m e c a s a f u n c t i o n o f a b s o l u t e t e m p e r a t u r e . o 00 ssnDo L j|p iM8U I"! TABLE I V Lineshapes as a function of temperature From Theory (Poole, 1967) * H h 0.5 1.5 2.0 Gaussian  h max 0.84 0.21 .06 Lorentzian max 0.80 0.30 0.20 From Figure 3.8 ******* I hmax] T=20°K [hmax] T=30.5°K [ hmax] T=62.5°K 0.5 1.5 2.0 0.88 0.18 0.05 Gaussian 0.81 0.26 0.17 Mixed 0.31 0.20 Lorentzian In f a c t , the l i n e shape changed from Lorentzian at 77°K to Gaussian at the lowest temperature studied, 20°K, with intermediate Voigt l i n e shapes at temperatures between. (Voigt l i n e shapes are a super-position of Lorentzian and Gaussian l i n e s of approximately equal width often used i n radioastronomy (see Posener, 1959).) The l i n e shape data are shown i n f i g u r e 3.8 and i n table IV f o r temperatures of 20°K, 30.5°K and 62.5°K. The Lorentzian l i n e described above should have a width independent of the microwave frequency used to observe the EPR s i g n a l . This explains the f i n d i n g of Morton (1971) i n a study of a horse heart cytochrome c s o l u t i o n at 77°K, at a frequency of 9 GHz, that the l i n e width f o r the g = 3 l i n e was 700 gauss. In contrast to the Lorentzian the Gaussian l i n e shape implies that the absorption at 20°K i s made up of a number of narrow l i n e s , randomly d i s t r i b u t e d about the g-value of 3.06. One source of such a d i s t r i b u t i o n i s unresolved hyperfine structure, but the evidence i s against t h i s . Hyperfine broadening i s independent of the microwave frequency used, but the X-band (9GHz) r e s u l t s (Salmeen and Palmer, 1968) at 20°K show that the g = 3 l i n e i s reduced i n width to 135 gauss from the K-band (24 GHz) value of 380 gauss. The r a t i o of the two l i n e widths (2.8) i s s i m i l a r to the r a t i o of the microwave frequencies (2.7). C a l c u l a t i o n s by Dr. B. R. Sreenathan, a member of our laboratory, suggest that with the f i v e nitrogen neighbors of the i r o n atom i n FIGURE 3.8 ( a ) E x a m p l e s o f h i g h t e m p e r a t u r e t u n a c y t o c h r o m e a b s o r p t i o n l i n e s . ( b ) N o t a t i o n u s e d i n T a b l e I V (b) c y t o c h r o m e c , t h e t o t a l s p l i t t i n g w o u l d be 60-100 g a u s s . F i n a l l y , r e c e n t w o r k b y S c h o l e s (1969) who c o - c r y s t a l l i z e d i s o l a t e d heme g r o u p s ( h e m i n ) i n p e r y l e n e , s h o w ed t h a t t h e h y p e r f i n e i n t e r a c t i o n o f f o u r n i t r o g e n s was a b o u t 50 g a u s s . We c o n c l u d e t h a t t h e h y p e r f i n e s p l i t t i n g i s t h e r e f o r e o b s c u r e d b y a n o t h e r b r o a d e n i n g p r o c e s s . 3.4.2 E x p l a n a t i o n o f l i n e s h a p e s a b o v e 50°K A p o s s i b l e e x p l a n a t i o n i s t h a t t h e b r o a d e n i n g r e p r e s e n t s a d i s t r i b u t i o n o f g - v a l u e s a r i s i n g f r o m v a r i a t i o n s i n t h e l i g a n d f i e l d p o t e n t i a l f r o m m o l e c u l e t o m o l e c u l e . E i s e n b e r g e r a n d P e r s h a n (1967) p o s t u l a t e d t h i s t o e x p l a i n t h e a n g u l a r v a r i a t i o n i n l i n e w i d t h s w h i c h t h e y o b s e r v e d i n s i n g l e c r y s t a l s o f m y o g l o b i n a z i d e , a n o t h e r l o w s p i n f e r r i c c ompound We h a v e a p p l i e d t h e i r t h e o r y t o o u r r e s u l t s a n d f i n d t h a t a ^11% v a r i a t i o n ( n o t -6% a s p u b l i s h e d ) i n t h e r h o m b i c p o t e n t i a l i s r e q u i r e d t o g i v e a g - v a l u e d i s t r i b u t i o n t h a t c o u l d e x p l a i n t h e 380 g a u s s l i n e s e e n . The same v a r i a -t i o n i n a x i a l p o t e n t i a l w o u l d c a u s e a l i n e w i d t h o f o n l y 40 g a u s s . E i s e n b e r g e r a n d P e r s h a n (1967) r e q u i r e d a r h o m b i c v a r i a t i o n o f ^ 4.5% t o e x p l a i n t h e i r o b s e r v a t i o n s . The l i n e w i d t h s a t t h e o t h e r g - v a l u e s c a n be c a l c u l a t e d on t h i s b a s i s a n d a g r e e r e a s o n a b l y w i t h m e a s u r e d v a l u e s . The g 2 l i n e — w h i c h h a s e x t r a b r o a d e n i n g due t o t h e g - v a l u e d i s t r i b u t i o n ( s e e f i g u r e 3.1 b ) h a d a c a l c u l a t e d w i d t h o f 1050 g a u s s a s c o m p a r e d t o t h e m e a s u r e d w i d t h ( p e a k - t o - p e a k o f d e r i v a t i v e ) of 1150 gauss. The g^ l i n e was c a l c u l a t e d to be 2800 gauss wide but the signal-to-noise r a t i o i n the spectrum was not high enough to permit a r e l i a b l e estimate of the width. Ext r a -p o l a t i n g from the width of Salmeen and Palmer's g^ l i n e suggests that at 24 GHz the g^ l i n e should be approximately 2500 gauss wide. A t h e o r e t i c a l treatment by Blumberg (1968) gives graphs of g-values as functions of a x i a l and rhombic p o t e n t i a l . These curves show that the desired g-value d i s t r i b u t i o n would r e s u l t from a v a r i a t i o n i n rhombic p o t e n t i a l s i m i l a r to that c a l c u l a t e d . This hypothesis can be checked by a sing l e c r y s t a l study of the l i n e width v a r i a t i o n with the o r i e n t a t i o n of the magnetic f i e l d . This study has been made and i s presented i n chapter 5. 3.5 CONCLUSIONS We have shown that the microwave power and tempera-ture dependence of the EPR spectrum obtained from tuna ferricytochrome c f i t the theory f o r spin 1/2 Kramers doublet s p l i t by a magnetic f i e l d . At temperatures below 20°K the EPR sig n a l has the form of an u n d i f f e r e n t i a t e d absorption l i n e . This a r i s e s from a combination of dynamic saturation and r a p i d passage e f f e c t s which we discuss more f u l l y i n chapter 6 and 7. A b o v e 20°K, we h a v e made u s e o f t h e o b s e r v a t i o n t h a t t h e p e a k s s e e n a t t h e e x t r e m e s o f t h e d e r i v a t i v e EPR s p e c t r u m o b t a i n e d i n s o l u t i o n a r e g o o d a p p r o x i m a t i o n s t o a n u n d i f f e r e n t i a t e d a b s o r p t i o n l i n e t o o b t a i n t h e l i n e s h a p e a s a f u n c t i o n o f t e m p e r a t u r e . The G a u s s i a n l i n e o f c o n s t a n t w i d t h b e l o w 50°K c a n be e x p l a i n e d a s a r i s i n g f r o m a s m a l l v a r i a t i o n (- 11%) i n t h e r h o m b i c s y m m e t r y o f t h e heme i r o n e n v i r o n m e n t . A t h i g h e r t e m p e r a t u r e s (= 77°K) t h e l i n e s h a p e b e c o m e s L o r e n t z i a n a n d a p p e a r s t o be d e t e r m i n e d b y t h e e l e c t r o n s p i n - l a t t i c e r e l a x a -t i o n t i m e . CHAPTER 4.0 S I N G L E CRYSTAL EPR - ORIENTATION OF G-AXES 4 . 1 INTRODUCTION The a i m o f t h i s c h a p t e r i s t o d e s c r i b e t h e r e s u l t s o f a n EPR s t u d y o f s i n g l e c r y s t a l s o f h o r s e h e a r t f e r r i c y t o -c h r o m e c . I n s e c t i o n s 4.2 a n d 4.3 we g i v e b r i e f d e s c r i p t i o n s o f t h e c y t o c h r o m e c c r y s t a l s , a n d t h e m e a s u r e m e n t t e c h n i q u e . T h e n we p r e s e n t t h e m e t h o d o f a n a l y s i s o f d a t a ( s e c t i o n 4 . 4 ) , f o l l o w e d b y t h e r e s u l t s t h e m s e l v e s ( s e c t i o n 4 . 5 ) . We d i s -c u s s , i n s e c t i o n 4.6, t h e e r r o r s i n g - a x i s o r i e n t a t i o n s o b t a i n e d , a n d c o m p a r e t h e r e s u l t s w i t h t h e t h r e e d i m e n s i o n a l X - r a y s t r u c t u r e a n a l y s i s on f e r r i c y t o c h r o m e c . 4 . 2 CRYSTAL PROPERTIES 4 . 2 . 1 C r y s t a l P r e p a r a t i o n s The h o r s e h e a r t f e r r i c y t o c h r o m e c c r y s t a l s w e r e a g i f t f r o m D r . R. E. D i c k e r s o n a n d D r . E. M a r g o l i a s h , who i n c o l l a b o r a t i o n , c a r r i e d o u t t h e t h r e e d i m e n s i o n a l X - r a y a n a l y s i s o f c y t o c h r o m e c t o 2.8& r e s o l u t i o n ( D i c k e r s o n e_t a l . , 1 9 7 1 ) . 93 The c r y s t a l s w e r e p r e p a r e d f r o m h o r s e h e a r t s u s i n g t h e m e t h o d o f M a r g o l i a s h a n d W a l a s e k ( 1 9 6 7 ) . B r i e f l y , t h e c y t o c h r o m e c was e x t r a c t e d f r o m h o r s e h e a r t s w i t h a l u m i n u m s u l p h a t e , t h e n p u r i f i e d b y c o l u m n c h r o m a -t o g r a p h y , a n d c r y s t a l l i s e d f r o m n e a r l y s a t u r a t e d ammonium s u l p h a t e s o l u t i o n a t pK 6 " t o 7. 4.2.2 C r y s t a l d a t a The c r y s t a l s a r e n e e d l e s o f r e c t a n g u l a r c r o s s s e c t i o n . X - r a y c r y s t a l l o g r a p h i c r e s u l t s ( D i c k e r s o n e t a l . , 1971) h a v e shown t h a t t h e s p a c e g r o u p i s P 4 3 . O n l y f a c e s o f t h e { l O O ] , j l O l ] , a n d { i l l } c l a s s e s a r e w e l l d e v e l o p e d . We h a v e l a b e l l e d t h e m u t u a l l y p e r p e n d i c u l a r a x e s a s shown i n f i g u r e 4.1. 4.3 TECHNIQUE' OF MEASUREMENT " " 4.3.1 C r y s t a l m o u n t i n g The c r y s t a l s w e r e m o u n t e d i n t o 3 mm o . d . , 2 mm i . d . q u a r t z t u b e s a s d e s c r i b e d i n t h e E x p e r i m e n t a l s e c t i o n ( 2 . 4 ) . We f o u n d t h a t t h e m o t h e r l i q u o r p r e s e n t i n t h e t u b e c o n t r i b u t e d a b r o a d b a c k g r o u n d EPR s o l u t i o n s i g n a l w h i c h t e n d e d t o o b s c u r e t h e s i n g l e c r y s t a l l i n e p o s i t i o n s . T h i s d i f f i c u l t y was o v e r c o m e b y s u s p e n d i n g t h e c r y s t a l s i n a n e a r l y s a t u r a t e d ammonium s u l p h a t e s o l u t i o n b e f o r e m o u n t i n g . F o r t h e h o r s e h e a r t c r y s t a l s no c o n c e n t r a t i o n o f ammonium s u l p h a t e was f o u n d i n w h i c h t h e c r y s t a l s d i d n o t e v e n t u a l l y d i s s o l v e . How-e v e r , i n 90% s a t u r a t e d s o l u t i o n t h e d i s s o l u t i o n t o o k a b o u t 2 h o u r s , l o n g e n o u g h t o g e t t h e c r y s t a l s m o u n t e d a n d f r o z e n . FIGURE 4.1 The l a b e l l i n g o f t h e a, b, a n d c a x e s i n a f e r r i c y t o c h r o m e c s i n g l e c r y s t a l . 95 4.3 e 2 EPR M e a s u r e m e n t s The c r y s t a l s w e r e r o t a t e d a b o u t a v e r t i c a l a x i s i n t h e c e n t r e o f t h e c y l i n d r i c a l m i c r o w a v e c a v i t y , a n d EPR s p e c t r a w e r e r u n a t 10° i n t e r v a l s o v e r 360°. The m i c r o w a v e p o w e r i n c i d e n t u p o n t h e c a v i t y was a p p r o x i m a t e l y 10 mW a n d t h e m a g n e t i c f i e l d m o d u l a t i o n was a t a f r e q u e n c y o f 1 0 0 k H z a n d a n a m p l i t u d e o f 1 g a u s s . The s a m p l e s w e r e i m m e r s e d i n l i q u i d h e l i u m a t 4.2°K. A t t h i s t e m p e r a t u r e a n d m o d u l a t i o n f r e q u e n c y t h e c o n d i t i o n s f o r f a s t p a s s a g e a r e f u l f i l l e d ( c h a p t e r 6) a n d t h e s i g n a l s h a d t h e s h a p e o f u n d i f f e r e n t i a t e d " a b s o r p t i o n " l i n e s ( f i g u r e 4 . 2 ) . 4.4 METHOD OF A N A L Y S I S 4.4.1 I n t r o d u c t i o n T h i s s e c t i o n d e a l s w i t h t h e m e t h o d s u s e d t o a n a l y s e t h e d a t a o b t a i n e d f r o m t h e e x p e r i m e n t a l s p e c t r u m , i n o r d e r t o f i n d t h e o r i e n t a t i o n o f t h e p r i n c i p a l g - a x e s r e l a t i v e t o t h e c r y s t a l l o g r a p h i c a x e s . We g i v e , i n s e c t i o n 4.4.2, a d e s c r i p t i o n o f how t h e e x p e r i m e n t a l g - v a l u e s o b t a i n e d f r o m r o t a t i o n o f t h e c r y s t a l a b o u t an a x i s p e r p e n d i c u l a r t o t h e m a g n e t i c f i e l d g i v e t h e d i r e c t i o n s o f t h e g - a x e s r e l a t i v e t o t h e r o t a t i o n a x i s . By c o m p a r i s o n o f t h e r e s u l t s o f two o r more o r i e n t a t i o n s o f t h e c r y s t a l we show t h a t t h e a n g l e s b e t w e e n t h e g - a x e s a n d c r y s t a l a x e s c a n b e o b t a i n e d ( s e c t i o n 4 . 4 . 3 ) . FIGURE 4.2 EPR s p e c t r u m f r o m a s i n g l e c r y s t a l o f h o r s e h e a r t f e r r i c y t o c h r o m e c a t 4.2°K. E x p e r i m e n t a l c o n d i t i o n s : ( i ) M i c r o w a v e p o w e r = 10 mW ( i i ) M o d u l a t i o n f r e q u e n c y = 100 KHz ( i i i ) M o d u l a t i o n a m p l i t u d e = 1 g a u s s cytochrome c lines 1 1 Magnetic Field - kilogauss 03 00 F i n a l l y i n s e c t i o n 4.4.4 t h e s t e r e o g r a p h i c p r o -j e c t i o n t e c h n i q u e u s e d t o d i s p l a y t h e r e s u l t s i s d e s c r i b e d . 4.4.2 T h e o r y o f q - v a l u e v a r i a t i o n The d i r e c t i o n o f p r i n c i p a l g - v a l u e s o f a m o l e c u l e i n a c r y s t a l c a n be o b t a i n e d f r o m m e a s u r e m e n t o f t h e g - v a l u e v a r i a t i o n i n p l a n e s w h i c h a r e d e f i n e d r e l a t i v e t o a known s e t o f a x e s i n t h e c r y s t a l . I f t h e m a g n e t i c f i e l d h a s d i r e c t i o n c o s i n e s j ? ^ , JL^i H ^  r e l a t i v e t o t h e s e a x e s , t h e s q u a r e o f t h e c o r r e s p o n d i n g g - v a l u e i s g i v e n b y ( P r y c e , 1950): 3 9 2 - 2 A i j 1 L / j = A j ± ) (4.1) i;i = i w h e r e t h e d e p e n d u p o n t h e c h o i c e o f r e f e r e n c e a x e s . F r o m e q u a t i o n (4.1) S c h o n l a n d (1959) h a s shown t h a t t h e g -y a l u e v a r i a t i o n i n t h e m a g n e t i c f i e l d p l a n e p e r p e n d i c u l a r t o t h e a x i s o f r o t a t i o n i s : g 2 = A + B c o s 2 ^ + C s i n 2 ^ (4.2) w h e r e ^ i s t h e a n g l e o f r o t a t i o n f r o m a n a r b i t r a r y z e r o a n d A, B a n d C a r e c o n s t a n t s w h i c h a r e f u n c t i o n s o f t h e d i r e c t i o n c o s i n e s o f t h e p l a n e o f m e a s u r e m e n t r e l a t i v e t o t h e r e f e r e n c e a x e s . We f i t o u r d a t a t o e q u a t i o n (4.2) b y a l e a s t s q u a r e s c o m p u t e r r o u t i n e , t h e n , u s i n g t h e v a l u e s o f A, B a n d C s o o b t a i n e d , c o m p u t e t h e g - v a l u e v a r i a t i o n a s a f u n c t i o n 100 of ^ . From t h i s we get the maximum (g +) and minimum (g_) g-values and the angles at which they occur ( + , \j/ ). From g + and g we can obtain the d i r e c t i o n cosines of the p r i n c i p a l g-axes r e l a t i v e to the axis of r o t a t i o n , using the following r e l a t i o n s (Schonland, 1959): g 2 + gf = g x 2 (l-£ 2) + g 2 2 (1-m2) + g 3 2 (1-n 2) (4.3a) 2 2 2 2 n 2 2 2 2 2 2 2 , - ^ . x g + . g_ = g 2 • q3 •^ + ^1 * ^3 * m + ^1 * ^2 * (4.3b) 1 = I 2 + m2 + n 2 (4.3c) where X- i s d i r e c t i o n cosine of g^ r e l a t i v e to the axis of r o t a t i o n m i s d i r e c t i o n cosine of g 2 r e l a t i v e to the axis of r o t a t i o n n i s d i r e c t i o n cosine of g^ r e l a t i v e to the axis of r o t a t i o n . The 3 g-axes are chosen to form a r i g h t handed t r i a d as shown i n f i g u r e 4.3. The d i r e c t i o n s of the g-axes r e l a t i v e to are e a s i l y c a l c u l a t e d . I f there i s more than one set of g-axes the angular r e l a t i o n s h i p between the sets can be found from the comparison of the + » 4.4.3 D i r e c t i o n of q-axes r e l a t i v e to c r y s t a l axes In section 4.4.2 we showed how to obtain the p r i n c i -pal g-axis d i r e c t i o n s r e l a t i v e to the r o t a t i o n axis f o r a p a r t i c u l a r o r i e n t a t i o n . We now wish to show how, by comparing the r e s u l t s from r o t a t i o n of the c r y s t a l about two d i f f e r e n t FIGURE 4.3 C o - o r d i n a t e s y s t e m u s e d t o d e s c r i b e t h e p o s i t i o n s o f p r i n c i p a l g - a x e s r e l a t i v e t o t h e l a b o r a t o r y m a g n e t i c f i e l d d i r e c t i o n . 102 a x e s , t h e g - a x e s d i r e c t i o n s r e l a t i v e t o t h e c r y s t a l a x e s c a n be o b t a i n e d . R o t a t i o n o f t h e c y t o c h r o m e c c r y s t a l a b o u t i t s c - a x i s w i l l g i v e t h e g - v a l u e v a r i a t i o n i n t h e ab p l a n e ( O r i e n t a t i o n I ) . S i m i l a r l y , r o t a t i o n a b o u t t h e b - a x i s w i l l g i v e t h e c o r r e s p o n d i n g v a r i a t i o n i n t h e a c p l a n e ( O r i e n t a -t i o n I I ) . I f t h e m a g n e t i c f i e l d i s d i r e c t e d a l o n g t h e c r y s t a l a o r b a x e s , t h e f o u r heme g r o u p s i n t h e u n i t c e l l o n l y p r o d u c e tv/o l i n e s . T h i s i s b e c a u s e e a c h member o f t h e p a i r o f hemes r e l a t e d b y a 180° r o t a t i o n i s e q u a l l y i n c l i n e d t o t h e m a g n e t i c f i e l d , a n d t h e r e f o r e h a s t h e same g - v a l u e . A s t h e a - a x i s i s common t o O r i e n t a t i o n I a n d O r i e n t a t i o n I I t h e r e w i l l b e a p o i n t on e a c h g - v a l u e v a r i a t i o n c u r v e w h e r e i d e n t i c a l g - v a l u e s o c c u r . Once t h e p o i n t common t o b o t h i s known, t h e a n g l e s t h e g - a x e s make w i t h a n y o t h e r a x i s c a n be d e t e r m i n e d . The a b o v e a n a l y s i s o f t h e d a t a i s made e a s i e r i f some m e t h o d o f d i s p l a y i n g t h e t h r e e d i m e n s i o n a l i n f o r m a t i o n i n t w o d i m e n s i o n s i s u s e d . We g i v e a d e s c r i p t i o n o f s u c h a m e t h o d i n t h e n e x t s e c t i o n . 4.4.4 S t e r e o g r a p h i c p r o j e c t i o n a n d t h e W u l f f n e t The o r i e n t a t i o n o f t h e p r i n c i p a l g - a x e s r e l a t i v e t o t h e m a g n e t i c f i e l d p l a n e a n d / o r t h e c r y s t a l l o g r a p h i c a x e s c a n be b e s t shown b y u s i n g s t e r e o g r a p h i c p r o j e c t i o n s ( B e n n e t t e t a l . , 1 9 5 7 ; P h i l l i p s , 1 9 6 2 ) . The m e t h o d u s e d t o r e p r e s e n t t h r e e d i m e n s i o n a l s p a c e i n two d i m e n s i o n s i s g i v e n i n f i g u r e 4.4. A s p h e r e i s i m a g i n e d t o e n c l o s e t h e c r y s t a l , a n d a n y d i r e c t i o n s o f i m p o r t a n c e a r e e x t r a p o l a t e d t o i n t e r s e c t t h e s u r f a c e o f t h e s p h e r e . T h e s e p o i n t s o n t h e s p h e r i c a l s u r f a c e a r e p r o j e c t e d o n t o t h e d i a m e t r a l p l a n e b y j o i n i n g t h e m t o t h e s o u t h p o l e ( P ) o f t h e s p h e r e , a n d n o t i n g w h e r e t h e s e l i n e s c u t t h e p l a n e . I n f i g u r e 4 , 4 a , f o r a c y t o c h r o m e c c r y s t a l a t t h e c e n t r e o f t h e s p h e r e p l a c e d w i t h i t s c ~ ( 0 0 1 ) a x i s v e r t i c a l , t h e a( 100) a n d b-(010) a x e s i n t e r s e c t t h e s p h e r i c a l s u r f a c e o n t h e e q u a t o r i a l c i r c l e , a s i n d i c a t e d i n 4,4b, A g e n e r a l p o i n t X($ }<£i) i s a l s o shown, w i t h i t s p r o j e c t i o n X'. I n o r d e r t o p l o t o u t r e s u l t s c o n v e n i e n t l y , we u s e a W u l f f n e t . T h i s i s shown i n f i g u r e 4,5, and i s a r e p r e s e n -t a t i o n o f " l i n e s o f l a t i t u d e a n d l o n g i t u d e " on t h e s u r f a c e o f t h e s p h e r e a t 2° i n t e r v a l s , p r o j e c t e d o n t o 'the d i a m e t r a l p l a n e , 4.5 RESULTS 4.5.1 I n t r o d u c t i o n We g i v e t h e r e s u l t s f o r t w o o r i e n t a t i o n s o f t h e h o r s e h e a r t f e r r i c y t o c h r o m e c c r y s t a l s . I n O r i e n t a t i o n I t h e h o r s e h e a r t f e r r i c y t o c h r o m e c c r y s t a l i s r o t a t e d w i t h i t s c - a x i s v e r t i c a l ( f i g u r e 4 . 1 ) . F o r O r i e n t a t i o n I I t h e b - a x i s i s v e r t i c a l . FIGURE 4.4 S t e r e o g r a p h i c p r o j e c t i o n o f c y t o c h r o m e c c r y s t a l ( a ) M e t h o d o f p r o j e c t i o n f r o m s p h e r i c a l s u r f a c e t o t h e d i a m e t r a l p l a n e . ( b ) The r e s u l t i n g s t e r e o g r a m . FIGURE 4 . 5 The W u l f f n e t P r o j e c t i o n s o f ' l i n e s o f l a t i t u d e a n d l o n g i t u d e ' o n t h e s u r f a c e o f a s p h e r e a t 2° i n t e r v a l s , p r o j e c t e d o n t o t h e d i a m e t r a l p l a n e . 108 TABLE V O r i e n t a t i o n I  R e s u l t s o f l e a s t s q u a r e s f i t t o e q u a t i o n (4.2) L i n e 1 g2. = 8.97 ^ + = 101.0 A = 6.38 gf = 3.79 l / / _ = 11-0 B = -2.40 C = 0.98 L i n e 2 gf = 8.99 {p+ = 10.25 A = 6.44 gf = 3.89 v//_ = 100.25 B = 2.39 C = 0.89 4 o 5 o 2 O r i e n t a t i o n I - c - a x i s v e r t i c a l The c y t o c h r o m e c was o r i e n t e d w i t h i t s l o n g c - a x i s n o t more t h a n 2° f r o m t h e v e r t i c a l . The c - a x i s i s a f o u r - f o l d s y m m e t r y a x i s a n d s i n c e t h e p a i r s o f heme p l a n e s i n t h e u n i t c e l l r e l a t e d b y 180° r o t a t i o n a r e e q u a l l y i n c l i n e d t o t h e m a g n e t i c f i e l d o n l y t w o l i n e s a r e s e e n ( o n e f r o m e a c h p a i r ) , i n s t e a d o f t h e f o u r t h e r e w o u l d be i n a n a r b i t r a r y o r i e n t a t i o n . The m a g n e t i c f i e l d o f t h e m a x i m a o f t h e " a b s o r p t i o n " l i n e s w e r e o b t a i n e d a n d , k n o w i n g t h e m i c r o w a v e f r e q u e n c y , t h e g - v a l u e s w e r e c a l c u l a t e d . A s t h e l i n e w i d t h v a r i e d s t r o n g l y w i t h g ( 4 0 0 g a u s s a t g = 3 t o 1000 g a u s s a t g = 2) t h e r e was c o n s i d e r a b l e l i n e o v e r l a p . I n a d d i t i o n , a t 4,2°K t h e q u a r t z t u b e s i g n a l a l s o s h o w e d a f a s t p a s s a g e s i g n a l , r e s u l t i n g i n a b r o a d a b s o r p t i o n l i n e s h a p e c e n t r e d o n g=2„ T h i s a n d t h e b r o a d c y t o c h r o m e c l i n e s , c a u s e d d i f f i c u l t y i n e s t i m a t i n g t h e t r u e p o s i t i o n o f t h e l i n e m a x i m a ( s e e s e c t i o n 4,6 f o r d i s c u s -s i o n o f e r r o r s ) . F o r t h i s o r i e n t a t i o n , h o w e v e r , t h e r e w e r e o n l y t w o l i n e s , s o i t was p o s s i b l e t o f o l l o w t h e m o t i o n o f e a c h l i n e o v e r a r o t a t i o n o f | 1 2 0 ° . The g - v a l u e s a n d r o t a t i o n a n g l e s , a t l e a s t 18 d a t a p o i n t s f o r e a c h l i n e , w e r e f i t t e d b y a t l e a s t s q u a r e s c o m p u t e r p r o g r a m t o t h e t h e o r e t i c a l l y e x p e c t e d d e p e n d e n c e o f g u p o n a n g l e - e q u a t i o n ( 4 , 2 ) , The c o m p u t e d r e s u l t s a r e g i v e n i n t a b l e V; t h e d a t a p o i n t s a n d b e s t f i t t e d c u r v e s a r e p l o t t e d i n f i g u r e 4,6. FIGURE 4.6 g - v a l u e a s a f u n c t i o n o f a n g l e o f f i e l d r o t a t i o n f o r O r i e n t a t i o n I . The d a t a p o i n t s a r e shown, t o g e t h e r w i t h t h e c u r v e s g i v i n g t h e b e s t l e a s t s q u a r e s f i t t o t h e s e d a t a p o i n t s u s i n g e q u a t i o n ( 4 . 2 ) . Orientation I c - a x i s ver t ica l \jt - Angle of Rotation in Degrees (Arbitrary zero) TABLE V I O r i e n t a t i o n I  D i r e c t i o n c o s i n e s a n d a n g l e s o f p r i n c i p a l q - a x e s r e l a t i v e t o r o t a t i o n a x i s L i n e 1 g - a x i s D i r e c t i o n C o s i n e A n g l e * -g x ( l ) = 1.25 JL1 = - 0.777 Ct = £ 38.9 g 2 ( l ) = 2.25 m x = ~ 0.575 ft = ± 54.9 g 3 ( l ) = 3 . 0 6 n l = " ° « 2 5 6 / j = - 75.2 L i n e 2 g - a x i s D i r e c t i o n C o s i n e A n g l e g x ( 2 ) = 1.25 i 2 = - 0.796 a 2 = ± 37.2 g 2 ( 2 ) = 2.25 m 2 = i 0.553 ^ 2 » ± 56.4 g 3 ( 2 ) = 3.06 n 2 = - 0.247 y 2 = ± 75.7 £ h e r e s y m b o l i s e t h a t t h e a n g l e t h e g - a x i s makes w i t h t h e r o t a t i o n a x i s c a n be eg. 0° - 38.9 o r 180° - 3 8 . 9 . 114 I n t a b l e V I we p r e s e n t t h e d i r e c t i o n c o s i n e s a n d a n g l e s o f t h e 3 p r i n c i p a l g - a x e s o f e a c h s e t , r e l a t i v e t o t h e a x i s o f r o t a t i o n ( t h e n o r m a l t o t h e m a g n e t i c f i e l d p l a n e ) . We now w i s h t o d e t e r m i n e t h e a n g u l a r p o s i t i o n s o f t h e t w o s e t s o f a x e s i n r e l a t i o n t o e a c h o t h e r . The g - v a l u e maximum w i l l c o i n c i d e w i t h t h e l a r g e s t p r i n c i p l e g - v a l u e d i r e c t i o n ( g ^ ) o n l y when t h i s a x i s h a p p e n s t o l i e i n t h e p l a n e o f r o t a t i o n . F o r a n y o t h e r ( k n o w n ) o r i e n t a t i o n o f t h e g - a x e s we c a n c a l c u l a t e t h e a n g l e s b e t w e e n t h e maximum g - v a l u e a n d t h e p r o j e c t i o n o f t h e g - a x e s i n t h e p l a n e o f r o t a t i o n . F o r o r i e n t a t i o n I , t h e d i f f e r e n c e s b e t w e e n \j/ a n d t h e g ^ - a x i s p r o j e c t i o n a r e s m a l l ( =2°); t h e p r o j e c t i o n s a r e s hown i n f i g u r e 4.7. The p r o j e c t i o n s o f t h e t w o g^ a x e s a r e a l m o s t 90° a p a r t : ^ ( 1 ) = 98.8° 9 d i f f e r e n c e = 90.7° Y>. g 3<2> - 8.1° a n d show t h e 4 f o l d s y m m e t r y p r e d i c t e d b y t h e X - r a y a n d o p t i c a l m e a s u r e m e n t s ( s e c t i o n - 4 . 2 . 2 ) . B e c a u s e o f t h e f o u r - f o l d s y m m e t r y a b o u t t h e c - a x i s we do n o t know t h e d i r e c t i o n s o f t h e a a n d b c r y s t a l a x e s r e l a t i v e t o t h e g - a x e s . H o w e v e r we do know t h a t w h e r e v e r , s a y , t h e a - a x i s i s when t h e m a g n e t i c f i e l d i s a l o n g t h i s a x i s 2 a t m o s t t w o l i n e s w i l l be s e e n , f o r w h i c h g l i e s b e t w e e n 8.98 a n d 3.79 ( t a b l e V ) . I f now we mount a c r y s t a l w i t h i t s FIGURE 4.7 S t e r e o g r a m o f O r i e n t a t i o n I N o t a t i o n : 1 ( 1 ) a r e t h e p r o j e c t i o n s o f t h e p r i n c i p a l g - v a l u e s o f l i n e 1 ( I « 1, 2, 3 ) . 1 ( 2 ) a r e t h e p r o j e c t i o n s o f t h e p r i n c i p a l g - v a l u e s o f l i n e 2 ( I = 1, 2, 3 ) . l / / + ( K ) i s t h e r o t a t i o n a n g l e a t w h i c h l i n e K (K = 1, 2) h a s i t s maximum g - v a l u e . */^(K) i s t h e r o t a t i o n a n g l e a t w h i c h p r i n c i p a l g - v a l u e g 3 l i e s f o r l i n e K (K = 1, 2 ) . 116 TABLE V I I O r i e n t a t i o n I I  R e s u l t s o f l e a s t s q u a r e s f i t t o e q u a t i o n ( 4 . 2 ) L i n e 1 .2 L i n e 3 2 2 g_ L i n e 4 .2 5.40 ^ = 45* A = 3.62 g g 2 = 1.85 \p _ = 135° B'= 0.06 C = 1.77 5.29 uV = 110° A = 3.65 + 2.02 v//_ = 20° B = - 1 . 2 1 C = - 1 . 1 0 8.18 ^ = 75° A •_- 5.98 + 3.79 ^ _ = 165° B = - 1 . 9 6 C = 0.99 g ^ = 8.15 v//+ = 80° A = 5.63 g f = 3.11 ^ = 170° B = -2.42 C = 0.72 I T ABLE V I I I O r i e n t a t i o n I I  D i r e c t i o n c o s i n e s a n d a n g l e s o f p r i n c i p a l q - a x e s  r e l a t i v e t o r o t a t i o n a x i s L i n e 1 D i r e c t i o n C o s i n e A n g l e * - 0 g 1 ( l ) = 1.25 i x = i 0.201 a 1 = i 78.4 g 2 ( l ) = 2.25 m1 = - 0.268 /3 = ~ 74.4 g 3 ( l ) = 3.06 n x = i 0.942 Y = i 1 9 . 6 L i n e 2 g - a x i s g x ( 2 ) = 1.25 l 2 = i 0.249 Q , = i 75.5 g 2 ( 2 ) = 2.25 m 2 = i 0.214 ft 2 = t 7 7 . 6 g 3 ( 2 ) = 3.06 n 2 = ± 0.944 / 2 = ± 1 9 . 2 3 L i n e 3 g x ( 3) £ 3 = i 0.735 a 3 = ± 48.7 g 2 ( 3 ) m 3 = i 0.513 / 3 3 = ± 59.1 g 3 ( 3 ) n 3 = - 0.443 Y $ = ~ 63.7 £ h e r e s y m b o l i s e t h a t t h e a n g l e t h e g - a x i s makes w i t h t h e r o t a t i o n a x i s c a n be e . g . 0° ~ 78.4 o r 180° - 7 8 . 4 . T a b l e V I I I c o n t i n u e d g - a x i s D i r e c t i o n C o s i n e A n g l e * - 0 g x(4) / 4 = i 0.611 a 4 = i 52.3 g 2(4) m4 = ± 0.632 - £ 4 = - 50.7 g 3(4) n 4 = ± 0.476 / 4 = ± 61.6 120 b - a x i s v e r t i c a l , t h e n t h e h o r i z o n t a l m a g n e t i c f i e l d p l a n e i n c l u d e s b o t h t h e a a n d c a x e s . I n a g e n e r a l d i r e c t i o n f o u r l i n e s w i l l b e s e e n b u t o n l y t w o l i n e s w i l l be o b s e r v e d when 2 t h e f i e l d i s a l o n g t h e a - a x i s . The p a i r o f l i n e s whose g v a l u e s l i e b e t w e e n t h e l i m i t s m e n t i o n e d ( 8 . 9 8 , 3.79) w i l l c o r r e s p o n d t o t h e a a x i s d i r e c t i o n . 4.5.3 O r i e n t a t i o n I I - b - a x i s v e r t i c a l T h e c y t o c h r o m e c c r y s t a l was o r i e n t e d w i t h i t s b r o a d f a c e a s c l o s e t o h o r i z o n t a l a s p o s s i b l e , a n d t h e n o r m a l t o t h i s f a c e , t a k e n a s t h e b - a x i s , was m e a s u r e d t o be 5° i 1° f r o m t h e v e r t i c a l . A s b e f o r e , s p e c t r a was t a k e n o v e r a s many p o i n t s a s p o s s i b l e i n a 360° r a n g e . A t f i r s t s i g h t o n l y t h r e e l i n e s w e r e s e e n ; h o w e v e r , c l o s e r e x a m i n a t i o n s h o w ed t h a t t h e r e w e r e i n f a c t f o u r l i n e s : o n e p a i r c l e a r l y s e p a r a t e a n d a n o t h e r p a i r l a r g e l y o v e r l a p p i n g . T h e p o s i t i o n o f t h e r e s o n a n c e maxima w e r e m e a s u r e d a n d t h e r e s u l t s o f t h e l i n e a s s i g n m e n t s t e s t e d b y t h e l e a s t s q u a r e s f i t t i n g p r o g r a m . The b e s t f i t s t o t h e d a t a g a v e t h e c o m p u t e d c u r v e s shown i n f i g u r e 4.8. The c o m p u t e d g-maxima, g - m i n i m a , a n d t h e p a r a m e t e r s d e f i n i n g t h e t h e o r e t i c a l c u r v e s a r e g i v e n i n t a b l e V I I , a n d t a b l e V I I I p r e s e n t s t h e a p p r o p r i a t e d i r e c t i o n c o s i n e s a n d a n g l e s o f t h e p r i n c i p a l g - a x e s f o r t h e 4 l i n e s . A f t e r c o m p u t i n g t h e a n g u l a r p o s i t i o n s o f t h e g - a x e s r e l a t i v e t o t h e g-maxima, we p l o t t h e p r o j e c t i o n s i n f i g u r e '4.9. FIGURE 4.8 g - v a l u e a s a f u n c t i o n o f a n g l e o f r o t a t i o n f o r O r i e n t a t i o n I I . The d a t a p o i n t s a r e shown, t o g e t h e r w i t h t h e c u r v e s g i v i n g t h e b e s t l e a s t s q u a r e s f i t t o e q u a t i o n ( 4 . 2 ) . Orientation I b-axis vertical — I 1 : 1 1 ' ' 270 300 330 360 30 60 90 120 ^ - Angle of Rototion in Degrees (Arbitrary zero) FIGURE 4.9 S t e r e o g r a m o f O r i e n t a t i o n I I N o t a t i o n : 1 ( 1 ) a r e t h e p r o j e c t i o n s o f t h e p r i n c i p a l g - v a l u e s o f l i n e 1, ( I = 1, 2, 3 ) . 1 ( 2 ) a r e t h e p r o j e c t i o n s o f t h e p r i n c i p a l g - v a l u e s o f l i n e 2, ( I = 1, 2, 3 ) . 1 ( 3 ) a r e t h e p r o j e c t i o n s o f t h e p r i n c i p a l g - v a l u e s o f l i n e 3, ( I = 1, 2, 3 ) . 1 ( 4 ) a r e t h e p r o j e c t i o n s o f t h e p r i n c i p a l g - v a l u e s o f l i n e 4, ( I = 1, 2, 3 ) . ^ i s t h e a n g l e o f r o t a t i o n 124 (b) 4.5.4 C r y s t a l l o q r a p h i c axes We now d e t e r m i n e t h e d i r e c t i o n s o f t h e c r y s t a l l o -g r a p h i c a x i s r e l a t i v e t o t h e p r i n c i p a l g -axes. As m e n t i o n e d above, w i t h t h e b - a x i s v e r t i c a l , t h e a - a x i s w i l l l i e a l o n g one o f t h e d i r e c t i o n s where f o u r l i n e s become two. From f i g u r e 4.8 t h i s o c c u r s a t t h e e x p e r i m e n t a l r o t a t i o n a n g l e s o f : 320° 348° 77° 2 g l i m i t s f r o m O r i e n t a t i o n I 2 g 4.6 3.25 8.15 8.98 9 2 1.85 3.0 4.3 3.79 The a - a x i s t h e r e f o r e l i e s a t ^ = 77° and we add t h i s o n t o t h e p r o j e c t i o n d i a g r a m o f f i g u r e 4.9. The c - a x i s i s a l s o shown, b e i n g 90° f r o m t h e a - a x i s . Now t h a t we have t h e g - v a i u e s when t h e m a g n e t i c f i e l d i s d i r e c t e d a l o n g t h e a - a x i s i n o r i e n t a t i o n I I , we can f i n d t h e c o r r e s p o n d i n g p o s i t i o n i n o r i e n t a t i o n I . I n f i g u r e 2 4.6, g v a l u e s o f 8.15 and 4.30 l i e i n t h e a n g u l a r r a n g e s 24°-34°, 77°-83°, and a t m u l t i p l e s o f 90° f r o m t h e s e . Because o f t h e f o u r f o l d symmetry o f t h i s o r i e n t a t i o n we do n o t know w h i c h o f t h e s e c o r r e s p o n d s t o t h e a - a x i s . T h i s a m b i g u i t y c a n n o t be r e s o l v e d by EPR and we must l o o k t o the t h r e e dimen-s i o n a l s t r u c t u r e d e t e r m i n e d by X - r a y d i f f r a c t i o n ( D i c k e r s o n e t al.> 1971) t o d e c i d e w h i c h c h o i c e i s the c o r r e c t one, and w h i c h i s t h e m i r r o r image. T h i s w i l l be d i s c u s s e d i n t h e n e x t s e c t i o n . 4.6 DISCUSSION OF ORIENTATION RESULTS 4.6.1 We d i s c u s s i n t h i s p o s s i b l e e r r o r s ( s e c t i o n 4.6.2) a n d t h e n c o m p a r e o u r r e s u l t s f o r t h e o r i e n t a t i o n o f t h e g -a x e s i n t h e c r y s t a l i n t h e l i g h t o f t h e 3 - d i m e n s i o n a l X - r a y r e s u l t s f o r t h e same c r y s t a l ( s e c t i o n 4 . 6 . 3 ) . 4.6.2 E x p e r i m e n t a l e r r o r s • -We i n t r o d u c e o u r d i s c u s s i o n o f t h e e x p e r i m e n t a l e r r o r s i n t h e m e a s u r e m e n t o f t h e o r i e n t a t i o n s o f t h e g - a x e s b y a s k i n g t h e f o l l o w i n g q u e s t i o n : C a n we t r a n s f o r m one o f t h e p r o j e c t i o n d i a g r a m s — e i t h e r f i g u r e 4,6 o r 4 . 9 — i n t o t h e o t h e r b y a 90° r o t a t i o n a b o u t t h e a - a x i s ? A n y d i s c r e p a n c i e s b e t w e e n t h e r o t a t e d v e r s i o n a n d t h e e x p e r i m e n t a l one c a n t h e n be e x a m i n e d t o s e e i f t h e y f a l l w i t h i n t h e e x p e r i m e n t a l e r r o r . B e c a u s e o f t h e f o u r f o l d s y m m e t r y o f o r i e n t a t i o n I , we c o u l d c h o o s e a n y one o f f o u r d i r e c t i o n s f o r t h e a - a x i s ( 2 7 - 3 5 , 7 7 - 8 3 , 1 1 7 - 1 2 5 ; 1 6 7 - 1 7 3 ) a n d r o t a t e b y 90° a b o u t t h i s a x i s . F i g u r e 4.10 i s t h e c o m p l e t e s t e r e o g r a p h i c p r o j e c t i o n f o r o r i e n t a t i o n I , s h o w i n g t h e f o u r s e t s o f g - a x e s . The b e s t f i t o f f i g u r e 4.9 t o f i g u r e 4.10 was g i v e n b y r o t a t i n g t h e m o l e c u l e w i t h c - a x i s v e r t i c a l 90° c o u n t e r c l o c k w i s e a b o u t t h e 117° d i r e c t i o n , a s shown i n f i g u r e 4 . 1 1 . The p o i n t s w i t h o u t l e t t e r s a r e t h o s e m e a s u r e d i n o r i e n t a t i o n I I . The d i s c r e p a n c i e s c a n be s u m m a r i z e d b y s a y i n g t h e g ^ a x e s p r o j e c t i o n s d i f f e r b y 4°-10° a n d t h e g^, g^ a x i s FIGURE 4 e 1 0 F u l l s t e r e o g r a m o f O r i e n t a t i o n I N o t a t i o n : A l ( I ss 1, 2, 3) a r e t h e p r o j e c t i o n s o f t h e 3 p r i n c i p a l g - a x e s o f t h e m o l e c u l e t h a t g i v e s l i n e 2 o f f i g u r e 4.7. B I ( I = 1, 2, 3) a r e t h e p r o j e c t i o n s o f t h e 3 p r i n c i p a l g - a x e s o f t h e m o l e c u l e t h a t g i v e s l i n e 1 o f f i g u r e 4.7. The C I a n d DI a r e t h e p r o j e c t i o n s o f t h e p r i n c i p a l g - a x e s o f t h e m o l e c u l e s r e l a t e d b y a 180° r o t a t i o n a b o u t t h e c - a x i s t o A l a n d B I r e s p e c t i v e l y . b-axis + 180" Orientation I FIGURE 4.11 C o m p a r i s o n o f O r i e n t a t i o n s I a n d I I A 90° c o u n t e r - c l o c k w i s e r o t a t i o n o f t h e s t e r e o -g r a p h i c p r o j e c t i o n o f O r i e n t a t i o n I a b o u t t h e a - a x i s ( f i g u r e 4.10) moves t h e d i r e c t i o n s o f t h e g - a x e s d e t e r m i n e d i n O r i e n t a t i o n I f o r c o m p a r i s o n w i t h t h o s e f r o m O r i e n t a t i o n I I . T h e a l p h a n u m e r i c s y m b o l s ( A l , e t c . ) c o r r e s p o n d t o t h e r o t a t e d a x e s o f O r i e n t a t i o n I . T h e o t h e r a x e s a r e t h o s e o f O r i e n t a t i o n I I ( f i g u r e 4 . 9 ) . p r o j e c t i o n s d i f f e r b y 5°-20°. The d i s a g r e e m e n t i s g r e a t e s t i n l i n e s 1 a n d 2 o f f i g u r e 4 . 1 1 a . To d e c i d e w h e t h e r t h i s d i s -a g r e e m e n t i s s i g n i f i c a n t we m u s t d i s c u s s t h e e x p e r i m e n t a l e r r o r s i n v o l v e d i n o b t a i n i n g t h e r e s u l t s . The s o u r c e s o f e r r o r s a r e : ( i ) m e a s u r e m e n t o f t h e p r i n c i p a l g - v a l u e s ( g ^ , g 2 and g^) ( i i ) r e a d i n g t h e m a g n e t i c f i e l d v a l u e a t t h e l i n e c e n t r e ( i i i ) s e t t i n g t h e a n g l e o f r o t a t i o n ( i v ) o r i e n t a t i o n o f t h e c r y s t a l i n t h e s a m p l e t u b e . We now d i s c u s s t h e s e i n t u r n : ( i ) The p r i n c i p a l g - v a l u e s w e r e o b t a i n e d f r o m m e a s u r e m e n t s on t h e m o t h e r l i q u o r o f t h e h o r s e h e a r t c r y s t a l s O n l y two v a l u e s c o u l d be o b t a i n e d , g 2 = 2.25 a n d g ^ = 3.06, a s t h e s o l u t i o n was s o d i l u t e (ImM) t h a t t h e h i g h f i e l d l i n e was n o t v i s i b l e . We t o o k g ^ t o b e 1.25, c o n s i s t e n t w i t h t h e r e s u l t s o f S a l m e e n a n d P a l m e r on b e e f h e a r t c y t o c h r o m e c . T h e e f f e c t o f u n c e r t a i n t i e s i n t h e p r i n c i p a l g -v a l u e s i s r e f l e c t e d i n t h e c a l c u l a t e d a n g l e s o f t h e g - a x e s r e l a t i v e t o t h e a x i s o f r o t a t i o n . When t h e c - a x i s i s v e r t i c a l , f o r e x a m p l e , a n d = 1.25, g 2 = 2.25 a n d g 3 i s a s shown b e l o w ( u s i n g t h e r o t a t i o n o f f i g u r e 4 . 3 ) : 9 3 . a P r 3.03 54.0° 38.1° 79.0° 3.06 54.9 38.9 75.2" 3.09 55.7 39.7 72.3 3.15 57.1 41.3 68.5 V a r i a t i o n s i n g 1 and g 2 have l e s s e r e f f e c t ; with g 2 = 2.24, and g~ = 3.06 we have f o r v a r i a t i o n s i n g.: «1 a r 1.20 38.3° 55.6° 75.3° 1.25 38.9 54.9 75.2 1.30 39.7 54.0 75.0 with s i m i l a r r e s u l t s f o r g 2 varying from 2.2 to 2.3. When the b-axis i s v e r t i c a l , the errors are of the same magnitude. For g^, we f i n d a value of 3.06 £ 0.02 and t h i s , with the other p r i n c i p a l values of g 2 = 2.25 - 0.025 and g^ = 1.25 - .05 gives a possible e r r o r i n angles with respect to the r o t a t i o n axis of - 1° f o r cLfj3 (angles of g^, g 2 axes) £ 2° f o r y (angle of g^ axes) from t h i s cause. TABLE I X Errors i n l e a s t squares f i t to equation (4.2) Orientation I (c-axis v e r t i c a l ) Error sum of squares Residual e r r o r i n g 2 Number of data points used Line 1 Line 2 0 . 1 8 0 . 3 4 0 . 0 9 0 . 1 4 2 2 2 0 Orient a t i o n I I (b-axis v e r t i c a l ) Line 1 Line 2 Line 3 Line 4 0 . 0 5 0 . 0 7 0 . 3 3 0 . 2 3 0 . 0 5 0 . 0 7 0 . 1 4 0 . 1 8 2 0 1 7 2 0 1 1 ( i i ) The d e t e r m i n a t i o n o f t h e p o s i t i o n o f t h e l i n e c e n t r e was d i f f i c u l t b e c a u s e o f t h e v e r y w i d e l i n e s . The u n c e r t a i n t y i s 10-20 g a u s s f o r t h e n a r r o w e s t l i n e s t o 100 t o 250 f o r t h e w i d e s t l i n e s . T h i s i s r e f l e c t e d i n t h e g o o d n e s s o f f i t o f t h e c o m p u t e d c u r v e s , a s shown i n t a b l e I X . 2 F o r t h e c - a x i s v e r t i c a l , t h e f i t t i n g e r r o r s i n g w o u l d l e a d t o a n e r r o r o f - 2-3° i n t h e a n g l e b e t w e e n g ^ a n d t h e v e r t i c a l . H o w e v e r , a s t h e h i g h e s t g - v a l u e s m e a s u r e d w e r e a l s o t h e m o s t p r e c i s e , m o s t o f t h i s p o s s i b l e e r r o r w i l l l i e i n t h e m inimum g - v a l u e , w h e r e i t p r o d u c e s an u n c e r t a i n t y i n t h e g ^ a n d d i r e c t i o n s o f - 5°. C o m p a r i s o n o f t h e a n g l e s f r o m t h e two s e t s o f r e s u l t s ( t a b l e V I ) shows t h a t t h e y a g r e e w e l l w i t h i n t h e s e l i m i t s . W i t h t h e b - a x i s v e r t i c a l , t h e r e s i d u a l e r r o r i n l i n e s 1 a n d 2 i s l e s s t h a n a n y o t h e r b e c a u s e we c o u l d m e a s u r e g - v a l u e s o v e r a w i d e r a n g e o f a n g l e s . F o r l i n e s 3 a n d 4, w h i c h w e r e l a r g e l y o v e r l a p p i n g , t h e r e s i d u a l e r r o r i s h i g h e r . T h i s s o u r c e o f e r r o r g i v e s : f o r o r i e n t a t i o n I - 5° f o r a,(3 • i n l i n e 1 a n d 2 i 1° f o r y o r i e n t a t i o n I I £ 1° f o r a,/3,y i n l i n e 1 a n d 2 i 3° f o r a,/3,/ i n l i n e 3 a n d 4 ( i i i ) T he r o t a t i o n a n g l e c o u l d be s e t t o w i t h i n 0.5°; r o t a t i o n s o f t h e s a m p l e w e r e made i n one d i r e c t i o n t o a v o i d b a c k l a s h . We a l s o r o t a t e d t h e m a g n e t , i n o r d e r t o c h a n g t h e a n g l e b e t w e e n t h e c r y s t a l a n d t h e d . c . f i e l d . No d i f -f e r e n c e i n r e s u l t s was o b t a i n e d w i t h t h e t w o r o t a t i o n m e t h o d s . A c o m b i n a t i o n o f s a m p l e r o t a t i o n a n d m a g n e t r o t a t i o n was q u i t e u s e f u l , w i t h t h e m a g n e t r o t a t i o n b e i n g u s e d a s a v e r n i e r f o r s m a l l a n g l e s . T h i s e r r o r i s s o s m a l l t h a t i t c a n be n e g l e c t e d i n c o m p a r i s o n w i t h ( i ) a n d ( i i ) . T h e sum o f t h e e r r o r s f r o m t h e s e t h r e e i n d e p e n d e n t c a u s e s g a v e t h e e r r o r i n t h e g ^ a n d g^ a x i s a s - 4-5° f o r b o t h o r i e n t a t i o n s , w i t h t h e g ^ a x i s h a v i n g a p o s s i b l e e r r o r o f 2-4°. E v e n a l l o w i n g f o r t h e s e e r r o r s , t h e r e i s s t i l l a d i s c r e p a n c y b e t w e e n t h e r o t a t e d c - a x i s r e s u l t s a n d t h e e x p e r i m e n t a l b - a x i s v e r t i c a l r e s u l t s . ( i v ) We m u s t t h e r e f o r e p o s t u l a t e t h a t o u r m e a s u r e -m e n t s o f t h e o r i e n t a t i o n o f t h e c r y s t a l i n t h e s a m p l e t u b e w e r e i n e r r o r . F o r t h e c - a x i s v e r t i c a l c a s e o u r e s t i m a t e d e r r o r was n o t more t h a n 2° i n t h e d i r e c t i o n o f t h e l o n g a x i s o f t h e c r y s t a l f r o m t h e v e r t i c a l . The e x p e r i m e n t a l g - v a l u e s a n d p a r t i c u l a r l y t h e d i r e c t i o n s o f t h e g ^ - a x e s show t h a t i n d e e d t h e c r y s t a l was a l i g n e d v e r t i c a l l y w i t h i n 0.5°. O t h e r w i s e t h e g ^ - a x e s a n g l e s w o u l d n o t h a v e b e e n s o n e a r l y e q u a l ( i . e . 7 5 . 2 ° , 7 5 . 7 ° ) . F o r t h e b - a x i s v e r t i c a l c a s e , t h e s e t t i n g u p e r r o r s w e r e g r e a t e r , t h e b - a x i s b e i n g e s t i m a t e d a s 5° o f f t h e v e r t i -c a l , w h i l e t h e c - a x i s was w i t h i n 2° o f h o r i z o n t a l . A s f i g u r e 4.11 s h o w s , t h e g r e a t e s t d i s c r e p a n c y b e t w e e n t h e r o t a t e d p r o j e c t i o n a n d t h e o b s e r v e d one was f o r l i n e s 1 a n d 2. We a t t e m p t e d t o r e m o v e t h e d i f f e r e n c e s f o r l i n e 1 b y a s m a l l r o t a t i o n o f t h e p r o j e c t i o n o f t h e o r i e n t a -t i o n I I a n g l e s . I t was f o u n d t h a t a n e x t r a r o t a t i o n o f 10° c o u n t e r c l o c k w i s e a b o u t an a x i s a t ^ - 99° d i d make t h e p r o j e c t i o n s o f l i n e 1 o v e r l a p w i t h i n t h e e x p e r i m e n t a l e r r o r a n d i m p r o v e d t h e p r o j e c t i o n s o f l i n e s 2, 3, a n d 4. . • - We t h e r e f o r e assume t h a t b e f o r e f r e e z i n g t h e c r y s t a l moved i n t h e t u b e b y a f e w d e g r e e s , t i p p i n g t h e b - a x i s a b o u t 10° f r o m t h e v e r t i c a l . t o w a r d s t h e b e p l a n e . We h a v e g i v e n t h i s f a i r l y l e n g t h y d i s c u s s i o n o f t h e p o s s i b l e e r r o r s t o s a t i s f y o u r s e l v e s t h a t we h a v e o b t a i n e d t h e o r i e n t a t i o n o f t h e g - a x e s as p r e c i s e l y a s p o s s i b l e . We s h o u l d p o i n t o u t t h a t i n f a c t , t h e c o m p l e t e o r i e n t a t i o n p i c t u r e c o u l d h a v e b e e n o b t a i n e d f r o m O r i e n t a t i o n I o n l y — t h a n k s t o t h e f o u r f o l d s y m m e t r y a b o u t t h e c - a x i s . I t i s e n c o u r a g i n g t h a t t h e r e s u l t s o f O r i e n t a t i o n I I , d e s p i t e t h e d i f f i c u l t i e s o f a n a l y s i s a n d m o u n t i n g , a r e c o n s i s t e n t w i t h t h o s e o f O r i e n t a t i o n I . TABLE X P r i n c i p a l q - a x i s d i r e c t i o n s r e l a t i v e t o t h e c r y s t a l a x e s ( a ) O u r c o - o r d i n a t e s y s t e m - a b c * - - i s V r a x i s a l a b c . 9 1 126 91 37 43 113 56 g 3 72 24 76 ( b ) D i c k e r s o n e t a l . (1971) c o - o r d i n a t e s y s t e m g - a x i s \ c r ^ f ^ a l * \ a x i s X Y z 68 118 143 96 33 122 24 72 76 ( c ) Heme p l a n e d i r e c t i o n s r e l a t i v e t o c r y s t a l a x e s * A x i s X y z N 2 - F e - N 4 86 46 135 N l - F e - N 3 68 133 129 S - F e - N 1 8 22 75 71.5 T h e s e d i r e c t i o n s a r e e s t i m a t e d f r o m t h e s t e r e o p i c t u r e s p r e s e n t e d i n D i c k e r s o n e t a l _ . ( 1 9 7 1 ) . 138 T o sum u p , t h e n , we h a v e d e t e r m i n e d t h e o r i e n t a -t i o n s o f t h e g ^ a n d g 2 a x e s t o w i t h i n 5° t o 10° a n d o f t h e g 3 - a x i s t o w i t h i n 2° t o 4 ° . I n t a b l e X a we p r e s e n t t h e a n g l e s b e t w e e n t h e c r y s t a l l o g r a p h i c a x e s a n d t h e g - a x e s e x p l i c i t l y . We c h o s e t h e s e t o f g - a x e s w h i c h , i n o u r s y s t e m o f n o t a t i o n , p u t t h e g ^ - a x e s I n t h e p o s i t i v e o c t a n t o f t h e a b c c o - o r d i n a t e s y s t e m o f f i g u r e 4 . 1 . I n c h a p t e r 5, i t w i l l be shown t h a t a s t u d y o f t h e EPR l i n e w i d t h a s a f u n c t i o n o f c r y s t a l r o t a t i o n a l s o s e r v e s a s a c h e c k o n t h e r e s u l t s p r e s e n t e d h e r e . 4.6.3 C o m p a r i s o n o f o u r r e s u l t s w i t h o p t i c a l a n d X - r a y d a t a B e f o r e t h e p u b l i c a t i o n o f t h e r e c e n t p a p e r b y D i c k e r s o n e t a l . ( 1 9 7 1 ) , t h e o n l y o t h e r d a t a a v a i l a b l e o n t h e o r i e n t a t i o n o f t h e heme r e l a t i v e t o c r y s t a l a x e s was t h a t f r o m t h e e l e c t r o n i c s p e c t r a o f s i n g l e c r y s t a l s o f c y t o c h r o m e c i n p o l a r i s e d l i g h t ( K a b a t , 1 9 6 7 ; E a t o n a n d H o c h s t r a s s e r , 1 9 6 7 ) . T h e s e w o r k e r s c o u l d o n l y o b t a i n t h e d i r e c t i o n o f t h e heme n o r m a l r e l a t i v e t o t h e c r y s t a l c - a x i s . T h e i r d a t a , t h e X - r a y d e t e r m i n e d o r i e n t a t i o n , a n d o u r r e s u l t s a r e : A n g l e o f heme n o r m a l t o c - a x i s E a t o n a n d H o c h s t r a s s e r 72 - 3° K a b a t 67.5 ± 2 . 5 ° T h i s w o r k ( g 3 ) 75 i 4° D i c k e r s o n e t a l . 71.5° We c a n t h e r e f o r e t a k e o u r g ^ - a x i s a s b e i n g d i r e c t e d c l o s e l y a l o n g t h e heme n o r m a l , w i t h t h e o t h e r t w o a x e s l y i n g i n t h e heme p l a n e e We now c o m p a r e t h e d i r e c t i o n o f t h e g^ a n d g ^ a x e s w i t h t h o s e o f t h e p y r r o l e n i t r o g e n - i r o n - p y r r o l e n i t r o g e n ( N - F e - N ) a x e s i n t h e p o r p h y r i n r i n g * F i r s t l y we c o n v e r t o u r a b c c o - o r d i n a t e s y s t e m i n t o t h e x y z s y s t e m u s e d b y D i c k e r s o n e t a l . ( 1 9 7 1 ) t o d e s c r i b e t h e i r X - r a y s t r u c t u r e r e s u l t s . The r e l a t i o n b e t w e e n t h e a x e s a r e : +a = +y; +b = +x; +c = - z . T a b l e X b shows t h e a n g l e s o f t h e EPR g - a x e s i n t h e x y z s y s t e m , a n d t a b l e X c g i v e s t h e d i r e c t i o n s o f v a r i o u s a x e s i n t h e heme r i n g a s e s t i m a t e d f r o m t h e p u b l i s h e d s t e r e o g r a p h i c X - r a y p i c t u r e s . The g 1 a n d g 2 a x e s l i e c l o s e t o t h e N-Fe-N a x e s d i r e c t i o n s a s i n d i c a t e d i n f i g u r e 4.12, a n d a r e c o p l a n a r w i t h t h e heme r i n g . We l a b e l l e d t h e n i t r o g e n a t o m s c o - o r d i n a t e d t o t h e i r o n a s shown; t h e p y r r o l e r i n g c o n t a i n i n g i s t h a t r i n g w h i c h i s b o u n d , v i a p h e n y l a l a n i n e - 4 6 , t o t h e a m i n o a c i d b a c k b o n e . 4.6.4, C o m p a r i s o n o f o u r r e s u l t s w i t h m y o g l o b i n a z i d e "~ . a n d c y a n i d e c o m p l e x e s The o n l y o t h e r l o w s p i n p r o t e i n c o m p l e x e s f o r w h i c h EPR a n d X - r a y s t r u c t u r e d a t a a r e a v a i l a b l e a r e met m y o g l o b i n FIGURE 4.12. P r o j e c t i o n o f g - a x e s o n t o heme p l a n e 141 a z i d e * ( H e l c k e e t a l , , 1 9 6 8 ) a n d m y o g l o b i n c y a n i d e (W, E. B l u m b e r g , p e r s o n a l c o m m u n i c a t i o n ) . B o t h o f t h e s e c ompounds h a v e t h e i r l a r g e s t g - v a l u e a x i s a l i g n e d a p p r o x i m a t e l y a l o n g t h e heme n o r m a l . I n t h e a z i d e t h e d e v i a t i o n i s 9 ° , a n d i n t h e c y a n i d e i t i s 1 3 ° . H e l c k e e t a l . ( 1 9 6 8 ) s p e c u l a t e t h a t t h e f i r s t n i t r o -g e n i n t h e a z i d e m o l e c u l e i s p r e v e n t e d f r o m e x a c t l y r e p l a c i n g t h e o x y g e n u s u a l l y b o u n d t o t h e p r o t e i n . T h e y e s t i m a t e t h a t o t h e n i t r o g e n a t o m w o u l d o n l y n e e d t o be d i s p l a c e d b y 0.4A t o e x p l a i n t h i s o b s e r v e d d i s c r e p a n c y i n a n g l e ; s u c h a s m a l l d i s p l a c e m e n t w o u l d n o t be v i s i b l e o n t h e X - r a y s t r u c t u r e ( S t r y e r e t a l . , 1 9 6 4 ) . O b v i o u s l y t h e same a r g u m e n t i s a p p l i -c a b l e t o t h e c y a n i d e compound, a l t h o u g h o n e m i g h t e x p e c t t h e d i s c r e p a n c y t o be l e s s , a s t h e c y a n i d e i s l e s s b u l k y t h a n t h e a z i d e . I n b o t h m y o g l o b i n a z i d e a n d m y o g l o b i n c y a n i d e t h e o t h e r t w o g - a x e s a r e a l i g n e d p a r a l l e l a n d p e r p e n d i c u l a r t o t h e p r o j e c t i o n o f t h e h i s t i d i n e - 1 8 i m i d a z o l e r i n g o n t h e p l a n e o f t h e heme. T h i s p r o j e c t i o n i s a p p r o x i m a t e l y t h e same i n a l l t h r e e m o l e c u l e s , p a s s i n g t h r o u g h m e t h e n e b r i d g e s o n o p p o s i t e s i d e s o f t h e r i n g b i s e c t i n g a d j a c e n t n i t r o g e n s — f i g u r e 4 . 1 2 . H e m o g l o b i n a z i d e h a s a l s o b e e n s t u d i e d b y EPR ( G i b s o n a n d I n g r a m , 1 9 5 7 ) a n d i t s s t r u c t u r e h a s b e e n d e t e r m i n e d b y X - r a y d i f f r a c t i o n ( P e r u t z a n d M a t h e w s , 1 9 6 6 ) . H o w e v e r , t h e d i r e c t i o n s o f g - a x e s h a v e n o t b e e n p u b l i s h e d . 143 I n t h e a z i d e i t i s t h e l o w e s t g - v a l u e ( g ^ ) t h a t i s p a r a l l e l t o t h e h i s t i d i n e p l a n e , w h i l e i n t h e c y a n i d e i t i s t h e m i d d l e g - v a l u e ( g 2 ) . A l s o , t h e c y a n i d e c o m p l e x i s d i f f e r e n t f r o m c y t o c h r o m e c a n d t h e a z i d e i n t h a t i t s g ^ - a x i s i s a p p r o x i m a t e l y 13° o u t o f t h e heme p l a n e , p o i n t i n g t o w a r d s t h e h i s t i d i n e i m i d a z o l e . I n b o t h t h e s e m y o g l o b i n c ompounds i t i s t h e p o s i t i o n o f t h e h i s t i d i n e i m i d a z o l e r i n g w h i c h d e t e r m i n e s t h e d i r e c t i o n o f t h e g - a x e s , p r e s u m a b l y t h r o u g h t h e a c t i o n o f t h e 77 -o r b i t a l s o f t h e S - n i t r o g e n a t o m b o u n d t o t h e i r o n . B l u m b e r g ( p e r s o n a l c o m m u n i c a t i o n ) s u g g e s t s " t h e f a c t t h a t g^ a n d g 2 h a v e b e e n i n t e r c h a n g e d ( i n t h e c y a n i d e ) a s c o m p a r e d t o t h e a z i d e c a s e , i s p r o b a b l y i n d i c a t i v e t h a t t h e c y a n i d e i o n i s p a r t i c i p a t i n g i n b a c k d o n a t i o n o f e l e c t r o n d e n s i t y , w h i l e t h e a z i d e i o n i s n o t " . H o w e v e r , i n c y t o c h r o m e c we c a n n o t assume t h a t t h e r h o m b i c f i e l d w h i c h s p l i t s t h e o r b i t a l s t o make g^ n o t e q u a l t o g 2 a r i s e s f r o m t h e h i s t i d i n e i m i d a z o l e , b e c a u s e t h e g -a x e s a r e a l m o s t e q u a l l y i n c l i n e d t o t h e p l a n e o f t h e h i s t i d i n e . A l s o , t h e amount o f r h o m b i c s p l i t t i n g b y t h i s i m i d a z o l e r i n g h a s b e e n c a l c u l a t e d b y K o t a n i ( 1 9 6 4 ) t o be 60 cm""" 1'—-far l e s s t h a n t h e amount r e q u i r e d t o g i v e t h e o b s e r v e d g - v a l u e s — = 900 cm" f o r t h e a z i d e , 500 "~ f o r c y t o c h r o m e c ( H a r r i s , 1 9 7 0 ) a n d = 400 cm"^ f o r t h e c y a n i d e ( B l u m b e r g , p e r s o n a l c o m m u n i c a t i o n ) . M i z u h a s h i ( 1 9 6 9 ) h a s a t t e m p t e d t o e x p l a i n t h e . a n i s o t r o p y o f g - v a l u e s i n h e m o g l o b i n a z i d e b y c a l c u l a t i n g t h e c o n t r i b u t i o n t o t h e r h o m b i c f i e l d f r o m t h e a z i d e i o n a n d f r o m t h e J a h n - T e l l e r d i s t o r t i o n o f t h e heme r i n g . H o w e v e r , he d i d n o t d i s c u s s t h e o r i e n t a t i o n o f t h e g - a x e s .in h i s t r e a t m e n t . He c o n c l u d e d t h a t t h e l a r g e a n i s o t r o p y s e e n was due t o t h e c o m b i n e d a c t i o n o f t h e d y n a m i c a l J a h n - T e l l e r i n t e r a c t i o n a n d t h e r h o m b i c f i e l d s o f a z i d e a n d i m i d a z o l e . We h a v e e x t e n d e d M i z u h a s h i ' s t r e a t m e n t t o c y t o -c h r o m e c t o a t t e m p t t o e x p l a i n t h e g - v a l u e a n i s o t r o p y . 4.6.5 J a h n - T e l l e r e f f e c t i n c y t o c h r o m e c The J a h n - T e l l e r t h e o r e m i s s t a t e d a s f o l l o w s : "A n o n l i n e a r p o l y a t o m i c m o l e c u l e i n a n o r b i t a l s t a t e w i t h o r b i t a l d e g e n e r a c y w i l l be u n s t a b l e i n i t s s y m m e t r i c a l e q u i l i b r i u m p o s i t i o n w i t h r e s p e c t t o d i s t o r t i o n s w h i c h d e s t r o y t h o s e e l e m e n t s o f s y m m e t r y r e s p o n s i b l e f o r t h e d e g e n e r a c y " ( J a h n a n d T e l l e r , 1 9 3 7 ) . I n p r a c t i c e , t h i s means s u c h a d e g e n e r a c y o c c u r s w h e n e v e r t h e r e i s a d o u b l y d e g e n e r a t e p a i r o f o r b i t a l s w i t h a n o d d number o f e l e c t r o n s I n t h e m . The o d d e l e c t r o n h a s a c h o i c e b e t w e e n e i t h e r o f t h e d e g e n e r a t e p a i r o f o r b i t a l s a n d t h e m o l e c u l e d i s t o r t s i n s u c h a way t h a t t h e d e g e n e r a c y i s r e m o v e d . I f t h e s t a b i l i s a t i o n e n e r g y g a i n e d b y t h e d i s t o r -t i o n i s v e r y much g r e a t e r t h a n t h e z e r o p o i n t e n e r g y o f t h e TABLE X I S i g n c o n v e n t i o n s u s e d f o r o r b i t a l s E i s e n b e r g e r a n d P e r s h a n ( 1 9 6 7 ) H a r r i s ( 1 9 7 0 ) M i z u h a s h i ( 1 9 6 9 ) A = 0.885 B = A = 0 . 8 8 5 B = a = 0.426 b = 0.174 C = 0.431 0.174 C = - 0 . 4 3 1 - 0 . 8 8 5 C = 0.174 ( T h e c o e f f i c i e n t s a r e f o r c y t o c h r o m e c ) n o r m a l mode o f t h e a p p r o p r i a t e v i b r a t i o n w h i c h d i s t o r t s t h e m o l e c u l e , t h e n t h e m o l e c u l e a d o p t s t h e d i s t o r t e d c o n f i g u r a -t i o n , a n d we h a v e t h e s t a t i c J a h n - T e l l e r e f f e e t * I f t h e s t a b i l i s a t i o n e n e r g y i s o f t h e same o r d e r , o r l e s s t h a n , t h e z e r o p o i n t e n e r g y o f t h e v i b r a t i o n t h e n t h e r e i s c o u p l i n g b e t w e e n t h e e l e c t r o n i c a n d n u c l e a r m o t i o n s t o f o r m a ' v i b r o n i c * s t a t e . The m o l e c u l e v i b r a t e s b e t w e e n i t s v a r i o u s d i s t o r t e d f o r m s . T h i s i s c a l l e d t h e d y n a m i c J a h n - T e l l e r e f f e c t . We now a s k — i s i t t h e s t a t i c o r d y n a m i c J a h n - T e l l e r e f f e c t t h a t i s o p e r a t i n g i n c y t o c h r o m e c ? F o l l o w i n g t h e t r e a t m e n t o f M i z u h a s h i ( 1 9 6 9 ) w i t h o u r g ~ v a l u e s , * we f i n d t h a t e n e r g y g a i n e d b y J a h n - T e l l e r i n t e r a c t i o n , n e g l e c t i n g n u c l e a r m o t i o n , i s a p p r o x i m a t e l y 225 c r a " ^ . The z e r o p o i n t e n e r g y o f t h e v i b r a t i o n a s s o c i a t e d w i t h t h e d i s t o r t i o n ( B 2 g ) i s a p p r o x i -m a t e l y 300 c m " 1 . The s t a t i c J a h n - T e l l e r e f f e c t i s t h e r e f o r e n o t o p e r a t i n g h e r e . We m u s t t h e r e f o r e s o l v e t h e H a m i l t o n i a n t a k i n g i n t o a c c o u n t t h e k i n e t i c e n e r g y o f t h e v i b r a t i o n . M i z u h a s h i c a r r i e d o u t t h i s c a l c u l a t i o n a n d o b t a i n e d a s e r i e s o f c u r v e s o f g - v a l u e a s a f u n c t i o n o f r h o m b i c f i e l d a n d o f t h e c o u p l i n g b e t w e e n t h e e l e c t r o n i c a n d n u c l e a r s y s t e m s . A r e p r o d u c t i o n o f h i s r e s u l t s f o r t h e g r o u n d d o u b l e t i s g i v e n i n f i g u r e 4 . 1 3 . • T h e n o t a t i o n u s e d b y M i z u h a s h i d i f f e r s f r o m o u r s — t h a t o f E i s e n b e r g e r a n d P e r s h a n ( 1 9 6 7 ) a n d f r o m t h a t o f H a r r i s ( 1 9 7 0 ) who p u b l i s h e d a r e v i e w o n t h e c h e m i s t r y o f l o w s p i n f e r r i c heme c o m p o u n d s . The r e l a t i o n s h i p b e t w e e n t h e v a r i o u s c o n -v e n t i o n s i s shown i n t a b l e X I . FIGURE 4 o l 3 g - v a l u e s a s a f u n c t i o n o f r h o m b i c f i e l d a n d J a h n - T e l l e r c o u p l i n g ( a f t e r M i z u h a s h i , 1 9 6 9 ) . 150 cm - i 220 440 660 880 Rhombic Field - cm F o r o u r g - v a l u e s ( 3 . 0 6 , 2.25, 1.25) a v a l u e o f r h o m b i c f i e l d s t r e n g t h o f 150 cm""1 a n d a c o u p l i n g o f X = 0.7 f i t b e s t . ( X i s d e f i n e d t o be t h e s q u a r e r o o t o f t h e r a t i o o f t h e e n e r g y g a i n e d b y t h e J . T . d i s t o r t i o n t o t h e z e r o p o i n t e n e r g y o f t h e d i s t o r t i n g v i b r a t i o n . ) T h e s e r e s u l t s show t h a t t h e d y n a m i c J a h n - T e l l e r e f f e c t c a n ' a m p l i f y * a s m a l l r h o m b i c s p l i t t i n g t o p r o d u c e t h e d i f f e r e n c e i n g - v a l u e s o b s e r v e d . I f t h e r e i s n o r h o m b i c s p l i t t i n g t h e n g^ = g 2 a n d t h e a n i s o t r o p y o f g - v a l u e s d o e s n o t o c c u r f o r a n y v i b r o n i c s t a t e . M i z u h a s h i a t t e m p t e d t o c a l c u l a t e X . He a s s u m e d t h a t t h e t 2 e l e c t r o n s w e r e s u b j e c t t o a v i b r a t i n g p o t e n t i a l f i e l d f r o m t h e n e i g h b o u r i n g p o r p h y r i n n i t r o g e n s . N u m e r i c a l e v a l u a t i o n , u s i n g H a r t r e e - F o c k wave f u n c t i o n s f o r t h e F e o r b i t a l s , g a v e X = 0 . 3 — i n r e a s o n a b l e a g r e e m e n t w i t h h i s v a l u e o f 0.8 f o r h e m o g l o b i n a z i d e a n d o u r s o f 0.7 f o r c y t o -c h r o m e c . The d i r e c t i o n s o f t h e g - a x e s w i l l be a f u n c t i o n o f b o t h t h e o r i e n t a t i o n s o f t h e i m i d a z o l e a n d a n y o t h e r l i g a n d s a n d o f t h e mode o f v i b r a t i o n o f t h e r i n g . The a n d B 2 q modes v i b r a t e a s : '2g N >N-Fe-N N N W-Ff-N* N B B 2g S i n c e a c c o r d i n g t o M i z u h a s h i t h e E^g mode c o m b i n e d w i t h t h e r h o m b i c c o n t r i b u t i o n s f r o m t h e i m i d a z o l e a n d a z i d e o p e r a t e t o g i v e t h e g ~ a x e s w i t h o r i e n t a t i o n s p a r a l l e l a n d p e r p e n d i c u l a r t o t h e h i s t i d i n e i m i d a z o l e a s s e e n i n m y o g l o b i n a z i d e , i n a s i m i l a r f a s h i o n t h e B 0 mode w i t h t h e i m i d a z o l e 2 g r h o m b i c f i e l d g i v e t h e r e s u l t s f o u n d f o r m y o g l o b i n c y a n i d e . We s u g g e s t t h a t i n c y t o c h r o m e c t h e mode ( i n t h e N-Fe-N d i r e c t i o n s ) w i t h t h e h i s t i d i n e r h o m b i c f i e l d g i v e s t h e p i c t u r e f o u n d b y u s — t h e g - a x e s l y i n g b e t w e e n t h e i m i d a z o l e a n d N-Fe-N p r o j e c t i o n s o n t h e heme p l a n e . 4.7 SUMMARY We h a v e p r e s e n t e d t h e m e t h o d s o f a n a l y s i s u s e d t o d e r i v e t h e d i r e c t i o n s o f t h e p r i n c i p a l g - a x e s , r e l a t i v e t o t h e c r y s t a l l o g r a p h i c a x e s , f r o m t h e EPR s p e c t r a o f h o r s e h e a r t f e r r i c y t o c h r o m e c . The r e s u l t s show t h a t t h e d i r e c t i o n o f t h e l a r g e s t p r i n c i p a l g - v a l u e ( g ^ = 3.06) l i e s w i t h i n 5° o f t h e heme n o r m a l d e t e r m i n e d f r o m X - r a y d i f f r a c t i o n m e a s u r e m e n t s a n d o t h e r d a t a . The o t h e r t w o g - a x e s l i e i n t h e heme p l a n e , a p p r o x i m a t e l y 15° f r o m t h e N-Fe-N d i r e c t i o n s i n t h e p o r p h y r i n . r i n g . M i z u h a s h i ' s t h e o r y , u s i n g t h e J a h n - T e l l e r e f f e c t t o a c c o u n t f o r t h e g - v a l u e s s e e n i n h e m o g l o b i n a z i d e , was a p p l i e d t o c y t o c h r o m e c . F r o m t h i s t h e o r y a r h o m b i c d i s t o r -t i o n o f 150 cm*"1 a n d d y n a m i c J . T . c o u p l i n g c o e f f i c i e n t ( Y) of 0.7 were needed to explain the g-values (3.06, 2.25, 1.25) observed experimentally. These are i n reasonable agreement with a 60 cm"1 rhombicity contribution, from the imidazole of h i s t i d i n e - 1 8 co-ordinated to the f i f t h p o s i t i o n of the i r o n ; and with a X of 0.3, predicted from a v i b r a t i n g point charge model of the 4 nitrogens i n the porphyrin r i n g system co-ordinated to the i r o n . CHAPTER 5.0 SI N G L E CRYSTAL EPR -LINEWIDTH VARIA T I O N AS A FUNCTION OF G-VALUE 5.1 INTRODUCTION I n t h i s c h a p t e r we p r e s e n t o u r r e s u l t s , a n d t h e t h e o r y t o e x p l a i n them, o f t h e c h a n g e s i n EPR l i n e w i d t h p r o d u c e d b y r o t a t i o n o f s i n g l e c r y s t a l s o f f e r r i c y t o c h r o m e c, r e l a t i v e t o t h e d . c . m a g n e t i c f i e l d d i r e c t i o n . M a t e r i a l s a n d m e t h o d s a r e t r e a t e d i n s e c t i o n 5.2. T h i s i s f o l l o w e d b y a s e c t i o n o n t h e c a u s e s o f l i n e b r o a d e n -i n g ( s e c t i o n 5 . 3 ) , t h e n o n t h e t h e o r y a p p l i c a b l e t o o u r s y s t e m ( s e c t i o n 5 . 4 ) . The r e s u l t s a n d d i s c u s s i o n f o l l o w ( s e c t i o n s 5.5 a n d 5.6) a n d o u r c o n c l u s i o n s a r e d r a w n i n s e c t i o n 5.7. 5.2 MATERIALS AND METHODS The m a t e r i a l s a n d m e t h o d s u s e d a r e l a r g e l y d e s c r i b e d i n c h a p t e r 4. M e a s u r e m e n t o f t h e w i d t h s o f a n y o f t h e l i n e s i s d i f f i c u l t f o r s e v e r a l r e a s o n s . A t l o w f i e l d s , c l o s e t o t h e maximum g - v a l u e s , w h e r e t h e l i n e s a r e r e l a t i v e l y n a r r o w , t h e y o v e r l a p w i t h t h e o t h e r s p r e s e n t , a n d w i t h a n i m p u r i t y 153 r e s o n a n c e c l o s e t o g = 2. T h i s makes d e t e r m i n a t i o n o f t h e l i n e s h a p e d i f f i c u l t . A t h i g h e r f i e l d s , w h e r e t h e l i n e s a r e w i d e r , t h e l o w e r e d s i g n a l - t o - n o i s e r a t i o a n d an u n c e r t a i n b a s e l i n e l e a d t o l a r g e r u n c e r t a i n t i e s i n t h e m e a s u r e m e n t s . 5.3 SOURCES OF L I N E BROADENING 5.3.1 M i n o r c o n t r i b u t i o n s t o l i n e w i d t h Many o f t h e u s u a l c a u s e s o f l i n e b r o a d e n i n g c a n be r u l e d o u t i m m e d i a t e l y a s t h e i r c o n t r i b u t i o n s a r e t o o s m a l l : / ( i ) The s i m p l e s t o n e - — t h a t o f r e l a x a t i o n t i m e b r o a d e n i n g — i s n o t t h e c a u s e , f o r t h e l i n e s h a p e s e e n i s G a u s s i a n , n o t L o r e n t z i a n , a n d t h e w i d t h i s i n d e p e n d e n t o f t e m p e r a t u r e b e t w e e n 4.2 a n d 50°K. ( A b o v e t h i s t e m p e r a t u r e , t h e l i n e s h a p e d o e s become L o r e n t z i a n — c h a p t e r 3). ( i i ) D i p o l a r a n d e x c h a n g e i n t e r a c t i o n s b e t w e e n i r o n a t o m s a r e n o t i m p o r t a n t h e r e s i n c e t h e s m a l l e s t p o s s i b l e - o d i s t a n c e b e t w e e n n e a r e s t n e i g h b o r s i s a p p r o x i m a t e l y 30 A. D i p o l a r i n t e r a c t i o n s b e t w e e n w a t e r p r o t o n s a n d t h e i r o n a t o m c o u l d c o n t r i b u t e t o t h e l i n e w i d t h . H o w e v e r , ENDOR ( e l e c t r o n -n u c l e a r d o u b l e r e s o n a n c e ) o n m y o g l o b i n a z i d e ( E i s e n b e r g e r a n d P e r s h a n , 1 9 6 7 ) s h o w e d t h a t t h e h y p e r f i n e c o u p l i n g t o t h e i r o n was o n l y o f t h e o r d e r o f 0.5 M H z — e q u i v a l e n t t o a l i n e w i d t h c o n t r i b u t i o n o f a b o u t 10 g a u s s . ( i i i ) B e c a u s e o f t h e p r e s e n c e o f f i v e n i t r o g e n a t o m s , e a c h w i t h n u c l e a r s p i n o f 1, i t m i g h t be e x p e c t e d t h a t t h e r e w o u l d be h y p e r f i n e b r o a d e n i n g f r o m t h i s s o u r c e . I n d e e d , S c h o l e s ( 1 9 6 9 ) h a s shown t h a t i n h e m i n c o - c r y s t a l l i z e d w i t h p e r y l e n e , t h e r e i s a h y p e r f i n e i n t e r a c t i o n s u f f i c i e n t t o p r o d u c e a l i n e o f a b o u t 50 g a u s s a t X - b a n d ( 9 G H z ) . S i n c e t h i s i n t e r a c t i o n i s f r e q u e n c y i n d e p e n d e n t i t w o u l d t h e r e f o r e p r o d u c e a s i m i l a r l i n e w i d t h a t o u r f r e q u e n c y , 24 GHz. The i n t e r a c t i o n w i t h t h e n i t r o g e n i s , t h e r e f o r e , t o o s m a l l t o a c c o u n t f o r t h e b r o a d l i n e w i d t h s o b s e r v e d . 5.3.2 M a j o r c a u s e s o f l i n e b r o a d e n i n g T h e m e c h a n i c a l s o f t n e s s a n d f r a g i l i t y o f p r o t e i n c r y s t a l s , t o g e t h e r w i t h t h e i r h i g h c o n t e n t o f w a t e r o f c r y s t a l l i s a t i o n ( 5 0 % i n c y t o c h r o m e c, D i c k e r s o n , 1 9 6 7 ) s u g -g e s t t h a t some d e g r e e o f d i s o r d e r m i g h t e x i s t i n t h e m . A s w i l l be shown, t h i s i s c a p a b l e o f c a u s i n g v e r y l a r g e l i n e b r o a d e n i n g . B a s i c a l l y , t h e s p e c t r a l l i n e s e e n i s t h e sum o f a s e r i e s o f l i n e s whose e f f e c t i v e g - v a l u e s a r e s p r e a d a b o u t a mean v a l u e ( t h e c e n t r e o f t h e m e a s u r e d s p e c t r u m ) . We know t h a t ( P r y c e , 1 9 5 0 ) : 2 ^ g = Z Ajj-Xj- J>i (Ajj = Ajj) ( 5 . D i,j = i t h e n , f o r m a l l y : A(g 2) = £ (AA,,)-^./, + 2 A ( i , ^ ) T h e f i r s t t e r m r e p r e s e n t s t h e e f f e c t o n t h e s p e c t r u m o f v a r i a t i o n i n g - t e n s o r ( A ) , i . e . a v a r i a t i o n i n t h e p r i n c i p a l g - v a l u e s . T h i s i s an i n t e r n a l m o l e c u l a r p r o p e r t y a n d a r i s e s f r o m s m a l l c h a n g e s i n t h e l i g a n d f i e l d s y m m e t r y f r o m m o l e c u l e 155 t o molecule« T h e s e c o n d t e r m r e p r e s e n t s a b r o a d e n i n g o f t h e s p e c t r u m f r o m a c h a n g e i n o r i e n t a t i o n o f t h e m o l e c u l e s w i t h r e s p e c t t o t h e m a g n e t i c f i e l d . I t i s w e l l k n own t h a t , i n g e n e r a l , c r y s t a l s a r e n o t t r u l y s i n g l e b u t c o n s i s t o f many s m a l l c r y s t a l l i t e s , e a c h w i t h s l i g h t l y d i f f e r e n t o r i e n t a t i o n . The g - t e n s o r i s t h e same i n e a c h c r y s t a l l i t e . The b r o a d e n i n g i s a minimum a l o n g t h e d i r e c t i o n s o f t h e p r i n c i p a l g - a x e s o f t h e c r y s t a l , w h e r e t h i s m i s o r i e n t a t i o n e f f e c t v a n i s h e s t o f i r s t o r d e r . We now g i v e t h e t h e o r y t o c a l c u l a t e t h e l i n e w i d t h c o n t r i b u t i o n s f r o m t h e s e t w o c a u s e s . 5,4 THEORY OP T.TME BROADENING 5.4.1 I n t r o d u c t i o n I f t h e e f f e c t i v e g - v a l u e i s a f u n c t i o n o f s e v e r a l p a r a m e t e r s — i , j , k s a y , — t h e n i n d e p e n d e n t s m a l l r a n d o m v a r i a t i o n s o f t h e s e p a r a m e t e r sAi, Aj,Ak, p r o d u c e t h e f o l -l o w i n g c h a n g e i n gs The t o t a l e f f e c t o f t h e s e i n d e p e n d e n t a n d r a n d o m v a r i a t i o n s w i l l be g i v e n b y t h e r o o t mean s q u a r e v a l u e o f / \ g , Ag = d\ d\ 1 dk & A i + a l A i + & A k ( 5 . 2 ) 2 d e f i n e d a s F o r i n d e p e n d e n t v a r i a b l e s , t h e s e c o n d t e r m i s z e r o , a n d < A g 2 > i = Z 0 f ( ( A i ) y ( 5 . 3 , <^ Ag a m e a s u r e o f t h e w i d t h o f t h e r e s o n a n c e l i n e , w h i l e t h e <^ Aj )> m e a s u r e t h e u n c e r t a i n t y i n t h e v a l u e s o f i , We now w i s h t o o b t a i n a n a l y t i c a l e x p r e s s i o n s f o r t h e p a r t i a l d e r i v a t i v e s )» a n d t o r e l a t e t h e s e t o t h e 0 i l i n e w i d t h s o b s e r v e d . 5 . 4 o 2 E v a l u a t i o n o f p a r t i a l d e r i v a t i v e s — l i n e w i d t h  due t o m i s o r i e n t a t i o n o f m o l e c u l e s When t h e d ec» m a g n e t i c f i e l d i s a t a p o s i t i o n w i t h d i r e c t i o n c o s i n e s x , m, n r e l a t i v e t o t h e m o l e c u l a r g - a x e s — 9 1» ^°—the g _ v a l u e i n t h a t d i r e c t i o n i s g i v e n b y 2 2^2 2 2 2 2 g = g, # + g 2 m + g 3 n I n p o l a r c o o r d i n a t e s we h a v e g 2 = g 2 sin2#cos2(£ + g 2 sin2#sin2<jf) + g 2 cos2(9 w h e n c e dg sin 2(9 d 0 2g | > « " o,)»in^ - (g, - g, j j I f we c a l l A 0 t h e r.«m«s. d e v i a t i o n i n t7 , t h e n t h e r.m.s, v a l u e o f g, Ag, i s A i A £ U s i n g h v - g$H t h e r e s u l t i n g r . m . s . v a l u e o f t h e l i n e W i d t h j AfHn» due t o t h e v a r i a t i o n i n Q i s g i v e n b y 8 157 A H, - hi/ A g 0 g (5..4) h i / sin 2$ r 2 2 2, 2 g A 0 We m u s t a l s o c o n s i d e r t h e b r o a d e n i n g p r o d u c e d b y a v a r i a t i o n i n , w h e r e <^> i s t h e a n g l e o f t h e d . c . f i e l d i n t h e g 2 p l a n e r e l a t i v e t o t h e g1 a x i s , a s shown i n f i g u r e 5.1b. One c a n o b t a i n a n e x p r e s s i o n s i m i l a r t o ( 5 . 4 ) f o r t h e l i n e w i d t h c o n t r i b u t i o n : A H w h e r e h * . 1 1 ^ . 3 ^ 0 . ( 0 * - e f t B g ActS ( 5 . 5 ) i s t h e r . m . s . d e v i a t i o n i n c u l a t e t h e l i n e w i d t h s i n m e t - m y o g l o b i n a z i d e . I n o r d e r t o f i t t h e i r d a t a t h e y c h o s e a n d a d d e d t h e two c o n t r i b u t i o n s l i n e a r l y . T h i s p r o c e d u r e , t o u s , seems i n c o r r e c t , f o r t h e v a r i a t i o n s i n Q a n d <^> a r e i n d e p e n d e n t a n d s h o u l d be c o m b i n e d b y s q u a r i n g a n d a d d i n g a n d » t h e n t a k i n g t h e s q u a r e r o o t t o o b t a i n t h e t o t a l l i n e w i d t h . I n a d d i t i o n , t h e u s e o f A<^> a s t h e m e a s u r e o f t h e r a n d o m v a r i a t i o n i s w r o n g . A r o u n d a n y d i r e c t i o n Q ^ — s e e f i g u r e 5 . 1 — t h e r e a r e a n u m b e r , N ( G ) , o f c r y s t a l l i t e s w hose d i r e c t i o n s a r e d i s -p l a c e d b y an a n g u l a r d i s t a n c e , q ° R andomness means t h a t : FIGURE 5*1 N o t a t i o n u s e d f o r l i n e w i d t h c a l c u l a t i o n s ( a ) C o n e o f s o l i d a n g l e , G. , i n g - a x i s c o - o r d i n a t e s y s t e m . ( b ) D e f i n i t i o n o f e l e m e n t o f a r e a i n p o l a r c o - o r d i n a t e s f o r a g e n e r a l p o i n t (Q*cf) i n t h e g - a x i s c o - o r d i n a t e s y s t e m . ( i ) N (Q) i s a G a u s s i a n f u n c t i o n o f N(a) a exp(-Y (a/cr) ) w h e r e O" i s t h e r . m . s . v a l u e o f Q ( i i ) N(cO i s i n d e p e n d e n t o f d i r e c t i o n a r o u n d On t h e u n i t s p h e r e , q = ds, an e l e m e n t o f l e n g t h o n t h e s u r f a c e a n d ( d s ) 2 = (d0)2+ sin ^  .(d<£)2 = + (du) C l e a r l y , a l s o cr2= <a2> = <ds2> = <d02>+<du2> . „ 2 / 2 X | 2 a n d <d£/ > = \dll / = ~z cr The c o r r e c t f o r m f o r e q u a t i o n ( 5 . 5 ) i s t h u s A H u sin2<ft- sing-Cg 2 - g 2 ) — - = r 2 I Au ( 5 . 6 ) 2-,-L 2 ( 5 . 7 ) a n d t h e t o t a l l i n e w i d t h , (A H ) m o s a i c i s AH mosaic * [ ( ^ ) 2 + (*"f] E q u a t i o n s ( 5 . 4 ) , ( 5 . 6 ) a n d ( 5 . 7 ) h a v e b e e n u s e d t o o b t a i n o u r t h e o r e t i c a l c u r v e s . 5 r 4 . 3 E v a l u a t i o n o f p a r t i a l d e r i v a t i v e s - q - v a l u e  d i s t r i b u t i o n d u e t o l i q a n d f i e l d v a r i a t i o n s H e r e we c a l c u l a t e t h e c o n t r i b u t i o n t o t h e l i n e w i d t h s o f a d i s t r i b u t i o n o f g - v a l u e s a r i s i n g f r o m a v a r i a t i o n i n l i g a n d f i e l d s f r o m m o l e c u l e t o m o l e c u l e . T h e m a g n i t u d e o f t h e s p l i t t i n g o f t h e g r o u n d s t a t e s p i n d o u b l e t ( a n d t h e r e f o r e t h e g - v a l u e s ) a t a n y d . c . f i e l d v a l u e i s a f u n c t i o n o f t h e r h o m b i c ( V ) a n d a x i a l (D) p o t e n t i a l s , , We w i s h t o c a l c u l a t e — — a n d — — t o o b t a i n _ • dv do AHy a n d AHD> t h e l i n e b r o a d e n i n g p r o d u c e d b y t h e r a n d o m d i s t r i b u t i o n s o f V a n d D a b o u t t h e i r mean v a l u e . A s b e f o r e , we s t a r t f r o m : 2 = \ a n d b y d i f f e r e n t i a t i o n g 2 sin2£cos2<£ + g2 sin2(9 sin2<£ + <? coszQ "fv" = 92 (9^/ s i n 2 ^ cos2^)+g 2^'sin2(9 sin2<£ * g ^ cos2£ w h e r e g .' = °®» a n d i = 1,2,3„ 1 dv T h e r e i s a s i m i l a r e x p r e s s i o n f o r D w i t h <3g, g i = do ' The d e r i v a t i o n o f t h e e x p l i c i t f o r m u l a e l e a d i n g • it t o g ^ a n d g^ a r e g i v e n i n A p p e n d i x I . The c o m p l e t e e x p r e s -s i o n f o r r h o m b i c v a r i a t i o n i n t e r m s o f t h e l i n e w i d t h , i s H v p g LavJ ( 5- 8 ) w h e r e A V i s t h e r.m.s. v a r i a t i o n i n r h o m b i c p o t e n t i a l . The e x p r e s s i o n f o r A Hp, t h e a x i a l l i n e w i d t h , i s s i m i l a r w i t h AD t h e v a r i a t i o n i n a x i a l p o t e n t i a l . We c a n a d d t h e s e ^ e x p r e s s i o n s a n d o b t a i n J_ r , — 2 2 n 2 ( 5 . 9 ) A H , = ( A H J + ( A H J symmetry L v V ' \ CK J We now c o m b i n e ( 5 . 7 ) a n d ( 5 . 9 ) t o o b t a i n t h e t o t a l l i n e w i d t h , ( A H ) t o t a l , 2 2 2 ~ '. A H t e t a l • [ • (S " * J *(AHi)j ( 5 - 1 0 ) A H j i s an i s o t r o p i c c o n t r i b u t i o n f r o m t h e m i n o r c a u s e s o f s e c t i o n 5 . 3 . 1 . 5.4.4 T r a n s f o r m a t i o n f r o m . l a b o r a t o r y s y s t e m t o q - a x e s c o o r d i n a t e s y s t e m T h e e x p r e s s i o n s g i v e n a b o v e f o r t h e v a r i o u s c o n -t r i b u t i o n s t o t h e l i n e w i d t h u s e a c o o r d i n a t e s y s t e m b a s e d o n t h e 3 g - a x e s ( f i g u r e 5 . 1 ) . O u r m e a s u r e m e n t s , h o w e v e r , a r e made b y a r o t a t i o n o f t h e c y t o c h r o m e c c r y s t a l a b o u t an a x i s p e r p e n d i c u l a r t o t h e m a g n e t i c f i e l d d i r e c t i o n ( f i g u r e 5 . 2 ) , i n t h e r i g h t h a n d e d c o - o r d i n a t e s y s t e m ( X , Y , Z ) . I n o u r c r y s t a l o r i e n t a t i o n s t u d y ( c h a p t e r 4) we o b t a i n e d t h e a n g l e s b e t w e e n ' t h e g - a x e s a n d t h e X,Y,Z l a b o r a t o r y s y s t e m . We u s e t h e s e a n g l e s , a n d s t a n d a r d f o r m u l a e f o r r o t a t i o n o f a x e s ( H a n d b o o k o f M a t h e m a t i c a l T a b l e s , 2 n d e d i t i o n , s u p p l e m e n t t o H a n d b o o k o f C h e m i s t r y , C h e m i c a l R u b b e r Company, p . 574) t o o b t a i n Q a n d i n t h e g - a x i s s y s t e m f o r a n y g i v e n \j/ i n t h e l a b o r a t o r y s y s t e m . I f t h e d i r e c t i o n c o s i n e s o f g^, g^ a n d g^ r e l a t i v e t o X a n d Y a r e ( i , fc ) , (m , m ) a n d ( n , n ) r e s p e c t i v e l y x y x y x y we o b t a i n 6 = a'rcos(n co-st + n s in^ ) • y = a r c t d n l ( m x + r r y t a n ^ / C £ x + / y tarW' ) FIGURE 5.2 C o - o r d i n a t e s y s t e m u s e d t o d e s c r i b e t h e p o s i t i o n s o f t h e p r i n c i p a l g - a x e s r e l a t i v e t o t h e l a b o r a t o r y m a g n e t i c f i e l d d i r e c t i o n (same a s f i g u r e 4 . 3 ) . 164 We make use o f t h e s e r e l a t i o n s i n o u r computer programs t o c a l c u l a t e t h e t h e o r e t i c a l l i n e w i d t h v a r i a t i o n on r o t a t i o n * 5.5 RESULTS 5.5.1 I n t r o d u c t i o n We g i v e t h e d a t a r e q u i r e d t o compute t h e t h e o r e t i -c a l l i n e w i d t h v a r i a t i o n i n s e c t i o n 5.5.2. I n s e c t i o n s 5.5.3, 5.5.4 a r e p r e s e n t e d t h e r e s u l t s o f t h e l i n e w i d t h s measured i n t h e same two o r i e n t a t i o n s u s e d i n c h a p t e r 4 ( v i z . , c - a x i s v e r t i c a l and b - a x i s v e r t i c a l ) . 5.5.2 D a t a f o r l i n e w i d t h c a l c u l a t i o n s I n o r d e r t o c a l c u l a t e t h e v a r i o u s c o n t r i b u t i o n s made by t h e mechanisms m e n t i o n e d above we r e q u i r e t h e f o l -l o w i n g i n f o r m a t i o n about t h e system: ( i ) t h e g - v a l u e s a l o n g t h e m a g n e t i c a x e s — g ^ , g 2» g ^ — t o s u b s t i t u t e i n t o t h e f o r m u l a e . These c a n be o b t a i n e d f r o m EPR o f t h e mother l i q u o r f r o m w h i c h t h e c r y s t a l s were grown. ( i i ) an e s t i m a t e o f t h e 'background' i s o t r o p i c c o n t r i b u t i o n s f r o m t h e c a u s e s m e n t i o n e d i n s e c t i o n 5.3.1. ( i i i ) t h e l i n e w i d t h a t t h e low f i e l d extremum o f t h e s o l u t i o n s p e c t r u m i n o r d e r t o c a l c u l a t e t h e magnitudes o f t h e v a r i a -t i o n s i n r h o m b i c and a x i a l p o t e n t i a l s (AV and ALT). We use an extreme g - v a l u e b e c a u s e , when t h e d.c. m a g n e t i c f i e l d i s a l o n g a g - a x i s t h e m i s o r i e n t a t i o n c o n t r i -b u t i o n v a n i s h e s t o f i r s t o r d e r . We choose t h e g ^ - a x i s 166 because there the linewidth is narrowest, and the signal is greatest. ( i v ) an estimate of the magnitudes of the variations in orientation {KB, A u k We give our values of these data below: (i) Only two g-values could be obtained from the mother liquor; g 2 = 2.25 and g^  = 3.06; we took g^^ to be 1.25. (i i) We assume that the AHj , the isotropic contribution is about 50 gauss, from our considerations in section 5.3.1. Most of this contribution is unresolved hyperfine structure. This will not contribute very much to the overall line width f o r at no angle of the magnetic field relative to the g-axes does AHjfcecome the most important contributor to(AH)total. ( i i i ) The linewidth at g 3 = 3.06 was found to be 400 - 20 gauss. For the d . c . magnetic field along the g 3 direction, the angle $ is zero and <^> is 90°. This makes(AH) mosaic equal to zero and therefore: > A H T O T O L * [ ( A H / * ( A H D ) 2 + (AHJ ) 2 ] 2 at 0 = 0° , = 9 0 ° and TABLE X I I D a t a f o r t h e o r e t i c a l c a l c u l a t i o n o f l i n e w i d t h s g - v a l u e s g^  = 1.25 g 2 = 2.25 g 3 = 3 . 0 6 r h o m b i c g^  = -0.522X"1 g 2 = 0.487 X"1 g 3 « 0.485X"1 a x i a l g^ = 0.138a" 1 g 2 = 0.154 X*1 g 3 = 0.059X"1 where X i s t h e s p i n - o r b i t c o u p l i n g c o n s t a n t ( s e e A p p e n d i x I ) r.m.s. d e v i a t i o n s m o s a i c AO = A u = 0.027 r a d i a n s symmetry AV = AD = 0.22 X 168 I f we now assume t h a t A v i s a p p r o x i m a t e l y e q u a l to AD, t h e n t h e m a g n i t u d e s o f A Hy a n d A HQ d e p e n d u p o n t h e i n i s i z e o f a n d g ^ . F o r o u r s y s t e m g ^ i s a p p r o x i m a t e l y t e n t i m e s g r e a t e r t h a n g^; t h e r e f o r e a l o n g t h i s g - a x i s t h e a x i a l l i n e w i d t h c o n t r i b u t i o n c a n be n e g l e c t e d r e l a t i v e t o t h e r h o m b i c , s o : " F o r a ( A H ) t o t a l o f 400 g a u s s a n d (AHj)of 50 g a u s s we o b t a i n A H V e q u a l t o 397 g a u s s . T h i s g i v e s AV= 0.22\. T h i s i s s i m i l a r t o t h a t f o r m y o g l o b i n a z i d e (AV = o.nX), ( E i s e n b e r g e r a n d P e r s h a n , 1 9 6 7 ) , w h e r e X i s t h e s p i n - o r b i t c o u p l i n g c o n s t a n t . ( i v ) A n e s t i m a t e o f t h e m o l e c u l a r m i s o r i e n t a t i o n c a n be o b t a i n e d f r o m t h e X - r a y c r y s t a l l o g r a p h i c d a t a . The d i f -f r a c t i o n p a t t e r n s o f p r o t e i n s f a d e o u t a t s p o t s c o r r e s p o n d i n g to l a t t i c e s p a c i n g s o f o r d e r 1 t o 2 A. I f , f o r a s p h e r i c a l p r o t e i n m o l e c u l e o f d i a m e t e r o 50 A, t h e r e e x i s t s a n u n c e r t a i n t y o f i t s o r i e n t a t i o n r e l a t i v e + 0 to i t s n e i g h b o u r s o f - 1.5 A, a t t h e p e r i p h e r y t h i s i s e q u i v a l e n t t o a n a n g u l a r u n c e r t a i n t y i n p o s i t i o n o f a b o u t - 1.5° ( o r - .027 r a d i a n s ) . T h e r e f o r e we t a k e All e q u a l to A 0 w i t h a v a l u e o f .027 r a d i a n s . T h e s e d a t a we s u m m a r i s e d i n t a b l e X I I . • 169 5.5•3 L i n e w i d t h v a r i a t i o n s i n a c r y s t a l w i t h b - a x i s  v e r t i c a l T h i s o r i e n t a t i o n c o r r e s p o n d s t o o r i e n t a t i o n I I o f c h a p t e r 4. We g i v e t h e r e s u l t s o f t h e v a r i a t i o n i n l i n e w i d t h o n r o t a t i o n o f t h e c r y s t a l f o r l i n e s 1 a n d 2. T h e s e l i n e s w e r e c h o s e n b e c a u s e t h e y a r e w e l l s e p a r a t e d due t o t h e r a n g e 2 o f g - v a l u e s ( g v a r i e s f r o m 5.40 t o 1.85) m a k i n g l i n e w i d t h m e a s u r e m e n t s e a s i e r . F i g u r e 5.3 p r e s e n t s t h e d a t a p o i n t s , t o g e t h e r w i t h t h e t h e o r e t i c a l c u r v e s c o m p u t e d f r o m t h e p a r a m e t e r s g i v e n i n t a b l e X I I . I n f i g u r e 5.4 i s shown t h e i n d i v i d u a l c o n t r i b u -t i o n s t o t h e t o t a l l i n e w i d t h f r o m t h e a x i a l , r h o m b i c , m o s a i c a n d i s o t r o p i c e f f e c t s f o r l i n e 2. T h e a g r e e m e n t i s f a i r , c o n s i d e r i n g t h e u n c e r t a i n -t i e s i n l i n e w i d t h m e a s u r e m e n t , e s t i m a t e d t o be - 50 g a u s s f o r w i d t h s l e s s t h a n 1000 g a u s s , i 200 g a u s s f o r t h e h i g h e r v a l u e s . A s m e n t i o n e d b e f o r e , t h e e r r o r s a r e c a u s e d b y t h e p r e s e n c e o f o t h e r ( o v e r l a p p i n g ) l i n e s , l o w s i g n a l - t o - n o i s e r a t i o a t h i g h f i e l d s a n d t h e p r e s e n c e o f t h e q u a r t z i m p u r i t y a t g = 2. F r o m f i g u r e 5.4, i t c a n b e s e e n t h a t t h e a n g u l a r v a r i a t i o n s o f t h e t h r e e i m p o r t a n t c o n t r i b u t i o n s a l l h a v e r o u g h l y t h e same f o r m , i . e . , t h e maximum a n d minimum l i n e -w i d t h v a l u e s o f a l l t h r e e o c c u r a t ^ ~ 2 0 a n d ^ ~ 1 1 0 ° r e s p e c t i v e l y . H e n c e t h i s o r i e n t a t i o n i s n o t v e r y u s e f u l f o r FIGURE 5.3 V a r i a t i o n i n w i d t h o f l i n e 1 a n d 2 o f O r i e n t a t i o n I I a s c r y s t a l i s r o t a t e d r e l a -t i v e t o d . c . m a g n e t i c f i e l d . T h e t h e o r e t i c a l c u r v e i s o b t a i n e d f r o m e q u a t i o n ( 5 . 1 0 ) . ^ - Rotation Angle of Crystal FIGURE 5.4 T h e o r e t i c a l v a r i a t i o n i n w i d t h o f l i n e 2 O r i e n t a t i o n I I s h o w i n g i n d i v i d u a l c o n t r i b u t i o n s . in 3 o o T3 4) 2000 + 1750 1500 1250 1000 30 60 90 120 150 180 210 </  - Rotation Angle of C rys ta l d e c i d i n g u p o n t h e r e l a t i v e v a l u e s o f A ^ . A u , AV , A D . The m a i n r e a s o n f o r t h i s i s 0 d o e s n o t c h a n g e b y more t h a n 40° i n a 180° r o t a t i o n o f t h e c r y s t a l . A b e t t e r o r i e n t a t i o n w o u l d be one i n w h i c h 0 a n d b o t h v a r y w i d e l y . A c r y s t a l o r i e n t e d w i t h i t s c - a x i s a p p r o x i m a t e l y v e r t i c a l w i l l g i v e 100° v a r i a t i o n i n G a n d ~ 1 8 0 ° v a r i a t i o n i n <p on a 180° r o t a t i o n o f t h e c r y s t a l . 5.5.4 L i n e w i d t h v a r i a t i o n f o r c r y s t a l w i t h c - a x i s  v e r t i c a l T h i s c o r r e s p o n d s t o o r i e n t a t i o n I o f c h a p t e r 4. The d a t a f o r t h e l i n e w i d t h v a r i a t i o n o f o n e l i n e i s shown i n f i g u r e 5.5, w i t h t h e c a l c u l a t e d c u r v e . The i n d i v i d u a l c o n -t r i b u t i o n s a r e p r e s e n t e d i n f i g u r e 5.6. O n ce a g a i n t h e a g r e e m e n t b e t w e e n c a l c u l a t e d a n d e x p e r i m e n t a l r e s u l t s i s r e a s o n a b l e , i n t h a t t h e r e i s a l a r g e r e g i o n o f r o u g h l y c o n s t a n t l i n e w i d t h o v e r a 90° r o t a t i o n ( 3 0-120°). A n i m p u r i t y r e s o n a n c e a r o u n d g = 2 g a v e a w i d e l i n e w h i c h made m e a s u r e m e n t o f t h e l i n e w i d t h s n e a r g . i m p o s s i b l e , a n d l e d t o t h e l a r g e u n c e r t a i n t i e s shown b y t h e e r r o r b a r s . 5.6 DISCUSSION 5.6.1 V a l u e o f p a r a m e t e r s T h e v a l u e s o f A 0 , A u . A v , a n d AD w e r e t h e same f o r b o t h o r i e n t a t i o n s . FIGURE 5.5 V a r i a t i o n i n w i d t h o f l i n e 1 o f O r i e n t a t i o n I a s c r y s t a l i s r o t a t e d r e l a t i v e t o . d . c . m a g n e t i c f i e l d . The t h e o r e t i c a l c u r v e i s o b t a i n e d f r o m e q u a t i o n ( 5 . 1 0 ) . Or ientat ion L i n e 1 Data * ° " l 8 ° ° D a t ° - * 180 - 360° ifr - Angle of Rotation in Degrees (Arbitrary zero) FIGURE 5 . 6 T h e o r e t i c a l v a r i a t i o n i n w i d t h o f l i n e 1 o f O r i e n t a t i o n I s h o w i n g i n d i v i d u a l c o n t r i b u t i o n s . 00 179. AS A v a n d AD a r e f i x e d b y t h e g - v a l u e s i n s o l u t i o n , t h e o n l y a d j u s t a b l e p a r a m e t e r s a r e A6andA j U . We t r i e d t o g e t b e t t e r f i t s t o t h e d a t a b y v a r y i n g t h e s e s y s t e m a t i c a l l y , b u t n o i m p r o v e m e n t was o b t a i n e d o v e r t h e v a l u e s g i v e n . T h i s m e r e l y r e f l e c t s t h a t u n c e r t a i n t i e s i n t h e l i n e w i d t h s m e a s u r e d a r e t o o l a r g e t o p e r m i t more p r e c i s e v a l u e s o f t h e s e p a r a m e t e r s t o b e o b t a i n e d . The o n l y o t h e r w o r k d o n e o n t h e l i n e w i d t h v a r i a t i o n s a s a f u n c t i o n o f o r i e n t a t i o n i s t h a t o f E i s e n b e r g e r and-P e r s h a n ( 1 9 6 7 ) a n d H e l c k e e t a l . ( 1 9 6 8 ) on a c i d - m e t - H b a n d met-Mb r e s p e c t i v e l y . ' A s i n d i c a t e d e a r l i e r , t h e t h e o r y we u s e d i s e s s e n -t i a l l y t h a t o f E i s e n b e r g e r a n d P e r s h a n ( 1 9 6 7 ) , b u t we h a v e e x t e n d e d i t t o i n c l u d e t h e c o n t r i b u t i o n s f r o m a d i s t r i b u t i o n o f a x i a l p o t e n t i a l . T h e s e a u t h o r s c a l c u l a t e d t h e l i n e w i d t h v a r i a t i o n d u e t o m i s o r i e n t a t i o n ( Ad ) a n d t o v a r i a t i o n s i n r h o m b i c p o t e n t i a l (AV) . T h e y h a d p r o b l e m s i n o b t a i n i n g a w i d e e n o u g h r a n g e o f d a t a t o t e s t t h e i r t h e o r y , b e c a u s e o f l i n e s o v e r -l a p p i n g , o r b e i n g t o o b r o a d t o b e v i s i b l e , o r b e i n g s p l i t . T h i s r e s u l t e d i n t h e i r b e i n g a b l e t o f i t t h e t h e o r y f o r r o t a t i o n o f t h e c r y s t a l i n t h e 92"" q3 P * - a n e ( t h e y z p l a n e I n t h e i r n o t a t i o n ) — c o r r e s p o n d i n g t o v a r y i n g 6 f r o m 0 t o 90° w i t h <f> = 9 0 ° . S i n c e <f> was c o n s t a n t t h e y w e r e u n a b l e t o t e s t t h e c o n t r i b u t i o n f r o m Au ( = s i n a l t h o u g h t h i s c a n c o n t r i b u t e a s much t o t h e l i n e w i d t h a s A 0 . The w i d e r r a n g e o f g - v a l u e s i n o u r s y s t e m ( 3 . 0 6 , 2.25, 1.25) a s c o m p a r e d w i t h t h e i r s ( 1 . 7 2 , 2.22, 2.80) a v o i d e d some o f t h e p r o b l e m s o f l i n e o v e r l a p b y s e p a r a t i n g l i n e s b e t t e r . T h e o t h e r p u b l i s h e d w o r k , t h a t o f H e l c k e e t a l . ( 1 9 6 8 ) , d o e s n o t e x p l a i n t h e o b s e r v e d l i n e w i d t h s n e a r l y a s w e l l a s E i s e n b e r g e r and. P e r s h a n ( 1 9 6 7 ) . H e l c k e n e g l e c t e d a n y c o n t r i b u t i o n f r o m r h o m b i c o r a x i a l v a r i a t i o n s i n p o t e n -t i a l , u s i n g a p u r e l y m i s o r i e n t a t i o n a l m o d e l . T h e y u s e d a s p a r a m e t e r s A 0 a n d f i n d i n g t h a t v a l u e s o f .027 r a d i a n s (1.5°) a n d .072 r a d i a n s (4°) w e r e r e q u i r e d t o f i t t h e i r d a t a . O u r r e s u l t s ( f i g u r e s 5.4 a n d 5 . 6 ) , a n d t h o s e o f E i s e n b e r g e r a n d P e r s h a n , show t h e c o n t r i b u t i o n s f r o m a x i a l a n d r h o m b i c v a r i a t i o n s i n p o t e n t i a l a r e j u s t a s g r e a t a s t h e m i s o r i e n t a -t i o n a l o n e . H e l c k e e t a l . a d d e d t h e i n d i v i d u a l c o n t r i b u t i o n s l i n e a r l y (AH ( ( > t s A + A ) , a p r o c e d u r e we c o n s i d e r i n c o r r e c t ( s e c t i o n 5 . 4 . 2 ) . A l s o , Au = sin 0A(^> ^ s t h e c o r r e c t v a r i a b l e t o d e s c r i b e t h e p h y s i c a l s i t u a t i o n , e s p e c i a l l y a s n o s p e c i a l c o n d i t i o n s , a p a r t f r o m r a n d o m n e s s , n e e d be i m p o s e d i n o u r c a l c u l a t i o n . We m e n t i o n h e r e t h a t t h e m o d e l o f t h e s y s t e m o b t a i n e d i n c h a p t e r 4 i s c h e c k e d b y a l i n e w i d t h s t u d y s u c h a s t h i s . I f one a s s i g n s t h e g - a x i s p r o j e c t i o n s t o a n i n -c o r r e c t q u a d r a n t , t h e n t h e c a l c u l a t e d g - v a l u e s a n d l i n e w i d t h s f o r a g i v e n v a l u e o f t h e m a g n e t i c f i e l d d i r e c t i o n do n o t 181 a g r e e w i t h t h e e x p e r i m e n t a l l y o b s e r v e d l i n e w i d t h a n d g - v a l u e a t t h a t f i e l d o r i e n t a t i o n . T h i s was e x t r e m e l y u s e f u l i n t h e e a r l y s t a g e s when we w e r e t e s t i n g v a r i o u s p r o j e c t i o n s . 5.6.2 P h y s i c a l r e a s o n s f o r t h e p r e s e n c e o f d i s t o r t i o n s  a n d m i s o r i e n t a t i o n We h a v e a l r e a d y a l l u d e d t o t h e s o f t n e s s a n d f r a g i l i t y o f p r o t e i n c r y s t a l s a s a p o s s i b l e r e a s o n f o r t h e o r i e n t a t i o n o f t h e m o l e c u l e s r e l a t i v e t o e a c h o t h e r t o v a r y . T h i s p o s s i b i l i t y i s s u p p o r t e d b y t h e f a c t t h a t X - r a y d i f -o f r a c t i o n p i c t u r e s f a d e o u t a t 1 t o 2 A c o r r e s p o n d i n g t o a n u n c e r t a i n t y i n p o s i t i o n o f 1 t o 2 ° . T h i s i s much g r e a t e r t h a n i n i n o r g a n i c c r y s t a l s w h e r e i t i s a b o u t 0.2° ( S t o u t a n d J e n s e n , 1 9 6 9 ) . The s t r u c t u r e o f p r o t e i n s a s d e d u c e d f r o m X - r a y d i f f r a c t i o n a n a l y s i s shows t h e p r e s e n c e o f s u b s t a n t i a l a m o u n t s o f w a t e r o f c r y s t a l l i s a t i o n ( u p t o 5 0 % b y v o l u m e i n c y t o c h r o m e c ) . T h i s means t h a t p r o t e i n c r y s t a l s a r e n o t s o much " c r y s t a l s " a s " o r d e r e d p o l y e l e c t r o y l y t e g e l s " ( D i c k e r s o n , 1 9 6 7 ) . I t i s t h e r e f o r e r e a s o n a b l e t h a t t h e m o l e c u l a r p a c k i n g i n t h e c r y s t a l w o u l d be l o o s e , l e a d i n g t o t h e m i s o r i e n t a t i o n : r e q u i r e d b y o u r r e s u l t s . T h e l i g a n d f i e l d d i s t r i b u t i o n s ( AV, AD p r e s e n t i n f r o z e n c r y s t a l s a n d s o l u t i o n s o f p r o t e i n s c o u l d b e a n o t h e r m a n i f e s t a t i o n o f t h e n o n - r i g i d s t r u c t u r e o f t h e p r o t e i n m o l e c u l e — d u e t o t h e d y n a m i c J a h n - T e l l e r e f f e c t . 1 8 2 A s d i s c u s s e d i n c h a p t e r 4, t h i s m e c h a n i s m shows t h a t t h e l o w f r e q u e n c y ( i n f r a - r e d - b r e a t h i n g ) v i b r a t i o n s o f t h e heme r i n g c a n c o u p l e w i t h t h e i r o n e l e c t r o n i c d - o r b i t a l s , l o w e r i n g t h e l i g a n d f i e l d s y m m e t r y . We s u g g e s t t h a t t h e r e i s a v a r i a t i o n i n v i b r a t i o n f r e q u e n c i e s f r o m m o l e c u l e t o m o l e c u l e , h a v i n g a r a n g e o f v a l u e s a b o u t some mean f r e q u e n c y . S u c h a d i s t r i b u t i o n c o u l d a r i s e f r o m s l i g h t c h a n g e s i n t h e c o v a l e n t b o n d i n g o f t h e heme g r o u p i n t o t h e a m i n o a c i d c h a i n , c a u s e d b y c h a n g e s i n t h e a t o m i c p o s i t i o n s f r o m m o l e c u l e t o m o l e c u l e . The v i b r a t i n g heme g r o u p w o u l d be ' w e i g h e d down' t o a g r e a t e r o r l e s s e r e x t e n t , t h e r e b y g i v i n g a s p r e a d o f f r e q u e n c i e s . We c a n a l s o a s k : w h a t c h a n g e s c a n o c c u r i n t h e e n v i r o n m e n t o f t h e heme when t h e m o l e c u l e i s f r o z e n ? Two e f f e c t s m i g h t be e x p e c t e d — d i s t o r t i o n s due t o t h e p r e s e n c e o f i c e c r y s t a l s a n d c h a n g e s i n t h e v o l u m e o f t h e i c e / p r o t e i n m a t r i x a s a f u n c t i o n o f t e m p e r a t u r e . Y o n e t a n i a n d S c h l e y e r ( 1 9 6 7 ) c a r r i e d o u t a n i n v e s -t i g a t i o n o n t h e e f f e c t s o f p h y s i c a l s t a t e — d r y i n g , r e p e a t e d f r e e z i n g a n d t h a w i n g — o n t h e EPR s i g n a l s f r o m f e r r i m y o g l o b i n a n d c y t o c h r o m e c p e r o x i d a s e c r y s t a l s . T h e y showed t h a t s i g n a l s w e r e h i g h l y s u s c e p t i b l e t o s u c h c h a n g e s i n s t a t e . The o r i e n t a t i o n s became r a n d o m o r t h e EPR s i g n a l s d i s a p p e a r e d . T h e y s u g g e s t t h a t i r r e v e r s i b l e c h a n g e s may h a v e o c c u r r e d e v e n a t t h e i n i t i a l f r e e z i n g s t e p . T h i s i s s u p p o r t e d b y t h e w o r k o f Low e t a l . ( 1 9 6 6 ) who s h o w e d t h a t t h e m o s a i c c h a r a c t e r o f i n s u l i n c r y s t a l s was i r r e v e r s i b l y e n h a n c e d on c o o l i n g t o -150°C. I t i s d i f f i c u l t , h o w e v e r , t o p u t a n u m e r i c a l e s t i -m a t e on s u c h c h a n g e s t o t e s t w h e t h e r t h e y e x p l a i n o u r d a t a . Some i d e a o f t h e e f f e c t o f v o l u m e c h a n g e s o n t h e f r o z e n s o l i d c a n b e o b t a i n e d f r o m s t u d i e s o f p r o t e i n s a t h i g h p r e s s u r e s . G r e n o b l e e t a l . ( 1 9 6 8 ) r e p o r t e d c h a n g e s i n t h e M o s s b a u e r s p e c t r a o f h e m i n when u p t o 100 k i l o b a r s p r e s s u r e was a p p l i e d . T h e y a t t r i b u t e d t h e s e c h a n g e s t o d i s -t o r t i o n s o f t h e o r b i t a l s — t h e m o s t l i k e l y o f w h i c h was a s p r e a d i n g o f t h e 3d o r b i t a l s . T h i s w o u l d a l t e r t h e s h i e l d i n g o f t h e i n n e r o r b i t a l s , t h e r e b y c h a n g i n g t h e i n t e r a c t i o n o f t h e s - o r b i t a l s w i t h t h e i r o n - 5 7 n u c l e u s . A n y d - o r b i t a l v a r i a t i o n s c o u l d be e x p e c t e d t o a f f e c t t h e EPR s i g n a l s , b u t i t i s d i f f i c u l t t o e s t i m a t e how l a r g e t h i s e f f e c t m i g h t b e . 5.7 CONCLUSIONS We h a v e shown i n t h i s c h a p t e r how t h e EPR l i n e -w i d t h s o f s i n g l e c r y s t a l s o f f e r r i c y t o c h r o m e c a n b e r e a s o n a b l y e x p l a i n e d b y a c o m b i n a t i o n o f c r y s t a l l i t e m i s -o r i e n t a t i o n a n d m o l e c u l a r d i s t o r t i o n . T h e m i s o r i e n t a t i o n t h e o r y p r o p o s e d b y H e l c k e e t a l . was shown t o b e i n a d e q u a t e , due t o a c o n c e p t u a l e r r o r , a n d was c o r r e c t e d . I n a d d i t i o n , we e x t e n d e d t h e t h e o r y o f E i s e n b e r g e r a n d P e r s h a n — d e r i v e d f o r r h o m b i c p o t e n t i a l v a r i a t i o n s — t o a x i a l p o t e n t i a l s . The b e s t t h e o r e t i c a l f i t t o t h e d a t a r e s u l t e d f r o m a mean a n g u l a r d i s t r i b u t i o n o f 1.5° a n d a r h o m b i c p o t e n t i a l v a r i a t i o n o f a b o u t 1 1 % ( a x i a l a n d i s o t r o p i c c o n -t r i b u t i o n s b e i n g s m a l l i n c o m p a r i s o n ) . Some p h y s i c a l c a u s e s f o r t h e s e e f f e c t s w e r e g i v e n . CHAPTER 6*0 CYTOCHROME C EPR - LINESHAPES AND RAPID PASSAGE 6.1 INTRODUCTION The aim of t h i s chapter i s to explain the shapes of the EPR l i n e s seen i n frozen s i n g l e c r y s t a l s of f e r r i -cytochrome c, using the theory developed by P o r t i s (1955). I n section 6.2 we present the behavior of the si g n a l s obtained from a ferricytochrome c c r y s t a l , discuss how i t d i f f e r s from 'normal* slow passage EPR, and suggest that r a p i d passage may be the cause. Following t h i s we describe the basic theory of r a p i d passage i n a s i n g l e spin packet (se c t i o n 6.3) and generalise t h i s f o r a d i s t r i b u t i o n of packets ( s e c t i o n 6.4) by using the theory of P o r t i s (1955). Section 6.5 describes a t e s t of the theory using a model system. I n section 6.6 the cytochrome c r e s u l t s are presented and explained. Also given i n t h i s section are data on the r e l a x a t i o n time and spin packet linewidth of cytochrome c at 4.2°K, determined from the f a s t passage r e s u l t s . 6.2 THE NATURE OF THE PROBLEM -6.2.1 R e s u l t s t o be e x p l a i n e d A s t u d y o f o n e o f t h e EPR l i n e s f r o m a s i n g l e c r y s t a l o f h o r s e h e a r t f e r r i c y t o c h r o m e c a t 4.2°K g a v e t h e f o l l o w i n g s e t o f r e s u l t s . ( i ) n o d e t e c t a b l e a b s o r p t i o n d e r i v a t i v e s i g n a l . ( i i ) a l a r g e s i g n a l when t h e d e t e c t o r was t u n e d t o o b s e r v e d i s p e r s i o n , w i t h t h e s h a p e o f a n u n d i f f e r e n t i a t e d G a u s s i a n a b s o r p t i o n l i n e ( f i g u r e 6.1). ( i i i ) a s t h e m i c r o w a v e p o w e r i n c i d e n t o n t h e c r y s t a l was i n c r e a s e d , t h e s i g n a l o f ( i i ) f i r s t i n c r e a s e d i n a m p l i t u d e , t h e n l e v e l l e d o f f ( f i g u r e 6.2). ( i v ) t h e s i g n a l o f ( i i ) l a g g e d t h e m o d u l a t i o n f i e l d i n p h a s e b y 157°, a s c o m p a r e d w i t h a s t a n d a r d (DPPH) when t h e m o d u l a t i o n f r e q u e n c y was 100 k H z . ( v ) t h e s i g n a l o f ( i i ) h a d a maximum a m p l i t u d e a t m a g n e t i c f i e l d m o d u l a t i o n o f o n e g a u s s , a l t h o u g h t h e l i n e w i d t h i s 400 g a u s s ( f i g u r e 6.3). 6.2.2 D i s c u s s i o n T h e a b s e n c e o f t h e e x p e c t e d s l o w p a s s a g e EPR a b s o r p t i o n s i g n a l d e r i v a t i v e ( o f t h e f o r m shown i n t h e i n s e t t o f i g u r e 6.1) s u g g e s t s t h a t we a r e s a t i s f y i n g t h e s a t u r a t i o n c o n d i t i o n : /H,T » | (6.1) FIGURE 6 0 1 EPR d i s p e r s i o n l i n e f r o m a s i n g l e c r y s t a l o f h o r s e h e a r t c y t o c h r o m e c a t 4«,2°K. I n s e t : EPR a b s o r p t i o n d e r i v a t i v e s p e c t r u m o f c h a r r e d d e x t r o s e s t a n d a r d . , FIGURE 6.2 M i c r o w a v e p o w e r d e p e n d e n c e o f c y t o c h r o m e c s i n g l e c r y s t a l d i s p e r s i o n s i g n a l h e i g h t . Signal Height - x'^j " Arbitrary Units G6T FIGURE 6.3 M a g n e t i c f i e l d m o d u l a t i o n d e p e n d e n c e o f c y t o c h r o m e c s i n g l e c r y s t a l d i s p e r s i o n s i g n a l h e i g h t . w h e r e y i s t h e g y r e - m a g n e t i c r a t i o o f t h e e l e c t r o n i s t h e a m p l i t u d e o f t h e m i c r o w a v e r - f f i e l d i n g a u s s T i s t h e mean r e l a x a t i o n t i m e — A / ^I* T2 i s t h e s p i n - l a t t i c e r e l a x a t i o n t i m e T 2 i s t h e s p i n - s p i n r e l a x a t i o n t i m e . I f t h i s w e r e t h e c a s e t h e n a d i s p e r s i o n s i g n a l s h o u l d be d e t e c t a b l e , s i n c e d i s p e r s i o n d o e s n o t show t h e same s a t u r a t i o n b e h a v i o u r a s a b s o r p t i o n . D i s p e r s i o n t e n d s t o a l i m i t a t h i g h m i c r o w a v e p o w e r s ( A b r a g a m , 1 9 6 1 ) a n d t h i s i s s e e n h e r e ( f i g u r e 6 . 2 ) . ^ F r o m o u r w o r k o n l i n e s h a p e s i n c h a p t e r 3 we know t h i s d i s p e r s i o n i s G a u s s i a n , b u t we w o u l d e x p e c t i t t o h a v e t h e d e r i v a t i v e s h a p e shown: w h e r e D/A = 3.5, P o o l e ( 1 9 6 7 , p . 5 3 1 ) . The n e g a t i v e s i d e ( A ) o f t h e d i s p e r s i o n s i g n a l w o u l d b e e a s i l y v i s i b l e i n f i g u r e 6.1 i f i t w e r e p r e s e n t . I n a d d i t i o n , a b s o r p t i o n a n d d i s p e r s i o n s i g n a l s a r e u s u a l l y i n p h a s e w i t h t h e m a g n e t i c f i e l d m o d u l a t i o n , n o t l a g g i n g a s o b s e r v e d . F i n a l l y , t h e m a g n e t i c f i e l d m o d u l a t i o n a m p l i t u d e r e q u i r e d t o p r o d u c e t h e maximum s i g n a l h e i g h t i s v e r y much l e s s t h a n e x p e c t e d . N o r m a l l y maximum s i g n a l h e i g h t i s n o t 194 a t t a i n e d i n a n EPR l i n e u n t i l t h e m o d u l a t i o n a m p l i t u d e i s o f t h e same o r d e r a s t h e l i n e w i d t h ( P o o l e , 1967, p. 4 0 6 ) - -h e r e a l i n e 400 g a u s s w i d e h a s i t s maximum h e i g h t a t o n e g a u s s m o d u l a t i o n , . We t h e r e f o r e , h a v e a d i s p e r s i o n s i g n a l o f u n u s u a l s h a p e ( a r i s i n g f r o m a G a u s s i a n d i s t r i b u t i o n o f l i n e s ) w h o s e h e i g h t i s v e r y much g r e a t e r t h a n common EPR o p e r a t i n g c o n -d i t i o n s w o u l d p r e d i c t . 6.2.3 R a p i d a d i a b a t i c p a s s a g e B l o c h (1946) e x a m i n e d t h e e q u a t i o n o f m o t i o n o f a s p i n s y s t e m w i t h m a g n e t i s a t i o n , M , p l a c e d i n a m a g n e t i c f i e l d , H w h o s e v a l u e c a n b e s w e p t . The e q u a t i o n i s 4M= y M"xTF (6.2) d t He o b t a i n e d t w o s o l u t i o n s f o r ( 6 . 2 ) ; o n e i n w h i c h t h e f i e l d i s v a r i e d s u f f i c i e n t l y s l o w l y t h a t a l l t i m e s s t e a d y s t a t e c o n d i t i o n s o c c u r , t h e o t h e r when t h e f i e l d i s a l t e r e d s o q u i c k l y t h a t r e l a x a t i o n p r o c e s s e s do n o t h a v e t i m e t o a c t . T he f i r s t s o l u t i o n g i v e s t h e more u s u a l ' s l o w p a s s a g e ' EPR w h i l e t h e s e c o n d g i v e s r i s e t o ' r a p i d a d i a b a t i c p a s s a g e ' s i g n a l s . I t i s t h e s e l a t t e r w h i c h a p p e a r i n c y t o c h r o m e c , b y v i r t u e o f t h e l o n g r e l a x a t i o n t i m e o f t h e p a r a m a g n e t i c c e n t r e o f t h e m o l e c u l e . I n t h e n e x t s e c t i o n ( 6 0 3 ) we d e s c r i b e r a p i d p a s s a g e i n a s i n g l e s p i n s y s t e m , a n d e x t e n d t h e t r e a t m e n t t o a d i s t r i b u t i o n o f s p i n s i n s e c t i o n 6 . 4 . 6.3 RAPID A D I A B A T I C PASSAGE I N A S I N G L E S P I N PACKET - ( B l o c h , 1 S 4 6 ) R a p i d a d i a b a t i c p a s s a g e r e q u i r e d a s p e c i a l s o l u t i o n o f t h e e q u a t i o n o f m o t i o n o f a s p i n s y s t e m ; dM. • v . M X H ( 6 . 2 ) ^ - = y. M x H We assume t h e s p i n s a r e s u b j e c t t o a s t e a d y m a g n e t i c f i e l d , K, a l o n g t h e z - a x i s o f a r i g h t h a n d e d c o o r d i n a t e s y s t e m , - —' -«=•».-OH VT ^^.^^ r. 1 ^  „ 4-h ^ n r n i l a r U W ^ C t«J. ^ l i <«tA« -»- -h_w.t.<-» **1 * £^  "*~ '— — — - - . -~ ^ — v e l o c i t y TJj i n t h e x - y p l a n e . T h e s p e c i a l s o l u t i o n d e s c r i b e s t h e f o r c e d p r e c e s s i o n o f t h e m a g n e t i s a t i o n w i t h t h e same v e l o c i t y a s t h a t o f t h e a p p l i e d r - f f i e l d . T he c o m p o n e n t s o f M a r e : M y = M0- s in Q. sincu t ( 6 . 3 a ) M x = M 0-sin(9«cos cut ( 6 . 3 b ) M z = M 0 - COS 9 ( 6 . 3 c ) T h e s e s a t i s f y t h e e q u a t i o n o f m o t i o n ( 6 . 2 ) p r o v i d e d tan0 = H,/(H -H") ( 6 . 4 ) w h e r e 196 I f we v i e w t h e s o l u t i o n f r o m a c o o r d i n a t e s y s t e m r o t a t i n g w i t h a n g u l a r v e l o c i t y a b o u t H, we s e e t h a t f o r H - . H * » H^, i . e . a b o v e r e s o n a n c e , Q ~ 0 ° H - H* ~ Hx a t r e s o n a n c e , Q ~ 17/2 H - H^^C b e l o w r e s o n a n c e Q ~ 77 T h e r e f o r e , a b o v e r e s o n a n c e t h e m a g n e t i s a t i o n i s a l m o s t p a r a l l e l t o K. I f H i s d e c r e a s e d s l o w l y , t h e a n g l e b e t w e e n M a n d H i n c r e a s e s , a t t a i n i n g 77/2 a t r e s o n a n c e a n d e v e n t u a l l y r e a c h i n g a l i m i t i n g v a l u e o f 77" f a r b e l o w r e s o n a n c e , w h e r e M i s a n t i p a r a l l e l t o H. T h i s s o l u t i o n i s s t r i c t l y v a l i d o n l y when H i s c o n s t a n t , b u t a l s o h o l d s i f H i s v a r i e d ' s l o w l y ' . A c h a n g e i n H i s c a l l e d ' s l o w ' i f t h e m a g n e t i s a t i o n , M, i s c l o s e a t a l l t i m e s t o H ( t h e v e c t o r sum o f f L a n d H - H * ) • e r r 1 T h i s l e a d s t o t h e a d i a b a t i c c o n d i t i o n : dH 2 — « r H | 2 (s.s) dH w h e r e i s t h e maximum r a t e o f c h a n g e o f H i n g a u s s p e r s e c o n d . F r o m t h e d e s c r i p t i o n g i v e n a b o v e o f t h e b e h a v i o r o f M i t i s i m p l i e d t h a t t h e sweep t h r o u g h r e s o n a n c e m u s t o c c u r i n a s u f f i c i e n t l y s h o r t t i m e t h a t r e l a x a t i o n p r o c e s s e s c a n n o t r e s t o r e M p a r a l l e l t o H ( t h e e q u i l i b r i u m p o s i t i o n o f M). T h i s r e q u i r e s t h a t : AH > H i / (6.6) d t T I n a d d i t i o n t h e s l o w e s t p r e c e s s i o n t i m e m u s t be f a s t e r t h a n t h e r e l a x a t i o n t i m e s o t h a t we a l s o r e q u i r e : y H \ > VT . ( 6 . 7 ) C o m b i n i n g t h e s e ' r a p i d ' c o n d i t i o n s , ( 6 . 6 ) a n d ( 6 . 7 ) , w i t h ( 6 c 5 ) g i v e s t h e r e q u i r e m e n t s f o r r a p i d a d i a b a t i c p a s s a g e y H 2 . dH . H, / i I T / T ( 6 - 8 > T h e s a t u r a t i o n c o n d i t i o n f o r s l o w p a s s a g e — YH^T > 1 — i s s e e n f r o m ( 6 . 8 ) a s a n e c e s s a r y , b u t n o t s u f f i c i e n t , c o n d i t i o n f o r r a p i d a d i a b a t i c p a s s a g e t o . o c c u r . H a v i n g i n t r o d u c e d t h e c o n d i t i o n s u n d e r w h i c h r a p i d a d i a b a t i c p a s s a g e c a n a r i s e i n a s i n g l e s p i n p a c k e t , we now show how t h e s o l u t i o n s o f t h e e q u a t i o n s o f m o t i o n m u s t b e m o d i f i e d when t h e r e i s a d i s t r i b u t i o n o f s p i n p a c k e t s . 6 . 4 R A P I D A D I A B A T I C PASSAGE I N A D I S T R I B U T I O N OF S P I N  PACKETS 6 . 4 . 1 T h e t r e a t m e n t f o l l o w s t h a t o f P o r t i s ( 1 9 5 5 ) , a n d Weger ( 1 9 6 0 ) . T h e p a p e r s b y Hyde ( 1 9 6 0 ) o n f a s t p a s s a g e i n i r r a d i a t e d L i F , a n d b y B u g a i ( 1 9 6 3 ) o n r a p i d p a s s a g e a t h i g h m o d u l a t i o n f r e q u e n c i e s , s h o u l d b e m e n t i o n e d a l s o . T h e y p r o -v i d e g o o d d e s c r i p t i o n s , i n n o n - m a t h e m a t i c a l t e r m s , o f t h e b e h a v i o r o f d i s t r i b u t i o n s o f s p i n s y s t e m s . 6e4<,2 P o r t i s ( 1 9 5 5 ) b e g i n s w i t h t h e s o l u t i o n s o f dM _ v ,15? v ^ . (Mo - M) T §g = | ( H X H) + w h e r e M 0 = X 0 H i s t h e e q u i l i b r i u m v a l u e o f t h e s p i n m a g n e t i s a t i o n , , T h e s o l u t i o n s a r e , i n t e r m s o f t h e r f s u s c e p t i b i l i t i e s ( (JJQ - UJ) T  X' ( w ) « 2 " X : w , T l + ( / H , ? } 2 + ( a > o - c j ) 2 r 2 < 6 - 1 0 a ) I X ( w ) = T^o^-7 : 2 2 — 2 - ( 6 . 1 0 b ) I + (/H.T) +(cuo- CO) r w h i c h , p r o v i d e d y H , T > > | , g i v e s > 2 - - J -x » * o ( 6 . 1 1 b ) We n o t e t h a t t h e d i s p e r s i o n i s a t a maximum a t t h e l i n e c e n t r e when CO s 0Jo . P o r t i s t h e n c a r r i e d o u t t h e i n t e g r a -t i o n o f t h e e q u a t i o n s ( 6 . 1 0 a ) a n d ( 6 . 1 0 b ) o v e r a d i s t r i b u t i o n o f r e s o n a n t f r e q u e n c i e s , h ( OJ - OJ ) , f o r an e x p r e s s i o n o f t h e F O R M CO P (cu - ojl) r • h(cu - oJo) §, \ y% (co) = — Y CO T f z T2~2" d ( a ; T ) 0 To c o r r e s p o n d t o t h e u s u a l c o n d i t i o n s o f r e s o n a n c e , we t a k e CD c o n s t a n t , a n d a p p l y a s w e p t d . c . f i e l d s H' (t) - d H / d f + Hmcos(comt> w h e r e COS ( c u m t ) ± s t h e m a g n e t i c f i e l d m o d u l a t i o n o f . f r e q u e n c y cu m . H m i s t h e m a g n e t i c f i e l d m o d u l a t i o n a m p l i t u d e dH -gg - i s t h e c o n s t a n t r a t e o f s w e e p i n g t h e mean m a g n e t i c f i e l d H c A s we a r e w o r k i n g i n t e r m s o f t h e f r e q u e n c y , H'(t), i s c o n v e r t e d t o cu'(t) b y w'Ct) = xH'(t) = y . d H / d t + X H m c o s ( a J mt) , P o r t i s d e v e l o p e d t h e i n t e g r a l s b y r e p e a t e d i n t e g r a -t i o n b y p a r t s , a n d t h e r e s u l t s w e r e a s e r i e s o f t e r m s c o n s i s t i n g o f o n l y t h o s e e x p r e s s i o n s p e r i o d i c i n t u m e We g i v e h e r e t h e f i r s t t w o t e r m s o f t h i s s e r i e s s CO ° 0 X' feu) = -^Xo-w-c-cos ^. sin(comt- <^ )- h(cj-cj0) w h e r e - f -X '«'/i-cos(cumt).r H m - ^ - [ h ( u J - c u 0)|(6.i3) € s twm H m T ) / H , ^ = T (dH/dt) /H, <£, " t a n - ' f t j r n r ) C o n v e r g e n c e r e q u i r e s t h a t fJL a n d €COS<£jbe l e s s t h a n u n i t y . F o r a s l o w s weep r a t e o f t h e d o C . f i e l d j±< I i s e a s y t o f u l f i l * F o r € COS<P I l e s s t h a n u n i t v H m u s t be l e s s t h a n - m H, w h e n e v e r < d m T » I : t h i s r e s t r i c t i o n o n H i s n o t i. m n e c e s s a r y when C U m T I , a n d H c a n t h e n e x c e e d H, . m m 1 6.4.3 T h r e e s p e c i a l c a s e s c a n now b e c o n s i d e r e d . ( i ) When t h e r e l a x a t i o n t i m e , T , i s v e r y s h o r t b u t yH.-f i s s t i l l g r e a t e r t h a n u n i t y t h e n H w m T < -rr- < I a n d e x p r e s s i o n ( 6 . 1 2 ) i s t h e d o m i n a n t c o n t r i b u t i o n t o t h e s u s c e p t i b i l i t y , a n d s o w h i c h h a s t h e s h a p e o f a d e r i v a t i v e o f d i s p e r s i o n a n d i s i n p h a s e w i t h t h e m a g n e t i c f i e l d m o d u l a t i o n . ( i i ) A s t h e mean r e l a x a t i o n t i m e i n t h e s y s t e m i n c r e a s e s , t h e n we o b t a i n H , / H r n < wmr < I a n d now t h e f i r s t t e r m i n ( 6 . 1 3 ) i s t h e d o m i n a n t o n e s o t h a t H i H e r e ^ i s s m a l l , s o t h a t cos<jfc>( ~ \ . The s i g n a l s h a p e i s t h a t o f t h e a b s o r p t i o n e n v e l o p e , h ( u J - Otfo) a n d l a g s t h e m o d u l a t i o n f i e l d b y 9 0 ° . ( i i i ) F o r H|/ H^< | < WmT r e s u l t i n g f r o m a n e v e n l o n g e r r e l a x a t i o n t i m e t h a n i n ( i i ) , t h e f i r s t t e r m o f X , ' ^ i s s t i l l d o m i n a n t a n d 'X'(w) - ^ > r ; w § n . w l hTC0s ^ | . s l n ( w m t - ^ 1 ) - h ' ( c u - u ; o ) H e r e O J m T > I a n d c o s <pi —«• 0 ( s i n c e (£, —*» 9 0 ° ) , a n d t h e p r o d u c t C u m T .cos<£>j —«» 1 s o t h a t : IT v H I H X'fo) - f X 0 " - ^ L cos(wmt)-h(w-cu0; H i , (6.1.6) T h i s i s t h e e n v e l o p e o f t h e d i s t r i b u t i o n o f l i n e s b u t 180° o u t o f p h a s e w i t h t h e m o d u l a t i o n f i e l d . A t i n t e r m e d i a t e v a l u e s o f C U m T , t h e p h a s e o f t h e s i g n a l w i l l v a r y b e t w e e n t h e s e t w o l i m i t s (90° - 180°) w i t h l i t t l e c h a n g e i n m a g n i t u d s i n c e WtnT-COS<pi r e m a i n s a p p r o x i m a t e l y c o n s t a n t a s C t ) m o r T v a r y . 6.4.4 A t h i g h m o d u l a t i o n f i e l d s w h e r e H > H . , t h e s e r i e s m l e x p a n s i o n m e t h o d u s e d a b o v e b r e a k s down. P o r t i s s o l v e d t h e e q u a t i o n s o f m o t i o n m a k i n g c e r t a i n a p p r o x i m a t i o n s a b o u t t h e a r g u m e n t s o f i n t e g r a l s u s e d , a n d o b t a i n e d t h e r e s u l t s t h a t , f o r H > H.: m l X'H, OC - H,-ln[^]-coscumt-h(w-aj p) X'H, a - H . l n j 2 ^ ^ ] . Sind;mt.h((u-(u0) a i m r » | ( 6 . 1 7 ) g i v i n g a s i g n a l w h i c h i n c r e a s e s o n l y s l o w l y a s i n c r e a s e s . 6 . 4 o 5 We c a n s u m m a r i z e 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 s s o f a r , a s f o l l o w s : ( i ) f o r a mean r e l a x a t i o n t i m e , r , s h o r t e r t h a n a l l o t h e r t i m e s e x c e p t l / ( y H^) we o b t a i n t h e d i s p e r s i o n d e r i v a t i v e , i d e n t i c a l w i t h t h a t i n s l o w p a s s a g e EPR. T h e s i g n a l a m p l i t u d e we d e t e c t i s p r o p o r t i o n a l t o X a n d t h e r e -f o r e t o H^ s ( i i ) a s T i n c r e a s e s we g e t a c h a n g e o v e r t o r a p i d p a s s a g e when t h e d i s p e r s i o n s i g n a l h a s t h e a p p e a r a n c e o f a n a b s o r p t i o n c u r v e w i t h a n a m p l i t u d e p r o p o r t i o n a l t o H^, a n d w h o s e p h a s e l a g s b e h i n d t h e m o d u l a t i o n f r e q u e n c y . F o r l a r g e H^, t h e s i g n a l h e i g h t b e c o mes l o g a r i t h m i c w i t h H^. A l s o i n t h i s c a s e X i s i n d e p e n d e n t o f H^. 6.5 TEST OF PORTIS EQUATIONS WITH MODEL SYSTEM 6.5.1 I n t r o d u c t i o n I n o r d e r t o t e s t t h e s e r e s u l t s ( e x p r e s s i o n ( 6 . 1 4 ) -( 6 . 1 8 ) ) we s t u d i e d r a p i d p a s s a g e i n a m o d e l s y s t e m , u s i n g charred dextrose (Pastor and Hoskins, 1960)• This i s i d e a l f o r our purposes as a wide range of x-elaxation times can be obtained by s u i t a b l e choice of charring temperature c 6»5.2 Materials Heating the dextrose to 200°C breaks down the carbon bonds, g i v i n g free r a d i c a l s . For chars formed below 300°C the density of the r a d i c a l s i s low (5xl0 1 7/cm 3) and they have a number of d i f f e r e n t l o c a l environments. Under these conditions EPR shows that the r e s u l t i n g l i n e has Gaussian shape—-implying inhomogeneous broadening-—and the system has a long r e l a x a t i o n time. At higher temperatures, up to 565°C, the spin density increases, the l i n e becomes narrower and changes shape to Lorentzian, and the r e l a x a t i o n time shortens. Above 565°C the dextrose becomes l a r g e l y graphite and the spin density r a p i d l y f a l l s . We used dextrose heated to approximately 250°C f o r t h i s study. 6.5.3 Methods and Results The r a p i d passage properties we p a r t i c u l a r l y wished to examine were: ( i ) The e f f e c t of varying H^. With 0Jm set to 2 IT x 100 kHz and H m = 1 gauss the i n c i d e n t microwave power was varied from one microwatt to ten m i l l i w a t t s . .r • A t t h e v e r y l o w e s t p o w e r , w h e r e i s a p p r o x i -m a t e l y O e 0 0 1 g a u s s , t h e c o n d i t i o n / H 1 r < l was a t t a i n e d a n d s l o w p a s s a g e a b s o r p t i o n a n d d i s p e r s i o n d e r i v a t i v e s w e r e s e e n ( f i g u r e 6.4a shows a b s o r p t i o n ) . A s t h e p o w e r was i n c r e a s e d t h e s i g n a l c h a n g e d f r o m a d e r i v a t i v e t o a n a b s o r p t i o n s h a p e a p p r o x i m a t e l y 170° o u t o f p h a s e w i t h t h e m a g n e t i c f i e l d m o d u l a t i o n ( f i g u r e 6.4b) w h o s e a m p l i t u d e i n c r e a s e d t h e n l e v e l l e d o f f ( f i g u r e 6 . 5 ) . T h i s i s i n a g r e e m e n t w i t h t h e b e h a v i o u r p r e d i c t e d f r o m e q u a t i o n ( 6 * 1 6 ) — t h e s i g n a l h e i g h t b e c o m e s i n d e p e n d e n t o f when a l l t h e c o n d i t i o n s f o r r a p i d a d i a b a t i c p a s s a g e a r e f u l f i l l e d . ( i i ) T he e f f e c t o f v a r y i n g t h e m a g n e t i c f i e l d m o d u l a t i o n a m p l i t u d e , H . m A t a m o d u l a t i o n f r e q u e n c y , OJ , o f 2 7T x 100 k H z , a n d a p o w e r l e v e l o f 10 mW, t h e m o d u l a t i o n a m p l i t u d e was v a r i e d f r o m 0 t o 5 g a u s s . The r e s u l t i n g v a r i a t i o n i n s i g n a l h e i g h t i s shown i n f i g u r e 6.6, a n d shows t h e b e h a v i o u r p r e -d i c t e d b y e q u a t i o n s ( 6 . 1 6 ) a n d ( 6 . 1 7 ) v i z . , a l i n e a r i n c r e a s e o f s i g n a l w i t h i n c r e a s e d H^, l e v e l l i n g o f f a t h i g h H^. I t s h o u l d b e n o t e d t h a t t h e l i n e a r p o r t i o n o f t h e c u r v e e x t e n d s u p t o H^ = 0.5 g a u s s . T h i s i s a b o u t f i v e t i m e s g r e a t e r t h a n H, a n d s h o w s t h a t e q u a t i o n ( 6 . 1 6 ) h o l d s f o r H >H, when C U m T J. m l " 1 i s g r e a t e r t h a n 1. T h i s i s a c o n s e q u e n c e o f t h e c o n v e r g e n c e c o n d i t i o n s m e n t i o n e d i n t h e d i s c u s s i o n f o l l o w i n g e q u a t i o n ( 6 . 1 3 ) a b o v e . I t i s p a r t i c u l a r l y u s e f u l t h a t t h i s o c c u r s , FIGURE 6„4 EPR s p e c t r a f r o m c h a r r e d d e x t r o s e t e s t s a m p l e , ( a ) s l o w p a s s a g e a b s o r p t i o n d e r i v a t i v e • ^ ( b ) r a p i d p a s s a g e d i s p e r s i o n 206 Slow Passage Absorption P cav • Fast Passage Dispersi on P = IOmW cav Modulation Frequency = 100 kHz H m = I gauss FIGURE 6.5 M i c r o w a v e p o w e r d e p e n d e n c e o f c h a r r e d d e x t r o s e d i s p e r s i o n s i g n a l h e i g h t . 802 FIGURE 6.6 M a g n e t i c f i e l d m o d u l a t i o n a m p l i t u d e d e p e n d e n c e o f c h a r r e d d e x t r o s e d i s p e r s i o n s i g n a l h e i g h t . 210 s i n c e s i g n a l s c a n be m e a s u r e d a t l a r g e H^, w i t h c o r r e s p o n d i n g i n c r e a s e i n s i g n a l a m p l i t u d e , m a k i n g a c c u r a t e p h a s e m e a s u r e -m e n t s e a s i e r . ( T h e s i g n a l d r o p s a t h i g h m o d u l a t i o n a m p l i t u d e s b e c a u s e t h e c o n d i t i o n s a r e t h e n s u c h a s t o p r o d u c e n o n -a d i a b a t i c f a s t p a s s a g e ) . ( i i i ) T h e e f f e c t o f v a r y i n g a » m, t h e m a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y . ' • _. ' A t a m i c r o w a v e p o w e r l e v e l o f 10 mW a n d m o d u l a t i o n a m p l i t u d e o f 0.5 g a u s s , t h e s i g n a l l a g was m e a s u r e d o v e r t h e r a n g e o f f r e q u e n c i e s , 5 k H z t o 100 k H z . The p h a s e s w e r e o b t a i n e d b y a n u l l i n g m e t h o d - — m a k i n g t h e s i g n a l z e r o b y t u n i n g t h e P.A.R. HR-8 p h a s e s e n s i t i v e d e t e c t o r 90° o u t o f p h a s e w i t h t h e s i g n a l . T h e ' z e r o ' o f p h a s e was o b t a i n e d f r o m t h e s i g n a l a t t h e l o w e s t p o w e r s w h e r e f a s t p a s s a g e e f f e c t s d o n o t o c c u r . T h e p h a s e l a g s o b t a i n e d , a n d t h e c o r r e s p o n d i n g 0 J m T a r e shown b e l o w a n d i n f i g u r e 6.7. - p h a s e l a g ( f r o m d e f i n i t i o n wm/277\ r e l a t i v e t o z e r o o f <f> ^ 10 5 170.5 i 5° 6.0 5 x 10 4 158.2 i 5° 4.8 3 x 10 4 153.5 i 5° 3.4 10 4 133.0 2 5° 0.9 5 x 10 3 130.0 i 5° 0.8 FIGURE 6.7 M a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y d e p e n d e n c e o f p h a s e l a g o f c h a r r e d d e x t r o s e d i s p e r s i o n s i g n a l a t r o o m t e m p e r a t u r e . The m e a s u r e d l a g , i s r e l a t e d t o t h e o r d i n a t e - W m T - b y <£ ( = tan 'C^nf) S e e t e x t , s e c t i o n 6.5.3. 6 . 5 . 4 D i s c u s s i o n F e h e r (1959) a n d B u g a i (1963) b o t h show t h a t t h e o n s e t o f t h e p l a t e a u i n s i g n a l h e i g h t a s a f u n c t i o n o f s i g n i f i e s t h a t t h e m o d u l a t i o n a m p l i t u d e i s g r e a t e r t h a n t h e i n d i v i d u a l l i n e w i d t h o f t h e s p i n p a c k e t s t h a t make u p t h e i n h o m o g e n e o u s l y b r o a d e n e d l i n e . ( T h e s e a u t h o r s u s e a s e m i -c l a s s i c a l t r e a t m e n t f o r t h e c a s e o f c u m T » 1 a n d H >H, . m m l Theix- d e r i v a t i o n r e q u i r e s t h a t s w eep t h r o u g h a l a r g e n u m b er o f s p i n p a c k e t s ) . C o n s e q u e n t l y we c a n s e e f r o m f i g u r e 6,6 t h a t t h e s p i n p a c k e t w i d t h i s a p p r o x i m a t e l y o n e g a u s s , e q u i v a l e n t t o —8 a s p i n - s p i n r e l a x a t i o n t i m e , T 2 o f 5 x 10" s e c . i n a g r e e m e n t —8 w i t h t h e c a l c u l a t i o n s o f T 2 (~10 s e c ) made b y P a s t o r a n d H o s k i n s ( 1 9 6 0 ) . T h e P.A.R. HR-8 p h a s e s e n s i t i v e d e t e c t o r u s e d i n t h e s e e x p e r i m e n t s h a s a p o s s i b l e e r r o r i n p h a s e s e t t i n g o f 5° on t h e 50 k H z - 100 k H z r a n g e a n d t h i s a c c o u n t s f o r t h e l a r g e e r r o r b a r s a t t h e h i g h e s t f r e q u e n c i e s . A n y e r r o r i s e x a g g e r a t e d a l s o b e c a u s e o f t h e r a p i d i n c r e a s e i n t h e t a n g e n t f u n c t i o n c l o s e t o 90°. A t l o w e r f r e q u e n c i e s 5° o f e r r o r h a s much l e s s e f f e c t . We u s e d t h e v a l u e o f w m T o b t a i n e d a t 10 k H z , t o r e d u c e t h e e f f e c t o f t h i s p o s s i b l e e r r o r . H e n c e , OJmT - 0.9 s o T = 0.9 / ( 2 x 1 0 4 ) = 1.5 x 1 0 ~ 5 s e c . 3 X 1 ( 1 y H , T = d « 7 * 1 0 7 ) x QO"* 1) x 9.5 x 1 0 " 5 ) = 23.5 a t 10 mW. A t 1 m i c r o w a t t , = 10"" g a u s s , a n d y H | T ~ 0.24 i m p l y i n g t h a t we s h o u l d be o u t o f t h e f a s t p a s s a g e c o n d i t i o n s a s o b s e r v e d , . D u r i n g t h e i n v e s t i g a t i o n a t l o w o» m, i t was n o t i c e d t h a t a t r u e n u l l c o u l d n o t be o b t a i n e d - - f i g u r e 6.8a shows t h e t r a c e . T h i s i s an a b s o r p t i o n d e r i v a t i v e s i g n a l i n p h a s e w i t h t h e m a g n e t i c f i e l d m o d u l a t i o n , a l t h o u g h t h e s p e c t r o m e t e r i s t u n e d t o d i s p e r s i o n . I t a r i s e s f r o m t h e s e c o n d t e r m i n e q u a t i o n ( 6 . 1 3 ) , w h i c h i s a f u n c t i o n o f t h e sweep r a t e o f t h e d . c . m a g n e t i c f i e l d , d H / d t . I t s p h a s e i s c h a n g e d b y 180° o n r e v e r s a l o f t h e m a g n e t i c f i e l d sweep d i r e c t i o n a s r e q u i r e d b y t h e P o r t i s t h e o r y ( s e c o n d t e r m o f e q u a t i o n 6 . 1 3 ) . T h i s t e r m b e c o m e s a p p r e c i a b l e a t t h e l o w e r f r e q u e n c i e s b e -c a u s e t h e r a t e o f s w e e p i n g f i e l d b y a . c . m o d u l a t i o n — CO^ H m — b e c o m e s c l o s e r t o t h e r a t e o f t h e d . c . f i e l d s w e e p — d H / d t . The p r e s e n c e o f t h i s s i g n a l i s f u r t h e r p r o o f o f t h e v a l i d i t y o f t h e P o r t i s e q u a t i o n s . 6.6 RESULTS FOR CYTOCHROME C 6.6.1 I n t r o d u c t i o n We a r e now a b l e t o d i s c u s s t h e c y t o c h r o m e c r e s u l t s p r e s e n t e d i n s e c t i o n 6.2. To r e c a p i t u l a t e : we o b t a i n e d n o d e t e c t a b l e a b s o r p -t i o n d e r i v a t i v e , b u t i n s t e a d a n a b s o r p t i o n l i n e o f G a u s s i a n s h a p e , l a g g i n g t h e m a g n e t i c f i e l d m o d u l a t i o n . T h i s s i g n a l a l s o l e v e l l e d o f f a t l o w m o d u l a t i o n a m p l i t u d e s a n d h i g h m i c r o w a v e p o w e r s . FIGURE 6.8 EPR d i s p e r s i o n s p e c t r a o f c h a r r e d d e x t r o s e f o r a s p e c i a l c a s e o f t h e P o r t i s t h e o r y . ( a ) T h e l a r g e r p a s s a g e s i g n a l i s p h a s e d o u t — l e a v i n g t h e s m a l l e r ' a b s o r p t i o n d e r i v a t i v e ' s i g n a l a r i s i n g f r o m s e c o n d t e r m o f e q u a t i o n ( 6 . 1 3 ) . ( b ) B o t h p a s s a g e s i g n a l s . (a) o — (b) f m = 10 kHz P.A.R. Phase = 180 + 8 0 ° P.A.R. Phase = 270 + 80° 6.6.2 D i s c u s s i o n o f r e s u l t s o f s e c t i o n 6.2 We s u g g e s t t h a t t h e l a c k o f d e t e c t a b l e a b s o r p t i o n d e r i v a t i v e i s b e c a u s e y H . / r > l . E q u a t i o n ( 6 . 1 1 ) s hows t h a t u n d e r t h i s c o n d i t i o n X ^ s v © r y s m a l l . B e c a u s e t h e l i n e l a g s t h e m o d u l a t i n g f i e l d b y 157° a t a m o d u l a t i o n f r e q u e n c y o f 1 00 k H z , t h e r e l a t i o n t U m T > l h o l d s a t t h i s f r e q u e n c y ( e q u a t i o n s 6.15 a n d 6 . 1 6 ) ) . S i n c e i n t h e s p e c t r o m e t e r we d e t e c t X * H| ' e q u a t i o n s ( 6 . 1 5 ) a n d ( 6 . 1 6 ) a l s o p r e d i c t t h a t t h e s i g n a l s e e n w i l l b e i n d e p e n d e n t o f H^ ( a n d t h e r e f o r e o f t h e m i c r o -wave p o w e r ) a n d t h i s i s o b s e r v e d a t h i g h m i c r o w a v e p o w e r s o f 10 mW ( f i g u r e 6 . 2 ) . The v a r i a t i o n o f t h e s i g n a l h e i g h t w i t h m o d u l a t i o n a m p l i t u d e , H^, f o l l o w s d i r e c t l y f r o m t h e P o r t i s e q u a t i o n s . 6.6.3 I n f o r m a t i o n d e r i v e d f r o m d a t a a s s u m i n g P o r t i s  t h e o r y i s c o r r e c t We c a n c a l c u l a t e : ( i ) t h e v a l u e o f t h e mean r e l a x a t i o n t i m e , T ( i i ) a n a p p r o x i m a t e v a l u e o f t h e l i n e w i d t h o f t h e i n d i v i d u a l s p i n p a c k e t s t h a t make up t h e i n h o m o g e n e o u s l y b r o a d e n e d l i n e . ( i ) E v a l u a t i o n o f t h e mean r e l a x a t i o n t i m e - T T h e m a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y was v a r i e d b e t w e e n 10 k H z a n d 100 k H z a n d oJmT o b t a i n e d f r o m t h e m e a s u r e d p h a s e l a g a s i n t h e d e x t r o s e e x p e r i m e n t s . S i n c e t h e s i g n a l s f r o m t h e s i n g l e c r y s t a l s w e r e n o t v i s i b l e a t v e r y l o w p o w e r s , t h e z e r o o f p h a s e was o b t a i n e d f r o m m e a s u r e -m e n t s o n a DPPH c r y s t a l . T h i s m a t e r i a l d i d n o t show p a s s a g e f f e c t s a t 4.2°K a n d h i g h p o w e r s ( 1 0 mW) s o t h a t a ' z e r o — p h a s e ' - s e t t i n g c o u l d be o b t a i n e d . The r e s u l t s a r e g i v e n i n f i g u r e 6 . 9 , a n d i n d i c a t e t h e v a l u e o f T t o be 3 . 8 x 1 0 " " ^ s e c o n d s . T h e c o r r e s p o n d i n g v a l u e o f X H ^ T i s : / ^ T = ( 1 . 7 x 1 0 7 ) x ( 1 0 " 1 - ) x ( 3 . 8 x 1 0 ~ 6 ) = 6.4 w h i c h i m p l i e s t h a t we w i l l be o u t o f t h e s a t u r a t i o n r e g i o n _2 i f we r e d u c e t o 1 0 g a u s s o r l e s s ( e q u i v a l e n t t o r e -d u c i n g t h e m i c r o w a v e p o w e r f r o m 1 0 mW t o l e s s t h a n 1 0 0 flVl). I n f a c t , f a s t p a s s a g e i s s e e n down t o p o w e r s o f l e s s t h a n 1 0/XW w h i c h s u g g e s t s t h a t t h e v a l u e o f T e s t i m a t e d i s t o o s h o r t . I t i s d i f f i c u l t t o a c c o u n t f o r t h i s d i s c r e p a n c y e s p e c i a l l y i n v i e w o f t h e r e a s o n a b l e d e x t r o s e r e s u l t s . I t i s p o s s i b l e t h a t t h e u s e o f a d i f f e r e n t s a m p l e h o l d e r a n d s a m p l e f o r t h e ' z e r o p h a s e ' m e a s u r e m e n t i s t h e c a u s e , e v e n a l t h o u g h t h e l a r g e s i g n a l g i v e n b y t h e DPPH s a m p l e r e s u l t e d i n a v e r y a c c u r a t e n u l l b e i n g o b t a i n e d . P o r t i s ( 1 9 5 5 ) m e n t i o n s t h a t , u n l e s s t h e t i m e o f p a s s a g e t h r o u g h a n i n d i v i d u a l r e s o n a n c e l i n e i s s h o r t com-p a r e d w i t h b o t h t h e s p i n - l a t t i c e r e l a x a t i o n t i m e , a n d t h e s p i n d i f f u s i o n t i m e , t h e n d e v i a t i o n s f r o m h i s f a s t p a s s a g e t h e o r y a r e t o b e e x p e c t e d . S p i n d i f f u s i o n means t h a t when o n l y some o f t h e s p i n s i n a s y s t e m a r e r e s o n a t i n g , t h e e n e r g y i s t r a n s f e r r e d FIGURE 6.9 M a g n e t i c f i e l d m o d u l a t i o n f r e q u e n c y d e p e n d e n c e o f p h a s e l a g o f c y t o c h r o m e c s i n g l e c r y s t a l d i s p e r s i o n s i g n a l a t 4.2 T h e m e a s u r e d l a g , , i s r e l a t e d t o t h e o r d i n a t e - WrnT- b y = tan'Cu; T) Cytochrome c IV) s l o w l y v i a s p i n - s p i n i n t e r a c t i o n f r o m t h e r e s o n a n t s p i n s t o t h e r e m a i n d e r . S w e e p i n g a n i n h o m o g e n e o u s l y b r o a d e n e d l i n e w i t h a l i n e a r d . c . f i e l d ( d H / d t ) o n w h i c h i s s u p e r -i m p o s e d a v a r y i n g a . c . m a g n e t i c f i e l d ( t O j p H n i ) , m o d u l a t i o n p r o d u c e s r e s o n a n c e o n l y i n a r e g i o n 2 H m w i d e . I n o u r e x p e r i m e n t s 2 H m ~ 2 g a u s s a n d a s t h e i n h o m o g e n e o u s l i n e -w i d t h i s ~ 4 0 0 g a u s s , s p i n d i f f u s i o n c o u l d b e t a k i n g p l a c e , —6 l i m i t i n g t h e o v e r a l l r e l a x a t i o n t i m e t o 3 e 8 x 1 0 " s e c o n d s . ( i i ) E s t i m a t i o n o f s p i n p a c k e t w i d t h . A s a l r e a d y m e n t i o n e d ( s e c t i o n 6 . 5 . 4 ) , F e h e r ( 1 9 5 9 ) a n d B u g a i ( 1 9 6 3 ) b o t h show t h a t t h e o n s e t o f t h e p l a t e a u i n s i g n a l h e i g h t a s a f u n c t i o n o f m o d u l a t i o n a m p l i t u d e , Hm, ( f i g u r e 6 . 3 ) s i g n i f i e s t h a t Hm i s g r e a t e r t h a n t h e l i n e w i d t h s o f t h e i n d i v i d u a l s p i n p a c k e t s w h i c h make up t h e i n h o m o g e -n e o u s l y b r o a d e n e d l i n e . F r o m f i g u r e 6 . 3 we c a n e s t i m a t e t h e p a c k e t w i d t h (A H^) t o be a p p r o x i m a t e l y o n e g a u s s . T h e r e l a x a t i o n t i m e c o r r e s p o n d i n g t o t h i s l i n e w i d t h i s a b o u t —8 5 x 1 0 " s e c o n d s u s i n g t h e f o r m u l a ( A l g e r , 1 9 6 8 , p . 4 5 ) : T — — sec . 9/3 We s h a l l p o s t p o n e a n y c o n s i d e r a t i o n s o f t h e c a u s e s o f t h i s r e l a x a t i o n t i m e u n t i l s e c t i o n 7 . 6 . 1 w h e r e we s h a l l d i s c u s s t h e c a u s e s o f a l l t h e x - e l a x a t i o n t i m e s o b s e r v e d i n more d e t a i l . 6.7 CONCLUSIONS I n t h i s c h a p t e r we h a v e p r e s e n t e d o u r EPR r e s u l t s o n a s i n g l e c r y s t a l o f c y t o c h r o m e c , n a m e l y : n o d e t e c t a b l e a b s o r p t i o n d e r i v a t i v e , b u t i n s t e a d a n a b s o r p t i o n l i n e o f G a u s s i a n s h a p e , 157° o u t o f p h a s e w i t h t h e m a g n e t i c f i e l d m o d u l a t i o n . The s i g n a l h e i g h t l e v e l l e d o f f a t h i g h m i c r o -w a v e p o w e r s a n d a t l o w m o d u l a t i o n a m p l i t u d e s ( a t much l o w e r a m p l i t u d e s t h a n t h e w i d t h o f t h e l i n e w o u l d i m p l y ) . We s u g g e s t e d t h a t r a p i d a d i a b a t i c p a s s a g e m i g h t b e t h e c a u s e a n d t h e n o u t l i n e d t h e c o n d i t i o n s u n d e r w h i c h p a s s a g e e f f e c t s c o u l d be s e e n g i v i n g a b r i e f d e s c r i p t i o n o f P o r t i s 1 t h e o r y f o r s u c h e f f e c t s i n i n h o m o g e n e o u s l y b r o a d e n e d l i n e s . T h e t h e o r y was t e s t e d b y s t u d y i n g t h e r a p i d p a s s a g e b e h a v i o u r o f a m o d e l s y s t e m o f c h a r r e d d e x t r o s e . T h e d e x t r o s e e x p e r i m e n t a l r e s u l t s w e r e s u c c e s s f u l l y e x p l a i n e d o n t h e b a s i s o f P o r t i s * t r e a t m e n t . T h u s e n c o u r a g e d , we d i s c u s s e d o u r c y t o c h r o m e c r e s u l t s i n t h e l i g h t o f h i s t h e o r y . F i n a l l y t h e o v e r a l l r e l a x a t i o n t i m e o f t h e s y s t e m a n d t h e s p i n p a c k e t l i n e w i d t h s w e r e e s t i m a t e d f o r t h e t e m p e r a t u r e o f 4.2°K, a n d f o u n d t o b e .3.8 x 10""^ s e c o n d s a n d AH, = 1 g a u s s . CHAPTER 7.0 RELAXATION TIME STUDY OF CYTOCHROME C 7.1 INTRODUCTION The e x p e r i m e n t s o n c y t o c h r o m e c i n t h e p r e v i o u s c h a p t e r w e r e c a r r i e d o u t a t 4.2°K. The f a s t p a s s a g e e f f e c t s o b s e r v e d r e m a i n a t t e m p e r a t u r e s u p t o a p p r o x i m a t e l y 20°K, w h e r e t h e o v e r a l l r e l a x a t i o n t i m e ( r ) becomes s u f f i c i e n t l y s h o r t t h a t t h e u s u a l s l o w p a s s a g e c o n d i t i o n s p r e v a i l . By u s e o f t h e P o r t i s ( 1 9 5 5 ) t h e o r y p r e s e n t e d i n t h e p r e v i o u s c h a p t e r we c a l c u l a t e t h e m a g n i t u d e o f t h e f a s t p a s s a g e s i g n a l h e i g h t a s a f u n c t i o n o f T ( s e c t i o n 7 . 2 ) . F r o m t h e e x p e r i m e n t a l r e s u l t s we o b t a i n t h e v a r i a t i o n i n s i g n a l h e i g h t a s t h e t e m p e r a t u r e i s a l t e r e d a n d we c o m b i n e t h e m w i t h t h e t h e o r y o f s e c t i o n 7.2 t o p r o d u c e a g r a p h o f r e l a x a t i o n t i m e a s a f u n c t i o n o f t e m p e r a t u r e i n t h e r a n g e 4.2°K t o 20°K ( s e c t i o n 7 . 3 ) . A b o v e 50°K, t h e r e l a x a t i o n t i m e c a n b e d e t e r m i n e d d i r e c t l y f r o m t h e e x p e r i m e n t a l c u r v e s ( s e c t i o n 7 . 4 ) . A f t e r a b r i e f r e v i e w o f r e l a x a t i o n t i m e t h e o r i e s ( s e c t i o n 7 . 5 ) , we d i s c u s s t h e e x p e r i m e n t a l d a t a i n t h e l i g h t o f t h e m ( s e c t i o n 7 . 6 ) . 7.2 CALCULATION OF RAPID PASSAGE SIGNAL AS A FUNCTION  OF RELAXATION TIME F r o m e q u a t i o n ( 6 . 1 3 ) i n t h e p r e v i o u s c h a p t e r we h a v e : X ' H i = ^ X ^ m H m C O S ^ . cjmr . sin(wmt-(^ )h(a;-aj0) ( 7 . 1 ) a t t h e o u t p u t o f t h e p h a s e s e n s i t i v e d e t e c t o r t h i s b e c o m e s X^ 'Hj = const-aJmT-COS <jf>( a t t h e l i n e c e n t r e ( 7 . 2 ) w h e r e <£ ( = t a n " 'w mT T h e s i g n a l p h a s e c h a n g e s w i t h v a r i a t i o n i n T due t o t h e s i n (to t -<p,) t e r m i n ( 7 . 1 ) . m ' 1 T h e r e i s n o t t i m e i n t h e e x p e r i m e n t t o f i n d t h e o p t i m u m p h a s e o f t h e s i g n a l a t e a c h p o i n t d u r i n g t h e w a r m i n g o f t h e s a m p l e , a s t h e t e m p e r a t u r e c h a n g e s t o o r a p i d l y . C o n s e q u e n t l y , we m u l t i p l y t h e t h e o r e t i c a l s i g n a l b y c o s C ^ g ^ q "tfrref^ w h e r e < £ s i q ^ s *" h e c a l c u l a t e d <^ f o r a g i v e n CU mT, a n d <prQf i s t h e a n g l e c o r r e s p o n d i n g t o t h e s t a r t i n g v a l u e ofCt> mT. The f i n a l e x p r e s s i o n i s ( 7 . 3 ) ^ X H , % a . c = C O n S t W m T - C O S ^ s i g - C O s C ^ j g - ^ ) T h i s e q u a t i o n i s p l o t t e d f o r a r a n g e o f C J m T i n f i g u r e 7 .1. FIGURE 7.1 R a p i d p a s s a g e s i g n a l h e i g h t a s a f u n c t i o n o f r e l a x a t i o n t i m e . The c u r v e i s a p l o t o f e q u a t i o n ( 7 . 3 ) d e r i v e d f r o m t h e P o r t i s e q u a t i o n s f o r r a p i d a d i a b a t i c p a s s a g e . 7.3'RELAXATION TIMS FROM FAST PASSAGE BETWEEN 4.2° AND 2Q°K The t e m p e r a t u r e r o s e i n t h e c a v i t y o n c e t h e h e l i u m h a d b o i l e d o f f , a n d t h e s i g n a l f r o m a s i n g l e c r y s t a l l i n e was o b t a i n e d a s a f u n c t i o n o f t e m p e r a t u r e . The s i g n a l d r o p s i n i n t e n s i t y n o t o n l y f r o m c h a n g e i n r e l a x a t i o n t i m e b u t a l s o f r o m t h e d e c r e a s e i n s p i n p o p u l a t i o n d i f f e r e n c e b e t w e e n t h e t w o e n e r g y l e v e l s . S i n c e t h e t e m p e r a t u r e o f e a c h s i g n a l i s known, t h e l a t t e r e f f e c t c a n b e a l l o w e d f o r , a n d t h e r e s u l t s o f s e v e r a l r u n s a r e shown i n f i g u r e 7.2 n o r m a l i z e d t o t h e p o p u l a t i o n d i f f e r e n c e a t 4.2°K. The s i g n a l s f r o m v a r i o u s c r y s t a l s v a r i e d due t o d i f f e r e n c e s i n c r y s t a l s i z e a n d p o s i t i o n o f r e s o n a n c e l i n e . A l l l i n e s w e r e i n t h e r e g i o n g s 2 . 5 t o 2 , 9 . F o r e a c h h e i g h t t h e r e c o r r e s p o n d v a l u e s o f T a n d t e m p e r a t u r e w h i c h we p l o t i n f i g u r e 7.3. The g r a p h i s d r a w n a s ( 1 / T) v e r s u s t e m p e r a t u r e o n a l o g - l o g s c a l e . T h e s o l i d l i n e o n t h e g r a p h i s t h a t o f a T^ d e p e n d e n c e o f ^ . 7.4 RELAXATION TIMES FROM EPR LINEWIDTHS ABOVE 20°K 7.4.1 I n t r o d u c t i o n A b o v e 20°K c o n v e n t i o n a l s l o w p a s s a g e s i g n a l s a r e p r e s e n t , d e t e c t e d a s d e r i v a t i v e s o f t h e a b s o r p t i o n . A s a l r e a d y m e n t i o n e d i n t h e l i n e w i d t h c h a p t e r s , when t h e t e m -p e r a t u r e i s g r e a t e r t h a n 50°K t h e G a u s s i a n l i n e s h a p e p r o d u c e d b y a d i s t r i b u t i o n o f s p i n p a c k e t s b e c o m e s b r o a d e n e d . T h i s i s FIGURE 7.2 R a p i d p a s s a g e s i g n a l h e i g h t a s a f u n c t i o n o f a b s o l u t e t e m p e r a t u r e * 1.0 0.01 r Slow Passage -i 1— 1 _i i _J ; „i • •• i_ 4 8 12 16 20 Temperature - ° K FIGURE 7.3 P l o t o f i n v e r s e r e l a x a t i o n t i m e v e r s u s t e m p e r a t u r e b e t w e e n 4.2°K a n d 20 °K. The d a t a p o i n t s a r e t a k e n f r o m f i g u r e s 7.1 a n d 7.2 ( s e e t e x t ) . TABLE X I I I Summation o f L o r e n t z i a n l i n e s w i t h a G a u s s i a n d i s t r i b u t i o n G a u s s i a n W i d t h - 300 gau s s P e a k - t o - P e a k L i n e w i d t h L o r e n t z i a n L i n e w i d t h T ^ a l ^ w i d t h AH . , r , (Gauss) A H h ( G a u s s ) P f cP < G a u s s > 50 100 200 312 340 400 260 280 320 300 400 500 480 560 640 350 380 430 d u e t o t h e i n d i v i d u a l s p i n p a c k e t s t h a t make up t h e d i s -t r i b u t i o n h a v i n g l i n e w i d t h s o f t h e same o r d e r o f m a g n i t u d e a s t h e w i d t h o f t h e d i s t r i b u t i o n i t s e l f , 7,4.2 A n a l y s i s B e l o w 20°K t h e f a s t p a s s a g e c o n d i t i o n g a v e a n • a b s o r p t i o n * l i n e s h a p e f r o m w h i c h t h e w i d t h a t h a l f h e i g h t c o u l d b e o b t a i n e d d i r e c t l y . A b o v e 20°K we h a v e c o n d i t i o n s o f s l o w p a s s a g e t h r o u g h t h e l i n e y i e l d i n g t h e c o n v e n t i o n a l d e r i v a t i v e d i s p l a y . The l i n e w i d t h m e a s u r e d i n t h e d e r i v a -t i v e d i s p l a y i s t h e p e a J c - t o - p e a k s e p a r a t i o n ( A Hp£p) a n < i t h i s d o e s n o t c o i n c i d e w i t h t h e w i d t h a t h a l f h e i g h t o f t h e a b s o r p t i o n c u r v e ( A H ^ ) . F o r a L o r e n t z i a n l i n e , * \ = 1 . 7 3 - * H p t p a n d f o r a G a u s s i a n l i n e A H , = 1.17- A H . % p t p I f t h e l i n e s h a p e i s a c o m b i n a t i o n o f t h e t w o , t h e m u l t i -p l y i n g f a c t o r l i e s b e t w e e n t h e s e t w o v a l u e s . We w r o t e a c o m p u t e r p r o g r a m t o sum t h e c o n t r i b u -t i o n s f r o m a G a u s s i a n d i s t r i b u t i o n o f L o r e n t z i a n l i n e s . T a b l e X I I I o p p o s i t e shows t h e r e s u l t s f o r a G a u s s i a n o f 300 g a u s s w i d t h , o n t o w h i c h h a v e b e e n f o l d e d L o r e n t z i a n s o f v a r i o u s w i d t h s . The p e a k - t o - p e a k l i n e w i d t h s a r e t h o s e a t t h e p o i n t s o f maximum s l o p e a n d t h e s e a r e u s e d f o r c o m p a r i s o n w i t h t h e e x p e r i m e n t a l r e s u l t s . T h e l i n e s h a p e i s a p p r o x i m a t e l y L o r e n t z i a n . T h i s i s r e a s o n a b l e s i n c e t h e w i d e w i n g s o f t h e L o r e n t z i a n s h a p e a d d t o b r o a d e n t h e d i s t r i b u t i o n e v e n when t h e i n d i v i d u a l L o r e n t z i a n l i n e w i d t h s aire s m a l l c . By m e a s u r i n g t h e p e a k - t o -p e a k d e r i v a t i v e , l i n e w i d t h s a s t h e t e m p e r a t u r e i n c r e a s e s a n d c o m p a r i n g t h e m w i t h t h e c a l c u l a t e d w i d t h s , o n e c a n o b t a i n t h e w i d t h o f t h e L o r e n t z i a n s a s a f u n c t i o n o f t e m -p e r a t u r e . 7,4.3 R e s u l t s T h e d a t a f r o m o n e c r y s t a l w h i c h h a d p a r t i c u l a r l y n a r r o w l i n e s ( 3 0 0 g a u s s a t 4.2°K) a r e p l o t t e d i n f i g u r e 7.4. T h e o p e n c i r c l e s a r e t h e L o r e n t z i a n l i n e w i d t h s w h i c h f i t t e d t h e e x p e r i m e n t a l l y o b s e r v e d d e r i v a t i v e w i d t h s ( a s s u m i n g a 300 g a u s s d i s t r i b u t i o n ) , t o g e t h e r w i t h t h e t o t a l o b s e r v e d w i d t h o f t h e a b s o r p t i o n l i n e s h a p e f r o m t h e c o m b i n a -t i o n o f L o r e n t z i a n a n d G a u s s i a n ( © ) a l l a s a f u n c t i o n o f t e m p e r a t u r e . F r o m t h e L o r e n t z i a n s we o b t a i n t h e i n v e r s e r e l a x a t i o n t i m e (1/T) i n s e c o n d s , c o r r e s p o n d i n g t o t h e i r l i n e w i d t h i n g a u s s , f r o m ( A l g e r , 1968, p . 4 5 ) : J - ( l i n e w i d t h i n g a u s s ) , m *\ T ° 5T1" x 1 0 - 8 — s e c o n d s ( 7 . 4 ) D o i n g t h i s j we o b t a i n : FIGURE 7.4 I n d i v i d u a l c o n t r i b u t i o n s t o t h e o v e r a l l l i n e w i d t h f r o m a G a u s s i a n d i s t r i b u t i o n 300 g a u s s w i d e a n d L o r e n t z i a n l i n e s b e t w e e n 30 a n d 77°K i n a c y t o c h r o m e c s i n g l e c r y s t a l ( s e e t e x t ) . 600| J , •-• Total / i (Lorentzian / » Gaussian ) / / f i o—o Lorentzian 400 \ ,/* f 1 Gaussian — ---o / I I 2 0 0 \ 0yo / / / o / / /° — + < 1 ( 1 . 1 — 30 50 70 90 Temperature - ° K I n v e r s e T°K L o r e n t z i a n W i d t h R e l a x a t i o n T i m e (1/ T ) 4 9 5 0 8 . 9 . 1 0 8 5 3 1 0 0 1 . 8 . 1 0 9 5 7 . 5 1 9 0 3 c 4 . 1 0 9 6 2 2 0 5 3 . 7 . 1 0 9 6 5 3 0 0 5 . 4 . 1 0 9 6 8 4 0 0 7 . 1 . 1 0 9 We u s e t h e s e r e l a x a t i o n t i m e s i n s e c t i o n 7 . 6 . 3 t o d e c i d e b e t w e e n t w o t h e o r e t i c a l m o d e l s o f t h e v a r i a t i o n o f r e l a x a t i o n t i m e w i t h t e m p e r a t u r e . 7 . 5 REVIEW OF THEORIES OF RELAXATION TIMES The mean r e l a x a t i o n t i m e o f t h i s s y s t e m , T , i s e q u a l t o ( T ^ T 2 ) w h e r e T^ i s t h e s p i n - l a t t i c e r e l a x a t i o n t i m e a n d i s t h e s p i n - s p i n r e l a x a t i o n t i m e . I n g e n e r a l , t h e s p i n - s p i n r e l a x a t i o n t i m e d o e s n o t c h a n g e r a p i d l y w i t h t e m p e r a t u r e , w h i l e t h e s p i n - l a t t i c e t i m e d o e s . A s we w i s h t o e x p l a i n a d e p e n d e n c e o f r e l a x a t i o n t i m e 7 o n t e m p e r a t u r e ( T ) o f T , o u r a t t e n t i o n w i l l b e f o c u s s e d o n t h e s p i n - l a t t i c e r e l a x a t i o n t i m e . T h e m o s t d i r e c t i n t e r a c t i o n o f t h e s p i n s y s t e m i s w i t h t h e t h e r m a l e l e c t r o - m a g n e t i c r a d i a t i o n b a t h — e n e r g y b e i n g l o s t f r o m t h e s y s t e m b y d o w n w a r d m a g n e t i c r e s o n a n c e t r a n s i t i o n s i n d u c e d b y p h o t o n s . H o w e v e r , t h e number o f p h o t o n s w i t h s u i t a b l e e n e r g i e s a t t h e f r e q u e n c y a n d t e m p e r a t u r e i n v o l v e d i s t o o s m a l l t o g i v e r e l a x a t i o n t i m e s s h o r t e n o u g h t o a g r e e w i t h e x p e r i m e n t * W a l l e r ( 1 9 3 2 ) p r o p o s e d t h a t , s i n c e t h e e n e r g y d e n s i t y o f l a t t i c e v i b r a t i o n s ( t h e p h o n o n r a d i a t i o n b a t h ) 15 was o f o r d e r 10 t i m e s g r e a t e r t h a n t h e p h o t o n r a d i a t i o n b a t h , m o d u l a t i o n o f t h e s p i n - s p i n i n t e r a c t i o n b y p h o n o n s c o u l d i n d u c e m a g n e t i c r e s o n a n c e t r a n s i t i o n s a n d t h u s p r o v i d e a r e l a x a t i o n p a t h w a y f o r t h e s p i n s y s t e m t o l o s e e n e r g y . He d i s t i n g u i s h e d t w o p r o c e s s e s : ( i ) D i r e c t i n t e r a c t i o n s - A p h o n o n o f t h e same e n e r g y a s t h e s p i n q u a n t u m r e q u i r e d f o r a r e s o n a n c e t r a n s i -t i o n i s e m i t t e d , a n d t h i s i s a c c o m p a n i e d b y a 'down' t r a n s i t i o n i n t h e s p i n s y s t e m . ( i i ) Raman i n t e r a c t i o n s - A p h o n o n o f a n y f r e q u e n c y (CU /277) i n t e r a c t s w i t h a s p i n ( u p o r down) c a u s i n g a t r a n s i t i o n w i t h i n t h e s p i n s y s t e m , t h e p h o n o n b e i n g s c a t -t e r e d w i t h a d i f f e r e n t f r e q u e n c y (.(JJ^/ZTT) + V , w h e r e V i s t h e m a g n e t i c r e s o n a n c e f r e q u e n c y . T h e Raman p r o c e s s d o m i n a t e s o v e r t h e d i r e c t p r o -c e s s a t t e m p e r a t u r e s a b o v e a f e w d e g r e e s K e l v i n . T h i s i s b e c a u s e a t t e m p e r a t u r e , T, w h e r e ( h i / / k T ) " " ! o n l y a s m a l l p r o p o r t i o n o f t h e p h o n o n s p r e s e n t l i e i n t h e f r e q u e n c y r a n g e V iA*/ w h i l e t h e r e a r e l a r g e n u m b e r s o f p h o n o n s w i t h e n e r g i e s d i f f e r i n g b y h i / . H o w e v e r , W a l l e r ' s t h e o r y o f p h o n o n m o d u l a t i o n o f t h e s p i n - s p i n i n t e r a c t i o n g a v e r e l a x a t i o n t i m e s many o r d e r s o f m a g n i t u d e t o o l o n g . K r o n i g a n d Bouwkamp ( 1 9 3 9 ) a n d V a n V l e c k ( 1 9 4 0 ) p o s t u l a t e d a n o t h e r r e l a x a t i o n m e c h a n i s m t h a t e m p l o y s m o d u l a t i o n b y p h o n o n s a n d w h i c h g i v e s much b e t t e r q u a n t i -t a t i v e a g r e e m e n t w i t h e x p e r i m e n t s . T h e y p r o p o s e d t h a t l a t t i c e v i b r a t i o n s m o d u l a t e d t h e l i g a n d f i e l d . T h i s p r o d u c e a v a r i a b l e e l e c t r i c f i e l d w h i c h c a n i n t e r a c t w i t h t h e e l e c t r o n i c o r b i t s , a n d , v i a s p i n - o r b i t c o u p l i n g , w i t h t h e e l e c t r o n s p i n s . T h e e f f e c t i v e n e s s o f t h i s m e c h a n i s m i s d e p e n d e n t o n t h e s t r e n g t h o f t h e s p i n - o r b i t c o u p l i n g — - w i t h s m a l l s p i n -o r b i t c o u p l i n g , i . e . g - v a l u e s c l o s e t o t h e f r e e s p i n v a l u e o f 2 . 0 0 2 3 , t h e r e l a x a t i o n t i m e s w i l l be l o n g . I n o u r c y t o -c h r o m e c s y s t e m , w h o s e g - v a l u e s d e v i a t e m a r k e d l y f r o m 2, we e x p e c t t h e r e l a x a t i o n t i m e s g i v e n b y t h i s m e c h a n i s m t o b e s h o r t , e x c e p t when t h e r e a r e f e w p h o n o n s p r e s e n t . We m u s t a l s o c o n s i d e r t h e O r b a c h p r o c e s s ( 1 9 6 1 ) : T h i s m e c h a n i s m r e q u i r e s t h e p r e s e n c e o f a l o w l y i n g s t a t e w i t h e n e r g y , A , a f e w t e n s o f i n v e r s e c e n t i m e t r e s a b o v e t h e m a g n e t i c s t a t e s , t o a n d f r o m w h i c h t r a n s i t i o n s c a n b e made. I n g e n e r a l t h e t e m p e r a t u r e d e p e n d e n c e o f t h e s p i n -l a t t i c e r e l a x a t i o n t i m e c a n be w r i t t e n a s '/j, = a"T + + c -exp( -A/T) ( 7 . 5 ) T h e f i r s t t e r m comes f r o m t h e o n e p h o n o n d i r e c t p r o c e s s , t h e n e x t a r i s e s f r o m t h e Raman p r o c e s s , a n d t h e l a s t f r o m 241 t h e Orbach p r o c e s s . A t v e r y h i g h t e m p e r a t u r e s t h e Raman p r o c e s s goes 2 o v e r t o a T dependence f o r 1/T^, a t t e m p e r a t u r e s above t h e Debye t e m p e r a t u r e o f the sample ( T p ) , ( A b r a g a m a n d B l e a n e y , 1970, p. 572). We now w i s h t o d i s c u s s o u r measurements i n t h e l i g h t o f t h e s e t h e o r i e s . 7.6 DISCUSSION 7.6.1 The mean r e l a x a t i o n t i m e o f t h i s h o r s e h e a r t c y t o c h r o m e c s y s t e m i s a p p r o x i m a t e l y c o n s t a n t f r o m 4.2 t o 9°K, t h e n s t a r t s t o r i s e f o l l o w i n g an a p p r o x i m a t e l y T dependence up t o 20°K. We s u g g e s t t h a t t h e p r o c e s s w h i c h g i v e s r i s e t o t h i s power t e m p e r a t u r e dependence i s a c o m b i n a t i o n o f t h e 9 T Raman s p i n - l a t t i c e r e l a x a t i o n p r o c e s s and a t e m p e r a t u r e i n s e n s i t i v e s p i n - s p i n r e l a x a t i o n p r o c e s s . The e v i d e n c e f o r t h i s model i s as f o l l o w s : ( i ) We have measured t h e w i d t h s o f t h e s p i n - p a c k e t s o f t h e i n d i v i d u a l l i n e s t o be a p p r o x i m a t e l y one g a u s s . The s i m p l e s t c ause o f a b r o a d e n i n g o f t h e n a t u r a l l i n e w i d t h t o 1 gauss i s d i p o l a r s p i n - s p i n i n t e r a c t i o n between t h e e l e c t r o n s o f n e i g h b o r i n g i r o n atoms. We know from X - r a y d a t a ( D i c k e r s o n e t a l . , 1971) o t h a t t h e heme i r o n atoms a r e a p p r o x i m a t e l y 25 A a p a r t i n c y t o c h r o m e c . The i n d u c e d f i e l d , B, due t o s i x n e a r e s t n e i g h b o u r s ( a s s u m i n g r o u g h l y c u b i c p a c k i n g ) i s : B - ^ L - ( 7 . 6 ) 4-77 w i t h B a B o h r m a g n e t o n ( 1 . 1 6 x 10"~^ 4 w e b e r / m e t r e ) r = 25 x 10~ 1 0 m e t r e s T h i s g i v e s B~3 g a u s s , s u p p o r t i n g o u r h y p o t h e s i s . We a l s o n o t e t h a t a n o t h e r s o u r c e o f b r o a d e n i n g c o u l d b e f r o m p r o t o n s c l o s e t o t h e i r o n . B o z a n i c e t a l . ( 1 9 6 9 ) s h o w e d b y s p i n - e c h o e x p e r i m e n t s t h a t t h e r e w e r e -7 r e l a x a t i o n t i m e s o f 10 s e c o n d s o r l e s s i n f r o z e n f e r r i -m y o g l o b i n s o l u t i o n s a t 4.2°K. T h e y a t t r i b u t e d t h e s e t o a d i p o l a r i n t e r a c t i o n o f a p r o t o n w i t h t h e i r o n . A s f e r r i -m y o g l o b i n i s i n a h i g h s p i n s t a t e , w i t h a w a t e r m o l e c u l e i n t h e s i x t h c o o r d i n a t i o n p o s i t i o n , t h i s i n t e r p r e t a t i o n i s q u i t e r e a s o n a b l e . H o w e v e r , t h i s m e c h a n i s m i f p r e s e n t w o u l d b e t h e m a j o r p a t h w a y f o r e n e r g y r e m o v a l f r o m t h e s p i n s y s t e m , b y -p a s s i n g t h e s p i n - l a t t i c e r e l a x a t i o n p r o c e s s . T h e v e r y s m a l l —7 r e l a x a t i o n t i m e s p r e d i c t e d b y t h i s m e c h a n i s m (<10 s e c ) a r e s o s h o r t t h a t n o f a s t p a s s a g e e f f e c t s w o u l d o c c u r . I n c y t o c h r o m e c , t h e heme i r o n i s b o u n d t o f i v e n i t r o g e n a t o m s a n d a s u l p h u r a t o m i n a h y d r o p h o b i c e n v i r o n -m e n t . I n t h i s s t r u c t u r e p r o t o n s a r e p r o b a b l y n o t c l o s e e n o u g h t o t h e i r o n a t o m f o r t h e i n v e r s e c u b i c t e r m i n t h e d i p o l a r i n t e r a c t i o n r e l a t i o n ( 7 . 6 ) t o o v e r c o m e t h e s m a l l p r o t o n d i p o l e moment a n d p r o d u c e t h e o b s e r v e d b r o a d e n i n g . A s s u m i n g a o n e g a u s s l i n e w i d t h — e q u i v a l e n t t o a 8 s p i n - s p i n r e l a x a t i o n t i m e ( T 2 ) o f 5 x 10 s e c o n d s — w e c a n o b t a i n 1/T 1 f r o m T u s i n g T = ( T . ^ ) ^ a n d t h e d a t a o f f i g u r e 7.3. O v e r t h e r e g i o n 10 t o 18°K, a s shown i n f i g u r e 7.5, 9 t h e d a t a c a n be f i t t e d b y e i t h e r a T c u r v e (Raman p r o c e s s ) o r b y a n e x p o n e n t i a l o f t h e f o r m e x p (- A / T ) ( O r b a c h p r o c e s s ) a s s u m i n g A = 166°K ( ^ l l S c r r f 1 ) . I n o r d e r t o d e c i d e b e t w e e n t h e s e t w o p o s s i b l e p r o c e s s e s we u s e t h e e x p e r i m e n t a l r e s u l t s o b t a i n e d a t t e m -p e r a t u r e s g r e a t e r t h a n 20°K ( s e c t i o n 7 . 4 ) . 7.6.2 T h e r e l a x a t i o n t i m e s i n t a b l e X I I I a r e p l o t t e d o n 5 f i g u r e 7.5 a n d h a v e an a p p r o x i m a t e T v a r i a t i o n o v e r t h e 9 r a n g e . N e i t h e r t h e o r e t i c a l c u r v e - — T o r e x p ( - A / T ) — f i t t h e h i g h t e m p e r a t u r e d a t a a s t h e y s t a n d . H o w e v e r , t h e o r y p r e d i c t 9 2 t h a t t h e Raman T p r o c e s s g o e s o v e r t o a T d e p e n d e n c e f o r 1/T^ a t t e m p e r a t u r e s a b o v e t h e D e b ye t e m p e r a t u r e ( T ^ ) o f t h e s a m p l e . I n t h e i n t e r m e d i a t e r e g i o n w h e r e T ~ T D we m i g h t 9 2 e x p e c t a d e p e n d e n c e b e t w e e n T a n d T . A s t h e d a t a show a n a p p r o x i m a t e T^ v a r i a t i o n , t h i s s u g g e s t s t h a t we a r e i n t h e t r a n s i t i o n r e g i o n . T h e e x p o n e n t i a l a l s o l e v e l s o f f a s t h e t e m p e r a t u r e i n c r e a s e s b u t i f t h e O r b a c h p r o c e s s i s d o m i n a n t , t h e s p i n l a t t i c e r e l a x a t i o n t i m e w o u l d h a v e t o be an o r d e r o f m a g n i -t u d e s h o r t e r t h a n i s m e a s u r e d f o r i t t o be t h e c o r r e c t m e c h a n i s m . FIGURE 7.5 P l o t o f i n v e r s e s p i n - l a t t i c e r e l a x a t i o n t i m e a s a f u n c t i o n o f t e m p e r a t u r e f r o m 4.2°K t o 70°K. Th e p o i n t s ( © ) a r e e s t i m a t e d f r o m f a s t p a s s a g e d a t a , w h i l e t h e p o i n t s ( A ) a r e o b t a i n e d f r o m L o r e n t z i a n l i n e w i d t h d a t a . The t e m p e r a t u r e 9 d e p e n d e n c e o f a Raman (T ) p r o c e s s a n d a n G r b a c h ( e x p - ( 1 6 6 / T ) ) shown f i t t h e d a t a b e t w e e n 10°K a n d 20°K ( s e e t e x t ) . ru -P I n 246 7.6.3 On t h e b a s i s o f t h e f o r e g o i n g c o n s i d e r a t i o n s , we h a v e o b t a i n e d a n e s t i m a t e o f t h e D e b y e t e m p e r a t u r e . C a n we make a n y o t h e r e s t i m a t e o f t h i s t e m p e r a t u r e ? A s a g e n e r a l r u l e , a t t h e Debye t e m p e r a t u r e t h e w a v e l e n g t h o f t h e l a t t i c e v i b r a t i o n s i s o f t h e same o r d e r a s t h e i n t e r i o n i c s e p a r a t i o n ( R o s e n b e r g , 1 9 6 5 ) . I f we knew t h e p h o n o n v e l o c i t y a n d t h e i n t e r i o n i c s p a c i n g we c o u l d c a l c u l a t e t h e a p p r o x i m a t e D e bye f r e q u e n c y a n d , h e n c e , t h e Debye t e m p e r a t u r e . A b r a g a m a n d B l e a n e y ( 1 9 7 0 ) q u o t e v a l u e s o f a p p r o x i -m a t e l y 5 x 1 0 ^ c m / s e c f o r t h e p h o n o n v e l o c i t y , i . e . t h e v e l o c i t y o f s o u n d i n p a r a m a g n e t i c c r y s t a l s . I n w a t e r , t h e 5 v e l o c i t y o f s o u n d i s 1.5 - 2 . 10 c m / s e c . T h e r e f o r e , l e t u s assume f o r o u r d i s c u s s i o n t h a t p h o n o n v e l o c i t y i s a p p r o x i m a t e l y 3 . 1 0 ^ c m / s e c i n c y t o c h r o m e c . L e t u s a l s o assume t h a t o n l y t h e h e a v i e s t a t o m s ( F e + ) d e t e r m i n e t h e u p p e r e n d o f t h e p h o n o n s p e c t r u m . T h e 3+ 0 i n t e r i o n i c s e p a r a t i o n o f t h e F e i s a b o u t 25 A i n c y t o c h r o m e s o t h a t : 3 • I 0 5 ~ 2 . 5 - 1 0 - 7 = 1 - 2 - I 0 ' 2 Hz then hi/Q = 1 . 2 - i o l 2 x 3 io"" = 4 2 cm"1 using hv0 = kT D we obtain T ~ — - = eo° K D 0.7 r\ w h i c h i s o f t h e c o r r e c t o r d e r — - u n e x p e c t e d l y c l o s e . i n v i e w o f . t h e a s s u m p t i o n s made. T h i s v a l u e i s l o w c o m p a r e d w i t h t h e D e b y e t e m p e r a t u r e s m e a s u r e d f o r m o s t e l e m e n t s ( e . g . , 400° f o r m e t a l l i c i r o n , 2000° f o r d i a m o n d ) . H o w e v e r , i n d e p e n d e n t e v i d e n c e f o r a Debye t e m p e r a t u r e o f o r d e r 100°K comes f r o m M o s s b a u e r w o r k o n f r o z e n s o l u t i o n s c o n t a i n i n g i r o n c o m p o u n d s . D e z s i e t a l . ( 1 9 6 8 ) o b t a i n e d T D ~ 1 0 0 ° K f o r F e 2 + i o n s i n f r o z e n a q u e o u s s o l u t i o n . By m e a s u r e m e n t s o n t h e s i z e o f t h e r e c o i l f r e e f r a c t i o n a t l o w t e m p e r a t u r e s , G o n s e r a n d G r a n t ( 1 9 6 5 ) s t u d i e d t h e t e m p e r a t u r e d e p e n d e n c e o f t h e r e c o i l f r a c t i o n i n r a t o x y - h e m o g l o b i n a n d o b t a i n e d a Debye t e m p e r a -t u r e o f a b o u t 180°K. T h e s e a u t h o r s s u g g e s t t h a t t h e s i m p l e Debye m o d e l , w h i c h c o n t a i n s o n l y a c c o u s t i c a l v i b r a t i o n a l modes, may n o t be s u f f i c i e n t t o d e s c r i b e t h e p h o n o n s p e c t r u m , a n d t h a t o p t i c a l modes s h o u l d a l s o be c o n s i d e r e d . T h e s e a r e c h a r a c t e r i s e d b y a s i n g l e v i b r a t i o n a l f r e q u e n c y ( o r s i n g l e e f f e c t i v e t e m p e r a t u r e , T E , t h e E i n s t e i n t e m p e r a t u r e ) . The same d e p e n d e n c e o f r e c o i l f r a c t i o n on t e m p e r a t u r e w o u l d b e s e e n i n h e m o g l o b i n w i t h ^ a T E ~ 1 2 0 ° K . G o n s e r a n d G r a n t ( 1 9 6 5 ) a l s o a t t e m p t e d t o e x p l a i n a n a n i s o t r o p y i n t h e q u a d r u p o l e s p l i t t i n g o f o x y - h e m o g l o b i n b y t h e p r e s e n c e o f an o p t i c a l v i b r a t i o n mode w i t h T E ~ 2 7 ° K v i b r a t i n g i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e heme p l a n e . H o w e v e r , t h e p r e s e n c e o f t h i s o p t i c a l mode h a s b e e n q u e s t i o n e d b y L a n g ( 1 9 7 0 ) who s u g g e s t e d t h a t t h e a s y m m e t r y may be due t o i m p u r i t i e s i n t h e p r e p a r a t i o n , s i n c e he h a s o b s e r v e d h i g h l y s y m m e t r i c s p e c t r a i n a t l e a s t some s a m p l e s o f r e d u c e d r a t h e m o g l o b i n o 7.7 CONCLUSION We h a v e shown t h a t t h e t e m p e r a t u r e d e p e n d e n c e o f t h e o v e r a l l r e l a x a t i o n t i m e c u r v e c a n b e f o l l o w e d f r o m 4.2°K t o 20°K u s i n g r a p i d a d i a b a t i c p a s s a g e t e c h n i q u e s a n d f r o m 50°K t o 77°K b y a n a l y s i s o f t h e s i g n a l l i n e w i d t h . We c o n c l u d e t h a t , b e t w e e n 10°K a n d 20°K t h e o v e r -a l l r e l a x a t i o n t i m e i s a c o m b i n a t i o n o f a t e m p e r a t u r e i n d e p e n d e n t s p i n - s p i n r e l a x a t i o n t i m e a r i s i n g - from d i p o l a r 9 i n t e r a c t i o n , c o m b i n e d w i t h a Raman T d e p e n d e n c e o f t h e s p i n l a t t i c e r e l a x a t i o n t i m e . A b o v e t h e s e t e m p e r a t u r e s t h e s p i n -l a t t i c e r e l a x a t i o n t i m e i s d o m i n a n t . The d e m o n s t r a t i o n t h a t t h e d e p e n d e n c e o f s p i n l a t t i c e r e l a x a t i o n t i m e h a s become a n a p p r o x i m a t e T"* f u n c t i o n b y 50° t o 60°K s u g g e s t s t h a t t h i s m i g h t e v e n t u a l l y become a 2 T f u n c t i o n a t h i g h e r t e m p e r a t u r e s . S u c h a d e p e n d e n c e o f s p i : l a t t i c e r e l a x a t i o n t i m e o n t h e s q u a r e o f t h e t e m p e r a t u r e i s t h e b e h a v i o u r p r o d u c e d b y a Raman r e l a x a t i o n p r o c e s s a t ' i n f i n i t e t e m p e r a t u r e s ' . We h a v e a l s o g i v e n a n e s t i m a t e f o r t h e Debye t e m p e r a t u r e ( T Q ) o f 60°K i n c y t o c h r o m e c a n d n o t e t h a t M o s s b a u e r d e t e r m i n a t i o n s o f i n a q u e o u s a n d p r o t e i n s o l u -t i o n s c o n t a i n i n g i r o n a t o m s a r e c o n s i s t e n t w i t h t h i s v a l u e . BIBLIOGRAPHY A b r a g a m , A. ( 1 9 6 1 ) " P r i n c i p l e s o f N u c l e a r M a g n e t i s m " ( O x f o r d U n i v e r s i t y P r e s s , L o n d o n ) . A b r a g a m , A c a n d B l e a n e y , B. ( 1 9 7 0 ) " E l e c t r o n P a r a m a g n e t i c R e s o n a n c e o f T r a n s i t i o n I o n s " ( O x f o r d U n i v e r s i t y P r e s s , L o n d o n ) . A l g e r , R.S. ( 1 9 6 8 ) " E l e c t r o n P a r a m a g n e t i c R e s o n a n c e : T e c h n i q u e s a n d A p p l i c a t i o n s " ( I n t e r s c i e n c e , New Y o r k ) . B a l l h a u s e n , C . J * ( 1 9 6 2 ) " I n t r o d u c t i o n t o L i g a n d F i e l d T h e o r y " ( M c G r a w - H i l l , New Y o r k ) . B e i n e r t , H. a n d P a l m e r , G. ( 1 9 6 5 ) " A d v a n c e s i n E n z y m o l o g y " e d . b y F . F . N o r d , V o l . X X V I I ( I n t e r s c i e n c e , New Y o r k ) p. 1 4 4 . B e n n e t t , J . E . , G i b s o n , J . F . a n d I n g r a m , D . J . E . ( 1 9 5 7 ) P r o c . R o y . S o c . A 2 4 0 , 6 7 . B e n n e t t , J . E . , G i b s o n , J . F . , I n g r a m , D.J.E., H a u g h t o n , T.M., K e r k u t , G.A. a n d Munday, K.A. ( 1 9 6 1 ) P r o c . R o y . 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L o n b e r g - H o l m ( A c a d e m i c P r e s s , New Y o r k ) p . 1 9 5 . B r i l l , A . S . a n d V e n a b l e , J r . J . H . ( 1 9 6 4 ) N a t u r e 2 0 3 , 7 5 2 . B u g a i , A.A. ( 1 9 6 3 ) S o v i e t P h y s i c s - S o l i d S t a t e 4, 2 2 1 8 . C h i e n , J.C.W. ( 1 9 6 9 ) J . Chem. P h y s . 5 1 , 4 2 2 0 . C o h n , M. ( 1 9 7 0 ) Q u a r t . R e v . B i o p h y s . 3,, 6 1 . D i c k e r s o n , R.E., K o p k a , M.L., B o r d e r s , J r . C.L., Va r n u m , J . , W e i n z e r l , J . E . a n d M a r g o l i a s h , E. ( 1 9 6 7 ) J . M o l . B i o l . 2 9 . 7 7 . D i c k e r s o n , R.E., T a k a n o , T., E i s e n b e r g , D., K a l l a i , O.B., Samson, L., C o o p e r , A. a n d M a r g o l i a s h , E. ( 1 9 7 1 ) J . B i o l . Chem. 246, 1 5 1 1 . D e z s i , I., K e s z t h e l y i , L . , M o l n a r , B., P o c s , L . ( 1 9 6 8 ) " H y p e r f i n e s t r u c t u r e a n d N u c l e a r R a d i a t i o n " e d . b y E. M a t t h i a s a n d D.A. 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( 1 9 6 1 ) " P r i n c i p l e s o f EPR I n s t r u m e n t a t i o n " 5 t h A n n u a l NMR-EPR V a r i a n W o r k s h o p , V a r i a n A s s o c i a t e s I n s t r u m e n t D i v i s i o n , P a l o A l t o , C a l i f o r n i a , USA. I n g r a m , D . J . E . ( 1 9 6 9 ) " B i o l o g i c a l a n d B i o c h e m i c a l A p p l i c a t i o n s o f E l e c t r o n S p i n R e s o n a n c e " (Adam H i l g e r , L o n d o n ) . I n g r a m , D . J . E . a n d B e n n e t t , J . E . ( 1 9 5 5 ) D i s c . F a r a d a y S o c . 1 9 , 1 4 0 . J a h n , H.A. a n d T e l l e r , E. ( 1 9 3 7 ) P r o c . R o y . S o c . A 1 6 1 , 2 2 0 . K a b a t , D. ( 1 9 6 7 ) B i o c h e m i s t r y 6, 3 4 4 3 . K n e u b u h l , F.K. a n d N a t t e r e r , B. ( 1 9 6 1 ) H e l v . P h y s . A c t a 34, 7 1 0 . K o h i n , R.P. a n d P o o l e , J r . C P . ( 1 9 5 8 ) B u l l . Am. P h y s . S o c . I I , 3, 8. 2 5 3 K o n , H. ( 1 9 6 9 ) B i o c h e n w B i o p h y s . R e s . Comm. 35_j 4 2 3 . K o t a n i , M. ( 1 9 6 1 ) S u p p . o f t h e P r o g , o f T h e o r e t . P h y s . 1 7 , 4. K o t a n i , M. ( 1 9 5 4 ) B i o p o l y m e r s Symp. 1,, 6 7 . K o t a n i , M. a n d M o r i m o t o , H. ( 1 9 6 7 ) " M a g n e t i c R e s o n a n c e i n B i o l o g i c a l S y s t e m s " e d . b y A. E h r e n b e r g , B.G. M a l m s t r o m a n d T. V a n n g a r d ( P e r g a m o n P r e s s , O x f o r d a n d New Y o r k ) p . 1 3 5 . K r o n i g , R. D e L . a n d Bouwkamp, C . J . ( 1 9 3 9 ) P h y s i c a . 6, 2 9 0 . K r o o n , D . J . ( 1 9 6 6 ) J . S c i . I n s t . 4 3 , 8 3 1 . L a n g , G. ( 1 9 7 0 ) Q u a r t . R e v . B i o p h y s . 3, 1. Low, B.W., C h e n , C.C.H., B e r g e r , J . E . , S i n g m a n , L . a n d P l e t c h e r , J . F . ( 1 9 6 6 ) P r o c . N a t . A c a d . S c i . ( U . S . ) 5_6, 1 7 4 6 . M a i l e r , C. a n d T a y l o r , C.P.S. ( 1 9 7 1 ) C a n . J . B i o c h e m . 4 9 , 6 9 5 . M a l m s t r o m , B.G. a n d V a n n g a r d , T. ( 1 9 6 0 ) J . M o l . B i o l . 2, 1 1 8 . M a r g o l i a s h , E. a n d S c h e t j e r , A. ( 1 9 6 6 ) A d v . P r o t . Chem. 21^, 1 1 4 . M a r g o l i a s h , E . a n d W a l a s e k , O.F. ( 1 9 6 7 ) " M e t h o d s i n E n z y m o l o g y " e d . b y R.W. E s t a b r o o k a n d M.E. P u l l m a n , v o l . X ( A c a d e m i c P r e s s , New Y o r k ) p . 3 3 9 . M i z u h a s h i , S. ( 1 9 6 9 ) J . Phys.^ S o c . ( J a p a n ) 2 6 , 4 6 8 . M o r t o n , R.A. a n d Boh a n , T .L. ( 1 9 7 1 ) C a n . J . B i o c h e m . 4 9 , 3 2 8 . O r b a c h , R. ( 1 9 6 1 ) P r o c . R o y . S o c . A 2 6 4 , 4 5 8 . P a l m e r , G., B r i n t z i n g e r , H., E s t a b r o o k , R.W. a n d S a n d s , R.H. ( 1 9 6 7 ) " M a g n e t i c R e s o n a n c e i n B i o l o g i c a l S y s t e m s " e d . b y A. E h r e n b e r g , B.G. M a l m s t r o m a n d T. V a n n g a r d ( P e r g a m o n P r e s s , O x f o r d a n d New Y o r k ) p . 1 5 9 . P a s t o r , R.C. a n d H o s k i n s , R.H. ( 1 9 6 0 ) J . Chem. P h y s . 3_2, 2 6 4 . P e i s a c h , J . , L e v i n e , W.G. a n d B l u m b e r g , W.E. ( 1 9 6 7 ) " M a g n e t i c R e s o n a n c e i n B i o l o g i c a l S y s t e m s " e d . b y A. E h r e n b e r g , B.G. M a l m s t r o m a n d T. V a n n g a r d ( P e r g a m o n P r e s s , O x f o r d a n d New Y o r k ) p . 1 9 9 . P e r u t z , M.F. ( 1 9 5 3 ) A c t a . C r y s t . 6, 8 5 9 . P e r u t z , M o F . a n d M a t h e w s , F . S . ( 1 9 6 6 ) J . M o l , B i o l . 2 1 , 1 9 9 . P h i l l i p s , F . C ( 1 9 6 2 ) "An I n t r o d u c t i o n t o C r y s t a l l o g r a p h y " 2 n d e d . ( L o n g m a n s , L o n d o n a n d New Y o r k ) p . 2 0 . P o o l e * J r . , C P . ( 1 9 6 7 ) " E l e c t r o n S p i n R e s o n a n c e : A C o m p r e h e n s i v e T r e a t i s e o n E x p e r i m e n t a l T e c h n i q u e s " ( I n t e r s c i e n c e , New Y o r k ) . P o r t i s , A.M. ( 1 9 5 5 ) T e c h n i c a l N o t e No. 1, S a r a h M e l l o n S c a r f e R a d i a t i o n L a b o r a t o r y , U n i v e r s i t y o f P i t t s b u r g . P o s e n e r , D.W. ( 1 9 5 9 ) A u s t r a l . J . P h y s . J L 2 , 1 8 4 . P o w e l l , R.L.., B u n c h , M.D. a n d C o r r u c c i n i , R . J . ( 1 9 6 1 ) C r y o g e n i c s 1_, 1 3 9 . P r y c e , M.H.L. ( 1 9 5 0 ) P r o c . P h y s . S o c . A 6 3 , 2 5 . R e i n , H., R i s t a u , 0. a n d J u n g , F . ( 1 9 6 8 ) E x p e r i e n t i a 24, 7 9 7 . R o s e n b e r g , H.M. ( 1 9 6 3 ) "Low T e m p e r a t u r e S o l i d S t a t e P h y s i c s " ( O x f o r d U n i v e r s i t y P r e s s , L o n d o n ) . S a l m e e n , I . a n d P a l m e r , G* ( 1 9 6 8 ) J . Chem. P h y s . 4 8 , 2 0 4 9 . S a n P i e t r o , A. ( 1 9 6 5 ) e d . "Non-Heme I r o n P r o t e i n s : R o l e i n E n e r g y C o n v e r s i o n " ( A n t i o c h P r e s s , Y e l l o w S p r i n g s , O h i o ) . S c h o l e s , C P . ( 1 9 6 9 ) P r o c . N a t . A c a d . S c i . ( U . S . ) 62_, 4 2 8 . S c h o n l a n d , D.S. ( 1 9 5 9 ) P r o c . P h y s . S o c . 73., 7 8 8 . S t e v e n s , K.W.H. ( 1 9 5 3 ) P r o c . R o y . S o c . A 2 1 9 , 5 4 2 . S t o u t , G.H. a n d J e n s e n , L . H . ( 1 9 6 9 ) " X - r a y S t r u c t u r e D e t e r m i n a t i o n : A . P r a c t i c a l G u i d e " ( M c M i l l a n , New Y o r k ) p. 6 6 . S t r y e r , L . , K e n d r e w , J . C . a n d W a t s o n , H . C ( 1 9 6 4 ) J . M o l . B i o l . 8, 9 6 . T a s a k i , A., O t s u k a , J . a n d K o t a n i , M. ( 1 9 6 7 ) B i o c h i m . B i o p h y s . A c t a 1 4 0, 2 8 4 . T s a i , H . J . a n d W i l l i a m s , G.R. ( 1 9 6 5 ) C a n . J . B i o c h e m . 4 3 , 1 4 0 9 . V a n V l e c k , J . H . ( 1 9 4 0 ) P h y s . R e v . 5_7, 4 2 6 . V e n a b l e , J r . , J . H . ( 1 9 6 5 ) " M a g n e t i c M e t h o d s f o r P r o t e i n S i n g l e C r y s t a l s : M e t a l B i n d i n g t o I n s u l i n " Ph.D. T h e s i s , Y a l e U n i v e r s i t y , New H a v e n , C o n n . , U.S.A. W a l l e r , I . ( 1 9 3 2 ) Z. P h y s . 7 9 , 3 7 0 . Weger, M. ( 1 9 6 0 ) B e l l S y s t e m T e c h . J . 39, 1 0 1 3 . W e i s s b l u t h , M. ( 1 9 6 6 ) S t r u c t u r e a n d B o n d i n g 2., 1. W h i t e , G.K. ( 1 9 6 8 ) " E x p e r i m e n t a l T e c h n i q u e s i n Low T e m p e r a t u r e P h y s i c s " 2 n d e d . ( O x f o r d U n i v e r s i t y P r e s s , L o n d o n ) p. 3 7 1 . W i l m s h u r s t , T.H. ( 1 9 6 7 ) " E l e c t r o n S p i n R e s o n a n c e S p e c t r o m e t e r s " (Adam H i l g e r , L o n d o n ) p . 1 6 8 . Y o n e t a n i , T. a n d S c h l e y e r , H. ( 1 9 6 7 ) J . B i o l . Chem. 242, 3 9 2 6 . Y o n e t a n i , R. a n d S c h l e y e r , H. ( 1 9 6 8 ) " S t r u c t u r e a n d F u n c t i o n o f C y t o c h r o m e s " e d . b y K. O k u n u k i , M. Kamen a n d I . S e k u z u ( U n i v . o f T o k y o P r e s s , T o k y o , J a p a n ) p . 5 3 5 . APPENDIX - A l CALCULATION OF EPR LINE BROADENING DUE TO VARIATIONS IN RHOMBIC AND AXIAL POTENTIAL Al«1 INTRODUCTION The method given by Eisenberger and Pershan (1967) w i l l be used© These authors obtained expressions f o r the l i n e widths i n terms of the energy of the a x i a l and rhombic s p l i t t i n g s that gave r i s e to the low spin g-values. Their p u b l i c a t i o n contained a number of e r r o r s , and so the c a l c u -l a t i o n i s rederived below. Some of the expressions have been found to have a simpler form than given by Eisenberger and Pershan, The c a l c u l a t i o n s t a r t s with the energy matrix of the ground state Kramers doublet, from which the c o e f f i c i e n t s of the o r b i t a l s making up the doublet can be obtained. With these c o e f f i c i e n t s , the g-value equations can be found and the d e r i v a t i v e s c a l c u l a t e d , A1.2 THE GROUND STATE KRAMERS DOUBLET 3 Low spin Fe has f i v e 3d electrons i n t 2 g o r b i t a l s For a system quantised along the heme normal these o r b i t a l s a r e ^ g ^ * ' fc2g+i? *~2g~^"* t e c m s o f ^ e P u r e d - o r b i t a l s c o n v e n t i o n a l l y c a l l e d d 2 , d _ 2 ? d^, d ^ ( B a l l h a u s e n , . 1 9 6 2 , p . 63) t h e t 0 a r e : •A ( d 2 - d . 2 ) f 2 g 2 g - I The r e a l f o r m o f t h e s e o r b i t a l s , d e n o t e d b y | d X y ^ > * J^y^> a n d I d \ ! x z / a r e '2 - 2 'V2 I n l o w s p i n ( S = l / 2 ) c y t o c h r o m e c t h e e n v i r o n m e n t o f t h e i r o n i s p r e d o m i n a n t l y c u b i c , b u t a l s o c o n t a i n s a x i a l a n d r h o m b i c c o m p o n e n t s . U n d e r c u b i c ( 0 ^ ) s y m m e t r y t h e t 2 o r b i t a l s a r e d e g e n e r a t e , r e d u c t i o n o f s y m m e t r y t o b y t h e a d d i t i o n o f an a x i a l p o t e n t i a l s e p a r a t e s | <*x ^ f r o m | d y z ^ > a n d | ^ x z ^ > » F u r t h e r r e d u c t i o n o f s y m m e t r y t o b y a r h o m b i c p o t e n t i a l s p l i t s |d^z^> a n d |d x z^> l i f t i n g t h e o r b i t a l d e g e n e r a c y c o m p l e t e l y . E a c h o r b i t a l s t i l l h a s a 2 - f o l d s p i n d e g e n e r a c y — t h e tv/o s t a t e s a r e d e n o t e d b y ( + ) f o r s p i n u p a n d (-) f o r s p i n down, e . g . |+l +^> a n d j + l y * • S p i n o r b i t c o u p l i n g m i x e s t h e s e O r b i t a l s w h i c h c o m b i n e i n t o t h r e e s e t s o f K r a m e r s d o u b l e t s . A p p l i c a t i o n o f a m a g n e t i c f i e l d s p l i t s t h e s e d o u b l e t s , a n d a n EPR e x p e r i m e n t w i l l o b s e r v e t h e t r a n s i t i o n s b e t w e e n t h e t w o l e v e l s o f t h e g r o u n d s t a t e * The b e h a v i o u r o f t h e f i v e e l e c t r o n s i n t h e t 0 o r b i t a l s i s t h e same a s a h o l e i n t h e t ~ ^ c l o s e d s h e l l , p r o v i d e d we i n v e r t t h e e n e r g y l e v e l s . I n t h i s t r e a t m e n t we n e g l e c t o r b i t a l r e d u c t i o n ( s e e S t e v e n s , 1 9 5 3 ) . F r o m G r i f f i t h ( 1 9 6 1 , p . 354) t h e t 2 g o r b i t a l s ( i n c l u d i n g s p i n ) f a c t o r i s e u n d e r s p i n - o r b i t c o u p l i n g o f - X L ' S i n t o t h e m a t r i x or -10 or 10 r> or B>or |r > x/2 V / 2 1 C > D 0 ->> y 2 0 ( A l . l ) w h e r e t h e c r y s t a l f i e l d e n e r g i e s c o r r e s p o n d i n g t o h o l e s i n t h e \dzx} * | dyz^> a n d l d x y ^ o r b i t a l s a r e g i v e n b y - V / 2 , a n d D r e s p e c t i v e l y : | — d x y D d . 1 / n -v i--V/z X A 1 . 2 ) i d zx The g r o u n d s t a t e K r a m e r s d o u b l e t h a s t h e f o r m r = A|l*> + B 0 + C |-l+> f " = A|-I^>- B + C ( A l . 3 a ) ( A 1 . 3 b ) a n d t h e e n e r g y , E, o f t h i s g r o u n d s t a t e r e l a t i v e t o t h e z e r o o f ( A l , 2 ) c a n be f o u n d f r o m t h e s e c u l a r d e t e r m i n a n t o f m a t r i x ( A l . l ) g i v i n g : (0.5X - E) A + ( X / V 2 ) B + ( V / 2 ) C = 0 U L * 4 A ) (Vv^)A • ( A - E ) B = 0 ( A 1 ' 4 B ) (V/2 ) A + (-0.5X - E)C = 0 ( A l . 4 c ) a n d t h e n o r m a l i s a t i o n c o n d i t i o n g i v e s A 2 + B 2 + C 2 = 1 ( A 1 . 5 ) T h e g r o u n d d o u b l e t g - - v a l u e s a r e g, = 2 (V2 A + B) -(V2 C - B) ( A i . 6 a ) g 9 = 2 ( V £ A >B)(J2C + B) ( A i . 6 b ) g ' = 2 ( 2 A 2 - B 2 ) ••(A1..6C) 3 We now w i s h t o o b t a i n t h e d e r i v a t i v e s o f t h e s e g - v a l u e s w i t h r e s p e c t t o t h e r h o m b i c p o t e n t i a l , V, a n d t h e a x i a l p o t e n t i a l , D. We d e f i n e g! = -^L and q . " = A^L with i = i , 2 , 3 A l . 3 EVALUATION OF RHOMBIC CONTRIBUTION i t t I n o r d e r t o o b t a i n t h e g^, we m u s t o b t a i n A , B , C a n d E . F r o m ( A 1 . 4 b ) R X A P = — (A1.7) Jz(E-D) D i f f e r e n t i a t e w i t h r e s p e c t t o V, a n d B' - ' = ( X / ^ ) [ A ' -(E-D)"' - A ( E - D ) " 2 E ' ] (X/V2) A•(E-D)",-[A,'/A - E'(E-D)] B\ = B [A'/A - E'(E-D ) " ] • ( A I 8 > s i m i l a r l y , f r o m ( A l e 4 c ) C - V-A 2 (0.5X + E) a n d d i f f e r e n t i a t i o n p l u s s i m p l i f i c a t i o n g i v e s C'• = C [ A V A * l/V - E ' ( E - 0 . 5 X)"|] C A i . 9 ) . i T o o b t a i n A , we u s e t h e n o r m a l i s a t i o n c o n d i t i o n ; d i f f e r e n -t i a t i n g we g e t 2 A-A' + 2B''B' + 2C- C = 0 s u b s t i t u t i o n o f ( A 1 . 7 ) a n d ( A l c 8 ) g i v e A' = A [c2 (E - 0 . 5 X)H - B 2 (D-E)E'-(C 2 / V) (Al»10) t i T o o b t a i n S , v;e d i f f e r e n t i a t e ( A 1 . 4 a ) , s u b s t i t u t e f o r A , t t B a n d C end e v e n t u a l l y p r o d u c e E' * AC ( A i . i i ) 263 The a n a l y t i c a l e x p r e s s i o n s f o r t h e g. a r e V « 2 [(VS ,A+B)-(V& ,-.B') * ( v ^ C - B)- (v^A 1 *B')] g2* -• 2 [ ( ^ A + B H V f c , + B') .+ (VS-C + B) (V? A' +Bfj (AI.I2) g e 2 [ 4 A A 1 - 2 B B ' ] i t We s u b s t i t u t e i n t o t h e s e r e l a t i o n s t h e v a l u e s o f A , B a n d i i C o b t a i n e d a b o v e . F i n a l l y , t h e s e g^ a r e p u t i n t o t h e e q u a t i o n t h a t g i v e s t h e c o n t r i b u t i o n t o t h e l i n e w i d t h f r o m t h e d i s t r i b u t i o n o f r h o m b i c p o t e n t i a l . A H V •- JUL. J. 0 ' 3 [9,9,' sin2#- cos 2 <f> + 9 2 9g sin 2 6 s i n 2 $ ( A 1 . 1 3 ) -+ ^ g g g C O S 2 ^ ] ' A V Al«4 EVALUATION OF A X I A L CONTRIBUTION By e x a c t l y s i m i l a r m e t h o d s we e x t e n d t h i s t r e a t -m ent t o t h e a x i a l c a s e t o o b t a i n t h e q . . Th e r e l a t i o n s h i p s a r e A [B 2 (I-E")(D-E)" , + C ?-E"(E-0 . 5 X ) " 1 ] B [A"/A - ( l~E , ,)(D-E)~'] C [A"/A - E"- (E - 0 .5 X)"'] X- AB- [v^"(D-EJ"'] A n d b y s u b s t i t u t i o n : AHQ = — ^ J g ^ ' s i n ^ c o s 2 ^ +' g 2 g 2 "s in 2 ^ S in 2 >^ A" = B" = C" = E" "« • gg "COS2^]AD 

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