Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Deuterium nuclear magnetic resonance in model membrane systems : an investigation of the interaction… Poulin, Neal M. 1985

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1985_A6_7 P68.pdf [ 4.77MB ]
Metadata
JSON: 831-1.0084980.json
JSON-LD: 831-1.0084980-ld.json
RDF/XML (Pretty): 831-1.0084980-rdf.xml
RDF/JSON: 831-1.0084980-rdf.json
Turtle: 831-1.0084980-turtle.txt
N-Triples: 831-1.0084980-rdf-ntriples.txt
Original Record: 831-1.0084980-source.json
Full Text
831-1.0084980-fulltext.txt
Citation
831-1.0084980.ris

Full Text

DEUTERIUM NUCLEAR MAGNETIC RESONANCE IN MODEL MEMBRANE SYSTEMS: AN INVESTIGATION OF THE INTERACTION OF A SYNTHETIC, AMPHIPHILIC POLYPEPTIDE WITH CHARGED LIPIDS by NEAL M. POULIN B.Sc, U n i v e r s i t y Of Winnipeg, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1985 © Neal M. Poulin, 1985 7 8 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s . or her r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Pl a c e Vancouver, Canada V6T 1W5 Date: September 1,1985 i i A b s t r a c t The theory of the quadrupole i n t e r a c t i o n i n n u c l e a r magnetic resonance spectroscopy and r e l a x a t i o n measurements i s presented i n d e t a i l , with a p p l i c a t i o n s to 2H-NMR s t u d i e s of order and dynamics i n b i l a y e r s of d e u t e r a t e d l i p i d s . I n v e s t i g a t i o n s of l i p i d - p r o t e i n i n t e r a c t i o n s i n r e c o n s t i t u t e d membrane systems and i n t a c t b i o l o g i c a l membranes are reviewed. An experimental program i s d e s c r i b e d which uses a s y n t h e t i c a m p h i p h i l i c p o l y p e p t i d e , with known geometry and v a r i a b l e l e n g t h , to i s o l a t e q u e s t i o n s about the ge o m e t r i c a l i n t e r p r e t a t i o n of o r i e n t a t i o n a l order i n l i p i d -p r o t e i n i n t e r a c t i o n s . A r e p o r t i s presented of an i n v e s t i g a t i o n of the e f f e c t s of t h i s p o l y p e p t i d e , L y s 2 - G l y - L e u 2 0 - L y s 2 - A l a - a m i d e , on the mixed b i l a y e r system: D i m y r i s t o y l p h o s p h a t i d y l c h o l i n e D i - p e r - d e u t e r i o - m y r i s t o y l p h o s p h a t i d i c A c i d . The a d d i t i o n of the pept i d e was found to have l i t t l e e f f e c t (^5%) on the f i r s t and second moments of the d i s t r i b u t i o n of quadrupole s p l i t t i n g s i n the l i q u i d c r y s t a l l i n e phase. S i m i l a r l y , the s p i n - l a t t i c e r e l a x a t i o n time c o n s t a n t s were a f f e c t e d by ^10% in the l i q u i d c r y s t a l l i n e phase. The time constant f o r the decay of the quadrupole echo decreased d r a m a t i c a l l y above the phase t r a n s i t i o n with the a d d i t i o n of p e p t i d e , a phenomenon which i s e x p l a i n e d i n terms of the presence of a new slow motion i n the l i p i d - p e p t i d e systems. A simple model of the slow motion induced by the p e p t i d e i s proposed, i n which the l i p i d molecules undergo a r a p i d exchange between boundary and bulk s i t e s . An e f f e c t i v e c o r r e l a t i o n time i s determined from an average over the r o t a t i o n s on each of these s i t e s . Using t h i s model, esti m a t e s are made of the change i n the second moment brought about by the onset of the r o t a t i o n s , and. of the number of b i n d i n g s i t e s on the pe p t i d e . These estimates are found to be i n agreement with independent measurements of the change i n the second moment, and the number of binding s i t e s i s w i t h i n the range p r e d i c t e d by simple c o n s i d e r a t i o n s of charge balance. The change i n the l i n e s h a p e with the v a r i a t i o n of the spacing of the p u l s e s i n the quadrupole echo experiment was i n v e s t i g a t e d , and i t was determined that the t r a n s v e r s e r e l a x a t i o n time constants have a s l i g h t o r i e n t a t i o n dependence. I t was a l s o determined that the a d d i t i o n of the peptid e has no s i g n i f i c a n t e f f e c t on the v a r i a t i o n of the l i n e s h a p e . Some experiments which could answer some of the qu e s t i o n s r a i s e d by these r e s u l t s are suggested. iv T able of Contents A b s t r a c t i i L i s t of Tables v i L i s t of F i g u r e s v i i Acknowledgement v i i i Chapter I INTRODUCTION 1 Chapter II QUADRUPOLE INTERACTIONS 4 2.1 THE ZEEMAN HAMILTONIAN 4 2.2 QUADRUPOLE INTERACTIONS 5 2.3 THE CLASSICAL QUADRUPOLE ENERGY 5 2.4 THE QUADRUPOLE HAMILTONIAN - THE CLASSICAL ANALOGY 7 2.5 MATRIX ELEMENTS OF THE QUADRUPOLE HAMILTONIAN 8 2.6 MOTIONAL AVERAGING - THE ORDER PARAMETER TENSOR ..12 2.7 THE ORDER PARAMETER TENSOR. IN LIPID BILAYERS 14 2.8 THE EFFECT OF BILAYER ORIENTATION 15 Chapter I I I DENSITY MATRIX FORMALISM 17 3.1 FICTITIOUS SPIN-1/2 OPERATORS 17 3.2 SPIN HAMILTONIANS ' 1 9i 3.3 THE QUADRUPOLE ECHO EXPERIMENT 23 Chapter IV RELAXATION 27 4.1 THE REDFIELD EQUATION 27 4.2 FINITE TEMPERATURE CORRECTION 31 4.3 OPERATOR FORM OF THE REDFIELD EQUATION 32 4.4 QUADRUPOLE RELAXATION 34 Chapter V THE QUADRUPOLE ECHO SPECTRUM 38 5.1 THE POWDER PATTERN SPECTRUM 38 5.2 SPECTRAL ANALYSIS 44 5.2.1 MOMENTS ANALYSIS 45 5.2.2 SPECTRAL DE-PAKE-ING 46 Chapter VI DEUTERIUM NMR STUDIES OF PHOSPHOLIPID BILAYERS 49 6.1 DEUTERIUM-NMR SPECTRA OF PHOSPHOLIPID MEMBRANES ..52 6.2 DEUTERI UM-NMR RELAXATION STUDIES ....55 6.2.1 ROTATIONAL DIFFUSION 56 6.2.2 DEFECT DIFFUSION 59 6.2.3 DECAY OF THE QUADRUPOLE ECHO 61 V Chapter VII LIPID-PROTEIN INTERACTIONS 63 7.1 LIPID-PROTEIN INTERACTIONS IN THE LIQUID CRYSTALLINE PHASE 66 7.2 RESULTS IN THE GEL PHASE 68 7.3 ORIENTATIONAL ORDER AND MEMBRANE STRUCTURE 69 7.4 A SYNTHETIC AMPHIPHILIC POLYPEPTIDE 70 Chapter VIII A DEUTERIUM NMR STUDY OF THE INTERACTION OF A SYNTHETIC AMPHIPHILIC POLYPEPTIDE WITH CHARGED LIPIDS 74 8.1 SAMPLE PREPARATION 74 8.2 NMR MEASUREMENTS AND CALCULATIONS 76 8.3 INITIAL INVESTIGATIONS OF THE POLYPEPTIDE/LIPID SYSTEM 77 8.4 LINESHAPE STUDY 108 8.5 CONCLUDING REMARKS 130 v i LIST OF TABLES I. MOMENTS: PRELIMINARY STUDY 87 I I . RELAXATION: T1F 95 I I I . RELAXATION: T1S 96 IV. RELAXATION: T1AV 97 V. RELAXATION: T2E 98 VI. MOMENTS: LINESHAPE STUDY 114 V I I . LINESHAPE STUDY: QUADRUPOLE SPLITTINGS 126 V I I I . LINESHAPE STUDY: LINEWIDTHS 127 LIST OF FIGURES 1. QUADRUPOLE INTERACTIONS 22 2. QUADRUPOLE ECHO PULSE SEQUENCE 26 3. THE POWDER PATTERN SPECTRUM 42 4. ORIENTATION-DEPENDENT BROADENING 43 5. DEPAKED SPECTRUM 48 6. SAMPLE SI: POWDER SPECTRA 80 7. SAMPLE S2: POWDER SPECTRA 81 8. SAMPLE S3: POWDER SPECTRA 82 9. THE EFFECT OF PEPTIDE CONCENTRATION: LIQUID CRYSTAL POWDER SPECTRA 83 10. THE EFFECT OF PEPTIDE CONCENTRATION: GEL POWDER SPECTRA 84 11. SPECTRAL MOMENTS: M1 85 12. SPECTRAL MOMENTS: M2 86 13. SPIN-LATTICE RELAXATION: T1F - FAST COMPONENT 89 14. SPIN-LATTICE RELAXATION: T1S - SLOW COMPONENT 90 15. SPIN-LATTICE RELAXATION: AVERAGE T1 91 16. DECAY OF THE QUADRUPOLE ECHO: T2E 92 17. RELAXATION: MINUIT ANALYSIS OF T1 MEASUREMENT 93 18. RELAXATION: MINUIT PARAMETERS 94 19. PARTIALLY RELAXED SPECTRUM 101 20. RELAXATION: MINUIT ANALYSES OF T2E MEASUREMENTS 104 21. RELAXATION: THE EFFECT OF PEPTIDE CONCENTRATION 105 22. LINESHAPE STUDY: POWDER SPECTRA FOR PURE LIPIDS 112 23. LINESHAPE STUDY: POWDER SPECTRA FOR LIPID-PEPTIDE SYSTEM 113 24. LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON ...116 25. LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON ...117 26. LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON ...118 27. LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON ...119 28. LINESHAPE STUDY: DE-PAKE-ED SPECTRA FOR PURE-LIPID SYSTEM 120 29. LINESHAPE STUDY: DE-PAKE-ED SPECTRA FOR LIPID-PEPTIDE SYSTEM 121 30. LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - POWDER SPECTRA 122 31. LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - POWDER SPECTRA 123 32. LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - DE-PAKE-ED SPECTRA 124 33. LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - DE-PAKE-ED SPECTRA 125 v i i i ACKNOWLEDGEMENT I would l i k e to thank Myer Bloom, Gina Hoatson, Alex MacKay, P h i l l i p e Devaux, Frank Volke, and other members of the Room 100 NMR l a b at UBC p h y s i c s f o r h e l p f u l d i s c u s s i o n s and i n d i s p e n s i b l e a d v i c e . S p e c i a l thanks to Myer Bloom f o r h i s support throughout the course of t h i s p r o j e c t ; to Gina Hoatson, who i n s t r u c t e d me on the use of the spectrometer, and who c o l l a b o r a t e d with me on the f i r s t p a r t of the experiment, and to Edward S t e r n i n f o r h i s h e l p with the l o c a l software on numerous o c c a s i o n s . I would a l s o l i k e to express my a p p r e c i a t i o n t o P i e t e r C u l l i s f o r h i s permissi o n to use the l a b f a c i l i t i e s at the bio c h e m i s t r y department. For p a r t of the time that t h i s r e s e a r c h was i n pro g r e s s , I was supported by a s c h o l a r s h i p from the N a t u r a l Sciences and En g i n e e r i n g Research C o u n c i l . 1 I. INTRODUCTION B i o l o g i c a l membranes are s e l e c t i v e l y permeable, s h e e t l i k e s t r u c t u r e s , which form c l o s e d boundaries between c e l l s or c e l l o r g a n e l l e s and t h e i r aqueous environments. They are composed p r i m a r i l y of l i p i d s and p r o t e i n s . The l i p i d s are a m p h i p h i l i c molecules ( i e . , having both hydrophobic and h y d r o p h i l i c p a r t s ( m o i e t i e s ) ) , spontaneously forming b i l a y e r e d l a m e l l a r s t r u c t u r e s i n aqueous d i s p e r s i o n s . These s t r u c t u r e s have l i q u i d - c r y s t a l l i n e p r o p e r t i e s at a p p r o p r i a t e temperatures, behaving s i m i l a r l y t o some of the l y o t r o p i c smectic mesophases. Membrane p r o t e i n s are the b i o l o g i c a l l y a c t i v e components of the membrane. While the f u n c t i o n of the l i p i d s i s understood to be p r i m a r i l y p a s s i v e - they form the boundaries, a c t i n g as b a r r i e r s to p o l a r molecules - i t i s the p r o t e i n s which mediate s p e c i f i c membrane a c t i v i t i e s . They are r e s p o n s i b l e f o r the t r a n s p o r t of i o n s , n u t r i e n t s , wastes, and other biomolecules ac r o s s the membrane, and can serve as chemical r e c e p t o r s , enzymes, and as energy t r a n s d u c e r s . The study of b i o l o g i c a l membranes i s then a problem i n de t e r m i n a t i o n of the s t r u c t u r e and f u n c t i o n of membrane p r o t e i n s , and i n r e l a t i n g these two p r o p e r t i e s . The l i p i d s , which provide an a p p r o p r i a t e environment f o r membrane p r o t e i n s , w i l l have a s i g n i f i c a n t e f f e c t on p r o t e i n s t r u c t u r e and f u n c t i o n , so that an important step in our 2 understanding i s the study of l i p i d - p r o t e i n i n t e r a c t i o n s . Because of the i n c r e d i b l e complexity of a c t u a l b i o l o g i c a l systems ( b i o l o g i c a l membranes can have a remarkable v a r i e t y of c o n s t i t u e n t l i p i d s and p r o t e i n s ) , and the i n h e r e n t d i f f i c u l t y i n studying i n t a c t membranes i n t h e i r p h y s i o l o g i c a l c o n d i t i o n s , researches have found i t f r u i t f u l to develop models f o r b i o l o g i c a l membranes. Model membranes i n c o r p o r a t e the main s t r u c t u r a l f e a t u r e of b i o l o g i c a l membranes: that of a l i p i d b i l a y e r with i n t e r c a l a t e d or s u r f a c e p r o t e i n s . In s t u d y i n g model membranes, i t i s p o s s i b l e , i n p r i n c i p l e , t o i s o l a t e s p e c i f i c a s p e c t s of l i p i d - p r o t e i n i n t e r a c t i o n s , by l i m i t i n g the number of d i f f e r e n t kinds of l i p i d and p r o t e i n components t o one or two. The d e s c r i p t i o n of the p h y s i c s of these systems becomes more manageable. Deuterium ( 2H) n u c l e a r magnetic resonance (NMR) has proven to be an important t o o l i n the i n v e s t i g a t i o n of the s t r u c t u r e and dynamics of model membrane systems. In t h i s study, the p r o p e r t i e s of the pure l i p i d membranes are i n v e s t i g a t e d , and the p r o t e i n i s inroduced i n t o the l i p i d system as a p e r t u r b i n g i n f l u e n c e . The l i p i d s are p e r d e u t e r a t e d - i e . , the hydrogen atoms on a l l segments of the f a t t y a c i d c h a i n s are exchanged with deuterium atoms. Deuterium NMR p r o v i d e s i n f o r m a t i o n on the average o r i e n t a t i o n a l order of the C-D bonds, as w e l l as d e t a i l e d i n f o r m a t i o n on molecular motions. T h i s t h e s i s has two main o b j e c t i v e s . F i r s t l y , i t i s 3 intended as a reasonably d e t a i l e d account of the theory and a p p l i c a t i o n s of deuterium NMR, i n c l u d i n g a d e s c r i p t i o n and d i s c u s s i o n of the types of measurements and the methods of data a n a l y s i s . Secondly, a r e p o r t w i l l be presented of a deuterium NMR study of a p a r t i c u l a r model membrane system. 4 I I . QUADRUPOLE INTERACTIONS 2.1 THE ZEEMAN HAMILTONIAN In bulk matter and i n the presence of an e x t e r n a l magnetic f i e l d H 0, n u c l e i with non-zero s p i n angular momentum, I, e x h i b i t a macroscopic magnetization along the d i r e c t i o n of the a p p l i e d f i e l d . T h i s phenomenon i s known as nucle a r paramagnetism, and i s to be understood quantum-mec h a n i c a l l y i n terms of the c o u p l i n g of the nuclear magnetic moments to the e x t e r n a l f i e l d . T h i s c o u p l i n g i s represented by the Zeeman Hamiltonian f o r the system, H = ~ 7 h H 0 I 2 , where 7 i s the gyromagnetic r a t i o of the n u c l e i , and the z - a x i s , the a x i s of q u a n t i z a t i o n , i s the d i r e c t i o n of the s t a t i c f i e l d . NMR spectroscopy c o n s i s t s of the study of the energy l e v e l s of t h i s Hamiltonian, which i n the presence of other i n t e r a c t i o n s have f i n e and h y p e r f i n e s t r u c t u r e s . The b a s i c theory of NMR measurements i s t r e a t e d i n many sources, and we r e f e r the reader to ( l ) a n d ( 2 ) f o r an i n t r o d u c t o r y understanding. In t h i s study we are concerned with the quadrupole i n t e r a c t i o n , which i s the dominant i n t e r a c t i o n f o r deuterium n u c l e i , being orders of magnitude l a r g e r than the d i p o l e - d i p o l e i n t e r a c t i o n . 5 2.2 QUADRUPOLE INTERACTIONS N u c l e i of spin 1^1 may possess an e l e c t r i c quadrupole moment, due to a n o n - s p h e r i c a l l y symmetric d i s t r i b u t i o n of charge. The e l e c t r o s t a t i c energy of t h i s d i s t r i b u t i o n depends on the o r i e n t a t i o n of the n u c l e i , and w i l l t h e r e f o r e have an a d d i t i o n a l e f f e c t on NMR measurements, i n v o l v i n g the s p l i t t i n g of the Zeeman energy l e v e l s of the n u c l e i . In our i n v e s t i g a t i o n of the quadrupole i n t e r a c t i o n , we w i l l d e f i n e - by c l a s s i c a l analogy - the quantum mechanical quadrupole o p e r a t o r s and the quadrupole H a m i l t o n i a n . We w i l l express these o p e r a t o r s i n terms of the angular momentum op e r a t o r s of the nucleus, so that we can demonstrate the quadrupole s p l i t t i n g of the Zeeman energy l e v e l s . T h i s r e q u i r e s a rather i n v o l v e d treatment, as f o l l o w s . 2.3 THE CLASSICAL QUADRUPOLE ENERGY Consider a nucleus (charge d i s t r i b u t i o n p) i n an e x t e r n a l f i e l d (of p o t e n t i a l V ) . C l a s s i c a l l y , the e l e c t r o s t a t i c energy i s giv e n by: E = J p ( r ) V ( r ) d r We expand V ( r ) i n a T a y l o r s e r i e s about the o r i g i n : V ( r ) = V(0) + I x«V„ + 1/2! Z Z xmxaV„. + . . . 6 Where V = 6> 2V/dX adX» ) i s the e l e c t r i c f i e l d g r a d i e n t . Then: E = V(0)/pdr + Z V^/x^pdr + 1/2 Z Z Jx.x^pdr + ... We choose the o r i g i n to be the centre of mass of the nucleus (which i s a l s o the c e n t r e of charge). Then the f i r s t term i s the energy of the nucleus taken as a p o i n t charge. The second term i n v o l v e s the e l e c t r i c d i p o l e moment of the nucleus, which i s zero. The t h i r d term i n v o l v e s the e l e c t r i c quadrupole moment. We d e f i n e : = J O x ^ - 6 ^ r 2 ) p d r Then the e l e c t r i c quadrupole energy i s : E = 1 / 2 1 I Jx.Xg pdr = 1/6 Z Z ( V ^ Q ^ + 6 ^ / r 2 p d r ) . Since v"2v = 0 (La p l a c e ' s e q u a t i o n ) , E = 1/6 Z Z V. Q_ « ft r i 7 2.4 THE QUADRUPOLE HAMILTONIAN - THE CLASSICAL ANALOGY T h i s c l a s s i c a l treatment can be extended to give a quantum mechanical e x p r e s s i o n f o r the quadrupole H a m i l t o n i a n . We r e p l a c e the c l a s s i c a l observables i n the preceeding by t h e i r c o r r e s p o n d i n g quantum mechanical o p e r a t o r s : r -> R p -> p ° M r ) = e I 6 ( r - f K ) , where the sum k runs over the protons i n the nucleus, Then the quadrupole operator i s : Q ^ = e I S(3Zm**fiK ~ 6, / 9R 2)6(R-R K)dr -  e I 3 X ~ « V - K p * D ' and the quadrupole Hamiltonian i s : 1. I t i s important t o note t h a t i n t h i s procedure, we t r e a t the e l e c t r i c f i e l d produced by the e l e c t r o n s as an e x t e r n a l , c l a s s i c a l f i e l d . A more r i g o r o u s d e r i v a t i o n would r e p l a c e the EFG tensor by quantum mechanical e l e c t r o n spin o p e r a t o r s , and proceed i n a somewhat more i n v o l v e d f a s h i o n . 8 I t turns out, as d i s c u s s e d i n Abragam (1), that i n bulk matter (with few e x c e p t i o n s , notably r a r e e a r t h s ) the e l e c t r o n wave f u n c t i o n s are unchanged by quadrupole i n t e r a c t i o n s . We c o u l d then r e p l a c e the e l e c t r o n s p i n operators by t h e i r e x p e c t a t i o n v a l u e s , the components of the c l a s s i c a l EFG tensor - our treatment here, i f s i m p l i f i e d , i s n e v e r t h e l e s s e q u i v a l e n t . 2.5 MATRIX ELEMENTS OF THE QUADRUPOLE HAMILTONIAN We wish to eva l u a t e the matrix elements of the quadrupole oper a t o r s i n the v e c t o r space of the angular momentum e i g e n s t a t e s of the nucleus. T h i s w i l l allow us to make a formal replacement of the p o s i t i o n o p e r a t o r s i n the quadrupole oper a t o r s by l i n e a r combinations of angular momentum o p e r a t o r s with p r o p o r t i o n a l matrix elements. I t can be shown (2) that the quadrupole o p e r a t o r s , i n v o l v i n g terms i n 3X., - R 2, are l i n e a r combinations of i r r e d u c i b l e tensor o p e r a t o r s , which are c h a r a c t e r i z e d by t h e i r t r a n s f o r m a t i o n p r o p e r t i e s under r o t a t i o n s . The Wigner-Ekart theorem (3) all o w s us to r e p l a c e the p o s i t i o n o perators by any p r o p o r t i o n a l i r r e d u c i b l e t e n s o r s : If we l e t the op e r a t o r s e Z x w x* correspond to ( I ^ I ^ + 1^ 1^ )72 i n the Wigner-Ekart theorem, we have: <I,M|Q^|I,M > = <I,M|e Z < 3 x ^ x r - 6^R 2)|I,M'> = <I ,M| 3/2(1^ 1^  + 1,1.,) - 6 I 2|I,M'> C, 9 Where C i s a constant f o r a giv e n I, independent of M, M', a, and (3, which we can e v a l u a t e f o r m=m =1 and a=/3=z: C = eQ/I(21-1), where eQ = <I,I|e Z (3z 2 - r K 2 ) 11 ,1 > . K The constant Q, depending on the nuclear s t r u c t u r e , i s c a l l e d the quadrupole moment of the nucleus. We see from equation 1 that we can w r i t e the quadrupole Hamiltonian thus: H Q - e Q / 6 I ( 2 I - l ) Z Z Vmp {3/2 (lm 1^  I„ ) - 6^ I 2 } It i s o f t e n convenient t o w r i t e the Hamiltonian i n the p r i n c i p l e a x i s frame of the e l e c t r i c f i e l d g r a d i e n t tensor (V =0 f o r a*/3) : H 6= e 2 q Q / 4 I ( 2 I - 1 ) { ( 3 I Z 2 - I 2 + T ? ( I 2 - I Y 2 ) } , = e 2 q Q / 4 I ( 2 I - 1 ) { ( 3 I 2 - I 2 2 + (r,/2)(I + 2 - I f ) } 2, where q = V Z 2 /e and r? = (V^ -V y y )/v 2 J > . In a p p l i c a t i o n s , the p r i n c i p l e a x i s of the EFG tensor i s not the same as the l a b o r a t o r y frame, the z - a x i s of which i s d e f i n e d as the a x i s of q u a n t i z a t i o n , the d i r e c t i o n of the a p p l i e d f i e l d . We accomplish the t r a n s f o r m a t i o n t o the lab frame (a r o t a t i o n ) by e x p r e s s i n g the EFG as an i r r e d u c i b l e t e n s o r , i n s p h e r i c a l c o o r d i n a t e s ( 1 ) , ( 2 ) , ( 4 ) : 10 V2,o = V „ V 2 J 1 = (2/3) ( V z y +/- i V z y ) V 2, S 2 = (2/3) { W 2 ( \ % -V y y ) +/- i V j t y } Def i n i n g : T 2 i 0 = 311 - 12 T 2, t, = -/+ ( 3 / 2 ) ' / * ( l t I x + I z I t ) T 2 | t 2 = (3/2)^(1*) 2 We have, from equation 2: H a = eQ/4I(2I-l) I (-1)""vj mT 2. w 3. In the p r i n c i p l e a x i s frame, V 2 j 0 = eq V 2 . 4 1 = 0 V2t2 = 1/2(2/3) 77eq Transforming to the l a b frame: where a,/3, and 7 are the E u l e r angles d e s c r i b i n g the o r i e n t a t i o n of the l a b frame with respect to the p r i n c i p l e a x i s frame. 11 Then: H Q = eQ/4I(2I-D Z ( - 1 )%,,„, Z, V 2yD 2._ m(a,0, 7). To f i r s t o r der: H Q = e 2 q Q / 4 l ( 2 I - 1 ) T 2 0 x {D 2 o 0(a,/3,7) +»?/2[D 2 2 O(a,/3,7)+D 2 2 0 (a,/3,7) 1} = e 2qQ/4I (21 - 1 ) (3I 2 2-I 2) x{P 2(cos/3)+ ( 7 j/2)sin 2/3cos2a} 4, where P 2 ( x ) = 1 / 2 ( 3 X 2 - 1 ) . The energy l e v e l s f o r a s i n g l e ' quantum t r a n s i t i o n (AI=0, Am= + / - 1 ) f o r deuterium n u c l e i (1 = 1 ) are separated by an amount: Av Q = ^ { P j (cos/3) + (T?/2)sin 20cos2a} 5, where co^ = 3e 2qQ/4h. 12 2.6 MOTIONAL AVERAGING - THE ORDER PARAMETER TENSOR The observed t r a n s i t i o n s c h a r a c t e r i z e d by the frequency L\V i n the preceeding s e c t i o n w i l l be a f f e c t e d by the presence of f l u c t u a t i o n s in the o r i e n t a t i o n of the EFG. If these f l u c t u a t i o n s occur on a time s c a l e short compared to the c h a r a c t e r i s t i c time we w i l l measure an average over a l l o r i e n t a t i o n s of the EFG t e n s o r . T h i s average can be d e s c r i b e d i n terms of an average o r i e n t a t i o n a l order e x i s t i n g w i t h i n the system, as we w i l l see i n the f o l l o w i n g development. In order t o d e f i n e the order parameter te n s o r , we co n s i d e r a system of N c l a s s i c a l , r i g i d molecules. Each molecule has a p o s i t i o n , r , and an o r i e n t a t i o n , u . The f a c t t hat the molecule i s r i g i d means that we can s p e c i f y i t ' s o r i e n t a t i o n with three parameters. In NMR, we are i n t e r e s t e d i n the averages of o r i e n t a t i o n - and p o s i t i o n -dependent o p e r a t o r s over f l u c t u a t i o n s o c c u r i n g over the c h a r a c t e r i s t i c time s c a l e of the measurement. The quadrupole i n t e r a c t i o n depends on the c o n f i g u r a t i o n of a s i n g l e p a r t i c l e , and the average of such an op e r a t o r , A(X,), i s given by: < A(X,) > = 1/n J d X ^ X , ) P 1 ( X , ) ; where P 1(X,) i s the d i s t r i b u t i o n of s i n g l e t c o n f i g u r a t i o n s . By i n s p e c t i o n , we see that P 1 i s of the form: 1 3 P 1 ( f , ,7J, ) = N < SvXVx'i ) > =N < 6 ( f , - r ( ) 6(af1-w,/) >,. where the average i s over the primed v a r i a b l e s . If the p h y s i c a l property under i n v e s t i g a t i o n i s t r a n s l a t i o n i n v a r i a n t , we can w r i t e : P 1 (r*, ) = N/V f (2*,), where f(uf,) i s the s i n g l e t o r i e n t a t i o n d i s t r i b u t i o n . The average of any o r i e n t a t i o n - d e p e n d e n t operator i s thus: < A(uf,) > = Jdw, f (D1, ) A {cf\ ) . In t h e o r e t i c a l c o n s i d e r a t i o n s , i t i s o f t e n most u s e f u l to express the d e l t a f u n c t i o n s i n equation 2 i n terms of the Wigner r o t a t i o n m a t r i c e s . In p r a c t i s e , i t i s convenient to express t(u) more e x p l i c i t l y , t a k i n g advantage of p a r t i c u l a r symmetries of the geometry of the system. For molecules i n a u n i a x i a l phase, we can d e f i n e C a r t e s i a n c o o r d i n a t e s - x,y,z, f i x e d i n the molecule, and a,b,c i n the l a b frame. The s i n g l e t o r i e n t a t i o n d i s t r i b u t i o n can be expanded i n terms of the d i r e c t i o n c o s i n e s of the d i r e c t o r d e s c r i b i n g the o r i e n t a t i o n of the molecule i n the l a b frame ( 5 ) : 14 87r 2f(cj) = 1 + 5 Z Z S cose . cosd,- + where S*? = 1/2 < 3cos0„; c o s ^ - 5 6- > 6 ( i , j = x , y , z , and a,/3=a,b,c), d e f i n e s the elements of the Saupe o r d e r i n g matrix, or the order parameter t e n s o r . R e c a l l that the average i s over the o r i e n t a t i o n a l f l u c t u a t i o n s . We note immediately from the d e f i n i t i o n t h a t : S"> = s"/5 =S.A", and I s!f = I s~? = 0, lJ ' l i 'j so that the order parameter tensor has 25 independent components. 2.7 THE ORDER PARAMETER TENSOR IN LIPID BILAYERS The l i q u i d c r y s t a l l i n e l i p i d b i l a y e r phase (which we w i l l assume i s smectic A - l i k e ) has an a x i s of symmetry along the b i l a y e r normal, and a plane of r e f l e c t i o n between the sheets of the b i l a y e r ( 4 ) . I f we d e f i n e the c a x i s ( l a b frame) to be the b i l a y e r normal, we have: S*.6" = S ^ and S*/1 =0 from the symmetry about the c a x i s ; and S c a = S c b =0 by r e f l e c t i o n . T h i s l e a v e s only the d i a g o n a l elements, (a=/3), and s i n c e the order parameter i s t r a c e l e s s , i t reduces t o a tensor with 1 5 f i v e independent components: S-. = = - 2 S . fcb ' 5 i R e turning to our e x p r e s s i o n 1 f o r the quadrupole s p l i t t i n g , we see that we can w r i t e the m o t i o n a l l y averaged q u a n t i t y : where we take the molecule f i x e d c o o r d i n a t e system as the p r i n c i p l e a x i s of the EFG t e n s o r , and the c a x i s of the l a b frame as the a x i s of q u a n t i z a t i o n . 2 . 8 THE EFFECT OF BILAYER ORIENTATION In the preceeding s e c t i o n , we s p e c i f i e d that the c a x i s of l a b frame was along the b i l a y e r normal. I f the b i l a y e r normal makes an angle 0 with the c a x i s , we must c o n s i d e r a f u r t h e r o r i e n t a t i o n a l dependence of the quadrupole s p l i t t i n g . We d e f i n e a molecule f i x e d a x i s : a,b,c, where c i s normal to the b i l a y e r . We l e t x,y,z be the p r i c i p l e a x i s frame of the EFG. We t r a n s f o r m the EFG to the l a b frame as f o l l o w s ( 4 ) , ( 6 ) : < A y B > = 2 u ( 0 { S , + ( T J / 3 ) ( S Z Z V 2 w » D 2 (a,/3,7> D 2 (a, 0 , 7 ) , 16 where a,0,7 s p e c i f y the o r i e n t a t i o n of the b i l a y e r normal system with respect to the EFG p r i n c i p l e a x i s system, and a, 0,7 s p e c i f y the o r i e n t a t i o n of the l a b frame with r e s p e c t to the b i l a y e r normal system. Proceeding analogously to the p r e v i o u s d e r i v a t i o n of the quadrupole s p l i t t i n g , we have: Hfl = e 2 q Q / 4 I ( 2 I - l ) T 2 0 Z I, V L^-J a, 0 , 7 )D 2, p ( a, 0, 7 ) . Now s i n c e the c a x i s i s an a x i s of symmetry, the r o t a t i o n s a and 7 have no e f f e c t on the Hamiltonian. T h e r e f o r e : H 4 = e 2qQ/4I (21 - 1 ) T 2 0 1 / 2 ( 3 C O S 2 0 - 1 ) I V ^ D j f . ^ a , 0 , 7 ) = 1 / 2 ( 3 C O S 2 0 - 1 )eQ/4I (21 - 1 ) ( 3 I Z 2 - I 2 ) x { ( e q / 2 ) ( 3 c o s 2 0 - 1 ) + ( i ? / 2 ) c o s 2 a s i n 2 0 } , and <A»» > = CJq/2 ( 3c o s 2 0 - 1 ) x { S X I + ( 7 ? / 3 ) ( S ^ - S y y ) } 8. 17 I I I . DENSITY MATRIX FORMALISM In the s t a t i s t i c a l mechanics of s e m i - c l a s s i c a l systems such as l i q u i d c r y s t a l s , we are concerned with the ensemble averages of the e x p e c t a t i o n v a l u e s of quantum mechanical o p e r a t o r s . We express the s t a t i s t i c a l nature of the ensemble of quantum mechanical systems i n terms of the d e n s i t y o p e r a t o r ; the d e n s i t y matrix formalism i s a method of r e p r e s e n t a t i o n of t h i s operator - a method which f a c i l i t a t e s the d e s c r i p t i o n of the dynamics of the ensemble. In t h i s chapter we w i l l present a formalism, due t o Vega and Pines ( 7 ) , which i s p a r t i c u l a r l y u s e f u l in many experimental s i t u a t i o n s i n NMR. 3.1 FICTITIOUS SPIN-1/2 OPERATORS We c o n s i d e r a system of non i n t e r a c t i n g s p i n s I i n an e x t e r n a l magnetic f i e l d . The eigenspace of the s p i n -H a m iltonian i s of dimension M=2I+1, and the o p e r a t o r s i n t h i s system are of dimension MxM. The b a s i s of the operator space i s of dimension N=M -1. For the case 1=1/2, the d e n s i t y matrix i s of dimension 2x2, and the operator space i s of dimension N=3. We see that the d e n s i t y operator can be expressed i n terms of the s p i n o p e r a t o r s - I B , I y , and I z - and can t h e r e f o r e be t r e a t e d as a magnetization v e c t o r which p r e c e s s e s i n the presence of magnetic f i e l d s . For I>l/2, the dimension of the operator space i s N>3, and t h i s simple vector 18 d e s c r i p t i o n i s no longer complete. For n u c l e i with 1=1, we need N=9 operators of dimension 3x3 to completely d e s c r i b e the d e n s i t y o p e r a t o r . In t h i s case, we can d e f i n e a set of o p e r a t o r s which i s p a r t i c u l a r l y convenient i n the d e s c r i p t i o n of quadrupole i n t e r a c t i o n s : Only e i g h t of these -operators are independent; we d e f i n e nine f o r reasons of symmetry. Together with the u n i t o p e r a t o r , these o p e r a t o r s span the operator space. One of the I P 3 o p e r a t o r s i s dependent on the others through: I p i = 1/2 I p p=x,y,z I p 2 = 1/2 (1^1,. + I r I % ) I p 3 = 1/2 ( I 2 - I r 2 ) 9, where p,q,r =x,y,z, or c y c l i c permutations. I + I + I P 3 = 0 We can w r i t e the d e n s i t y matrix, then, as f o l l o w s : o(t) 10. The u t i l i t y of t h i s spanning set becomes apparent when we c o n s i d e r the commutation r e l a t i o n s and t r a n s f o r m a t i o n p r o p e r t i e s . Most i m p o r t a n t l y , we have, f o r a l l p=x,y,z: 19 [ I f 1 r l j . i l = 1 I?3 11, or c y c l i c permutations of 1,2,3. If we d e f i n e U^ ,; (6) = e x p ( i 0 I p ; ), then from the commutation r e l a t i o n above we have: U t p : ( e ) I p ; U p ; ( 0 ) = cos(6) I p j + s i n ( 0 ) IfK . So i n each subspace p, we have three o p e r a t o r s - I p l f I p 2 » a n d 1^,3, - which behave analogously, to the angular momentum ope r a t o r s - Ix , I y , and I 2 - f o r s p i n 1/2. For t h i s reason, these o p e r a t o r s are c a l l e d " F i c t i t i o u s Spin-1/2 Operators." We w i l l see p r e s e n t l y how t r a n s i t i o n s i n t h i s 3 - l e v e l system can be d e s c r i b e d i n terms of r o t a t i o n s of the d e n s i t y operator i n subspaces of these o p e r a t o r s . 3.2 SPIN HAMILTONIANS The Hamiltonian of a system with quadrupole i n t e r a c t i o n s i s : H = - w 0 l z + w e ( 3 I | - I 2 ) / 3 , where w Q = 3e 2qQ [(3cos 2/3 - 1) + 7?sin2/3 c o s 2 2 a ] / 8 l (1 + 1 ) and <j0 i s the energy of the Zeeman i n t e r a c t i o n . In NMR, we i r r a d i a t e the system with a r a d i o - f r e q u e n c y (RF) pulse, along the x - d i r e c t i o n , and the Hamiltonian becomes, in the 20 r o t a t i n g frame of the p u l s e : H = - ( A w ) i z - + " f l ( 3 I * - I 2 ) / 3 . where u> i s the frequency of the pu l s e 2a;, i s the amplitude of the pulse and the o f f s e t frequency i s : ACJ = (u0-u>) We w i l l c o n s i d e r two s p e c i a l cases of t h i s H a m i l t o n i a n : a) At resonance, no p u l s e (Ao> = 0, w,=0) H = 2/3 w „ < I „ 3 " Iy3> = ^<jl* 3 " UQ (Iy 3 - I Z3 )/3 = - a ) f i I y 3 - ua(IZ3 - lu3)/3 12 These three e q u i v a l e n t e x p r e s s i o n s allow us to c a l c u l a t e the time-dependence of the d e n s i t y matrix f o r d i f f e r e n t i n i t i a l s t a t e s . i ) o (0) = a r 1 I 2 1 do/At = -[H,a(0)] = a 2 , [ I , ,1 3 - I 3 ] = 0 So o(t) = a ( 0 ) . 21 i i ) o ( 0 ) = a x , I X , da/dt = a y 1 { [ I K l , W g I x 3 ] - [ l < i ^ ( I y 3 - I z 3 ) / 3 ] } Noting that the l a s t commutator i s zero, we can w r i t e : o(t) = U t y 3 ( ^ t ) a X 1 I Y , U j t a t ^ t ) = a x 1 ( c o s w a t IJCI + s i n u ^ t I Y 2 ) We see t h a t the quadrupole i n t e r a c t i o n i s e q u i v a l e n t to r o t a t i o n s i n the x and y subspaces of the f i c t i t i o u s s p i n -1/2 o p e r a t o r s , as shown i n f i g u r e ( 1 ) with ACJ=0. b) Off resonance, no pulse (ACJ*0, CL>,=0) We t r a n s f o r m the Hamiltonian a c c o r d i n g t o : H = Ut 22(»/2) H U z 2 U / 2 ) = 2 A C J I 2 3 + ( 2 / 3 ) w a ( I x 3 - I Y 3 ) = - (cja + A w ) I Y 3 - { ( l / 3 ) a ) f i - & J } ( I z 3 - i M ) = + (o>& - Aw)I* a - { ( i / 3 ) w + A ^ } ( I y 3 - l 2 3 ) 13 T h i s s i t u a t i o n i s represented i n f i g u r e ( 1 ) . Figure 1 - QUADRUPOLE INTERACTIONS The quadrupole i n t e r a c t i o n written in each of the x,y, and z subspaces of the f i c t i t i o u s spin-1/2 operators. Au> i s the o f f s e t frequency. The elements of the density operator, in the absence of other i n t e r a c t i o n s , rotate in the appropriate subspace analogously to the spin-1/2 operators. 23 3.3 THE QUADRUPOLE ECHO EXPERIMENT The quadrupole echo experiment (8) uses the f o l l o w i n g pulse sequence (as shown i n f i g u r e ( 2 ) ) : a) 90), a pu l s e along the x d i r e c t i o n of amplitude and d u r a t i o n such that the z-magnetization i s r o t a t e d by 90 degrees. We denote the Hamiltonian of the system d u r i n g t h i s p ulse as H x ; b) - the system i s allowed to evolve i n the absence of RF r a d i a t i o n f o r a time under the i n f l u e n c e of the fr e e H a miltonian, H^; c) 90) y a second 90 degree p u l s e i s a p p l i e d along the y d i r e c t i o n . The Hamiltonian d u r i n g t h i s time i s H ; d) The system again e v o l v e s f r e e l y i n time. We note that the p u l s e s used i n the quadrupole echo experiment are "hard" p u l s e s - i . e . of extremely short d u r a t i o n and l a r g e amplitude (w, >> ), so t h a t we can ignore the quadrupole i n t e r a c t i o n d u r i n g t h e i r a p p l i c a t i o n . We w i l l assume that the p u l s e s are a p p l i e d e x a c t l y at 24 resonance (Ao>=0). We co n s i d e r the e v o l u t i o n of the system under the i n f l u e n c e of each Hamiltonian: a) 90) x : a(0) = M 0 I = 2M 0I Z, H x = ~2u> 1 I x 1 a ( t ) = 2M0 (cosu^t I 7 , - sinu),t I y i ) a(t 9 o ) = 2M 01yi, Where 1 9 0 = tf/2a>, . b) H = -a)^Iy 3 ( n e g l e c t i n g commuting terms) a(0) = 2 M 0 I y 1 , a( t ) = 2M 0(coscj f it I y 1 - sinu„t I y 2 ) c ) a (0 ) = o ( r) t ' = t + T = 2M 0(COSW^T I y i - sinoj^T I y 2 ) H y — ~2u) i I yl . a ( t 9 0 ) = 2M 0(COSU) ?T I y 1 + sina>flr I y 2 ) d) H^ = -a> I y 3 a(0) = • a ( T + t 9 0 ) O U + T ) = 2M0 [cosa>CIT(cosa>£t/I y 1 - s i n u ^ t l y 2) + s i n w £ I T ( c o s a ) a t ' l y 2 + s i n a j ^ t l y , ) ] O(2T) = 2 M 0 I Y i . 25 We see that there i s indeed an echo of the t r a n s v e r s e magnetization, as a r e s u l t of the quadrupole i n t e r a c t i o n . In the case of a d i s t r i b u t i o n of f r e q u e n c i e s u , we see t h a t , as i n f i g u r e (2), the s i g n a l w i l l be a f r e e - i n d u c t i o n decay. But as t h i s treatment i s v a l i d f o r a l l values of CJ , each monochromat w i l l be refocussed at t=2r. F i g u r e 2 - QUADRUPOLE ECHO PULSE SEQUENCE The quadrupole echo p u l s e sequence, denoted 90) -T-90) ; the d i s t r i b u t i o n of the quadrupole f r e q u e n c i e s produces a f r e e i n d u c t i o n decay, part of which occurs in the region of r e c e i v e r dead time (shaded). The FID i s r e f o c u s s e d at t = 2 t , o u t s i d e t h i s r e g i o n . 27 IV. RELAXATION In the study of r e l a x a t i o n phenomena, we are concerned with the dynamics of systems of n u c l e a r s p i n s . The time-e v o l u t i o n of these systems i s i n f l u e n c e d by the i n t e r a c t i o n s between the s p i n s , and by the i n t e r a c t i o n s of the s p i n s with the l a t t i c e , which a c t s as a thermal bath. In s o l i d s , where the s p i n - s p i n c o u p l i n g s are l a r g e compared to the s p i n - l a t t i c e c o u p l i n g s , i t i s a p p r o p r i a t e to c o n s i d e r the s p i n s c o l l e c t i v e l y as a s i n g l e system with many degrees of freedom. Due to the strong c o u p l i n g between the s p i n s , i t i s p o s s i b l e to d e f i n e a s p i n temperature, and to d e s c r i b e the i n t e r a c t i o n with the l a t t i c e as i n f l u e n c i n g t h i s temperature. In l i q u i d c r y s t a l s , where the s p i n - s p i n c o u p l i n g s are c o n s i d e r a b l y reduced by the r a p i d motions, the s p i n - l a t t i c e c o u p l i n g i s dominant. We c o n s i d e r each s p i n to be e f f e c t i v e l y i s o l a t e d from the o t h e r s , i n t e r a c t i n g independently with the l a t t i c e . In t h i s approach, we develop the time-dependent s t a t i s t i c a l p r o p e r t i e s of the s p i n / l a t t i c e system u s i n g the d e n s i t y matrix formalism. 4.1 THE REDFIELD EQUATION Our s t a r t i n g p o i n t f o r the i n v e s t i g a t i o n of r e l a x a t i o n i n l i q u i d c r y s t a l s i s the second order approximation to the equation of motion of the d e n s i t y matrix. From time dependent p e r t u r b a t i o n theory, we have the standard r e s u l t 28 ( 2 ) , ( 3 ) : da'/dt = i [ a ' ( 0 ) f H , ' ( t ) ] + i 2 / [ [ o ' ( 0 ) , H 1 ' . ( t ' ) ] , H 1 ' ( t ) ] d t ' ..14, o where the operators are given i n t h e i r i n t e r a c t i o n r e p r e s e n t a t i o n s : a'(t) = e x p ( - i H 0 t ) ff(0) e x p ( i H 0 t ) For s p i n - l a t t i c e r e l a x a t i o n , we w i l l assume that the p e r t u r b a t i o n H,'(t) - which d e s c r i b e s the mechanisms of s p i n - l a t t i c e c o u p l i n g - i s a stationary,'random f u n c t i o n of time. T h i s assumption r e q u i r e s a f u r t h e r averaging of the terms in 14 over the f l u c t u a t i o n s i n the values of H , ' ( t ) . Thus: da'/dt = i [ a ' ( 0 ) , H , ' ( t ) ] + i * /[ [ o' ( 0 ) ,H, ' (t') ] ,H, ' ( t ) ]dt', o where the bar denotes the aforementioned average. We wish to compute the matrix elements: do^./dt =i<a|[o'(0),H,'(t)]|a> + i 2 /<a| [ [ a ' ( 0 ) , H 1 ' ( t ' ) ] , H 1 ' ( t ) ] |a'>dt 15 Expanding the products i n the f i r s t term, we have terms i n : 29 <a|a'(0) |bxb|H,' (t) |a>; s i n c e H,' i s a random f u n c t i o n , a l l these terms are equal to 0, and the f i r s t term has no c o n t r i b u t i o n . D e f i n i n g r = t-t', and expanding the p r o d u c t s i n the second term of 15, we get terms l i k e : a' (0)/<a |H,' ( t - T ) |bxb'|H, ' (t) |a'>exp{ i (a-b) r }exp{ i (a-b+b-o a') t }dr. We d e f i n e the c o r r e l a t i o n f u n c t i o n ( 2 ) : G^'(T) = <a|H,'(t) |bxb'|H 1'(t + r) |a> 16, and the s p e c t r a l d e n s i t y of t h i s c o r r e l a t i o n ( 2 ) : J a a ' ( w ) s ; G ^ ( T ) e x p ( - i w t ) d r .17. Since H,'(t) i s a s t a t i o n a r y random f u n c t i o n , the c o r r e l a t i o n f u n c t i o n i s symmetric and depends only on the value of T. T y p i c a l l y , we d e f i n e a c o r r e l a t i o n time of the i n t e r a c t i o n , r c , such t h a t : G ^ ' ( T ) = 0 f o r T > T C . We can then , c o n f i n i n g our a t t e n t i o n t o times t > T , extend the l i m i t s of i n t e g r a t i o n i n 15 to i n f i n i t y . 30 These c o n s i d e r a t i o n s s i m p l i f y equation 15 somewhat da' ,/dt = I R ^ e x p l i (a-a-b+b')t}a' (0) 18, where the c o e f f i c i e n t s R a a, i n v o l v e the s p e c t r a l d e n s i t i e s of the c o r r e l a t i o n f u n c t i o n s : C = l/2[J.?'(a-b) +j"Aa-b) - 6^, I J ^ i c b ) - 5 c b ? ^ ' ( c - b ' ) ] We wish to write equation 18 as a d i f f e r e n t i a l equation i n o ' ( t ) . If we have a'(0) = o ' ( t ) , we can simply r e p l a c e a'(0) on the r i g h t hand s i d e of 18. The i n c r e a s e : | [ o ' ( t ) - o ' ( 0 ) ] / o ' ( 0 ) | , i s on the order of ( 1 ) : 11H, 2 | TC . So i f we have, sim u l t a n e o u s l y , t>>r c and |H,2| << 1/tr c , we can w r i t e : da',/dt = I R^ b.exp{i(a-a-b+b')t}a',(t) 19, eta kk' N e g l e c t i n g the r a p i d l y - o s c i l l a t i n g terms i n the e x p o n e n t i a l , we have d y / d t = Z*%o[hAt) 20, 31 (where the sum i s over values of b and b such that a-b=a' -b ' ) . which i s R e d f i e l d ' s equation, or the g e n e r a l i z e d master equation. 4.2 FINITE TEMPERATURE CORRECTION Let us c o n s i d e r the R e d f i e l d equation f o r the matrix element a' : aa. &-0. , aa t> b . . bt By i n s p e c t i o n , we see that the c o e f f i c i e n t s RA<t = Wi(L f o r a*b where WfcA i s the p r o b a b l i l i t y of inducing a t r a n s i t i o n from s t a t e lb> to s t a t e la>. So we have: If we assume that the p o p u l a t i o n s of each s t a t e obey the simple equation: dN t/dt= I ( N ^ - N ^ ) , where N t i s the p o p u l a t i o n of the s t a t e b, then at e q u i l i b r i u m : 32 / W a A(N 4-N f t) = 0, so Na=Nfc f o r a l l a,b. Thi s i s r e l a x a t i o n to an i n f i n i t e temperature, where a l l the po p u l a t i o n s are equal. T h i s i n f i n i t e temperature i s an obvious r e s u l t of the f a c t that the d e n s i t y matrix i n equation 6 c o n t a i n s no inform a t i o n about the s t a t e of the l a t t i c e , which w i l l a f f e c t the t r a n s i t i o n p r o b a b i l i t i e s . We can apply a ri g o r o u s c o r r e c t i o n to our s e m i - c l a s s i c a l d e s c r i p t i o n of the l a t t i c e by an e x p l i c i t c o n s i d e r a t i o n of the l a t t i c e v a r i a b l e s i n the development of the R e d f i e l d equation (1). T h i s amounts to an equation ( f o r high temperatures) which i s s l i g h t l y m o d i f i e d : da aV/dt = Lt R^expU (a-a'-b+b')t} { o ^ U J - o ^ / f T ) 21a, where ofc'fc,(T) = exp{-ihb/kT}/Tr{exp(-iH/kT)} 21b i s the thermal e q u i l i b r i u m value of the d e n s i t y matrix element. 4.3 OPERATOR FORM OF THE REDFIELD EQUATION It i s o f t e n e a s i e s t to ev a l u a t e the e x p e c t a t a t i o n values of operators of i n t e r e s t by w r i t i n g equation 7 in operator form. We can always w r i t e : H, (t) = 2 ( - 1 )*F(t) A 0 2 2 , 33 where F ( t ) are s t a t i o n a r y random f u n c t i o n s of time, and A are s p i n o p e r a t o r s . Then the c o r r e l a t i o n f u n c t i o n s are: GbY(r) = I (-l) V*<a|A' |bxb'|A' |a> F, ( t ) F # / ( t + r) 23. We can d e f i n e s p e c t r a l d e n s i t i e s ( 1 ) : J ,(CJ) = (-1) J F # ( t ) F - ( t + T ) e x p ( - i u T ) d r 24, We d e f i n e d p r e v i o u s l y , i n equation 16, the s p e c t r a l d e n s i t y : Ja.fc ^ ~ $ ('")exp ( - i c j T)dr = I (-1) <a|A' |bxb'|A' |a> JM.,(u) 25. In order to write the e v o l u t i o n equation i n operator form, we c o n s i d e r the second order approximation: da'/dt = - / [ [ a ' ( 0 ) , H 1 ' ( t ) ] , H , ' ( t + T ) ] d T . We can w r i t e : H,'(t) = I (-1)*FjA^ = I (-1) F 0 A„ exp(iw, p t ) , 34 where i s the frequency of the t r a n s i t i o n s between the energy l e v e l s of the system, and the sum p runs over these f r e q u e n c i e s . By convention we choose A. such t h a t A =A , so that Uyp=-cj^p. Now, suppressing the r a p i d l y o s c i l l a t i n g terms i n the e x p o n e n t i a l , we set o>y=-a^ , so that p=p/, and q=-q' I f we f u r t h e r assume that J^Au) - ^y^^.^iu) > which i s encountered in almost a l l cases (1), we can write f i n a l l y : 26, do/dt = - L (-1)*J• (u0)[A , [A , o ]] which i s the operator form of R e d f i e l d ' s equation. 4.4 QUADRUPOLE RELAXATION We can write the Hamiltonian of a system with quadrupole r e l a x a t i o n as f o l l o w s : H = H z+ H f t(t) + {H^U) - H Q ( t ) } where H z i s the Zeeman Hamiltonian, H,j(t) i s the quadrupole Hamiltonian, and H^(t) i s the s t a t i c p a r t of the quadrupole H a m i l t o n i a n . From our ex p r e s s i o n i n chapter two f o r the quadrupole Hamiltonian, i t i s c l e a r t h a t : H,(t) = ( H Q ( t ) - H ^ ( t ) ) = CQL (-1 f T ^ V ^ J t ) ..27, where Cg i s a constant of p r o p o r t i o n a l i t y ; the s t a t i c part 35 of H a r i s e s from the time average of the V ( t ) . The s p e c t r a l d e n s i t i e s are given in the l a b frame by: J1f'{u}m) = %'{u*m) = exp(ima)T) ( t ) V a ^ ( t + T)dT. In order to e v a l u a t e these s p e c t r a l d e n s i t i e s , we must f i n d the components of the EFG tensor in the l a b frame. The EFG, i n s p h e r i c a l c o o r d i n a t e s , transforms i n terms of s p h e r i c a l harmonics a c c o r d i n g to ( 4 ) : V ^ ( t ) = ( 4 T T / 5 ) V 2 0 LY2rn(8l<t>)a 2n(p') 2 8 , where /3' i s the angle between the b i l a y e r normal and the l a b frame, 6 and <t> are the p o l a r and azimuthal angles d e s c r i b i n g the o r i e n t a t i o n of the EFG p r i n c i p l e a x i s with respect to the b i l a y e r normal (we assume, f o r s i m p l i c i t y , that the EFG tensor i s a x i a l l y symmetric), and d ^ (0) i s p a r t of the Wigner r o t a t i o n matrix ( 6 ) . The s p e c t r a l d e n s i t i e s are then given by: J„(mo>) = (-1 f (4*/5) ( V 0 0 ) 2 ) x z z ^ d ^ n a j n r U ' ) x / exp(inujr) Y 2„'(e(t) ,<t>) Y2m.( 6 ( t - T ), *) 29 The time dependence i s now c o ntained i n the averages of the s p h e r i c a l harmonics ( i g n o r i n g f l u c t u a t i o n s i n the angle /3 ), 3 6 i n the f l u c t u a t i o n s of the angles s p e c i f y i n g the o r i e n t a t i o n of the EFG. We postpone the development of a model which would s p e c i f y the form of these f l u c t u a t i o n s f o r the moment, and r e t u r n to the equation of motion of the d e n s i t y o p e r a t o r . If we express the d e n s i t y operator i n terms of the f i c t i t i o u s spin-1/2 o p e r a t o r s , we f i n d : Where a p i (0) i s the e q u i l i b r i u m value of that component of the d e n s i t y o p e r a t o r . E v a l u a t i n g the commutators i n the equation 30, we o b t a i n the f o l l o w i n g equations ( 4 ) , ( 9 ) : d a K 1 / d t = -cj^a^z ~ a x 1 /T 2 t t d a y 1 / d t = w 6 a y 2 - a y 1 / T 2 b da„ 2/dt = w&ax , - a * 2 / T 2 a d a y 2 / d t = - ^ a y 1 - a y 2 / T 2 b 31 I ? ; dap-t /dt = \ E ( _ 1 } ^ " ^ o X a p ; - a p ; (0)) *I T*«'I Ta,--' Ipl ] ] • • • • 3 0 where: 1 / T 2 a = ( e Q / h ) 2 / 8 { 3 L o ( 0 ) / 2 + 5 L , ( « 0 ) / 2 + L 2 ( 2 C J 0 ) } 32a 1/T 2 b = ( e Q / h ) 2 / 8 { 3 L O ( 0 ) / 2 + L , ( « 0 ) / 2 + L 2 ( 2 C J 0 ) } 32b, When we s o l v e these coupled equations f o r the t r a n s v e r s e magnetization, we f i n d that i t decays e x p o n e n t i a l l y to zero with c h a r a c t e r i s t i c time T 2, where: 37 1/T2 = 1/2(1/T 2 < L+ 1/T 2 6) 32c Furthermore, f o r the z-subspace, we have: da z , / d t = - ( a ^ - a J W T , 33, where i s the thermal e q u i l i b r i u m value of t h i s component (assumed to be the only non-zero thermal e q u i l i b r i u m element), and 1/T 1 2 = (eQ/h) 2 / 8 { L , (w 0) + 4L 2(2w 0)} 34. We w i l l r e t u r n to the problem of e v a l u a t i o n of the s p e c t r a l d e n s i t i e s i n these equations i n chapter s i x , where we c o n s i d e r s p e c i f i c examples of the motions r e s p o n s i b l e f o r r e l a x a t i o n i n d i f f e r e n t c i r c u m s t a n c e s . 38 V. THE QUADRUPOLE ECHO SPECTRUM 5.1 THE POWDER PATTERN SPECTRUM In chapter two we d e r i v e d a formula f o r the quadrupole s p l i t t i n g of the deuterium resonance s i g n a l f o r an o r i e n t e d sample: <Ava> = CJ^(3COS 2/3'-1 ) { S 2 2 +(TJ/3) ( S x y - S y y ) }/2, where i s the angle between the b i l a y e r normal and the a x i s of q u a n t i z a t i o n . In the experiments performed i n t h i s study, the NMR sample c o n s i s t s of l i p i d v e s i c l e s , where the b i l a y e r normal i s randomly o r i e n t e d with r e s p e c t to the a x i s of q u a n t i z a t i o n . The observed resonance spectrum i s t h e r e f o r e a s u p e r p o s i t i o n of i n d i v i d u a l quadrupole s p l i t t i n g s , and i s c a l l e d a powder p a t t e r n . I t i s p o s s i b l e , using the above e x p r e s s i o n , to c a l c u l a t e the lineshape f o r the powder p a t t e r n . In the case where the asymmetry parameter of the EFG i s zero, a simple treatment w i l l s u f f i c e t o c a l c u l a t e the li n e s h a p e of the powder p a t t e r n spectrum. We can c o n s i d e r the random d i s t r i b u t i o n of o r i e n t a t i o n s of the b i l a y e r normal as a uniform d i s t r i b u t i o n of N n u c l e i over the s u r f a c e of a sphere ( r a d i u s R), where the p o s i t i o n of the nucleus on the sphere corresponds to the o r i e n t a t i o n of the molecule with which i t i s a s s o c i a t e d (6). The s u r f a c e d e n s i t y of n u c l e i i s : 39 a = N / ( 4 T T R 2 ) . The number of n u c l e i o r i e n t e d with p o l a r angle between 8 and 6+d8 i s dN = a(2?rR sin0)R dfl. The p r o b a b i l i t y that a nucleus w i l l have t h i s o r i e n t a t i o n i s : p(0)d0 = dN/N = s i n e 6 6 / 2 Now the resonance frequency of any nucleus i s ( f o r 7?=0): hu>0 +/-= hw0 +/- w aS 2 Z ( 3 c o s 2 0 -1) 35. We d e f i n e a reduced resonance frequency: z+ = (*>Q ~ ha) 0)/("«S I Z ) = +/-(3cos 20 -1 )/2 36. where z + e [-1/2,1] z. e [-1,1/2] We can d e f i n e a p r o b a b i l i t y d i s t r i b u t i o n of the reduced 40 resonant f r e q u e n c i e s : p ( z ) d z , the p r o b a b i l i t y that a nucleus w i l l have give a resonance s i g n a l between z and z+dz. We a l s o d e f i n e the d i s t r i b u t i o n s p(z+) such t h a t : p(z) = p(z*) + p(z_) i f p(z) = p ( z + ) i f p(z) = p(z.) i f ze(-1/2,1/2) ze[1/2,1] z e [ - 1 , - 1 / 2 ] . Now p ( z t ) = p(0)d0/dz t = ( - l / 2 ) d c o s 0 / d z ± , and s i n c e dz + = +/-3cos0dcos0, l A we have: p(z«.) = |C/cos6| = C/(+/- 2z+ + 1) 37 where C i s a constant of p r o p o r t i o n a l i t y . The i n t e n s i t y of each resonance peak i s r e l a t e d to t h i s p r o b a b i l i t y d i s t r i b u t i o n by the t r a n s i t i o n p r o b a b i l i t y I ( z ). I f we assume that the t r a n s i t i o n p r o b a b i l i t y i s independent of o r i e n t a t i o n and equal f o r both t r a n s i t i o n s , the i n t e n s i t y i s given by: S(z) = Ip(z) Note that the spectrum d i v e r g e s f o r z = +/- 1/2, r e p r e s e n t i n g the most probable o r i e n t a t i o n , 6=90'. The spacing between these peaks i s Az =1, so that a . J ^ = O ) ^ S I 2 . For the case where TJ*0, the treatment i s more comp l i c a t e d , as we can no longer simply r e l a t e the p r o b a b i l i t y d i s t r i b u t i o n of o r i e n t a t i o n s to the quadrupole s p l i t t i n g . 41 We w i l l not present the d e r i v a t i o n here, but simply r e f e r the reader to r e f e r e n c e s (4),. (10), and (11). In a c t u a l s p e c t r a , the t h e o r e t i c a l powder p a t t e r n i s mo d i f i e d by motions and i n t e r a c t i o n s between the n u c l e i , an e f f e c t which can be accounted f o r by c o n v o l u t i n g the spectrum with a broadening f u n c t i o n . A s i n g l e resonance l i n e , with no broadening, i s given by a d e l t a f u n c t i o n . If t h i s l i n e i s broadened through d i p o l a r i n t e r a c t i o n s , c e n t e r e d about some frequency z , i t can be approximated by the normalized Gaussian f u n c t i o n : G(z-z') = exp{-(z-z' ) 2 / ( 2 a 2 ) }/(2w) a 38a, •with a full-width-half-maximum (FWHM) of Az = 2.35a. A L o r e n t z i a n l i n e s h a p e i s another commonly used approximation: L ( z - z ) = Uo[1 + ( z - z ' )2/o2)}' 1 38b, with a FWHM of Az = 2a. The powder p a t t e r n spectrum i s given by the c o n v o l u t i o n : S(z) = / B(z-z' ) p ( z ' ) d z ' , where B i s the broadening f u n c t i o n . F i g u r e (3) g i v e s the powder s p e c t r a f o r a Gaussian broadening f u n c t i o n , for v a r i o u s values of the parameter a. F i g u r e 3 - THE POWDER PATTERN SPECTRUM a ) T h e o r e t i c a l powder p a t t e r n , with small broadening (a=l0" 5) i n c l u d e d to a v o i d d i s c o n t i n u i t i e s . Peaks occur at 0=90'. b ) , c ) , d ) , e ) : Gaussian broadening f o r values of a=.01,.025,.05,.75. to -i 1 1 1.44 1.92 2.4 R E D U C E D F K E t f U E M C y F i g u r e 4 - ORIENTATION-DEPENDENT BROADENING Orientation-dependent Gaussian broadening for:a) o 0=.025, a,=0.; b) o o=.025, o,=.0l; c) o o=.025, o,=.025; d) o o=.025, o,=.05: e) (To=.025, cr.=.075. 1 04 192 2.4 1 •« 0.0 0.48 0.96 REDUCED FRe<4.U6WCV 44 I t i s p o s s i b l e t h a t , i n the case of d i p o l a r i n t e r a c t i o n s , the magnitude of the broadening may vary a c c o r d i n g to the o r i e n t a t i o n of the nucleus. In t h i s case, the value of the parameter a w i l l change with the resonance frequency i n the above treatment. To approximate t h i s e f f e c t , we i n t r o d u c e an o r i e n t a t i o n dependent f u n c t i o n i n the Gaussian d i s t r i b u t i o n : a = 0 o + cTi (3cos 20-1 ) 39 The r e s u l t s of t h i s broadening are shown i n f i g u r e ( 4 ) . 5.2 SPECTRAL ANALYSIS Although we can o b t a i n exact e x p r e s s i o n s f o r the powder p a t t e r n l i n e s h a p e s , i n v o l v i n g s u p e r p o s i t i o n s of quadrupole s p l i t t i n g s f o r d i f f e r e n t n u c l e i , the broadening f u n c t i o n s used i n the approximations u s u a l l y r e s u l t i n imperfect agreement between r e a l and simulated s p e c t r a . The q u a n t i t a t i v e analyses of NMR s p e c t r a makes use of a number of methods. We can measure the quadrupole s p l i t t i n g s d i r e c t l y from the s p e c t r a , a method which i s s u b j e c t to s y s t e m a t i c e r r o r due to the broadening, as p o i n t e d out i n (12). U sing the l i n e s h a p e formula i n the preceeding s e c t i o n , and a d j u s t i n g the parameters i n the broadening f u n c t i o n s , we can attempt to simulate the s p e c t r a . T h i s i n v o l v e s some method of f i t t i n g the experimental data by 45 a d j u s t i n g the values of the quadrupole s p l i t t i n g s , and r e q u i r e s that we make assumptions about the form of the broadening f u n c t i o n and the dynamics of the system. S p e c t r a l s i m u l a t i o n s have been d e s c r i b e d i n d e t a i l i n (12) and (13). Moments a n a l y s i s and s p e c t r a l "de-Pake-ing", two f a i r l y simple numerical techniques, are perhaps among the more t r a c t a b l e of the methods of s p e c t r a l a n a l y s i s , and r e q u i r e no a p r i o r i assumptions about the dynamics of the i n t e r a c t i o n . 5.2.1 MOMENTS ANALYSIS The moments of the quadrupole echo spectrum can be c a l c u l a t e d a c c u r a t e l y and s y s t e m a t i c a l l y , y i e l d i n g a d e s c r i p t i o n of the d i s t r i b u t i o n of the quadrupole s p l i t t i n g s of the system. For our purposes we d e f i n e the nth moment of the spectrum, M , as f o l l o w s : M r = / co r if(a))dcj/ / f ( u ) d u 40. 0 0 I t can be shown (14) that the nth moment of a spectrum with one quadrupole s p l i t t i n g , i n the case of a x i a l symmetry, i s given by: M „ = A n ( A ^ n ) 41, where A„ i s a constant. For a d i s t r i b u t i o n of quadrupole s p l i t t i n g s , we see that the nth moments are given by: 46 M = A (2TT) / (&v ) p(Ap )d(Av ) 42, where p(Ai> ) i s the d i s t r i b u t i o n of the quadrupole s p l i t t i n g s ; we note that the integ r a n d i s the nth moment of t h i s d i s t r i b u t i o n . The f r a c t i o n a l mean-squared width of the d i s t r i b u t i o n of quadrupole f r e q u e n c i e s i s given by the parameter, A 2, which has been d e s c r i b e d in (4) and (14). In the case of a x i a l symmetry: A 2 = (<S 2> - <S > 2)/<S > 2). = (M2/1 .35M, 2 )-1 43. 5.2.2 SPECTRAL DE-PAKE-ING The essence of the method of "de-Pake-ing" i s that i t i s p o s s i b l e i n p r i n c i p l e to completely d e s c r i b e the d i s t r i b u t i o n of quadrupole s p l i t t i n g s , so that we can c a l c u l a t e the o r i e n t e d spectrum - i n the case of a x i a l symmetry, i t i s p o s s i b l e to e x t r a c t t h i s i n f o r m a t i o n d i r e c t l y . We r e f e r the reader to (15) and (16) f o r d e t a i l e d accounts of t h i s proceedure. In f i g u r e (5), we present a comparison of the o r i e n t e d spectrum and the powder spectrum f o r one of the samples used i n the experiment i n t h i s t h e s i s . The advantage of the o r i e n t e d spectrum i s that i t rep r e s e n t s the d i s t r i b u t i o n of quadrupole s p l i t t i n g s e x p l i c i t l y , o f t e n a l l o w i n g us to detect s u b t l e f e a t u r e s which would be obscured by the s u p e r p o s i t i o n s i n the powder spectrum. 47 In t h i s r e p o r t , a l l of the de-Pake-ed s p e c t r a are presented as they would be f o r samples o r i e n t e d with the magnetic f i e l d p a r a l l e l to the plane of the membrane; i . e . , at 90 degrees with respect to the b i l a y e r normal. T h i s has the advantage of making the s p e c t r a l f r e q u e n c i e s of the o r i e n t e d s p e c t r a c o i n c i d e with those of the s p e c t r a l "edges" of the powder s p e c t r a . F i g u r e 5 - DEPAKED SPECTRUM A comparison of the "de-Pake-ed" spectrum, (A), with the powder spectrum, (B), showing d i r e c t l y the d i s t r i b u t i o n of order parameters. Spectra are f o r v e s i c l e s of perdeuterated DMPA ( D i - M y r i s t o y l P h o s p h a t i d i c Acid) with DMPC ( D i - M y r i s t o y l Phosphatidal Choline) in a molar r a t i o of DMPA:DMPC=5:1, i n the l i q u i d c r y s t a l l i n e phase at 44°C. " t 1 1 1 1 1 1 1 r r 1 0.0 4.0 a.O 12.0 IB.O 20.0 24.0 20.0 S2.0 36.0 40.0 FREQUENCY (KHz) 49 VI. DEUTERIUM NMR STUDIES OF PHOSPHOLIPID BILAYERS P h o s p h o l i p i d b i l a y e r s , above t h e i r s o - c a l l e d g e l - l i q u i d c r y s t a l phase t r a n s i t i o n , behave l i k e l y o t r o p i c l i q u i d c r y s t a l s . L i k e a l l l i q u i d c r y s t a l s , they are c h a r a c t e r i z e d by somewhat p a r a d o x i c a l p r o p e r t i e s : i n one d i r e c t i o n , the plane t r a n s v e r s e to the membrane, they are h i g h l y ordered -they behave l i k e o p t i c a l l y u n i a x i a l c r y s t a l s , with the o p t i c a l a x i s along the b i l a y e r normal. In other d i r e c t i o n s , they e x h i b i t the r a p i d t r a n s l a t i o n a l , r o t a t i o n a l , and f l e x i n g motions t y p i c a l of the f l u i d phase. The d i f f u s i o n tensor i s thus h i g h l y a n i s o t r o p i c : l a t e r a l d i f f u s i o n r a t e s can be up to ten o rders of magnitude higher than t r a n s v e r s e d i f f u s i o n r a t e s (17),(18). In b i o l o g i c a l membranes, t h i s f l u i d i t y i n the plane of the membrane serves many b i o l o g i c a l f u n c t i o n s , such as s u r f a c e p l a s t i c i t y , s e l f - s e a l i n g of the membrane, c e l l f u s i o n and endo/exocytosis, and o t h e r s . The u n i a x i a l o r d e r i n g in the t r a n s v e r s e plane makes these systems a t t r a c t i v e to i n v e s t i g a t i o n , s i n c e the r o t a t i o n a l motions and c o n f o r m a t i o n a l changes, i f f a s t on the time s c a l e of the measurement, w i l l be c h a r a c t e r i z e d by a x i a l symmetry - the asymmetric components of the motion w i l l be " m o t i o n a l l y averaged" to zero about the a x i s of the b i l a y e r normal. The c o r r e l a t i o n times a s s o c i a t e d with these molecular motions vary over a wide range, with estimated values from 10~ 7 - 1 0 " 8 S f o r r o t a t i o n s about the long a x i s and l a t e r a l 50 d i f f u s i o n , t o 10" 1 1 f o r c o n f o r m a t i o n a l changes. 2H-NMR i s an extremely powerful t o o l . i n the study of these motions, and p r e s e n t s many advantages over other techniques used i n l i p i d systems. In proton ('H)'H-NMR, the a n i s o t r o p y of the motion r e s u l t s i n l a r g e d i p o l a r i n t e r a c t i o n s , r e s u l t i n g i n a broad, f e a t u r e l e s s spectrum. With 2H-NMR, where the quadrupole i n t e r a c t i o n i s orders of magnitude l a r g e r than the d i p o l a r i n t e r a c t i o n , the quadrupole s p l i t t i n g s are sh a r p l y d e f i n e d . The quadrupole i n t e r a c t i o n depends only on the o r i e n t a t i o n of a s i n g l e n u c l e a r s i t e , and removes the co m p l i c a t i o n s inherent i n i n t e r p r e t a t i o n of both i n t e r - and i n t r a - m o l e c u l a r i n t e r a c t i o n s . B i o s y n t h e t i c techniques f o r the d e u t e r a t i o n of l i p i d s ( s u b s t i t u t i o n of 1H atoms with deuterium) have been p e r f e c t e d to the p o i n t where i t i s p o s s i b l e to s y s t e m a t i c a l l y l a b e l s p e c i f i c p o s i t i o n s on the f a t t y a c i d c h a i n s of the l i p i d . The enormous advantage of t h i s technique i s that deuterium, with a Van der Waals r a d i u s i d e n t i c a l to that of the hydrogen, leaves the membrane system v i r t u a l l y unperturbed. We c o n t r a s t t h i s s i t u a t i o n to the method of s p i n - l a b e l e l e c t r o n s p i n resonance, which r e q u i r e s the presence of a bulky s p i n l a b e l group, and present s a s i g n i f i c a n t p e r t u r b a t i o n . The almost n e g i l i b l e abundance of deuterium i n b i o l o g i c a l systems e l i m i n a t e s the ov e r l a p p i n g of resonances from u n l a b e l l e d p a r t s of the molecule; t h i s becomes important i n s e l e c t i v e l a b e l l i n g experiments, where the only s i g n a l i n the spectrum i s from 51 the l a b e l l e d segment ( u n l i k e 1H- and 1 3C-NMR). The c h a r a c t e r i s t i c time s c a l e of 2H-NMR spectroscopy i s determined by the range of quadrupole s p l i t t i n g s a s s o c i a t e d with d i f f e r e n t o r i e n t a t i o n s of the EFG at the nucl e a r s i t e . The second moment of the spectrum, M 2 - the root-mean-square quadrupole s p l i t t i n g - i s a u s e f u l measure of t h i s range. If the molecular motions, c h a r a c t e r i z e d by the c o r r e l a t i o n time T , are such t h a t : M 2 r 2 >>1, the motion i s f a s t on the time s c a l e of the measurement, and the spectrum w i l l be " m o t i o n a l l y narrowed"(1),(2); i . e . , the quadrupole i n t e r a c t i o n s w i l l be averaged over the motions. The c h a r a c t e r i s t i c time s c a l e of 2H-NMR i s on the order of microseconds, so that the spectrum i s extremely s e n s i t i v e to motions o c u r r i n g with c o r r e l a t i o n times i n t h i s regime, but can give no d e t a i l e d i n f o r m a t i o n on much f a s t e r motions. Measurements of the time constant f o r the decay of the quadrupole echo, T 2 ,have a l s o been found (19) to be s e n s i t i v e to r e l a t i v e l y slow motions, with c o r r e l a t i o n times T > 10~ 7 seconds. S p i n - l a t t i c e r e l a x a t i o n measurements, however, are s e n s i t i v e to much f a s t e r motions. The s p i n - l a t t i c e r e l a x a t i o n time, T,, i s determined by the s p e c t r a l d e n s i t y of the f l u c t u a t i o n s of the quadrupole i n t e r a c t i o n which occur i n the v i c i n i t y of the resonant frequency of the nucleus (4). Since 2H-NMR resonant f r e q u e n c i e s are on the order of tens of MHz, T, measurements are s e n s i t i v e to motions with c o r r e l a t i o n times on the order of 10" 9 seconds. 52 6.1 DEUTERIUM-NMR SPECTRA OF PHOSPHOLIPID MEMBRANES Aqueous d i s p e r s i o n s of p h o s p h o l i p i d s spontaneously form l i q u i d c r y s t a l l i n e b i l a y e r s t r u c t u r e s at c e r t a i n temperatures, and undergo a phase t r a n s i t i o n , to a more h i g h l y ordered g e l phase as they a re c o o l e d . T y p i c a l 2H-NMR powder s p e c t r a of these two phases are shown i n f i g u r e ( 6 ) (pp. 80) . We note that the s p e c t r a are r e p r e s e n t a t i v e of an asymmetry parameter 17=0 f o r the l i q u i d - c r y s t a l l i n e spectrum, and that the spectrum i s ve r y broad with 779*0 f o r the g e l spectrum. T h i s i s to be understood i n terms of the f a c t that the molecular motions are c o n s i d e r a b l y reduced i n the g e l phase: the spectrum i s su b j e c t to l e s s motional narrowing, as the c o r r e l a t i o n times f o r the motion become longer. The asymmetry parameter of the C-D bond i s n e g l i g i b l e i n the l i q u i d c r y s t a l l i n e s p e c t r a ( 7 7 S . O 6 ) ; s i n c e in the g e l s p e c t r a we observe much l a r g e r v a l u e s , we conclude that the averaging i s over a x i a l l y symmetric motions i n the l i q u i d c r y s t a l , and that the motions i n the ge l phase are not a x i a l l y symmetric. The systems s t u d i e d most tho r o u g h l y with 2H-NMR are the di a c y l - p h o s p h o g l y c e r i d e s , where the s t r a t e g y has been to deuterate the a c y l chains of the f a t t y a c i d groups. The motion of the l i p i d s can then be s t u d i e d as a f u n c t i o n of depth in the membrane by p e r d e u t e r a t i o n ( s u b s t i t u t i o n of a l l 1H atoms on the c h a i n ) , or by - a s e r i e s of s e l e c t i v e d e u t e r a t i o n s . For an o r i e n t e d sample of perdeuterated l i p i d 53 in the l i q u i d - c r y s t a l l i n e regime, the 2H-NMR spectrum c o n s i s t s of a s u p e r p o s i t i o n of quadrupole s p l i t t i n g s c orresponding to each deuteron on the c h a i n s . The EFG i s averaged such that the asymmetry parameter i s zero; the quadrupole s p l i t t i n g i s thus d i r e c t l y p r o p o r t i o n a l to the order parameter of the C-D bond. The powder p a t t e r n spectrum of the same l i p i d , e x p e r i m e n t a l l y e a s i e r and l e s s time-consuming to o b t a i n , would c o n s i s t of a s u p e r p o s i t i o n of the powder p a t t e r n s of each deuteron. S e l e c t i v e d e u t e r a t i o n has shown (6) that i t i s p o s s i b l e to a s s i g n the quadrupole s p l i t t i n g s (order parameters) to s p e c i f i c deuterons without ambiguity. In the l i q u i d c r y s t a l l i n e r e g i o n , the f o l l o w i n g general r u l e s h o l d : the order parameters remain roughly constant f o r the f i r s t s e v e r a l p o s i t i o n s down the c h a i n from the headgroup ( t h i s i s the s o - c a l l e d "plateau r e g i o n " ) , and decrease from that p o i n t on to a minimum at the t e r m i n a l methyl group. T h i s d i s t r i b u t i o n of order parameters i s r e f l e c t e d i n the powder p a t t e r n spectrum f o r perdeuterated samples in the l a r g e i n t e n s i t y at the edge of the spectrum, a s s o c i a t e d with a g r e a t e r number of deuterons i n the p l a t e a u r e g i o n , with the l a r g e s t order parameters, and i n the s m a l l e s t order parameter a s s o c i a t e d with the methyl groups at the end of the c h a i n (6,14). T h i s means that there i s a f l e x i b i l i t y g r a d i e n t along the a c y l chains of the l i p i d molecule, a phenomenon which can be e x p l a i n e d at a molecular l e v e l i n terms of a simple 54 q u a l i t a t i v e argument (17). The p o l a r headgroups of the l i p i d molecules are suspended at the i n t e r f a c e between the water and the hydrophobic c h a i n s , and are packed c l o s e l y together i n t h i s p lane. T h i s packing imposes s t e r i c c o n s t r a i n t on the co n f o r m a t i o n a l freedom of the a c y l c h a i n s , r e s u l t i n g i n a measurable o r d e r i n g of the C-D bonds, corresponding to a p a r t i a l i n h i b i t i o n of the angular motions of the C-D bonds. The c o n s t r a i n t s are a p p r e c i a b l y r e l i e v e d some d i s t a n c e down the c h a i n . Towards the cente r of the b i l a y e r , the region of the f r e e ends of the c h a i n , the s t e r i c c o n s t r a i n t s approach a minimum, and the order parameter decreases. The f l e x i b i l i t y of the hydrocarbon chains i s a r e s u l t of r a p i d exchanges between the t r a n s and gauche conformal isomers with respect t o r o t a t i o n s about the carbon-carbon bonds. A ch a i n i n the a l l - t r a n s conformation r e s u l t s i n the longest and t h i n n e s t conformer, and i s hence the most e a s i l y packed (and t h e r e f o r e the most h i g h l y - o r d e r e d ) conformation. I s o l a t e d gauche conformations r e s u l t i n a bend i n the e n t i r e chain of approximately 120°; the order parameters of a l l deuterons subsequent to t h i s conformation w i l l be decreased by a f a c t o r P 2 ( c o s 6 0 " ) . Kinks i n the cha i n are l o c a l d e f e c t s i n the a l l - t r a n s conformation, are the r e s u l t of the co n f o r m a t i o n a l sequence: gauche -trans-gauche . I f we assume that t h i s d e f e c t d i f f u s e s along the c h a i n , with equal p r o b a b i l i t y of oc c u r i n g at any cha i n p o s i t i o n , and with a c o r r e l a t i o n time short on the 2H-NMR time s c a l e , we see that 55 i t w i l l have no e f f e c t on the order parameter p r o f i l e (17). I t i s important to note t h a t , i n our d e f i n i t i o n of the order parameter in chapter two, we assumed that the molecule was r i g i d " . The treatment for f l e x i b l e molecules i s much more i n v o l v e d . In i n t e r p r e t i n g the order parameters in the NMR s p e c t r a , we must be c a r e f u l to d i s t i n g u i s h between molecular r e o r i e n t a t i o n s and c o n formational i n t e r c o n v e r s i o n s , and we must a l s o c o n s i d e r the f a c t that these motions must n e c e s s a r i l y be i n t e r r e l a t e d . Whether or not a complete determination of molecular order i s p o s s i b l e from t h i s s t a n d p o i n t , the order parameters are s e n s i t i v e to the changes i n molecular motions, and give v a l u a b l e i n s i g h t i n t o the molecular o r i e n t a t i o n , conformation, and dynamics of l i p i d systems. 6.2 DEUTERIUM-NMR RELAXATION STUDIES In the general theory of r e l a x a t i o n d i s c u s s e d in chapter f o u r , we went only so f a r as to express the r e l a x a t i o n times T, and T 2 i n terms of the s p e c t r a l d e n s i t i e s of the f l u c t u a t i n g quadrupole i n t e r a c t i o n . We saw that the s p e c t r a l d e n s i t i e s c o u l d be expressed i n terms of f l u c t u a t i o n s of the o r i e n t a t i o n of the EFG p r i n c i p l e a x i s system with respect to the b i l a y e r normal. There are c u r r e n t l y s e v e r a l t h e o r e t i c a l and experimental d i f f i c u l t i e s a s s o c i a t e d with i n t e r p r e t a t i o n of r e l a x a t i o n measurements in l i p i d systems. The t h e o r e t i c a l d i f f i c u l t i e s a r i s e from the l a c k of a p p r o p r i a t e models f o r 5 6 the motions r e s p o n s i b l e f o r r e l a x a t i o n - there i s a c e r t a i n amount of ambiguity and disagreement i n v o l v e d i n the s p e c i f i c a t i o n of models in d i f f e r e n t dynamical regimes. In the context of a given model, the h e t e r o g e n e i t y of the dynamics of l i p i d systems p r e c l u d e s the r i g o r o u s d e r i v a t i o n of a simple r e l a t i o n s h i p between the c o r r e l a t i o n times of the motions, T , and the t r a n s v e r s e and s p i n - l a t t i c e r e l a x a t i o n times, T, and T 2 . The experimental d i f f i c u l t i e s i n the i n t e r p r e t a t i o n i n v o l v e the measurement of the e f f e c t of the order parameter of the molecular segment under c o n s i d e r a t i o n on the r e l a x a t i o n times. Although t h i s dependence can be s t u d i e d i n d e t a i l with s e l e c t i v e d e u t e r a t i o n , as in (20), the time r e q u i r e d and the expense of t h i s approach are s i g n i f i c a n t drawbacks. Perdeuterated samples seem much more a t t r a c t i v e i n t h i s r e s p e c t , e s p e c i a l l y i n l i g h t of the f a c t that r e l a x a t i o n measurements are s t i l l e s s e n t i a l l y q u a l i t a t i v e i n d i c a t o r s of molecular dynamics. With t h i s c a u t i o n i n mind, we proceed to examine some of the models which have been proposed. 6.2.1 ROTATIONAL DIFFUSION A molecular d e s c r i p t i o n of r o t a t i o n a l motion i n l i q u i d s can be approached, i n the d i f f u s i o n l i m i t , by c o n s i d e r i n g the motion as a s e r i e s of random walks over s m a l l angular displacements. The d i f f u s i o n l i m i t i s a p p r o p r i a t e i n most l i q u i d s , where the angular v e l o c i t i e s of the motion are 57 q u i c k l y clamped by the medium, and the angular displacements are t h e r e f o r e not h i g h l y c o r r e l a t e d . A n a t u r a l f i r s t approximation to the motions i n the g e l phase i s to assume that the methylene groups on the f a t t y a c i d chains are simply r o t a t i n g about the b i l a y e r normal, and that the b i l a y e r normal i s an a x i s of symmetry, so that the angle between i t and the C-D bond i s const a n t , we can evalu a t e the s p e c t r a l d e n s i t i e s as i n ( 4 ) : J i ( u 0 ) = ( V 2 0 ) 2 [ 1 - C O S « 0 ] S e x p ( i a j T ) g 2 ( r ) d T ...44a J 2 ( 2 C J 0 ) = ( V 2 0 ) 2 [1+COS 4 /3 + 6cos 2 | 3 ] J exp ( 2 icor ) g 2 ( T ) dr /4 44b where g 2 ( r ) = exp( 2i <j> (t) ) exp(-2 i </> ( t - r ) 44c i s the c o r r e l a t i o n f u n c t i o n of the r o t a t i o n . If we assume that the r o t a t i o n can be d e s c r i b e d as a random walk with c o r r e l a t i o n time TC , we have: g 2 ( r ) = exp( | T | A C ) , and the r e l a x a t i o n time i s given by: 1/T, = u Q { [ 1 - C O S ^ ] ( 2 T 6 /1+W 2 T 6 2 ) + [ 1+cos*/3+6cos 2 0 ] (2r e /l.+ 4w 2T c a ) } 45, 58 which i s equal to 1/T 2 i n the extreme narrowing l i m i t , AM 2 T 2 « 1 . In the crude hydrodynamic d e s c r i p t i o n of r o t a t i o n a l d i f f u s i o n , assuming that the molecule i s c y l i n d r i c a l , and undergoing i s o t r o p i c r o t a t i o n s in a medium of l o c a l v i s c o s i t y 17, the c o r r e l a t i o n time i s given by (21): T = T J V / K T , where V i s the volume of the c y l i n d e r , and kT i s the thermal energy of the molecule. T h i s model p r e d i c t s a strong dependence of T, on the o r i e n t a t i o n of the b i l a y e r normal. In the presence of f u r t h e r mechanisms of motional averaging f a s t on the NMR time s c a l e , such as l a t e r a l d i f f u s i o n and/or v e s s i c l e tumbling, t h i s dependence would be removed by an average over a l l o r i e n t a t i o n s of the b i l a y e r normal: 1/T, = ^ { ( 2 r £ /1+C J 2 T / ) + (8T c /l+4o ; 2 T E 2 ) } 46 In the extreme narrowing l i m i t , t h i s e x p r e s s i o n reduces to the form: 1/T, = 1/T 2, p r o p o r t i o n a l to r t . We assumed i n t h i s d e r i v a t i o n that the p o l a r angle 8, d e s c r i b i n g the o r i e n t a t i o n of the C-D bond w i t h respect to the b i l a y e r normal, was constant, and equal to 90* (corresponding to S c o=1, a h i g h l y ordered s t a t e ) . In g e n e r a l , we expect that the s p e c t r a l d e n s i t i e s w i l l i n v o l v e 59 terms i n the order parameter of the segment, r e p r e s e n t i n g an averaging over the f l u c t u a t i o n s i n the angle 6. Using the extreme narrowing l i m i t , Brown (21) p r e s e n t s a s i m i l a r d e r i v a t i o n of the s p i n - l a t t i c e r e l a x a t i o n time: 1/T, = {u>0/2)l^ - P 2(cos/3 ) S C D - S C 2 P (1-P 2(cos0 ) ) ] T C 47 Where SCD = <P 2(cos0)>, where the average i s over the f l u c t u a t i o n s i n the angle 8. T h i s reduces, i n the case of r a p i d i s o t r o p i c l a t e r a l d i f f u s i o n and/or v e s i c l e tumbling, t o : 1/T, = (w a/2) 0 - S c D 2 ) T C . 6.2.2 DEFECT DIFFUSION In the l i q u i d c r y s t a l l i n e r e g i o n , the s p i n - l a t t i c e r e l a x a t i o n time f o r the a c y l c h a i n s i n l i p i d b i l a y e r s has i n general been found to be p r o p o r t i o n a l to CJ 0 ; t h i s phenomenon i s u s u a l l y e x p l a i n e d i n terms of c o n f o r m a t i o n a l changes as the dominant mechanism f o r r e l a x a t i o n . In p a r t i c u l a r , Kimmich (23) d e r i v e s a formula f o r the s p e c t r a l d e n s i t i e s of the f l u c t u a t i o n s due t o d e f e c t d i f f u s i o n i n the short c o r r e l a t i o n time regime: J ( u 0 ) = ( ^ ^ { [ ( r , ) * - 2 ( r , / 1 ]/[T A - r^ ] 2 x u 0 6 0 48 i f O) 0T << 1 , where: T„ = b 2/2D, v = d 2/2D, b i s the l e n g t h of the d e f e c t , d i s the width of the b i l a y e r , D = ( 2 r ) " 1 b 2 i s the one-dimensional d i f f u s i o n c onstant, and T = 7 0exp(E/RT) i s the c o r r e l a t i o n time f o r d e f e c t d i f f u s i o n . T h i s model p r e d i c t s a r e l a x a t i o n time: 1/T, = ( 3 . 8 / 3 ) A M 2 { [ ( 7 t / V ) * - 2 ] / [ ( r 4 / T , y * - l ] } x (u/coofa 49, where AM2 i s the change i n the second moment brought about by t h i s motion. I t should a l s o been noted that Brown (23) has suggested a model which g i v e s the r e q u i r e d T, dependence which i n v o l v e s a treatment of " c o l l e c t i v e motions" of the l i p i d s . 61 6.2.3 DECAY OF THE QUADRUPOLE ECHO For l a r g e ( n o n - d i f f u s i o n a l ) molecular r e o r i e n t a t i o n s in the extreme narrowing l i m i t , the theory of motional averaging f o r i s o t r o p i c motions (1) g i v e s the r e s u l t : 1/T 2 = A M 2 T c 50, where AM 2 i s the change i n the second moment which i s produced by the motion. For r o t a t i o n a l motion about the long a x i s of the molecule, AM 2 = C J 2 [ 1 -P 2 (cos0) ] 2 . In the d i f f u s i o n l i m i t , where the molecular r e o r i e n t a t i o n s are s m a l l , we have 1/T 2 = 1/T, f o r A M 2 r c 2 <<1 , as d e s c r i b e d p r e v i o u s l y . In the slow motion regime, AM 2 T E 2 >>1, and f o r l a r g e molecular r e o r i e n t a t i o n s , i t can be shown (24) t h a t : T2 = P^c ' W I T H P- 1 5 1 • In the echo experiment, we i n t e r p r e t t h i s r e s u l t as f o l l o w s . We c o n s i d e r a molecular jump o c c u r i n g such that the resonant frequency of the nucleus changes randomly by an amount > ( A M 2 ) . Within the time 2T between the f i r s t p u l s e and the top of the echo, the s i g n a l w i l l accumulate random phase i n t h i s manner, which means t h a t . p a r t of the FID w i l l not be re f o c u s s e d in the echo. The i n t e n s i t y of the echo w i l l be p r o p o r t i o n a l t o : exp(-2T/pr e ), where p corresponds to the "phase memory" of 62 the system f o l l o w i n g the jump. I t has been shown (25).that i n the case of r o t a t i o n a l d i f f u s i o n i n the slow motion regime, T 2 has the same d i r e c t p r o p o r t i n a l i t y to T c as i n t h i s treatment. Given the behavior of T 2 i n the l i m i t i n g cases of slow and f a s t motion, we see that i t must approach a minimum value i n some intermediate regime. On t h i s b a s i s , f o r l a r g e molecular r e o r i e n t a t i o n s , i t has been proposed (19) that T 2 may be represented e m p i r i c a l l y f o r AM2 =1: by the f o l l o w i n g i n t e r p o l a t i o n formula: 1/T2 = AM 2r c /( 1+pAM2 ) 52. T h i s equation reduces to the a p p r o p r i a t e form f o r each domain, and has a minimum f o r T 2 = 2(p/AM 2) , with c o r r e l a t i o n time r = (pAM 2) 63 V I I . LIPID-PROTEIN INTERACTIONS A v a r i e t y of techniques have been d i r e c t e d towards the study of spontaneous a s s o c i a t i o n s between membrane p r o t e i n s and l i p i d s , i n c l u d i n g X-Ray d i f f r a c t i o n , d i f f e r e n t i a l -scanning c a l o r i m e t r y , c i r c u l a r d i c h r o i s m , f r e e z e - f r a c t u r e e l e c t r o n microscopy, s p i n l a b e l ESR, NMR, and o t h e r s . T h i s has r e s u l t e d i n a wealth of i n f o r m a t i o n i n t h i s very broad s u b j e c t , i n v o l v i n g complimentary and c o o p e r a t i v e e f f o r t s from d i v e r s e f i e l d s i n the b i o l o g i c a l and p h y s i c a l s c i e n c e s . While a u n i f y i n g d i s c u s s i o n of t h i s i n f o r m a t i o n i s a formidable task, w e l l beyond the scope of t h i s paper, a general overview of some of the important f e a t u r e s of the s t r u c t u r e and o r g a n i z a t i o n of membranes i s v a l u a b l e at t h i s p o i n t . We r e f e r the reader to (13),(17), (18), and (26), for comprehensive reviews of 2H-NMR of l i p i d - p r o t e i n i n t e r a c t i o n s . B i o l o g i c a l membranes, even those with much lower than p h y s i o l o g i c a l 1 i p i d / p r o t e i n (L/P) r a t i o s , p r e s e r v e the main s t r u c t u r a l and dynamical f e a t u r e s of pure l i p i d membranes: the l i p i d b i l a y e r i s the dominant, i f not the only s t r u c t u r e of b i o l o g i c a l membranes; b i o l o g i c a l membranes e x h i b i t c h a r a c t e r i s t i c p h o s p h o l i p i d phase t r a n s i t i o n s , and t h e i r v i s c o s i t i e s and d i f f u s i o n t e n s o r s are comparable to those of pure l i p i d membranes. S u r p r i s i n g l y , the l i p i d - p r o t e i n i n t e r a c t i o n s must be seen i n l i g h t of these f a c t s as p e r t u r b a t i o n s to the p r o p e r t i e s of the l i p i d s . 6 4 Membrane p r o t e i n s can be c l a s s i f i e d a c c o r d i n g to t h e i r a s s o c i a t i o n s with the l i p i d s : i n t e g r a l ( i n t r i n s i c ) membrane p r o t e i n s span the l i p i d b i l a y e r - they are a m p h i p h i l i c , and i n t e r a c t with both of the l i p i d m o i e t i e s ; p e r i p h e r a l ( e x t r i n s i c ) membrane p r o t e i n s are p o l a r molecules, i n t e r a c t i n g d i r e c t l y only with the headgroups of the l i p i d s , at the l i p i d - w a t e r i n t e r f a c e . We can make the f o l l o w i n g g e n e r a l i z a t i o n s about i n t e g r a l membrane p r o t e i n s : There must be a s i g n i f i c a n t f r a c t i o n of the p h o s p h o l i p i d i n d i r e c t c o n t a c t with the i n t r i n s i c p r o t e i n - hydrophobic i n t e r a c t i o n s are expected to play a s i g n i f i c a n t r o l e . (28) There i s l i t t l e evidence to suggest that l i p i d -p r o t e i n i n t e r a c t i o n s are s p e c i f i c (some e x t r i n s i c p r o t e i n s have been shown to bind p r e f e r e n t i a l l y to n e g a t i v e l y charged l i p i d s ) (18); t h i s i s c o n s i s t e n t with the view that hydrophobic i n t e r a c t i o n s are dominant. - There i s no evidence to suggest that l i p i d - p r o t e i n i n t e r a c t i o n s occur v i a the formation of s t a b l e l i p o -p r o t e i n complexes (18). The dominant intramembrane s t r u c t u r e of the pol y p e p t i d e components of i n t e g r a l membrane p r o t e i n s i s the a l p h a - h e l i x (26). 65 We. have seen .that deuterium NMR g i v e s d e t a i l e d i n f o r m a t i o n on the order and dynamics of l i p i d s i n membrane s t r u c t u r e s , and we expect that the i n t r o d u c t i o n of p r o t e i n s i n t o the membrane w i l l have a s i g n i f i c a n t e f f e c t on these p r o p e r t i e s . NMR. s t u d i e s have mostly i n v o l v e d the r e c o n s t i t u t i o n of p u r i f i e d membrane p r o t e i n s with deuterated l i p i d systems. Some s t u d i e s have been c a r r i e d out by the i n c o r p o r a t i o n of deuterated l i p i d s or f a t t y a c i d s as probes i n i n t a c t membranes. In both cases, the p r o p e r t i e s of the l i p i d - p r o t e i n system are compared with those of d i s p e r s i o n s of the l i p i d s a l o n e . A v a r i e t y of l i p i d - p r o t e i n i n t e r a c t i o n s have been i n v e s t i g a t e d (see (18) f o r t a b u l a t e d r e f e r e n c e s to systems s t u d i e d up to 1982); f o l l o w i n g e a r l i e r p o s i t i v e ESR experiments, the f i r s t attempts i n these i n v e s t i g a t i o n s c o n c e n t r a t e d on the search f o r two d i f f e r e n t kinds of p h o s p h o l i p i d s i g n a l s : one corresponding to "bulk" l i p i d s , those not i n contact with p r o t e i n , and another corresponding to "boundary", or "annular" l i p i d s , those i n c o n t a c t with the p r o t e i n s . I t was expected that the order parameters of the l i p i d s i n the l i q u i d c r y s t a l l i n e phase, normally q u i t e low, would be i n c r e a s e d s u b s t a n t i a l l y by a s s o c i a t i o n with the c o m p a r i t i v e l y r i g i d p r o t e i n s t r u c t u r e s . The boundary l i p i d s would e x h i b i t some degree of i m m o b i l i z a t i o n , r e s u l t i n g i n i n c r e a s e d i n t e n s i t y i n the wings (outer edges) of the spectrum, corresponding to l i p i d s with a higher order parameter. 66 7.1 LIPID-PROTEIN INTERACTIONS IN THE LIQUID CRYSTALLINE  PHASE The r e s u l t s of these i n v e s t i g a t i o n s showed t h a t , i n the l i q u i d c r y s t a l regime, there was no evidence f o r the e x i s t e n c e of bulk and boundary l i p i d domains on the time s c a l e of the experiment ( l O " 5 s ) . T h i s corresponds to the p i c t u r e of the l i p i d molecules exchanging between the d i f f e r e n t domains, r a p i d l y on the NMR time s c a l e , but slowly on the ESR time s c a l e ( i n l i g h t of the p o s i t i v e ESR r e s u l t ) . The l i q u i d c r y s t a l l i n e s p e c t r a of these systems are broadened with the a d d i t i o n of p r o t e i n , but the average order parameter i s , s t r i k i n g l y , unchanged. The broadening can be e x p l a i n e d i n terms of a random d i s t r i b u t i o n of p r o t e i n , so that a l a t e r a l l y - d i f f u s i n g l i p i d encounters a d i s t r i b u t i o n of inhomogeneous r e g i o n s . The f a c t that the average order parameter of the l i p i d molecule i s unchanged remains p u z z l i n g - i t has been suggested (28) that the p r o t e i n s are not i n f a c t r i g i d , but are "squishy" i n nature, l e n d i n g themselves to p e r f e c t mechanical matching with the l i p i d s . The study of the d i s t r i b u t i o n of order parameters as a f u n c t i o n of p r o t e i n c o n c e n t r a t i o n at temperatures w e l l above the p h o s p h o l i p i d phase t r a n s i t i o n g i v e s the f o l l o w i n g g eneral r e s u l t s : the apparent quadrupole s p l i t t i n g s reduced as the L/P r a t i o decreased - t h i s i s not to say that the a c t u a l quadrupole s p l i t t i n g s became s m a l l e r , but that the 6 7 spectrum begins to c o l l a p s e i n t o a s i n g l e , broadened l i n e -for s u f f i c i e n t l y low values of L/P, the s p l i t t i n g s c o l l a p s e d , g i v i n g r i s e to a spectrum c h a r a c t e r i s t i c of a p r o t e i n - a s s o c i a t e d phase. The average value of the quadrupole s p l i t t i n g of the p r o t e i n - a s s o c i a t e d phase s p e c t r a are i n agreement with that f o r the pure p h o s p h o l i p i d l i q u i d c r y s t a l l i n e phase. T h i s suggests t h a t , i f we can c o n s i d e r the s p e c t r a to be a x i a l l y symmetric, l i p i d s i n the p r o t e i n -a s s o c i a t e d phase have the same average order parameter as those i n the pure l i p i d l i q u i d c r y s t a l . We turn now to a c o n s i d e r a t i o n of the r e l a x a t i o n s t u d i e s i n the l i q u i d c r y s t a l l i n e phase, which can be c h a r a c t e r i z e d by two g e n e r a l r e s u l t s : The value of T, are not changed by more than 10-30% by the presence of p r o t e i n s . - The values of T 2 decrease d r a m a t i c a l l y with the a d d i t i o n of p r o t e i n s , even f o r high L/P r a t i o s . In l i p i d b i l a y e r s above the phase t r a n s i t i o n , as p o i n t e d out p r e v i o u s l y , the value of T, e x h i b i t s a c h a r a c t e r i s t i c o;0 dependence. Brown (23) has demonstrated that the mechanism of " c o l l e c t i v e motions" g i v e s t h i s dependence. I t has been shown (4) that d e f e c t d i f f u s i o n can a l s o e x p l a i n t h i s behavior,as seen in equation 49. In view of the f a c t that T, i s not d r a m a t i c a l l y a l t e r e d by the 68 presence of the p r o t e i n , i t would seem that d e f e c t d i f f u s i o n (which we expect w i l l not be. d r a m a t i c a l l y i n f l u e n c e d by the presence of p r o t e i n ) i s r e s p o n s i b l e f o r the r e l a x a t i o n . C o l l e c t i v e motions, i t would be expected, are much more l i k e l y to be d i s r u p t e d by the presence of the p r o t e i n . The T 2 r e s u l t suggests that the p r o t e i n i n t r o d u c e s a slow motion i n the p h o s p h o l i p i d s ; t h i s motion would have to be too slow to s e v e r e l y i n f l u e n c e T 1 f yet introduce s u f f i c i e n t modulation of the quadrupole f r e q u e n c i e s to decrease T 2 . V a r i o u s models have been proposed to e x p l a i n t h i s r e s u l t , i n c l u d i n g : Two s i t e exchange between bulk and boundary l i p i d domains (26), - R o t a t i o n of the p r o t e i n (29), - R e o r i e n t a t i o n of the boundary l i p i d s (26), and - Change in the s i z e or shape of the v e s i c l e s due to the p r o t e i n (30). 7.2 RESULTS IN THE GEL PHASE The e f f e c t of the l i p i d - p r o t e i n i n t e r a c t i o n i n the g e l phase i s to reduce the average order parameter - t h i s i s o f t e n r e f e r r e d to as the " f l u i d i z a t i o n " of the a c y l c h a i n . For high v a l u e s of L/P, the g e l s p e c t r a can be r i g o r o u s l y decomposed i n t o two s p e c t r a : one i s i d e n t i c a l to that of the pure l i p i d g e l phase; the other i s i d e n t i c a l t o the p r o t e i n -a s s o c i a t e d l i p i d phase i n the r e g i o n above the phase 69 t r a n s i t i o n (26),(31). F r e e z e - f r a c t u r e methods (18) have shown that t h i s corresponds to a macroscopic s p a t i a l s e p a r a t i o n of these two phases. The f l u i d i z a t i o n of the membrane i s a s s o c i a t e d with the p r o t e i n - a s s o c i a t e d r e g i o n s . 7.3 ORIENTATIONAL ORDER AND MEMBRANE STRUCTURE We r e t u r n now to one of the c e n t r a l q u e s t i o n s r a i s e d by these i n v e s t i g a t i o n s : How can we e x p l a i n the f a c t that the average order parameter of p r o t e i n - a s s o c i a t e d l i p i d s i s the same as that f o r the pure l i p i d s i n the l i q u i d c r y s t a l l i n e phase? A complete answer to t h i s q u e s t i o n depends on a more d e t a i l e d c o n s i d e r a t i o n of p r o t e i n s t r u c t u r e . Mouritson and Bloom (28), i n "The Mattress Model of L i p i d - P r o t e i n I n t e r a c t i o n s " , present a d e t a i l e d t h e o r e t i c a l a n a l y s i s of the e f f e c t of hydrophobic c o n s t r a i n t s on membrane s t r u c t u r e , and give a geom e t r i c a l i n t e r p r e t a t i o n of o r i e n t a t i o n a l order in membranes. They demonstrate t h a t the order parameter of the a c y l c h a i n i s c o r r e l a t e d with the width of the b i l a y e r , and argue that the membrane must s t r e t c h or c o n t r a c t a c c o r d i n g to the len g t h of the hydrophobic part of the p r o t e i n . The systems s t u d i e d to date would be expected to have been h y d r o p h o b i c a l l y matched, they argue, s i n c e the s t r a t e g y has been to study p r o t e i n s with l i p i d systems t y p i c a l to those found i n the n a t u r a l membrane of the p r o t e i n s . Under such circumstances, we would not expect the average order of the a c y l c h a i n s to be a f f e c t e d by the a d d i t i o n of p r o t e i n . 70 These c o n s i d e r a t i o n s l e d d i r e c t l y to the c o n c e p t i o n of an experimental program designed to measure the e f f e c t s of hydrophobic mismatch, by i n c o r p o r a t i n g v a r y i n g l e n g t h s of s y n t h e t i c p o l y p e p t i d e i n t o the membrane. 7.4 A SYNTHETIC AMPHIPHILIC POLYPEPTIDE The p o l y p e p t i d e L y s 2 - G l y - L e u -Lys 2-Ala-amide (K 2GLK 2A-amide) was s y n t h e s i z e d by s o l i d - p h a s e peptide s y n t h e s i s , as d e s c r i b e d i n (32), f o r values of n= 16, 20, 24. T h i s p e p t i d e has a t o t a l of 5 p o s i t i v e charges at n e u t r a l pH: two-on e i t h e r p o l a r end c o n t r i b u t e d by the l e u c i n e r e s i d u e s , and one at one end c o n t r i b u t e d by the amide group. I n i t i a l s t u d i e s of the peptide i n c o r p o r a t e d i n t o DPPC ( d i - p a l m i t o y l p h o s p h a t i d y l c h o l i n e ) and pottasium p a l m i t a t e b i l a y e r s (32), (33), and (34), have shown promising r e s u l t s . C i r c u l a r d i c h r o i s m measurements show that the peptide i s a s t r o n g a-h e l i x formaer in l i p i d b i l a y e r s , as expected. With the knowledge of the geometry of the p e p t i d e , and the a b i l i t y to modify the length of the hydrophobic r e g i o n , we can i s o l a t e s p e c i f i c q u e s t i o n s concerning l i p i d - p r o t e i n i n t e r a c t i o n s . As p r e d i c t e d by the mattress model, a d d i t i o n of the p e p t i d e f o r with a value of n=24, c o r r e s p o n d i n g to a hydrophobic length g r e a t e r than the hydrophobic t h i c k n e s s of the b i l a y e r , produces an i n c r e a s e i n the average order parameter of the a c y l c h a i n s of the l i p i d s . X-ray d i f f r a c t i o n measurements show that t h i s i n c r e a s e i n the 71 order parameter corresponds to an i n c r e a s e i n the t h i c k n e s s of the b i l a y e r . E f f o r t s are c u r r e n t l y i n pr o g r e s s to map the complete phase diagram of these systems, with temperature and peptide c o n c e n t r a t i o n as the independent var i a b l e s (33). A study of the dynamics of the p o l y p e p t i d e i n the same l i p i d system was performed using 2H-NMR measurements of the exchangeable hydrogen s i t e s on the peptide (19). Below the phase t r a n s i t i o n of the l i p i d - p e p t i d e system, s p e c t r a were obtained which were q u i t e s i m i l a r to the spectrum of the s o l i d d e u t e r a t e d peptide spectrum at room temperature; The pepti d e was immobilized, on the time s c a l e of the measurement. Through the phase t r a n s i t i o n , the s p e c t r a corresponded to the onset of a r a p i d r e o r i e n t a t i o n of the pepti d e about the long a x i s of the a - h e l i x . Measurements of T 2 showed a decrease over the phase t r a n s i t i o n r e g i o n , to a minimum at 35°C. T h i s has been i n t e r p r e t e d as due to the presence of i n c r e a s i n g f r a c t i o n s of f l u i d phase l i p i d domains, and the c o r r e l a t i o n time at which a minimum value of T 2 would occur was c a l c u l a t e d , a c c o r d i n g to equation (8), to be (r ) = 9p MS. In the l i q u i d c r y s t a l l i n e r e g i o n , T was c a l c u l a t e d i n the range of f a s t motions to be T = 2X10~ 7 S., g i v i n g a v a l u e f o r the membrane v i s c o s i t y of 77 = 1 . 1 p o i s e . T h i s value i s in agreement with independent measurements of membrane v i s c o s i t y f o r r e c o n s t i t u t e d p r o t e i n systems. The l i p i d s used i n these s t u d i e s were n e u t r a l l y charged 72 z w i t t e r i o n s , r e s u l t i n g i n a net p o s i t i v e charge on the membrane. T h i s c h o i c e i s reasonable f o r the i n v e s t i g a t i o n of hydrophobic i n t e r a c t i o n s . Given that the peptide i s charged, however, i t i s n a t u r a l to ask, f o l l o w i n g v a r i o u s r e p o r t s of p r e f e r e n t i a l b i n d i n g of charged e x t r i n s i c p r o t e i n s , about the nature of the i n t e r a c t i o n of the peptide with n e g a t i v e l y charged l i p i d s . The experiment r e p o r t e d here was conc e i v e d as an exten s i o n of the work on the pept i d e , i n l i g h t of recent s t u d i e s by Devaux et a l (35) on the bi n d i n g of cytochrome-C to the charged headgroups of DMPS ( d i - m y r i s t o y l p h o s p h a t i d y l s e r i n e ) and DMPA ( d i - m y r i s t o y l p h o s p h a t i d i c a c i d ) ( u n p u b l i s h e d ) . I t was intended as a study of the i n t e r a c t i o n of DMPA with the pe p t i d e , o r i g i n a l l y i n terms of both the Coulomb i n t e r a c t i o n of the headgroups of the DMPA with the p o l a r m o i e t i e s of the p e p t i d e , and of the e f f e c t s of hydrophobic mismatch. The i n i t i a l s t r a t e g y was again to search f o r an immobilized l i p i d component - the r e l a t i v e s t r e n g t h and long range of the Coulomb i n t e r a c t i o n would presumably i n c r e a s e the l i f e t i m e of DMPA/peptide complexes, i n c r e a s i n g the l i k e l i h o o d of obs e r v i n g an immobilized component on the NMR time s c a l e . In the absence of an immoblized component, we would at l e a s t expect that the e f f e c t s of the l i p i d - p r o t e i n i n t e r a c t i o n would be more pronounced than i n the z w i t t e r i o n i n c case. In order to observe any h y p o t h e t i c a l immobilized 73 component, or in the case of f a s t exchange to observe the maximum e f f e c t of the Coulomb i n t e r a c t i o n , a two-component l i p i d system was used, c o n s i s t i n g of n e u t r a l , non-deuterated DMPC, and perdeuterated, n e g a t i v e l y charged DMPA used as a probe. The from the i n t e r a c t i n g l i p i d s would not then be overwhelmed by a l a r g e s i g n a l from bulk l i p i d s . T h i s experiment i s d e s c r i b e d i n d e t a i l i n the next chapter. 74 V I I I . A, DEUTERIUM NMR STUDY OF THE INTERACTION OF A  SYNTHETIC AMPHIPHILIC POLYPEPTIDE WITH CHARGED LIPIDS 8.1 SAMPLE PREPARATION The l i p i d s used i n t h i s experiment were t e s t e d f o r p u r i t y with t h i n l a y e r chromatography(TLC), a method which separates d i s s o l v e d l i p i d s and the products of t h e i r degradation on the b a s i s of d i f f e r e n t i a l s o l u b i l i t y . Each l i p i d gave one spot with TLC before and a f t e r the NMR measurements, i n d i c a t i n g that i t had not decomposed a p p r e c i a b l y . A l l s o l v e n t s and other m a t e r i a l s used were of reagent grade. The l i p i d - p o l y p e p t i d e d i s p e r s i o n s were prepared i n the f o l l o w i n g manner: 1) -the l i p i d s and p o l y p e p t i d e were c o d i s s o l v e d i n methanol i n a round-bottom f l a s k ; 2) -the methanol was then evaporated from the f l a s k using a ROTOVAP, a vacuum-evaporation device which r o t a t e s the f l a s k i n a warm water bath; the r e s u l t was a t h i n homogeneous f i l m d e p o s i t e d on the bottom of the f l a s k ; 3) -the f l a s k was then placed i n a high-vacuum at room temperature and l e f t for a s u f f i c i e n t l y long p e r i o d of time so t h a t the remaining s o l v e n t was d r i v e n o f f (minimum 10 h o u r s ) . 4) -a q u a n t i t y of b u f f e r s o l u t i o n was then added to the f l a s k ; the s o l u t i o n was warmed, and a g i t a t e d m i l d l y on 75 a v o r t e x - v i b r a t o r ; the r e s u l t of t h i s a g i t a t i o n was the formation of m u l t i l a y e r e d v e s i c l e s , which c o n s i s t of s e v e r a l c l o s e d b i l a y e r s arranged l i k e the l a y e r s of an onion. 5) -these d i s p e r s i o n s were then s u b j e c t e d to s e v e r a l freeze-thaw c y c l e s , where the f l a s k was immersed, a l t e r n a t e l y , i n l i q u i d n i t r o g e n and warm water - t h i s has the e f f e c t of c r e a t i n g a homogeneity in the s i z e of the v e s i c l e s , by f u s i n g them together and s e p a r a t i n g them a number of times. 6 ) -the d i s p e r s i o n was then c e n t r i f u g e d at u l t r a - h i g h speed for a minimum of four hours, r e s u l t i n g i n the formation of a dry p e l l e t of compressed l i p i d -p o l y p e p t i d e v e s i c l e s as the natent. 7) -the p e l l e t was p l a c e d i n the NMR sample tube, and hydrated i n excess of 100 weight % with deuterium-d e p l e t e d b u f f e r s o l u t i o n . The b u f f e r s o l u t i o n was deuterium d e p l e t e d so that there would be no water s i g n a l i n the deuterium NMR spectrum. The samples were s e a l e d with a cork, and s t o r e d i n l i q u i d n i t r o g e n between experimental runs. When removed from the n i t r o g e n , they were warmed to room temperature, and vortexed thoroughly, i n order to ensure homogeneity of the d i s p e r s i o n . The b u f f e r s o l u t i o n used in t h i s study was at pH=7.5, 76 and had the f o l l o w i n g composition: 40 mM Hepes (N-2-Hydroxyethylpiperazine-N -2-e t h a n e s u l f o n i c a c i d ) 50 mM NaCl 1 mM EDTA ( E t h y l e n e d i a m i n e t e t r a a c e t i c a c i d ) EDTA i s a compound which complexes with c a l c i u m ions, which have a s i g n i f i c a n t e f f e c t on the behavior of the DMPA headgroup. By adding EDTA i n t h i s amount, we e f f e c t i v e l y removed t h i s v a r i a b l e . 8.2 NMR MEASUREMENTS AND CALCULATIONS Much of the NMR methodology used i n t h i s study has been reviewed and d e s c r i b e d by Davis ( 4 ) . A l l NMR measurements i n t h i s experiment were performed on the D60 35 MHz deuterium spectrometer c o n s t r u c t e d by the e l e c t r o n i c s shop at the UBC p h y s i c s department, using the quadrupole echo pul s e sequence. Quadrupole echo s p e c t r a were c o l l e c t e d with p u l s e lengths of 5 microseconds (M S),pulse s e p a r a t i o n 40 us (unless otherwise s p e c i f i e d ) , and s i g n a l averaged with a r e c y c l e time of 250 m i l l i s e c o n d s over 100,000 scans. R e l a x a t i o n data were c o l l e c t e d using i n v e r s i o n - r e c o v e r y p u l s e sequence f o r T, , and using the quadrupole echo pulse sequence with v a r y i n g pulse spacing f o r T 2 . The echo s p e c t r a were c o l l e c t e d i n one channel i n phase with the a p p l i e d RF frequency, and f o r quadrature F o u r i e r 77 t r a n s f o r m a t i o n , the out of phase channel was assumed to be zero (experience has shown, t h a t , with c a r e f u l phase adjustment, t h i s i s an a p p r o p r i a t e assumption), and t h e r e f o r e was not c o l l e c t e d f o r reasons of speed in data c o l l e c t i o n . In the case where the data p o i n t s d e f i n i n g the echo were asymmetric about t=2r, the p o i n t s were s h i f t e d to the po i n t where they were symmetric, using a polynomial f i t t i n g program. The F o u r i e r transform was then performed using a f a s t - F o u r i e r transform r o u t i n e , g i v i n g the frequency spectrum of the quadrupole i n t e r a c t i o n . These c a l c u l a t i o n s , as w e l l as the moments an a l y s e s , were performed on an I n t e l MDS-230 microcomputer. A l l other c a l c u l a t i o n s were performed on the Amdahl VS/470 mainframe computer of the Computing Centre of the U n i v e r s i t y of B r i t i s h Columbia. The spe c t r a were "de-Pak-ed" using a program w r i t t e n i n t h i s l a b o r a t o r y (16). The r e l a x a t i o n data were f i t t e d to exp o n e n t i a l and b i - e x p o n e n t i a l f u n c t i o n s using the MINUIT r o u t i n e s from the TRIUMF l i b r a r y . 8.3 INITIAL INVESTIGATIONS OF THE POLYPEPTIDE/LIPID SYSTEM The l i p i d system used i n t h i s study c o n s i s t e d of the two l i p i d s : DMPC ( L - a - P h o s p h a t i d y l c h o l i n e , D i m y r i s t o y l ) and per-deuterated DMPA (L-a-P h o s p h a t i d i c A c i d , D i - p e r - d e u t e r i o -m y r i s t o y l ) , in a molar r a t i o of DMPC:DMPA=5:1. I n i t i a l c a l o r i m e t r i c measurements on the pure l i p i d system systems showed a broad phase t r a n s i t i o n c e ntered at approximately 78 27°C, with the system w e l l i n t o the l i q u i d c r y s t a l l i n e regime at p h y s i o l o g i c a l temperatures. We note t h a t , at pH=7.5, DMPC i s a z w i t t e r i o n , but that DMPA has two negative charges on the headgroup. The p o l y p e p t i d e , K 2GL 2 0K 2A-amide, as d i s c u s s e d p r e v i o u s l y , has f i v e p o s i t i v e charges at t h i s pH. An e l e c t r i c a l l y n e u t r a l l i p i d - p e p t i d e mixture would thus correspond to a molar r a t i o of DMPA:peptide = 2.5:1. If t h i s peptide were to completely immobilize the DMPA molecules, we would expect to observe the maximum s i g n a l from immobilized DMPA with a molar r a t i o of N:1, where N i s the number of b i n d i n g s i t e s on the p e p t i d e . C a l o r i m e t r i c s t u d i e s i n d i c a t e d t h a t the phase behavior of the l i p i d / p e p t i d e system was f a i r l y i n s e n s i t i v e to the a d d i t i o n of pe p t i d e . The phase t r a n s i t i o n was s i m i l a r to that f o r the pure l i p i d system, with some broadening of the t r a n s i t i o n r e g i o n , and no s h i f t i n the peak p o s i t i o n . In p a r t i c u l a r , there was no i n d i c a t i o n that the l i p i d s were s e g r e g a t i n g i n t o d i s t i n c t phases. We decided to approach the study of t h i s system by c o n s i d e r i n g two v a r i a b l e s : the c o n c e n t r a t i o n of pepti d e i n the l i p i d system, and the temperature. Three samples were made up, S1 with no p e p t i d e , and the others with molar r a t i o s of DMPA:peptide as f o l l o w s : S2- 10:1, S3- 5:1, correspo n d i n g to i n c r e a s i n g c o n c e n t r a t i o n s of p e p t i d e . NMR measurements were performed on these samples at the f o l l o w i n g temperatures: 10'C, 19°C, 34°C, and 44'C. 79 For each sample, the two hig h e s t temperatures corresponded to s p e c t r a c h a r a c t e r i s t i c of the l i q u i d c r y s t a l l i n e phase, and the lowest temperature corresponded to s p e c t r a c h a r a c t e r i s t i c of the g e l phase.In samples S1 and S2, the s p e c t r a at 19°C are i n the g e l phase, but i n sample S3, the spectrum at 19"C was a p p a r e n t l y intermediate between the two phases. T h i s spectrum i s not a s u p e r p o s i t i o n of the two c o e x i s t i n g phases, s i n c e t h i s would be i n d i c a t e d by a s p l i t t i n g of the c e n t r a l peaks. T h i s suggests that the molecules were i n r a p i d exchange between g e l - and f l u i d - l i k e phases. See f i g u r e s ( 6 ) , ( 7 ) , and (8) f o r stacked p l o t s of these s p e c t r a . F i g u r e s (9) and (10) show s p e c t r a f o r d i f f e r e n t samples at the same temperature, and represent the e f f e c t of i n c r e a s i n g c o n c e n t r a t i o n s of p e p t i d e , i n the l i q u i d c r y s t a l and g e l phases. The moments of these s p e c t r a are t a b u l a t e d i n Table I, and graphed as a f u n c t i o n of temperature i n f i g u r e s (11) and (12). The s p e c t r a e x h i b i t the same t y p i c a l v a r i a t i o n over the phase t r a n s i t i o n f o r each c o n c e n t r a t i o n of the pe p t i d e . From an immediate i n s p e c t i o n of f i g u r e (9), there seems to be very l i t t l e change i n the gross f e a t u r e s of the l i q u i d c r y s t a l s p e c t r a with i n c r e a s i n g c o n c e n t r a t i o n of p e p t i d e . We c e r t a i n l y see no new components which would corespond to the e x i s t e n c e of an immobilized f r a c t i o n of the DMPA. In l i g h t of p r e v i o u s i n v e s t i g a t i o n s , t h i s i s not s u r p r i s i n g . The l i q u i d c r y s t a l s p e c t r a w i l l be examined and d i s c u s s e d i n some d e t a i l i n l a t e r s e c t i o n s . LEAF 80 MISSED IN NUMBERING. F i g u r e 7 - SAMPLE S2: POWDER SPECTRA Powder s p e c t r a f o r the sample S2, with PA:PEPT=10:1, at the same temperatures as the pure l i p i d s p e c t r a : ahO'C, b)l9'C, c)34"C, d)44*C. F i g u r e 8 - SAMPLE S3: POWDER SPECTRA Powder sp e c t r a f o r the sample S3, with PA:PEPT=10:1, at the temperatures: a) 10°C, b) 19°C, c) 34°C, d) 44°C. F i g u r e 9 - THE EFFECT OF PEPTIDE CONCENTRATION: LIQUID CRYSTAL POWDER SPECTRA Powder s p e c t r a at f f ' C , f o r i n c r e a s i n g c o n c e n t r a t i o n s of peptide: a)PA:PEPT=1:0, b)PA:PEPT=10:1, c)PA:PEPT=5:1, showing no obvious v a r i a t i o n of lineshape with a d d i t i o n of p e p t i d e . F i g u r e 10 - THE EFFECT OF PEPTIDE CONCENTRATION: GEL POWDER SPECTRA Powder sp e c t r a at 19 "C, f o r i n c r e a s i n g c o n c e n t r a t i o n s of pe p t i d e : a)PA:PEPT=1:-, b)PA:PEPT=10:1, c)PA:PEPT=5:1, showing a decrease i n the apparent s p l i t t i n g s f o r PA:PEPT=5:1. F i g u r e 11 - SPECTRAL MOMENTS: Ml The f i r s t moments of the spectra i n f i g u r e s (5), (6), and (7), p l o t t e d f o r each sample as a f u n c t i o n of temperature: a)PA:PEPT=1:0, b)PA:PEPT=lO:1, c)PA:PEPT=5:1. MOMENTS: M1 vs T 1180 WJ2 0.04 u \ 3 7.78 f — 6.48 3.20 \ 0.0 __L J L._ W.2 oo CJI TEMP.(deq. C) F i g u r e 12 - SPECTRAL MOMENTS: M2 The second moments of the s p e c t r a in f i g u r e s (5), (6), and (7), p l o t t e d f o r each sample as a f u n c t i o n of temperature: a)PA:PEPT=1:0, b)PA:PEPT=lO:1, c)PA:PEPT=5:1. MOMENTS: M2 vs T ,1 I L J L J J L__.J L...-J 0.0 18.2 23.4 30.6 37.8 4S.0 TEMP.(d«g. C) 87 Table I - MOMENTS: PRELIMINARY STUDY The f i r s t and second moments, and the parameter A 2 (D2) f o r each of the s p e c t r a i n f i g u r e s ( 6 ) , (7), and (8) . PA:PEPT TEMP M1 M2 D2 'C (xlO" s" 1 ) ( X10 9 s-2) ( X10- 2) 10 1 1 .57 21.23 17.35 1 :0 19 10.71 18.16 1 7.27 34 5.775 4.626 2.745 44 5.445 4.016 .3242 10 10.99 20.00 22.50 10:1 19 10.36 18.06 24.77 34 5.825 4.849 5.843 44 5.471 4.442 9.918 10 10.93 20.67 28. 17 5:1 19 7.420 8.812 18.55 34 6.238 4.958 -5.63 44 5.320 3.840 .4822 88 The g e l phase s p e c t r a do show some i n t e r e s t i n g f e a t u r e s . We see i n f i g u r e (10) that the a d d i t i o n of peptide r e s u l t s i n a narrowing of the spectrum, coresponding to the " f l u i d i z a t i o n " of the a c y l c h a i n s d i s c u s s e d i n chapter 7. Because of the p r e l i m i n a r y nature of t h i s study, no attempt was made to f i n d the L/P r a t i o corresponding to the " p e p t i d e - a s s o c i a t e d " phase, and so we can say very l i t t l e about t h i s phenomenon. The moments analyses of these s p e c t r a bear out the prev i o u s d i s c u s s i o n . In p a r t i c u l a r , we note the very small change i n the value of M 2 f o r s p e c t r a i n the l i q u i d c r y s t a l l i n e r e g i o n . At the lower temperatures, we see that the values of M, and M 2 decrease with the a d d i t i o n of the pe p t i d e , most d r a m a t i c a l l y i n the sp e c t r a at 19.'C. R e l a x a t i o n data are t a b u l a t e d in Tables II.-V and graphed as f u n c t i o n s of temperature i n f i g u r e s (13)—(18). The v a l u e s f o r T, are obtained from a f i t of the experimental data to the b i - e x p o n e n t i a l : E ( t ) = Aexp(-t/T1F) + Bexp(-t/T1S), where E ( t ) i s the value of the z magnetization a time t a f t e r i t ' s i n v e r s i o n . Although t h i s i s a s i m p l i s i t i c approach to the problem of r e l a x a t i o n , i n v o l v i n g two d i f f e r e n t kinds of s i t e s to d e s c r i b e the r e l a x a t i o n of 54 d i f f e r e n t deuterons, we see i n f i g u r e (17) that t h i s F i g u r e 13 - SPIN-LATTICE RELAXATION: TIF - FAST COMPONENT A p l o t of the f a s t component of T,, as determined from minuit a n a l y s i s , f o r each sample, as a f u n c t i o n of temperature: a)PA:PEPT=1:0, b)PA:PEPT=10:1, c)PA:PEPT=5:1. The dashed l i n e s , as e x p l a i n e d i n the t e x t , are not intended to approximate any i n t e r p o l a t i o n between the intermediate p o i n t s . F i g u r e 14 - SPIN-LATTICE RELAXATION: T1S - SLOW COMPONENT A p l o t of the slow component of T,, as determined from minuit a n a l y s i s , f o r each sample, as a f u n c t i o n of temperature: a)PA:PEPT=1:0, b)PA:PEPT=10:1, c)PA:PEPT=5:1. TEMPERATURE (dag. C) F i g u r e 15 - SPIN-LATTICE RELAXATION: AVERAGE T1 A p l o t of the average value of T,, determined from the weighted sum of f a s t and slow r e l a x i n g components, f o r each sample, as a f u n c t i o n of temperature. a)PA:PEPT=1:0, b)PA:PEPT=lO:1, c)PA:PEPT=5:1. RELAXATION: T1(AV) vs T 40.0 33.8 rv \ \ \ \ \ i L \ \ / 27.4 23.2 \ c ) \ > / a) 19.0 0.0 J I L J J l J 16.2 23.4 30.6 TEMP (deg. C) 37.8 43.0 F i g u r e 16 - DECAY OF THE QUADRUPOLE ECHO: T2E A p l o t of the time constant f o r the decay of the quadrupole echo, T2E f o r each sample, as a f u n c t i o n of temperature. a)PA:PEPT=1:0 b)PA:PEPT=10:1, c)PA:PEPT=5:1. TEMPERATURE (deg. C) F i g u r e 17 - RELAXATION: MINUIT ANALYSIS OF T1 MEASUREMENT A r e p r e s e n t a t i v e f i t of the experimental p o i n t s i n the i n v e r s i o n -recovery experiment to the equation E ( t ) = A exp(-t/T1F) + B exp(-t/TIS) using the MINUIT c u r v e - f i t t i n g r o u t i n e s . The graph i s semi l o g a r i t h m i c . F i g u r e 18 - RELAXATION: MINUIT PARAMETERS The f r a c t i o n A/(A+B), which represents the f r a c t i o n of the r e l a x a t i o n governed by the f a s t - r e l a x i n g component, i s p l o t t e d as a fun c t i o n of temperature f o r each sample. PA= PEP =. a ) 1:0 , ID) S'- l 10 • \ RELAXATION: A / (A+B) vs T TEMP. (deq. C) CO 95 Table II - RELAXATION: T1F The f a s t - r e l a x i n g component of the s p i n - l a t t i c e r e l a x a t i o n time f o r each sample, with the corresponding parameter A from the b i - e x p o n e n t i a l f i t . PA:PEPT TEMP A T1F C O (mV) (msec.) 10 24.87 24.59 1 :0 19 20.04 16.02 34 53.31 24.90 44 50.73 35.05 10 4.470 21.44 10:1 19 10.13 17.53 34 24.27 28.30 44 7.600 33.04 10 17.84 26 .98 5:1 19 19 .80 19 .57 34 27 .50 21 .22 44 36.37 31 .02 96 Table III - RELAXATION: T1S Slow-relaxing component of T,, with parameter B from bi-exponential f i t . PA:PEPT TEMP B T1S C O (mV) (msec.) 10 9.972 94.51 1:0 19 6.612 60.05 34 8.840 228.4 44 7.264 340.5 10 3.970 111.6 10:1 IS 1 .961 201 .6 34 3.915 239.8 44 i 1 . 190 333.7 i 10 17.09 80.00 5:1 19 8.320 168.4 34 6.901 200.8 44 8.430 290.5 97 Table IV - RELAXATION: T1AV F r a c t i o n f of s p i n s r e l a x i n g with T,=T1F : f = A/(A+B), i n a s i m p l i s t i c t w o - s i t e model of r e l a x a t i o n , and: 1/T,(AV) = f/T1F+(1-f)/T1S PA:PEPT TEMP C O A(mV) B(mV) A/(A+B) T1AV(msec) 10.000 14.870 9.972 0.599 34.98 1 :0 19.000 14.040 6.612 0.680 20.93 34.000 56.31.0 8.840 0.864 28.32 44.000 50.730 7.264 0.875 39.47 10.000 4.470 3.970 0.530 34.58 10:1 19.000 10.130 1 .961 0.774 22.09 34.000 24.270 3.915 0.861 32.25 44.000 7.600 1 . 190 0.865 37.63 10 .000 17.840 7 .090 0.511 39.93 5:1 19 .000 19 .800 8.320 0.704 26.50 34.000 27 .500 6.901 0.799 25.86 44.000 36.370 8.430 0.812 37.29 Table V - RELAXATION: T2E Time constant f o r the decay of the quadrupolar echo T2E, with parameter C from s i n g l e e x p o n e n t i a l f i t . PA:PEPT TEMP C T2E ( *C) (mv) (msec.) 10 35.20 173.9 1 :0 1 9 30.74 60.75 34 19.26 262.9 44 29.66 342.7 10 8.030 101.0 10:1 19 14.14 65.49 34 12.56 1 53. 1 44 5.475 239. 1 10 41 .43 94.00 5:1 19 30.27 78.80 34 37.24 148.8 44 31 .34 170.6 99 assumption f i t s the data reasonably w e l l . In t h i s model, the parameter A/(A+B) r e p r e s e n t s the f r a c t i o n of the t o t a l s i g n a l which r e l a x e s with the f a s t T,. T h i s f r a c t i o n i s p l o t t e d as a f u n c t i o n of temperature f o r each sample i n f i g u r e (18). In the short c o r r e l a t i o n time l i m i t , the f a s t - r e l a x i n g component i s a s s o c i a t e d with the h i g h e s t quadrupole s p l i t t i n g (13). In the l i q u i d c r y s t a l l i n e domain, t h i s corresponds to the p l a t e a u r e g i o n ; we see that the f r a c t i o n of f a s t - r e l a x i n g s p i n s i s indeed h i g h i n the l i q u i d c r y s t a l s , corresponding to a l a r g e p l a t e a u . That t h i s i s the case has been w e l l e s t a b l i s h e d e x p e r i m e n t a l l y , as summarized by Brown (20). We demonstrate t h i s i n f i g u r e (19), which compares the normal spectrum of the pure l i p i d sample to i t s p a r t i a l l y r e l a x e d spectrum. The p a r t i a l l y r e l a x e d spectrum corresponds to a spectrum where the s i g n a l from the f a s t e r component has been e l i m i n a t e d . The p a r t i a l l y r e l a x e d s p e c t r a are c o l l e c t e d with a p u l s e sequence which f i r s t i n v e r t s the z-magnetization; a f t e r a time , the quadrupole echo sequence i s performed; the f a s t components w i l l have r e l a x e d to t h e i r e q u i l i b r i u m value i f the time TK i s chosen a p p r o p r i a t e l y . T h i s s i g n a l i s then s u b t r a c t e d from an echo where the i n v e r s i o n i s not performed, g i v i n g r i s e to a spectrum echo from only the s l o w l y - r e l a x i n g components of the m a g n e t i z a t i o n . The value of T„is 40 msec. in the spectrum here. From the f i g u r e , we see that the f a s t components a r i s e from the p l a t e a u region 100 of the spectrum. In the g e l phase, the f r a c t i o n of f a s t r e l a x i n g component decreases, although i n a l l samples t h i s f r a c t i o n i s over 1/2. We expect that the r e l a x a t i o n i n the g e l phase w i l l have more of an o r i e n t a t i o n dependence (the order parameters are up by a f a c t o r of 5 from the l i q u i d c r y s t a l l i n e s p e c t r a ) , but the dynamics i n the g e l phase are p o o r l y understood. In p a r t i c u l a r , i t i s impossible on the b a s i s of the i n f o r m a t i o n shown here to a s s o c i a t e the f a s t and slow components with d i f f e r e n t p a r t s of the molecule. Examining the graphs of the f a s t component of T,, we see, f i r s t of a l l , that the a d d i t i o n of peptide has l i t t l e e f f e c t on the value of the f a s t component of T,. T h i s i s in agreement with p r e v i o u s r e l a x a t i o n s t u d i e s of l i p i d - p r o t e i n i n t e r a c t i o n s . Secondly, we note an i n t e r e s t i n g f e a t u r e of a l l of the p l o t s : the value of T, goes through a minimum at a temperature somewhere i n the g e l - l i q u i d c r y s t a l phase t r a n s i t i o n r e g i o n . In the s p i r i t of our p r e v i o u s d i s c u s s i o n of r e l a x a t i o n minima, we see that t h i s i s a r e s u l t of T, being determined by two d i f f e r e n t c o r r e l a t i o n time regimes i n the d i f f e r e n t phases. In the case of T 2 , the c o r r e l a t i o n time regimes were determined by the second moment of the spectrum; with T,, we saw e a r l i e r that the c o n t r i b u t i o n s to the r e l a x a t i o n depend on the s p e c t r a l d e n s i t y of the i n t e r a c t i o n at f r e q u e n c i e s on the order of CJ 0 . The short c o r r e l a t i o n time regime i s thus determined by: CJ 2T 2<<1; in t h i s r e g i o n , T, i s F i g u r e 19 - PARTIALLY RELAXED SPECTRUM The p a r t i a l l y r e l a x e d spectrum of the pure l i p i d system, with p a r t i a l r e l a x a t i o n time T=40msec, (a) i s compared to the normal spectrum of the system ( b ) . Both s p e c t r a were c o l l e c t e d at 44 C. 1 1 1 — 60.0 -4B.0 -36.0 -24.0 -12.0 1 02 p r o p o r t i o n a l to T ~ 1 , and thus i n c r e a s e s with i n c r e a s i n g temperature. In the slow motion regime, CJ 2T 2>>1, and T, i s p r o p o r t i o n a l to r . In the gel-phase, T, decreases with Temp., i n d i c a t i n g that the motions are i n the slow motion regime. In the l i q u i d c r y s t a l l i n e regime, T, i n c r e a s e s with temp, and the motions are i n the short c o r r e l a t i o n time regime. The l i n e s j o i n i n g the p o i n t s i n f i g u r e (13) are simply to a i d the eye; we expect that T, w i l l vary smoothly through the f i r s t and l a s t p a i r s of v a l u e s . The l i n e s between the i n t e r m e d i a t e p o i n t s are dashed, s i n c e we can make no p r e d i c t i o n about the shape of the minimum on the b a s i s of the sparse data c o l l e c t e d i n these p r e l i m i n a r y experiments. From our knowledge of other s t u d i e s of T, minima, however, we f u l l y expect that there w i l l be a d i s c o n t i n u o u s jump in the graph of T,, at the phase t r a n s i t i o n temperature. T h i s e f f e c t c o u l d be averaged over c o e x i s t i n g g e l and l i q u i d c r y s t a l domains, g i v i n g a smooth minimum. Observing that the v a l u e s of the slow component of T, in f i g u r e (14) are an order of magnitude l a r g e r than those of the f a s t component, we suggest that a d i f f e r e n t motion must be r e s p o n s i b l e f o r t h i s component. Due to the f a c t t h a t the f r a c t i o n of • s l o w l y - r e l a x i n g deuterons changes s i g n i f i c a n t l y i n the g e l phase and through the t r a n s i t i o n , i t i s d i f f i c u l t to i n t e r p r e t t h i s parameter i n t h i s r e g i o n . A l i k e l y candidate f o r the slow component of t h e . s p i n -l a t t i c e r e l a x a t i o n would be the very r a p i d f r e e r o t a t i o n of 1 03 the t e r m i n a l methyl groups. In f i g u r e (15), we p l o t the average of T, over these motions, which i s given by: <1/T,> = (A/T1F+B/T1S)/(A+B) 53. Thi s p l o t shows that the dominant T, r e l a x a t i o n mechanism i s the f a s t component, with the minimum pre s e r v e d i n the averaging. The decay of the quadrupole echo for the perd e u t e r a t e d l i p i d can be represented a c c u r a t e l y by a s i n g l e e x p o n e n t i a l , as we see i n f i g u r e (20). The decrease i n the val u e of T 2 with peptide c o n c e n t r a t i o n i s dramatic, as shown i n f i g u r e (16); t h i s r e l a t i o n s h i p i s p l o t t e d i n f i g u r e (21) f o r the temperature 44.'C. For the higher temperature, we see that the graph of 1/T2 versus the c o n c e n t r a t i o n of pepti d e can be f i t t e d by a s t r a i g h t l i n e . We can account f o r t h i s r e l a t i o n s h i p with the f o l l o w i n g model: We assume that the DMPA molecules jump on and. o f f the peptid e with an exchange time f a s t on the 2H-NMR ti m e s c a l e . We f u r t h e r assume that T 2 i s produced by r o t a t i o n a l motions while the DMPA molecules are on and o f f the peptide, but that there i s no c o n t r i b u t i o n from the exchange between these domains; i . e . the motional averaging i s e q u i v a l e n t i n each case, g i v i n g r i s e to a AM2 which i s the same f o r each s i t e . T h i s assumption corresponds p h y s i c a l l y to the assumption that the F i g u r e 2 0 - RELAXATION: MINUIT ANALYSES OF T 2 E MEASUREMENTS A r e p r e s e n t a t i v e f i t of the f i t of the the experimental p o i n t s i n the echo decay experiment t o a s i n g l e e x p o n e n t i a l with time constant T 2 E . F i g u r e 21 - RELAXATION: THE EFFECT OF PEPTIDE CONCENTRATION A p l o t demonstrating the inverse p r o p o r t i o n a l i t y of T 2 E to the c o n c e n t r a t i o n of pept i d e at HH-'C. RELAXATION: 1/T2E vs [PEPT]/[DMPA] [PEPT]/[DMPA] ' 106 p o l y p e p t i d e s are h y d r o p h o b i c a l l y matched to the b i l a y e r t h i c k n e s s . T h i s assumption would presumably not h o l d i n g eneral for v a r y i n g lengths of p o l y p e p t i d e , but i n t h i s case, we can w r i t e : 1/T 2 = AM2 Tei>f . . b 54, where T9ff i s an average value of the c o r r e l a t i o n time. At any given time, there w i l l be a f r a c t i o n f of the DMPA bound to the p e p t i d e , with c o r r e l a t i o n time TPep ' s o t h a t Te*t = fV/> + ( 1 _ £ )rl>»lk 55' where r i i ( / ^ i s the c o r r e l a t i o n time of the bulk l i p i d s . We c o n s i d e r that the f r a c t i o n f i s given by: f = n[PEPT]/[PA], where n i s the number of b i n d i n g s i t e s . We expect that 2.5 < n ^ 5. From the graph of 1 0 6/T 2 vs f/n = [PEPT]/[PA], we can w r i t e : 1/T 2 = Af/n +B 56, where A = m2n(Tpef> -r6olK ), and B = AM2r, „. 57. so t h a t : A/B = n ( r ^ -rh„IH)/rkulk 1 07 In the d i f f u s i o n l i m i t (small r o t a t i o n a l jumps) (see pp. 56-58): A/B = n ( 2 R 2 ^ - R ^ l/R^w 58, where Rj>#p i^-bulk a r e the r a d i i of the c y l i n d e r s used to approximate the molecules. The f a c t o r of two a r i s e s from the assumption that the l e n g t h of the p e p t i d e i s simply twice that of the l i p i d s . From the study of the peptide i n DPPC, we expect that > 2x10" 7 (the value i n DPPC), s i n c e the presence of the Coulomb i n t e r a c t i o n with DMPA would tend to make the membrane more v i s c o u s than DPPC/PEPTIDE membranes. T h i s p r o v i d e s us with an important check of our numbers. At 44.'C, we f i n d t h a t : A = 14.39 x I 0 3 s e c " 1 B = 2.835 x I 0 3 s e c " 1 From estimates i n the l i t e r a t u r e (19),(28), we use the rough v a l u e s : R/>*f = 16.5 A ° 2 R2bu)M = 16.0 A' 2 A/B = 5.076 = 1.06n, so that n=4.7, which i s w i t h i n the l i m i t s d e s c r i b e d e a r l i e r . From these estimates, with Tpap > 2x10" 7, we can s o l v e f o r AM2: 108 AM 2 < 26.x10 9sec" 2, which i s on the order of the change i n M 2 going through the g e l - l i q u i d c r y s t a l phase t r a n s i t i o n ( A M 2 = 15x10 s s e c " 2 ) In t h i s p r e l i m i n a r y study, we demonstrated some of the main f e a t u r e s of the phase behavior of t h i s system. The r e s u l t s were c o n s i s t e n t with previous f i n d i n g s i n i n v e s t i g a t i o n s of l i p i d - p r o t e i n i n t e r a c t i o n s . The c o m p l e x i t i e s of the dynamics i n the g e l region and in the t r a n s i t i o n r e gion are not w e l l understood; the behavior of the value of T 2 i n the l i q u i d c r y s t a l r e g i o n can be d e s c r i b e d f a i r l y w e l l by the simple model we proposed above. Since t h i s parameter i s the most s e n s i t i v e to the motions i n t r o d u c e d by the pep t i d e , we were l e d . to focus our a t t e n t i o n on the e f f e c t s of of a d d i t i o n of pe p t i d e to the l i q u i d c r y s t a l l i n e l i n e s h a p e . 8.4 LINESHAPE STUDY We c o n s i d e r a general arguement to show t h a t , depending on the motions r e s p o n s i b l e f o r the decay of the t r a n s v e r s e magnetization, the quadrupole echo spectrum v a r i e s a c c o r d i n g to the spacing T between the 90 degree p u l s e s . F i r s t l y , we review some b a s i c f a c t s about the quadrupole echo experiment. The FID of the magnetization for a pure quadrupole i n t e r a c t i o n f o l l o w i n g a 90 degree pulse ( n e g e l e c t i n g f i e l d inhomegeneities) i s due to a d i s t r i b u t i o n of quadrupole s p l i t t i n g s APU , due to d i f f e r e n t o r i e n t a t i o n s of the EFG. 109 Each s p i n has a d i f f e r e n t quadrupole frequency, and precesses i n the r o t a t i n g r e f e r e n c e frame at t h i s frequency. The r e s u l t i s that the spins a c q u i r e phase at d i f f e r e n t r a t e s - the net magnetization dephases, and the s i g n a l d i e s out. The second pulse e f f e c t i v e l y reverses time f o r the spin system, by i n v e r t i n g the phase accumulated due to the p r e c e s s i o n s , so that i n the absence of r e l a x a t i o n mechanisms, a l l the spins refocus at a time 2T, and the FID i s repeated. In f a c t , a l l the spin s are not refocussed i n the echo: the amplitude of the echo decays as the pulse spacing i s inc r e a s e d , i n d i c a t i n g that the motions i n the system p a r t i a l l y d i s r u p t the "phase memory" of the s p i n s . The time constant f o r the decay of the quadrupole echo as a f u n c t i o n of the pul s e spacing, , T 2 , i s a measure of t h i s "phase memory." The T 2 measurements made i n these experiments were of perdeuterated, unoriented systems, and the value of T 2 was obtained from a f i t to a s i n g l e e x p o n e n t i a l . Although T 2 depends, i n g e n e r a l , on the deuteron chain p o s i t i o n and on d i r e c t o r o r i e n t a t i o n , the data were adequately f i t t e d by a s i n g l e e x p o n e n t i a l . Thus, the v a r i a t i o n of T 2 with the o r i e n t a t i o n and the chain p o s i t i o n was i n s u f f i c i e n t to give non-exponential behavior over the range of 2T used i n the experiments. Th e r e f o r e , the measured T 2 r e p r e s e n t s an average over d i f f e r e n t deuterons on the same molecule, and an average over the d i r e c t o r o r i e n t a t i o n s ; i n the case of 1 10 the l i p i d - p e p t i d e system, we must a l s o i n c l u d e the p o s s i b i l i t y of an averaging over d i f f e r e n t p o s s i b l e motional regimes - one a s s o c i a t e d with the boundary l i p i d s , and the other a s s o c i a t e d with bulk l i p i d s . In order to study the t r a n s v e r s e r e l a x a t i o n more thoroughly, we study the decay of each p a r t of the spectrum as a f u n c t i o n of the pulse spac i n g . When the c o r r e l a t i o n times of the motions r e s p o n s i b l e f o r the r e l a x a t i o n are on the order of the pulse s e p a r a t i o n , i t i s p o s s i b l e to observe dramatic changes i n the powder p a t t e r n spectrum. The powder s p e c t r a have been modelled f o r t w o - s i t e exchange r e l a x a t i o n processes by S p i e s s and S i l l e s c u i n (37), where the authors present a study of the t h e o r e t i c a l v a r i a t i o n i n the l i n e s h a p e f o r v a r y i n g pulse spacings f o r t h i s c o r r e l a t i o n time r e g i o n . In t h i s case, p a r t s of the spectrum can have v a l u e s of T 2 much s h o r t e r than T, so that they disappear completely from the spectrum. The study of the l i n e s h a p e with v a r y i n g pulse spacings i s then a study of the p a r t i a l l y r e l a x e d spectrum. In p a r t i c u l a r , we can determine the dependence of T 2 on the o r i e n t a t i o n of the d i r e c t o r , as r e p o r t e d by Volke (38). S i m u l a t i o n s of l i n e s h a p e v a r i a t i o n s have a l s o been performed and d i s c u s s e d by G r i f f i n (11). The importance of these s t u d i e s i n the study of l i p i d -p r o t e i n i n t e r a c t i o n s i s due to the f a c t t h at the peptide i n t r o d u c e s slow motions to the system - i t changes the value of T 2 d r a m a t i c a l l y , yet leaves the s p e c t r a at r=40jxsec. 111 v i r t u a l l y unchanged. We can suppose, as in the model in the pr e v i o u s s e c t i o n , that the l i p i d s exchange between boundary and bulk motional regimes, and that the c o r r e l a t i o n times f o r the motions i n these regimes are d i f f e r e n t . If the exchange time were longer than the pu l s e spacing of 40Msec., the o b s e r v a t i o n that the average value of T 2 over these s i t e s decreases with the a d d i t i o n of peptide would suggest that d i f f e r e n c e s i n the r e l a x a t i o n f o r bound l i p i d s would show up i n a change i n the shape and/or width of the p a r t i a l l y r e l a x e d s p e c t r a . New samples were prepared f o r t h i s p a r t of the experiment, using the techniques o u t l i n e d i n d e t a i l i n the f i r s t s e c t i o n of t h i s c h a p t e r . The same c o n c e n t r a t i o n s were used. S p e c t r a were c o l l e c t e d f o r each sample, with pulse spacings:. T = iOixsec, 5 0 / i s e c , and 200Msec, at the temperatures i n the l i q u i d c r y s t a l l i n e regime used p r e v i o u s l y . In f i g u r e s (22) and (23), we d i s p l a y these s p e c t r a f o r the h i g h e s t temperature f o r two samples: one without p e p t i d e , and one with DMPA:PEPT=5:1. In Table VI, we give the v a l u e s of the f i r s t and second moments of a l l s p e c t r a accumulated i n t h i s study. On immediate i n s p e c t i o n , we see that the s p e c t r a with and without p e p t i d e have the same s o r t of l i n e s h a p e dependence on the pu l s e spacing: the d i f f e r e n c e between the l i n e s h a p e at T=40MS. and T=50*IS. i s n e g l i g i b l e . At 200MS., we see that the r e l a t i v e i n t e n s i t i e s of two p a r t s of the spectrum decrease. We can i d e n t i f y F i g u r e 22 - LINESHAPE STUDY: POWDER SPECTRA FOR PURE LIPIDS Stacked p l o t s of powder sp e c t r a f o r p u r e - l i p i d system with spacing between the p u l s e s i n the quadrupole echo experiment of T * :A) 40«isec, B) 50 / i s e c , and C) 200/usec. A l l spectra are for the sample at temperature 44?C. FREQUENCY (KHz) F i g u r e 23 - LINESHAPE STUDY: POWDER SPECTRA FOR LIPID-PEPTIDE SYSTEM Stacked p l o t s of powder s p e c t r a for system with DMPA:PEPTIDE * 5:1 (molar), with spacing between the pulses in the quadrupole echo experiment of T = :A) 4 0 * i s e c , B) 5 0 j i s e c , and C) 200/isec. A l l s p e c t r a are f o r the sample at temperature 44*C. FREQUENCY (KHz) Table VI - MOMENTS: LINESHAPE STUDY The f i r s t and second moments, and the parameter A2 (D2), f o r s p e c t r a c o l l e c t e d with v a r y i n g v a l u e s of the time between the pu l s e spacing, T, i n the quadrupole echo experiment. PA:PEPT TEMP. M1 M2 D2 C O (Msec. ) ( X 1 0 " S - 1 ) ( X 1 0 9 S " 2 ) ( X 1 0 - 2 ) 40 6.329 5.581 3.204 3 4 50 6.267 5.466 3.091 200 6.213 5.400 3.625 1 :0 40 5.632 4.372 2.121 4 4 50 5.404 4.036 2.384 200 5.516 4.014 2.267 40 5.685 5.432 24.51 3 4 50 4.727 4.465 4.802 200 5.472 5.054 25.02 5:1 40 5.324 4.114 7.528 4 4 50 5. 122 4.058 14.59 200 3.241 2.433 71 .58 1 15 three p a r t s of the spectrum i n order to measure t h i s e f f e c t : i ) the trough - the minimum value of the c e n t r a l p a r t of the spectrum; i i ) the 90 degree edge - the p l a t e a u r e g i o n ; and i i i ) the shoulder - the 0 degree edge. Comparing the i n t e n s i t i e s , we f i n d that the edges of the spectrum remain unchanged, but that both the troughs and the shoulders drop in i n t e n s i t y by a f a c t o r of about 1/2. T h i s phenomenon i s observed i n every sample, r e g a r d l e s s of peptide c o n c e n t r a t i o n . The drop i n i n t e n s i t y i n the shoulders i s the most u n c e r t a i n measurement, l i m i t e d by the s i g n a l - n o i s e r a t i o . As we w i l l see i n more d e t a i l i n the comparisons of the s p e c t r a f o r samples with and without p e p t i d e , t h i s phenomenon does not . seem to have anything to do with the presence of the p e p t i d e , but can be a s c r i b e d to an orientation-dependence of the T 2 f o r the pure l i p i d system -a dependence which i s not a p p r e c i a b l y a f f e c t e d by the presence of p e p t i d e . F i g u r e s (24)-(27) give the comparison of the de-Pake-ed spectra with the powder s p e c t r a f o r these samples. The i n t e g r a t e d i n t e n s i t y of the de-Paked s p e c t r a (not shown here) g i v e s an estimate of the t o t a l s p e c t r a l d e n s i t y a s s o c i a t e d with each peak. Within the l i m i t s of the e r r o r a s s o c i a t e d with the i n h e r e n t l y poor s i g n a l - n o i s e r a t i o of the de-Pake-ed s p e c t r a (down by at l e a s t a f a c t o r of 5 from the powder s p e c t r a ) , we were able to d i s t i n g u i s h four peaks in each spectrum. In order of i n c r e a s i n g s p l i t t i n g , the r e l a t i v e i n t e n s i t i e s and p o s i t i o n s of the peaks suggest the F i g u r e 24 - LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON Comparison of de-Pake-ed spectrum (A) with the powder spectrum (B) f the p u r e - l i p i d sample f o r pulse spacing T = 40«xsec. f o r sample temperature 44°C. A I.I ? o 12.0 ^ r a a».o Sii 2.o » . o F i g u r e 25 - LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON Comparison of de-Pake-ed spectrum (A) with the powder spectrum (B) f o r the p u r e - l i p i d sample, with pulse spacing T = 200 j isec. for sample at temperature 44 6C. 1 1 1 1 1 1 r 1 1 1 0.0 4.0 0.0 12.0 16.0 20.0 24.0 20.0 32.0 30.0 40.0 FREQUENCY (KHz) F i g u r e 26 - LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON Comparison of de-Pake-ed spectrum (A) with the powder spectrum (B) f o r the sample with DMPA:PEPT =5:1, with pulse spacing T = 40/isec. f o r the sample at temperature 44°C. Co ~f 1 1 I I 1 T 1 I I 1 9.0 4.0 1.0 12.0 IB.O 20.0 24.0 2B.0 32.0 38.0 40.0 FREQUENCY (KHz) F i g u r e 27 - LINESHAPE STUDY: DE-PAKE-ED SPECTRA - COMPARISON Comparison of de-Pake-ed spectrum (A) with the powder spectrum (B) f o r the sample with DMPA:PEPT =5:1, with pulse spacing r = 200/isec. f o r the sample at temperature 44°C. F i g u r e 28 - LINESHAPE STUDY: DE-PAKE-ED SPECTRA FOR PURE-LI PID SYSTEM Stacked p l o t of de-Pake-ed spectra f o r pure l i p i d system with spacing between the p u l s e s in the quadrupole echo experiment of T = :A) 4 0 n s e c , B) 50Msec, and C) 200MSec. A l l sp e c t r a are f o r the sample at temperature 44°C. -\ 1 1 1 r r r 1 i 1 1 3.0 S.O 10.0 19.0 20.0 29.0 30.0 39.0 40.0 «.0 50.0 FREQUENCY (KHz) F i g u r e 29 - LINESHAPE STUDY: DE-PAKE-ED SPECTRA FOR LIPID-PEPTIDE SYSTEM Stacked p l o t of de-Pake-ed spectra f o r sample with DMPA:PEPT = 5:1 with s p a c i n g between the pulses i n the quadrupole echo experiment of of T — :A) 4 0 j j s e c , B) 5 0 y s e c , and C) 200«isec. A l l spectra are f o r the sample at temperature 44°C. to 3.0 S.O 10.0 19.0 20.0 23.0 30.0 33.0 43.0 45.0 30.0 FREQUENCY (KHz) F i g u r e 30 - LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - POWDER SPECTRA A comparison of the powder spectra of A) p u r e - l i p i d sample with B) sample with DMPA:PEPT = 5:1 . for pulse spacing r = 40 usee. The sample i s at temperature 44°C. FREQUENCY (KHz) F i g u r e 31 - LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - POWDER SPECTRA A comparison of the powder s p e c t r a of A) p u r e - l i p i d sample with B) sample with DMPA:PEPT =5:1, for pulse spacing T = 200 Msec. The sample i s at temperature 44°C. GOO -4B.0 -36.0 -24.0 -12.0 0.0 12.0 24.0 36.0 4B.0 80.0 FREQUENCY (KHz) F i g u r e 32 - LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - DE-PAKE-ED SPECTRA A comparison of the de-Pake-ed s p e c t r a of A) p u r e - l i p i d sample with B) sample with DMPA:PEPT = 5 : 1 , f o r pulse spacing r = 40 usee. The sample i s at temperature 44°C. B o.o 1— s.o 10.0 13.0 20.0 23.0 30.0 FREQUENCY (KHz) 33.0 40.0 43.0 30.0 F i g u r e 33 - LINESHAPE STUDY: THE EFFECT OF THE PEPTIDE - DE-PAKE-ED SPECTRA A comparison of the de-Pake-ed s p e c t r a of A) p u r e - l i p i d sample with B) sample with DMPA:PEPT = 5:1, f o r pulse spacing r = 200 /isec. The sample i s at temperature 44°C. .0 FREQUENCY (KHz) Table VII - LINESHAPE STUDY: QUADRUPOLE SPLITTINGS FROM DE-PAKE-ED SPECTRA A t a b u l a t i o n of the mean quadrupole s p l i t t i n g s f o r each of the four l i n e s (LI,L2,L3,L4) d i s t i n g u i s h e d from the de-Pake-ed s p e c t r a i n the l i n e s h a p e study. Data are f o r two samples, one with [PA]:[PEPT]=5:1, at temperatures i n the l i q u i d c r y s t a l l i n e regime, f o r puls e spacings T. SAMPLE TEMP QUADRUPOLE SPLITTINGS (KHz.) C s L1 L2 L3 L4 40 3.69 11.6 15.0 21 .6 34 50 3.69 10.8 14.8 20.3 pure 200 3.17 10.3 14.2 20.6 l i p i d 40 3.17 8.71 11.3 18.2 44 50 2.64 8.97 12. 1 18.2 200 3.01 8.89 11.4 18.0 40 3.69 11.6 15.0 21 .6 34 50 3.69 1 1.9 15.3 21.6 with 200 3.69 11.6 15.3 22.4 pept. 40 2.64 9.5 12.1 19.0 44 50 2.90 9.5 12.1 18.7 200 2.64 9.5 12.1 18.5 127 Table VIII - LINESHAPE STUDY: LINEWIDTHS FROM DE-PAKE-ED SPECTRA A t a b u l a t i o n of the l i n e w i d t h s (FWHM) f o r each of the four l i n e s (L1,L2,L3,L4) d i s t i n g u i s h e d from the de-Pake-ed s p e c t r a i n the l i n e s h a p e study. Data are f o r two samples, one with [PA]:[PEPT]=5:1, at temperatures i n the l i q u i d c r y s t a l l i n e regime, f o r p u l s e spacings SAMPLE TEMP LINEWIDTHS (KHz.) C s L1 L2 L3 L4 40 0 . 8 3.2 2.6 3.4 pure 34 50 1 . 1 3.2 3.2 3.2 l i p i d 200 1 . 1 3.1 2.6 2.6 40 0 . 8 2.4 2.4 2.4 44 50 0 . 8 2.3 2.1 2.3 200 0 . 8 2.1 3.2 2.1 40 1 . 1 2.6 2.6 4.2 34 50 1 . 1 2.9 2.1 3.7 with 200 1 . 1 2.9 2.6 4.2 pept. 40 0 . 8 2.1 2.1 3.2 44 5 0 0 . 8 2.1 2.1 3.2 200 2.6 3.7 2.6 128 f o l l o w i n g t e n t a t i v e assignments: the t e r m i n a l methyl peak, corresponding to 6 deuterons two peaks c o r r e s p o n d i n g to 4 deuterons each, presumably a s s o c i a t e d with the f i r s t and second carbon p o s i t i o n s up the cha i n s from the te r m i n a l methyl group - one peak corresponding t o 40 deuterons, a s s o c i a t e d with the p l a t e a u r e g i o n . In t a b l e s VII and V I I I , the quadrupole s p l i t t i n g s and the l i n e w i d t h s a s s o c i a t e d with each of these peaks are given for samples with and without p e p t i d e . F i g u r e s (28) and (29) are stacked p l o t s of the depaked s p e c t r a , showing, f o r each sample, no d e t e c t a b l e d i f f e r e n c e as the pul s e spacing i s in c r e a s e d . t h i s i s c o n s i s t e n t with the assumption that the v a r i a t i o n i n l i n e width i n the powder spectrum i s due to o r i e n t a t i o n dependence alone, s i n c e the depaked s p e c t r a are given f o r the 90 degree o r i e n t a t i o n . We c o n s i d e r i n d e t a i l the comparisons of the s p e c t r a with and without p e p t i d e at two values of T: f i g u r e s (30) and (31) show the comparisons of the powder s p e c t r a , and f i g u r e s (32) and (33) show the comparisons of the de-Paked s p e c t r a . In f i g u r e (30), with p u l s e s p a c i n g 40;xsec., we see that we l o s e i n t e n s i t y i n the trough of the c e n t r a l p o r t i o n with the a d d i t i o n of p e p t i d e . T h i s c o u l d be i n t e r p r e t e d as an e f f e c t of an exchange between bound and unbound s i t e s . At 129 the edge of the spectrum of the sample with p e p t i d e , we see that the peak seems to be s p l i t i n t o two components, and we are l e d to ask whether t h i s c o u l d be a s i g n a l from bound l i p i d s . A c o n s i d e r a t i o n of the the depaked s p e c t r a f o r t h i s p ulse spacing, f i g u r e (32), shows t h i s s p l i t t i n g i n the o r i e n t e d sample. The h y p o t h e t i c a l bound component appears on the i n s i d e edge of the l a r g e peak, at approximately 13 KHz. From the i n t e g r a t e d i n t e n s i t i e s of these s p e c t r a , which we emphasize are only a c c u r a t e to w i t h i n 20%, i t i s i m p o s s i b l e to determine whether t h i s i n f a c t corresponds to a d i s t i n c t , " r e a l " peak. At t h i s p o i n t , we can only i n d i c a t e t h i s f e a t u r e , and remark that i t i s c e r t a i n l y suggest i v e . In f i g u r e s (31) and (33), the same comparisons are given f o r T=200ysec, and we see that t h i s f e a t u r e no longer appears. If we were to a s s o c i a t e t h i s f e a t u r e with bound l i p i d s , we could e x p l a i n i t s disappearance at the longer p u l s e spacing by supposing that the T 2 f o r the bound l i p i d s i s short compared to the p u l s e spacing, and that t h i s part of the spectrum i s not r e f o c u s s e d i n the echo. In summary, we are f o r c e d to conclude that the l i n e s h a p e experiment y i e l d s a negative r e s u l t : There i s evidence f o r a s l i g h t o r i e n t a t i o n dependence of T 2 , but the presence of the p e p t i d e does not s i g n i f i c a n t l y a l t e r the l i n e s h a p e at any of the p u l s e spacings i n v e s t i g a t e d here -the orientation-dependence of T 2 i s the same f o r each c o n c e n t r a t i o n of p e p t i d e s t u d i e d as i n the p u r e - l i p i d 1 30 system. We are unable to conclude that the h y p o t h e t i c a l peak in f i g u r e s (30) and (32) corresponds to an immobilized f r a c t i o n of the l i p i d s . 8.5 CONCLUDING REMARKS At t h i s p o i n t , we decided to end our i n v e s t i g a t i o n of t h i s system f o r the purposes of t h i s r e p o r t , although these experiments r a i s e some i n t e r e s t i n g and c h a l l e n g i n g q u e s t i o n s . We saw, i n the p r e l i m i n a r y s e c t i o n , that the presence of the p e p t i d e had l i t t l e e f f e c t on the parameters M1, M2, and T1, but that the v a l u e s of T2 were decreased d r a m a t i c a l l y i n the l i q u i d c r y s t a l l i n e phase. The s p e c t r a i n the l i q u i d c r y s t a l l i n e phase seem to be u n a f f e c t e d by the a d d i t i o n of the p e p t i d e : we can account f o r the change in the values of T 2 by assuming a model where the l i p i d s jump between bulk and boundary s i t e s - from t h i s model, we c a l c u l a t e d the change in the second moment brought about by the onset of the new motion i n t r o d u c e d i n t o the system by the p e p t i d e , and found i t to be on the order of the d i f f e r e n c e in the second moments of the g e l and l i q u i d c r y s t a l l i n e s p e c t r a . We a l s o c a l c u l a t e d the number of b i n d i n g s i t e s on the peptide to be n=3, w i t h i n the l i m i t s expected from a c o n s i d e r a t i o n of the charges on the l i p i d s and p e p t i d e s . In the l i n e s h a p e study, we found that the presence of the peptide had l i t t l e e f f e c t on the shape of the powder p a t t e r n spectrum at any of the pulse s p a c i n g s . In 131 p a r t i c u l a r , we were unable to r e s o l v e any new s p e c t r a l f e a t u r e s that we c o u l d a s s o c i a t e with the presence of long-l i v e d boundary l i p i d s i t e s . Nor were we able to d i s t i n g u i s h any d i f f e r e n c e i n the t r a n s v e r s e r e l a x a t i o n r a t e s of d i f f e r e n t p a r t s of the spectrum due to the presence of p e p t i d e . The main q u e s t i o n that remains to be answered i s t h i s : How can we r e c o n c i l e the f a c t that T 2 decreases d r a m a t i c a l l y with the a d d i t i o n of peptide with the f a c t t h a t the spectra are i n s e n s i t i v e to the a d d i t i o n ? There are a number of experiments that c o u l d be performed in order to answer t h i s q u e s t i o n . One p o s s i b l e e x p l a n a t i o n of these r e s u l t s i s that we simply do not observe a s i g n a l from the boundary l i p i d s . A l l other t h i n g s being equal, the s i g n a l from a sample c o n t a i n i n g peptide in t h i s case would be down i n maximum i n t e n s i t y by a f a c t o r f = n[PEPT]/[DMPA] compared to the s i g n a l from a pure l i p i d sample, where as i n s e c t i o n 2, n i s the number of b i n d i n g s i t e s . An a c c u r a t e spin-count experiment c o u l d be c a r r i e d out on samples where the number of deuterons i s known with p r e c i s i o n . The s i g n a l from samples with and without peptide c o u l d be c a l i b r a t e d with a standard, and we would be able to determine the f a c t o r f i n the above. A more d e t a i l e d study of t h i s model membrane system would i n v o l v e a more thorough i n v e s t i g a t i o n of i t s phase behavior; l a r g e r c o n c e n t r a t i o n s of p e p t i d e should be 1 32 i n v e s t i g a t e d , and the temperature dependence of the v a r i o u s parameters should be s t u d i e d more c a r e f u l l y . I t would be i n t e r e s t i n g to study the e f f e c t s of hydrophobic mismatch, with d i f f e r e n t lengths of p o l y p e p t i d e , as o r i g n a l l y intended. It would a l s o be i n s t r u c t i v e to c o n s i d e r s t u d y i n g the behavior of the other l i p i d i n the system, DMPC. Samples c o u l d be prepared with d e u t e r a t e d DMPC, and undeuterated DMPA, i n order to i s o l a t e the i n t e r a c t i o n of the pe p t i d e with DMPC. F i n a l l y , i t should be noted that t h i s i n v e s t i g a t i o n was n e c e s s a r i l y l i m i t e d i n scope, and that the experiments presented some d i f f i c u l t c o m p l i c a t i o n s . A more complete study of t h i s model membrane system would r e q u i r e a s u b s t a n t i a l e f f o r t . 133 BIBLIOGRAPHY 1. Abragham, A. (1961) The P r i n c i p l e s of Nuclear Magnetism. Oxford U n i v e r s i t y Press, London 2. S l i c h t e r , C P . (1978) P r i n c i p l e s of Magnetic Resonance. S p r i n g e r - V e r l a g B e r l i n H e i d e l b e r g , New York 3. Messiah, A. Quantum Mechanics - Volume I I . John Wiley and Sons, Inc., New York 4. Da v i s , J.H., (1983) The D e s c r i p t i o n of Membrane L i p i d Conformation, Order, and Dynamics by Deuterium-NMR, Biochim. Biophys. Acta 737: 117,171 5. Zannoni, C. (1985) Q u a n t i t a t i v e D e s c r i p t i o n of O r i e n t a t i o n a l Order: R i g i d M olecules. In: Emsley,J.W.(ed.) Nuclear Magnetic Resonance of L i q u i d C r y s t a l s , D. R e i d e l P u b l i s h i n g Company, Dordrecht, Holland,pp 1-31 6. S e e l i g , J . (1977) Deuterium Magnetic Resonance: Theory and A p p l i c a t i o n s to L i p i d Membranes. Qu. Rev Biphys. 10: 353-418 7. Vegas,S. and Pines,A. (1977) J . Chem. Phys. 67: 1752-1758 8. Davis J . H . , J e f f r y K.R.,Bloom M., V a l i c M.I., Higgs T.P. (1976) Quadrupole Echo Deuteron Magnetic Resonance Spectroscopy i n Ordered Hydrocarbon Chains. Chem. Phys. L e t t e r s 42: 390-394. 9. J e f f r y , K.R. (1981) B u l l . Magnetic Resonance 3: 69-82 Cohen M.J. and Re i f F. (1957) Quadrupole E f f e c t s i n Nuclear Magnetic Resonance Studi e s of S o l i d s . In: S e i t z , F. and T u r n b u l l , D.(eds.) S o l i d S tate Physics V o l . 5, Academic Press, New York 10. Bloembergen, N. and Rowland, T.U. (1953) Acta M e t a l l u r g i c a 1: 731-746 11. G r i f f i n , R.G. (1981) S o l i d S tate Nuclear Magnetic Resonance i n L i p i d B i l a y e r s . Methods Enzymol. 72: 108-174 12. Rang S.Y., Gutowski H.S., O l d f i e l d E. (1979a) NMR I n v e s t i g a t i o n of the Cytochrome oxidase - P h o s p h o l i p i d I n t e r a c t i o n . Biochemistry 18: 3257-3267 134 13. Davis, J.H.0979) J . Biophys. 27: 339-358 14. Paddy M.R., D a h l q u i s t F.W., Davis J.H., Bloom M. (1981) Dynamical and Temperature-Dependent E f f e c t s of L i p i d - P r o t e i n I n t e r a c t i o n s . B iochemistry 20: 3152-3162 15. Bloom M., Davis J.H., MacKay A.L. (1981) D i r e c t Determination of the O r i e n t e d Sample NMR Spectrum from the Powder Spectrum f o r Systems with L o c a l A x i a l Symmetry. Chem. Phys. L e t t e r s 80: 198-202 16. S t e r n i n E., Bloom M., MacKay A.L. (1983) De-Pake-ing of NMR Spectra. J . Magn. Res. 55: 274-282 17. S e e l i g J . , S e e l i g A., Tamm L. (1981) Nuclear Magnetic Resonance and L i p i d - P r o t e i n I n t e r a c t i o n s . In: L i p i d -P r o t e i n I n t e r a c t i o n s V o l . 2 . John Wiley and Sons, Toronto, pp 127-149 18. Devaux, P.F. (1983) ESR and NMR S t u d i e s of L i p i d -P r o t e i n I n t e r a c t i o n s i n Membranes. In: B e r l i n e r , L . J . and Reuben,J. (eds.) B i o l o g i c a l Magnetic Resonance V o l . 5. Plenum P r e s s , New York, pp 183-299 19. Pauls K.P., MacKay A.L., Soderman 0., Bloom M., Tanjea A.K., Hodges R.S. (1985) European Biophys J . 12: 1-11 20. Brown M.F. and S e e l i g J . (1979) S t r u c t u r a l Dynamics i n P h o s p h o l i p i d B i l a y e r s from Deuterium S p i n - L a t t i c e R e l a x a t i o n Time Measurements. J . Chem. Phys. 70: 5045-5053 21. Huntress, J r . W.T. (1970) The Study of A n i s o t r o p i c R o t a t i o n of Molecu l e s i n L i q u i d s by NMR Quadrupolar R e l a x a t i o n . In: Waugh, J.S. (ed.) Advances i n Magnetic Resonance V o l . 4, Acadameic Press, New York, pp 2-35 22. Kimmich R., Schuur G., Scheurmann A. (1983) Spin-L a t t i c e R e l a x a t i o n and Lineshape Parameters i n NMR of Lammelar L i p i d Systems. Chem. Phys. L i p i d s 32: 271-322 23. Brown M.F. (1982) Theory of S p i n - L a t t i c e R e l a x a t i o n i n L i p i d B i l a y e r s and B i o l o g i c a l Membranes. J . Chem. Phys. 77: 1576-1599 1 3 5 24. Bloom M., Reeves L.W., Wells E . J . (1965) Spin Echoes and Chemical Exchange. J . Chem. Phys. 42: 1615-1624 25. Woessner D.E., Snowden J r . B.S., Meyer G.H. (1969) C a l c u l a t i o n of NMR Free I n d u c t i o n S i g n a l s f o r N u c l e i of Molecules i n a H i g h l y V i s c o u s Medium or a S o l i d -L i q u i d System. J . Chem. Phys 51: 2968-2975 26. Bloom M. and Smith I. (1985) M a n i f e s t a t i o n s of L i p i d - P r o t e i n I n t e r a c t i o n s i n Deuterium NMR. In: Watts, Dupont (eds.) Progress i n P r o t e i n - L i p i d I n t e r a c t i o n s . E l s e v i e r Science P u b l i s h e r s B.V. 27. S t r y e r , L. (1981) B i o c h e m i s t r y . W.H. Freeman and Company, San F r a n s i s c o . 28. Mouritsen M. and Bloom M. (1984) Mattress Model of L i p i d - P r o t e i n I n t e r a c t i o n s i n Membranes. Biophys. J . 46: 141-153 29. Unwin, P.N.T. and Henderson, R. (1984) The S t r u c t u r e of P r o t e i n s i n B i o l o g i c a l Membranes. S c i . Am. 250: 78-94 30. B u r n e l l E.E., C u l l i s P.R., and de K r u i j f f B. (1980) The E f f e c t s of Tumbling and L a t e r a l D i f f u s i o n on P h o s p h a t i d y l c h o l i n e Model Membranes Biochim. Biophys. Acta 603: 63-69 31. Bienvenue A., Bloom M., Davis J.H., Devaux P.F. (1982) Evidence f o r P r o t e i n - A s s o c i a t e d L i p i d s from Deuterium NMR S t u d i e s of Rhodopsin-D i m y r i s t o y l p h o s p h a t i d y l c h o l i n e Recombinants. J . B i o l . Chem. 257: 3032-3038 32. Davis J.H., Hodges R.S., Bloom M.(1982) The I n t e r a c t i o n Between a S y n t h e t i c Amphiphylic P o l y p e p t i d e and L i p i d s . Biophys. J . 37: 170-171 33. Davis J.H., C l a r e D.M.,Hodges R.S., Bloom M.(1983) I n t e r a c t i o n of a S y n t h e t i c Amphiphylic P o l y p e p t i d e and L i p i d s i n a B i l a y e r S t r u c t u r e . B i o c h e m i s t r y 22: 5298-5305 34. H u s c h i l t J.C., Hodges R.S., Davis J.H.(1985) Phase E q u i l i b r i a i n an Amphiphylic P e p t i d e - P h o s p h o l i p i d Model Membrane by Deuterium NMR D i f f e r e n c e Spectroscopy Biochemistry 24: 1377-1386 35. Devaux P.F, Hoatson G.L., Favre E., Fellman P., F a r r e n , B., MacKay A., Bloom M.(1985) I n t e r a c t i o n of Cytochrome c with Mixed D i m y r i s t o y l p h o s p h a t i d y l c h o l i n e 136 - D i m y r i s t o y l p h o s p h a t i d y l s e r i n e B i l a y e r s : A Deuterium Nuclear Magnetic Resonance Study. Submitted to B i o c h e m i s t r y . 36. Chapman, D. (1975) Phase T r a n s i t i o n s and F l u i d i t y C h a r a c t e r i s t i c s of L i p i d s and C e l l Membranes.Q. Rev. Biophys. 8: 185-235 37. Sp e i s s H.W. and S i l l e s c u H. ( 1 9 8 3 ) S o l i d Echoes i n the Slow Motion Regime. J . Magn. Res. 42: 381-389 38. Volke, F. (1984) The O r i e n t a t i o n Dependence of Deuterium Transverse R e l a x a t i o n Rates Obtained from Unoriented Model Membranes Systems. Chem. Phys. L e t t . 112-6: 551-554 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0084980/manifest

Comment

Related Items