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UBC Theses and Dissertations

Deuterium nuclear magnetic resonance study of water in model and biological membrane systems Wei, C.M. 1979

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DEUTERIUM NUCLEAR MAGNETIC RESONANCE STUDY OF WATER IN MODEL AND BIOLOGICAL MEMBRANE SYSTEMS b y C. M. WEI A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE DEPARTMENT OF PHYSICS IF THE FACULTY OF GRADUATE STUDIES We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA October, 1979 (DfeC.M. Wei, 1979 In present ing th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree l y ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 G i i A b s t r a c t A deuteron magnetic resonance study of water has been c a r r i e d out i n the l a m e l l a r phase of the egg y o l k l e c i t h i n - w a t e r and outer membrane of E. Col i - w a t e r systems i n excess water (abbreviated to EYL/EXCD2O and EC/EXCD 20, r e s p e c t i v e l y ) and egg y o l k l e c i t h i n - w a t e r i n 2 2 % (by weight) water (EYL/22%WD20). Spectra of these systems were taken as a f u n c t i o n of temperature, and t h e i r moments were c a l c u l a t e d . A n a l y s i s of the i n t e g r a t e d s i g n a l i n t e n s i t i e s revealed that the excess free (bulk) water i n the EC/EXCD 20 and EYL/EXCD^O froze a t -1°C and -2°C, r e s p e c t i v e l y (pure _ 2© freezes at 4°C). The water from b i l a y e r s i n the two excess water systems was frozen immediately a f t e r i t was squeezed out, whereas the water squeezed out from b i l a y e r s i n the EYL/22%WD20 remained unfrozen down to -10°C. A l l the squeezed out water i n the EYL/22%WD 20 was frozen at -15°C. The bound water i n that system was unfreezable i n the region under study. The amount of water frozen out i n the EC/EXCD20 at -2°C and i n the EYL/EXCE^O at -3°C was found to be approximately 85% of the t o t a l water content of the systems. The water frozen out i n the EYL/22%WD 20 at -15°C was determined to be 50% of the t o t a l water i n that-;system. A minimum i n the second moment vs temperature of the EYL/22%WD20 was observed and asc r i b e d to the presence of i s o t r o p i c free water squeezed out from the b i l a y e r s . Proton magnetic resonance r e s u l t s showed that there was no l i p i d phase t r a n s i t i o n i n the EYL/22%WD 0 i n the region from -10°C to 48°C. i i i Table of Contents Page Abs t r a c t i i L i s t of Figures v i Acknowledgements i x Chapter 1 I n t r o d u c t i o n 1.1. Functions of B i o l o g i c a l Membranes 1 1.2. S t r u c t u r e of B i o l o g i c a l Membranes 1 1.3. Model Membranes '5 1.4. Membrane F l u i d i t y and Transport C a r r i e r s 5 1.5. Involvement of L i p i d Component of Membrane i n Phase T r a n s i t i o n 6 1.6. B i o l o g i c a l S i g n i f i c a n c e of Phase T r a n s i t i o n i n Membrane System 7 1.7. Study of Membrane System by NMR Technique 8 1.8. Water i n Membrane System 9 1.9. M o t i v a t i o n f o r the Research P r o j e c t 11 2 NMR Theory For N u c l e i In L y o t r o p i c A m p h i p h i l i c L i q u i d C r y s t a l / Water System 2.1. Ba s i c P r i n c i p l e s of Nuclear Magnetic Resonance 12 2.2. Quadrupole . I n t e r a c t i o n s and The F i r s t Order P e r t u r b a t i o n 12 2.3. Deuterium Magnetic Resonance 15 a) Deuteron on the Hydrocarbon Chain 15 b) DMR on D~0 21 2.4. Proton Magnetic Resonance 22 a) A Two-Spin System 23 XV Page b) PMR on a p r o t i a t e d L i p i d System 25 3 Experimental 3.1. The M a t e r i a l s 27 3.2. Samples P r e p a r a t i o n 27 3.3. NMR Apparatus 27 3.4. NMR Measurements 28 4 The Results 4.1. DMR Results a) Quadrupole S p l i t t i n g s 3 0 b) Evaluations of the DMR sp e c t r a i n terms of t h e i r moments 3Q 4.2. PMR Results 31 4.3. Sources of E r r o r 32 5 D i s c u s s i o n And Conclusion 5.1. DMR Spectra of D-0 r\ a) Spectra of D~0 i n the EYL/EXCD.0 and EC/EXCD20 systems 33 b) Spectra of D-0 i n the EYL/22%WD20 system 39 5.2. The Moments of the DMR Spectra a) The i n t e g r a t e d s i g n a l i n t e n s i t i e s (M ) of the DMR spectra ". of D-0 i n the EYL/EXCD-0 and EC/EXCD20 systems 42 b) The i n t e g r a t e d i n t e n s i t y of the DMR spectra of D 20 i n the EYL/22%WD20 system 44 c) The f i r s t and the second moments of the DMR spectra of D 20 i n the EYL/EXCD-0 and EC/EXCD20 systems 44 d) The f i r s t and second moments of the DMR spectra of D20 i n the EYL/22ZWD-0 55 2 e) The temperature dependence of the A 2 and the M^/M2 f o r the EYL/22%WD20, EYL/EXCD20 and EC/EXCD20 60 Page 5.3. Comparison With Other Work 64 5.4. PMR Results For The EYL/22%WD20 66 Appendix A Moments Of Nuclear Magnetic Resonance Spectra 71 Appendix B C o n t r i b u t i o n s To The Second Moment 76 References 78 v i L i s t of Figures Figure Page 1 A schematic r e p r e s e n t a t i o n of d i p o l m i t o y l - 3 - s n -p h o s p h a t i d y l c h o l i n e l i p i d molecule. 2 2 A schematic r e p r e s e n t a t i o n of l a m e l l a r and v e s i c u l a r s t r u c t u r e s i n a l i p i d - w a t e r system. 4 3. A schematic r e p r e s e n t a t i o n of the geometry and the coordinate system i n a l i p i d b i l a y e r . 14 4 T h e o r e t i c a l powder p a t t e r n f o r a deuterium nucleus. 20 5. Energy l e v e l s and the corresponding spectrum of a two-spin system o r i e n t e d a t an angle 8. 24 6 Logarithmic PMR lineshape as derived by Bloom et a l . 26 7 DMR spectra of D 20 i n the E. Coli/EXCD 20 obtained a t (a) 20°C, (b) 4°C, (c) -1°C, (d) -2°C. 37 8 DMR spectra of D 20 i n the EYL/EXCT>20 obtained a t (a) 18°C, (b) 4°C, (c) -2°C, (d) -3°C. 38 DMR spectra of D 20 i n the EYL/22%WD20 obtained at (a) 25°C, (b) 4°C, (c) -3°C, (d) -4°C, (e) -5°C, ( f ) -10°C, (g) -15°C, (h) -20°C, ( i ) -25°C. 41 10 Temperature dependence of the i n t e g r a t e d s i g n a l i n t e n s i t y , M Q, of DMR of D 20 i n the EYL/EXCD20 and the E. Coli/EXCD 20. 43 11 Temperature dependence of the in t e g r a t e d s i g n a l i n t e n s i t y , M Q, of DMR of D 20 i n the EYL/22%WD20. 45 12 Temperature dependence of the f i r s t moment M^ of the DMR spectrum of D 20 i n the EYL/EXCD20 and E. Coli/EXCD 20. 50 13 Temperature dependence of the second moment M 2 of the DMR spectrum of D 20 i n the EYL/EXCD20 and E. Coli/EXCD 20. 51 Second moments of the DMR spectra of D20 in the EYL/EXCD20 plotted against that in the EYL/22%WD20. Temperature dependence of the line width A v of the DMR spectrum of D20 in the E. Coli/EXCD20 and EYL/EXCD20. DMR Temperature dependence of the fourth moment M^  of the spectrum of D20 in the EYL,/EXCD20 and E. Coli/EXCD20. Temperature dependence of the f i r s t and the fourth moments, M 1 S M^ , of the DMR spectrum of D20 in the EYL/22%WD20. Temperature dependence of the second moment M2 of the DMR spectrum of D20 in the EYL/22%WD20. 2 Temperature dependence of the ratio M^ /M2 and the relative mean square deviation A 2 of the DMR spectrum of D20 in the E. Coli/EXCD20. 2 Temperature dependence of the ratio M^ /M,, and the relative mean square deviation A 2 of the DMR spectrum of D20 in the EYL/EXCD20. Temperature dependence of the relative mean square deviation 2 A 2 and the ratio K^/M^ of the DMR spectrum of D20 in the EYL/22%WD20. Temperature dependence of the quadrupole splitting A v which is measured directly from the DMR spectrum of D20 in the EYL/22%WD20. 2 Temperature dependence of the ratio M^/M2 and the second moment of the PMR spectrum of the EYL/22%WD20. Temperature dependence of the second moment of PMR spectrum of (a) the outer membrane of E. C o l i i n the EC/EXCD-O, (b) the EYL i n the EYL/EXCD-0, ( C ) temperature dependence of M^/M2 of the PMR spectrum of the EYL i n the EYL/EXCD-0. (a) Second moments vs temperature of the DMR spectra of outer membranes of E. C o l i grown on a medium c o n t a i n i n g perdeuterated p a l m i t i c a c i d and o l e i c a c i d as w e l l as perdeuterated p a l m i t i c a c i d . (b) The parameter Av vs temperature of the DMR spectra of the outer membranes of E. C o l i grown on a medium c o n t a i n i n g o l e i c a c i d as w e l l as Perdeuterated p a l m i t i c a c i d . i x A c k n o w l e d g e m e n t s F i r s t and f o r e m o s t , I w i s h t o a c k n o w l e d g e t h e p a t i e n c e and e n t h u s i a s m o f my s u p e r v i s o r , D r . Myer B l o o m , who has g r e a t l y e n r i c h e d my m a s t e r s e x p e r i e n c e . I have b e n e f i t e d g r e a t l y f r o m h i s i n s t r u c t i o n and t h e many d i s c u s s i o n s we have h a d . I am v e r y g r a t e f u l t o Dr. A l e x Mackay and Dr. James H. D a v i s f o r t h e t e c h n i c a l c o n s u l t a t i o n s and h e l p i n t h e e x p e r i m e n t s and f o r t h e many f r u i t f u l d i s c u s s i o n s and u s e f u l s u g g e s t i o n s . I n t h i s r e g a r d t h e c o n t i n u e d s u p p o r t o f A l e x i n many o t h e r r e s p e c t s i s g r e a t l y a p p r e c i a t e d . F i n a l l y , I t h a n k my w i f e , May S a n , f o r t y p i n g t h e t h e s i s and f o r h e r p a t i e n c e and c o n t i n u i n g e n c o u r a g e m e n t and m o r a l s u p p o r t . 1 Chapter 1 In t r o d u c t i o n 1.1. Functions of B i o l o g i c a l Membranes Every l i v i n g c e l l i s enclosed by a membrane that serves not only as a sturdy enclosure i n s i d e which the c e l l can f u n c t i o n , but i t can a l s o perform a l a r g e number of b i o l o g i c a l f u n c t i o n s . For in s t a n c e , i t can f u n c t i o n as a d i s c r i m i n a t i n g gate, enabling n u t r i e n t s and other essen-t i a l agents to enter and waste products to leave. The cytoplasmic c e l l membrane can "pump" substances from one side of the membrane where such substance's concentration i s low to the other where i t i s much higher. Thus the cytoplasmic membrane s e l e c t i v e l y r e g u l a t e s the f l u x of n u t r i -ents and ions between the c e l l and i t s e x t e r n a l environment. The c e l l s of higher organisms have, i n a d d i t i o n to a cytoplasmic membrane, a number of i n t e r n a l membranes that i s o l a t e the s t r u c t u r e s termed o r g a n e l l e s , which play v a r i o u s s p e c i a l i z e d r o l e s ( 1). 1.2. S t r u c t u r e of B i o l o g i c a l Membranes A l l of the b i o l o g i c a l membranes mentioned above are remarkably s i m i -l a r i n t h e i r b a s i c s t r u c t u r a l f e a t u r e s . I t i s c l e a r that any model formu-l a t e d to describe membrane s t r u c t u r e s must be able to account f o r such an ex t r a o r d i n a r y range of f u n c t i o n s . Membranes are composed almost e n t i r e l y of two cl a s s e s of molecules: p r o t e i n s and l i p i d s (polysaccharides e t c . are a l s o a s s o c i a t e d w i t h b i o l o g -i c a l membranes). The p r o t e i n s provide the membrane w i t h i t s d i s t i n c t i v e f u n c t i o n a l p r o p e r t i e s , whereas the l i p i d s give the gross s t r u c t u r a l prop-e r t i e s of the membrane ( 1 ) . A d i p a l m i t o y l - 3 - s n - p h o s p h a t i d y l c h o l i n e (DPPC) l i p i d molecule i s shown i n F i g . 1. The l i p i d s found i n membranes are 2 N +(CH3)3 Polar head group ( h y d r o p h i l i c ) Chain 1 Chain 2 (Hydrophobic t a i l ) Figure 1. A schematic representation of Dipalmitoyl-3-sn-phosphatidylcholine l i p i d molecule. 3 amphipathic, meaning th a t one end of the molecule i s hydrophobic, or i n s o l u b l e i n water, and the other end h y d r o p h i l i c , or wat e r - s o l u b l e . The h y d r o p h i l i c region of the l i p i d molecule i s p o l a r , and the hydrophobic region i s non-p o l a r . In most membrane l i p i d s the nonpolar region c o n s i s t s of hydrocarbon chains of f a t t y a c i d s : hydrocarbon molecules w i t h a carb o x y l group (COOH) at one end. In a t y p i c a l membrane l i p i d two f a t t y a c i d molecules are chemically bonded through t h e i r carboxyl ends to a backbone of g l y c e r o l . The g l y c e r o l backbone, i n t u r n , i s attached to a p o l a r head group c o n s i s t i n g of phosphate and other groups. Phosphate-containing l i p i d s o f t h i s type are c a l l e d p h o s p h o l i p i d s , which are found i n a l l membranes. Because the two pa r t s of a membrane l i p i d have incompatible s o l u b i l -i t i e s , the l i p i d molecules i n the presence of water spontaneously organize themselves i n the form of a v a r i e t y of l y o t r o p i c mesophases c h a r a c t e r i z e d by the existence of long range order and short range d i s o r d e r (1-4). The convincing evidence f o r the existence of these mesophases have been p r o v i d -ed by X-ray s t u d i e s ( 5 ) , nuclear magnetic resonance (NMR) st u d i e s (6-7), and other physicochemical techniques (8-12). Of p a r t i c u l a r i n t e r e s t i s the l a m e l l a r l i q u i d c r y s t a l (L^) phase where the l i p i d molecules form b i l a y e r s and a l t e r n a t e i n a re g u l a r l a t t i c e w i t h l a y e r s of water and counter ions as shown i n F i g . 2. In t h i s way the hydrophobic region of each molecule i s s h i e l d e d from water, whereas the h y d r o p h i l i c p o l a r head groups f i n d themselves i n a lower e l e c t r o s t a t i c energy by a s s o c i a t i n g i n t i m a t e l y w i t h water. The hydrocarbon chains asso-c i a t e themselves i n the form of b i l a y e r s as a r e s u l t of the above i n t e r -a c t i o n s and the a t t r a c t i v e Van der Waal's f o r c e s . Most b i l a y e r s i n l i p i d - w a t e r system i n l a m e l l a r l i q u i d c r y s t a l l i n e phase n a t u r a l l y arrange themselves i n t o an onion r i n g c o n f i g u r a t i o n , so 4 (b) r~S-n Water r^TZ. Water mm Figure 2. (a) The pho s p h o l i p i d b i l a y e r forms the fundamental s t r u c t u r a l matrix of a membrane. The l i p i d s are arranged chain to chain so that only the h y d r o p h i l i c p o l a r heads (the c i r c l e s ) are exposed to the aqueous s o l u t i o n on both sides of the membrane. L i p i d molecules can d i f f u s e l a t e r a l l y at a high frequency, but can r a r e l y execute a f l i p - f l o p t r a n s i t i o n from one l a y e r to the other. The wiggly l i n e s represent the hydrocarbon chain. (b) S p h e r i c a l b i l a y e r ( v e s i c l e ) . 5 that nowhere i n the b i l a y e r s the hydrophobic t a i l s are i n contact w i t h wa-t e r . These b i l a y e r s form the b a s i c s t r u c t u r e of most b i o l o g i c a l membranes. 1.3. Model Membranes^ Due to the complexity of r e a l b i o l o g i c a l membranes, the NMR spectrum obtained i s a s u p e r p o s i t i o n of a l a r g e number of powder patterns which are not w e l l r e s o l v e d . Thus, i n order to understand the fundamental p h y s i c a l p r o p e r t i e s which determine the proper f u n c t i o n of membranes, workers i n _ the f i e l d look f o r simpler membrane systems that could be used as "models". Since l i p i d s are the fundamental " b u i l d i n g b l o c k s " o f . t h e - c e l l membrane, model membrane systems have been made w i t h l i p i d s obtained from n a t u r a l sources such as egg y o l k l e c i t h i n (13-14) as w e l l as from s y n t h e t i c tech-niques such as DPPC (15). These model membranes are s i m i l a r (not i d e n t i -c a l ) i n t h e i r s t r u c t u r a l p r o p e r t i e s to b i o l o g i c a l membranes and they can be much b e t t e r c h a r a c t e r i z e d p h y s i c a l l y and chemically than membranes from l i v i n g c e l l s , which have a complicated assortment of d i f f e r e n t l i p i d s . The phase t r a n s i t i o n s are a l s o much sharper w i t h p u r i f i e d l i p i d s of a s i n g l e homogenous type, t h i s makes q u a n t i t a t i v e measurement and t h e o r e t i c a l i n t e r p r e t a t i o n much e a s i e r . Furthermore, NMR spectra given by model mem-branes are simpler to i n t e r p r e t than those given by n a t u r a l ones. There have been numerous s t u d i e s on model and b i o l o g i c a l membranes concerning the importance of the hydrophobic parts of the p h o s p h o l i p i d molecules i n determining the general p r o p e r t i e s of the i n t e r i o r of the p h o s p h o l i p i d b i -l a y e r (5-7, 16).. 1.4. Membrane F l u i d i t y and Transport C a r r i e r s Since the phenomenon of l i f e occurs i n s i d e the c e l l , which i s enclosed by a membrane, how do substances pass through the membrane to provide the c e l l w i t h l i f e - s u p p o r t i n g n u t r i e n t s ? The hydrophobic nonpolar f a t t y a c i d 6 hydrocarbon chain region of a pho s p h o l i p i d b i l a y e r i s p h y s i c a l l y incompat-i b l e w i t h small water-soluble substances such as metal i o n s , sugars and amino a c i d s , and thus acts as a b a r r i e r through which they cannot flow ".• f r e e l y . Bangham et a l of the A g r i c u l t u r a l Research C o u n c i l i n Cambridge, England, and Chappell et a l of the U n i v e r s i t y of Cambridge (17) measured the r a t e at which glucose passes through the p h o s p h o l i p i d - b i l a y e r w a l l s of liposomes ( v e s i c l e s ) and found that i t was f a r too low to account f o r the ra t e at which glucose penetrates b i o l o g i c a l membranes. They demonstrated that a h i g h l y s e l e c t i v e c a r r i e r p r o t e i n e x i s t s i n b i o l o g i c a l membranes to f a c i l i t a t e the passage of metal ions and small p o l a r molecules through the pe r m e a b i l i t y b a r r i e r presented by the phospholipid b i l a y e r . Since transport c a r r i e r s must be mobile i n order to move substances from one si d e of the c e l l membrane to the other, i t i s necessary f o r the region c o n t a i n i n g the f a t t y a c i d chains to have a high degree of f l u i d i t y i n which each b i l a y e r behaves as a two dimensional f l u i d w i t h the l i p i d chains p r e f e r e n t i a l l y o r i e n t e d along the normal to the b i l a y e r surface. Within the b i l a y e r the hydrocarbon chains of the l i p i d molecules are f l e x -i b l e (melted) and the molecules undergo r a p i d l a t e r a l d i f f u s i o n and r o t a -t i o n about t h e i r long a x i s . D i f f e r e n t p a r t s of the hydrocarbon chain can als o undergo small and r a p i d angular excursions such as bending, t w i s t i n g and f l o p p i n g perpendicular to the molecular a x i s (the long a x i s of the mol-ecule) . X-ray d i f f r a c t i o n patterns of membrane systems above the t r a n s i t t i o n temperature are d i f f u s e and q u i t e s i m i l a r to those obtained from long-chain l i q u i d hydrocarbons found i n p a r a f f i n , which i n d i c a t e s that the f a t t y acids of membranes are i n f a c t disordered at p h y s i o l o g i c a l temperature ( 5 ) . 1.5. Involvement of L i p i d Component of Membrane i n Phase T r a n s i t i o n Experiments have c o n c l u s i v e l y demonstrated that the phase t r a n s i t i o n 7 i n membrane systems (such as Acholeplasma l a i d l a w i i and model membranes) i s due e x c l u s i v e l y to the l i p i d component of the membrane (18-19). There-fore physical techniques that elucidate structure, and theories that deal with phases are appropriate tools to study t h i s phenomenon. The basic nature of the phase t r a n s i t i o n i n m u l t i b i l a y e r dispersions i s revealed by calorimetry, X-ray d i f f r a c t i o n and nuclear magnetic resonance (16-18). Calorimetric data (18-19) showed that the t r a n s i t i o n s i n l i p i d b i l a y e r s mainly involve hydrocarbon-chain disordering (phase t r a n s i t i o n s i n v o l v i n g s t r u c t u r a l changes i n l i p i d b i l a y e r s such as lamellar to hexagonal phase t r a n s i t i o n had been observed by X-ray technique (5)). X-ray d i f f r a c t i o n and other phy s i c a l techniques studies show that, with low and varying water concentration, l i p i d s have a r i c h v a r i e t y of phase behaviour i n addition to the phase t r a n s i t i o n i n the presence of excess water, which i s of primary b i o l o g i c a l i n t e r e s t (18). 1.6. B i o l o g i c a l Significance of Phase T r a n s i t i o n i n Membrane System The order-disorder change i n the dynamical state of hydrocarbon chains i n b i l a y e r membranes gives r i s e to a phase t r a n s i t i o n that occurs i n many c e l l membranes as well as model membranes. Much work (18) has been done on the cytoplasmic membrane of Acholeplasma l a i d l a w i i , which i s a prim i -t i v e orgamism with a large surface-to-volume r a t i o . A f t e r the c e l l was grown, at a p a r t i c u l a r temperature T , the membrane wascextracted and ca l o r i m e t r i c measurements were made. Data show a s p e c i f i c heat anomaly, about 20°C broad, centered near or s l i g h t l y below T , When these c e l l s were grown at d i f f e r e n t growth temperatures T ', the c a l o r i m e t r i c anomaly i s s h i f t e d towards the temperature T '. This suggests that the phase t r a n s i t i o n i s not j u s t some ph y s i c a l phenomenon that happens to occur, but that i t has r e a l b i o l o g i c a l relevance. There are many other examples 8 of p h y s i c a l l y induced b i o l o g i c a l changes r e l a t e d to the membrane phase t r a n s i t i o n . Organisms that e x i s t i n c o l d environments have membrane compo-nents g i v i n g r i s e to reduced p h a s e - t r a n s i t i o n temperatures. 1.7. Study of Membrane System by NMR Technique Nuclear magnetic resonance i s a very u s e f u l technique f o r studying the s t r u c t u r e and the dynamical s t a t e of the membrane systems (3, 20). By s t r u c t u r e , we mean the average o r i e n t a t i o n of the hydrocarbon chains, the p o l a r groups, and the amplitudes of f l u c t u a t i o n of the l i p i d segments. The study of the dynamical s t a t e of a membrane system i s concerned w i t h the r a t e of segmental motions and the r a t e of d i f f u s i o n of the l i p i d s w i t h i n each monolayer. In terms of experimental NMR parameters, the problems of membrane s t r u c t u r e and dynamical p r o p e r t i e s of the b i l a y e r are solved i f the complete set of segmental second rank order parameter tensors (see Sec-t i o n 2.3a, Chapter 2) and the r e l a x a t i o n times are measured, and i f a con-s i s t e n t molecular i n t e r p r e t a t i o n of the experimental data can be provided. The order parameter S ^ of a deuterium bond vec t o r and, f o r a s p e c i a l case, S of a H-H bond v e c t o r can be measured from the quadrupole s p l i t t i n g , tin A VQ, i n the deuterium nuclear magnetic resonance (DMR) spectrum and from the nuclear d i p o l a r s p l i t t i n g i n the proton magnetic resonance (PMR) spec-trum r e s p e c t i v e l y , s i n c e they are d i r e c t l y p r o p o r t i o n a l to the s p l i t t i n g s (Appendix A). The order parameter can then be i n t e r p r e t e d i n terms of s t a t i s t i c a l models f o r the b i l a y e r s t r u c t u r e . The onset of motions or the phase t r a n s i t i o n from one form of l a t t i c e s t r u c t u r e to the other with a higher degree of l a t t i c e symmetry (such as a cubic phase) w i l l , due to motional averaging, reduce the order parameters S and S , and conse-CD HH quently the s p l i t t i n g s i n the s p e c t r a . The other NMR parameters such as the l i n e width of a s i n g l e t absorption spectrum are a l s o reduced due to 9 motional averaging of d i p o l a r or quadrupolar i n t e r a c t i o n s . Therefore, phase t r a n s i t i o n s or onset of motions i n these systems are detected by the abrupt change i n the NMR parameters as the temperature i s v a r i e d . Much work has been done on the conformations and motions of the l i p i d molecules i n membrane system (21-30). Davis et a l (23) st u d i e d the hydrocarbon chain d i s o r d e r i n the potassium palmitate-water system. In the l i q u i d c r y s t a l l i n e phase the C-D order parameters of the f i r s t few meth-ylene chain segments were found to increase w i t h i n c r e a s i n g temperature to a maximun of 100°C and then decrease at higher temperatures. In c o n t r a s t , the C-D order parameters f o r the r e s t of the methylene chain segments de-creased w i t h i n c r e a s i n g temperature. In the same system Higg and Mackay (30) have determined the complete order parameter tensors f o r the a-meth-ylene group by measuring the a-CH 2 d i p o l a r s p l i t t i n g s i n an otherwise perdeuterated chain and the a-CD 2 s p l i t t i n g s (23) i n the s p e c i f i c a l l y deuterated chains. The temperature dependence of the a-CH 2 s p l i t t i n g s was s i m i l a r to that of the a-Ci>2 s p l i t t i n g s . B u r n e l l et a l (31) st u d i e d the ord e r i n g of water i n the potassium palmitate/D 20 system. They found that the order parameter of the deuterium i n D 20 had the same temperature dependence as the f i r s t few methylene p a i r s as e s t a b l i s h e d by Davis et a l described p r e v i o u s l y . This c o r r e l a t i o n was asc r i b e d to the l i p i d - w a t e r i n t e r a c t i o n v i a hydrogen bonding between the water and the p o l a r heads near the l i p i d - w a t e r i n t e r f a c e . 1.8. Water i n Membrane System Water (H 20) c o n s t i t u t e s a major component of c e l l s of a l l l i v i n g o r -ganism, and plays an important r o l e i n l i f e process at the c e l l u l a r l e v e l . The o r d e r i n g of water at the l i p i d - w a t e r (D 20) i n t e r f a c e can be observed because of the i n t e r a c t i o n s between the nuclear quadrupole moments of 10 deuterium i n B^O and the e l e c t r i c f i e l d gradients at the nuclear s i t e s . The anisotropy experienced by water molecules i n the l i p i d - w a t e r system i s f a r smaller than that i n the hydrocarbon chains, but i s s t i l l l a r g e enough to give a w e l l re s o l v e d quadrupole s p l i t t i n g f o r water deuterons i n D2O. In the DMR spec t r a of D^O i n l a m e l l a r phases of egg y o l k l e c i t h i n , egg phosphatidylethanolamine, ox b r a i n sodium p h o s p h a t i d y l s e r i n e (32) and 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 (33), a sharp c e n t r a l l i n e was observed. F i n e r et a l as c r i b e d t h i s sharp c e n t r a l l i n e to the presence of i s o t r o p i c water. One should be cautious of t h i s e x p l a nation. As pointed out by Wennerstrom et a l (34) and Lindblbm et a l (35) that f o r some samples, the sharp c e n t r a l l i n e was independent of the added water c o n c e n t r a t i o n , and may be due to double quantum t r a n s i t i o n s , which i s unobservable at low RF power, while i t i s much more intense than the powder p a t t e r n at high RF f i e l d s t rengths. Thus i t i s very easy to determine the nature of the peak by i n v e s t i g a t i n g the dependence of the s i g n a l i n t e n s i t y on the RF f i e l d s t r e n g t h . A p p l i c a t i o n of F i n e r ' s method of a n a l y s i s (36), F i n e r and Darke were able to d i s t i n g u i s h 2, 3 and 4 d i f f e r e n t kinds of water f o r egg phosphatidylethanolamine, egg l e c i t h i n and sodium p h o s p h a t i d y l s e r i n e respec-t i v e l y . These water types were i d e n t i f i e d as t i g h t l y bound inner h y d r a t i o n s h e l l , weakly bound water, trapped water, w i t h exchange between them being r a p i d on the NMR time s c a l e , and f r e e water. The c h a r a c t e r i s t i c s p l i t t i n g of the main hy d r a t i o n s h e l l of egg yo l k l e c i t h i n i s 0.37 kHz, and that of the inner h y d r a t i o n s h e l l or the most t i g h t l y bound s h e l l of the same system i s 6.9 kHz, which are consider a b l y smaller t h a n the s p l i t t i n g of 170 kHz t y p i c a l l y found f o r p o l y c r y s t a l l i n e i c e and hydrates (56). 11 1.9. M o t i v a t i o n f o r the Research P r o j e c t Deuterium magnetic resonance of D^ O i n EYL/D^O system has been done by F i n e r et a l (32) and others. A l l of them measured the quadrupole s p l i t t i n g s of DMR spectra of water (D 20) as a f u n c t i o n of temperature. This method of e v a l u a t i n g DMR spectra i s subjected to a systematic e r r o r and i s d i f f i c u l t to apply when the sp e c t r a are broadened. In t h i s research, the moments (Appendix A) w i l l be c a l c u l a t e d from the DMR spectra of V^O i n the EYL/D^O and E. C o l i / D 2 0 systems, and the s p l i t t i n g s from the moments thus determined. The l a t t e r w i l l be compared to the s p l i t t i n g s d i r e c t l y obtained from measuring the separations between the doublet peaks of the spe c t r a . According to the data (Alex Mackay, unpublished) obtained from PMR st u d i e s of the p r o t i a t e d outer membrane of E. C o l i / D 2 0 and EYL/D^O i n excess water, the temperature dependence of the second moments and the 2 M^ /M^  c a l c u l a t e d from the PMR spec t r a of the two systems showed an anomalous d i s c o n t i n u i t y at 4°C that i s not observed i n Davis' data (37) obtained from DMR study of the deuterated outer membranes of E. C o l i . Since PMR i s s e n s i -t i v e to i n t e r m o l e c u l a r motions while DMR i s not, the data seem to suggest that the anomalous d i s c o n t i n u i t y was due to onset of l a t e r a l d i f f u s i o n of the phospholipid molecules. Because pure D 20 freezes at 4°C, the l a t e r a l o d i f f u s i o n below 4 C might have been stopped by f r e e z i n g of the water. The p o s s i b l e c o r r e l a t i o n between the disappearance of the l a t e r a l d i f f u s i o n and f r e e z i n g of the water w i l l be confirmed or dismissed by the r e s u l t s of my experiments. 12 Chapter 2 NMR Theory For N u c l e i In Lipid/Water System 2.1. Basic P r i n c i p l e s of Nuclear Magnetic Resonance N u c l e i having a non-zero nuclear s p i n possess a magnetic d i p o l e moment •]_, which i s r e l a t e d to the nuclear s p i n j by-(38) ' — - -y = y f i l { 2 . i } where y I s t n e gyromagnetic r a t i o , and I i s a ve c t o r operator. In a s t a t i c magnetic f i e l d H^, a magnetic nucleus may e x i s t i n one of the 2 1 + 1 s p i n s t a t e s w i t h s p i n quantum number m = I , I - 1, ... - I and energy l e v e l s E_(m) = -yhHom = -fico^m, which are the eigenvalues of the Zeeman Hamiltonian H Z = -v-h - ^ V z { 2 - 2 } where I i s the Z-component of the nuclear s p i n operator I . Thus, when ZJ an appropriate r a d i o frequency electromagnetic r a d i a t i o n (RF f i e l d ) i s a p p l i e d to the nucleus perpendicular to the a p p l i e d magnetic f i e l d H q , a t r a n s i t i o n between two s p i n s t a t e s i s induced by a coupling of the magnetic d i p o l e moment to the RF f i e l d , and the phenomenon of nuclear magnetic resonance occurs. 2.2. Quadrupolar I n t e r a c t i o n s and The F i r s t Order P e r t u r b a t i o n In a d d i t i o n to a magnetic d i p o l e moment, a nucleus of s p i n I > _ possesses an e l e c t r i c quadrupole moment which has i t s o r i g i n i n a nonspheri-c a l l y symmetric nuclear charge d i s t r i b u t i o n . Consequently, the nucleus has an e l e c t r o s t a t i c i n t e r a c t i o n w i t h i t s environment when i t i s i n an e l e c t r o -s t a t i c f i e l d gradient (EFG) which does not possess too high a degree of symmetry. This i n t e r a c t i o n depends on the o r i e n t a t i o n of the nuclear s p i n . H e n c e , , t h e t o t a l H a m i l t o n i a n f o r a n u c l e u s w i t h s p i n I > \ i n a n a p p l i e d m a g n e t i c f i e l d H q i s g i v e n b y H = H z + H Q { 2 . 3 } w h e r e H i s t h e Z e e m a n H a m i l t o n i a n , a n d H i s t h e q u a d r u p o l a r H a m i l t o n i a n z Q d u e t o t h e i n t e r a c t i o n b e t w e e n t h e q u a d r u p o l e m o m e n t a n d t h e e l e c t r i c f i e l d g r a d i e n t e x i s t i n g a t t h e n u c l e a r s i t e . H e r e w e h a v e i g n o r e d c h e m i c a l s h i f t a n d d i p o l e - d i p o l e i n t e r a c t i o n t e r m s e t c . I f t h e c o u p l i n g b e t w e e n t h e n u c l e a r q u a d r u p o l e m o m e n t a n d E F G i s m u c h s m a l l e r t h a n t h e c o u p l i n g o f t h e n u c l e a r m a g n e t i c d i p o l e m o m e n t t o t h e a p p l i e d s t a t i c m a g n e t i c f i e l d , i t c a n b e s h o w n ( 3 8 - 4 1 ) t h a t t h e f i r s t o r d e r p e r t u r b a t i o n t o t h e Z e e m a n e n e r g y l e v e l s d u e t o H ^ a r e g i v e n i n f r e q u e n c y u n i t s b y E m 1 } = 4 h I ( 2 I - l ) { 3 C Q 2 2 9 " 1 + 2 S i n 2 e c o s 2 $ } { 3 m 2 - I ( I + l ) } { 2 , 4 } w h e r e t h e a n g l e s 6, $ s p e c i f y t h e m a g n e t i c f i e l d d i r e c t i o n w i t h r e s p e c t t o t h e p r i n c i p a l c o o r d i n a t e s y s t e m o f t h e E F G a s s h o w n i n F i g . 3 a , m i s t h e m a g n e t i c q u a n t u m n u m b e r i n t h e r e p r e s e n t a t i o n w h e r e I i s d i a g o n a l , a n d Li 2 e q Q / h i s t h e q u a d r u p o l e c o u p l i n g c o n s t a n t i n v o l v i n g t h e p r o d u c t o f t h e Z Z c o m p o n e n t o f t h e E F G q = V' /'e i n t h e p r i n c i p a l c o o r d i n a t e s y s t e m , a n d t h e q u a d r u p o l e m o m e n t eQ a s s o c i a t e d w i t h t h e s p i n I n u c l e u s . T h e q u a n t i t y : n ={|Vyy| - | V x J } / | V z J { 2 . 5 } 2 2 i s c a l l e d t h e a s y m m e t r y p a r a m e t e r , w h e r e V = 9 V / 9 x e t c . , a n d V i s t h e X X n e t e l e c t r o s t a t i c p o t e n t i a l a t t h e n u c l e a r s i t e . C o n v e n t i o n a l l y , t h e E F G p r i n c i p a l a x e s a r e o r d e r e d s o t h a t IV I IV I > | V I { 2 . 6 } 1 z z = 1 y y ' — x x ' i ^ . u j . 14 (a) (b) Figure 3. (a) O r i e n t a t i o n of the s t a t i c magnetic f i e l d w i t h respect to the p r i n c i p a l coordinate system of the e l e c t r i c f i e l d g r a d i e n t . (b) Schematic r e p r e s e n t a t i o n of the geometry and the coordinate system i n a l i p i d b i l a y e r . 6 i s the angle between the magnetic f i e l d H q and the b i l a y e r normal f i , 6 i s the angle between the C-D bond d i r e c t i o n and H , and 9 i s the angle between the C-D o n bond d i r e c t i o n and n. / 15 and,^consequently, 0 <_ n <_ 1. In I t s p r i n c i p a l axis system, the EFG tensor at the nuclear s i t e i s determined by the parameters q and n. The asymmetry parameter ri i s a measure of the deviation of the EFG from a x i a l symmetry. The parameter Q , which i s a measure of the deviation of the nuclear charge d i s t r i b u t i o n from s p h e r i c a l symmetry, i s the property of the nucleus alone, and i s the same for a l l compounds i n which a given nucleus i s found. For a prolate spheroidal d i s t r i b u t i o n (foot b a l l l i k e ) Q i s p o s i t i v e , for an oblate spheroid ( f l a t t e n e d at the poles and bulging at the equator) Q i s negative. Q vanishes for a s p h e r i c a l l y symmetric charge d i s t r i b u t i o n . Thus the energy l e v e l s of the t o t a l Hamiltonian i n frequency units as a r e s u l t of the f i r s t order perturbation s o l u t i o n of eqn. {2.3} are given by E = E ( 0 ) + E ( 1 ) {2.7} m m m where E ^ = E_(m)/27Tn = •(—YH /_TT)_=-v m, and V i s the Larmor frequency. m Z o o o ^ J Let us define a parameter as v - _ - __ _ e 2qQ ( 2 o) V Q 4'hI(2I-l) 2TT 4 h l ( 2 l - l ) t Z - O J Then eqn. {2.7} i s written as E = -v m + \ 3 c o s Q-1 + 5s in 2 ecos2$}{3m 2-I(I+l)} {2.9} m o J z z 2.3. Deuterium Magnetic Resonance (DMR) a) Deuteron on the hydrocarbon chain. Assume that the amphiphilic membrane system i s i n the r i g i d l a t t i c e phase so that the chain motion i s suppressed. Let us consider one.of the th deuterons i n the n p o s i t i o n of the hydrocarbon chain as depicted i n F i g . 3b. Since the deuterium nucleus has a spin 1 = 1 , i t s energy l e v e l s as given by the general perturbation r e s u l t expressed i n eqn. {2.9} are 16 2 •• M - V q + i v ( 3 c o s 9-1 + n s i n 2 Q c o s 2 $ ) {2.10} n E o = Z i v ( 3 c p s V l + n s . n 2 9 c o s 2 $ ) { 2 > 1 1 } 2 T-. _ ^ 1 /3cos 9- 1 , n . 2 /o i o \ E + 1 = - V Q + ( 2 + 2 s l n 9cos2$) {2.12} TI and the corresponding resonance frequencies are 2 E - - E = V + V A ( 3 c°s 9-1 + 3 s i n 26cos2$) {2.13} -1 o o Q 2 2 n 2 E - E x = V - v. ( 3 C°^ 6 - 1 + 5 s i n 2 e c o s 2 $ ) {2.14} O +1 o u z z x n - 1 3 2 where V = (2rr) ^ e q nQ/h, and q n i s the ZZ component of the EFG at the th deuteron s i t e i n the n p o s i t i o n of the cha i n . Thus the NMR spectrum a r i s i n g from the p a r t i c u l a r deuteron w i l l c o n s i s t of two sharp peaks centered about the c e n t r a l frequency V q and separated by 2 2 A v = v_ (3cos 9-1) + n v _ s i n 9 Cos2$ {2.15a} n n Notice t h a t , i n the absence of c y l i n d r i c a l symmetry ( n i- 0) the resonance frequencies and the s p l i t t i n g depend on the angles 9 and $. The e f f e c t of non -zero n on the powder p a t t e r n spectrum has been explored i n great d e t a i l by Barnes (41) and Cohen et a l (40). Suppose that the membrane system i n a! l a m e l l a r l i q u i d c r y s t a l l i n e phase i s o r i e n t e d so that a l l the o p t i c a l axes ( i n a l i p i d b i l a y e r they are the normals to the b i l a y e r surfaces) of the microdomains i n the macroscopic sample are p a r a l l e l to each other, making an angle 9 w i t h the a p p l i e d -»-magnetic f i e l d H^. I f the l i p i d molecules i n the o r i e n t e d sample undergo a r a p i d a n i s o t r o p i c motion w i t h a c o r r e l a t i o n time T c much shor t e r than the 17. . inv e r s e of the s t a t i c quadrupolar s p l i t t i n g s , then the r e o r i e n t a t i o n of the C - D bond w i l l modulate 0 and hence 6_ as depicted i n F i g . 3b, and the angular dependent f a c t o r s i n eqn. (2.15a}.is taken as the time average. Thus: 2 2 Av = vn <3cos 6-l> + T)\)n <sin Gcos2$> {2.15b} n Q Q n n The EFG at a deuteron s i t e on a hydrocarbon chain i n l i q u i d c r y s t a l l i n e phase has, to a good approximation, a c y l i n d r i c a l symmetry*, thus the asymmetry parameter r) i s p r a c t i c a l l y zero. Therefore, n e g l e c t i n g n, eqn. {10b} i s reduced to a much simpler form: 2 A V n = VQ < 3 c o s 9 _ 1 > ^2.15c} th The a n i s o t r o p i c motion of the C D 2 i n the n p o s i t i o n of the hydrocarbon chain can be separated i n t o two independent components: (a) r e o r i e n t a t i o n of the C D ^ group about a symmetry a x i s and (b) f l u c t u a t i o n s of the d i r e c t i o n of t h i s a x i s (the average d i r e c t i o n of t h i s a x i s has been shown to be normal to the l a m e l l a i n r e l a t e d systems (42)). With the motions separated i n t o these two independent components, and using the w e l l known a d d i t i o n theorem 2 f o r s p h e r i c a l harmonics (Abragam book, p. 454), <3cos 8-l> can be w r i t t e n 2 ? 3cos 6-1 „ <3cos 9-l> = < ^-2—>(3cos 0.-1) {2.16} and the f i r s t order quadrupole s p l i t t i n g i n the DMR spectrum a r i s i n g from t i l the deuteron i n the n p o s i t i o n of the hydrocarbon chain i s given by 3cos 26 -1 Av = V n < _ n _ > ( 3 c o s z a - l ) {2-.1.7} n Q 2 ^n *The Z p r i n c i p a l a x i s of the EFG at the s i t e of a deuterium nucleus on the hydrocarbon chain i s almost always w i t h i n a few degrees of the C - D covalent bond d i r e c t i o n and that the asymmetry parameter i s r\ <^ 0.05 (41) 18 where the angles 6 n and Q are defined i n F i g . 3. The formulae {2.15a}-.and {2.17}are the fundamental equation's f o r the e v a l u a t i o n of D M R s p e c t r a . We define a C - D bond order parameter S as 2 n 3cos 6 -1 SCD = < r 5 " * { 2 - 1 8 } n t h where the s u b s c r i p t n on D denotes the deuteron i n the n p o s i t i o n along the hydrocarbon chain. Then eqn. {2.17} i s s i m p l i f i e d to Av = Vn S „ _ (3cos 2e-l) {2.19} n U LD — n n To a good approximation, a l l the hydrocarbon chain - CD^ deuterons are assumed to be chemically equivalent and hence w i l l have the same quadru pole c o u p l i n g constant. Furthermore, s i n c e the quadrupole c o u p l i n g V n of the deuteron on the hydrocarbon chain i s predominantly of i n t r a m o l e c u l a r o r i g i n ( C - D bond), i t depends mainly on the s t a t e of the covalent C - D bond. Thus V depends on temperature through bond v i b r a t i o n . However, i n the n temperature range of i n t e r e s t i n t h i s study, the temperature v a r i a t i o n of V i s expected to be n e g l i g i b l y s m a l l , and thus can be assumed constant i n that range. Consequently, the order parameter f o r the methylene deuterons can be obtained d i r e c t l y from the measured quadrupole s p l i t t i n g s using eqn. {2.19} i f V i s known. An o r i e n t e d sample i s d i f f i c u l t to prepare, so i n s t e a d one u s u a l l y uses a powder sample made of many small c r y s t a l s o r i e n t e d randomly w i t h respect to the l a b o r a t o r y frame whose Z-axis i s o r i e n t e d along the a p p l i e d s t a t i c magnetic f i e l d d i r e c t i o n . In our case, the b i l a y e r normals are ran-domly o r i e n t e d w i t h equal p r o b a b i l i t y f o r a l l d i r e c t i o n s , and the super-p o s i t i o n of the l i n e s a r i s i n g from the d i f f e r e n t o r i e n t a t i o n s gives r i s e 19 to a broad absorption curve c h a r a c t e r i s t i c of a powder p a t t e r n of the form f (w) = |{g_(w) + g_ <-<*>)} '{2.20} 2 U B n where _ Aw - J Aw -_ Aw n 0 < w < - 7 T -n 1 Aw - 1 = ^ <HT + W ) A ^ n , . n — 7 T - < w < Aw 2 — — n = 0 otherwise {2.21} 3 e 2 0 A W n = 4 ^ l i SCD = 2 7 T V0 SCD ' a n d SCD ± S g ± V e n b y e q i U { 2- 1 8^' T h e n n n spectrum has two intense peaks separated by an amount given by the s p l i t t i n g i n the DMR spectrum of the corresponding sample o r i e n t e d at 0 = 90°, i . e . the peaks se p a r a t i o n i n the powder pattern, i s Aw n = 2ir|v "S | {2.22} TI n as given by eqn.{2*19} w i t h 0 =90°. This f i r s t order powder p a t t e r n spectrum i s shown i n F i g . 4a. When d i p o l a r l i n e broadening i s taken i n t o account, the powder p a t t e r n assumes the form (43) AN Tf -1 f (w) = -=A f'M-^-y + {to+2Trvn S__ P (cos9)} ) T2n ° T0,.2 Q n C°n 2 2n -1 + ( ~ ^ _ - +' {W-2TTV S C D P 2 ( c o s 0 ) } 2 ) }sin0d0 {2.23} T„ n n 2n t _1 where N^ i s the number of deuterons i n the n p o s i t i o n of the hydrocarbon chains, T 0 i s the s p i n - s p i n r e l a x a t i o n time of the deuterons i n that zn p o s i t i o n , and A i s a n o r m a l i z a t i o n constant. The f u n c t i o n P 2(cos0) i s the 20 l> ll Frequency ( i n kHz) s t Figure 4. (a) T h e o r e t i c a l 1 order powder p a t t e r n f o r a deuterium nucleus i n a symmetric e l e c t r i c f i e l d gradient (n = 0 ) . The dashed l i n e s show the i n d i v i d u a l components of the m = -1 f-> m = 0 and m = 0 •*-»- m = +1 t r a n s i t i o n s , w h i l e the s o l i d l i n e i n d i c a t e s the sum of the two components. Note that the s e p a r a t i o n of the 180° "shoulders" i s twice the doublet s e p a r a t i o n Av. (b) The same spectrum as (a) but quadrupolar and d i p o l a r l i n e broadening has been taken i n t o account. The dotted curve i s the unbroadened spectrum i n ( a ) . 21 2 Legendre polynomial defined as P2(cos0) = (3cos 9-1)/2. Notice that the i n t r i n s i c s p l i t t i n g i n t h i s broadened powder p a t t e r n i s Acon = 2TT|V | . % n This broadened f i r s t order powder p a t t e r n i s depicted i n F i g . 4b. For a perdeuterated l i p i d system c o n s i s t i n g of M non-equivalent deuteron p o s i t i o n s along the chains, the r e s u l t a n t 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 M broad absorption curves given by (43) f ( U ) = X ^ C o . ) each w i t h two sharp edges ( 9 0 ° ) separated by an angular frequency Au> = 2rr|v S C D | . n n In an i s o t r o p i c l i q u i d , i f the motion i s f a s t compared to the s t a t i c s p l i t t i n g frequency, the order parameter i s averaged to zero and no s p l i t t i n g i s observed. However, s i n c e the motion i n a l a m e l l a r l i q u i d c r y s -t a l system i s i n general a n i s o t r o p i c , S 4 0 and a s p l i t t i n g of the f i r s t n order spectrum w i l l u s u a l l y be observed, b) DMR on D20. The p r i n c i p l e s of deuteron magnetic resonance of D20 are the same as those o u t l i n e d i n the l a s t s e c t i o n , except that i n deuteron s i t e s , although the major c o n t r i b u t i o n to the e l e c t r i c f i e l d g radient i s from the i n t r a -molecular 0-D bond, there can be c o n t r i b u t i o n s which are of i n t e r m o l e c u l a r o r i g i n such as the charge d i s t r i b u t i o n i n the v i c i n i t y of the p o l a r head groups as w e l l as other o r i g i n s . Furthermore, a decrease of w i t h i n c r e a s i n g hydrogen-bond strength has been observed (44) . Therefore the quadrupole c o u p l i n g constant could be d i f f e r e n t from s i t e to s i t e . Further c o m p l i c a t i o n a l s o a r i s e s due to chemical exchange between deuterons i n water molecules i n d i f f e r e n t s i t e s . I f the chemical exchange 22 r a t e i s much more r a p i d than the quadrupole s p l i t t i n g , the observed s p l i t t i n g i s a weighted average over the d i f f e r e n t s i t e s and i s given by (36, 45-46) Av = I ^ P ^ S J {2.24} where P^ i s the p r o b a b i l i t y (defined by the f r a c t i o n of n u c l e i i n s i t e i ) that a nucleus i s i n s i t e i w i t h a c h a r a c t e r i s t i c quadrupole c o u p l i n g con-i i st a n t VQ and order parameter S^. The l a r g e s t c o n t r i b u t i o n to V ^ i s assumed to come from the i n t r a m o l e c u l a r 0-D bond (44) , which means that V ^ 1 remains approximately the same f o r the d i f f e r e n t s i t e s . I f V Q 1 does not vary s i g n i f i c a n t l y w i t h temperature e i t h e r , then, as shown by eqn. {2.24}, the measured s p l i t t i n g s are approximately p r o p o r t i o n a l to an average order parameter <S> = Z.P.S. {2.25} wi t h a constant of p r o p o r t i o n a l i t y equal to V ^ - ^ Q 1 * Thus Av can then be w r i t t e n as Av = l l . P . V . V l = IvJlE.P.S.I {2.26} 1 l x Q x 1 1 Q' 1 x x x 1 which can provide u s e f u l i n f o r m a t i o n f o r a q u a l i t a t i v e i n t e r p r e t a t i o n of the quadrupole s p l i t t i n g s of the f i r s t order DMR spectrum of water system. 2.4. Proton Magnetic Resonance For the purpose of comparing the r e s u l t s of B^O i n the p r o t i a t e d EYL/^O sample ( i n 22% water by weight) obtained through DMR technique to those obtained from the proton magnetic resonance of EYL i n the same sample, a b r i e f - r e v i e w of.PMR theory i s presented here. The t o t a l Hamiltonian of N protons system i s given by (39) 23 H = H z + H d •} {2.27> where the f i r s t term is the Zeeman energy and the second term i s the sum of the pairwise interaction energy between two magnetic dipoles y. and y,. 3 k When the protons in the N-body spin system form small groups within which the proton separations are distinctly smaller than those between two neighbouring groups, to a f i r s t order approximation one may consider such a group as an isolated system and calculate i t s energy levels in the presence of an applied f i e l d H o > treating the rest of the protons in the system as being a perturbation on these energy levels, because the dipole-dipole interaction decreases rapidly with distance, a) A two-spin system For a pair of strongly coupled protons on a a-CH^ group in an other-wise deuterated hydrocarbon chain in a sample oriented at an angle 6, and assuming that the la t t i c e is r i g i d and dipolar broading by the neighbouring protons i s neglected, solution of eqn. {2.2-7} gives a doublet of i n f i n i t e l y sharp peaks centered about the Larmor frequency U J q = yH Q with separation between them given by each of which are broadened by the dipolar interaction with the neighbouring dipoles. The dipolar energy levels of the two-spin system oriented at 6 and the corresponding f i r s t order spectrum are given in Fig. 5. For a powder sample, a similar argument as that in Section 2.3 on DMR (47-48) w i l l give a well-known Pake doublet, which is identical.with that A(JJ = TT (l-3cos 2 0 ) {2.28} 2 3 r 24 Zeeman Dipolar Coupling E = YhH o m = -1 E = -YhH m = +1 2.2 (1 - 3cos"6) / 3 4r (a) » = 0 v " TTa ~ 3 c O S 6> 2 2 + ^ - ( 1 - 3cos29) 4r r r Figure 5. (a) Zeeman energy (left) and dipolar coupling (right) of the trip l e t states of a two-spin system oriented at an angle 6. (b) The f i r s t order spectrum of the two-spin system oriented at 0 whose energy levels are given by (a). The dashed lines are the unbroadened resonance lines and the solid curves are dipolar broadened doublets. 25 3 "Y^ri shown i n F i g . 4 w i t h the peak s e p a r a t i o n r e p l a c e d by AOJ = -z -^—Is I, where L J Hn r S i s the component of the second order, symmetric, t r a c e l e s s order rlri parameter tensor along the H-H d i r e c t i o n . b) PMR on a p r o t i a t e d l i p i d system For a perdeuterated l i p i d system, the a-deuterium NMR s i g n a l i s e a s i l y d i s t i n g u i s h e d from the other deuteron resonances , cdue_to th e . l a r g e a-deuteron quadrupolar s p l i t t i n g . In c o n t r a s t , i n a non-deuterated l i p i d system i n l i q u i d c r y s t a l l i n e phase, the intra-methylene and inter-methylene proton d i p o l a r i n t e r a c t i o n s are comparable, and are much l a r g e r than the range of proton chemical s h i f t s . Consequently, the s p i n systems have complex l i n e -shapes (30). For i n s t a n c e , f o r p h o s p h o l i p i d b i l a y e r v e s i c l e s the observed PMR lineshape i s a s u p e r p o s i t i o n of L o r e n t z i a n curves (49-50), and f o r the l a m e l l a r b i l a y e r i n l i q u i d c r y s t a l l i n e mesophase the PMR lineshape i s c h a r a c t e r i z e d by extremely broad wings w i t h a s i n g u l a r i t y at the c e n t r a l (Larmor) frequency (51). Theories have been developed by Bloom et a l (52) and Wennerstrom (53) to e x p l a i n the lineshape observed i n l i q u i d c r y s t a l l a m e l l a r b i l a y e r . The l o g a r i t h m i c lineshape given by the theory i s shown i n F i g . 6, curve d. 26 Figure 6. PMR lineshapes f o r values of m„(0)/oj. (0) to be (a) 10 , - 2 - 1 (b) 10 , (c) 10 and (d) 1 (Bloom, et a l (52)). Spectrum (d) i s the type of absorption lineshape f i r s t observed by Lawson and F l a u t t (51). Chapter 3 Experimental 3.1. The M a t e r i a l s a) The D 20 (99.7% enrichment) was purchased from Merck Sharpe and Dohme (Montreal). b) The p r o t i a t e d EYL (egg y o l k l e c i t h i n ) , type 3E was purchased from Sigma Corporation and used without f u r t h e r p u r i f i c a t i o n . 3.2. Samples P r e p a r a t i o n a) To get r i d of the water i n the EYL, we d i s s o l v e d the m a t e r i a l i n benzene and r e c r y s t a l l i z e d i t by blowing n i t r o g e n through the s o l u t i o n at room temperature a t a r a t e f a s t enough to minimize exposure to l i g h t and prevent i t from o x i d a t i o n . The sample was then kept under vacuum at room temperature overnight to e l i m i n a t e the l a s t traces of benzene. The p r o t i a t e d EYL sample was made by weighing the corresponding molar amount of V^O i n t o the sample tube which contained the d r i e d EYL. The sample was mixed by s t i r r i n g w i t h a sp a t u l a and sealed. I t was then wrapped i n t i n f o i l and kept i n a r e f r i g e r a t o r at -20°C. b) D e s c r i p t i o n of the p r e p a r a t i o n of E. C o l i sample has been given by Davis et a l (37). 3.3. NMR Apparatus a) The deuterium s p e c t r a were taken at 34.44 MHz i n a high r e s o l u t i o n superconducting s o l e n o i d s u p p l i e d by Nalorac, Inc., Corcord, C a l i f o r n i a w i t h a Bruker SXP4-100 NMR spectrometer. The t r a n s i e n t d i g i t i z a t i o n and averaging was accomplished w i t h a N i c o l e t 1090AR d i g i t a l o s c i l l o s c o p e i n t e r f a c e d to an I n t e l 8080A microprocessor-based data a c q u i s i t i o n system. The F o u r i e r transforms and moment c a l c u l a t i o n s were done w i t h a BNC-12 minicomputer. The 28 BNC-12 computer i s equipped w i t h a Diablo Disk Drive ( s e r i e s 31 s i n g l e density) and was used f o r storage and a n a l y s i s of the data. The Spectro-meter i s capable of p u t t i n g out a t r a i n of up to 4 RF pulses of c o n t r o l l e d amplitude and whose phases and lengths can be v a r i e d independently. A programable timer ( N i c o l e t 293 I/O c o n t r o l l e r ) i n t e r f a c e d to the computer was used to automate the NMR measurements. Thus the t r i g g e r i n g of the i n d i v i d u a l RF p u l s e s , the spacing between them, and the r e p e t i t i o n r a t e were computer c o n t r o l l e d . The probe c o n s i s t e d of a RF c o i l i n t o which the sample was i n s e r t e d . The c o i l (together w i t h the sample) was enclosed i n a c y l i n d r i c a l copper oven. Cooling was achieved by blowing coolant ( l i q u i d nitrogen) through the oven. The temperature gradient across the sample volume was estimated to be considerably l e s s than 1°C. 3.4. NMR Measurements The conventional method of o b t a i n i n g NMR s p e c t r a c o n s i s t s of applying a 90° RF pulse and then F o u r i e r transforming the f r e e i n d u c t i o n decay (FID). During the a p p l i c a t i o n of the RF p u l s e , the r e c e i v e r of the NMR spectrometer gets saturated and a c e r t a i n length of time ( c a l l e d the recovery time or dead time) has to elapse before i t returns to i t s normal operating c o n d i t i o n s . Therefore the e a r l y p a r t of the FID cannot be observed due to the recovery time of the r e c e i v e r . The usual method, d e l a y i n g data a c q u i s i t i o n u n t i l the r e c e i v e r has recovered, r e s u l t s i n l o s s of the i n f o r m a t i o n contained i n the e a r l y part of the FID (which i s very important f o r wide l i n e s ) and i n v a r i a b l y leads to d i s t o r t i o n of the spectrum (54). I t a l s o introduces f i r s t order phase s h i f t s and a p o o r l y defined base l i n e . To circumvent t h i s problem the NMR spectra were obtained using the s o l i d echo by the method of Davis et a l (54). This method c o n s i s t s of applying a 90° pulse (whose phase 29 i s 0° w i t h respect to the reference frequency) followed by another 90° pulse whose phase i s s h i f t e d by 90° w i t h respect to the f i r s t pulse at a time x ( t y p i c a l l y 100-200 Us) l a t e r . An echo i s formed at 2x due to the r e f o c u s i n g of the nuclear magnetization. By F o u r i e r transforming the echo s t a r t i n g at t = 2T the f u l l spectrum i s obtained. To enhance s i g n a l to n o i s e , an a l t e r n a t i n g phase pulse sequence • ( 9 O O ) E _ O - T - ( 9 O O ) e _ g o o - T r - ( 9 O O ) e _ 1 8 O o - T - ( 9 O O ) 0 _ 9 O o was used, where the second quadrupolar echo pulse sequence was phase s h i f t e d by 180° w i t h respect to the reference frequency. F i v e hundred scans f o r DMR on D^O and 1000 scans f o r PMR on EYL are s u f f i c i e n t to o b t a i n good s i g n a l averaging. The s i g n a l s were detected i n quadrature. 30 Chapter 4 The Results 4.1. DMR Results For convenience, the egg y o l k l e c i t h i n / D ^ O i n 22% (by weight) of water (D 20) , egg y o l k l e c i t h i n / ^ O and E. C o l i / ^ O i n excess water are r e s p e c t i v e l y abbreviated to EYL/22%WD20, EYL/EXCD20 and EC/EXCD20. A l l the r e s u l t s mentioned here w i l l be discussed more f u l l y i n Chapter 5 where the f i g u r e s are placed. a) Quadrupole s p l i t t i n g s When l i n e broadening i s absent, the quadrupole s p l i t t i n g s can be obtained from the peak p o s i t i o n s i n the DMR s p e c t r a . However, i n the pres-ence of d i p o l a r broadening, the p o s i t i o n s of the maximum i n t e n s i t y are no longer c o i n c i d e n t w i t h the p o s i t i o n s of the 90° edges of the quadrupole powder p a t t e r n (Appendix A). In t h i s s i t u a t i o n , the " s p l i t t i n g s " are e m p i r i c a l l y defined as the average over two measurements; namely, the sep a r a t i o n between the peaks and the se p a r a t i o n between two p o i n t s on the outer edges of the powder p a t t e r n at 75% of the peak amplitude of the experimental DMR s p e c t r a . By the method described above, quadrupole " s p l i t t i n g s " were measured d i r e c t l y from the DMR spe c t r a of the EYL/22%WD20 sample, the r e p r e s e n t a t i v e s of which are shown i n F i g . 9. In the region between -6°C and -10°C, the lineshapes are s i n g l e t s , and only the l i n e widths at half-maximum i n t e n s i t y were measured. The r e s u l t s of the measure-ments are presented i n F i g . 22. b) E v a l u a t i o n of the DMR s p e c t r a i n terms of t h e i r moments The values of the moments M n (Appendix A) can r e a d i l y be c a l c u l a t e d from th_e DMR s p e c t r a such as those shown i n F i g . 7-9. Although i n the 31 experiment the f i r s t eight mements were routinely calculated by integrating over the spectrum, only the f i r s t few moments of the spectra (M_ to M^ ) are reliable in the liquid crystalline phase, and at lower temperature only M Q - M ^ can be accurately determined, because the accuracy of the higher moments depends c r i t i c a l l y on the f i d e l i t y of the broad part of the spectrum. The results of the calculations of -the integrated signal intensities , the f i r s t , the second and the' fourth moments for the three samples (namely, the EYL/22%WD20, the EYL/EXCD^O and the EC/EXCD^O) are given in Fig. 10-13 and Fig. 16-18. In order to compare the results of quadrupole splittings obtained by direct measurements from the DMR spectra of the EYL/22%WD20 sample (Fig. 22), the f i r s t moment M^  (which gives the average splitting) as given by eqn. A.6 in Appendix A, namely, 4 it- . M = —T <Av> 1 3/3 was used to calculate the quadrupole splittings i n the DMR spectra of the EYL/22%WD20 i n the region from 5°C to 25°C. Below 5°C line broadening may become too severe to render validity to the equation given above. The results of the calculations are shown in Fig. 22. + 4.2. PMR Results The second and the fourth moments were calculated from the PMR spectrum of the EYL/22%WD20 system, and the results are shown in Fig. 23. 4.3. Sources of Error There is no severe problem of distortion in the DMR spectrum of D20 due to 90° pulse length, because the spectral width of D20 is small (as 32 compared to that of PMR and DMR spectrum of the l i p i d s ) . For our PMR experiment, we obtained a pulse length of 0.9 microsecond f o r the 90° p u l s e , which i s short enough to r o t a t e a l l magnetizations i n a l l p a r ts of the NMR spectrum through the same angle. One of the s i g n i f i c a n t systematic e r r o r i n the DMR experiment f o r B^O was magnetic inhomogeneity, which cannot be t o t a l l y e l i m i n a t e d . Another source of e r r o r arose from image formation due to ina c c u r a t e quadruture phasing. For i n s t a n c e , the asymmetries i n the base of the s p e c t r a of the EC/EXCD20 at -1°C to 10°C as shown i n F i g . 7 were due to the formation of images. This work w,as done i n c o l l a b o r a t i o n w i t h Dr. A. L. Mackay. 33 Chapter 5 Disc u s s i o n And Conclusion 5.1. DMR Spectra of D 20 a) Spectra of D 20 i n the EYL/EXCD20 and EC/EXCD20 systems As shown i n F i g . 7-8, the DMR sp e c t r a of D 20 i n the EYL/EXCD20 (whose water concentration l i e s between 50% and 60%) are q u a l i t a t i v e l y s i m i l a r to those i n the EC/EXCD20 (whose water concentration > 90%). U n l i k e the DMR sp e c t r a of D 20 i n the EYL/22%WD20, which c o n s i s t of powder p a t t e r n doublets w i t h quadrupole " s p l i t t i n g s " i n the order of 1 kHz i n the temperature range between -3°C and 25°C as shown i n F i g . 9, the quadrupole " s p l i t t i n g s " i n these s p e c t r a are absent. For T >_ -2°C, the s p e c t r a of the EYL/EXCD20 c o n s i s t of an extremely narrow s i n g l e t c h a r a c t e r i s t i c of i s o t r o p i c water. L i k e the EYL/EXCD 20, the s p e c t r a of the EC/EXCD20 f o r T _> -1°C a l s o c o n s i s t of a very narrow s i n g l e t . Q u a n t i t a t i v e l y , the s p e c t r a of D 20 i n the EYL/EXCD20 and that i n the EC/EXCD20 are d i f f e r e n t . The observed l i n e widths of the s p e c t r a of the EYL/EXCD20 range from 81 Hz at -2°C to 110 Hz at 20°C, wh i l e those of the EC/EXCD20 stay constant at 30 Hz f o r T >^  -1°C. Thus,Lthe l i n e widths of the DMR s p e c t r a of the EC/EXCD20 are smaller than those of the EYL/EXCD20 by a f a c t o r of 3 to 4. The r e d u c t i o n i n quadrupole s p l i t t i n g s from 1 kHz f o r the s p e c t r a of the EYL/22%WD20 ( F i g . 9) to the zero s p l i t t i n g i n the sp e c t r a of the EYL/EXCD20 ( F i g . 8) can be understood as f o l l o w s : Case I . Assume that the system i s i n water concentration C j< the maximum hy d r a t i o n (40%). A l l concentrations are expressed by grams of water per gram of l i p i d and water mixture. For a crude approximation, the two-s i t e model proposed by Wennerstrom et a l (34) i s adopted f o r the system. The two 34 s i t e s correspond to the trapped water (water which i s i n c o r p o r a t e d between b i l a y e r s , but i s weakly a s s o c i a t e d to the p o l a r head groups (34b)) and bound water (water which i s s t r o n g l y bound to the surfaces formed by the p o l a r head groups). The s p l i t t i n g i n the DMR spectrum of trapped water cannot be r e -solved, w h i l e that f o r the bound water has a value i n the order of s e v e r a l kHz. Since the maximum water incor p o r a t e d between b i l a y e r s i n the l i q u i d c r y s t a l l i n e l a m e l l a r phase i s 40% by weight (16), a l l the water i n the EYL/22%WD20 system i n l i q u i d c r y s t a l l i n e l a m e l l a r phase i s incorporated i n t o the b i l a y e r s , and there i s p r a c t i c a l l y no bulk water (f r e e i s o t r o p i c water that i s not i n c o r p o r a t e d between b i l a y e r s and exchanges s l o w l y w i t h water that i s i ncorporated between b i l a y e r s (32)). In the l a m e l l a r phase where C _< 40%, the b i l a y e r s s w e l l i n two ways (namely, increase i n thickness of the b i l a y e r s and moving-apart of the p o l a r head groups) as water i s added, so that a l l the water, trapped or bound, i s increased. However, i n our crude model, we assume that the change i n the bound water i s n e g l i g i b l e so that only the amount of trapped water i s increased when the t o t a l water content of the system i s increased (54). On the b a s i s of t h i s model, and, s i n c e the exchange between the two s i t e s i s f a s t as i n d i c a t e d by the ex i s t e n c e of only a s i n g l e powder p a t t e r n ( r a t h e r than two or more superimposed spectra) shown i n F i g . 9a, the quadrupole s p l i t t i n g A v i n the observed DMR spectrum of water deuteron i n the EYL/22%WD„0 i s given by eqn. {2.24}: where A v « A v ; W and W are the mole f r a c t i o n s of the t°tal D-0 content of t b b 2 the system and bound water, r e s p e c t i v e l y ; Av , = V ^ s , i s the c h a r a c t e r i s t i c b Q b s p l i t t i n g i n the DMR spectrum of D„0 when i t i s s t r o n g l y bound to the i n t e r f a c e I - — ^ - A v + T ^ A V ^ | 1 W t W b 1 W - W, W, A v = {5.1} i=t,b 35 formed by p o l a r head groups, w h i l e A v f c = VQ t^ t 1 S the c h a r a c t e r i s t i c s p l i t t i n g i n the DMR spectrum of D^ O when i t i s i n the trapped water s i t e . The above equation f o r A v demonstrates t h a t , f o r low water c o n c e n t r a t i o n , the observed s p l i t t i n g i s predominantly that of Av , , s i n c e A v « Av , . As more water i s b t b added to the system, the f r a c t i o n a l c o n t r i b u t i o n from Av , ( Av , may depend on b b water c o n c e n t r a t i o n , but here we assume i t to be constant) becomes s m a l l e r . Consequently, an o v e r a l l reduction i n the s p l i t t i n g of the observed spectrum occurs w i t h i n c r e a s i n g water concentration u n t i l the s p l i t t i n g cannot be resolved. In a more r e a l i s t i c model, more than one type of bound s i t e should be assumed. Case I I . For C > 40%, two phases of D^ O are formed: the maximum amount of water that i s incorpo r a t e d i n t o the b i l a y e r s , and bulk or i s o t r o p i c excess free water i n a separate phase exchanging sl o w l y w i t h the maximum water of hydr a t i o n . In t h i s s i t u a t i o n , the observed spectrum i s a s u p e r p o s i t i o n of the two s p e c t r a a r i s i n g from the water w i t h i n the b i l a y e r s and the bulk water. The second moment of the observed spectrum i s given by (Appendix B) r.i H „, EX M 2 " 1 + A E X / A H + 1 + V / A E X { 5 ' 2 ' H EX Here and are r e s p e c t i v e l y the second moments of the s p e c t r a of the water of maximum h y d r a t i o n and the excess f r e e water. A^ and A ^ are the i n t e g r a t e d i n t e n s i t i e s of the sp e c t r a of the water of maximum h y d r a t i o n f H and of the excess f r e e water f„„: EX 00 AJJ = / f H ( f l ) dft {5.3} \ x = / f£XW) « {5.4} 36 For C -> 0 0, A_ x -»• 0 0 and, consequently, eqn. {5.2} i s reduced to M 2 = M 2 E X i . e . the second moments and the l i n e widths are mainly that of the i s o t r o p i c excess free water as observed i n the DMR spectrum of the EC/EXCD20 system ( F i g . 7, 13 and 15). Notice t h a t the second moments and the l i n e widths f o r T 2_ -1°C shown i n F i g . 13 and 15 are mainly those of the magnet inhomogeneity (~30 Hz ± 20%). For T < -2°C, the DMR spectrum of D 20 i n the EC/EXCD20 c o n s i s t s of a broad and s t r u c t u r e l e s s l i n e whose l i n e width v a r i e s from 240 to 700 Hz. The spectrum f o r the EYL/EXCD20 i n the region T <_ -3°C i s a l s o broad and i t s l i n e width v a r i e s from 380 to 400 Hz. Noti c e that the enormous drop i n the DMR s i g n a l i n t e n s i t y shown i n F i g . 10 i s due to disappearance of the DMR s i g n a l of a l l the excess free water as a r e s u l t of the f r e e z i n g out of t h i s water ( t h i s w i l l be discussed l a t e r on). The spectrum of the frozen excess free f D 20 (the i c e ) i s too broad to be detectable . Thus, the DMR spec t r a observed i n t h i s temperature region a r i s e s o l e l y from the water i n c o r p o r a t e d between the b i l a y e r s . The observed broad l i n e s can be understood i n terms of eqn. {5.2}. When the excess free water has been frozen out, A_„ = 0, and, because the i c e s i g n a l cannot be detected, the equation i s e f f e c t i v e l y reduced to M 2 - M 2 H H EX M 2 i s expected t o be l a r g e r than M 2 , and consequently a broader l i n e width t The i c e spectrum has a quadrupole s p l i t t i n g of 160 kHz. To observe the e n t i r e DMR spectrum of p o l y c r y s t a l l i n e i c e , a s p e c t r a l width of at l e a s t 300 kHz i s required. The s p e c t r a l width used f o r my water DMR experiments was only 10 kHz f o r the EC/EXCD 0 and the EYL/EXCD.O systems, and 20 kHz f o r the EYL/22%WD 0. z 2 37 Figure 7. Deuterium nuclear magnetic resonance s p e c t r a of D 0 i n the E. C o l i / D 2 0 system i n excess water 90% by weignt) obtained at (a) 20°C, (b) 4°C, (c) -1°C, (d) -2°C and 34.42 MHz, 300 scans using the quadrupolar echo and quadrature d e t e c t i o n method. R e p e t i t i o n r a t e = 1 second, s p e c t r a l w i t h = 10 kHz. A l l the s p e c t r a are 5 kHz p l o t . EYL/D 0 IN EXCESS WATER 5 kHz PLOT 38 Figure 8. Deuterium nuclear magnetic resonance spectra of D^ O in the egg yolk lecithin/^O system in excess water concentration (>>40%) obtained at (a) 18°C, (b) 4°C, (c) -2°C, (d) -3°C and 34.42 MHz, 500 scans using the quadrupolar echo and quadrature detection method. Spectral width = 10 kHz. A l l the spectra are 5 kHz plot. 39 t t below the f r e e z i n g p o i n t . The absence of s t r u c t u r e i n the observed spectrum i n t h i s r e gion may be caused by inhomogeneous broadening; the presence of i c e i n t h i s temperature region may have created d i f f e r e n t regions i n t o which the unfreezable water i s d i s t r i b u t e d . Water i n d i f f e r e n t regions exchanges slow-l y . Each of these regions may have the same T^, but the order parameter of the 0-D bond d i r e c t i o n i n the D^ O molecules v a r i e s from r e g i o n to r e g i o n , so that water i n d i f f e r e n t regions gives r i s e to DMR spectrum w i t h d i f f e r e n t quadrupole s p l i t t i n g . A s u p e r p o s i t i o n of a l l of these s p e c t r a a r i s i n g from water i n a l l the regions y i e l d s a spectrum whose s t r u c t u r e cannot be res o l v e d . b) Spectra of D 20 i n the EYL/22%WD20 For temperatures between 25°C and -3°C, the DMR spectrum of D 20 i n the EYL/22%WD20 c l e a r l y e x h i b i t s the w e l l known doublet powder p a t t e r n w i t h quadrupole " s p l i t t i n g " v a r y i n g from 1.3 kHz f o r the system i n l a m e l l a r l i q u i d c r y s t a l l i n e phase ( L a ) to 0.72 kHz at -3°C. The quadrupole s p l i t t i n g ( i n the order of 1 kHz) i n the doublet powder p a t t e r n obtained f o r t h i s system i s much l e s s than that t y p i c a l l y found (55) f o r p o l y c r y s t a l l i n e i c e and hydrates ( i n the order of 150 kHz). The l a r g e r e d u c t i o n i n the s p l i t t i n g i s an i n d i c a t i o n of the greater time averaging of the e l e c t r i c f i e l d g radients due to a r a p i d but a n i s o t r o p i c r e o r i e n t a t i o n of the water molecules a s s o c i a t e d w i t h the l e c i t h i n p o l a r groups w h i l e d i f f u s i n g along the i n t e r f a c e . The v a l i d i t y of St r u c t u r e i n the sp e c t r a i n the region I < -2 C f o r the EC/EXCT^O and I < -3 C f o r the EYL/EXCD20 i s expected, s i n c e , as the trapped water decreases, and, as w i l l be shown l a t e r , the squeezed out water i n these systems i s frozen as soon as i t i s squeezed out, eqn. {5.1} i n t h i s chapter p r e d i c t s that the DMR s i g n a l of the bound water, which has c h a r a c t e r i s t i c s p l i t t i n g of s e v e r a l kHz, should predominate. 40 t h i s i n t e r p r e t a t i o n i s f u r t h e r supported by the work of F i n e r et a l (32) and C h a r v o l i n et a l (56). The presence of sharp edges i n the doublet powder patterns shows that the asymmetry parameter n = 0. The growth of the s i g n a l a t the centre of the spectrum i s d i s c e r n i b l e at -3°C, which becomes more pronounced i n the spectrum at -4°C. The spectrum at -5°C i s d e f i n i t e l y a s u p e r p o s i t i o n of two s p e c t r a : one i s the powder p a t t e r n doublet due to the t i g h t l y bound water and the other i s c h a r a c t e r i s t i c of free i s o t r o p i c water. In the spectrum at -10°C, the i d e n t i t y of the powder p a t t e r n doublet i s l o s t . This spectrum c o n s i s t s of a narrow s i n g l e t w i t h an i n t r i n s i c l i n e width of 90 Hz. The sharp c e n t r a l component of the spectrum at -5°C a r i s e s from the i s o t r o p i c "free", water which has been squeezed out from the b i l a y e r s . As shown i n F i g . 9 g - i , the zero degree shoulders and 90 degree edges are absent i n the powder patterns f o r T £ -15°C. The s p l i t t i n g s i n these s p e c t r a range from 160 to 360 Hz, which are very smail values. However, due to l a c k of experimental c e r t a i n t y about the o r i g i n of the broadening mechanism, we s h a l l not attempt t o i n t e r -p ret the observed s p e c t r a i n t h i s region. Note that the drop i n s i g n a l to noise i n the spectrum f o r the EYL/22%WD20 at -25°C i s not due t o drop i n the DMR s i g n a l i n t e n s i t y as can be seen from F i g . 11, but due to the broadening of the l i n e . The area of the spectrum M q ( i n t e g r a t e d s i g n a l i n t e n s i t y ) , which i s p r o p o r t i o n a l to the s p i n p o p u l a t i o n , should be constant except f o r a small change i n the Boltzmann f a c t o r , because the s p i n p o p u l a t i o n i s unchanged during the experiment. Figure 9. DMR sp e c t r a of i n the Ei*L/22%tfD 0 obtained at (a) 25°C, (b) 4°C, (c) -3°C, "(d) -4bC, (e) -5°C, ( f ) -10°C, (g) -15°C, (h) -20°C, ( i ) -25°C and 34.46 MHz, 1000 scans accumulation f o r s i g n a l averaging using the quadrupolar echo and quadrature d e t e c t i o n method. Pvepetition r a t e = % second, s p e c t r a l width = 20 kHz. A l l the sp e c t r a are 5 kHz p l o t . Spectrum i n Figure 10-j i s the same as that i n Figure 1 0 - i except i t i s a 10 kHz p l o t . 42 5.2. The Moments of the DMR Spectra a) The i n t e g r a t e d s i g n a l i n t e n s i t i e s (M q ) of the DMR s p e c t r a of D^ O i n the EYL/EXCD20 and EC/EXCD 0 The i n t e g r a t e d s i g n a l i n t e n s i t y (M q) VS T f o r D 20 i n the two systems are shown i n F i g . 10. I n the temperature range defined by T > -2°C, the s i g n a l i n t e n s i t y of D 20 i n both systems increases as temperature decreases. This i s p a r t l y due to the Boltzmann f a c t o r ( s i g n a l i n t e n s i t y i s p r o p o r t i o n a l to the population d i f f e r e n c e i n s p i n s t a t e s , which i s i n turn p r o p o r t i o n a l to the Boltzmann f a c t o r ) , and p a r t l y due t o the f a c t that s i g n a l i n t e n s i t y i s s e n s i t i v e to change i n Q of the RF c o i l c o n t a i n i n g the sample, because as T decreases, Q i n c r e a s e s , so does the s i g n a l i n t e n s i t y . I t i s a l s o s e n s i t i v e to the change i n the d i e l e c t r i c constant of the water and the l i p i d s . The enormous drop i n i n t e n s i t y o f the DMR s i g n a l of D 20 i n the EYL/EXCD20 i s due to f r e e z i n g out of a l l the excess f r e e water i n the system. Note t h a t , w i t h i n experimental e r r o r , the M Q VS T i s l i n e a r f o r T >^  -2°C, thus i t i s pe r m i s s i b l e to e x t r a p o l a t e the M Q VS T l i n e down to T = -3°C. The r a t i o of the value of M q a c t u a l l y measured at T = -3°C to the e x t r a p o l a t e d value of M Q at the same temperature i s 0.15. This means that the water that remained un-frozen at -3°C comprises 15% of the o r i g i n a l t o t a l water content of the sample. Furthermore, f o r T <_ -3°C, there i s systematic and gradual reduction i n s i g n a l i n t e n s i t y as temperature decreases, as shown by the smooth curve on the l e f t of the M q vs T d i s c o n t i n u i t y i n F i g . 10. That i s an i n d i c a t i o n of f u r t h e r f r e e z i n g out of water. As w i l l be s u b s t a n t i a t e d i n Secti o n 5.2d, i t i s the f r e e z i n g out of the water squeezed out from the b i l a y e r s that leads to the gradual decrease i n the s i g n a l i n t e n s i t y i n the region below the f r e e z i n g p o i n t as temperature decreases. The temperature dependence of the DMR s i g n a l i n t e n s i t y of D„0 i n the 43 44 EC/EXCD20 i s s i m i l a r to that i n the EYL/EXCD^O. The excess f r e e water freezes at -1°C. This f r e e z i n g p o i n t and that of 1>20 i n the EYL/EXCD 20 (which freezes at -2°C) d i f f e r only by 1°C. The f r a c t i o n of water that remained unfrozen at -2°C i s 16% of the t o t a l water. b) The i n t e g r a t e d i n t e n s i t y of the DMR spectrum of D 20 i n the EYL/22%WD20 system The general temperature dependence of the DMR s i g n a l i n t e n s i t y of D 20 i n the EYL/22%WD20 f o r T >_-10°C ( F i g . 11) i s s i m i l a r to those mentioned e a r l i e r . There i s an enormous drop i n i n t e n s i t y as temperature changes from -10°C to -15°C, i n d i c a t i n g that the water squeezed out from the b i l a y e r s freezes between -10°C and -15°C. The f r a c t i o n of water remaining unfrozen at -15°C i s 0.50. Note that the f r e e z i n g p o i n t of the squeezed out i s o t r o p i c water i n t h i s system i s 8°C lower than that of D 20 i n the other two systems. c) The f i r s t and the second moments of the DMR s p e c t r a of D 20 i n the EYL/EXCD20 and EC/EXCD20 systems The temperature dependence of M 2 of the DMR spectrum of D 20 i n the EYL/EXCD20 i s shown i n F i g . 13. To r a t i o n a l i z e the observed M 2 f o r the system EYL/EXCD 20, the model proposed i n Appendix B i s assumed f o r the d i s t r i b u t i o n of water i n t h i s system. Since water concentration i n the EYL/EXCD20 l i e s between 50% and 60% ( r e c a l l that water concentration c i s expressed as c = W/(W+L), where L and W are r e s p e c t i v e l y the weight of the l i p i d s and of the water) , a maximum amount W of the water which gives W17lJ/(W +L) = 0.4 i s incorpo r a t e d i n t o the l i p i d b i l a y e r s i n l i q u i d c r y s t a l l i n e l a m e l l a r phase, forming the water of maximum h y d r a t i o n . The r e s t of the water forms the i s o t r o p i c excess f r e e water e x i s t i n g i n a separate phase and exchanging sl o w l y w i t h the water incor p o r a t e d between the b i l a y e r s . In such a model, the second moment, M 2(EYL/EXCD 20), of the observed NMR spectrum of the. water i n the EYL/EXCD20 i s 8 -x10 _ 6h 2h i I i i i i u -25 -10 0 10 25 T ( ° C ) Figure 11. Temperature dependence of the integrated signal intensity, M o of deuteron magnetic resonance of DO in the EYL/22%WDo0. 46 given by eqn. {B.l} i n Appendix B: M 2(EYL/EXCD 20) = f M ^ + (1 - f ) M 2 H {5.5} where i s the f r a c t i o n of the water o u t s i d e the b i l a y e r s . Here f i s assumed to be EX H independent of temperature. M 2 , M 2 are r e s p e c t i v e l y the second moments of the i s o t r o p i c excess free water and the water of maximum h y d r a t i o n as defined i n Appendix B. Assume that = pM 2 (EYL/22%WD20) {5.7} where M2(EYL/22%WD20) i s the second moment of the DMR spectrum of D 20 i n the EYL/22%WD20, and that p i s independent of temperature. S u b s t i t u t i n g eqn. {5.7} i n t o eqn. {5.5} gives M 2(EYL/EXCD 20) = f M ^ + (1 - f)pM 2(EYL/22%WD 20) {5.8} EX Since the observed second moment M 2 of the free water i s v i r t u a l l y that of the EX magnet inhomogeneity, M 2 i s i n s e n s i t i v e to change i n temperature as can be seen i n the vs T p l o t f o r the EC/EXCD20 shown i n F i g . 13 where the spectrum i s dominated by the excess f r e e water f o r T _> -1°C. I f the model under c o n s i d e r a t i o n i s v a l i d , then, according to eqn. {5.8}, we expect to o b t a i n a s t r a i g h t l i n e f o r a p l o t of the M 2(EYL/EXCD 20) versus EX M2(EYL/22%WD20) w i t h slope (1 - f ) p and i n t e r c e p t fM2 . The p l o t of the M 2(EYL/EXCD 20) f o r T _> -2°C (represented by p o i n t s marked by o i n F i g . 13) against the M2(EYL/22%WD20) i n the same te m p e r a t u r e range (presented i n F i g . 18) i s shown i n F i g . 14 where the M2 (EYL/EXCD 20) i s indeed l i n e a r l y r e l a t e d to the M 2(EYL/22%WD 20). The slope and the i n t e r c e p t o f the s t r a i g h t l i n e obtained by 47 the p r i n c i p l e of l e a s t squares f i t to the experimental data are determined to be: (1 - f ) p = 0.073 ± 0.0036 {5.9} f M 2 E X = 0.63 x 10 6 s e c " 2 ± 0.038 x 10 6 s e c " 2 {5.10} mx Let W, be the amount of water found i n the bound s i t e , and W. be the b i s o maximum amount of water added to the EYL/22%WD20 to make i t f u l l y hydrated, mx i . e . W,-, = W, + W. i s the amount of water so th a t C = W_. T/( W™ + L) = 40%. FH b i s o FH FH The upper and lower bounds f o r the p r o p o r t i o n a l constant p can be c a l c u l a t e d on the b a s i s of the tw o - s i t e model proposed i n Se c t i o n 5.1a and the f o l l o w i n g assumptions: (a) The surface area per p o l a r head group stays constant when more water i s added to the system EYL/22%WD20, so that remained constant and that a l l the water added to t h i s system c o n t r i b u t e s only to the i s o t r o p i c s i t e . (b) There i s only one value of order parameter i n the bound s i t e , and thusoonly one value of s p l i t t i n g Av , f o r t h i s s i t e . The s p l i t t i n g i n the : D i s o t r o p i c s i t e i s zero. Consequently, the average o v e r a l l s p l i t t i n g s observed i n the systems EYL/22%WD20 and EYL/40%WD2O are r e s p e c t i v e l y given by eqn. {5.1}: W, W <Av> = A v 5 - ^ r - Av , = r f - A v , {5.11} + W22 b W22 b b i s o W W <Av> = A v = k Av , E -f- A v , {5.12} + wmx,, b W r a b b i s o Here W„. = W, + W.2 i s the t o t a l water i n the EYL/22%WDo0 and W22 ' i s the 22 b i s o 2 i s o mx i s o t r o p i c water between the b i l a y e r s . W „ T = W, + W. i s the t o t a l amount of J FH b i s o 48 water i n the EYL/40%WD20. The second moment M 2 as given by eqn. A.3 or A.6 i n Appendix A can be w r i t t e n as M 2 = K <(AV) 2> = K {Av} 2 {5.13} where K i s a constant. Eqn. {5.13} gives M2(EYL/22%WD20) and = M2(EYL/40%WD2O) the f o l l o w i n g expressions: W M_(EYL/22%WDo0) = K {r^~ Av,} 2 {5.14} 2 Z W 2 2 b M „ H = K & - AV. } 2 {5.15} 2 FH b From eqn. {5.7}, p i s given by M 2 H _ W22 2 P " M„(EYL/22%WDo0) ~ ( W~7 } {5.16} Z Z r r i Note that as assumption (a) i s r e l a x e d , tends to increase w i t h i n c r e a s i n g c, so that p given by eqn. {5.16} represents the lower bound, P-J^J ^ O R P' From the expressions f o r the water concentrations of the systems EYL/22%WL>20 and EYL/40%WD2O: W ? ? WFH = 0.22 and T T , _ = 0.4 {5.17} W22 + L W f r + L W„/W was found to be 0.42 and p,. = 0.18. The assumption that the water added Z Z FH i b to the EYL/22%WD20 does not c o n t r i b u t e to the second moment, but i t makes the average environment more i s o t r o p i c puts an upper bound on p so that 0.18 < p < 1 {5.18} Eqn. {5.9} and i n e q u a l i t y {5.18} give 0.59 ± 0.02 < f < 0.93 ± 0.0036 {5.19} and from {5.10} and {5.9}, the upper and the lower bounds on M 2 E X are given by: (0.68 ± 0.014) x 1 0 6 s e c " 2 < M E X < (1.07 ± 0. 0 7 4 ) x l 0 6 s e c " 2 {5.20} From eqn. {5.5} or {5.8}, i t i s c l e a r that the increase i n f o r the EYL/EXCD2O w i t h i n c r e a s i n g temperature ( t h i s i s a l s o observed i n EYL/22%WD20 as shown i n F i g . 18), a behaviour unexpected when compared w i t h most s t u d i e s of motional narrowing, i s due to l i p i d - w a t e r i n t e r a c t i o n , which imposes a c o n s t r a i n t on the ordering of the time average efg at the l i p i d - w a t e r i n t e r f a c e . The ordering e f f e c t i s assumed to be such that at low temperatures, the p r i n c i p a l a x i s e f g , which i s along the 0-D bond d i r e c t i o n , assumes an o r i e n t a t i o n w i t h -r respect to the b i l a y e r normal n, g i v i n g an average 0-D bond order parameter which i s smaller than that f o r the high-temperature c o n f i g u r a t i o n . As temperatur< changes from -2°C to -3°C, there i s a sharp increase i n the value of M 2 by a f a c t o r of three, corresponding to the broadening of the DMR absorption l i n e by the mechanism e l u c i d a t e d i n Section 5.1a, namely, the sudden disappearance of the domination by the i s o t r o p i c excess f r e e water. The v a r i a t i o n of the M 2 of the DMR spectrum of D 20 i n the EC/EXCT>20 as a f u n c t i o n of temperature i s s i m i l a r to that f o r the EYL/EXCD 20, except that f o r T 2_ -1°C, the M 2 i s independent of temperature because the observed DMR s i g n a l i s dominantly that of the i s o t r o p i c excess f r e e water as discussed i n Sect i o n 5.1a. As shown i n F i g . 13, the value of M 2 i n the region T >^  -1°C f o r the EC/EXCD20, which i s dominantly that of the magnet inhomogeneity, i s found to 6 —^  6 —2 be 0.45 x 10 sec *" ± 0.15 x 10 sec , which i s smaller than the lower bound of the M 2 E X f o r the EYL/EXCD20 given by the i n e q u a l i t y {5.20}. This i s not FX unexpected, because M 2 depends on the sample and i t s geometry and the s t a t e of the magnet inhomogeneity i n the space occupied by the sample. Again, i n t h i s system, the d i s c o n t i n u i t y i n the M 2 occurs at the temperature at which the excess f r e e water i s frozen out. The f i v e - f o l d increase i n the M 2 as temperature changes from -1°C to -2°C r e f l e c t s the 51 Figure 13. Temperature dependence of the second moment M 2 of the DMR spectrum of D 20 i n the EYL/EXCD20 ( c i r c l e s ) and the EC/EXCD„0 ( t r i a n g l e s ) . 52 x10 M 2 ( EYL/22°/oWD 20) Figure 14. Second moments, M 2(EYL/EXCD 20), of the DMR spectra of D 20 i n the system EYL/EXCD20 p l o t t e d against the second moments, M 2(EYL/22%WD 20), of D 20 i n the EYL/22%WD20. The s o l i d l i n e i s the l e a s t squares f i t to the experimental data ( c i r c l e s ) where a two parameter f i t of the form M 2(EYL/EXCD 20) = f M 2 E X + (1 --f)pM2(EYL/22%WD-0) was used. 53 8 2 x10 L N 6 -0 -10 A A AAA i*r oooooooo O O O Q T Q O O 0 T 10 (°C) 20 Figure 15. Temperature dependence of the l i n e width Av of the DMR spectrum of D 20 i n the E. Coli/EXCD 20 ( c i r c l e s ) and . EYL/EXCD20 ( t r i a n g l e s ) . 54 15 -13 -X10 -T (°C) Figure 16. Temperature dependence of the f o u r t h moment M. of the DMR 4 spectrum of D^O i n the EYL/EXCD20 ( c i r c l e s ) and E. Coli/EXCD 90 ( t r i a n g l e s ) . 55 disappearance of the domination by the excess f r e e water due to f r e e z i n g out of the water. S i m i l a r remarks can be made about the temperature dependence of the f i r s t moment M ( F i g . 12) and the l i n e width Av x ( F i g . 15) of the D M R I 2 spectrum of the E Y L / E X C L > 2 0 and E C / E X C D 2 0 . d) The f i r s t and second moments of the D M R s p e c t r a of D 2 0 i n the E Y L / 22% W D 20 Comparison of the M 2 vs T f o r the E Y L / 2 2 % W D 2 0 w i t h that f o r the E Y L / E X C D 2 0 presented i n F i g . 13 and 18 shows that the values of the M 2 f o r the f i r s t system are 6 (at - 2 ° C ) to 9 (at 2 0 ° C ) times l a r g e r than those f o r the second. This d i f f e r e n c e i s expected, because when more water i s added to a system w i t h maximum h y d r a t i o n , the amount of i s o t r o p i c water i n c r e a s e s , and thus, as pointed out i n Sect i o n 5.1a and Appendix B, the average environment f o r the D 2 0 molecules becomes more i s o t r o p i c due to the increased domination by the excess f r e e water, l e a d i n g to the smaller M 2 as d i c t a t e d by eqn. {B.1} i n Appendix B. The observation of the phenomenon of l i p i d - w a t e r i n t e r a c t i o n i n t h i s system ( E Y L / 2 2 % W D 2 0 ) i s much more pronounced than that i n the E Y L / E X C D 2 0 and E C / E X C D 2 0 . In a system where the water c o n c e n t r a t i o n i s w e l l below the maximum hy d r a t i o n , eqn.{5.l} derived from the two - s i t e model i n t h i s chapter i n d i c a t e s that the bound water, which i s i n v o l v e d i n the l i p i d - w a t e r i n t e r a c t i o n , dominates the observed D M R s i g n a l , g i v i n g r i s e to sp e c t r a w i t h s p l i t t i n g s i n the order of s e v e r a l kHz. Consequently, the M 2 of the spectrum i s very s e n s i t i v e to the 0 - D bond o r i e n t a t i o n a l order parameter whose magnitude de-pends on the l i p i d - w a t e r i n t e r a c t i o n . In the region - 1 0 ° C _< T _< - 3 ° C , the most s t r i k i n g c o n t r a s t between the behaviour of the M 2 f o r t h i s system ( E Y L / 2 2 % W D 2 0 ) and the E Y L / E X C D 2 0 i s t h a t , as temperature decreases, the former decreases whereas the l a t t e r i n c r e a s e s . Furthermore, as temperature decreases, the M 2 (and correspondingly the l i n e width as shown i n Fig.22) f o r the EYL/22%WD20 i n the region decreases much more r a p i d l y than that f o r the same system i n the r e g i o n T >^  -3°C. The tem-perature dependence of the M 2 f o r the EYL/22%wT>20 i n t h i s r e gion i s not unt expected, because the trapped water i n t h i s r e g i o n i s g r a d u a l l y squeezed out from the b i l a y e r s . The squeezed out i s o t r o p i c water remains unfrozen down to -10°C as shown i n Sect i o n 5.2b, and the DMR s i g n a l of that i s o t r o p i c water becomes more and more dominant as temperature decreases, l e a d i n g to a d e c l i n e i n the M 2 much more r a p i d than that expected from l i p i d - w a t e r i n t e r a c t i o n i n the r e g i o n T >^  -3°C. On the other hand, as shown i n F i g . 10, a l l the i s o t r o p i c excess f r e e water i n the EYL/EXCD20 has been f r o z e n out i n t h i s r e g i o n , and, i n the range -3°C _< T <^  -2°C, the l i p i d i s f u l l y hydrated. Thus, f o r -10°C <^  T <_ -3°C, the trapped water i s expected to be squeezed out from the b i l a y e r s g r a d u a l l y as temperature i s lowered. However, the behaviour of the M 2 f o r the EYL/EXCD20 i s obviously incompatible w i t h the existence of such i s o t r o p i c water. This means that the i s o t r o p i c water i s f r o z e n as soon as i t i s squeezed out from the b i l a y e r s . This hypothesis i s supported by the evidence provided by the decrease i n the DMS s i g n a l i n t e n s i t y f o r the EYL/ EXCD20 i n t h i s r e gion as shown i n F i g . 1Q. Thus, we expect that the M 2 of the spectrum f o r the EYL/EXCD20 at -10°C should be c o n s i d e r a b l y l a r g e r than that of the spectrum f o r the EYL/22%WD20 at the same temperature. This i s indeed the case as shown i n F i g . 1-3 and F i g . 18, where the r a t i o of M 2 f o r the EYL/EXCD20 to that f o r the EYL/22%WD20 i s 53:5 or 11. The increase i n M 2 of the spectrum f o r the EYL/EXCD20 i n the range from -3°C to -10°C as tempera-ture decreases may be a s c r i b e d to the f o l l o w i n g mechanisms: (a) The squeezing out of the trapped water r e s u l t s i n a p o p u l a t i o n r e d u c t i o n i n the i s o t r o p i c s i t e s between the b i l a y e r s , w h i le the p o p u l a t i o n i n the a n i s o t r o p i c s i t e s may 57 remain f i x e d . Consequently, according to eqn,{5.l} i n t h i s chapter, the Av^ i n that equation gains more and more weight, w h i l e the c o n t r i b u t i o n from the Av^ fades away, r e s u l t i n g i n the increase i n as temperature decreases, (b) As temperature i s lowered, the average 0-D bond order parameter at each s i t e i n c r e a s e s , g i v i n g r i s e to a broader l i n e w idth, (c) When temperature i s decreased, the motions of the l i p i d p o l a r head group, to which the water molecules are bound, are reduced (36), r e s u l t i n g i n a l e s s e r motional averaging of the e l e c t r i c f i e l d gradients and the d i p o l a r i n t e r a c t i o n , so that the s p i n -s p i n r e l a x a t i o n time T 2 i s shortened. Consequently, a l i f e time l i n e broaden-ing occurs. I t i s very d i f f i c u l t to d i s t i n g u i s h between the process described i n (a) and that i n ( b ) . The M 2 vs T p l o t f o r the EC/EXCD20 ( F i g . 13) i n the r e g i o n between -3°C and -10°C i s s i m i l a r to that f o r the EYL/EXCD 20, and i s a l s o d i c t a t e d by the same mechanism as supported by the experimental f a c t s shown i n F i g . 10. The f r e e z i n g of the squeezed out water i n the EYL/EXCD20 arid EC/EXCD20 i n the re g i o n mentioned above i s i n great c o n t r a s t to the squeezed out water i n the EYL/22%WD20, which d i d not freeze u n t i l below -10°C. This d i f f e r e n c e i s not s u r p r i s i n g , because, i n the neighborhood of -2°C, l a r g e surfaces of i c e are formed due to the f r e e z i n g of the excess f r e e water i n the EYL/EXCD20 and EC/EXCD20. These surfaces enhance f r e e z i n g of water pushed out from the b i l a y e r s i n the temperature r e g i o n from -2°C to -10°C. In the EYL/22%WD20 sample, no such i c e surfaces e x i s t at -2°C, so that there i s no surface of c r y s t a l l i z a t i o n i n the re g i o n . As temperature decreases, the M,> f o r the EYL/22%WD20 reaches a minimum at -10°C. From the spec t r a of t h i s system i n the re g i o n from -3°C to -10°C as shown i n F i g . 9c-f, i t i s evident that the presence of i s o t r o p i c water squeezed out from the b i l a y e r s i s one of the most l i k e l y mechanisms ,58 Figure 17. Temperature dependence of the f i r s t ( t r i a n g l e s ) and the f o u r t h ( c i r c l e s ) moments, M^, M^, of the DMR spectrum of B^O i n the EYL/22%WD20. 60 re s p o n s i b l e f o r the minumum. A complete phase diagram of egg y o l k l e c i t h i n -water system has been provided by Small and Chapman (57) and Reiss-Husson (58). However, si n c e the phase l i n e i s i l l - d e f i n e d , i t i s dangerous to draw any concl u s i o n regarding the phase behaviour of our sample (the EYL/22%WD20) s o l e l y on the b a s i s of t h e i r phase diagram. U n f o r t u n a t e l y , our PMR experiment o d i d not cover the re g i o n below -10 C, so that no i n f o r m a t i o n concerning the phase behaviour of our EYL/22%WD20 sample i s a v a i l a b l e . Due to l a c k of experimental c e r t a i n t y , i t i s very d i f f i c u l t to know whether a phase t r a n s i t i o n i s a l s o i n v o l v e d a t -10°C. 2 e) The temperature dependence of the and the M^ /M^  f o r the EYL/ 22%WD20, EYL/EXCD20 and EC/EXCD 0 2 The temperature dependence of the parameters Ag and M^/M2 f o r the three systems are p l o t t e d i n F i g . 19-21. 2 The shapes of the A„ vs T and M./M„ vs T f o r the EYL/EXCDo0 and 2 4 2 2 2 EC/EXCD^O are very s i m i l a r . A sharp drop i n A 2 and m^ /M^  occurs as temperature changes from -2°C to -3°C and -1°C to -2°C f o r the EYL/EXCD20 and EC/EXCD20, r e s p e c t i v e l y . The values of these parameters f o r temperatures below the ': f r e e z i n g p o i n t are smaller than those at temperatures above i t . 2 The shapes of the A 2 vs T and M^/M2 vs T f o r the EYL/22%WD20 are very d i f f e r e n t from those f o r the other two systems. A sharp anomalous increase i n the valuesoof A„ and M./M 2 occurs around -10°C. Note that the anomalous 2 4 2 2 increase i n e i t h e r A 2 or M^ /M^  (Appendix A) does not correspond to the co-exis t e n c e of i c e and water phases, s i n c e i c e does not c o n t r i b u t e to the moments of the spectrum. However, si n c e the s t a t e of water depends on the environment (the l i p i d s ) i n which the water e x i s t s , i t i s very l i k e l y that the anomalous behaviour of the two parameters f o r the EYL/22%WD20 at -10°C corresponds to the coexistence of d i f f e r e n t phases of water. 61 T (°C) 2 Figure 19. Temperature dependence of the r a t i o M^ /M^  ( c i r c l e s ) and the r e l a t i v e mean square d e v i a t i o n A 2 ( t r i a n g l e s ) of the DMR spectrum of D o0 i n the E. Coli/EXCD o0. 62 Figure 20. Temperature dependence of the r a t i o M^/M^ ( c i r c l e s ) and the r e l a t i v e mean square d e v i a t i o n A,, ( t r i a n g l e s ) of the DMR spectrum of D„0 i n the EYL/EXCD„0. 63 -25 -10 0 10 25 T ( ° C ) Figure 21. Temperature dependence of the r e l a t i v e mean square d e v i a t i o n 2 ( c i r c l e s ) and the r a t i o M^ /M^  ( t r i a n g l e s ) of the DMR spectrum of Do0 i n the EYL/22%WDo0. .64 5.3. Comparison With Other Work The temperature dependence of the f i r s t moment (which gives the average quadrupole s p l i t t i n g , and hence average order parameter), the second moment and the quadrupole s p l i t t i n g measured d i r e c t l y from the DMR spectrum of D^ O i n the EYL/22%WD20 as shown i n F i g . 18 and 22 have the same general shape as the quadrupole s p l i t t i n g vs temperature of the DMR spectrum f o r the EYL/QD^O (19% by weight of water concentration) presented by F i n e r et a l (32) as shown i n F i g . 22, but they do not agree i n d e t a i l . For convenience, l e t us denote EYL/9D 20 as EYL/19%WD20. At -20°C, the spectrum of the EYL/19%WD20 (Fig.22). has a quadrupole s p l i t t i n g of 1.8 kHz, w h i l e that of the EYL/22%WD20 ( F i g . 22) has a much small s p l i t t i n g of only 0.37 kHz, a d i f f e r e n c e by a f a c t o r of f i v e . The s p l i t t i n g at -10°C i n the spectrum of the EYL/19%WD20 i s 0.81 kHz, and that of the EYL/22%WD20 i s only 0.1 kHz, a f a c t o r of eight s m a l l e r . But the s p l i t t i n g i n the spec t r a at 25°C f o r these two systems are n e a r l y i d e n t i c a l ; namely, 1.2 kHz f o r the s p l i t t i n g i n the spectrum of the EYL/19%WD20, and that of the EYL/22%WD20 i s 1.02 kHz. A minimum i n the Ay vs T of the EYL/19%WD20 and EYL/22%WD20 occurs at -2°C and at -10°C, r e s p e c t i v e l y , w i t h a minimal value of 0.52 kHz f o r the former and 0.12 kHz f o r the l a t t e r . The phase diagrams provided by Small (57) and F. Reiss-Husson (58) i n d i c a t e that the phase behav-i o u r of the EYL/water system i s s e n s i t i v e to small change i n water concentra-:'. i o n , so the d i f f e r e n c e i n the temperature at which the minimum i n the Av vs T of the two systems occurs i s not unexpected. F i n e r a s c r i b e d the minimum to the onset of chain m e l t i n g . However, no such t r a n s i t i o n i n the EYL/22%WD20 i s observed i n the range from -10°C to 48°C as demonstrated by our PMR data given i n F i g . 23. 65 Figure 22. (a) Temperature dependence of the quadrupole s p l i t t i n g Av which i s measured d i r e c t l y from the DMR spectrum of D^ O i n the EYL/22%WD 0 ( c i r c l e s ) . The po i n t s marked by t r i a n g l e s are the quadrupole s p l i t t i n g s Av^ c a l c u l a t e d from the f i r s t moment M^ of the DMR spectrum of D^ O i n the s a i d system. Note that the c a l c u l a t e d s p l i t t i n g s are s y s t e m a t i c a l l y l a r g e r than those obtained by d i r e c t measurement, i n d i c a t i n g the existence of l i n e broadening. I f l i n e broadening i s absent, the two curves should be c o i n c i d e n t , (b) The i n s e r t shows the quadrupole s p l i t t i n g vs T i n the DMR spectra of egg yo l k phosphatidylethanolamine/21%WD 20 ( t r i a n g l e s ) and EYL/19%WD20 = EYL/9P20 ( c i r c l e s ) presented by F i n e r et a l (32). 66 5.4. PMR Results for the EYL/22%WD20 2 Temperature dependence of the M2 and M^ /M2 of PMR spectrum of EYL In the EYL/22%WD20 system are given in Fig. 23. The M2 decreases monotonically and smoothly as temperature rises from -10°C to 48°C. The value of M2 at 48°C is smaller by a factor of five as compared to that at -10°C. The reduction in M2 at higher temperature region arises from motional narrowing due to increase in time averaging of the dipole-dipole interaction. 2 Within experimental error, the M^/M2 is independent of temperature. If we assume the existence of translational diffusion of the phospholipid molecules along the bilayer surfaces and rotation about the normal to the bilayers, and, furthermore, i f the powder pattern lineshape is assumed to be given by a super-position of the lineshape f(co,9) = {27TM 2(P 2(cos6)) 2} _ l sexp{-(a)-uJ o) 2 /2M 2(P 2(cos0) }2} {5.21} by summing over the orientation angle 8 (52), we obtain 6.45 for the value of 2 the M^ /M2 , which is consistent with the data for Fig. 23 within experimental 2 error. Thus, this particular value of M^/M2 may suggest that there are s t i l l a lot of motions at a temperature as low as -10°C. Furthermore, since M^/M22 is very sensitive to coexistence of phases (Appendix A), the absence of anomalous increase in that parameter indicates that there is no phase transition in the entire region under investigation. However, as shown in Fig. 24, the preliminary PMR results for EYL in the EYL/EXC_20 (Alex Mackay, unpublished) exhibit some unusual behaviour near 4°C, This work was done in collaboration with Dr. A. L. Mackay. 67 which could be as s o c i a t e d w i t h a change i n the ordering of the water molecules. This phenomenon a l s o occurs i n the EC/EXCT^O at 4°C as shown i n the same f i g u r e 2 (the M^/M2 f o r t h i s system i s not shown). Since the DMR data on the E. C o l i outer membrane (37) does not show any anomaly at 4°C ( F i g . 25), and no anomaly i s observed i n the EYL/22%WD20 as shown above, the phenomenon observed by A l e x t Mackay i n the two systems i n excess water has been a s c r i b e d to the onset of t r a n s l a t i o n a l d i f f u s i o n of the phospholipid molecules. The p o s s i b l e c o r r e l a t i o n between t h i s phenomenon and the f r e e z i n g out of water i n the two systems i n excess water has been r u l e d out on the evidence provided by my DMR r e s u l t s presented i n t h i s t h e s i s . When the w r i t e up of t h i s t h e s i s i s completed, Alex Mackay repeated the PMR experiment f o r the EYL/EXCD20 sample, and found that the r e s u l t s are not re p r o d u c i b l e . 68 Figure 23. Temperature dependence of the r a t i o M^ /M^  ( c i r c l e s ) and the second moment ( t r i a n g l e s ) of the PMR spectrum of the EYL/22%WD 0. 69 -5 0 10 20 35 T (°C) Figure 24. Temperature dependence of the second moment IL^ °f P M R spectrum of (a) the outer membrane of E. C o l i i n the EC/EXCD20 ( c i r c l e s ) , (b) the EYL i n the EYL/EXCT^O ( t r i a n g l e s ) , (c) temperature dependence 2 of M^/M2 of the PMR spectrum of the EYL i n the EYL/EXCD20 (squares) (Alex Mackay's unpublished data). 7 0 (a) 0 1 0 2 0 3 0 4 0 0 1 0 2 0 3 0 4 0 T ( ° C ) T (°C) F i g u r e 2 5 . ( a ) S e c o n d m o m e n t s v s t e m p e r a t u r e o f t h e DMR s p e c t r u m o f o u t e r m e m b r a n e s o f E . C o l i g r o w n o n a m e d i u m c o n t a i n i n g p e r d e u t e r a t e d p a l m i t i c a c i d ( t r i a n g l e s ) . ( b ) T h e p a r a m e t e r A 2 v s t e m p e r a t u r e o f t h e DMR s p e c t r a o f t h e o u t e r m e m b r a n e s o f E . C o l i g r o w n o n a m e d i u m c o n t a i n i n g o l e i c a c i d a s w e l l a s p e r d e u t e r a t e d p a l m i t i c a c i d ( 3 7 ) . 71 Appendix A Moments Of Nuclear Magnetic Resonance Spectra When l i n e broadening i n NMR a b s o r p t i o n spectra becomes s u b s t a n t i a l , the sharp s p e c t r a l features disappear. Thus, the true or i n t r i n s i c quadru-pole -J s p l i t t i n g s (see S e c t i o n 2.3a, Chapter 2 ) , and consequently the o r i e n t a t i o n a l order parameters cannot be determined p r e c i s e l y from the usual spectroscopic technique; namely by measuring the separation between the peaks (peak s p l i t t i n g ) i n a spectrum d i r e c t l y . That i s , the i n t r i n s i c s p l i t t i n g s can no longer be i d e n t i f i e d w i t h the peak s p l i t t i n g s . However, the i n t r i n s i c s p l i t t i n g s can be determined by matching each experimental spectrum w i t h a computer simulated spectrum (Bloom et a l , 1978a, unpublish-ed) , where a known broadening i s introduced i n t o a powder p a t t e r n of known s p l i t t i n g s . A l t e r n a t i v e l y , the moments of a NMR absorption spectrum provide a systematic way of determining the order parameter d i s t r i b u t i o n , th The n moment of a spectrum i s defined as where f (co-oo^) i s the lineshape of the NMR absorption spectrum centered about the Larmor frequency u) ( i n angular frequency u n i t s ) . Moments of Deuterium Nuclear Magnetic Resonance Spectrum For a f i r s t order quadrupole powder p a t t e r n , the DMR spectrum f(oo-0) o) i s symmetric about the angular Larmor frequency, ( J J q , i . e . f (] to—u>o | ) = f (-1 ( J O - U ) ^ | ) (Abragam, 1961), consequently the odd moments of t h i s lineshape v a n i s h . I t i s convenient to use the moments of the half-spectrum defined as oo r i oo M = / (w-w ) f(w-w ) dw / / f(w-w ) dco n w o o w o o o oo n °° = / n f(Q) dn / / of(ft) d^ { A i 2} where = w-w . In this case both odd and even moments exist. Notice that o for even moments, eqn.fA.l) is reduced to eqn.{A.2} because of the symmetry in f(fi). The DMR lineshape f(fl) of a simple system having only one type of deuterium site (and hence one value of order parameter) is a single quadru-pole powder pattern given by eqns. {14} and {15} in Section 2.3a, Chapter 2, and the moments of this spectrum as defined by eqn.{A.2} hassbeen shown (Bloom et a l , 1978a) to be M _ A (- e 2 q Q ) n S n n V 4 fi } bCD = A_(2Tr) n (Av) n { A > 3 } when line broadening can be neglected, i.e. when there i s no line broadening or the intrinsic line width i s much smaller than the quadrupole split t i n g . The coefficient A N can be calculated from the expression for the spin 1 powder pattern linshape (Bloom et a l , 1978a). The f i r s t two coefficients are given as A ± = 2/3/3 and A 2 = 1/5. In eqn{A.3}, i s the C-D bond .. order parameter previously defined (Chapter 2), and Av is the quadrupole splitting in the powder spectrum. Thus, for a single spin 1 = 1 quadru-pole.. powder pattern, when-broadening is negligible, the moments of the spectrum are functions only of the quadrupole . sp l i t t i n g Aw = 2TTAV, and the measurement of any one of the moments is equivalent to a measurement of the quadrupole sp l i t t i n g . In deuterium-labelled phospholipids.'.in biological membranes, large numbers of inequivalent deuterium sites may exist, because most biological 73 membranes have a v a r i e t y of phospholipids corresponding to d i f f e r e n t p o l a r head groups and combinations of p a i r s of a c y l chains having v a r i o u s lengths and degrees of s a t u r a t i o n . These d i f f e r e n c e s may give r i s e to v a r i a t i o n s i n the value of C-D bond order parameter among the d i f f e r e n t l i p i d s even i f a l l l i p i d s were l a b e l l e d at the same p o s i t i o n on a hydrocarbon chain. Larger v a r i a t i o n s i n S occur between g e l and l i q u i d c r y s t a l regions of the sample. At p h y s i o l o g i c a l temperature, these d i f f e r e n t thermodynamic phases are expected to c o e x i s t i n b i o l o g i c a l membranes, which are r e l a t i v e -l y inhomogeneous si n c e they c o n s i s t of mixtures of l i p i d s and p r o t e i n s . Furthermore, the l i p i d - p r o t e i n i n t e r f a c e i t s e l f i n a b i o l o g i c a l membrane system may give r i s e to observable v a r i a t i o n s i n S . Thus, the informa-t i o n r e q u i r e d to c h a r a c t e r i z e the o r i e n t a t i o n a l order of such a complex system i s not simply an order parameter S, but r a t h e r an order parameter d i s t r i b u t i o n f u n c t i o n P ( S ) . th The n moment of the order parameter d i s t r i b u t i o n P(S) i s defined as S = f° S nP(S) dS {A-.4} n o where, f o r a quasi-continuous d i s t r i b u t i o n of S , P(S)dS i s the p r o b a b i l i t y of f i n d i n g an o r i e n t a t i o n a l order parameter between S and S + dS f o r the deuterium s i t e s of the complex system. For a d i s t r i b u t i o n of order parameters c h a r a c t e r i z e d by P ( S ) , the r e s u l t a n t DMR spectrum i s a s u p e r p o s i t i o n of powder p a t t e r n s . I t has been shown (Bloom et a l , 1978, unpublished) that 74 M = A ( r 4 ^ S n n 4 n = A (2TTV n) n <S. n n> n Q CD = A (2rr) n < ( A v ) n > {A.5} n where V 3 e2gQ Q 4 h A V = VQ SCD For the f i r s t two moments, eqn.{A.5} gives M l = 1 7 3 < A V > = 3 7 3 VQ < S C D > 2 2 4TT 2 4ir 2 2 = ^ < ( A v ) 2 > = ^ V Q 2 <S C D 2> { A . 6 } ' The r e l a t i v e meain square d e v i a t i o n from the mean A^ of the d i s t r i b u t i o n of o r i e n t a t i o n a l order parameters i s defined as 2 2 2 . S2 - S l < SCD > - < S C D > 2 " s 2 ' <s >2 1 CD M2 - 1 {A.7} 1.35M 1 2 A simple system having only a s i n g l e order parameter (and consequently, i t s DMR spectrum c o n s i s t s of only a s i n g l e powder pattern) has the property that S = S_ n = S„_ n g i v i n g A „ = 0 ( n e g l e c t i n g l i n e broadening). In t h i s n 1 CD z simple case, eqn. iA.5} i s reduced to eqn. {A.3}. The parameter A 2 i s a very u s e f u l one i n that i t c h a r a c t e r i z e s the width 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 . This parameter i s very s e n s i t i v e to inhomogeneity i n the k i n d of sample under c o n s i d e r a t i o n such as 75 the coexistence of phases (59). Proton Magnetic Resonance U n l i k e deuterium-labelled phospholipids i n model and b i o l o g i c a l membrane systems where deuteron-deuteron d i p o l a r i n t e r a c t i o n s are n e g l i g i b l e , A 2 f o r s p e c i f i c a l l y protonated or p r o t i a t e d phospholipids i n membrane system i s d i f f i c u l t to i n t e r p r e t , because i n t h i s system proton-proton d i p o l a r i n t e r -a c t i o n s are l a r g e and cannot be ignored. However, the r a t i o of the f o u r t h to the square of the second moments of a PMR spectrum i s s t i l l u s e f u l and has p h y s i c a l meaning. For example, consider a p r o t i a t e d system i n a s t a t e i n which there i s n e i t h e r t r a n s l a t i o n a l d i f f u s i o n nor r o t a t i o n of the phospholipid molecules, then the PMR lineshape i s assumed to be a simple Gaussian given by which give s M^/M2 =3. In a l a m e l l a r l i q u i d c r y s t a l l i n e phase, both of these motions are present. I f the lineshape of the o r i e n t e d sample i s given by (52) where 0 i s the angle between the magnetic f i e l d H q and the a x i s of symmetry f o r the motion (the normal to the b i l a y e r ) , P 2(cos0) i s the second order Legendre 2 polynomial (3cos 0 - l ) / 2 , and M2(0) i s the second moment of the spectrum when the sample i s o r i e n t e d at 8 = 0. The second and the f o u r t h moments of the powder p a t t e r n a r i s i n g from eqn. {A.9} are r e s p e c t i v e l y ^M2(0) and -~M^(0), 2 2 consequently, M^/M2 = 6.45. The parameter M^/M2 i s very s e n s i t i v e to coexistence of phases such as g e l and l i q u i d c r y s t a l l i n e phases. An anomalous 2 behaviour i n the temperature dependence of M^/M2 over the r e g i o n of co-existence of phases has been observed i n the DPL/D 20 system (Alex Mackay, unpublished). {A.8} 2 {A.9} Appendix B C o n t r i b u t i o n s To The Second Moment Let us consider a l i p i d / w a t e r system i n a water c o n c e n t r a t i o n , c, which i s much higher than 40% by weight (the maximum h y d r a t i o n ) . In t h i s system, a maximum amount (by weight), W , of the water which give s W /(W +Lipids) = 40% i n r H r H i s incorporated i n t o the b i l a y e r s (33), w h i l e the r e s t of the water forms what i s c a l l e d bulk or i s o t r o p i c excess f r e e water e x i s t i n g i n separate phase and exchanging slowly w i t h the water incorporated between the b i l a y e r s (32-33, 36, 61). Thus, the observed NMR spectrum of the water, F( f t ) , i s a s u p e r p o s i t i o n of two sp e c t r a : a spectrum ( f u ) a r i s e s from the water a s s o c i a t e d w i t h the l i p i d , n and the other (f„„) i s the NMR absorption spectrum of the excess f r e e water;: that i s : F(ft) = f R(„) + f E X ( f i ) {B.l} From the d e f i n i t i o n of moments given i n Appendix A, oo 9 oo M 2 = ft F(ft) dft / /_ F(ft) dft /ft2f„(ft) dft + / f t 2 f fft) dft n hX  / f (ft) dft + /f„v(ft) dft M_H M „ E X  2 + ? {B.2} 1 + AEX / h 1 + ^ I ^ = f M 2 E X + (1 - f ) M 2 H {B.3} where M.H = / f t 2 f u ( f t ) dft / /£_(„) dft {B.4} 77 M 2 E X = /^fgjCfi) dti I / f E X ( ^ ) dtt {B.5} = /f H ( f l ) dft {B.6} f = 1 / {1 + Ag / A^} {B.8} For c > 40% by weight, addition or reduction in the amount of water in the system w i l l not alter the amount of water incorporated between the bilayers, consequently, A^ and are constant. 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