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The lipid-water interaction in lyotropic measophases : an NMR study Abdolall, Khaled 1978-02-26

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THE LIPID-WATER INTERACTION IN LYOTROPIC MEASOPHASES AN NMR STUDY by KHALED ABDOLALL B. Sc., University of Waterloo, 1972 M. Sc. University of British Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA in the Department of PHYSICS In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date V 57 t 6 Abstract A nuclear magnetic resonance study of the lipid-water interaction has been carried out in the lamellar mesophase of the sodium laurate-water system. Deuterium quadrupole splittings and spin lattice relaxation time measurements of perdeuterated fatty acid chains and the quadrupole splittings of water (D2O) and the sodium counter ion are used to study the effects of this interaction. The results indicate that the lipid-water interaction has a strong influence on the conformations and motions of the. lipid chains, particularly those chain segments near the lipid water interface. The details of this interaction are not included in the theories which attempt to explain hydrocarbon chain ordering in bilayer membranes in terms of chain-chain (or lipid-lipid) interactions only. A thermodynamic analysis of the results indicates that a description of the ordering of the hydrocarbon chains entirely in terms of chain-chain interactions is not complete, and that a complete theory should include the lipid-water interaction explicitly. The first experimental evidence for a spin lattice relaxation mechanism between water protons and lipid protons in a lipid water system is also reported. The effect of isotopic modification of the methylene hydrogen nuclei on the proton spin lattice relaxation rate in l^O and also of the effect of the isotope modification of the water on the relaxation rates of the lipid protons is investigated. Although the measurements show that protons deep in the bilayer make a substantial contribution to the spin-lattice relaxation rate of the water protons, a detailed theore tical analysis demonstrates that the experimental results can be accounted for without invoking deep penetration of the water in the bilayer. Table of Contents Page Abstract List of Figures i List of Tables 111 Acknowledgements iv Chapter 1 Introduction 1 2 Theory 2.1 Quadrupolar Interactions 13 2.1.1 Deuterium Magnetic Resonance 5 a) Chain deuterons b) Deuterium in D2O 7 2.1.2 23Na NMR 12.2 Spin-Lattice Relaxation 19 2.2.1 Deuterium Spin-Lattice Relaxation 23 2.2.2 Chain Protons. (Lipid^O mixtures) 4 2.2.3 Chain Protons and H2O Protons (Lipid/H20 mixtures) 26 3 Experimental 27 3.1 Fatty acids3.2 D20• 3.3 Deuteration of the fatty acids 27 3.4 Proton labelling of deuterated fatty acids 23.5 Samples 29 3.6 NMRApparatus. 30 A) The SpectrometerB) Probehead and Variable Temperature Oven 30 3.8 NMR Measurements 32 A) SpectroscopyB) Relaxation Measurements 33 4 Results 4.1 Quadrupole Splittings 35 4.2 Spin-Lattice Relaxation Times 36 A) Deuterium ResultsB) a-Protons 8 C) H20 Results 34.3 Sources of Error5 The Lipid-Water Interaction ... -. A Microscopic Interpretation 52 5.1 Quadrupole Splittings5.2 Spin-Lattice Relaxation (Perdeuterated Chains) 56 5.3 A Model for Molecular Motions Mediated by the Lipid-Water Interaction 58 5.4 Isotope Effects 59 6 The Lipid-Water Interaction?Anylysis In terms of 62 , Macroscopic Variables '• ' i 6.1 Quadrupole Splittings . 62 Discussion in relation to existing theories 77 6.2 Spin-Lattice Relaxation 78 7 Spin-Lattice Relaxation between Water protons and Lipid Protons 84 7.1 Analysis and Discussion of the H20 results 84 The Model 85 7.2 a-CH2 Results 92 Appendix A. The temperature Dependence of Water and Counter Ion order in Soap-Water Mesophases. A deuterium and Sodium NMR study .96 Appendix B. Determination of The equation of State for the Sodium Laurate-Water system Using low angle X-ray Scattering 104 Appendix C. Water self Diffusion and spin-spin relaxation in Sodium Laurate/H20 113 References 117 i List of Figures Figure Page 1 A Schematic representation of the lamellar liquid crystal phase of a lipid-water system. 3 2 A schematic representation of the geometry in a lipid bilayer. 14 3 Experimental arrangement for the deuteration of the fatty acids 28 4 Variable temperature oven and sample holder. 31 5 Representative deuterium partially relaxed spectra 34 6 Representative deuterium, proton and ^3^a JJ^R spectra . 40 7 vn versus T 41 8 Vn versus n 2 9 vn versus C 3 10 Quadrupole splittings versus temperature for D2O , 23fla and the first few positions on the hydrocarbon chain 44 11 1/Ti as a function.of position on the hydrocarbon chain 45 12 1/Tln Versus 103/T 46 13 versus n 7 14 1/Ti versus C 48 15 1/Ti versus 103/T for the ct-CD2 in d__C._-Na/6H20 and d23C12-Na/6D20 ^ 1 49 16 '1/Ti versus 103/T for the C1-CH2 protons in d„ C12-Na/6H20 and d2]C12-Na/6D20 ZX " 50 17 1/Ti versus 103/T for H20 in d0C12-Na/6H20 , d21C12-Na/6H20 and d23C12-Na/6H20 51 18 l/vn Ovn/8C)T versus n 5 ii Figure Page 19 1/Tln versus n 61 20(a,b) (a) Order parameter curves obtained for two different lamellar phases of dC^K-I^O » having the same A value, (b) Order parameter curves variations for different a values in the lamellar phase of dC^2K-H20 . 65 20C Quadrupole splittings curves for two samples of dC^2~Na/H20 having the same A value. 66 21 The fractional variation of the quadrupole splitting with temperature keeping the area per polar head constant and keeping the water concentration constant. 71 22 The fractional variation of the quadrupole splittings with temperature keeping the area per polar head constant. 73 23 The fractional variation of the quadrupole splittings with water concentration keeping the area per polar head constant. , 24 The fractional variation of the quadrupole splitting with water concentration keeping the area per polar head constant for different C values at a fixed temperature. 25 The ration of the change in Vn with T keeping A fixed to the change in vn with T keeping C fixed. 75 26 A semilog plot of roo-rn . 76 27 Ratio of the activation energies for the chain deuterons at constant surface area per polar head to that at constant water concentration. 81 28 Effect of systematic errors due to the X-ray : measurements 29 Schematic representation of the different spatial regions, in which water protous and lipid protons move 83 86 30 Log A versus Log C for the sodium laurate water system. 110 31 The dependence of AQ on temperature 111 32 The dependence of A on temperature for the sodium laurate-water system. 112 33 Log, (S/SQ) + 2T/T2 versus T3 116 ) III List of Tables Table Page Ratio of the change in the spin-lattice relaxation rate of protons due to the -CH2 protons to the change of the relaxation rate of the -CH2 protons due to the water protons in dj-jC^-Na/SI^O . 94 Temperature dependence of the lamellar repeat distance, thickness of the bilayer and the mean area per polar head for the sodium laurate water system. 109 Dependence on the water concentration of the lamellar repeat distance, the bilayer thickness and the area per polar head. 109 Ratio D/D0 of the self diffusion coefficient of water in the sodium laurate-water system to that of pure water at 100 °C for several G values. 115 Quadrupole splittings of chain deuterons in the sodium laurate water system; dependence on temperature and water concentration. 1-20 Spin-lattice relaxation times of chain deuterons in the sodium laurate water system; dependence on temperature and water concentration. 121 iv Acknowledgements I am very grateful to my research supervisor Professor Myer Bloom who provided the stimulus and continuing help through-out the course of this work. I have benefited greatly from his instruction and the many discussions we had. I thank my wife Khadija for typing the thesis and for her patience and continuing encouragement and moral support. I express my sincere gratitude to the following people who through individual help have contributed to this work: Dr. Elliot Burnell of the Chemistry Department for many discussions and useful suggestions and for providing the help and equipment in f sample preperation. Dr. J. Charvolin of the Laboratoire de Physique de Solides, Orsay for the useful discussions we had and for suggesting the work on the sodium laurate-water system. Dr. Alex Mackay, Dr J. Davis and Dr M. I. Valic for the technical consultations and help in the experiments and for the many discussions and useful suggestions. In this regard the continued support of Alex in many other respects is greatly appreciated. Dr. T. P.Higgs for his assistance in preparing the specifically protonated samples. Dr. W. N. Hardy for the useful suggestions he gave for explaining the dependence of the deuterium relaxation rates on water concentration. Dr. K. Jeffrey of the University of Guelph for providing the assistance and facilities to do the X-ray measurements. 1 Chapter 1 Introduction Water is a major component of cells and tissues of all living organisms. The importance of its role in life processes on the cellu lar level is well recognised but has not yet been fully understood. Many of the properties of the cell membrane involve water directly or indirectly. For example most of the transport mechanisms in the cell membrane are mediated by water. Another example is the membrane action of some anesthetics which is believed by some to involve membrane associated water in a physical way rather than causing chemical changes in the cell membrane. From such examples it is, apparent that an under standing of how the water interacts with the "building blocks" of the cell membrane is very important to the understanding of membrane processes. The complexity of real biological membranes has motivated scientists in the field to look for simpler systems that could be used as "models". Since lipids form a large fraction of the "building blocks" of the cell membrane, lipid bilayers have been used as such models because of their similarity in structure to real biological membranes. The purpose of this work is to contribute to the understanding of the mechanism of the lipid-water interaction in a lipid bilayer membrane. In the presence of water, lipid molecules form a variety of lyotropic mesophases characterized by the existence of long range order and short range disorder. These phases have been identified by X-ray studies (1), 2 nuclear magnetic resonance (2,3) and other techniques (4). Of partic ular interest is the lamellar liquid crystal (LQ) phase where the lipid molecules form bilayers of indefinite extent and alternate in a regular lattice with layers of water and counter ions as shown in fig. 1 . Each bilayer can be thought of as a two dimensional fluid with the lipid chains preferentially oriented along the normal to the bilayer surface. Within the bilayer the hydrocarbon chains of the molecules are flexible (melted) and the molecules undergo rapid lateral diffusion (5) and the rotation about their long axis. Different parts of hydrocarbon chain can also undergo small and rapid angular excursions (such as bending, twisting and flopping) perpendicular to the molecular axis (the long axis of the molecule). The ordering of the hydrocarbon chains within the bilayer is then described in terms of averages over the fast molecular motions. A measure of this ordering will thus give information on the physical state and fluidity of the bilayer. There have been numerous studies on model and biological membranes using X-rays (1) nuclear magnetic resonance (2,3), and other techniques (4). With few exceptions (6,Appendix A) most of these studies are either concerned with the structure and properties of water in these systems per se (7-11), or with the dynamics and structure of the amphilic region. The influence of the interaction between the water and the lipid on the structure and dynamics of the two regions has not yet been developed appreciably. The ability of water to form hydrogen bonds which is simply under stood as an electrostatic attraction between the electropositive hydrogen of one water molecule and the electronegative oxygen of another water or Figure 1. The lamellar liquid crystal phase of a lipid-water system. The circles represent the polar portions (polar heads) of the molecules, and the zig-zag lines represent the hydro carbon chains. 4 lipid molecule, can have profound influence on the ordering and dynamics of the hydrocarbon chains within the bilayer. Thus lipid and water mutually affect each other via hydrogen bonding (Appendix A). The details of such an interaction have not been included in many of the theories which attempt to explain hydrocarbon chain disorder in bilayer membranes (12, 13, 14). The central questions that will be highlighted in this work are: how does the water between the bilayers influence the ordering of the hydrocarbon chains and how deeply does it penetrate into the bilayer? To answer such questions it is necessary to use local probes that are sensitive to their environment as well as to the dynamics of the system. An NMR study has been carried out on the lamellar liquid crystal (L^) phase of the sodium laurate/water system. There are two reasons for choosing this system: (i) a very detailed phase diagram is available (15), and (ii) one can obtain complementary information on the sodium counter ion. Interactions between the nuclear quadrupole moments of deuterium 23 in D^O or Na and the electric field gradients at the nuclear sites will provide information on the charge distribution in the vicinity of the sodium and/or the structure and ordering of water at the lipid-water interface. The conformations and. motion of the lipidrmolecules can be studied by measuring the quadrupole splittings in the deuterium magnetic resonance spectrum for deuterons along the hydrocarbon chain of a perdeuterated molecule (16-18), or the proton dipolar splittings in a a-CH^ group (19) in an otherwise deuterated hydrocarbon chain. The deuterium splittings are related to order parameters Srn which give a 5 measure of the orientational order of the C-D bond direction in a methy lene group while the proton dipolar splittings are related to order parameters S which give a measure of the orientational order of the proton-proton vector in a 01-CH2 group. Thus a knowledge of and will completely specify the orientational order of the methylene chain segment. Information on the lipid water interaction can also be obtained by relaxation measurements which can yield information regarding the nature and the strength of the interaction of the nuclear spin system with its molecular environment. Parameters such as correlation times (x ) associated with the molecular motions can be determined from relaxation measurements. Complementary information on the lipid water interaction can also be obtained from correlations in the quadrupole splittings of 23 Na, water (D2O), and chain (Appendix A) deuterons. Deuterium magnetic resonance has been used by several workers in the study of the conformation and motion of the lipid molecules in bilayer membranes (16-18, 20-26). The most prominent of these studies are those by Charvolin et al (17) and Mely et al (18) on the potassium laurate-water system, Seelig and Niederberger (23) on sodium deconate-water, Seelig and Seelig (23, 26) on phospholipid bilayers, and Davis and Jeffrey (19) on the potassium palmitate-water system. In these studies the order parameter profile for the hydrocarbon chains in the liquid crystal phase was determined by measuring the quadrupole splittings for deuterons on specifically deuterated (24) or perdeuterated chains (16, 17, 18). A characteristic feature common to all of these results is the appearance of a "plateau" region where about half of the chain methylenes after the a position have virtually the same order parameter. This plateau 6 disappears at higher temperature and water concentration with a decrease in the quadrupole splittings. The plateau was believed to have some origins in the steric repulsions between neighbouring chains (27, 28). Some models have been proposed where the steric interactions are taken into account in terms of the crosssectional area of a given chain conformation (13). This crosssectional area is characterized by the lateral space occupied by a lipid molecule, or the mean area (A) per polar head at the lipid water interface. Mely et al (17) have studied the order of the lipid chains in potassium laurate-water mesophases as a function of the temperature in samples having constant water concentrat ion and as a function of water concentration at constant temperature. They have found that the deuterium quadrupole splittings decreased with increasing temperature or water concentration. The same authors have also measured the quadrupole splittings for samples having the same area polar head but at different water concentration and temperature. The splittings for such samples}though not identical,were similar. On the basis of these results the authors concluded that A is a "good parameter to represent the average microscopic order". They believe that the variat ion of the quadrupole splittings with temperature and/or water concentrat-ion depends primarily on the variation of A with these parameters Davis and Jeffrey (18) studied the hydrocarbon chain disorder in the potassium palmitate-water system. In the liquid crystalline phase the C-D order parameters of the first methylene chain segments were found to increase with increasing temperature to a maximum of 100°C and.then decrease at higher temperature. In contrast the C-D order parameters for the rest of the methylene chain segments decreased with increasing temperature. In the same system Higgs and Mackay (19) have determined the complete ordering tensor for the a-methylene group by measuring the CI-CH2 dipolar splittings in an otherwise perdeuterated chain and the 0.-CD2 splittings in specifically deuterated chains. The temperature dependence of the a-CI^ splittings was similar to that of the a-CD^ splittings. However the order parameters and S^, difffered by 0-20% LU tin in the temperature range studied (40-100°C), indicating that the motion of this particular segment is not truly axially symmetric around the normal to the bilayer. From these studies the behaviour of the methylene chain segment near the lipid water interface was believed to be due to a lipid water interaction. Abdolall, Burnell and Valic (Appendix A) 23 studied the hydrocarbon chain, D20 and Na counter ion order in the potassium palmitate/D20 and sodium palmitate/D20 systems. There was a striking correlation between the ordering of the first few methylenes of 23 the hydrocarbon chain, deuterium in D20 and the Na counter ion. This correlation was ascribed to the structuring effect of the water via hydrogen bonding with the polar heads. A model consistent with the experimental results was proposed (Appendix A). In terms of this model the lipid water structure at low temperatures imposes a direction for which all the order parameters are smaller than for the higher temperature structure for purely geometric reasons. At higher temperatures the structuring effect of water decreases and there is an "apparent" increase in order until the intrinsic decrease in order parameters resulting from the thermal excitations dominates at still higher temperatures. There have been many theoretical studies on the chain ordering in liquid crystals and bilayer membranes (12-14, 29-32). One of the most successful of these is Marceljas (13) molecular field calculation. In the mean field approximation the interaction energy of a single chain in the molecular field is given by the sum of the internal energy of the single chain, the Vander Waals interactions of the chain with its neighbors via the molecular field and a lateral pressure term which is proportional to the cross sectional area per polar head. The lateral pressure term takes into account the steric repulsions between the hard cores of the atoms. Statistical averages are then calculated by summing over all conformations of a single chain in the molecular field of its neighbours. This model has been successfully u ed to interpret the deuterium NMR results for the sodium-deconate-deconol-water (13) and other NMR data (33). However the results do depend on the assumed orientation of the initial chain segment, which implies that a descrip tion of the ordering of the hydrocarbon chains in a bilayer membrane entirely in terms of chain-chain interactions is incomplete. What has been ignored in these theoretical calculations is the lipid-water inter action. Empirical evidence that such a lipid water interaction has a strong influence on the orientational order of the hydrocarbon chains in a lipid bilayer, especially those chain segments near the polar head is , as already mentioned, demonstrated by the striking similarity in the temperature dependence of the quadrupole splittings of the first few 23 methylene chain deuterons, and Na counter ion in the palmitates-water systems (Appendix A) and in the sodium laurate-water system studied in this thesis. 9 In order to investigate the interactions that are most important in determining the orientational ordering of the hydrocarbon chains in the lipid bilayer in a quantitative way we have made a systematic study on the sodium laurate-water system in the liquid crystal (L^) phase. The quadrupole splittings of the methylene chain deuterons were measured as a function of temperature and water concentration. An equation of state relating the area per polar head, temperature and water concentrat ion was determined by X-rays for this system. Using elementary thermodynamics the results were analysed by examining the sensitivity of the quadrupole splittings to variations in the different thermodynamic variables. The results of the analysis indicate that a description of the disorder of the hydrocarbon chain entirely >'n terms of chain-chain interactions is indeed not complete. A complete theory should include the lipid water interaction explicitly. Spin-lattice relaxation measurements in lipid bilayer membranes using deuterium NMR is fairly recent. Using a series of fatty acid probes of different lengths and labelled at several positions Stockton et al (34) showed thac the molecular motions within the phosphotldyl-choline bilayers increase rapidly with distance from the lipid water interface. More recently Davis, Bloom and Jeffrey (35) measured the spin-lattice relaxation times as a function of temperature and position on the hydrocarbon chain for the methylene deuterons in perdeuterated chains of the potassium palmitate-water system. In the liquid crystal line (k^) phase, their results indicate the presence of complex molecular motions of the different methylene chain segments. A simple model 10 proposing two different types of motions (a fast and a slow motion) with different correlation times was considered. In terms of this model they were able to explain in a qualitative way the change in relaxation times profile with increasing temperature. In the present thesis the spin-lattice relaxation rates for methylene chain deuterons of perdeuterate d chains in the liquid crystal (L^) phase of the sodium laurate-water system were measured as a function of temperature, water concentration and position on the hydrocarbon chain. The results obtained suggest that complex molecular motions mediated by a lipid-water inter action must be taken into account. A model is proposed to explain the influence of the lipid water interaction on the relaxation rates of the methylene chain deuterons. While deuterium quadrupole splittings and spin lattice relaxation time measurements can be used to study the effect of the lipid-water interaction on the orientational ordering and mobility of the hydro -carbon chains, it is not possible to obtain from such measurements information on the extent of water penetration into the lipid bilayer. In order to obtain such information it is necessary to measure quantities that depend on the spatial location of the local probe used and on the strength of the interaction between the probe and its molecular environment. The spin lattice relaxation rate for a proton due to dipolar couplings (which in most cases are the main relaxation mechanisms for protons) with another nuclear spin depends on the inverse of the 6tn power of the separation between the spins and on the product of the square of their magnetogyric ratios. Thus a study of the effect of isotope modification of the lipid region on the spin lattice relaxation rates of the water protons should in principle give useful information on the extent of water penetration into the bilayer. Complementary information can also be obtained by studing the effect of isotope modification of the water on the spin lattice relaxation rates of proton spin labels in otherwise perdeuterated chains. We have performed such studies on the sodium laurate-water system. The results obtained suggest at first glance that water penetrates much further into the bilayer than the a-position. However, a detailed theoretical analysis, shows that the experimental results can be accounted for without invoking deep penetration of the water in the bilayer. Information on the extent of water penetration in the bilayer can also be obtained from neutron diffraction measurements. Just recently Buldt et al (37) reported some results on phospholipid bilayers. The authors conclude that water penetrates into the bilayer up to the glycerol back bone of the lipid molecules. Earlier studies by Schoenborn (36) on a similar system did not have sufficient resolution to detect such water penetration. 12 Thesis Outline This thesis will consist of 6 more chapters. The second chapter will include the relevant NMR theory in lyotropic mesophases . Experimental details and results are given in chapters 3 and 4. In chapter 5 an interpretation of the results will be discussed in terms of a microscopic model for the lipid-water interaction. In chapter 6 a thermodynamic analysis of the results is used to investigate the interactions that are important in determining the state of the lipid chains in a lipid water system. The last chapter deals with the mechanism of spin lattice relaxation between water protons and methyl ene protons in a lipid water system and the problem of water penetration into the bilayer. 13 Chapter 2 Theory 2.1 Quadrupolar Interactions The total Hamiltonian for a nucleus with spin in an applied magnetic field H is given by (ignoring chemical shift terms etc.) H = Hz + HQ [l] where Hz is the Zeeman Hamiltonian and HQ is the quadrupolar Hamiltonian due to the interaction of the quadrupole moment eQ associated with the spin I and the electric field gradients (efg) existing at the site of the nucleus. , If HQ <(<( Hz it can be shown (38) that the first order perturbat ion to the Zeeman energy levels due to HQ are given by (in frequency units) where the angles G , $ specify the magnetic field direction relative to the principal coordinate system of the efg as shown in fig. 2b ,m is the magnetic quantum number in the represention where Iz is diagonal, e^qQ is the quadrupole coupling constant and n is an asymmetry parameter h f , defined such that 0<n<l and is a measure of the deviation of the efg from axial symmetry. If r|=0 then the energy levels of the total Hamiltonian in frequency units are E . E<°> • E <» m m T m 14 Figure 2. (a) Schematic representation of the geometry in a lipid bilayer. n is the normal to the bilayer. is the angle between the magnetic field H* and n, 0 is tlje angle between the C-D bon§ direction and H and 9 is the angle between the C-D bond direction and n. (b) Orientation of the magnetic field direction relative to the principal coordinate system of the electric field gradient. 15 --v + F(i£24^)(3"2-i«+i)) w where v. is the Larmor frequency YH and vn _ 3e^qQ L 2n 4 h2I(2I-l) 2.1.1 Deuterium Magnetic Resonance. a) chain deuterons. The deuterium nucleus has a nuclear spin 1=1. If we concentrate on one deuteron on the n position of the hydrocarbon chain and assume for the moment that the chain is not moving (rigid lattice) then the energy levels as given by eqn. [5] are n - vT + VQ_ (3cos20-lA -1 6 \ 2 ) E = I 3cos20-l 0 3 V 2 E. = -VT + VQ /3cos20-l  1 L 6 1 2 The corresponding resonance frequencies are E0-Ex = VL _ Vp. ^3cos20-l ^ and the NMR spectrum will consist of two sharp peaks separated by. AV, = VQ ^3cos2e-l^ [4] The efg at the site of a deuterium nucleus on the hydrocarbon chain is along the C-D bond direction. If the molecules are undergoing fast anisotropic motion at frequencies much greater than those of the quadru-polar splittings, then the reorientation of the C-D bond will modulate 16 2 0 and a time average of 3cos Q-l has to be considered. For the 2 molecules in a lipid bilayer, the symmetry axis of the motion is the normal n to the bilayer. Thus for a deuteron on the ntn position of the hydrocarbon chain eqn. [4] will give for the quadrupole splittings S / 3cos2ft-l\ £5] US / 3cos2jM\ Qn [ — J where as shown in fig. 2a on page \t\ Qn is the angle between the C-D bond direction and the normal n to the bilayer is the angle between n and the magnetic field direction and the quantity g z 3cos2Vl\ 0] 3 = /3cos-Qn-l y is defined as the order parameter, normally denoted in the * . literature by SQJJ • All the hydrocarbon chain -CD2 deuterons are chemically equivalent and therefore will have the same VQ. Since the bilayers are randomly oriented, the different values of cosft are equally probable and the superposition of the lines arising from the different orientations gives rise to a broad absorption curve characteristic of a powder pattern of the form (39) g(v) = / d cosft [7] with two intense peaks separated by 17 = |—|vQSn) [8] For a perdeuterated hydrocarbon chain, the spectrum will consist of a number of overlapping powder patterns arising from the various deuterons situated along the hydrocarbon chain. A representative spectrum is shown in fig.6a on page 40 • Since VQ is of intramolecular origin (C-D bond) it is not expected to be temperature dependent. Consequently the order parameters for the methylene chain deuterons can be obtained directly from the measured quadrupole splittings using equation [8] . b) Deuterium in D20 For water deuterons although the main contribution to the efg is from the intramolecular 0-D bond, there can be other contributions of intermolecular origin such as the charge distribution near the polar heads and other interactions (Appendix A ). Moreover, hydrogen bonding can reduce the contribution of the 0-D bond from 312 KHz (as measured in the gas phase) to 213 KHz (40). For such reasons VQ could be different for different sites. Also chemical exchange can take place between nuclei in different environments. If the exchange rate is much faster than the splitting difference, the observed splitting is a weigh ted average over the different sites and is given by. D,o - i lpi vQi sil M V ;2' where Pj_ is the fraction of nuclei in site i with associated quadrupole coupling constant | VQ± and order parameter S± defined by 18 S± . 3cos291 -1 [10] 2 where is the angle between ri and the efg principal axis and the bar denotes a time average. As discussed in Appendix A the separation of the various terms in eqn. [9] is not possible without making certain assumptions. The largest contribution to VQ^ is assumed to come from the intramolecular 0-D bond. This could mean that VQ^ remains roughly the same for the different sites. If in addition to Vqi being independent of i it is further assumed that does not vary significantly with tempers ture then the measured splittings are roughly prportional to an average order parameter S=£ P^S^ which can provide useful information in a i qualitative way. 2.1.2 23Na NMR The Na nucleus has a spin 1= ^ • From eqn. £3] the resonance frequencies are given by E-3/2 ~E-l/2 " *L + VQ (3cos2e-l^ E-l/2 "El/2 = VL and the corresponding NMR spectrum will consist of 3 peaks separated by AvNa Since rapid exchange can take place between the sodium ions in 19 different sites in the aqueous region then as outlined in the previous section the observed splitting is a weighted average given by ^ i The efg at the site of a sodium nucleus is of intermolecular origin and is largely due to the charge distribution near the polar head groups and the asymmetric distribution of waters of hydration (see appendix A page 77)• Thus it is expected that there will be a distribution of Vqi , p^ and S^ that can be quite temperature dependent making the 23 interpretation of the Na NMR splittings in terms of an order parameter rather difficult. However if correlations exist in the temperature 23 dependence of the methylene chain deuterons 1 . and , changes in 23 the Na splittings will give a measure of the order of the surrounding charge groups. 2.2 Spin -Lattice Relaxation The general problem of spin-lattice relaxation in lyotropic mesophases is complicated and not completely understood. Only the general aspects of the relevant theory will be discussed here. The total Hamiltonian for a nuclear spin system is, in most cases, given by H = Hz + H±(t) [12] where Hz is the Zeeman Hamiltonian and H^(t) is a time dependent Hamiltonian corresponding to quadrupole or dipolar couplings (for some 19 nuclei, such as F , large chemical shift terms must also be included) 20 These couplings are modulated by the lattice: The quadrupolar couplings depend on the electric field gradients at the site of a nucleus and dipolar couplings depend on the relative positions of the spins. H^(t) can always be decomposed into an average and a fluctuation about the average as follows (41) Hi(t) = <H±> + (H^t)-^^ ) = <Hi> + HiCt) [13] For an anisotropic environment is nonzero and causes splittings in the NMR spectrum and H^(t) is a spin lattice coupling responsible for the relaxation of the spin system toward thermal equilibrium with the lattice. H^t) can be written (38) H±(t) = ^F(m)(t)A(m) [14] m=-2 m (m) where the F (t) are random functions of time and the A are operators acting on the spin variables. F^m^ and A^ transform under rotations as the spherical harmonics of order two. The spin-lattice relaxation rate - which describes the rate of energy Tl transfer from the nuclear spin system to the lattice may be expressed in the form 2 [15] i =<o*i\zy <% j(mo)0) m=0 where ^ l^i^^P^ Is tne 1116311 squared value of the spin-lattice coupling, are numerical factors, OJQ is the Larmor frequency and Jn/^o) is tne Fourier transform of the reduced correlation function gjjjCx) defined by gjx) = GM(T) _ <FW(t)>2 [16] Gm(0) - <F(m)(t)>2 Gm(x) is called the correlation function of F^(t) defined by G^x) = <F(m)(t) F(m)(t+T)> [17] and describes how F^m^(t) at time t is correlated to its value at some later time t+T . The time variation of F^m\t) is due to some physical motions in the system. For times x much shorter than some critical time TC (called the correlation time), the motions are negligible and F^ (t) = F^(t+x). For times ' uch greater than TC there is no correlation between F^ (t) and F^ (t+x) and Gm(x) = ^F^(t) y 2. Therefore the reduced correlation function gm(x) has a maximum value of unity at x=0 and falls off to zero for T »TC . The correlation time xc may then be used as a measure or time scale for the motions. Thus if there is a model for the molecular motions, the correlation function can be calculated, which in turn would allow the calculation of the relaxation rates. On the other hand, measurements of spin lattice relaxation rates allow the determination of the reduced spectral density functions which can be used to test models for the molecular motions. In most cases the calculation of g^x) is nontrivial. A crude assumption that is often made is to assume that gm(x) decays exponentially, i. e. gM(T) - e -T/T* [18] 22 then, the reduced spectral density jm(u) is oo -OO 2?c [19] c 2 For very short correlation times, a) xf 1 and all the spectral density functions jm(mu)0) are independent of frequency and equal., to JiCo) = J2(o) = j (o) = 2TC [20] In this case the expression for ^ as given by eqn. 15 reduces to a Tl product of an intensity factor and a correlation time Tc • For example the spin-lattice relaxation rate for a pair of dipolar coupled protons undergoing isotropic motion is given by (38) i - aL±y?T, [21] Another example is the quadrupole relaxation through isotropic molecular reorientation. For a nuclear spin with 1=1, the relaxation rate jj- is (for an axially symmetric electric field gradient) Tl 1 ( is&Y T [22] Ti °\~ir-j "c In equations 21 and 22 , the quantities - -^-5- and e qQ are the dipole-2 r h dipole and the quadrupolar coupling constants respectively. If these coupling constants are known the measurements will provide Tc . If the fluctuations in the interaction H^(t) arise from molecular motion that varies with temperature, relaxation measurements can be used to study the temperature variation of T . Often there is a 23 "barrier" to motion and an activation energy Ea such that Tc . xweVkT [23] where Too is the correlation time at infinite temperature. Thus the temperature variation of ^ should give a measure of the activation energy. 2.2.1 Deuterium Spin-Lattice Relaxation. For the deterium nucleus, the coupling Hq(t) of the nuclear quadrupole moment with the fluctuating electric field gradients at the nuclear site is almost always the main relaxation mechanism. Since the effectiveness of a relaxation mechanism depends on the magnitude (or intensity) of the corresponding spin-lattice coupling (see eqn. 15 ), the magnetic dipolar couplings, which are very effective for protons make a negligible contribution to the deuterium relaxation rates. This is due to the fact that the dipolar interaction between a deuterium neucleus and another nucleus S depends on the product 2 2" Y-r Y » where Y^ and Y are the magnetogyric ratios of the deuterium I S I S nucleus and the S nucleus respectively (Abragam p.2^5"). Thus the 2 deuteron-proton dipolar interactions are weaker by _3_(YH/YD^ — .24 8 than the corresponding proton-proton dipolar interaction. Therefore considering the large value of the deuterium quadrupole coupling constant (^170 KHz) in the systems studied here dipolar interactions make at most a minor contribution to the deuterium relaxation rates. 1/T^ for deuterons is then given by 24 TL <I"QI2> ( H*o) + J2(2«O)) PQ where ^ |HQ | 2^> is the mean square value of Hg(t) and JJ^Q) and ^(^Uq) are the reduced spectral density functions associated with Hq(t). For deuterons on the hydrocarbon chain these spectral density functions are expected to be fairly complicated receiving contributions from the reorientation of the C-D bond around a given chain segment, single molecule motions relative to the director as well as collective motions of many molecules which are associated with the motions of the director itself about its equilibrium orientation. For a single type of motion it can be shown that, in the short correlation time limit (W2T.2 « 1 » where T„ is the correlation time On " characteristic of the motion of the n*"*1 methylene chain segment)* 1 t"h the relaxation rates ^ for deuterons on the n position of the Xln hydrocarbon chain are given by (35) 2 where Sn is the order parameter defined by eqn. 6 . The factor 1-Sn takes into account the anisotropy of the system. For Sn=0 eqn. 25 becomes the usual expression (eqn. 22) for deuterium quadrupole relaxation in isotropic fluids (38). 2.2.2 Chain Protons. (Lipid/D20 mixtures) For protons on the hydrocarbon chain, the problem is less simple than that for deuterons. In this case the spin lattice coupling responsible for relaxation is 25 HdU> = Hd (t) - <Hd> [26] Where H<j(t) is the time dependent dipolar Hamiltonian for the spin system. Due to the large magnetic moment of the proton, the dipolar interactions between protons on the neighbouring methylenes of the hydrocarbon chain are quite strong and tBnd to cause a fast establish ment of a common spin temperature for all the protons on the hydro carbon chain. This makes the interpretation of proton relaxation measurements rather difficult. This difficulty can be made easier by having protons only in one position in an otherwise perdeuterated chain. This essentially eliminates the inter .-CH2 dipolar contribution to the relaxation rates for methylenes on the same chain. The contribution of the inter molecular dipolar interactions to the relaxation rates of the methylene protons is expected to be much weaker than that due to the intra molecular dipolar interactions. This is because the magnitude of the dipolar interactions depends on r (where r is the proton-proton distance so that the contribution to ^ falls off very rapidly (r~^ ) with increasing r. Thus in the short correlation time limit, the main relaxation mechanism for a pair of methylene protons in an otherwise perdeuterated chain, is mainly due to the reorientation of the H-H vector joining the proton pair. The proton relaxation rate in this case is, in analogy with the deuterium case, given by where Sut. is an order parameter defined by tin « v 3cos20-l\ [28] HH \ 2 / where 0 is the angle between the H-H vector and the normal n to the bilayer. 2.2.3 Chain protons and protons (Lipid/H^O mixtures). The theory of spin-lattice relaxation for water and chain protons in lipid/H20 samples prepared with nondeuterated, perdeuterated and specifically protonated chain will be dealt with in chapter 7 . 27 Chapter 3 Experimental. 3.1 Fatty acids The fatty acids (reagent grade) were purchased from the Eastman Kodak Co. and used without further purification. 3.2 P20 The (99.7% enrichment) was purchased from Merck Sharpe and Dohme (Montreal). 3.3 Deuteration of the fatty acids. The procedure of deuterating the fatty acids is the same as that of Hsiao et al (42). The fatty acid and palladium on charcoal, as a catalyst, in the ratio of 5:1 by weight, are placed in a two neck flask as shown in fig. 3 and heated-to 180°C, with deuterium gas (obtained by electrolysis of D^O) passing over the surf ace continually at the rate of 35 cc. per. minute for 1 week. The palladium on charcoal was removed by dissolving the mixture of fatty acid and palladium on charcoal in chloroform, filtering through a celite column, and then evaporating the chloroform using a rotary evaporator. Mass spectral analysis revealed better than 99.2% deuteration. 3.4 Proton labelling of deuterated fatty acids. Laurie ($,-W)d2i acid having a -CH2 group at the. a position in an otherwise deuterated chain was prepared by exchange with KOH/^O at 230°C. Equimolar amounts of the fatty acid and KOH (reagent grade) with .25 moles/litre of water excess KOH were dissolved in ^0 and heated for 24 hours at 220°C in a sealed stainless steel tube. The 28 cold water D2 gas ELHYGEN MARK IV Milton Roy Figure 3. Experimental arrangement for the deuteration of the fatty acids. 29 exhanged fatty acid salt solution was acidified with concentrated HC1 to precipitate the fatty acid. Separation of the fatty acid was accomplished by shaking-with diethyl ether in a separating funnel. The ether layer was dried over anhydrous sodium sulfate and the ether was then evaporated in a rotary evaporator. Further purification of the fatty acid was accomplished by silica gel chromotography. Mass spectral and NMR analysis of the d23 and d2j acids indicated better than 99% H atthea position and 99.2%D at the (3-w) position of the lauric-d21 acid sample. 3.5 Samples The fatty acid salts were prepared by dissolving equimolar amounts of the fatty acid and the corresponding base (KOH or NaOH) in ethanol and slowly crystallizing the fatty acid salts. After filtration and washing with ethanol, the precipitated salt was recrystallized, washed with ethanol, and dried under vacuum at 140°C . The samples were made by weighing the corresponding molar amounts of the dry salt and H20 or D20 and sealed in a glass tube. Mixing was accomplished by centrifuging back and forth through a constriction in the glass tube. The samples were further homogenized by leaving them 4 days in an oven at 120°C. Notation; Different samples will be referred to by the number of < deuterated positions on the hydrocarbon chain, the concentration of water and whether.it is prepared with H20 or D20. For example d2iCi2-Na/6H20 stands for a sample prepared to have 6 moles of water per 1 mole of sodium laurate with one pair of protons at the a position in an 30 otherwise perdeuterated chain; similarly c^-jC^2_Na/6D20 stands for a sample with 6 moles of D2O per 1 mole of perdeuterated sodium laurate. 3.6 NMR Apparatus. A) The Spectrometer. The NMR measurements were carried out on a Bruker SXP4-100 NMR pulse spectrometer with a Nicolet BNC-12 computer. The spectrometer is capable of putting out a train of up to 4 RF pulses of controlled amplitude and whose phases and lengths could be'varied independently. The computer is equipped with a Diablo Disk Drive (series 31 single density) and was used for storage and analysis of the /. data. A programable timer (Nicolet 293 I/O controller) interfaced to the computer was used to automate the NMR measurements. Thus the triggering of the individual RF pulses, the spacing between theia, the repetition rate as well as the changing of the sample temperature was computer controlled. Automation of the measurements, especially the measurement of the spin lattice relaxation rates resulted in a tremendous saving of time. The accumulation of the thesis data would have other wise taken another 2VJ years. B) Probehead and Variable Temperature Oven. The probehead and the air flow heating system provided with the spectrometer were found to be unsatisfactory due to the large temperature gradient across the sample. To circumvent this problem a variable temperature oven was built. The diagram is shown in fig. (4). The oven was connected to the temperature control unit supplied with the 31 RF connector RF to probe arm Figure A . Variable temperature oven and sample holder. (a) Heater arrangement, (b) Oven with sarnie holder. On the inside of the copper block there is a shield (not shown in the diagram )made up of afihe brass screen to prevent ringing and eddy current effects after the application of an RF pulse. 32 spectrometer. Automatic temperature control was also made possible by interfacing the temperature control unit to the computer. Temperature gradients over a sample space of 1 cm diameter and 3 cm in height were undetectable (less than .2°C at 100°C ) The temperature stability was within 1°C over a 24 hour period. The time required for the sample to reach thermal equilibrium was less than 20 minutes for an increment in temperature of 10°C . 3.8 NMR Measurements A) Spectroscopy. The conventional method of obtaining NMR spectra consists of applying a 90° RF pulse and then Fourier transforming the free induction decay (FID). During the application of the R.F.. pulse, the receiver of the NMR spectrometer gets saturated and a certain time (called the recovery time or dead time) has to elapse before it returns to its normal operating condition. Therefore the early part of the FID cannot be observed due to the recovery time of the receiver. The usual method, delaying data acquisition until the receiver has recovered, results in the loss of the information contained in the early part of the FID (which is very important for wide lines) and invariably leads to distortion of the spectrum. It also introduces first order phase shifts and a poorly defined base line. To circumvent this problem the NMR spectra were obtained using the solid echo by the simple method of Davis et al (43). This method consists of applying a 90^ 0 pulse followed by another 90° pulse whose phase is shifted by 90 with respect to the first pulse at a time (typically 100-200V|s) later.An echo is formed at 2T due to the refocusing of the nuclear 33 magnetization. By Fourier transforming the echo starting at t=2x the full spectrum is obtained. B) Relaxation measurements. Spin-lattice relaxation times, Tj, were measured using a 180- T-solid echo pulse sequence where the 180° pulse was applied on only every second cycle and alternate scans were subtracted from the computer memory. The intensities of the individual peaks of the Fourier transformed spectra decay according to M0-Mz(t) = 2MQe _T/Tln where MQ is theequilibrium magnetization • and n is the position on the hydrocarbon chain. Fig. 5 shows partially relaxed Fourier transformed spectra for different x values. The repetition rate , or time between pulse . sequences, was chosen to be at least 5 times longer than the longest Tj in the spectrum. The Tj's were obtained from semilog plots of T versus peak amplitudes. For H20 T^ measurement the conventional 180-T-90 pulse sequence was used. Figure 5. Representative deuterium partially relaxed spectra for sodium laurate/water at 13.8 MHz and 90 C. Repetition rate = 10 seconds; 90 pulse length = 4.5 us; number of scans = 200. The spectra were obtained using the quadrupolar echo method as described in the text. 35 Chapter 4 Results 4.1 Quadrupole Splittings. The deuterium quadrupole splittings of the (-CT>^) groups on perdeuterated hydrocarbon chains in the liquid Crystal (La) phase of the sodium laurate-water system, were measured as a function of temperature , position on the hydrocarbon chain and water concentration. Fig. 6a is a representative spectrum fronr.which the splittings are obtained. It consists of 11 overlapping powder patterns arising from deuterons at the different positions on the hydrocarbon chain. At high temperatures and water concentration it is possible to resolve 10 of the 11 peaks in the spectrum. The assignment of the peak positions was t distance , made assuming that the order decreases with ^ from the head groups (39)i Fig. 7 is a representative diagram showing the temperature dependence of the quadrupole splittings for '^^j^-^/GH 0, '^ie quadrupole splittings for the 01-CD2 and the 3,4 positions show a temperature dependence that is quite different from the rest of methyl ene chain deuterons: The splittings increase with temperature reach a maximum at /vL25°C and then show a slight decrease at higher temperatures. In contrast, the splittings for the rest of the chain deuterons decrease with increasing temperature. Fig 8 . shows the quadrupole splittings as a function of position on the hydrocarbon chain at two different temperatures. These splittings are large for the a, and the first few methylenes and become progressively 36 smaller for the methylene pairs at the tail of the hydrocarbon chain. In the same figure it is interesting to note the absence of the "plateau" observed in other systems ; for example (17, 18) . A representative diagram for the dependence of the quadrupole splittings on water concentration is shown in fig. .9 . At low water concentrations the splittings are large for the first few hydrocarbon chain segments and decrease rapidly with increasing water concentration. Near the tail of the hydrocarbon chain the splittings are progressively smaller and seem to have a somewhat weaker dependence on water . concentration. The effect of water isotope composition on the quadrupole splitting of the chain deuterons was investigated for two samples prepared with 1^0 and D20 respectively but having the same molar ratio of water to fatty acid salt. There was no observable difference in the measured quadrupole splittings of the two samples to within the experimental error. The quadrupole splittings as a function of temperature for 23 deuterium in D£0 , Na and the first 3 positions on the chain are shown in fig. 10 . They all. have a similar temperature dependence suggesting a correlation in the ordering of water, counter ion and the hydrocarbon chain segments close to the polar head. A similar behaviour was observed for the potassium palmitate-water system (see appendix A) . A.2 Spin-Lattice Relaxation Times, A) Deuterium Results: The spin lattice relaxation time Tj, for each-CD2 group of a perdeuterated fatty acid chain in the sodium laurate-water 37 system were measured at 13.8 MHz as a function of temperature and water concentration. Fig. 11 shows 1/Tp as a function of chain position at 105°C » The relaxation rates ) are large for the methylenes close to the rln polar region and progressively get smaller towards the centre of the bilayer. In fig. 12 the temperature dependence of -j- is shown. It is Tln clear from fig. 12 that all the T^'s are characterised by an activation energy. Except for the methyl (-CD3) group, the activation energies are approximately the same. Fig. .13 is a plot of the activation energy versus chain position • The dependence of 1/T-j, on water concentration is shown in fig. 14 for the 2 and 10 positions and for the CD^-group. The relaxation rates for the 3-9" positions (not shown in the diagram) also increase with increasing water concentration. The relaxation rates for the methylenes near the polar head seem to be more sensitive to the water concentration than the methylenes near the centre of the bilayer. In fact ^ for the CD3 is almost independent of water concentration. The effect of water isotope composition on the relaxation rates of the methylene chain segments was studied for two samples prepared with H2O and D2O respectively but with same molar ratio of H2O or D2O to the fatty acid salt. The results are shown in fig. 15 for the 01-CD2 . The relaxation rates for the chain deuterons were not measurably affected by changing the isotope composition of the water • 38 B) ct-protons: The spin lattice relaxation time for the cc-C^ in an other wise perdeuterated chain were measured as a function of temperature for two samples prepared with H^O and D^O respectively but with the same molar ratio of water to fatty acid salt. For brevity they will be refered to as d„ C -Na/6H 0 and d C -Na/6D 0 . T here was a small but 21 12 2 21 12 2 measureable difference in the T 's of the a protons of the two samples . 1 the results are shown in fig. 16 . To ensure that the state of the two samples was the same, the quadrupole splittings of the chain deuterons in the two samples were measured and were found to be identical, c) H^O Results The effect of isotope composition on the H^O T^ in the sodium laurate-water system was measured for in 3 different samples: H^O with perdeuterated chains, H^O with orprotonated chains and H^O with all protonated chains. These samples will be refered to by d,,C, -Ka/6H0, (JB-U) d„,C -Na/6H 0 and d C -Na/6H 0 . Where the 23 12 2 21 12 2 o 12 2 subscript on the d stands for the number of deuterated positions on the hydrocarbon chain. The results are shown in fig .17 -. For all the temperatures studied the relaxation rate (1/Tj) for the H20 protons increase with increasing number of protons on the hydrocarbon chain. 4.3 Sources of error Quadrupole splittings were obtained from the peak positions in the NMR spectra. In the presence of dipolar broadening of the quadrupolar spectra, the positions of maximum intensity are no longer coincident with the positions of the 90° edges of the powder patterns. This introduces a small decrease in the measured quadrupole splittings. 39 Ih addition for spectra consisting of a superposition of quadrupolar powder patterns, e.g. of the perdeuterated chains, the overlap between neighbouring patterns causes a small apparent increase in the measured quadrupole splittings. Due to the small magnetic moment of the deuteron and in view of the large quadrupole splittings considered here such effects were assumed to introduce only a negligible systematic error in the measurements. The accuracy of determining the peak positions of maximum intensity is essentially limited by the spectral resolution of the computer, which is 12.5 Hz for a 50 KHz spectrum. However, the signal to noise ratio is different for the different positions on the hydrocarbon chain and ranges from about 400:1 for the CD3 to 50:1 for the a position. Therefore it is more difficult to locate the positions of maximum intensity for the 2-4 positions, thus introducing an additional error of about 50 Hz in the splittings of those positions. The main sources of error in the measurements of the deuterium relaxation rates are the reduction in the signal to noise ratio at long T values and interference between overlapping powder patterns. Since the deuterium magnetic resonance spectrum is a superposition of powder patterns, then the peak intensity of a certain position will contain contributions from those powder patterns with the larger splittings. These contributions are only significant for short xvalues because the relaxation rates increase with increasing splittings. For this reason the T^'s for the overlapping peaks were obtained from the plots of T versus peak amplitudes at longer T values. 40 Figure 6. (a) A deuterium magnetic resonance spectrum for d23C12~Na^6H2° obtained at 120°C and 13.8 MHz using the quadrupolar echo method. (b) A proton magnetic resonance spectrum of the a-protons in a d2^-laurate water sample obtained at 90 MHz and 90 C using the solid echo and quadrature detection. The central peak is due to water and residual protons on the chain. (c£ A 23Na spectrum of d23C12-Na/6H20 at 23.8 MHz and 86 C obtained using the echo method. See text for abbreviations. 80 100 120 140 TEMPERATURE(°C ) Figure 7. Temperature dependence of the quadrupole splittings for d23Cj2-Na/6H20. The numbers beside the curves denote the positions on the hydrocarbon chain. The solid curves are least square fits to the experimental data (dots and circles) where a three parameter fit of the form v(T) = a<n) + a5n)(T - T ) + a n o 1 v o (n) (T - T ) o was used. 42 16.0 IM X 6.0 4.0 2.0 1 — A 1 — • at 105 °C — A A at 135 °C — • A 9 —. A • — A - • A I . .1.1.1 1 A 1 2 (3,4) 6 8 10 12 n (CARBON NUMBER) Figure 8. The quadrupole splittings as a function of position on the hydrocarbon chain for d22Cj2~^a/^2^* 43 3 4 5 6 7 C ( MOLES OF H^IMOLE OF dC-Na) Figure 9. Dependence of the quadrupole splittings on water concentrations at 120 C. The solid curves are least square fits to the experimental data (circles) where a fit of the form . »2 v (c) = b + b- (c - c ) + b_ x-, n o 1 oil. was used. 44 TEMPERATURE (°C) Figure 10. The quadrupole splittings as a function of temperature for deuterium in D^O (triangles), 23Na (squares), the a (open circles) and the 3 and 4 positions (solid circles). The D20 splittings are for d23Ci2-Na/6D20 and the rest are for d23Ci2~Na/^H2^* 45 2 (3,4) 6 8 10 12 n (CARBON NUMBER) Figure 11. T as a function of position on the hydrocarbon chain at 13.8 MHz and 105°C for d23C12-Na/6H20. 2.5 2.6 2.7 2.8 I03/T (KH) Figure 12. Temperature dependence of the relaxation rates of the chain deuterons in d23C12-Na/6H20 at 13.8 MHz. The solid lines are the least square fits to the experimental data where a fit of the form Log Lln = a + b -J n n T was used. The numbers appearing beside the solid lines indicate the position n on the hydrocarbon chain. 47 4.0 — • -< I ? 3.0 ro g . 1 T T i c o LU 2.0 — T I 1.0 1 1 1 1 1,1,1 2 (3,4) (5,6) 8 10 12 n (CARBON NUMBER) Figure 13. Activation energy versus chain position for the deuterium relaxation rates in dOQC, o-Na/6Ho0. 48 l O LU CO 14.0 — o 2 12.0 IQO o 8.0 — 6.0 • 4.0 • • o © 10 2.0 -a i ft X i i A 1 A CD3 I 3 4 5 6 7 C (MOLES OF H20/1M0LE OF C)2-Na) Figure 14. Dependence of the relaxation rates for the chain deuterons on water concentration at 13.8 MHz and 105°C. Positions 3-9 are not shown in the diagram (see text) . 49 Figure 15 . Dependence of the relaxation rates on inverse temperature for the 01-CD2 in d23Ci2_Na/6H2° (open circles) and d23c12~Na/6D2° ( triangles ) at 13.8 MHz . 50 1°- (K) Figure 16 . Dependence of the relaxation rates of the (X-CH2 protons on inverse temperature in ($-w)d2]C12~Na/6H2° (open circles) and (B-w)d2iC12"Na/6l)20 (solid circles) at 90 MHz . 51 Figure 17 . Dependence of the relaxation rates on inverse temperature for H20 in d0Ci2-Na/6H2O (solid circles), (3-w)d2iC12-Na/6H2° (triangles) and d23Ci2-Na/6H20 (open circles) at 90 MHz 52 Chapter 5 The Lipid-Water Interaction A Microscopic Interpretation. 5.1 Quadrupole Splittings, It was pointed out earlier that the quadrupole splittings are proportional to order parameters which can provide inform-tion on the conformations and motion of the lipid molecules within the bilayer as well as on the ordering of water and counter ions at the lipid-water interface. The results shown in fig. 10 on page indicate a correlation between the quadrupole splittings of the deuterons on the first few methylene chain segments, the deuterons in D20 and the 2%a counter ion. The splittings increase with temperature, reach a maximum o at 125 C and then decrease at higher temperature. In contrast the splittings for the rest of the hydrocarbon chain segments shown in fig. 7 on page 41 decrease with increasing temperature. A similar behaviour was observed for the potassium palmitate-water system where this correlation was ascribed to the structuring effect of the water via hydrogen bonding. A model consistent with the experimental results (see Appendix A for details) proposes two configurations which inter change rapidly compared with NMR splittings. At lower temperatures, the water, via some complicated hydrogen bonded structures with the oxygens of the lipid carboxyl groups, imposes a constraint on the first C-C bond direction causing it to be parallel to the normal n to the bilayer, leaving the tail on the average some what tilted. This, for 53 purely geometric reasons, results in smaller quadrupole splittings than those at the high temperatures (see table 1 Appendix A ). Since the lipid-water interaction tends to tilt the hydrocarbon chain, it is in competition with chain-chain (i.e. lipid-lipid) interactions whose influence is to cause the hydrocarbon chain to be parallel to n . When the molecular axis is parallel to n , the quadrupole splittings are larger than for the structure imposed by the lipid-water interaction. Therefore when the hydrogen bonded structures tend to break up at higher temperatures and the chain-chain interactions become more dominant, the quadrupole splittings for deuterons near the head of the chain increase even though the order of the system as a whole decreases. The D20 and 2%a results could also be explained ir terms of this model. For the deuterons in D2O and the 2%a counter ion the average environment in the low temperature configuration are such that their principal parameters which are smaller than those for the high temperature configuration (see table 1 appendix A'). The dependence of the quadrupole splittings on water concentration is shown in fig. 9 on page 43 . There is a marked decrease in the quadrupole splittings with increasing water concentration. It is important to note that the fractional variation in the quadrupole splittings at high water concentrations shown in fig. 18 is greater for the methylene chain segments close to the polar head than those near the centre of the bilayer. This result is easily accounted for in terms of the proposed model. Near the surface of the bilayer it is the electric field gradients (ej .es with n giving average order 54 lipid-water interaction that is responsible for the variation in the order parameters. The more water there is between the bilayers the more degrees of freedom there are for forming different hydrogen bonded structures which would result in an average conformation of the hydro carbon chains that would lead to a reduction in the quadrupole splittings. On the other hand, near the centre of the bilayer, it is the interaction between the chains, which is less dependent on water concentration, that controls the variation of the quadrupole splittings. Samples prepared with H2O and D2O respectively but having the same molar ratio of water to fatty acid salt show no observable difference in the measured quadrupole splittings of the two samples. This indicates that the average o f (3cos20-l) over n thylene motions is not 2 measurably affected by modification of the isotope composition of the hydrogen in the water. 2 (3,4) 6 8 10 12 n CARBON NUMBER Figure 18. Fractional variations of the quadrupole splittings of.the chain deuterons with water concentration at fixed temperature (120 C). Partial derivatives are evaluated at C= 6 56 5.2 Spin-Lattice Relaxation (Perdeuterated Chains) While deuterium quadrupole splittings give information on the orientational ordering and degree of motion of the lipid chains, spin-lattice relaxation measurements allow the determination of a time scale and an intensity factor for the molecular motions. In chapter 2 it has been already stated that for chain deuterons quadrupolar interactions are the main relaxation mechanism. The spin-lattice relaxation rate }• , was expressed in the form (eqn. 24 chapter 2) Al Tl = ON* > ( jl(uo> + J2(2o)o) z where <^|HQ|^ is the mean square value of the spin lattice coupling Hq(t) due to the quadrupolar interactions and Ji(w0) , ,J2(2w0) are the reduced spectral density functions associated with Hq(t) . Thus 1 /-jj- measurements allow the determination of Ji(w0) and J2(2co0) which can be used to define a correlation time for the molecular motions occurring at the Larmor frequency w0 and at 2u)0 . Figure'11 on page 45 shows ^ as a function of position on the hydrocarbon chain. The relaxation rates are large for the -CD2 groups near the polar head and become progressively smaller towards the centre of the bilayer. This indicates that the motions of the chain segments near the lipid water interface are considerably different from those of the methylene chain segments of the tail of the hydrocarbon chain. The temperature dependence of ^ is shown in fig. 12 on page 46 Aln for the different positions on the hydrocarbon chain. All the relaxation 57 rates are characterized by an activation energy. Except for the methyl (-CD3) group, all the activation energies are roughly the same as shown in fig. 13 on page 47 . The decrease in the relaxation rates with increasing temperatures indicates that the short correlation time 2 2 limit (w0T<s.l ; where TN is the correlation time characterizing the motion of the n*"*1 methylene chain segment) is satisfied. Further evidence that this limit is satisfied comes from the direct comparison, as will be discussed below, of the relaxation rates of the a-CH2 at 90 • MHz and the a-CD2 at 13.8 MHz . The detection of distinct types of molecular motion as manifested by different correlation times requires a study of the dependence of ^ on the nuclear Larmor frequency w0 . The larg r proton magnetic moment would permit ^ measurements at much higher values of uQ thus making 1 possible the detection of shorter correlation times. However, in the short correlation time limit, the relaxation rates are independent of frequency (see chapter 2, section 2.2 ) and the expression for ^ reduces to a product of an intensity factor and a correlation time. From the experimental data of figures 12 and 16 , the relaxation rates of the a-CD2 at 13.8 MHz and of the a-CH2 at 90 MHz were found to be in the ratio of 17.8 as compared with 18.7 for the ratio of the square of the coupling constants.This implies that the correlation times associated with the motions occuring at these frequncies are short For a single type of fast motion, it was previously shown (eqn. 25 chapter 2 ) that in the short correlation time limit, the relaxation rates are given by 58 where Sn is the order parameter for the nth C-D bond defined by equation, 2 6 in chapter 2 and the factor 1-Sn takes into account the anisotropy of the system. A logarithmic plot of ^ versus vn (recall the V_ is proportional to Sn ) is shown in fig. 19 for 4 different temperatures. Except for the first few positions, the dependence of ^ on Vn seems 1 ln to obey a power law ( = ^ ) where p = 1.1 at 90°C and becomes smaller In at higher temperatures. It is likely that more complex motions must be taken into account. If the above equation is valid .however,the factor 1-Snhardly changes since Sn~.2 . Therefore changes in the relaxation rates ~ are expected to be largely due to changes in the correlation iln times xn . The dependence of J- on water concentration is shown in fig. 14 In on page 48 . The relaxation rates for the methylenes near the polar haad increase with increasing amounts of water more than the relaxation rates of the methylenes near the centre of the bilayer. For the -CD^ ^ is almost independent of water concentration. In contrast, the quadrupole splittings all decrease with increasing water concentration. This is further evidence that complex molecular motion mediated by the lipid-water interaction must be taken into account. A simple model explaining the influence of the water on the relaxation rates of the methylene chain deuterons will be discussed below. 5.3 A Model for Molecular Motions Mediated by The lipid-Water Interaction. The increase in the relaxation rates with increasing water 59 concentration maybe due to the interaction of the water with the polar heads via hydrogen bonding. Increasing the water concentration will result in greater formation of hydrogen bonded structures (lipid-molecules and the water engaged in hydrogen bonding with the polar heads). Therefore, it is the motion of the combination (lipid +water hydrogen bonded to the polar heads) that has to be taken into account. It is quite possible that increasing water concentration will increase the effective mass of the hydrogen bonded structures and result in slowing down of the rotational motion as well as the twisting, bending and wiggling motions about the long axis of the molecules and thus give rise to higher relaxation rates. In addition , increasing water concen tration will increase the area per polar head (£ee appendix B fig. 29). This will increase the available space occupied by the hydrocarbon chains which would allow for single molecule motions of larger amplitude also around the directorvresulting in larger relaxation rates. However, an increase in the area per polar head would also give more freedom of movement for the individual chain segments causing a reduction in the observed quadrupole splittings (see fig. 9 on page 43 ) 5.4. Isotope effects. The isotope modification of the water may influence the relaxation rates of the chain deuterons in two ways: (i) by changing the spectral densities (or correlation times) through modification of the motions or (ii) by affecting the contribution of the dipolar inter actions to the relaxation rates. In the previous section it has been pointed out that, due to the 60 hydrogen bonding of the water with the polar heads, the motion of the combination (lipid + water hydrogen bonded to the polar heads) should be considered. Since the deuteron is heavier than the proton, then replacing H2O with D20 would result in an increase in the mass of the hydrogen bonded structures resulting in the slowing down of the motions (local bending and twisting around the chain segments as well as the overall translational and rotational motion of the hydrogen bonded structure). This would increase the correlation times and therefore result in larger relaxation rates. However, an examination of the relaxation data for two samples prepared with H20 and D20 respectively but with the same molar ratio of water to fatty acid salt, reveals that this effect is negligible, since there was no measurable difference (within the experimental error) in the relaxation rates of the two samples as shown in fig. 15 on page 49 for the a-CD2 . The results also indicate that the contribution to the relaxation rates due to dipolar interactions is also negligible. This result is not surprising since, as was discussed earlier, the quadrupolar interactions of the chain deuterons are much larger than deuteron-proton dipolar interactions and hence are much more effective in spin-lattice relaxation. We -conclude that just as isotope modification of water does not change the average quadrupolar interaction, as discussed earlier, the fluctuations of the quadrupolar interactions about the average which are responsible for spin lattice-relaxation are not affected significantly. 1 2 4 6 4vn(kHz) Figure 19. A log-log plot of the relaxation rates versus quadrupole splittings for the chain deuterons in d23C12-Na/6H20 at 80°C (solid dots), 105°C (open circles), 125°C (triangles), and 135 C (squares). 62 Chapter 6 The Lipid-Water Interaction Analysis in terms of Macroscopic Variables 6.1 Quadrupole Splittings The order parameters of the C-H bonds along a hydrocarbon chain in a lipid bilayer give a quantitative measure of the nature of the fluidity of the bilayer. These order parameters are proportional to the deuterium quadrupole splittings of the -CD2 groups on the different positions of the hydrocarbon chain. In this chapter a systematic investigation is made of the dependence of these local order parameters on the macroscopic thermodynamic parameters which characterise the lipid-water system. It has been suggested by Mely et al (17) that A , the mean area per polar head, is a "good parameter to represent the average microscopic order" in the bilayer. Their conclusion is based on measurements on the potassium laurate -water system which show that where-as the profile of the quadrupole splittings changes appreciably when the temperature T is varied with the water concentration C kept constant and also when C is varied with T kept fixed, the variation is much less than when both T and C are varied in such a way as to keep A constant.As may be seen from their data which are reproduced in fig. 20 (a.b)the quadrupole splittings in the samples having identical surface area are not identical. Rather the profiles of Sn versus n for these samples resemble each other more closely in a qualitative sense than do the profiles in which C and T are separately varied without keeping A constant . As shown in fig.20C the curves for the quadrupole splittings 63 X -X 30 5= <J 20 \A at c "3." 10 w» u a 9 £X •o a s 3 a A*32.8A' \ . :247.HjO-50*C . :21V.HjO-74*C _l_ JL 1_ 2 4 S 3 10~ 12 Carbon number (from polar head) -JO 02 2 i 6 8 10 12 Carbon number (from polar head) Figure 20 . (a) Order parameter curves obtained for two different lamellar samples of dC^K-R^O,having the same A value. (b) Order parameter curves variations as the area per polar head is increased in the lamellar phase of dC12K-H20 (• :21%-31°C;A :24%-50°C;§ :30%-51°C;-r- :30%-110 C). (The white dots come from computer simulation of the unresolved lines.) Reproduced from reference (17) . 64 16 14 N i 12 CM \10 8 0 9 O o c = 6 j T = 105°C — © C = 5 j T=13 5C: HIM - o • — o — 8 o Q S o • _ i I i I I . I I i i i 2 (3,4) 6 8 10 12 n(CARBON NUMBER) Figure 20C . Quadrupole splittings curves for two samples of dC..-Na/R^O having the same A value. 65 for two samples of dC^-Na/l^O having the same A are also similar. The hypothesis of Mely et al that the quadrupole splittings v^CC.T) depend on C and T in such a way that they vary only if the surface area varies, i.e. vn(C,T) is a function of A(C,T) makes good physical sense if the dominant interactions responsible for the orientational order of the hydrocarbon chain segments are the chain-chain interactions. These inter actions are expected to be sensitive to the average separation of the hydrocarbon chains, which would be simply related to the surface area per polar head. In view of the fact that we have strong empirical evidence that the lipid water interaction has a strong influence on the orientional order of the hydrocarbon chains, especially those chain segments near the polar head, it is of great interest to examine the validity of their hypothesis in a systematic manner for another system. In order to do this, it is not necessary to compare samples having the same values of A at different C and T as has been done by Mely et al . Since A is related to C and T through an equation of state, as will be discussed later, it is a simple procedure, using elementary thermodynamics, to examine the dependence of Vn for any specified variation of the thermodynamic variables so long as data is available over a wide range of any set of independent variables. In this section we carry through such a thermodynamic analysis using the measured values of Vn over a range of C and T which were presented in chapter 3 and the experimentally determined empirical equation of state presented in Appendix B. Thermodynamic Formulation. If the quadrupole splitting of the -CD2 groups on the ntQ 66 position of a hydrocarbon chain is a function of several independent thermodynamic variables {Qj} where Qj could be water concentration, temperature, PH, pressure or ionic species. A differential change dv, in Vn is given by : \ f3vn(...tQ±,..) \ i In this work the NMR measurements were carried out on samples of different water concentration at different temperatures but always under the constraint of the equilibrium water vapor pressure because the samples were sealed. Therefore it will be assumed that there are only two independent thermodynamic variables that have to be considered in this analysis: the water concentration and the temperature. Eqn. [l] then simplifies to. where T and C are the temperature and water concentration respectively. In order to test for the sensitivity of the quadrupole splittings Vn to changes in the different thermodynamic variables, it is sufficient to examine the partial derivatives of vn with T and C keeping A the surface area per polar head constant. That is, to determine whether vn depends on T and C only through A . If these variations are different from zero then A is not necessarily "a good parameter" which [1] 67 characterises the state of the bilayer as proposed by Mely et al . From [2] the partial derivatives of vn with respect to T and C keeping A, the surface area constant are given by All the partial derivatives on the right of equations [3]»[4] are obtained from the NMR experiments except for OC/3T)Aand OT/3C)A which have to be determined independently from an equation of state. In appendix B the equation of state for the Cj^-Na/H^O system was found to be A = AC(T)CP [5] where A is the surface area per polar head, C is the water concentration, p is a constant equal to .24 and AQ(T) is the area per polar head in the limit of zero water concentration and is only a function of temperature. From [3], [4] and [5] ..we obtain: / 3Vn \ - C . 1 / 9Vn \ [6] \3T )c P Ab dT V8C ;T /^L \ . P.AQ.J_ 7\\ [7] V3C ) C dAo hi / N XT dT C All the partial derivatives [ \ and ( ^n j were obtained V a T jc \ a c JT from linear least square fits to the experimental data as indicated in. 68 the captions of figures 7 and 9 on pages 41 and 43. Due to the wide variation in the quadrupole splittings of the methy lene deuterons on the hydrocarbon chain, it is the relative quantities 1 (9vn \ , 1 ( 3vn \ that are meaningful. 311(1 v£\Tc)i 1 / 8vn \ i f 3vn \ Figure 21 is a plot of ^ [ ]A and ~ [jT IC Versus position number n on the hydrocarbon chain. It is clear that 1 ( 9vn \ — I -^"Y" K does not vanish except for the n=7 position which may' be just accidental. In fact keeping A constant seems to be less effective in controlling the fractional variation of the quadrupole splittings of the first few -CD2 groups than keeping C constant. In fig. 22 a plot of 1 (3vn \ \T~ \1TT~ JA versus n the position number on the hydrocarbon chain is shown for four different temperatures. It can be seen that the fractional varation of vn with T keeping A constant for the first 6 positions depends on temperature but is independent of temperature for positions (7-12) inclusive. A similar observation can be made for 1 ( 3Vn \ 1 / 3vn \ — ^ JA. AS SNOWN IN ^S* 23.Fig 24 shows that \ Xc~ /A depends on C only for the first 6 positions. This suggests that the ordering of the hydrocarbon chain segments close to the polar head is influenced by the constraints which water imposes on the first C-C bond via hydrogen bonding. The influence of the water gets weaker towards the centre of the bilayer. This effect can be seen more clearly in fig. 25 where the ratio 69 (_3Vn_\ V 9T /C is plotted aganist the position number for different temperatures. Again this ratio is a function of temperature for the first few chain segments but is independent of temperature for the chain segments close to the centre of the bilayer and asymptotically approaches a constant value of .7. This indicates that the variation with temperature in the ordering of the hydrocarbon chain segments near the lipid-water interface keeping A constant is much greater than it is keeping C constant. However in the centre of the bilayer A tends to be slightly more effective than C in controlling the variation in the ordering of the methylene chain segments. This variation is consistent with the model mentioned in the preceding chapter. At lower temperatures hydrogen bonding of the water with the polar head groups is more favourable than at higher temperature where the hydrogen bonding structures tend to break up. "An effective range" <1JI> which is a measure of the extent of the persistence of the influence of the water can be obtained from fig 25. A semilog plot of r -r„ versus n is shown in fig. 26 where r = limit r ° oo « o n->oo- n r corresponds to an infinitely long chain. It was fitted by inspection to a value of .8 assuming an exponential dependence of rQ on n and an effective range of 4<v5 chain segments was obtained. The above observations imply that insofar as the system studied in this thesis is concerned, A is not the best parameter to characterise the state of the bilayer particularly for the methylene chain segments that are close to the lipid water interface . The 70 conformations of these chain segments are dictated by the hydrogen bonded structures of water at the surface. However in the centre of the bilayer keeping A constant but varying T or C does not influence the variation of the order parameters significantly. This is in agreement with the intuitive physical picture where in the centre of the bilayer it is chain chain interactions and the available, area per polar head which determines the ordering and conformations of the hydrocarbon chains, whereas in the vicinity of the polar head it is the lipid water inter action via hydrogen bonding that controls the variation of the order parameters. 71 CVJ I O x -2 >-4 TO ;?~6 -8 e o Q = A o A Q = C -A • _ — A 9 — A • A A • • - A • • A — A e — A 1 i 1 1 1 1 1 • 1 (3,4) 6 8 10 n (CARBON NUMBER ) 12 Figure 21. The fractional variation of the quadrupole splittings with temperature keeping the area per polar head constant (open circles), and keeping the water concen trations constant (triangles) .Partial derivatives are evaluated at T=105°C and C= 6 . 72 I o o CM I o <r C 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 5 1 -o-2 C =6 l 1 9 86 °C A105°C °12 5°C A135°C 1 o A A 9 O 6 i 1 2 3,4 6 8 10 12 n (CARBON NUMBER) Figure 22. The fractional variation of the quadrupole splittings with temperature keeping the area per polar head constant. 73 0.3 0.2 0.1 c -0.1 -0.2 • 8 6 °C o 105°C n X A 12 5 °C a 8 • 1 3 5 °C A ° — • s 8 e @ - 0 o o • C =6 • 1 I I i i i 2 (3,4) 6 8 10 12 n (CARBON NUMBER) Figure 23. The fractional variation of the quadrupole splittings with water concentration keeping the area per polar head constant. 74 c O -1^ 2 3,4 6 8 10 12 n (CARBON NUMBER) Figure 24. The fractional variation of the quadrupole splittings with water concentration keeping the area per polar head constant. 75 c <T5 C 6.0 4.0 2.0 2.0 -4.0 -6.0 -8.0 — -°-o — _ X -x-C = 6 — A .a a a a 4 ° m -i-—x— * o 1 A "T X *86°C o 105°C A 12 5°C — d 135°C i i . 1 , I.I.I (3,4) 6 8 10 n(CARBON NUMBER) 12 Figure 25. The ratio of the change in.v with T keeping A fixed to the change in Vr with T keeping G fixed. 76 Figure 26. A semilog plot of r abbreviations. r . See text for n 77 Discussion in relation to existing theories. There have been many theoretical calculations on the chain ordering in liquid crystals and bilayer membranes (12-14, 29-32). The most successful of these is Marcelja's (13) molecular field calculation. In the mean field approximation, the interaction energy of a single chain in the molecular field is given by. E - E. fc + E+ PA int disp where E. is the internal energy of a single chain and depends on int the particular conformation, E^ is due to Vander Waal's interactions of the chain with its neighbors via the molecular field. The last term is due to the lateral pressure on each chain and stems from the steric repulsions among the hard coresf'of the atoms. It is proportional to the pressure P and the average cross sectional area A of the chain. Statistical averages are then evaluated by summing over all conformations of a single chain in the molecular field of its neighbors. By adjusting P and assuming an initial orientation for the first C-C bond this model is able to predict the order parameter profile of some NMR (13, 33)data. of However the results do depend on the assumed orientation^the initial chain segment, which implies that a description of the ordering of the hydro carbon chain in bilayer membranes entirely in terms of chain chain interactions is incomplete. Instead of fixing the orientation of the first C-C bond in an ad hoc way a more complete theory should, as could be seen from the thermodynamic analysis, include the lipid water interaction explicitly. 78 6.2 Spin-Lattice Relaxation In the previous section it was found that the lipid-water interaction played an important role in controlling the ordering and fluidity of the hydrocarbon chains within the bilayer. In this section, the same thermodynamic approach will be used to see to what extent the lipid-water interaction affects the dynamics of the lipid chains. Using the same formalism of the previous section, the relaxation rate Rn of the -CH2 group in the ntn position of the hydrocarbon chain is assumed to be a function of several thermodynamic variables Qj Since all the relaxation rates in this work were found to have an activation energy, it would bei more convenient to consider log RQ instead of Rn (since the activation energies a* proportional to (31og Rn/31/T) and are independent of temperature) . A differential change in log Rn is then given by If the same assumptions of the previous section are made i.e. the only independent thermodynamic parameters that have to be cons idered are the temperature and water concentration then 9 reduces to [9] [10] where 3 = - From (10) we immediately obtain 3 79 All the partial derivatives on the right of equation (11) are obtained from the NMR results except for (3C/33)A which is readily obtained from the equation of state and is given by \ 3B/A \ 38/C/ \dC /3 = T2 C I dAo [12] P A dT o Substituting 12 into 11 gives 31°g M = f^iM , T2_C I ^/3 logJR^X 33 /A \ 33 /C P AodT \ 3C / 3 In 13 we identify {^\°^ ^k and {^^~~^)c by activation energies (E ). and .(E )r and. 13 becomes. (Ean}A - <%m>C * P AQ d T \ 3 C /g L1* J where Ean is the activation energy for the relaxation process fo'r the deuterons on the ntn position of the hydrocarbon chain. Analysis of the results. In order to test for the sensitivity of the dynamical state of the hydrocarbon chain on the different thermodynamic parameters we examine the following ratio ( y 2 C . 1 . ^0 f 3 log *n) r15l (Ean}A = * P A0 dT V 3 C / ft L J (Ean)c (^an)c The quantity ^3 log ^n\ was obtained from the dependence of \ 3 C /8 the relaxation rates on water concentration. (E„n)-, was obtained from 80 semilog plots of versus - for each position on the chain. Fig. 27 shows (Ean)A/(Ean)c for the different positions on the hydrocarbon chain. This ratio is greater than 1. for the first few positions but tends towards unity in the centre of the bilayer. ,This result is somewhat similar to those obtained for the order parameters. Near the polar head it is the lipid-water interaction that controls the dynamics of that part of the chain but in the centre of the bilayer it is chain chain interaction that is the controlling factor which in turn is dependent on the available area per polar head. 81 2 (3,4) (5,6) 8 10 12 n (CARBON NUMBER) Figure 27. Ratio of the activation energies for the chain deuterons at constant surface area per polar head to that at constant water concentration. 82 Error Analysis In estimating the error associated with the various Vn V 3 T A vn V 9 C .... . u • . * , I* i , ••• etc, the standard quantities —[ — ^ — formula for calculating the error in a function f of several variables Xj was used, 2 2 6Xj where e is the error in f and 6Xj is the error in Xj . It should be noted that only the errors in the quantites obtained from the NMR / 3vn\ /3vn \ measurements such as \"g"Y"/ a \3~C~ / **' etc» were included in C T computing the error in the derived quantities where the equation of state (eqn. 5 ) was used. This was done because errors arising from the use of the equation of state (i.e. error in p, A , dAo ) as dT determined by X-rays are propagated only in a systematic manner. This is illustrated in fig. 28 where as an example the value of _1_ dAo P A0 dT has been increased by 1 standard error in calculting the quantity 1 / Hi \ — ^ -jpjT J . As can be seen from the comparison of figures 22 28 such errors only shift the whole family of curves without changing the qualitative variation with n and T. 83 • • © 86 °C A A A 105°C o12 5°C — o A ° A« A135°C -A — 4 A 1 A O ' '•"» A • A O — A • 1 i ill 1 1 1 • I (3,4) 6 8 n(CARBON 10 12 NUMBER ) Figure 28. Effect of systematic errors due to the X-ray measurements on the fractional variation of the quadrupole splittings with temperature keeping the area per polar head constant. The parameters P, Ab and dAe/dT obtained from the X-ray measurements have been increased by 1 standard error. A comparison of the data shown above with those of fig. 22 shows that the effect of this systematic error is to shift the whole family of curves without changing the qualitative variation with n and T. 84 Chapter 7 Spin-Lattice Relaxation between Water Protons and Lipid Protons In the previous chapters it was found out that water does play an important role in the ordering of the hydrocarbon chains within the bilayer via hydrogen bonding with the polar heads. The remaining question to be answered is : does water penetrate into the bilayer and if so how deeply? In an attempt to answer this question, the effect of isotopic modification of the methylene hydrogen nuclei on the proton spin-lattice relaxation rate in H2O and also of the effect c" the isotope modification of the water on the methylene protons on the hydrocarbon chain was studied. 7.1 Analysis and Discussion of the H2O results. Fig. 17 on page 51 shows the temperature dependence of ^ for the protons in H20 in d23C12~Na/6H20 , (3-w)d21C12-Na/6H20 and d^C^-Na^R^O , where the first sample contains no methylene protons, the second has -CH2 groups only in the a-position and the third has protons all along the hydrocarbon chain. The subscript on the d indicates the number of deuterons on the hydrocarbon chain. It can be seen that for all the temperatures studied, the relaxation rate ^ for the H20 protons increase with increasing number of protons. This result indicates that the dipolar interactions between the -CrL, and H20 protons contribute appreciably to the relaxation rate of the protons in the R^O V t This work was done in collaboration with Dr. A.L. Mackay and professor M. Bloom. 85 In fact the increase in the relaxation rate of the H2O protons A(i ) in the d sample due to the -CH2 protons and the increase in rl_CH2 0 l the relaxation rate of the R~0 protons A(——V in the doi sample due to the a-CH2 protons were found to be in the ratio R - ^4^CH* " 6 W A(f1)a-CH2 As discussed in the theoretical section, the contribution to the relaxation rate due to dipolar interactions between a pair of spins is proportional to r-^ for two spins at a fixed separation r . It would appear, therfore, that in order to account for the large contribution to \ of the HoO protons due the lipid protons far from the lipid water 1 interface, appreciable penetration of the bilayer by H20 molecules would have to be involved. However, a detailed theoretical analysis (M. Bloom private communication) shows that this experimental result can be accounted for without invoking deep penetration of the water in the bilayer. The mathematical details of this analysis are rather lengthy and will not be presented here. Only the results and a description of the model used and the different assumptions made will be discussed. The Model. Fig. 29 shows a schematic representation of the different spatial regions of the water and lipid separated by a plane at the lipid-water interface. The relaxation rates are proportional to the spectral density 86 Region 1 (water region) Region 2 (lipid region) Region 2 (lipid region) Region 1 (water region) 1 Figure 29 . Schematic representation of the different spatial regions in which a spin of type l(a water proton ) and a spin of type 2 (a lipid proton ) move. The position of the two spins is specified by the space cordinates {z,p ,<f> , £ } in a coord inate system having the z axis normal to the planes separating regions 1 and 2 . Lj is the thickness of the water layer,L2 is the thickness of the lipid bilayer, I is the thickness of a proton layer in in the lipid region , dg is the lamellar repeat spacing and d is the distance of closest approach between spins 1 and 2 . Note that z + £ , p, <J> are the relative positions of the two spins in cylindrical cordinates. The cordinate <J> is not shown in the diagram. 87 functions of the spatial part of the dipolar interaction between the two spins diffusing in the different regions. These spectral density functions are the Fouler transforms of the correlation functions defined by Gm(T) = / Y2m(r0) Y2m(r(t)) \ ^ r3(o) r3(t) / [2] where the Y2m's are the spherical harmonics of order 2. The following assumptions are made in this model: 1) A spin of type 2 (a proton on the hydrocarbon chain) has a uniform probability of being in any of the regions of volume 2 and is given by r pdpdfldg , d<C<d+& L2-d-i<l Q otherwise P2(p, <j>, O = =<(' A£ ' ' L2-d-£<£<L2-d A spin of type 1 (a water proton) will have a uniform probability of of being in any of the regions of volume 1 dz 0<Z<L. p1(z)dz = <; Lj O otherwise 2) The conditional probability that the two spins are in the position specified by the space coordinates Z,p,<{>,£ at time t , knowing that they were in the position specified by Z0,P0»<l>o ?o at time zero is given by. • ' ' -P(p,<j>,C ,z,t;p0,<J>0,C0,ze) = P(C,?0,t)P(p,4.,p0,4>o»t)P(z,z0,t) [3] where P(C,C0,t) = 6(C-C0) ' 88 and is equivalent to assuming that type 2 spins do not move perpendicular to the bilayer, PCP.^.PQ.^Q'O is the solution to the two dimensional diffusion equation with a diffusion coeffiencient DH = Dl + D2|( M given by the sum of the diffusion coefficients of spins 1 and 2 parallel to the plane of the bilayer and P(z,z0,t) is the solution to the diffusion equation characterised by a diffusion cefficient Dj_ for the diffusive motion perpendicular to the plane of the bilayer with reflections at the Z=0 and Z=L^ boundaries. Mathematical procedures for P(p,(j>,p0,<|>0,t) and P(z,zQ,t) can be found in (44). We first consider some limiting cases which are never met in practice but nevertheless will give some insight into the nature of the problem. In all of the cases that will be considered the short 2 2 correlation time limit (oariT«l) will be assumed. The experimental u c result that ratio of the relaxation rates of the 0C-CH2 at 90 MHz and of the 01-CD2 at 13.8 MHz are in the ratio of the coupling constants squared (see chapter 2 ) together with the fact that the relaxation rates decrease with increasing temperature support this assumption. Case i : A lipid bilayer of infinite extent in contact with an infinite reservoir of water. This corresponds to the case where Lj»d . The spins of type 1 (water protons) are assumed to have a diffusion coefficient much greater than the diffusion coefficient of the lipid.(Djj> 10 D2y ) and that D j^. Under these assumption the spectral density 89 functions at zero frequency (short correlation time limit) were found to be M where the am's are normalization constants given by a* = 5/16TT , a\ = 5/24TT , a* = and is the number of spins per unit volume in region 2. To calculate the ratio R defined by eqn. 1 it is sufficient to compare the ratio of two Jm(o) values for &=.8A° and £=9.2A° respect ively. The two different values of Z correspond to the thickness of the layer of protons in the d2l and dQ samples and were estimated- from the thickness of the bilayer and the mean area per polar head as determined by X-ray measurements (Appendix B). A value of 5.75 was obtained for the ratio R, which is in very close agreement with the experimental value of ^6 . However a rough calculation for the absolute values of ^ , by putting 2A° for d ; 1.73xlO~"5cm2/see for and 2 2 ^ 1 7.08.x 10 /cmJ = •—- for the density of protons in the lipid containing a-CH2,gave results that were an order of magnitude too small (.014 sec*~l as compared with the experimental value of .08 sec-"'" for 01-CH2). The results for this case were insensitive to the ratio of D,,/^ Case ii Parallel diffusion of water molecules in a finite water layer. Jn(o) 3TTP a 2 m 8 LlD1 Lo g(—j-) L2-d-Jl •)! 90 In this case the water molecules are assumed to be diffusing in a finite region 0 z L^with a constant diffusion coefficient in the parallel direction, i. e D,^(z)=D|j= constant and D^c 0 . The expressions for the spectral density function are then given by where now the contribution from the successive bilayers has been included. From 6 a ratio R=497 was obtained. Again the calculations for the absolute values of ~ gave results that are an order of magnitude *1 shorter than the experimental values (.02 sec--'- as. compared with the ('•> experimental value of .08;sec ). Case iii Slow diffusion of a boundary water layer. In view of the fact that the model in the previous two cases could predict the correct value for R but gave results that are an order of magnitude too small for the absolute value of the relaxation rates suggests that other mechanisms that could be effective in relaxing the water protons should be considered. One such mechanism could be that a small fraction f of the water binds to the lipid at the water-lipid interface for a long time compared with the correlation times considered here (<~10~^see) and thus would diffuse more slowly than the bulk water with a diffusion coefficient . By incorporating this into the theory the contribution to the relaxation rates from the boundary layer 91 can be shown to be proportional to m ds°b !Cotir(A)-.CotTr(d±^-)4cotTr(y± )-cotTr(Li!^+i) ds dS ds . [9] where a delta function was used for the probability distribution of the bound water layer. In order for this model to predict the correct absolute value for the relaxation rates and also for the ratio R , Djj has to be about thirty times less than the measured diffusion coefficient for the water (1.73 xlO cnr/sec ; Appendix C ). A lower limit for Db is the diffusion coefficient for the lipid molecules themselves. The diffusion coefficient of the lipid chains in the liquid crystal phase of potassium laurate ±s^2.^Xl6^ cm2/sec(45)-As far as we know the diffusion coefficient of the lipid molecules in the sodium laurate-wacer system has not yet been measured. Such a measurement, together with the measurement of the diffusion coefficient of H2O as well as the relaxation rate as a function of its concentration should be useful in formulating a successful theory. From the preceeding analysis it seems that the question of whether water does penetrate into the bilayer or not remains to be resolved. However, the ratio R=6 does not imply water penetration. Further theoretical and experimental work is required to resolve this problem. It should be noted that information on the extent of water penetration into the bilayer can also be obtained from neutron diffraction 92 measurements. Buldt et al (37) have recently reported some measurement on a phospholipid water system where they conclude that water penetrates into the bilayer up to the glycerol backbone of the lipid molecules. Earlier measurements by Schoenborn on a similar system did not have sufficient resolution to detect such water penenetration. 7.2 CC-CH2 Results The temperature dependence of the relaxation rate for -^CR^ in the d2j-laurate for two samples containing R^O and D2O respectively is shown in fig. 16 on page 50 . There is a measurable difference in the CA-CH2 relaxation rates for the two samples, indicating the influence of the dipolar interaction between the H2O protons and OC-CH2 protons. The fact that the a-CH2 protons and the H20 protons still have distinct T^ values when they interact with each other excludes the possibility of long correlation times for the dipolar interaction between the -CH2 protons which would otherwise bring the two spin systems to a common temperature. Short correlation times for the dipolar interactions between H2O and CV-CH2 protons are expected because of rapid diffusion of water and also of the hydrocarbon chains. The model used to interpret the relaxation of the H2O protons due to the dipolar interactions with1 the lipid protons can also be used to calculate the relaxation rate of the lipid protons due to the same interaction. Denoting the number of H2O and -CH2 protons by and NB respectively, the model assumes each of the A spins to be equivalent and independent of each other and the same for each of the B spins. It is then easy to show that for small perturbations to the spin-lattice 93 relaxation rate of the A system (H20) , A(1/T1)A is related, to that of the B system (CH2) by / M "l^' ~ NB R = " ~ % since the gyromagnetic ratios and spins of the A and B systems are exactly the same. For the d2i sample used here NB/NA = 1/6 . In a previous publication ( 46 ) we have reported a value for R of .. 6 ± . 3 which was the average value calculated from the data of figures 16 and 17 over the temperature range 80-115 °C . It was initially argued that this large discrepency could not be attributable to sample differences or systematic errors in the experiment. This argument was based on the fact that the measured deuterium quadrupole splittings for the (3-u) positions of the d„ C -Na/6H 0 , d C -Na/6D 0 and 21 12 2 21 12 2 d C -Na/6H 0 were found to be identical, implying that the hydrocarbon chains have the same flexibility gradient and therefore it would be unlikely that the molecular motions which cause spin-lattice relaxation in each of the samples are appreciably different. Since this paper was written we have made a critical estimate of the relaxation data which has led us to the conclusion that the difference between the experimental and predicted values of R may not be inconsistent with each other within the experimental error. From figures 16 and 17 the measured values of R* are shown in table 1 for several temperatures. In the high temperat ure region it is difficult to make any conclusions about this ratio due Temperature (*C ) 120 115 105 90 80 Table. (1) Ratio of the change in of H2O protons due to a-protons ti the a-protons due to the water pr< / R 2.6 + 2.2 1.0 + .7 .7 + .3 .6 + .3 • k ± .1 the spin lattice relaxation rate rate < the change of the relaxation^of tons in d2]C-Na/6H20 . 95 to the large uncertainties involved. At lower temperatures this ratio is almost twice the predicted value of J_ within the experimental 6 uncertainties. These large uncertainties arise mainly from the fact that A[ -j- ) and A( ^ ) are obtained by subtracting large number of comparable magnitudes. Moreover systematic errors in the T^ measure ments cannot be exluded. In view of the large uncertainties in It'we believe that further experimental work is needed. Such work should involve experiments where the contribution to the spin lattice relaxation rates due to the intra molecular dipolar interactions are reduced. For the 1^0 molecule the intra molecular dipolar interaction between the two protons makes an important contribution to spin-lattice relaxation: The proton Tj for HDO in D20 is much longer than that for D20 . Therefore the contribution of the intramolecular dipolar interaction to the water relaxation can be eliminated by carrying out experiments analogous to those described above using samples which contain a small amount of HDO in D20 . A study of R.' for higher values of should also be carried out, since this would give a larger effect which would be easitr to measure. It should be noted that the value of NB/NA predicted by the model calculations for R does not take into account the correlated motions of the two CH2 protons. General coupled equations for an AB2 system are available, for example (47,48 ). The parameters thereby introduced maybe examined by measuring quantities such as the dipolar relaxation of the CH2 spin system (49 ) or by the equivalent methods of "selective inversion recovery" ( 50 )• Summary and Conclusions The flow chart shown below summarizes the areas of investigation in the thesis and the conclusions reached. PROBLEM: To understand the lipid-water interaction How deep does water penetrate into the bilayer 7 i QUESTIONS How does the lipid-water interaction influence the conformations and motions of the lipid chains ? Fluidity of the bilayer NMR relaxation measurements of H.O and lipid protons;effeet oflsotopic modification on the relaxation rates I-Water ( %c a, H20 with H20 with HO with all pro-perdeut a-proto-erated nated tonated chains chains chains -protonatd chains H20 with D20 (b) (c) (d) (e) Spin-lattice relaxation between water protons and lipid protons; Protons deep in the bilayer make a substantial contribution to the spin lattice relaxation rates; Theoretical analysis accounts for the results without Invoking deep penetration of water In the bilayer . -{ NMR experiments}-1 Spin-lattice relaxation —>Information on the dynamics of the lipid chains l/TJn(C,T) The nnodynamic Vn<C,T) analysis Spectroscopy Quadrupole splittings —^Information on the ordering of the lipid chains,water'(D20) and counter ions Perdeuterated chains D20 23Na Correlations.empirical evidence for a lipid-water Interaction The lipid-water Interaction has a strong Influence on the conformations and motions of the lipid chains Microscopic model for the llpld-water interaction V A description of the state of the bilayer entirely in terms of chain-chain Interactions ,as done in existing theories,Is not complete. A complete theory should include the lipid water Interaction explicitly. APPENDIX A Chemistry and Physics of Lipids 20 (1977) 115-129 ©Elsevier/North-Holland Scientific Publishers, Ltd. THE TEMPERATURE DEPENDENCE OF WATER AND COUNTER ION ORDER IN SOAP-WATER MESOPHASES. A DEUTERIUM AND SODIUM NMR STUDY* K. ABDOLALL, E.E. BURNELL3 and M.I. VALIC Department of Physics and Department of Chemistry", University of British Columbia, Vancouver, B.C., Canada, V6T1W5 Received January 3, 1977 accepted March 23, 1977 Deuterium magnetic resonance (DMR) spectra of the water and hydrocarbon chains in potas sium and sodium palmitates and sodium magnetic resonance spectra of sodium in sodium palmitate demonstrate correlations between water, hydrocarbon chain and counter ion order. In the lamellar phase of potassium palmitate the order parameters inferred from DMR splittings of D,0 and the first few methylene chain segments initially increase and then decrease with increasing temperature. This is explained in terms of a model where the lipid-water structure at low temperatures imposes a direction for which all the order parameters are smaller than for the higher temperature structure for purely geometric reasons. As temperature increases the structuring effect of water decreases and there is an "apparent" increase in order until at even higher temperatures there is an intrinsic decrease in order parameter. In addition, for potassium palmitate the DMR splittings of D,0 and the first few methylene segments indicate a "phase transition" within the liquid crystalline phase. I. Introduction In order to understand the role played by water near biological membranes, it is essential to understand the mechanism of lipid-water interaction. In the presence of water, lipid molecules form a variety of lyotropic mesophases characterized by the existence of long range order and short range disorder. These phases have been identif ied by X-ray studies [1 ], nuclear magnetic resonance [2,3] and other techniques [4]. Their description has received considerable attention in the literature [5] and will not be elaborated on here. Of particular interest are the three lamellar phases: liquid crystal (La), gel (L^) and coagel. In these phases the soap molecules form bilayers which are stacked parellel to one another and separated from each other by water and counter ions. In the La phase the chains are flexible (melted) and the soap molecules undergo rapid lateral diffusion and rotations abcu» their long axis, while in the lower tempera-* Research supported by the National Research Council of Canada and a special Killam-Canada Council Interdisciplinary Grant. US A'. Ahdolall el al.. NMR studies in potassium ami solium palmilales lure 1.0 phase (he chains arc slil'f(frozen), fully cxlcndcd, lightly packed, and interdig-ilaled. In the cnagcl phase Ihc lamellar structure is relained but waler is squeezed out into pockets of hulk or "lice" waler. To obtain infoiniation about ilic ordering of cither the water at the lipid water interface or the lipid molecules within the bilayer, it is necessary to use microprobes that are sensitive to their local environment as well as to the dynamics of the system. Interactions between nuclear quadrupole moments and the electric field gradients arising from the surrounding charges affect the NMR spectra in a dear-cut way and enable such information to be obtained. From such studies there is considerable evid ence that water near model and biological membranes possesses significantly more strucluic than bulk water |(>|. This is paper III of a series of three papers on the NMR of soap water mesophascs. Paper I |7| discusses hydrocarbon chain disorder as observed using DMR, and paper II |8| discusses the proton magnetic resonance (I'MK) from potassium palmitate (/}Lo)-dj». In this paper, results as a function of temperature of an NMR study of waler,sodium counter ion and hydrocarbon chains in potassium and sodium palmitates are reported. In particular, we demonstrate a striking correlation between the ordering of water and the first few methylenes of the hydrocarbon chain. An explanation for this correlation in terms of a simple model is given in section IV. We also report a new -fluid fluid "phase transition" within the lamellar phase of a potassium palmitatc-water system. This ttansition manifests itself in the order parameters of the water and the methylenes near the polar end of the hydrocarbon chain. II. Experimental Palmilic(C|6)acid was purchased from Hormel Institute or Calbiochem (California), and was used without further purification. Dcuterated compounds were prepared in our laboratory. The fatty acid salt was prepared by dissolving in ethanol stoichiometric amounts of dry fatty acid and sodium or potassium hydroxide, and slowly crystallizing the fatty acid salt. After filtration and washing with ethanol the precipitated soap was dried at I00°C under vacuum for several hours. The soap-water mixtures were made by weighing the corresponding amounts of dry soap and D20 or HjO into a glass tube and scaling under dry nitrogen gas. For the sodium soap a quartz tube was used. The samples were mixed by ccntrifuging back and forth through a constriction in the tube. Before each cenlrifugation the samples were heated well above the gel-liquid crystalline phase transition temperature. Further homogenizalion was accomplished by leaving the sample in an oven for several days at a temperature above its gel-fluid phase transition. It was found that the variable temperature unit (gas flow system) suppled with the spectrometer was not adequate when dealing with soap-water mixtures. Therefore, a temperature controlled oven (paper 1) with greatly reduced temperature gradients in the sample was constructed and used for the measurements. K. Abdokll el al., NMR studies in potassium and sodium palmilales 117 The NMR measurements were carried out on a Bruker SXP 4-100 pulsed NMR spectrometer. The quadrupole splittings were obtained using the quadrupolar echo by the method of Davis et al. |9|. 111. Results ^The interaction between nuclear quadrupole moments and electric field giadients efg at nuclei gives rise to splittings in deuterium and sodium magnetic resonance spectra. For water deuterons chemical exchange takes place between nuclei in different sites and. if the exchange is much faster than the splitting difference, the observed splitting is a weighted average over the different sites |10] .For an oriented sample, where the normal n to the bilayer makes a uniform angle fl with the magnetic field direction, the high field deuterium (or sodium) NMR spectrum consists of two (or three) peaks separated by Av=l£ PiK)iSj (3 cosJ a- 1) I (I) where p, is the fraction of nuclei occupying site i with associated quadrupole coupling constant VQ and order parameter Sj (discussed more fully in paper II). For an axially symmetric efg. St is given by Si = 3/2cosJ iJj-1/2, (2) where i>; is the angle between n and the efg principal axis and the bar denotes a time average. Hq. (I) can be applied to (he case of hydrocarbon chain and water deuterons and counter ions. However, there are differences among these three cases. For example, all hydrocarbon chain CD2 deuterons are chemically equivalent and will therefore have the same I>Q, . Furthermore, since I>Q is of intramolecular origin (C-D bond) it is not expected to vary with temperature. Consequently, the splittings Ai> yield order para meters directly. If the efg is axially symmetric about the C-D direction, ^ equals SQJ, the order parameter for the bond direction. The case of water and counter ions is more difficult because pjr I*Q and Sj may all differ from site to site, hence the separation of the various terms in eq. (I) is not possible without assumptions. For DjO the largest contribution to I>Q is from the intramolecular 0-1) bond, and its value is close to that for ice. Therefore, all fg. are similar and are not expected to vary significantly with temperature. The measured splittings are thus roughly proportional to an average order parameter S = ?P,S,, enab ling a qualitative comparison of the experimental Ac with model calculations. Also, we do not expect p; to vary greatly with temperature in the narrow temperature region of interest, and hence the temperature dependence of the observed splittings reflects mainly changes in order parameter. v© 118 K. Ahdolall ttal., NMR studies In potassium and sodium pahnltates In contrast, theefg at a counter ion (e.g. sodium) nucleus arises from intermolccular sources (polar head group changes and water dipoles, i.e., via the asymmetric distribu tion of waters of hydration). Therefore, one expects to have a distribution of fy (as well as pj and S,) which can be quite temperature dependent, thus making the inlcrpre-tation of Av in terms of definite order parameters rather difficult. However, should the counter ion splittings correlate with those of hydrocarbon chain and water deuterons then llie temperature changes of counter ion Av should give an indication of changes in order parameters. It will be shown that this is the case for the sodium splittings of perdeuterated sodium palmitate dC16Na. A. Potassium palmitate Non-oriented samples yield essentially the same information as oriented samples. For an unoriented sample there is a random distribution of all values of and the deuterium magnetic resonance spectrum (see fig. I for representative half spectra) consists of a broad absorption curve with major peaks separated by Avj2 (paper I and Bloom el al. (111). The spectra of fig. I show"the presence of residual quadrupolar splittings indicating that the water molecules are located in an anisotropic environment in which they experience non-zero time averageikelectric field gradients and are said to be partially oriented. The anisotropy is imposed by the presence of the interface and the charge distribution near it, as well as by the hydrogen bonding of the water mole cules and hydialed ions with the polar heads. For the liquid crystalline (fig. I a) and the gel (fig. lb) phases the D20 deuterium spectrum is the usual "power doublet", the Fig. 1. Representative half spectra for D,0 In C„K In (•) lamellar phase, (b) gel phase and (c) mixture of ge! and coagel phases (see text). K. Abdolall tt al., NMR studies In potassium and sodium pclmilalet 119 TEMPERATURE ("CI TEMPERATURE l"C) Fig. 2a.The temperature dependence of the D,0 and hydrocarbon chain deuteion quadrupole splittings for a perdeuterated, dC„ K sample. In this case the splittings were measured from the centre to the corresponding peaks in the absorption spectrum. It should be noted that the experi ments reported in Fig. 2a were performed to study lipid-water correlation effects, and that the signal to noise ratio was not as good as that for the spectra reported in paper 1. Quantitative diff erences in the CDj order parameters between Fig. 2 and paper I should therefore not be taken seriously. Fig. 2b. Temperature dependence of the quadrupole splittings of D,0 and a-CD, in the perdeuter ated dCltK sample. The insert is the temperature dependence of the D,0 splittings in a protorutcd C,,K sample. lower temperature spectrum of fig. lc contains in addition a central peak due to the isotropic "bulk" (free) water squeezed out from the water layers of the lamellar phase. This isotropic water is indicative of a coagel phase, and, in principle, the number of water molecules associated with different phases can be determined from NMR inten sity measurements. It is likely that fig. Ic is a spectrum of a sample containing a mix ture of gel and coagel phases. Alternatively, this spectrum could represent different phase separation zones within the gel region. More work, including X-ray studies, ii needed for a complete understanding of soap structures in the solid phase. The quadrupole splittings, Av, as a function of temperature for the water deuterons as measured from spectra similar to those in fig. I are shown in fig. 2 for perdeuterated potassium palmitate dC,„K. The splittings are appreciable in the low temperature coagel and gel phases and the sudden decrease in splitting denotes the coagel or gel to liquid crystalline phase transition. This decrease may be due to lesser intrinsic order of the - ) 120 K. Abdotatl et at., NMR studies In potassium and sodium palmitate* water at the lamellar surface, changes in the average direction of water ordering, and reduction of long-range order throughout the water layer. As shown in fig. 2 the splittings for DjO as a function of temperature in the liquid crystalline phase increase with temperature at first, exhibit a maximum between 70— I00°C. and then decrease. The deuterium splittings of the chain deuterons were also measured as a function of temperature. The results, shown in fig. 2a, indicate that the splittings of the first few methylene pairs exhibit a temperature dependence similar to that of D20. This suggests a strong correlation between the ordering of the water mol ecules and the first few links of the hydrocarbon chain. A model explaining this correl ation will be discussed below. The results in fig. 2a at ~ 50-60°C indicate a discontinuity in the splittings which is ascribed to a liquid crystalline-liquid crystalline "phase transition". The discontinuity occurs only for the water and the first several CD2's of the hydrocarbon chain and is thus suggestive of changes in the lipid-water structure and possibly a conformational change in the hydrocarbon chain near the lipid-water interface. Since this transition is a new phenomenon, the experiment was repeated and the temperature was -iried in smaller intervals. The results for the o-CDj and DjO are displayed in fig. 2b. The insert in the same figure shows the results for a protonated C,6K sample. It is clear that the Fig. 3a. Spectra taken at three different temperatures showing the D,0 splittings near the "fluid-fluid" phase transition in the dC„K sample. The two smaller peaks are due to -CD, groups. The shoulders of the D,0 signal are also visible. Fig. 3b. Representative half spectra at three different temperatures for the perdeuterated dC„K sample showing the water peak and the peaks due to the methylene pairs on the hydrocarbon chain. The right hand part of the spectra is expanded vertically 16-limes. The arrows Indicate the splittings from the a-CD,. K. Abdolatl tt at., NMR studies In potassium and sodium palmitates 121 fluid-fluid transition occurs at somewhat different temperatures in different samples; these differences may be due to sample preparation techniques and/or to isotope effects. Representative spectra for D20 and chain deuterons of dC„,K in the region of this fluid-fluid phase transition are shown in figs. 3a and b. These spectra show the water line (fig. 3a) as well as the lines due to all chain deuterons (fig. 3b). Note thai the or-CD2 and the D20 lines are each split near 50°C indicating the co-existence of two fluid phases. (See also the results of paper 1 where the a-CD2 splitting is more clearly shown, and paper II). B. Sodium palmitate Sodium has a nuclear spin of 3/2 and the high field NMR spectrum for an unoriented sample is similar to that for deuterium with two major peaks separated by Ac but in cludes a sharp orientation independent central line (if chemical shift anisotropy is neglected); fig. 4 is a plot of the quadrupole splittings of D20, "Na counter ion, and chain deuterons as a function of temperature for dC,6Na. As seen in fig. 4 the temper ature dependence of the Na, D20, and first several CD2 exhibit the same slope which demonstrates a definite correlation between the water, "Na counter ion and the first few hydrocarbon chain segments. All of these splittings decrease less rapidly with tem perature than those from CD2's near the methyl end of the chain; this agrees with pre dictions of the Marcelja mean field calculations for chain order parameters [12] be-10 i W i •op I I I 100 110 120 OO TFMPF.RATURE (*C ) Fig 4 Temperature dependence or the quadrupolar splittings for D,0. "Na counter ion and deuterons on the hydrocarbon chain in a dC,.Na sample. (The C,. results were obta.ned from a H,0 * dC„ Na sample because the -CD, line overlaps with that of D,0 making the separation or the two very difficult.). 122 cause increased temperature yields increased area per polar head and decreased lateral pressure. In contrast toCltK no fluid-fluid phase transition is detected,and no increase in splittings with temperature is observed in the lamellar phasefor the DjO, Na* or CDj's. Below the coagel-lamcllar phase transition temperature (~ 83°C), we were unable to observe quadrupolar split lines for Na' or D,0. contrary to the case for dC)6K. Only central Na* and D30 lines were observed. Further work is needed on this phase. IV. Discussion Water exists as various hydrogen bounded structures, and also interacts with the polar head groups of fatty acid salts via hydrogen bonding and other electrostatic in teractions. These phenomena give rise to structuring effects: i.e. lipid and water struc tures mutually affect each other (13]. The details of such an interaction have not been included in many of the theories [12, 14] which attempt to explain hydrocarbon chain order parameters. The interaction between lipid and water is demonstrated in a particularly interesting way by the results of this and papers I and II which show that the NMR splittings of DjO, a-Cllj, a-CDj and the first few CD2's of the alkyl chain for the lamellar fluid phase of C|SK all increase with temperature to a maximum value, and then slowly decrease (figs. 2, 3 of I and fig. 2 of II). In general, such an increase in splitting might be though of as an increase in order. However, this is not necessarily true; a strong interaction between lipid and water might well impose a constraint where all the observed order parameters (eq. (2)) are small for purely geometric reasons. A decrease in the strength of this interaction at higher temperatures could lead to larger observed values of S even though the system may be inherently less ordered. To dem onstrate how such ideas lead to new information about lipid structure and lipid-water interaction, we present here a simple model which explains qualitatively the results of this and the preceding two papers. A. A model consistent with the experimental results of papers Our model proposes two rapidly (compared with NMR splittings) interchanging configurations, A and B, for the lipid-water interface [15]. The main idea here is that at lower temperatures (configuration A) the lipid-water interaction imposes ordering effects near the polar head region. In particular, the first C-C bond is more or less normal to the interface leaving the tail on the average somewhat tilted. At higher tem peratures (configuration B) the lipid-water structuring effects are reduced and the dominant steric interaction imposes lipid ordering such that the chain is more or less parallel to n, the normal to the lamellar surface, leaving the first C-C bond at an angle to the surface. The two configurations are now described in more detail. 1. Configuration A As an example, suppose that at low temperatures a lipid-water structure similar to 123 Fig. 5. Model for the lipid-water Interfac* 1A\ B ^ • B) P-domta.,, connpuratlon ' *^ > aCTT^""''0" " low "m'™'u"-'•on, with one puch* rotation are .hownTn a1,1 ?' l" confo'°»<™ «* conform.-See text To, further expiation. "" ,he °"CH. 'HH '» Perpendicular to n. 124 K. Ahdotall et al. NMR studies in potassium and sodium palmltates that shown in fig. 5a (where two of the molecular conformations of the first five car bons are shown) is energetically favoured. For this configuration: (i)The first C-C bond vector rcf. is aligned at some angle to n, such that for the o>CII} both the proton-proton vector r(l(| and the carbon-deuteron vector rcn make an angle some what less than 00° with n. (For simplicity the diagram shows the first C-C bond par allel to nr. a tilt out of the plane of the paper must be introduced to explain the Q-CHJ results), (ii) The average environment of the water deuterons (and of the sodium coun ter ions) are such that their principal elg axes make angles with n giving average order parameters which are smaller than those for configuration B. The diagram is drawn with the water deuteron-deuteron vector rDD perpcndicrhr to n and the bisector of the angle DOD parallel to n. 2. Configuration B At higher temperatures the extra thermal energy allows fewer hydrogen bonds lead ing to a decrease in the lipid-water interaction. As an example we suppose (fig. 5b) that the long axis of the hydrocarbon tail for the all-trans conformation lies along n such that for the a-carhon both r(p and are perpendicular to n. This configuration should lead to minimum free energy if hydrocarbon intermolecular interactions are dominant. For the purposes of illustrative calculation we allow isotropic rotation of DjO about the hydrogen bonded OD bond shown in fig. 5b. Two lipid molecular con formations are shown. B. Simple calculations based on the model To illustrate the effects of different structures on the observed NMR splittings, we have listed in table I calculated values of the various order parameters (eq. (2)) for each of configurations A and B. The a-Crl^ has not been included in this simplified model; however, similar qualitative results will obtain for all order parameters (includ ing cr-CH)) if the first C-C bond makes some appropriate angle with n. The values in table I are averaged over the various molecular conformations of the hydrocarbon chain. The following assumptions have been made for the calculations: (i) All CCC and.CCD angles are tetrahedral. For water, angle MOM is 105°. and the angle between lone pair electrons is 120° (16). For ease of calculation, conformer rotations of 120° are used rather than the more likely value of 112.5° |I7]; this does not significantly modify the calculated order parameters, (ii) There arc rapid (at least threefold symmetric) motions of lipids, water and counter ions about n such that the anisotropic interactions are motionally narrowed and the residual projected along n |I0, 18]. (iii) The average order parameter is given by eq. 2, with 0 being the angle between n and the vector of interest (tIM, T(-n, r00, qn). All efg tensors are assumed to be axially symmetric along the bond directions. For sodium the efg used is Ihat^ from the oriented water lone pair electrons .with the principal axis qn indicated as efg in fig. 5. (iv)pji (pji) is the probability that the bond between carbons i and j is trans (gauche) to the precedingCj-2-Cj-| bond, with pjJ + 2p'J = 1. K. A bdohtt et al., NMR studies In potassium and sodium palmltates 125 Table I Calculated order parameters2 Configuration A Configuration B 1 1 "3 *2 3 F' 3 V 2 Pl D,0 006 0.15d efg -i -0.18d "Calculations are averaged over conformations t^i^t^, tafl'M**' WATS* *ni W&U-I'or A g^j and tQ0 give equal contributions to order parameters. b Assumes that the C-C bond is at an angle 35V,° to n. Motions about COO-C0 axis are not con sidered (i.e. p°f - 0). cFor gauche conformation S • Vi (P, cos (90) + P, 00s (35M)) • 0. ''Assumes hydrogen bond parallel to first C-C bond and free rotation about 0. . . . D-0 axis. C. Discussion 1. . ie lamellar La phase As shown in Fig. 2 (and papers I and II) the experimental order parameters for DjO, or-CHj and the first several CDj's of the hydrocarbon chain in the liquid crystalline phase of CmK increase with temperature at first to a maximum and then gradually decrease. On the basis of ourmodel these results can be explained in the following way. We note that the calculated order parameters (table I) associated with configuration B are all of larger absolute value than those for A. According to the model the observed order parameters are the average over rapid exchange between both configurations, A being favoured at low, and B at higher temperatures. Hence, as temperature is raised we expect the observed order parameters to increase with (he increasing probability of structure B, until such a point that the probability of configuration A becomes small. In addition, with increasing temperature the thermal motions lead to an overall intrinsic decrease of the "molecular" order parameters, as observed for nematic liquid crystals (19). Thus, the model indeed predicts the experimentally observed temperature dep endence of order parameters. Incidentally, for dC|6Na (fig. 4) no increase in order parameters is observed, suggesting that configuration B already dominates at temper atures just above the coagcl-lamellar phase transition. 126 K. Abdolcll etal.NMR ttudiet in potassium and sodium palmitates 2. "Plateau" The experimental order parameters for -CDj's along the hydrocarbon chains in soap-water systems (paper I [7, 20)) exhibit a constant value for the first several —CDj's at low temperatures in the L0 phase. This observation, referred to as the "plateau" in the literature, is usually associated with the influence of "kinks" [21 ], but it can also be accounted for with the proposed model if the lipid-water interaction in the liquid crystalline phase affects to a progressively lesser extent those CD] groups further removed from the polar head region. Let us assume that there is minimum free energy for the intermolecular forces between hydrocarbon chains when the long axis of the hydrocarbon tail lies along n(a situation facilitating the rapid rotational motions about n in the lamellar phase (10, 18]).For configuration B where, as discussed earlier, the water has little or no ordering influence, the steric forces are dominant. However, for configuration A where the first C-C bond lies along (or close) to n. the long axis of the hydrocarbon tail for the all-trans conformation does not lie along n. Since the influence of the surface becomes progressively less as we go down the chain, the steric forces will tend to align parallel to n, that part of the chain which is further from the polar head. This could involve molecular conformations which are slightly distorted from normal trans and gauche. Thus, the orientation of the molecular axis gradually changes from that of configuration A to that like configuration B, and according to table 1 this corresponds to an increase in the observed SCD- On the other hand,because of the extra conformations available to those carbons far removed from the polar head, a general decrease in order parameters is expected as we go down the chain. Thus, the increase predicted above and the general decrease act oppositely and cancel each other in the region of the first few - CD2 links leaving a constant value for the SCD as ob served experimentally. 3. "Odd-even"effect Measured values (figs 2 and 4 of this paper and fig. 3 of paper I) of the order par ameters Sen at lower temperatures in the L;, phase show that Sjj = Sy > Sa * Se, etc. This applies all along the chain except for those -CDj's near the -CD3 end where each successive -CD] has lower order parameter. This low temperature behaviour of SCD'S is referred to in the literature as the "odd-even" effect (7, 20]. With increasing temp erature, the "odd-even" effect is observed to disappear progressively in the tail-head direction until at sufficiently high temperatures each successive -CD2 has lower order parameter (fig. 3 of paper I shows that all except (J and y peaks have separated at ~90°C). This effect is readily accounted for by our model. Indeed, if the high energy conformationsg*g* are neglected [17], thenconfiguration A predicts equal values for Sfl and Sj. However, ISjjl and ISy I are also predicted to be smaller than ISQI due to the term 2/3 p^7 (table 1). Similar results would obtain for Sj and Se (if the confor: mation gy5'{egf j is neglected), and for each pair of carbons down the hydrocarbon chain to a progressively lesser extent (depending on the probabilities of certain confor mations for which the order parameters for both members of the pair are different), thus giving rise to the "odd-even" effect. On the other hand configuration B predicts K. Abdolatl etal., NMR studies in potassium and sodium palmitates 127 (table 1) that each successive CDj will have lower order parameter. Hence, as temper ature increases the experimental ratio Sg/SE increases from the low temperature (con figuration A) value of 1, where the 6-CD2 and e-CD2 peaks overlap completely, to a higher value where separate peaks are observed. This is consistent with configuration B becoming more probable at higher temperature. Further down the chain the ratio differs from 1 for all temperatures; this is consistent with the long molecular axis be coming aligned along n as one leaves the polar region. 4. The gel phase The experimental order parameters for the Q-deuterons and a-protons in the gel phase are of slightly lower value thart-those for other chain deuterons [papers 1,11). Furthermore, the experimental valuesare such that ISHH ' K slightly smaller than ISrjD both being < M. However, in the lamellar phase ISHH I is equal to or slightly greater than ISCD The results for the gel phase can be explained by the model if a gauche rotation near the polar head (between 0- and 7-carbons, say) is invoked. Such a rotation might accomodate more readily the water and counter ion structure and leave the rest of the tail all-trans and roughly parallel to n, hence facilitating a rotation about a symmetry axis m (not necessarily 1 n). As a result, the first C-C bond will be inclined to n in such a way that both TQ) and ri(H for the cr-position make an angle with n f somewhat less than 90" in agreement with the experimental values ISHH I ~ ISCD V4-Assuming that in the gel phase the soap molecules are rigid and consist of two rapidly interconverting g|j to g^g conformations* and that the only other motion is rapid rotation about m then the results of paper I and II allow the calculation of the angles 0, that various vectors r make with m. The values SQ-CHJ = - u-34 (fig. 5, paper II) gives = 109° for the a-CH2;the value Sa<:D} = -036 (paper I) in the conjunc tion with SO>CH2 gives Syy = - 0.40 where y is thehisector of angle HCH. This value of "yy gives Oy = 105°. Using the values for angles CCC = 112°, CCH = HCH = 109° [17) and the result that the value for Sct>, for most of the chain is -0.44 (paper I) leads to a rotation of either 21° or 76° about the Cg-C7 bond for formation of the gauche con formation. The latter is close to the "most probable" value of 112.5° for the dihedral angle, and distortions from this value are not unexpected [I7J. The above calculations are crude, and a more plausible picture might be a rotation about C^-C-y of somewhat more that 90° followed by rotations of a degree or so in the opposite sense about bonds further down the chain, hence yielding a distribution of order parameters for different CD2's as indicated by the results of paper I. The fact that for the cr-position the order parameter SHH is less than SCD in 'he gel phase, while in the lamellar phase SHT| is larger than or equal to Sr-ry, depends on ho-v close to n the first C-C bond lies and on the particular motions involved in the averaging. In general, a hydrocarbon chain, even in the all-trans conformation, has no three-fold or •Cfl-Cy is the closest C-C bond to the o-carbon around which a gauche rotation will leave the molecular axis roughly parallel to m. We neglect other possible fluctuatiunJ or the molecular axis about m in this rigid molecule calculation. 128 K. Abdotatl tt al., NMR audits in potassium and sodium palmltatts greater axis of symmetry, and hence there is no reason to expect that SHU De equal to SCD (See paper II for further discussion on this point). V. Conclusions This paper has demonstrated that the magnetic resonance spectra of deuterium in water and hydrocarbon chains of potassium and sodium palmitates and of sodium in sodium palmitate indicate correlations between water, hydrocarbon chain and counter ion order. For the lamellar phase of potassium palmitate the order parameters of DaO and the first few methylene chain segments are shown to increase and then decrease with increasing temperature. This has been explained in terms of a model where the lipid-water structure at low temperatures imposes a direction on the first C—C bond for which all the order parameters are small for purely geometric reasons. At higher temperatures the structuring effect of water is decreased and there is an increase in the splittings. Details of the lipid-water structure at lower temperatures may involve con siderations of a-CH2 -COO conformation and complicated hydrogen bonded structures such as two lipid molecules being bridged by one water molecule, and hydrogen bond- • ingof more than one water per lipid oxygen. Also, some water superstructure (possibly tetrahcdral) is likely to persist throughout the water layer. The above qualitative arguments should be checked with a more complete theory, using for example an intermolecular force mean field to account for fluctuations of the molecular axis about n for each conformation. In such calculations the lipid-water interaction should not be neglected and attention should be given to the first C-C bond orientation which might well depend on both conformation and water structure. A fluid-fluid phase transition is detected within the lamellar phase of C,6K and involves the water and first several segments of the hydrocarbon chain. This transition may well indicate a change in the water superstructure, or a conformational change fas trans to gauche) near the polar head. The presence of such a gauche conformation in the gel phase is suggested by a rigid molecule calcul:.-ion. Furthermore, these studies demonstrate that NMR is a very sensitive method of detecting different phases and phase transitions in lipid-water systems. Acknowledgements We thank Dr. T.P. Higgs for synthesizing the deuterated fatty acids and for assisting in sample preparation. We are grateful to the authors of papers I and II for careful criticism of our work. We are indebted to Myer Bloom for many fruitful discussions. References |1| V. Luzzati, in: Biological Membranes, ed. by D. Chapman. Academic Pre*. N.Y. (1948) K. A bdohtll tt al, NMR studies in potassium and sodium palmltatts 129 (2) A. Johansson and B. Lindman, in: Lipuid Crystals and Plastic Crystals, Vol. 2, ed. by G.W. Cray and P.A. Winsor, Kliss llorwood, Chichester (1974); G.J.T. Tiddy, in: Specialist Periodical Reports on NMR, Vol. 16 (1977) [ 31 B. Mely and J. Charvolin, Chem. Phys. Lipids (in press) |4| B.D. Landbionke and D. Chapman,Chem. Phys. Lipids 3 (1969) 304 |5| P.G. de Genncs, in: The Physics of Liquid Crystals, Clarendon Press. Oxford (1974) |6] II.A. Rcsing, A.N. Garroway and K.R. roster, in: Magnetic Resonance in Colloid and Interface Science, ed. by II.A. Rising and C.G. Wade, ACS Symposium, 34 (1976) 516 (7| J.ll. Davis and K.R. Jeffrey, Chem. Phys. Lipids 20 (1977) 87 [8] A.L. MacKay and T.P. Higgs, ChJm. Phys. Lipids 20 (1977) 105 |9| J.ll. Davis, K.R. Jeffrey, M. Bloom, M.I. Valic and T.P. Higgs, Chem. Phys. Letters 42 (1976) 390 [10} G. Lindblom. N.O. Persson and B. Arvidson. in: Lyotropic Liquid Crystals and the Structure of Biomembranes, Chapter 9, ed. by S. Friberg, Adv. Chem. Set. 1976; H. Wennerstrdm, G. Lindblom and B. Lindman, Chem. Scripta 6 (1974) 97 (III M. Bloom, EE. Burncll. S.B.W. Roeder and M.I. Valic, J. Chem. Phys. (in press) 112) S. MarCelja, Biochim. Biophys. Acta 367 (1974) 165 [13] F.Y. Fujiwaraand L.W. Reeves, J. Am. Chem. Soc. 98 (1976) 6790 |14| J.F. Nagle, J. Chem. Phys. 58 (1973) 252; P. Bothorel, J. Belle and B. Lamaire. Chem. Phyt. Lipids 12 (1974 ) 96; H. Schindler and J. Seelig, Biochemistry 14 (1975) 2283: ML. Scott. J. Chem. Phys. 62 (1975) I 347; RE. Jacobs, B. Hudson and II.C. Anderson, Proc. Nad. Acad. Sci. U.S.A., 72 (1975) 3993; J.A. McCammon and J.M. Deutch, J. Am. Chem. Soc. 97 (1975) 6675: MB. Jackson, Biochemistry 15 (1976) 2555 (15) A two site model has been used previously to explain the NMR of acetylene [P. Diehl. S-Sykora.W. Niedcrbcrger and E.E. Bumell, J. Mag. Res. 14 (1974) 260] and methyl fluoride |E.E. Bumell, JR. Council andS.E. Ulrich, Chem. Phys. Letters, 31 (1975) 3951 dissolved in nematic solvents |16| D. F.isenberg and W. Kauzmann, in: The Structure and Properties of Water, Oxford University Press (1969) [17] P.J. Flory.in: Statistical Mechanics of Chain Molecules. Interscicnce, N.Y. (1969) [18] M. Bloom, in: Proc. First Specialized "Colloque Ampere1' ed. by J.W. Hennel. Krakow, Poland (1973), p. 80 M. Bloom, E.E. Bumell, M.l. Valic and G. Weeks, Chem. Phys. Lipids 14 (1975) 107 (19| W.Maierand A. Saupe.Z. Naturforsch. A13 (1958)564; AM, (1959) 882; A15 (1960)287 [20] B. Mely, J. Charvolin and P. Keller, Chem. Phys. Lipids 15 (1975) 161; J. Charvolin, P. Manneville and B. Deloche. Chem. Phys. Letters 23 (1973) 345 [ 211 A. Seelig and J. Seelig. Biochemistry 13 (1974) 4839 104 APPENDIX B Determination of The Equation of State for The sodium Laurate-Water system Using low Angle X-ray Scattering^ In order to determine the equation of state relating the different thermodynamic parameters (mean area/polar head, temperature and water concentration) for the lamellar liquid crystal (La) of the sodium laurate-water system the lamellar repeat distance for this system was measured using low angle X-ray scattering as a function of temperature and water concentration. Following the procedure of Gallot and Skoulious (51) the area per polar head was derived from the measured lamellar repeat spacings. Experimental Results and Discussion Due to the random orientations of the liquid crystal domains of bilayer of the sodium laurate-water system in the lamellar liquid crystal phase in the samples used the X-ray diffraction patterns are concentric rings whose diameters are related to the lamellar repeat spacing by Bragg's law mA = 2 dsin0 [l] where m = 0, 1, 2, .... d is the lamellar repeat distance, X Is the wavelength of the X-ray radiation used (1.54A ) and 0 is the scattering angle. If d^ is the measured diameter of the mtn ring , f is distance between the sample "''The X-ray measurements were carried out in collaboration with L. Wood and K. Jeffrey at the University of Guelph Ontario. 105 and the film the 0 is given by tan 20 = ^ ' i [2l 2 f 1 J Due to instirumental limitations only the rings corresponding to 1/2 , 1/3 , ... of the fundamental spacing (i.e for m-2,3,...) were observed. Using equations 1 and 2 the fundamental spacing was obtained from the rings corresponding to m=2 and 3 . Table 2 shows the temperature dependence for the lamellar repeat spacing in the temperature range (85^145*0 ) for a sample having 6 moles of water/1 mole of sodium laurate. It remains roughly the same between 86 and 125 C and then shows a decrease at higher temperatures. Table 3 shows the concentration dependence for d at two different temperatures. There is a slight increase in d with increasing water concentration. Calculation of the bilayer thicVjiess da and the area per polar head A In the model used by Gallot and Skoulious the simplifying assumption that water does not penetrate into the bilayer is made .'. implicitly. If va and ve are the specific volumes of soap and water respectively, ca and ce = l-ca are the fractional concentrations (per unit mass) then d c v + (1-c )v -a a a e which gives -1 w 106 va has only been measured for the potassium Myristate-water system (51). For the other soaps va was derived by assuming the additivity of partial molar volumes in the following way. If a soap molecule CnX of the ion type X with n carbons on the hydrocarbon chain then its molar volume Vn(X) is given by. Vn(X) = VCH3 + (n-2) VCH2 + [5] Where VcR$ is tne partial molar volume of the methyl group, Is the partial molar voulme of one..methylene group and V^Q^ is the volume of the carboxyl group and the X ion. In terms of the molar volume for 0^-K eqn. 5 can be rewritten., as VX) = VU(K) + (n-14)VCR2 +(VX-VK) [>] Where VX-VK is the difference in partial molar volumes of ion X and the potassium ion. The sepecific volume va is then given by the ,\ relation Vn(X) =Mava where Ma is the molecular weight of the soap. Gallot and Skpulious (51) have verified that eqn. 6 yields results that are.in good agreementwith experimental values for va . For the Cj^-Na eqn. 6 becomes V12(Na) „ V1A(K) - 2VCHz + (VNa-V^) [l\ The value of VcH2 at a given temperature was arrived at by comparing the specific volumes of the normal alkanes as a function of the number of the carbon atoms (52). VNa-VK was put equal to the difference of the molar volumes of NaCl and KC1 in water (53). 107 Using the calculated va values as outlined above and the specific volume of water in equilibium with its own vapour pressure for ve , da was calculated using 4 . The results are shown in table 2 To calculate the mean surface are per polar head the following equation was used. where N is Avogadro's number. Fig.30 is a log-log plot of the dependence of A on water concentration for three temperatures. The results at 86°C are those of Gallot and Skoulious (51). Like all the other sodium soaps studied in (51) the following empirical relation O was found to hold. C is the water concentration in moles of water/1 mole of soap and P is a constant = .24. Fig.31 is a plot of AQ versus T. In fig.32 the temperature dependence of A is shown for 0=6 it increases linearly with temperature in the range 86-120 °C and then increases at a faster rate at higher temperatures with a break in the slope at-125 °C. We believe that this is probably an artifact due to slight perturbations in the experimental arrangement. Furthermore there were no anomalies in the deuterium quadrupole splittings of the perdeuterated sample of the same composition. The fact that da is almost independent of temperature and depends very weakly on concentration can imply two things :(i) water penetration A Where AQ(T) is a function of temperature only ,T is the temperature. 108 in between the chains or (ii) shrinkage of the chains resulting from bending and twisting motionsi In the model used for calculations the first possibility has been excluded. It must be kept in mind, however, that the possibility of some penetration of the water into the bilayers cannot be excluded on the basis of our experiments. This will in no doubt introduce some uncertainty in the experimental results. 109 o :rature (C) d(A) da(A) A (A ) 86 30.6 20.4 36.2 91 30.8 20.2 36.6 96 30.4 20.3 36.6 105 30.6 20.4 36.9 112 30.4 20.3 37.3 117 30.5 20.4 37.3 122 30.5 20.4 37.5 127 29.7 19.8 38.8 132 29.1 19.4 39.8 137 28.4 18.9 41.0 142.5 29.4 18.9 41.2 Table 2 . Temperature dependence of the lamellar repeat distance d, the thickness of the bilayer da and the mean area per polar head A for the sodium laurate-water system. The water concentration is 6 moles of water/1 mole of sodium laurate. (§105 °C @135°C °c d(A) da(A) A(A 3 29.6 23.7 31.8 4 29.9 22.4 33.5 5 30.3 21.4 35.2 6 28.8 19.2 36.9 7 28.1 17.7 39.4 2 30.0 25.8 30.0 3 29.1 23.3 33.3 4 29.1 21.8 35.5 5 30.3 21.4 36.2 6 27.1 18.1 40.0 7 28.2 17.8 44.0 Table 3 . Dependence on the water concentration C(moles of water/ 1 mole of sodium laurate) of the lamellar repeat spacing d, the bilayer thickness da and the mean are per polar head A at 105 °C and 135 °C . 110 0.3 0.4 0.5 0.6 0.7 0.8 0.9 LOG C (MOLES OF H20/1-M0LE OF C^-Na) Figure 30 . A log-log plot of the mean area per polar head versus water concentration for the sodium laurate-water system at 86 C (open circles, obtained from ref.51 ), 105 C (solid dots) and 135°C (triangles). Figure 31 The dependence of Aq on temperature. 112 Figure 32. The dependence of the mean area polar head on temperature at a fixed concentration for the sodium laurate-water system. 113 Appendix C Water self Diffusion and Spin-Spin relaxation In sodium laurate/H20 . The self-diffusion coefficient and the spin-spin relaxation time T2 of H20 were measured, in the sodium laurate water system. The spin echo method with an externally applied field gradient (38) was used. Theory For an isotropic liquid sample the amplitude of a spin echo in an NMR experiment is given by S(2T) = S(0) exp (-2T/T2), exp (- IYVDT3) [l] 3 ' where T is the spacing between the 90° and 180° pulses, G is the applied magnetic field gradient and D is the diffusion coefficient. The term involving D in equation 1 takes, into account the extra damping to the transverse nuclear magnetization due to the change in the Larmor frequency as a result of translational diffusion of the molecules across an inhomogenous applied magnetic field. If G and T2 are known D can be obtained from the slope of a semilog plot of log (S(2T) + 2T/T2) versus T For water in an anisotropic enviroment eqn. [lj has to be modified. In the sodium laurate water system the diffusion of water between the bilayers is mainly parallel to the bilayers. For an oriented sample it can be shown (54) that the angular dependence in the lab frame of the diffusion tensor is given by Dll - <D1> + (<D..> -<Dj>)sin20 DO 114 where ^Dx^> and are the components of the diffusion tensor : in the frame of the bilayer and 9 is the angle between the normal ri to gradient the bilayer and the magnetic field direction. Since the water molecules are confined to narrow regions between the bilayer, the diffusive motion perpendicular to the lamellae does not transport the water molecules very far and therefore it will be assumed that <^D^— 0 . Equation 2 then becomes % - <Dn > sin2Q = d (i_u2) c3 J where D = <^D^and y = cos 0 . and the echo amplitude will be given by: S(2x) = SQ exp (-2T/T2). exp (- | Y2G2D(l-y2)) [4] In randomly oriented samples all values of y=cos0 are equally probable and an average over all arientations has to be considered giving S(2T) = SQ exp (-2T/T2) /dy exp (- | y2G2D(l-y2)x3) = S0exp(-2x/T2). exp(- 1 Y2G2DT3)X (\/| YVDT) [5] 3 J 1 y 2 where x(y) = ~ f dx. exp (x ). / 2 2 2 3 — Y G D. T ) is a correction term for lamellar systems to equation 1 , the expression for S(2T) for isotropic systems. Experimental Results Using a Bruker B-KR 300Z18 field gradient unit, rtie echo Amplitude was measured for several T values at different values of G for the water in the sodium laurate water sample. To calibrate G the same measurements were made on pure water (R^O) • The sample sizes and diameters of the tubes (.5cm) were chosen to be identical. T2 for 115 rLjO in the sample was measured using the 90°-T-180 pulse sequency in the absence of a field gradient and was found to be 30. msec. To calculate D equation 5 was fitted to the data by a 2 parameter non linear least squares fit where the integral was evaluated numerically. Fig. 33 is a representative fit for eqn. 5 . The ratio of D to the self diffusion coefficient in pure water D/DQ , is shown in table 4 for different G values. 2 2 2 9 2 2 " . • G | y G DQ 3YGD D/DQ _0 _2 q —2 (Gauss/cm) (msec )xl0 (msec-J)xlO 19.4 1.55. .30 .19 27.3 3.08: ' .59 .19 34.4 4.90 1.03 .21 41.0 6.93 1.50 .22 Table 4. Ratio D/Da of the self, diffusion coefficient of water in the sodium laurate water system to that of pure water at 100°C for several G values. D/D0 was obtained from the ratio of 2y2G2D/3 calculated by fitting eqn. 5 to the experimental data (column 3), to the slope of log S(2x) + 2T/T2) versus x3 plot for pure water (column.-2). 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P. 120 1-3 CD cf M CD d23C12-Na/2H20 Vn (kHz ) 2 •yn 2 3,4 5 6 7 8 9 10 11 12 100 20.45 16.05 14.83 *..: 7 13.02' 10.53 9.12 5.93 2.06 105 20.84 16.44 15.42 13. 90 11.36 9.80 6.43 2.26 108 20.90 16.46 15.41 13.83 11.32 9.68 6.38 2.22 110 20.85 16.33 15.23 13.67 11.21 9.59 6.32 2.17 111 20.83 16...33 15.27 13.66 11.16 9.62 6.29 2.19 112 20.80 16.26 15.19 13. 61 11.07 9.51 6.24 2.14 115 20.78 16.11 15.05 13.42 10.93 9.36 6.12 2.09 116 20.78 16.17 15.10 13. 45 10.95 9.34 6.12 2.11 117 20.78 16.13 14.98 13.31 10.84 6.03 2.08 120 20.64 16.04 14.94 13.26 10.72 9.18 5.98 2.05 125 20.65 15.99 14.76 13. 16 10.57 9.01 5.83 1.97 130 20.52 15.88 14.65 13.10 12.71 10.39 8.87 5.74 1.90 135 20.40 15.72 14.45 12.85 12.41 10.14 8.64 5.58 1.84 140 20.36 15.66 14.31 12.67 12.28 9.99 8.56 5.46 1.78 145 20.28 15.52 14.17 12.51 12.11 9.84 8.36 5.33 1.75 150 20.19 15.36 14.01 12.38 11.99 9.67 8.23 5.20 1.67 d23C12_Na/3H2° vn (kHz ) 2 2 3,4 5 6 7 8 9 10 11 12 100 19.63 14.89 13.24 11.39 10,71 8.70 7.35 4.82 1.61 102 19.54 14.86 13.16 11.38 10.62 8.62 7.30 4.74 1.59 103 19.57 14.86 "".13.15"1 7" 11.33 10760 8.59 7.24 4.72 1.57 105 19.59 14.78 13.10 11.22 10.51 8.46 7.18 4.64 1.56 110 19.45 14.67 12.93 11.11 10.34 8.36 6.99 4.53 1.50 115 19.36 14.53 12.81 10.86 10.13 8.15 6.82 4.41 1.43 120 19.25 14.47 12.63 10.10 9.97 8.04 6.68 4.27 1.37 125 19.17 14.38 12.50 10.62 9.84 7.89 6.52 4.15 1.33 130 19.06 14.23 12.32 10.46 9.68 7.74 6.38 4.06 1.29 135 18.97 14.16 12.16 10.28 9.51 7.51 6.25 3.96 1.22 140 18.85 14.01 12.04 10.14 9.35 7.40 6.10 3.86 1.17 145 ' 18.79 13.93 11.91 10.00 9.23 7.25 5.98 3.77 1.12 150 18.67 13.82 11.77 9.84 9.08 7.13 5.86 3.69 1. 11 pj £ ct- fO CO 4 2 << o CD I—1 ct- CD CO B CO M PL H-CO c+ •d ct-CD H-0 CS ci, rjq CD m ti o o CO Hj o o & to c+ H" CD 0 & CD CD 4 0 c+ CD ^ 2 CD t3 03 P3 pu 3 s; t+ P tr et- CD CO l-J CO O O O M-O £ o B CD ti i—1 <+ fo 4 0 ps H ct- 03 H- ct-O CD 3 d25C12-Na/6H20 ^SL (kHz ) 2 n 2 3,4 5 6 7 8 9 10 11 12 C 70 14.49 10.58 9 .55 8.42 7.91 6.59 5.66 3.91 1.34 75 14.66 10.73 9 .53 8.29 7.80 6.43 5.51 3.78 1.29 80 14.84 10.79 9 .54 8.20 7.69 6.25 5.37 3.64 1.23 85 14.99 10.89 9 .52 8.15 7.52 6.10 5.18 3.52 1.18 90 15.10 10.99 9 .47 8.06 7.42 5.96 5.03 3.42 1.15 95 15.25 11.10 9 .39 7.93 7.24 5.80 4.87 3.30 1.10 100 15.38 11.08 9.67 9.33 7.86 7.18 5.71 4.74 3.22 1.06 105' 15.43 11C13 9.67 9.18 7.71 7.08 5.37 4.54 3.13 1.03 110 15.43 11.13 9.57 9.18 7.67 6.93 5.42 4.49 3.03 .98 115 15.52 .11.17 9.51 9.07 7.56 6.82 5.31 4.38 2.92 .95 120 15.58 11.18 9.55 9.05 7.45 6.65 5.15 4.27 2.86 .92 125 15.58 11.18 9.42 8.94 7.40 6.57 5.04 4.16 2.75 .87 130 15.59 11.17 9.46 8.86 7.29 6.45 4.96 4.05 2.70 .85 135 15.50 11.11 9.38 8.81 7.20 6.35 4.87 3.99 2.60 .78 140 15.45 11.01 9.25 8.67 7.06 6.18 4.71 3.83 2.50 .74 145 15.42 11.00 9.24 8.61 6.95 6.09 4.60 3.75 2.44 .73 150 15.33 10.94 9.13 8.50 6.84 5.96 4.49 3.66 2.39 .70 d23C12-Na/7H20 (kHz ) 2 n 2 3,4 5 6 7 8 9 10 11 12 C 70 12.18 8.86 8. 20 7.39 7.02 5.90 5.10 3.58 1.20 73 12.21 8.90 8. 20 7.32 6.95 5.83 5.03 3.52 1.17 75 12.38 8.98 8. 19 7.30 6.90 5.77 4.97 3.48 1.16 81 12.38 8.98 8. 15 7.20 6.79 5.62 4.81 3.34 1.11 91 12.51 9.11 8. 09 7.03 6.52 5.33 4.53 3.13 1.03 95 12.66 9.14 . 8. 04 6.95 6.41 5.21 4.41 3.03 .99 100 12.77 9.20 8. 01 6.84 6.32 5.10 4.30 2.93 .95 106 12.83 9.22 8.21 7.92 6.77 6.19 4.97 4.16 2.82 .90 110 12.92 9.28 8.15 7.89 6.70 6.09 4.85 4.06 2.75 .88 120 13.00 9.34 8.09 7. 76 6.53 5.87 4.63 3.83 2.59 .81 130 13.06 9.38 8.03 7.67 6.35 5.66 4.42 3.64 2.43 .74 141 13.05 9.36 7.97 7.51 6.19 5.48 4.21 3.43 2.26 .67 150 13.09 9.33 7.91 7.40 6.05 5.30 4.04 3.25 2.11 .61 P. 17 d23C12-Na/4H20 (kHz ) 2 n 2 3,4 5 6 7 8 9 10 11 12 °c > 90 18.49 13.78 11. 73 9.83 9.03 7.18 6.03 3.96 1.34 92 18.52 13.76 11. 69 9.79 8.96 7.10 5.93 3.93 1.33 95 18.47 13.71 11. 58 9.70 8.87 6.99 5.85 3.85 1.32 100 18.51 13.62 11. 45 9.62 8.74 6.88 5.74 3.78 1.26 105 18.35 13.48 11. 30 9.42 8.59 6.74 5.62 3.66 1.21 110 18.32 13.46 11. 16 9.33 8,50 6.62 5.52 3.61 1. 12 120 18.24 13.26 10. 91 9.11 8.24 6.40 5.27 3.42 1.10 126 18.02 13.18 11. 23 10. 79 8.91 8.08 6.23 5.11 3.31 1.05 130 17.93 13.07 11. 17 10. 73 8.87 8.02 6.19 5.08 3.27 1.03 135 17.81 13.06 11. 11 10. 64 8.79 7.91 6.05 4.97 3.20 .98 140 17.74 12.93 10. 93 10. 49 8.70 7.73 5.97 4.87 3.14 .94 145 17.66 12.87 10. 89 10. 41 8.61 7.68 5.87 4.76 3.05 .90 150 , 17.64 12.78 10. 80 10. 31 8.48 7.56 5.80 4.68 2.99 .88 d23C12_Na/5H2° ( kHz ) 2 2 3,4 5 6 7 8 9 10 11 12 C 80 17.32 12.72 10. 74 8.98 8.20 6.51 5.48 3. 70 1.22 85 17.38 12.66 " 10. 62 8.85 8.04 6.34 5.33 3.58 1.18 90 17.33 12.61 11.01 10.47 8.72 7.89 6.18 5.20 3.47 1.14 95 17.37 12.64 10.91 10.44 8.61 7.76 6.09 5.09 3.42 1.11 100 17.27 12.62 10.85 10.34 8.53 7.67 5.97 4.97 3.32 1.07 105 17.30 12.54 10.68 10.22 8.44 7.48 5.82 4.87 3.21 1.04 110 17.26 12.50 10.64 10.11 8.30 7.37 5.71 4.74 3.14 1.01 120 17.09 12.33 10.47 9.89 8.07 7.17 5.49 4.51 2.98 .95 128 16.86 12.10 10.14 9.56 7.73 6.87 5.19 4.21 2.77 .87 130 16.99 12.16 10.27 9.70 7.89 6.96 5.30 4.35 2.86 .89 135 16.89 12. 18 10.18 9.59 7.76 6.86 5.19 4.22 2.77 .85 140 16.76 12.05 10.08 9.59 7.69 6.74 5.09 4.14 2.70 .84 145 16.70 11.96 10.03 9.40 7.59 6.66 5.03 4.06 2.64 .81 150 16.67 11.87 9.94 9.35 7.54 6.57 4.93 3.98 2.59 .78 Table 6. Spin-lattice relaxation times of chain deuterons in the sodium laurate water system; dependence on temperature and •Water concentration. OO o CO IN CM TJ w c CM CO cd cu J25 cn 1 *—s vo CN a •—1 t—i in u H co co m >-i CM O CM vO co in o IN m •—< <t in N m in i^-CO <r CM vo m co IN IN IN 00 o m • 00 i—i vO CM m ON «* i-H i-H CM CM CM CO m vO CO ON <* TJ rN o IN m O o CM co o C IN r-H m ON i-H m i-H CM O r—1 i-H i-H CM CM CO a o CO OJ "N. CO cd 1—^ S3 d o IN 1 r-H vO CO CM <r m in IN i-H CM H I—1 CM o •—i m ON CM CO I—1 in i-H •—i CM CM CM CM • u -CO CM TJ O ON CO CO rN |N in o IN r-H i-H O O i—1 VO CM CO CO i-H r-H CM CM CM CO mom vo H CM CO CO oo <r vo vo co CM r*» vo o "-H co -vi vo vO • • • • r-« m m r~- O oo oo co oi H oo CM CM o- m fN IN m m IN f"» co o oo CM co ON m co i-H CM co co <f m N ci vo rs r> ffl O Csl 0> H <f •-H CM CM CM CO -sT CO 00 vO CM o m 1—' CM CM m m vo *-H r-H CM CM CM CO C / m m m / H r» CO / U -H -< m m m m OOOHNCO C_) 00 CJ> i-H i-H i-H i-H o m O CM o CM vO ON i-H CM CO vo 00 ON • • • o vo o co <f <r ON I IN vo m <• o o ON I CM co -o- m vo IN m co IN IN o r~-oo vo es IN <r H r-H CM co co <j" m CM O CM i-H <r m 00 i-H IN r-H fN vO i-H CM CM CO CO vT |N CM m O fN ID oo | co oo CM m co <r i-H i-H CM CM CO CO 00 TJ C IN O o CM O EC cu CO '—. N-/ eg c S r-H VO I H CM m r-H o CO CM TJ <r CO rN CM m CM O O CM vO O -J" CO CM •-H r-H CM CM CM CO o m m m CM in o m vo o •-H CM CM CM tO o ON IN m m o CM m ON in IN o r-H rH r-H CM CM CO o vo m co IN in co vo o m oo m r-H i-H CM CM CM CO / m m m m o o o -H CM CO CJ 00 ON rH i—I r-H r-H o P.- nt*. d^C12-Na/5H20 (seconds) n o 2 3,4 5,6 7 8 9 10 11 12 C 80 . 0822 C9096 .097 .0977 .117 .148 .187 .290 .750 90 .116 .135 .137 . 138 . 155 .184 .243 .335 .815 105 . 173 . 183 .177 .200 .220 .325 .327 .370 1.01 115 .220 .224 .228 .242 .267 .320 .375 .537 1.24 125 .260 .273 .266 .285 .312 .370 .445 .614 1.19 135 .342 .309 .310 .345 .380 .437 .510 .735 1.42 d -T -23^12 •Na/6H20 T. (seconds) in >n o 2 3,4 5,6 7 8 9 10 11 12 C 80 .057 .067 .060 .067 .070 .090 . 107 .158 .68 90 .066 .077 .080 .088 .095 . 123 .150 .200 .76 105 .097 .110 . 117 .126 .137 .172 .210 .277 .89 115 .118 .140 .150 .167 .178 .230 .275 .385 1.05 125 .168 . 160 .173 .212 .230 .315 .370 .493 1.22 135 .204 .212 .257 .295 .310 .405 .470 .595 1.36 d23C12-•Na/7H20 T, (seconds) In . n 2 3,4 5,6 7 8 9 10 11 12 C 80 .043 .048 .048 .056 .057 .076 .100 .163 .600 90 .052 .062 .064 .073 .083 .112 .140 .235 . 760 105 .076 .085 .097 .107 .118 .147 .210 .329 .946 115 .098 . 112 . 120 .135 .145 .187 .245 .353 1.02 125 .113 . 138 .147 .163 . 182 .250 .317 .458 1.15 135 .144 . 156 .187 .223 .210 .280 .368 .555 1.31 

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