Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Sternin, Edward
Doctor of Philosophy - PhD
University of British Columbia
Experimental proof is presented that some of the motions responsible for transverse relaxation (T₂) in deuterium magnetic resonance (²H NMR) experiments on acyl chains of a model membrane in the liquid crystalline phase are extremely slow on the ²H NMR time scale being characterized by a correlation time T₂ > ѡq⁻¹. The experiments used to investigate these slow motions involve a form of the Carr-Purcell-Meiboom-Gill pulse sequence modified so as to be suitable for ²H NMR (q-CPMG). The most plausible mechanism responsible for T₂ relaxation is the gradual change in the average molecular orientation due to lateral diffusion of the phospholipid molecules along curved membrane surfaces. Presence of such diffusion is directly established by a selective inversion recovery experiment in which magnetization transfer across the spectrum is seen. The results of the T₂ relaxation as measured in the q-CPMG experiments are fitted to an average correlation time, T₂ ≈ 62 ms, yielding an estimate of the average effective radius of curvature of 1.2 µm for a typical model membrane system, in good agreement with other methods of measurement. The implications of this main result are examined for a number of model membranes; in particular, considerable changes are seen in the character of molecular motions in systems containing small concentrations of sterols. Similarly, changes caused by the topological differences between the lamellar L∝ and hexagonal H₁₁ phases are examined in a model membrane system which undergoes a L∝ to H₁₁ phase transition. A novel way of quantifying the differences in the orientational order parameters across the phase transition is used; the observed differences are consistent with the different symmetry properties of the two phases. Perdeuteriated polycrystalline hexamethylbenzene is used to demonstrate various methods of measuring ²H NMR relaxation. In addition, some aspects of orientation dependence of the relaxation rates are examined, and found to agree with the theory.
S O M E M E C H A N I S M S O F T R A N S V E R S E N U C L E A R M A G N E T I C R E L A X A T I O N I N M O D E L M E M B R A N E S Edward Sternin B. Sc. (Mathematics and Physics) University of British Columbia M. Sc. (Physics) University of British Columbia A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1988 @ Edward Sternin 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 refer-ence 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia 1956 Main Mall Vancouver, Canada Date: Abstract Experimental proof is presented that some of the motions responsible for transverse relaxation ( T2) in deuterium magnetic resonance ( 2 H NMR) experiments on acyl chains of a model membrane in the liquid crystalline phase are extremely slow on the 2 H NMR time scale being characterized by a correlation time Ti > u;" 1 . The experiments used to investigate these slow motions involve a form of the Carr-Purcell-Meiboom-Gill pulse sequence modified so as to be suitable for 2 H NMR (q-CPMG). The most plausible mechanism responsible for T-i relaxation is the gradual change in the average molecular orientation due to lateral diffusion of the phospholipid molecules along curved membrane surfaces. Presence of such diffusion is directly established by a selective inversion recovery experiment in which magnetization transfer across the spectrum is seen. The results of the T2 relaxation as measured in the q-CPMG experiments are fitted to an average correlation time, T2 ~ 62 ms, yielding an estimate of the average effective radius of curvature of 1.2 fim for a typical model membrane system, in good agreement with other methods of measurement. The implications of this main result are examined for a number of model membranes; in particular, considerable changes are seen in the character of molecular motions in systems con-taining small concentrations of sterols. Similarly, changes caused by the topological differences between the lamellar La and hexagonal Hn phases are examined in a model membrane system which undergoes a La to Hn phase transition. A novel way of quantifying the differences in the orientational order parameters across the phase transition is used; the observed differences are consistent with the different symmetry properties of the two phases. Perdeuteriated polycrystalline hexamethylbenzene is used to demonstrate various methods of measuring 2 H NMR relaxation. In addition, some aspects of orientation dependence of the relaxation rates are examined, and found to agree with the theory. ii Table of Contents Abstract i i List of Tables vi List of Figures v i i Acknowledgements ix 1 Review of N M R Theory 1 1.1 Relaxation 1 1.2 General Spin System 2 1.3 Specifics of Spin-1 Systems 5 1.4 Perturbation Theory Treatment of Spin Relaxation 10 1.5 Quadrupolar Relaxation 12 2 Experimental Methods of Measuring Relaxation 15 2.1 Free Induction Decay 15 2.2 Quadrupolar Echo 17 2.3 Inversion Recovery 21 2.4 Stimulated Echo 21 2.5 Jeener-Broekaert Echo 24 2.6 The Pake Doublet Spectrum 27 2.7 Selective Inversion Recovery 31 2.8 Multiple Echo Techniques 32 iii 3 Model Membranes 35 3.1 Motions and Order Parameter 36 3.2 Aspects of NMR Unique to Membranes 38 3.3 Diffusion as a Slow Motion 39 4 Performing N M R Experiments 41 4.1 Equipment 41 4.1.1 General Principles 42 4.1.2 Basic System Parameters 44 4.1.3 Devices 48 4.1.4 Phase Shifter 49 4.2 Materials 51 4.3 Data Acquisition during Multiple Pulse Trains 53 4.4 Fidelity 53 4.5 Software 55 5 Results and Discussion 57 5.1 Presence of Lateral Diffusion 58 5.2 Hexamethylbenzene as a Test Compound 67 5.3 T| e Measurements in Hexamethylbenzene 69 5.4 Tf Measurements in D P P C - d 2 / H 2 0 Model Membrane 73 5.5 Using Q-CPMG to Separate the Effects of Lateral Diffusion 76 5.6 Application to Model Membranes Containing Sterols 82 5.7 Hexagonal Hn Phase Measurements 85 5.7.1 Changes in the Orientational Order Parameter 85 5.7.2 Relaxation and Spectral Changes due to the Extra Motion 92 6 Summary and Concluding Remarks 98 iv Bibliography 104 A Orientation Dependence of 2 H N M R Relaxation 110 A . l Coordinate Systems and Euler Angles 110 A.2 Rotation of the Spherical Harmonics I l l A.3 Averaging over Anisotropic Motions: General Remarks I l l A.4 Detailed Calculation of the Anisotropic Averaging 112 A.5 Interpretation of the Experimental Results 116 v List of Tables 1.1 Commutators of spin-^ basis operators 4 1.2 Commutators of spin-1 basis operators 6 2.1 Free induction decay 16 2.2 Quadrupolar echo 18 2.3 Relaxation effects measured by the quadrupolar echo 20 2.4 Using inversion recovery to measure T\z 22 2.5 Stimulated echo 23 2.6 A mixture of quadrupolar and double-quantum order 26 2.7 Quadrupolar Carr-Purcell-Meiboom-Gill (q-CPMG) echo train 33 5.1 Comparison of transverse relaxation rates in DPPC-d3i 80 A . l Values of ^|^2m"(^"?^")| 2) f l„ ^„ expressed in terms of 52 and 54 112 A.2 Values of (|F m » m ( / 3 , 7 )| 2 )^ 113 A.3 Comparison of the experimental values of T\z(f3) with the theoretical predictions. 118 A.4 Comparison of the experimental values oiT\q{(3) with the theoretical predictions. 119 vi List of Figures 1.1 Precession diagram for a spin-^ in a static magnetic field (Larmor precession). . 4 1.2 Precession diagram for a spin-1 in a static magnetic field 7 1.3 Precession diagram for a spin-1 under an axially symmetric quadrupolar inter-action 9 2.1 Quadrupolar echo 19 2.2 Stimulated echo 25 2.3 Jeener-Broekaert echo 28 2.4 Selective inversion recovery. 30 2.5 Quadrupolar Carr-Purcell-Meiboom-Gill (q-CPMG) echo train 34 4.1 A block diagram of a typical data acquisition and processing system 42 4.2 A block diagram of the NMR data acquisition and processing system 45 4.3 A block diagram of the digital phase shifter 50 4.4 Selective data acquisition scheme for CPMG 54 5.1 DPPC-d2: non-selective inversion recovery 59 5.2 DPPC-d2: selective inversion recovery 61 5.3 DPPC-d2: T\z in selected spectral windows 63 5.4 Hexamethylbenzene: selective inversion recovery 65 5.5 Hexamethylbenzene: orientation dependence of T\z 66 vii 5.6 Hexamethylbenzene: orientation dependence of T%e 70 5.7 Hexamethylbenzene: orientation dependence of effective transverse relaxation rate. 72 5.8 DPPC-d2: orientation dependence of T 2 74 5.9 Comparison of transverse relaxation rates in DPPC-d3i 78 5.10 DPPC-d3i: Initial slopes of the relaxation curves 81 5.11 DPPC-d3i/lanosterol mixtures: Initial slopes of the q-CPMG relaxation curves. 84 5.12 Comparison of the La and En phases of phospholipid-water dispersions 86 5.13 La to En phase transition as seen from the 2 H NMR spectra 88 5.14 Comparison of the distribution of the fractional order parameter in the two phases. 89 5.15 Order parameter profile of La and En 91 5.16 TD in POPE/H2O: comparison of the relaxation rates in La phase 93 5.17 TD in P O P E / H 2 0 : comparison of the partially relaxed spectra in the La phase. 94 5.18 TD in POPE/H2O: comparison of the relaxation rates in Eji phase 95 5.19 TD in POPE/H2O: comparison of the partially relaxed spectra in the En phase. 96 viii Acknowledgements I thank with pleasure my supervisor Professor Myer Bloom for for his guidance, encouragement, and financial support over the years; and for making me do it right. I would also like to thank Professor Alexander L. MacKay for his valuable help, and to acknowledge his contribution, as well as that of Professors Elliott Burnell and James H. Davis, in teaching me the art of experimental NMR. I learned about biological membranes from Prof. P. Cullis, and was inspired by the work of his group. I am grateful to all of my teachers at UBC; and especially Prof. F. Kaempffer — for making physics exciting, in more than one language, and for making it fit on a single page. Prof. R. Cushley and the researchers of his group at SFU were very generous with the samples, and adventurous enough to act as a testing ground for some of my creations. Professors E. Evans and P. Martin as members of my advisory committee contributed a great deal to keeping me on track. I have greatly benefitted from interactions with my fellow inhabitants of Room 100 and its environs. It is impossible to mention all — as it is impossible to forget any — of them, but my very special thanks are extended to Dr. E. James Delikatny, Mark A. LeGros, Ulrike Narger, Dr. John C.T. Rendell, Julia C. Wallace for their generous help and their friendship over the years; and to Clare Morrison who helped enormously at the most critical time. I collaborated with Bernard Fine, Alastair Martin, Dr. Michel Roux, and Dr. Colin P.S. Tilcock at the various stages of my project. Without them, and the excellent facilities and staff of the Department of Physics, my task would have been considerably more difficult. Credit also goes to Dr. R. Lomnes, Gail Shroeder, I. and T. Goldenstein, Stan Knotek, and Domenic DiTomaso for helping me to design and build the hardware. I have acquired a great deal of computer expertise, which would have been impossible ix without the help of Edmund Bacon, Eduardo Castillo, Mary Ann Potts, Kenneth Whittall, and many of the members of the UBC Computing Centre, TRIUMF Computing Services, and the UBC computer community at large, too numerous to mention here. My ten years in Vancouver would not have been the same without Mrs. Edith Allen of the UBC Registrar's office, Tom Steele of the Manpower and Immigration Canada, | Isaac | and Sophie Waldman, who all helped at the time of need; the "dogwooders", the "shadlings", the "whoees" — where I belong; Eleanor who led the way. Thanks, dear friends. Finally, I wish to thank my parents, Maya and Vulf Sternin, for their encouragement, understanding, patience, and support, always; and for the courage and wisdom of the decision we made in the Spring of 1977, in Priedaine. x Chapter 1 Review of NMR Theory 1.1 Relaxation The relaxation behaviour of a system of N noninteracting spins / = \ is commonly described by the so-called Bloch's phenomenological equation dM Mxi + Mv] Mz- M0l , x ¥ = 7 M x H - ^ - ^ i k (1.1) first proposed by Felix Bloch in 1946 [1]. Here M is the ensemble nuclear magnetization, H is the applied magnetic field, 7 is the gyromagnetic ratio, and i, j, and k are the unit vectors in the laboratory frame of reference. According to Curie's law, Mz = Mo is the equilibrium value of magnetization in a static applied field Hz = Ho, ^ . ^ w o * ( , 2 ) T\ is the longitudinal relaxation time, and it describes the trend of nuclear spins toward an equilibrium with their surroundings, or lattice. T 2 is the transverse relaxation time, and it describes how the component of the nuclear magnetization perpendicular to the applied field decays due to various local fields caused by the fact that the spins are actually not free but interact with each other. Hence the often used names "spin-lattice" and "spin-spin" relaxation times for T\ and Ti, respectively. A single time constant is not usually sufficient to describe the evolution of the nuclear spins towards equilibrium with the lattice. Similarly, in a complex system of interacting spins, a single value for the transverse relaxation rate is rarely adequate. In this context, T\ and T 2 values are used to give at least the time scale of the processes involved. The general problem 1 Chapter 1. Review of NMR Theory 2 of evolution of the spin system is wholly dependent on the motions and interactions present in the system and its surroundings, in short — on its Hamiltonian. 1.2 General Spin System We shall follow the standard density operator formalism to describe the general dynamical properties of spin systems [2,3]. In the presence of a Hamiltonian TiH, the time evolution of the density operator, a, of a general system of noninteracting spins is given by the equation of motion ^ = -«'[*,*]. (1.3) In the space of Hilbert operators one can always select a complete (i. e. spanning the space) set of orthonormal basis operators, {p,-,i = 0,1...n}, Tr{pi,pj} = 8{j, and to expand any operator, O, in terms of this basis: 0 = ]T c,p t. (1.4) t For the density operator, it is convenient to include a unit operator, 1, in the basis: a - c 0 l + 5ZC'P»- ( L 5) i If the number of spins in the system is conserved, Co is independent of time and we shall, without loss of generality, assume that CQ = 1. In terms of these basis operators, the equation of motion (Equation 1.3) becomes ^ = - £ < ^ • (1-6) 3 where Qij = -Qji = -iTr^pi [Pj,n] }. (1.7) We are left with a set of coupled linear differential equations for the coefficients c,-(<), which contain all of the information about the time evolution of the system. The difficulty now is in finding the set of coupling coefficients, QtJ, which are determined by the Hamiltonian H. Chapter 1. Review of NMR Theory 3 However, the Hamiltonian itself can also be expanded in terms of the {p,}. The problem is therefore further reduced to the calculation of the commutators of [pt',Pj]- After a table of commutators for a given set of basis operators is established, the equations of motion for a particular % can be written down by inspection. It is instructive to consider the simplest possible example of the system of noninteracting spin-| nuclei. A convenient choice of basis operators is the Cartesian representation of the angular momentum, Po = 1, Pi = P2 = ~j^Jv> P 3 " T^ 2' where cr's denote the Pauli matrices. For a magnetic field applied along the z-direction the Hamiltonian is H = —W0P3, where UIQ is the Larmor frequency. Taking into account the usual commutation relations of the Pauli matrices (see Table 1.1), we can immediately write dc! dc2 dc3 -dT = - ^ -d7 = UoCl' -dT = 0- (L9) Solving these equations yields the Larmor precession c\{t) — ci(0) coswotf — 02(0) sin u>o£ c2(t) = ci(0) sinwoi + c2(0) coso>0^ (1.10) c3(<) = £3(0) = const We can represent the solutions given by Equations 1.10 in a pictorial form of a so-called precession diagram [3], as shown in Figure 1.1. It is no surprise that the picture we arrived at in this manner is that of the classical precession of the magnetization vector in the xj/-plane. It should be noted, however, that the diagram of Figure 1.1 is a shorthand representation of the solutions of a purely quantum mechanical problem. The power of the precession diagram notation becomes apparent when applied to the more complex spin systems, where the precession need not take place in the space of vectors. The hand-waving methods of visualization familiar to all students of NMR usually fail when the Chapter 1. Review of NMR Theory 4 Figure 1.1: Precession diagram for a spin—| in a static magnetic field. Evolution of <r{i) — 1 + -^Ci(t)ax + -^C2{t)ay + -^C3(t)(Tz under the Hamiltonian ~H = ^wo<7z (Larmor precession). Coupling between Ci(t) and c 2(i) is shown; C3 is a constant. Table 1.1: Commutators of spin—| basis operators. The values shown are for [p,-,pj]. Here pi = ~^axi P2 = ~^2ay> P3 — ~^piaz- ^ n e trivial po = 1 is omitted for brevity. Pi P2 P3 Pi 0 «P3 -zp 2 P2 -«P3 0 ipi P3 «P2 -ipi 0 Chapter 1. Review of NMR Theory 5 time evolution takes place in a multidimensional space of second rank tensors. On the contrary, the precession diagrams remain as transparent as they were in the vector case, in analogy with some other popular methods of shorthand, such as coherence transfer pathways [4]. Also, the precession diagrams contain a complete and exact description of the system, albeit a pictorial one. For now, we shall treat the expansion of Equation 1.5 in a formal way, without addressing the physical reasons for making this particular choice of {pt}. In the Liouville formalism [2] — and that is what we adopted here — the mechanics of solving the problem are separated from the particulars of the model. In this way, the formal part of the solution does not require duplication of the effort whenever a new model is considered. This should become apparent as we look at the simplification the Liouville formalism brings to the case of a spin-1 system. 1.3 Specifics of Spin—1 Systems For spin I — 1 the above treatment remains formally identical, except that we now need a set of nine basis operators to span the entire operator space. Accordingly, the size of the commutator table is now 9x9. The equations of motion (Equation 1.6) have been solved for a number of different basis sets, selected to match the Hamiltonian of the specific problem under consideration. Following [3], let us select the following basis operators: po - 1, Pi - ^Ix, p 2 = ^Iy, P3 - Jjlz, P 4 = ^=(37* - 2), P5 = ^ ( i * / , + /,/*), P6 = ^{Iyh + Uy), (1.11) P 7 = ^ ( 4 2 - / 2 ) , P 8 = ±(IxIy + IyIx). We can immediately calculate the commutator table for these operators, shown1 in Table 1.2. If the Hamiltonian is that of a magnetic field applied along the z-axis, Hz = —UQIz = —V^woP3, we can represent the solutions by a set of precession diagrams shown in Figure 1.2a. 1This table corrects two small errors that inadvertently appeared in both [3] and [5]. Chapter 1. Review of NMR Theory 6 Table 1.2: Commutators of spin—1 basis operators. The values shown are for [pt, Pj], with p, defined by Equations 1.11. The trivial po = 1 is again omitted for brevity. All values in the table must be multiplied by -4=. Pi P2 P3 P4 P5 P6 P7 P8 Pi 0 P3 - P 2 -V^Pe -P8 v/3p 4 + P7 -P6 P5 P2 -P3 0 Pi \Z3ps -\/3p4 + P7 P8 -P5 P6 P3 P2 - P i 0 0 P6 -P5 2p 8 - 2 P 7 P4 V^Pe -\/3~P5 0 0 -V3pi 0 0 P5 P8 \/3p4~P7 -P6 - \ / 3 p 2 0 P3 P2 - P i P6 - \ / 3 p 4-P7 -P8 P5 \ / 3 p i -P3 0 Pi P2 P7 P6 P5 -2p 8 0 - P 2 - P i 0 2p 3 P8 -P5 P6 2p 7 0 Pi - P 2 2p 3 0 (Figures 1.2b and 1.2c are the precession diagrams for 7YX and Hy, respectively and are related to Figure 1.2a through a cyclical permutation of indices). As can be seen from Table 1.2, p 3 and p4 commute with the Hamiltonian and are thus invariants of the motion. Therefore, C 3 and C 4 are constants. Obtaining the rest of the solutions of the equations of motion is exactly analogous to the spin-| case. To gain some insight into the solutions of Figure 1.2a we should note that there is a direct analogy between our basis operators, { p j , and the spherical harmonics Yjm(0,</>). In fact, Pi? P2 5 and p 3 transform under rotations in the same way as linear combinations of Y~i m; P4 transforms under rotations as Y20; P5, P6 and P7, p§ as symmetric and anti-symmetric linear combinations of >2±i and Y2±2, respectively. The first of the three precession diagrams of Figure 1.2a corresponds to the classical Larmor precession of the magnetization vector with an angular frequency UQ about the z-axis. The other two diagrams represent evolution of the higher order components which do not have classical vector analogs. er 1. Review of NMR Theory b Invariants: P i . - T P 4 - T P 7 c K=—s/2uy p 2 Invariants: P2. T P 4 + T P 7 Figure 1.2: Precession diagram for a spin—1 in a static magnetic field. Evolution of a{t) under the Hamiltonians (a) H = Hz = — \/2^oP3 (Larmor precession); (b) H = Hx = -\/2wiPi (rotation induced by an RF pulse along the z-axis); and (c) % = Jiy — — \/2wiP2 (rotation about the j/-axis). See Equations 1.11 for the definitions of pt-. Chapter 1. Review of NMR Theory 8 In the above trivial example we ignored all interactions besides the (Zeeman) interaction with the external magnetic field. We shall now examine the consequences of the so-called quadrupolar interaction. A nucleus with spin 1 = 1 may possess an electric quadrupole mo-ment, eQ. This produces a shifting of the Zeeman energy levels of the system whenever there is an electric field gradient (EFG) at the nucleus. Such an EFG due to the C— 2H bond in a hydrocarbon molecule, where hydrogen atoms have been replaced with deuterium, results in a quadrupolar coupling constant, e ^ « 167kHz. By contrast, in the magnetic field of a few Tesla the Larmor frequency is measured in tens of MHz. Thus we can treat the quadrupolar interaction as a first-order perturbation on the Zeeman interaction. The dipolar and chemi-cal shift interactions will be neglected in this treatment, as in the model membrane systems which are of main interest to us here, they are typically much smaller than the quadrupolar interactions. A more detailed discussion will be presented in a later chapter. The EFG tensor is a traceless, symmetric tensor of rank two, Vj-j. When expressed in its principal axis system it has only two independent components, usually denoted by Vzz = eq and T] = (Vxx — Vyy)/Vzz, assuming \VXX\ < \Vyy\ < \VZZ\ and so 0 < rj < 1. We can write the quadrupolar Hamiltonian as e2qQ n0 = [zil - / * ) } . (1.12) ' 4/(2/ - 1) In an axially symmetric case, i.e. when 77 = 0, Equation 1.12 is reduced for spin / = 1 to H 9 = ^ ( 3 / 2 - 2 ) = v/|a; gp 4, (1.13) where uq = | € ^ . Using Table 1.2 in the familiar fashion, we can easily obtain the re-sult shown in Figure 1.3, describing the evolution of the system under an axially symmetric quadrupolar Hamiltonian. In this case, P3, p 4, p7, and ps are invariants of the motion and there are two precessing pairs, {pi,pe} and {p2,Ps}-Chapter 1. Review of NMR Theory 9 Figure 1.3: Precession diagram for a spin-1 under an axially symmetric quadrupolar interaction. P3, p4, p7, and ps are invariants of the motion. Chapter 1. Review of NMR Theory 10 1.4 Perturbation Theory Treatment of Spin Relaxation The above examples and their combinations provide us with the tools necessary to easily design and analyze various pulse configurations. We shall use these tools later; for the time being we shall concentrate on the implications of the quadrupolar interaction for relaxation. We shall closely follow the approach of [6], which is summarized below. Let us examine the time evolution of the density operator, both in terms of its response to static interactions and in terms of its relaxation, or return to equilibrium. We shall assume that the Hamiltonian of the system consists of two parts: the static interaction, Ho, and a randomly varying time-dependent part, H\{i). Later we shall identify the two parts as Zeeman and quadrupolar interactions, respectively. In general, we shall examine how random fluctuations of the time-dependent part of the Hamiltonian lead to relaxation. We can write n = Ho +nx(t), (1.14) where the average value of the time-dependent part vanishes, Hi(t) — 0. Following the standard treatment of first-order perturbation theory, we can transform into the interaction represen-tation (into the "rotating frame", for the Zeeman interaction) by replacing each operator p with p(t) = e inotpe- iHot. (1.15) It immediately follows that Equation 1.3 transforms into da(t) dt Wi(<),^ (<) (1.16) We integrate Equation 1.16 from zero to a time t such that a(i) does not differ much from tr(0), to get, to first order, a(f) ~ £(0) - t-jf [Wi(i'),£(0) dt'. (1.17) Substitution of Equation 1.17 into Equation 1.16 yields da(t) dt Wi(t),af(0)] - f [Wi(*),[Wi(i')>*(*0 dt', (1.18) Chapter 1. Review of NMR Theory 11 where the bar indicates that we took the average over the random perturbation 7i\. Since 7~L\{i) vanishes, we are left with only the second term on the right-hand side, da(t) dt = - J* [wi(0,[Wi(<,),5(*')]]*'- (1.19) We can express the perturbation Hi in terms of spin operators Va and random functions of time Fa(t): n1(t) = ^2VaFa(t). (1.20) The random functions Fa(t) can be selected so as to have zero average value, Fa(t) = 0, and their correlation functions Fa(t)Fp(t') = Ga>p(\t — t'\). Furthermore, we select Va in such a way that Va(t) = e iHot Va e- iHot = e iwot Va. ' (1.21) It can be shown [6] that in Equation 1.20 the terms appear in pairs {a, —a}. Furthermore, for weak perturbation, ? W - ^ « l , (1.22) o-(O) and in the high temperature limit, the density operator <r(0) is replaced by2 <r(0) ~ a(t) = a. (1.23) With these two assumptions we can obtain the master equation for the density operator from Equations 1.19 and 1.20, da ~dt (1.24) Here the spectral densities, Ja(ua), are the Fourier transforms of the correlation functions, Gott — oti Ja(u)= I" Ga,-a(T)e iuTdT. (1.25) Jo 2More generally, <x(0) is replaced by a — 7req, where acq is the density operator at thermal equilibrium with the lattice. Chapter 1. Review of NMR Theory 12 Finally, using the property of traces, Tr^A[B,C]| = Tr|[j4,jB]c|, it follows from Equa-tion 1.24 that for an observable associated with an arbitrary operator, p, the basic equation for relaxation is d , . d ^ . _ r dcr i - = - r r {p . , } = r , { p - } = - s r r { [ [ p ' v » l > v - « ] * } • ' • ( " « ) • ( L 2 6 > We are now prepared to consider the particular case when the relaxation is dominated by the fluctuations in the quadrupolar interaction. 1.5 Quadrupolar Relaxation If random fluctuations in the quadrupolar interaction dominate the relaxation then the role of the perturbation 7i\ is played by the time-dependent part of the quadrupolar Hamiltonian. The static, or time-independent part is included, together with the Zeeman Hamiltonian, in the unperturbed static Hamiltonian, Tio in the notation of the previous section. Thus, the average value of the perturbation vanishes, as required. We shall now modify slightly our set of basis operators to bring out the second rank spherical tensor symmetry, implicit in Figure 1.2. Very generally speaking, we are focussing our attention on the symmetry properties of the interaction Hamiltonian itself, but for our present purposes it suffices to say that in the modified set of basis operators we substitute the linear combinations V±i = T 2 ,±i = T ^ ( P 5 ± ipe) in place of p 5 and p 6 , and V±2 = T2,±i = ^(P7 ± ips) in place of p7 and ps- Also, let Vo = T^o = P4- The notation T2,±a reflects the fact that the resulting operators have the symmetry properties of the spherical tensors of rank two. As in Equation 1.20, we can expand the quadrupolar Hamiltonian of Equation 1.12 in terms of these new basis operators. For spin I = 1 we obtain [7] ot=+2 n - i . 5 3 nq= Y , v ^ FQ = {-\)aJ^-^-Y2,_a{e^), (i.27) where Y2,-a(6,<j>) are the spherical harmonics. The time dependence is implicit in the random fluctuations of 0(t) and <j>(t). It is common to assume the stationary nature of these random Chapter 1. Review of NMR Theory 13 processes; for a stationary random function Fa(t) we can introduce the reduced correlation function, g(r), by Ga,_a(|r|) = Fa(t)F.a(t-r) = \F^tjf g(\r\). (1-28) Physically, this reduced (fif(O) = 1) correlation function defines the time scale of the interaction under consideration, g(r)dT = r c, with r c being the correlation time of the interaction. Note that the approximations we made in deriving the Equation 1.24 correspond to u^r* <C 1, i.e. we assume that the time scale of the random fluctuations in the quadrupolar interaction is much shorter than that determined by the width of our distribution of quadrupolar frequencies, l/max(u>q). 2 1 For an isotropic system \Y2t-a(9, <j>)\ = ^p, and the expression for the spectral densities (Equation 1.25) becomes Ja(u) = \Ftfj(u>) = (1-29) where = lie | ^ ° ° g(r) e t W d r | (1.30) is the reduced spectral density. As an example let us consider P3 = (Zeeman interaction) as the operator of interest in Equation 1.26. Calculating the commutators in the usual fashion, 1 [ [ P 3 , F + 1 ] , y _ p 3 , V _ i ,V+i :P3, (1.31) and similarly for a = ± 2 , we find [7] the contributions from the terms with a = ± 1 and with a = ± 2 to be, respectively, \ J(OJO) and 2j(2a;o) each. Thus, the relaxation behavior of p3 is described by ^(P3) = - ^ ( P 3 ) , lz 15 U q j(lj0) + 4j(2uj0) (1.32) Appropriately, the T\z in the above expression is called Zeeman or "longitudinal" relaxation time. Expressions for relaxations of other basis operators can be derived in exactly the same way. For pi = ~j^Ix or p2 = ~^Iy the "transverse" relaxation rate is (1.33) T2z 5 W ' i(0) + |j(wo) + |j(2wo) Chapter 1. Review of NMR Theory 14 for P4 = ^ ( 3 / 2 — 2) the "quadrupolar" relaxation rate is 1 2 2 •( , 7^ = 5 W ' j W (1.34) for ps or p6 (or for V±\, their linear combinations) the "quadrupolar transverse" relaxation rate is 1 1 „ r 1 9 1 (1.35) I2q 1 2 — UJ 5 " j(0) + p(uo) + |j(2wb) and, finally, for p 7 or p 8 (or for V± 2) the "double quantum" relaxation rate is i(wo) + 2j(2wo) . 1 2 2 = -u Tdq 5 ' (1.36) Equations 1.32—1.36 have been derived in [7], and also in [8,9] in a slightly different context. These equations represent the relaxation rates of the observables associated with a particular set of basis operators, chosen in such a way as to represent the various interactions under consideration. It is an idealized picture, however; the observed relaxation rate in a given experiment may have contributions due to the evolution of more than one basis operator. For example, from Figure 1.3 we can see that quadrupolar interaction couples p 2 and ps (also: Pi and pe). Thus, we have to solve two coupled differential equations together to get the transverse relaxation rate in the presence of quadrupolar precession; usually, UqT2 ~> 1 and the cross-terms can be neglected. Then- the solution is reduced [9] to a single relaxation rate for both p 2 and ps: T~2 1 / 1 1 + 2\T' 2z l2q 1 2 —U) 5 « j(0) + j(uo) + - i(2wo) (1.37) We can now identify the pulse techniques used to measure each of the various relaxation times of spin-1. Chapter 2 Experimental Methods of Measuring Relaxation 2.1 Free Induction Decay In this chapter we shall use the operator formalism of the previous chapter to examine a variety of pulse techniques used to measure various relaxation times in a non-interacting spin-1 system. A state of a spin system can at all times be described by a set of coefficients c,(i) (see Equation 1.6). This can be thought of as a time-dependent vector c(f) = (co(t),Ci(t),C2(t)...) in the space spanned by the basis operators for the problem. For example, for a system at equilibrium with the external magnetic field and in the space defined by the basis operators of Equation 1.11 this vector is c = (1,0,0, io,0,0,0,0,0), or Co = 1, C 3 = Jo, and all other Ci = 0. Here IQ = —y/2huJo/kT, \Io\ < 1 in the high temperature approximation1. In the rotating frame the Zeeman Hamiltonian is simply Hz — —OJQIZ = —\/2u>oP3, and since P3 commutes with the Hamiltonian, c3(2) = c3(0) = io and the state of the system does not change with time. One can say that the system is Zeeman-ordered. If we now apply the transverse magnetic field in the rotating frame, i.e. a radio frequency (RF) pulse, described by the Hamiltonian Hi = —uily = — v/2u>iP2, we couple P3 with pi. In the space of operators, the vector c(0) = (l,0,0,io> 0,0,0,0,0) gets rotated into c(r u ;) = (1, Io sin 6,0, Io cos 0,0,0,0,0,0), where 6 = W\TU represents the angle of rotation by the RF pulse of duration TW. If 0 = TT/2 we say that a "90° pulse" has been applied; in this case C(TU) = (l,/o,0,0,0,0,0,0,0). If after the pulse the evolution of the system is governed by the quadrupolar interaction, then at time t after the pulse C{Tw + t) = (l,7ocos(o;g£),0,0,0,0,/osin(aV),0,0), as evident J N o t e that using the normalization that we chose, = y/2 Io at equilibrium. 15 Chapter 2. Experimental Methods of Measuring Relaxation 16 Table 2.1: Free induction decay. The evolution of the vector c(t) in the space spanned by the basis operators p, due to a single RF pulse, 0y, along the y-axis and of arbitrary length. Co is omitted for brevity. All values in the table should be multiplied by IQ. Operators p, are included for reference; see Introduction for details. P. Ci nz Hq r e l a x a t i o n C l 0 s i n # s i n 8 C O S ( C J 9 < ) s i n t 9 e - < / T 2 c o s ( u ; ^ ) 1 / C2 0 0 0 0 C3 1 c o s 0 c o s 0 1 - (1 - c o s 6)e-tlTi* ^ ~ 2) C4 0 0 0 0 ^{IJZ + IZIX) C5 0 0 0 0 ^(Iyh+Ijy) C6 0 0 s i n 9 sin(u)qt) s i n 0 e - ' / 7 2 sm(u)qt) ~ l2y) C7 0 0 0 0 Cg 0 0 0 0 from the precession diagrams of Figure 1.3. The effects of relaxation can also be included; as discussed at the end of the Introduction we solve two coupled differential equations to get C(Tw + t) = ( l , i o e - < / T 2 cos(u;gt),0,0,0,0,ioe~f/r2 sin(w^),0,0), where the relaxation time T2 is determined by Equation 1.37. This expression is valid for any value of uq, and it is the ensemble average of ci(i) over the spectrum of u;g's that determines the nature of the observed NMR signal. This signal is usually called the "free induction decay" (FID), where the word "decay" refers to the decrease in the amplitude of the observed signal caused by the dephasing of the components with different u)qfs. The characteristic time of this decay, often referred to as T2 , is determined by the width of the distribution of o;9's, i.e. by the width of the spectrum. A summary of the above discussion for an RF pulse, 0y, applied along the j/-axis and of arbitrary length, is presented in Table 2.1. Chapter 2. Experimental Methods of Measuring Relaxation 17 2.2 Quadrupolar Echo Very often one of the most serious instrumental limitations in NMR is the so-called "dead time" of the receiver. For a typical RF coil with Q « 102 it takes several microseconds for the signal due to a strong RF pulse to ring down to the noise level. During this time the NMR signal is obscured by this ring down signal and cannot be observed. Reducing the Q of the coil to shorten this dead time is not acceptable since it also reduces the sensitivity of the circuit. At the same time, the initial part of the signal contains valuable information about the lineshape and ignoring this part of the signal would cause severe spectral distortion. A solution to this problem for a spin-1 system is provided by the quadrupolar echo pulse sequence [10], consisting of two pulses separated by a time r and 90° out of phase with each other. We can schematically represent such a pulse sequence by 0y — r — <f>x — t—2. If we now use the precession diagrams of Figures 1.2 and 1.3 to trace the evolution of the vector c(t), we obtain, in the absence of relaxation, the results presented in Table 2.2. One can immediately see that the sign of the phase accumulated by the various components is reversed by the second pulse, and at time t = T later all of the components are aligned the same way they were immediately following the first pulse. In particular, for <f> = 90°, ci(f) = sin0 cosLJq(t — r), and c2(i) = — cos0 cos(uqt). The observed signal is proportional to the ensemble average of ci(t) for the in-phase signal, and C2(t) for the quadrature signal, over the range of wg's. In general, components with different quadrupolar frequencies accumulate phase at different rates; in the ensemble average, these components interfere destructively (on the time scale of T£)• However, at time t = r the value of ci(t) becomes independent of uq. Consequently, all of the components with different u>g's have the same phase and interfere constructively, producing an echo. This echo signal is maximum when both 0 = 90° and <f> = 90°: c\{t = r) = 1 and c2(t = r) = 0. In this case, the refocussing is complete, and the state of the system at t = r is exactly equivalent to that immediately following the first 90y pulse, which is the FID discussed in the previous section. Effectively, the quadrupolar echo sequence displaces the FID signal to the new 2 I n this notation, t is measured from the <f>x pulse. Chapter 2. Experimental Methods of Measuring Relaxation 18 Table 2.2: Quadrupolar echo. The evolution of the vector c(t) due to the pulse sequence 6y — r — <j>x — t—. Here we assume no relaxation, and set Io — 1. To maintain the compatibility with the precession diagrams of Figures 1.2 and 1.3 we calculate the time-dependent coefficients for the linear combinations ^ P 4 + |P7 (coupled to p 6 under 7ix) and |p 4 - ^ P 7 (invariant under Hx), which we denote C 4 + 7 and C 4 _ 7 , respectively. c, By — T — 4>x -t-Cl 0 sin# s'mO cos(uqT) sin 0 cos(u;gr) sin 0 cos(u;gr) cos(u;g£)— sin 0 cos(2</>) sin(a;gr) sin(u>gf) C2 0 0 0 — cos 0 sin <f> — cos 0 sin cos(a;gf) C3 1 cos 0 COS0 COS 0 COS (j) cos 0 cos C 4 0 0 0 ^ 4 + 7 ^ 4 + 7 C5 0 0 0 0 — cos 0 sin c/> sin(u;g7;) C6 0 0 sin 0 sin(a;gr) sin 0 sin(u>gr) cos(2</>) sin 0 cos(uqT) sin(a;g^ ) + sin 0 cos(2</>) sin(u>?r) cos(cjgf) C7 0 0 0 2C4+7 2C4+7 C8 0 0 0 0 0 C4+7 0 0 0 sin 0 sin(wgr) sin(2</>) sin 0 sin(wgr) sin(2</>) C4-7 0 0 0 0 0 Chapter 2. Experimental Methods of Measuring Relaxation 19 time origin at the top of the echo. However, there is one important difference: if r is chosen to be sufficiently long, then the echo takes place outside of the dead time of the receiver, and the subsequent NMR signal is not affected by the ring down. If the Fourier transform is now performed starting at the top of the echo, one obtains a spectrum which is free of distortion. Figure 2.1 illustrates a typical application of the quadrupolar echo. I I l I I l 0.0 0.5 1.0 1.5 2.0 2.5 Time (ms) Figure 2.1: The quadrupolar echo pulse sequence, 90j, — r — 90^ — t—, displaces the time origin of the FID signal to a point outside of the dead time of the receiver. Echoes for two values of r are shown for deuteriated hexamethylbenzene. Variation of the echo intensity with r provides a measure of T29e. As we see from Figure 2.1 the amplitude of the echo decreases with increasing r, due to the irreversible loss of phase memory. Equations 1.32-1.36 contain all of the possible relaxation rates; we can now include the relaxation effects essentially by inspection. For simplicity, let us restrict ourselves to 9 = 90°. Table 2.3 summarizes the effects of various relaxations. In particular, if we set (f> — 90° and neglect the longitudinal relaxation (typically, T\z >• T 2), Chapter 2. Experimental Methods of Measuring Relaxation 20 Table 2.3: Relaxation effects measured by the quadrupolar echo. The evolution of the vector c(t) due to the pulse sequence 90y — r - <f>x — t-, taking relaxation into account. Again, we set Io — 1. 90y — T — <t>x - Z -C l 1 e-T/T2 c o s ( u ; g r ) e - ( T + < ) / T 2 [ c o s ( u ; 9 7 - ) c o s ( u ; , i ) -cos(2c/>) s i n ( o ; g r ) s i n ( a ; 9 i ) ] C2 0 0 -(1 - e _ T / T l * ) s i n < / > -(1 - e _ T / r i z ) s i n ( / ) c o s ( w g / ) e - ' / T 2 C3 0 1 - e-TlTl* (1 - e - T / r i * ) c o s < £ 1 - [1 - (1 - e-TlT^)cos,<t>]e-tlT^ c 4 0 0 ^ c - T / T 2 s i n ( u ; , r ) s i n ( 2 < / ) ) &e-TlT* s i n ( w , r ) s i n ^ e - ' / 7 ^ C5 0 0 0 -(1 - e - T / T l * ) s i n < £ s i n ( u ; , 0 e _ 1 / T 2 C6 0 e - r / T 2 s i n ^ 9 r ) e - T / T 2 s i n f ^ j - ) cos(2c/>) e - ( r + < ) / T 2 [ c o s ( a ; , r ) s i n ( a ; , < ) - | -cos(2</>) s i n ( a ; g r ) c o s ( a ; g Z)] C7 0 0 i e _ T / T 2 sin(w g r)sin(2</>) | e - T / T 2 s i n ( a ; , r ) s i n ( 2 ( / ! > ) e - ' / r o g C8 0 0 0 0 C4+7 0 0 e _ T / T 2 s i n ( u > , r ) sin(2c/>) e _ T / r 2 s i n ( w , r ) s i n (2 ( /> ) X C4-7 0 0 0 e _ T / T 2 s i n ( w , r ) sin(2</>) X &(e-t/Tlq _ e-t/TDQ^ Chapter 2. Experimental Methods of Measuring Relaxation 21 we obtain an echo at t = r: c\{t = r) = cosuq(t — r)e~( t+TWT2 = e~2r/T2, independent of uq. This is why measuring the amplitude of the quadrupolar echo as a function of the pulse separation, r, is commonly used to obtain T2. To emphasize this point, and to distinguish from other methods of measuring T2, we shall use the notation T%e • 2.3 Inversion Recovery A usual way of measuring T\z is the inversion recovery pulse sequence, 180j, — r — 90y — t—. The initial 180° pulse inverts the sign of the C 3 component, and the 90° pulse simply produces an FID whose amplitude will depend on r, as the system returns to equilibrium. The amplitude of the FID immediately following the 90° pulse should be 1 - 2e~TlTlz. The quadrupolar echo provides us with a reliable way of getting around the limitations imposed by the dead time of the receiver. Thus we can use it in place of the single 90° pulse in the inversion recovery sequence. The modified pulse sequence is then 180^ — T\ — 90y — r 2 — 90^ — t—, and time evolution of the system is given in Table 2.4. The immediate consequence is that the echo amplitude at time t = r2 is C\(t = T2) = (1 — 2e~Tl/Tlz)e~2T2/T7. If we choose r2 sufficiently long so that the echo occurs outside the dead time (yet as short as possible to avoid an overall decrease in amplitude due to T2 relaxation), and then keep it constant while measuring the echo amplitude as a function of T\, we immediately obtain T\z. 2.4 Stimulated Echo Let us now consider the evolution of the spin system due to the two-pulse sequence in which the second pulse is in phase with the first one, 90,, — r — 90y — t—. This case differs from the quadrupolar echo sequence in that there is a transfer of order into a coherent mixture of Zeeman and double-quantum coherence ( C 3 and c$), as seen in Table 2.5. In fact, if we apply another 90y pulse we can bring that order back, in a so-called "stimulated echo" experiment [11,12], 90^ — Ti — 90^ — T2 — 90y — t—. After the last 90^ pulse, the observed signal is the ensemble Chapter 2. Experimental Methods of Measuring Relaxation 22 Table 2.4: Using inversion recovery to measure T\z. The pulse sequence is 180,, - r i - 90,,-r2-90^-2-. The 180,, pulse inverts the sign of the equilib-rium 03(0), and during the time T\ it decays to A = 1 — 2e~T1/Tlz. The rest of the time evolution is exactly analogous to the one presented in Table 2.3, with 03(0) = A. Measuring the final echo amplitude as a function of T\ immediately yields T\z, since c\(t = r 2) = (1 — 2e~Tl /Tlz)e~2T2/T2. 180y - n - 90y - r 2 - 90 x -t-Cl Ae-T2IT2 cos(uqT2) A e _ T 2 / T 2 COS(O;0T2) ^e-(T2+*)/T 2 C O S 4 J g ( f _ T 2 ) C2 0 _(1 _ g-Tj/Tix) -(1 - e _ T 2/ T l z)cos(a;,i)e-'/ : r 2 C3 1 - e - T 2 l T l z 0 1 -C 4 0 0 0 C5 0 0 _(1 _ e - ^ / ^ ^ s i n ^ ^ e - ' / 7 2 C6 ^4 e-T 2/T 2 s i n ^ ^ ) -Ae-^/ 7 2 sin(u;?r2) Ae-(T2+tVT2 sinuq(t - r 2) C7 0 0 0 Cg 0 0 0 Chapter 2. Experimental Methods of Measuring Relaxation 23 Table 2.5: Stimulated echo. Preparation of a coherent mixture of Zeeman and double-quantum coherence. c , - 90y - T i - 90y - r 2 - 90y C l 1 _ c - n / r i , (1 - e - n / r i . ) c - ^ / r a COS(U,T 2) l - [ l + e - T 1 / T 2 cos(u}qT1)]e-T2/T^ C2 0 0 0 C3 -e _ T l/T 2cos(a; gr 1) l - [ l + e _ T l / r 2 cos(a;gri)]e-T2/Tlz -(1 - e-Tl/Tl*)e-T*/T* cos(u}QT2) C 4 0 0 0 C5 0 0 0 C6 0 (1 - e - ^ i / ^ i ^ e - ^ / 3 2 sin(wgT2) _ e - r 1 / r 2 sin(w,r 1 )e- ' r 2 / T £ , « C7 0 0 0 e g e - T i / T 2 s j n ^ ^ ) e - n / T 2 s i n ( a , 9 T - 1 ) e -T2 / T D g (1 - e - n / r i ^ g - r a / T j s i n^jvj) average over a distribution of uq values of C l(i) = | 1 - (l + e _ T l / T 2cos(a;,ri))e-' r 2 / : r i i cos(o;g<)+ e - T l / T 2 sin(u; g Ti)e- T 2 / T D « sin(o; go|e"' / r 2 = (l -e - T 2 / T l z )e- ' / T 2 cos(a; g i ) -r-_ ^ e - T 2 / T l z _ e-r2/TDQ^ C Q S U ^ t _ ri) - ( e - T 2 / T l 2 + e _ T 2 / : r D « ) cosa;g(* + n) I e -(r 1 + t) /T 2 (2.1) Here, the first term is exactly analogous to the inversion recovery experiment (see Table 2.4); it contributes an FID-like component to the signal after the last pulse. The terms containing cosw?(2 T\) are responsible for two negative echoes at t — ±Ti: for either term the phase accumulated by different components is proportional to u>q causing destructive interference in the ensemble average, except at t = ±Ti, respectively, where all of the components have the same phase and interfere constructively. As discussed before, this dephasing of the components Chapter 2. Experimental Methods of Measuring Relaxation 24 with various wg's takes place over time T%; thus if we choose 2T\ > the signal can be thought of as two separate echoes, since in this case the contribution due to cosuq(t + T\) is negligible at t = T\, and vice versa. In order that causality is not violated we can only observe the signal for t > 0, so that the echo at t = — T\ remains "invisible". However, if we modify slightly the stimulated echo pulse sequence by applying an additional 90x at t = Tz > T\ we can bring back both of these echoes, much the same way as we refocus the FID signal in a quadrupolar echo sequence. Effectively, this last pulse displaces the location of point t = 0 to a time 73 later, t = 2T3, and if it is chosen so that 7-3 — T\ > 0 then both echoes are observed, attenuated by e-2r 1 / T 2 T m g i g s n o w n i n Figure 2.2. As seen from Equation 2.1, varying r 2 and measuring the amplitude of the two echoes allows us to obtain both T\z and Tng in a single experiment. The standard stimulated echo experiment, where we can only see the echo at t = T\, provides us only with a measure of (e-T2/TDQ _ e-v2/Tizy rj<0 s e p a r a t e the two unknowns, we would need additional information, such as an independent measurement of T\z from an inversion recovery experiment described in the previous section. 2.5 Jeener—Broekaert Echo From the general result of Table 2.3 we can observe that the pulse sequence 90^ — r — <f>x transfers the original Zeeman order into a mixture of quadrupolar order and double-quantum coherence, as seen from the non-zero value of C 4 + 7 after this sequence. In fact, this transfer of order is maximized when <f> = 45°. In the usual manner we can summarize this case by Table 2.6. As in the case of the stimulated echo, we do not have a direct way of observing either quadrupolar order ( C 4 ) or the double-quantum coherence (07). However, we can apply another pulse to transfer the order back into our observable, C\. The optimal way to achieve this is by using another 45^ pulse. This is the so-called "Jeener-Broekaert echo sequence" [13], 90y — T\ — \§x — T2 — 45x — t—. After the last pulse, the observed signal is the ensemble average, Chapter 2. Experimental Methods of Measuring Relaxation l I i i 1 1 1 1 - . 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t ( m s ) . Figure 2.2: Stimulated echo. The standard stimulated echo pulse sequence is modified to 90^ — T\ — 90y — r 2 — 90^ — r 3 — 90x — t—. Only the last two pulses are shown here. Here we use T\ = 400/is, r 2 = 3ms, and T3 = 1ms. The contributions due to all three terms in Equation 2.1 are seen: an FID signal ~ (1 — eT2/Tl2) at t = 0 and its quadrupolar echo at t = 2T3; a negative echo ~ (e _ T 2/T l* — e~T2/Tl>Q) at t — T\ and its echo at t = 2 r 3 - ri . The "invisible" echo ~ ( e - ^ / r i 2 + e-T2/TDQ^ A T T _ _ N is only seen when refocussed at t = 2T3 + T\, and not at t < 0 (before the third 90^ pulse). Note how the amplitudes of all of the echoes are attenuated by T 2 relaxation. Chapter 2. Experimental Methods of Measuring Relaxation 26 Table 2.6: A mixture of quadrupolar and double—quantum order. 9 0 y — 7 i — 4 5 * — r 2 - 45x C l e -(Ti+ T 2) /T 2 c o s ( a ; g r i ) c o s ( a ; , r 2 ) C - ( T I + T 2 ) / T 2 c o s ^ ^ ) cos(u;,r2) C2 -^5(1 - e- Ti/ T^)cos(u;,r 2)e-^/ r 2 - e - T ' / T ^ ) [ c o s ( o ; g r 2 ) e - T 2 / : r 2 -C 3 1 - [1 - (^1 - c - n / T i , ) ] c - 7 a / T i , - e-r^T^)[cos(ujqT2)e-^lT^ + C 4 ^ e - r i / r 2 s i n ^ r ^ e - ^ / T i , ^ C 4 + 7 + 5 C 4 - 7 C 5 -^ (1 - e - T i / r i 0 s i n ( u ; g r 2 ) e - T 2 / r 2 - e - ^ / ^ s m ^ r ^ e - ^ / 7 2 C 6 e - ( T i + T 2 ) / T 2 c o s ( w g r i ) s i n ( a ; , r 2 ) - i e - T 1 / T 2 s i n ( o; gri)(3e- T 2/ T l'' + e-T2/Tl}Q) C 7 I e - T i/r 2 s i n ( u ; 9 r i ) e - T 2 / T D « 1 \ / 3 2 c 4 + 7 - 2 C 4 - 7 C 8 0 1(1 _ e-TilT^)sm(uqT2)e-T*lT* C 4 + 7 I e - n / T 2 sin(a;gri)(3e-T2/:ri« + e - 7 * / 7 " ^ ) e - ( T i + T 2 ) / T 2 c o s ^ r i ) s i n ( o ; g r 2 ) C 4 - 7 ^ e - T ! / T 2 S i n ( U , g T l ) ( e -T 2 / T 1 , _ e-T2/TDQ) i ^ e - r i / r 2 sm(wgri)(e-T2/:ri'» - e-T^TDQ) Chapter 2. Experimental Methods of Measuring Relaxation 27 over a distribution of wg's, of C l ( t ) = e-(r 1+r2)/T2 c o s ^ ^ ) c o s ( o ; , r 2 ) c o s ( w , t ) + i e - T l / T 2 s in(o; ( 7 r 1 ) (3e-^ / T l « + e^/Tog) sin(«,t) = c-(Ti+*)/ra [ ( i e - T 2 / T 2 cos(u;gr2) + | e - ^ / r . , + I c-<*/TI>Q) C O S A ; G (* _ T L) + e-t/T2 (_e-r2/r2 c o s ( u ; g r 2 ) _ _ e - r 2 / r 1 , _ _ e - r 2 / r D Q ^ c o s ^ t + T j ) _ ( 2 < 2 ) Thus we again obtain a superposition of two Jeener-Broekaert echoes3 at / = ± T I , which can be brought back by applying an additional 90x at t = T3 > Ti. This is shown in Figure 2.3. Notice that the two echoes are of opposite sign in this case, while in the case of the stimulated echo experiment of the previous section, both echoes were negative. As seen from Equation 2.2, varying r 2 and measuring the amplitude of the two echoes allows us to obtain both T\Q and TDQ. Again, note that the standard Jeener-Broekaert experiment re-quires additional information to separate the two, for example, TDQ as measured independently by a stimulated echo experiment. 2.6 The Pake Doublet Spectrum As described in a previous section, the quadrupolar echo pulse sequence allows one to obtain an undistorted spectrum. If the sample consists of a large number of domains where the molecular axes are randomly oriented with respect to the external magnetic field, the 2 H NMR spectrum is a superposition of doublets with quadrupolar splittings representing all possible orientations, usually referred to as the "powder pattern" spectrum. The dependence of the quadrupolar splitting on orientation with respect to the external magnetic field arises from the fact that for 2 H atoms the primary contributions to the electric field gradient (EFG) are due to the molecular electronic distribution. Thus the principal axis system of the EFG tensor is a molecule-fixed system [8]. Since the direction of quantization is defined in the laboratory frame by the direction of the external magnetic field, Ho, we need to 3 I n 2 H N M R literature they are often called "spin alignment" echoes [14]. Chapter 2. Experimental Methods of Measuring Relaxation f_ ! ! ! ( ! j - . 5 0.0 0.5 1.0 1.5 2.0 2.5 t ( m s ) Figure 2.3: Jeener-Broekaert echo. The standard Jeener-Broekaert pulse sequence is modified to 90^ — r i _ 45x — r 2 — 45x — T3 — 90x — i—. This allows the "invisible" echo at t = —T\ (see Equation 2.2) to be observed by displacing the point t = 0 to t = 2r3. The amplitudes of the echoes are attenuated by T 2 relaxation. Here, T\ = 300/xs, r 2 = 4ms, and r3 = 700/zs. We use phase alternation of the 90^ pulse to eliminate other echoes arising from the static dipolar interaction. Chapter 2. Experimental Methods of Measuring Relaxation 29 transform the quadrupolar Hamiltonian, Hq, defined by Equation 1.12 from the principal axis system of the EFG tensor to the laboratory frame. Using Wigner rotation matrices in the usual fashion we can show [8] that the quadrupolar splitting depends on orientation as 3 e2aO r i Avq = - — p - [(3cos2/3 - 1) + r) cos2a sin2/?j, (2.3) where a and /? are the Euler angles [15] specifying the orientation of the E F G tensor with respect to Ho. It follows from Equation 2.3 that the quadrupolar splitting is a function of the orientation, and if the sample consists of a large number of randomly oriented domains, the 2 H spectrum is a superposition of doublets with quadrupolar splittings representing all possible orientations. In the axially symmetric (rj = 0) case the quadrupolar splitting depends only on the angle (5 and the integrated intensity of such "powder sample" can be written [8] as / ( * ) = { (3 + 6a;)-1/2 \<x<\ (2.4) (3 + 6a:)-1/2 + (3 - 6a;)"1/2 0 < x < \ where /S.v = ^—^*-2x. This is the powder pattern first observed by G.E. Pake [16] in the systems containing dipolar coupled spin-| pairs. In those systems, each spin in the pair had its effective local magnetic field augmented or diminished by its neighbor, resulting in a pair of spectral lines with a splitting dependent on the angle between the external field and the vector connecting the spins. In our case, it is the presence of a quadrupolar interaction of spin-1 with the gradient of the local electric field that is responsible for producing the splitting; yet both systems have the same functional form of orientation dependence (Equation 2.4) and thus the same characteristic shape of the spectrum. Equation 2.4 has a singularity at x = \ corresponding to the (3 = 90° orientation. In the experimental lineshape, a certain amount of inherent line broadening is expected. When calculating powder lineshapes one usually assumes either a Lorentzian broadening characteristic of motional narrowing as found in isotropic fluids, or a Gaussian broadening appropriate for Chapter 2. Experimental Methods of Measuring Relaxation 30 the dipolar coupled solid systems. Such line broadening would have an effect of smoothing over the singularity, producing the so-called "Pake doublet" lineshape. An example of such Pake doublet is shown in Figure 2.4a. The singularities get replaced by the maxima ("edges") -30 - 2 0 -10 o 10 20 30 F r e q u e n c y ( k H z ) Figure 2.4: Selective inversion recovery introduces an ^-dependence into the amplitude of the signal, producing a "windowing" effect, or a non-uniform spin temperature distribution. Deuteriated hexa-methylbenzene spectrum in (a) was obtained by using a quadrupolar echo pulse sequence, 90y — T — 90x — t—, with r = 50 [is. In (b) the same quadrupolar echo sequence was preceded by the two selectively inverting pulses, 90y-Ti - 90,,-r2 -90y-Ts- 90^ — /-, and the result subtracted from the spectrum in (a). Here T\ = 55/is, r 2 = 1ms, and T 3 = 50 us. corresponding to the orientations perpendicular to the magnetic field which are more abundant than the ones parallel with the field ("shoulders"). Chapter 2. Experimental Methods of Measuring Relaxation 31 2.7 Selective Inversion Recovery The stimulated echo experiment described earlier has another use. In the limit T\ —> 0, the pulse sequence 90y — T\ — 90y — r 2 — 90y — t— becomes identical to the inversion recovery pulse sequence, 180y — r — 90y — t—. This is also evident from the result we derived earlier for the stimulated echo; if we set T\ = 0 in Equation 2.1, we obtain ci(t) = (1 - 2e~T2lTl*) e~t/T2 cos(uqt), (2.5) which is identical to the expression we obtained for the inversion recovery pulse sequence. All frequencies uq are affected in exactly the same way, and thus to the extent that a single 180y pulse can be approximated by a two-pulse combination, 90y — ri — 90y, we gained no new information. Suppose however, that T\ ^ 0. For simplicity, let us assume r 2 > TDQ; this assumption may not be appropriate in some experimental cases, but it is instructive to see that Equation 2.1 is then reduced to d(t) 1 - ( l + e~T l / T 2 cos(a;gr1))e-T 2 / T l z e~t/T2 cos(uqt). (2.6) Comparison of the two expressions (Equations 2.5 and 2.6) indicates that we introduce an uq dependence into the amplitude of the signal, representing the non-uniformity of the Zeeman order in the spin system. This is sometimes referred to as a "non-uniform spin temperature" distribution, as opposed to the uniform spin temperature in the inversion recovery experiment. In fact, with a proper choice of T\ we can selectively invert the order for some spins without greatly affecting the rest; hence the name "selective inversion recovery". A Fourier transform of C\{t) can be thought of in this case as a convolution of the spectrum with a frequency-selective windowing function, as illustrated in Figure 2.4b. Here we replace the last 90y pulse by the usual quadrupolar echo pair; the actual pulse sequence is therefore 90y — T\ — 90y — r 2 — 90y — 7 3 — 90r — t—. Figure 2.4b shows the spectrum of deuteriated hexamethylbenzene obtained by Fourier transform of the echo signal. For reference, compare this spectrum to the standard Pake doublet (see previous section) shown in Figure 2.4a. The intensity of the frequency components Chapter 2. Experimental Methods of Measuring Relaxation 32 near the "edges" of the powder pattern is seen to be almost completely eliminated by the selective inversion method. At the same time, the spectrum near the "shoulders" of the powder pattern has not been affected. This was achieved by adjusting the separation of the two selective inversion pulses (T\) SO that the frequencies near the edges are not inverted while most others are; and by subtracting the resulting spectrum from the spectrum shown in Figure 2.4a. The described simple method of selective inversion can be viewed as a particular case of the so-called "DANTE pulse sequence" [17,18], which in turn is a simplification of more general methods of creating an arbitrary pattern of irradiation of an NMR spectrum by manipulation of the phase and amplitude of the RF pulses [19]. 2.8 Multiple Echo Techniques One can extend the quadrupolar echo technique in the following way. If we consider the state of the system at time t = r after the second pulse, i.e. after the pulse sequence 90,, — r — 90a; — T—, the last column of the Table 2.2 yields c(r) = (l,e _ 2 T / T : 2,0,0,0,0,0,0,0), if we do not take T\z relaxation into account. This is identical to the state of the system immediately after the first 90,, pulse, save for a decrease in C\ by a factor of e~2rlT2. Thus if we repeat the (r — 90x — T) part of the sequence again, we can immediately predict that the state of the system will be c(r) = (l,e - 4 T/T 2,0,0,0,0,0,0,0). In general, the state of the system following the pulse sequence 90y — (r — 90x — r) n will be (1, e~2nTlT2,0,0,0,0,0,0,0). If we measure the strength of the signal, proportional to c\, at times t = 2nr from the first 90,, pulse, we can obtain in a single experiment a series of values which give us T 2 . By comparison, using a single quadrupolar echo, we would need to repeat the experiment a number of times, varying r. This, in effect, is the so-called "Carr-Purcell-Meiboom-Gill (CPMG) echo train" [20], modified to be suitable for the spin-1 case. We shall use the notation "q-CPMG" for this pulse sequence, although some authors refer to it as "MW-4 sequence" [21,3]. It can be shown that the above remains true even if we include the effects of T\z relaxation. We illustrate this in Table 2.7 where we calculate the state of the system at t = 2T and t = 4r taking relaxation into account, Chapter 2. Experimental Methods of Measuring Relaxation 33 Table 2.7: Quadrupolar Carr-Purcell-Meiboom-Gill ( q - C P M G ) echo train. The pulse sequence is 90y — (TI — 90a; — r) n . In a single experiment, a number of values of ci(t — 2nr) ~ e-2nT/r2 -1S obtained, yielding T2. The results for n = 1 and n = 2 are shown taking the relaxation effects into account. c; * - 2r * = 4r Cl e-2r/T 2 e-4r/T 2 C2 -(1 - e - T / T » * ) e - T / r 2 cos(a;,r) _(1 _ e-2T/7iz) e - r / r 2 cos(a;<,r) C3 1 - e ~ r ^ 1 - [1 + (1 - e-TlT^) e-2TlT* cos(2a;(,r)]e-T/T^ C4 0 0 c5 _(1 _ C - T / T H ) C - T / T 2 Sin(a;gr) _(1 _ e-2r/TiM) e-r/Ti sin^qT) c6 0 0 C7 0 0 C8 0 0 and in Figure 2.5 where the first few echoes of a q- CPMG experiment are shown. Thus we have another method of measuring T2. To distinguish from the result we obtain by performing a series of quadrupolar echo experiments, varying r, which we denoted T 2 , e earlier, we shall use the notation t%~CPMG for the T2 measured by a q-CPMG echo train. Note that within the confines of the theoretical model we developed so far, T| e = t%~CPMG. Chapter 2. Experimental Methods of Measuring Relaxation t (ms) 3 Figure 2.5: Quadrupolar Carr-Purcell-Meiboom-Gill ( q - C P M G ) echo train. The FID and the first four echoes of a q-CPMG pulse sequence, 90^ — (r — 90x - r ) n are shown for deuteriated hexamethylbenzene. The signal at t = 2nr is proportional to e ~ 2 n T / T 2 (see Table 2.7), allowing us to measure T | - C 7 M G in a single experiment. Chapter 3 Model Membranes A simple model membrane system is formed by an aqueous dispersion of a diacyl phosphatidyl-choline in water. Because the phospholipid has both hydrophilic and hydrophobic regions, it forms a lamellar (bilayer) liquid crystalline ( La) phase above a certain temperature (~ 41.2° for DPPC), with choline headgroups positioned at the lipid-water interface and the acyl chains extending into the central region of the bilayer. Typically, multilamellar vesicles (MLV's) are formed spontaneously, with the outside wall consisting of several bilayers arranged concentri-cally as the skins of an onion. The distribution of sizes and shapes of these MLV's is fairly broad as determined by the freeze-fracture microphotography [22]. The MLV's are ruptured each time the dispersion is frozen, and get re-formed upon thawing. Multiple freeze-thaw cycles can be used to produce a more homogeneous distribution of MLV sizes. Rapid extrusion through poly-carbonate filters has been shown to produce unilamellar vesicles of uniform size [23] but this technique has not yet come into universal usage. The attraction in using these systems is that the order and dynamics of independent hydro-carbon chains in a lamellar structure can be studied, ultimately leading to better understanding of processes in the biological cells. To use 2 H NMR it is necessary to synthesize 2H-labelled phospholipid molecules in which some (or all) of the hydrogen atoms on the acyl chain have been replaced with deuterium atoms. The techniques for making both specifically and completely deuteriated saturated fatty acids are by now well known (see [8], and references therein). Due to the nature of MLV's the model membrane samples are usually "powder samples", in the sense that different molecules are oriented randomly with respect to the external magnetic field1. Thus the spectrum of a specifically deuteriated sample for which the deuterium atoms ' i t is possible to produce macroscopically oriented samples by squeezing small amount of lipid/water dispersion 35 Chapter 3. Model Membranes 36 on all molecules in the sample are equivalent has the usual Pake doublet lineshape. In the case of many inequivalent deuterium positions, where differences in the local orientational order or in the motions of different parts of the molecule may result in different quadrupolar splittings, the experimental spectrum is a superposition of many such Pake doublets. 3.1 Motions and Order Parameter Rapid molecular motions modulate the Euler angles specifying the orientation of the EFG tensor with respect to Ho, denoted a and f3 in Equation 2.3. If these motions occur on the time scale short compared to ( A i ^ ) - 1 the observed quadrupolar splitting is simply (Avq) where the average is over the range of these molecular motions. If these rapid motions possess an axial symmetry (3-fold or higher) then it is useful to perform the transformation between the laboratory frame and the molecule-bound one of the principal axes of the EFG tensor, in two steps. First, the transformation to the reference frame associated with the symmetry axes of the rapid motion (Euler angles 9 and </>), and second, the transformation from that reference frame to the molecule-bound one (angles a' and /?'). For an axially symmetric motion the first transformation is characterized by a single angle 9 between the axis of symmetry of the motion and the external magnetic field, while the angles a! and /3' can be thought of as the instantaneous angles of the axis of symmetry in the time-varying principal axis coordinate system. It has become customary [3] in the 2 H NMR literature to to use an "orientational order parameter", S, to describe the effects of the averaging over the range of the rapid motions: (3.1) 3e2qQ (3 cos 29-l)S, (3.2) 4 h where (3.3) between glass plates. However, these samples are much harder to work with. Chapter 3. Model Membranes 37 and where the average is again over the range of the rapid motion2. This effect of reduction of the quadrupolar splittings caused by the rapid motions is called "motional averaging"; it is responsible for the narrow, "high-resolution" NMR lines in ordinary liquids where the rapid motion is isotropic and motional narrowing of the spectral lines is complete. In liquid crystalline systems, such as model membranes, the motional averaging due to an anisotropic (but axially symmetric) motion is not complete and is usually quantified by the reduction in the second moment of the spectrum, AM2 [8]. This immediately defines the inherent time scale via the characteristic correlation time of this motion, TM = (AM2)~ 1^ 2• Different parts of the molecule may move differently, with different mean orientations and degrees of reorientation, and may, therefore, have different order parameters. As a consequence of Equation 3.1, these correspond to different quadrupolar splittings and can thus be measured directly from the spectra. Using isotopic substitution it may be possible to measure the orien-tational order parameters for each rigid part of a flexible molecule; this provides a way to look at the changes in molecular orientation and dynamics. It can be also seen from Equation 3.1 that a rapid uniaxial motion projects the molecule-fixed interactions along the principal axes of the EFG onto the symmetry axis of the motion, or the local director. The result is that spectral contributions from the domains of different orientations of the local director are simply related by a P2(cos0) scaling in quadrupolar split-ting, where 9 is the angle between the local director and the external magnetic field, Ho, which defines the laboratory frame. For a powder sample consisting of domains of randomly oriented local directors, the spectrum is a superposition of all such contributions. Thus in an experi-mental spectrum the observed lineshape is determined by the distribution of orientational order parameters in the molecule as well as by the random (powder) distribution of orientations of the local director. However, it has been shown that a numerical procedure ("de-Pakeing") can be reliably used to separate the two factors [24,25,26]. 2 I n general, S is a tensor with many independent components [8] needed to fully describe the molecule. The number of the components is reduced due to the symmetry of the molecule and of the motion, in this case to a single "order parameter". Chapter 3. Model Membranes 38 3.2 Aspects of N M R Unique to Membranes A commonly accepted model for the motions in the La phase of the phospholipid/water disper-sions includes: diffusion in the plane of the bilayer (with a diffusion constant of ~ 10 - 7 cm2/s), a rapid (on the NMR time scale, w^T1) rotational reorientation about the long axis of an essen-tially cylindrical phospholipid molecule, and a rapid conformational trans-gauche isomerization of the C—C bonds. In addition, it has been shown [27,28] that the normal to the bilayer is the axis of symmetry for the motions on the NMR time scale. These properties have interesting consequences for the 2 H NMR spectra of the model membranes. By using specifically labelled lipids the characteristic variation of the orientational order parameter with depth has been measured [29,30] to be roughly constant, S « 0.2, for the half of the acyl chain closer to the polar headgroup ("plateau"), and falling rapidly to much smaller values in the membrane interior. Similar results have since been obtained using one-chain perdeuteriated DPPC-d3i [31]. The resulting order profile effectively measures the increase in the acyl chain flexibility towards the end of the chain; these measurements provided the basis for a number of theoretical studies [32,33,30,34]. The changes in this characteristic signature of a phospholipid bilayer upon addition of cholesterol or membrane proteins have been used to characterize lipid—cholesterol or lipid—protein interactions3. In addition to the local orientational order, the dynamical properties of the model mem-branes can be determined from the 2 H NMR measurements of the longitudinal ( T\z) and transverse ( T 2 z ) relaxation times. Experimentally, T\z > T 2 z as observed by 2 H NMR [8], implying that there must exist motions with correlation times T\ <; OJQ1 that are responsible for T\z relaxation, and other motions with correlation times r 2 >^ u^1 that dominate T 2 2 relaxation. The implied hierarchy of correlation times then is ri £ "o1 < T2- (3.4) Measured relaxation rates typically yield T\ < 10 - 1 0s, while UQ 1 « 5 X 10 - 9 (for a Larmor 3 T h e unfortunate term "fluidity" to mean orientational disorder has been used in this context; see comments in [3,31,35]. Chapter 3. Model Membranes 39 frequency of « 35 MHz), and T2 in the range 10 - 7-10 - 8s, which is certainly consistent with the above time scale. Traditionally, it has been assumed that the relaxation is produced by a rapidly fluctuating (on the time scale of motional averaging, TM) interaction, whose correlation time r c <C TM — (AM2)~ 1^ 2• This places T\J at the upper end of the above time scale, n £ «o 1 < T2 < TM, (3.5) as confirmed by an experimental estimate of TM ~ 10 - 5s [30]. However, as seen from Chapter 1 (Equations 1.32 and 1.33) T\z relaxation is sensitive to the spectral density, j(u>), of the fluctuating quadrupolar interactions at u = OJQ and at u = 2u>o, while the expression for T2z also contains the term j(0). This dependence on the low-frequency components of the spectral density makes T2z very sensitive to slow motions, and we have indeed shown in a recent study [36] that contrary to previous interpretations, a large fraction of the 2 H NMR transverse relaxation in model membranes is due to molecular motions with r c >• TM. 3.3 Diffusion as a Slow Motion The motion which is the prime candidate for such long correlation times is the lateral diffusion of lipids along the curved bilayer membrane of a MLV, or the equivalent effects due to the rotational tumbling of the vesicle as a whole. For molecules diffusing on the surface of a sphere of radius R with a diffusion constant D, the characteristic correlation time is [37, pages 298-300] n = (3.6) For vesicles of R « 1 fim [22] and the diffusion constant D « 4 x 10 _ 1 2 m 2 s _ 1 [38] this yields T2 w 42 ms. The actual MLV samples may contain a wide distribution of sizes, and the shapes of the vesicles need not be spherical, so that this estimate for T2 is only valid to within an order of magnitude. Even so, these motions are too slow to contribute to motional averaging, and thus should not produce any non-adiabatic effects and, consequently, no relaxation. However, since Chapter 3. Model Membranes 40 the membrane is curved, the diffusion along the membrane can result in a slow reorientation of the local director and thus modulate the observed quadrupolar splitting (see Equation 3.1), by making the angle 6 between the local director and the external magnetic field a random function of time. The problem of rotational diffusion along a spherical surface has been shown [39] to produce an exponential decay of the amplitude of the quadrupolar echo (90y — r — 90x—), with the exponent proportional to —r3 sin22# in the limit of slow motions, appropriate to a highly viscous medium or, as was seen experimentally, for a "solid-liquid system" of D 2 0 on the surface of clay platelets. In this study we examined whether similar effects are seen in the nuclear magnetic relaxation measurements in model membranes and whether their explanation in terms of slow reorientational diffusion of phospholipid molecules along the bilayer can account for the observed lineshapes of the NMR spectra. Chapter 4 Performing N M R Experiments 4.1 Equipment During the course of this study a considerable amount of time was spent on improving and modifying the instrumentation. The NMR spectrometers available to the author, both com-mercial (Bruker SXP-100) and home-made by a previous generation of workers [40], needed major modifications of their radio frequency components due to their advancing age. In addi-tion to these more traditional aspects of magnetic resonance, it was quickly realised that the data acquisition and control equipment was rapidly getting out of date. When designed a few years previously, the systems represented a pioneering effort in the use of microcomputers in NMR spectrometers, but due to the explosive growth of the microcomputer industry in the intervening years they were nearly obsolete. Under these circumstances it was appropriate to perform a major revision of the entire data acquisition and processing scheme in the laboratory. In particular, it was deemed essential to aim for a unified environment capable of accommodating a wide range of spectroscopic equipment and flexible enough to allow for growth and expansion in the future. The existing methods of transferring data between various spectrometers and data processing computers bordered-on the anecdotal, and that problem also demanded immediate attention. Thus, a new nuclear magnetic resonance data acquisition and processing system was de-signed [41]. The approach we took proved successful, witnessed by the relatively painless evolution of the system over the few years since the original design was put into place, as a whole new generation of equipment was successfully brought on line. The particulars of various pieces of equipment may already be out of date, yet a number of general considerations and the 41 Chapter 4. Performing NMR Experiments 42 description of the principles and the criteria used in designing a laboratory data system are of general interest. Therefore, the following two introductory sections from [41] are worth quoting here, with only minimal revisions. 4.1.1 General Principles Generally speaking, the environment of a typical scientific laboratory can be described in terms of the following major constituents: experimental apparatus, control of the experiment, data acquisition, and data analysis. Without identifying ourselves with any particular experimental field we can describe these by a simple block-diagram of Figure 4.1. A number of general Figure 4.1: A block diagram of a typical data acquisition and processing system. features is evident from this diagram. In a well-designed data acquisition system the highest priority is given to the movement of the experimental data with minimum delay. Depending on the particulars of the Experiment, Data Reduction can be very simple or very sophisticated with Chapter 4. Performing NMR Experiments 43 a number of feedbacks into the Control, but in all cases the integrity and the efficiency of the data path from the experiment to the Mass Storage is of primary importance. As a result, the complexity and, consequently, the cost of the data acquisition hardware is usually determined by the need to satisfy the timing and bandwidth limitations of this data path. Except in the extreme cases, other tasks like Data Analysis and Control have either the bandwidth or the timing requirements somewhat relaxed and thus represent a much smaller part of the overall cost of the system. In addition, parts of the control system may already be built into some of the instruments used in the experiment, and some data processing equipment may already be available, forcing the experimenters into accepting the difficulties caused by inferior or outdated equipment in order to reduce the total cost. Similarly, high quality of Support Devices and Services, including the important but all too often neglected data backup and a comprehensive range of peripherals, is often sacrificed to obtain the best possible data acquisition capability within the constraints of a budget. Traditionally, access to the state-of-the-art computers to assist the researchers in the task of data acquisition and data analysis was only possible when an overwhelming volume of data was to be processed and when the cost of an automated data acquisition and processing system constituted only a minor fraction of the total expense of conducting the experiment, such as in particle physics. However, this situation is being drastically altered by a number of recent changes in the field of micro- and minicomputers. A short historical note is appropriate here. When microprocessors became widely available in the mid-1970's, a variety of intelligent laboratory instruments appeared, making use of that new technology. Usually they were designed to perform a relatively narrow range of operations, appropriate for devices intended primarily for routine measurements. This approach, however, led to difficulties in adapting these instruments to tasks not anticipated by the original design. In spite of being inherently programmable, the first generation microprocessors were relatively slow and unsophisticated and their modification usually required learning the details of their architecture and their low-level machine code conventions. As the hardware rapidly evolved this effort had to be duplicated each time it was required to incorporate the latest technological Chapter 4. Performing NMR Experiments 44 improvements into the instrumentation. The software written in hardware-dependent machine codes usually had to be completely re-written. In contrast, this was not the route that the evolution of the general purpose micro- and minicomputers took. A great effort was spent to ensure compatibility with the previous generations of software, so that sometimes the old microprocessors were literally included in the products, side-by-side with the new ones. At the same time, availability of, inter alia, larger and cheaper memories, encouraged an increase in the sophistication and the proverbial "user-friendliness" of the microcomputers and their operating systems. As a result the dividing line between microcomputers and their mainframe counterparts became increasingly vague, culminating with the recent appearance of a number of "supermicrocomputers" that bridge that gap. Using these computers in an experimental laboratory can provide a number of dramatic changes in the complexity of tools and methods available at a small research facility. To take advantage of these developments a new data acquisition and processing system was recently built at the NMR Laboratory of the Department of Physics of the University of British Columbia. While the technical details are of interest mostly to the NMR specialists and may not be directly applicable to other fields of research, it is hoped that the overview of concepts and principles involved in the design of this system will prove useful to the researchers in other disciplines as well. 4.1.2 Basic System Parameters A number of important factors influence both the overall structure and the choice of particular components of the system whose block diagram is presented in Figure 4.2. The system is de-signed to be an open-architecture system in order to satisfy the requirement of "unbundling" of the hardware. Most of the commercially available systems are examples of packaged "turn-key" systems, optimized for a specific range of tasks. Often these systems use non-standard com-puter hardware and software, such as proprietary interface buses and support routines, locking the user into relying on a single source of supply for all expansions, upgrades and ancillaries. Chapter 4. Performing NMR Experiments P u l s e P r o g r a m m e r P u l s e S e q u e n c e D ig i ta l S c o p e B u f f e r M e m o r y A / D C o n v e r t e r S l a v e C o m p u t e r D a t a A c q u i s i t i o n P r o c e s s S i g n a l A v e r a g i n g M e m o r y C e n t r a l C o m p u t e r U s e r ( s ) Figure 4.2: A block diagram of the N M R data acquisition and process-ing system. Chapter 4. Performing NMR Experiments 46 The great convenience of the initial installation is more than offset by the difficulties of incor-porating devices or features other than the ones anticipated by the manufacturer. Invariably, these systems are very expensive, not in line with the rapidly dropping prices of the equip-ment mass-produced by large specialized companies, reflecting the long lead time required to develop a complex custom turn-key system. An open-architecture system allows the use of the best possible equipment from various manufacturers so that the criteria of cost, availability of technical service and support, and expandability can be applied to the choosing of individual components rather than the system as a whole. This results in substantial savings and in better compatibility with industry standards. For example, addition or replacement of a component can be done without any major changes to the rest of such a loosely coupled system and it does not require any additional expenditures, save the cost of the component being added. Also, high demand put on certain components during various stages of data acquisition and process-ing does not affect the rest of the system, as each component is required to rely on itself. To satisfy the requirement of unbundling of the hardware, the system uses a serial communication protocol (RS-232) for all non-critical links between the individual components of the system. To maintain the overall system performance at a satisfactory level, each of the components is expected to be able to operate as a stand-alone device without constant service from an intelligent master. This is not a very strict limitation for the majority of the devices available today. On the other hand the choice of the serial communication protocol provides a sufficiently low common denominator to make the integration of individual components into the system trivial. In practice, serial links represent by far the simplest and the cheapest interface protocol supported by virtually every manufacturer of electronic devices. A single multi-tasking computer at the center of the star-like system is capable of maintain-ing adequate level of support in a shared-resource environment with quite a large number of peripheral devices if all of them need to be attended to only from time to time. A small number of processes that may occasionally require a higher level of service (such as data acquisition processes) can be accommodated by giving them a priority higher than that of the data analysis Chapter 4. Performing NMR Experiments 47 processes. To a certain degree this compensates for the physical bandwidth limitations of the serial interface, maintaining an adequate level of system support without having to adopt the communication conventions between devices that attempt to satisfy the worst-case scenarios. The level of service to other processes may temporarily drop, but since all of them are expected to be able to deal with such a temporary delay of service, the calculations of the bandwidth required for various intra-system links can be made on the basis of the average rather then the peak data flow rate. When the duty cycle is taken into consideration the serial communication becomes a viable alternative to other protocols with much higher unit cost. In the simplified diagram of Figure 4.2 only one NMR spectrometer, one control module, one data acquisition subsystem, and one data analysis process are shown, but there is no difficulty in duplicating any of these modules without any structural changes to the system. This is indicated on the diagram by the block labelled "Other Processes". At the present time we are only limited by the total number of ports on the serial multiplexer board of our central computer (16) which can be easily increased as need arises, and, in fact, we have two identical data acquisition systems associated with the two NMR spectrometers in our laboratory and a number of data processing terminals, all connected to the central computer. A slightly modified version of this data acquisition system has recently been build for a Differential Scanning Calorimeter. Another important feature that can be seen from the block diagram is that the data ac-quisition and the data analysis tasks are completely separated and, as indicated by the dashed line, can be performed concurrently by different users from physically different terminals or by the same user from the same terminal, but as two separate tasks. Moreover, since the Data Acquisition Supervisor Process need not be attended to once it is started, it can be submitted to the system as a batch mode process, and the terminal used to do that freed up for other tasks or users. It also should be pointed out that these separate processes need not explicitly communicate with each other for they both are resident to the same system and all of the system's resources are common to them; in particular, no data transfers need take place. This represents a marked departure from some other systems where separation of data acquisition Chapter 4. Performing NMR Experiments 48 from data analysis required separation in hardware with time-consuming and not always auto-matic shipping of data from one computer to another. Due to the multi-user, multi-tasking capabilities of the central computer it is possible to isolate the user from the details of the hardware by providing a uniform environment, whether during data acquisition or during data analysis. The high volume of raw NMR data requiring a real-time response, however simple (e.g., signal averaging), that would overwhelm the central computer still calls for a Slave Com-puter dedicated to each data source but this slave computer does not require any peripherals or mass storage devices of its own, thereby reducing its cost and complexity. Also, the user need not know of its existence as all the communications with this computer, involving only standard serial input/output routines, are performed transparently to the user by a simple supervisor process. This improves data integrity and simplifies the training of new personnel. 4.1.3 Devices The developments in the field of scientific instrumentation occur at a very rapid pace. Thus the choices of the particular devices [41] that we made in the Spring of 1984 are not optimal any more. In fact, some of the important pieces have since been upgraded or replaced, most notably the Central Computer. Suffice it to say that some of the devices were inherited from the existing system (Nicolet Explorer digital oscilloscopes), some were purchased "off-the-shelf' (MicroVAX central computer from the Digital Equipment Corporation; DB32016 and DB32032 single-board slave computers from the National Semiconductor Corporation), and some were designed and built as the need arose (a custom interface board, a pulse programmer). Following the successful re-design of the data acquisition and processing system, the same general principles were applied to modifying the NMR spectrometers: there are now two spectrometers in operation in the laboratory for which several of important components are custom designed (16-bit digital phase shifters), while some have been purchased off-the-shelf (PTS frequency synthesizer, ENI power amplifier), and some salvaged from the old equipment (Bruker SXP-100 power transmitter). Attention was paid to making the design as modular as possible, and thus most of the important Chapter 4. Performing NMR Experiments 49 components are completely interchangeable between the spectrometers, and are easily replaced and upgraded as needed. We adopted and successfully implemented a modular, "unbundled", design of both an NMR spectrometer and the associated data acquisition and processing system. This approach makes it easy to rearrange the separate elements of the system to suit a new experiment, or to replace some of them as the need arises. Thus it is fitting that the current state of the system is best described by the most recent hardware manuals, constantly under revision [42]. There were three devices designed and built to fill in the roles for which a commercial device could not be obtained. Those were: a custom interface board, a pulse programmer (both described in [41]), and a digital phase shifter. 4.1.4 Phase Shifter In a typical NMR spectrometer the radio frequency (RF) pulses are created, with the desired relative phases, at some intermediate frequency, which is then mixed up or down to the desired resonance (Larmor) frequency. For the reasons of simplicity and flexibility, this so-called super-heterodyne design was abandoned in favor of a straight-through design where the RF pulses are defined, both in phase and duration, at the actual resonance frequency. This makes the design much simpler conceptually but imposes the requirement that the entire RF section of the spectrometer be wide band. Wide band power amplifiers are not uncommon, while the wide band devices for controlling the phase and the duration of the RF pulses suitable for NMR (splitters/combiners, phase shifters, RF gates) are rare and very expensive. We use the name "phase shifter" to refer to the device that we built; in fact it controls the duration of an RF pulse as well as its phase, and provides a convenient way of adjusting the reference phase. Also, it incorporates some protection against catastrophic events (human errors or malfunctions of the control equipment) by monitoring the ratio of the on/off times of the RF transmitter: power amplifiers used in pulse NMR are capable of the output levels which can easily damage the sample and/or the coil of the spectrometer probe. To minimize the cost, we Chapter 4. Performing NMR Experiments 50 made use of a modular design of the phase shifter in which all of the frequency-dependent parts were combined in a single removable module with a certain guaranteed bandwidth. A selection of such modules covering most of the range 10-300 MHz was built, enabling us to switch frequency relatively easily without having to resort to the more complex super-heterodyne design. Figure 4.3 is a block-diagram of the digital phase shifter. This design was implemented by R F In Q -| Mu l t i p l y ing I D A C | • < A d d e r (16) COS i i C o m p l e x | P h a s o r j M o d u l a t o r * ) i T' •| — D u t y C y c l e I L i m i t e r I i j-v^/v — o D u t y c y c l e j S p l i t t e r * | J 3 u a d P h a s e S e l e c t o r ^ R F G a t e R e f e r e n c e P h a s e ( p u s h w h e e l s w i t c h e s ) C o n t r o l l ines f r o m P u l s e P r o g r a m m e r - O R e s e t o A d j u s t m e n t G a t e d R F O u t Figure 4.3: A block diagram of the digital phase shifter. The plug-in units marked (*) are frequency-dependent. For simplic-ity, the amplifiers used to compensate for the RF power loss across the various components are not shown. S. Knotek (engineering) and D. DiTomaso (construction) of the Electronics Shop of the Physics Department [43]. The device uses components from Natel Engineering Co. (16-bit multiplying sin/cos digital-to-analog converter), and from Olektron Corp. (complex phasor modulator). A more detailed description is found in [42]; for a complete set of diagrams see [43]. The principle of the design is fairly straightforward: a 16-bit word from a controlling device Chapter 4. Performing NMR Experiments 51 (a pulse programmer) is added to a 16-bit word from the set of front-panel pushwheel switches which provide the reference phase shift. The resulting word (9) is converted by a multiplying digital-to-analog converter (DAC) to two voltages, proportional to the (sin 9) and (cos 9). These voltages are applied to a complex phasor modulator, which produces a signal on its output with the square of its amplitude proportional to the sum of the squares of the two control voltages, and the phase shift (relative to the phase of the input signal) proportional to the [arctan of the] ratio of those voltages. Because of the sin/cos relationship of the two voltages in our device, the amplitude of the output signal remains constant and the phase shift is directly proportional to the incoming 16-bit word. The signal settles to a new phase shift in about 20 fxs. Since faster switching time may be required in many applications, an additional four-way phase selector is used to produce relative phase shifts of 0°, 90°, 180°, and 270°. In the present design the two stages of the phase shifter are connected in series, as we use the digital phase shifter to provide both an arbitrary computer-controlled phase shift and the manual reference phase shift; we do not use any phase-shifting devices in the receiver arm of the spectrometer. However, one can imagine a different configuration, with the digital phase shifter module being connected in parallel to the four-way splitter and providing an arbitrary phase shift pathway in addition to the usual four. This would require another way of varying the reference phase. Due to the modular construction we can easily change to any desired configuration, as needed. The device has proven to be linear in phase beyond our ability to calibrate it; in principle, however, a look-up table correcting for its non-linearity could be constructed and referenced by the pulse programmer to define an arbitrary phase angle with high precision. 4.2 Materials This study involved a wide range of model membrane samples. Most of the work was done on the phospholipid-water dispersions, sometimes with inclusion of sterols. Two deuteriated phospho-lipids were kindly provided by Dr. R. Cushley: sn-2-(2H3i)dipalmitoylphosphatidylcholine (one chain perdeuteriated DPPC, DPPC-d 3 i) and sn-2-(2H2)4,4-dipalmitoylphosphatidylcholine Chapter 4. Performing NMR Experiments 52 (DPPC specifically labelled in the 4-position, DPPC-d2). Other chemicals were purchased from the usual commercial sources. A standard method of preparing multilamellar phospholipid-water dispersions was followed throughout this study. An effort was made to maintain the uniformity of the sample preparation over the years. Typically, the required amounts of dry lipid and, when appropriate, of sterol were mixed and then dissolved in an organic solvent. By rotating the flask in a warm bath while the solvent was pumped off a thin homogeneous film was deposited on the walls. To remove the remaining solvent the samples were then dried under high vacuum for a minimum of 8 hours. Aqueous dispersions were formed by adding excess buffer (pH 7.0) to the films, vigorously mixing, and freeze-thawing a number of times. The resulting samples were centrifuged down to a pellet size, excess buffer removed and the pellets transferred into sample tubes of ~ 0.5cm3. Deuterium depleted buffer was added in excess, samples sealed and additional freeze-thaw cycles were performed. Samples were vortexed every time they were thawed, including immediately before the experiment. Additional details may be found in [31]. Early on in the project, an attempt was made to compare the samples prepared in pure H 2 O , rather than the buffer solution at pH 7.0, but no difference was observed and all subsequent samples were prepared using the buffer solution, to minimize the deterioration of the samples. When samples had to be stored for an extended period of time, a deep-freezer (—20°) was used. Degradation of the samples was monitored by TLC; none was observed. One of the samples was prepared so as to provide an osmotic pressure gradient by forming the lipid vesicles in a ~ 100 mM salt solution. After the salt solution was sealed inside the vesicles, distilled water was added to the outside. This way an excess pressure was maintained on the inside of the vesicles, providing on average a more spherical vesicle shape. One sample was prepared by a rapid extrusion through a polycarbonate filter with 0.8 /im pore size, to create a more uniform size distribution of the vesicles [23]. Chapter 4. Performing NMR Experiments 53 4.3 Data Acquisition during Multiple Pulse Trains Data acquisition of the time domain signal in Fourier Transform (FT) NMR is traditionally performed by converting the analog output of the radio frequency amplifier into a series of digital values produced by an analog-to-digital (A/D) converter at a constant rate, once every so-called "dwell time". Once triggered, the A / D converter usually uses its own free-running time base to determine when the next data point is taken relative to the previous one. The shorter the dwell time the greater is the spread of spectral frequencies which can be detected. However, for a fixed size of the computer memory decreasing the dwell time also shortens the total time over which the data is collected. Due to the nature of FT NMR the signal must decay to zero due to destructive interference of the various signal components before the data collection is terminated, otherwise a truncated data set is obtained in which components of similar frequency may not be resolved. Thus for a data set of a fixed size the requirements of decreasing the dwell time to improve the spectral bandwidth and increasing the total time of the data acquisition to improve spectral resolution are contradictory. In the case of a CPMG pulse sequence the signal is refocussed many times and the difficulty of acquiring a complete data set rises accordingly. With the help of the new pulse program-mer [41] we were able to control the time base of the A / D converter externally and to selectively acquire the data points only in the regions of interest. For a CPMG pulse sequence these re-gions are in the vicinity of the peaks of the echoes occuring at times 2nr, n = 1,2, . . . , i V . This is schematically indicated in Figure 4.4 [36]. In fact, this efficient use of the computer memory enabled us to obtain in the same experiment the partially relaxed spectra, in addition to measuring the ^ ~ C P M G ? by switching to a free-running time base at the TY-th echo. 4.4 Fidelity A great effort was made to achieve high fidelity of the NMR spectra. Problems posed by the dead time of the receiver were addressed by using the quadrupolar echo technique, described in Chapter 4. Performing NMR Experiments 54 i 1 11 n i it 0.0 0.2 0.4 0.6 0.8 1.0 t ( m s ) Figure 4.4: (a) Schematic representation of the selective data acquisition scheme used in the quadrupolar CPMG experiments, 90j, — (r — 90a; - T—)N, is shown by the dashed line. Only the data points in the regions of interest — near the peaks of the echoes — were recorded, as indicated by the solid line. At the last echo the digitizer is switched into a free-running mode and the entire echo is recorded. For illustrative purposes we chose N = 5. (b) An example of an actual experimental data set acquired as in (a), using a sample of D P P C - d 3 1 at 44°C. Here, N = 32 and r = 100 fisec. The time scale is only shown starting at the last echo where it becomes continuous. Chapter 4. Performing NMR Experiments 55 the Introduction. The time domain signals were digitized by the Explorer model 2090 digital oscilloscope (Nicolet Instruments Corp.) with the dwell time that varied from 500 ns to a few p,s, with a small (< 10ns) sample time uncertainty. The radio frequency pulse length was kept to a minimum, typically below 3 fis for a 90° pulse. Sometimes magneto-acoustic ringing effects set off by the RF pulse were observed; usually they could be eliminated by re-designing the coil. The shape of the ringing signal depended on the phase and the length of the RF pulse, and was reproducible. Thus, these effects could be eliminated by an appropriate phase alternation scheme or, alternatively, averaged out if on every second scan the pulse sequence being used was preceded by a a non-selective 180° inversion pulse, and the consecutive scans were in turn added and subtracted during signal averaging. This was especially important for the selectively deuteriated samples with their low signal-to-noise ratio. When necessary, a five-point interpolation algorithm [8] was used to adjust the data so that there always was a data point at the top of the echo (i.e. at t = 0). In all of the experiments various forms of CYCLOPS phase alternation were used to get rid of some of the imperfection in the phases and the lengths of the radio frequency pulses [44,45]. At least a four-cycle of (y, x, —y, —x) was always used to correct for the instrumental imper-fections, such as phase and amplitude imbalance in the two channels of the receiver. The signal was always collected in quadrature and rarely needed a small zero-order phase shift. No first order phase correction was applied. Analog filters of no less than 200 kHz in bandwidth were used on the receiver output, prior to digitization. No digital filtering (including exponential multiplication) was used, except on the oriented spectra calculated by the depakeing program. 4.5 Software Data acquisition was controlled by the custom software co-written with E. Bacon. Data analysis was performed using mostly the custom software package, DANS, co-written with E. Castillo (Wren Design Group). The FTNMR program by D. Hare was also used. A DePakeing pro-gram [25] was modified to improve the user interface, and a version suitable for a VAX/VMS Chapter 4. Performing NMR Experiments 56 computer system was written. Fitting and other sundry data manipulation was done with OPDATA and MINUIT software packages; these programs were obtained from TRIUMF, as was the graphics software relied upon for most of the hardcopy output, including the figures in this manuscript. The manuscript itself was prepared using Tr^ X typesetting language by D.E. Knuth, with IATEX macros. Chapter 5 Results and Discussion Traditionally, the relaxation in model membranes is interpreted in terms of the motions fast on the NMR time scale, in the manner described in Chapter 1. Often, a single correlation time model is assumed for simplicity, and the effects due to slow motions are ignored. Model membranes are complex molecular systems, with a variety of motions characterized by different values of r c. The correlation times for trans-gauche isomerizations of the acyl chains in the liquid crystalline phase are generally believed to be very short, W Q T 2 <C 1, while those of rotational diffusion of the phospholipid molecules about the bilayer normal, and translational diffusion over distances comparable to the distances between molecules ( « 5 A) are intermediate to long, TC ~ 1 0 - 8 s. As a consequence of Equation 3.6, translational diffusion along the surface of the bilayer over distances of order 100 A or tumbling of small vesicles of radius 100 A have TC « 10~6 s, corresponding to UQT2 > 1 but satisfying u^r,? <C 1. Diffusion over distances of order 1 fim corresponding to the size of a small biological cell leads to values of r c of order tens to hundreds of milliseconds. Considerable theoretical and experimental work has been done on the frequency dependence of longitudinal relaxation time, T\z(u$), during the past ten years (see, for example [46,47,48]). These studies have shown that the experimental data cannot be explained in terms of a single correlation time. This led to the introduction of motional models involving a quasi-continuous distribution of correlation times extending over a wide range of rc values. One type of motion which is postulated to lead to this behaviour is slow collective motion such as, for example, the synchronized swinging of the chains of a large number of neighbouring phospholipid molecules or lateral diffusion along ripples in the membrane surface [49]. Though the range of correlation 57 Chapter 5. Results and Discussion 58 times considered thus far in the interpretation of spin-lattice relaxation time in membranes extends from fast (<JQT2 <C 1) to slow (U>QT2 > 1) in comparison with Larmor frequency, it has been implicitly assumed through the use of theoretical expressions such as those reviewed in Chapter 1 that all the important motions are fast on the time scale associated with the spin-lattice interaction energy. For quadrupolar interaction this corresponds to u^r 2 <C 1. As seen from Equations 1.32-1.37, T\z measurements are sensitive to j(uo) and j(2uo), while Ti also depends on j(0), making T 2 sensitive to a different range of correlation times, much longer than those responsible for the T\z relaxation. Thus we decided to measure the transverse relaxation rates of a number of typical model membrane systems and to examine more critically the role of slow motions in relaxation. As we shall see, the experimental results yielded important contributions to T 2 from motions having much longer values of r c than were anticipated from previous studies. We shall propose that the origin of these long correlation times is diffusion of phospholipid molecules along surfaces having very large radii of curvature. The present study is not an exhaustive relaxation study of any one particular system. Instead, we examine selected aspects of the relaxation in a variety of model membrane systems in an attempt to establish the importance of ultra slow motions for nuclear magnetic relaxation. A complete consideration of each of the systems studied would be a major undertaking on its own. Our interest here lies in the aspects of relaxation common to all of them. To establish this commonality, we also consider a system of perdeuteriated crystalline hexamethylbenzene in which the slow translational diffusion of individual molecules is absent, thus isolating the implications of such motion for the model membrane systems. 5.1 Presence of Lateral Diffusion As described in Chapter 3 the quadrupolar splitting observed in 2 H NMR of a phospholipid-water model membrane system depends both on the local orientational order parameter, which varies with the carbon position along the acyl chain of a phospholipid molecule, and on the orientation of the local axis of symmetry of the molecular motion (the director) with respect to Chapter 5. Results and Discussion 59 the external magnetic field. To separate the two contributions we shall first consider a relatively simple system of 4,4-DPPC-d2. Here only the two equivalent hydrogen atoms at the carbon position 4 on one of the two acyl chains of the dipalmitoylphosphatidylcholine molecule are replaced by the deuterium atoms. Thus, the variation in the quadrupolar splitting is due only to the different orientation of various parts of the sample. The quadrupolar echo pulse sequence (90y — T — 90x — t—, see Chapter 2) can be used to measure the spectrum; the resulting powder pattern is shown in Figure 5.1a and is an example of a "Pake doublet". We use the inversion - 5 0 - 4 0 - 3 0 - 2 0 -10 O 10 20 30 40 F r e q u e n c y ( k H z ) Figure 5.1: DPPC—d 2 : non—selective inversion recovery. (a) A spectrum of multilamellar 4,4-DPPC-d2/H20 dispersion at 44°C obtained by using a standard quadrupolar echo sequence, 90^ — T-90x-t-, withr = 50//s, iVscans = 1000. (b), (c), and (d) are the difference spectra between (a) and the ones obtained using a non-selective inversion recovery sequence, 180,, — T\ — 90y — T2 — 90^ — t—, for various values of T\, with r2 = 50fis, iVSCans = 1000 for (b)and (c) and i V s c a n s = 50,000 for (d). All spectra acquired at the Larmor frequency of 35.49 MHz. recovery pulse sequence (180^ — T\ — 9Qy — r 2 — 90^ — t—) to obtain T\z = 24.9 ± 0.2 ms, as Chapter 5. Results and Discussion 60 measured from the exponential decay of the intensity of the quadrupolar echo with T\. The lineshape appears to be independent of T\ as witnessed by the spectra in Figure 5.1b, c, and d 1 . It would be incorrect, however, to conclude that T\z is independent of orientation since the diffusion of the individual phospholipid molecules along the membrane surface would change the orientation of their directors, providing an effective exchange mechanism between the domains of different orientation. In the presence of such an exchange the observed relaxation rate will be averaged over all orientations. To examine further why the measured T\z does not seem to depend on orientation one needs to prepare the system in a state of non-uniform spin temperature distribution by using a selective inversion recovery pulse sequence. The non-selective "hard" 180y pulse is sufficiently short so that it inverts all of the frequencies in the spectrum producing a uniform spin tem-perature for the domains of all orientations; one possibility is to make the 180,, pulse "soft", i.e. to apply a weak radio frequency field for a long time. The spread of the spectral frequen-cies affected by an RF pulse is roughly inversely proportional to the pulse length; making the pulse sufficiently long will enable us to selectively invert only the frequencies near the middle of the spectrum, without affecting the rest. Such an experiment has been reported [50] for the 4,4-DPPC-d 2/H20 dispersion at 51°. The observed relaxation rate near the center of the spec-trum is " . . .non-exponential. The initial slope is due to the transfer of magnetization across the spectrum, while, at long times (ri > 10 ms) the amplitude near the center of the spectrum decays at the relaxation rate, 1/T\Z, averaged over all bilayer orientations". Another possible selective inversion scheme is the one described in Section 2.6, where the non-selective hard 180y pulse is replaced by a pair of 90^ pulses separated by a short time: 90y — T\ — 90y — r 2 — 90y — T3 — 90a; — t—. The advantage of this method is that by choosing an appropriate value of T\ we can preferentially invert, at least partially, all frequencies except near the so-called "edge" of the Pake doublet. The edge corresponds to the orientation of the local ' B y convention, the spectra shown are actually the difference spectra between the standard uninverted spec-trum (shown in a) and the ones obtained in the respective inversion recovery experiments; this way the spectral intensity asymptotically approaches zero for large values of r 2 . Chapter 5. Results and Discussion 61 director of 0 = 90° with respect to the external magnetic field; here the intensity of the spectrum is large and easy to monitor accurately. Also, in the vicinity of the edge the quadrupolar splitting, proportional to P2(cos#), varies very slowly with orientation and thus one can hope to extend the time that it takes for the diffusion to equilibrate the spin temperature across the spectrum, as individual molecules have to diffuse through a larger angle. In contrast, inverting the magnetization near the center of the spectrum corresponds to the "magic angle", where P2(cos0) tx 0 changes much more rapidly. Figure 5.2 shows the results of a selective inversion experiment in which the edges of the Pake doublet were not inverted. The spectra shown —50 - 4 0 - 3 0 - 2 0 -10 0 10 20 30 40 50 F r e q u e n c y ( k H z ) Figure 5.2: DPPC—d2: selective inversion recovery. (a) A spectrum of multilamellar 4 ,4-DPPC-d 2 /H 2 0 dispersion at 44°C obtained by using a standard quadrupolar echo sequence, 90^ — T — 90x — t—, with r = 50ns. (b), (c), and (d) are the difference spectra between (a) and the ones obtained using a selective inversion recovery sequence, 90y — T\ — 90j, — r2 — 90^ — Tz — 90x — t—, for various values of T2, with T\ = 32/is and Tz = 50 /xs. A s^cans = 20,000 for (a), (b) and (c) and A^cans = 60,000 for (d). All spectra acquired at the Larmor frequency of 46.175 MHz. Chapter 5. Results and Discussion 62 correspond to the difference between the spectra of the standard quadrupolar echo (without the preceding pair of 90y pulses) and that of an appropriate selectively inverted spectrum; at short T2 the resulting spectrum is seen to have two "holes" at the edge frequencies (as compared to the spectrum in Figure 5.2a; this spectrum is similar to the one shown previously in Figure 5.1a). As T2 is increased the holes are "filled in" by the diffusive transfer of magnetization across the spectrum. Note that even for the shortest of the r2 values shown, r 2 = 1 ms, some transfer has already taken place, and the edge intensity has already recovered somewhat. To present a clearer picture of this result we can plot the intensity of different parts of the spectrum as a function of r2. We shall concentrate our attention on three different spectral windows: one near the shoulder of the powder pattern (6 = 5 — 10°), one near the center of the spectrum (6 = 45 - 50°), and one near the edge (6 « 90°), as seen in Figure 5.3. Calculating the average intensity in the shoulder window presents no difficulties; for the other two windows, there is a slight complication due to the symmetry of the quadrupolar spectrum: the Pake doublet can be thought of as a superposition of two powder patterns, one for each of the lines in the quadrupolar doublet. Between the edges the two patterns overlap and thus we need to separate the observed average intensity into the two components, one for each of the patterns. This is schematically shown in the insert of Figure 5.3. For the middle window we simply assume that the intensity is separated into two equal contributions, for reasons of symmetry. Unfortunately, it is impossible to separate unambiguously the two contributions to the observed intensity in the edge window, namely the ones from 6 « 90° and from 6 « 32.2602. We can consider two limiting cases: at short times, both of the contributions are affected equally, and one way to interpret the measured intensity would be to simply divide it in the proportion of their intensities in the normal, non-relaxed, spectrum. As discussed above, the expected relaxation rate is much faster for the contribution from 6 = 32.26° and thus at longer times that ratio of the two contributions would be incorrect. In fact, the other limiting case would be to assume that the contribution from 6 = 32.26° is fully recovered, to recalculate it using 2 Since |P2(cos 9 0 ° ) | = |P2(cos 3 2 . 2 6 ° ) | = 1/2 Chapter 5. Results and Discussion O 10 20 30 40 50 60 T 2 ( m s ) Figure 5.3: D P P C — d 2 : T\z in selected spectral windows near the edge, shoulder, and in the middle of the spectra of Figure 5.2. The solid lines are the least squares' fits to a single exponential, for each win-dow. Insert: As discussed in text, the intensity in the shoulder window is used to calculate the contributions from the two overlap-ping powder patterns measured in the other windows. Chapter 5. Results and Discussion 64 the intensity in the shoulder window, and to assign the remaining intensity to the contribution from 0 = 90°. This method is more appropriate for the longer values of r 2 and produces errors at short times. In both cases, the overall shape of the relaxation curve is roughly the same, and is also similar to the plot of the overall intensity in the window, without subdividing it into the two contributions. For clarity, only the latter method is presented here, shown by the dashed line connecting the points in Figure 5.3; it is essentially an estimate of the lower bound, and produces the most dramatic effect. More precisely, we extrapolate the intensity from the shoulder window to 0 = 32.26° assuming the powder pattern lineshape to be [8] / ( » ) = 3 + 6P2(cos 0)] 1 / 2 1 < P2(cos 0) < 1 3 + 6P2(cos0)] 1 / 2 + [3 - 6P2(cos 0)1 1 / 2 0 < P2(cos0) < \ -1/2 Thus f(9 = 32.26°) = y/3J2 f(9 = 0°) and the difference between the observed total intensity in the edge window and the intensity in the shoulder window multiplied by \/3/2 gives us an estimate of the intensity due to the orientation near 0 = 90°. The intensity in the shoulder window is seen to be exponentially decreasing with a relax-ation rate comparable to the average T\z measured by the non-selective inversion recovery experiment, 24.9ms. In fact, for long values of r2, the same is true for all three windows, as seen from the least squares' fits to a single exponential shown by the solid lines. However, for short r2's the observed relaxation in the edge window is non-exponential: initially (r2 < 10 ms) the intensity is actually seen to increase. This agrees with the interpretation proposed earlier: on the one hand, the spectral intensity eliminated by the selective inversion pulses is recovering due to the magnetization transfer across the spectrum induced by the lateral diffusion; on the other hand, the spin-lattice relaxation is causing an overall reduction of intensity. The orien-tation dependence of the spin-lattice relaxation rate, T\z, if any, is again not seen because of the lateral diffusion; only the orientationally averaged relaxation rate is measured. In a typical multilamellar dispersion where the average radius of curvature, R, is of order 1 /jm, the diffusion of a DPPC molecule leads to a sizable angular displacement of order r/R ~ Chapter 5. Results and Discussion 65 0.4 rad in about 10 ms, as estimated from the relation (r2) « ADt with D ~ 4 x l 0 - 1 2 m 2 s _ 1 [50]. If we compare this time to the time scale seen in Figure 5.3 it becomes clear that lateral diffusion is the most likely mechanism for the observed magnetization transfer across the spectrum. As a confirmation, we can perform an identical set of measurements on a system in which such diffusion is absent. Figure 5.4 presents a set of partially relaxed spectra of perdeuteriated polycrystalline hexamethylbenzene, in which the spectral intensity near the edge has been suppressed by a selective inversion pulse sequence. This figure is completely analogous to - 3 0 - 2 0 -10 O 10 20 30 F r e q u e n c y ( k H z ) Figure 5.4: Hexamethylbenzene: selective inversion recovery. A standard quadrupolar echo spectrum of deuteriated hexamethyl-benzene (a: T = 50 fis) and the spectra obtained using a selective inversion recovery sequence (T\ = 55 fis, T 3 = 50 fis; b: r 2 = lms, c: T 2 = 8 ms, d: r 2 = 64 ms). The lineshape does not vary with r2, unlike in Figure 5.2; there is no magnetization transfer across the spectrum. Figure 5.2; however, as r 2 is varied the lineshape of the spectrum does not change: the edge intensity remains suppressed, and there is no evidence of magnetization transfer across the Chapter 5. Results and Discussion 66 spectrum. As an immediate consequence of this, we can measure the orientation dependence of T\z in this system; since there is no magnetization transfer between different frequencies, for best results we can perform a series of non-selective inversion recovery experiments (not shown). As we again calculate the intensity in the three characteristic windows, in analogy with Figure 5.3, relaxation in each window is seen to be exponential, with the relaxation time being the longest at the edge and the shortest at the shoulder. The result is shown in Figure 5.5. 6 5 CD A 2 0 1 1 1 1 1 1 (average) J- 1z = 5 3 . 2 m s — (edge) 1 1z = 7 0 . 1 m s — -(center) 1 1Z = 4 5 . 7 m s -(shoulder) = 3 0 . 7 m s 1 1 1 1 1 1 10 2 0 3 0 4 0 5 0 6 0 7 0 T ( m s ) Figure 5.5: Hexamethylbenzene: orientation dependence of T\z. Spectral intensity in the selected windows near the edge, shoulder, and in the middle of the spectra of deuteriated hexamethylbenzene. The intensity in the edge window is a superposition of contributions from two overlapping powder patterns. The solid lines are the least squares' fits to a single exponential, for each window. A detailed discussion of the orientation dependence of T\z is given in the Ap-pendix. The results presented in this section can be summarized as follows. As we saw, longitudinal Chapter 5. Results and Discussion 67 ( T\z) relaxation in hexamethylbenzene depends on the orientation with respect to the external magnetic field, as evident from the variation across the spectrum of the decay rate of the intensity in a few selected spectral windows, in a series of non-selective inversion recovery experiments. The nature of this dependence is determined by the character of the molecular motions and is not of great importance here; a more detailed account is presented in the Appendix. By contrast, in a model membrane system, T\z is measured to be independent of the orientation. This can be explained in terms of a motional exchange, whereby the individual spins can move between the domains of different orientations. As a result, the magnetization is transferred across the spectrum. To test this hypothesis, we prepare the spin system in a state of non-uniform spin temperature, by performing a series of selective inversion recovery experiments. The above exchange mechanism, when present, causes a rapid restoration of the original shape of the spectrum; this is seen in the case of a model membrane system. In hexamethylbenzene, on the other hand, there is no evidence for such magnetization transfer. The time scale over which this transfer is taking place in a model membrane system seems to agree with the estimated time scale for lateral diffusion along the surface of the bilayer, characteristic of all model membrane systems. 5.2 Hexamethylbenzene as a Test Compound There is a number of important similarities between perdeuteriated polycrystalline hexamethyl-benzene and our model membrane system (DPPC-d2 in H 2 O ) . Both give rise to a classic NMR powder pattern: the model membrane system is selectively deuteriated in carbon position 4 along the acyl chain with no variation in the order parameter between the two equivalent 2 H nuclei. In hexamethylbenzene, all eighteen 2 H nuclei are equivalent. The asymmetry param-eter, 77, is below our ability to measure for the model membrane system, and is very small, 77 = 0.087, for hexamethylbenzene, as determined from the depaked spectrum (qe, r = 50 fis). Thus to a good approximation we can treat both systems as axially symmetric. The observed quadrupolar splitting is 16.85 kHz, corresponding to the order parameter slightly below ^ in Chapter 5. Results and Discussion 68 hexamethylbenzene, and 27.4 kHz for the model membrane. There are rapid molecular motions in both systems. In model membranes, these are the trans-gauche isomerizations of the acyl chains and rotation of the phospholipid molecule about its long axis which is on average oriented along the bilayer normal. A hexamethylbenzene molecule in a crystal also participates in two motions: rotation of each -C 2 H3 group about its axis of symmetry, and rotation of the entire molecule about its six-fold axis3. These motions are fast on the NMR time scale, causing the entire quadrupolar interaction, whose strength is 6 ^ = 2ir x 167 kHz, to fluctuate with a correlation time r c such that w 2 r 2 «C 1. The simple theory presented in Chapter 1 (cf. Equations 1.32 and 1.34) predicts that Zk = I Tlz 3 For a single exponential correlation function, J'(w) = i ,T°2 2 ' 1 + UJLT£ so that Equation 5.1 becomes Tll_l 5 + 8a;2r2 Tl2 ~ 3 1 + 4a>2r2 ' ^ predicting that the ratio Tlq/Tiz varies between ^ for u^r 2 < 1, and | for W Q T 2 > 1. The measured ratios 27.4 ms/24.9 ms for the model membrane system and 60.2ms/53.2ms for the hexamethylbenzene, indicate the presence of molecular motions having correlation times such that UQT2 ~ 1. Therefore, hexamethylbenzene acts as a convenient "model" system for the model mem-branes. The difference between the two systems arises because the crystalline sample of hex-amethylbenzene lacks one important characteristic of a model membrane system, namely the translational diffusion of the individual molecules. This affects profoundly the nature of the transverse relaxation seen in the two systems. In hexamethylbenzene, both of the fast rotational motions are reorientations about fixed centers, which are known to be ineffective in completely 3 I f these were the only motions, we would expect the order parameter of exactly j x | = | , and r) = 0. 1 + 4j(2^o) i(^o) (5.1) Chapter 5. Results and Discussion 69 removing dipolar couplings, causing the time-averaged value of the static dipolar interaction to be non-zero. This is because the internuclear vectors connecting two 2 H nuclei on different molecules do not undergo a spherical average over the motion, a condition necessary for the static part of the dipolar Hamiltonian to be averaged out [51]. Yet from a comparison between the calculated rigid lattice moment of a [non-deuteriated] hexamethylbenzene and the measure-ments obtained by 1 H NMR, it is evident that approximately half of the dipolar contribution to the second moment of the spectrum of non-deuteriated hexamethylbenzene is due to the intermolecular dipolar interactions. Thus one anticipates that the static dipolar interaction dominates the T 2 ? e of hexamethylbenzene. On the other hand, model membrane systems have motions with much longer correlation times, such as the lateral diffusion along the plane of the bilayer. Since T2 depends on j(0), one expects it to be very sensitive to such motions. Therefore, a comparison between the transverse relaxation measurements in the two systems makes it possible to isolate the role played by these slow motions in model membrane systems. 5.3 T| e Measurements in Hexamethylbenzene The transverse relaxation of deuteriated hexamethylbenzene can be measured by a series of quadrupolar echo experiments, as shown in Figure 5.6. The relaxation at short times is seen to be non-exponential, with the overall relaxation rate increasing with the distance from the center of the spectrum, while remaining much shorter than the T\z. The non-exponential shape of the relaxation curves in the individual spectral windows, as well as that of the overall echo decay, as seen in Figure 5.6, suggests that the sample not only has a distribution of relaxation times depending on orientation, but that even the individual orientations relax non-exponentially. If transverse relaxation is dominated by the motions fast on the NMR time scale, OJQT2 <C 1, we expect T2z « T\z. However, the experimental values of T| e are much shorter than T\z, implying that contributions from other relaxation mechanisms have to be considered. For example, it is expected that the dipolar contribution is due mainly to the interaction between Chapter 5. Results and Discussion 1— 1.5 t ( m s ) Figure 5.6: Hexamethylbenzene: orientation dependence of T| e . Average intensity in each individual spectral window is plotted against the time of the quadrupolar echo, t « 2r. The solid lines represent fits to Equation 5.3, as discussed in text. The average value shown is simply the intensity of the quadrupolar echo, in the time domain. Note that this corresponds to the average being taken over the distribution of relaxation rates, i.e. (1/T 2 ) - 1 - Since the dominant term in the relaxation rate is expected to be proportional to |P2(cos#)| = Au/Aumax, only the total intensity in each spectral window is used (without separating it into the two contributions as we did earlier) since the relaxation rate is expected to be the same for both orientations. Chapter 5. Results and Discussion 71 two nuclei on the same -C 2 H3 group. Since the quadrupolar interaction is associated with the C — 2 H bond of the -C 2 H3 group, the molecular motions affect both dipolar and quadrupolar interactions in a similar way, and the dipolar and quadrupolar order parameters are not expected to differ greatly, aside from the geometrical factor of 2/3 = ^(cos 109.47°)/P2(cos90°). Thus we can immediately calculate [52] the decay time of the free induction decay due to the spread of frequencies caused by the dipolar interaction to be Tj*(dipolar) ~ 2.9 ms as compared to ^(quadrupolar) ~ 17 fis. This order-of-magnitude estimate is in reasonable agreement with the relaxation curves of Figure 5.6. As discussed in the previous section, we anticipate that T| e relaxation is dominated by the static dipolar interaction. With this in mind, the relaxation curves of Figure 5.6, can be characterized by means of an expansion S(t) = 5(0) l - f * 2 + £ * < - . . (5.3) where t sa 2r is the time of the echo in a quadrupolar echo experiment. This choice of notation is meant to draw the analogy to the moments expansion, commonly used to characterize complex spectral lineshapes. It should be emphasized that the above "moments" A \ , etc. are not the conventional moments M2,M4,etc. of the dipolar spectrum. The results of such fits are shown as solid lines in Figure 5.6. The values of A2, A4, etc. can be used in place of a single parameter, T%e, to describe the orientation dependence of transverse relaxation. In particular, we can plot A2, proportional to the square of the effective relaxation rate A2 ~ (^")2? f ° r e a c h of the spectral windows against the square of the distance from the center of the spectrum where the window is located, as presented in Figure 5.7. We renormalize that distance to the maximum quadrupolar splitting, Aumax = 16.85 kHz, which corresponds to the 6 = 0° orientation; thus AvjAvmax = |P2(cos0)|. Indeed, the transverse relaxation curves of Figure 5.6 fit well to the Equation 5.3, in agree-ment with the experimental results typical of polycrystalline samples [53]. The detailed in-terpretation of the "moments" A2,A4,etc. is obviously dependent on the model; one such calculation in terms of a simple two-spin model [54] predicts A2 = QM^v, etc., where M%V Chapter 5. Results and Discussion 2.5 2.0 -CNJ I m O a O h-2 Figure 5.8: D P E orientation dependence of Tg Thecs©lid Iint9s2are thieast squares' fit8<4o a single exponential, f©i* each window. O n l y the total inttnsiCy^SiSeXch window is reported, without separating it into the two contributions as in the case of T\z. T h e average value shown is simply the intensity of the quadrupolar echo, i n time domain, which corresponds to the average being taken over the distribution of relaxation rates, i.e. ( I / T 2 ) - 1 . Chapter 5. Results and Discussion 73 is the van Vleck second moment of the dipolar spectrum [55]. Qualitatively, the orientation dependence of the transverse relaxation is expected to be of the form where A 2 n t e r and A 2 n t T a denote the inter- and intra-molecular dipolar contributions, respec-tively. The experimental values of A 2 (as determined by the fits to Equation 5.3) shown in Figure 5.7 agree with Equation 5.4. In fact, the solid line is the result of a fit to Equation 5.4, with A 2 n t e r = 0.58 x 10 6 s - 2 and A 2 n t r a = 1.98 X 10 6 s - 2 . Taking into account that the average value (|P2(cos #)|2) = \ , the fact that A 2 n t e r « \ A 2 n t T a confirms that the inter-molecular dipo-lar contribution is of the same order of magnitude as the infra-molecular one, as was discussed in the previous section. Both contributions would have to be taken into consideration if the above "moments" A 2 , A ^ e t c . are to be interpreted correctly, a task which goes beyond the scope of the present study. 5.4 T 2 9 e Measurements in DPPC-d 2/H 20 Model Membrane Figure 5.8 shows the orientation dependence of T2e as measured by the decay of spectral intensity in the three selected windows near the edge, shoulder, and in the middle of the spectra in a series of quadrupolar echo experiments on the D P P C - d 2 / H 2 0 sample (spectra not shown). The relaxation is seen to be exponential, and dependent on the orientation. The longest relaxation time is associated with the edge, the shortest — with the shoulder of the spectrum. The average relaxation time, measured by the decay of the intensity at the top of the echo in the time domain, is the T2e as denned in Chapter 2. It is immediately clear that we can rule out the static dipolar interaction as the relaxation mechanism: not only are the relaxation times much shorter than in the case of hexamethylbenzene, but the relaxation has a completely different orientation dependence. This is consistent with the fact that X H - 2 H dipolar coupling has been found to have only a minor effect on T2e in the liquid crystalline phase of model membranes, as measured by 2 H NMR of a selectively deuteriated phospholipid system with and without 1 H decoupling [56]. (5.4) Chapter 5. Results and Discussion 74 Figure 5.8: DPPC—0*2: orientation dependence of T2. The solid lines are the least squares' fits to a single exponential, for each window. Only the total intensity in each window is reported, without separating it into the two contributions as in the case of T\z. The average value shown is simply the intensity of the quadrupolar echo, in time domain, which corresponds to the average being taken over the distribution of relaxation rates, i.e. {1/T 2) _ 1. Chapter 5. Results and Discussion 75 At first glance, the exponential decay observed experimentally seems to be consistent with the predictions of the model derived in Chapter 1. In fact, we are now in a position to try to reconcile all three relaxation measurements with this model. Earlier we presented the inversion recovery measurement of T\Z = 24.9 ms; in a similar manner we obtain T\Q = 27.4 ms in a series of Jeener-Broekaert echo experiments. Now we can solve the three equations, 1.32, 1.34 and 1.37, for the three "unknowns", j(0), J(UJQ) and j(2vo) to obtain a rough idea of the frequency dependence of the reduced correlation function, j(u). The above experimental values correspond to j(0) : j(^o) : j(2u;o) ~ 242 : 140 : 1. These values are not sufficient to define the details of the frequency dependence of the correlation function; it is best to think of them as constraints on the functional form of j(co). At this stage, the usual theoretical approach is to take the simplest form of the correlation function which fits the experimental values. However, the above constraints are not satisfied if we try to use a single correlation time model, It is obvious that if we allow more than one correlation time in the model, we will be able to satisfy those constraints: after all, we only have three "experimental" points to fit. It is also clear that the solution would probably not be unique. Of particular interest here are the models which allow for motions slow on the NMR time scale, (JJQT2 >• 1. One such model which was proposed earlier [46,47] to explain the frequency dependence of the longitudinal relaxation time, T\Z{U}Q), introduced the concept of "collective motions", or "order director fluctuations" to explain the observed dependence [49, equation 39] Here the term B denotes the contributions due to the motions of a single molecule, and A corresponds to the collective motions. The above relationship holds for a certain form of the distribution of correlation form, including some motions which are fast and some which are slow on the time scale of OJQ. Chapter 5. Results and Discussion 76 The advantage of including the transverse relaxation measurements into the model is that it is sensitive to the slow motions, allowing one to probe the upper limit of the range of the correlation times, to which T\z and T\q are not sensitive. In fact, one can test whether the slow motion regime extends to the time scale of uq, i.e. whether OJ2T2 < 1 or not. If the "slow" motions remain fast on the time scale oiuq one expects that rf^~CPMG ~ y|e_ However, if there are ultra slow processes with correlation times OJ^T2 > 1 which contribute to the relaxation, it is possible that some of the experimental time intervals, say r of the quadrupolar echo experiment, become comparable to r c, causing the measured relaxation rate to depend on the details of the experiment. In particular, one would expect r£^~CPMG ^ j1^. 5.5 Using Q-CPMG to Separate the Effects of Lateral Diffusion One way of separating the various relaxation mechanisms in model membranes is to eliminate the random modulation of the local director orientation, which results when an individual lipid molecule diffuses along the surface of the bilayer, by using macroscopically oriented samples [50]. This is typically achieved by squeezing a small amount of sample between glass plates. Stacks of such glass plates can then be oriented at an arbitrary angle with respect to the external magnetic field. While this procedure does eliminate the reorientation due to lateral diffusion as a relaxation mechanism, some questions remain as to whether the samples so prepared are a reliable model membrane system. In addition, the need for the glass plates greatly reduces the signal-to-noise ratio in such samples, making them much harder to work with. Another way of distinguishing between the relaxation mechanisms according to their corre-lation times is to use a form of the Carr-Purcell-Meiboom-Gill pulse sequence appropriate to 2 H NMR, which we denoted "q-CPMG" in Chapter 2: 90j, - (r - 90x - r)N [36]. The echoes appear at times 2nr, n = 1,2,..., N following the initial 90y pulse. The amplitude of the n-th echo is given by 2nr S{2nr) = 5(0) exp- y g _ C P M G , (5.5) Chapter 5. Results and Discussion 77 using the notation we introduced earlier. Let us assume that the transverse relaxation is dom-inated by a single molecular motion which modulates a portion, AM2, of the second moment. This motion is characterized by its correlation time, r 2. With this notation, one can write down the following expression for T^~CPMG [57] 1 - ^ t a n h ( ^ ) ] . (5.6) -CPMG = A M 2 T 2 This result is identical to the expression for the CPMG relaxation rate of spin-| nuclei under-going chemical exchange between sites having different chemical shifts [58]. For a rapidly fluctuating quadrupolar interaction, r 2 <C TM — ( A M 2 ) - 1 / 2 , the experimental limit r r 2 is invariably satisfied, and Equation 5.6 reduces to rpq-CPMG = &M2T2, r 2 < TM. (5.7) This is exactly the result that has been calculated [59] in the case of a single quadrupolar echo; in this limit, Tl~CPMG ~ T f . However, it is the opposite limit, r 2 > TM, that is of main interest to us here, since it allows us to consider the relaxation effects due to slow modulations of the quadrupolar interaction caused by the lateral diffusion along the surface of the membrane. In this case, it is possible to use the experimental values of r <C T2- Equation 5.6 then predicts 1 A M 2 r 2 1 / r „, T,-CPMG - + j7. T 2 > T M (5.8) where T'2 is the transverse relaxation time of processes having correlation times T2 <C TM [36,57]. By comparison, in the case of a single quadrupolar echo one expects [59] T^e oc T2 when T2 ^> TM-Thus, we can use quadrupolar CPMG experiments to determine whether the slow modulations of the quadrupolar interaction caused by the lateral diffusion along the surface of the membrane have an effect on transverse relaxation. Experimentally, this effect is quite dramatic, and is observed in a variety of systems. For example, Figure 5.9 presents a comparison of transverse relaxation rates as measured by the quadrupolar echo (qe) and by the quadrupolar CPMG (q-CPMG) methods [36, Figure 2]. To Chapter 5. Results and Discussion 78 O 1 2 3 4 5 "t ( m s ) Figure 5.9: Comparison of transverse relaxation rates in a model mem-brane system (DPPC-dai/H^O) at 44°C. Intensity of the echo peak is plotted (on the logarithmic scale) against the actual time the echo oc-curs, t « 2r for the quadrupolar echo (qe) experiments, and t ss 2nr for the quadrupolar CPMG (q-CPMG) experiments. Results for a number of q-CPMG experiments with different values of r are shown; the points are connected for clarity. The solid line through the qe points is the result of a least squares' fit to a single exponential, yielding T| e = 493 ± 12 /J,S. Chapter 5. Results and Discussion 79 improve the signal-to-noise ratio we use here a preparation of one chain perdeuteriated DPPC (DPPC-d3i) in H 2 O . The local orientational order parameter is not the same for the deuterium atoms attached to different carbon positions along the acyl chain; consequently, instead of a single Pake doublet the spectrum consists of a superposition of several Pake doublets. However, the relaxation behaviour exhibits the same essential features as that of a selectively deuteriated system. When the loss of phase memory due to diffusion is eliminated by the refocussing pulses of the q-CPMG sequence, a dramatic decrease in the relaxation rate is observed. One should emphasize that it is the relaxation due to slow motion that is being eliminated; the "long" (compared to T2e) relaxation times observed in a q-CPMG experiment are those characteristic of the faster motions. Note that the first echo of a q-CPMG experiment is exactly equivalent to the appropriate qe experiment; thus for all r values the q-CPMG echo curves should start from the qe line, as they do to within an experimental error. In all of the q-CPMG experiments 32 echoes were recorded but only the echoes occurring at times shorter than 4 ms are presented in Figure 5.9. As can be seen, the data from the qe experiments fits well to a single exponent; a least squares' fit yields T2e = 493 ± 12p,s.4 The data from the q-CPMG experiments is strongly non-exponential. At short times (first few echoes), the relaxation rates exhibit a strong r dependence, while at longer times the relaxation rates appear to become independent of r. The results of least squares' fits to the initial few echoes, and to the echoes falling into the interval between 2.5 ms and 4 ms are presented in Table 5.1 [36, Table 1]. Equation 5.8 provides a framework for understanding these results. The sample is expected to have a distribution of MLV sizes, and thus the results cannot be interpreted in terms of a single correlation time, as implied by Equation 3.6. However, the initial slopes of the relaxation curves of Figure 5.9 are exactly equal to q-cPMG > w n e r e • • • denotes the average over the distribution of effective local radii of curvature as well as over the quadrupolar splittings (which are in turn determined for this sample by both the order parameter and the orientation 4 D u e to an oversight, this value was quoted as 214(is in [36], 493 = 214 X ln(10). Table 5.1 is similarly affected. Chapter 5. Results and Discussion 80 Table 5.1: Comparison of transverse relaxation rates in a DPPC-CI31/H2C) model mem-brane system at 44°C, as determined by the least squares' fits to the initial slopes of the q-CPMG echo envelopes (first few echoes) in Figure 5.9, and to the same echo envelopes at a later time ( 2 . 5 m ^ t ^ 4 m « ) . Measurements for various values of r are presented. By compari-son, T 2 ' e = 493 ± 12 [is. All values shown are in fis. T initial slopes 2.5ms<lt<l4ms rpq-CPMG 12 fitted to rpq-CPMG fitted to 50 1,480 ± 3 5 3 pts 3,044 ± 5 5 12 pts 80 1,403 3 pts 2,940 9 pts 100 1,270 3 pts 2,970 8 pts 130 1,111 2 pts 2,878 7 pts 150 1,012 2 pts 2,892 5 pts of the local director). The plot of the initial slopes of the relaxation curves of Figure 5.9 versus r 2 is shown in Figure 5.10 to be linear as predicted by Equation 5.8. The lateral diffusion along the surface of the membrane is too slow to contribute to motional averaging, it is only capable of gradually modulating the quadrupolar splittings which are measured in the 2 H NMR spectrum. For spherical bilayers we may, therefore, identify ( A M 2 ) in Equation 5.8 with the "residual second moment", M2r, of the 2 H NMR spectrum [38,40]. Using Equation 3.6 and denning an effective radius for diffusional relaxation, Rejj, by R~fj = R~2, we obtain — i 2 ^ > + W (5.9) rpq-CPMG ~ D2 'T1/ The linear plot in Figure 5.10 gives 2 M 2 r D = 161 x 108sec~3 and W = . For lipid Rgjj 12 1.6ms bilayers having a diffusion constant D w 4 x 10" 1 2 m 2 s _ 1 [38] and a value of M2r which we measured to be M2r = 3.0 X 10 9s - 2 , in agreement with [40], we obtain Reff = 1.2^m and, substituting into Equation 3.6, r2 ~ 62 ms. Chapter 5. Results and Discussion Figure 5.10: D P P C - C I 3 1 : Initial slopes of the relaxation curves of Fig-ure 5.9 are plotted versus r 2 . The actual values of r £ ^ ~ C P M G a r e listed in Table 5.1. The solid line is the result of a least squares' fit to Equation 5.9, yielding {^k^j = \ § m s an<i T2 ~ 62 ms. Chapter 5. Results and Discussion 82 An independent confirmation of the average size of the dispersions in this DPPC -d3 i /H 2 0 sample was performed by means of light scattering. To avoid multiple scatterings, a small fraction of the sample was diluted in H 2 0 until almost completely transparent. The results agreed with our estimate of Rejj = 1.2/xm well within the experimental error due to the aggregation of multiple MLV's during the light scattering experiment. Independently, this value is also in good agreement with the freeze-fracture electron microphotography measurements reported by many workers; multilamellar phospholipid dispersions prepared in the same way are described to "exhibit broad size distributions centered around diameters of a micron or more" [22]. Using the quadrupolar CPMG pulse sequence it is possible to eliminate the slow modulation of the quadrupolar interactions due to the lateral diffusion along the curved surface of the membrane, thus effectively extending the experimental time scale. It is somewhat surprising that the motions with a correlation time of r 2 « 62 ms are responsible for the loss of two orders of magnitude in the amplitude of the signal on the time scale of sa 1 ms, as seen in Figure 5.9. However, the results agree well with the known physical parameters of model membrane systems, confirming the proposed model of diffusion along curved membrane surface. 5.6 Application to Model Membranes Containing Sterols We applied the above method to several model membrane systems. One example is a somewhat more complex, ternary system of phospholipid/sterol/water. Addition of sterols such as choles-terol is known to influence the physical properties of model membranes [60,61]; even in small concentrations the sterols affect in a non-trivial way the phase behaviour of phospholipid/water systems [62,63]. The sterols are also thought to have great physiological significance, and to play a role in the evolution of life [35]. Many studies of different phospholipid/sterol/water systems have been performed [31, and references therein], using 2 H NMR, differential scanning calorimetry, and other techniques. In particular, T| e relaxation time has been shown to increase by a factor of two upon addition of Chapter 5. Results and Discussion 83 relatively small amounts of cholesterol to a DPPC-C131/H2O dispersion [31]. Since T2 depends on j(0) (see Equation 1.37), this has been interpreted as an indication of the speeding up of the slowest motion of the phospholipid. A similar but smaller effect was seen when lanosterol was used instead of cholesterol. Since the only major structural difference between the molecules of cholesterol and lanosterol is an a-methyl group protruding from the a-face of lanosterol, it was argued, that the smoothness of the cholesterol molecule is allowing the phospholipid molecules to slide by, while the protrusion on the lanosterol molecule hinders the movement of the phospholipid. As described in the previous section, comparison of T 2 ? e and x^~CPMG allows one to separate the influence of slow molecular motions, which are expected to be responsible for the changes in transverse relaxation upon addition of sterol. Therefore, the relaxation behaviour of the DPPC/lanosterol/water system was examined. We used two samples, with different lanosterol concentrations (5Mol% and 15Mol%), both prepared in excess H2O. The trend towards longer T| e upon addition of lanosterol was again observed; however, the actual T| e values were different by ~ 25% from the ones quoted in [31]. This was not surprising considering the relatively low (70%) purity of the lanosterol used in the preparation of the samples, and the differences in their thermal history. The results of T | _ C P M G measurements in these DPPC/lanosterol/water mixtures are summarized in Figure 5.11. As compared to the pure DPPC-d3i/water sample, the value of \rh) appears to increase with lanosterol concentration. The initial slopes of the q-CPMG relaxation curves do not exhibit the r 2 dependence characteristic of the pure DPPC/water sample, in the observed range of r values. The quality of the samples precluded a more thorough investigation; however, even these preliminary results indicate that a considerable change is taking place in the system: the presence of even a small amount of sterol greatly affects the motions with correlation times T~2 < TM, responsible for the T'2 relaxation. This is in agreement with the prediction made in [31] regarding the nature of the molecular interactions in such systems; however, the fact that the apparent value of T2 is decreased is in direct contradiction with the T| e measurements, 2 Chapter 5. Results and Discussion Figure 5.11: DPPC-d.3i/Ianosterol mixtures: Initial slopes of the q -C P M G relaxation curves are plotted versus r 2 for two concen-trations of lanosterol. The points are connected for clarity. The results for the pure DPPC-d3i/water sample, shown in Figure 5.10, are presented again for comparison. Chapter 5. Results and Discussion 85 and requires a further investigation. We expect that there is a difference in the distribution of vesicle sizes between the two samples. In fact, this is the most likely cause of the difficulties in reproducing the T^e values, and it could also be responsible for the breakdown of the r 2 dependence predicted by Equation 5.8, since in smaller vesicles the condition r < r 2 may be more difficult to satisfy. It is also known that the mechanical strength of the membrane is greatly increased upon addition of sterol [60,61], while the rate of lateral diffusion remains high [64,65]. In some sense, the presence of sterol seems to make the membrane stronger without making it less "fluid". As seen in Figure 5.11, the range of r values over which the relaxation rate depends on r is smaller as compared to the pure DPPC-d3i sample, suggesting that the time scale of diffusion is reduced in the presence of lanosterol, perhaps through a decrease in the effective local radius of curvature of the membrane caused by surface ripples or waves. 5.7 Hexagonal Hn Phase Measurements 5.7.1 Changes in the Orientational Order Parameter Phospholipid-water systems are "polymorphic", i.e. they may assume several structurally distinct phases. All of the results presented so far were obtained with the model membrane systems in their lamellar ( La) or bilayer phase. Also of interest, and of possible physiological significance [66] is the hexagonal ( Hn) phase. In this phase, the water is found on the inside of long parallel cylinders which are hexagonally coordinated in the plane perpendicular to their axes of symmetry as shown schematically in Figure 5.12. The hydrophobic acyl chains fill the space between the cylinders. One of the most significant characteristics of Hn phase is a rapid (on the NMR time scale) diffusion of the lipid molecules around the cylinder axes, which is well established from phosphorus (31P) NMR measurements [67]. This diffusion affects the distribution of the local orientational order parameter of the system, as well as its relaxation behaviour. The symmetry properties of the Hn phase, and in particular, the anticipated nature of the acyl chain packing in the regions between the cylinders leads one to expect that the Chapter 5. Results and Discussion 86 Figure 5.12: Comparison of the La (on the left) and hexagonal Hn (on the right) phases of phospholipid-water dispersions. A schematic representation of their geometry is shown at the top. Due to the rapid diffusion around the cylinder axes in the HJJ phase, both the 3 1 P NMR (middle) and the 2 H NMR (bottom) spectra are scaled by a factor of— 1/2. The 3 1 P NMR spectra can yield both the absolute value of S and its sign. Because of their symmetry, the 2 H NMR spectra do not provide the information about the sign of S. For illustrative purposes, the simulated powder patterns are shown on arbitrary horizontal and vertical scales, with U)Q indicating the Larmor frequency for each nucleus. Chapter 5. Results and Discussion 87 dependence of orientational order parameter, S, on the chain position, n, be qualitatively different from the "fluid bilayer signature"5 characteristic of the La phase [30]. This expectation is supported by measurements made by Jarrell and coworkers [68,69] which give the changes in orientational order between La and Hn phases for some selected chain positions. In a recent paper [70], we presented a systematic study of the variation of S with n, which enabled us to analyze quantitatively substantial differences in chain packing between the La and Hn phases. There, a ternary mixture of l-palmitoyl-2-oleoyl-phosphatidylethanolamine (POPE), 20mol% perdeuteriated tetradecanol (TD), and water (in excess) was used. Adding alcohol to the lipid-water mixtures has a strong effect on the temperature of the La to Hn phase transition; yet TD molecules are pinned down to a certain extent at the lipid-water interface and are found to report faithfully the orientational order of the host phospholipid. The 2 H NMR spectra of 20 mol% TD in POPE at varying temperatures are shown in Figure 5.13. Below the La to Hn transition (as determined from the 3 1 P NMR) the 2 H NMR spectra exhibit the usual lamellar lineshape for a perdeuteriated chain [40,71]. Above the transition the lineshape is significantly different. In the immediate vicinity of the transition, spectra containing mixtures of both lineshapes are obtained. The observed difference between the lineshapes indicates that across the phase transition the various S(n) values along the TD chain do not scale by a common factor. Also, measuring the average quadrupolar splitting as a function of temperature gives a decrease by considerably more than a factor of —1/2, expected if the rapid diffusion about the cylinder axis were the only difference between LQ and Hn phases. These two observations agree with our expectations since the geometry of the Hn phase allows more conformational freedom for the ends of the chains than that of the La phase. An important and much discussed feature of the variation of \S(n)\ in the La phase is the "plateau". In the depaked spectrum of the La phase shown in Figure 5.13d, the plateau corresponds to the largest splitting, which accounts for almost half of the total intensity. In 5 T h e often used term "plateau" refers in this context to a very small variation of S with n for the part of the acyl chain closer to the glycerol backbone, followed by a rapid decrease towards the end of the chain. Chapter 5. Results and Discussion 88 -50 ^25 0 25 50 -50 , -25 0 25 50 F r e q u e n c y ( k H z ) F r e q u e n c y ( k H z ) Figure 5.13: La to Hn phase transition as seen from the 2 H N M R spectra. The results of dePakeing the powder spectra on the left, (a) through (c), are shown on the right, (d) through (f), for three different temperatures. The spectrum in (a), and also in (d), exhibits the "plateau" lineshape characteristic of the La phase; the lineshape of (c), and of (f), is completely different. In the im-mediate vicinity of the phase transition, (b) and (e), a mixture of both lineshapes is observed. The integral intensity of all spectra is normalized to 1, but for clarity the spectrum in (a) is shown at twice the vertical scale of those in (b) and (c). By convention, the depaked spectra are shown for the orientation corresponding to 6 = 0 ("shoulder"). Binomial smoothing over ± 3 points was used on the depaked spectra. Chapter 5. Results and Discussion 89 the Hn phase, shown in Figure 5.13f, a relatively larger fraction of the total intensity appears at smaller splittings, indicating a diminishing importance of the plateau. This point is more readily seen from the renormalized plots of Figures 5.13d and 5.13f shown in Figure 5.14. Here L Him 11 i i h H 1 in 1 nun H n I HIM I I I I I I 1- H 1—I I I M HUM I 1 1 1 1 1 1 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1.0 1.5 CT=S(n)/Smox Figure 5.14: Comparison of the distribution of the fractional order parameter in the two phases. La phase, shown by the dotted line, exhibits the characteristic "plateau" lineshape where a major fraction of the total inten-sity is in the outermost peaks of the spectrum. In contrast, in the HJI phase, shown by the solid line, the importance of the plateau is diminished. Both spectra are normalized to have the total intensity of 27. The fractional order parameter for every carbon position is assigned by calculating the midpoints of inter-vals weighted 3-2-2-2-..., as discussed in text. Their locations are shown for both phases, immediately above (for La) and and below (for Hn) the spectra. Notice, that the tick marks corre-sponding to these midpoints are more evenly spaced for the Hn phase. we plot the spectral intensity against the variable a = S/Smax, where Smax is the maximum value of the local chain orientational order parameter in a given phase, as measured from the maximum observed splittings in Figures 5.13d and 5.13f. In addition, the area of each spectrum has been normalized to 27, the number of 2 H nuclei per TD chain contributing to the 2 H NMR Chapter 5. Results and Discussion 90 spectra. This corresponds to twelve methylene (position one is not deuteriated because of the nature of the synthetic procedure [70]) and one methyl group. Therefore, each curve can be viewed as a plot of the number of deuterons per unit interval of a. The plots show clearly that the En phase no longer has a plateau near a= 1 and that the distribution of a is more heavily weighted at lower values in the HJJ phase relative to the LA phase. If one now assumes that a varies monotonically along the chain, one can assign an average fractional order parameter in each phase to every carbon position n. This is denoted by oiXn) for the La phase and anin) for the Hn phase. In practice, we separately assign the peaks associated with the smallest values of a in Figure 5.14 to the methyl deuterons. The remaining area under the curve corresponds to 24 deuterons and is divided into 12 equal parts. The value of a at the center of each of these parts determines the value of ai,(n) or 07/(n) corresponding to the appropriate methylene group in the chain. The plots of the average cr(n) values for each phase assigned in this manner are shown in Figure 5.15. The observed change in the orientational order profile can be analyzed quantitatively [70] and provides a measure of the increase in conformational freedom available for different parts of the TD chain when the system undergoes the La to Hn phase transition. Note that the ratio of the maximum splittings, which we associate with n = 2, differs only slightly from 1/2. This demonstrates that the conformational averaging of the polar headgroup region of the phospholipid molecules is not appreciably affected by the curvature of the cylindrical lipid-water interface in the Hn phase. Equivalently, one can say that the local orientational order parameter for the polar headgroup is not affected by this curvature. The factor of -1/2 is attributable to the rapid diffusive motion of the lipid molecules around the cylinder axes. By contrast, we have shown that the local orientational order of the acyl chains, as sensed by 2 H NMR of the TD molecules in our samples, is systematically lower in the Hn phase than in the La phase. For the n = 2 position, the ratio of the quadrupolar splittings in the Hn phase to that of the La phase is 1/2.4; near the end of the chain this ratio decreases to about 1/5.0. The fact that the ratio of 1/2.4 is within 20% of the factor of -1/2 demonstrates that the polar end of the TD molecule spends Chapter 5. Results and Discussion 91 1.0 - i 0 .8 -X E 0 0 0 .6 -C IS) 0 . 4 -II b 0.2 -•A 0 .0 0 2 4 4 6 8 10 r. n, carbon position 12 14 16 Figure 5.15: Order parameter profile of LA (A) and HJI (•). We use the assumption that a decreases monotonically with car-bon position to divide the integrated intensity, normalized to 27, into the intervals of weight 3-2-2-2-..., and assign the midpoints of these intervals to the average fractional order parameter asso-ciated with the appropriate carbon position. In this way, Smax in each phase is the one associated with the outermost such inter-val which we assign to the carbon position n = 2. Consequently, <TL(2) = aH{2) = 1. Chapter 5. Results and Discussion 92 most of its time close to the lipid-water interface, while the systematic additional decrease of 5#(n) down the chain indicates significantly greater angular freedom deep in the hydrophobic region of the HJJ phase as compared to the LA phase. 5.7.2 Relaxation and Spectral Changes due to the Extra Motion In the LA phase the diffusion along the membrane surface acts as a slow motion, capable of modulating the quadrupolar interaction because of the change of orientation of the local director associated with the diffusion along a curved surface. By using a q-CPMG pulse sequence, it is possible to keep refocussing the magnetization on the time scale short compared to the r c of diffusion and thus to eliminate the relaxation effects of such slow diffusive motion. The overall observed relaxation rate is greatly extended, as was shown for a D P P C - d 3 x / H 2 0 model membrane in Figure 5.9. A similar behaviour is seen in Figure 5.16 for the case of 20mol% TD in P O P E / H 2 0 . A more detailed account of the differences between the two methods of measuring the relaxation (qe and q-CPMG) can be obtained by comparing the partially relaxed spectra, as shown in Figure 5.17. Here the spectra shown are obtained, as usual, by a Fourier transform of the time domain signal, starting at the top of the echo. The plots in Figure 5.17a and 5.17b show the depaked quadrupolar echo spectra for two different values of r. Figure 5.17c is a depaked spectrum obtained in a q-CPMG experiment in which the echo train was terminated after 11 echoes (N = 11 in the notation of Chapter 2) and the 12-th echo was digitized in its entirety by switching the digitizer into a free-running mode. Similarly, Figure 5.17d is the depaked spectrum of the 24-th echo of a q-CPMG experiment with r = 50 fis. The reason for selecting N = 12 and N = 24 is that as seen in Figure 5.16 the amplitude of a single qe with r = 180 fis falls between the amplitudes of the 12-th and the 24-th echoes of a q-CPMG train with r = 50^s, providing a direct comparison between the spectra of Figure 5.17b, 5.17c, and 5.17d. Each depaked spectrum can be looked upon as a direct measure of the variation of the local orientational order parameter with carbon position, S(n). All four spectra are very Chapter 5. Results and Discussion Figure 5.16: T D in P O P E / H 2 O : comparison of the relaxation rates in La phase, at 33°C. In analogy with Figure 5.9, the decay of a single qe with t « 2r is compared with the decay of the consecutive echoes of q-CPMG experiments, at i w 2nr. The data points are connected for clarity. Chapter 5. Results and Discussion - 5 0 - 4 0 - 3 0 - 2 0 -10 0 10 20 30 40 50 F r e q u e n c y ( k H z ) Figure 5.17: T D in P O P E / H 2 O : comparison of the partially relaxed spec-tra in the La phase, at 33°C. (a) qe, r = 50 /is; (b) qe, r = 180 fis; (c) q-CPMG, r = 50 fis, a spectrum at the top of the 12-th echo; (d) q-CPMG, r = 50 fis, a spectrum at the top of the 24-th echo. All spectra are depaked with ~ 10 iterations each, and are normalized to a unit integral intensity, excluding the innermost peaks which we attribute to the terminal methyl group. Binomial smoothing over ± 2 points is used. Chapter 5. Results and Discussion 95 similar, suggesting that the character of the motions responsible for the relaxation does not vary greatly with carbon position, with the sole exception of the terminal methyl group. This applies both to the slow diffusive motions along the surface of the membrane, as well as to the faster motions responsible for the \ jT'2 relaxation (see Equation 5.8). The geometry of the HJJ phase is quite different from that of the LA phase. Lateral diffusion along the surface of the membrane has its analog in the diffusion along the axial direction of the cylinders, while the diffusion about the axes of the cylinders becomes a very fast motion. The effects of the slow diffusive motion along the axis remain significant as seen in Figure 5.18 where applying a q-CPMG echo sequence greatly extends the time scale of the CD o 2 -N = 12 N=24 q e n 1 r 2 3 4 t ( m s ) Figure 5.18: T D in P O P E / H 2 O : comparison of the relaxation rates in Eu phase, at 48°C. The decay of a single qe with t « IT is compared with the decay of the consecutive echoes of q-CPMG experiments, at t « 2nr. The data points are connected for clarity. relaxation. The overall behaviour is very similar to the LA phase. In particular, the q-CPMG Chapter 5. Results and Discussion 96 curve acquired with r = 50 fis is almost indistinguishable from its counterpart in Figure 5.16. This suggests that the character of the diffusive motion of the molecule as a whole does not differ much between the two phases. However, there is a significant difference between the lineshapes of the partially relaxed spectra between the two phases, as seen from a comparison of Figures 5.17 ( La phase) and 5.19 ( Eu phase). In the En phase a significant change is seen in the shape of the depaked - 2 5 - 2 0 -15 -10 - 5 0 5 10 15 20 25 F r e q u e n c y ( k H z ) Figure 5.19: T D in P O P E / H 2 O : comparison of the partially relaxed spec-tra in the En phase, at 48°C. (a) qe, r = 50 fis; (b) qe, r = 180 fis; (c) q-CPMG, r = 50 fis, a spectrum at the top of the 12-th echo; (d) q-CPMG, r = 50 fis, a spectrum at the top of the 24-th echo. All spectra are depaked with ~ 10 iterations each, and are normalized to a unit integral intensity, excluding the innermost peaks which we attribute to the terminal methyl group. Binomial smoothing over ± 2 points is used. spectra acquired along the q-CPMG relaxation curve, and hence, in the orientational order profiles. It appears that while the relaxation due to slow diffusive motions is independent of the Chapter 5. Results and Discussion 97 carbon position, the character of the faster motions is changed as compared to the La phase. The transverse relaxation rate near the beginning of the acyl chain is increased relative to the relaxation rate near the end. Chapter 6 Summary and Concluding Remarks In this work, it was established that motions with very long correlation times strongly influence 2 H NMR relaxation in model membranes. Traditionally, the hierarchy of time scales measured by 2 H NMR is assumed to be: Ti ~ 1(T 1 0 s < a;"1 ~ 5 X 10 - 9 s < r 2 ~ 10 - 7 - 10 - 8 s < u~x ~ 1(T5 a , (6.1) where T\ and r 2 are the correlation times of the processes responsible for the T\ and T 2 relax-ation, respectively; u>o is the Larmor frequency; u>q is the frequency of the dominant interaction, i.e. the quadrupolar frequency in the case of 2 H NMR. A wide range of motions is present in model membranes in their liquid crystalline state, from very fast trans-gauche isomeriza-tions of the acyl chains of phospholipid molecules, to slower rotations of these molecules about the bilayer normal, to very slow translational diffusion along the bilayer. To date, all of the experimental results supported the above hierarchy of time scales. In particular, fast motions, i.e. motions with correlation times r c such that r c <C w" 1 , have been extensively studied through the longitudinal relaxation rates, T\z~ l and T\q~x. As seen from the theory presented in the Appendix, the UQ 1 "milestone" can be used to further restrict the time scale of these fast motions by testing whether the correlation time is fast or slow on the time scale associated with the Larmor frequency, i.e. whether rc «C 1 or TC ^ ^cT1) respectively. As an example, we considered a sample of perdeuteriated polycrystalline hexamethylbenzene (HMB-di 8). As expected, we found that the experimental results could be explained by the model in which the condition r c <C w" 1 was satisfied for both the rotation of a C H 3 group about its axis of symmetry and the rotation of the entire molecule about its C6 axis, in agreement with previously published work. In model membranes, similar measurements 98 Chapter 6. Summary and Concluding Remarks 99 of the longitudinal relaxation rates had previously demonstrated that the range of correlation times is extended to longer times, TC~£UIQ 1 . In particular, the concept of "collective motions" has - 1/2 been introduced to explain the observed frequency dependence of T\z ~ w0 . Additional degrees of freedom introduced by extending the single correlation time model to a distribution of correlation times, allowed the experimental data to be fitted. However, the form of this distribution was not unique, as the data could be explained by a number of possible models. Nevertheless, the introduction of longer correlation times raised the important question of how long the longest of them can be. In particular, one needs to test the implicit assumption that only the motions which are fast on the interaction time scale, r c <C w" 1 , contribute to the nuclear magnetic relaxation. As we discussed in Chapter 1, the transverse relaxation rate, T2~x is sensitive to j(0), as opposed to the the longitudinal relaxation rates which depend only on j(wo) and j(2wo), making it the natural choice for examining the longer correlation times. Surprisingly, very few measurements of T2 have been performed in model membrane systems. One of the reasons for this, as we knew from work done previously in our laboratory, was a poor reproducibility of such measurements. Even in the simple two-component (phospholipid and water) model membrane systems prepared with synthetic phospholipids where most of the chemical variables could be eliminated, there remained the question of variability of the physical parameters, such as the shapes and sizes of the vesicles which spontaneously form in phospholipid/water mixtures. In order for these parameters to matter, the relaxation has to be affected by the motions with correlation times comparable to the time it takes a phospholipid molecule to change its time-averaged (by the fast rotation about the bilayer normal) orientation in the magnetic field (9), thus introducing a P2(cos 9) modulation into the quadrupolar splitting. For a typical vesicle size the time it takes a phospholipid molecule to change its orientation due to the diffusion along a curved surface of the vesicle is extremely long (rc;> 10 ms), extending not only to r c > WQ 1 but also beyond the other "milestone" on the time scale, namely that determined by the strength of the interaction, rc > u;" 1 . Chapter 6. Summary and Concluding Remarks 100 In this work, we established that such ultra slow motions are, indeed, of great importance in the 2 H NMR relaxation in model membranes. We examined a number of systems, both selectively and fully deuteriated. We did not attempt an exhaustive study of any one single system, but rather, tried to establish some important features common to all model membranes. In the simplest case of a selectively deuteriated system, the T%e relaxation is exponential (see Figure 5.8) and not ~ r 3 sin2 29, successfully used previously to explain, in the limit of slow motions r c 3> u~x, the consequences of lateral diffusion of water along the surface of clay particles [39]. Also, we examined the orientation dependence of the relaxation rate: it is the longest near the edges (9 = 90°) of the so-called "Pake doublet" powder pattern. This is, in fact, a non-trivial result; it is due to the fact that near the edges the above expression goes to zero, and other relaxation mechanisms become important. For example, 1 H - 2 H static dipolar interaction has been shown to contribute no more than 20% to T| e in the liquid crystalline phase of model membranes, as measured by 2 H NMR of a selectively deuteriated phospholipid system with and without 1 H decoupling. This contribution becomes more important near the edges. For perdeuteriated molecules, the experimental spectrum is a superposition of many Pake doublets, with different quadrupolar splittings caused by the variation of the orientational order parameter between different sites on the molecule. The shape of T| e relaxation curves in this case is consistent with a superposition of many relaxation rates. To establish the importance of the ultra slow motions, we compared these measurements of r| e to the results of a Carr-Purcell-Meiboom-Gill experiment modified as to be suitable for 2 H NMR, which was referred to as "q-CPMG". The experimental values ofT^~CPMG showed a dramatic decrease in the apparent relaxation rate, as compared to the values of T| e , thus confirming the existence of such motions. The extremely long correlation time implied by the rpq-CPMG m e a s u r e m e n t s meant that the lateral diffusion was the most likely candidate. Its presence was unambiguously confirmed by a selective inversion recovery experiment in which magnetization transfer across the spectrum was seen. We used a selective inversion scheme in which the edges of the Pake doublet were preferentially affected, unlike previously reported Chapter 6. Summary and Concluding Remarks 101 experiments in which a narrow range in the center of the pattern was selected. The edges correspond to the largest spectral intensity, making it easier to accurately monitor its recovery. In addition, in the vicinity of the edge the quadrupolar splitting, proportional to P2(cos 0) varies more slowly than near the center of the spectrum and thus one could hope to extend the time scale of the experiment, as individual molecules have to diffuse through a larger angle. Having established the presence of the lateral diffusion along the membrane surface, we could fit the results of the q-CPMG relaxation measurements to a model [57] which assumes a single molecular motion and describes the remaining contributions, most likely those due to the slowest of the conformational motions, by a single effective relaxation rate, l /T^. Thus we estimated the correlation time for the slow motion of order 62 ms. Assuming for simplicity that a single effective radius of curvature, R~jj = (R~2), is sufficient to characterize the distribution of sizes and shapes in a typical multilamellar phospholipid/water dispersion, we estimated that r this correlation time corresponds to Rejf « 1.2 fim, in good agreement with other methods of measurement. For the first time, this confirmed the importance of ultra slow motions for 2 H NMR relaxation, and established lateral diffusion of phospholipid molecules along the plane of the bilayer as the most likely candidate. The implications of this main result were examined for a number of model membrane sys-tems, for example, systems containing sterols such as cholesterol or lanosterol, which exhibit a dramatic increase in T| e upon addition of a small amount of sterol. We measured the trans-verse relaxation (both T| e and rr^-CPMG^ f o r t w o lanosterol concentrations. The problems in reproducing the exact values cited in [31] are most likely due to the differences in the vesicle size distribution in different samples, which make it difficult to quantify the prediction regarding the nature of the molecular interactions made there. However, some preliminary mea-surements of x^~CPMG indicate that when the contribution due to the slow lateral diffusion is eliminated by the q-CPMG sequence, there is, indeed, a substantial change — upon addition of sterol — in the character of the faster motions, assumed to be the rotational diffusion of phos-pholipid molecules about the normal to the bilayer, and/or the slowest of the conformational Chapter 6. Summary and Concluding Remarks 102 motions. Presence of impurities in the samples prevented us from making a more quantitative interpretation. Phospholipid/water dispersions can form spatial structures, or phases, other than the bilayer ( La) phase. In particular, in the so-called hexagonal Hn phase, the nature of the molecular motions is expected to be quite different from that of the La phase, due to the differences in their topology. We used a ternary POPE/tetradecanol/water sample both above and below the La to Hn transition temperature to examine these differences. To characterize changes in the fast conformational motions, we used a novel way of quantifying the differences in the distribution of the local orientational order along the chain between the two phases. We found an almost linear (with carbon position along the acyl chain) increase in the orientational freedom available to the acyl chains [70], consistent with their spatial arrangement in the Hn phase. At the same time, the local orientational order of the headgroup region is decreased by a factor of ~ 1/2 as compared to the La phase, which can be accounted for by the rapid diffusive motion around the axis of the cylinder. This indicates that the local conformational motions near the lipid-water interface are not greatly affected by the La to Hn phase transition. In contrast, the conformational motions deep inside the hydrophobic region are quite different in the two phases, as established by the differences in the shape of the partially relaxed spectra obtained in the qe and q-CPMG experiments. In terms of a single effective radius of curvature which we assign to the model membrane, it is impossible to distinguish between the lateral diffusion of the phospholipid molecules along the surface of the bilayer, and the rotational diffusion (tumbling) of the vesicle as a whole. This is a limitation of the proposed technique; it can be circumvented by repeating the r £ ^ ~ C P M G relaxation measurements at different temperatures, since the temperature dependence of the the two types of motion is expected to be different. A careful study of such temperature dependence may also allow one to separate the effects due to surface ripples or waves. Deuteriated hexamethylbenzene was used initially for demonstration purposes but proved to be itself a very instructive "test" sample for comparing with model membranes. The complex Chapter 6. Summary and Concluding Remarks 103 interplay of various motions and interactions in hexamethylbenzene was not examined in great detail, but some of the predictions of the theory of orientation dependence of the relaxation rates, presented in the Appendix, were confirmed. Additional work is being done in our labo-ratory. In particular, application of the multiple pulse trains similar to q-CPMG we used here to such a system is of great interest, and some preliminary measurements of the systematic variation of the relaxation as a function of the rotation angle of the refocussing pulses suggest that such methods may have implications for analyzing multiple spin correlations in solids. Prof. H.W. Spiess and coworkers [72] have recently reported a two-dimensional (2D) NMR method of measuring the rotational angles of molecular reorientations, which they applied to a variety of systems. Some of the aspects of the diffusive motions in model membranes lend themselves very well to this method, and should be examined. Bibliography [1] F. Bloch. Phys. Rev., 70:460-474, 1946. [2] A. Abragam and M. Goldman. Nuclear Magnetism: Order and Disorder. Clarendon Press, Oxford, 1982. [3] M. Bloom. NMR studies of membranes and whole cells. In Enrico Fermi International School on the Physics of Magnetic Resonance in Biology and Medicine, Societa Italiana di Fisica, Elsevier Science Publishers, The Netherlands1, 1987. [4] R. Ernst, G. Bodenhausen, and A. Wokaun. Principles of NMR in One and Two Dimen-sions. Oxford University Press, London, 1986. [5] M. Bloom and M. A. LeGros. Can. J. Phys., 64:1522-1528, 1986. [6] M. Goldman. The physics of NMR spectroscopy in biology and medicine. 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Weeks. Biochem-istry, 17:5750-5762, 1978. [39] D.E. Woessner, Jr. B.S. Snowden, and G.H. Meyer. J. Chem. Phys., 51:2968-2976, 1969. [40] J. H. Davis. Biophys. J., 27:339-358, 1979. [41] E. Sternin. Rev. Sci. Instrum., 56:2043-2049, 1985. [42] C. Morrison. Introduction to RmlOO. In preparation. [43] D60 NMR Spectrometer. Instructions Manual. Electronics Shop, Department of Physics, University of British Columbia. Bibliography 107 [44 [45; [46 [47 [48; [49; [50 [51 [52 [53 [54 [55 [56 [57; [58 [59 [60 [61 D. I. Hoult and R.E. Richards. Proc. Roy. Soc, 344:311-340, 1975. M. Ranee and R. A. Byrd. J. Magn. Reson., 52:221-240, 1983. M. F. Brown, A. A. Ribeiro, and G. W. Williams. Proc. Natl. Acad. Sci., 80:4325-4329, 1983. M. F. Brown. J. Chem. Phys., 80:2832-2836, 1984. R. Kimmich, G. Shuur, and A. Scheurmann. Chem. Phys. Lipids, 32:271-322, 1983. K.R. Jeffrey, T .C . Wong, E.E. Burnell, M.J. Thompson, T.P. Higgs, and N.R. Chapman. J. Magn. Reson., 36:151-171, 1979. M.F. Brown and J.H. Davis. Chem. Phys. Let., 79:431-435, 1981. W. P. Rothwell and J. S. Waugh. J. Chem. Phys., 74:2721-2732, 1981. M. Bloom. Private communication. N. Boden and Y. K. Levine. J. Magn. Reson., 30:327-342, 1978. N. Boden and P. K. Kahol. Mol. Phys., 40:1117-1135, 1980. J. H. van Vleck. Phys. Rev., 74:1168-1183, 1948. K. P. Datema, P. A. de Jager, and M. A. Hemminga. J. Magn. Reson., 1988. In print. J. S. Blicharski. Can. J. Phys., 64:733-735, 1986. Z. Luz and S. Meiboom. J. Chem. Phys., 39:366-370, 1963. K. P. Pauls, A. L. MacKay, O. Soderman, M. Bloom, A. K. Taneja, and R. S. Hodges. Eur. Biophys. J., 12:1-11, 1985. E. Evans and D. Needham. Faraday Discuss. Chem. Soc., 81:267-280, 1986. E. Evans and D. Needham. J. Phys. Chem., 91:4219-4228, 1987. Bibliography 108 [62] M. Vist. Partial Phase Behaviour of Perdeuteriated Dipalmitoy[phosphatidylcholine— Cholesterol Model Membranes. M.Sc. thesis, Department of Physics, University of Guelph, 1984. [63] J. H. Davis. NMR studies of cholesterol orientational order and dynamics, and the phase equilibria of cholesterol/phospholipid mixtures. In Enrico Fermi International School on the Physics of Magnetic Resonance in Biology and Medicine, Societa Italiana di Fisica, Elsevier Science Publishers, The Netherlands1, 1987. [64] N.L. Thompson and D. Axelrod. Biochim. Biophys. Acta, 597:155, 1980. [65] G. Lindblom, L.B.A. Johansson, and G. Arvidson. Biochemistry, 20:2204-2207, 1981. [66] P. R. Cullis, M. J. Hope, and Colin P. S. Tilcock. Chem. Phys. Lipids, 40:127-144, 1986. [67] P. R. Cullis and B. de Kruijff. Biochim. Biophys. Acta, 559:399-420, 1979. [68] H. C. Jarrell, J. B. Giziewicz, and I. C. Smith. Biochemistry, 25:3950-3957, 1986. [69] B. Perly, I. C. Smith, and H. C. Jarrell. Biochemistry, 24:1055-1063, 1985. [70] E. Sternin, B. Fine, M. Bloom, C.P.S. Tilcock, K.F. Wong, and P.R. Cullis. Biophys. J., 1988. In print. [71] J. L. Thewalt, S. R. Wassal, H. Gorissen, and R. S. Cushley. Biochim. Biophys. Acta, 817:355-365, 1985. [72] C. Schmidt, S. Wefing, B. Bliimich, and H. W. Spiess. Chem. Phys. Let., 130:84-90, 1986. [73] L. J. Schwartz, E. Meirovitch, J. A. Ripmeister, and J. H. Freed. J. Phys. Chem., 87:4453, 1983. [74] J. Tang, L. Sterna, and A. Pines. J. Magn. Reson., 41:389-394, 1980. 1 T o o k place at V i l l a Monastero, in Varenna sul Lago di Como, Italy, in July, 1986 Bibliography 109 [75] P. S. Allen and A. Cowking. J. Chem. Phys., 47:4286-4289, 1967. [76] M. Tinkham. Group Theory in Quantum Mechanics. McGraw-Hill Book Company, New York, 1964. Appendix A Orientation Dependence of 2 H N M R Relaxation in Perdeuteriated Hexamet hy lbenzene* A large number of previous NMR studies of hexamethylbenzene, Ce(CH3)6, have established that two important internal motions are present at room temperature in hexamethylbenzene (HMB) solid [73,74,75]. The methyl groups rotate rapidly about their C3 symmetry axis with a correlation time rx and the entire HMB molecule rotates about its Ce axis with a correlation time T2- It is well established that at room temperature, u^r 2 <C 1, T\ <C T2, and u^To2 <C 1. In this thesis, it has been demonstrated that the values of T\z and T\q in 2 H NMR studies of perdeuteriated HMB are dependent on the angle, /3, between Ho and the HMB C6 axis. The angular dependence of the relaxation rates is calculated in this Appendix in terms of molecular geometry, u0, uq, r i , and r 2. A . l Coordinate Systems and Euler Angles —• A Cartesian system (x,y, z) is chosen such that z is along HQ and the CQ symmetry axis, denoted by the vector n is in the xy plane. The (a;', ?/, z') coordinate system has z' along n and x' along one of the C3 symmetry axes. The Euler angles which rotate (x,y,z) into (x',y',z') are (0,/3,7). The (x",y",z") axes have z" along x' and are obtained from (x',y',z') via Euler angles (0 ,§ ,0). * T h e material presented in this Appendix was kindly provided by Prof. M . Bloom. 110 Appendix A. Orientation Dependence of 2H NMR Relaxation 111 A.2 Rotation of the Spherical Harmonics Let (0, <f>) be the spherical polar coordinates of a specific C — 2 H bond direction in (x,y,z) and (9",<f>") be the corresponding coordinates in (x",y",z"). Then, using the properties of Wigner rotation matrices, Dm,m(ct ft7) = e~m'a dm,m(/3) e -" 7 1 7 , and noting especially the convention for expressing functions such as spherical harmonics Y2m(9,<f>) in terms of the rotated axes as described clearly in [76, Table 5-1, page 112], we obtain Y2m(6,<f>)= F 2 „ m ( / ? , 7 ) Y 2 m „ ( 0 " , < A (A.l) m"=-2 where *£»mGS,7) = £ e _ , W 7 Cm(0) d2M ( § ) • (A.2) A convenient tabulation of the d^^fl) is given in [21, page 294]. A.3 Averaging over Anisotropic Motions: General Remarks In order to generalize the expressions for relaxation rates derived in Equations 1.32-1.37 for isotropic molecular reorientations to include the effects of anisotropic reorientations as in the case of HMB, we first note that the symbols j(muo) stand for reduced spectral densities of the Y2Tn(t9,0), where m takes on values of 0,1 and 2. In deriving Equations 1.32-1.37, use has been made of the average values (Y2m(0,<f>)) = 0 and ^|y2m(0, <f>)\2^ = for isotropic motions. In calculating the relaxation rates for anisotropic motions, we must calculate the dependence on /? of both these quantities — in general, these averages will depend on m — and to replace (\Y2m(6,<f>)\2^ by the mean squared fluctuation of Y2m(0,<f>) about its average value, i.e. (\Y2m(0, 4>)\2) = ^ is replaced by (\Y2m(9, <j>)\2) - \(Y2m(9, <f>))\\ The dependence on (3 is conveniently expressed in terms of the Legendre polynomials P2(/J) = 2(3/x2 — 1) and P4(/j) = - ±^n2 + |, where ft = cos/3. Appendix A. Orientation Dependence of 2H NMR Relaxation 112 A.4 Detailed Calculation of the Anisotropic Averaging We wish to calculate averages over two types of motions. In the first motion, characterized by T\, the stochastic variables are 8", <f>" and we make use of (Y 2 m „(0" ,<£")) = ( ^ ) S 2 Sm..0 (A.3) where S, = (fi ( c o 8 (A.4) Similarly, ^|Y2m»(#",</>")|2^ ) ( may be expressed in terms of 5 2 and 54. For a fixed value of 8" as in the case of a C H 3 group rotating about its symmetry axis with no wobble, (cos40") = (cos20")2 so that these averages may be expressed in terms of 5 2 and 5 2 , but in the general case the relevant parameters are 5 2 and 54. The values of (\Y2m»(8", ^ ")|2)e„ ^ „ for the general case and for fixed 6" are both summarized in Table A . l . Table A . l : Values of (\Y2m..(8",<t>")\2)g expressed in terms of 5 2 and 54 for the general case and in terms of 5 2 and 5 2 for the case of fixed 8". In the latter case, use has been made of the relationship for Legendre polynomials P 2 = | + |P 2 -f 1|P 4. m" General Results Results for fixed 6" 0 _ L ( I + mS2 + fs4) 5 r-2 Air3! ±1 _ L ( 1 + f 5 2 - ^ 5 4 ) 1 / 5 ,5(i 1 0 c2\ + 3^ ~ 3 b2) ± 2 _ mS2 +154) 1/5 5 c 1 5c!l 4^6 ~ 3^ 2 + 6*2; In the second type of motion, characterized by r 2, the stochastic variable is 7 and we use Appendix A. Orientation Dependence of 2H NMR Relaxation 113 where Similarly, the results for (\Fm»m(f3, j)\2^ are given Table A.2. (A.5) (A.6) Table A.2: Values of (\Fm„m((3,7)\2) . The values are independent of the signs of m" and m, and of permutation of the indices. They satisfy the sum rules £ (|euA7)f> = E ( | ^ » m ( / 5 , 7 ) | 2 ) =1-m"=-2 m=-2 (m", m) (0,0) j { l - f f t ( M ) + gft(/*)} (0,1) l { l _ i ^ ) - A p 4 ( „ ) } (0,2) i { l + fft0*) + iP4(M)} (1,1) Ap 2 (^+fP 4 ( / i ) } (2,2) i { l -Sp a (A*) + I f 5 P 4 0i)} (1,2) I { l + A P 2 ( M ) - 4 P 4 ( M ) } In order to separate the influences of the two types of motion, we express the mean squared fluctuations of 1^(0, <t>) as a sum of the mean squared fluctuations of the two types of motions: - (r 2 r a(M)> „ '2 T,8",4>" = G ( l ) + G < 2 > (A.7) Appendix A. Orientation Dependence of 2H NMR Relaxation 114 where (A.8) (A.9) The justification for this decomposition is that HMB consists of six C H 3 groups at angles <y(p) = p(2£), where p = 1,2,... ,6. The experimental relaxation rates even in the absence of any fluctuations in 7 are averages over all C H 3 groups. This is equivalent, for a second rank tensor, to averaging over a uniform distribution of values of 7. Using Equations A . l to A.9 and Tables A . l and A.2 we write the expressions necessary to obtain Gm : Tfi",4>" 2 = ^ S\ fm(l3), (A.10) where MP) = 5 ' 2 { l + yP 2(/x) + yP4(M)} !±m = S'2 {l + - yP 4(M)} 10 „ , , 3 (A.ll) f*M = s'2{i-jP2(ri) + ^p4(ri)} and S'2 = \ for the C 6 motion in HMB. The use of S' in Equations A . l l allows us to generalize the theory to other types of motions. Similarly, where (|<5M^ )W-f) = ^ Si hm(/3), hotf) = l-yP 2(M) + |^(M) h±2(P) = l + yP2(/x) + A p 4 ( / i ) (A.12) (A.13) Therefore, using Equations A.8 to A.13, g $ = ^ si (hm - /m), (A.14) Appendix A. Orientation Dependence of 2H NMR Relaxation 115 where ho-fo = ( l - 5 ' 2 ) - ^ ( l + 25'2)P2(/i) + | ( l - ^ ' 2 ) P 4 ( M ) * ± i - / ± i = ( l - 5 ' 2 ) - ^ ( l + 2 5 ' 2 ) P 2 ( / i ) - ^ ( l - ^ 5 ' 2 ) P 4 ( M ) (A.15) h±2-f±2 = ( l - 5 ' 2 ) + y ( l + 25'2)P2(M) + ^ ( l - ^ 5 ' 2 ) P 4 ( M ) The evaluation of Gm' requires the use of the following equation which follows from Equa-tions A.1-A.4 (Y2m(0,<j>)2) = £ l\Fmm„{M)2) (Y2m.,(0",<t>")2) Using Equation A.16 and Tables A . l and A.2, it may be shown that ( | y 2 m ( M ) 2 \ =^-Hm{(i) (A.16) (A.17) where Ho({3) = 1 - ^ S 2 P 2 ( M ) + | S 4 P 4 ( M ) H±1(/3) = 1 - ^5 2P 2(/x) - ^5 4 P 4 ( / i ) H±2(P) = l + ^ 2 P 2 ( M ) + ^5 4 P 4 ( / i ) It follows from Equations A.8, A.12 and A.17 that (A.18) Gfi> = ^ [Hm((3) - S22 hm((3)}, (A.19) where, from Equations A.13 and A.18, 27 H0(/3)-S2h0((3) = ( l - 5 2 ) - - ( 5 2 - 5 2 2 ) P 2 ( ^ ) + - ( 5 4 - ^ ) P 4 ( M ) H±1(P)-S2h0((3) = ( i _ 5 2 2 ) - ^ ( 5 2 - 5 2 ) P 2 ( / x ) - ^ ( 5 4 - 5 2 ) P 4 ( / i ) (A.20) H±2(P)-S2h0(l3) = (l-S2) + ^(S2-S2)P2(ri) + ^(S4-S2)P4(fi) We can now summarize in compact form the influence of anisotropic motions on relaxation in HMB due to quadrupolar interactions. Defining reduced spectral densities associated with Appendix A. Orientation Dependence of 2H NMR Relaxation 116 motions 1 and 2 by ji(u>) and j2(u), respectively, where for exponential correlation functions (A.21) 3a = l + w 2 r 2 the generalized expressions for relaxation rates given by Equations 1.32 to 1.37 are obtained by replacing the spectral densities j(mwo) by their generalized expressions as follows j(mw0) — \Hm{L3) - S22 hm((3)] ji(mw0) + S22 [hm(p) - fm((3)} J2(mw0). (A.22) In order to interpret the average relaxation for a "powder", it is useful to calculate the average spectral density over all orientations for a random distribution of orientations. Since (Pi(n))pow(ier = 0, this result is independent of the index m, i.e. (j(mu0)) powder ^ [Hm(P) - S2 hm(P)] } ji(mo;o) + (s2 [hm{fl) - /m(/?)] ^ j2(mu0) = ( l - ^ ) j i M + ^ ( l - S 2 2 ) j 2 H (A.23) A.5 Interpretation of the Experimental Results In Section 5.2, it was shown that the ratio of the orientationally averaged 2 H NMR relaxation rates T\z~x and T i 0 - 1 in HMB solid at room temperature is T\q/T\z = 1.13. Assuming that the relaxation is due to a single type of motion characterized by exponential correlation functions, Equation 5.2 gives UQTC « 0.54. For a Larmor frequency of 46 MHz, this corresponds to a value of rc = 1.9 X 10 _ 9s, which is close to the value r c = r 2 anticipated for reorientation about the C6 axis of HMB on the basis of previous studies [73,74,75]. Since at room temperature T\ < 10 _ 3 r 2 [75], we may conclude with confidence that spin-lattice relaxation in HMB-dis is dominated by the C6 rotation. Using Equation 1.32 and the theory of the previous section, we obtain t z = JI^252 {lhi{(3) - h { u o ) + 4 - / 2 ( / ? ) l h { 2 u J o ) } ( A - 2 4 ) where 5 2 = | for C H 3 rotation about its symmetry axis. The quantities hm(/3) — fm{P) which contain information on the angular dependence of the spectral densities are given by Appendix A. Orientation Dependence of 2 if NMR Relaxation 117 Equations A. 15 for S'2 = 7 to be ho - fo h±i - f±i = -4 h±2 - f±2 = 7 4 1D 1 l - y W + yiM/*) l - f p a O O - f w 1 + y + P4(M) 45 32 2\2 ( 1 - ^ ) = ^ ( 1 + 6^ + /^) (A.25) Substitution of Equations A.25 into A.24 gives 1 _1 7F~ = ^ wJi2(wo)5| where J2(2u;o) _ 1 + u^TJ J2M 1 + 4o;2r22 r 2 = (A.26) (A.27) This result is in agreement with that obtained previously by Tang et al. [74]. In the short correlation time limit, o>Qr| <C 1, r2 = 1, and Equation A.26 gives « ^ 2 r 2 < ? 2 [ l + P 2 ( M ) ] . (A.28) For HMB at room temperature, we put U0T2 = 0.54 and S2 = |£ 2 . The coefficient £ is used to fit P 1 2 - 1 to the average relaxation rate. It takes into account that the experimental value of the 2 H NMR order parameter in HMB at room temperature is about 20% lower than the value of S = g predicted by the superposition of the C 3 and C6 rotational motions; i.e. we consider a value of £ in the range 0.8 < £ < 1.0 to represent a satisfactory fit for the average relaxation rate. Substitution of these values into Equation A.26 gives = 3 4 £ 2 [l + 0.79P2(/0 - 0.03P4(/*)] (A.29) The average relaxation rate of (53 ms) - 1 gives 34f2 = Tlz~r = (0.053)-1 and, hence, £ = 0.74. Considering the approximations made in modelling the molecular reorientation of HMB, this represents excellent agreement of the model with experiment. Appendix A. Orientation Dependence of 2 Jf NMR Relaxation 118 Table A.3: Comparison of the experimental values of T\z(ft) with the theoretical predictions. The experimental value for the window near the edge is a super-position of the contributions from two orientations, ft = 90° and ft = 32.26°. Spectral window Experimental T\z [ms] Predicted T\z [ms] shoulder, ft w 5° - 10° 30.7 7.5° 30.0 center, 0 « 45° - 50° 45.7 47.5° 46.2 edge, ft « 90°, 32.26° 70.1 90° 88.0 32.26° 38.0 Using this value of £, we can compare the observed dependence of T\z on orientation with the predictions of Equation A.29. This is shown in Table A.3. Similarly, we can examine the orientation dependence of T\q. Using Equations 1.34, A.22 and A.25, we obtain 2 •19 i ^qer2 hi(f3)-fi(P) 30 1 + wgr2 16.56 i - j W - - PM 20.7 ( 1 - M 4 ) (A.30) The angular dependence of Tiq, unlike that of T\z, is predicted to have a maximum at ft = 0° and a minimum at ft = 90°. As may be seen from Table A.4, the measured angular variation of T\q is in agreement with this prediction. It is clear that an additional relaxation mechanism must be invoked to explain quantitatively the value of T\q near ft = 0° where Equation A.30 predicts that T i , - 1 —• 0. endix A. Orientation Dependence of 2H NMR Relaxation Table A.4: Comparison of the experimental values of T\q{f3) with the theoretical predictions. The experimental value for the window near the edge is a super-position of the contributions from two orientations, (3 = 90° and 13 = 32.26°. Spectral window Experimental T\q [ms] P Predicted T\q [ms] shoulder, (3 « 5° - 10° 128.7 7.5° 1420 center, (3 « 45° - 50° 55.4 47.5° 61.0 edge, (3 « 90°, 32.26° 41.9 90° 48.2 32.26° 98.9
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Some mechanisms of transverse nuclear magnetic relaxation in model membranes
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