Let us now consider defined as
I Di,jl, the determinant of the matrix
I
(‘414)
+
N (-l)‘kj+ujid,kl
Di,,jkl
(along “jth row”) (A15a)
Dik,jjl
(down “ith cohmn”)
k= 1
(kfi)
and in the second
ID.‘.J.I =
N
1 (-
1 )‘ki+uUdkil
k= 1 (kzi)
(A15b)
Now from the symmetry symmetries
References (1) (a) Camille and Henry Dreyfus Foundation Fellow; author to whom corre-
W e shall evaluate this determinant in two ways, once by expansion along the “jth row”, then again down the “ith column”; by ‘7th row” we mean the row whose elements are dj!, dj2. . . . , though this will actually occur in row j 1 of D ~ ifJ j > i, and similar comments apply to the “ith column”. In the first case we get =
= AjjJDjl
which, in view of (A7), is the desired result.
matrix formed from D by striking Di,j = row i and column j
IDi,jl
Aijl Djl
(dij
=
dji)
ID.., 1J Jk I = I Djk.ijl = I Dkj.ijl9 ’
of D and the obvious
I Dikjil = I Dkijil
(A 16)
of the definition (AlO), one can recognize from (A13a) and (A15a) that AijlDj( = (-1)uij+uji+l
IDi,jl
(A17a)
I DijI
(A17b)
and from (A13b) and (A15b) that Ajil Di) = (-l)u,i+Oij+l
from which it follows that
spondence should be addressed at Theoretical Chemistry Institute. University of Wisonsin, Madison, Wis. 53706:(b) National Science Foundation Predoctoral Fellow. (2)L. C. Pauling, J. Am. Chem. SOC., 53, 1367 (1931);“The Nature of the Chemical Bond”. Cornell University Press, Ithaca, N.Y., 1960,p 108 see also J. C. Slater, Phys. Rev., 37,481 (1931);38,325, 1109 (1931); R. S.Mulllken, ibid., 41,49 (1932); J. N. Murrell, J. Chem. Phys., 32,767 (1960);T. L. Glibertand P. G. Lykos, ibid., 34,2199 (1961). (3)See, e.g., C. A. Coulson, “Valence”, Oxford University Press, New York, N.Y., 1961,p 76;A. Streitweiser. “Molecular Orbital Theory for Organic Chemists”, Wiley, New York. N.Y., 1961,p 13. (4)See, e.g., M. Wolfsberg and L. Helmholtz, J. Chem. Phys., 20,837(1952); R. Hoffmann, ibid., 39, 1397 (1963); L. L. Lohr, Jr., and W. N. Lipscomb, ibid., 38, 1607 (1963);J. Jordan, H. W. Smith, L. L. Lohr. Jr., and W. N. C. J. Ballhausen and H. B. Lipscomb, J. Am. Chem. SOC.,85,846(1963); Gray, Jnorg. Chem., 1, 1 1 1 (1962). (5)For simplicity, overlap is neglected in the secular determinant, as is common in simple valence theory. (6) See, e.g., R . Hoffmann, Acc. Chem. Res., 4, l(1971). J. K. L. MacDo(7)E. A. Hylleraas and B. Undheim, Z.Phys., 85,759 (1930); nald. Phys. Rev., 43,830 (1933). (8) Of course, such a statement implies a certain phase convention, but no set of orbital sign changes can disguise the uniqueness of the orbital of “odd” overlap. (9)T. K. Brunck and F. Weinhold, J. Am. Chem. Soc.. submitted for publication. (IO)The detailed computational procedure for obtaining MO’s in the LCBO framework (e.g., the construction of unitary matrices relating AO’s to BO’s. and subsequent transformation of the SCF equations into the BO basis) will be described elsewhere. In the present application, idealized geometries and orbital hybridizations were employed throughout, and contributions from antlbonding BO’s were ignored (as suggested by ref 9), but the qualitative conclusions should be insensitive to these specific choices. (1 1) See, e.g., J. A. Pople and D. L. Beveridge, “Approximate Molecular Orbital Theory”, McGraw-Hill, New York, N.Y.. 1970. (12)Professor R. Hoffmann has kindly indicated (private communication) how the overlap phase effects can be derived in higher orders of perturbation theory. Perturbative expansions of ci2/c,2can also be developed directly from eq 10 (J. Diamond, private communication), showing that the phase effects contribute in 3d, 5th, and successively higher orders.
19FNuclear Magnetic Resonance Studies of Lipid Bilayer Systems. 1 M. P. N. Gent, I. M. Armitage,’ and J. H. Prestegard* Contributionfrom the Department of Chemistry and from the Section of Physical Sciences of the Medical School, Yale University, New Haven, Connecticut 06520. Received J u n e 28, 1975
Abstract: A fluorinated lipid, l-palmitoyl-2-8,8-difluoropalmitoyl-sn-glycero-3-phosphorylcholine, has been synthesized and the dynamic properties of lipid bilayer systems containing this molecule have been studied using fluorine- 19 N M R . Spin-lattice relaxation rates and nuclear Overhauser effects have been measured over a range of temperatures and the results have been interpreted in terms of correlation times for specific motions involving the gem-difluoromethylene group. The correlation times are shown to be consistent with ‘H and I3Crelaxation data of similar lipid bilayer systems. The data, however, prove to be particularly valuable in characterizing a motion on the time scale of translational diffusion.
Introduction In recent years much effort has been devoted to the characterization of the hydrocarbon chain mobility of phospholipids in bilayer membranes. The subject is of interest because the bilayer is an indigenous component of virtually all biological membranes.2 The phospholipid motions are intimately linked to the activity of proteins and the transport of metabolites within and through the membrane.2 Characterization of the
anisotropic motion in the liquid crystalline lipid bilayer has also proven to be a challenging problem in physical chemistry. EPR nitroxide spin-label ~ t u d i e s and ~ - ~nuclear magnetic resonance relaxation s t ~ d i e s ~have - ’ ~ yielded valuable results concerning the segmental rotational motions and translational diffusion of the phospholipids, but both methods have been subjected to some criticism. Spin labels can perturb the structure of the bilayer. This is shown by the fact that in identical lipid bilayer systems, even after correction for time
Gent, Armitage, Prestegard
1
19FN M R Studies of Lipid Bilayer Systems
3750 scale differences, the order parameter for a nitroxide probe, dipole-dipole mechanism, there are many possible interactions measured by EPR, is different than that for the C-D bond, that could contribute to relaxation. These include dipolar inmeasured by deuterium NMR." In addition, the methylene teractions with the geminal nucleus, with vicinal and other resonance in 'H and I3C N M R does not correspond to a single intrachain nuclei, and with nuclei on adjacent chains. There defined methylene group, but to an envelope of resonances is a J ( W H - W F ) term in the H-F spin-lattice relaxation from different methylenes with a distribution of correlation equation that is not present in either the H-H or F-F relaxatimes.'* Thus, the N M R relaxation cannot be interpreted in tion equations. This term indicates that dipolar interactions terms of the motion and magnetic interactions of one specific modulated by slow motions, such as diffusion motions with T~ nucleus. Because of the small contribution to 'H relaxation = 5 X lo-* s, will be much more efficient in causing I9F rerates from diffusional motions, and because of experimental laxation than 'H relaxation. problems with the deuterium dilution technique used to reIn order to gain information about the interactions that solve this contribution, the diffusion constants measured by cause relaxation, and thus to determine how much of each of IH N M R have low accuracy. the above considerations affect the results, several experiments I9FN M R relaxation experiments can be designed to avoid can be done. These include studies of the H-F nuclear Ovsome of the deficiences of the techniques mentioned above. erhauser enhancement, (NOE) (7 l ) , of the temperature First, fluorine can be inserted into a specific position of the dependence of relaxation, and of the effect of deuterium phospholipid so that the motion of a single defined nucleus is dilution. represented in the relaxation behavior. Second, experiments It is possible to experimentally separate the H-F relaxation can be designed to be sensitive to specific types of motion, such due to intermethylene and interchain dipolar interactions from as translational diffusion. Finally the CF2 group is similar to contributions due to F-F geminal interactions by examination the CH2 group in terms of its size, geometry, and physical of the H-F N O E . The N O E is a measure of the increase in c h a r a c t e r i ~ t i c s . l ~Thus ~ ' ~ fluorine containing lipids will not intensity of the fluorine resonance due to saturation of the greatly perturb the structure of the bilayer membrane. Because proton resonances.16 It reaches a maximum value, the magof these advantages we have studied the I9FN M R relaxation nitude of which depends on the correlation time of the interof I-palmitoyl-2-8,8-difluoropalmitoyl-sn-glycero-3- action causing r e l a ~ a t i o n ,if' ~all the fluorine relaxation is due phosphorylcholine (8,8F-PC) incorporated into lipid bilayer to H-F dipolar interactions. At a resonance frequency of 84 membranes. M H z for fluorine, it can be calculated that the H-F N O E maximum varies from 1.532for T~ 1 X s.17The equation to predict the N O E if there all the relaxation mechanisms or the contributions to relaxation are several contributions to relaxation is from different magnetic interactions must be resolved. Because N O E -1 = C (NOEmax -1)(R1i/Rl total) (1) of the complexity of molecular motion, the results are not easy i to interpret. Previous I9F relaxation studies of amphipathic where Rli is the spin-lattice relaxation contribution due to the molecules in bilayer membranes assumed that the similar ith H-F interaction modulated a t a rate described by rei. If gyromagnetic ratios of I9F and 'H nuclei and the similar the correlation time for I9F spin lattice relaxation is faster than structures of the CF2 and CH2 groups should lead to the same s, which holds true for ' H and I3C relaxation; then the relaxation mechanisms for ' H and I9F n u ~ l e i . ' ~ In . ' ~fact, amount of relaxation due to H-F interactions is given by similar relaxation rates and activation energies are observed for ' H and I9Fnuclei incorporated into the methylenes of hyobserved N O E - 1 Ci Rli = R I total (2) drocarbon chains in membra ne^.'^,'^ However, a careful 0.532 consideration of the contributions to 19F relaxation shows that If the correlation times are slower than 1O-Io s, then the septhe relation between IH and I9F relaxation is more complicated aration of relaxation contributions is not straight forward. than it appears to be. However, the calculation of the N O E using eq 1 is a useful Proton N M R studies of phospholipids in lipid bilayer experimental check of any postulated set of relaxation conmembranes show that the relaxation is due to dipole-dipole tributions. interactions modulated by molecular motions. Interactions The temperature dependence of the spin-lattice relaxation with the geminal-bonded hydrogen make a large contribuand of the N O E can be used to characterize the motions tion,'.I0 but intermolecular effects account for 20-40% of the causing relaxation. If T~ < s then the N O E will increase r e l a ~ a t i o nThe . ~ temperature dependence of the spin-lattice or the R1 will decrease with temperature, but for T~ > s relaxation rate, R1, indicates a correlation time faster than R1 will increase with temperature. The activation energy exs . However ~ R1 # R2, so this fast motion must be antracted from the temperature dependence is also useful in deisotropic. The spin-spin relaxation, R2, is due to slower isotermining which molecular motions cause relaxation. tropic averaging of the residual dipolar interactions.I0 That part of the H-F spin-lattice relaxation due to difFor several reasons the I9F relaxation could be due to very fusional motions can be measured by diluting the fluorine ladifferent interactions than those that govern 'H relaxation. The beled lipids with lipids having highly deuterated hydrocarbon mean squared value of the geminal-bonded dipolar interaction chains. Since the F-D dipolar interactions are small due to the is a factor of four smaller for the CF2 group than for the CH2 group. This difference is due to the longer C-F bond, 1.35 8, small magnetic moment of deuterium, the measured R1 will decrease in proportion to the interchain contribution to R1 .9 relative to the C-H bond, 1. I A, and the slightly smaller fluOnce the contributions to relaxation have been separated orine magnetogyric ratio. There are minor chemical differences by the methods described above, the rate of molecular motions between protons and fluorine that could change the rate of can be calculated. Equations have been developed to relate the motion of the CF2 group and cause more efficient relaxation. relaxation rate to the magnitude of magnetic interactions and Fluorine-induced changes in the C-C bonds and the steric the correlation time for the motions that modulate the interhindrance of the slightly larger CF2 group compared to the actions.Is These equations can be expressed as CH2 group could affect the motion of the methylene. Magnetic field fluctuations due to chemical shift anisotropy and spinrotation interactions are much larger for fluorine than for protons and could make a large contribution to relaxation. Even if the 19F relaxation is only due to the nuclear magnetic
+
Journal of the American Chemical Society
/
98.1 3
/ June 23, 1976
3751 fluorinated using Fluoreze-M, the MoF6 reagent supplied by P.C.R., Gainsville, Fla., according to the method of Mathey and Bensoam.21 A crude product, distilled from the reaction mixture a t
