Jan 27, 1992 - high-resolution electron microscopy (Dorset, Pang- born & Hancock, 1983 .... York State Department of Health in Albany, New. York (Fig. lb).
778
J. AppL Cryst. (1993).
26, 778-786
Electron Diffraction from Phospholipids- an Approximate Correction for Dynamical Scattering and Tests for a Correct Phase Determination By D. L. DORSET AND M. P. McCOURT
Electron Diffraction Department, Medical Foundation of Buffalo, Inc., 73 High Street, Buffalo, N Y 14203-1196, USA AND W. F. TIVOL AND J. N. TURNER
Wadsworth Center for Laboratories and Research, New York State Department of Health and School of Public Health, The University at Albany, Box 509, Albany, N Y 12201-0509, USA (Received 27 January
1992;
accepted 27 May
1993)
Abstract
suggested by X-ray crystal structures of near analogs An approximate experimental correction of electron were used to construct molecular conformers. These, diffraction intensities from an epitaxically crystallized in turn, were used in a translational search for an phospholipid bilayer for dynamical scattering is de- R-factor minimum (Dorset, Massalski & Fryer, 1987). scribed. This correction, which is useful for certain A similar analysis of X-ray data from a phosphatidyllow-angle centrosymmetric data sets, compares in- ethanolamine has also been reported (Hitchcock, tensities recorded at high and low electron-accelera- Mason & Shipley, 1975). The problem with such model-based structure ting voltages to ascertain which reflections are most analyses is twofold. First, it is not known whether the affected by n-beam interactions. When applied to assumption of an unchanged molecular conformation experimental intensity data from 1,2-dihexadecyl-sncan be made when polymethylene chain lengths are glycerophosphoethanolamine (DHPE), the correction altered or, more importantly, when chain linkages to facilitates a direct phase determination based on the glycerol are changed. Second, if the conformational probabilistic estimate of three-phase invariants bemodel is valid, it is not known that a translational cause a more accurate estimate of the hierarchy of IE~l search will find a unique solution, especially since the values is obtained. When a multisolution technique is statistical significance of the crystallographic R factor used, incorporating algebraic unknowns for certain is poor when the number of measured diffraction data phase values, the best phase assignment can be asis small (Hamilton, 1964). sessed by comparison of the single convolution of After these uncertainties were made clear, it was phased structure factors to the observed structurehoped that a more direct means of crystallographic factor magnitudes for the low-voltage data. This apphase determination could be found, which would be proach exploits an approximate analogy made earlier independent of any assumed molecular model. by Moodie between the Sayre equation and the phase grating series and is valid as long as the single convo- High-resolution (6/~) electron micrographs were then lution adequately models experimental low-voltage obtained from the phospholipid 1,2-dihexadecyl-snglycerophosphoethanolamine (DHPE) (Dorset, Beckdata (a condition favored by light-atom structures in mann & Zemlin, 1990). The crystallographic phases a low-angle region of reciprocal space). In real space, found after image analysis (Fourier peak filtration) the correct structure can also be readily identified as seemed to match those obtained from the earlier the one having the smoothest density profile. search with a molecular model (Dorset, Massalski & Fryer, 1987). Furthermore, direct methods, based on Introduction the evaluation of three-phase structure invariants Numerous polymethylene compounds, including (Hauptman, 1972), could be used to extend the image phospholipids, can be epitaxically oriented on organic phases to the 3.4/~ limit of the electron-diffraction substrates to permit study by electron diffraction and pattern (Dorset, Beckmann & Zemlin, 1990). Alternahigh-resolution electron microscopy (Dorset, Pang- tively, direct phasing could be used without the image born & Hancock, 1983; Fryer & Dorset, 1987). information to find values for 13 of 16 reflections for Electron-diffraction intensity data have been collected the D H P E structure, with similar success in the from phospholipids with different headgroups in order analysis of other epitaxically oriented phospholipids to determine their layer structures. Initially, models (Dorset, 1990a). © 1993 International Union of Crystallography Journal of Applied Crystallography Printed in Great Britain - all rights reserved
ISSN 0021-8898
© 1993
D. L. DORSET, M. P. McCOURT, W. F. TIVOL AND J. N. TURNER
If no electron-microscope images were available, then complete phase sets could be obtained from the partial determination by real-space refinement based on the density-flattening technique used in protein crystallography (Wang, 1985). Figures of merit, evaluating the density profiles of the potential maps, were also proposed to identify the most likely phase solution (Dorset, 1991a). Although this approach seemed to work reasonably well for simulated lipid bilayer diffraction data and also experimental X-ray data from phospholipids (Dorset, 1991a,b), problems were experienced in the analysis of the electrondiffraction data, apparently caused by some sort of multiple-scattering perturbation to the intensities (Dorset, 1991b). Initially, the perturbation of intensities was ascribed to secondary scattering (Dorset, 1990a), seemingly supported by a correction based on a weighted sum of convolved intensities (Cowley, Rees & Spink, 1951). However, this paper demonstrates that n-beam dynamical scattering is a more probable cause. An approximate experimental correction to the observed data for this dynamical scattering is proposed, furthermore, which can also be used to verify the correctness of the phase assignment.
779
Before the analysis of experimental data could be carried out, it was necessary to evaluate simulated diffraction data from a model phospholipid lamella,
viz 1,2-dimyristoyl-rac-glycerophosphoethanolamine (DL-DMPE). This is to ascertain whether the proposed correction is applicable to the experimental problem. As described in an earlier paper (Dorset, 1988), the crystal structure of the dilauroyl homolog, determined as its acetic acid solvate (Hitchcock, Mason, Thomas & Shipley, 1974), was lengthened by two methylene units and the solvate molecule was removed. As shown by Hitchcock, Mason & Shipley (1975) for X-ray data, this conformational model could then be translated to a unit-cell position where an acceptable model for the experimental intensities could be found. Monoclinic unit-cell constants are a = 9.95, b = 7.77, c = 50.25 A and fl = 92.0°; the space group is P2~/a. The fractional phosphorus position is at z = 0.042 and the outermost chain carbon at z = 0.505. Thermal parameters were adjusted so that the phosphorus was assigned Bp = 3.0 A 2, with a graduation of other values dependent upon the atomic z position, so that, for the outermost carbon, Bc = 26.3 A 2. From the original model before the indicated translation, B = (4.0 + 108z 2) A 2, where
Materials and methods Lath-like crystals of 1,2-dihexadecyl-sn-glycerophosphoethanolamine were grown by epitaxic orientation on the (001) face of naphthalene as described previously (Dorset, 1990b) and the naphthalene was subsequently removed by sublimation in vacuo. Selected-area electron-diffraction patterns from these crystals were obtained at 40 and 100 kV on JEOL JEM-100B7 or JEM-100CXII electron microscopes (Fig. la). A few patterns were also obtained at 1000 kV with the AEI EM-7 high-voltage electron microscope at the Biological Microscopy and Image Reconstruction Resource at the Wadsworth Center of the New York State Department of Health in Albany, New York (Fig. lb). Radiation exposure of the specimen was minimized by control of the electron-beam illumination and the use of suitably sensitive photographic films (Kodak DEF-5, Dupont Cronex). Intensities were measured from scans of the electron-diffraction films on a Joyce-Loebl MklIIC flat-bed microdensitometer by integrating the peak areas. Because of the curvilinear distortion to the lipid lamellae, causing arcing of the diffraction peaks on the film, a phenomenological Lorentz correction was applied for the calculation of observed structurefactor magnitudes: IFo(00/)l = {[_Io(00l)I]}1/2, where I is the order of the lamellar reflection.
(a)
(b) Fig. 1. Electron diffraction patterns from epitaxically oriented DHPE lamella¢. (a) 40 kV, (b) 1000 kV. Comparing the relative intensities of these lamellar patterns, one can readily see that the 002 and 004 reflections have markedly higher values for low-voltage electron diffraction patterns.
780
ELECTRON DIFFRACTION FROM PHOSPHOLIPIDS
z represents the fractional coordinates of nonphosphorus atoms along the molecular length. For the structure-factor calculation, used in subsequent n-beam dynamical-scattering computations by the multislice method (Cowley, 1981), the model crystal structure was projected down a [1, 10,0] zone axis so that only 00l reflections were excited to 3 ,~ resolution. This models experimental electron-diffraction data (e.g. Fig. 1) where true hOl patterns are never observed owing to the paracrystalline disorder (Dorset, Massalski & Fryer, 1987). Nevertheless, the projected repeat is still near b ~- 7.77 ,~ but only the 001 reflections should be considered for dynamicalscattering contributions.
Correction for dynamical scattering In the initial appraisal of which multiple-scattering effect was most responsible for affecting observed electron-diffraction intensities from DHPE, n-beam dynamical scattering had been rejected because the conformational model used to determine the layer structure did not allow convergence with the experimental data (Dorset, 1990a). This points to a basic problem with making corrections for n-beam dynamical scattering, since one ordinarily needs to know the crystal structure before it is determined to match the deviation from kinematical diffraction. Hence, if the actual layer structure were different from the assumed conformational model, there may be little chance in identifying this difference to enable this correction to be made. That is to say, in the worst case, it may be impossible to construct a correct model a p r i o r i for a completely unknown structure. Fortunately, it has been found recently, in our analysis of a paraffin structure (Dorset, 1992), that some estimate of the intensities most changed by dynamical electron scattering (Dorset, 1976a) can, in some special cases, be made experimentally, providing that the phase-grating approximation is a valid scattering model for all accelerating voltages. This assumption is particularly appropriate for the low-angle scattering region considered in this paper. As discussed in a previous paper (Dorset, 1992), neither the multislice model nor the phase-grating approximation are strictly accurate for simulating dynamical scattering in typical selected-area diffraction experiments because the former assumes a fiat crystal and the latter a 'flat' Ewald sphere. With experimental electron-diffraction data from monolamellar n-paraffin crystals taken at various voltages (Dorset, 1976b), it has been demonstrated that the resolution limit of the diffraction patterns at low voltage (e.g. 40kV) is n o t determined by the Ewald-sphere curvature (Dorset, 1992). This is because the rather large crystal area (ca 10~tm diameter) illuminated to obtain typical electron-
diffraction patterns is elastically bent (Dorset, 1980). In a suitable low-angle region, both multislice and phase-grating calculations for dynamical scattering adequately model the experimental intensities but, especially at low voltage, only the phase-grating expression will produce a reasonable match to the experimental data at the observed resolution. This is not to say, however, that the phase-grating calculation provides a rigorous explanation of the observed diffraction data; a rigorous calculation would require an average of multislice calculations over a number of crystal orientations dictated by the elastic bend deformation, as described by Turner & Cowley (1969), and such a calculation would be rather tedious and time consuming. We assumed, therefore, the phasegrating model to be suitable for our purposes so long as the most significant changes to the diffraction intensities occur at a suitably low angle. In our development below of an empirical method for detecting which intensities are most changed by dynamical scattering, we use the phase-grating expression merely to describe the f o r m of the correction. We wish to find which reflections are most changed and by what approximate amount so that we can make a correction based on the subtraction of an appropriately weighted term. For the phase-grating expression, the exit electron wave function from the crystal is written (Cowley, 1981) q(x,
y) = exp [--itr'qg(x, y)],
(1)
where the interaction term t r ' = (2re/E2)[1 + (1 - f 1 2 ) l / 2 ] - l ( 4 7 . 8 7 / f 2 ) n t includes the electron accelerating voltage E, the electron wavelength 2 with a relativistic correction and the number n of unit slices through the crystal, each of thickness t. The third term, including the unitcell volume f2, converts structure-factor values Fhk t to potentials Vhu. The Fourier transform of (1) includes imaginary terms Im (/) = F l -- (tr'E/3!)Ft, F l * f ~ + ' " ,
(2)
where Fl is the kinematical structure factor and • denotes convolution. The series also includes real terms Re (/) -- (cr'/2!)Fl * Fl -- (tr'a/4!)Fl * Fl * El * Fl + "'" •
(3t It is important, moreover, that the projected structure is centrosymmetric so that F l is real [see Cowley & Moodie (1959) for a discussion of the noncentrosymmetric case] to preserve the identity of Friedel-related intensities. Convergence of these series depends on the magnitude of tr', which, in turn, depends on the value
D. L. DORSET, M. P. M c C O U R T , W. F. TIVOL A N D J. N. T U R N E R
781
Table 1. Data sets f o r D H P E
of E and/or nt. Thus, if the electron accelerating voltage is high or the crystal thickness is small, the numerically dominant term for the two series is Ft. If, on the other hand, one obtains data at some suitable lower voltage, it may be possible that the first term of Re (/) will be dominant, as pointed out by Cowley (1981) and, in addition, the convolution term Ft * Ft * Ft will dominate the imaginary series. If, as is found experimentally for some light-atom structures (Dorset, 1992), Ft * Fz "~ Ft * Ft * Ft is also approximately true, then these measured low-voltage structure-factor magnitudes may be closer in value to the single convolution of kinematical structure factors than the kinematical structure-factor amplitudes themselves. As we show later, for structures where it is valid, this approximation can be exploited further for structure identification. A correction for dynamical scattering, therefore, would depend on one's ability to measure experimentally adequate approximations for the two terms in the expression
Making appropriate substitutions to (4) as described, we seek improved structure factors by assuming
IF~Y"I = [Re (/)2 + Im (02] 1/2
IFki"l "-~ [(FtnV)2 -- mtFLV)2] 1;2,
(4)
We postulate, first of all, that there is a very high voltage where I m ( / ) = F ki", i.e. the phase-object approximation is rigorously true. Suppose, at some other measurable high voltage, we find IFHVl = IF dy"I, the data set we wish to correct. Suppose also, at some lower voltage, we can detect something approximating m l F L V l - IRe (I)1 -~ let * Ftl. (5) How can we justify this statement? Obviously, at some value of a', which can be adjusted experimentally, we know Re (/) _~ (a'/2!)F • F,
(6)
but this is not the same thing as saying IF Lv] ~IRe(/)]. We also rely on the dominant term of the other series to be Im (/) - (ty'2/3!)F • F • F.
(7)
Again, as stated above, if Ft * Ft ~- kFz * Fl * Ft is also approximately true, as it seems to be for some light-atom structures (Dorset, 1992), at an appropriate a' value, then m lFLV[ ~ [Ft* Ft[ should be a useful approximation so long as we do not require high accuracy. That is to say, IF~Vl = [Re (/)z + Im (/)z]t/2 [(aF t * Fl) 2 + (bFt * F t * Ft)2] 1/2 ~- {(aF t • Ft) 2 + [(b/k)Ft * Ft] 2} 1/2
= {[a 2 + (b2/kZ)](Fl, Fz)2}1/2 (1/w)lFt *
F,I.
(Here, it is apparent that a
= tr'2/2!,
b
= tr'3/3!.)
I 1 2
3 4
5 6 7
8 9
10 11 12 13 14
15 16
HV
IFootl 1.49 0.62 1.42 0.82 1.00 0.52 0.80 0.40 0.31 1.02 1.04 1.80 1.55 1.84 1.27 1.06
LV
IFoo~l 1.69 1.29 1.39 1.10 0.89 0.64 0.82 0.53 0.58 0.87 1.05 1.29 1.41 1.44 1.12 0.87
FTkin
-oot 1.33 0.34 1.30 0.61 0.93 0.45 0.73 0.34 0.21 0.96 0.95 1.72 1.45 1.75 1.19 1.00
(8)
where m = w 2 is a refinable parameter. The values of the low-voltage intensities weighted by m will, therefore, determine which reflections are most perturbed by dynamical scattering. In the earlier study of an empirical dynamical correction for n-paraffins (Dorset, 1992), it was found that the first term of the series in (3) was indeed an adequate approximation for experimental low-voltage data, a proposition which needs to be evaluated here for a phospholipid. The derivation used to arrive at (8), on the other hand, is recognized to be somewhat overly simplistic in assuming that Im (l) and Re (/) correspond only to single respective terms in each series. Even the approximation that kl IIm (I)1 = k2lRe (I)[ = lEt * Ft[ holds only under certain special conditions. Nevertheless, for the types of problems analyzed, (5) is found empirically to be a useful approximation, as is justified below with a model calculation for the type of structure considered in this paper. In general, it should be emphatically stressed that this methodology is not generally applicable to all structural problems but seems to be valid for predominantly light-atom molecules packing in centrosymmetric projections and especially in the low-angle scattering region. Independent tests (Dorset, unpublished) indicate its utility for correcting data from polymers, such as polyethylene sulfide, but not for compounds with many heavy atoms, such as copper perchlorophthalocyanine. For epitaxically oriented DHPE, experimental intensity data obtained from a suitably thin crystal at 100 kV were found to match data sets obtained at 1000 kV, i.e. the values IF~Vl listed in Table 1. (The high-voltage experiment, therefore, demonstrates that
782
ELECTRON DIFFRACTION FROM PHOSPHOLIPIDS
T a b l e 2. Correction o f model structure factor values from DL-DMPE for dynamical scattering using (2); t = 200 ,~, m = 0.16 1
IF~°°°kVl
IF~°kVI
IH°"I
Fkin
71.27 27.35 10.78 28.94 8.73 12.26 9.20 12.15 24.29 22.92 37.79 29.74 38.70 19.18 10.10
45.51 3.24 2.14 28.44 11.82 15.49 13.43 16.35 26.32 26.13 43.73 34.96 51.36 28.96 15.52
48.99 - 2.66 - 0.92 - 28.38 - 11.66 - 15.47 - 13.39 - 16.23 - 28.42 - 26.11 -43.60 - 34.24 - 50.30 - 28.06 - 14.96
51.86 28.11 13.81 30.86 11.58 14.08 10.74 11.83 22.43 20.35 41.06 28.71 43.61 23.52 10.78
48.81 4.89 2.80 28.23 11.60 15.35 13.19 15.97 28.09 25.79 42.31 34.00 49.85 27.77 14.74
48.99 - 2.66 - 0.92 - 28.38 - 11.66 - 15.47 - 13.39 - 16.23 - 28.42 - 26.11 - 43.60 - 34.24 - 50.30 - 28.06 - 14.96
(a) Multislice calculation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
49.49 11.31 4.71 27.80 11.08 14.61 12.49 15.32 27.10 24.61 41.48 32.72 48.16 26.72 13.96
(b) Phase-grating calculation
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
48.56 11.97 6.02 28.24 11.42 14.92 12.63 15.14 26.84 24.61 41.48 32.72 48.16 26.72 13.96
the 100 kV d a t a are also n e a r the single-scattering a p p r o x i m a t i o n . ) T w o d a t a sets o b t a i n e d at 40 kV were a v e r a g e d to p r o d u c e the set IFLVl. Setting m = 0.14, a corrected d a t a set IF~'i"l was p r o d u c e d using (8) after scaling such that )-" IFtnVl = k ~ l F L V I. These values are used for s u b s e q u e n t structure analysis.
~IF~I}I = k ~ IFI2}I applies.) If the p h a s e - g r a t i n g a p p r o x i m a t i o n is used, this R = 0.078 a n d the difference between the t w o a p p r o x i m a t i o n s is m e a s u r e d by R = 0.016. These high-voltage d y n a m i cal d a t a sets can be used as the trial intensities to be corrected. If some lower voltage is selected for this crystal thickness to find intensities IFLVl 2 m o r e p e r t u r b e d by d y n a m i c a l scattering, then the results of multislice a n d p h a s e - g r a t i n g calculations can again be c o m p a r e d by the c r y s t a l l o g r a p h i c residual: at 100 kV, R = 0.063; at 40 kV, R = 0.14. This indicates that either c o m p u t a t i o n a l m e t h o d might be used to calculate d y n a m i c a l intensities, since we are restricting o u r d a t a resolution to a region of reciprocal space (3 A) where the deviation of the E w a l d sphere f r o m reflection centers is not t o o large. At the m o s t extreme conditions tested, i.e. t = 200 A, E = 40 kV, 99.2% of the original intensity is left in the diffraction pattern. At this lower voltage, the m a t c h of the first single-convolution term to the multislice IF~Vl term in (5) is R = 0.23. At 40 kV, this m a t c h is generally a d e q u a t e for crystal thicknesses < 2 0 0 A in steps of 50 A. The single c o n v o l u t i o n of k i n e m a t i c a l structure factors can be c o m p a r e d with the d y n a m i c a l structurefactor m a g n i t u d e s calculated by either the phaseg r a t i n g or multislice m e t h o d s (Fig. 2). A d e q u a c y of the a p p r o x i m a t i o n given in (6) is also d e m o n s t r a t e d in Fig. 2 but b e y o n d 100 A the a p p r o x i m a t i o n m a d e in (7) is not useful at 40 kV. D e s p i t e this deviation, the a p p r o x i m a t i o n given by (5) still holds. Thus, a n d m o s t i m p o r t a n t l y , the c o r r e c t i o n predicts which intensities are m o s t affected by d y n a m i c a l scattering so that these will have the m o s t relative significance in the correction with (8). If the multislice values are used (Table 2), where H V = 1000 a n d LV = 40 kV, then a c o r r e c t i o n by (8) i m p r o v e s the m a t c h to the k i n e m a t i c a l d a t a by lowering R from 0.071 to 0.031 when m = 0.16. The
o IF~OI us I F , FI • IRe~lvs IF * FI ,, IImpG I vs IF * F{
Justification of the dynamical correction The a d e q u a c y of the a p p r o p r i a t e ab initio correction for d y n a m i c a l scattering m a d e to the e x p e r i m e n t a l d a t a m u s t n o w be justified by m o d e l calculations for an a n a l o g o u s structure. The kinematical structurefactor calculation for the D L - D M P E model described a b o v e yields the values listed in Table 2. These are the only reflections excited in the diffraction e x p e r i m e n t (as e x p e r i m e n t a l l y observed). If the m o d e l crystal thickness is as su m e d , m o r e o v e r , to be 200 A, then a multislice d y n a m i c a l calculation will yield structurefactor m a g n i t u d e s IF~Vl deviating from the kinematical values by R = 0.071. ( F o r c o m p u t a t i o n of R factors it is a l w a y s a s s u m e d that the scaling
/
0.5
lb~
~ 50
/
c
/ 100
, ~5o
~
~o
t(~,)
Fig. 2. Test of different functions against IF * F[ for DL-DMPE at 40kV and various thicknesses nt. (a) rdy, for multislice --MS calculation; (b) Fdr"POfor phase-grating approximation; (c) IRepc [ real part of phase-grating series; (d) IlmpG[ imaginary part of phase-grating series.
783
D. L. DORSET, M. P. McCOURT, W. F. TIVOL AND J. N. T U R N E R
Table 3. Hierarchy of l Eli for D H P E This analysis 0,0,14 1.767 0,0,12 1.654 0,0,13 1.439 0,0,15 1.247 0,0,16 1.085 001 1.061 003 !.051 0,0,11 0.884 0,0,10 0.868 005 0.764 007 0.638 004 0.494 006 0.375 008 0.299 002 0.278 009 0.178
Previous analysis 0,0,14 1.554 0,0,12 1.524 0,0,13 1.513 0,0,15 1.486 0,0,16 1.244 003 0.964 0,0,11 0.932 001 0.920 0,0,10 0.836 005 0.722 007 0.591 004 0.547 009 0.523 002 0.469 006 0.434 008 0.165
comparable correction, assuming the phase-grating approximation to be adequate, lowers R from 0.078 to 0.023. As seen by inspection of Table 2, the reflections most affected by dynamical scattering are the lowest intensity values, exactly what is found for the experimental data from DHPE.
Direct phase determination Using the corrected data in Table 1, normalized structure-factor magnitudes ]E~] were calculated, assuming that the pseudocentrosymmetric unit cell (a = 9.95, b = 7.77, c = 55.2 A, ~,/7 and ~ _ 90 °, where b = t is the slice thickness) used in (1) contains two molecules of DHPE. (For bilayers containing a single phospholipid, this assumed centrosymmetry of the density profile is generally valid, even if the lipid
IEhl 1.5"
molecules are chiral.) Thus, IEll2=lFpi"121~if2 , where the f~ are the electron scattering factors (Doyle & Turner, 1968) for this structure. This leads to a hierarchy of normalized structure-factor magnitudes (Table 3) that is somewhat different from the ones used in our previous analysis (Dorset, Beckmann & Zemlin, 1990). Note that [Eoo2l, IEoo41 and IEoo91 are lower than previously estimated. Evaluation of three-phase structure invariants in space group P1 (Hauptman, 1972),
where the M i l l e r - i n d e x sum is constrained to ~ i li - O, leads to the phase assignments in Fig. 3 after the unit-cell origin is defined by setting qSo01 = 0. These assignments are based on the 2"1 and '~2 triples listed in Table 4. Only the value of qJoo9 remains undefined. However, since this occurs near a node between two homogeneous phase domains and corresponds to a weak structure-factor magnitude, we assigned it an arbitrary value of ft. The one-dimensional potential map from this phase assignment is shown in Fig. 4. It should be noted here that this calculation is made with a value for Fooo obtained from an earlier structurefactor calculation (with an appropriate scale for the observed values in Table 1). This value is also used for all subsequent calculations incorporating the convolution operation F t • Fz etc.
Proof of phase assignment The phase set obtained in this analysis differs from the previous one, since phases ~oo2 = ~oo4 = ~oo6 = ~'0o7 = rt were determined earlier from the electronmicroscope images (Dorset, Beckmann & Zemlin, 1990). How can we justify this new phase assignment? One way is to use a multisolution approach, assuming values qJool = ~/003 = 1/1005 = 0 that are common to
]
;"
origin v 1.0-
o t7
?
0.5-
I"1
tVi
! i
o.lo
o:2o
It ##
o.~ d'~ 4)
Fig. 3. Schematic representation of direct phase determination on
-
-
\',,
O /
DHPE using ]Footl values calculated from corrected structure-
factor magnitudes. Solid lines represent reflections related through the origin definition (~/'ool = 0) via Z2 triples. Specific reflections are phased by negative 2"1 triples (and, in one case, a 2"2 triple). Legend for phase: • = 0; [] = n.
Fig. 4. One-dimensional potential map for D H P E (dashed line). A comparison with the map obtained earlier (solid line) shows that, although features of the headgroup density are similar, the hydrocarbon-chain region is flatter for this newer solution.
784
ELECTRON DIFFRACTION FROM PHOSPHOLIPIDS
both phase assignments and then generating 2* = 16 phase sets for the unknowns 0002 = a, 000, = b, 0006 = c and 0007 = d. These phases are assigned to the corrected structure-factor values in Table l and a convolution IF t • Ftl is calculated for each phase set to be compared with the 40 kV structure factors, since we assume that m. . . Il rr 4o O o tk V [ -~ iFoo t , Foot[ above. Of the 16 possible phase permutations, only five have residuals R < 0.20 and none of these corresponds to the phase set deduced in the original determination with uncorrected data. The original phase assignment for Foot is shown to be expressed again by all but one of the convolution products for the respective permutations. Of the phase sets analyzed, three with R < 0.17 include one which is the Babinet set of the original determination for lamellar reflections (001)--*(007) (Dorset, Beckmann & Zemlin, 1990), after an origin change is made for the subset of indices. To describe which of these three solutions is most likely, one-dimensional potential maps were calculated (including the Foo o term) and the smoothness of the density function p was evaluated by the average ([t3p/t3xl), a figure of merit used previously for simulated and experimental X-ray data sets (Dorset, 1991a,b). Only one solution represents a salient minimum for this function and this corresponds to a=b=c=d=0, in agreement with our directphasing analysis. Most of this solution can be obtained in yet another way. Computer programs that assemble '~2 triples as the various contributors to a given phase in the tangent formula (Karle & Hauptman, 1956) can be used to evaluate the phases of high-lE[ reflections in DHPE. After definition of tPoo t = 0, it is clearly indicated by these contributors that all of the first six reflections have the same phase. Only the tPoo7 value determined by this method is uncertain. If the data correction and phase determination are valid, a dynamical calculation should converge to the observed experimental structure-factor values measured, e.g. at 40, 100 and 1000kV. With the phase-grating expression (justified by the comparison of models above) used to compute the dynamical structure-factor magnitudes at the three voltages, this convergence can be demonstrated to correspond to a crystal thickness of c a 200 A. R = 0.12, 0.12 and 0.05 for intensity data observed at 40, 100 and 1000 kV, respectively. It can also be shown that the fit of the dynamical data to the observed structure factors is also sensitive to the phase values used in the original structural model. Discussion
The approximate experimental correction of observed intensities for dynamical scattering discussed above is
beneficial for the ensuing structure analysis in a number of ways. First, compared to the earlier determination made on the uncorrected data set (Dorset, 1990a), it is clear that the ordering of X 1 and ,~V"2 triplet-invariant relationships from the most to the least probable is more accurate because the sequence of IEtl values is closer to the kinematical hierarchy. Thus, 15 of the 16 reflections in the diffraction pattern can be assigned phase values directly (rather than the 13 found earlier). It is apparent that the sequence of IEtl is most important for structure determination based on very small data sets. The low-voltage data, moreover, serve as an approximate estimate of the single-convolution product of these phased structure factors and can be used to screen the number of possible phase sets chosen as likely solutions if a multisolution approach to phase determination is needed. The above procedure exploits the similarity of this convolution product to the Sayre (1952) equation, as noted earlier by Moodie (1965). Although model calculations with the DL-DMPE model demonstrate that m F L v ~ _ Ft* Ft is not rigorously correct, the agreement appears, nevertheless, to be good enough to enable one to detect possible structural solutions. A dynamical calculation based on the phased structure factors v i a the phase-grating approximation matches well the observed diffraction data at three voltages and thus serves to test for convergence at a single reasonable crystal thickness. As demonstrated above, the phase-grating approximation is well suited to the low-angle range considered here and enables one to carry out a dynamical calculation with corrected data once the crystallographic phases are found-even though the structure itself is not resolved to atomic detail. Finally, the appearance of the best potential map, which is assessed by a test for density smoothness, now resembles those calculated with model data (Dorset, 1990a) in that the hydrocarbonchain region has an overall flatness not observed in the original analysis of the uncorrected data set (Fig. 4). The success of this phase analysis in finding a solution different from the low-angle determination made from high-resolution electron micrographs, however, requires some further explanation. In the initial phase determination from experimental image data (Dorset, Beckmann & Zemlin, 1990), it was assumed that the phase-contrast envelope of the electron-microscope objective lens had a negative sign for orders one to six and that it reversed contrast between the sixth and seventh diffraction orders. The assumed phase-contrast transfer function was based on a plot of IFimi/lFedi v e r s u s d*, using data from an n-paraffin image photographed under similar conditions (Dorset & Zemlin, 1990). (Here, IFim i is the wave amplitude of the image transform and IFed I is the
D. L. DORSET, M. P. McCOURT, W. F. TIVOL AND J. N. TURNER
corresponding electron-diffraction structure-factor amplitude.) Unfortunately, it is difficult to know exactly what this contrast envelope is when carrying out low-dose experiments, even when a known defocus value, Af, is applied to a nearby area before the crystal lattice is photographed. This uncertainty is caused by bend distortions common for such crystals and support films. For example, Uyeda, Kobayashi, Suto, Harada & Watanabe (1972) have demonstrated for phthalocyanines that the image contrast of nearby crystal areas can change sign because of the effective local change in Af. The solution found in the phase determination described above is actually a Babinet (reverse-contrast) set for the first six lamellar reflections in the previous determination. The seventh order would also be passed by the transfer function with the same phase sign as the other six reflections. The likelihood of such uncertainty in the actual transfer function must always be taken into account when extracting phase information from such images, especially when only a single reciprocal-lattice row of low-angle reflections is being analyzed. Finally, although the deviations from kinematical electron scattering are seemingly well explained by n-beam dynamical theory, it is also necessary to demonstrate why a correction for secondary scattering also produced a better fit between observed and calculated data in an earlier study (Dorset, 1990a). For either multiple-scattering perturbation, the form of the correction states that /]ET'°bs~2 tltTkinh2 -~- mKool, ~,zO01! -~- ~001! where Koot is a correction parameter and m is an empirical variable to improve the fit. Although in general, Ioo ~• loo t :~ (Foo t • Foot) 2, either substitution for Koo t actually changes the relative values of weakest reflections in the diffraction pattern the most. The effect, therefore, can be very similar. With the corrected structure factors described above, it is possible to adjust for secondary scattering such that the fit to the 100kV data represents a residual R = 0.10. The choice of which correction is appropriate, therefore, depends on which is physically most reasonable. Epitaxic orientation of the phospholipid results in a crystal array with thicknesses distributed around a mean value. The marked effect of accelerating voltage on the intensity distribution of the diffraction pattern seems to be most consistent with the results expected from dynamical scattering, since the correction converges to the same mean thickness for three voltages. While a similar agreement can also be shown, for example, for 40 and 100 kV data using the secondary scattering correction, a very different weighting term m must be applied (m = 1.0 and 0.10 for 40 and 100 kV, respectively) to I . I for the different voltages. However, the model for secondary scattering assumes that a sequence of stacked crystalline layers are scattering incoherently
785
from one another (Cowley, Rees & Spink, 1951). Thus, for a given mean crystal thickness composed of very thin crystalline slices, the fractional contribution of secondary-scattering events should be somewhat voltage independent since any large relative difference must depend on the crystal texture. For example, if the separate crystalline layers were only 47 A thick (six unit cells), then a multislice calculation demonstrates that the agreement of [FTY"l to lEnin[ is R = 0.027 at 100 kV and 0.032 at 40 kV. Thus, the weighting term m should be nearly the same for the two voltages. Hence, the experimental observations favor an n-beam dynamical model as the primary perturbation to the intensities, even though some secondary scattering must also be present (Hu, Dorset & Moss, 1989). Research was supported by a grant from the National Institute of General Medical Sciences (GM46733). Time on the high-voltage electron microscope was supported by Biotechnological Resource grant RR01219, awarded by the National Center for Research Resources, Department of Health and Human Services/Public Health Service, to support the Wadsworth Center of the NY State Department of Health's Biological Microscopy and Image Reconstruction Facility as a National BiDtechnological Resource. References
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