Structure determination of asymmetric membrane profiles using an iterative Fourier method

STRUCTURE DETERMINATION OF ASYMMETRIC MEMB RANE PROFILES USING AN ITERATIVE FOURIER METHOD ROBERT M. STROUD AND DAVID A. AGARD, Department o Biochemistry and Biophysics, University ofCalifornia, San Francisco, California 94143 U.S.A. ABSTRACT An iterative Fourier method is applied to solving and refining the electron density profile projected onto the line perpendicular to a membrane surface. Solutions to the continuous X-ray scattering pattern derived from swelling of multilayer systems or from membrane dispersions can be obtained by this technique. The method deals directly with the observed structure factors and does not rely on deconvolution of the Patterson function. We used this method previously to derive the electron density profile for acetylcholine re- ceptor membranes (Ross et al., 1977). The present paper is an analysis of the theoretical basis for the procedure. In addition, the technique is tested on artificially generated con- tinuous-scattering data, on the data for frog sciatic nerve myelin derived from swelling experiments by Worthington and McIntosh (1974), and on the data for purple membrane (Blaurock and Stoeckenius, 1971). Although the method applies to asymmetric membranes, the special case of centrosymmetric profiles is also shown to be solvable by the same tech- nique. The limitations of the method and the boundary conditions that limit the degeneracy of the solution are analyzed. INTRODUCTION The principles of X-ray crystallography were first developed to determine structure in re- peating systems. Three-dimensional order in a crystalline specimen permits a great enhance- ment of the X-ray scattering expected from a single molecule or dispersion of molecules, although the scattering function is consequently sampled only at the positions determined by Bragg's law. The components of biological membranes are inherently closely packed in two dimensions, although crystalline ordering of membrane-bound protein, even in domains con- taining several hundred molecules, is rare. Most specialized biological membranes are in- herently asymmetric structures, i.e. the distribution of protein (or other material) in a direc- tion perpendicular to the plane of a lipid bilayer is asymmetric. X-ray diffraction has played a major role in establishing the nature of phospholipid bi- layers (Levine and Wilkins, 1971). Although phospholipid molecules in the leaflet were shown to possess no regular two-dimensional crystalline order at temperatures above the phase transition, ordering and structural repetition could be obtained in the third dimen- sion, perpendicular to the membrane surface, by close packing of the equivalent and sym- metric layers. As a result, the diffraction pattern defined along a reciprocal lattice direction s, where s is normal to the membrane surface, is contained in discrete Bragg reflections. Robert M. Stroud is an Alfred P. Sloan Foundation Fellow. David A. Agard is a National Science Fouiidation Pre- doctoral Fellow at the California Institute of Technology, Pasadena, Calif. 91125. BIoPHYs J. @Biophysical Society * 0006-3495/79/03/495/18 $1.00 495 Volume 25 March 1979 495-512 Since a small number of biological membranes, generated by folding over a cell envelope such as the myelin sheath, or rod outer segment, contain a preexistent center of symmetry, the theories of diffraction from centrosymmetric structures could be applied. Solution of the phase problem, for the signs of successive reflections in such cases, is often by deconvolu- tion of the Patterson function or by trial, since experimental methods, such as isomorphic re- placement of one component of the sample by another, have not generally been successful for membranes. Often the diffraction data can only be phased to relatively low resolution, as the correct phase choice relies to some extent on agreement with a reasonable structural model. There is no precedent for use of the finer modulations of structure (which depend on the higher resolution data) as a means of choosing between the possible signs associated with higher order terms. In some centrosymmetric membrane structures, such as myelin sheath, the membrane multilayers may be reversibly swollen (Robertson, 1958; Moody, 1963; Blaurock, 1971; Caspar and Kirschner, 1971; Worthington and McIntosh, 1974). If the structural unit re- mains intact during swelling, i.e., if the function p(r) (referred to as the membrane profile) does not change within the limited thickness of the leaflets and only the repeat or multilayer spacing d is assumed to vary, then the observed transform is sampled at different positions defined by sh = Xh/d, and the continuous transform can be mapped. The positions of zeros in the transform, i.e., where F(s)1 2 - I(s) = 0, provide a powerful constraint useful in de- fining similarities of signs for adjacent reflections that lie between the zeros in the con- tinuous transform. However, the function F(s) need not change sign at one of these nodes. For frog sciatic myelin (Rana pipiens), the mapping of the continuous transform by Worth- ington and McIntosh (1974) reduced the number of possible sign combinations for diffrac- tion orders h = 1-12, from 2,048 to 64. Difficult as it is to define the phases for a centrosymmetric structure, determination of general phases for asymmetric membranes is much more difficult, and this is a primary con- cern here. Generally, when asymmetric membranes are closely aligned one against the other, there will be no true repetition in the direction perpendicular to the sheets. The statistical distribution of membranes with alternate surfaces up or down gives rise to large statistical fluctuations in the true repeat distances in the specimen. As a result, there is a pillar of continuous scatter perpendicular to the membrane surface due to the product of the true transform of one membrane sheet, F(s), and the transform of a statistical stacking function. The continuous scatter may show slight evidence for the minimum possible repeat distance in the stacking direction arising from the chance stacking of several layers in the same direction. Where sheets are almost symmetric in terms of the distribution of density through the membrane, the pattern may show a reasonably well-defined minimum repeat distance. This is the case for the asymmetric purple membrane from Halobacterium halobium (Blaurock and Stoeckenius, 1971; Henderson, 1975). But in all cases, the pattern obtained from closely packed asymmetric membrane systems is made only more complex by the close packing of successive leaflets, as it is the product of the continuous transform from one layer F(s) and a sampling function, which may not depend on random statistics of stacking for obvious physical reasons. Since the second function is extremely difficult to derive (Burge and Draper, 1967), the first is equally difficult to extract. A more easily obtained scattering pattern is the one from membrane dispersions or 496 BIOPHYSICAL JOURNAL VOLUME 25 1979 moderately close-packed membrane sheets, where the scattering pattern Ob.(S) sin2 0 more nearly represents the square of the Fourier transform of a single membrane sheet F(s) (Wilkins et al., 1971). It is also possible to orient the membrane sheets centrifugally while maintaining only moderate close packing. This allows for separation of the normal and in-plane diffraction, which is important if there is significant in-plane lattice structure. The information present in continuous X-ray scattering patterns of these types is in one sense greater, since the transform is a continuous function often observed to moderate resolution. In many cases the transform can be shown to be unsampled by the statistical stacking disorder function (disorder of the second kind of Hosemann and Bagchi, 1962) by comparison with the diffraction pattern from membrane vesicles, for example. The con- tinuity of the observed transform places restrictions on the solution. The main difficulty in interpreting these patterns is again determination of the phases +(s) of the observed con- tinuous transform F(s)I. In the case of sampled diffraction patterns, as in crystallography, Fourier refinement methods generally require a good approximation to the real solution before refinement is at all meaningful. The continuity of the observed Fourier transform partially removes this condition. One further piece of information, namely knowledge that the structure is of finite rather than infinite thickness, is sufficient to restrict the number of possible solutions to a very small number, and often to just one. The method of refinement discussed in this paper imposes the criterion of limited physical extent (constrained thickness) during successive cycles of Fourier refinement of an electron density model to the observed continuous transform IFO(s) 1. The starting trial model used to initiate the procedure is demonstrably unimportant when a unique solution exists, since the method always leads to convergence on a solution. Likewise, the constraining thickness is theoretically unimportant as regards obtaining a solution, providing it is larger than the actual thickness of the membrane structure. No con- verged solution can be obtained if the thickness used for refinement is too small. Practically, a correct choice for the constraining thickness assists in achieving more rapid convergence to a solution, and in minimizing problems associated with errors in the data, including the fact that data are necessarily of limited resolution. Resolution is limited theoretically by the wavelength of the radiation to X/2, and practically limited by experimental conditions, or by signal-to-noise ratio. This application of Fourier refinement can determine and refine possible solutions for the electron density function, although absolute certainty that all solutions have been obtained would require testing of a large number of possible alternatives. In practice, the number of real alternatives is limited to those consistent with the maximum and minimum electron density allowed by the chemical constituents of the system, i.e., to chemically reasonable alternatives that can in principle all be tested. The more difficult question is whether a de- termined solution is necessarily the correct one. If more than one solution is obtainable, identification of the correct one requires experimental evidence of a different type, such as isomorphic replacement. Thus, in reporting membrane profiles, many solutions should be sought and reported where obtainable. To test the efficacy of the iterative refinement method, several transforms were generated from artificial model structures. The character and dimensions of the models were of the STROUD AND AGARD Membrane Profile Determination 497 kind that might be expected for genuine membranes, so that discussions of resolution limits, etc., may be compared with real cases. The iterative method was then applied to the generated transform data for hypothetical membrane-like structures to assess the power of boundary conditions and limitations of the method. The method was also applied to solu- tion of the density profiles from the continuous purple membrane data of Blaurock and Stoeckenius (1971) and from the myelin data of Worthington and McIntosh (1974).' The primary motivation for evaluation of the method is to apply the technique to the lo- cation of biochemically characterized components in specialized membranes. THEORY OF THE METHOD FOR ASYMMETRIC AND SYMMETRIC STRUCTURES Let the electron density perpendicular to the membrane be p'(r) and let p, be the constant electron density of the solvent. For simplicity in the ensuing discussion, it is convenient to define a solvent relative profile po(r) = p'(r) - p, since the transform of po(r) is observed in a diffraction experiment. The transform of the constant density p3 is contained in the direct beam. For a real asymmetric structure, the choice of origin is arbitrary. Thus if the membrane is of width w, then po(r) may be nonzero only between the limits r = a and r = b, where w = b - a. The function po(r) is then said to be constrained to lie within these lim- its. Although po(r) will generally be largely positive, it may also be negative where, as in the center of a bilayer for example, p'(r) is less than the electron density of the solvent (Levine and Wilkins, 1971). The Fourier transform of po(r) will be Fo(s), where the reciprocal space vector s is parallel to r. Then the observed diffraction pattern, if it can be shown to be an unsampled con- tinuous pattern (or is the continuous data extracted from swelling experiments), will be, after suitable corrections for the experimental arrangement are applied, Io(s) = IFo(s) 12. Hav- ing derived the function Fo(s) from the diffracted intensity, the problem is to determine solutions for p(r) that obey the normal Fourier inversion relationship: p(r) = f -s, Fo(s) exp (-27rir-s)ds, (1) where Fo(s) = p(r) exp (27rir * s)dr, (2) and where s, defines the highest resolution (dmin = X/s,) to which the data are measured. Data obscured by the beam stop can be interpolated by using either the sampling theorem or by curve fitting. Since the observed transform is continuous (unlike the crystallographic case), Eqs. 1 and 2 are linear integrals of a continuous function. Any random phase function +(s) associated with IFo(s) I, such that Fo(s) = IFo(s) exp (27ri/(s)), when transformed according to Eq. 1, will yield a function p(r) that when I It should be clearly noted that the objectives in these cases were to test the method against solutions obtained by these authors by using their particular data sets. The results are therefore entirely dependent on the data set, and the tests were set up this way so that solutions obtained by this method could be directly compared with those ob- tained by these authors. 498 4BiOPHYSICAL JOURNAL VOLUME 25 1979 transformed by Eq. 2 will necessarily generate the same F(s). This is true only so long as the integration in Eq. 2 is carried out over all space (r = - 00 to r = + oX). The re- sulting p(r) will be a function of infinite extent; i.e. p(r) may be nonzero for all r (- X < r < 00). The solution p(r) so obtained clearly has no physical meaning. Since the phases +(s) were chosen at random, there are an infinite number of such solutions. Without additional information, there is no basis for determining the phases of FO(s) from the observed values of IFO(s) 1. Since p(r) must correspond to a real object, it cannot be of infinite extent or thickness. This alone restricts the number of solutions, no matter how they are obtained, to a small number, all of which are necessarily of the correct actual membrane thickness, without any assumption of what that thickness may be. This can be seen in the following way. The transform of FO(s) * FO*(s) is the Patterson function SI P(r) = f 1FO(s)I 2exp (27rir s)ds, (3) and will always be equal to the convolution p(r) p(-r) (see Lipson and Taylor, 1958, for ex- ample). This convolution is termed the autocorrelation function, Q(r) = j r+ p(r')p(r' + r)dr'. (4) Thus P(r) Q(r), whatever the phase set that is applied to IFO(s) in Eq. 1, providing the integrals in Eqs. 2 and 4 are carried out from r = -0 to r = +0. Now consider the autocorrelation function computed from a p(r) of finite extent, i.e. p(r) = 0 for r < a or r > b. The autocorrelation function Q(r) = f r+X p(r')p(r + r')dr' will bezero for all values ofirI > w, (where w = b - a) since at least one of the terms p(r'), p(r + r') must be zero when rI > w. Conversely, if Q(r) = 0 for rI > w,then any function p(r) that is a solution to Eq. 4, can only be nonzero within an interval Ar = w provided that p(r) is of finite extent. If the actual (true) thickness of the membrane leaflet is w, the autocorrelation function for the structure QO(r) [- PO(r)] will be zero for all rI > w. PO(r) is determined by the inten- sity distribution Fo(s)l2 alone, and not by the phases assigned to Fo(s)l. Any function p(r) of thickness t > w will necessarily produce an autocorrelation function Q(r) that has non- zero components2 for w < I rI < t. Therefore Q(r) . PO(r), at least for w < rI < t, and so p(r) cannot be a function derived from the observed moduli FO(s) (Eq. 1). Thus no function p(r) of finite width t > w(and by similar arguments I < w) can be a solution to Eqs. 1 and 2. It follows that imposition of any constraining thickness, which al- ters Eq. 2 to rb Fo(s) = f p(r)exp(2irir.s)dr, (5) 2In general Q(r) will similarly be nonzero for all r < w. However, there may be particular values of r for which all terms in Eq. 4 cancel to give a zero in Q(r). STROUD AND AGARD Membrane Profile Determination 499 where b - a > w, only allows solutions, p(r), of thickness w. In the Fourier refinement method, solutions that obey Eqs. 1 and 5 are determined in an iterative process. By analogy with usual Fourier refinement procedures, an initial structure pi(r) is trans- formed according to Eq. 5, i.e., between some limits r = a to r = b to generate an ini- tial calculated transform Fj(s). The phases Xc(s) derived from this model are then associated with the "observed" Fo(s) and transformed back by using Eq. 1 to obtain a new function j4(r). The entire process is cycled until convergence is reached (Fig. 1). All refined solu- tions, p(r), will necessarily be of the same thickness. If multiple solutions, p(r), exist that are compatible with a thickness of w, then these will form a set of possible solutions (strictly a homometric set of solutions) that cannot be eliminated without further information by any method of treating F(s) I. The choice of starting model pi(r) in these instances will tend to select for the solution closest to the start- ing model. Consequently, this approach can be used to test hypothetical structures. Ob- viously, if only one p(r) of finite thickness exists, this procedure will converge upon that structure. In practice it seems that when multiple solutions do exist, they are often qualita- tively quite similar to one another. The boundary limits r = a, r = b are unimportant as regards achieving a converged so- lution. The choice of a constraining thickness w that is close to the correct value for con- verged solutions greatly speeds up convergence. The constrained thickness may be estimated in several ways and, therefore, is not difficult to determine accurately enough to be effective. It may be estimated: from electron micros- copy of isolated protein molecules, or of thin sections cut perpendicular to close-packed membrane sheets; from the minimum close-packing distance found in X-ray scattering from densely packed specimens; or from the Patterson function, which identifies (within resolu- tion limits) the largest vectors in the structure. So far in our considerations we have assumed that Fo(s) contains no errors. In any real case the values of IFo(s) will be extracted with care to try to achieve this condition (see, for example, Ross et al., 1977). If there is independent evidence for the thickness of the leaflet FIGURE 1 Schematic ofthe refinement procedure. 500 BIOPHYSICAL JOURNAL VOLUME 25 1979 (of the kind mentioned above), then imposition of the boundary thickness can mini- mize errors in the profile introduced by errors in Fo(s) I. If such errors in the data exist, they will be assayed by the final agreement between Fo(s) and Fe(s) for the constrained struc- ture. Systematic errors in the refined profile arise when the data are cut off at some particular resolution, s,, such that IFO(s) is nonzero for s > s,. This is to be expected since the "true" resolution limited structure (computed with true phases and limited in resolution to sl) will contain Fourier series termination ripples of frequency equal to s,. The presence of termination ripples is inconsistent with the refinement procedure used here [p(r) is con- strained to be zero outside the boundary thickness]. Therefore the phases of the terms F(s) with s closest to s, will be modified, and so changed away from their true phase to best accommodate the termination ripples. In any real case, data to the highest observable resolu- tion should be included, such that IFO(s) - 0 at the resolution cutoff limit. The im- position of an artificial temperature factor exp (- Bs2) is useful when this condition is not met. Criterion for Acceptance ofa Solution The criterion for accepting a refined solution, p(r), is that the amplitude of the transform I Fe(s) computed in Eq. 5 should agree with the observed transform Fo(s) I. The degree to which IFc(s) approximates IFO(s) characterizes the extent of convergence, and indicates the degree to which the diffraction data are consistent with a structure of width w < b - a in thickness. The agreement index, -Fc(s) IFo(s)Ids R = °(6) rs II Fo(s)lds is typically 0.1%O-1% for a solution to artificially created and purple membrane data, or about 1%Y-5% for measured IFo(s)l data from acetylcholine receptor membranes, for ex- ample, (Ross et al., 1977). Since for any solution, p(r), there will be three other trivial solutions, -p(r), p(- r), and -p(- r), only one solution of the set for which J p(r) dr > 0 will be considered. TESTS OF THE METHOD An Asymmetric Structure A supposed (initial) electron density function shown in Fig. 2 A was used to generate the function F(s) I, which was then regarded as an observed transform. The back transform of F(s) at 10 A resolution, the correct resolution-limited structure, is also shown in Figure 2 A. Attempts were then made to derive the electron density function p(r) starting only from the moduli IF(s) by application of the refinement method. Numerous different trial models were used to initiate the procedure by using data to 10 A resolution. These ranged from a single delta function, to several step functions, to various continuous functions. In each case convergence on the same final structure was achieved (Fig. 2 C). The boundary thick- ness chosen varied from 210 to 250 A. STROUD AND AGARD Membrane Profle Determination 501  'U-k A 'FI\ - - f - -i a' K I I S a F .OM% Ge. I- wgaw iL - 4. 00% at C .0 . ,., e I t , 4 I -150 -1W0 .*50, 0 So 100 .1.50 OA v.W 75 15s m .( R (A) It {). R (A) FIGURE 2 Results of refinement of an asymmetric "membrane" profile using synthetic data to 10 A. The model used to generate the observed structure factors is shown in A. Using a ramp function for the trial structure results in the profiles depicted in B and C (1 and 30 cycles of refinement, respectively). Cor- responding observed and calculated Fourier amplitudes and calculated phases are shown in D and E. A typical plot of both Q(r) and P(r) is shown in F. RF and RP factors in all refined cases are less than 1.5%. When the structure was refined at 20 A resolution, series termination ripples in p(r) at each successive stage were forced into the structure by the refinement procedure (which eliminates ripples outside the structure) (Fig. 3 B). Later addition of higher resolution data (20-10 A) did not immediately lead to further refinement of the 20 A data, since sub- sequent refinement first tends to minimize ripples outside the boundary that at this stage are produced only by the additional data (Fig. 3 C). In this situation the additional data from 502 BIOPHYSICAL JOURNAL VOLUME 25 1979  A AnI < 0o-l x 1i' m- A 0 TN 50 0 50 R (A) B , I S (A1) 0 z B < o- x zL i c 0 I 50 50 .a R (A) -1 50 -1 0 -5o 0 50 16o 150 0 R (A) s (A-) FIGURE 3 FIGURE 4 FIGURE 3 Refinement of the same profile as in Fig. 2 but at lower resolution. The true structure is de- picted in A. The converged solution using 20 A data after 20 refinement cycles appears in B. Although similar to the profile obtained with 10 A data (Fig. 2 C), there are distinct differences resulting from forc- ing the series termination ripples to be accommodated within the structure. These ripples are not imme- diately removed by a later inclusion of the higher resolution data, as shown in C. FIGURE 4 The two solutions (A, B) to the purple membrane continuous scattering data of Blaurock and King (obtained after 30 cycles of Fourier refinement). The left side of the figure shows the derived phase functions and the equivalence of Fo(s) and Fe(s) (shown superimposed). The R-factor in both cases was below 0.15%. On the right are the corresponding electron density profiles, showing the asymmetry in the protein distribution across the membrane. Many different starting models were used; however, only these two solutions could be obtained. STROUD AND AGARD Membrane Profile Determination 503 20 to 10 A resolution are initially being treated as an independent data set. Only as suc- cessive changes in structure begin to affect the phases of the lower resolution data does the pressure to refine the lower resolution phases develop. When the starting trial model pi(r) is centrosymmetric, convergence to the correct structure, which is asymmetric, requires that the centrosymmetric phase relationships (which are signs if the origin is chosen at the symmetric center) be broken. If the refined model is constrained to be concentric with the centrosymmetric starting model, it is impossible to achieve even qualitative agreement with the observed data. Convergence on the final solu- tion is possible when the constraint to symmetry is removed by imposing a boundary size whose limits are not concentric with the starting model. The asymmetric truncation of series termination ripples allows a degree of freedom for the phases and so allows refinement to continue. Several different initial structures were used to generate observed values of F(s) . The experience with the case described above was shared in other similar cases. As the com- plexity of the profile increased to the point where variations in p(r) were of the same order as the resolution cutoff limit applied to the data, multiple solutions were obtained. These so- lutions were similar with differences accountable as alternative ways of phasing the lower in- tensity data at higher resolution while maintaining a width w. This is a consequence of the - errors introduced by using data of limited resolution (see Theory section), and the lower pressure to phase weak-intensity data, whose contribution to p(r) is least. Since the lowest intensities are found in general at highest resolution, phase errors will correspondingly be greater at higher resolution. As another test, data from suspensions of purple membrane from Halobacterium halobium, kindly provided by Allen Blaurock (California Institute of Technology), were re- fined by using this method. Various starting models were used and only two different solu- tions could be obtained (Fig. 4). These solutions are identical to the two reported by Blaurock and King, 1977. The two solutions correspond to different senses of phase change (do/ds) between maxima in the observed transform Fo(s) I. Centrosymmetric Structures A more difficult question is whether an initially centrosymmetric model p(r) used to generate a transform or a centrosymmetric membrane structure will also be determined correctly, since the final phase angles must accommodate discontinuities between regions of opposite sign in the transform F(s). This was tested first by calculation of observed transforms from symmetric models (see Fig. 5). Refinement against the data shown in Fig. 5 always led to convergence on the correct structure (Fig. 5 D). The phases changed most rapidly at the positions of sign change in the observed transform (where the computed IFO(s) are small- est) and accommodated a phase change of 1800 within a small range of As. In a second test of the method to determine centrosymmetric structures, the continuous transform for frog sciatic nerve myelin mapped from a series of swelling experiments by Worthington and McIntosh (1974) was used. The data were extracted from their Fig. 9 and digitized at intervals of 0.001 A-' in reciprocal spacing to a resolution of s = 1/14 A'. Different data sets reported elsewhere, or different interpretations of what constitutes the continuous transform of myelin (considered extensively by Blaurock, 1976), would clearly give rise to different electron density profiles. Our objective was to see how closely the 504 5BioPHYSICAL JOURNAL VOLUME 25 1979 (a). - b)W '(c)-. l. I coo 0-03 '- .'1 I(A FC 9 *(,) . _IV.; . :s.!| *. 4-t .A . . . -IQ- ----l- )O S: ;: _q. 1-.-- t s7- V w ftZ\ o. .: & 11' ---i-l- .-.r \ V. y \-ffi-1 -L ). pd,r)---- .. . .. p (r) -- :- -- ; | 0. I r- () .( FIGURE 5 Test of solution at IoA resolution for data generated from a centrosymmetric structure is shown in part 2 of A. The trial model used to initiate refinement was a box function (solid line in part I of A). Fourier transformation of the Fe(s) l, X(s) generated from the trial model (A), and cut off at 10 A (-------- in A) contains series termination ripples. At the first cycle (B) observed Fo(s) ( ) were coupled with phases calculated from the trial model (A) to generate a new model po(r). Thi's' model was used to generate amplitudes Fe(s) (------ ) and phases Xc(s). (ps(s) is computed from Fe(s) .kc(s)). The procedure was iterated, and results after 4 cycles (C), and after 14 cycles (D), show convergence onto an essentially correct structure, which may be compared with A above. The agreement between the Pat- terson and autocorrelation function is essentially perfect (E). STROUD AND AGARD Membrane Profile Determination 505  solution using the Fourier method fitted the conclusions and the ambiguities derived by Worthington and McIntosh from these data. The myelin data also represent an ideal system for illustrating the theoretical basis for the multiple solution problem. The normal frog sciatic nerve myelin structure gives rise to Bragg reflections that lie on maxima determined by the minimum multilayer repeat of d = 171 A. A width of 180-190 A was chosen as an estimate of the thickness of a single re- peated unit. A width slightly larger than the true repeat Bragg spacing was chosen so as to allow a smooth truncation of series termination ripples within the boundary thickness. The Fourier refinement method was applied to several different starting models bearing no re- lationship to the solutions obtained by Worthington and McIntosh (1974). One solution summarized in Fig. 6 again indicates that convergence on an almost centrosymmetric struc- (a) [. . 8 X _*s i J PC@~~~~~~~~~0()-- ''! cyclek I( R 130.4% / Ra3SX% (b) R- 5.0% R * 1.94% (C) 25% c& A sL&3) rc)* r (h R-S.4% a : - FIGURE 6 Test of solution using the continuous dataI Fo(s) for frog sciatic nerve myelin (Worthington and McIntosh, 1974) to a resolution of s = 1/ 14 A - 1. The trial model for initiating refinement was pC(r) (-------- in A); and at cycle 50 (B), are shown (left). The central figures show the calculated model gen- erated by transformation of Fo(s) I, Xc(s) [po(r)] and of Fe(s) I, 0,(s) [p,(r)] at corresponding cycles. Agreement between P(r) and Q(r) is indicated at the right side. Values for the residual R at each stage are indicated below each data summary. The residual values underneath the Patterson summary are Rp values. The structure po(r) shown in B is not completely centrosymmetric, and the two sides of the struc- ture were subjected to weighted averaging to achieve pressure toward centrosymmetricity. Weighting was 0.75 of po(r) + 0.25 of po(ro - r) with respect to a pseudocenter at the center of mass (ro). Results are shown in C. The signs indicated for the zero order, and the six regions of the transform between zeros are (+) - + - + - + (the "alternate" structure of Worthington and McIntosh, 1974). 506 5BOPHYSICAL JOURNAL VOLUME 25 1979  ture is achieved. Further refinement cycles tend to generate more perfectly centrosymmetric structures. The resulting structure (Fig. 6 B) was similar to one of the solutions obtained by Worthington and McIntosh (1974)-their "alternate" structure. It was possible to incorporate knowledge of centrosymmetry in the final stages of refine- ment by defining a new p(r) composed of 75% of the original value and 25% of the sym- metrically related value of p(r) (Fig. 6 C). This constraint allows the phases to refine to the correct values, whereas forcing them to be signs, + or -, would not. In some cases (see Fig. 7) the myelin refinement yielded noncentrosymmetric solutions. This is to be expected in light of the greater degrees of freedom available with general phases. Application of the real-space averaging procedure (above) resulted in an alterna- tive symmetric solution to that of Fig. 6. In all, four different centrosymmetric solutions for the myelin structure were obtained that matched our criteria for an acceptable solution. They correspond to the four possible permutations of - or + sign choices for regions V and VI of the observed transform (see below), and were similarly identified as possible am- biguities by Worthington and McIntosh (1974); two of these solutions were favored. This raises the question as to why there should be four homometric solutions, and ultimately shows that two of these structures cannot be distinguished without more information than is contained in the one continuous transform. The other two structures can in theory be (b) cFb ~~ ~~~~~~~-0 -50 0 |- lo 160 200l\ 30 -~~~~- Rs4% Rt-.3% (c) 0. FIGURE 7 Another of the four solutions to the myelin transform, (a), after 40 cycles of refinement, is noncentrosymmetric, although the structure retains symmetric features of the structure permuted in cyclic order. Pressure toward centrosymmetricity (using 0.75/0.25 weights) leads to a symmetric struc- ture with somewhat better agreement between P(r) and Q(r) (B), (R p = 1.3%). The phases derived from refinement (C) indicate a sign choice (+) - + - + + - for the zero-order component, and the six trans- form regions between zeros. STROUD AND AGARD Membrane Profile Determination 507 eliminated, but this requires essentially perfect data, or data from different species as was used by Worthington and McIntosh in their analysis. Worthington and McIntosh (1974) identified six zeros (I F(s) 2 = 0.0) in the observed transform out to a resolution of 7 A (s = 1/ 14 A). They identified seven regions of the transform separated by the six zeros, which were labeled regions 0, I, II, III, IV, V, and VI. The ambiguities in solution by the iterative Fourier method are directly associated with the phases (or signs) of regions V and VI. Sign changes in the latter region cannot be discrim- inated without additional information beyond that contained in a single continuous trans- form from one species. Each region of the continuous transform contributes a Fourier component to the electron density synthesis, and in each case, the contribution of just one region is a wave packet that resembles a cosine curve of frequency determined by the s value where the local value of F(s) is maximum multiplied by a smooth envelope that is qualitatively like either a Gaussian or a zero-order Bessel function. Looked at another way, each region of the trans- form F(s) is a delta function at the s value of the center of the peak, convoluted with the profile of the region. The contribution to electron density synthesis is thus the product of a cosine wave with a Gaussian-like envelope. The width of the envelope is inversely pro- portional to the width of the transform region. Each one of the regions 0-V of the transform is sufficiently narrow (<0.013 A- l) so that its contribution to the density synthesis extends beyond the constrained thickness of the structure. Region VI, on the other hand, is at least 0.02 A-1 in width, and its contribution to the resulting electron density synthesis is confined well within the thickness (180 A) of the profile (Fig. 6). There is no pressure on the refine- ment process to phase this region, since its contribution to the profile outside the bound- ary is too small to be significant. As a direct consequence of this fact, the autocorrela- tion function is the same as the Patterson synthesis for both structures when computed using either sign for the data of region VI. Worthington and McIntosh (1974) presented two structures for frog sciatic nerve myelin that differed in the same way-namely, in the choice of sign for region VI of the profile. Their results were obtained by direct methods of deconvoluting the Patterson function, based upon the theory of Hosemann and Bagchi (1962) and reconstruction methods based on the Shannon (1949) sampling theory applied to the sampled transform data. The transform of region V is approxi- mately the same size as the total width of the double bilayer structure and because it is rather small, its sign cannot practicably be determined from a single specimen scattering pattern alone. Different sign choices from those we found (the same as discussed by Worthington and McIntosh, 1974) have been proposed earlier by Caspar and Kirschner (1971), and by Blaurock and Worthington (1969) using different data. Our analysis (which would not allow these solutions) applies strictly to the data set used. DISCUSSION From the applications of the Fourier method to continuous scattering data, the following generalizations can be made: (a) The starting model for initiating refinement is arbitrary, and its only function is to generate a set of phases, 0(s), which roughly correspond to a model centered somewhere near the arbitrary ro. The initial model may be useful in testing different types of structure. 508 BIOPHYSICAL JOURNAL VOLUME 25 1979  (b) Successive cycles of refinement from the starting model always lead to convergence on a solution, usually after 15-40 cycles. (c) In every case we tested from artificially synthesized data, F(s) I, one solution of a very small number was correct, suggesting that in general the number of solutions will be small. The difference between different solutions was often small and probably reflects the lower pressure to phase the lowest amplitudes in the transform F(s), since they contribute least to the p(r) synthesis (Eq. 1). Often a unique solution is found. (d) If the assigned boundary condition, w, is chosen to be too small, refinement can never adequately lead to fitting even the first minimum of the transform. If w is too large (several times the correct thickness) then refinement is slower, since the pressure to produce alterations in the structure at each cycle is diminished. (e) Alternate structures can be identified by the Fourier method where they exist, whereas they cannot necessarily be found by direct methods (King, 1975), or by normal methods of deconvolution of the Patterson function. In those cases where more than one solution exist, one cannot be certain that all possible solutions have been obtained when using only a limited set of starting models. It is possible, however, to test various classes of solutions by appropriate choice of starting models. Since most asymmetric membranes are based on a bilayer structure, chemical reasonableness may be a basis for preference rather than proof. Nevertheless, our concern was primarily with the location of asymmetric protein or other components at relatively low resolution in a simple structure based on a bilayer. Is the protein localized only on one side of a membrane, for example, or does the protein traverse the lipid bilayer in any particular case? The method can be extremely powerful in determin- ing the range of possible solutions in these cases. The essential component of the Fourier refinement method applied to continuous diffrac- tion is the assumption that the scattering object is of finite, rather than infinite, dimension. This condition is applied by the imposition of a finite size to the density profile. This re- straint allows convergence. It also explains the pressure to converge on a definite solution, since it forces all Fourier components used to calculate p(r) at each cycle to account cor- rectly for a constant solvent density outside the boundary limits. Since the calculated am- plitudes Fe(s) must match the observations Fo(s) , the phases must account for the fea- tureless solvent region. Thus, the method is essentially refinement of the solvent density for r = - toa;bto c. Alternative Approaches to Solution With the exception of our own studies of acetylcholine receptor membranes (Ross et al., 1977) and the work on purple membrane that is only slightly asymmetric (Blaurock and King, 1977), solutions to membrane profiles, p(r), have so far only been attempted for centro- symmetric structures, i.e., cases in which the repeating unit consists of a pair of membranes generated by the folding over of a cell envelope or artificial lipid bilayers without protein. The problems associated with phasing of lamellar diffraction from symmetric structures have been approached in several different ways. A number of methods for deriving the elec- tron density profile have been proposed, and most of these depend on a deconvolution of the Patterson function P(r). Several developments of the deconvolution method of Hosemann and Bagchi (1962) have been used. These and other methods have been critically analyzed by Worthington, et al. (1973). Since then Moody (1974) has proposed other methods for STROUD AND AGARD Membrane Profle Determination 509 achieving deconvolution in the particular case where swelling experiments can be used to map the continuous transform from multilayer systems. Luzzati et al. (1972) also proposed a pattern recognition approach based on interpretation of even higher order autoconvolutions of F(h) in terms of higher powers of p(r). In the case of asymmetric membrane profile analysis, general deconvolution of the Patter- son function has been proposed, but to this point remains untested. A direct method based on the Hilbert transform (Papoulis, 1962; King, 1975) has recently been applied successfully to the purple membrane profile, but only one of the two possible solutions could be found (Blaurock and King, 1977). A general and severe problem with these and other methods providing a single solution from a single data set is the uncertainty that always remains as to whether other solutions exist, and if so, how many might equally well fit the data. Blaurock and King (1977) also presented a trial and error method which, as with our method, uses knowledge of the membrane thickness. The continuous transform was sampled at intervals determined by a hypothetical unit cell. By using all possible phase combinations for these "reflections," the continuous transform was reconstructed by the Shannon method (1949) and compared to the observed transform. The observed and reconstructed functions would be identical (within data errors) for a correct phase choice. This process is equivalent to choosing a combination of phases that will generate a p(r) contained entirely within the "unit cell" (our constraining thickness, w). As our and their approaches employ the same physical criterion for determining a solution, they would provide identical solutions if the trial phases were sampled at infinitesimally small intervals. The method of Blaurock and King (1977) requires that all possible phase choices be tried, and thus in principle, it can obtain all solutions. Within the bounds of practicality, all phase combinations cannot be tried. In an effort to circumvent this problem, the authors chose the rather coarse interval of 7r/8 between trial phases to limit the number of tests. Even so, this results in 16n Shannon reconstructions for n reflections. For purple membrane where a 50 A "unit cell" was chosen, only two reflections were involved at the observed resolution (196 trials). When larger structures such as the acetylcholine receptor (110 A actual thickness, Ross et al., 1977) are considered, where higher resolutions are required, or where smaller phase intervals are used, the computation involved would be prohibitive. Application to the acetylcholine receptor, for example, would require 169 = 6.9 . 10'0 trials and is clearly impractical. In our Fourier refinement technique, consistent sets of phases are jointly developed in a convergent stochastic process, and are not restricted to be multiples of Zr/8. In our approach it is also possible to incorporate other knowledge about the structure (e.g. limits on allowable den- sities, centrosymmetricity, etc.) whenever it is available. Asymmetric Membranes and Continuous Diffraction It is possible in many cases to prepare asymmetric membrane fractions from specialized tissue, which contain a highly enriched biochemical function. Membranes containing acetylcholine receptors, for example, have been prepared from the electric organ of the elec- tric fish Torpedo californica. Other fractions prepared from the same organ show that separate patches contain other biochemical functions such as acetylcholine esterase and ATPase. In this system, at least, it would seem that specialized functions are contained within specialized regions of the synapse (Cohen et al., 1972; Duguid and Raftery, 1973). In the case of acetylcholine receptor membranes, it is clear that there is a structural asymmetry 510 BIOPHYSICAL JOURNAL VOLUME 25 1979 to the membrane and that receptor molecules are all aligned in the same way, and sometimes organized into an array within each sheet (Ross et al., 1977). This type of situation is also known in bacterial systems, where the purple membrane of Halobacterium halobium has been shown to contain an organized, directionally polar array of bacteriorhodopsin molecules (Henderson, 1975). Thus we expect there will be a large number of systems such as these where the Fourier method of structure determination will initially provide valuable informa- tion on the distribution of macromolecules in a membrane. The necessary X-ray data may be generated from vesicle dispersions. Such X-ray scattering patterns have been acquired in several cases for membranes isolated from cells or organelles (Engleman, 1971; Wilkins et al. 1971; Lesslauer, et al., 1971; Lesslauer and Blasie, 1972; and Blaurock, 1973). In some cases centrifugation or drying of vesicles has been used to record lamellar diffraction (Finean and Burge, 1963; Dupont, et al., 1973; and Worthington and Liu, 1973), and some of the complex problems associated with the statis- tical arrangement of stacked asymmetric structures were detailed in the latter presentation. The iterative Fourier method can be simply applied to continuous scattering data and avoids the complex problems of stacking disorder. It is a simple procedure and can identify and refine possible solutions for the electron density profile from either asymmetric or sym- metric membranes. Since this Fourier method deals directly with IFo(s)j, rather than higher powers of Fo(s)I, as in deconvolution of the Patterson function by the method of moments or recur- sion, the solution is less sensitive to errors in the data. In a much broader context it is clear that any solution to structural problems using diffraction techniques can be improved when physical and chemical constraints or non- crystallographic symmetry relationships can be applied. Such constraints can best be applied in the electron density domain. Working in the Patterson synthesis, or with intensi- ties F(s) 12, or with higher order convolutions of p(r) inevitably makes the analysis more complex and does not easily allow for iterative improvements in the result. 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