The Structure of Human-Plasma Low-Density Lipoprotein B : An X-Ray Small-Angle Scattering Study

Fur. J . Riochem. 82. 73-90 (1978)

The Structure of Human-Plasma Low-Density Lipoprotein B An X-Ray Small-Angle Scattering Study

Karl MULLER and Peter LAGGNER

Institut fur Rontgenfeinstrukturforschung der Osterreichischen Akademie der Wissenschaften und des Forschungszentrums Graz

Otto CLATTER

Institut fur Physikalische Chemie der Universitat Graz

Gerhard KOSTNER

Institut fur Medizinische Biochemie der Universitat Graz

(Received April 8. 1977)

1 . The X-ray small-angle scattering of human plasma lipoprotein B of the low-density fraction ( Q = 1.016- 1.060 g . ~ m - ~ ) has been recorded to high precision at different electron density contrasts.

2. The overall structure of the particles is characterized by a quasi-spherical shape and radial symmetry. A maximum diameter of 23 nm and a molecular weight of 2.4 x lo6 have been determined.

3. The internal structure is described in terms of a model consisting of spherical layers with different electron densities indicating that the neutral lipids are arranged in the core of the molecule up to a radius of about 8 nm surrounded by a monolayer of free cholesterol, phospholipids and protein. The neutral lipids are shown to be in an ordered, liquid crystalline state at 4 “C and to undergo a thermotropic transition into a disordered state at higher temperatures.

Low-density lipoproteins play a central role in lipid metabolism as the major transport vehicles of cholesterol in the human organism [1,2]. Their apparent pathophysiological significance in the develop- ment of atherosclerosis [3 - 51 underlines the need for a thorough characterization of this macromolecular 1ipid.protein complex to provide a basis for the establishment of structure-function relationships.

Numerous physico-chemical studies (for recent reviews see [6 - 91) have led to the generally accepted idea that low-density lipoproteins form highly isotropic, sphere-like particles of about 22-nm diameter. More controversial are the current ideas with respect to the internal structure of low-density lipoproteins. In attempts to define the spatial relationship of the individual lipid and the protein constituents within the complex. various techniques have been applied and the results provided a wealth of information about the microenvironment of the molecular components. However, they are difficult to integrate into one consistent model of the molecular architecture of low- density lipoprotein without resorting to speculation.

The method of X-ray small-angle scattering [lo, 1 I ] in principle provides both information on the overall shape of particles as well as their internal structure when the scattering functions are recorded at different electron density contrasts [12]. From the fact that low- density lipoproteins are macromolecular complexes consisting of discrete molecular components of different electron densities the scattering information is highly complex and not easily interpreted unequivo- cally; all the former X-ray studies [13 - 151 on low- density lipoprotein structure have introduced certain a priori assumptions in the data analysis with respect to either, the ‘allowed’ electron density levels within the particles [15] or the overall shape [14]. By adopting a suitable set of theoretical considerations and conditions [16] Tardicu et al. arrived at a quasispherical structure of low-density lipoprotein [15,17], in which the surface shows convolutions between long spikes (8 nm). A maximum diameter of 29 nm and a solvent penetration reaching to 5.5 nm far from the center were reported for this structure. Apart from suggesting considerable internal electron density fluctuations these

74 Structure of Low-Density Lipoproteins

authors leave the structural organization within the core of low-density lipoprotein largely open to question. By applying small-angle neutron scattering which offers a much wider range for contrast variation by choosing different H,O/’H,O mixtures as solvents, Stuhrmann et al. [I81 were also able to establish the quasi-spherical shape of low-density lipoproteins and the coincidence of the centers of gravity of the hydrocarbon and the polar regions. However, the resolution of structural details for the particle interior was less desriptive than in [14]. The results of both Tardieu et a/. [15] and Stuhrmann et al. [18] were in strong contrast to the previously postulated spherical lipid bilayer model of Mateu et al. [13] with some protein in the center of the particle, and also to the model based upon NMR spectroscopic data by Finer et nl. [19]. Deckelbaum et 01. [20,21] first reported the observation of a reversible calorimetric transition of low- density lipoproteins which correlates to changes in the X-ray small-angle scattering pattern. A quantitative analysis of the temperature dependence of the scattering curves carried out independently by Atkin- son et 01. [22] and by our group [23] showed that this effect relates to a liquid crystalline transition of cholesteryl esters in the particle core.

In this article we present an extensive set of experimental data which permit upon careful analysis a detailed structural interpretation of the molecular architecture of low-density lipoproteins. In contrast to previous X-ray studies radial symmetry of the entire structure was not only assumed but is in fact shown to be the predominant principle of the molecular arrangement in low-density lipoproteins.

Lipoprotein B is the dominant lipoprotein species of the low-density lipoprotein class and. in addition to being a narrow cut of the low-density lipoprotein spectrum, is homogeneous with respect to its immuno- chemical characteristics [24]. Its X-ray small-angle scattering curves were measured to high precision in solvents of different electron density.

Two different approaches were taken in the data analysis. Firstly, the comparison of several characteristics of the experimental data with chemically and physically plausible model structures. A novel method of calculating the real-space distance distribution function p ( r ) directly from the experimental data [25] proved to be a powerful tool for this approach. Secondly, the separation of the characteristic scattering functions of the excluded volume reflecting the particle shape and the internal electron density distribution [12]. These two conceptionally different methods were used to extract the maximum amount of reliable information from the data. The advantages and limitations of both approaches were critically evaluated and the particular usefulness of the p ( r ) function in X-ray small-angle scattering experiments has been thus demonstrated.

EXPERIMENTAL PROCEDURE

Isolation and Characterization of Lipoprotein B

Lipoprotein B from pooled sera of 10 male volun- teers in the age range between 19 and 24 years was isolated by sequential ultracentrifugation at increasing densities. The individuals were normolipemic and apparently healthy. All sera used for the preparation of lipoprotein B were negative or only very weakly positive for lipoprotein (a), tested as described earlier [26]. Immediately after sampling the blood was allowed to clot at room temperature for 6 h and centri- fuged at 3000 xg. 1 mg NaN, and 0.5 mg Na,EDTA were added per 1 ml serum and the density adjusted to 1.030 g/ml by adding solid NaC1. Very-low-density lipoproteins and low-density lipoprotein-( 1) were separated by spinning at 120000 x g for 20 h at 16 “C (Beckman preparative ultracentrifuge L-111 50 with fixed-angle rotors of type 50 Ti or 60 Ti). Floating lipoproteins were removed by tube slicing and the in- franatant adjusted to a density of 1.065 glml by adding solid NaCI. The low-density lipoproteins were isolated by ultracentrifugating for 22 h at 125000 x g and 16 “C. The floating low-density lipoproteins were washed once at a solvent density of 1.025 g/ml and once at a solvent density of 1.055 g/ml. Following the last flotation, the low-density lipoprotein fraction was dialyzed against a 0.15 M NaCl solution buffered to pH 8.0 with 0.05 M Tris-HCI. This fraction was further purified by agarose gel column chromatography. 10-ml aliquots containing 150 -200 mg of lipoprotein B were passed over a column (100 x 2.5 cm) packed with BioGel A-5m (Bio Rad, Richmond, Cal.). Elution was performed with 0.1 M NaCl solution, pH 8.2 buffered with Tris-HCI. The material of several runs eluting in the main peak as indicated in Fig. 1 was combined and concentrated by dialysis against poly- ethylene glycol ( M , 20000, Fluka, Buchs, Switzerland) to a concentration of 50-70 mg/ml. This fraction was then dialyzed exhaustively against the working buffer (9 g NaCI, 1 g NaN,, 0.5 g Na,EDTA, 50 ml 3 M sodium phosphate buffer pH 7.5, to 1000 ml with bidistilled H,O).

The isolated lipoprotein fraction was analyzed in the analytical ultracentrifuge. In order to check for the possible presence of aggregates the 5 lipoprotein B solution was sedimented in working buffer or floated in a NaCl solution of Q 1.063 g/ml using the 4“ sector cell with a thickness of 3 mm. One single symmetrical peak in both cases was observed. Fig. 2 shows the schlieren pattern of lipoprotein B (c=15 mg/ml) in NaCl solution (e = 1.063 g/ml).

The composition of lipoprotein B isolated according to the outlined procedure is summarized in Table 1.

To further confirm the purity of lipoprotein B, agarose gel electrophoresis and electrophoresis in 3.75% polyacrylamide gel were performed. In both

K. Miiller, P. Laggner, 0. Clatter, and C. Kostner 75

0.2 1 4

4: : I 0.1

0 \ 0 100 200

ELuate ( m l )

Fig. 1. Elution paltern oJ low-ilensrty lipoproteins from Biogel A-5m column. 10 ml of the density fraction 1.025 - 1.055 g/ml containing approximately 150 mg of lipoprotein were applied to the column ( 1 0 0 ~ 2 . 5 cm) and eluted with 0.15 M NaCl solution buffered with Tris-HCI to pH 8.0.

Fig. 2 . Pattern qf lipoprotein B in the analytical ultracentrifuge. Lipoprotein B (15 mglrnl) was dialyzed against a NaCl solution of e=1.0630 g.cnC3. Speed: 52000 rev./min; the pictures were taken 60 rnin after the start at an angle (schlieren angle) of 65" for four different preparations.

cases only one band in the appropriate position was observed and no material remained at the origin. The lipoprotein B sample was also tested immunochemi- cally for the presence of different apolipoproteins using specific antisera to lipoprotein A, lipoprotein B and lipoprotein C polypeptides. The antisera were the same as used in previous investigations [24,27].

Lipoprotein B was applied to the immunodiffusion plates at different concentrations. The only reaction observed was that with antiserum to lipoprotein B. None could be seen with antisera to apoAI, apoAII, apoAIII, apoCI, apoCII, apoCIII, 'arginine-rich' poly- peptide or lipoprotein (a).

Physicochemical Data

The flotation coefficient sf of lipoprotein B was 4.5 f0.2 S, corrected for viscosity; the molecular weight as obtained from ultracentrifuge analysis according to Yphantis [28] yielded 2.28 x lo6. The partial specific volume of lipoprotein B was determined by highprecision density measurements [29,30] using both a commercially available digital density meter DMA 50 (Anton Paar, Graz, Austria) and a modified DMA 601 density meter with two oscillators operating in the reference mode [31].

The apparent partial specific volumes in the working buffer measured within the concentration range of 0.2 > c > 0.01 g/cm3 were extrapolated to zero concentration and the values obtained were 0.952 and 0.977 cm3.g-', at 4 "C and 37 "C respectively. Recent experiments indicated a considerable dependence of this parameter on solvent composition (Laggner et al., unpublished).

The values in solvents of increasing sucrose concentration used in the present work are summarized in Table 2.

Samplesjkr the X-Ray Measurements

The samples for the actual X-ray measurements were prepared from a stock solution of lipoprotein B containing 0.208 g lipoprotein B/ml in the working buffer. Lipoprotein concentration was determined by dry weight after the solution was dialyzed exhaustively against an unbuffered NaCl solution (0.15 M). This solution was directly used for the measurements as well as dilutions of 0.104 g/ml, 0.050 g/ml, 0.025 g/ml and 0.012 g/ml. The solvent electron density was adjusted by adding adequate amounts of a saturated solution of sucrose to the stock solution. The concentrations of these solutions were below the original concentration, as determined by protein estimation according to Lowry et al. [32], standardized by dry weight of lipoprotein B.

High-precision X-ray measurements were carried out in solutions with a content of 0, 5, 11, 25 and 55 "/I


Table 1 . Molecular data of lipoprotein B The proportions of the components were determined by chemical analysis of lipoprotein B. The total molecular weight was determined by absolute intensity measurement. N,=number of lipid molecules in one molecule lipoprotein B. V,=volurne of the single lipid molecules, derived from the individual partial specific volumes; V,=total volume of each component; the total volume given for lipoprotein B does not allow for any solvent interaction or hydration. u2 =partial specific volume at 20 "C; the specific volume of pure cholesterol is 0.938 cm3.g-' (Handbook of Chemistry & Physics, 1971) but its apparent partial specific volume in a hydrocarbon environment is substantially higher [61]; 1'2 for triglyceride is the experimentally determined value for triolein at 20 "C. uz for total lipoprotein is the weighted average of the partial volumes of the individual components. The value agrees with our experimental findings when 02 is extrapolated to zero salt concentration from an NaCl concentralion series. zi is the electron concentration; N, is the total number of electrons; e, is the electron density calculated according to Eqn (4). The mean electron density of lipoprotein B was calculated as a weighted (volume fraction) average of the individual values (considering no solvent interaction and hydration).

Component Proportion M , Individual Ni V , Vl Fraction zi Ne e, M ,

O; (wiw) nm3 "d (v/v) cm3,g-' g - e . g- ' e . nm13

Protein 20.8 5.0 xi05 - - - 597 15.7 0.72 [59] 0.535 2.68 x lo5 447 Phospholipids 21.6 5.2 1 0 5 780 670 1.26 844 22.2 0.98 [60] 0.553 2.88 x lo5 340 Cholesterol 10.6 2.5 x 105 387 645 0.635 409 10.7 0.97 0.558 1 . 3 9 ~ 10s 347 Cholesterol ester 42.4 10.0 x 105 650 1540 1.14 1759 46.2 1.04 [60] 0.560 5 . 6 0 ~ lo5 309 Triglyceride 4.6 1.1 xi05 880 125 1.60 199 5.2 1.09 0.560 0.62 x lo5 309

~ _ _ 100

~ ____ ~ .- .- ~. .

Total lipoprotein 100 2 . 4 0 ~ lo6 - 3000 - 3800 100 0.956 0.553 1 . 3 2 ~ lo5 348

(w/w) of sucrose. These solvents are subsequently referred to by use of the letters a, b, c, d and e, respectively. In addition, for the determination of the radius of gyration, R , and the scattering intensity at zero angle, experiments were performed in solvents containing 4, 12, 20, 35 and 50% (w/w) sucrose. For the corresponding electron density levels of these solvents expressed in (electrons per cubic nano- meter) see Table 2.

X - Ray Measurements

The measuring procedure was performed in the same way as previously described [14,33] using a Rigaku Denki RU 500 rotating anode generator (Cu target, 50 kV, 400 mA) specially modified [34] for the use with a slit collimation system [35]. Scattering intensities were recorded by a proportional counter with pulse-height discrimination operated by an elec- tronic step scanner [36] in the angular range of 8 x < h < 2.0 nm-' (17 = 4n/2 sin 0, 0 = half scattering angle, i = 0.154 nm) at about 100 individual angles with a constant number of 5 x lo5 pulses accumulated at each angle. The blank consisting of the solvent-filled capillary was measured to the same accuracy and subtracted from the sample scattering. Samples were kept at constant temperature during the measurement by means of an electronically controlled Peltier cuvette [ 3 7 ] .

RESULTS

Fig. 3 shows the scattering curves of lipoprotein B at various contrasts. These curves have been corrected

for wavelength, slit length and slit width effects using the new indirect Fourier transformation method described in the Appendix.

The most significant features of these curves are as follows.

a) A number of well-resolved side maxima indicated that the lipoprotein B solutions were sufficiently monodisperse, and that the particles have a highly isometric shape.

b) The relative intensity of the first side maximum is extraordinarily high as compared to the main maximum at zero angle (1 5 "/, in solvent a) suggesting considerable electron density fluctuations in the interior of the particles.

c) The intensity of the outer scattering region and in particular the broad maximum at h=1 .7 nm-' is practically independent of the solvent electron density. This indicates that the scattering in this range is predominantly caused by internal electron density fluctuations whereas the contribution from the overall shape is negligible.

Since the total scattering is in principle a composite effect of the overall shape and the internal structure, a dissection of these different scattering contributions and the separate analysis of each would be the most refined approach for obtaining detailed structural information (see below).

Prior to such attempts, however, it is highly in- formative to discuss the distance distribution functions p ( r ) at different electron density contrasts. This function merits priority in the course of data analysis since it contains the entire real space information that can be extracted from a scattering curve, and because it is directly obtained from the experimental data by

K. Miiller. P. Laggner, 0. Glatter. and G. Kostner

A 0 C

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0 1 2 0 2 0 1 2

0 1 1 2 2 0 1 2 0

1 o4

10'

1 oo md 0 2 0 1 2 0 1 2

1 o4

1 0' 1 0'

loo loo 1 oo

e Tsl 0 1 2 0 1 2 0 1 2

Fig. 3. The e-xperimental scattering curi'es ( A j are compared with the theoretical scuttering.functions oj'the radial model ( B j and the correspond- ingfunctions o f the subunit model ( C ) a f uarious contrasts IN, h. c . d , e j , The intensities are given in relative units, normalized to equal primary energy. solute concentration. absorption coefficient. thickness of the sample etc. thus being directly comparable. h = ( 4 n ' i ) sin 0, where A = wavelength of the radiation (CuK a : 0.154 nm) and I) = half scattering angle

Fourier transformation thus being unaffected by errors that necessarily arise in intermediate data mani- pulations, such as the separation into characteristic scattering function (see below and Appendix). A certain disadvantage of the p ( r ) function lies in its rather abstract nature since it is also a composite function of particle shape and internal structure analogous to the scattering curves themselves. Never- theless, a thorough discussion of p ( r ) at different contrasts allows important insights into the particle structure.

Oivrall Structure and Pnrticle Size

Fig. 4 shows the p ( r ) functions of lipoprotein B at different electron density contrasts. The maximum particle dimension can. by definition, be inferred from

the point at which p ( r ) finally vanishes. However, this holds only for exact scattering functions. In the prac- tical case, fluctuations about the abscissa of varying amplitudes are observed at high values of r . One of the possible causes for these fluctuations, the termination error, is largely suppressed by the indirect Fourier transformation procedure described in the Appendix. Other errors with similar effects can be caused by the concentration dependence of the scattering curves at low angles, due to interparticle interferences. Routi- nely, these portions of the curves were measured at several dilutions and extrapolated to zero concentration. There remains, however, some uncertainty in this process. With the help of the indirect Fourier transformation these problems can be reduced by cutting off the concentration-dependent portions of the curves, which in the case of lipoprotein B are rather short


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Fig. 4. Distance distribution frincrioru p(r) c~cilrulated tq. ineans oJ indirect Fourier transformation procedure. The p ( r ) functions derived from the experimental data (A) are compared with those calculated for both the radial (B) and the subunit (C) model at various contrasts (a -e)

(h<0.2 nm-'). This way the p ( r ) functions remain essentially unchanged, but the oscillations at larger distances are considerably reduced. By this method, using the results obtained at different contrasts a maximum intraparticle distance of 23 - 25 nm can be defined.

Alternatively, information about the overall particle dimensions is obtained from the radius ofgyration at different contrasts. This parameter was obtained in two different ways: (a) from the p ( r ) function according to Equation (15) (Appendix) and (b) by the Guinier approximation [38] :

(1)

from the slopes in the very lowest angular range of the scattering curves in the lg I vs h2 plot, according to

kh) = I, . e-'4R2hL lam h + O

In particular at very low contrasts, in solutions b and c, where very large absolute values of R are obtained, the evaluation of R from the p ( r ) function is far more reliable than the graphic method according to Eqn (2), since in these cases the Guinier approximation holds only for a very limited angular range'. The resulting R values as a function of contrast are graphically displayed in Fig. 5A. The numerical values obtained by the two different methods are compared in Table 2 . The strong dependence of R on the electron density contrast A@ makes it impossible, without any further knowledge, to use any given R value as a direct

~~ ~ ' Systematic model calculations with spherical model structures have shown that R can be estimated from Guinier approximation withanaccuracyofbetterthan f 10";onlyin therangeof(h.R)< 1.0 and for an accuracy of better than 504 (h .R) must be lower than 0.7. Our experimental curves start with a value of ( h . R ) = 1 . 5 in case b.

K. Miiller, P. Laggner. 0. Clatter, and C. Kostner 79

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2 . 1 0 ~ B

a 0 a

a A P N

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I a,

4.10' I C

I \---k3 Fig. 5 . Dependence of the radius of gyration R on the solvent electron density e , or the contrast A @ , respecrively. in different plots. (A) R vs e,; the perpendicular dotted line allows a rough estimation of the mean electron density of lipoprotein B P o , since R = k m when e ,=Qo ( A Q = 0). The horizontal assymptotic trends (horizontal dotted line) indicate R at infinite contrast, R , , which is a direct measure of the particle size. (B) R z . A e vs A @ ; the full line corresponds to the radial model based upon Fourier transformation of the scattering amplitudes with altering signs. Dotted line=subunit model. (C) R2 vs ( A @ ) - ' ; thc course of this function is controlled by the R values at low contrast which are of less accuracy

measure of particle size. However, by extrapolation to infinite contrast according to the equation

(3)

the radius of gyration of the excluded volume R, can be estimated. This extrapolation yields a value of R,=9 nm (Fig. 5C) which is in good agreement

with the value reported bv Tardieu [15], but somewhat higher than the value obtained from neutron scattering experiments [18]. Assuming spherical overall shape for the particle, this value corresponds to a radius of 11.6 nm.

This result bears some importance in considerations regarding particle shape, since a spherical particle would have the smallest maximum dimension consistent with the results. Estimation of the maximum dimension of lipoprotein B from the p ( r ) function (see Fig. 4) has yielded values between 23 and 25 nm. This comparison thus shows that lipoprotein B must be a highly isotropic, sphere-like particle.

Evidence in support of a quasi-spherical shape of the lipoprotein B particle is also obtained from an analysis of the p ( r ) function at high electron density contrast (solution e). Whereas at lower contrasts p ( r ) is dominated by the internal electron density distribution as reflected by the strong fluctuations at low values of r, at 550/, sucrose concentration p ( r ) is a smooth function indicating that at this contrast the internal structure does not contribute significantly to the scattering. In fact, the p ( r ) function of lipoprotein B at high contrast is practically coincident with the theoretical p ( r ) function of a homogeneous sphere. It should be noted here that the p ( r ) function is not extremely sensitive to small deviations from ideal sphericity. The results exclude, however, the possibility that the structure deviates from a sphere by more than 5 % in the axial ratio. Deep convolutions of the surface as postulated by Tardieu [15] are not consistent with the present results since they would invariably lead to a shift of the p ( r ) maximum towards smaller values of r.

Internal Structure

The strong fluctuations in the p ( r ) function between positive and negative values at low electron density contrast and their striking dependence on the solvent electron density, as well as the contrast-dependence of the R values, indicate a distinct compart- mentation of regions with different electron densities within the lipoprotein B particle. Furthermore the fact that the experimental scattering curves at larger angles are almost independent of the contrast shows that the non-contrast-dependent term of Eqn (5) (see next section), I s , dominates the total scattering at least at larger angles. An exact evaluation of the internal structure requires the precise knowledge of this function I , . This function is equal to the scattering curve recorded at zero contrast, or extrapolated to zero contrast, respectively. Since this is difficult to achieve (see also next section), it seemed worthwhile to con- sider the scattering of significant model structures and to compare their theoretical scattering functions, the p ( r ) functions and the R values to the experimental


Table 2. Experiniental X - r y &a e, is the electron density of the individual sucrose solutions at 4 "C. u2 is the partial specific volume of lipoprotein B in the actual solvent at 4 "C; the asterisk indicates values obtained by extrapolation. & is the mean electron density of lipoprotein B (4 "C). calculated according to Eqn (4). 3Q is the effective contrast at 4 "C. R, the radius of gyration, was determined from the experimental data using Eqns (15) or (2). r,") = scattering intensity at zero angle determined according to Eqn (16). A ( " ) = scattering amplitude at zero angle: A ( " , = rt 1%; the actual sign corresponds to the sign of the net contrast

Solvent Sucrose e, "2 60 A e R from I 0 4 0 , R calc. for .~ -~ Eqn (15) Eqn ( 2 ) radial subunit

model model

0 334.5 4 339.2

5 340.5

11 347.7 12 348.9 20 359.1

25 365.5 35 379.4 50 403.0

55 412.0

~~ _ _ _ _ _ ~ ~

cm3 . g-1

0.952 0.957

0.959

0.967 0.968 0.976

0.981 0.986* 0.991*

0.994*

~

e . nm-3

349.7 +15.2 347.8 +8.6

347.1 +6.6

~ ~ -

344.2 -3.5 343 9 -5.0 341.1 -18.0

339.3 -26.2 337.6 -41.8 3359 -67.1

334.9 -77.1

nm

13.8 13.2 440 15.9 15.9 139

18.9 15.5 72

11.8 8.0 21 2.4 2.0 50 4.0 5.1 630

6.0 6.2 1340 7.7 7.2 3 360 8.4 8.0 8 740

8.7 8.1 11 700

+21.0 +11.8

+ 8.5

- 4.9 - 7.0 -25.0

- 36.6 - 58.0 -93.5

- 108.1

nm

13.71 12.72 16.08 15.46

17.50 17.00

17.41 18.78 13.57 15.03 3.92 3.93

6.33 3.35 7.73 5.64 8.54 6.73

8.70 6.93

results at different contrasts. It is to be expected that, particularly in solvents b and c where de approaches zero, thep(r) functions should provide sensitive criteria for different internal structures.

In any case a spherical overall shape and a diameter of 23 nm will be assumed according to the above results. The inhomogeneous internal arrangement will be approximated by two different types of models: (a) models of an arbitrary distribution of regions with different electron density, and (b) models of radial symmetry also with respect to the internal electron density distribution. Furthermore the models must be constructed in accordance with the mean electron density of native lipoprotein B.

The Mean Electron Densit?. of Lipoprotein B. The mean electron density of a particle, given in e . nm-3, can be calculated from 5" (electron concentrstion in g- e . g - ' ) and the partial specific volume 112 according to

(4)

According to the chemical composition of lipoprotein B, Zo takes a value of 0.553 e .g- ' . It must be considered that v2 is dependent on solvent composition and varies from 0.952 cm3. g -' in the working buffer to 0.994 cm3. g-' in a 557; (w/w) sucrose solution. Consequently lipoprotein B takes different values of & in different solvents which will be important for the model calculations. Another method to obtain information about the mean electron density is to extrapolate the scattering amplitudes at zero angle recorded

at various contrasts to zero contrast. This method will be described in more detail in the Appendix.

The mean electron densities as obtained independently from these two methods are in good agreement and are summarized in Table 2.

Models with Arbitrary Internal Arrangement. Scat- tering functions of spherical models with 23 nm outer diameter have been calculated for the contrast using the &, and Q s values listed in Table 2. Regions of different electron density corresponding to the different polar and apolar constituents of lipoprotein B (see Table 1) have been approximated by spherical sub- structures of various size and electron density. By this method a wide variety of models with different non- radial electron density distributions have been calculated. The resulting scattering functions, p(r) functions and R values have been compared critically with the experimental scattering curves at the different contrasts. In general it was not possible to achieve a satisfactory fit with the experimental data; in particular, the relative height of the broad maximum at h = 1.7 nm-' could not be reproduced by these models. Also thep(r) functions, as well as the radii of gyration dependent on the contrast, showed only very little resemblance to the experimental results. The fit was found to be the worse the more asymmetrical the internal arrangement. Out of these models the one to fit best to the experimental scattering function consisted of a spherical core of 5-nm radius surrounded by 60 subunits of 1.7-nm radius in a regular arrangement (for a detailed description see Appendix). The model scattering functions and the p(r) functions are plotted

K. Miiller, P. Laggner. 0. Glatter, and G. Kostner 81

C d

0 5 10 300 mk 0 5 10

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-7

-10

0 1 1 0 1 0 1 0 1

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0 1 0 1 0 1 0 1 0 1

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Fig. 6. T/IC rc.su// o/ / / l o FourjcJr /r(~tlS/i,,’mtl/iOfI of’rhr scm/rring arnplitirdrs A(h) tire .sunrmcirized. The upper row shows ~ l i c d i a l l y averaged electron density distribution of lipoprotein B at 4 “C on an absolute scale. The dotted lines represent the solvent electron densities e,. In the middle, the scattering amplitudes are displayed. Below, the scattering functions calculated from the individual electron density profiles (which remain zero beyond the maximum radius) are compared with the experimental scattering curves

in Fig. 3 and Fig. 4. The R values are listed in Table 2 (see also Fig. 5 B).

Models with Radially Symmetric Internal Arrange- ment. These models are also characterized by spherical overall shape, 23 nm in diameter and mean electron densities at various contrasts as summarized in Table 2. However, in this case different electron density levels are confined to spherical concentric shells. By computerized optimization, models consisting of six zones of different electron density were found to be in good agreement with the experimental data. The optimal model has the following specifications: 0- 2 nm, 280 2-3.5 nm, 380 e-nm-3; 3.5-5.5 nm, 280 e .nm-3; 5.5-7 nm, 360 e.nrn-’; 7-9 nm, 310 e . r ~ m - ~ ; 9-11.5 nm, 370 e - r ~ m - ~ . The scattering functions, as well as thep(r) functions, at the different contrast show good agreement with the experimental results. Also the fit of the radii of gyration at different contrasts is satisfactory and the overall agreement with the experimental results is significantly better than for the models considered in the previous section. This result is further supported by Fourier transformation of the scattering amplitudes. The radial electron density distribution functions e ( r ) obtained are

plotted on an absolute scale in Fig. 6, for the different contrasts. It was found that these functions are in almost perfect agreement with the optimum model found by trial and error. This suggests strongly that the internal structure of lipoprotein B is dominated by radially symmetric electron density fluctuations. The details regarding the Fourier transformation of the scattering amplitudes are given in the Appendix.

The Characteristic Scattering Functions

According to Eqn (5)

I ( / ) , A @ ) = 1s + IFS. d@ + I F . (A@)’ ( 5 ) it is possible to decompose the scattering function /(/z, A Q ) of a particle with non-homogeneous internal electron density distribution into three characteristic terms: IF which is the scattering function of the excluded volume, I , which reflects the internal structure and the mixed term I,, [39].

IF and I, are scattering functions at infinite and at zero contrast, respectively, which can be analyzed independently, resulting in a separate description of surface structure and the internal structure.


A

0 1 2 3 i h (nm-')

Fig. 7. ( A ) The experimental scattering funcrion recorded to h = 5 nm is compared with the theorericai scattering ,function of the rudial model (dotted line). ( B ) The theoretical scattering curve of a spherical subunit model consisting of 60 identical, spherical sub- unirs of 1.75-nm radius with an orerall diameter n/ 23 nm is shown. The mutual distance of the centers of these subunits equals 3.9 nm. The entire scattering is caused by the subunits (scattering weight of the core=zero). The arrows D, indicate the regular nearest neighbor of the centers of the subunits and D, the radius of the subunit ( r=4 .5 ,hDl ) (581

We calculated the characteristic functions IF and I, by a least-square algorithm, taking into account the error limits of the experimental data. For the latter we used both the experimental scattering curves point by point at any contrast as well as the solution coefficients c, of the indirect Fourier transformation approximation (see Appendix). The latter method was found to provide the more reliable results. The resulting functions are shown in Fig. 8 together with the experimental scattering functions for solvents e and c which are comparable to the functions I , and I , .

The most important information from these calculations is that the function I s is practically identical with the experimental results in solvent c, and also agrees within reasonable error limits with the theoretical function I, of the radially symmetrical model. The excluded volume, function IF, deviates from the spherical model at larger angles. It is difficult to decide whether these differences are significant since in this region of low intensity the experimental errors are of comparable magnitude. Such differences in the outer part of the curves could be also accounted for by minute errors in the blank subtraction. It is clear,

1 .o 15 2.0 h (nm-')

Fig. 8. Churucreristic scarrering funcrions ( A ) I , and ( B ) I,. Full lines: theoretical course of the radial model; broken lines: I , and I , calculated from the experimental curves; circles: experimental curves in solution c and e.

however, that the maximum around 12 = I .7 nm-' is predominantly due to the radially symmetrical internal structure.

Consequently we did not attempt a further interpretation of the characteristic scattering functions.

Molecular Weight

The molecular weight of a particle can be determined from X-ray small-angle scattering data using the formula

M,=const. . I ( , ) . ( A Z ) - ~ (6)

where I(") is the relative scattering intensity at zero angle and d r the number of excess gram-electrons per gram of the substance compared with the solvent [40]. The constant is defined by the expression

const.=(I,. N , . P , . c . D . C 2 ) - '

where I , = Thomson factor of the single electron scattering = 7.90 x N A = Avogadro constant = 6.02 x loz3, Po = intensity of the integral primary beam normalized to the same units as I,,,, measured according to [41- 431, c= concentration of the dissolved sample in g . ~ m - ~ , D = thickness of the irradiated sample in cm, a = distance between the sample and the plane of registration in cm. Using the values of 2!2 (for the calculation of dz) and I, summarized in Table 2

K . Miiller, P. Laggner, 0. Clatter. and G . Kostner 83

we obtained an average molecular weight of 2.4 x lo6. The molecular weight of this preparation of lipoprotein B as determined by ultracentrifugation was 2.28 x lo6.

DISCUSSION

Since the characteristic scattering functions I , and I , obtained from the present results appear to be too unreliable to warrant their further structural interpretation, the following discussion will be focused upon the models considered in Results; an attempt will be made to decide in favour of either the radial model or the subunit model. This decision will be based upon a detailed comparison and discussion of the properties of their scattering functions, p ( r ) functions and radii of gyration at different contrasts.

The main features of the theoretical scattering curves (Fig. 3 ) of both models, i.e. the angular positions of maxima and minima, are in some qualitative agreement with the experimental curves, at least at low contrast. The relative intensities, especially at high contrast and in the region of the h = 1.7 nm-' maximum are, however, in considerable disagreement with the experimental data in the case of the subunit model; this maximum deviates to intensities which are 6 - 60 times lower than the experimentally determined intensities.

These differences are illustrated even more strongly in the corresponding p ( r ) functions (Fig. 4), where even at low contrast the subunit model function shows very little resemblance to the experimental data. Con- versely, at high contrast the p ( r ) functions of both models show the characteristics of a spherical particle. This is not unexpected since there the influence of the internal structure is already very low.

In Table 2 the contrast dependence of the radii of gyration for both models are compared. Apart from deviations at very low contrasts, the radial model is in good agreement with the experimental values whereas significant differences are observed for the subunit model along the entire range of contrasts (see Fig. 5B).

A further difference is observed with respect to an extrapolation of the scattering functions to larger angles. As described in the Appendix this is possible by means of the indirect Fourier transformation method within a certain angular range. However, it can be shown by numerous test calculations that the accuracy of the approximation of the extrapolated region to the theoretical course of the curve depends considerably on the model parameters. Extrapolation works best for particles of spherical symmetry, whereas deviations towards less isotropic overall shape as well as deviations in the internal structure reduce the possibility to extrapolate (Fig. 9). The experimental scattering curves have been found to behave very similarly

0 1 2

I h (nm-')

1 o2

H

1 oo

0 1 2

h (nm-'1

1 o4

L i lo2

roo

0 1 2

h (nm-')

Fig. 9. E?rrrupoltrrro~i o/ rhc scurrrring ,/unctions by rneuns of indirect Fourirr t l .u)~.?/~~.) i~uf io/ i . The dotted lines display the extrapolated course, the circles the exact course. The full spot marks the angular position of curve termination. The lower plot demonstrates the result of the same procedure for another spherical model consisting of only two regions of different electron density

to those of the radial model. The trend throughout these comparisons indicates that a subunit-like structure is much less capable of approximating the experimentally found scattering of lipoprotein B than a radial model. We thus conclude that the general structural principle of lipoprotein B is best represented by an overall radial symmetry.

To further confirm this concept, we computed the scattering functions and p ( r ) functions of numerous models with space-angle-dependent internal structure varying the model described in Results (subunits inside the particle, subunits of different electron densities etc.). Without exception we found even greater disagreement with the experimental data, unless the absolute electron densities of the subunits were raised to values (1000


e .nm-3) which cannot be reached by any chemical component of lipoprotein B. Moreover, even in the extreme case of infinite electron density for the subunits (scattering contribution of the remaining volume equal to zero, see Fig. 7B) the maximum at h=1.7 nm..' fails to reach the experimentally observed relative intensity.

Consequently, the lack of resemblance of the subunit unit model to the experimental data cannot be taken as resulting from the arbitrary choice of an inappropriate model but rat her is generally applicable for all models with space-angle-depedent internal electron density fluctuations. Although the presence of subunits or any other ((p,9) structure cannot be totally excluded, their contribution to the total scattering must be of low influence.

The foregoing considerations suggest a molecular model of lipoprotein B based upon the radial electron density distribution function e(r ) . Since the electron density levels are given on an absolute scale a direct relationship to the molecular components of lipoprotein B can be obtained. The electron density values of the individual constituents are summarized in Table 1.

From these considerations it may reasonably be deduced that the regions of low electron density (0-2.0 nm and 3.5-5.5 nm) dropping down to 280 e . nm-3 are occupied by the apolar hydrocarbon chains of the lipid constituents. The regions between 2.0 - 3.5 nm and 5.5 - 7.0 nm with electron densities of 380 and 360 e . nmM3, respectively, contain the sterol moieties and ester groupings of the cholesteryl esters and the ester groupings of the triglycerides. These compounds are obviously in a relatively dense packing as indicated by the absolute electron density levels of these zones. The thickness of these shells of 1.5 nm is in good agreement with the dimension of the sterol-ring system. The presence of protein or phospholipid head groups in these regions of the particle is highly improbable, since the electron densities of these components in an unhydrated state are significantly higher than the values found by Fourier transformation. This is in agreement with the previous findings that both protein and phospholipid polar headgroups are largely acces- sible to interactions from the solvent phase [44-461. Further, any penetration of solvent to decrease the electron density by hydration is doubtful because of the presence of the hydrophobic layer with an electron density of 310 e.nm-3 reaching from 7.0-9.5 nm. This latter zone could either contain the 17-hydrocarbon chain of the cholesteryl esters or the fatty acid chains of conversely arranged cholesteryl esters inter- acting in a bilayer-like arrangement with the fatty acid chains of the phospholipids.

Thus the present results confirm the previous sug- gestion [14,20 - 231 that cholesterylesters form the core of the particle by two succeeding spherical layers in a considerably ordered state. The precise molecular

4oc 37%

Fig. 10. Molecular model of lipoprotein B at 4."C and 37 'C. The corresponding functions e(r ) , I ( h ) and p ( r ) are displayed below. The ordered arrangement of neutral lipids at 4 "C in the form of two succeeding spherical layers is shown schematically. Since it is unknown whether the cholesteryl esters in the second layer are orien- tated with their sterol moiety towards the surface or towards the center, both possibilities are considered in the model. (a) Protein; (b) phospholipid; (c) cholesterol; (d) cholesteryl ester; (e) triglyceride

arrangement in this state cannot be discussed on the basis of the present experimental resolution. A decision can possibly be expected from studies of cholesteryl ester crystals [47] or mixed lipid model systems.

On the whole, the volume provided for the hydrophobic constituents of lipoprotein B by the model is in agreement with the spacial demand of these sub- stances. Apart from the polar head groups of the phospholipids and the polar parts of the protein all constituents are accomodated within a sphere of 9.0-9.5-nm radius. The volume of this space of 3.0-3.5 x lo3 nm3 is in agreement with the volume calculated from the physico-chemical data (Table l) , which is approximately 3.3 x lo3 nm3 (half of the protein assumed to be immersed in this region) or 3.0 x I @ nm3 (assuming no protein within 9.5-nm radius).

The packing density within the hydrophobic-hy- drophilic interface is obviously rather low. Even con-

K . Miiller, P. Laggner, 0. Glatter, and G. Kostner 85

sidering the hydration of the polar head groups and the protein, these components scarcely occupy 50% of the volume available. This is confirmed by the experimentally observed electron density which is only 370 e.nm-3 as compared with about 460 e .nm-3 for unhydrated head groups, indicating that more than 50% of the volume is occupied by the solvent. (The electron density is calculated as the average of 460 e .nm-3 for protein and head groups and 334 e.nm-3 for the solvent, each weighted with its individual volume fraction in this outermost shell.)

Thedemonstrated deviation ofthe function R 2 / d e-' (Fig. 5C) from the linear course, may be considered as a further indication for high solvent content in this outermost shell of lipoprotein B.

This plot (Fig.5C) is very sensitive to errors in the R values at low contrast, which have been shown to be less accurate than the values determined for higher contrasts, and because of this uncertainty we have refrained from a quantitative analysis of this deviation. Nevertheless, model calculations allowing for solvent penetration in the outermost shell have shown best agreement to the experimentally observed function R 2 / A Q - l in the case of 407; solvent in this region

An important supplement to the present findings is provided by studying the temperature dependence of lipoprotein B small angle scattering. It has been shown previously [20- 231, that the scattering curve of lipoprotein B is strongly temperature-dependent within the temperature range of 15 "C to 40 "C, particularly with respect to the maximum at 11 = 1.7 nm-'. This finding was interpreted in terms of order-disorder changes of the internal structure. In fact, the corresponding functions I(h), p ( r ) and e,,, (Fig. 10) indicate a thermal transition of the lipid core from an ordered to a liquid-like random distribution with an average electron density of 310 e.nm-3.

On the whole, the structural features of lipoprotein B confirm the apparent general principle of lipoprotein structure consisting of a central core containing the neutral lipids surrounded by a shell of cholesterol, phospholipid and protein, as has been already reported for some representatives of the high-density class [14, 48 - 511. The primary difference for the case of low- density lipoprotein structure is the presence of a more complex internal structure as a consequence of larger particle size.

Some incertainty remains as to the surface structure of lipoprotein B. Tardieu et al. [15] have reported a deeply convoluted surface, covered with long spikes (8.0 nm) reaching to a maximum intraparticle distance of 29 nm, with the solvent penetrating as far as 5.5 nm from the center. The present results do not confirm this postulation. Since the actual experimental scattering curves leading to this result have not been published we cannot discuss the possible reasons for

~ 4 1 .

this discrepancy. Nevertheless, the fact that the shell covering the range 9.5 - 11.5-nm radius is occupied by phospholipid head groups and protein to a maximum of 5 0 x of the available volume, would allow for a surface structure in which these constituents are arranged in the form of clusters or patches at the surface, thus forming something like subunits as considered in Results.

It can be expected that more information relevant to the surface structure of low-density lipoproteins will emerge from our current studies on samples which have been exposed to specific proteolytic enzyme action [62].

APPENDIX

In this section some more specialized considerations with respect to X-ray small-angle scattering data evaluation and data interpretation are presented. First, a short survey of the indirect Fourier transformation method is given, since a large part of the foregoing evaluations was based on an interpretation of the resulting distance distribution function p ( r ) .

Indirect Fourier Transformation

The method solves the problems of smoothing, desmearing and Fourier transformation of X-ray and neutron small-angle scattering data simultaneously [25]. This method can only be applied to particle scattering curves, i.e. for scattering media whose distance distributions are zero beyond a certain value D . The distance distribution function p ( r ) can be calculated from the correlation function y(r), or 'characteristic' of a particle, [52] by multiplying by 3 [38]

In a dilute system of particles, a value D can be defined for which there applies a relation :

p ( r ) = O for r > D . (8)

From this, it can be seen that the function p ( r ) differs from zero only in a certain limited region of the real space, namely between 0 and D, where D corresponds to the maximum distance in the particle. We underly the assumption that the distance distribution p(r> can be approximated by a linear combination of a finite number of cubic B splines cp,.(r) [53,54]

N P A ( r ) = C c,, . cp,(r). (9 )

v = l

where N usually takes values of about 20-30. The expansion coefficients c, are the unknowns. The particular functions (P, are Fourier-transformed into reci-


procal space. The resulting functions which represent scattering intensities are given by

These functions are defined in the entire reciprocal space. They can be 'smeared' according to the experimental conditions (wavelengths, slitlength and slit- width effect).

~ d h ) =.I'.I'S Q(,) * 4t) . . $"(A)

The resulting function x U ( / t ) represents an optimal set of functions for a smooth approximation of the experimental data. The accuracy of the data is taken into account by a weighted least-squares fit of these functions to the data points, according to

where M is the number of data points, othi) is the variance at the i-th point.

The smooth approximation curve is given by N

v = l

The desmeared scattering curve is defined by the equation

N

IA(h)= 1 cv ' $ v ( h ) (14) v = l

while the solution in real space is given by Eqn (9). A stabilization routine with self-determination of the necessary stabilizing parameter prevents artificial oscillations inherent in least-squares techniques for relative large numbers of unknowns. Although the range of approximation is limited to h, dhdh, the functions 7 ~ ( h ) and ZA(h) are defined in the whole reciprocal space. This is the range of definition for Xv(h)

and $"(h). This fact can be used for an extrapolation of the scattering curve. The several functions used in this procedure are shown schematically in Fig. 11.

Radius of Gyration and Zero Intensity

This method also offers a new possibility for estimation of the radius of gyration which is often more reliable than the Guinier approximation [38] by using the relation

D

s PA(r) . 9 . dr

2 j P.&) dr (1 5 )

0 R:=

0

This approximation has the advantage that the radius of gyration is calculated from the smeared data points using the whole scattering curve and using the information that the particle is not larger than D. The Fourier theorem which shows that Eqn (15) is equi- valent to the Guinier approximation holds only for very small scattering angles and full integration limits.

An approximate value of zero intensity can be computed in a similar way by the formula

D

IA(0) = 4 n j PA(r) . dr . (16) 0

In most actual experiments the value of D will be unknown and hence an arbitrary value must be chosen. In the following, the dependence of the results of Eqn (15) from the chosen value of D is demonstrated by a test calculation, assuming a single spherical model with a radial electron density distribution @(r) of the specifications :

@ ( r ) = -@I at O<r< 8.0nm @(,)= +@z at 8 .0Grd10 .0 nm (17) Q ( r , = o at r > 10.0 nm.

The exact value of R of this model is 11.47 nm. The theoretical scattering curve is smeared accord-

ing to the profiles used in the experiments with lipoprotein B. The data points were computed by adding statistical fluctuations of about the same magnitude as in the experiment. The first data point was at h, =0.14 nm-' (h1R=l.6), the last at h,=1.76 nm-' (h, R z 20) in agreement with the experimental conditions.

The largest distance of the model is Dexact=2 rmax =20 nm. The distance D was varied from 16.0 nm to 40.0 nm.

As shown in Fig. 12, the dependence of both R and I(o) is practically negligible over a range of D,,,,, < D <(1.75 x Dexac,). Thus, in practice, a value of D about 50% larger than that expected becomes most useful. In the case of lipoprotein B we chose a value of 30.0 nm.

Using Guinier's approximation, the error would be larger than 40% for the smeared curve and larger than 18% for the desmeared curve. Since the function system $ v ( h ) is defined in the whole reciprocal space, as stated above, this method can also be used for an extrapolation of the scattering curve beyond the limits of the measuring range. A demonstration of the quality of this extrapolation is shown in Fig. 9. The scattering function was given for 0 d h < h, = 1.76 nm-'. The function I,,,, (Eqn 14) was calculated for OGhGh, =2.4 nm-'. The extrapolated ZA(h) curve is in good agreement with the theoretical curve.

Partial Speclfic Volume and Mean Electron Density

By high-precision digital densimetry [29] the partial specific volume u, of lipoprotein B in solvents of

K. Muller, P. Laggner, 0. Glatter, and G. Kostner

7 1 7 1

6 c

0 l L 0 1 2

b (nm-')

.c\ 5

c $ 4

3

2

1

0

I h

87

r (nm)

I A' t

Smearing Fourier transform Fig. 1 1. Sc/rcwro/ic my^^ r ! / / / rc i / rd; rc~t Fourier transformation procedure. A set 01' qlinc. l.unclion> 117, ( A ' ) was Fourier-transformed yielding the functions $ ). (B') and 'smeared' yielding the functions x. (C'). The coefficients cu were computed in order to result in the function t, being a weighted least-square approximation to the experimental measuring points, Iexp. Knowing cy the desmeared curve IA(h) was obtained from $" and the corresponding function PA(;) from cp.. respectively

-7 t

I

3

-1

-2 - Dh ag 0 -

-3 - -4

-5

-6

-7

Fig. 12. The relative deviation from the exact values R and I,,, plotted as a function of D. Within the range of Demc, < D<(1.75 x Dex.c,) the deviations are below 0.5 %

various sucrose content was determined. The actual values are summarized in Table 2. A nearly linear in- crease in u, is observed with increasing solvent density. Since this observation is essential for model calculation

as well as for the calculation of the characteristic scattering functions, the results were tested by the X-ray data. The scattering intensity at zero angle, I,, depends only on the electron density contrast between


20

0

-20 - - -60

<

-100

420

400

- 7

E 380 b

d -

360

340 . . .

0 10 20 . 30 40 50 Sucrose (o lo ,w /w)

Fig. 13. Graphical determination oJ' the mean electron density ijo of lipoprotein B at zero contrast. (A) A ( 0 ) (scattering amplitude at zero angle) vs Q- A ( 0 ) becomes zero, when Q, = 00. The course of the experimentally obtained function ( A - - -A) is influenced by the changes of the partial specific volume, u 2 . Correction for these changes relative to the iso-electron density shifts the position of the experimental points on the e, scale. The corrected values (0-0) lie on a straight line. (B) Q~ and $0 [calculated according to Eqn (4)] as a function of the solvent composition. e,=eo at 8.8% (w/w) sucrose concentration, corresponding to 345 e 'nm- '

the scattering particles and the surrounding solvent and the partial specific volume of the particles, provided that the parameters of the constant in Equation (1 8) remain unchanged.

to , = const. . u: . (A@)' const.=M;N,.I;c. D'P, , . r rC' . (18)

(Symbols are defined in Results.) l I ( O ) , the scattering amplitude at zero contrast A(O), should thus be a linear function of the electron density contrast A@ in the case of constant partial specific volume. Fig. 13 presents A o , as a function of the solvent electron density e , . The function shows a significant curvature clearly indicating a variable value of u 2 .

At the intersection with the abscissa the mean electron density of lipoprotein B equals to the solvent electron density. A value of 345 e .nm-3 is found corresponding to a solvent containing 8.87; sucrose. Using Eqn (4) this value leads to a u2 of 0.965 cm3 .g-' in this particular solvent. The density measurements yield 0.964 cm3.g-' in an 8.8x sucrose solution at

4 "C. These results are graphically shown in Fig. 13 B. The partial specific volume determined in this way could be termed an 'isoelectron-density' value of u2 , in analogy to the 'iso-density' value which can be determined by ultracentrifugation [ 5 5 ] .

To further confirm these findings the density scale of Fig. 13A was corrected for the changing mean electron density of lipoprotein B. In this case a linear function for the dependence of A!,,) on e, is obtained as should be expected by considering Eqn (18).

Contrast Variation and Characteristic Scattering Functions

functions we employed Eqn (19) For the calculation of the characteristic scattering

(19)

in addition to Eqn (5). This relation holds for the scattering functions I ~ ( l i ) as solutions from the indirect Fourier transformation (see above).

c,, = C$ + eF . A @ + C% . (A@)'

IA(I7) = IX(l1) + ZzF(/l) AQ + IAF(h) ' (A@)' . (20)

This method requires less computation since the expansion into polynomials of second degree need not to be done for each point of the scattering curve but only for N ( 2 0 - 2 5 ) solution coefficients co. A further advantage is the quality of the estimation of the zero intensity as well as the scattering region from zero to the first scattering point. This is of particular interest for the calculation of I, , which ought to reach zero at zero angle.

Nevertheless, it was not possible to achieve suffi- cient separation of the different scattering terms. In the following we shall point out some possible reasons for this failure.

The required accuracy of the absolute scattered intensities at different contrasts is much higher than for conventional analysis and may be achieved only with a high expenditure of time using even the most sophisticated technical aids presently available.

Normalization of the curves recorded at different contrast to equal primary intensities and absorption coefficients for both the solution and the blank will be especially important as well as the manner in which a possible background is subtracted. These factors pre- sumably will not seriously influence the absolute intensities at small angles. They may, however, cause considerable differences in the outer parts of the scattering curves recorded at different contrasts, which are most significant for the internal structure.

The absolute electron density contrast was shown in this work to be considerably dependent on the actual solvent composition from changes in the partial specific volume, which consequently must be measured carefully in any solvent. These changes of u2 further-

K . Miiller, P. Laggner, 0. Glatter, and G. Kostner 89

more question the method in general, since Eqn (8) depends entirely upon the assumption that the excluded volume and the structure of the particle remain unchanged in the different solvents. A further assumption is that the particle is impenetrable to the solvent, which is probably not the case for lipoprotein B (see Discussion). Better results are, of course, expected for a higher number of contrasts, the effects mentioned above, however, cannot be circumvented.

Choice and Calculation of Model Scattering Functions

In the previous sections little was said about the way in which representative models were selected from the infinite number of possible structures. For these considerations the assumption of a sphere for the overall shape is not further discussed since it has been derived unambigously in the first section of Results.

Of the two kinds of models considered in the foregoing the radial symmetric model is the simplier one. Scattering functions of such bodies were calculated by Fourier transformation of electron density profiles consisting of step functions with variable number, size and heights of steps. The choice of models with six electron density levels was found by trial and error, using an optimization procedure resulting in the smallest number of steps consistent with the experimental results. On the other hand the electron density distribution function as obtained from the experimental curves led also to this model, which was found to give an optimal fit to the experimental findings. The calculation of ec,, according to Eqn (21)

was based on an indirect transformation method [48, 561 in order to minimize termination and background errors. The scattering amplitudes A(h) were extrapolated to zero in the minima. Zeros are not to be expected for experimental scattering curves for numerous reasons, e.g. deviations from ideal sphericity, non- ideal monodispersity, imperfections in blank scatteqing subtraction, imperfections in desmearing and moto- chromatisation procedure etc. By termination of the experimental scattering function measured up to h=5 nm-' (Fig. 7A) at several points between 0.2<h < 0.5 it was proved that e(v) is practically free form termination effects.

The examination of the non-centrosymmetric models was more difficult because of the by far greater variability. Since in this case the characteristics of the experimental scattering functions are suggested to be related to optional space-angle-dependent as well as radius-dependent inhomogeneities, hints for model construction are obtained by an alternative interpretation of the scattering curves measured up to h = 5

nm-'. The 3.7-nm Bra@ spacing correlated to the broad scattering maximum at h = 1.7 nm-' was now interpreted as a regular center-to-center distance of scattering units inside the particle with electron densities differing from their environment. Size, shape and electron density levels of these units are unknown. However, the shape is only of little importance, provided these units are small as compared with the entire particle. The units were thus approximated by spherical subunits and the actual scattering functions were calculated according to [57,58]. Information about the possible size of the subunits might be obtained from the relatively sharp minimum at h = 2.6 nm-'. Provided that there are indeed subunits present this minimum would indicate a radius of 1.73 nm for these subunits [%I. Based upon these considerations, numerous models were calculated. It was found that all these modelsdiffer from the experimental findings. Variations towards altered center-to-center distances or subunit sizes led expectedly to even worse agreement. The best fit was achieved by a model with a homogeneous, spherical core of 8-nm radius surrounded by 60 identical subunits of 1.7-nm radius each and a center-to- center distance of 3.8 nm. However, in this case satisfactory approximation of the experimental scattering function is observed only for extraordinarily high ratios of subunit electron density to core electron density. Fig. 7B shows the scattering function of this model for subunit scattering only (scattering weight of the core equal to zero). Such extreme electron density distribution, however, is not possible from the chemical view, since the core must be assumed to contain the hydrophobic constituents with an average of 310 e . nm-3. Consequently the electron density of the subunits cannot exceed a value of 400 e . nm-3 considering the measured mean electron density of lipoprotein B. Hence the optimum subunit model of lipoprotein B was constructed with 310 e . nm-3 electron density in the core and a value of 400 e . nm-3 for the subunit shell. All curves and data assigned to the 'subunit model' correspond to this model.

We gratefully acknowledge the support by the ijstrrrric./ii.c(./rc,,. Fonds zur Forderung der Wissenschaftlichen Forschung.

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0. Clatter. Institut fur Physikalische Chemie der Karl-Franzens-Universitat Graz, HeinrichstraOe 28, A-8010 Graz, Austria

G . Kostner, Institut fur Medizinische Biochemie der Karl-Franzens-Universitat Graz, Herrechgasse 21, A-8010 Ciraz, Austria

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