Proc. Nati. Acad. Sci. USA Vol. 74, No. 6, pp. 2316-2320, June 1977
Biochemistry
Shape of the 50S subunit of Escherichia coli ribosomes (neutron low-angle scattering/contrast variation/spherical harmonics)
H. B. STUHRMANNtt, M. H. J. KOCHt, R. PARFAIT§, J. HAAS0, K. IBELI, AND R. R. CRICHTON§ t Institut fur Physikalische Chemie der Universitat Mainz, 6500 Mainz, Germany; § Unit6 de Biochimie, Universit6 Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium; and I Institut Max von Laue-Paul Langevin, BP 156, Centre de Tri, 38042 Grenoble, France
Communicated by C. B. Anfinsen, March 18,1977
ABSTRACT Extrapolation of a series of low-angle neutron scattering curves to infinitely high contrast gives a scattering function IC(K) which is dependent on the shape of the solute molecule. For the 50S subunit of E. coil ribosomes, the first part of the structure determination by neutron scattering, namely the determination of the molecular shape from IC(K), is reported. The result is in good agreement with models of the SOS subunit determined by electron microscopy.
With the advent of high-flux reactors, neutron scattering has become a powerful technique in the investigation of biological structure. The usefulness of neutrons in scattering methods is apparent, especially for structures that have chemically distinct components and exhibit pronounced intramolecular fluctuations of scattering density (e.g., ribosomes, lipoproteins, viruses, and chromatin). Neutron small-angle scattering curves in H20/D20 mixtures correspond to various views of the same molecule in solution. If the mean scattering density of the solute is close to that of the solvent (which is the situation at low contrast), intramolecular fluctuations of scattering density will predominantly contribute to the scattering curve. At higher contrasts, the scattering curve essentially depends on the overall shape of the solute particle. Extrapolation of a series of scattering curves obtained in different solvents to infinitely high contrast yields a scattering function IC(K) that is entirely dependent on the shape of the solute molecule. At the start of a structure determination, the evaluation of IC(K) is very helpful because the shape is described by a smaller number of parameters than the whole structure. Moreover, the subsequent determination of the detailed internal structure is then greatly facilitated by the boundary conditions given by the shape. In this paper we present the first part of the structure determination of the larger subunit of Escherichia coli ribosomes by neutron scattering-the determination of the molecular shape from IC(K). These results are compared with models of the 50S ribosomal subunit as determined by electron micros-
olation to zero concentration, we measured a series of five concentrations from 5 to 25 mg/ml for each contrast. The sedimentation profiles of H20 and D20 samples were analyzed for homogeneity, and were reanalyzed after neutron scattering measurements in order to detect possible alteration of the material. The samples were found to be unchanged. The activity of the 50S preparations in poly(U)-directed polyphenylalanine synthesis in the presence of 30S particles according to Fahnestock et al. (2) was also tested. All preparations showed normal activity. Neutron scattering experiments were done at the high flux beam reactor of the Institute Max von Laue-Paul Langevin by using the small angle scattering device Dli (3). The scattering curves were measured in three parts to obtain a sufficiently large range of momentum transfer K (K = 47r sin O/X, in which X = wavelength, and 2 0 = scattering angle). With distances of 10.40, 2.40, and 0.70 m and wavelengths of 8.0 A, 8.0 A, and 5.5 A, respectively, the total range of momentum transfers extends from 4.0 X 10-3 to 0.5 A-1. The average measuring time was 15 min at each position. Transmission measurements were made for each sample and used to determine the exact H20 content. Two sets of data obtained from independent preparations were collected at different temperatures. The first set in TMK buffer was measured at room temperature and the second, in the scattering buffer, was measured at 50 to ensure conditions identical to those used by Moore et al. (4). The discussion which is given below applies to both data sets. MATHEMATICAL BACKGROUND The success of the interpretation of small angle scattering of dissolved particles depends on the constraints which can be applied to excess scattering density p(i). This becomes clear if one considers the multipole expansion of an arbitrary p(i). 1o
1
[1] Y, m=-I E plm (r) Ylm (W) r) = 1=0 in which Ylm (w) are spherical harmonics, and w is a unit vector. The coefficients plm(r) characterize the scattering density, p(i), of the solute molecule. They are calculated from p(i) by means of the orthogonality relation of Ylm(() [21 plm(r) = fp(i)Yjm*(w)dw. is the Similarly, amplitude given by: p
copy.
METHODS
The preparation of the large ribosomal subunit from E. coli B (strain MRE600) used the methods described (1). Briefly, two batches of 50S particles were dialyzed to equilibrium against TMK buffer (30 mM Tris-CCl, pH 7.4/20 mM magnesium acetate-350 mM KCl) or scattering buffer (10 mM Tris-HCl, pH 7.4/0.5 mM magnesium acetate 50 mM KCI) in H20 or D20. Intermediate contrasts were obtained by mixing appropriate amounts of these solutions. To obtain a reliable extrap-
c
A(K)
=
I
E E Alm(K)Ylm(Q)
1=O m=-I
[31
in which Q is a unit vector in reciprocal space, k = (K,Q), and the coefficients Ajm(K) can be obtained from
Laboratory Outstation, c/o Deutsches Elektronen-Synchrotron, 2000 Hamburg 52,
t Present address: European Molecular Biology
Alm(K) =
Germany. 2316
(-)1/2 2
S
plm(r)jl(Kr)r2dr
[41
Proc. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Stuhrmann et al. in which ji are spherical Bessel functions. It follows from the small angle scattering function (5)
Table 1. Modified coefficients of the power series of the shape scattering function IC(K)
[5]
n
bn
that a unique interpretation of I(K) cannot be given, i.e., there is a large number of structures p(i) which correspond to the same I(K) (6). Now we include simplifications of p(rj) that are given by contrast variation. As the contrast tends to infinity, fluctuations of the scattering density of the internal structure become negligibly small. The particle appears as a region of uniform density inaccessible to solvent molecules. The shape of the excluded volume is described by pc(r) [pc(r) = 1 inside the volume and zero elsewhere]. For pc(9), the question arises whether the condition of homogeneity, i.e., the absence of any internal structure is sufficient to obtain a unique pc(r) from IC(K). The constraints introduced by the shape function pc(r) lead to a strong correlation of the multipoles plm (r) in real space. A corresponding correlation exists in reciprocal space. Therefore, the sum z z 1 Alm (K) 12 can no longer be constructed in an arbitrary way to
0 1 2 3
78.307 105.709 118.559 128.655 136.895 143.774 149.612 154.631 158.993 162.821 166.210
= I0(K)
m1=0 m=-l
I Alm(K) 12
give IC(K). A wide class of shapes can be represented by: = 1 0 < r < F(w)
into Eq. 4 and expands F(w) as a series of spherical harmonics:
1=0 m=-I
-E f(oYIo(W) + 1=0
fimYim(W)
m=l UlmReIYlm(C)I E
+
VimIMlIYlm()WI [ [8]
we get A(C)lm(K)
1/
=
7r
E
ip f m(1+2p+ 3) K++2p + 3fmi2+)i2
pol +2p
[9]
in which
5 6 7 8 9 10
bn coefficients as defined in Eq. 12. bo is the radius of a sphere with a volume equal to the volume of the 50S particle.
The coefficients an of the power series of IC(K) are conveniently calculated from the moments of the distance distribution D(u) of IC(K) ,
n=O
du D(U) sinKU KU
a,,K2n
u=O co
=0 elsewhere. More general structures can be constructed by adding further shape functions. If one introduces the power series of the spherical Bessel function jj: -~~~~c (1P (Kr)P+ 2p (Kr) E d(Kr) E+pop![2(l 2P + p) + 1]!! [7]
F(() = E
4
IC(K) =
[6]
2317
1_)n
coX-2
nEo (2n + 1)! S= D(u)u2duK2n [11] in which u = - i ' connects two volume elements of the shape. As the an vary over many orders of mangitude we consider ^ r -
bn
=
(2n +
I)I aJ
.ao
1/2n
[12]
This series of coefficients tends to the maximum intramolecular distance of the 50S ribosome subunit (dmax = 215 A) as n tends to infinity (Table 1). bo is the radius of an equivalent sphere with the same volume as the 50S particle. Unfortunately, there is no direct way for the calculation of the Jim from the set of bn. Therefore, an iterative method has to be used to approximate the bn obtained from the experimental IC(K) by the fim of the tentative model. Details of our Algol program will be published elsewhere. RESULTS AND DISCUSSION The shape-scattering function IC(K) of the 50S subunit was extrapolated from a series of small angle scattering curves obtained in twelve H20/D20 mixtures (Fig. 1) by a least squares fit (6, 7). The shape of the particle pc(), as defined in the mathematical background, can be determined by using Eq. 10.
fiM (q) JF(5)jqy1M (w)dw. The sum of the absolute squares of A (C)lm(K) yields IC (K), which is now given as a power series. I
c
n
n-Ii
A = E K2n E IC(K) = 1=0 m=-I E A(C)lm(K)12 n=O 1=0 p=O m=-l X
dipdinpfim (I+ 2p + 3) ffim (2n + 1-2p + 31* [101 (1 + 2p + 3)(2n + 1-2p + 3)
Thus, Eq. 10 gives the relationship between the parameters of the molecular shape, Jim, and the coefficients an of the power series of the shape function IC(K) derived from the experimental data (6, 8).
The search for a model was performed by a stepwise increase of the number of coefficients taken into account, and thus of the resolution of the model (6). The simplest model is given by foo alone, which represents a solid sphere. Taking into account the next higher multipole component, no solution can be found from bo and bl. Ellipsoid-like models are calculated by including the multipoles f2m of a quadrupole. Models with rotational symmetry (column 2 in Table 2) with the symmetry of an ellipsoid with three different axes (column 3 in Table 2) can be calculated from (bo, b2) and (bo, bl, b2), respectively. If a higher resolution of the structure is desired, the remaining multipoles Jim and f2rn are not suitable for a better fit of the experimental bn (column 4 in Table 2), i.e., an ellipsoid-like structure is no longer a good solution. The convergence of the iteration is considerably faster, and the approximation is much better, if the shape coefficients f3m are included. With 11
2318
Biochemistry: Stuhrmann et al.
OIC(K)
Proc. Natl. Acad. Sci. USA 74 (1977)
i
/
0.01
-0.01
-0.02
0.02
0.04 0.06 0.1 0.08 Momentum transfer K[A-'I FIG. 1. Comparison of the experimental shape scattering function [AIC(K) = IC(K) - exp (-R2K2/3) (0)] with the AIc scattering functions
of the models of the 50S ribosomal subunit. IC(K) has been normalized to 1. *-*, exp (-R2K2/3). The radius of gyration R is 74.5 A. The vertical lines are error bars. Homogeneous model ---, and inhomogeneous model -, described by spherical harmonics Yim with I = 0,1,2,3. ..., The ellipsoid with the axial ratio 0.75:1:1.8 approximates the maximum at = 0.08 A-' [Hill et al. (14)]. -----, An ellipsoid with lower axial ratio 0.75:1:1.4 or (not shown here) a prolate ellipsoid 1:1:1.6 is a good approximation of the experimental Ic(K) up to K = 0.04 A-i.
coefficients bn, the shape parameters fim with 1 = 0, 1, 2, 3 have been determined (column 6 in Table 2). Very different initial structures have been used in order to show the uniqueness of the resulting model. Though the numerical values of the coefficients of the final model vary considerably between runs due to the displacement of the centre of mass and symmetry transformations, the models all look very similar. Approximately 100 computer runs have been made and this may serve as a measure of the probability of the model proposed in Fig. 2. The choice of the initial values of fim has been almost random as long as the number of variables was lower than six. Thus, the symmetry of the model could be roughly established and further fim were included in order to refine the model given in column 6 of Table 2. The resolution of the model is 100 A, as can be calculated from the K value 0.06 A-1, in which the power series with 11 coefficients becomes divergent. Inspection of the fit between the experimental values of IC(K) and the values calculated from the model of Fig. 2 shows a major discrepancy at K values higher than 0.05 A-'. The cal-
culated curve is higher than the experimental. A possible cause of this lack of fit is the inhomogeneity of pc(f), which we have neglected until now. As the protein part of the 50S particle is closer to the surface than the rRNA part (1), the decrease of pc(), due to H/D exchange, will be more important in the outer region of the 50S subunit. As we do not yet know more about the distribution of dissociating protons in the larger subunit, we assumed a simple two-phase model of pc(r) in order to estimate the effect of such an inhomogeneity on IC(K). The calculation becomes easy, if we postulate a relative decrease of pc(r) by a factor q at the distances between pF(w) and F(w) from the origin (= white area of the cross section in Fig. 3). This model approximates slightly better the given bn with q = 0.8-0.9 and p = 0.2-0.5. (column 7 in Table 2). Still more important is the fact that the resulting scattering curve of the model follows the experimental IC(K) quite well. There is hardly any change of the resulting shape when the low density shell model is introduced (Fig. 3). At K> 0.09 A-1, the scattering curves of both the homoge-
Proc. Natl. Acad. SC{. USA 74 (1977)
Biochemistry: Stuhrmann et al.
2319
y
FIG. 3. Cross section of the 50S ribosome subunit. -, Homogeneous model; ----, inhomogeneous model. The shaded area is the assumed region of higher density.
bn with higher n strongly depends on the assumption of a maximum distance and on its precision. The latter is given by dmax. = 215 i 5 A. The model does not depend very much on dmax as long as the number of variables is relatively low (e.g., Table 2. Iterative modeling of the 50S subunit
FIG. 2. Two views of the model for the 50S ribosome subunit from neutron scattering.
neous and inhomogeneous models fall below the experimental IC(K) (Fig. 1). This indicates that the specific surface (surface/ volume) (9) of each model is still too low and needs refinement by including higher multipoles of F(cw) with I > 3 that would give a more detailed surface. It should be emphasized that although the proposed model of the 50S subunit appears to be rather detailed its main features are governed by a few fim only. The less important coefficients V31., V32., V33, U22, and V22 could be omitted without a major change of the model. The shape has been kept as "smooth" as possible by taking into account the lowest possible indices 1 and m. This strategy does not necessarily give the best model, and a more refined calculation with higher fim might resolve the molecular shape in a better way. A severe problem is the question whether the coefficients of the power series of the experimental IC(K) are accurate enough to justify the present calculation and, even more so, its extension to higher resolution. The evaluation of the coefficients
foo
1
2
3
Models 4
282
241
235
244
53
fio [20
-111 142
6
7
240 52
228 63 -52 61 25 28 57 56 17
254 59 -55 53 21 15 55 50 81 46
58 64
U 21
U22
5
60
44
59
V22
f3o
64 56 59 61
U 31
U32 U33 V31
V32 V33
A
0
0
0.56
1.66
0.41
47 0 15 2
-11 14 5
0.25
0.15
Jlm coefficients and errors for the various models are described in the text. Uii, VII, V2l are kept zero in order to define an orientation of the model (5). 1
N
N1 1~F_ I bn (exp) =
-
b (model)1
2320
Proe. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Stuhrmann et al.
less than six). However, any reliable improvement of the model on the grounds of small angle scattering data that includes a larger number of bn and of film appears to us a prohibitively difficult task at the moment. More complicated surfaces with internal holes and concave indentations produced by multivalued F(w) do not offer a better approximation at the resolution actually achieved. The shape of the 50S particle presented here (Fig. 3) agrees fairly well with the results obtained from electron microscopy at least if the comparison is made at low resolution. From electron microscopy studies, the 50S subunit was first described as a somewhat excentric cone with rounded edges (10). This kind of symmetry is present in our model and is essentially due to foo and fso, which have relatively large values. Later, the models presented by Wabl (11) and Tischendorf et al. (12) consisted of three crests on a half sphere. The presence of fm, in our model describes this modification. Both electron microscopy and neutron scattering show that the three crests are not identical. At this stage of resolution our findings are qualitatively in agreement with those of Tischendorf et al. (12) and Lake (13). The costs of publication of this article were defrayed in part by the payment of page charges from funds made available to support the research which is the subject of the article. This article must therefore
be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact. 1. Stuhrmann, H. B., Haas, J., Ibel, K., Koch, M. H. J., De Wolf, B., Parfait, R. & Crichton, R. R. (1976) Proc. Natl. Acad. Sci. USA
73,2379-2383.
2. Fahnestock, S., Erdmann, V. & Nomura, M. (1974) Methods Enzymol. 30, 554-562. 3. Ibel, K. (1976) J. Appl. Crystallogr. 9,296-309.
4. Moore, P. B., Engelman, D. M. & Schoenborn, B. P. (1975) J. Mol. Biol. 91, 101-120. 5. Stuhrmann, H. B. (1970) Acta Crystallogr. Sect. A 26, 297-
506.
6. Stuhrmann, H. B. (1970) Z. Physik. Chem. 72, 185-198. 7. Stuhrmann, H. B. (1974) J. Appl. Crystallogr. 7, 173-178. 8. Stuhrmann, H. B. (1975) "Neutron scattering for the analysis of Biological Structures," Brookhaven Symp. Biol. 27, IV-3, IV19. 9. Porod, G. (1951) Kollid Z. 124,83-114. 10. Damaschun, G., Muller, J. J. & Bielka, H. (1975) Acta Biol. Med. Ger. 34,229-239. 11. Wabl, M. R. (1974) J. Mol. Biol. 84,241-247. 12. Tischendorf, G. W., Zeichhardt, H. & St6ffler, G. (1975) Proc. Nat!. Acad. Sci. USA 72,4820-4824. 13. Lake, J. (1976) J. Mol. Biol. 165, 131-159. 14. Hill, W. E., Thompson, J. D. & Anderegg, J. W. (1969) J. Mol. Biol. 44, 89-102.
Biochemistry
Shape of the 50S subunit of Escherichia coli ribosomes (neutron low-angle scattering/contrast variation/spherical harmonics)
H. B. STUHRMANNtt, M. H. J. KOCHt, R. PARFAIT§, J. HAAS0, K. IBELI, AND R. R. CRICHTON§ t Institut fur Physikalische Chemie der Universitat Mainz, 6500 Mainz, Germany; § Unit6 de Biochimie, Universit6 Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium; and I Institut Max von Laue-Paul Langevin, BP 156, Centre de Tri, 38042 Grenoble, France
Communicated by C. B. Anfinsen, March 18,1977
ABSTRACT Extrapolation of a series of low-angle neutron scattering curves to infinitely high contrast gives a scattering function IC(K) which is dependent on the shape of the solute molecule. For the 50S subunit of E. coil ribosomes, the first part of the structure determination by neutron scattering, namely the determination of the molecular shape from IC(K), is reported. The result is in good agreement with models of the SOS subunit determined by electron microscopy.
With the advent of high-flux reactors, neutron scattering has become a powerful technique in the investigation of biological structure. The usefulness of neutrons in scattering methods is apparent, especially for structures that have chemically distinct components and exhibit pronounced intramolecular fluctuations of scattering density (e.g., ribosomes, lipoproteins, viruses, and chromatin). Neutron small-angle scattering curves in H20/D20 mixtures correspond to various views of the same molecule in solution. If the mean scattering density of the solute is close to that of the solvent (which is the situation at low contrast), intramolecular fluctuations of scattering density will predominantly contribute to the scattering curve. At higher contrasts, the scattering curve essentially depends on the overall shape of the solute particle. Extrapolation of a series of scattering curves obtained in different solvents to infinitely high contrast yields a scattering function IC(K) that is entirely dependent on the shape of the solute molecule. At the start of a structure determination, the evaluation of IC(K) is very helpful because the shape is described by a smaller number of parameters than the whole structure. Moreover, the subsequent determination of the detailed internal structure is then greatly facilitated by the boundary conditions given by the shape. In this paper we present the first part of the structure determination of the larger subunit of Escherichia coli ribosomes by neutron scattering-the determination of the molecular shape from IC(K). These results are compared with models of the 50S ribosomal subunit as determined by electron micros-
olation to zero concentration, we measured a series of five concentrations from 5 to 25 mg/ml for each contrast. The sedimentation profiles of H20 and D20 samples were analyzed for homogeneity, and were reanalyzed after neutron scattering measurements in order to detect possible alteration of the material. The samples were found to be unchanged. The activity of the 50S preparations in poly(U)-directed polyphenylalanine synthesis in the presence of 30S particles according to Fahnestock et al. (2) was also tested. All preparations showed normal activity. Neutron scattering experiments were done at the high flux beam reactor of the Institute Max von Laue-Paul Langevin by using the small angle scattering device Dli (3). The scattering curves were measured in three parts to obtain a sufficiently large range of momentum transfer K (K = 47r sin O/X, in which X = wavelength, and 2 0 = scattering angle). With distances of 10.40, 2.40, and 0.70 m and wavelengths of 8.0 A, 8.0 A, and 5.5 A, respectively, the total range of momentum transfers extends from 4.0 X 10-3 to 0.5 A-1. The average measuring time was 15 min at each position. Transmission measurements were made for each sample and used to determine the exact H20 content. Two sets of data obtained from independent preparations were collected at different temperatures. The first set in TMK buffer was measured at room temperature and the second, in the scattering buffer, was measured at 50 to ensure conditions identical to those used by Moore et al. (4). The discussion which is given below applies to both data sets. MATHEMATICAL BACKGROUND The success of the interpretation of small angle scattering of dissolved particles depends on the constraints which can be applied to excess scattering density p(i). This becomes clear if one considers the multipole expansion of an arbitrary p(i). 1o
1
[1] Y, m=-I E plm (r) Ylm (W) r) = 1=0 in which Ylm (w) are spherical harmonics, and w is a unit vector. The coefficients plm(r) characterize the scattering density, p(i), of the solute molecule. They are calculated from p(i) by means of the orthogonality relation of Ylm(() [21 plm(r) = fp(i)Yjm*(w)dw. is the Similarly, amplitude given by: p
copy.
METHODS
The preparation of the large ribosomal subunit from E. coli B (strain MRE600) used the methods described (1). Briefly, two batches of 50S particles were dialyzed to equilibrium against TMK buffer (30 mM Tris-CCl, pH 7.4/20 mM magnesium acetate-350 mM KCl) or scattering buffer (10 mM Tris-HCl, pH 7.4/0.5 mM magnesium acetate 50 mM KCI) in H20 or D20. Intermediate contrasts were obtained by mixing appropriate amounts of these solutions. To obtain a reliable extrap-
c
A(K)
=
I
E E Alm(K)Ylm(Q)
1=O m=-I
[31
in which Q is a unit vector in reciprocal space, k = (K,Q), and the coefficients Ajm(K) can be obtained from
Laboratory Outstation, c/o Deutsches Elektronen-Synchrotron, 2000 Hamburg 52,
t Present address: European Molecular Biology
Alm(K) =
Germany. 2316
(-)1/2 2
S
plm(r)jl(Kr)r2dr
[41
Proc. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Stuhrmann et al. in which ji are spherical Bessel functions. It follows from the small angle scattering function (5)
Table 1. Modified coefficients of the power series of the shape scattering function IC(K)
[5]
n
bn
that a unique interpretation of I(K) cannot be given, i.e., there is a large number of structures p(i) which correspond to the same I(K) (6). Now we include simplifications of p(rj) that are given by contrast variation. As the contrast tends to infinity, fluctuations of the scattering density of the internal structure become negligibly small. The particle appears as a region of uniform density inaccessible to solvent molecules. The shape of the excluded volume is described by pc(r) [pc(r) = 1 inside the volume and zero elsewhere]. For pc(9), the question arises whether the condition of homogeneity, i.e., the absence of any internal structure is sufficient to obtain a unique pc(r) from IC(K). The constraints introduced by the shape function pc(r) lead to a strong correlation of the multipoles plm (r) in real space. A corresponding correlation exists in reciprocal space. Therefore, the sum z z 1 Alm (K) 12 can no longer be constructed in an arbitrary way to
0 1 2 3
78.307 105.709 118.559 128.655 136.895 143.774 149.612 154.631 158.993 162.821 166.210
= I0(K)
m1=0 m=-l
I Alm(K) 12
give IC(K). A wide class of shapes can be represented by: = 1 0 < r < F(w)
into Eq. 4 and expands F(w) as a series of spherical harmonics:
1=0 m=-I
-E f(oYIo(W) + 1=0
fimYim(W)
m=l UlmReIYlm(C)I E
+
VimIMlIYlm()WI [ [8]
we get A(C)lm(K)
1/
=
7r
E
ip f m(1+2p+ 3) K++2p + 3fmi2+)i2
pol +2p
[9]
in which
5 6 7 8 9 10
bn coefficients as defined in Eq. 12. bo is the radius of a sphere with a volume equal to the volume of the 50S particle.
The coefficients an of the power series of IC(K) are conveniently calculated from the moments of the distance distribution D(u) of IC(K) ,
n=O
du D(U) sinKU KU
a,,K2n
u=O co
=0 elsewhere. More general structures can be constructed by adding further shape functions. If one introduces the power series of the spherical Bessel function jj: -~~~~c (1P (Kr)P+ 2p (Kr) E d(Kr) E+pop![2(l 2P + p) + 1]!! [7]
F(() = E
4
IC(K) =
[6]
2317
1_)n
coX-2
nEo (2n + 1)! S= D(u)u2duK2n [11] in which u = - i ' connects two volume elements of the shape. As the an vary over many orders of mangitude we consider ^ r -
bn
=
(2n +
I)I aJ
.ao
1/2n
[12]
This series of coefficients tends to the maximum intramolecular distance of the 50S ribosome subunit (dmax = 215 A) as n tends to infinity (Table 1). bo is the radius of an equivalent sphere with the same volume as the 50S particle. Unfortunately, there is no direct way for the calculation of the Jim from the set of bn. Therefore, an iterative method has to be used to approximate the bn obtained from the experimental IC(K) by the fim of the tentative model. Details of our Algol program will be published elsewhere. RESULTS AND DISCUSSION The shape-scattering function IC(K) of the 50S subunit was extrapolated from a series of small angle scattering curves obtained in twelve H20/D20 mixtures (Fig. 1) by a least squares fit (6, 7). The shape of the particle pc(), as defined in the mathematical background, can be determined by using Eq. 10.
fiM (q) JF(5)jqy1M (w)dw. The sum of the absolute squares of A (C)lm(K) yields IC (K), which is now given as a power series. I
c
n
n-Ii
A = E K2n E IC(K) = 1=0 m=-I E A(C)lm(K)12 n=O 1=0 p=O m=-l X
dipdinpfim (I+ 2p + 3) ffim (2n + 1-2p + 31* [101 (1 + 2p + 3)(2n + 1-2p + 3)
Thus, Eq. 10 gives the relationship between the parameters of the molecular shape, Jim, and the coefficients an of the power series of the shape function IC(K) derived from the experimental data (6, 8).
The search for a model was performed by a stepwise increase of the number of coefficients taken into account, and thus of the resolution of the model (6). The simplest model is given by foo alone, which represents a solid sphere. Taking into account the next higher multipole component, no solution can be found from bo and bl. Ellipsoid-like models are calculated by including the multipoles f2m of a quadrupole. Models with rotational symmetry (column 2 in Table 2) with the symmetry of an ellipsoid with three different axes (column 3 in Table 2) can be calculated from (bo, b2) and (bo, bl, b2), respectively. If a higher resolution of the structure is desired, the remaining multipoles Jim and f2rn are not suitable for a better fit of the experimental bn (column 4 in Table 2), i.e., an ellipsoid-like structure is no longer a good solution. The convergence of the iteration is considerably faster, and the approximation is much better, if the shape coefficients f3m are included. With 11
2318
Biochemistry: Stuhrmann et al.
OIC(K)
Proc. Natl. Acad. Sci. USA 74 (1977)
i
/
0.01
-0.01
-0.02
0.02
0.04 0.06 0.1 0.08 Momentum transfer K[A-'I FIG. 1. Comparison of the experimental shape scattering function [AIC(K) = IC(K) - exp (-R2K2/3) (0)] with the AIc scattering functions
of the models of the 50S ribosomal subunit. IC(K) has been normalized to 1. *-*, exp (-R2K2/3). The radius of gyration R is 74.5 A. The vertical lines are error bars. Homogeneous model ---, and inhomogeneous model -, described by spherical harmonics Yim with I = 0,1,2,3. ..., The ellipsoid with the axial ratio 0.75:1:1.8 approximates the maximum at = 0.08 A-' [Hill et al. (14)]. -----, An ellipsoid with lower axial ratio 0.75:1:1.4 or (not shown here) a prolate ellipsoid 1:1:1.6 is a good approximation of the experimental Ic(K) up to K = 0.04 A-i.
coefficients bn, the shape parameters fim with 1 = 0, 1, 2, 3 have been determined (column 6 in Table 2). Very different initial structures have been used in order to show the uniqueness of the resulting model. Though the numerical values of the coefficients of the final model vary considerably between runs due to the displacement of the centre of mass and symmetry transformations, the models all look very similar. Approximately 100 computer runs have been made and this may serve as a measure of the probability of the model proposed in Fig. 2. The choice of the initial values of fim has been almost random as long as the number of variables was lower than six. Thus, the symmetry of the model could be roughly established and further fim were included in order to refine the model given in column 6 of Table 2. The resolution of the model is 100 A, as can be calculated from the K value 0.06 A-1, in which the power series with 11 coefficients becomes divergent. Inspection of the fit between the experimental values of IC(K) and the values calculated from the model of Fig. 2 shows a major discrepancy at K values higher than 0.05 A-'. The cal-
culated curve is higher than the experimental. A possible cause of this lack of fit is the inhomogeneity of pc(f), which we have neglected until now. As the protein part of the 50S particle is closer to the surface than the rRNA part (1), the decrease of pc(), due to H/D exchange, will be more important in the outer region of the 50S subunit. As we do not yet know more about the distribution of dissociating protons in the larger subunit, we assumed a simple two-phase model of pc(r) in order to estimate the effect of such an inhomogeneity on IC(K). The calculation becomes easy, if we postulate a relative decrease of pc(r) by a factor q at the distances between pF(w) and F(w) from the origin (= white area of the cross section in Fig. 3). This model approximates slightly better the given bn with q = 0.8-0.9 and p = 0.2-0.5. (column 7 in Table 2). Still more important is the fact that the resulting scattering curve of the model follows the experimental IC(K) quite well. There is hardly any change of the resulting shape when the low density shell model is introduced (Fig. 3). At K> 0.09 A-1, the scattering curves of both the homoge-
Proc. Natl. Acad. SC{. USA 74 (1977)
Biochemistry: Stuhrmann et al.
2319
y
FIG. 3. Cross section of the 50S ribosome subunit. -, Homogeneous model; ----, inhomogeneous model. The shaded area is the assumed region of higher density.
bn with higher n strongly depends on the assumption of a maximum distance and on its precision. The latter is given by dmax. = 215 i 5 A. The model does not depend very much on dmax as long as the number of variables is relatively low (e.g., Table 2. Iterative modeling of the 50S subunit
FIG. 2. Two views of the model for the 50S ribosome subunit from neutron scattering.
neous and inhomogeneous models fall below the experimental IC(K) (Fig. 1). This indicates that the specific surface (surface/ volume) (9) of each model is still too low and needs refinement by including higher multipoles of F(cw) with I > 3 that would give a more detailed surface. It should be emphasized that although the proposed model of the 50S subunit appears to be rather detailed its main features are governed by a few fim only. The less important coefficients V31., V32., V33, U22, and V22 could be omitted without a major change of the model. The shape has been kept as "smooth" as possible by taking into account the lowest possible indices 1 and m. This strategy does not necessarily give the best model, and a more refined calculation with higher fim might resolve the molecular shape in a better way. A severe problem is the question whether the coefficients of the power series of the experimental IC(K) are accurate enough to justify the present calculation and, even more so, its extension to higher resolution. The evaluation of the coefficients
foo
1
2
3
Models 4
282
241
235
244
53
fio [20
-111 142
6
7
240 52
228 63 -52 61 25 28 57 56 17
254 59 -55 53 21 15 55 50 81 46
58 64
U 21
U22
5
60
44
59
V22
f3o
64 56 59 61
U 31
U32 U33 V31
V32 V33
A
0
0
0.56
1.66
0.41
47 0 15 2
-11 14 5
0.25
0.15
Jlm coefficients and errors for the various models are described in the text. Uii, VII, V2l are kept zero in order to define an orientation of the model (5). 1
N
N1 1~F_ I bn (exp) =
-
b (model)1
2320
Proe. Natl. Acad. Sci. USA 74 (1977)
Biochemistry: Stuhrmann et al.
less than six). However, any reliable improvement of the model on the grounds of small angle scattering data that includes a larger number of bn and of film appears to us a prohibitively difficult task at the moment. More complicated surfaces with internal holes and concave indentations produced by multivalued F(w) do not offer a better approximation at the resolution actually achieved. The shape of the 50S particle presented here (Fig. 3) agrees fairly well with the results obtained from electron microscopy at least if the comparison is made at low resolution. From electron microscopy studies, the 50S subunit was first described as a somewhat excentric cone with rounded edges (10). This kind of symmetry is present in our model and is essentially due to foo and fso, which have relatively large values. Later, the models presented by Wabl (11) and Tischendorf et al. (12) consisted of three crests on a half sphere. The presence of fm, in our model describes this modification. Both electron microscopy and neutron scattering show that the three crests are not identical. At this stage of resolution our findings are qualitatively in agreement with those of Tischendorf et al. (12) and Lake (13). The costs of publication of this article were defrayed in part by the payment of page charges from funds made available to support the research which is the subject of the article. This article must therefore
be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact. 1. Stuhrmann, H. B., Haas, J., Ibel, K., Koch, M. H. J., De Wolf, B., Parfait, R. & Crichton, R. R. (1976) Proc. Natl. Acad. Sci. USA
73,2379-2383.
2. Fahnestock, S., Erdmann, V. & Nomura, M. (1974) Methods Enzymol. 30, 554-562. 3. Ibel, K. (1976) J. Appl. Crystallogr. 9,296-309.
4. Moore, P. B., Engelman, D. M. & Schoenborn, B. P. (1975) J. Mol. Biol. 91, 101-120. 5. Stuhrmann, H. B. (1970) Acta Crystallogr. Sect. A 26, 297-
506.
6. Stuhrmann, H. B. (1970) Z. Physik. Chem. 72, 185-198. 7. Stuhrmann, H. B. (1974) J. Appl. Crystallogr. 7, 173-178. 8. Stuhrmann, H. B. (1975) "Neutron scattering for the analysis of Biological Structures," Brookhaven Symp. Biol. 27, IV-3, IV19. 9. Porod, G. (1951) Kollid Z. 124,83-114. 10. Damaschun, G., Muller, J. J. & Bielka, H. (1975) Acta Biol. Med. Ger. 34,229-239. 11. Wabl, M. R. (1974) J. Mol. Biol. 84,241-247. 12. Tischendorf, G. W., Zeichhardt, H. & St6ffler, G. (1975) Proc. Nat!. Acad. Sci. USA 72,4820-4824. 13. Lake, J. (1976) J. Mol. Biol. 165, 131-159. 14. Hill, W. E., Thompson, J. D. & Anderegg, J. W. (1969) J. Mol. Biol. 44, 89-102.
