The crystal structure of insulin

J. Mol. Biol. (1966) 16, 227-241

The Crystal Structure of Insulin

m.

Evidence for a 2-fold Axis in Rhombohedral Zinc Insulin

ELEANOR DODSON, MARJORIE M. HARDINGt, DOROTHY CROWFOOT HODGKIN

Chemical Crystallography Laboratory, South Parks Road, Oxford, England AND MICHAEL

G.

ROSSMANNt

M.R.C. Laboratory of Molecular Biology, Cambridge, England (Received 8 November 1965) The existence of non-space-group symmetry elements in rhombohedral 2 Zn insulin and 4 Zn insulin crystals has been investigated. A rotation function shows the existence of a 2-fold axis perpendicular to the crystallographic c-axis (3.fold) and making an angle of 44° with the a-axis. A translation function, and independent arguments from the Patterson series, show that this is a 2·fold axis, without translation parallel to its length. In 2 Zn insulin the 2-fold axes pass through or within 0'5 A of a 3-fold or, less probably, a 3·fold screw axis; in 4 Zn insulin they are about 1 A from it.

1. Introduction Both 2 Zn and 4 Zn insulin rhombohedral crystals and also the orthorhombic series of insulin salts have asymmetric units which contain two protein monomers of molecular weight about 6000. Insulin dimers also occur in many insulin solutions and may well be identical with the asymmetric unit of these crystals. In 1960, Low & Einstein published evidence for the presence, in the crystals of orthorhombic insulin sulphate, of non-space-group 2-foldaxes of symmetry which relate pairs of monomers within each insulin dimer; these require the two molecules in anyone dimer to be identical in configuration. They also suggested, and their suggestion has been discussed in some detail by Marcker & Graae (1962), that in the rhombohedral zinccontaining crystals similar 2-fold axes might exist within each insulin dimer, oriented perpendicular to the 3-fold axis in the crystal structure and intersecting it. The rotation and translation functions recently developed (Rossmann & Blow, 1962; Rossmann, Blow, Harding & Coller, 1964) make it possible to investigate in a precise way the relation between subunits within any crystal. We have therefore applied these functions to the data for 2 Zn and 4 Zn insulin and have re-examined the Patterson distribution described in the preceding paper (Harding, Hodgkin, Kennedy, O'Connor & Weitzmann, 1966) in the light of our results. Our evidence shows the existence and orientation of 2-fold axes in the crystals and imposes limits on their position.

t Present address: Chemistry Department, University of Edinburgh, West Mains Road, Edinburgh 9, Scotland. ~ Present address: Purdue University, Lafayette, Ind., U.S.A. 227

228 E. DODSON, M. M. HARDING, D. C. HODGKIN & M. G. ROSSMANN

As described earlier, 2 Zn and 4 Zn pig insulin crystals both have the space group R3; the dimensions of the equivalent hexagonal cells are a = 82,5, C = 34·0 A; and a = 80·7, c = 37·6 A, respectively.

2. The Rotation Function If there are two identical molecules in the asymmetric unit of a crystal, it is possible to make every atom of molecule I coincide with the corresponding atom of molecule II by a rotation of molecule I about some axis and a translation in some direction. The self.Patterson of molecule I, i.e. the array of vectors between atoms in this molecule, can be made to coincide with the self-Patterson of molecule II by the same rotation about an axis which is in the same direction but passes through the origin of Patterson space. The Patterson distribution will consist of these two self-Pattersons together with all the vectors between atoms in different molecules; the cross vectors are not, in the general case, related by the rotation axis through the origin. The orientation relationship between the two molecules can be found by rotating the Patterson function until the rotated and original functions are brought into maximum coineidence. Rossmann & Blow (1962) have shown that this is most conveniently done in practice by calculating-a rotation function R(K,If,(P) in reciprocal space, using (F)2 values. The operation is equivalent to taking a sphere of radius R around the origin of the Patterson distribution, rotating this through KO about an axis defined by the polar eo-ordinates If and c/> (see Fig. 1), superimposing this rotated Patterson on the unrotated one, forming the product of the two vector densities at all points and then integrating over the whole sphere. c

FIG. 1. The relation between the hexagonal crystallographic axes and the angles the rotation function.

rp, rf>,

used in

In the space group R3, an asymmetric unit of the rotation function lies within the range K = 0 to 180°, If = 0 to 180° and c/> = 0 to 60°; other rotations and axes are related by the space group symmetry.

THE CRYSTAL STRUCTURE OF INSULIN.

III

229

The method of calculation described by Rossmann & Blow (1962) was followed. Initially, intensity data for 2 Zn insulin for planes with spacings> 10 A were used to calculate R(K,if,,p) over a coarse grid with intervals of 20° in each angle (Fig. 2).

K"'80°

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FIG. 2. Stereograms representing the results of the rotation function calculated for 2 Zn insulin at 10 A resolution, self-Patterson radius taken as 30 A. The "origin" peak at K = 120°, .p = 0° represents the crystallographic 3-fold rotation axis. The dotted contour line represents the rotation function at a level of 25% of the origin peak. The full lines are contours at intervals corresponding to 10% of the origin peak.

Intensity data to 6A resolution (a little over 200 independent reflections) were then used to calculate R (180°, if, ,p) for both 2 Zn and 4 Zn insulin, which showed large maxima in the same position for both crystal forms (Fig. 3(a) and (bj), The peak maxima in both compounds were found to be at K = 180°, if = 90°, ,p = 44°. Each of the angles is probably correct to about 2°. Finally, 4·5 A data for 2 Zn insulin were used to calculate R (180°, 90°, ,p) which shows an even larger maximum (Fig. 3(d)}. Figure 3(c} demonstrated that the maximum lies accurately at K = 180° by mapping R (K, 90°, 44°). This means that a rotation of 180°, i.e. a 2-fold axis, perpendicular to c and at 44° to a, is required to make the self-Pattersons coincide. Since 2-fold axes in the same crystallographic direction had been found in both 2 Zn and 4 Zn insulin, a further calculation was made to examine the relations between the two structures. A rotation function R (M, 90°, 44°) illustrated in Fig. 4, was computed to compare a sphere of 2 Zn insulin Patterson with 4 Zn insulin Patterson. The maxima are at K = 0° and 180°, which suggests that there is very little, if any, relative rotation of the molecules in the two structures. The peak heights of the rotation functions at K = 0° and 180° are given in Table 1. These show that the 2-fold axes in the crystal are quite powerful, relative to the 3-fold axes; 88% of the vectors, as observed at 6 A resolution, in the 2 Zn crystal and 76% in the 4 Zn crystal conform with 2-fold symmetry.

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Fro. 3. Stereograms represent R (180 °, .p, .p) for (a) 2 Zn insulin and (b) 4 Zn insulin at 6 A resolution and radius of integration = 30 A; (c) shows R(K , 90°, 44°) for 2 Zn insulin at 6 A resolution using a 35 A radius self-Patterson sphere; (d) shows R(180°, 90°, .p) for 2 Zn ins ulin at 4·5 A resolution using a 30 A radius self- Patterson sphere.

THE CRYSTAL STRUCTURE OF INSULIN.

III

231

2400

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FIG. 4. Rotation function R(K, 90°, 44°) at 6 and 4 Zn insulin.

A resolution showing relationship between 2 Zn

TABLE

1

Relative heights of 3-fold and 2-fold rotation function peaks R(O, 0, 0)

2 Zn insulin 4 Zn insulin 2 Zn-4 Zn comparison

100 100 100

88 76

96

3. The Translation Function The observation that there are 2-fold axes in the 2 Zn and 4 Zn insulin crystals perpendicular to the 3-fold axes formally leaves open the question of their position relative to the symmetry elements of the crystal, 3·fold and 3-fold screw axes. It also formally leaves undetermined whether the 2-fold axes are 2.fold rotation or screw axes. Figure 5 illustrates the four conceivable types of arrangement of 2-fold axes and 2-fold screw axes relative to 3·fold axes. The translation function described by Rossmann et al. (1964) is designed to find the vector distances, A, between two molecules related by a 2-fold screw axis. A has a precise component, t, parallel to the 2·fold screw axis; its other components perpendicular to this which relate the molecular centres cannot be precisely defined when the molecule is complicated and irregular in shape. A rotation of the Patterson function about the direction of the axis, combined with a translation, 2t, parallel to the axis, brings the cross vector patterns of the two identical molecules into coincidence. In practice, the operation of rotation and translation can be carried out by the calculation of an appropriate Fourier series starting with (F2) terms.

232

E. DODSON, M. M. HARDING, D. C. HODGKIN & M. G. ROSSMANN

Within a rhombohedral unit cell (Fig. 5) there are three screw axis relations between pairs of identical molecules, namely Al and B v A2 and B 3 , and A3 and B 2 •

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These screw translations are related: if p is the perpendicular distance of the 2-fold axis relating Al to B I from the origin 3-fold axis, as in Fig. 6, and t the screw translation from Al to B I , then the other screw translations are: All '" B 3 t A 2B , A3

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THE CRYSTAL STRUCTURE OF INSULIN.

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233

Sections of T (A) have been calculated for 2 Zn and 4 Zn insulin with different values of r, the radius inside which all cross vectors can be expected and with and without the origin peak and its surroundings removed to allow for the extra effect of the self vectors in this region (Rossmann et al., 1964). Sections at Z= 0, perpendicular to the c axis are shown in Fig. 7. The rotation axis used for all these functions was if1 = 90°, ~ = 44°, K = 180°. T (A) is plotted in terms of the crystallographic hex. agonal axes and the direction of t, parallel to the rotation axis, has been drawn in afterwards. As might be expected, the peaks are sharp in the direction of t but extended in the plane perpendicular to it. As shown they provide evidence on the relative arrangement of the molecules as seen in projection only in Fig. 5. The main maxima consist of long ridges passing through the origin and at the rhombohedral lattice repeats. On the 2 Zn insulin map at high resolution, (Fig. 7(a)), they are almost the only feature and strongly suggest that here tAiBi' tA 2B 3 and tA 3B 2 are all equal and very close to 0, i.e. that the insulin molecules are related by 2-fold axes that pass through a 3-fold axis as in Fig. 5(d). The small slope of the main maximum to the t = 0 line might be due to the 2-fold rotation axis having an orientation angle slightly greater, about 45°, to a, than the angle used in the calculation, or a position very slightly displaced, probably less than 0·3 A from the 3-fold axis. At lower resolution there is a marked tendency for peaks to occur roughly midway between the main maxima, which could indicate an alternative position for the 2-fold axis, displaced by half a unit cell from the 3-fold axis. It seems most likely that these peaks are in the nature of a series termination effect; at low resolution the largely unresolved Patterson peak density superimposes on itself through rotation and translation in this region. The effect is enhanced, as might be expected, by the removal of the inner part of the Patterson function in the calculation (compare Fig. 7(b) and (c)). The 4 Zn patterns at low resolution are very similar to those of 2 Zn insulin, indicating that the molecular positions are not greatly different. In the high resolution pattern (Fig. 7(d)), the long ridges running through the origin and equivalent positions are each clearly broken into two maxima. One set of these corresponds closely with the maxima in the 2 Zn pattern and here too suggests that the 2-fold axis is at 45° to a. The second set of peaks marks the differentiation of the cross Patterson vectors and strongly suggests that the 2-fold axis passes close to, but not through, the 3-fold axis. From the distance apart of the two peaks at the same slevel, the distance of the 2-fold axis from the 3-fold axis can be estimated as about l'6jV3, i.e. about 0·9 A.

4. Discussion The conclusions drawn from the rotation and translation functions for 2 Zn and 4 Zn insulin are supported by a direct examination of the Patterson functions. These show very clearly the features to be expected for Patterson maps derived from structures in which there are 2-fold axes running (1) through, and (2) at a small distance from, a 3-fold axis as in Figs 8 and 9. The type of arrangement found for 2 Zn insulin (Fig. 8(a)) is one of high symmetry. All the vectors between atoms within one group of six molecules should have the symmetry 32m, as result of adding the Patterson centre of symmetry to the molecular symmetry 32. The mirror planes and 2-foldaxes in the Patterson distribution, since they are not space group symmetry elements, are limited in their extent by the

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236 E. DODSON, M. M. HARDING, D. C. HODGKIN & M. G. ROSSMANN

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interpenetration of the vector patterns around neighbouring lattice points. In the 6 A resolution Patterson maps (Harding et al., 1966) all the strong peaks actually lie on the mirror planes or midway between them. The very simplicity of the maps calls for some caution in their interpretation; the pattern could be produced by the existence of one prominent chain direction in the insulin asymmetric unit and was indeed interpreted in this way in early attempts to understand the distribution (compare Lindley & Rollett, 1955). However, both the rotation and translation functions and direct inspection of the Patterson maps at high resolution show that the observed high symmetry persists as far as the data extend. The mirror planes, for example, can be traced as in Fig. 10 in individual sections of the 2·2 A sharpened Patterson distributions to a distance of about 30 A (compare the streak length in Fig. 7(a)); approximate mirror planes can also be seen directly in single reciprocal lattice layers extending to the limit of the reflections observed. In 4 Zn insulin, particularly in the high-resolution maps, the effects of additional symmetry are less marked than in 2 Zn insulin, and conform quite well to the situation indicated in Fig. 9(b). One can trace, as in Fig. 11, with perhaps some over optimism, lines of peaks parallel to the mirror planes through the origin of the Patterson, in the position of the displaced mirror planes to be expected for a structure in which the 2-fold axis lies at about 1 A from the 3-fold axis. The preferred position is perhaps best identified by the comparison of the Patterson projections for 2 Zn and 4 Zn insulin illustrated in Fig. 12. Probably the best general evidence for the position of the 2-fold axes in the insulin crystals is that derived not so much from the translation relations as from calculations of relative peak heights in the rotation function. The number of vector peaks which should be related by a 2-fold axis through the origin are very different in the four

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240

E. DODSON, M. M. HARDING, D. C. HODGKIN & M. G. ROSSMANN

examples pictured in Fig. 5. They are given below, where n is the number of atoms in each molecule and therefore 36 n 2 vectors arise in the six molecules. (b) 8n 2

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(where n is large and n (n - 1) ~ n 2 ) The observed situation in 2 Zn insulin recorded in Table 1, where the 2-fold axis rotation peak is nearly as large as the origin peak, clearly corresponds to case (d); the corresponding peak in 4 Zn insulin is only a little lower, again suggesting that the six monomers conform to the 2-fold axis less well in the latter crystal than in the former. One problem remains. All the formal symmetry relations so far described are shown by structures in which a 2-fold axis passes through or near a 3-fold screw axis, as well as by those in which a 2-fold axis passes through or near a 3-fold axis. The ambiguity is due to the difficulty of isolating individual groups of symmetry related vectors in the complicated vector patterns observed for insulin. Strictly therefore, by the arguments so far discussed, we can only define the translation relations as seen in projections along c. We may illustrate the arrangement found up to this point, for 2 Zn insulin by Fig. 13, where the 3-fold axis may lie at either position, I, II, or III.

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It is possible, tentatively to carry the argument a stage further. The comparison of the wet and air-dried 2 Zn insulin Patterson distributions shows that there is very close similarity of the vector patterns around each origin, provided one pattern is turned 6° around the 3-fold axis relative to the other. The relations suggest that here one can distinguish the effect on the vector pattern of moving apart and rotating hexamer units which are essentially coherent around the 3-fold axis. The symmetry of the coherent vector pattern is 32m, and correspondingly it seems likely that the

THE CRYSTAL STRUCTURE OF INSULIN.

III

241

hexamers have the symmetry 32, i.e. that the 3-fold axis is at I in Fig. 13. The argument is a little weakened by the diffuse character of the low-resolution maps. However, in the 4 Zn insulin maps one can certainly begin to distinguish vectors between dimers and within one dimer unit. Here the hexamer around the 3-fold axis appears to have the symmetry 3, approximating to 32. The relations between the two crystals can be easily understood if there are, in both crystals, identical dimers co-ordinated around zinc ions. In 2 Zn insulin an arrangement is suggested in which ligands from three insulin molecules are attached to each zinc ion; in 4 Zn insulin, three of the zinc ions may surround the 3-fold axis and each be attached to only one insulin molecule. The type of arrangement found in 2 Zn insulin is that proposed by Low & Einstein and by Marcker & Graae. It is perhaps worth adding, since Marcker & Graae expressed some concern over this problem, that it is not inconsistent with the asymmetric growth of insulin crystals observed by Schlichtkrull (1957). If the effective unit is a hexamer of 32 symmetry, the crystals may well grow by contact around one or other of the 3-fold screw axes. These are both asymmetric in character and are differently related to the two insulin molecules in a dimer. Further evidence on the existence of insulin hexamers is described in a forthcoming paper on monoclinic insulin crystals (G. Ferguson and M. G. Rossmann, manuscript in preparation). We are most grateful to Dr M. F. Perutz for suggesting the application of the rotation and translation functions to the X-ray data on insulin and to Dr David Blow for frequent, very helpful and stimulating discussions. We wish to thank the Cambridge University Mathematical Laboratory, IBM United Kingdom Ltd., and the Oxford University Computing Laboratory, for all the computing we have been allowed to do. We also thank Miss Jill Collard, Miss Angela Campbell (at Cambridge) and Mrs Sabina Cole, Miss Gwen Humphries and Miss Ann Cropper (at Oxford) for assistance in various stages of the work. We gratefully acknowledge the financial assistance of the Rockefeller Foundation.

REFERENCES Harding, M. M., Hodgkin, D. C., Kennedy, A. F., O'Connor, A. & Weitzmann, P. D. J. (1966). J. Mol. Biol. 16, 212. Lindley, H. & Rollett, J. S. (1955). Biochim. biophsj«. Acta, 18, 183. Low, B. W. & Einstein, J. R. (1960). Nature, 186, 470. Marcker, K. & Graae, J. (1962). Acta. Chem, Scand. 16, 41. Rossmann, M. G. & Blow, D. M. (1962). Acta Cryst. 15, 24. Rossmann, M. G., Blow, D. M., Harding, M. M. & Coller, E. (1964). Acta Cryst. 17, 338. Schlichtkrull, J. (1957). Acta Ohem, Scand., II, 484.

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