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Orientation of the EFG Tensor with Respect to the Heme Group of Deoxymyoglobin

Y . M A E D A * , T . H A R A M I * , A . T R A U T W E I N * * , a n d U . G O N S E R * *

* Research Reactor Institute, Kyoto University, Kumatori-cho, Osaka, Japan ** Fachbereich Angewandte Physik, Universität des Saarlandes, Saarbrücken

(Z. Naturforsch. 31b, 487-490 [1976]; received July 30, 1975)

EFG Tensor, Herne Group, Deoxymyoglobin

Following Zimmermann's method of correlating intensity tensors from single crystal Mössbauer spectra with electric field gradient (EFG) tensors we are able to derive a manyfold of solutions for the orientation of the principal axis system of the local EFG 0£y£ with respect to the heme coordinate system in deoxymyoglobin. Comparing these results with theoretical estimates of the asymmetry parameter r\ we are able to limit the range of possible orientations of 0£y£, and we find to be rather close to the heme plane with a deviation from that of at most 14°.

I. Introduction Deoxymyoglobin single crystals are of the mono-

clinic A-type and contain two molecules per unit cell. Recently, a Mössbauer investigation of such single crystals has been performed, and the orientation of the principal axis of the electric field gradient (EFG) was derived assuming identical local EFG tensors for the two molecules in the unit cell, and further assuming rj = 0 1 .

Recently, R. Z I M M E R M A N N 2 has described a new method for evaluating single crystal Fe57 Mössbauer spectra which exhibit quadrupole splitting. There2

it is shown that the quadrupole line intensities can be characterized by an intensity tensor, which is proportional to the macroscopic EFG tensor. In the case of two Fe57 lattice sites per unit cell the macro-scopic EFG tensor VPq<m) (p =x, y, z) is the average of two local EFG tensors Vpq<loc x> and Vp(1<loc 2> associated with the two Fe lattice sites:

Vpq(rn) = (Vpq(locl) + Vpq(loc2)). (1)

The problem is thus reduced to the determination of the local EFG tensor from the macroscopic EFG tensor.

Requests for reprints should be sent to Prof. Dr. A. T R A U T W E IN .Fachbereich Angewandte Physik, Univer-sität des Saarlandes, D-6600 Saarbrücken 11.

II. Application of Zimmermann's Method to Deoxymyoglobin

Applying this method to deoxymyoglobin single crystals we are concerned with the specific symmetry properties of monoclinic crystals. For monoclinic crystals with a C2 axis along the b axis of the macro-scopic EFG tensor this axis has to be one of the principal axes because of symmetry considerations, and we define this axis to be the z axis of the

Fig. 1. Crystallographic frame of deoxymyoglobin single crystal with one of the two hemes per unit cell indicated by the pyramid. The b axis is a C2 axis and therefore choosen as z axis of the macroscopic PAS Oxyz (see text). The orientation of x and y within the a'c plane follows from the macroscopic intensity tensor, as derived from Mössbauer single crystal spectra of

deoxymyoglobin1.

488 Y. MAEDA ET AL. • ORIENTATION OF THE EFG TENSOR

macroscopic principal axes system (PAS) Qxyz-Thus, the other two axes (x and y) have to lie in the a'c plane, where we define the a' axis to be mutual orthogonal with the crystallographic b and c axes (Fig. 1). Since VPq (m) is diagonal in the macroscopic PAS, one derives for the local EFG contribution in the present case:

Vpp(ioc2> = Vpp(ioci)j p = x , y , z Y x y { 100» = V ^ d o c i ) , (2)

V (loc 2) _ V (loci) y xz * xz > v (loc 2) — V (loci) y yz v yz (By rotation of 180° around z we are concerned

with the transformations z—>z, x—>—x, and y - > —y. ) From equation (2) it is obvious that only the four local EFG components (Vpp( locl> = VPP<m> and Va;2/( locl) = 0 = V,r2/<m> can be derived from the macroscopic EFG tensor, whereas the components V x 2 ( l o c l ) and V^1 0 0 2* remain undetermined. Fol-lowing Zimmermann we define an angle xp, which describes the manyfold of solutions for Vp(1<locl> in deoxymyoglobin:

xp = arc tan (V^iooD/y^uoci)). (3)

Using equation (3) we are now able to specify the local EFG tensor V p q ( l o c l ) in terms of the para-meter xp. After diagonalization of VPq( l0Cl) we derive

the orientation of the local PAS with respect to the macroscopic PAS Ox y z in terms of the Euler angles a, ß, y. 0xyz is related to the crystallographic frame a b c as shown in Fig. 1. For the study of deoxymyoglobin we get the intensity tensor from a least square fit of the experimental results from our former work1 and derive a manyfold of solutions for the asymmetry parameter rj, and the angles a, ß, y as shown in Fig. 2. The values of xp in Fig. 2 are restricted to 0 <xp < n/2; the solutions for the corresponding rp in the other quadrants n—xp, ti -f- xp, and 2 n—xp are obtained by reflecting the local EFG at the yz-, xy-, and rcz-planes, respec-tively. The Euler angles for n/2 < xp < 2 n result then from the values of Fig. 2 by using the trans-formations

(a,ß,y) -> (n—a, ß, n—y) for xp -» n— xp, (a, ß, y) - > (a, n—ß, ti—y) for xp n + xp, (a, ß, y) (ti—a, n—ß, y) for xp 2 n — xp.

III. Discussion

From three separate theoretical studies of the electronic structure in deoxymyoglobin and deoxy-hemoglobin3 -5 rj was suggested to be in the range 0.3 <rj < 0.5, and was found to be positive. With this result we are able to limit the range of possible values of xp from Fig. 2 and the trans-formations above to 15° < xp < 29°, 151° < xp < 165°, 195° < xp < 209°, and 331° < xp < 345°.

In order to locate the local PAS with respect to the heme coordinate system we make the following

transformations. The local PAS p = ^f^ is expressed

in terms of the macroscopic PAS p = (f) by the coordinate transformation matrix A (a, ß, y) with a, ß, y being the Euler angles from Fig. 2:

?) = A p . (4)

The macroscopic PAS p is transformed into the heme coordinate system p ' = (y )̂ of Fig. 3 by the coordinate transformation matrix B :

p ' = B p . (5)

The coordinates of the four heme nitrogens which define p ' are taken from acid metmyoglobin, the derivative for which the high resolution X-ray structural information (1.4 Ä) is available6. The error involved in the definition of p ' for deoxymyo-globin by taking heme coordinates from acid metmyoglobin is believed to be very small as

[degrees]

Fig. 2. Sign of V|| ( > 0 for dotted rj-line, and < 0 for dashed ??-line) asymmetry parameter ij, and Euler angles a, ß, y of the local PAS Oxyz with respect to the macroscopic PAS Oxyz as a function of the parameter xp, which describes the manyfold of solutions in mono-clinic crystals. The angle ß is identical with the angle

between Viz and the b axis.

489 Y. MAEDA ET AL. • ORIENTATION OF THE EFG TENSOR

Fig. 3. Heme coordinate system p'={%,') as derived from the X-ray structure analysis of acid-metmyo-globin6 together with the orientation of the local PAS p with respect to p' (The angle between N1-N3-and N2-N4-axes is 89.12°. The numbering of heme nitrogens is the same as that in Fig. 1). By rotation of p' around z' by a we get xa and ya. The proceeding rotation around ya by ß' transforms z' into I and xa

into xß. Finally by rotation around 2 by y' we obtain the local PAS p with respect to p'.

difference Fourier analysis attests7'8. For the final location of the local PAS p within the heme coordinate system p ' we substitute p from equation (5) into equation (4) and get:

p = A B " 1 p ' (6)

The matrix product C = A B - 1 in equation (6) is a function of new Euler angles a', ß', y', which are given in terms of xp within the range 0 < xp < 71/2 in Fig. 4. The angles a', ß', y' define direction cosines of the local PAS within the heme coordinate system; they are summarized in Table I for the four possible ranges of ^-values given above.

Assuming (i) that the heme plane has fourfold symmetry, and (ii) that the proximale histidine group gives rise to lower the symmetry of the whole heme group to only C2Vf we are within the approxi-mation of the theoretical studies3-5 mentioned above. This is a reasonable assumption and allows

Table I. Direction cosines of the local PAS within the heme coordinate system for the four possible ranges of

^-values defined in the text.

Quadrant xp Direction cosines*

4> [degrees]

Fig. 4. Sign of Vzz ( > 0 for dotted 77-line, and < 0 for dashed r]-line) asymmetry parameter rj, and Euler angles a', ß', y' of the local PAS Oxyz with respect to the heme coordinate system Ox'y'z' as a function to

the parameter xp.

1 15° — 0.8763 0.4281 0.2212 — 0.2813 — 0.1013 --0.9542 — 0.3873 — 0.8976 0.2107

25° — 0.8830 0.4683 0.0315 — 0.1377 — 0.2014 --0.9698 — 0.4484 — 0.8601 0.2432

2 165° 0.6003 0.1775 --0.7798 0.1203 — 0.9827 --0.1412

—0.7925 — 0.0082 --0.6098 155° 0.5105 0.3493 --0.7357

0.1970 — 0.9358 --0.2922 — 0.8381 — 0.0050 --0.5456

3 195° —0.5545 0.7334 0.3933 —0.5307 — 0.6693 C.5200

0.6431 0.0802 0.7616 205° — 0.4972 0.8344 0.2378

— 0.6324 — 0.5342 0.5553 0.5394 0.1248 0.7902

4 345° 0.2786 — 0.1279 --0.9519 — 0.9323 0.2121 --0.2930

0.2378 0.9695 --0.0588 335° 0.1246 — 0.0168 --0.9921

—0.9721 0.2003 --0.1222 0.1998 0.9798 0.0094

* The components of the local PAS X, Y , Z with respect to the heme coordinate system are defined by the direction cosine matrix

X x ' Xy ' X z ' Y x ' Yy' Y z ' Zx ' Zy' Zz ' .

490 Y. MAEDA ET AL. • ORIENTATION OF THE EFG TENSOR

us to further limit the range of possible solutions given in Table I. With the heme normal being nearly a twofold symmetry axis each of the vectors X , Y, Z must have one component close to one. From inspection of Table I we find that this condition is comparable with the direction cosines for ^-values in the quadrants 1 and 4. From Fig. 4 we finally find 113.5° < a' < 119°, 76° < ß' <78° , 88.5° < y' < 103°, and from a corresponding plot for the quadrant 4 we get 76° < a' < 79°, 93° < ß' < 88°, 4° < y' < 17°. These angles indicate that the main component \**z ° f the local EFG tensor is oriented rather close to the heme plane, and to one of the two N-N-axis (y'). This result seems to be specific for deoxymyoglobin (and deoxyhemoglobin) where iron is pentacoordinated8-9. The deviation of V**

1 A . TRATTTWEIN, Y . M A E D A , U . G O N S E R , F . P A R A K , and H . F O R M A N E K , Proc. of the 5th Intern. Conf. on Mössbauer Spectroscopy, Bratislava (CSSR), Sept. 1 9 7 3 ; U . G O N S E R , Y . M A E D A , A . T R A T J T W E I N , F . P A R A K , and H . F O R M A N E K , Z . Naturforsch. 29b, 241 [1974].

2 R . Z I M M E R M A N N , Nucl. Instr. Meth., in press. 3 B . H . H U Y N H , G . C . P A P A E F T H Y M I O U , C . S . Y E N ,

J. L. G R O V E S , and C. S. Wu, J. Chem. Phys. 61, 3750 [1974].

4 A . TRATTTWEIN, R . Z I M M E R M A N N , a n d F . E . H A R R I S , Theor. Chim. Acta 37, 89 [1975].

from being totally "in plane" by 90°—ß' is probably due to the whole heme group which is not an exact plane10. The additional slight deviation of the Y'/z

direction from the y'z'-plane by a'—90° might be ascribed to the influence of the proximale histidine (imidazole) group, the plane of which also deviates from both, the orientation of the x'z' plane and that of the y'z' plane11.

We wish to thank Dr. R . Z I M M E R M A N N for providing us with his manuscript (ref.2) prior to its publication. One of us (A.T.) likes to acknowledge the invitation by Prof. H I G A S H I M U R A to the Research Reactor Institute, Kyoto University, where this work was completed. We are thankful for the sup-port of the Stiftung Volkswagenwerk and the Deutsche Forschungsgemeinschaft.

5 H. E I C H E R , D . B A D E , and F. P A R A K , to be published. 6 J . C . K E N D R E W , H . C . W A T S O N , a n d P H I L L I P S , u n -

published. 7 E . A N T O N I N I and M. BRTJNORI , in "Hemoglobin and

Myoglobin in their Functions with Ligands", p 89, North Holland, Amsterdam 1971.

8 C . L . N O B B S , H . C . W A T S O N , a n d J . C . K E N D R E W , Nature 2 0 9 , 3 3 9 [ 1 9 6 6 ] .

9 H. M U I R H E A D and J . G R E E R , Nature 2 2 8 , 5 1 6 [ 1 9 7 0 ] . 1 0 D . F . K O E N I G , Acta Crystallogr. 1 8 , 6 6 3 [ 1 9 6 5 ] . 1 1 J . C . K E N D R E W , R . E . D I C K E R S O N , B . E . S T R A N D -

B E R G , and E . R . D A VIES , Nature 1 8 5 , 4 2 2 [ I 9 6 0 ] .