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Paramagnetic anisotropy measurements on a single crystal of deoxyhemoglobin
BIOCHIMICA ET BIOPHYSICA ACTA
355
SBA 36209 P A R A M A G N E T I C A N I S O T R O P Y M E A S U R E M E N T S ON A S I N G L E CRYSTAL OF D E O X Y H E M O G L O B I N
N O R I H I T O NAKANO, J I N Y A OTSUKA AND A K I R A TASAKI
Faculty of Engineering Science, Osaka University, Toyonaka, Osaka (Japan) (Received April 5th, 1972)
SUMMARY
The temperature dependence of the paramagnetic anisotropy of a single deoxyhemoglobin crystal (Form II) was measured by the torque method with a sensitivity of lO -4 dyne. cm in the temperature range of 2 to 77 °K. Torque curves were measured in the three perpendicular planes of the crystal b y a rotating magnetic field. The torque curves obtained were of a sinusoidal shape per half rotation of the magnetic field. The phases of the torque curves as well as the amplitudes changed with temperature. This means that the three principal values of magnetic susceptibility of heine depended on temperature and these temperature dependencies were not the same. The amplitude of the observed torque was very small when the magnetic field was rotated almost parallel to the heme planes in the unit cell, while it was very large when the magnetic field was rotated perpendicular to the heme planes. The directions of the magnetic field which gave zero torque values were nearly perpendicular and parallel to the heme plane. From these results, it was concluded that the magnetic susceptibility of heme in deoxyhemoglobin was fairly axially symmetrical, taking the axis to be nearly perpendicular to the plane of the heme, though in detail a small a s y m m e t r y in the heine plane existed. The temperature dependence of amplitude measured over a wide temperature range was analyzed on the basis of the spin Hamiltonian; 9ff ~-- DS~ + E(S~ -- Sy) + /3-SgH. Experimental data were best represented by the theoretical curves which were calculated from the values of D = 5.3 and E = 0. 9 cm -~. This study made it clear that the previously reported result of D = -- 18 cm -~ was incorrect (M. Kotani, Ann. N . Y . Acad. Sci., 158 (1969) 20).
INTRODUCTION
Previously we have reported several magnetic studies and a theoretical analysis of the electronic state of the iron ion in hemoproteins 1-3. Since E P R methods were applicable to the ferric form only, our magnetic studies were started also from this ferric form1, ~. The main results obtained from these studies can be summarized as Biochim. Biophys. Aeta, 278 (1972) 355-371
356
N. NAKANO el al.
follows: (I) The fine structure of the lowest energy level (6A1) for the high-spin iron ion in methemoglobin and metmyoglobin is well represented by the spin Hamiltonian of D S z when D was chosen to be IO cm -1. (2) In the case of acidic Imethemoglobin, a considerable amount of heine iron was of the low-spin type even at very low temperatures ( 7 7 - 2 °K), and this low-spin compound showed a different E P R spectrum from that of the low-spin type in alkaline pH. For the ferrous compounds of hemoproteins, however, sufficiently accurate magnetic studies have not yet been made. This may be due to the fact that E P R methods are not usually applicable to ferrous ion, which has an even number of d-electrons in its outer orbit. Lang and Marshall have studied the M6ssbauer effect of deoxyhemoglobin and they reported that the magnetic field dependence of the M6ssbauer spectra could be explained by D = 5 cm-t if the fine structure of d-electronic energy level was represented by the spin Hamiltonian of D S z (ref. 4). In the previous work 3, we have measured the paramagnetic susceptibility of deoxyhemoglobin and deoxymyoglobin and also reported that the temperature dependence of the paramagnetic susceptibility could be well represented by the spin Hamiltonian D S z and D 5 cm-1. =
='
T
=
Sz=t2
/
312
$z7--+1
D -
_
Sz=_+O
Fig. I. The fine s t r u c t u r e of the d-electronic energy level of the ferrous high spin iron ion in hemoprotein. This s t r u c t u r e of the energy level corresponds to the case of D > o a n d E << D.
In order to obtain detailed information of the electronic state and to confirm the expected energy scheme shown in Fig. I, measurements of the paramagnetic anisotropy of a single crystal of deoxyhemoglobin were required. An experiment on this line has already been reported by Kotani 5, who analyzed the preliminary experimental data of lizuka. The estimated value of D from this experiment was --18 cm -1. Since this value was inconsistent with results of M6ssbauer effect and with those of the magnetic susceptibility previously reported by us, another precise experiment on the temperature dependence of the paramagnetic anisotropy was performed. In the present paper, we report the results of this paramagnetic anisotropy study. SAMPLE
Hemoglobin was prepared from human red cells by lysis with toluene. The single crystal of deoxyhemoglobin was grown by dialyzing a hemoglobin solution (1% hemoglobin in I.I M (NH4)2SO, solution, pH 6.5) against 2.5 M (NHa)2SO4 solution at pH 6. 5 with a small amount of EDTA. These inner and outer solutions were deoxygenated by adding dithionite. The completeness of deoxygenation was checked by measuring absorption spectra in the region from 400 to 650 nm. The single crystal of about 2 mm 3 volume was used as a sample. No contamination by methemoglobin in the sample crystal was checked by measuring E P R spectra at Biochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC ANISOTROPYOF DEOXYHEMOGLOBIN
357
liquid N 2 temperature before the paramaguetic anisotropy measurement. It is well known that deoxyhemoglobin gives no E P R signal, while methemoglobin exhibits a sharp absorption line of E P R spectra in both high- and low-spin state. The E P R signal of methemoglobin could not be observed even with very high sensitivity. The E P R absorption spectrum of the methemoglobin solution in the same quantity of hemoglobin as the sample crystal was measured. From the amplitude of the E P R signal, it can be calculated that the contamination by methemoglobin of the sample crystal is less than o.1%. Deoxyhemoglobin of normal adult human crystallizes in three different forms 6. In the present study, we used the Form I I crystal, since the nature of this form has been elucidated crystallographically b y X - r a y diffraction studies 7-9. The character of this crystal is monoclinic with a space group of P21, where the dyad screw axis corresponds to the b axis. Unit cell dimensions are given as a ---- 62.5 A, b = 83.2 A, c ---- 52.8 ,~ and fl -- 98°. There are two hemoglobin molecules in the unit cell, and therefore eight hemes. The clinographic projection of this crystal form is illustrated in Fig. 2. After paramagnetic anisotropy measurement, the crystal was quantitatively analyzed b y spectrophotometry in solution by converting it to pyridine hemochromogen. We used a millimolar extinction coefficient of e m M ---~ 34.7 at 556 nm.
bj
.,TIO
/010
110
Fig. 2. The clinographic projection of the deoxyhemoglobin single crystal (Form II). The crystal character is monoclinic and the space group P21. The b axis is dyad screw axis. The unit cell dimensions are a = 62.5 .&-,b = 83.2 A, c = 52.8 ~k and ~ = 98°. There are two hemoglobin molecules per unit cell. MEASUREMENT
The measurement of the paramagnetic anisotropy of the single crystal is carried out b y measuring the torque exerted on the crystal, when the crystal is subjected to the homogeneous magnetic field. The direction of magnetic field is rotated in a certain plane of the sample and thus the angular dependence of the torque value is obtained. The equipment prepared for the present experiment is illustrated in Fig. 3. The sample, attached to the lower tip of the quartz bar, is placed between the two poles of the magnet. The upper end of the quartz bar is fixed to a compensation coil as shown in Fig. 3. Sample, quartz bar and compensation coil make a single rotating unit, and this unit is suspended b y a thin phosphor bronze ribbon to enable it to rotate Biochim. Biophys. Acta, 278 (1972) 355-371
358
NAKANO x. et al. . . . . *~....... b a c k line
3sphor e ribbon I mirror net
~oto cell 3ensation coil
tz b a r
thermometer
Fig. 3. The equipment for the measurement of the temperature dependence of torque.
freely. A small mirror is attached to the compensation coil, and the small rotation of the unit is detected by the deflection of this mirror. The deflection is received by a twin photo-cell and the resulting signal is amplified and fed back into the compensation coil. Thus, the balance of the torque meter can be automatically controlled. The sensitivity is enhanced when the weight of the rotating unit is reduced. In the present apparatus, this weight is about IO g, which gave a sensitivity of lO -4 dyne.cm. The procedure for calibration of the torque meter was carried out according to the method of Morimoto et al. x°. In order to obtain low temperatures, we used liquid N 2, H , and He, and in addition we obtained intermediate temperatures between these constant temperatures by slight vaporization of these liquids. The temperature of the sample was measured by a carbon resistor. Since the magnetic ion is dilute in the protein sample and, moreover, the sample crystal is very small (2 ram3), a high accuracy of torque measurement is required and therefore the blank torque value should be minimized. The attachBiochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
359
ment of the thermometer to the sample increases the blank torque value, thus instead, it was placed as near the sample as possible. The sample and the thermometer were then surrounded b y a copper sheet which rendered stability and uniformity of temperature around the sample. Thus, we were able to achieve accurate values of I / T to within + o . o i . The magnetic field was measured with Hall's probe, which was calibrated by proton NMR.
THEORETICAL BACKGROUND TO TORQUE MEASUREMENT
Let us consider the general case where the three principal values of the magnetic susceptibility tensor of a heme are different to each other. These principal directions of the tensor are taken to be x, y and z, and the corresponding principal values denoted as Zx, Zy and Zz. Since the sample crystal is of monoclinic type and the dyad screw axis corresponds to be b axis, the paramagnetic susceptibility tensor per unit cell m a y be expressed by the following form with the (a ,b, c*) coordinate system. =
~'CtO Zbb ~ \Z-~* o Z~%*/
(I)
Elements of its tensor are written in terms of the three principal values (Xx, Xy, Zz) as follows : + Uaa,y'Zy
+ Uaa,z'Y~z
+ Ubb,y'Z~ + Ubb,z'Zz Zc*c* = Uc*c',x'Zx + Uc*o*,y'Zy + U ~* *o ,z'Zz Zae* = Uac*x'Zx + Uae*r'Zy + Ua~*z'Zz
ZDb
~ Ubb,x" Zx
]
J
(2)
The eight hemes in the unit cell are assumed to be magnetically independent and identical. The coefficients Ua~,x, Ubb,y etc. are obtained as the sum of unitary transformations for these eight hemes. These values are constants determined by the directions of the three principal axes of magnetic susceptibility in each berne, and they satisfy the following general relations: Uaa,x
-~-
Uaa,y
-~- Uaa, z
Ubb,x
+ Ubb,y
+ Ubb,z
Uc*e*,x + U *c*o ,y + Uc*c*,z Uae*x + Uae*y + Uae*z
= 8 ]
8
8 o
(3)
We now define the new coordinate system (X, Y, Z), where X and Y axes are in the same plane as the rotating magnetic field, and the Z axis is perpendicular to this plane. The relation between this new coordinate system and the (a, b, c*) coordinate system is described as follows using Eulerian angles. The (a, b, c*) coordinate system is rotated around the c* axis b y an angle of 9, to obtain an (a', b', c*) coordinate system. The (a', b', c*) coordinate system is then rotated around the b' axis by an angle of 0, to obtain an (a", b', c*') coordinate system and finally this (a", b', c*') coordinate system is rotated around the c*' axis by an angle of ~v. Thus the (X, Y, Z) coordinate system is obtained. I f the Y axis is taken to be in the direction of magnetic field, the rotated angle of the magnetic field corresponds to ~p. The magnetic field vector in the (a, b, c*) coordinate system can be written by the Eulerian angles as follows: Biochim. Biophys. Acta, 278 (1972) 355-371
360
N. NAKANO et al. = H (--cos 0.cos ~o.sin ~o--sin ~0.cos ,p, --cos 0.sin ~0.sin ~p +cos q~.cos % sin 0" sin ~p)
(4)
The static magnetic energy per unit cell is generally expressed as (5)
E ~ --1/2HzH
Then the torque value L(~o) along the direction of the rotating magnetic field can be obtained b y differentiating the above magnetic energy with respect to ~o. Thus, using Eqns 1 - 5, the torque value per unit cell is described as follows: L(~p)
dL dip
IH2 {( Zz
2
Z.,:+Zy) -2
(klsinz~+k2c°s2~p}+(Zx--ZY) (kssinztP +
+ k4cos2~) }
(6)
This is the basic equation along quantitative analysis of torque value, because it is -epresented in terms of the anisotropy of the paramagnetic susceptibility (Zz -(%x + %y)/2) and (Zx -- Zy). The temperature dependence of torque values is attributed only to these anisotropy terms. The coefficients k 1, k 2, k 3 and k, are functions of the orientation of each heme in unit cell, and of the direction of the rotating magnetic field plane (~, 0). The coefficients are m a t h e m a t i c a l l y expressed as follows: kl ~ Uaa,z (cos2 0cos 2 9--sin~ qg) + Ubb,z (cos2 0sin 2 9--c0s2 qo) + Uc*c*,zSinz O--2Uac*,z cos Osin Ocos~p h2 ~ U~a,z cos Ocos qosinqo-- Ubb,zcos Ocos9sin ¢p-- Uac* zSin Osin ¢p k3 = (Uaa,y-- Uaa,~) (sin2 ~--cos 20cos 2 ¢p) + (Ubb,x-- Ubb,y) (COS20sin 2 qg--cos2 ¢p) + ( Uc*c',x-- Ue*c*,y) sin 2 0 -- 2 (Uac*,x-- Uac*,y) cos 0sin 0cos ~0 ka = (Uaa,x-- Uaa,v)cos 0cos 9sin q~--(Ubb,x-- Ubb,y) cos 0cos 9sin 9 -- (Uac*x-- Uac*,y) sin 0sin 9
(7)
RESULTS In the present measurements, torque curves were measured in the three different planes of the sample crystal. One plane is perpendicular to a axis (case A), the second parallel to (Oil) plane (case B), and the last perpendicular to the above (A, B) two planes (case C). Thus the planes are perpendicular to each other. Fig. 4 shows experimental d a t a of torque curves (case B) which were measured under various strengths of magnetic field at liquid He temperature. All torque curves are of sinusoidal shape with one period for each half rotation of the magnetic field (Fig. 4a), and these amplitudes are proportional to the square of magnetic field strength H (Fig. 4b), which is consistent with E q n 6. The linear relation between the amplitude and H 2 means t h a t the paramagnetic saturation effect does not occur at 4.2 °K even under a magnetic field of 12 kOe. The temperature dependence of torque curves was compared under a constant magnetic field strength. For example, two torque curves measured at liquid-N 2 and liquid-He temperatures are shown in Fig. 5 (case B). These two torque curves are quite different in their amplitudes, and moreover to some extent in their phases. As can be easily seen from E q n 6, the difference in amplitude of the torque curves is due to the temperature dependence of each of the three principal values of the magnetic susceptibility tensor, but the difference in phase is due to the differences in Biochim. Biophys. Acta, 278 (1972) 355-371
361
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
15
/
0
1
~IO
~\\~ / / I
/X \ ~ . ~
\\\\~-///t9.3~
"- 8.45 -'-
l\\-./P ,.o
.j" ~ o 4 dyno.~m
/
/.
/
./
~5
'fin \ 4.84
/
/"
/'
/ /
/
\\W/~ ..8 00
051108
I
10.108
I
1.540
8
H2 (Oe2) Fig. 4. (a) The magnetic field dependence of the torque curves measured at liquid-He temperature. The rotating magnetic field is parallel to the (oTI) plane (case B). (b) The magnetic field dependency of the peak-to-peak value in the above torque curves. The vertical axis is represented in arbitrary unit.
(lig.-Ne temp.) x ¼
. ~ 2 dyne .cm
Fig. 5. The temperature dependence of the torque curves measured in liquid-N 2 and -He temperatures (case B). Magnetic field strength is 22oo Oe. The difference in the phase between these torque curves is 8 °.
temperature dependencies of these three principal values. If, as in the case of acid methemoglobin in the high-spin state, the magnetic ion is of axial type and the two principal values of magnetic susceptibility tensor are equal to each other n, the phase of torque curve is not shifted by the change of temperature. In the case of C, the temperature dependence of phase as well as amplitude was also observed to same extent as in the case of B, which is shown in Fig. 6. In the case of A, however, the phase shift of torque curve could not be observed and the torque values were always zero at any temperatures, when the direction of magnetic field was parallel to the b and c* axes, as shown in Fig. 7. This tendency can Biochim. Biophys. Acta, 278 (1972) 355-371
362
N. NAKANO et al.
~ [
(hg.-He temp.)x{
\
/(lig.-N2temp.)
]
\
Fig. 6. T h e temperature dependence of the torque curves measured in liquid-N~ and -He temperatures (case B). Magnetic field strength is 22oo Oe. The difference in the phase b e t w e e n these torque curves is 7 ° .
/
/
/"~1(OT[)Ptane /
/
'~
Fig. 7. T h e temperature d e p e n d e n c e of t h e torque curves measured in liquid N~ and H e temperatures (case A). Magnetic field strength is 6ooo Oe.
be explained by Eqns 6 and 7, where the direction of magnetic field is in the bc* plane, so that 9 = o and 0 ---- ~/2. As easily calculated from Eqn 7, the coefficients k 2 and k 4 b e c o m e zero, and so Eqn 6 is reduced to the form :
-I H 2 { kl ( •z 2
Zx --+ Zy.) + ks(Zx 2
Zy) } sin 2 ~p
where ~p corresponds to the angle between b axis and the direction of the magnetic Biochim. Biophys. Acta, 278 (1972) 355-371
363
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
field. In this case the differences in the temperature dependencies of principal values do not affect the phase of torque curve. It follows that the magnetic susceptibility of deoxyhemoglobin is not of axial type, unlike acid methemoglobin in the high spin state.
3C
2~
20
•
C
"
z/(n 2) 15
10
I
o.1
o:2
o:3
I
o.4
o:~
1/T (°K'I) Fig. 8. The t e m p e r a t u r e dependence of the t o r q u e value per h e m e for t h e three cases (case A, B, C). B o t h t h e direction a n d t h e s t r e n g t h of m a g n e t i c field were k e p t to be constant. The vertical axis represents 3kT]~2.2LIHL Solid lines r e p r e s e n t the theoretical curves for the pair of F = 5.5 cm-1 a n d sin a = o.28.
The temperature dependencies of torque values were also measured continuously over a wide range of temperature (2-77 °K) keeping both the strength and the direction of the magnetic field constant. The direction of magnetic field was chosen so that the torque gave a maximum value at liquid He temperature in each case (case A, B, C). These results are shown in Fig. 8, where the torque values per heine are plotted in terms of (3kT[tS~) • (2L/H2). According to Eqn 6, this quantity corresponds to ([71(~z2 nx2 ~-nY2")--2+ C2(nx* -- ny~) -A(n*) (8) with the effective Bohr magneton numbers nx, n~ and nz defined by Zx = (nx*fl*)/3kT, •y ~-(ny*fl2)/3kT and Zz----(nz*fl*)/3kT, respectively. The parameters C 1 and Ca are constant but take different values in each case (case A, B, C). Fig. 8 exhibits the Biochim. Biophys. Aaa, 278 (1972) 355-371
N. NAKANOt3t al.
364
following tendencies of temperature dependence of A(n2). At the high temperature limit (I/T = o), the values of A(n ~) for all three cases converge to zero, so that nx, ny and nz become equal to each other at this temperature limit. With the lowering of temperature, the values of A(n ~) increase sharply, and give m a x i m u m values in the vicinity of I / T = o.12, and then decrease gradually as temperature is lowered further. These temperature dependencies of A (n 2) are due to the fine structure of d-electronic energy level of heme iron ion, which suggests that the lowest energy level of this fine structure should be Sz = o and the other excited levels are higher by about 6 em -1 corresponding to I/T = o.12. This means that the value of D is positive and of several units (cm-1). TABLE I THE
RELATIVE
VALUES
O F A($~ 2) F O R T H E
THREE
CASES AT SEVERAL
TEMPERATURES
The value of A(n 2) at I/T = o.12 is normalized to unity.
1/T
AA
AB
AC
0.03 0.07
0.490 0.868
0.593 o.917
0.596 o.917
0.12
I
I
I
o.24 0.32 0. 4
o.943 o.877 o.811
o.926 o.852 0.835
o.917 o.835 o.761
I f the axial character of the magnetic susceptibility tensor is considered, according to Eqn 8, the temperature dependence of A (n 2) in any directions of magnetic field becomes homologous. These relative values of A (n 2) at several temperatures are shown in Table I, wherein the values ofA (n 2) at I/T = 0.12 are normalized to unity. The following tendencies among these values are observed: At higher temperatures ( I / T < 0.I2), the relative values of A(n 2) are in the order o f A C > AB > AA, while in the lower temperature region (I/T > 0.I2), this order becomes AC < AB < AA. These relationships imply that A (n ~) exhibits the strongest dependence on temperature in the case of C and the weakest dependence in the case of A. The non-homologous temperature dependence of A(n 2) is consistent with the phenomenon of the phase shift produced by change in temperature. Both of them suggest that it is necessary to consider the a s y m m e t r y in heine plane, though the results of magnetic susceptibility measurement in solution or polycrystals can be explained without this a s y m m e t r y as reported in the previous paper 3. ANALYSIS In the previous paper 3, the general form of spin Hamiltonian including the a s y m m e t r y in heme plane was described as
~e = DS: + E(S~- S~) + S ' ~ . H
(9)
the coefficients (D, E and g) of this spin Hamiltonian were represented by the functions of spin orbit interaction and the energy differences between sublevels of the dlevel. Taking account of only two sublevels of the ground state (SB~) and the nearest
Biochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC
ANISOTROPY
365
OF DEOXYHEMOGLOBIN
excited state (BE), the coefficients of the spin Hamiltonian E q n 9 were represented as follows : D
~
-
-
32
e(~E,)
a2 {
gx = 2 +
-
e(~B~)+ ~(~E,~)
I -
E
32
-
e(SE~)
e(SB2)
a
(~B~)
I -
s("En)
}
e(SB2)
I
0o)
2 e(SEn) - - e(SB,~) a
i
gy~2+ 2 e ( S E , ) - - e(SB~) gz ~2
where a stands for the coefficient of spin orbit interaction. I n order to consider the E term of E q n 9 we use new parameters F (F > o) and a instead of D and E, which are defined as D = F . c o s a and E = F - s i n a/x/3. Then the spin Hamiltonian E q n 9 becomes ~a
= F.cos
a . S z~ +
--
it
$
F.sina],v/3(Sx _ Sy) + ~g~H
(II)
The temperature dependence of three principal values of magnetic susceptibility are calculated on the basis of E q n I I . I f the magnetic field (H) is zero, we can easily diagonize this spin Hamiltonian using Eigen functions of Sz (S ---- 2). Eigen values and Eigen functions obtained are as follows: 21 ~ --2F 22 ~ - - F c o s a - - ~ / ~ F s i n a ~tit ~ - - F c o s a + ~ / ~ F s i n a '~a ~ 2 F c o s a 25 = 2 F
(I2)
and ~//1 =
I --II)
I -- --[
V~
- - I)
V~
I
,p~=--II) V~
I
+--J-I) V~
I
I
~.~ = - - 12) - - - I - 2) V~ V~ sin a ~4
~/cos a + I 12)
2 Vcos a + I %
(~3)
sina
2VI - cosa
sin a IO) +
~/~ 12) +
~/i --cosa
V~
-
2)
-
2)
2~/cosa + I Io) ~-
sin a
2VI - cosa
I n the present experiments, the magnetic field is 8000 Oe in the high temperature region and 300o Oe in the low temperature region, so t h a t the Zeeman term becomes less t h a n 0.8 c m - L Since from previous magnetic susceptibility measurements, the value of D was found to be 5 cm-a, it can be expected t h a t the value of F falls to around t~iochim. Biophys. Acta,
278 (I972) 355-371
366
N. NAKANOet al.
this value. Therefore, this Zeeman term m a y be treated as a perturbation. As the paramagnetic saturation effect cannot be observed, each energy level (),'i; i z 1-5) is calculated up to the second order of magnetic field strength. Components of the magnetic m o m e n t s mix : - - ~ ] ( i / 6 H x , rely : - - 6~'l/~Hy miz = -- 6~t'l/6Hz for each level are obtained from eigen values 2'i (i : I-5) of the spin Hamiltonian E q n xi. B y averaging these m o m e n t s with the use of the B o l t z m a n n factor, we obtain the x component of magnetic susceptibility per heme as follows: 2: Mx
Xx = .
I
. . Hx
ml×exp
--
i = 1
.
(I4) Hx
Z
exp
--
Expressions for Zy and •z are also obtained in the same way. The explicit forms of Xx, Xy and Zz are shown in the Appendix. Each component of the effective Bohr magneton n u m b e r (nx, ny, nz) is calculated from the above expressions of magnetic susceptibility using the relation of n ~ = 3kT/fl2. Z. Generally speaking, each c o m p o n e n t of the effective Bohr m a g n e t o n n u m b e r is a function of temperature (T) and the coefficients (F, a, gx, gy, gz) of the spin Hamiltonian E q n I I . According to E q n io, however, these coefficients are the functions of three independent u n k n o w n quantities of e(SE~) -- e(SB~), e(6E~) -- e(SB2) and a. Since a stands for the coefficient of spin orbit interaction, it can be assumed to be 4oo cm -1 with reference to ferrous free iron ion TM. Therefore the effective Bohr magneton numbers become the functions of two independent u n k n o w n quantities. Here we adopt F and a as these quantities. Thus gx and gy in E q n IO can be rewritten using F and a as follows: 8 gx = 2 +-"
F(cosa--
sina/~/3)
a
8 gy = 2 -r - ' F (cosa
(~5) + sin a / ~ / 3 )
Thus, the curve of (A(n ~) vs I / T ) is a function o f F , a, C 1 and C 2. Its temperature dependence is determined b y the values of F and a, and its magnitude depends on the values C~ and C2. Since the intramolecular structure of deoxyhemoglobin has alr e a d y been elucidated b y X - r a y diffraction studiesS, 9, the orientations of the heroes in the unit cell are known from the molecular coordinates. The principal axes of magnetic susceptibility, however, cannot be obtained from molecular structure. In the present analysis, therefore, the values of C1 and C2 are taken to be u n k n o w n quantities. For a set of definite values of F and a, the values of C 1 and C 2 are used as adjustable parameters which minimize the deviation between the experimental data and the calculated curve. The same procedure for various pairs of F and a can be carried out and the value of m i n i m u m deviation (8) can be obtained for each pair of F and a. These treatments were carried out b y an electronic computer, and we obtained F ~- 5.5 cm-~ and sin a ~ o.28, the theoretical curves of which are in best agreement with the experimental results. This pair gives the values of D = 5.3 cm-1 and E = o. 9 cm -1. The dependence of m i n i m u m deviation (8) on the values of various F and a, is represented in Fig. 9. The solid lines in Fig. 8 represent the calculated Biochim. Biophys. Acta, 2 7 8 ( 1 9 7 2 ) 3 5 5 - 3 7 1
367
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
15 /
lC 6
~
5
5
I
c
I
0.1
0.2 sing
m
I
0.3
1
I
0.4
Fig. 9. T h e F a n d a d e p e n d e n c e of t h e m i n i m u m d e v i a t i o n b e t w e e n t h e e x p e r i m e n t a l a n d t h e o r e t i c a l v alues. T h e v a l u e on each c u r v e i n d i c a t e s F va l ue . TABLE II THEORETICAL
V A L U E S OF T H E C O E F F I C I E N T S
C 1 AND C2 FOR THE THREE CASES WHEN
F
=
5.5 c m -1
AND sin a = o.28 T h e s e v a l u e s are in b e s t a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a .
Case
C1
C,
CJC1
A B C
0.442 i.o 7 1.36
o.15o o.332 o.423
o.339 o.31o o.312
curves for this pair of F and a. The values of C 1 and C 2 for three experimental results are shown in Table II. These values, together with the knowledge of the molecular structure, will contribute to determination of the principal axes of magnetic susceptibility of heme, which will be discussed in the future. DISCUSSION
The spin Hamiltonian parameters (D ---- 5.3 cm-1 and E ----0. 9 cm-1), obtained from the present study of paramagnetic anisotropy, are in fairly good agreement with those parameters of magnetic susceptibility (D ---- 5 cm-1 and E << D) reported in the previous paper 3. The preliminary result (D = - - I 8 cm -1) of the paramagnetic anisotropy measurement, which has been reported b y Kotani, is inconsistent with the present result as well as the previous one. The preliminary work of Kotani was carried out at three constant temperatures (liquid N2, H2, He temperatures), and the direction of the rotating magnetic field corresponded to the case B of the present work. Biochim. Biophas. Acta, 278 (1972) 355-371
368
N. NAKANOJ al.
D=-18cm
20
.
'
11~ [II I
0.1
I E=O
m 1
i
i
i
0.2
0.3
0.4
1/r (°K 4) Fig. IO. T h e c o m p a r i s o n of t h e v a l u e s of z~(n 2) b e t w e e n t h e p r e s e n t (case B) a n d t h e experim e n t a l d a t a r e p o r t e d preliminarily. T h e signs of × r e p r e s e n t t h e p r e l i m i n a r y e x p e r i m e n t a l d a t a , a n d t h e d o t t e d line r e p r e s e n t s t h e e x p e r i m e n t a l d a t a o b t a i n e d in t h e p r e s e n t work. T h e r e p o r t e d d a t a are m u l t i p l i e d b y a c o n s t a n t factor (o.418) as e x p l a i n e d in t h e D i s c u s s i o n section. Solid lines r e p r e s e n t t h e t h e o r e t i c a l c u r v e s c o r r e s p o n d i n g to t h e cases of (D = - - 1 8 c m -1, E = o) a n d (D = 5.3 cm-1, E = 0.9 cm-1).
The reported values of A(n ~) were 46.4, 40.8, 14.8 for the temperatures of liquid He, H~, N 2, respectively. Present data of the case B at these temperatures are 19.4, 16.8, 6.1. I f reported values are multiplied by a common factor of o.418, which is the ratio of two values at liquid He temperature, the reported values become 19. 4 , 17.1, 6.2. These values are in good agreement with the present values (19. 4 , 16.8, 6.1). The reported results and present results of the case B are compared in Fig. IO, where the reported results were multiplied by the constant factor (o.418). In the present experiments, the temperature dependence of paramagnetic anisotropy was measured in the intermediate temperature regions, in addition to the constant temperatures of liquid N2, H~, He. The experimental data cannot be explained completely by the theoretical curve calculated from D = --18 cm -1, although only three points of experimental data can be represented by this theoretical curve. Therefore, it seems that the reported result (D = --18 cm -1) is incorrect. Now, the values of A(n l) for the three cases are very different to each other. This fact indicates that the eight hemes in unit cell are almost parallel. In fact, Perutz et al. ~ have pointed out that the two hemoglobin molecules in the unit cell of Form I I crystals are nearly parallel to each other. The relationships between the orientation of each heme and the directions of the rotating magnetic field are illustrated in Fig. I I, by stereographic projection. The points of alI, allI, etc. represent the perpendicular directions to each heme plane, which were calculated from the results of X - r a y diffraction studies 8& The numbers I and I I refer to the two hemoglobin molecules in the unit cell. The figure reveals that each perpendicular direction to the heme planes Biochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
369
-a
Fig. I I. T h e s t e r e o g r a p h of t h e h e m e o r i e n t a t i o n s a n d t h e t o r q u e curves. Solid lines r e p r e s e n t t h e p l a n e s of t h e r o t a t i n g m a g n e t i c field. T h e signs of • r e p r e s e n t t h e p e r p e n d i c u l a r direction to t h e h e m e planes. T h e directions of a, b, c a n d c* indicate t h e c r y s t a l axes. × ; t h e a v e r a g e direction w h i c h gives zero t o r q u e value. - + ; t h e direction of t h e p h a s e s h i f t w h i c h is c a u s e d b y lowering temperature.
is nearly parallel to the a axis. Solid lines represent the planes of the rotating magnetic field. The direction of the magnetic field in case A is rotated almost parallel to the planes of the hemes, while the direction in case C is almost perpendicular to the planes of the hemes. Case A gave very small torque values and case C gave very large one, which indicate that the a s y m m e t r y in the heme plane is small but the a s y m m e t r y between perpendicular and parallel directions to heine plane is large. I t is therefore considered that the hemes in deoxyhemoglobin still conserve almost axial s y m m e t r y of magnetic susceptibility with the axis of the perpendicular direction to the heme plane, although a small a s y m m e t r y exists in the heine plane. This is consistent with the values of D ~ 5.3 cm-1 and E = 0. 9 cm -1 estimated from the temperature dependence of torque value. Though the phase shifts are brought about b y the change of temperature, the changes in this phase shift are not very large (see Figs 5, 6). Therefore, we can choose the approximate directions of magnetic field at which the torque values become zero. The signs of × in Fig. I I represent these directions, corresponding to the average Biochim. Biophys. Acta, 278 (1972) 355-371
37 °
N. NAKANO et al.
direction between the liquid Ne and He temperatures. In the cases of both B and C, these directions are nearly parallel and perpendicular to the heine plane. That is, all three of the principal axes of the magnetic susceptibility tensor about heine are almost either perpendicular and parallel to the heme plane. According to the derivation of the spin Hamiltonian Eqn 9, it is expected that the z axis corresponds to this perpendicular direction. Roughly speaking, the terms of D and E mainly contribute to the two anisotropy terms (gz -- (Zx + Zy)/2) and (Zx -- Zy), respectively. Since the E value is sufficiently small compared with the D value, it can be shown that (Zx ~ Zy) exhibits weaker temperature dependence than (%z -- (Zx + Zy)/2), according to the detailed calculation of magnetic susceptibility using these D and E values. Since in the case A the ratio of C2/C 1, which implies the ratio of coefficients of (Zx -- Zy) and (Zz - - (Zx @- Zy)/2), is larger than the others, it is reasonably expected that the case A gives weaker temperature dependence of A (n l) than in the other cases. It is well known that the derivatives of methemoglobin and metmyoglobin in the low-spin state also exhibit the a s y m m e t r y in the heine plane 13. However, the sixth ligand position is believed to remain vacant in the case of deoxyhemoglobin, in contrast to the case of methemoglobin in the low-spin state. Therefore, the a s y m m e t r y of the heme in deoxyhemoglobin should be attributed to effects of the imidazole ring of the proximal or distal histidine residue, or a distortion of the heine plane, though we lack more detailed information. APPENDIX %z = 8fl2 F(-I ~- cosa) exp ~ -
+
sinh
-
V~3Fsin a
(Fcosa~
× exp \ ~ - - - /
\
kT
2(1 + cos a) ( + F(I cosa) sinh ( F ( I -~ cosa)]/ exp
I --CO81~
F(I + cosa)]
( 2Fcos(/)] kT
F(I + cosa) exp
/@
LF(2 -- cosa + ~/3 sina)
k-T
F(cosa + ~/3 sina) )
+
kT
exp F(3cos a + ~/3 sin a) f
_ [
2+cosa--~/3sina
_
2--cosa+~/3sina
! F(2--cosa+@3sina)
F(2+cosa--~/3sina)
,xp( 2F~cosa~!
F(3 cos a + %/7 sin a) 2 -- cosa + ~ / 3 s i n a
_ exp
F(2 + cos a -- ~/3 sin a)
Biochim. Biophys. Acta, 278 (1972) 355-371
/
(
~-
/(~)
}
i
371
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
ZY = 2g~'fl~ " ~
I
+- c°s~a +_ f3sina_
exp
~-
[_F(2 -- cos ct -- %/3 sin a)
] 2+~o~+V'3sin~
_
2--cos~,--V'~sin~
t F(2--cosa--X/3sina) 1
+
F ( 2 + c o s a + V Z 3 s ina)
exp
exp
(
F(cos a + V'3 sin a)l kT
)
(F(cosa-~/3sina)) ~/7 "
F(3 cos a -- X/3 sin a)
F(3 c o s a -- ~ / 3 sina) __
kT
2 -- c o s a -- X/3sinct
exp
( 22)] --
/®
F(2 + cos ct + ~¢/3 sin a) (~)=exp(~-T) + 2exp \
[ F cos a ~ +2expk~]c°sh kT
cosh
(
1/3 F sin a kT ) kT
]
REFERENCES I 2 3 4 5 6 7 8 9 io 1i I2 13
A. Tasaki, J. O t s u k a a n d M. Kotani, Biochim. Biophys. Acta, 14o (1967) 284. N. Nakano, K. N a k a n o a n d A. Tasaki, Biochim. Biophys. Acta, 251 (1971) 3o3 • N. Nakano, J. O t s u k a a n d A. Tasaki, Biochim. Biophys. Acta, 236 (I97 I) 2z2. G. L a n g a n d W. Marshall, Proc. Phys. Soc., 87 (1966) 3. M. Kotani, Ann. N . Y . Aead. Sci., I58 (1969) 2o. M. F. Perutz, I. F. Trotter, E. R. Howells a n d D. W. Green, Acta Crystallogr., 8 (I955) 241. H. Muirhead, J. M. Cox, L. Mazzarella a n d M. F. Perutz, J. Mol. Biol., 28 (I967) t I 7. W. Bolton, J. M. Cox a n d M. F. Perutz, J. Mol. Biol., 33 (1968) 283. H. M u i r h e a d a n d J. Greer, Nature, 228 (i97 o) 516. H. Morimoto, T. Iizuka, J. O t s u k a a n d M. Kotani, Biochim. Biophys. Acta, lO2 (I965) 624. J. F. Gibson, D. J. E. I n g r a m a n d D. Schonland, Disc. Faraday Soc., 26 (1958) 72. 13. B l e a n e y a n d K. W. H. Stevevs, Rep. Prog. Phys., i6 (I953) io8. H. }{off, Biochim. Biophys. Acta, 251 (I97 I) 227.
Biochim. Biophys. Acta, 278 (1972) 355-371
355
SBA 36209 P A R A M A G N E T I C A N I S O T R O P Y M E A S U R E M E N T S ON A S I N G L E CRYSTAL OF D E O X Y H E M O G L O B I N
N O R I H I T O NAKANO, J I N Y A OTSUKA AND A K I R A TASAKI
Faculty of Engineering Science, Osaka University, Toyonaka, Osaka (Japan) (Received April 5th, 1972)
SUMMARY
The temperature dependence of the paramagnetic anisotropy of a single deoxyhemoglobin crystal (Form II) was measured by the torque method with a sensitivity of lO -4 dyne. cm in the temperature range of 2 to 77 °K. Torque curves were measured in the three perpendicular planes of the crystal b y a rotating magnetic field. The torque curves obtained were of a sinusoidal shape per half rotation of the magnetic field. The phases of the torque curves as well as the amplitudes changed with temperature. This means that the three principal values of magnetic susceptibility of heine depended on temperature and these temperature dependencies were not the same. The amplitude of the observed torque was very small when the magnetic field was rotated almost parallel to the heme planes in the unit cell, while it was very large when the magnetic field was rotated perpendicular to the heme planes. The directions of the magnetic field which gave zero torque values were nearly perpendicular and parallel to the heme plane. From these results, it was concluded that the magnetic susceptibility of heme in deoxyhemoglobin was fairly axially symmetrical, taking the axis to be nearly perpendicular to the plane of the heme, though in detail a small a s y m m e t r y in the heine plane existed. The temperature dependence of amplitude measured over a wide temperature range was analyzed on the basis of the spin Hamiltonian; 9ff ~-- DS~ + E(S~ -- Sy) + /3-SgH. Experimental data were best represented by the theoretical curves which were calculated from the values of D = 5.3 and E = 0. 9 cm -~. This study made it clear that the previously reported result of D = -- 18 cm -~ was incorrect (M. Kotani, Ann. N . Y . Acad. Sci., 158 (1969) 20).
INTRODUCTION
Previously we have reported several magnetic studies and a theoretical analysis of the electronic state of the iron ion in hemoproteins 1-3. Since E P R methods were applicable to the ferric form only, our magnetic studies were started also from this ferric form1, ~. The main results obtained from these studies can be summarized as Biochim. Biophys. Aeta, 278 (1972) 355-371
356
N. NAKANO el al.
follows: (I) The fine structure of the lowest energy level (6A1) for the high-spin iron ion in methemoglobin and metmyoglobin is well represented by the spin Hamiltonian of D S z when D was chosen to be IO cm -1. (2) In the case of acidic Imethemoglobin, a considerable amount of heine iron was of the low-spin type even at very low temperatures ( 7 7 - 2 °K), and this low-spin compound showed a different E P R spectrum from that of the low-spin type in alkaline pH. For the ferrous compounds of hemoproteins, however, sufficiently accurate magnetic studies have not yet been made. This may be due to the fact that E P R methods are not usually applicable to ferrous ion, which has an even number of d-electrons in its outer orbit. Lang and Marshall have studied the M6ssbauer effect of deoxyhemoglobin and they reported that the magnetic field dependence of the M6ssbauer spectra could be explained by D = 5 cm-t if the fine structure of d-electronic energy level was represented by the spin Hamiltonian of D S z (ref. 4). In the previous work 3, we have measured the paramagnetic susceptibility of deoxyhemoglobin and deoxymyoglobin and also reported that the temperature dependence of the paramagnetic susceptibility could be well represented by the spin Hamiltonian D S z and D 5 cm-1. =
='
T
=
Sz=t2
/
312
$z7--+1
D -
_
Sz=_+O
Fig. I. The fine s t r u c t u r e of the d-electronic energy level of the ferrous high spin iron ion in hemoprotein. This s t r u c t u r e of the energy level corresponds to the case of D > o a n d E << D.
In order to obtain detailed information of the electronic state and to confirm the expected energy scheme shown in Fig. I, measurements of the paramagnetic anisotropy of a single crystal of deoxyhemoglobin were required. An experiment on this line has already been reported by Kotani 5, who analyzed the preliminary experimental data of lizuka. The estimated value of D from this experiment was --18 cm -1. Since this value was inconsistent with results of M6ssbauer effect and with those of the magnetic susceptibility previously reported by us, another precise experiment on the temperature dependence of the paramagnetic anisotropy was performed. In the present paper, we report the results of this paramagnetic anisotropy study. SAMPLE
Hemoglobin was prepared from human red cells by lysis with toluene. The single crystal of deoxyhemoglobin was grown by dialyzing a hemoglobin solution (1% hemoglobin in I.I M (NH4)2SO, solution, pH 6.5) against 2.5 M (NHa)2SO4 solution at pH 6. 5 with a small amount of EDTA. These inner and outer solutions were deoxygenated by adding dithionite. The completeness of deoxygenation was checked by measuring absorption spectra in the region from 400 to 650 nm. The single crystal of about 2 mm 3 volume was used as a sample. No contamination by methemoglobin in the sample crystal was checked by measuring E P R spectra at Biochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC ANISOTROPYOF DEOXYHEMOGLOBIN
357
liquid N 2 temperature before the paramaguetic anisotropy measurement. It is well known that deoxyhemoglobin gives no E P R signal, while methemoglobin exhibits a sharp absorption line of E P R spectra in both high- and low-spin state. The E P R signal of methemoglobin could not be observed even with very high sensitivity. The E P R absorption spectrum of the methemoglobin solution in the same quantity of hemoglobin as the sample crystal was measured. From the amplitude of the E P R signal, it can be calculated that the contamination by methemoglobin of the sample crystal is less than o.1%. Deoxyhemoglobin of normal adult human crystallizes in three different forms 6. In the present study, we used the Form I I crystal, since the nature of this form has been elucidated crystallographically b y X - r a y diffraction studies 7-9. The character of this crystal is monoclinic with a space group of P21, where the dyad screw axis corresponds to the b axis. Unit cell dimensions are given as a ---- 62.5 A, b = 83.2 A, c ---- 52.8 ,~ and fl -- 98°. There are two hemoglobin molecules in the unit cell, and therefore eight hemes. The clinographic projection of this crystal form is illustrated in Fig. 2. After paramagnetic anisotropy measurement, the crystal was quantitatively analyzed b y spectrophotometry in solution by converting it to pyridine hemochromogen. We used a millimolar extinction coefficient of e m M ---~ 34.7 at 556 nm.
bj
.,TIO
/010
110
Fig. 2. The clinographic projection of the deoxyhemoglobin single crystal (Form II). The crystal character is monoclinic and the space group P21. The b axis is dyad screw axis. The unit cell dimensions are a = 62.5 .&-,b = 83.2 A, c = 52.8 ~k and ~ = 98°. There are two hemoglobin molecules per unit cell. MEASUREMENT
The measurement of the paramagnetic anisotropy of the single crystal is carried out b y measuring the torque exerted on the crystal, when the crystal is subjected to the homogeneous magnetic field. The direction of magnetic field is rotated in a certain plane of the sample and thus the angular dependence of the torque value is obtained. The equipment prepared for the present experiment is illustrated in Fig. 3. The sample, attached to the lower tip of the quartz bar, is placed between the two poles of the magnet. The upper end of the quartz bar is fixed to a compensation coil as shown in Fig. 3. Sample, quartz bar and compensation coil make a single rotating unit, and this unit is suspended b y a thin phosphor bronze ribbon to enable it to rotate Biochim. Biophys. Acta, 278 (1972) 355-371
358
NAKANO x. et al. . . . . *~....... b a c k line
3sphor e ribbon I mirror net
~oto cell 3ensation coil
tz b a r
thermometer
Fig. 3. The equipment for the measurement of the temperature dependence of torque.
freely. A small mirror is attached to the compensation coil, and the small rotation of the unit is detected by the deflection of this mirror. The deflection is received by a twin photo-cell and the resulting signal is amplified and fed back into the compensation coil. Thus, the balance of the torque meter can be automatically controlled. The sensitivity is enhanced when the weight of the rotating unit is reduced. In the present apparatus, this weight is about IO g, which gave a sensitivity of lO -4 dyne.cm. The procedure for calibration of the torque meter was carried out according to the method of Morimoto et al. x°. In order to obtain low temperatures, we used liquid N 2, H , and He, and in addition we obtained intermediate temperatures between these constant temperatures by slight vaporization of these liquids. The temperature of the sample was measured by a carbon resistor. Since the magnetic ion is dilute in the protein sample and, moreover, the sample crystal is very small (2 ram3), a high accuracy of torque measurement is required and therefore the blank torque value should be minimized. The attachBiochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
359
ment of the thermometer to the sample increases the blank torque value, thus instead, it was placed as near the sample as possible. The sample and the thermometer were then surrounded b y a copper sheet which rendered stability and uniformity of temperature around the sample. Thus, we were able to achieve accurate values of I / T to within + o . o i . The magnetic field was measured with Hall's probe, which was calibrated by proton NMR.
THEORETICAL BACKGROUND TO TORQUE MEASUREMENT
Let us consider the general case where the three principal values of the magnetic susceptibility tensor of a heme are different to each other. These principal directions of the tensor are taken to be x, y and z, and the corresponding principal values denoted as Zx, Zy and Zz. Since the sample crystal is of monoclinic type and the dyad screw axis corresponds to be b axis, the paramagnetic susceptibility tensor per unit cell m a y be expressed by the following form with the (a ,b, c*) coordinate system. =
~'CtO Zbb ~ \Z-~* o Z~%*/
(I)
Elements of its tensor are written in terms of the three principal values (Xx, Xy, Zz) as follows : + Uaa,y'Zy
+ Uaa,z'Y~z
+ Ubb,y'Z~ + Ubb,z'Zz Zc*c* = Uc*c',x'Zx + Uc*o*,y'Zy + U ~* *o ,z'Zz Zae* = Uac*x'Zx + Uae*r'Zy + Ua~*z'Zz
ZDb
~ Ubb,x" Zx
]
J
(2)
The eight hemes in the unit cell are assumed to be magnetically independent and identical. The coefficients Ua~,x, Ubb,y etc. are obtained as the sum of unitary transformations for these eight hemes. These values are constants determined by the directions of the three principal axes of magnetic susceptibility in each berne, and they satisfy the following general relations: Uaa,x
-~-
Uaa,y
-~- Uaa, z
Ubb,x
+ Ubb,y
+ Ubb,z
Uc*e*,x + U *c*o ,y + Uc*c*,z Uae*x + Uae*y + Uae*z
= 8 ]
8
8 o
(3)
We now define the new coordinate system (X, Y, Z), where X and Y axes are in the same plane as the rotating magnetic field, and the Z axis is perpendicular to this plane. The relation between this new coordinate system and the (a, b, c*) coordinate system is described as follows using Eulerian angles. The (a, b, c*) coordinate system is rotated around the c* axis b y an angle of 9, to obtain an (a', b', c*) coordinate system. The (a', b', c*) coordinate system is then rotated around the b' axis by an angle of 0, to obtain an (a", b', c*') coordinate system and finally this (a", b', c*') coordinate system is rotated around the c*' axis by an angle of ~v. Thus the (X, Y, Z) coordinate system is obtained. I f the Y axis is taken to be in the direction of magnetic field, the rotated angle of the magnetic field corresponds to ~p. The magnetic field vector in the (a, b, c*) coordinate system can be written by the Eulerian angles as follows: Biochim. Biophys. Acta, 278 (1972) 355-371
360
N. NAKANO et al. = H (--cos 0.cos ~o.sin ~o--sin ~0.cos ,p, --cos 0.sin ~0.sin ~p +cos q~.cos % sin 0" sin ~p)
(4)
The static magnetic energy per unit cell is generally expressed as (5)
E ~ --1/2HzH
Then the torque value L(~o) along the direction of the rotating magnetic field can be obtained b y differentiating the above magnetic energy with respect to ~o. Thus, using Eqns 1 - 5, the torque value per unit cell is described as follows: L(~p)
dL dip
IH2 {( Zz
2
Z.,:+Zy) -2
(klsinz~+k2c°s2~p}+(Zx--ZY) (kssinztP +
+ k4cos2~) }
(6)
This is the basic equation along quantitative analysis of torque value, because it is -epresented in terms of the anisotropy of the paramagnetic susceptibility (Zz -(%x + %y)/2) and (Zx -- Zy). The temperature dependence of torque values is attributed only to these anisotropy terms. The coefficients k 1, k 2, k 3 and k, are functions of the orientation of each heme in unit cell, and of the direction of the rotating magnetic field plane (~, 0). The coefficients are m a t h e m a t i c a l l y expressed as follows: kl ~ Uaa,z (cos2 0cos 2 9--sin~ qg) + Ubb,z (cos2 0sin 2 9--c0s2 qo) + Uc*c*,zSinz O--2Uac*,z cos Osin Ocos~p h2 ~ U~a,z cos Ocos qosinqo-- Ubb,zcos Ocos9sin ¢p-- Uac* zSin Osin ¢p k3 = (Uaa,y-- Uaa,~) (sin2 ~--cos 20cos 2 ¢p) + (Ubb,x-- Ubb,y) (COS20sin 2 qg--cos2 ¢p) + ( Uc*c',x-- Ue*c*,y) sin 2 0 -- 2 (Uac*,x-- Uac*,y) cos 0sin 0cos ~0 ka = (Uaa,x-- Uaa,v)cos 0cos 9sin q~--(Ubb,x-- Ubb,y) cos 0cos 9sin 9 -- (Uac*x-- Uac*,y) sin 0sin 9
(7)
RESULTS In the present measurements, torque curves were measured in the three different planes of the sample crystal. One plane is perpendicular to a axis (case A), the second parallel to (Oil) plane (case B), and the last perpendicular to the above (A, B) two planes (case C). Thus the planes are perpendicular to each other. Fig. 4 shows experimental d a t a of torque curves (case B) which were measured under various strengths of magnetic field at liquid He temperature. All torque curves are of sinusoidal shape with one period for each half rotation of the magnetic field (Fig. 4a), and these amplitudes are proportional to the square of magnetic field strength H (Fig. 4b), which is consistent with E q n 6. The linear relation between the amplitude and H 2 means t h a t the paramagnetic saturation effect does not occur at 4.2 °K even under a magnetic field of 12 kOe. The temperature dependence of torque curves was compared under a constant magnetic field strength. For example, two torque curves measured at liquid-N 2 and liquid-He temperatures are shown in Fig. 5 (case B). These two torque curves are quite different in their amplitudes, and moreover to some extent in their phases. As can be easily seen from E q n 6, the difference in amplitude of the torque curves is due to the temperature dependence of each of the three principal values of the magnetic susceptibility tensor, but the difference in phase is due to the differences in Biochim. Biophys. Acta, 278 (1972) 355-371
361
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
15
/
0
1
~IO
~\\~ / / I
/X \ ~ . ~
\\\\~-///t9.3~
"- 8.45 -'-
l\\-./P ,.o
.j" ~ o 4 dyno.~m
/
/.
/
./
~5
'fin \ 4.84
/
/"
/'
/ /
/
\\W/~ ..8 00
051108
I
10.108
I
1.540
8
H2 (Oe2) Fig. 4. (a) The magnetic field dependence of the torque curves measured at liquid-He temperature. The rotating magnetic field is parallel to the (oTI) plane (case B). (b) The magnetic field dependency of the peak-to-peak value in the above torque curves. The vertical axis is represented in arbitrary unit.
(lig.-Ne temp.) x ¼
. ~ 2 dyne .cm
Fig. 5. The temperature dependence of the torque curves measured in liquid-N 2 and -He temperatures (case B). Magnetic field strength is 22oo Oe. The difference in the phase between these torque curves is 8 °.
temperature dependencies of these three principal values. If, as in the case of acid methemoglobin in the high-spin state, the magnetic ion is of axial type and the two principal values of magnetic susceptibility tensor are equal to each other n, the phase of torque curve is not shifted by the change of temperature. In the case of C, the temperature dependence of phase as well as amplitude was also observed to same extent as in the case of B, which is shown in Fig. 6. In the case of A, however, the phase shift of torque curve could not be observed and the torque values were always zero at any temperatures, when the direction of magnetic field was parallel to the b and c* axes, as shown in Fig. 7. This tendency can Biochim. Biophys. Acta, 278 (1972) 355-371
362
N. NAKANO et al.
~ [
(hg.-He temp.)x{
\
/(lig.-N2temp.)
]
\
Fig. 6. T h e temperature dependence of the torque curves measured in liquid-N~ and -He temperatures (case B). Magnetic field strength is 22oo Oe. The difference in the phase b e t w e e n these torque curves is 7 ° .
/
/
/"~1(OT[)Ptane /
/
'~
Fig. 7. T h e temperature d e p e n d e n c e of t h e torque curves measured in liquid N~ and H e temperatures (case A). Magnetic field strength is 6ooo Oe.
be explained by Eqns 6 and 7, where the direction of magnetic field is in the bc* plane, so that 9 = o and 0 ---- ~/2. As easily calculated from Eqn 7, the coefficients k 2 and k 4 b e c o m e zero, and so Eqn 6 is reduced to the form :
-I H 2 { kl ( •z 2
Zx --+ Zy.) + ks(Zx 2
Zy) } sin 2 ~p
where ~p corresponds to the angle between b axis and the direction of the magnetic Biochim. Biophys. Acta, 278 (1972) 355-371
363
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
field. In this case the differences in the temperature dependencies of principal values do not affect the phase of torque curve. It follows that the magnetic susceptibility of deoxyhemoglobin is not of axial type, unlike acid methemoglobin in the high spin state.
3C
2~
20
•
C
"
z/(n 2) 15
10
I
o.1
o:2
o:3
I
o.4
o:~
1/T (°K'I) Fig. 8. The t e m p e r a t u r e dependence of the t o r q u e value per h e m e for t h e three cases (case A, B, C). B o t h t h e direction a n d t h e s t r e n g t h of m a g n e t i c field were k e p t to be constant. The vertical axis represents 3kT]~2.2LIHL Solid lines r e p r e s e n t the theoretical curves for the pair of F = 5.5 cm-1 a n d sin a = o.28.
The temperature dependencies of torque values were also measured continuously over a wide range of temperature (2-77 °K) keeping both the strength and the direction of the magnetic field constant. The direction of magnetic field was chosen so that the torque gave a maximum value at liquid He temperature in each case (case A, B, C). These results are shown in Fig. 8, where the torque values per heine are plotted in terms of (3kT[tS~) • (2L/H2). According to Eqn 6, this quantity corresponds to ([71(~z2 nx2 ~-nY2")--2+ C2(nx* -- ny~) -A(n*) (8) with the effective Bohr magneton numbers nx, n~ and nz defined by Zx = (nx*fl*)/3kT, •y ~-(ny*fl2)/3kT and Zz----(nz*fl*)/3kT, respectively. The parameters C 1 and Ca are constant but take different values in each case (case A, B, C). Fig. 8 exhibits the Biochim. Biophys. Aaa, 278 (1972) 355-371
N. NAKANOt3t al.
364
following tendencies of temperature dependence of A(n2). At the high temperature limit (I/T = o), the values of A(n ~) for all three cases converge to zero, so that nx, ny and nz become equal to each other at this temperature limit. With the lowering of temperature, the values of A(n ~) increase sharply, and give m a x i m u m values in the vicinity of I / T = o.12, and then decrease gradually as temperature is lowered further. These temperature dependencies of A (n 2) are due to the fine structure of d-electronic energy level of heme iron ion, which suggests that the lowest energy level of this fine structure should be Sz = o and the other excited levels are higher by about 6 em -1 corresponding to I/T = o.12. This means that the value of D is positive and of several units (cm-1). TABLE I THE
RELATIVE
VALUES
O F A($~ 2) F O R T H E
THREE
CASES AT SEVERAL
TEMPERATURES
The value of A(n 2) at I/T = o.12 is normalized to unity.
1/T
AA
AB
AC
0.03 0.07
0.490 0.868
0.593 o.917
0.596 o.917
0.12
I
I
I
o.24 0.32 0. 4
o.943 o.877 o.811
o.926 o.852 0.835
o.917 o.835 o.761
I f the axial character of the magnetic susceptibility tensor is considered, according to Eqn 8, the temperature dependence of A (n 2) in any directions of magnetic field becomes homologous. These relative values of A (n 2) at several temperatures are shown in Table I, wherein the values ofA (n 2) at I/T = 0.12 are normalized to unity. The following tendencies among these values are observed: At higher temperatures ( I / T < 0.I2), the relative values of A(n 2) are in the order o f A C > AB > AA, while in the lower temperature region (I/T > 0.I2), this order becomes AC < AB < AA. These relationships imply that A (n ~) exhibits the strongest dependence on temperature in the case of C and the weakest dependence in the case of A. The non-homologous temperature dependence of A(n 2) is consistent with the phenomenon of the phase shift produced by change in temperature. Both of them suggest that it is necessary to consider the a s y m m e t r y in heine plane, though the results of magnetic susceptibility measurement in solution or polycrystals can be explained without this a s y m m e t r y as reported in the previous paper 3. ANALYSIS In the previous paper 3, the general form of spin Hamiltonian including the a s y m m e t r y in heme plane was described as
~e = DS: + E(S~- S~) + S ' ~ . H
(9)
the coefficients (D, E and g) of this spin Hamiltonian were represented by the functions of spin orbit interaction and the energy differences between sublevels of the dlevel. Taking account of only two sublevels of the ground state (SB~) and the nearest
Biochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC
ANISOTROPY
365
OF DEOXYHEMOGLOBIN
excited state (BE), the coefficients of the spin Hamiltonian E q n 9 were represented as follows : D
~
-
-
32
e(~E,)
a2 {
gx = 2 +
-
e(~B~)+ ~(~E,~)
I -
E
32
-
e(SE~)
e(SB2)
a
(~B~)
I -
s("En)
}
e(SB2)
I
0o)
2 e(SEn) - - e(SB,~) a
i
gy~2+ 2 e ( S E , ) - - e(SB~) gz ~2
where a stands for the coefficient of spin orbit interaction. I n order to consider the E term of E q n 9 we use new parameters F (F > o) and a instead of D and E, which are defined as D = F . c o s a and E = F - s i n a/x/3. Then the spin Hamiltonian E q n 9 becomes ~a
= F.cos
a . S z~ +
--
it
$
F.sina],v/3(Sx _ Sy) + ~g~H
(II)
The temperature dependence of three principal values of magnetic susceptibility are calculated on the basis of E q n I I . I f the magnetic field (H) is zero, we can easily diagonize this spin Hamiltonian using Eigen functions of Sz (S ---- 2). Eigen values and Eigen functions obtained are as follows: 21 ~ --2F 22 ~ - - F c o s a - - ~ / ~ F s i n a ~tit ~ - - F c o s a + ~ / ~ F s i n a '~a ~ 2 F c o s a 25 = 2 F
(I2)
and ~//1 =
I --II)
I -- --[
V~
- - I)
V~
I
,p~=--II) V~
I
+--J-I) V~
I
I
~.~ = - - 12) - - - I - 2) V~ V~ sin a ~4
~/cos a + I 12)
2 Vcos a + I %
(~3)
sina
2VI - cosa
sin a IO) +
~/~ 12) +
~/i --cosa
V~
-
2)
-
2)
2~/cosa + I Io) ~-
sin a
2VI - cosa
I n the present experiments, the magnetic field is 8000 Oe in the high temperature region and 300o Oe in the low temperature region, so t h a t the Zeeman term becomes less t h a n 0.8 c m - L Since from previous magnetic susceptibility measurements, the value of D was found to be 5 cm-a, it can be expected t h a t the value of F falls to around t~iochim. Biophys. Acta,
278 (I972) 355-371
366
N. NAKANOet al.
this value. Therefore, this Zeeman term m a y be treated as a perturbation. As the paramagnetic saturation effect cannot be observed, each energy level (),'i; i z 1-5) is calculated up to the second order of magnetic field strength. Components of the magnetic m o m e n t s mix : - - ~ ] ( i / 6 H x , rely : - - 6~'l/~Hy miz = -- 6~t'l/6Hz for each level are obtained from eigen values 2'i (i : I-5) of the spin Hamiltonian E q n xi. B y averaging these m o m e n t s with the use of the B o l t z m a n n factor, we obtain the x component of magnetic susceptibility per heme as follows: 2: Mx
Xx = .
I
. . Hx
ml×exp
--
i = 1
.
(I4) Hx
Z
exp
--
Expressions for Zy and •z are also obtained in the same way. The explicit forms of Xx, Xy and Zz are shown in the Appendix. Each component of the effective Bohr magneton n u m b e r (nx, ny, nz) is calculated from the above expressions of magnetic susceptibility using the relation of n ~ = 3kT/fl2. Z. Generally speaking, each c o m p o n e n t of the effective Bohr m a g n e t o n n u m b e r is a function of temperature (T) and the coefficients (F, a, gx, gy, gz) of the spin Hamiltonian E q n I I . According to E q n io, however, these coefficients are the functions of three independent u n k n o w n quantities of e(SE~) -- e(SB~), e(6E~) -- e(SB2) and a. Since a stands for the coefficient of spin orbit interaction, it can be assumed to be 4oo cm -1 with reference to ferrous free iron ion TM. Therefore the effective Bohr magneton numbers become the functions of two independent u n k n o w n quantities. Here we adopt F and a as these quantities. Thus gx and gy in E q n IO can be rewritten using F and a as follows: 8 gx = 2 +-"
F(cosa--
sina/~/3)
a
8 gy = 2 -r - ' F (cosa
(~5) + sin a / ~ / 3 )
Thus, the curve of (A(n ~) vs I / T ) is a function o f F , a, C 1 and C 2. Its temperature dependence is determined b y the values of F and a, and its magnitude depends on the values C~ and C2. Since the intramolecular structure of deoxyhemoglobin has alr e a d y been elucidated b y X - r a y diffraction studiesS, 9, the orientations of the heroes in the unit cell are known from the molecular coordinates. The principal axes of magnetic susceptibility, however, cannot be obtained from molecular structure. In the present analysis, therefore, the values of C1 and C2 are taken to be u n k n o w n quantities. For a set of definite values of F and a, the values of C 1 and C 2 are used as adjustable parameters which minimize the deviation between the experimental data and the calculated curve. The same procedure for various pairs of F and a can be carried out and the value of m i n i m u m deviation (8) can be obtained for each pair of F and a. These treatments were carried out b y an electronic computer, and we obtained F ~- 5.5 cm-~ and sin a ~ o.28, the theoretical curves of which are in best agreement with the experimental results. This pair gives the values of D = 5.3 cm-1 and E = o. 9 cm -1. The dependence of m i n i m u m deviation (8) on the values of various F and a, is represented in Fig. 9. The solid lines in Fig. 8 represent the calculated Biochim. Biophys. Acta, 2 7 8 ( 1 9 7 2 ) 3 5 5 - 3 7 1
367
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
15 /
lC 6
~
5
5
I
c
I
0.1
0.2 sing
m
I
0.3
1
I
0.4
Fig. 9. T h e F a n d a d e p e n d e n c e of t h e m i n i m u m d e v i a t i o n b e t w e e n t h e e x p e r i m e n t a l a n d t h e o r e t i c a l v alues. T h e v a l u e on each c u r v e i n d i c a t e s F va l ue . TABLE II THEORETICAL
V A L U E S OF T H E C O E F F I C I E N T S
C 1 AND C2 FOR THE THREE CASES WHEN
F
=
5.5 c m -1
AND sin a = o.28 T h e s e v a l u e s are in b e s t a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a .
Case
C1
C,
CJC1
A B C
0.442 i.o 7 1.36
o.15o o.332 o.423
o.339 o.31o o.312
curves for this pair of F and a. The values of C 1 and C 2 for three experimental results are shown in Table II. These values, together with the knowledge of the molecular structure, will contribute to determination of the principal axes of magnetic susceptibility of heme, which will be discussed in the future. DISCUSSION
The spin Hamiltonian parameters (D ---- 5.3 cm-1 and E ----0. 9 cm-1), obtained from the present study of paramagnetic anisotropy, are in fairly good agreement with those parameters of magnetic susceptibility (D ---- 5 cm-1 and E << D) reported in the previous paper 3. The preliminary result (D = - - I 8 cm -1) of the paramagnetic anisotropy measurement, which has been reported b y Kotani, is inconsistent with the present result as well as the previous one. The preliminary work of Kotani was carried out at three constant temperatures (liquid N2, H2, He temperatures), and the direction of the rotating magnetic field corresponded to the case B of the present work. Biochim. Biophas. Acta, 278 (1972) 355-371
368
N. NAKANOJ al.
D=-18cm
20
.
'
11~ [II I
0.1
I E=O
m 1
i
i
i
0.2
0.3
0.4
1/r (°K 4) Fig. IO. T h e c o m p a r i s o n of t h e v a l u e s of z~(n 2) b e t w e e n t h e p r e s e n t (case B) a n d t h e experim e n t a l d a t a r e p o r t e d preliminarily. T h e signs of × r e p r e s e n t t h e p r e l i m i n a r y e x p e r i m e n t a l d a t a , a n d t h e d o t t e d line r e p r e s e n t s t h e e x p e r i m e n t a l d a t a o b t a i n e d in t h e p r e s e n t work. T h e r e p o r t e d d a t a are m u l t i p l i e d b y a c o n s t a n t factor (o.418) as e x p l a i n e d in t h e D i s c u s s i o n section. Solid lines r e p r e s e n t t h e t h e o r e t i c a l c u r v e s c o r r e s p o n d i n g to t h e cases of (D = - - 1 8 c m -1, E = o) a n d (D = 5.3 cm-1, E = 0.9 cm-1).
The reported values of A(n ~) were 46.4, 40.8, 14.8 for the temperatures of liquid He, H~, N 2, respectively. Present data of the case B at these temperatures are 19.4, 16.8, 6.1. I f reported values are multiplied by a common factor of o.418, which is the ratio of two values at liquid He temperature, the reported values become 19. 4 , 17.1, 6.2. These values are in good agreement with the present values (19. 4 , 16.8, 6.1). The reported results and present results of the case B are compared in Fig. IO, where the reported results were multiplied by the constant factor (o.418). In the present experiments, the temperature dependence of paramagnetic anisotropy was measured in the intermediate temperature regions, in addition to the constant temperatures of liquid N2, H~, He. The experimental data cannot be explained completely by the theoretical curve calculated from D = --18 cm -1, although only three points of experimental data can be represented by this theoretical curve. Therefore, it seems that the reported result (D = --18 cm -1) is incorrect. Now, the values of A(n l) for the three cases are very different to each other. This fact indicates that the eight hemes in unit cell are almost parallel. In fact, Perutz et al. ~ have pointed out that the two hemoglobin molecules in the unit cell of Form I I crystals are nearly parallel to each other. The relationships between the orientation of each heme and the directions of the rotating magnetic field are illustrated in Fig. I I, by stereographic projection. The points of alI, allI, etc. represent the perpendicular directions to each heme plane, which were calculated from the results of X - r a y diffraction studies 8& The numbers I and I I refer to the two hemoglobin molecules in the unit cell. The figure reveals that each perpendicular direction to the heme planes Biochim. Biophys. Acta, 278 (1972) 355-371
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
369
-a
Fig. I I. T h e s t e r e o g r a p h of t h e h e m e o r i e n t a t i o n s a n d t h e t o r q u e curves. Solid lines r e p r e s e n t t h e p l a n e s of t h e r o t a t i n g m a g n e t i c field. T h e signs of • r e p r e s e n t t h e p e r p e n d i c u l a r direction to t h e h e m e planes. T h e directions of a, b, c a n d c* indicate t h e c r y s t a l axes. × ; t h e a v e r a g e direction w h i c h gives zero t o r q u e value. - + ; t h e direction of t h e p h a s e s h i f t w h i c h is c a u s e d b y lowering temperature.
is nearly parallel to the a axis. Solid lines represent the planes of the rotating magnetic field. The direction of the magnetic field in case A is rotated almost parallel to the planes of the hemes, while the direction in case C is almost perpendicular to the planes of the hemes. Case A gave very small torque values and case C gave very large one, which indicate that the a s y m m e t r y in the heme plane is small but the a s y m m e t r y between perpendicular and parallel directions to heine plane is large. I t is therefore considered that the hemes in deoxyhemoglobin still conserve almost axial s y m m e t r y of magnetic susceptibility with the axis of the perpendicular direction to the heme plane, although a small a s y m m e t r y exists in the heine plane. This is consistent with the values of D ~ 5.3 cm-1 and E = 0. 9 cm -1 estimated from the temperature dependence of torque value. Though the phase shifts are brought about b y the change of temperature, the changes in this phase shift are not very large (see Figs 5, 6). Therefore, we can choose the approximate directions of magnetic field at which the torque values become zero. The signs of × in Fig. I I represent these directions, corresponding to the average Biochim. Biophys. Acta, 278 (1972) 355-371
37 °
N. NAKANO et al.
direction between the liquid Ne and He temperatures. In the cases of both B and C, these directions are nearly parallel and perpendicular to the heine plane. That is, all three of the principal axes of the magnetic susceptibility tensor about heine are almost either perpendicular and parallel to the heme plane. According to the derivation of the spin Hamiltonian Eqn 9, it is expected that the z axis corresponds to this perpendicular direction. Roughly speaking, the terms of D and E mainly contribute to the two anisotropy terms (gz -- (Zx + Zy)/2) and (Zx -- Zy), respectively. Since the E value is sufficiently small compared with the D value, it can be shown that (Zx ~ Zy) exhibits weaker temperature dependence than (%z -- (Zx + Zy)/2), according to the detailed calculation of magnetic susceptibility using these D and E values. Since in the case A the ratio of C2/C 1, which implies the ratio of coefficients of (Zx -- Zy) and (Zz - - (Zx @- Zy)/2), is larger than the others, it is reasonably expected that the case A gives weaker temperature dependence of A (n l) than in the other cases. It is well known that the derivatives of methemoglobin and metmyoglobin in the low-spin state also exhibit the a s y m m e t r y in the heine plane 13. However, the sixth ligand position is believed to remain vacant in the case of deoxyhemoglobin, in contrast to the case of methemoglobin in the low-spin state. Therefore, the a s y m m e t r y of the heme in deoxyhemoglobin should be attributed to effects of the imidazole ring of the proximal or distal histidine residue, or a distortion of the heine plane, though we lack more detailed information. APPENDIX %z = 8fl2 F(-I ~- cosa) exp ~ -
+
sinh
-
V~3Fsin a
(Fcosa~
× exp \ ~ - - - /
\
kT
2(1 + cos a) ( + F(I cosa) sinh ( F ( I -~ cosa)]/ exp
I --CO81~
F(I + cosa)]
( 2Fcos(/)] kT
F(I + cosa) exp
/@
LF(2 -- cosa + ~/3 sina)
k-T
F(cosa + ~/3 sina) )
+
kT
exp F(3cos a + ~/3 sin a) f
_ [
2+cosa--~/3sina
_
2--cosa+~/3sina
! F(2--cosa+@3sina)
F(2+cosa--~/3sina)
,xp( 2F~cosa~!
F(3 cos a + %/7 sin a) 2 -- cosa + ~ / 3 s i n a
_ exp
F(2 + cos a -- ~/3 sin a)
Biochim. Biophys. Acta, 278 (1972) 355-371
/
(
~-
/(~)
}
i
371
PARAMAGNETIC ANISOTROPY OF DEOXYHEMOGLOBIN
ZY = 2g~'fl~ " ~
I
+- c°s~a +_ f3sina_
exp
~-
[_F(2 -- cos ct -- %/3 sin a)
] 2+~o~+V'3sin~
_
2--cos~,--V'~sin~
t F(2--cosa--X/3sina) 1
+
F ( 2 + c o s a + V Z 3 s ina)
exp
exp
(
F(cos a + V'3 sin a)l kT
)
(F(cosa-~/3sina)) ~/7 "
F(3 cos a -- X/3 sin a)
F(3 c o s a -- ~ / 3 sina) __
kT
2 -- c o s a -- X/3sinct
exp
( 22)] --
/®
F(2 + cos ct + ~¢/3 sin a) (~)=exp(~-T) + 2exp \
[ F cos a ~ +2expk~]c°sh kT
cosh
(
1/3 F sin a kT ) kT
]
REFERENCES I 2 3 4 5 6 7 8 9 io 1i I2 13
A. Tasaki, J. O t s u k a a n d M. Kotani, Biochim. Biophys. Acta, 14o (1967) 284. N. Nakano, K. N a k a n o a n d A. Tasaki, Biochim. Biophys. Acta, 251 (1971) 3o3 • N. Nakano, J. O t s u k a a n d A. Tasaki, Biochim. Biophys. Acta, 236 (I97 I) 2z2. G. L a n g a n d W. Marshall, Proc. Phys. Soc., 87 (1966) 3. M. Kotani, Ann. N . Y . Aead. Sci., I58 (1969) 2o. M. F. Perutz, I. F. Trotter, E. R. Howells a n d D. W. Green, Acta Crystallogr., 8 (I955) 241. H. Muirhead, J. M. Cox, L. Mazzarella a n d M. F. Perutz, J. Mol. Biol., 28 (I967) t I 7. W. Bolton, J. M. Cox a n d M. F. Perutz, J. Mol. Biol., 33 (1968) 283. H. M u i r h e a d a n d J. Greer, Nature, 228 (i97 o) 516. H. Morimoto, T. Iizuka, J. O t s u k a a n d M. Kotani, Biochim. Biophys. Acta, lO2 (I965) 624. J. F. Gibson, D. J. E. I n g r a m a n d D. Schonland, Disc. Faraday Soc., 26 (1958) 72. 13. B l e a n e y a n d K. W. H. Stevevs, Rep. Prog. Phys., i6 (I953) io8. H. }{off, Biochim. Biophys. Acta, 251 (I97 I) 227.
Biochim. Biophys. Acta, 278 (1972) 355-371