WILLIAMH.ORTTUNC
2908
that reach the geometric area of the sublimed PbS deposit on the quartz wall of the reaction vessel. At low surface coverages, small sticking factors of this sort may arise because of activated adsorption, steric factors, inefficiencies of energy transfer, surface heterogeneities, or impurities. I n the present case the results are not sufficient to permit a choice among these. Although the adsorption process is slow in the experimental pressure range, the first-order dependence on oxygen pressure used to treat experimental data implies a substantial rate of adsorption at atmospheric pressure. If this long extrapolation is valid at a pressure of oxygen equivalent to the partial pressure in air, about 150 Torr, the adsorption at room temperature would be expected to reach O/Pb values of 0.5 after about 7 min exposure and should approach 1.0 after about 2.5 hr. If valid, these results would indicate that any polycrystalline sample of PbS that required handling in air would have
acquired a substantial film of chemisorbed oxygen. Since some incorporation of the oxygen into the lattice is expected and since some SO2 may evolve, the adsorption cannot be assumed to be reversed during any subsequent evacuation. This irreversible adsorption should be carefully considered in any future research with carefully prepared films of PbS. Acknowledgments. The author wishes to thank Battelle Memorial Institute for the time and funds needed to carry out this research and to acknowledge the assistance of a number of individuals who made the study possible. Especially, thanks are due to Dr. F. J. Reid and Mr. Richard Buist for their help in carrying out the Hall coefficient experiment, to Dr. Harvey Goering and Dr. Albert Beer for assistance offered throughout the research program, and to Dr. Ronald Lee for many discussions of the problems and for review of this paper before publication.
The Kerr Effect Optical Dispersion of Hemoglobin. A Molecular Interpretation1 by William H. Orttung Department of Chemistry, University of Cal.iforna, Riverside, Calijornia 926'02
(Received January 87, 1969)
Kerr constant increments of horse oxy and met hemoglobin are reported at 546, 436, and 365 mp. The data are interpreted in terms of theoretical estimates of the molecular dielectric and optical parameters. Dielectric parameters were calculated by extensions of the Kirkwood-Shumaker and Tanford-Kirkwood theories, using site coordinates from a model of horse oxy hemoglobin. The contribution of the heme spectrum to the polarizability anisotropy was calculated from the Kramers-Kronig relation and geometrical information. The theoretically predicted Kerr constant increments were found to be very sensitive not only to the heme orientations and small x,.y splittings of the Soret and visible bands, but also to the details of the proton-fluctuation anisotropy. Significant pH dependence was also predicted. The theory and data are in qualitative agreement, although more extensive experimental results are desirable.
Introduction The success of X-ray crystallographers in determining atomic coordinates of protein molecules in the crystalline state has important consequences for the study of protein structure in solution. Although solution experiments cannot provide enough data to describe a protein structure, it should be possible to predict many solution properties with the aid of crystal data. Such predictions can be compared with solution measurements to yield significant information about structure and structural changes in solution. To initiate a program of this type, the Kerr effect of The Journal of Physical Chemistry
hemoglobin solutions has been investigated both theoretically and experimentally. The Kerr effect of a globular protein solution depends on the relative orientations of the static and transition dipole moments. The contributions of different chromophores may be isolated by studying the wavelength dependence. Hemoglobin is a protein whose structure has been revealed by X-ray investigations.2 It also has prominent (1) This investigation was supported in part by Public Health Service Research Grant GM 11683 from the Division of General Medical Sciences. (2) M. F. Perutz, H. Muirhead, J. M. Cox, and L. C. G. Goaman,
Nature, 219, 131 (1968).
KERREFFECTOPTICALDISPERSION OF HEMOGLOBIN chromophores (the heme groups) whose transition dipole orientations are approximately known. Kerreffect results for hemoglobin should therefore be sensitive to the protein structure in solution, and should be subject to molecular interpretation by comparison with calculations based on the structural data. In a preliminary report,a the heme groups were assumed to be circular absorbers and the proton fluctuations were assumed to be isotropic. The data then implied that the molecule had a net dipole moment of a t least 100 debyes parallel to the 2-fold axis. With regard to the circular absorber assumption, the possibility of an in-plane (x,y) splitting of the heme optical transitions has not yet been examined by crystal dichroism experiments, but is considered in the analysis presented below. To examine the importance of the second assumption in the above analysis, atomic models of the cy and P chains of horse oxy hemoglobin were constructed following the description of Perutz.4 The Tanford-Kirkwood theory of protein titration curves and the Kirkwood-Shumaker theory of dipole-moment fluctuations were then extended to allow the calculation of protonfluctuation anisotropy effects.6 Coordinates of the proton-binding sites were obtained from the models, and the theory was fitted to the experimental titration curve.6 The mean-square dipole-moment magnitude was then calculated and compared with estimates from solution dielectric increments.? The proton-fluctuation anisotropy effectswere obtained from the same calculation.7 The present paper constitutes a theoretical analysis of the available data using structural, spectroscopic, and dielectric information. The results suggest that the method will be useful as a probe of protein structure in solution, and the problems for future investigation are clearly indicated.
Experimental Section The general methods and procedures used to obtain the Kerr effect data have been described.s The apparatus8 has been significantly improved with regard to optical and electronic components, as described below. A new optical bench was set up with a Hanovia XeHg 528B 1000-W arc lamp, followed by a Hilger & Watts D292 grating monochromator and a microscope slide beam splitter leading to a monitoring RCA 1P21 photomultiplier. The monochromator exit slit was focused through the polarizer to the center of the cell by an achromatic lens. Crystal Optics Company Glan prisms were used as polarizer and analyzer. Following the analyzer, a second achromatic lens focused the beam through a slit to eliminate stray light from the signaldetecting RCA 3P21 photomultiplier. The rotation of the analyzer was controlled by a micrometer-driven arm, and was readable to AO.01". The thermostated brass cell holders was retained
2909
and used with l-cm cells consisting of microscope slide windows held to a glass "U" by Apiezon W. Dipping electrodes were constructed by previously described methods.8 The electrode spacing, d, was 1.47 mm and the length was 7.04 mm. A correction, d / n , was added to the measured length.* A l-mm slit was attached to the electrode assembly at a position immediately preceding the front window of the cell. The pulse generating circuits8 were not modified, but a Tektronix Type 551 dual-beam oscilloscope was incorporated. The pulse voltages were displayed on the upper beam by a Type Z plug-in unit and the signalmonitor difference was displayed on the lower beam by a Type 0 plug-in unit used as a difference amplifier. The simultaneous pulse and signal traces were recorded by a Tektronix Type C 12 Polaroid camera. Single pulses of 3000-5000 V amplitude and 80 psec duration were used. The treatment of data followed earlier methodss except that analyzer rotations replaced neutral filter readings in determining Io. No residual birefringence was observed in the new apparatus (presumably, because of improved slit construction), so that the data analysis was simplified. Since the experiments provide better estimates of the solution/solvent Kerr-constant ratio, BIBo, than of either B or Bo separately, a Kerr-constant increment, k , was introduced as follows
B = Bo(1
+ ks)
(1)
where g is the solute concentration in g/l. Oxy and met hemoglobin were prepared from freshly drawn horse blood. The method of preparing and analyzing the samples has been described.O
Results All measurements were carried out a t a temperature of 15". Water measurements preceded each set of hemoglobin measurements and are summarized in Table I. The concentration range used for met and oxy hemoglobin was 1.1-1.4 and 0.6-1.0 g/L, respectively, resulting in per cent transmittances in the range 450%. There was too little transmittance to obtain data with met hemoglobin at 365 mp. Each experimental value shown in Table I is the average of three or four duplicate determinations. The uncertainties shown are an average of experimental average deviations and slightly larger theoretical estimates.
(3) W.H.Orttung, J. Amer. Chem. SOC.,87, 924 (1965). (4) M.F. Perutz, J . Mol. Biol., 13, 646 (1965). (5) W. H.Orttung, J. Phys. Chem., 7 2 , 4058,4066 (1968). (6) W.H.Orttung, J . Amer. Chem. Soc., 91, 162 (1969); Nature, 220,1122 (1968). (7) W. H. Orttung, J. Phys. Chem., 7 3 , 418 (1969). (8) W. H. Orttung and J. A. Meyers, ibid., 67, 1905 (1963). (9) W.H.Orttung and J. Warner, ibid., 69, 3188 (1965). Volume '78,Number 9 September 1960
WILLIAMH. ORTTUNG
29 10 Table I: Experimental Results and Molecular Parameters at 15” A, mr 436
7
546
10’Bo
IC
3.58 f 0.1 -0.31 3t 0.03 -0.14 f 0.06
Met OXY
1OSsh
4.36 =k 0.1 -0.03 i 0.03 0.05 & 0.10
-13.7 & 2 - 6 . 2 f3
Met OXY
-1.3 & 2 2.1 f 4
Data Reduction If induced-moment orientation and hyperpolarizability effectslo may be ignored, the Kerr constant of a dilute solution of a globular protein may be expressed as
where n is the refractive index of the solution, X is the wavelength of the light, and hi, is the number of molecules of type j per mL5 In the following discussion, j = 1, 2 is used for solvent and solute, respectively. If the molecular environment is assumed to be a real spherical cavity in a uniform dielectric,” then
where D is the dielectric constant of the solution, and cyap and are the optical polarizability tensor and dipole moment of a molecule of type j as it exists in the solution.12 The model underlying eq 3 should be suitable for dilute solutions of globular proteins, although corrections may be required for nonspherical shapes. In any case, it is desirable to reduce the data to bz values, which may then be interpreted in terms of molecular parameters. If we introduce Q = 2 n ( n 2 2)2/(90Xnk2T2),eq 2 may be written for pure solvent and solution as
+
Bo = QoNobo B
&(Nib1
+
Nzb2)
(6) NA Q o where Mz is the solute molecular weight and N A is Avogadro’s number. To evaluate eq 6, the refractive index of water a t 15’ is adequately represented by
(7)
where v = 1/X. The values of b2 obtained from the data, using Mz = 64,500, are given in Table I. The Journal of Phusical Chemistry
... -0.16 =k 0.08
- 6 . 6 zk 5
Theoretical Analysis The bz molecular parameter of eq 3 must be interpreted in terms of the optical polarizability and dipole moment tensors, a+ and ( p a p p ) ~ ,of the hemoglobin molecule. If the molecule has a 2-fold rotation axis, each tensor is specified by four parameters. If there is no symmetry, six parameters are required. The dipole tensor has been evaluated from theory and structural model^,^ so that we need only consider the polarizability tensor. The statistical possibility of different residues in positions B5a1 and B5az, or in E9a1 and E9az,19 may be assumed to have a negligible effect on the 2-fold symmetry of the optical polarizability tensor. Polarizability Contributions. The tensor aa8 is composed of contributions from all transitions occurring in the absorption spectrum. In hemoglobin, the spectrum in the 3000-7000-A wavelength range is due to the four heme groups in the molecule. I n the range 25003000 8, tyrosine, tryptophan, and phenylalanine also contribute. At shorter wavelengths absorption is eventually observed for all electrons of the molecule. For present purposes, it is useful to separate the 25007000-A spectral contribution of the heme from all other contributions. The contribution of the spectrum of a solute to its refractive-index increment in the same solution may be calculated from the Kramers-Kronig relation in the following form
(5)
bz=-- lOOOMz Bo
+ 0.00300 X 10-8~2
5 . 0 5 f 0.1
(4)
where subscripts 0, 1, and 2 refer to pure solvent, solvent in solution, and solute, respectively. In the limit of low concentration, and assuming that bo and bl are in the ratio of the dielectric-constant factors of eq 3, we obtain bz in terms of the Kerr-constant increment defined in eq l
no = 1.3250
7
365
where E ( V ) is the molar extinction coefficient a t v = 1/X, Mz is the corresponding solute molecular weight, and P indicates the principal value of the integral.9 The contribution to the optical polarizability of the solute may be obtained from the refractive-index increment by employing the Lorenz-Lorentz equation
(9) (10) A. D. Buckingham and J. A. Pople, Proc. Phys. SOC.,A68, 905 (1955). (11) L. Onsager, J. Amer. Chem. SOC.,58, 1486 (1936). (12) The convention for Greek subscripts is the following: If a subscript appears twice in the same symbol or in a product of symbols, summation over that index is implied. (13) “Atlas of Protein Sequence and Structure 1967-68,” M. 0. Dayhoff and R. V. Eck, Ed., The National Biomedical Research Foundation, Silver Spring, Md., 1968.
KERREFFECTOPTICAL DISPERSION OF HEMOGLOBIN where a1 and ~2 are the orientationally averaged polarizabilities of solvent and solute molecules, N1 and Na are the number of molecules per ml, and rz is the refractive index of the solution. Equation 9 may be rearranged to give an expression for 3 2 valid in the limit of low solute concentration
where c is in g/ml, no is the refractive index of the pure solvent, and ~ 7 2is the partial specific volume of the solute. It then follows that the spectral contribution to zzis given by
To obtain a theoretical estimate of the asaptensor components, it is necessary to consider its relationship to the quantum-mechanical transition dipoles responsible for the absorption spectrum. The molar extinc, be related t o the tion coefficient in solution, E ~ may transition dipole mkoabetween sets of states o and IC by the equation
2911 If tia8 is a matrix in the molecule system (123 = xyx = c*ab) whose columns are unit vectors along the axes of the ith heme coordinate system, then for the ith heme, the tensor ezas in the molecule system is obtained from the corresponding tensor in the heme coordinate system by the traneformation tzay diratzap-’. If all of the heme transitions between 2500 and 7000 are assumed t o be polarized along x’,y’, or x’, then we may speak of the polarizability contributions az,,aut, as!from each type of heme polarization. For a circular absorber, we would expect that a1. = a y ~corresponding , to ex! = eyr at each wavelength of the spectrum. However, if the bands are not doubly degenerate in the x‘y‘ plane, ax!and ayt would be expected to have different wavelength dependence. For this reason, we may take an additional decomposition, such that
+
+
(16) where the terms on the right-hand side give the spectral contribution of each type of heme polarization to the polarizability tensor of the whole molecule. To separate the wavelength dependence from the geometrical dependence, we may introduce d-matrices defined by aX’Ct8
a6Ct0
a,’ap
Qz’ap
(17)
(~xf,p= 4/3dz*apaxf
with analogous expressions for and The factor of four in eq 17 reflects the fact that there are four hemes per undissociated tetrameric hemoglobin molecule. We may also introduce dnnp such that dztup,
$tug
where n is the refractive index of the solution, v is the frequency in wave numbers, and &o(v> describes the distribution of absorption frequencies for each transition.I4 ea is a unit vector parallel to the electric field of the light wave and the average is over all orientations of the molecules relative to the field. The form of eq 12 suggests the introduction of a tensor ea,g defined as
Thus if we imagine all molecules of the solution to be oriented in a given direction and use polarized incident light, eq 12 becomes eM‘ = eapeaej3
(14)
If eaB is used for e in eq 8, and the resulting An/c in eq 11, we obtain the spectral contribution to the polarizability tensor, asap,of the molecule. The polarizabili t y tensor of the molecule may then be divided into two parts aap
asap
+
anup
(15)
where asupis the contribution from the heme spectrum between 2500 and 7000 A, and anapis the ‘‘normal” contribution from all other transitions of the heme and globin components. To describe the orientation of the transition dipoles relative to the heme groups, we may introduce a heme coordinate system, x‘y’x’, in which the x’ and y’ axes are in the porphyrin plane, and x’ is normal to the plane.
(18)
anaB = dnapnm
where E~ = anaa/3. The heme-related tensors dxta8, etc., depend only on the heme orientations and are independent of wavelength if all transitions in the range 2500-7000 have only, d,y’, or x’ polarization. The d,, matrix is also effectively independent of wavelength for our purposes. Substituting eq 16-18 in eq 15 and rearranging, we obtain aap
=
dpaoap
+
+
+
drnap~m
where a p = 4(azr
dnadanap
+ a y ’ ) / 3 ,am = 4(ax!-
+ duad/2
d,a, = = (dxap
- d,ap)/6
+
+
(19) aut), and (20)
(21) Substitution of eq 19 in eq 3 gives a result of the form &ap
bz
cpc~p
Cmam
C~W
+ c*&
(22)
where
(14) W. Kauzmann, “Quantum Chemistry,” Academic Press, New York, N. Y.,1957, Chapter 15.
Volume 73, Number 9 September 1969
WILLIAMH. ORTTUNG
2912 X (mp)-r 200
a
330
250 ~
~
1
,
SqO 690
3qO 490 ,
1
1
1
1
,
BOO
1
I
1
100 "OI
-2L '
-31
'
40
36
32
28 24 20 10-3, (ern-')
16
12
and the analogous expressions for cm, cz, and cn are functions of the geometrical d-matrices and of the dipolemoment tensor components. Equation 22 provides a useful decomposition of the theoretical contributions to
b2. Evaluation of the Polarizabilities. The spectra of met and oxy h e m ~ g l o b i nfrom ~ ~ 2500 to 7000 were plotted os. v in Figure 1 after separating the contribution of tryptophan, tyrosine, and phenylalanine.13Je The integrations indicated in eq 8 were then carried out on an IBM 7040 computer, using MZ = 16,700. The results of the calculations are shown in Figure 2. a,/4 was then obtained from ( A ~ / C ) ~using ,, eq 11with Mz = 16,125 as shown in Figure 3. The normal dispersion contribution to An/c in the range 500-700 mp has been fitted to an equation of the formg
+
Combination of eq 7, 10, and 24, using and M B= 64,500, yielded loa4& = 6370
I
36
I
'
32
I
'
28
I
" I
24
20
I
16
'
J 12
we have represented the x' and y f polarized bands by Gaussians of the type e(v) =
eoe-4(1n
9)(v-vde/ya
(26)
and have calculated the effect of a 100-cm-l splitting of the two polarizations. For met and oxy hemoglobin, the parameters (BO, yoz', v o B t , y) were taken as (240,000, 24,700, 24,600, 1400) and (220,000, 24,100, 24,000, 1400), respectively.g Values of azt - ayt obtained from eq 8 and 11 are shown in Figure 3. In all the following considerations, it has been assumed that azf = 0 for met and oxy hemoglobin between 2500 and 7000 A. If this assumption is true, then it follows that ap = as. Evaluation of the Dipole-Geometrical Coeficients. The dipole moment averages required by eq 23 and its analogs have been estimated.' The protein environment was assumed to be a real spherical cavity of molecular size in a uniform die1e~tric.l~For this model
where
(24) 82 =
+ 70 X 10-*v2
0.75 cc/g (25)
The heme group is thought to be a circular absorb:r in met and oxy hemoglobin in the range 2500-7000 A, although a small splitting of the Soret band has been observed in cytochrome-^'^ and met myoglobin.18 To estimate the effect of a small splitting of the Soret band, The Journal of Physical Chemistry
'
Figure 2. The spectral contributions to An/c for the spectra of Figure 1. The results were obtained by evaluating the Kramers-Kronig integral.
Figure 1. The solution spectra of met and oxy hemoglobin. The tryptophan-tyrosine-phenylalanine contribution has been subtracted out and is shown separately. The curves have been extrapolated to zero above 40,000 cm-1.
( A ~ / C = ) ~0.174 ~ ~ ~ 0.0025 X
40
(16) R. Lemberg and J. W. Legge, "Hematin Compounds and Bile Pigments," Interscience Publishers, Inc., New York, N. Y.,1949, p 749.
(16) J. S,Fruton and S. Simmonds, "General Biochemistry," 2nd ed, John Wiley and Sons, New York, N. Y.,1960,p 73. (17) W.A. Eaton and R. M. Hochstrasser, J. Chem. Phys., 46, 2533 (1967). (18) W. A. Eaton and R. M. Hochstrasser, ibid., 49,986 (1968). (19) L. Onsager, J. Amer. Chem. SOC.,58, 1486 (1936).
KERREFFECT OPTICAL DISPERSION OF HEMOGLOBIN
2913 11
22
33
12
dpa5 = 1.467 dma5 = -0.343
0.466 0.076
1.067 0.267
-0.128 0.214
46,618
33,218
10,111
tYp
=
the results
- t g
200
w
If, in addition 10S6A(peafieg)= 30,886
and (p,& = -365 D, corresponding to the theoretical dielectric parameters (neglecting correlations) for the best fit of the titration curve of horse oxy hemoglobin a t pH 6-76,' then we obtain from eq 23 and its analogs 1036c, = 1803 and 10a4c, = 1149, The senW
5 -loot IU
-150l4;'
I
36
'
I
32
I
28
I
I
24
'
20
I
'
16
'
'
12
Figure 3. The spectral contribution to the mean polarizability of the hemoglobin molecule, as caluclated from the results of Figure 2. The effect of a 100-cm-1 splitting of the Soret band is also shown as CY,^ aut.
':i I2
-
in which D, is the effective dielectric constant of the protein, kua is the vacuum dipole moment of the molecule if all proton binding site charges were removed, A(keageg) is the proton fluctuation contribution,s and
is the embedded dipole moment arising from the nat proton binding sites.K The tiag matrices describing the rotation of the ith heme coordinate system from the molecule coordinate system may be specified by the Euler angles +t, et, $t, where e2 and - 90" are the polar and azimuth angles of the zt' axis, and $1 is the rotation of the xi' axis out of the xy plane toward yt', The calculation of tiaa from the Euler angles is straightforward.20 +t, et values may be estimated from electron-spin resonance results for crystalline horse met hemoglobin.21 For the al-chain, 4, 0 = 165.5, 57.0", and for the pl-chain, +, 0 = 180, 122.0°, all angles 1". For a2 and pt, the cp values are 180" larger than for 011 and PI, respectively. Although $t values are not needed to evaluate the c, coefficient of eq 22 and 23, estimates are required in connection with the c, coefficient. If the origin of the heme coordinate system is taken at the iron-atom projection on the porphyrin plane, and if we assume (for illustrative calculations) that x' is positive toward the bridging carbon CL, and y' is positive toward CU (see ref 22 for heme-atomlabeling conventions), then, according to our model,6 $ = 84 and -61" for the a and ,8 chains, respectively. Using the heme axis orientations described above we obtain for
-' I
-20
4
5
6
7
8
9
1
0
PH Figure 4. Theoretical pH dependence of c p for various estimates of the dielectric parameters of horse oxy hemoglobin:7 (a) best fit of the titration curvea using the individual site average approximation with no correlations; (a-10) including correlation averages of sites within 10 A of each other; (b) same parameters as (a), but using the group average approximation; (c) same parameters as (a), but assuming both E9a sites to be Lys rather than Gln; (c-1) calculation (a), modifitbd by an estimate of the effect of adding one Lys E9a.T The small rectangle illustrates the range of values inferred from the data. (20) H. Goldstein, "Classical Mechanics," Addison-Wesley, Cambridge, Mass., 1953,Chapter 4. (21) J. E. Bennett, J. F. Gibson, and D. J. E. Ingram, Proc. Roy. SOC.,A240, 67 (1967). (22) B. P.Schoenborn, H. C. Watson, and J. C. Kendrew, Nature, 207, 28 (1965).
Volume 79, Number 9 September 1969
WILLIAMH. ORTTUNG
2914
+
1 1 sitivity of these results to 1" increments in the Euler angles specifying the heme orientations, i.e. , to ea
4,
$a
00
40
$0
211 -1
0
-4158 -6
421 0
0 -20
24 0
1
200
i:
is as follows A(lOaecp)= 4068 ~ ( 1 0 8 4= ~ ~ )13
4
It is readily seen that cp is very sensitive to 8, and e,, and that small uncertainties of the other angles are not of particular importance. c, is sensitive to pH and to modifications of the dipole moment tensor, as illustrated in Figures 4 and 5 . The corresponding pH dependence of c, is shown in Figure 6. All of the dielectric parameters used in Figures 4-6 have been described previou~ly.~ A realistic calculation of d,, is not possible because the net polarization of the transitions involved is unknown. A very crude estimate of the tensor may be obtained by assuming the molecule to be intrinsically isotropic and considering only the effect of its deviation from spherical shape. According to the low resolution X-ray dataJ23the met or oxy form of the molecule is roughly ellipsoidal with axes of 64, 55 and 50 A coinciding approximately with a*, c, and b crystal directions, respectively. According to Scholte's model of a uniform isotropic ellipsoidal m ~ l e c u l e ~ ~ ~ ~ ~
where d ' l n a p is expressed in the ellipsoid coordinate system (x"y"x" = ca*b), n and n2 are the refractive indices of solution and solute, and A , is the ellipsoidal form factor ( A , = 1/3 for a sphere).*6 A rough esti-
ox
rl
I-
I""[ 120
E
40
PH Figure 6. Theoretical pH dependence of c, for an assumed x', y' splitting orientation. The parameters of curves a, a-10,
b, c, and c-1 are described in Figure 4; (d) horse met hemoglobin; (e) human oxy hemoglobin; (f) horse deoxy hemoglobin.
+
mate of n2 is obtained from 8, = r3((nS2- l ) / ( n z 2), where r3 is the product of the three half-axes. According to eq 25, = 6650 X at 500 mp, so that nz2~ 13.30. ! The form factors are calculated to be 0.34, 0.28, and 0.38 for x", y", and x", and we may also use n = 1.34. Equation 30 then gives d'lnap = 0.995, 1.035, and 0.970 for ab = 11, 22, and 33, respectively. In the 123 = zyx = c*ab system, d,, = 1.000, 1.030, 0.970, and 0.013 for cy@ = 11, 22, 33, and 12, respectively. The pH dependence of c, is shown in Figure 7. The absolute magnitude of these results is not as interesting as the marked sensitivity to pH and hemoglobin modification. A curve for deoxy hemoglobin is not shown because the shape ellipsoid axes must be quite different from those of the oxy and met species.
Comparison with Data The theoretical analysis presented in the preceding section suggested that the cp, cm, and cn terms of eq 22 may all be of importance. However, since the data are limited and sinbe we do not yet have reliable information about x', y' splitting and its orientation, the b2 parameters of Table I have been fitted by the c, and Cn terms of eq 22, assuming azt = cyy#, and using the leastsquares method for the three oxy hemoglobin points. The results obtained were Met OXY
PH Figure 5. Theoretical estimates of the pH dependence of c p for various modifications of hemoglobin: (a) best fit of horse oxy hemoglobin, as in Figure 4; (b) horse met hemoglobin; (c) human oxy hemoglobin; (d) horse deoxy hemoglobin. The Journal of Physical Chemistry
ioaec,
ioaec,
550 150
-35.6 -9.6
and the fit is shown in Figure 8 for each species. If the x', y' splitting of the Soret band is similar to that as(23) M. F. Peruta, M. G. Rossmann, A. F. Cullis, H. Muirhead, G. Will, and A. C. T. North, Nature, 185, 416 (1960). (24) T.G.Scholte, Physica, 15, 436 (1949). (25) W. H. Orttung and J. A. Meyers, J . Phy6. Chem., 67, 1911 (1963).
I~ER EFFECT R O P T I C A L DISPERSION O F HEMOGLOBIN
2915
-
0
-'Ob
-I2l
-16 -I4[
v
3
Figure 7. Theoretical pH dependence of c,. The curves are labeled as in Figure 6 . Curve (f) is not shown. The horizontal arrow denotes the value inferred from the data on met and oxy hemoglobin.
sumed in the preceding section, then the c, parameters would be larger than obtained by the two-term fit. A small rectangular area has been placed on Figure 4 to indicate the c, value inferred from the data. This value is consistent with the weighted average of curves a, c-1, and c of Figure 4, although the very strong pH dependence of these curves seems inconsistent with the fact that reproducible data could be obtained with relatively unbuffered solutions. This apparent discrepancy is undoubtedly explained by the neglect of correlations in curves a, c-1, and c. Curves b and a-10 of Figure 4 do include a measure of the correlation effect and have their maxima shifted t o the pH range in which the experiments were performed. The very great vertical sensitivity of the curves to the tilt of the heme norper degree) premals from the %fold axis (-4 X cludes a more quantitative comparison with the available data, but does suggest that the Kerr effect of hemoglobin solutions should be very sensitive to small changes of the heme normal directions. The c, parameters suggested by the data are of the same sign but much smaller than the values predicted in Figure 7. The discrepancy is not surprising in view of the crude assumptions used in the theoretical prediction. The molecule is not well represented by an ellipsoidal contour, and may not be intrinsically isotropic, as was assumed.
Discussion The preceding analysis of the Kerr effect in hemoglobin solutions involved several steps. The data were first reduced to a set of molecular parameters using the Onsager model of a liquid dielectric. The theoretical
-
A
10-
-3019
I
-40
4 40
36
32
28
24
20
16
12
C I O - ~ ~ (crn-1)
Figure 8. Two-parameter fit of the met and oxy hemoglobin molecular Kerr-eff ect parameters, assuming only normal dispersion and circular-absorber heme spectral contributions. Both the total fit (cPap c n ~ Jand the normal dispersion contributions ( c , ~ , ) are shown for each molecule.
+
predictions of these parameters mere then found to be quite sensitive to many of the input variables such as heme orientation, 5, y splitting of the heme spectrum, dipole moment, and proton-fluctuation anisotropy. Since the input variables have not been precisely established by other experiments, the original hope of testing the theory on hemoglobin has had to be deferred until more extensive data become available. At this point, it can only be said that the theory and the available data are qualitatively consistent. From a different point of view, the theoretical results are of considerable interest in themselves, because the sensitivity to the input variables suggests that the Kerreffect dispersion method should be a sensitive probe of the molecular condition of proteins in solution. The preceding calculations illustrate that a rational and straightforward analysis can be carried out if information on the protein structure and chromophore-spectrum polarization is available. In the preliminary analysis of the hemoglobin Kerreffect dispersion dataj3it was assumed that the proton fluctuations were isotropic and that the heme was a circular absorber. With these assumptions, eq 23 simplifies to
and c, = czr = 0. Xumerical values from eq 23 and 31 differ greatly, illustrating the importance of protonfluctuation anisotropy effects. The qualitative conclusion of the earlier analysis3that the molecule orients its 2-fold axis parallel to the applied field is still valid, although the estimates of ( & ,obtained by using eq 31 were too small. Volume 78, Number 9 September 1969
