418
WILLIAMH. ORTTUNG
Calculation of the Mean-Square Dipole Moment and Proton Fluctuation Anisotropy of Hemoglobin at Low Ionic Strength' by William H. Orttung Department of Chemistry, University of California, Riverside, California (Received August IS, 1068)
The mean-square dipole moment and proton fluctuation anisotropy of horse oxy hemoglobin at low ionic strength have been calculated from the Kirkwood-Shumaker and extended Tanford-Kirkwood theories. Site coordinates were taken from a model built according to Perutd 1965 description. Results are presented for the pH range 4.5-9.0. The mean moment is considerably more sensitive to the parameters and approximations used than is the fluctuation moment, and the correlation averages were found to make a significant contribution to the fluctuation moment. The anisotropy of the fluctuations is appreciable and varies with pH. Calculations for met hemoglobin and for hypothetical models of deoxy and human oxy hemoglobin are also presented. Good agreement was obtained with data for horse oxy hemoglobin, and small but unexplained discrepancies with the available data were noted for the other modifications.
At low ionic strength, the dielectric constant of a protein solution is largely determined by the charges bound to the protein molecules. A single molecular parameter, the root-mean-square (rms) dipole moment, may be deduced from measurements at low frequencies and studies of the frequency dependence may yield information about, the relaxation tinies associated with certain degrees of freedom of the molecule. The pertinent available data on hemoglobin may be briefly summarized. Oncley2 measured a low frequency dielectric increment, AD/g, of 0.33 per g/l. and a single relaxation time of 8.4 X sec for horse carboxyhemoglobin solutions a t 26". No frequency dependence was observed below 500 kHz to the lowest frequency measured (25 kHz). I n a more recent study at 15" and a frequency of 1 MHz, Takashima3 reported dielectric increments of 0.25, 0.35, and 0.58 for horse oxy, carboxy, and met hemoglobin, respectively. Goebel and Voge14 studied met homoglobin solutions between 100 kHz and 10 MEIz and also reported p H dependence near the isoionic point. They obtained a dielectric increment of 0.34 at 25" and a pH dependence of approximately +0.1 per pH unit a t pH 7. Hanss and Banerjee6 found that no observable change occurred in the dielectric increment or relaxation of horse or human hemoglobin in the range 0-100% oxygenation, and that the dielectric increment of human hemoglobin was significantly smaller than that of horse hemoglobin (0.28 us. 0.32). Oncleyz used the Clausius-Mosotti equation to estimate an rrns dipole moment of 470 D for horse carboxyhemoglobin, and attributed the dielectric dispersion to rotational relaxation in the applied field. Kirkwood and Shumaker6 pointed out that proton fluctuations among the available binding sites make a major contribution to the rrns dipole moment of proteins. I n order The Journal of Physical Chemistry
to test the Kirkwood-Shumalrer theory, Takashirna7 collected data on the p H dependence of the dielectric increment and dispersion of ovalbumin and bovine serum albumin solutions. However, a quantitative test of the theory is not possible until the site coordinates are known for these proteins. Lumry and Yues considered recent data on the kinetics of proton transfer processes and concluded that proton redistribution rates in the neutral pH region should be very slow relative to the 100 kHz to 1 MHz frequencies used in dielectric measurements. However, it does not follow that proton fluctuations do not contribute to the dielectric increment in this frequency range. At a given instant of time, all possible fluctuations exist on different molecules in the solution between the electrodes. When a high-frequency field is applied, each fluctuation acts as a rigid moment and the dispersion reflects only the rotational relaxation. The lack of observed dispersion below 500 kHz and the calculated magnitudes of the dielectric increment presented below constitute support for this point of view. The Tanford-Kirkwood theory of protein titration curvesg has recently been extendedlOJ1to allow calcula(1) This investigation was supported in part by Public Health Service Research Grant GM 11683 from the Division of General Medical Sciences. (2) J. L. Onoley, J , Amer. Chem. Soc., 60, 1115 (1938). (3) S. Takashima, ibid., 78, 541 (1956). (4) W.Goebel and H. Vogel, Z. Naturforsch., 19b,292 (1964). (5) M.Hanss and R. Banerjee, Biopolymers, 5, 879 (1967). (6) J. G. Kirkwood and J. B. Shumaker, Proc. Nat. Acad. Sei. U.S. 38, 865 (1952). (7) S. Takashima, J . Phys. Chem., 69, 2281 (1965). (8) R. Lumry and R. H. Yue, J . Phys. Chem., 69, 1162 (1965). (9) C. Tanford and J. G. Kirkwood, J. Amer. Chem. SOC.,79, 5333 (1957). (10) W.H.Orttung, J. Phys. Chem., 72, 4058 (1908). (11) W.H. Orttung, ibid., 72, 4066 (1908).
419
MEAN-SQUARE DIPOLEMOMENTOF HEMOGLOBIN tion of the site occupation averages required by the Kirkwood-Shumaker theory, and site coordinates may now be estimated from three-dimensional structural models of proteins such as hemoglobin.12 This approach has yielded a detailed interpretation of the titration curve of hemoglobin13 and is applied to the problem of the dielectric constant in the present paper.
Theoretical Relationships If the environment of solvent and solute molecules is assumed to be a real spherical cavity of molecular size in a uniform dielectric,14then
where D is the dielectric constant of the solution, aojap and F~~ are the electrostatic polarizability tensor and dipole moment of a molecule of type j as it exists in the s0lution,~5and N j is the number of molecules of type j per milliliter. The average is over all rotational and internal coordinates describing the dipole moment of the molecule in the limit of zero applied field. The convention j = 1, 2 for solvent and solute is used below, In the Onsager theory,15as applied to a globular protein Pa
2D+1 D,+2 D, 3
= 20
+
[-
+ /*ea]
~ v a
(2)
where D, is the effective dielectric constant of the protein, pva is the vacuum dipole moment of the molecule if all proton binding site charges were removed, and
C(el + ezt)Rl, 1=1 nat
Pea
=
(3)
is the embedded dipole moment arising from the nBt proton binding sites. If we introduce
then, to a good approximation at low solute concentration
from which we may deduce that
+ bz(Da- I)]
(6)
in the limit of low protein concentration, where M z is the molecular weight of the solute, N A is Avogadro's number, ?jz is the partial specific volume of the solute, and Do is the dielectric constant of the solvent. If ( ~ is negligible, ~ 2 ~ then ~ eq 4 and 6 allow estimation of ( ~ 2 ~ l . 1 from 2 ~ ) ~ the data. The theoretical prediction of a2 is similarly given in
terms of (P,Pa)O
= 4dO(Pa)O
+ A(PaPa)
(7)
where the subscript j = 2 is not explicitly shown, and it has been assumed that aOZaais negligible. A ( P ~ M ~ ) is the fluctuation contribution to (~app)o.10 From eq 2
If fluctuations of
pva are
negligible, then we also have
The theoretical prediction therefore requires the calculation of ( p v J 0 , (p,J0, and A ( p e a p e p ) . The method by which these quantities are evaluated below has been described.'Otl' I?aput Parameters for the Calculation. The model of horse oxy hemoglobin from which the site coordinates were obtained has been described.13 The same model was used to estimate the center of mass and intrinsic dipole moment. The computer-generated coordinates of all nonhydrogen atoms were used to calculate the center of mass. The masses of hydrogen atoms were combined with the masses of the atoms binding them. The center of mass was then calculated to be at x = -1.52 A in the crystal related coordinate system, 123 = xyx = c"ab. The intrinsic dipole moment was calculated on the assumption that only the planar peptide groups make a net contribution. Using a dipole magnitude of 3.7 D oriented at 57" to the CN bontl,16 dipole moment magnitudes of 87 and 80 D were obtained for the CY and p chains. The vector components for the 011 and 01 chains were (pz, py, p z ) = (- 62, 59, - 12) and (77, 23, -2), respectively. The net moment of the tetramer was directed along the x axis, with pz = -28 D. The intrinsic pK's, site depth, protein dielectric constant, and masking inference were obtained by fitting the titration curve.13 To obtain site coordinates for deoxy hemoglobin, the a and 0chains were rotated as rigid units." The center of mass shifted to x = -1.51 A, and the net dipole moment due to the planar peptides changed to pz = -15 D. The three sites, Arg FG4a, Arg CSp, and His FG4P were also assumed to be unmasked in the deoxy structure.13 To represent met hemoglobin, sites were introduced at the iron atoms having zero charge with no proton (12) >I. F. Perutz, J . XoE. EioE., 13, 646 (1965). (13) W. H. Ort-tung, J . gmer. Chem. Soc., 91, 162 (1969); .Vatwe, in press. (14) L. Onsager, J . Amer. Chem. SOC.,5 8 , 1486 (1936). (15) 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 applied. (16) D. A. Brant and P. J. Flory, J. Am,er. Chern. SOC.,87, 2791 (1965) (17) H. Muirhead, J. 1LI. Cox, L. Mazzarella, and M. F. Perutz, J. Mol. Biol., 28, 117 (1967). I
Volume '79, Number 2 February 1060
420
WILLIAM H. ORTTUNG
bound. A pK of 8.7 (suggested by spectrophotometric titration datal8) was tried in a first calculation, and the sites were found t o be fully occupied up to pH 9. I n a second calculation, the sites were treated as histidine residues (intrinsic pK of 6.013) and the apparent pK resulting from the calculation was about 7.8. It is thus clear that an intrinsic pK of about 7.0 should give an apparent pK of about 8.7 for these sites. Unfortunately, however, the available computer programs handle only singly excited distributions, while such a calculation would require consideration of multiply excited distributions. The best representation of met hemoglobin is therefore the first calculation (pKint of 8.7), which would only be expected to be correct up to a pH of about 6.5 at which the FeOHz+ site begins titrating. The calculated titration curve between pH 4.5 and 6.5 is almost identical with, but slightly higher than, the calculated best fit for horse o~yhemoglobin.'~ A representation of human oxy hemoglobin was obtained by removing His E13& His 20& Asp E13& Glu B20, and Glu H3P and adding His Na2p.le The conversion of Arg G18P to His was not made because Arg Gl8p had already been assumed to be masked in the horse oxy hemoglobin titration calculations.*a All other sites and their coordinates were assumed to be the same as in horse oxy hemoglobin. The calculated titration curve is shown in Figure 1. The shift of the isoionic pH by almost 0.2 unit is in the same direction as the experimentally observed shift of 0.4 unit.20 20
l
f
r
'
' 5
I
,
,
,
l
f
(
l
,
L a
-5004
5
6
7
8
9
1
0
PH Figure 2. Theoretical mean embedded dipole moment of horse oxy hemoglobin due to the charges on proton binding sites: a, independent site approximation; b, group average approximation; c, individual site average approximation. The parameters used in the calculation gave the best fit of the titration curve.13
,
-10 -
-
-204
' 6
I
7
I
8
'
HORSE I
9
I
1
I
'0
loo[
a
PH Figure 1. Comparison of the theoretical curves of horse and human oxy hemoglobin at zero ionic strength. The curve for horse hemoglobin is the best fit of the data obtained previously.13
Calculations Detailed theoretical results for horse oxy hemoglobin are presented in Figures 2-6. The parameters used in the calculations were the same ones that gave the best fit of the individual site average approximation to the titration curve.13 The large difference between the group and individual site average approximations in Figure 2 emphasizes the importance of variations among sites of a given type. In Figure 3 it is seen that site variations me not as important in the fluctuation moment contribution. The group average calculation gives lower results than the individual site average The Journal of Physical Chemistry
Figure 3. Theoretical r m embedded fluctuation dipole moment magnitude of horse oxy hemoglobin due to the charges on proton binding sites. The labels of curves a-c are the same ae in Figure 2. The effect of including correlation averages for sites within 6, 8, and 10 A of each other are also shown for limited pH intervals.
approximation. This difference is expected because the group average approximation includes a measure of the correlations between sites of the same type.1° The effect of including correlations in the individual (18) P.George and G. Hanania, Biochem. J., 55, 236 (1953). (19) "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. (20) C. Tanford and Y. Nosaki, J. Biol. Chem., 241, 2832 (1966).
MEAN-SQUARE DIPOLEMOMENT
4 '
5
6
7
8
9
IO
PH
Figure 4. Theoretical rms principal values of the mean square fluctuation moment ellipsoid of horse oxy hemoglobin due to the charges on proton binding sites according to the individual site average approximation. The labels X, Y associate the curves with the closest axes x (c*), y (a). Z' includes the mean embedded moment contribution of the site charges. The effect of including correlation averages for sites within 6, 8, and 10 A of each other is shown for limited pH intervals.
Ex 0
1
3001:
I
PH Figure 5. The curves are analogous to those of Figure 4, except that the group average calculation has been used.
-
42 1
OF HEMOGLOBIN
lations would require an excessively long time on our present IBM 7040 computer. The anisotropy of the fluctuation moment is shown in Figure 4 for the individual site average and in Figure 5 for the group average approximation. It is clear that correlations have the strongest depressing effect on the fluctuations parallel to the long ( w a * ) axis of the molecule. The rotation of the largest principal value in the xy (or ac) plane is shown in Figure 6. This result shows that the alignment of a protein in an electric field may vary considerably with pH. It will be shown that the anisotropy effects of Figures 4-6 are of considerable importance in the interpretation of the Kerr effect of hemoglobinsz1 Figures 7 and 8 show the mean moment and fluctuation moment due to the bound protons and their sites for the various hemoglobin species that are closely related to horse oxy hemoglobin. The variations in the mean moment are much greater than the variations in the fluctuation moment. Additional details, of interest for Kerr effect interpretation, are shown in Table I. Figure 9 shows the root mean square moments obtained from eq 7-9. These values represent the most realistic calculations that could be performed at the present time. Deviations from the data could arise froin several sources, including (1) neglect of correlations, (2) inaccuracies in the site coordinates, and (3) contributions to p v a in addition to that of the peptide units. It should also be remembered that the calculations are for zero ionic strength.
Comparison with Data Horse Oxy Hemoglobin. The experimental value of AD/g a t 25" is 0.33 per g/l. at the isoionic pH (6.7-
50 I 0o
o
-
-
r
2
-1 -100
I-
4
Figure 6. Theoretical rotation of the major axis labeled Y in Figures 4 and 5 from the g (a) axis toward the x ( c * ) axis. The labels of curves a-c are the same as in Figures 2 and 3. The effect of including correlation averages for sites within 6, 8, and 10 if of each other is also shown for limited pH intervals.
site average approximation is also shown in Figure 3. Correlations between sites separated by more than 10 are expected to be of i m p o r t a n ~ e , ' ~but ? ' ~ such calcu-
1
-6004
5
6
7
8
9
1
0
PH Figure 7. Theoretical mean embedded dipole moment due to the charges on proton binding sites: a, horse oxy hemoglobin, best fit of the titration curve; b, best fit (a), with Lys EQa; c, best fit (a), with FG4q C6p, and FG4p unmasked; d, same as (c), but with deoxy coordinates; e, met hemoglobin; f, human oxy hemoglobin. (21) W. H. Orttung, t o be published. Volume Y3, Number I Februarv 1960
422
WILLIAMH. ORTTUNG
400
a0
0 x
Table I: Theoretical Embedded Dipole Moment Averages for the Bound Protons and Their Sites of Hemoglobin Modifications at p H 7
-
-340
-
300
-
. ..
2004-
PH
Figure 8. Theoretical rms embedded fluctuation dipole moment magnitude due to the charges on proton binding sites. The curves are labeled as in Figure 7 and do not include the effect of correlations.
600 -
m
-251 -269 -426 -282 -133
29,641 28 ,735 27 ,780 23,263 17,338 30,625 27,178 29,519 27,999 28,963
41 ,597 40 ,272 33 ,760 32,137 28,828 35,956 35 ,430 37,951 36,912 33 , 837
33 , 144 32 ,940 32 ,380 30,444 33 ,977 32 ,994 34,202 33,231 32 ,702 31 ,494
9 ,493 9,638 7,451 6 ,988 6 ,898 10,095 6 ,856 8,723 9,805 8,605
The configurations for the various calculations were as follows: (a) best fit of the horse oxy hemoglobin titration curve, using the individual site average approximation; (b) same as (a), but using the group average approximation; (c) same as (a), with sites F G ~ c uC6p, , and FG4p unmasked; (d) same as (c), with subunits rotated to the deoxy positions17; (e) same as (a), with both ESCUsites Lys rather than Gln; (f) same as (a), but positive charge on the Fe atoms, approximating met hemoglobin; (g) sequence modification of (a) to approximate human oxy hemoglobin.
0 500Y
2 300 200
l O4
.5
o6
7
8 09
1
0 l
DH Figure 9. Theoretical estimates of the rms dipole moments of hemoglobin species; 8, horse oxy hemoglobin, corresponding to the best fit of the titration curve; b, best fit (a), with Lys E9a; c, horse deoxy hemoglobin; d, horse met hemoglobin; e, human oxy hemoglobin. None of the curves includes the effect of correlations.
6.8).2J If M z = 64,500, ~2 = 0.75 cc/g, and Do = If the polariza78.4, then from eq 6, a2 = 4.1 X bility term of eq 4 is negligible with respect to the dipole term, we obtain ((pa2)>3'/' = 523 D. To obtain a comparative value from the theoretical results, two complicating factors must be considered: (1) correlations have not been fully considered in the calculations; and (2) in any given molecule, residues B5a may be either Phe or Tyr and residues E9a may be either Lys (slow component) or Gln (fast component).lg To bracket the correlation problem, we may first consider the individual site average calculation with no correlations (and Gln at Ego(). Using D, = 4 and pH 6.75 in eq 7-9, ( ( p a Z ) ~ ) ' / '= 526 D. If we substitute the Tho Journal
gf
Phgsical Chemistrid
group average approximation for the fluctuation contribution, A ( p e a 2 ) ,the above becomes 501 D, suggesting that correlations depress the theoretical moment by about 25 D. The B5a substitution is probably not of importance for dielectric properties since neither residue bears a charge at neutral pH. The E9a substitution should be quite important, however, since Lys bears a positive charge and Gln is uncharged at neutral pH. Four molecular species should exist in solution having 0, 1, 1, and 2 Lys E9a groups. The molecules having only one Lys E9a do not have a twofold electric symmetry axis, and could not be subjected t o calculations because of computer time considerations. From the individual site average calculsation with no correlations and two Lys E9a, we obtain ((p,2)o)'/z = 592 D. According t o our model, the coordinates of Eys E9a1 relativ? to the center of mass are (z, y, x ) = (23, 20, - 15) A, corresponding to an embedded dipole moment, era, of (111,94, -72) D. As may be seenfrom Figure 7, the change in the embedded dipole moment upon adding two Lys E9a residues is only -85 D rather than the 2 X -72 = -144 D that would result by simply adding the sites without affecting other charge binding equilibria. If the above observation is used as a correction factor, the change in embedded moment upon adding one Lys E9a may be taken as (65, 56, -43) D. If the change in the fluctuation moment is assumed to be negligible, then we obtain the estimate ((po12),J1'2 = 545 D for the species containing only one Lys
423
NMRSTUDYOF MESOPORPHYRIN IX DIMETHYL ESTER
E9a, ignoring correlation effects in the individual site calculation. If Lys and Gln are present in equal amounts, the relative concentrations of the 0, 1, 2 Lys l/q, respectively. A weighted species will be average of the above moments (526, 545, and 592 D) then gives 552 D. If correlations reduce this value by 25 D, as suggested above, we obtain a theoretical estimate of 527 D, in surprisingly good agreement with the “experimental” value of 523 D. Other ModiJications. The experimental dielectric increments suggest that horse met,4 deoxy,6 and oxy hemoglobin have essentially identical rms dipole moments, but that human oxy hemoglobin is 5-10% lower.6 The theoretical comparison shown in Figure 9 predicts that human oxy should be about 30% lower and that horse met and deoxy should be about 10 and 20% lower than horse oxy hemoglobin. The pH dependence of met hemoglobin is also predicted to be negative, in disagreement with the available data on fairly concentrated solutions (30-40 g/L) .4 An explanation for these discrepancies is not apparent at this time. It is possible that structural differences occur which were not recognized by the theoretical calculations.
Discussion The calculations presented in this paper suggest that it is possible to achieve a detailed molecular interpretation of the dielectric properties of protein solutions. To pursue the subject further, it will be necessary to have more extensive experimental results, particularly including pH dependence. The availability of more reliable structural information will also help t o remove some of the uncertainties in the present analysis, and the use of larger computers should allow the effect of correlations between the proton binding sites to be considered in a more satisfactory way. The original objective of these calculations was t o provide a basis for a molecular interpretation of Kerr effect optical dispersion data on hemoglobin.22 The Kerr effect is more difficult to interpret than the dielectric constant since it depends on both optical and electrostatic parameters. However, it also has a greater potential as a method for studying protein structure in solution. The results presented here provide a necessary input for the Kerr effect analysis.21 (22) W.H.Orttung, J. Amer. Chem. Soc., 87, 924 (1965).
A Nuclear Magnetic Resonance Study of the Association of Porphyrins
in Chloroform Solution. Mesoporphyrin IX Dimethyl Ester and Its Nickel Chelate
by Daryl A. Doughty and C. W. Dwiggins, Jr. Bureau of Mines, Bartlesville Petroleum Research Center, U.S. Department of the Interior, Bartleauille, Oklahoma (Received August $6, 1068)
The proton nmr spectra of chloroform solutions of mesoporphyrin I X dimethyl ester and its nickel chelate exhibit an appreciable concentration dependence, all peaks shifting downfield upon dilution. This dilution shift is greater for the nickel chelate. The spectra of both porphyrins exhibit additional fine structure a t higher concentrations. These results are interpreted in terms of porphyrin association with equilibrium between monomer and dimer forms. I n the dimer two porphyrin molecules are stacked one tbove the other with the planes of the molecules parallel and separated by 10.0 in the mesoporphyrin and 7.9 A in the chelate. I n the nickel chelate, association is enhanced by the slight affinity of the nickel for further coordination. The fine structure can be accounted for by a lateral displacement of one molecule in the dimer relative to the other. This displacement is caused by steric interference of the longer side chains. KOattempt was made to calculate the magnitude of the displacement for either porphyrin.
Introduction During an investigation of mctalloporphyrins contained in petroleums, a proton nmr study of mesoporphyrin IX dimethyl ester and its nickel and vanadyl chelates was undertaken. Preliminary work Soon
established the impracticability of using high-resolution nmr on the VanadYl chelate. The broadeniW of the peaks in the spectra of this chelate, an effect due to the strong Paramagnetism of the VaIladYl complex,’ Pro(1) D. Kivelson and s . - ~ Lee, . J. Chem. Phys., 41, 1896 (1964).
Volume 73!Number 8 February 1969
